Limits and Derivatives Class 11 NCERT Solutions | Exercise-wise Answers

Updated on Jul 28, 2025 14:46 IST

By Pallavi Pathak, Assistant Manager Content

The subject matter experts at Shiksha created the Class 11 Limits and Derivatives NCERT solutions. The chapter introduces Calculus, a branch of mathematics that studies the change in the value of a function as the points in the domain change. Limits and Derivatives class 11 Chapter 12 Maths includes the definitions of limits and derivatives and some algebra involving these concepts.
The step-by-step solutions help students to understand the concepts clearly and also provide them the confidence to appear in their school exam, CBSE Board exam, and other entrance tests later on. These solutions are accurate, and students can depend on them for their exam preparation. If they practiced only from these solutions, after going through the NCERT chapter, it would be more than sufficient for their examination preparation.
For comprehensive subject-wise NCERT notes of Class 11 and Class 12, Check Here. You will get the reliable notes for quick revision.

Class 11 Maths Chapter 12 Snapshot: Limits and Derivatives

Here is a quick summary of the Limit and Derivatives Class 11 Maths:

According to Chapter 12 of Class 11 Maths, $lim_{x \to a^{-}} f (x)$ is the expected value of f, which is called the left-hand limit of f at a, when x=a. Here, it is given that the values of f near x to the left of a.
To find the right-hand limit of f(x) at a, mathematically we show as - $lim_{x \to a^{+}} f (x)$ . It is the expected value of f at x = a, given the values of f near x to the right of a.
There is also the concept of common value when the right and left-hand limits coincide. It is denoted by $lim_{x \to a} f (x)$ .

To get access to the chapter-wise Class 11 Maths NCERT Notes that offer solved examples and also provide the important topics, students must check here.

Class 11 Math Limits and Derivatives: Key Topics, Weightage

Limits Class 11 Maths Chapter 12 is the part of the broader "Limits, Continuity & Differentiability" topic in the JEE Main exam. Hence, the students must focus on the core concepts of the chapter. Before starting the preparation of the chapter, it is wise to know the topics of this chapter. See below the topics covered in this chapter:

Exercise	Topics Covered
12.1	Introduction
12.2	Intuitive Idea of Derivatives
12.3	Limits

Class 11 Limits and Derivatives Weightage in JEE Main Exam

Exam	Number of Questions	Weightage
JEE Main	1 to 3 questions	3.3% to 10%

Important Formulas of Class 11 Limits and Derivatives

Important Formulae of Limits and Derivatives for CBSE and Competitive Exams

First Principle of Differentiation $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Power Functions: $\frac{d}{dx} (x^n) = n x^{n-1}$
Trigonometric Functions:
- $\frac{d}{dx} (\sin x) = \cos x$
- $\frac{d}{dx} (\cos x) = -\sin x$
- $\frac{d}{dx} (\tan x) = \sec^2 x$
- $\frac{d}{dx} (\cot x) = -\csc^2 x$
- $\frac{d}{dx} (\sec x) = \sec x \tan x$
- $\frac{d}{dx} (\csc x) = -\csc x \cot x$
Exponential & Logarithmic Functions:
- $\frac{d}{dx} (e^x) = e^x$
- $\frac{d}{dx} (a^x) = a^x \ln a$
- $\frac{d}{dx} (\log_a x) = \frac{1}{x \ln a}$
Rules of Differentiation
- Sum Rule: $(f + g)' = f' + g'$
- Difference Rule: $(f - g)' = f' - g'$
- Product Rule: $(uv)' = u'v + uv'$
- Quotient Rule: $\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}$
- Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Class 11 Maths Chapter 12 NCERT Solutions – Limits and Derivatives PDF Free

If students download the free Limits and Derivatives Class 11 PDF from the link given below, they will get the well-structured solutions of all the NCERT questions of chapter 12 Class 11 Maths. It will help them to do well in school exam, CBSE, and the JEE Main exam.

Chapter 5 Limits and Derivatives Class 11 NCERT Solutions: Download PDF for Free

Class 11 Math Limits and Derivatives Exercise-wise Solution

Class 11th Math Limits and Derivatives helps students understand the fundamentals of calculus unit, including differentiation, application of derivatives, integral and more. There are various important topics in this chapter such as checking differentiability, continuity, derivatives of different functions, and more. Limits and Derivatives Class 11 Solutions will include solution of all the exercises in lucid language. Ex 13.1 Class 11 will include problems related to the basic concepts of Limits and Derivatives such as using limit, finding continuity and differentiability. Class 11 Math Ex 13.2 includes more concepts such as finding derivatives of various functions and using them as application. Miscellaneous exercise covers various topics covered throughout the chapter. Students can check all the solutions below;

Class 11 Math Limits and Derivatives Exercise 13.1 Solution

Limits and Derivatives Exercise 13.1 focuses on fundamentals of the Limits. Student will be able to understand the application of limit to find first derivative of the given function for a given range. Ex 13.1 Class 11 Math in the Limits and Derivatives includes problems based on trigonometric functions, algebraic functions and exponential functions and finding their first derivative using the limits. Class 11 Math Ex 13.1 solution will contain solution of all 32 short and long answer type questions. Students can check the solutions below;

Class 11 Math Limits and Derivatives Exercise 13.1 NCERT Solution

Q1. $\underset{x \to 3}{l i m}$ x+ 3

A.1. $\underset{x \to 3}{l i m}$ x + 3 = 3 + 3 = 6.

Q2. $\lim_{x \to π} (x - \frac{22}{7})$

A.2. $\underset{x \to π}{l i m} (x - \frac{22}{7}) = π - \frac{22}{7}$

Q3. $\lim_{r \to 1} π r^{2}$

A.3. $\underset{r \to 1}{l i m} π r^{2} = π \cdot {(1)}^{2} = π$

Q4. $\lim_{x \to 4} \frac{4 x + 3}{x - 2}$

A4. $\underset{x \to 4}{l i m} \frac{4 x + 3}{x - 2} = \frac{4 \times 4 + 3}{4 - 2} = \frac{16 + 3}{2} = \frac{19}{2}$

Commonly asked questions

40. Find the derivative of $\frac{x^{n} - a^{n}}{x - a}$ for some constant a.

40. Given, f(x)= $\frac{x^{n} - a^{n}}{x - a} .$

So, $f^{'} (x) = \frac{(x - a) \frac{d}{d x} (x^{n} - a^{3}) - \frac{d}{d x} (x - a) \cdot (x^{n} - a^{n})}{{(x - a)}^{2}} = \frac{(x - a) \cdot n x^{n - 1} - (x^{n} - a^{n})}{{(x - a)}^{2}}$

$= \frac{(x - a) \cdot n x^{n - 1} - (x^{n} - a^{n})}{{(x - a)}^{2}}$

$= \frac{n x^{n - 1} \cdot x - n a x^{n - 1} - x^{n} + a^{n}}{{(x - a)}^{2}}$

$= \frac{n x^{n} - x^{x} - n a x^{n - 1} + a^{n}}{{(x - a)}^{2}}$

42. Find the derivative of the following functions:

(i)sin x cos x (ii) $s e c x$ (iii)5 sec x + 4 cosx

(iv) $c o s e c x$ (v)3cot x + 5 cosec x

(vi) $5 s i n x - 6 c o s x + 7$ (vii) $2 t a n x - 7 s e c x$

(i) f(x)=sin x cos x

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{s i n (x + h) c o s (x + h) ? s i n x c o s x}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} \times [2 s i n (x + h) c o s (x + h) ? 2 s i n x c o s x]$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} [s i n 2 (x + h) ? s i n 2 x]$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} [2 c o s \frac{2 (x + h) + 2 x}{2} s i n \frac{2 (x + h) ? 2 x}{2}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [c o s (2 x + h) s i n h]$

$= \underset{h ? 0}{l i m} c o s (2 x + h) \times \underset{h ? 0}{l i m} \frac{s i n h}{h}$

$= c o s (2 x + 0)$

$= c o s 2 x$

$(ii) f (x) = s e c x$

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{h} [s e c (x + h) ? s e c x]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{1}{c o s (x + h)} ? \frac{1}{c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{c o s x ? c o s (x + h)}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{? 2 s i n (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [? \frac{2 s i n (\frac{2 x + h}{2}) s i n (? h / 2)}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{(? 1 s i n (\frac{2 x + h}{2})}{c o s (x + h) c o s x} \times \underset{h ? 0}{l i m} (? 1) \frac{s i n h / 2}{h / 2}$

$= \frac{s i n x}{c o s x ? c o s x} \times 1$

$= t a n x ? s e c x .$

(iii) Given f(x)=5 sec x+4 cosx.

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$\begin{array}{l} = \underset{h ? 0}{l i m} \frac{5 s e c (x + h) + 4 c o s (x + h) ? [5 s e c x + 4 c o s x]}{h} . \end{array}$

$= \underset{h ? 0}{l i m} \frac{5}{h} [s e c (x + h) ? s e c x] + \underset{h ? 0}{l i m} \frac{4}{h} [c o s (x + h) ? c o s x]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [\frac{1}{c o s (x + h)} ? \frac{1}{c o s x}] + \underset{h ? 0}{l i m} \frac{4}{h} [? 2 s i n (\frac{x + h + x}{2}) s i n (\frac{x + h ? x}{2})]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [\frac{c o s x ? c o s (x + h)}{c o s (x + h) (c o s x)}] + \underset{h ? 0}{l i m} \frac{4}{h} [? 2 s i n (\frac{2 x + h}{2}) s i n \frac{h}{2}]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [? \frac{2 s i n (\frac{2 x + h}{2}) s i n (? h / 2)}{c o s (x + h) c o s x}] ? 4 \underset{h ? 0}{l i m} s i n (\frac{2 x + h}{2}) \underset{h ? 0}{l i m} \frac{s i n h / 2}{h / 2}$

$= \frac{s i n (\frac{2 x + 0}{2})}{c o s (x + 0) c o s x} \times 1 ? 4 s i n (\frac{2 x}{2})$

$= 5 \frac{s i n x}{c o s x} ? \frac{1}{c o s x} ? 4 s i n x$

$= 5 t a n x ? s e c x ? 4 s i n x$

$(iv) Given f (x) = c o s e c x$

$f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{h} [c o s e c (x + h) ? c o s e c x]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{1}{s i n (x + h)} ? \frac{1}{s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{s i n x ? s i n (x + h)}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})]}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{2 x + h}{2}) s i n (? h / 2)]}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{c o s (\frac{2 x + h}{2})}{s i n (x + h) s i n x} \times \frac{(? 1) s i n (2)}{h / 2})$

$= \frac{c o s (\frac{2 x + 0}{2})}{s i n (x + 0) s i n x} \times (? 1)$

$= ? \frac{c o s x}{s i n x} \times \frac{1}{s i n x}$

$= ? c o t x ? c o s e c x$

(v) Given,f(x)=3 cot x+5cosecx.

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$ $= \frac{2}{c o s (x + 0) c o s x} \times 1 + 7 s i n (\frac{2 x + 0}{2}) \times (? 1)$

$= \underset{h ? 0}{l i m} 3 c o t (x + h) + 5 c o s e c (x + h) ? [3 c o t x + 5 c o s x)$

$_{h ? 0} \frac{3}{h} [c o t (x + h) ? c o t x] + \underset{h ? 0}{l i m}$

$= ? \frac{5}{h} [c o s e c (x + h) ? c o s e c x]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{c o s (x + h)}{s i n (x + h)} ? \frac{c o s x}{s i n x}] + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{1}{s i n (x + h)} ? \frac{1}{s i n x}]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{c o s (x + h) s i n x ? c o s x s i n (x + h)}{s i n (x + h) s i n x}] + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{s i n x ? s i n (x + h)}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{s i n (x ? (x + h))}{s i n (x + h) s i n x} + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{s i n (x + h) s i n x}$

$= \underset{h ? 0}{l i m} \frac{3}{s i n (x + h) s i n x} \times \underset{h ? 0}{l i m} (? 1) \frac{s i n h}{h} + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{2 c o s (\frac{2 x + h}{2}) s i n (? h / 2)}{s i n (x + h) s i n x}$

$= \frac{3}{s i n x ? s i n x} \times (? 1) + 5 \underset{hQ: 52. Kindly consider the following \frac{p x^{2} + q x + r}{a x + b}A: 52. Given, f (x) = \frac{p x^{2} + q x + r}{a x + b} f^{'} (x) = \frac{(a x + b) \frac{d}{d x} (p x^{2} + q x + r) - (p x^{2} + q x + r) \frac{d}{d x} (a x + b)}{{(a x + b)}^{2}} = \frac{(a x + b) (2 x p + q) - (p x^{2} + q x + r) (a)}{{(a x + b)}^{2}} = \frac{2 a p x^{2} + 2 b p x + a q x + b q - a p x^{2} - a q x - a r}{{(a x + b)}^{2}} = \frac{a p x^{2} + 2 b p x + b q - a r}{{(a x + b)}^{2}}Q: 59. Kindly consider the following \frac{c o s x}{1 + s i n x}A: 59. Given, f (x) = \frac{c o s x}{1 + s i n x} So, f?(x) = \frac{(1 + s i n x) \frac{d}{d x} (c o s x) - c o s x \frac{d}{d x} (1 + s i n x)}{{(1 + s i n x)}^{2}} Putting (2) and (3) in (1) we get, f^{'} (x) = \frac{(1 + s i n x) (- s i n x) - c o s x (c o s x)}{{(1 + s i n x)}^{2}} = \frac{s i n x - s i n^{2} x - c o s^{2} x}{{(1 + s i n x)}^{2}} = \frac{- s i n x - (s i n^{2} x + c o s^{2} x)}{{(1 + s i n x)}^{2}} = \frac{- (s i n x + 1)}{{(1 + s i n x)}^{2}} = - \frac{1}{1 + s i n x} .Q: 66. Kindly consider the following (x 2 + 1) cos xA: 66. Given, f (x) = (x 2 + 1) cos x f? (x) = (x 2 + 1) \frac{d}{d x} c o s x + c o s x \frac{d}{d x} (x^{2} + 1) = (x^{2} + 1) (- s i n x) + c o s x (2 x + 0) [? \frac{d}{d x} c o s x = - s i n x] = x 2 sin x sin x + 2x cos x.Q: Kindly consider the following 68. (x + cos x) (x tan x)A: 68. Given, f (x) = (x + cos x) (x tan x) So, f?(x) = (x + cos x) \frac{d}{d x} - t a n x) + \frac{d}{d x} (x + c o s x) \cdot (x - t a n x) = (x + c o s x) (\frac{d x}{d x} - \frac{d}{d x} t a n x) + (\frac{d}{d x} + \frac{d c o s x}{d x}) (x - t a n x) Let g (x) = tan x. So, g?(x) = = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h)}{c o s (x + h)} - \frac{s i n x}{c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h) c o s x - s i n x c o s (x + h)}{c o s (x + h) \cdot c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h - x)}{c o s x \cdot c o s (x + h)}] = \underset{h \to 0}{l i m} \frac{1}{c o s x \cdot c o} s (x + h) \times \underset{h \to 0}{l i m} \frac{s i n h}{h} = \frac{1}{c o s^{2} x} \times 1 = sec 2 x______ (2) Put (2) in (1) we get, f ?(x) = (x + cos x) (1 - sec 2 x) +(1 - sin x) (x - tan x) We know that, 1 + tan 2 x = sec 2 x Þ 1 - sec 2 x = - tan 2 x So, f ?(x) = - tan 2 x (x + cos x) +(x - tan x) (1 - sin x).Q: 17. Kindly consider the followingA: 17. Kindly go through the solution = = \underset{x \to 0}{l i m} \frac{2 s i n^{2} x}{2 s i n^{2} \frac{x}{2}} = \frac{\underset{x \to 0}{l i m} {(\frac{s i n x}{x})}^{2} \times x^{2}}{\underset{x \to 0}{l i m} {(\frac{s i n^{} \frac{x}{2}}{\frac{x}{2}})}^{2} {\frac{x}{2}}^{2}} = \frac{{(1)}^{2} \times x^{2} \times 4}{{(1)}^{2} \times x^{2}} = 4Q: 30. If f (x) = {\begin{matrix} | x | + 1, & x < 0 & 0, \\ x = 0 & | x | - 1, & x > 0 \end{matrix} For what value (s) of a does \underset{x \to a}{l i m} f (x) exists?A: 30. Given, f (x) = {\begin{matrix} | x | + 1, & x < 0 & 0, \\ x = 0 & | x | - 1, & x > 0 \end{matrix} \underset{x \to a}{l i m} f (x) = ? As | x | = {\begin{cases} - x, x < 0 \\ x, x > 0 \end{cases} We can rewrite f (x) = \begin{array}{l} {\begin{cases} - x + 1, x < 0 \\ 0, x = 0 \end{cases} \\ x - 1, x > 0 . \end{array} Case 1: when a<0, \underset{x \to a}{l i m} f (x) = \underset{x \to a}{l i m} (- x + 1) = - a + 1 So, \underset{x \to a}{l i m} f (x) = exist such that a< 0 Case II when a> 0, \underset{x \to a}{l i m} f (x) = \underset{x \to a}{l i m} (x - 1) = a - 1 So, \underset{x \to a}{l i m} f (x) exist such that a>0. Case III when a = 0. L.H.L = \underset{x \to a^{-}}{l i m} f (x) = \underset{x \to a^{-}}{l i m} (- x + 1) = \underset{x \to 0^{-}}{l i m} (- x + 1) = 1 R.H.L = \underset{x \to a^{+}}{l i m} f (x) = \underset{x \to a^{+}}{l i m} (x - 1) = \underset{x \to 0^{+}}{l i m} (x - 1) = - 1 Thus, \underset{x \to a^{-}}{l i m} f (x) \neq \underset{x \to a^{+}}{l i m} f (x) So, \underset{x \to a}{l i m} f (x) does not exist at a = 0.Q: 22. \underset{x \to \frac{π}{2}}{l i m} \frac{t a n 2 x}{x - \frac{π}{2}}A: 22. \underset{x \to \frac{x}{2}}{l i m} \frac{t a n 2 x}{x - \frac{π}{2}} Put y = x \frac{π}{2} . So that as y 0 cos \frac{π}{2}Q: 12. \lim_{x \to - 2} \frac{\frac{1}{x} + \frac{1}{2}}{x + 2}A: Kindly go through the solution = \frac{1}{2 \times (- 2)} = - \frac{1}{4} .Q: 10 \lim_{z \to 1} \frac{z^{\frac{1}{3}} - 1}{z^{\frac{1}{6}} - 1}A: 10. = \underset{z \to 1}{l i m} [\frac{z^{\frac{1}{3}} - 1^{\frac{1}{3}}}{z - 1}] \div \underset{z \to 1}{l i m} [\frac{z^{\frac{1}{6}} - 1^{\frac{1}{6}}}{z - 1}] = \underset{z \to 1}{l i m} [\frac{z^{\frac{1}{3}} - 1^{\frac{1}{3}}}{z - 1}] \div \underset{z \to 1}{l i m} [\frac{z^{\frac{1}{6}} - 1^{\frac{1}{6}}}{z - 1}] We know that, \underset{x \to a}{l i m} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1} So, = \frac{1}{3} \times 6 = 2Q: 6. \lim_{x \to 0} \frac{{(x + 1)}^{5} - 1}{x}A: 6. \underset{x \to 0}{l i m} \frac{{(x + 1)}^{5} - 1}{x} = \underset{x \to 0}{l i m} x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1 - 1 Q6. \lim_{x \to 0} \frac{{(x + 1)}^{5} - 1}{x} 6. \underset{x \to 0}{l i m} \frac{{(x + 1)}^{5} - 1}{x} = \underset{x \to 0}{l i m} x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1 - 1 = \underset{x \to 0}{l i m} x^{4} + 5 x^{3} + 10 x^{2} + 10 x + 5 . (0)4 + 5(0)3 + 10(0)2 + 10(0) + 5 = 5.Q: Kindly consider the following 71. \frac{x}{1 + t a n x}A: 71. Given, f (x) = \frac{x}{1 + t a n x} = \frac{x}{1 + \frac{s i n x}{c o s x}} = \frac{c o s x \cdot x}{c o s x + s i n x} S o, f^{'} (x) = \frac{(c o s x + s i n x) \frac{d}{d x} (x \cdot c o s x) - (x \cdot c o s x) \frac{d}{d x} (c o s x + s i n x)}{{(c o s x + s i n x)}^{2}} = \frac{(c o s x + s i n x) (- x s i n x + c o s x) - (- x \cdot s i n x c o s x + x c o s^{2} x)}{{(c o s x + s i n x)}^{2}} = \frac{- x c o s x s i n x + c o s^{2} x - x s i n^{2} x + s i n x c o s x + x s i n x c o s x - x c o s^{2} x}{{(c o s x + s i n x)}^{2}} = \frac{c o s^{2} x + s i n x c o s x - x (s i n^{2} x + c o s^{2} x)}{{(c o s x + c s i n x)}^{2}} Dividing numerator and denominator by cos2x we get, = \frac{\frac{c o s^{2} x}{c o s^{2} x} + \frac{s i n x}{c o s x} \cdot \frac{c o s x}{c o s x} - x \frac{1}{c o s^{2} x}}{{(\frac{c o s x}{c o s x} + \frac{s i n x}{c o s x})}^{2}} = \frac{1 + t a n x - x s e c^{2} x}{{(1 + t a n x)}^{2}}Q: 1. l \underset{x \to 3}{i m} x+ 3A: Exercise 13.1 1. 1. \underset{x \to 3}{l i m} x + 3 = 3 + 3 = 6.Q: 7. \lim_{x \to 2} \frac{3 x^{2} - x - 10}{x^{2} - 4}A: 7. = \underset{x \to 2}{l i m} \frac{x (3 x + 5) - 2 (3 x + 5)}{(x - 2) (x + 2)} = \underset{x \to 2}{l i m} \frac{(x - 2) (3 x + 5)}{(x - 2) (x + 2)} = \underset{x \to 2}{l i m} \frac{3 x + 5}{x + 2} = \frac{3 \times 2 + 5}{2 + 2} = \frac{6 + 5}{4} = \frac{11}{4} .Q: 9. Li m_{x \to 0} \frac{a x + b}{c x + 1}A: Kindly go through the solutionQ: 18. \underset{x \to 0}{l i m} \frac{a x + x c o s x}{b s i n x}A: 18. \underset{x \to 0}{l i m} \frac{a x + x c o s x}{b s i n x} = \underset{x \to 0}{l i m} \frac{x (a + c o s x)}{b s i n x} = \frac{1}{b} \times \frac{(a + c o s 0)}{1} = \frac{a + 1}{b} View other answersQ: 16. \underset{x \to 0}{l i m} \frac{c o s x}{π - x}A: 16. \underset{x \to 0}{l i m} \frac{c o s x}{π - x} = \frac{c o s 0}{π - 0} = \frac{1}{π}Q: 15. \underset{x \to π}{l i m} \frac{s i n (π - x)}{π (π - x)}A: 15. \underset{x \to π}{l i m} \frac{s i n (π - x)}{π (π - x)} = \frac{1}{π} \times \underset{x \to π}{l i m} \frac{s i n (π - x)}{π - x} = \frac{1}{π} .Q: 8. \lim_{x \to 3} \frac{x^{4} - 81}{2 x^{2} - 5 x - 3}A: 8. = \underset{x \to 3}{l i m} \frac{(x + 3) (x^{2} + 9)}{2 x + 1 .} = \frac{(3 + 3) (3^{2} + 9)}{2 \times 3 + 1} = \frac{6 \times 18}{6 + 1} = \frac{108}{7}Q: 14. \underset{x \to 0}{l i m} \frac{s i n a x}{s i n b x}, a, b \neq 0A: 14. \underset{x \to 0}{l i m} \frac{s i n a x}{s i n b x} = \underset{x \to 0}{l i m} \frac{\frac{s i n a x}{a x} \times a x}{\frac{s i n b x}{b x} \times b x} = \frac{a}{b}Q: 19. \underset{x \to 0}{l i m} x s e c xA: 19. \underset{x \to 0}{l i m} x s e c x = \underset{x \to 0}{l i m} \frac{x}{c o s x} = \frac{0}{c o s 0} = \frac{0}{1} = 0.Q: 4. \lim_{x \to 4} \frac{4 x + 3}{x - 2}A: 4. \underset{x \to 4}{l i m} \frac{4 x + 3}{x - 2} = \frac{4 \times 4 + 3}{4 - 2} = \frac{16 + 3}{2} = \frac{19}{2}Q: 5. \lim_{x \to - 1} \frac{x^{10} + x^{5} + 1}{x - 1}A: 5. \underset{x \to - 1}{l i m} \frac{x^{10} + x^{5} + 1}{x - 1} = \frac{{(- 1)}^{10} + {(- 1)}^{5} + 1}{(- 1) - 1} = \frac{1 - 1 + 1}{- 2} = - \frac{1}{2}Q: 21. \underset{x \to 0}{l i m} (c o s e c x - c o t x)A: 21. \underset{x \to 0}{l i m} (c o s e c x - c o t x) = \underset{x \to 0}{l i m} (\frac{1}{s i n x} - \frac{c o s x}{s i n x}) = \underset{x \to 0}{l i m} (\frac{1 - c o s x}{s i n x}) = \underset{x \to 0}{l i m} \frac{2 s i {n^{2}}^{\frac{x}{2}}}{2 s i n^{\frac{x}{2}} c o s^{\frac{x}{2}}} {\begin{matrix} \end{matrix} ? c o s 2 x = 1 - 2 s i n^{2} x \Rightarrow 1 - c o s 2 x = 2 s i n^{2} x 1 - c o s x = 2 s i n^{2} \frac{x}{2} ? s i n 2 x = 2 s i n x c o s x s i n x = 2 s i n \frac{x}{2} c o s \frac{x}{2} {= \underset{x \to 0}{l i m} \frac{x}{2} = t a n \frac{0}{2} = 0 .Q: 2. \lim_{x \to π} (x - \frac{22}{7})A: 2. \underset{x \to π}{l i m} (x - \frac{22}{7}) = π - \frac{22}{7}Q: 3. \lim_{r \to 1} π r^{2}A: 3 . \underset{r \to 1}{l i m} π r^{2} = π \cdot {(1)}^{2} = πQ: 11. \lim_{x \to 1} \frac{a x^{2} + b x + c}{c x^{2} + b x + a} = a + b + c \neq 0A: 11. \underset{x \to 1}{l i m} \frac{a x^{2} + b x + c}{c x^{2} + b x + a} = \frac{a + b + c}{c + b + a} = 1Q: 13. \lim_{x \to 0} \frac{a x}{b x}A: 13. \underset{x \to 0}{l i m} \frac{s i n a x}{b x} = \frac{1}{b} \underset{x \to 0}{l i m} \frac{s i n a x}{a x} \times a = \frac{a}{b} \underset{x \to 0}{l i m} \frac{s i n a x}{a x} = \frac{a}{b}Q: 20. \underset{x \to 0}{l i m} \frac{s i n a x + b x}{a x + s i n b x} a, b, a + b \neq 0A: 20. \underset{x \to 0}{l i m} \frac{s i n a x + b x}{a x + s i n b x} = \frac{\underset{x \to 0}{l i m} \frac{s i n a x}{a x} \times a x + b x}{\underset{x \to 0}{l i m} a x + \frac{s i n b x}{b x} \times b x} . \frac{\underset{x \to 0}{l i m} \frac{s i n a x}{a x} \times \underset{x \to 0}{l i m} a x + \underset{x \to 0}{l i m} b x}{\underset{x \to 0}{l i m} a x + \underset{x \to 0}{l i m} \frac{s i n b x}{b x} \times \underset{x \to 0}{l i m} b x} = \frac{\underset{x \to 0}{l i m} a x + \underset{x \to 0}{l i m} b x}{\underset{x \to 0}{l i m} a x + \underset{x \to 0}{l i m} b x} = \underset{x \to 0}{l i m} \frac{a x + b x}{a x + b x} = \underset{x \to 0}{l i m} 1 = 1.Q: 23. Find \underset{x \to 0}{l i m} f (x) and \underset{x \to 1}{l i m} f (x), where f (x) {\begin{array}{l} 2 x + 3, & x \leq 0 \\ 3 (x + 1), & x > 0 \end{array}}A: 23. Given f (x) {\begin{array}{l} 2 x + 3, & x \leq 0 \\ 3 (x + 1), & x > 0 \end{array}} for \underset{x \to 0}{l i m} f (x), left hand limit, L.H.S = \underset{x \to 0^{}}{l i m} f (x) = \underset{x \to 0^{}}{l i m} (2 x + 3) = 2 0 + 3 = 3. Right hand limit, R.H.L = \underset{x \to 0^{+}}{l i m} f (x) = \underset{x \to 0^{+}}{l i m} 3 (x + 1) = (0 + 1) = 3 1 = 3. Thus, \underset{x \to 0^{-}}{l i m} f (x) = \underset{x \to 0^{+}}{l i m} f (x) = \underset{x \to 0^{-}}{l i m} f (x) = 3 For \underset{x \to 1}{l i m} f (x), L.H.L = \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{-}}{l i m} 3 (x + 1) = 3 (1 + 1) = 3 \times 2 = 6 R.H.L = \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} 3 (x + 1) = 3 (1 + 1) = 3 \times 2 = 6 . Thus, \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1}{l i m} f (x) = 6 .Q: 24. Find \underset{x \to 1}{l i m} f (x), where f (x) = {\begin{matrix} x^{2} - 1, x \leq 1 & - x^{2} - 1, x > 1 \end{matrix}A: 24. Given f (x) = {\begin{matrix} x^{2} - 1, x \leq 1 & - x^{2} - 1, x > 1 \end{matrix}, \underset{x \to 1}{l i m} f (x) = ? Now, L.H.L = \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{-}}{l i m} (x^{2} - 1) 12 - 1 = 0 And R.H.L = \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} (- x^{2} - 1) (1)2 1 = 1 = 2. Thus, \underset{x \to 1^{-}}{l i m} f (x) \neq \underset{x \to 1^{+}}{l i m} f (x) So, \underset{x \to 1}{l i m} f (x) does not exist.Q: 25. Evaluate \underset{x \to 0}{l i m} f (x), where f (x) = {\begin{matrix} \frac{| x |}{x}, x \neq 0 & 0, x = 0 \end{matrix}A: 25. Given f (x) = {\begin{matrix} \frac{| x |}{x}, x \neq 0 & 0, x = 0 \end{matrix}, \underset{x \to 0}{l i m} f (x) = ? We know that, | x | = {\begin{matrix} x, & x \geq 0 \\ - x, & x < 0 \end{matrix} Now, L.H.L = \underset{x \to 0^{-}}{l i m} f (x) = \underset{x \to 0^{-}}{l i m} \frac{- x}{x} = \underset{x \to 0^{-}}{l i m} - 1 = - 1 and R.H.L = \underset{x \to 0^{+}}{l i m} f (x) = \underset{x \to 0^{+}}{l i m} \frac{x}{x} = \underset{x \to 0^{+}}{l i m} 1 = 1 Thus, \underset{x \to 0^{-}}{l i m} f (x) \neq \underset{x \to 0^{+}}{l i m} f (x) i e, \underset{x \to 0}{l i m} f (x) does not exist.Q: 26. Find \underset{x \to 0}{l i m} f (x), wheref (x) {\begin{array}{l} \frac{x}{| x |}, & x \neq 0 \\ 0, & x = 0 \end{array}A: 26. Given, f (x) {\begin{array}{l} \frac{x}{| x |}, & x \neq 0 \\ 0, & x = 0 \end{array} L.H.S = \underset{x \to 0^{-}}{l i m} f (x) = \underset{x \to 0^{-}}{l i m} \frac{x}{- x} = \underset{x \to 0^{-}}{l i m} - 1 = - 1 R.H.L \underset{x \to 0^{+}}{l i m} f (x) = \underset{x \to 0^{+}}{l i m} \frac{x}{x} = \underset{x \to 0^{+}}{l i m} 1 = 1 Thus, \underset{x \to 0^{-}}{l i m} f (x) \neq \underset{x \to 0^{+}}{l i m} f (x) i e, \underset{x \to 0}{l i m} f (x) does not exist.Q: 27.Find \underset{x \to 5}{l i m} f (x), w h e r e f (x) = | x | - 5A: 27. \underset{x \to 5}{l i m} f (x) = \underset{x \to 5}{l i m} | x | - 5 = | 5 | - 5 = 5 - 5 = 0Q: 28. Supposef (x) = {\begin{cases} a + b x, x < 1 \\ 4, x = 1 \\ b - a x, x > 1 \end{cases} and if \underset{x \to 1}{l i m} f (x) = f (1) what are possible values of a and b?A: 28. Given, f (x) = \begin{array}{l} {\begin{cases} a + b x, x < 1 \\ 4, x = 1 \end{cases} \\ b - a x, x > 1 \end{array} Since we need \underset{x \to 1}{l i m} f (x) we need, LHL = \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{-}}{l i m} (a + b x) = a + b × 1 = a + b and RHL = \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} (b - a x) = b - a × 1 = b - a Given, \underset{x \to 1}{l i m} f (x) = f (1) : we have the following equations a + b = 4 ____ (1) b - a = 4 ____ (2) Adding (1) and (2) we get, 2b = 8 b = 4 Putting b = 4 in (1) we get a + 4 = 4 a = 0Q: 31. If the function f(x) satisfies \underset{x \to 1}{l i m} \frac{f (x) - 2}{x^{2} - 1} = π, evaluate \underset{x \to 1}{l i m} f (x) .A: 31 . Given, \underset{x \to 1}{l i m} \frac{f (x) - 2}{x^{2} - 1} = π \frac{\underset{x \to 1}{l i m} f (x) - 2}{\underset{x \to 1}{l i m} x^{2} - 1} = π \Rightarrow \underset{x \to 1^{}}{l i m} [f (x) - 2] = π \underset{x \to 1}{l i m} (x^{2} - 1) \Rightarrow \underset{x \to 1}{l i m} f (x) - \underset{x \to 1}{l i m} 2 = π (1^{2} - 1) \Rightarrow \underset{x \to 1}{l i m} f (x) - 2 = 0 . \Rightarrow \underset{x \to 1}{l i m} f (x) = 2Q: 39. For some constants aand b, find the derivative of (i) (x- a) + (x - b) (ii) {(a x^{2} + b)}^{2} (iii) \frac{x - 9}{x - b} Read moreA: (c) Given, f(x)=(x-a)(x-b) where a and b are constants. So, f^{'} (x) = (x - a) \frac{d}{d x} (x - b) + (x - b) \frac{d}{d x} (x - a) =(x-a)+(x-b) = 2x– a- b. (ii) Given f(x)= {(a x^{2} + b)}^{2} . where ab are constant So, f^{'} (x) = \frac{d}{d x} {(a x^{2} + b)}^{2} = \frac{d}{d x} (a^{2} x^{4} + b^{2} + 2 a x^{2} b) = \frac{d}{d x} a^{2} x^{4} + \frac{d}{d x} b^{2} + \frac{d}{d x} 2 a x^{2} b = 4 a^{2} x^{3} + 0 + 4 a x b = 4 a x (a x^{2} + b) . (iii) Given, f(x)= \frac{x - 9}{x - b} where a and bare constants So, f (x) = \frac{(x - b) \frac{d}{d x} (x - a) - (x - a) \frac{d}{d x} (x - b)}{{(x - b)}^{2}} = \frac{(x - b) - (x - a)}{{(x - b)}^{2}} = \frac{a - b}{{(x - b)}^{2}}Q: 41. Find the derivative of (i) 2 x - \frac{3}{4} (ii) (5 x^{3} + 3 x - 1) (x - 1) (iii) x^{- 3} (5 + 3 x) (iv) x^{5} (3 - 6 x^{- 9}) (v) x^{- 4} (3 - 4 x^{- 5}) (vi) \frac{2}{x + 1} - \frac{x^{2}}{3 x - 1} Read moreA: 41. (i) f (x) = 2 x - \frac{3}{4} f^{'} (x) = \frac{d}{d x} (2 x - \frac{3}{4}) = 2 \frac{d x}{d x} - 0 =2. (ii) Given, f(x)= (5 x^{3} + 3 x - 1) (x - 1) So, f (x) = (5 x^{3} + 3 x - 1) \frac{d}{d x} (x - 1) + (x - 1) \frac{d}{d x} (5 x^{3} + 3 x - 1) = (5 x^{3} + 3 x - 1) . 1 + (x - 1) (15 x^{2} + 3) = 5 x^{3} + 3 x - 1 + i 5 x^{3} + 3 x - 15 x^{2} - 3 = 20 x^{3} - 15 x^{2} + 6 x - 4 . (iii) Given, f(x) = x^{- 3} (5 + 3 x) So, \begin{array}{l} f^{'} (x) = x^{- 3} \frac{d}{d x} (5 + 3 x) + (5 + 3 x) \frac{d x^{- 3}}{d x} \\ = x^{- 3} \cdot 3 + (5 + 3 x) \cdot (- 3) x^{- 4} \end{array} = \frac{3}{x^{3}} - \frac{15}{x^{4}} - \frac{9}{x^{3}} = \frac{- 15}{x^{4}} - \frac{6}{x^{3}} = - \frac{3}{x^{4}} (5 + 2 x) . (iv) Given, f(x)= x^{5} (3 - 6 x^{- 9}) . f^{'} (x) = x^{5} \frac{d}{d x} (3 - 6 x^{- 9}) + (3 - 6 x^{- 9}) \frac{d}{d x} x^{5} . = x^{5} (- 6 x - 9 x^{- 10}) + (3 - 6 x^{- 9}) \cdot 5 x^{4} = x^{5} (54 x^{- 10}) + 15 x^{4} - 30 x^{- 5} = 54 x^{- 5} - 30 x^{- 5} + 15 x^{4} = \frac{24}{x^{5}} + 15 x^{4} (v) Given, f(x)= x^{- 4} (3 - 4 x^{- 5}) . So, f^{'} (x) = x^{- 4} \cdot \frac{d}{d x} (3 - 4 x^{- 5}) + (3 - 4 x^{- 5}) \frac{d}{d x} x^{- 4} = x^{- 4} \cdot (- 4 x - 5 \times x^{- 6}) + (3 - 4 x^{- 5}) \cdot (- 4 x^{- 5}) = 20 x^{- 10} - 12 x^{- 5} + 16 x^{- 10} . = 36 x^{- 10} - 12 x^{- 5} . = \frac{36}{x^{10}} - \frac{12}{x^{5}} . (vi) Given, f(x)= \frac{2}{x + 1} - \frac{x^{2}}{3 x - 1} So, f^{'} (x) = \frac{d}{d x} (\frac{2}{x + 1}) - \frac{d}{d x} (\frac{x^{2}}{3 x - 1}) = \frac{(x + 1) \frac{d}{d x} - 2 \frac{d}{d x} (x + 1)}{{(x + 1)}^{2}} - \frac{(3 x - 1) \frac{d x^{2}}{d x} - x^{2} \frac{d}{d x} (3 x - 1)}{{(3 x - 1)}^{2}} = \frac{- 2}{{(x + 1)}^{2}} - \frac{2 x (3 x - 1) - 3 x^{2}}{{(3 x - 1)}^{2}} = - \frac{2}{{(x + 1)}^{2}} - \frac{6 x^{2} - 2 x - 3 x^{2}}{{(3 x - 1)}^{2}} = - \frac{2}{{(x + 1)}^{2}} - \frac{3 x^{2} - 2 x}{{(3 x - 1)}^{2}} = - \frac{2}{{(x + 1)}^{2}} - \frac{x (3 x - 2)}{{(3 x - 1)}^{2}}Q: 44. Find the derivative of the following functions from first principle: (i) x (ii) (x)1 (iii) sin(x + 1) (iv) cos (x - \frac{π}{8}) Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Read moreA: 44. = \underset{h \to 0}{l i m} \frac{- x - h + x}{h} = \underset{h \to 0}{l i m} \frac{- h}{h} = \underset{h \to 0}{l i m} - 1 = - 1. (ii) Given, f (x) = (- x) - 1 by first principle, f(x) \underset{h \to 0}{l i m} \frac{[-] {(x + h)}^{- 1} - {(- x)}^{- 1}}{h} (iii) Given, f(x) = sin(x + 1) By first principle, f’(x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{s i n (x + h + 1) - s i n (x + 1)}{h} = \underset{h \to 0}{l i m} \frac{2}{h} \cdot c o s (\frac{x + h + 1 + x + 1}{2}) \cdot s i n (\frac{x + h + 1 - (x + 1)}{2}) = \underset{h \to 0}{l i m} \frac{2}{h} c o s (\frac{2 x + 2 + h}{2}) s i n (\frac{h}{2}) . = \underset{h \to 0}{l i m} c o s (\frac{2 x + 2 + h}{2}) \underset{h \to 0}{l i m} \frac{s i n \frac{h}{2}}{\frac{h}{2}} = c o s (\frac{2 x + 2 + 0}{2}) \times 1 . = cos (x + 1) (iv) Given, f(x) = cos (x - \frac{π}{8}) By first principle, f(x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{c o s (x + h - \frac{π}{8}) - c o s (x - \frac{π}{8})}{h}Q: 48. Kindly consider the following \frac{1}{a x^{2} + b x + c}A: 48. Given, f (x) = \frac{1}{a x^{2} + b x + c} So, f? (x) = \frac{(a x^{2} + b x + c) \frac{d}{d x} (1) - 1 \cdot \frac{d}{d x} (a x^{2} + b x + c)}{{(a x^{2} + b x + c)}^{2}} = \frac{(a x^{2} + b x + c) (0) - (2 a x + b)}{{(a x^{2} + b x + c)}^{2}} = - \frac{(2 a x + b)}{{(a x^{2} + b x + c)}^{2}}Q: 54. Kindly consider the followingA: 54. So f? (x) = \frac{d}{d x} (4 - 2) = \frac{d}{d x} (4) - \frac{d}{d x} (2) = 4 \frac{d}{d x} (x \frac{1}{2}) - 0 = 4 \times \frac{1}{2} {x^{\frac{1}{2}}}^{- 1} = 2 x^{- \frac{1}{2}} = 2Q: 60. Kindly consider the following \frac{s i n x + c o s x}{s i n x - c o s x}A: 60. Given, f (x) = \frac{s i n x + c o s x}{s i n x - c o s x} So, f?(x) = \frac{s i n x - c o s x \frac{d}{d x} (s i n x + c o s x) - (s i n x + c o s 2) \frac{d}{d x} (s i n x - c o s x)}{{(s i n x - c o s x)}^{2}} Let g(x) = cos x and p(x) = sin x. {from so g’(x) A ) (upto equation 3) Let g(x) = cos2 and p(x) = sin x. So, g?(x) = \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [c o s (x + h) - c o s x] = \underset{h \to 0}{l i m} \frac{1}{h} [- 2 \cdot s i n (\frac{x + h + x}{2}) s i n (\frac{x + h - x}{2})] = \underset{h \to 0}{l i m} \frac{1}{h} [- 2 s i n (\frac{2 x + h}{2}) s i n (\frac{h}{2})] = - s i n (\frac{2 x + 0}{2}) \times 1 = -sin x ______ (2) And p?(x) = \underset{h \to 0}{l i m} \frac{p (x + h) - p (x)}{h} = \underset{h \to 0}{l i m} \frac{s i n (x + h) - s i n x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} \cdot 2 c o s (\frac{2 x + h}{2}) s i n (\frac{h}{2}) = \underset{h \to 0}{l i m} c o s (\frac{2 x + h}{2}) \times \underset{h \to 0}{l i m} \frac{s i n (\frac{h}{2})}{(\frac{h}{2})} = cos x _____ (3) Putting (2) and (3) in (1) we get, f^{'} (x) = \frac{(s i n x - c o s x) [c o s x - s i n x] - (c s i n x + c o s x) [c o s x + s i n x]}{{(s i n x - c o s x)}^{2}} = \frac{- {(s i n x - c o s x)}^{2} - {(s i n x + c o s x)}^{2}}{{(s i n x - c o s x)}^{2}} = \frac{- (s i n^{2} x + c o s^{2} x) + 2 s i n x c o s x - (s i n^{2} x + c o s^{2} x) - 2 s i n x c o s x}{{(s i n x - c o s x)}^{2}} = \frac{- 1 - 1}{{(s i n x - c o s x)}^{2}} = \frac{- 2}{{(s i n x - c o s x)}^{2}}Q: 61. Kindly consider the following \frac{s e c x - 1}{s e c x + 1}A: 61. Given, f (x) = \frac{s e c x - 1}{s e c x + 1} = \frac{\frac{1}{c o s x} - 1}{\frac{1}{c o s x} + 1} = \frac{1 - c o s x}{1 + c o s x} So, f?(x) = \frac{(1 + c o s x) \frac{d}{d x} (1 - c o s x) - (1 - c o s x) \frac{d}{d x} (1 + c o s x)}{{(1 + c o s x)}^{2}} = \frac{(1 + c o s x) (- 1) \frac{d}{d x} (- c o s x) - (1 - c o s x) \frac{d}{d x} (c o s x)}{{(1 + c o s x)}^{2}} Let g(x) = cos x. So, g?(x) = \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = \underset{h \to 0}{l i m} \frac{c o s (x + h) - c o s x}{h} = \underset{h \to 0}{l i m} - \frac{2}{h} s i n (\frac{x + h + x}{2}) s i n (\frac{x + h - x}{2}) = \underset{h \to 0}{l i m} \frac{- 2}{h} s i n (\frac{2 x + h}{2}) s i n (\frac{h}{2}) = \underset{h \to 0}{l i m} - s i n (\frac{2 x + h}{2}) \times \underset{h \to 0}{l i m} \frac{s i n \frac{h}{2}}{\frac{h}{2}} = - s i n (\frac{2 x + 0}{2}) \times 1 = -sin x. So, f?(x) \frac{(1 + c o s x) (- 1) (- s i n x) - (1 - c o s x) (- s i n x)}{{(1 + c o s x)}^{2}} = \frac{s i n x + c o s x s i n x + s i n x - s i n x c o s x}{{(1 + c o s x)}^{2}} = \frac{2 s i n x}{{(1 + c o s x)}^{2}} = \frac{2 s i n x}{{(1 + \frac{1}{s e c x})}^{2}} = \frac{2 s i n x \times s e c^{2} x}{{(s e c x + 1)}^{2}} = \frac{2 s e c x \cdot s i n x}{{(s i n x + 1)}^{2} \cdot c o s x} . = \frac{2 s e c x \cdot t a n x}{{(s i n x + 1)}^{2}}Q: 64. Kindly consider the following \frac{s i n (x + a)}{c o s x}A: 64. Given, f (x) = \frac{s i n (x + a)}{c o s x} f^{'} (x) = \frac{c o s x \frac{d}{d x} s i n (x + a) - s i n (x + a) \frac{d}{d x} c o s x}{c o s^{2} x} Let g?(x) = sin (x + a) So, g?(x) = \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = cos (x + a) And P(x) = cos x So, P?(x) = \underset{h \to 0}{l i m} \frac{p (x + h) - p (x)}{h} Thus, f?(x) = \frac{c o s x \cdot c o s (x + a) - s i n (x + a) (- s i n x)}{c o s^{2} x} = \frac{c o s x \cdot c o s (x + a) + s i n (x + a) s i n x}{c o s^{2} x} = \frac{c o s (x + a - x)}{c o s^{2} x} = \frac{c o s a}{c o s^{2} x}Q: 66. Kindly consider the following x 4 (5 sin x 3 cos x)A: 65. Given, f (x) = x 4 . (5 sin x 3 cos x) f^{'} (x) = x^{4} \frac{d}{d x} (5 s i n x - 3 c o s x) + (5 s i n x - 3 c o s x) \frac{d x^{4}}{d x} . = x^{4} [5 \frac{d}{d x} s i n x - 3 \cdot \frac{d c o s x}{d x}] + [5 s i n x - 3 c o s x] \cdot 4 x^{3} As \frac{d}{d x} s i n x = c o s x and \frac{d}{d x} c o s x = - s i n x Thus, f^{'} (x) = x^{4} [5 c o s x + 3 s i n x] + [5 s i n x - 3 c o s x] \cdot 4 x^{3} = x^{3} [5 c o s x + 3 x s i n x + 20 s i n x - 12 c o s x] = x^{3} [5 x c o s x + 3 x s i n x + 20 s i n x - 12 c o s x] .Q: Kindly consider the following 70. \frac{x^{2} c o s (\frac{π}{4})}{s i n x}A: 70. Given, f (x) = \frac{x^{2} c o s (\frac{π}{4})}{s i n x} = \frac{x^{2}}{s i n x} \times c o s \frac{π}{4} So, f? (x) = = c o s \frac{π}{4} [\frac{2 \cdot x s i n x - x^{2} c o s x}{s i n^{2} x}] = x c o s \frac{π}{4} [\frac{2 s i n x - x c o s x}{s i n^{2} x}] .Q: Kindly consider the following 72. (x + sec x) (x - tan xA: 72. Given, f (x) = (x + sec x) (x tan x) So, f?(x) = (x + see x) \frac{d}{d x} (x - t a n x) + (x - t a n x) \frac{d}{d x} (x + s e c x) . = (x + s e c x) (\frac{d x}{d x} - \frac{d}{d x} t a n x) + (x - t a n x) (\frac{d x}{d x} + \frac{d}{d x} s e c x) Let g(x) = tan x. So, g?(x) = = \underset{h \to 0}{l i m} \frac{1}{h} [t a n (x + h) - t a n x] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h)}{c o s (x + h)} - \frac{s i n x}{c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h) c o s x - s i n x c o s (x + h)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h - x)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{c o s (x + h) c o s x} \times \underset{h \to 0}{l i m} \frac{s i n h}{h} = \frac{1}{c o s^{2} x} = s e c^{2} x . _____ (2) And P(x) = see x. = \underset{h \to 0}{l i m} \frac{s e c (x + h) - s e c x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{1}{c o s (x + h)} - \frac{1}{c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{c o s x - c o s (x + h)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{- 2 s i n (\frac{x + x + h}{2}) s i n (\frac{x - (x + h)}{2})}{c o s (x + h) c o s x}] = \frac{s i n x}{c o s^{2} x} = tan x sec x. ____ (3) Putting (2) and (3) in (1) we get, f ?(x) = (x + sec x) (1 - sec 2 x) + (x - tan x) (1 + tan x sec x)Q: Kindly consider the following 73. \frac{x}{s i n^{n} x}A: 73. Given, f (x) = \frac{x}{s i n^{n} x} So, f?(x) = \frac{(s i n^{x} x \frac{d x}{d x}) - (x \frac{d}{d x} s i n^{x} x)}{{(s i n^{x} x)}^{2}} = \frac{s i n^{x} x - x \cdot n {(s i n x)}^{x - 1} \frac{d}{d x} s i n x}{{(s i n x)}^{2 x}} = \frac{{(s i n x)}^{n} - x \cdot x {(s i n x)}^{n - 1} \cdot c o s x}{{(s i n x)}^{2 n}} = \frac{{(s i n x)}^{n - 1} [s i n x - n x \cdot c o s x]}{{(s i n x)}^{2 n}} = \frac{s i n x - n \cdot x \cdot c o s x}{{(s i n x)}^{2 n - (n - 1)}} = \frac{s i n x - n x c o s x}{{(s i n x)}^{n + 1}}Q: 32. If f (x) = {\begin{cases} \begin{array}{l} m x^{2} + n, & x < 0 \\ n x + m, & 0 \leq x \leq 1 \end{array} \\ n x^{3} + m, x > 1 \end{cases} . For what integers m and n does both \underset{x \to 0}{l i m} f (x) and \underset{x \to 1}{l i m} f (x) exist?A: 32. Given, f (x) = \begin{array}{l} {\begin{array}{l} m x^{2} + n, & x < 0 \\ n x + m, & 0 \leq x \leq 1 \end{array} \\ n x^{3} + m, x > 1 . \end{array} For \underset{x \to 0}{l i m} f (x) \underset{x \to 0^{-}}{l i m,} f (x) = \underset{x \to 0^{+}}{l i m} f (x) \Rightarrow \underset{x \to 0^{-}}{l i m} (m x^{2} + n) = \underset{x \to 0^{+}}{l i m} (m x^{3} + m) n = m So, \underset{x \to 0}{l i m} f (x) exist for n = m. Again, \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{-}}{l i m} n x + m = n + m . \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} n x^{3} + m = n + m So, \underset{x \to 1^{-}}{l i m} f (x) = \underset{x \to 1^{+}}{l i m} f (x) = \underset{x \to 1^{-}}{l i m} f (x) = n + m, Thus, \underset{x \to 1}{l i m} f (x) exist for any integral value of m and n.Q: 33. Find the derivativeA: 33. Given, f (x)=x 2 - 2 ., f^{'} (10) = ? We have, f^{'} (10) = \underset{h \to 0}{l i m} \frac{f (10 + h) - f (10)}{h} = \underset{h \to 0}{l i m} \frac{[{(10 + h)}^{2} - 2] - [1 0^{2} - 2]}{h} = \underset{h \to 0}{l i m} \frac{1 0^{2} + h^{2} + 20 h - 2 - 1 0^{2} + 2}{h} = \underset{h \to 0}{l i m} \frac{h (h + 20)}{h} \Rightarrow \underset{h \to 0}{l i m} h + 20 =20Q: 34. Find the derivative of x at x = 1.A: 34. Given, f (x)=x, f (1)=? We have, f^{'} (1) = \underset{h \to 0}{l i m} \frac{f (1 + h) - f (1)}{h} = \underset{h \to 0}{l i m} \frac{1 + h - 1}{h} = \underset{h \to 0}{l i m} \frac{h}{h} = \underset{h \to 0}{l i m} 1 =1Q: 35. Find the derivative of 99x at x = l00A: 35. Given, f (x)= 99x, f (100)=? So, f (100)=_{h \to 0} \frac{f (100 + h) - f (100)}{h} = \underset{h \to 0}{l i m} \frac{99 (100 + h) - 99 (100)}{h} = \underset{h \to 0}{l i m} \frac{99 \times 100 + 99 \times h - 99 \times 100}{h} = \underset{h \to 0}{l i m} \frac{99 h}{h} = \underset{h \to 0}{l i m} 99 =99Q: 36. Find the derivative of the following functions from first principle.A: 36. (i) x^{3} - 27 (ii) (x -1)(x-2) (iii) \frac{1}{x^{2}} (iv) \frac{x + 1}{x - 1} A.4.(i) Given, f (x) = x^{3} - 27 . So, f^{'} (x) = \frac{\underset{h \to 0}{l i m} f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{[{(x + h)}^{3} - 27] - [x^{3} - 27]}{h} = \underset{h \to 0}{l i m} \frac{x^{3} + h^{3} + 3 x h (x + h) - 27 - x^{3} + 27}{h} = \underset{h \to 0}{l i m} \frac{h (h^{2} + 3 x (x + h))}{h} = \underset{h \to 0}{l i m} h^{2} + 3 x (x + h) =0+3 x(x+ 0) =3 x 2 (ii) Given, f(x) =(x-1)(x-2) = x 2 - 3x+2 So, f^{'} (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} \underset{h \to 0}{l i m} \frac{[{(x + h)}^{2} - 3 (x + h) + 2] - [x^{2} - 3 x + 2]}{h} = \underset{h \to 0}{l i m} \frac{x^{2} + h^{2} + 2 x h - 3 x - 3 h + 2 - x^{2} + 3 x - 2}{h} = \underset{h \to 0}{l i m} \frac{h (h + 2 x - 3)}{h} = \underset{h \to 0}{l i m} h + 2 x - 3 = 2x – 3. (iii) Given, f(x)= \frac{1}{x^{2}} So, f^{'} (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}}{h} = \frac{\underset{h \to 0}{l i m} \frac{x^{2} - {(x + h)}^{2}}{{(x + h)}^{2} - x^{2}}}{h} = \underset{h \to 0}{l i m} \frac{x^{2} - x^{2} - h^{2} - 2 x h}{h x^{2} (x + h)^{2}} = \underset{h \to 0}{l i m} \frac{h (- h - 2 x)}{h x^{2} (x + h)^{2}} = \underset{h \to 0}{l i m} \frac{- h - 2 x}{x^{2} (x + h)^{2}} = \frac{- 0 - 2 x}{x^{2} (x + 0)^{2}} = \frac{- 2 x}{x^{4}} = \frac{- 2}{x^{3}} (iv) Given, f(x)= \frac{x + 1}{x - 1} f (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{f x + h + 1}{x + h - 1} - \frac{x + 1}{x - 1}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{(x + h + 1) (x - 1) - (x + 1) (x + h - 1)}{(x + h - 1) (x - 1)}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{x^{2} - x + h x - h + x - 1 - x^{2} - h x + x - x - h + 1}{(x - 1) (x + h - 1)}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{- 2 h}{(x - 1) (x + h - 1)}] = \underset{h \to 0}{l i m} \frac{- 2}{(x - 1) (x + h - 1)} = \frac{- 2}{(x - 1) (x - 1)} = \frac{- 2}{{(x - 1)}^{2}}Q: 37. For the function f (x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^{2}}{2} + x + 1 Prove that f^{'} (1) = 100 f^{'} (0)A: 37. Given f (x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + ? + \frac{x^{2}}{2} + x + 1 \underset{h \to 0}{l i m} \frac{{(x + h)}^{100} - x^{100}}{100 \cdot h} + \underset{h \to 0}{l i m} \frac{{(x + h)}^{99} - x^{99}}{99 \cdot h} + \dots + \underset{x \to 0}{l i m} \frac{{(x + h)}^{2} - x^{2}}{2 h} + \underset{x \to 0}{l i m} x + \underset{x \to 0}{l i m} 1 = \frac{100 \cdot x^{99}}{100} + \frac{99 x^{98}}{99} + ? + \frac{2 x}{2} + 0 + 1 = x^{99} + x^{98} + ? + x + 1 At x=0, f(X) =1. and at x=1, f^{'} (1) = 1^{99} + 1^{98} + \dots + 1^{2} + 1 + 1 =100 \times 1 =100 \times f^{'} (0) Hence, f^{'} (1) = 100 f^{'} (0)Q: 38. Find the derivative of x^{n} + a x^{n - 1} + a^{2} x^{n - 2} + \dots + a^{n - 1} x + a^{n} for some fixed real number a.A: 38. Given, f (x) = x^{n} + a x^{n - 1} + a^{2} x^{n - 2} + ? + a^{n - 1} x + a^{n} . We know that, \frac{d}{d x} (x^{x}) = n x^{x - 1} So, f^{'} (x) = \frac{d}{d x} x^{x} + \frac{d}{d x} a x^{x - 1} + \frac{d}{d x} a^{2} x^{x - 2} + ? + \frac{d}{d x} a^{x - 1} x + \frac{d}{d x} a^{x} = n x^{n - 1} + a (n - 1) x^{n - 2} + a^{2} (n - 2) x^{n - 3} + ? + a^{n - 1} + 0 . \begin{array}{l} (? \frac{d a x}{d x} = a \frac{d x}{d x} \frac{and d a}{d x} = 0whereaisconstant \end{array} \begin{array}{l} \end{array} \begin{array}{l} \end{array}Q: 42. Find the derivative of cos x from first principle.A: 42. Given, f(x) = cos x f (x + h) = c o s (x + h) By first principle, f^{'} (x) = \underset{x \to h}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{x \to h}{l i m} \frac{1}{h} [c o s (x + h) - c o s x] = \underset{x \to h}{l i m} \frac{1}{h} {[- 2 s i n (\frac{x + h + x}{2}) s i n (\frac{x + h - x}{2})]}^{'} = \underset{x \to h}{l i m} \frac{1}{h} [- 2 s i n (\frac{2 x + h}{2}) s i n (\frac{h}{2})] = \underset{x \to h}{l i m} \frac{1}{h} [- 2 s i n (\frac{2 x + h}{2}) s i n (\frac{h}{2})] = \underset{x \to h}{l i m} - s i n (\frac{2 x + h}{2}) \times \underset{x \to h}{l i m} \frac{s i n h / 2}{h / 2} = - s i n (\frac{2 x + 0}{2}) \times 1 = - s i n x .Q: 47. (ax + b) (cx + d) 2A: 47. Given, f (x) = (ax + b) (cx + d) 2 So, f?(x) = (ax +b) \frac{d}{d x} {(c x + d)}^{2} + {(c x + d)}^{2} \frac{d}{d x} (a x + b) = (a x + b) \frac{d}{d x} [(c + d) (c x + d)] + {(c x + d)}^{2} a = (a x + b) [(c + d) \frac{d}{d x} (x + d) + (c + d) \frac{d}{d x} (c + d)] + a {(c x + d)}^{2} = (a x + b) [c (c x + d) + c (c x + d)] + a {(c x + d)}^{2} = (a x + b) \cdot 2 \cdot c (c x + d) + a {(c x + d)}^{2}Q: 48. Kindly consider the following \frac{a x + b}{c x + d}A: 48. Given, f (x) = \frac{a x + b}{c x + d} So, f (x) = \frac{(c x + d) \frac{d}{d x} (a x + b) - (a x + b) \frac{d}{d x} (c x + d)}{({(x + d)}^{2}} = \frac{(c x + d) a - (a x + b) c}{{(c x + d)}^{2}} = \frac{a c x + a d - a c x - c b}{{(c x + d)}^{2}} = \frac{a d - b c}{{(c x + d)}^{2}}Q: 49. Kindly consider the following \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}A: 49. Given, f (x) = \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} = \frac{\frac{x + 1}{x}}{\frac{x - 1}{x}} = \frac{x + 1}{x - 1} So, f (x) = \frac{(x - 1) \frac{d}{d x} (x + 1) - (x + 1) \frac{d}{d x} (x - 1)}{{(x - 1)}^{2}} = \frac{(x - 1) - (x + 1)}{{(x - 1)}^{2}} = \frac{x - 1 - x - 1}{{(x - 1)}^{2}} = \frac{- 2}{{(x - 1)}^{2}}Q: 51. Kindly consider the following \frac{(a x + b)}{p x^{2} + q x + r}A: 51. Given, f (x) = \frac{(a x + b)}{p x^{2} + q x + r} So, f(x) = \frac{(p x^{2} + q x + r) \frac{d}{d x} (a x + b) - (a x + b) \frac{d}{d x} (p x^{2} + q x + r)}{{(p x^{2} + q x + r)}^{2}} = \frac{(p x^{2} + q x + r) a - (a x + b) (2 x p + q)}{{(p x^{2} + q x + r)}^{2}} = \frac{a p x^{2} + a q x + a r - 2 a p x^{2} - a q x - 2 b p x - b}{{(p x^{2} + q x + 9)}^{2}} q . = \frac{- a p x^{2} - 2 b p x + a r - b q}{{(p x^{2} + q x + r)}^{2}}Q: 53. Kindly consider the following \frac{a}{x^{4}} - \frac{b}{x} + c o s xA: 53. Given, f (x) = = \frac{a}{x^{4}} - \frac{b}{x} + c o s x So, f(x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{a}{{(x + n)}^{4}} - \frac{b}{{(x + h)}^{2}} + c o s (x + h) - (\frac{a}{x^{4}} - \frac{b}{x^{2}} + c o s x)] = \underset{h \to 0}{l i m} \frac{a}{h} [\frac{1}{{(x + h)}^{4}} - \frac{1}{x^{4}}] - \underset{h \to 0}{l i m} \frac{b}{h} [\frac{1}{{(x + h)}^{2}} - \frac{1}{x^{2}}] + \underset{h \to 0}{l i m} \frac{1}{h} [c o s (x + h) - c o s x] = \underset{h \to 0}{l i m} \frac{a}{h} [\frac{x^{4} - {(x + h)}^{4}}{{(x + h)}^{4} \cdot x^{4}}] - \underset{h \to 0}{l i m} \frac{b}{h} [\frac{x^{2} - {(x + h)}^{2}}{{(x + h)}^{2} \cdot x^{2}}] + \underset{h \to 0}{l i m} \frac{1}{h} [- 2 s i n (\frac{x + h + x}{2}) s i n (\frac{x + h - x}{2})] = \underset{h \to 0}{l i m} \frac{a}{h} [\frac{x^{4} - x^{4} - 4 x^{3} h - 6 x^{2} h^{2} - 4 x h^{3} - h^{3}}{x^{4} \cdot {(x + h)}^{4}}] - \underset{h \to 0}{l i m} \frac{b}{h} [\frac{x^{2} - x^{2} - h^{2} - 2 x h}{x^{2} (x + h)^{2}}] + \underset{h \to 0}{l i m} \frac{a}{h} [- 2 s i n (\frac{2 x + h}{2}) s i n \frac{h}{2}] = \underset{h \to 0}{l i m} \frac{a}{h} [\frac{h (- 4 x^{3} - 6 x^{2} h - 4 x h^{2} - h^{2})}{x^{4} (x + h)^{4}}] - \underset{h \to 0}{l i m} \frac{b}{h} [\frac{h (- h - 2 x)}{x^{2} (x + h)^{2}}] - \underset{h \to 0}{l i m} s i n (\frac{2 x + h}{2}) \underset{h \to 0}{l i m} \frac{s i n \frac{h}{2}}{\frac{h}{2}} = \underset{h \to 0}{l i m} \frac{a (- 4 x^{3} - 6 x^{2} h - 4 x h^{2} - h^{2})}{x^{4} (x + h)^{4}} - \underset{h \to 0}{l i m} \frac{b (- h - 2 x)}{x^{2} (x + h)^{2}} s i n (\frac{2 x + 0}{2}) \times 1 = \frac{a (- 4 x^{3})}{x^{4} \cdot x^{4}} - \frac{b (- 2 x)}{x^{2} \cdot x^{2}} - s i n x . = - \frac{4 a}{x^{5}} + \frac{2 b}{x^{3}} - s i n xQ: 55. Kindly consider the following (ax + b) nA: 55. Given, f (x) = (ax + b) n Chain rule, \frac{d}{d x} u {(x)}^{n} = n u {(x)}^{n - 1} \frac{d u}{d x} u (x) where u (x) is a function of x. So, f (x) = \frac{d}{d x} {(a x + b)}^{n} = n {(a x + b)}^{n - 1} \frac{d}{d x} (a x + b) = n a {(a x + b)}^{n - 1}Q: 56. (ax + b) n (cx + d) mA: 56. Given, f (x) = (ax + b) n (cx + d) m So, f’(x) = {(a x + b)}^{n} \frac{d}{d x} {(c x + d)}^{n} + {(c x + d)}^{m} \frac{d}{d x} {(a x + b)}^{n} = {(a x + b)}^{x} \cdot m c {(c x + d)}^{m - 1} + {(c x + d)}^{m} n a {(a x + b)}^{x - 1} = {(a x + b)}^{n - 1} (c x + d)^{m - 1} [(a x + b) \cdot m c + (c x + d) n a] = {(a x + b)}^{n - 1} (c x + d)^{m - 1} [m c (a x + b) + n a (c x + d)]Q: 57. Kindly consider the following sin(x + a)A: 57. Given, f (x) = sin (x + a) So, f (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h} = \underset{h \to 0}{l i m} s i n \frac{(x + h + a) - s i n (x + a)}{h} = \underset{h \to 0}{l i m} \frac{1}{h} \cdot 2 c o s (\frac{x + h + a + x + a}{2}) s i n (\frac{x + h + a - (x + a)}{2}) = \underset{h \to 0}{l i m} c o s (\frac{2 x + 2 a + h}{2}) \underset{h \to 0}{l i m} \frac{s i n (\frac{h}{2})}{\frac{h}{2}} = c o s \frac{(2 x + 2 a + 0)}{2} \times 1 = cos (x + a)Q: 58. Kindly consider the following cosec x. cot xA: 58. Given f (x) = cosec x. cot x. By Leibnitz product rule, So, g(x) = \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = \underset{h \to 0}{l i m} \frac{c o t (x + h) - c o t x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{c o s (x + h)}{s i n (x + h)} - \frac{c o s x}{s i n x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n x c o s (x + h) - c o s x s i n (x + h)}{s i n x s i n (x + h)}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x - (x + h))}{s i n x s i n (x + h)}] [\begin{matrix} ? csin (A - B) = & s i n A c o s B - c o s A s i n B \end{matrix}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (- h)}{s i n x . s i n (x + h)}] = \underset{h \to 0}{l i m} \frac{1}{s i n x s i n (x + h)} \times (- 1) \underset{h \to 0}{l i m} \frac{s i n h}{h} = \frac{1}{s i n x s i n (x + 0)} \times (- 1) = -cosec 2 x.______(2) And hx) = \underset{h \to 0}{l i m} \frac{h (x + h) - h (x)}{h} = \underset{h \to 0}{l i m} \frac{c o s e c (x + h) - c o s e c x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{1}{s i n (x + h)} - \frac{1}{s i n x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n x - s i n (x + h)}{s i n x \cdot s i n (x + h)}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x - (x + h)}{2})}{s i n x s i n (x + h) .}]Q: 63. Kindly consider the following \frac{a + b s i n x}{c + d c o s x}A: 63. Given, f (x) = \frac{a + b s i n x}{c + d c o s x} f?(x) = \frac{(c + d c o s x) \frac{d}{d x} (b s i n x) - (a + b s i n x) \frac{d}{d x} (c + d c o s x)}{{(c + d c o s x)}^{2}} f^{'} (x) = \frac{(c + d c o s x) \cdot b \cdot \frac{d}{d x} (s i n x) - (a + b s i n x) \cdot d \cdot \frac{d}{d x} (c o s x)}{{(c + d c o s x)}^{2}}_____(1) {Copy (A)} So, g?(x) = \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [c o s (x + h) - c o s x] = \underset{h \to 0}{l i m} \frac{1}{h} [- 2 \cdot s i n (\frac{x + h + x}{2}) s i n (\frac{x + h - x}{2})] = \underset{h \to 0}{l i m} \frac{1}{h} [- 2 s i n (\frac{2 x + h}{2}) s i n (\frac{h}{2})] = - s i n (\frac{2 x + 0}{2}) \times 1 = sin x ______ (2) And p?(x) = \underset{h \to 0}{l i m} \frac{p (x + h) - p (x)}{h} = \underset{h \to 0}{l i m} \frac{s i n (x + h) - s i n x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} \cdot 2 c o s (\frac{2 x + h}{2}) s i n (\frac{h}{2}) = \underset{h \to 0}{l i m} c o s (\frac{2 x + h}{2}) \times \underset{h \to 0}{l i m} \frac{s i n (\frac{h}{2})}{(\frac{h}{2})} = cos x _____ (3) So, put (2) and (3) in (1) we get, f^{'} (x) = \frac{(c + d c o s x) (b \cdot c o s x) - (a + b s i n x) (- d \cdot s i n x)}{{(c + d c o s x)}^{2}} = \frac{b e c c o s x + b d c o s^{2} x + a d s i n x + b d s i n^{2} x}{{(c + d c o s x)}^{2}} = \frac{b c c o s x + a d s i n x + b d (c o s^{2} x + s i n^{2} x)}{{(c + d c o s x)}^{2}} = \frac{b c c o s x + a d s i n x + b d}{{(c + d c o s x)}^{2}}Q: 62. Kindly consider the following sin n xA: 62. Given, f (x) =sin n x By chain rule, f? (x) = n (sin x) n -1 \frac{d}{d h} sin x Let (gx) = sinx So, g? (x) \underset{h \to 0}{l i m} \frac{g (x + h) - g (x)}{h} = \underset{h \to 0}{l i m} \frac{s i n (x + h) - s i n x}{h} = \underset{h \to 0}{l i m} \frac{2}{h} c o s (\frac{2 a + h}{2}) s i n (\frac{h}{2}) = \underset{h \to 0}{l i m} c o s (\frac{2 x + h}{2}) \times \underset{h \to 0}{l i m} \frac{s i n (\frac{h}{2})}{\frac{h}{2}} = c o s \frac{(2 x + 0)}{2} \times 1 = cos x. So, f? (x) = n (sin x) n -1 cos x.Q: 67. Kindly consider the following (ax 2 + sin x) (p +q cos x)A: 67. Given, f (x) = (ax 2 + sin x) (p +q cos x). So, f? (x) = (ax 2 + sin x) \frac{d}{d x} (p + q c o s x) + (p + q c o s x) \frac{d}{d x} (a x^{2} + s i n x) = (a x^{2} + s i n x) (0 + q \frac{d}{d x} c o s x) + (p + q c o s x) (a \frac{d}{d x} (x^{2}) + \frac{d}{d x} s i n x) = (a x^{2} + s i n x) (q (- s i n x) + (p + q c o s x) (a \cdot 2 x + c o s x) = q sin x (ax 2 + sin x) + (p + q cos x) (2ax + cos x)Q: Kindly consider the following 69. \frac{4 x + 5 s i n x}{3 x + 7 c o s x}A: 69. Given, f (x) = \frac{4 x + 5 s i n x}{3 x + 7 c o s x} So, f^{'} (x) = \frac{(3 x + 7 c o s x) \frac{d}{d x} (4 x + 5 s i n x) - (4 x + 5 s i n x) \frac{d}{d x} (3 x + 7 c o s x)}{{(3 x + 7 c o s x)}^{2}} = \frac{(3 x + 7 c o s x) (4 x + 5 c o s x) - (4 x + 5 s i n x) (3 - 7 s i n x)}{{(3 x + 7 c o s x)}^{2}} \begin{array}{l} \underline{= x + x x + x +^{} x - x + x x - x + 35 s i n^{2} x} \\ {(3 x + 7 c o s x)}^{2} \end{array} = \frac{15 x c o s x + 28 c o s x + 28 x s i n x - 15 s i n x + 35 (c o s^{2} x + s i n^{2} x)}{{(3 x + 7 c o s x)}^{2}} = \frac{35 + 15 x c o s x + 28 c o s x + 28 x s i n x - 15 s i n x}{{(3 x + 7 c o s x)}^{2}}Q: 29. Kindly consider the followingA: 29. Given, f (x) = (x - a 1) (x - a 2)… (x - a n) So, \underset{x \to a_{1}}{l i m} f (x) = \underset{x \to a_{1}}{l i m} (x - a_{1}) \underset{x \to a_{1}}{l i m} (x - a_{2}) ? \underset{x \to a_{1}}{l i m} (x - a^{n}) = (a 1 - a 1) (a 1 - a 2) … (a 1 - a n) = 0 (a 1 - a 2) … (a 1 - a n) = 0 And \underset{x \to a}{l i m} f (x) = \underset{x \to a}{l i m} (x - a_{1}) \underset{x \to a}{l i m} (x - a_{2}) \dots \underset{x \to a}{l i m} (x - a n) = (a - a 1) (a - a 2) … (a - a n)Explore exams which ask questions on Maths Ncert Solutions class 11th Select your preferred stream Engineering Boards View more examsMaths Ncert Solutions class 11th Exam Student Forum Anything you would want to ask experts? Write here...Other Similar chapters for you Binomial Theorem Probability Statistics Mathematical Reasoning Limits and Derivatives Introduction to Three Dimensional Geometry Conic Sections Straight Lines Sequences and Series NCERT Maths 11th Permutations and Combinations Linear Inequalities Complex Numbers and Quadratic Equations Principle of Mathematical Induction Trigonometric Functions Popular Courses After 12th Science Commerce Arts Bachelor of Technology (BE/BTech) Exams accepted JEE Main | BITSAT | WBJEE Bachelor of Architecture (BArch) Exams accepted NATA | AAT | ITM NEST Bachelor of Science (BSc) Exams accepted CUET | UPCATET | IIT JAM Bachelor of Computer Applications (BCA) Exams accepted BUMAT | GSAT | SET Bachelor of Medicine & Bachelor of Surgery (MBBS) Exams accepted NEET | AYUSH | FMGE View all courses after 12th Chartered Accountancy (CA) Exams accepted CA Foundation Company Secretary (CS) Exams accepted ICSI Exam BCom in Accounting & Commerce Exams accepted BHU UET | GLAET | GD Goenka Test Bachelor of Business Administration & Bachelor of Law Exams accepted CLAT | LSAT India | AIBE BBA/BMS Exams accepted IPMAT | NMIMS - NPAT | SET View all courses after 12th BA in Humanities & Social Sciences Exams accepted SUAT | JNUEE | BHU UET Bachelor of Fine Arts (BFA) Exams accepted BHU UET | KUK Entrance Exam | JMI Entrance Exam Bachelor of Design in Animation (BDes) Exams accepted UCEED | NIFT Entrance Exam | NID Entrance Exam BA LLB (Bachelor of Arts + Bachelor of Laws) Exams accepted CLAT | AILET | LSAT India Bachelor of Journalism & Mass Communication (BJMC) Exams accepted LUACMAT | SRMHCAT | GD Goenka Test View all courses after 12th News & Updates Latest News Popular News GATE 2026 Last Date to Apply TODAY with Late Fee Live: Link, Steps, Fees, Status, and Exam Registration Details UPASANA PRADHAN Oct 13, 2025 IBPS PO Mains Exam Analysis 2025 OUT LIVE Updates: Check Difficulty Level, Good Attempts & Review Shikha Goyal Oct 13, 2025 JEE Main 2026 Registration Live: NTA Form, Date, Syllabus, Pattern, Eligibility, jeemain.nta.nic.in Latest News Samridhi Mishra Oct 13, 2025 SSLC, Plus Two Time Table 2026 Kerala Board Soon; Live Updates on Kerala SSLC, Plus Two Exam Time Table Salviya Antony Oct 13, 2025 UP Board Exam Date Sheet 2026 Class 10, 12 (Soon) Live Updates; Check UPMSP High School, Inter Time Table Md Shahzad Oct 13, 2025 1 Comment JEE Main Marks vs Percentile vs Rank 2025: Calculate Percentile and Rank Using Marks Satyendra Singh Oct 4, 2025 5.4M View s List of NIT Colleges in India 2025 - NIRF Ranking, Courses, Seats, Cutoff, Fees and Placement Satyendra Singh Sep 4, 2025 3.6M View s 205 Comment s CBSE 10th Maths Question Paper 2025, 2024, 2023, 2022, 2020, 2019 PDF Download Mayank Uniyal Sep 13, 2025 3.4M View s NEET Chapter Wise Weightage 2026 PDF by NTA: Physics, Chemistry & Biology Sreetama Datta Roy Oct 8, 2025 3.1M View s 115 Comment s List of Top IITs in India 2025: NIRF Ranking, Courses, Seats, Fees & Placement Satyendra Singh Sep 4, 2025 2.6M View s 41 Comment sGet our experts to answer your questions within 24 Hrs Write your questions here Ask Question MBA MBA Top MBA Colleges MBA Colleges Executive MBA Colleges MBA Exams CAT MAT Online MBA MBA College Predictors Engineering Engineering Top Engineering Colleges Engineering Colleges Engineering Exams JEE Main JEE Advanced Engineering College Predictors Medicine NEET UG NEET PG NEET SS NEET MDS INI CET FMGE AIAPGET Top Medical Colleges Medical Colleges Medical Exams Other Courses Animation B.Com B.Sc BBA CA Fashion Designing Hotel Management Law Mass Communication & Media MBBS Sarkari Exams RRB Group D RRB NTPC CTET UPTET UGC NET DSSSB SSC CGL SSC CHSL SSC GD NDA Resources Careers after 12th Courses After 12th Ask a Question Write a college review Articles Law College Predictors Hospitality College Predictor NCERT Solutions NCERT Solutions Class 12 Maths NCERT Solutions Class 12 Physics Important Updates XAT 2026 Exam CAT Form Correction 2025 CAT Syllabus 2025 CAT Score vs Percentile CAT 2025 JEE Main 2026 Trigonometric Table NEET 2026 JEE Main PYQ Chemical Bonding and Molecular Structure Study Abroad Study Abroad Home BTech abroad MBA abroad MS abroad GRE GMAT SAT IELTS TOEFL Shiksha Online Free Online courses Free Government courses Data Science Courses Artificial Intelligence Courses Machine Learning Courses Cloud Computing Courses AWS Certification Python Courses Cybersecurity Courses Digital Marketing Courses + 1 More Shiksha Online Blog Get App, Its faster and better Get on Play Store Get on App Store About Shiksha About Us Management Team Careers Shiksha Authors FAQs Contact Us Our Group Infoedge.in Naukri.com Naukri Campus Naukrigulf 99acres.com Jeevansathi.com AmbitionBox.com Shiksha Online Shiksha Abroad Job Hai Enterprise Client Login Advertising/Sales Enquiries Add Colleges Get in Touch 8826978461 Contribute Write Review Follow Us A Group Company Grievances Notices / Summons Privacy Sitemap Terms & Conditions Trade Marks belong to the respective owners. Copyright © 2025 Info edge India Ltd. All rights reserved.function lazyLoadJs(e,a,r,l){for(var s,t,c=[],d=e.scripts[0];s=r.shift();)t=e.createElement(a),l&&(t.type = "text/partytown"),"async"in d?(t.async=1,e.head.appendChild(t)):d.readyState?(c.push(t),t.onreadystatechange=n):e.write("<"+a+' src="'+s+'" defer></'+a+">"),t.src=s}

var addLazyLoadClass = function(){
var isExist = document.querySelector('body.lazyLoad');
if(isExist==null)
{
document.querySelector('body').classList.add('lazyLoad');
}
}

function insertHeadSection(file) {
var scrp = document.createElement('link');
scrp.rel = 'preload';
scrp.as = 'script';
scrp.href = file;
document.head.appendChild(scrp);
}

var referrer = '';
var validDomains = [];

validDomains.push('@gmail.com');

validDomains.push('@yahoo.com');

validDomains.push('@yahoo.co.in');

validDomains.push('@rediffmail.com');

validDomains.push('@hotmail.com');

document.addEventListener('click', function(e){
addLazyLoadClass();
}, false);
document.addEventListener('scroll', function(e){
addLazyLoadClass();
}, { passive: true });

/*if (typeof extra_css == 'string') {
if (document.getElementById('extraCss')) {
document.getElementById('extraCss').innerHTML = extra_css;
}
}*/
if(!('requestIdleCallback' in window)) {
window.requestIdleCallback = function(callback) {callback()};
}window.__PRELOADED_STATE__ = {"config":{"PROTOCOL":"https://","EXAM_SERVICE_PORT":9005,"LISTING_SERVICE_PORT":9002,"PRED_SERVICE_PORT":9012,"DEFAULT_PORT":9014,"API_SERVER":"apis.shiksha.jsb9.net","API_SERVER_CALL":true,"CLP_PRODUCT_ID":"0010004","S3_MEDIA_DOMAIN":"https://shiksha-static-mediadata.s3.ap-southeast-1.amazonaws.com","TRUECALLER_APP_SA_KEY":"cU4fO2163eaf49c46439b879cdb0de3008f01","TRUECALLER_APP_SA_NAME":"Shiksha Study Abroad","GOOGLE_CLIENT_ID":"570999533947-fdh4s99044jpealotgpq4r0274bqtcbd.apps.googleusercontent.com","SHIKSHA_HOME":"https://www.shiksha.com","BEACON_SHIKSHA_HOME":"https://track.shiksha.com","SHIKSHA_STUDYABROAD_HOME":"https://www.shiksha.com/studyabroad","SHIKSHA_ASK_HOME":"https://ask.shiksha.com","BEACON_TRACK_URL":"https://track.shiksha.com/trackinggateway/trackingApi/v1/beacon/trackpageviews","BEACON_INDEX_TRACK_URL":"https://track.shiksha.com/beacon/Beacon/index","IMAGES_SHIKSHA":"https://images.shiksha.com","IMAGES_DOMAIN":"https://images.shiksha.ws","COOKIE_DOMAIN":".shiksha.com","JS_DOMAIN":"https://js.shiksha.ws","CSS_DOMAIN":"https://css.shiksha.ws","IMG_DOMAIN":"https://images.shiksha.ws","FONT_DOMAIN":"https://images.shiksha.ws","CMS_DOMAIN":"https://internal.shiksha.com","OLD_SHIKSHA_STUDYABROAD_HOME":"https://studyabroad.shiksha.com"},"hpdata":{},"courseData":{},"instituteData":{},"userDetails":{},"rankingPageData":{},"rankingData":{},"countryHomePageData":{},"applyPageData":{},"categoryData":{},"imageData":{},"searchPageData":{},"importantDatesAmp":{},"alumniData":{},"campusRepData":{},"dfpParams":{},"trendingData":{},"courseHomePageData":{},"courseHomePageLoaderData":{},"footerLinks":{"footer1":[{"group":"MBA","gnbFooterLinksData":{"MBA":[{"linksText":"MBA","linksUrl":"https://www.shiksha.com/mba-masters-of-business-administration-chp","target":null,"scope":null},{"linksText":"Top MBA Colleges","linksUrl":"https://www.shiksha.com/mba/ranking/top-mba-colleges-in-india/2-2-0-0-0","target":null,"scope":null},{"linksText":"MBA Colleges","linksUrl":"https://www.shiksha.com/mba/colleges/mba-colleges-india","target":null,"scope":null},{"linksText":"Executive MBA Colleges","linksUrl":"https://www.shiksha.com/business-management-studies/colleges/executive-mba-colleges-india","target":null,"scope":null},{"linksText":"MBA Exams","linksUrl":"https://www.shiksha.com/mba/exams-pc-101","target":null,"scope":null},{"linksText":"CAT","linksUrl":"https://www.shiksha.com/mba/cat-exam","target":null,"scope":null},{"linksText":"MAT","linksUrl":"https://www.shiksha.com/mba/mat-exam","target":null,"scope":null},{"linksText":"Online MBA","linksUrl":"https://www.shiksha.com/online-mba-chp","target":null,"scope":null},{"linksText":"MBA College Predictors","linksUrl":"https://www.shiksha.com/mba/college-predictor","target":null,"scope":null}]}},{"group":"Engineering","gnbFooterLinksData":{"Engineering":[{"linksText":"Engineering","linksUrl":"https://www.shiksha.com/engineering-chp","target":null,"scope":null},{"linksText":"Top Engineering Colleges","linksUrl":"https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india/44-2-0-0-0","target":null,"scope":null},{"linksText":"Engineering Colleges","linksUrl":"https://www.shiksha.com/engineering/colleges/colleges-india","target":null,"scope":null},{"linksText":"Engineering Exams","linksUrl":"https://www.shiksha.com/engineering/exams-st-2","target":null,"scope":null},{"linksText":"JEE Main","linksUrl":"https://www.shiksha.com/engineering/jee-main-exam","target":null,"scope":null},{"linksText":"JEE Advanced","linksUrl":"https://www.shiksha.com/engineering/jee-advanced-exam","target":null,"scope":null},{"linksText":"Engineering College Predictors","linksUrl":"https://www.shiksha.com/engineering/college-predictor","target":null,"scope":null}]}},{"group":"Medicine","gnbFooterLinksData":{"Medicine":[{"linksText":"NEET UG","linksUrl":"https://www.shiksha.com/medicine-health-sciences/neet-exam","target":null,"scope":null},{"linksText":"NEET PG","linksUrl":"https://www.shiksha.com/medicine-health-sciences/neet-pg-exam","target":null,"scope":null},{"linksText":"NEET SS","linksUrl":"https://www.shiksha.com/medicine-health-sciences/neet-ss-exam","target":null,"scope":null},{"linksText":"NEET MDS","linksUrl":"https://www.shiksha.com/medicine-health-sciences/dental/neet-mds-exam","target":null,"scope":null},{"linksText":"INI CET","linksUrl":"https://www.shiksha.com/medicine-health-sciences/ini-cet-exam","target":null,"scope":null},{"linksText":"FMGE","linksUrl":"https://www.shiksha.com/medicine-health-sciences/fmge-exam","target":null,"scope":null},{"linksText":"AIAPGET","linksUrl":"https://www.shiksha.com/medicine-health-sciences/aiapget-exam","target":null,"scope":null},{"linksText":"Top Medical Colleges","linksUrl":"https://www.shiksha.com/medicine-health-sciences/ranking/top-medical-colleges-in-india/100-2-0-0-0","target":null,"scope":null},{"linksText":"Medical Colleges","linksUrl":"https://www.shiksha.com/medicine-health-sciences/colleges/colleges-india","target":null,"scope":null},{"linksText":"Medical Exams","linksUrl":"https://www.shiksha.com/medicine-health-sciences/exams-st-18","target":null,"scope":null}]}},{"group":"Other_Courses","gnbFooterLinksData":{"Other_Courses":[{"linksText":"Animation","linksUrl":"https://www.shiksha.com/animation-chp","target":null,"scope":null},{"linksText":"B.Com","linksUrl":"https://www.shiksha.com/b-com-chp","target":null,"scope":null},{"linksText":"B.Sc","linksUrl":"https://www.shiksha.com/b-sc-chp","target":null,"scope":null},{"linksText":"BBA","linksUrl":"https://www.shiksha.com/bba-bachelor-of-business-administration-chp","target":null,"scope":null},{"linksText":"CA","linksUrl":"https://www.shiksha.com/accounting-commerce/chartered-accountancy-chp","target":null,"scope":null},{"linksText":"Fashion Designing","linksUrl":"https://www.shiksha.com/design/fashion-designing-chp","target":null,"scope":null},{"linksText":"Hotel Management","linksUrl":"https://www.shiksha.com/hospitality-travel/hotel-management-chp","target":null,"scope":null},{"linksText":"Law","linksUrl":"https://www.shiksha.com/law-chp","target":null,"scope":null},{"linksText":"Mass Communication & Media","linksUrl":"https://www.shiksha.com/mass-communication-media-chp","target":null,"scope":null},{"linksText":"MBBS","linksUrl":"https://www.shiksha.com/mbbs-chp","target":null,"scope":null}]}},{"group":"Sarkari_Exams","gnbFooterLinksData":{"Sarkari_Exams":[{"linksText":"RRB Group D","linksUrl":"https://www.shiksha.com/exams/rrb-group-d-exam","target":null,"scope":null},{"linksText":"RRB NTPC","linksUrl":"https://www.shiksha.com/exams/rrb-ntpc-exam","target":null,"scope":null},{"linksText":"CTET","linksUrl":"https://www.shiksha.com/exams/ctet-exam","target":null,"scope":null},{"linksText":"UPTET","linksUrl":"https://www.shiksha.com/exams/uptet-exam","target":null,"scope":null},{"linksText":"UGC NET","linksUrl":"https://www.shiksha.com/exams/ugc-net-exam","target":null,"scope":null},{"linksText":"DSSSB","linksUrl":"https://www.shiksha.com/exams/dsssb-exam","target":null,"scope":null},{"linksText":"SSC CGL","linksUrl":"https://www.shiksha.com/exams/ssc-cgl-exam","target":null,"scope":null},{"linksText":"SSC CHSL","linksUrl":"https://www.shiksha.com/exams/ssc-chsl-exam","target":null,"scope":null},{"linksText":"SSC GD","linksUrl":"https://www.shiksha.com/exams/ssc-gd-exam","target":null,"scope":null},{"linksText":"NDA","linksUrl":"https://www.shiksha.com/exams/nda-exam","target":null,"scope":null}]}},{"group":"Resources","gnbFooterLinksData":{"Resources":[{"linksText":"Careers after 12th","linksUrl":"https://www.shiksha.com/careers","target":null,"scope":null},{"linksText":"Courses After 12th","linksUrl":"https://www.shiksha.com/courses-after-12th","target":null,"scope":null},{"linksText":"Ask a Question","linksUrl":"https://ask.shiksha.com/questions","target":null,"scope":null},{"linksText":"Write a college review","linksUrl":"https://www.shiksha.com/college-review-rating-form","target":null,"scope":null},{"linksText":"Articles","linksUrl":"https://www.shiksha.com/articles-all","target":null,"scope":null},{"linksText":"Law College Predictors","linksUrl":"https://www.shiksha.com/law/college-predictor","target":null,"scope":null},{"linksText":"Hospitality College Predictor","linksUrl":"https://www.shiksha.com/hospitality-and-travel/college-predictor","target":null,"scope":null},{"linksText":"NCERT Solutions","linksUrl":"https://www.shiksha.com/ncert-solutions-preparation","target":null,"scope":null},{"linksText":"NCERT Solutions Class 12 Maths","linksUrl":"https://www.shiksha.com/preparation/maths-ncert-solutions-class-12th-ns","target":null,"scope":null},{"linksText":"NCERT Solutions Class 12 Physics","linksUrl":"https://www.shiksha.com/preparation/physics-ncert-solutions-class-12th-ns","target":null,"scope":null}]}},{"group":"Important_Updates","gnbFooterLinksData":{"Important_Updates":[{"linksText":"XAT 2026 Exam","linksUrl":"https://www.shiksha.com/mba/articles/xat-2026-exam-blogId-188974","target":null,"scope":null},{"linksText":"CAT Form Correction 2025","linksUrl":"https://www.shiksha.com/mba/cat-exam-registration","target":null,"scope":null},{"linksText":"CAT Syllabus 2025","linksUrl":"https://www.shiksha.com/mba/cat-exam-syllabus","target":null,"scope":null},{"linksText":"CAT Score vs Percentile","linksUrl":"https://www.shiksha.com/mba/articles/cat-score-vs-percentile-blogId-19573","target":null,"scope":null},{"linksText":"CAT 2025","linksUrl":"https://www.shiksha.com/mba/cat-exam","target":null,"scope":null},{"linksText":"JEE Main 2026","linksUrl":"https://www.shiksha.com/engineering/jee-main-exam","target":null,"scope":null},{"linksText":"Trigonometric Table","linksUrl":"https://www.shiksha.com/preparation/maths-trigonometry-trigonometry-table-3433-tp","target":null,"scope":null},{"linksText":"NEET 2026","linksUrl":"https://www.shiksha.com/medicine-health-sciences/articles/nta-neet-2026-blogId-199362","target":null,"scope":null},{"linksText":"JEE Main PYQ","linksUrl":"https://www.shiksha.com/engineering/jee-main-exam-question-papers","target":null,"scope":null},{"linksText":"Chemical Bonding and Molecular Structure","linksUrl":"https://www.shiksha.com/preparation/chemistry-chemical-bonding-and-molecular-structure-tp","target":null,"scope":null}]}},{"group":"Study_Abroad","gnbFooterLinksData":{"Study_Abroad":[{"linksText":"Study Abroad Home","linksUrl":"https://www.shiksha.com/studyabroad","target":null,"scope":null},{"linksText":"BTech abroad","linksUrl":"https://www.shiksha.com/studyabroad/be-btech-in-abroad-dc11510","target":null,"scope":null},{"linksText":"MBA abroad","linksUrl":"https://www.shiksha.com/studyabroad/mba-in-abroad-dc11508","target":null,"scope":null},{"linksText":"MS abroad","linksUrl":"https://www.shiksha.com/studyabroad/ms-in-abroad-dc11509","target":null,"scope":null},{"linksText":"GRE","linksUrl":"https://www.shiksha.com/studyabroad/exams/gre","target":null,"scope":null},{"linksText":"GMAT","linksUrl":"https://www.shiksha.com/studyabroad/exams/gmat","target":null,"scope":null},{"linksText":"SAT","linksUrl":"https://www.shiksha.com/studyabroad/exams/sat","target":null,"scope":null},{"linksText":"IELTS","linksUrl":"https://www.shiksha.com/studyabroad/exams/ielts","target":null,"scope":null},{"linksText":"TOEFL","linksUrl":"https://www.shiksha.com/studyabroad/exams/toefl","target":null,"scope":null}]}},{"group":"Shiksha_Online","gnbFooterLinksData":{"Shiksha_Online":[{"linksText":"Free Online courses","linksUrl":"https://www.shiksha.com/online-courses/free-online-courses-certificates","target":null,"scope":null},{"linksText":"Free Government courses","linksUrl":"https://www.shiksha.com/online-courses/government-certification","target":null,"scope":null},{"linksText":"Data Science Courses","linksUrl":"https://www.shiksha.com/online-courses/data-science-courses-certification-training-ct123","target":null,"scope":null},{"linksText":"Artificial Intelligence Courses","linksUrl":"https://www.shiksha.com/online-courses/artificial-intelligence-courses-certification-training-st551-tg1405","target":null,"scope":null},{"linksText":"Machine Learning Courses","linksUrl":"https://www.shiksha.com/online-courses/machine-learning-courses-certification-training-st553","target":null,"scope":null},{"linksText":"Cloud Computing Courses","linksUrl":"https://www.shiksha.com/online-courses/cloud-computing-courses-certification-training-st607-tg185","target":null,"scope":null},{"linksText":"AWS Certification","linksUrl":"https://www.shiksha.com/online-courses/aws-certification-courses-certification-training-st607-tg105","target":null,"scope":null},{"linksText":"Python Courses","linksUrl":"https://www.shiksha.com/online-courses/python-courses-certification-training-st619-tg21","target":null,"scope":null},{"linksText":"Cybersecurity Courses","linksUrl":"https://www.shiksha.com/online-courses/cybersecurity-courses-certification-training-st603","target":null,"scope":null},{"linksText":"Digital Marketing Courses","linksUrl":"https://www.shiksha.com/online-courses/digital-marketing-courses-certification-training-st593-tg301","target":null,"scope":null},{"linksText":"Shiksha Online Blog","linksUrl":"https://www.shiksha.com/online-courses/articles/","target":null,"scope":null}]}}],"footer2":[{"group":"About_Shiksha","gnbFooterLinksData":{"About_Shiksha":[{"linksText":"About Us","linksUrl":"https://shiksha.com/about","target":null,"scope":null},{"linksText":"Management Team","linksUrl":"https://www.shiksha.com/team","target":null,"scope":null},{"linksText":"Careers","linksUrl":"https://careers.shiksha.com/","target":null,"scope":null},{"linksText":"Shiksha Authors","linksUrl":"https://www.shiksha.com/shiksha-authors","target":null,"scope":null},{"linksText":"FAQs","linksUrl":"https://faq.shiksha.com/","target":null,"scope":null},{"linksText":"Contact Us","linksUrl":"https://www.shiksha.com/contact","target":null,"scope":null},{"linksText":"Our Vision Statement","linksUrl":"https://www.shiksha.com/studyabroad/our-vision","target":null,"scope":"abroad"}]}},{"group":"Our_Group","gnbFooterLinksData":{"Our_Group":[{"linksText":"Infoedge.in","linksUrl":"http://www.infoedge.in/","target":"tab","scope":null},{"linksText":"Naukri.com","linksUrl":"https://www.naukri.com/","target":"tab","scope":null},{"linksText":"Naukri Campus","linksUrl":"https://www.naukri.com/campus/","target":"tab","scope":null},{"linksText":"Naukrigulf","linksUrl":"https://www.naukrigulf.com/","target":"tab","scope":null},{"linksText":"99acres.com","linksUrl":"https://www.99acres.com/","target":"tab","scope":null},{"linksText":"Jeevansathi.com","linksUrl":"https://www.jeevansathi.com/","target":"tab","scope":null},{"linksText":"AmbitionBox.com","linksUrl":"https://www.ambitionbox.com/","target":"tab","scope":null},{"linksText":"Shiksha Online","linksUrl":"https://www.shiksha.com/online-courses/","target":"tab","scope":null},{"linksText":"Shiksha Abroad","linksUrl":"https://www.shiksha.com/studyabroad","target":"tab","scope":null},{"linksText":"Job Hai","linksUrl":"https://www.jobhai.com","target":"tab","scope":null}]}},{"group":"Enterprise","gnbFooterLinksData":{"Enterprise":[{"linksText":"Client Login","linksUrl":"https://enterprise.shiksha.com/enterprise/Enterprise/loginEnterprise","target":null,"scope":null},{"linksText":"Advertising/Sales Enquiries","linksUrl":"mailto:sales@shiksha.com","target":null,"scope":null},{"linksText":"Add Colleges","linksUrl":"https://enterprise.shiksha.com/enterprise/Enterprise/loginEnterprise","target":null,"scope":null}]}}]},"saFooterLinks":{},"gnbLinks":{"config":{"key":"GNB_DATA","level":0,"type":"","url":"","urlHost":null,"child":[[[{"key":"MBA","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Top Ranked Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top MBA Colleges in India","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-india/2-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Private MBA Colleges in India","level":3,"type":"url","url":"/mba/ranking/top-private-mba-colleges-in-india/125-2-0-0-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Bangalore","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-bangalore/2-2-0-278-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Mumbai","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-mumbai/2-2-0-151-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Pune","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-pune/2-2-0-174-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Hyderabad","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-hyderabad/2-2-0-702-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Delhi","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-delhi/2-2-0-74-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Chennai","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-chennai/2-2-0-64-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Maharashtra","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-maharashtra/2-2-114-0-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Kolkata","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-kolkata/2-2-0-130-0","urlHost":"relative","child":null},{"key":"Top MBA Colleges in Kerala","level":3,"type":"url","url":"/mba/ranking/top-mba-colleges-in-kerala/2-2-107-0-0","urlHost":"relative","child":null}],[{"key":"Featured Colleges","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"International Institute of Management Studies (IIMS Pune)","level":3,"type":"url","url":"/college/international-institute-of-management-studies-iims-pune-hinjewadi-31240/course-post-graduate-diploma-in-management-pgdm-124473","urlHost":"national","child":null}]]]},{"key":"Popular Courses","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"MBA/PGDM","level":3,"type":"url","url":"/mba-masters-of-business-administration-chp","urlHost":"relative","child":null},{"key":"Executive MBA","level":3,"type":"url","url":"/executive-mba-pgdm-chp","urlHost":"relative","child":null},{"key":"Distance MBA","level":3,"type":"url","url":"/distance-mba-mba-distance-education-chp","urlHost":"relative","child":null},{"key":"Online MBA","level":3,"type":"url","url":"/online-mba-chp","urlHost":"relative","child":null},{"key":"Part-Time MBA","level":3,"type":"url","url":"/part-time-mba-pgdm-chp","urlHost":"relative","child":null}]]]},{"key":"Popular Specializations","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"MBA in Finance","level":3,"type":"url","url":"/mba-in-finance-chp","urlHost":"relative","child":null},{"key":"MBA in Healthcare Management","level":3,"type":"url","url":"/business-management-studies/mba-in-healthcare-management-chp","urlHost":"relative","child":null},{"key":"MBA in HR","level":3,"type":"url","url":"/mba-in-hr-human-resource-chp","urlHost":"relative","child":null},{"key":"MBA in IT","level":3,"type":"url","url":"/mba-in-it-systems-chp","urlHost":"relative","child":null},{"key":"MBA in Operations Management","level":3,"type":"url","url":"/mba-in-operations-chp","urlHost":"relative","child":null},{"key":"MBA in Marketing","level":3,"type":"url","url":"/mba-in-sales-marketing-chp","urlHost":"relative","child":null},{"key":"MBA in International Business","level":3,"type":"url","url":"/mba-in-international-business-chp","urlHost":"relative","child":null},{"key":"MBA in Pharmaceutical Management","level":3,"type":"url","url":"/business-management-studies/mba-in-pharmaceutical-management-chp","urlHost":"relative","child":null},{"key":"MBA in Digital Marketing","level":3,"type":"url","url":"/business-management-studies/mba-in-digital-marketing-chp","urlHost":"relative","child":null},{"key":"MBA in Data Analytics","level":3,"type":"url","url":"/business-management-studies/mba-in-data-analytics-chp","urlHost":"relative","child":null}],[{"key":"MBA in Entrepreneurship","level":3,"type":"url","url":"/mba-in-entrepreneurship-chp","urlHost":"relative","child":null},{"key":"MBA in Family Managed Business","level":3,"type":"url","url":"/mba-in-family-business-chp","urlHost":"relative","child":null},{"key":"MBA in Agriculture","level":3,"type":"url","url":"/business-management-studies/mba-in-agriculture-chp","urlHost":"relative","child":null},{"key":"MBA in product management","level":3,"type":"url","url":"/business-management-studies/mba-product-management-chp","urlHost":"relative","child":null},{"key":"MBA in General Management","level":3,"type":"url","url":"/business-management-studies/mba-general-management-chp","urlHost":"relative","child":null},{"key":"MBA in Data Science","level":3,"type":"url","url":"/business-management-studies/mba-in-data-science-chp","urlHost":"relative","child":null}]]]},{"key":"Exams","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Exams","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"CAT","level":3,"type":"url","url":"/mba/cat-exam","urlHost":"relative","child":null},{"key":"CMAT","level":3,"type":"url","url":"/mba/cmat-exam","urlHost":"relative","child":null},{"key":"SNAP","level":3,"type":"url","url":"/mba/snap-exam","urlHost":"relative","child":null},{"key":"XAT","level":3,"type":"url","url":"/mba/xat-exam","urlHost":"relative","child":null},{"key":"MAT","level":3,"type":"url","url":"/mba/mat-exam","urlHost":"relative","child":null},{"key":"ATMA","level":3,"type":"url","url":"/mba/atma-exam","urlHost":"relative","child":null},{"key":"NMAT by GMAC","level":3,"type":"url","url":"/mba/nmat-by-gmac-exam","urlHost":"relative","child":null},{"key":"IBSAT","level":3,"type":"url","url":"/mba/ibsat-exam","urlHost":"relative","child":null},{"key":"KIITEE Management","level":3,"type":"url","url":"/mba/kiitee-management-exam","urlHost":"relative","child":null},{"key":"UPCET","level":3,"type":"url","url":"/mba/upcet-exam","urlHost":"relative","child":null},{"key":"All MBA Exams","level":3,"type":"urlWithAnchor","url":"/mba/exams-pc-101","urlHost":"relative","child":null},{"key":"Exam Calendar","level":3,"type":"urlWithAnchor","url":"/mba/resources/exam-calendar","urlHost":"national","child":null}],[]]]},{"key":"Colleges by Location","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"MBA Colleges in India","level":3,"type":"url","url":"/mba/colleges/mba-colleges-india","urlHost":"relative","child":null},{"key":"MBA Colleges in Bangalore","level":3,"type":"url","url":"/mba/colleges/mba-colleges-bangalore","urlHost":"relative","child":null},{"key":"MBA Colleges in Chennai","level":3,"type":"url","url":"/mba/colleges/mba-colleges-chennai","urlHost":"relative","child":null},{"key":"MBA colleges in Delhi-NCR","level":3,"type":"url","url":"/mba/colleges/mba-colleges-delhi-ncr","urlHost":"relative","child":null},{"key":"MBA Colleges in Hyderabad","level":3,"type":"url","url":"/mba/colleges/mba-colleges-hyderabad","urlHost":"relative","child":null},{"key":"MBA Colleges in Kolkata","level":3,"type":"url","url":"/mba/colleges/mba-colleges-kolkata","urlHost":"relative","child":null},{"key":"MBA Colleges in Mumbai","level":3,"type":"url","url":"/mba/colleges/mba-colleges-mumbai-all","urlHost":"relative","child":null},{"key":"MBA Colleges in Pune","level":3,"type":"url","url":"/mba/colleges/mba-colleges-pune","urlHost":"relative","child":null},{"key":"All Locations","level":3,"type":"all","url":"/mba/colleges/mba-colleges-india","urlHost":"relative","child":null}]]]},{"key":"Compare Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Comparisons","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"IIM Ahmedabad Vs IIM Bangalore","level":3,"type":"url","url":"/compare-iim-ahmedabad-vs-iim-bangalore-id-1653-1688","urlHost":"relative","child":null},{"key":"IIM Ahmedabad Vs IIM Calcutta","level":3,"type":"url","url":"/compare-iim-ahmedabad-vs-iim-calcutta-id-1653-22931","urlHost":"relative","child":null},{"key":"SIBM Pune Vs SCMHRD Pune","level":3,"type":"url","url":"/compare-sibm-pune-vs-scmhrd-pune-id-13280-164869","urlHost":"relative","child":null},{"key":"SP Jain (SPJIMR) Vs MDI Gurgaon","level":3,"type":"url","url":"/compare-sp-jain-spjimr-vs-mdi-gurgaon-id-165223-9938","urlHost":"relative","child":null},{"key":"NMIMS SBM Mumbai Vs SP Jain (SPJIMR)","level":3,"type":"url","url":"/compare-nmims-sbm-mumbai-vs-sp-jain-spjimr-id-228082-165223","urlHost":"relative","child":null},{"key":"Compare other MBA colleges","level":3,"type":"urlWithAnchor","url":"/compare","urlHost":"national","child":null}]]]},{"key":"College Reviews","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"IIM Ahmedabad Reviews","level":3,"type":"url","url":"/college/iim-ahmedabad-indian-institute-of-management-vastrapur-307/reviews","urlHost":"relative","child":null},{"key":"IIM Bangalore Reviews","level":3,"type":"url","url":"/college/iim-bangalore-indian-institute-of-management-bannerghatta-road-318/reviews","urlHost":"relative","child":null},{"key":"IIM Calcutta Reviews","level":3,"type":"url","url":"/college/iim-calcutta-indian-institute-of-management-joka-kolkata-20190/reviews","urlHost":"relative","child":null},{"key":"IIM Lucknow Reviews","level":3,"type":"url","url":"/college/iim-lucknow-indian-institute-of-management-333/reviews","urlHost":"relative","child":null},{"key":"IIM Kozhikode Reviews","level":3,"type":"url","url":"/college/iim-kozhikode-indian-institute-of-management-20188/reviews","urlHost":"relative","child":null},{"key":"IIM Indore Reviews","level":3,"type":"url","url":"/college/iim-indore-indian-institute-of-management-29623/reviews","urlHost":"relative","child":null},{"key":"FMS Delhi Reviews","level":3,"type":"url","url":"/college/fms-faculty-of-management-studies-delhi-university-north-campus-28361/reviews","urlHost":"relative","child":null},{"key":"SP Jain Reviews","level":3,"type":"url","url":"/college/spjimr-s-p-jain-institute-of-management-and-research-andheri-west-mumbai-3489/reviews","urlHost":"relative","child":null},{"key":"MDI Gurgaon Reviews","level":3,"type":"url","url":"/college/mdi-gurgaon-management-development-institute-4223/reviews","urlHost":"relative","child":null},{"key":"Write a review","level":3,"type":"urlWithAnchor","url":"/college-review-form","urlHost":"relative","child":null}]]]},{"key":"CAT Percentile Predictor","level":2,"type":"subItem","url":"/mba/cat-percentile-predictor","urlHost":"relative","child":[[[]]]},{"key":"College Predictors","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"IIM & Non IIM Call Predictor","level":3,"type":"url","url":"/mba/cat-college-predictor","urlHost":"relative","child":null},{"key":"CAT College Predictor","level":3,"type":"url","url":"/mba/cat-college-predictor","urlHost":"relative","child":null},{"key":"MAH CET College predictor","level":3,"type":"url","url":"/mba/mah-cet-college-predictor","urlHost":"relative","child":null},{"key":"XAT College/ Call Predictor","level":3,"type":"url","url":"/mba/xat-college-predictor","urlHost":"relative","child":null},{"key":"IIFT College predictor","level":3,"type":"url","url":"/mba/iift-college-predictor","urlHost":"relative","child":null},{"key":"NMAT College predictor","level":3,"type":"url","url":"/mba/nmat-college-predictor","urlHost":"relative","child":null},{"key":"SNAP College and Call Predictor","level":3,"type":"url","url":"/mba/snap-college-predictor","urlHost":"relative","child":null},{"key":"CMAT College predictor","level":3,"type":"url","url":"/mba/cmat-college-predictor","urlHost":"relative","child":null},{"key":"MBA College predictor","level":3,"type":"urlWithAnchor","url":"/mba/college-predictor","urlHost":"relative","child":null}],[{"key":"MAT College predictor","level":3,"type":"url","url":"/mba/mat-college-predictor","urlHost":"relative","child":null},{"key":"KMAT College predictor","level":3,"type":"url","url":"/mba/kmat-college-predictor","urlHost":"relative","child":null},{"key":"TANCET MBA College predictor","level":3,"type":"url","url":"/mba/tancet-mba-college-predictor","urlHost":"relative","child":null},{"key":"TSICET College predictor","level":3,"type":"url","url":"/mba/tsicet-college-predictor","urlHost":"relative","child":null},{"key":"IBSAT College predictor","level":3,"type":"url","url":"/mba/ibsat-college-predictor","urlHost":"relative","child":null},{"key":"UPCET College predictor","level":3,"type":"url","url":"/mba/upcet-college-predictor","urlHost":"relative","child":null}]]]},{"key":"Ask Current MBA Students","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Colleges","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"XIME Bangalore","level":3,"type":"url","url":"/college/xavier-institute-of-management-and-entrepreneurship-bangalore-hosur-road-28230/questions","urlHost":"national","child":null},{"key":"SIBM Pune","level":3,"type":"url","url":"/college/symbiosis-institute-of-business-management-lavale-village-pune-3959/questions","urlHost":"national","child":null},{"key":"JBIMS Mumbai","level":3,"type":"url","url":"/college/jamnalal-bajaj-institute-of-management-studies-churchgate-mumbai-344/questions","urlHost":"national","child":null},{"key":"FMS","level":3,"type":"url","url":"/college/faculty-of-management-studies-university-of-delhi-north-campus-28361/questions","urlHost":"national","child":null},{"key":"IIM Ahmedabad","level":3,"type":"url","url":"/college/indian-institute-of-management-ahmedabad-vastrapur-307/questions","urlHost":"national","child":null},{"key":"NMIMS","level":3,"type":"url","url":"/college/nmims-school-of-business-management-mumbai-narsee-monjee-vile-parle-west-46487/questions","urlHost":"national","child":null},{"key":"Other MBA colleges","level":3,"type":"urlWithAnchor","url":"/business-management-studies/resources/campus-connect-program-1","urlHost":"national","child":null}]]]},{"key":"Resources","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"MBA Alumni Salary Data","level":3,"type":"url","url":"/mba/resources/mba-alumni-data","urlHost":"national","child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/mba-tdp-422","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/mba-tdp-422?type=discussion","urlHost":"national","child":null},{"key":"MBA News","level":3,"type":"url","url":"/news/mba-pc-101","urlHost":"national","child":null},{"key":"MBA Articles","level":3,"type":"url","url":"/mba/articles-pc-101","urlHost":"relative","child":null},{"key":"Apply to colleges","level":3,"type":"url","url":"/mba/resources/application-forms","urlHost":"national","child":null},{"key":"Trends in MBA","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/base_course/101","urlHost":"national","child":null}]]]}]]]},{"key":"Engineering","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Top Ranked Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Engineering Colleges in India","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-india/44-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Private Engineering Colleges in India","level":3,"type":"url","url":"/engineering/ranking/top-private-engineering-colleges-in-india/109-2-0-0-0","urlHost":"relative","child":null},{"key":"Top IITs in India","level":3,"type":"url","url":"/engineering/ranking/top-iits-colleges-in-india/113-2-0-0-0","urlHost":"relative","child":null},{"key":"Top NITs in India","level":3,"type":"url","url":"/engineering/ranking/top-nits-colleges-in-india/115-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Bangalore","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-bangalore/44-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Karnataka","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-karnataka/44-2-106-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Hyderabad","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-hyderabad/44-2-0-702-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Pune","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-pune/44-2-0-174-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Mumbai","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-mumbai/44-2-0-151-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Maharashtra","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-maharashtra/44-2-114-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Chennai","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-chennai/44-2-0-64-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Kerala","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-kerala/44-2-107-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Delhi","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-delhi/44-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Telangana","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-telangana/44-2-413-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in Gujarat","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-gujarat/44-2-109-0-0","urlHost":"relative","child":null},{"key":"Top Engineering Colleges in West Bengal","level":3,"type":"url","url":"/engineering/ranking/top-engineering-colleges-in-west-bengal/44-2-127-0-0","urlHost":"relative","child":null}],[{"key":"Featured Colleges","level":0,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"NHCE Bangalore - New Horizon College of Engineering","level":3,"type":"url","url":"/college/nhce-bangalore-new-horizon-college-of-engineering-marathahalli-46973/course-b-e-in-computer-science-and-engineering-232149","urlHost":"national","child":null},{"key":"Dev Bhoomi School of Engineering, Dev Bhoomi Uttarakhand University","level":3,"type":"url","url":"/college/dev-bhoomi-school-of-engineering-dev-bhoomi-uttarakhand-university-dehradun-21757/course-b-tech-in-computer-science-and-engineering-176034","urlHost":"national","child":null}]]]},{"key":"Popular Courses","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"B.E/B.Tech","level":3,"type":"url","url":"/b-tech-bachelor-of-technology-chp","urlHost":"relative","child":null},{"key":"M.E/M.Tech","level":3,"type":"url","url":"/m-e-m-tech-chp","urlHost":"relative","child":null},{"key":"Ph.D.","level":3,"type":"url","url":"/ph-d-chp","urlHost":"relative","child":null},{"key":"Diploma Courses","level":3,"type":"url","url":"/engineering/colleges/diploma-courses-india","urlHost":"relative","child":null},{"key":"Distance Diploma Courses","level":3,"type":"url","url":"/engineering/colleges/distance-correspondence-diploma-courses-india","urlHost":"relative","child":null},{"key":"Distance B.Tech","level":3,"type":"url","url":"/distance-b-e-b-tech-chp","urlHost":"relative","child":null},{"key":"All Engineering Courses","level":3,"type":"urlWithAnchor","url":"/engineering/colleges/colleges-india","urlHost":"relative","child":null}]]]},{"key":"Popular Specializations","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Computer Science Engineering","level":3,"type":"url","url":"/engineering/computer-science-engineering-chp","urlHost":"relative","child":null},{"key":"Mechanical Engineering","level":3,"type":"url","url":"/engineering/mechanical-engineering-chp","urlHost":"relative","child":null},{"key":"Civil Engineering","level":3,"type":"url","url":"/engineering/civil-engineering-chp","urlHost":"relative","child":null},{"key":"Electronics & Communication Engineering","level":3,"type":"url","url":"/engineering/electronics-communication-engineering-chp","urlHost":"relative","child":null},{"key":"Aeronautical Engineering","level":3,"type":"url","url":"/engineering/aeronautical-engineering-chp","urlHost":"relative","child":null},{"key":"Aerospace Engineering","level":3,"type":"url","url":"/engineering/aerospace-engineering-chp","urlHost":"relative","child":null},{"key":"Information Technology","level":3,"type":"url","url":"/engineering/information-technology-chp","urlHost":"relative","child":null},{"key":"Electrical Engineering","level":3,"type":"url","url":"/engineering/electrical-engineering-chp","urlHost":"relative","child":null},{"key":"Electronics Engineering","level":3,"type":"url","url":"/engineering/electronics-engineering-chp","urlHost":"relative","child":null},{"key":"Nanotechnology","level":3,"type":"url","url":"/engineering/nanotechnology-chp","urlHost":"relative","child":null},{"key":"Chemical Engineering","level":3,"type":"url","url":"/engineering/chemical-engineering-chp","urlHost":"relative","child":null},{"key":"Automobile Engineering","level":3,"type":"url","url":"/engineering/automobile-engineering-chp","urlHost":"relative","child":null},{"key":"Biomedical Engineering","level":3,"type":"url","url":"/engineering/biomedical-engineering-chp","urlHost":"relative","child":null},{"key":"Construction Engineering","level":3,"type":"url","url":"/engineering/construction-engineering-chp","urlHost":"relative","child":null},{"key":"Pulp & Paper Technology","level":3,"type":"url","url":"/engineering/pulp-paper-technology-chp","urlHost":"relative","child":null}],[{"key":"Marine Engineering","level":3,"type":"url","url":"/engineering/marine-engineering-chp","urlHost":"relative","child":null},{"key":"Genetic Engineering","level":3,"type":"url","url":"/engineering/genetic-engineering-chp","urlHost":"relative","child":null},{"key":"Food Technology","level":3,"type":"url","url":"/engineering/food-technology-chp","urlHost":"relative","child":null},{"key":"Petroleum Engineering","level":3,"type":"url","url":"/engineering/petroleum-engineering-chp","urlHost":"relative","child":null},{"key":"Control Systems","level":3,"type":"url","url":"/engineering/control-systems-chp","urlHost":"relative","child":null},{"key":"Industrial Engineering","level":3,"type":"url","url":"/engineering/industrial-engineering-chp","urlHost":"relative","child":null},{"key":"Production Engineering","level":3,"type":"url","url":"/engineering/production-engineering-chp","urlHost":"relative","child":null},{"key":"Environmental Engineering","level":3,"type":"url","url":"/engineering/environmental-engineering-chp","urlHost":"relative","child":null},{"key":"Robotics Engineering","level":3,"type":"url","url":"/engineering/robotics-engineering-chp","urlHost":"relative","child":null},{"key":"Telecommunication Engineering","level":3,"type":"url","url":"/engineering/telecommunication-engineering-chp","urlHost":"relative","child":null},{"key":"Materials Science","level":3,"type":"url","url":"/engineering/materials-science-chp","urlHost":"relative","child":null},{"key":"Structural Engineering","level":3,"type":"url","url":"/engineering/structural-engineering-chp","urlHost":"relative","child":null},{"key":"Aircraft Maintenance","level":3,"type":"url","url":"/aviation/aircraft-maintenance-engineering-chp","urlHost":"relative","child":null},{"key":"RF & Microwave Engineering","level":3,"type":"url","url":"/engineering/rf-microwave-engineering-chp","urlHost":"relative","child":null}],[{"key":"VLSI Design","level":3,"type":"url","url":"/engineering/vlsi-design-chp","urlHost":"relative","child":null},{"key":"Mechatronics Engineering","level":3,"type":"url","url":"/engineering/mechatronics-engineering-chp","urlHost":"relative","child":null},{"key":"Mining Engineering","level":3,"type":"url","url":"/engineering/mining-engineering-chp","urlHost":"relative","child":null},{"key":"Biotechnology Engineering","level":3,"type":"url","url":"/engineering/biotechnology-engineering-chp","urlHost":"relative","child":null},{"key":"Transportation Engineering","level":3,"type":"url","url":"/engineering/transportation-engineering-chp","urlHost":"relative","child":null},{"key":"Metallurgical Engineering","level":3,"type":"url","url":"/engineering/metallurgical-engineering-chp","urlHost":"relative","child":null},{"key":"Textile Engineering","level":3,"type":"url","url":"/engineering/textile-engineering-chp","urlHost":"relative","child":null},{"key":"Naval Architecture","level":3,"type":"url","url":"/engineering/naval-architecture-chp","urlHost":"relative","child":null},{"key":"Power Engineering","level":3,"type":"url","url":"/engineering/power-engineering-chp","urlHost":"relative","child":null},{"key":"Dairy Technology","level":3,"type":"url","url":"/engineering/dairy-technology-chp","urlHost":"relative","child":null},{"key":"Microelectronics","level":3,"type":"url","url":"/engineering/microelectronics-chp","urlHost":"relative","child":null},{"key":"Communications Engineering","level":3,"type":"url","url":"/engineering/communications-engineering-chp","urlHost":"relative","child":null},{"key":"Tool Engineering","level":3,"type":"url","url":"/engineering/tool-engineering-chp","urlHost":"relative","child":null},{"key":"Ceramic Engineering","level":3,"type":"url","url":"/engineering/ceramic-engineering-chp","urlHost":"relative","child":null},{"key":"Jute & Fiber Technology","level":3,"type":"url","url":"/engineering/jute-fiber-technology-chp","urlHost":"relative","child":null}]]]},{"key":"Exams","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"JEE Main","level":3,"type":"url","url":"/engineering/jee-main-exam","urlHost":"relative","child":null},{"key":"COMEDK","level":3,"type":"url","url":"/engineering/comedk-uget-exam","urlHost":"relative","child":null},{"key":"BITSAT","level":3,"type":"url","url":"/engineering/bitsat-exam","urlHost":"relative","child":null},{"key":"WBJEE","level":3,"type":"url","url":"/engineering/wbjee-exam","urlHost":"relative","child":null},{"key":"JEE Advanced","level":3,"type":"url","url":"/engineering/jee-advanced-exam","urlHost":"relative","child":null},{"key":"GATE","level":3,"type":"url","url":"/engineering/gate-exam","urlHost":"relative","child":null},{"key":"LPU-NEST","level":3,"type":"url","url":"/engineering/lpu-nest-exam","urlHost":"relative","child":null},{"key":"JET","level":3,"type":"url","url":"/engineering/jet-exam","urlHost":"relative","child":null},{"key":"CUCET Chandigarh University","level":3,"type":"url","url":"/university/chandigarh-university-cu/cucet-chandigarh-university-exam","urlHost":"relative","child":null},{"key":"CGCUET","level":3,"type":"url","url":"/exams/cgcuet-exam","urlHost":"relative","child":null},{"key":"All Engineering Exams","level":3,"type":"urlWithAnchor","url":"/engineering/exams-st-2","urlHost":"relative","child":null},{"key":"Exam Calendar","level":3,"type":"urlWithAnchor","url":"/engineering/resources/exam-calendar","urlHost":"national","child":null}],[]]]},{"key":"Colleges by Location","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Engineering Colleges in India","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-india","urlHost":"relative","child":null},{"key":"Engineering Colleges in Bangalore","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-bangalore","urlHost":"relative","child":null},{"key":"Engineering Colleges in Chennai","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-chennai","urlHost":"relative","child":null},{"key":"Engineering Colleges in Delhi-NCR","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-delhi-ncr","urlHost":"relative","child":null},{"key":"Engineering Colleges in Kolkata","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-kolkata","urlHost":"relative","child":null},{"key":"Engineering Colleges in Mumbai","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-mumbai-all","urlHost":"relative","child":null},{"key":"Engineering Colleges in Pune","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-pune","urlHost":"relative","child":null},{"key":"Engineering Colleges in Hyderabad","level":3,"type":"url","url":"/engineering/colleges/b-tech-colleges-hyderabad","urlHost":"relative","child":null},{"key":"All Locations","level":3,"type":"all","url":"/engineering/colleges/b-tech-colleges-india","urlHost":"relative","child":null}]]]},{"key":"Compare Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Comparisons","level":0,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"IIT Madras Vs IIT Kanpur","level":3,"type":"url","url":"/compare-iit-madras-vs-iit-kanpur-id-8158-109925","urlHost":"relative","child":null},{"key":"VNIT Nagpur Vs NIT Rourkela","level":3,"type":"url","url":"/compare-vnit-nagpur-vs-nit-rourkela-id-61703-112801","urlHost":"relative","child":null},{"key":"Alliance University Vs Christ University","level":3,"type":"url","url":"/compare-alliance-college-of-engineering-and-design-vs-christ-kengeri-campus-id-155426-110241","urlHost":"relative","child":null},{"key":"IIT Bombay Vs IIT Delhi","level":3,"type":"url","url":"/compare-iit-bombay-vs-iit-delhi-id-109575-110582","urlHost":"relative","child":null},{"key":"BITS Pilani Vs DTU Delhi","level":3,"type":"url","url":"/compare-bits-pilani-vs-dtu-delhi-id-119869-52813","urlHost":"relative","child":null},{"key":"Compare other colleges","level":3,"type":"urlWithAnchor","url":"/compare","urlHost":"national","child":null}]]]},{"key":"Rank Predictors","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"COMEDK UGET Rank Predictor","level":3,"type":"url","url":"/engineering/comedk-uget-rank-predictor","urlHost":"relative","child":null},{"key":"JEE Advanced Rank predictor","level":3,"type":"url","url":"/engineering/jee-advanced-rank-predictor","urlHost":"relative","child":null},{"key":"JEE MAIN Rank predictor","level":3,"type":"url","url":"/engineering/jee-main-rank-predictor","urlHost":"relative","child":null}]]]},{"key":"College Predictors","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"CG PET College Predictor","level":3,"type":"url","url":"/engineering/cg-pet-college-predictor","urlHost":"relative","child":null},{"key":"COMEDK UGET College Predictor","level":3,"type":"url","url":"/engineering/comedk-uget-college-predictor","urlHost":"relative","child":null},{"key":"JEE MAIN College predictor","level":3,"type":"url","url":"/engineering/jee-main-college-predictor","urlHost":"relative","child":null},{"key":"JEE Advanced College predictor","level":3,"type":"url","url":"/engineering/jee-advanced-college-predictor","urlHost":"relative","child":null},{"key":"KCET College predictor","level":3,"type":"url","url":"/engineering/kcet-college-predictor","urlHost":"relative","child":null},{"key":"KEAM College predictor","level":3,"type":"url","url":"/engineering/keam-college-predictor","urlHost":"relative","child":null},{"key":"MHT CET College predictor","level":3,"type":"url","url":"/engineering/mht-cet-college-predictor","urlHost":"relative","child":null},{"key":"MP BE College Predictor","level":3,"type":"url","url":"/engineering/mp-be-college-predictor","urlHost":"relative","child":null},{"key":"PTU BTech College predictor","level":3,"type":"url","url":"/engineering/ptu-btech-college-predictor","urlHost":"relative","child":null},{"key":"JAC Chandigarh College predictor","level":3,"type":"url","url":"/engineering/jac-chandigarh-college-predictor","urlHost":"relative","child":null},{"key":"GATE College Predictor","level":3,"type":"url","url":"/engineering/gate-college-predictor","urlHost":"relative","child":null},{"key":"Engineering College Predictor","level":3,"type":"urlWithAnchor","url":"/engineering/college-predictor","urlHost":"relative","child":null}],[{"key":"TNEA College predictor","level":3,"type":"url","url":"/engineering/tnea-college-predictor","urlHost":"relative","child":null},{"key":"WBJEE College predictor","level":3,"type":"url","url":"/engineering/wbjee-college-predictor","urlHost":"relative","child":null},{"key":"AP EAMCET College predictor","level":3,"type":"url","url":"/engineering/ap-eamcet-college-predictor","urlHost":"relative","child":null},{"key":"TS EAMCET College predictor","level":3,"type":"url","url":"/engineering/ts-eamcet-college-predictor","urlHost":"relative","child":null},{"key":"BITSAT College predictor","level":3,"type":"url","url":"/engineering/bitsat-college-predictor","urlHost":"relative","child":null},{"key":"IPU CET College predictor","level":3,"type":"url","url":"/engineering/ipu-cet-college-predictor","urlHost":"relative","child":null},{"key":"OJEE College predictor","level":3,"type":"url","url":"/engineering/ojee-college-predictor","urlHost":"relative","child":null},{"key":"GUJCET College predictor","level":3,"type":"url","url":"/engineering/gujcet-college-predictor","urlHost":"relative","child":null},{"key":"VITEEE College predictor","level":3,"type":"url","url":"/engineering/viteee-college-predictor","urlHost":"relative","child":null},{"key":"SRMJEEE College predictor","level":3,"type":"url","url":"/engineering/srmjeee-college-predictor","urlHost":"relative","child":null},{"key":"JAC Delhi College predictor","level":3,"type":"url","url":"/engineering/jac-delhi-college-predictor","urlHost":"relative","child":null}]]]},{"key":"College Reviews","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"IIT Bombay Reviews","level":3,"type":"url","url":"/university/iit-bombay-indian-institute-of-technology-mumbai-54212/reviews","urlHost":"relative","child":null},{"key":"IIT Delhi Reviews","level":3,"type":"url","url":"/university/iit-delhi-indian-institute-of-technology-53938/reviews","urlHost":"relative","child":null},{"key":"IIT Kanpur Reviews","level":3,"type":"url","url":"/college/iit-kanpur-indian-institute-of-technology-25116/reviews","urlHost":"relative","child":null},{"key":"IIIT Hyderabad Reviews","level":3,"type":"url","url":"/university/iiit-hyderabad-international-institute-of-information-technology-53885/reviews","urlHost":"relative","child":null},{"key":"IIT Kharagpur Reviews","level":3,"type":"url","url":"/college/iit-kharagpur-indian-institute-of-technology-2999/reviews","urlHost":"relative","child":null},{"key":"NIT Trichy Reviews","level":3,"type":"url","url":"/university/nit-trichy-national-institute-of-technology-tiruchirappalli-2996/reviews","urlHost":"relative","child":null},{"key":"NIT Warangal Reviews","level":3,"type":"url","url":"/college/nit-warangal-national-institute-of-technology-25425/reviews","urlHost":"relative","child":null},{"key":"IIT Madras Reviews","level":3,"type":"url","url":"/college/iit-madras-indian-institute-of-technology-adyar-chennai-3031/reviews","urlHost":"relative","child":null},{"key":"BITS Pilani Reviews","level":3,"type":"url","url":"/university/bits-pilani-birla-institute-of-technology-and-science-467/reviews","urlHost":"relative","child":null},{"key":"IIT Hyderabad Reviews","level":3,"type":"url","url":"/college/iit-hyderabad-indian-institute-of-technology-32726/reviews","urlHost":"relative","child":null},{"key":"Write a review","level":3,"type":"urlWithAnchor","url":"/college-review-form","urlHost":"relative","child":null}]]]},{"key":"Resources","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Ask a Question","level":3,"type":"url","url":"/tags/engineering-tdp-20","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/engineering-tdp-20?type=discussion","urlHost":"national","child":null},{"key":"Engineering News","level":3,"type":"url","url":"/news/engineering-st-2","urlHost":"national","child":null},{"key":"Engineering Articles","level":3,"type":"url","url":"/engineering/articles-st-2","urlHost":"relative","child":null},{"key":"Apply to colleges","level":3,"type":"url","url":"/btech/resources/application-forms","urlHost":"national","child":null},{"key":"Trends in BTech","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/base_course/10","urlHost":"national","child":null},{"key":"Trends in Engineering","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/2","urlHost":"national","child":null}]]]}]]]},{"key":"Medical","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Top Ranked Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Medical Colleges in India","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-india/100-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Medical Colleges in Karnataka","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-karnataka/100-2-106-0-0","urlHost":"relative","child":null},{"key":"Top Pharmacy Colleges in India","level":3,"type":"url","url":"/medicine-health-sciences/pharmacy/ranking/top-pharmacy-colleges-in-india/117-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Medical Colleges in Bangalore","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-bangalore/100-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Dental Colleges in India","level":3,"type":"url","url":"/medicine-health-sciences/dental/ranking/top-dental-colleges-in-india/119-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Medical Colleges in Maharashtra","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-maharashtra/100-2-114-0-0","urlHost":"relative","child":null},{"key":"Top Medical Colleges in Mumbai","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-mumbai/100-2-0-151-0","urlHost":"relative","child":null},{"key":"Top Medical Colleges in Delhi","level":3,"type":"url","url":"/medicine-health-sciences/ranking/top-medical-colleges-in-delhi/100-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Pharmacy Colleges in Maharashtra","level":3,"type":"url","url":"/medicine-health-sciences/pharmacy/ranking/top-pharmacy-colleges-in-maharashtra/117-2-114-0-0","urlHost":"relative","child":null}]]]},{"key":"Popular Courses","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"MBBS","level":3,"type":"url","url":"/mbbs-chp","urlHost":"relative","child":null},{"key":"MD","level":3,"type":"url","url":"/md-doctor-of-medicine-chp","urlHost":"relative","child":null},{"key":"BMLT","level":3,"type":"url","url":"/bmlt-bachelor-in-medical-laboratory-technology-chp","urlHost":"relative","child":null},{"key":"MPT","level":3,"type":"url","url":"/mpt-master-of-physiotherapy-chp","urlHost":"relative","child":null},{"key":"MPH","level":3,"type":"url","url":"/mph-master-of-public-health-chp","urlHost":"relative","child":null},{"key":"All Medical Courses","level":3,"type":"link","url":"/medicine-health-sciences/colleges/colleges-india","urlHost":"relative","child":null}]]]},{"key":"Popular Specializations","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Alternative Medicine","level":3,"type":"url","url":"/medicine-health-sciences/alternative-medicine-chp","urlHost":"relative","child":null},{"key":"Dental","level":3,"type":"url","url":"/medicine-health-sciences/dental-chp","urlHost":"relative","child":null},{"key":"Dietics & Nutrition","level":3,"type":"url","url":"/medicine-health-sciences/dietics-nutrition-chp","urlHost":"relative","child":null},{"key":"Medicine","level":3,"type":"url","url":"/medicine-health-sciences/medicine-chp","urlHost":"relative","child":null},{"key":"Paramedical","level":3,"type":"url","url":"/medicine-health-sciences/paramedical-chp","urlHost":"relative","child":null},{"key":"Pharmacy","level":3,"type":"url","url":"/medicine-health-sciences/pharmacy-chp","urlHost":"relative","child":null},{"key":"Physiotherapy","level":3,"type":"url","url":"/medicine-health-sciences/physiotherapy-chp","urlHost":"relative","child":null},{"key":"Public Health & Management","level":3,"type":"url","url":"/medicine-health-sciences/public-health-management-chp","urlHost":"relative","child":null},{"key":"All Medical Specializations","level":3,"type":"link","url":"/medicine-health-sciences/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Clinical Psychology","level":3,"type":"url","url":"/medicine-health-sciences/clinical-psychology-chp","urlHost":"relative","child":null},{"key":"Clinical Research","level":3,"type":"url","url":"/medicine-health-sciences/clinical-research-chp","urlHost":"relative","child":null}],[]]]},{"key":"Exams","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"NEET UG","level":3,"type":"url","url":"/medicine-health-sciences/neet-exam","urlHost":"relative","child":null},{"key":"NEET PG","level":3,"type":"url","url":"/medicine-health-sciences/neet-pg-exam","urlHost":"relative","child":null},{"key":"NEET SS","level":3,"type":"url","url":"/medicine-health-sciences/neet-ss-exam","urlHost":"relative","child":null},{"key":"NEET MDS","level":3,"type":"url","url":"/medicine-health-sciences/dental/neet-mds-exam","urlHost":"relative","child":null},{"key":"INI CET","level":3,"type":"url","url":"/medicine-health-sciences/ini-cet-exam","urlHost":"relative","child":null},{"key":"FMGE","level":3,"type":"url","url":"/medicine-health-sciences/fmge-exam","urlHost":"relative","child":null},{"key":"AIAPGET","level":3,"type":"url","url":"/medicine-health-sciences/aiapget-exam","urlHost":"relative","child":null},{"key":"All Medicine Exams","level":3,"type":"link","url":"/medicine-health-sciences/exams-st-18","urlHost":"relative","child":null}]]]},{"key":"Colleges By Location","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Medical Colleges in India","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-india","urlHost":"relative","child":null},{"key":"Medical Colleges in Delhi","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-delhi","urlHost":"relative","child":null},{"key":"Medical Colleges in Bangalore","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-bangalore","urlHost":"relative","child":null},{"key":"Medical Colleges in Chennai","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-chennai","urlHost":"relative","child":null},{"key":"Medical Colleges in Hyderabad","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-hyderabad","urlHost":"relative","child":null},{"key":"Medical Colleges in Mumbai","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-mumbai","urlHost":"relative","child":null},{"key":"Medical Colleges in Kolkata","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-kolkata","urlHost":"relative","child":null},{"key":"Medical Colleges in Pune","level":3,"type":"url","url":"/medicine-health-sciences/colleges/mbbs-colleges-pune","urlHost":"relative","child":null}]]]},{"key":"College Predictors","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"NEET 2022 College predictor","level":3,"type":"url","url":"/medicine-and-health-sciences/neet-college-predictor","urlHost":"relative","child":null},{"key":"NEET PG College Predictor","level":3,"type":"url","url":"/medicine-and-health-sciences/neet-pg-college-predictor","urlHost":"relative","child":null},{"key":"Medicine College predictor","level":3,"type":"urlWithAnchor","url":"/medicine-and-health-sciences/college-predictor","urlHost":"relative","child":null}]]]},{"key":"Resources","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Ask a Question","level":3,"type":"url","url":"/tags/medicine-health-sciences-tdp-430824","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/medicine-health-sciences-tdp-430824?type=discussion","urlHost":"national","child":null},{"key":"Medical News","level":3,"type":"url","url":"/news/medical-st-18","urlHost":"national","child":null},{"key":"Medical Articles","level":3,"type":"url","url":"/medical/articles-st-18","urlHost":"relative","child":null},{"key":"Trends in Medicine & Health Sciences","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/18","urlHost":"national","child":null}]]]}]]]},{"key":"Design","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Top Ranked Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Fashion Designing Colleges in India","level":3,"type":"url","url":"/design/fashion-design/ranking/top-fashion-design-colleges-in-india/94-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Fashion Designing Colleges in Bangalore","level":3,"type":"url","url":"/design/fashion-design/ranking/top-fashion-design-colleges-in-bangalore/94-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Fashion Designing Colleges in Delhi/NCR","level":3,"type":"url","url":"/design/fashion-design/ranking/top-fashion-design-colleges-in-delhi-ncr/94-2-0-10223-0","urlHost":"relative","child":null}],[]]]},{"key":"Popular Specializations","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Fashion Designing","level":3,"type":"url","url":"/design/fashion-designing-chp","urlHost":"relative","child":null},{"key":"Interior Design","level":3,"type":"url","url":"/design/interior-designing-chp","urlHost":"relative","child":null},{"key":"Graphic Design","level":3,"type":"url","url":"/design/graphic-design-chp","urlHost":"relative","child":null},{"key":"Jewellery Design","level":3,"type":"url","url":"/design/jewellery-design-chp","urlHost":"relative","child":null},{"key":"Web Design","level":3,"type":"url","url":"/design/web-design-chp","urlHost":"relative","child":null},{"key":"Furniture Design","level":3,"type":"url","url":"/design/furniture-design-chp","urlHost":"relative","child":null},{"key":"Game Design","level":3,"type":"url","url":"/it-software/game-design-chp","urlHost":"relative","child":null},{"key":"Product Design","level":3,"type":"url","url":"/design/product-design-chp","urlHost":"relative","child":null},{"key":"Textile Design","level":3,"type":"url","url":"/design/textile-design-chp","urlHost":"relative","child":null},{"key":"Visual Merchandising","level":3,"type":"url","url":"/design/visual-merchandising-chp","urlHost":"relative","child":null},{"key":"Ceramic & Glass Design","level":3,"type":"url","url":"/design/ceramic-glass-design-chp","urlHost":"relative","child":null},{"key":"Film & Video Design","level":3,"type":"url","url":"/design/film-video-design-chp","urlHost":"relative","child":null}],[{"key":"UI / UX","level":3,"type":"url","url":"/design/ui-ux-chp","urlHost":"relative","child":null},{"key":"Footwear Design","level":3,"type":"url","url":"/design/footwear-design-chp","urlHost":"relative","child":null},{"key":"Automotive Design","level":3,"type":"url","url":"/design/automotive-design-chp","urlHost":"relative","child":null},{"key":"Communication Design","level":3,"type":"url","url":"/design/communication-design-chp","urlHost":"relative","child":null},{"key":"Apparel Design","level":3,"type":"url","url":"/design/apparel-design-chp","urlHost":"relative","child":null},{"key":"Exhibition Design","level":3,"type":"url","url":"/design/exhibition-design-chp","urlHost":"relative","child":null},{"key":"Information Design","level":3,"type":"url","url":"/design/information-design-chp","urlHost":"relative","child":null},{"key":"Knitwear Design","level":3,"type":"url","url":"/design/knitwear-design-chp","urlHost":"relative","child":null},{"key":"Leather Design","level":3,"type":"url","url":"/design/leather-design-chp","urlHost":"relative","child":null},{"key":"Toy Design","level":3,"type":"url","url":"/design/toy-design-chp","urlHost":"relative","child":null},{"key":"Lifestyle Accesory Design","level":3,"type":"url","url":"/design/lifestyle-accessory-design-chp","urlHost":"relative","child":null},{"key":"All Design Specializations","level":3,"type":"urlWithAnchor","url":"/design/colleges/colleges-india","urlHost":"relative","child":null}]]]},{"key":"Popular Courses","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"B.Des","level":3,"type":"url","url":"/b-des-bachelor-of-design-chp","urlHost":"relative","child":null},{"key":"M.Des","level":3,"type":"url","url":"/m-des-chp","urlHost":"relative","child":null},{"key":"B.Des in Fashion Design","level":3,"type":"url","url":"/design/b-des-in-fashion-design-chp","urlHost":"relative","child":null},{"key":"B.Des in Interior Design","level":3,"type":"url","url":"/design/b-des-in-interior-design-chp","urlHost":"relative","child":null},{"key":"B.Sc in Fashion Design","level":3,"type":"url","url":"/design/b-sc-in-fashion-design-chp","urlHost":"relative","child":null},{"key":"B.Sc in Interior Design","level":3,"type":"url","url":"/design/b-sc-in-interior-design-chp","urlHost":"relative","child":null},{"key":"All Design Courses","level":3,"type":"urlWithAnchor","url":"/design/colleges/colleges-india","urlHost":"relative","child":null}]]]},{"key":"Exams","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Popular Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"WUD Aptitude Test","level":3,"type":"url","url":"/college/world-university-of-design/wud-aptitude-test-exam","urlHost":"relative","child":null},{"key":"Pearl Academy Entrance Exam","level":3,"type":"url","url":"/design/fashion-design/pearl-academy-entrance-exam","urlHost":"relative","child":null},{"key":"CEED","level":3,"type":"url","url":"/design/industrial-product-design/ceed-exam","urlHost":"relative","child":null},{"key":"NID Entrance Exam","level":3,"type":"url","url":"/design/communication-design/nid-entrance-exam","urlHost":"relative","child":null},{"key":"NIFT Entrance Exam","level":3,"type":"url","url":"/design/nift-entrance-exam","urlHost":"relative","child":null},{"key":"UCEED","level":3,"type":"url","url":"/design/uceed-exam","urlHost":"relative","child":null},{"key":"All Design Exams","level":3,"type":"urlWithAnchor","url":"/design/exams-st-3","urlHost":"relative","child":null}],[]]]},{"key":"College Predictors","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"NID College predictor","level":3,"type":"url","url":"/design/nid-entrance-exam-college-predictor","urlHost":"relative","child":null},{"key":"NIFT College predictor","level":3,"type":"url","url":"/design/nift-entrance-exam-college-predictor","urlHost":"relative","child":null},{"key":"Design College predictor","level":3,"type":"urlWithAnchor","url":"/design/college-predictor","urlHost":"relative","child":null}]]]},{"key":"Colleges by Location","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Design Colleges in India","level":3,"type":"url","url":"/design/colleges/colleges-india","urlHost":"relative","child":null},{"key":"Design Colleges in Maharashtra","level":3,"type":"url","url":"/design/colleges/colleges-maharashtra","urlHost":"relative","child":null},{"key":"Design Colleges in Delhi","level":3,"type":"url","url":"/design/colleges/colleges-delhi","urlHost":"relative","child":null},{"key":"Design Colleges in Karnataka","level":3,"type":"url","url":"/design/colleges/colleges-karnataka","urlHost":"relative","child":null},{"key":"Design Colleges in Punjab","level":3,"type":"url","url":"/design/colleges/colleges-punjab","urlHost":"relative","child":null},{"key":"Design Colleges in Telangana","level":3,"type":"url","url":"/design/colleges/colleges-telangana","urlHost":"relative","child":null},{"key":"Design Colleges in Gujarat","level":3,"type":"url","url":"/design/colleges/colleges-gujarat","urlHost":"relative","child":null},{"key":"Design Colleges in Chandigarh","level":3,"type":"url","url":"/design/colleges/colleges-chandigarh","urlHost":"relative","child":null},{"key":"Design Colleges in Rajasthan","level":3,"type":"url","url":"/design/colleges/colleges-rajasthan","urlHost":"relative","child":null},{"key":"Design Colleges in Madhya Pradesh","level":3,"type":"url","url":"/design/colleges/colleges-madhya-pradesh","urlHost":"relative","child":null},{"key":"Design Colleges in Uttar Pradesh","level":3,"type":"url","url":"/design/colleges/colleges-uttar-pradesh","urlHost":"relative","child":null},{"key":"Design Colleges in Tamil Nadu","level":3,"type":"url","url":"/design/colleges/colleges-tamil-nadu ","urlHost":"relative","child":null}],[{"key":"Design Colleges in Pune","level":3,"type":"url","url":"/design/colleges/colleges-pune","urlHost":"relative","child":null},{"key":"Design Colleges in Mumbai","level":3,"type":"url","url":"/design/colleges/colleges-mumbai","urlHost":"relative","child":null},{"key":"Design Colleges in Bangalore","level":3,"type":"url","url":"/design/colleges/colleges-bangalore","urlHost":"relative","child":null},{"key":"Design Colleges in Hyderabad","level":3,"type":"url","url":"/design/colleges/colleges-hyderabad","urlHost":"relative","child":null},{"key":"Design Colleges in Ahmedabad","level":3,"type":"url","url":"/design/colleges/colleges-ahmedabad","urlHost":"relative","child":null},{"key":"Design Colleges in Ludhiana","level":3,"type":"url","url":"/design/colleges/colleges-ludhiana","urlHost":"relative","child":null},{"key":"Design Colleges in Jalandhar","level":3,"type":"url","url":"/design/colleges/colleges-jalandhar","urlHost":"relative","child":null},{"key":"Design Colleges in Jaipur","level":3,"type":"url","url":"/design/colleges/colleges-jaipur","urlHost":"relative","child":null},{"key":"Design Colleges in Indore","level":3,"type":"url","url":"/design/colleges/colleges-indore","urlHost":"relative","child":null},{"key":"Design Colleges in Gurgaon","level":3,"type":"url","url":"/design/colleges/colleges-gurgaon","urlHost":"relative","child":null}]]]},{"key":"Resources","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Ask a Question","level":3,"type":"url","url":"/tags/design-tdp-10","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/design-tdp-10?type=discussion","urlHost":"national","child":null},{"key":"Design News","level":3,"type":"url","url":"/news/design-st-3","urlHost":"national","child":null},{"key":"Design Articles","level":3,"type":"url","url":"/design/articles-st-3","urlHost":"relative","child":null},{"key":"Trends in Design","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/3","urlHost":"national","child":null}]]]}]]]},{"key":"More","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Sarkari Exams","level":2,"type":"subItem","url":"/sarkari-exams/exams-st-21","urlHost":"relative","child":[[[{"key":"Banking","level":0,"type":"heading","url":"","urlHost":null,"child":null},{"key":"IBPS Clerk","level":3,"type":"url","url":"/exams/ibps-clerk-exam","urlHost":"relative","child":null},{"key":"IBPS PO","level":3,"type":"url","url":"/exams/ibps-po-exam","urlHost":"relative","child":null},{"key":"SBI Clerk","level":3,"type":"url","url":"/exams/sbi-clerk-exam","urlHost":"relative","child":null},{"key":"SBI PO","level":3,"type":"url","url":"/exams/sbi-po-exam","urlHost":"relative","child":null},{"key":"IBPS RRB","level":3,"type":"url","url":"/exams/ibps-rrb-exam","urlHost":"relative","child":null},{"key":"All Banking Exams","level":3,"type":"link","url":"/sarkari-exams/banking/exams-sb-21-107","urlHost":"relative","child":null}],[{"key":"Teaching","level":0,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CTET","level":3,"type":"url","url":"/exams/ctet-exam","urlHost":"relative","child":null},{"key":"UPTET","level":3,"type":"url","url":"/exams/uptet-exam","urlHost":"relative","child":null},{"key":"UGC NET","level":3,"type":"url","url":"/exams/ugc-net-exam","urlHost":"relative","child":null},{"key":"CSIR NET","level":3,"type":"url","url":"/exams/csir-net-exam","urlHost":"relative","child":null},{"key":"APSET","level":3,"type":"url","url":"/exams/apset-exam ","urlHost":"relative","child":null},{"key":"All Teaching Exams","level":3,"type":"link","url":"/sarkari-exams/teaching/exams-sb-21-109","urlHost":"relative","child":null}],[{"key":"SSC","level":0,"type":"heading","url":"","urlHost":null,"child":null},{"key":"SSC CGL","level":3,"type":"url","url":"/exams/ssc-cgl-exam","urlHost":"relative","child":null},{"key":"SSC JE","level":3,"type":"url","url":"/exams/ssc-je-exam","urlHost":"relative","child":null},{"key":"SSC CHSL","level":3,"type":"url","url":"/exams/ssc-chsl-exam","urlHost":"relative","child":null},{"key":"SSC GD","level":3,"type":"url","url":"/exams/ssc-gd-exam","urlHost":"relative","child":null},{"key":"SSC JHT","level":3,"type":"url","url":"/exams/ssc-jht-exam","urlHost":"relative","child":null},{"key":"All SSC Exams","level":3,"type":"link","url":"/sarkari-exams/ssc/exams-sb-21-111","urlHost":"relative","child":null}]],[[{"key":"Defence","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"NDA","level":3,"type":"url","url":"/exams/nda-exam","urlHost":"relative","child":null},{"key":"AFCAT","level":3,"type":"url","url":"/exams/afcat-exam","urlHost":"relative","child":null},{"key":"CDS","level":3,"type":"url","url":"/exams/cds-exam","urlHost":"relative","child":null},{"key":"DRDO CEPTAM","level":3,"type":"url","url":"/exams/drdo-ceptam-exam","urlHost":"relative","child":null},{"key":"RPF SI","level":3,"type":"url","url":"/exams/rpf-si-exam","urlHost":"relative","child":null},{"key":"All Defence Exams","level":3,"type":"link","url":"/sarkari-exams/defence/exams-sb-21-113","urlHost":"relative","child":null}],[{"key":"Railway","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"RRB Group D","level":3,"type":"url","url":"/exams/rrb-group-d-exam","urlHost":"relative","child":null},{"key":"RRB NTPC","level":3,"type":"url","url":"/exams/rrb-ntpc-exam","urlHost":"relative","child":null},{"key":"RRB JE","level":3,"type":"url","url":"/exams/rrb-je-exam","urlHost":"relative","child":null},{"key":"RPF Constable","level":3,"type":"url","url":"/exams/rpf-constable-exam","urlHost":"relative","child":null},{"key":"All Railway Exams","level":3,"type":"link","url":"/sarkari-exams/railway/exams-sb-21-115","urlHost":"relative","child":null}],[{"key":"All Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"All UPSC Exams","level":3,"type":"url","url":"/sarkari-exams/upsc/exams-sb-21-119","urlHost":"relative","child":null},{"key":"All State PSC Exams","level":3,"type":"url","url":"/sarkari-exams/state-psc/exams-sb-21-121","urlHost":"relative","child":null},{"key":"All Scholarship Exams","level":3,"type":"url","url":"/sarkari-exams/scholarship/exams-sb-21-123","urlHost":"relative","child":null},{"key":"All PSU Exams","level":3,"type":"url","url":"/sarkari-exams/psu/exams-sb-21-129","urlHost":"relative","child":null},{"key":"All State Exams","level":3,"type":"url","url":"/sarkari-exams/state-exams/exams-sb-21-131","urlHost":"relative","child":null},{"key":"All Insurance Exams","level":3,"type":"url","url":"/sarkari-exams/insurance/exams-sb-21-125","urlHost":"relative","child":null},{"key":"All Police Exams","level":3,"type":"url","url":"/sarkari-exams/police/exams-sb-21-135","urlHost":"relative","child":null},{"key":"All Sarkari Exams","level":3,"type":"url","url":"/sarkari-exams/exams-st-21","urlHost":"relative","child":null}]]]},{"key":"Law","level":2,"type":"subItem","url":"/law-chp","urlHost":"relative","child":[[[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Law Colleges in India","level":3,"type":"url","url":"/law/ranking/top-law-colleges-in-india/56-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Law Colleges in Bangalore","level":3,"type":"url","url":"/law/ranking/top-law-colleges-in-bangalore/56-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Law Colleges in Delhi","level":3,"type":"url","url":"/law/ranking/top-law-colleges-in-delhi/56-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Law Colleges in Pune","level":3,"type":"url","url":"/law/ranking/top-law-colleges-in-pune/56-2-0-174-0","urlHost":"relative","child":null},{"key":"Top Law Colleges in Hyderabad","level":3,"type":"url","url":"/law/ranking/top-law-colleges-in-hyderabad/56-2-0-702-0","urlHost":"relative","child":null}],[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.A. LL.B.","level":3,"type":"url","url":"/b-a-ll-b-chp","urlHost":"relative","child":null},{"key":"BBA LL.B.","level":3,"type":"url","url":"/bba-ll-b-chp","urlHost":"relative","child":null},{"key":"LL.B.","level":3,"type":"url","url":"/ll-b-chp","urlHost":"relative","child":null},{"key":"LL.M.","level":3,"type":"url","url":"/ll-m-chp","urlHost":"relative","child":null},{"key":"B.Sc. LL.B","level":3,"type":"url","url":"/b-sc-ll-b-chp","urlHost":"relative","child":null},{"key":"B.Com LL.B","level":3,"type":"url","url":"/b-com-ll-b-chp","urlHost":"relative","child":null},{"key":"B.L.S. LL.B.","level":3,"type":"url","url":"/b-l-s-ll-b-bachelor-of-legal-science-bachelor-of-law-chp","urlHost":"relative","child":null},{"key":"All Law Courses","level":3,"type":"link","url":"/law/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Company Law","level":3,"type":"url","url":"/law/company-law-chp","urlHost":"relative","child":null},{"key":"Business Law","level":3,"type":"url","url":"/law/business-law-chp","urlHost":"relative","child":null},{"key":"Cyber Law","level":3,"type":"url","url":"/law/cyber-law-chp","urlHost":"relative","child":null},{"key":"Corporate Law","level":3,"type":"url","url":"/law/corporate-law-chp","urlHost":"relative","child":null},{"key":"Criminal Law","level":3,"type":"url","url":"/law/criminal-law-chp","urlHost":"relative","child":null},{"key":"Administrative Law","level":3,"type":"url","url":"/law/administrative-law-chp","urlHost":"relative","child":null},{"key":"Family Law","level":3,"type":"url","url":"/law/family-law-chp","urlHost":"relative","child":null},{"key":"Constitutional Law","level":3,"type":"url","url":"/law/constitutional-law-chp","urlHost":"relative","child":null},{"key":"Environmental Law","level":3,"type":"url","url":"/law/environmental-law-chp","urlHost":"relative","child":null},{"key":"Intellectual Property Law","level":3,"type":"url","url":"/law/intellectual-property-law-chp","urlHost":"relative","child":null},{"key":"Banking Law","level":3,"type":"url","url":"/law/banking-law-chp","urlHost":"relative","child":null},{"key":"Competition Law","level":3,"type":"url","url":"/law/competition-law-chp","urlHost":"relative","child":null},{"key":"Commercial Law","level":3,"type":"url","url":"/law/commercial-law-chp","urlHost":"relative","child":null},{"key":"Immigration Law","level":3,"type":"url","url":"/law/immigration-law-chp","urlHost":"relative","child":null},{"key":"Tax Law","level":3,"type":"url","url":"/law/tax-law-chp","urlHost":"relative","child":null},{"key":"Insurance Law","level":3,"type":"url","url":"/law/insurance-law-chp","urlHost":"relative","child":null},{"key":"Energy Law","level":3,"type":"url","url":"/law/energy-law-chp","urlHost":"relative","child":null},{"key":"International Trade Law","level":3,"type":"url","url":"/law/international-trade-law-chp","urlHost":"relative","child":null},{"key":"Consumer Law","level":3,"type":"url","url":"/law/consumer-law-chp","urlHost":"relative","child":null},{"key":"Arbitration Law","level":3,"type":"url","url":"/law/arbitration-law-chp","urlHost":"relative","child":null},{"key":"Real Estate / Infrastructure Law","level":3,"type":"url","url":"/law/real-estate-infrastructure-law-chp","urlHost":"relative","child":null},{"key":"Information Technology Law","level":3,"type":"url","url":"/law/information-technology-law-chp","urlHost":"relative","child":null},{"key":"Healthcare Law","level":3,"type":"url","url":"/law/healthcare-law-chp","urlHost":"relative","child":null},{"key":"Labor & Employment Law","level":3,"type":"url","url":"/law/labor-employment-law-chp","urlHost":"relative","child":null},{"key":"Air & Space Law","level":3,"type":"url","url":"/law/air-space-law-chp","urlHost":"relative","child":null},{"key":"Nuclear Law","level":3,"type":"url","url":"/law/nuclear-law-chp","urlHost":"relative","child":null},{"key":"Human Rights & International Humanitarian Law","level":3,"type":"url","url":"/law/human-rights-international-humanitarian-law-chp","urlHost":"relative","child":null},{"key":"Security & Investment Law","level":3,"type":"url","url":"/law/security-investment-law-chp","urlHost":"relative","child":null},{"key":"Entertainment & Media Law","level":3,"type":"url","url":"/law/entertainment-media-law-chp","urlHost":"relative","child":null}]],[[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CLAT","level":3,"type":"url","url":"/law/clat-exam","urlHost":"relative","child":null},{"key":"LSAT India","level":3,"type":"url","url":"/law/lsat-india-exam","urlHost":"relative","child":null},{"key":"AILET","level":3,"type":"url","url":"/law/ailet-exam","urlHost":"relative","child":null},{"key":"AIBE","level":3,"type":"url","url":"/law/aibe-exam","urlHost":"relative","child":null},{"key":"DU LLB Exam","level":3,"type":"url","url":"/university/delhi-university-du/du-llb-entrance-exam","urlHost":"relative","child":null},{"key":"AMU Law Entrance Exam","level":3,"type":"url","url":"/university/aligarh-muslim-university/amu-entrance-exam","urlHost":"relative","child":null},{"key":"ACLAT","level":3,"type":"url","url":"/university/alliance-university-bangalore/aclat-exam","urlHost":"relative","child":null},{"key":"All Law Exams","level":3,"type":"link","url":"/law/exams-st-5","urlHost":"relative","child":null}],[{"key":"Colleges by Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Law Colleges in India","level":3,"type":"url","url":"/law/colleges/colleges-india","urlHost":"relative","child":null},{"key":"Law Colleges in Punjab","level":3,"type":"url","url":"/law/colleges/colleges-punjab","urlHost":"relative","child":null},{"key":"Law Colleges in Delhi","level":3,"type":"url","url":"/law/colleges/colleges-delhi","urlHost":"relative","child":null},{"key":"Law Colleges in Chandigarh","level":3,"type":"url","url":"/law/colleges/colleges-chandigarh","urlHost":"relative","child":null},{"key":"Law Colleges in Maharashtra","level":3,"type":"url","url":"/law/colleges/colleges-maharashtra","urlHost":"relative","child":null},{"key":"Law Colleges in Orissa","level":3,"type":"url","url":"/law/colleges/colleges-orissa","urlHost":"relative","child":null},{"key":"Law Colleges in Uttarakhand","level":3,"type":"url","url":"/law/colleges/colleges-uttarakhand","urlHost":"relative","child":null},{"key":"Law Colleges in West Bengal","level":3,"type":"url","url":"/law/colleges/colleges-west-bengal","urlHost":"relative","child":null},{"key":"Law Colleges in Karnataka","level":3,"type":"url","url":"/law/colleges/colleges-karnataka","urlHost":"relative","child":null},{"key":"Law Colleges in Ludhiana","level":3,"type":"url","url":"/law/colleges/colleges-ludhiana","urlHost":"relative","child":null},{"key":"Law Colleges in Pune","level":3,"type":"url","url":"/law/colleges/colleges-pune","urlHost":"relative","child":null},{"key":"Law Colleges in Jalandhar","level":3,"type":"url","url":"/law/colleges/colleges-jalandhar","urlHost":"relative","child":null},{"key":"Law Colleges in Bhubaneswar","level":3,"type":"url","url":"/law/colleges/colleges-bhubaneswar","urlHost":"relative","child":null},{"key":"Law Colleges in Roorkee","level":3,"type":"url","url":"/law/colleges/colleges-roorkee","urlHost":"relative","child":null},{"key":"Law Colleges in Kolkata","level":3,"type":"url","url":"/law/colleges/colleges-kolkata","urlHost":"relative","child":null},{"key":"Law Colleges in Udupi","level":3,"type":"url","url":"/law/colleges/colleges-udupi","urlHost":"relative","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/law-tdp-15","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/law-tdp-15?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/law/articles-st-5","urlHost":"relative","child":null},{"key":"Trends in Law","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/5","urlHost":"national","child":null},{"key":"CLAT College Predictor","level":3,"type":"url","url":"/law/clat-college-predictor","urlHost":"relative","child":null},{"key":"AILET (B.A. LL.B) College predictor","level":3,"type":"url","url":"/law/ailet-college-predictor?sg[]=264","urlHost":"relative","child":null},{"key":"AILET (LL.M) College predictor","level":3,"type":"url","url":"/law/ailet-college-predictor?sg[]=419","urlHost":"relative","child":null},{"key":"Law College predictor","level":3,"type":"urlWithAnchor","url":"/law/college-predictor","urlHost":"relative","child":null}]]]},{"key":"Hospitality & Travel","level":2,"type":"subItem","url":"/hospitality-travel-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BHM","level":3,"type":"url","url":"/bhm-bachelor-of-hotel-management-chp","urlHost":"relative","child":null},{"key":"Diploma in Hotel Management","level":3,"type":"url","url":"/hospitality-travel/diploma-in-hotel-hospitality-management-chp","urlHost":"relative","child":null},{"key":"B.Sc. In Hotel Management","level":3,"type":"url","url":"/hospitality-travel/b-sc-in-hotel-hospitality-management-chp","urlHost":"relative","child":null},{"key":"All Hospitality Courses","level":3,"type":"link","url":"/hospitality-travel/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Hotel Management Colleges in India","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/ranking/top-hotel-management-colleges-in-india/98-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Hotel Management Colleges in Hyderabad","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/ranking/top-hotel-management-colleges-in-hyderabad/98-2-0-702-0","urlHost":"relative","child":null},{"key":"Top Hotel Management Colleges in Delhi","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/ranking/top-hotel-management-colleges-in-delhi/98-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Hotel Management Colleges in Mumbai","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/ranking/top-hotel-management-colleges-in-mumbai/98-2-0-151-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"IIHM eCHAT","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/iihm-echat-exam","urlHost":"relative","child":null},{"key":"NCHMCT JEE","level":3,"type":"url","url":"/hospitality-travel/hotel-hospitality-management/nchmct-jee-exam","urlHost":"relative","child":null},{"key":"All Hospitality Exams","level":3,"type":"link","url":"/hospitality-travel/exams-st-4","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Catering","level":3,"type":"url","url":"/hospitality-travel/catering-courses-chp","urlHost":"relative","child":null},{"key":"Culinary Arts","level":3,"type":"url","url":"/hospitality-travel/culinary-arts-chp","urlHost":"relative","child":null},{"key":"Event Management","level":3,"type":"url","url":"/hospitality-travel/event-management-chp","urlHost":"relative","child":null},{"key":"Fares & Ticketing","level":3,"type":"url","url":"/hospitality-travel/fares-ticketing-courses-chp","urlHost":"relative","child":null},{"key":"Hotel / Hospitality Management","level":3,"type":"url","url":"/hospitality-travel/hotel-management-chp","urlHost":"relative","child":null},{"key":"CBS","level":3,"type":"url","url":"/cbs-chp","urlHost":"relative","child":null},{"key":"Travel & Tourism","level":3,"type":"url","url":"/hospitality-travel/travel-tourism-chp","urlHost":"relative","child":null},{"key":"Corporate Banking","level":3,"type":"url","url":"/banking-finance-insurance/corporate-banking-chp","urlHost":"relative","child":null},{"key":"Health Insurance","level":3,"type":"url","url":"/banking-finance-insurance/health-insurance-chp","urlHost":"relative","child":null},{"key":"Health Insurance","level":3,"type":"url","url":"/banking-finance-insurance/health-insurance-chp","urlHost":"relative","child":null},{"key":"Investment Banking","level":3,"type":"url","url":"/banking-finance-insurance/investment-banking-chp","urlHost":"relative","child":null},{"key":"Capital Markets","level":3,"type":"url","url":"/banking-finance-insurance/capital-markets-chp","urlHost":"relative","child":null},{"key":"All Hospitality Specializations","level":3,"type":"link","url":"/hospitality-travel/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Colleges by Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Culinary Arts colleges in India","level":3,"type":"url","url":"/hospitality-travel/culinary-arts/colleges/colleges-india","urlHost":"relative","child":null},{"key":"Travel & Tourism colleges in India","level":3,"type":"url","url":"/hospitality-travel/travel-tourism/colleges/colleges-india","urlHost":"relative","child":null},{"key":"Event Management colleges in India","level":3,"type":"url","url":"/hospitality-travel/event-management/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"NCHMCT College predictor","level":3,"type":"url","url":"/hospitality-and-travel/nchmct-jee-college-predictor","urlHost":"relative","child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/hotel-management-tdp-22","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/hotel-management-tdp-22?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/hospitality-travel/articles-st-4","urlHost":"relative","child":null},{"key":"Trends in Hospitality & Travel","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/4","urlHost":"national","child":null}]]]},{"key":"Animation","level":2,"type":"subItem","url":"/animation-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Sc. in Animation","level":3,"type":"url","url":"/animation/b-sc-in-animation-chp","urlHost":"relative","child":null},{"key":"M.Sc. in Animation","level":3,"type":"url","url":"/animation/m-sc-in-animation-chp","urlHost":"relative","child":null},{"key":"Diploma in Web/Graphic Design","level":3,"type":"url","url":"/animation/diploma-in-graphic-web-design-chp","urlHost":"relative","child":null},{"key":"Diploma in VFX","level":3,"type":"url","url":"/animation/diploma-in-visual-effects-vfx-chp","urlHost":"relative","child":null},{"key":"Diploma in Animation","level":3,"type":"url","url":"/diploma-in-animation-chp","urlHost":"relative","child":null},{"key":"All Animation Courses","level":3,"type":"urlWithAnchor","url":"/animation/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MAAC","level":3,"type":"url","url":"/college/maya-academy-of-advanced-cinematics-camp-camp-pune-31765","urlHost":"national","child":null},{"key":"Arena Animation","level":3,"type":"url","url":"/college/arena-animation-andheri-west-mumbai-29986","urlHost":"national","child":null},{"key":"FX School","level":3,"type":"url","url":"/college/arena-animation-borivali-west-mumbai-29986","urlHost":"national","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/animation-tdp-365","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/animation-tdp-365?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/animation/articles-st-6","urlHost":"relative","child":null},{"key":"Trends in Animation","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/6","urlHost":"national","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"2D/3D Animation","level":3,"type":"url","url":"/animation/2d-3d-animation-chp","urlHost":"relative","child":null},{"key":"Animation Film Making","level":3,"type":"url","url":"/animation/animation-film-making-chp","urlHost":"relative","child":null},{"key":"Graphic / Web Design","level":3,"type":"url","url":"/animation/graphic-web-design-chp","urlHost":"relative","child":null},{"key":"Sound & Video Editing","level":3,"type":"url","url":"/animation/sound-video-editing-chp","urlHost":"relative","child":null},{"key":"VFX","level":3,"type":"url","url":"/animation/visual-effects-vfx-chp","urlHost":"relative","child":null},{"key":"Game Design","level":3,"type":"url","url":"/it-software/game-design-chp","urlHost":"relative","child":null},{"key":"All Animation Specializations","level":3,"type":"link","url":"/animation/colleges/colleges-india","urlHost":"relative","child":null}],[]]]},{"key":"Mass Communication & Media","level":2,"type":"subItem","url":"/mass-communication-media-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.J.","level":3,"type":"url","url":"/b-j-bachelor-of-journalism-chp","urlHost":"relative","child":null},{"key":"B.J.M.C.","level":3,"type":"url","url":"/b-j-m-c-bachelor-of-journalism-mass-communication-chp","urlHost":"relative","child":null},{"key":"B.M.M.","level":3,"type":"url","url":"/b-m-m-bachelor-of-mass-media-chp","urlHost":"relative","child":null},{"key":"M.A.","level":3,"type":"url","url":"/m-a-master-of-arts-chp","urlHost":"relative","child":null},{"key":"Diploma in Journalism","level":3,"type":"url","url":"/mass-communication-media/diploma-in-journalism-chp","urlHost":"relative","child":null},{"key":"All Mass Comm & Media Courses","level":3,"type":"link","url":"/mass-communication-media/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Mass Communication Colleges in India","level":3,"type":"url","url":"/mass-communication-media/ranking/top-mass-communication-colleges-in-india/99-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Mass Communication Colleges in Delhi","level":3,"type":"url","url":"/mass-communication-media/ranking/top-mass-communication-colleges-in-delhi/99-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Mass Communication Colleges in Mumbai","level":3,"type":"url","url":"/mass-communication-media/ranking/top-mass-communication-colleges-in-mumbai/99-2-0-151-0","urlHost":"relative","child":null},{"key":"Top Mass Communication Colleges in Kolkata","level":3,"type":"url","url":"/mass-communication-media/ranking/top-mass-communication-colleges-in-kolkata/99-2-0-130-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"IIMC Entrance Exam","level":3,"type":"url","url":"/college/indian-institute-of-mass-communication-new-delhi/iimc-entrance-exam","urlHost":"relative","child":null},{"key":"JMI Entrance Exam","level":3,"type":"url","url":"/university/jamia-millia-islamia/jmi-entrance-exam","urlHost":"relative","child":null},{"key":"ACJ Entrance Exam","level":3,"type":"url","url":"/college/asian-college-of-journalism-acj/acj-entrance-exam","urlHost":"relative","child":null},{"key":"All Mass Comm & Media Exams","level":3,"type":"link","url":"/mass-communication-media/exams-st-7","urlHost":"relative","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.J.M.C. colleges in India","level":3,"type":"url","url":"/mass-communication-media/colleges/b-j-m-c-colleges-india","urlHost":"relative","child":null},{"key":"Journalism colleges in India","level":3,"type":"url","url":"/mass-communication-media/journalism/colleges/colleges-india","urlHost":"relative","child":null},{"key":"M.A. in Mass Communications in India","level":3,"type":"url","url":"/mass-communication-media/colleges/m-a-colleges-india","urlHost":"relative","child":null},{"key":"Public Relations colleges in India","level":3,"type":"url","url":"/mass-communication-media/public-relations/colleges/colleges-india","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Advertising","level":3,"type":"url","url":"/mass-communication-media/advertising-chp","urlHost":"relative","child":null},{"key":"Event Management","level":3,"type":"url","url":"/hospitality-travel/event-management-chp","urlHost":"relative","child":null},{"key":"Film & TV","level":3,"type":"url","url":"/mass-communication-media/film-tv-chp","urlHost":"relative","child":null},{"key":"Journalism","level":3,"type":"url","url":"/mass-communication-media/journalism-chp","urlHost":"relative","child":null},{"key":"Music & Sound Production","level":3,"type":"url","url":"/mass-communication-media/music-sound-production-chp","urlHost":"relative","child":null},{"key":"Public Relations","level":3,"type":"url","url":"/mass-communication-media/public-relations-chp","urlHost":"relative","child":null},{"key":"Radio","level":3,"type":"url","url":"/mass-communication-media/radio-chp","urlHost":"relative","child":null},{"key":"All Mass Comm & Media Specializations","level":3,"type":"link","url":"/mass-communication-media/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/mass-communication-media-tdp-14","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/mass-communication-media-tdp-14?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/mass-communication/articles-st-7","urlHost":"relative","child":null},{"key":"Trends in Mass Communication & Media","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/7","urlHost":"national","child":null},{"key":"Delhi University Cut-Offs 2020","level":3,"type":"url","url":"/university/university-of-delhi-24642/cutoff","urlHost":"national","child":null}]]]},{"key":"Business & Management Studies","level":2,"type":"subItem","url":"/business-management-studies-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BBA","level":3,"type":"url","url":"/bba-bachelor-of-business-administration-chp","urlHost":"relative","child":null},{"key":"Management Certifications","level":3,"type":"url","url":"/certificate-in-business-management-studies-chp","urlHost":"relative","child":null},{"key":"MBA/PGDM","level":3,"type":"url","url":"/mba-masters-of-business-administration-chp","urlHost":"relative","child":null},{"key":"Executive MBA/PGDM","level":3,"type":"url","url":"/executive-mba-chp","urlHost":"relative","child":null},{"key":"Distance MBA","level":3,"type":"url","url":"/distance-mba-mba-distance-education-chp","urlHost":"relative","child":null},{"key":"Online MBA","level":3,"type":"url","url":"/online-mba-chp","urlHost":"relative","child":null},{"key":"Part-Time MBA","level":3,"type":"url","url":"/part-time-mba-pgdm-chp","urlHost":"relative","child":null},{"key":"All Management Courses","level":3,"type":"link","url":"/business-management-studies/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top BBA Colleges in India","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-india/93-2-0-0-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Delhi","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-delhi/93-2-0-74-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Bangalore","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-bangalore/93-2-0-278-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Hyderabad","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-hyderabad/93-2-0-702-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Pune","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-pune/93-2-0-174-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Kolkata","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-kolkata/93-2-0-130-0","urlHost":"relative","child":null},{"key":"Top BBA Colleges in Mumbai","level":3,"type":"url","url":"/bba/ranking/top-bba-colleges-in-mumbai/93-2-0-151-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"SET Exam","level":3,"type":"url","url":"/bba/set-exam","urlHost":"relative","child":null},{"key":"NPAT","level":3,"type":"url","url":"/bba/nmims-npat-exam","urlHost":"relative","child":null},{"key":"SUAT BBA","level":3,"type":"url","url":"/university/sharda-university/suat-bba-exam","urlHost":"relative","child":null},{"key":"DU JAT","level":3,"type":"url","url":"/bba/du-jat-exam","urlHost":"relative","child":null},{"key":"All Management Exams","level":3,"type":"link","url":"/business-management-studies/exams-st-1","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Finance","level":3,"type":"url","url":"/business-management-studies/finance-chp","urlHost":"relative","child":null},{"key":"Sales & Marketing","level":3,"type":"url","url":"/business-management-studies/sales-marketing-chp","urlHost":"relative","child":null},{"key":"Entrepreneurship","level":3,"type":"url","url":"/business-management-studies/entrepreneurship-courses-chp","urlHost":"relative","child":null},{"key":"Digital Marketing","level":3,"type":"url","url":"/business-management-studies/digital-marketing-chp","urlHost":"relative","child":null},{"key":"Operations","level":3,"type":"url","url":"/business-management-studies/operations-chp","urlHost":"relative","child":null},{"key":"Infrastructure","level":3,"type":"url","url":"/business-management-studies/infrastructure-courses-chp","urlHost":"relative","child":null},{"key":"Human Resources","level":3,"type":"url","url":"/business-management-studies/human-resources-chp","urlHost":"relative","child":null},{"key":"Telecom","level":3,"type":"url","url":"/business-management-studies/telecom-courses-chp","urlHost":"relative","child":null},{"key":"Business Analytics","level":3,"type":"url","url":"/business-management-studies/business-analytics-chp","urlHost":"relative","child":null},{"key":"Supply Chain","level":3,"type":"url","url":"/business-management-studies/supply-chain-chp","urlHost":"relative","child":null},{"key":"Retail","level":3,"type":"url","url":"/business-management-studies/retail-courses-chp","urlHost":"relative","child":null},{"key":"Import & Export","level":3,"type":"url","url":"/business-management-studies/import-export-chp","urlHost":"relative","child":null},{"key":"International Business","level":3,"type":"url","url":"/business-management-studies/international-business-chp","urlHost":"relative","child":null},{"key":"Business Economics","level":3,"type":"url","url":"/business-management-studies/business-economics-chp","urlHost":"relative","child":null},{"key":"Family Business","level":3,"type":"url","url":"/business-management-studies/family-business-courses-chp","urlHost":"relative","child":null},{"key":"Materials Management","level":3,"type":"url","url":"/business-management-studies/materials-management-chp","urlHost":"relative","child":null},{"key":"Rural Management","level":3,"type":"url","url":"/business-management-studies/rural-management-chp","urlHost":"relative","child":null},{"key":"Textile Management","level":3,"type":"url","url":"/business-management-studies/textile-management-chp","urlHost":"relative","child":null},{"key":"Agriculture & Food Business","level":3,"type":"url","url":"/business-management-studies/agriculture-food-business-chp","urlHost":"relative","child":null},{"key":"All Management Specializations","level":3,"type":"urlWithAnchor","url":"/business-management-studies/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/business-management-studies-tdp-17","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/business-management-studies-tdp-17?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/business-management-studies/articles-st-1","urlHost":"relative","child":null},{"key":"Trends in Business & Management Studies","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/1","urlHost":"national","child":null}]]]},{"key":"IT & Software","level":2,"type":"subItem","url":"/it-software-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BCA","level":3,"type":"url","url":"/bca-bachelor-of-computer-applications-chp","urlHost":"relative","child":null},{"key":"B.Sc. in IT & Software","level":3,"type":"url","url":"/it-software/b-sc-it-information-technology-chp","urlHost":"relative","child":null},{"key":"Distance BCA","level":3,"type":"url","url":"/distance-correspondence-bca-chp","urlHost":"relative","child":null},{"key":"MCA","level":3,"type":"url","url":"/mca-master-of-computer-applications-chp","urlHost":"relative","child":null},{"key":"M.Sc. in IT & Software","level":3,"type":"url","url":"/it-software/m-sc-in-it-software-chp","urlHost":"relative","child":null},{"key":"Part-Time MCA","level":3,"type":"url","url":"/part-time-mca-chp","urlHost":"relative","child":null},{"key":"Distance MCA","level":3,"type":"url","url":"/distance-correspondence-mca-chp","urlHost":"relative","child":null},{"key":"CCNA","level":3,"type":"url","url":"/ccna-chp","urlHost":"relative","child":null},{"key":"DOEACC O Level","level":3,"type":"url","url":"/nielit-doeacc-o-level-chp","urlHost":"relative","child":null},{"key":"All IT Courses","level":3,"type":"link","url":"/it-software/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top BCA Colleges in India","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-india","urlHost":"relative","child":null},{"key":"Top BCA Colleges in Delhi","level":3,"type":"url","url":"/it-software/ranking/top-bca-colleges-in-delhi/96-2-0-74-0","urlHost":"relative","child":null},{"key":"Top BCA Colleges in Bangalore","level":3,"type":"url","url":"/it-software/ranking/top-bca-colleges-in-bangalore/96-2-0-278-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"NIMCET","level":3,"type":"url","url":"/it-software/nimcet-exam","urlHost":"relative","child":null},{"key":"MAH MCA CET","level":3,"type":"url","url":"/it-software/mah-mca-cet-exam","urlHost":"relative","child":null},{"key":"WBJEE JECA","level":3,"type":"url","url":"/it-software/wbjee-jeca-exam","urlHost":"relative","child":null},{"key":"All IT & Software Exams","level":3,"type":"link","url":"/it-software/exams-st-8","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"AI & Robotics","level":3,"type":"url","url":"/it-software/ai-robotics-chp","urlHost":"relative","child":null},{"key":"Data Analytics","level":3,"type":"url","url":"/it-software/data-analytics-chp","urlHost":"relative","child":null},{"key":"CAD / CAM / CAE","level":3,"type":"url","url":"/it-software/cad-cam-cae-chp","urlHost":"relative","child":null},{"key":"Cloud Computing","level":3,"type":"url","url":"/it-software/cloud-computing-chp","urlHost":"relative","child":null},{"key":"CRM","level":3,"type":"url","url":"/it-software/crm-customer-relationship-management-chp","urlHost":"relative","child":null},{"key":"Databases","level":3,"type":"url","url":"/it-software/databases-chp","urlHost":"relative","child":null},{"key":"ERP","level":3,"type":"url","url":"/it-software/erp-enterprise-resource-planning-chp","urlHost":"relative","child":null},{"key":"Game Design","level":3,"type":"url","url":"/it-software/game-design-chp","urlHost":"relative","child":null},{"key":"IT Service Management","level":3,"type":"url","url":"/it-software/it-service-management-chp","urlHost":"relative","child":null},{"key":"Mobile App Development","level":3,"type":"url","url":"/it-software/mobile-app-development-chp","urlHost":"relative","child":null},{"key":"Networking, Hardware & Security","level":3,"type":"url","url":"/it-software/networking-hardware-security-chp","urlHost":"relative","child":null},{"key":"Office Suite","level":3,"type":"url","url":"/it-software/office-suite-chp","urlHost":"relative","child":null},{"key":"Operating Systems","level":3,"type":"url","url":"/it-software/operating-systems-chp","urlHost":"relative","child":null},{"key":"Programming","level":3,"type":"url","url":"/it-software/programming-chp","urlHost":"relative","child":null},{"key":"Project Management","level":3,"type":"url","url":"/it-software/project-management-chp","urlHost":"relative","child":null},{"key":"Quality Assurance & Testing","level":3,"type":"url","url":"/it-software/quality-assurance-testing-chp","urlHost":"relative","child":null},{"key":"All IT Specializations","level":3,"type":"link","url":"/it-software/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Colleges by Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MCA Colleges in India","level":3,"type":"url","url":"/it-software/colleges/mca-colleges-india","urlHost":"relative","child":null},{"key":"MCA Colleges in Bangalore","level":3,"type":"url","url":"/it-software/colleges/mca-colleges-bangalore","urlHost":"relative","child":null},{"key":"MCA Colleges in Pune","level":3,"type":"url","url":"/it-software/colleges/mca-colleges-pune","urlHost":"relative","child":null},{"key":"MCA Colleges in Hyderabad","level":3,"type":"url","url":"/it-software/colleges/mca-colleges-hyderabad","urlHost":"relative","child":null},{"key":"BCA Colleges in Pune","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-pune","urlHost":"relative","child":null},{"key":"BCA Colleges in Delhi NCR","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-delhi-ncr","urlHost":"relative","child":null},{"key":"BCA Colleges in India","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-india","urlHost":"relative","child":null},{"key":"BCA Colleges in Mumbai","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-mumbai-all","urlHost":"relative","child":null},{"key":"BCA Colleges in Kolkata","level":3,"type":"url","url":"/it-software/colleges/bca-colleges-kolkata","urlHost":"relative","child":null},{"key":"BSc Colleges in Mumbai","level":3,"type":"url","url":"/it-software/colleges/b-sc-colleges-mumbai-all","urlHost":"relative","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/information-technology-tdp-266","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/information-technology-tdp-266?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/information-technology/articles-st-8","urlHost":"relative","child":null},{"key":"Trends in IT & Software","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/8","urlHost":"national","child":null}]]]},{"key":"Humanities & Social Sciences","level":2,"type":"subItem","url":"/humanities-social-sciences-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.A.","level":3,"type":"url","url":"/b-a-bachelor-of-arts-chp","urlHost":"relative","child":null},{"key":"B.Sc. in Humanities & Social Sciences","level":3,"type":"url","url":"/humanities-social-sciences/b-sc-in-humanities-social-sciences-chp","urlHost":"relative","child":null},{"key":"B.S.W.","level":3,"type":"url","url":"/bsw-bachelor-of-social-work-chp","urlHost":"relative","child":null},{"key":"M.A.","level":3,"type":"url","url":"/m-a-master-of-arts-chp","urlHost":"relative","child":null},{"key":"M.Phil.","level":3,"type":"url","url":"/m-phil-master-of-philosophy-chp","urlHost":"relative","child":null},{"key":"M.Sc. in Humanities & Social Sciences","level":3,"type":"url","url":"/humanities-social-sciences/m-sc-in-humanities-social-sciences-chp","urlHost":"relative","child":null},{"key":"MSW","level":3,"type":"url","url":"/msw-master-of-social-work-chp","urlHost":"relative","child":null},{"key":"All Humanities Courses","level":3,"type":"link","url":"/humanities-social-sciences/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Loyola College","level":3,"type":"url","url":"/college/loyola-college-nungambakkam-chennai-1108","urlHost":"national","child":null},{"key":"Fergusson College","level":3,"type":"url","url":"/college/fergusson-college-fergusson-college-road-pune-25748","urlHost":"national","child":null},{"key":"Hansraj College","level":3,"type":"url","url":"/college/hansraj-college-university-of-delhi-malka-ganj-3062","urlHost":"national","child":null},{"key":"Arts & Humanities College Rankings","level":3,"type":"link","url":"/humanities-social-sciences/ranking/top-arts-colleges-in-india/95-2-0-0-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"JNUEE","level":3,"type":"url","url":"/university/jawaharlal-nehru-university/jnuee-exam","urlHost":"relative","child":null},{"key":"DUET (CUET)","level":3,"type":"url","url":"/university/delhi-university-du/duet-exam","urlHost":"relative","child":null},{"key":"PUBDET","level":3,"type":"url","url":"/university/presidency-university-kolkata/pubdet-exam","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Anthropology","level":3,"type":"url","url":"/humanities-social-sciences/anthropology-chp","urlHost":"relative","child":null},{"key":"Archaeology","level":3,"type":"url","url":"/humanities-social-sciences/archaeology-chp","urlHost":"relative","child":null},{"key":"Communication Studies","level":3,"type":"url","url":"/humanities-social-sciences/communication-studies-chp","urlHost":"relative","child":null},{"key":"Economics","level":3,"type":"url","url":"/humanities-social-sciences/economics-chp","urlHost":"relative","child":null},{"key":"Geography","level":3,"type":"url","url":"/humanities-social-sciences/geography-chp","urlHost":"relative","child":null},{"key":"History","level":3,"type":"url","url":"/humanities-social-sciences/history-chp","urlHost":"relative","child":null},{"key":"Languages","level":3,"type":"url","url":"/humanities-social-sciences/languages-chp","urlHost":"relative","child":null},{"key":"Library & Information Science","level":3,"type":"url","url":"/humanities-social-sciences/library-information-science-chp","urlHost":"relative","child":null},{"key":"Linguistics","level":3,"type":"url","url":"/humanities-social-sciences/linguistics-chp","urlHost":"relative","child":null},{"key":"Literature","level":3,"type":"url","url":"/humanities-social-sciences/literature-chp","urlHost":"relative","child":null},{"key":"Philosophy","level":3,"type":"url","url":"/humanities-social-sciences/philosophy-chp","urlHost":"relative","child":null},{"key":"Political Science","level":3,"type":"url","url":"/humanities-social-sciences/political-science-chp","urlHost":"relative","child":null},{"key":"Psychology","level":3,"type":"url","url":"/humanities-social-sciences/psychology-chp","urlHost":"relative","child":null},{"key":"Religious Studies","level":3,"type":"url","url":"/humanities-social-sciences/religious-studies-chp","urlHost":"relative","child":null},{"key":"Rural Studies","level":3,"type":"url","url":"/humanities-social-sciences/rural-studies-chp","urlHost":"relative","child":null},{"key":"Social Work","level":3,"type":"url","url":"/humanities-social-sciences/social-work-chp","urlHost":"relative","child":null},{"key":"Sociology","level":3,"type":"url","url":"/humanities-social-sciences/sociology-chp","urlHost":"relative","child":null},{"key":"All Humanities Specializations","level":3,"type":"link","url":"/humanities-social-sciences/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Colleges by Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MSW Colleges in India","level":3,"type":"url","url":"/humanities-social-sciences/colleges/msw-colleges-india","urlHost":"relative","child":null},{"key":"Psychology colleges in India","level":3,"type":"url","url":"/humanities-social-sciences/psychology/colleges/colleges-india","urlHost":"relative","child":null},{"key":"B.A. Colleges in India","level":3,"type":"url","url":"/humanities-social-sciences/colleges/b-a-colleges-india","urlHost":"relative","child":null},{"key":"M.A. Colleges in India","level":3,"type":"url","url":"/humanities-social-sciences/colleges/m-a-colleges-india ","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/humanities-social-sciences-tdp-429459","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/humanities-social-sciences-tdp-429459?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/arts-humanities/articles-st-9","urlHost":"relative","child":null},{"key":"Trends in Humanities & Social Sciences","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/9","urlHost":"national","child":null},{"key":"Delhi University Cut-Offs 2020","level":3,"type":"url","url":"/university/university-of-delhi-24642/cutoff","urlHost":"national","child":null}]]]},{"key":"Arts (Fine/Visual/Performing)","level":2,"type":"subItem","url":"/arts-fine-visual-performing-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BFA","level":3,"type":"url","url":"/bfa-bachelor-of-fine-arts-chp","urlHost":"relative","child":null},{"key":"MFA","level":3,"type":"url","url":"/mfa-master-of-fine-arts-chp","urlHost":"relative","child":null},{"key":"All Arts Courses","level":3,"type":"link","url":"/arts-fine-visual-performing/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Arts Colleges in India","level":3,"type":"url","url":"/humanities-social-sciences/ranking/top-arts-colleges-in-india/95-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Arts Colleges in Chennai","level":3,"type":"url","url":"/humanities-social-sciences/ranking/top-arts-colleges-in-chennai/95-2-0-64-0","urlHost":"relative","child":null},{"key":"Top Arts Colleges in Mumbai","level":3,"type":"url","url":"/humanities-social-sciences/ranking/top-arts-colleges-in-mumbai/95-2-0-151-0","urlHost":"relative","child":null},{"key":"Top Colleges in Delhi for Arts","level":3,"type":"url","url":"/humanities-social-sciences/ranking/top-arts-colleges-in-delhi/95-2-0-74-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"JMI Exam","level":3,"type":"url","url":"/university/jamia-millia-islamia/jmi-entrance-exam","urlHost":"relative","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BFA Colleges in India","level":3,"type":"url","url":"/arts-fine-visual-performing/colleges/bfa-colleges-india","urlHost":"relative","child":null},{"key":"MFA Colleges in India","level":3,"type":"url","url":"/arts-fine-visual-performing/colleges/mfa-colleges-india","urlHost":"relative","child":null},{"key":"Music colleges in India","level":3,"type":"url","url":"/arts-fine-visual-performing/music/colleges/colleges-india","urlHost":"relative","child":null},{"key":"Photography colleges in India","level":3,"type":"url","url":"/arts-fine-visual-performing/photography/colleges/colleges-india","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Applied Arts","level":3,"type":"url","url":"/arts-fine-visual-performing/applied-arts-chp","urlHost":"relative","child":null},{"key":"Art History & Aesthetics","level":3,"type":"url","url":"/arts-fine-visual-performing/art-history-aesthetics-chp","urlHost":"relative","child":null},{"key":"Ceramics","level":3,"type":"url","url":"/arts-fine-visual-performing/ceramics-designer-chp","urlHost":"relative","child":null},{"key":"Dance & Choreography","level":3,"type":"url","url":"/arts-fine-visual-performing/dance-choreography-chp","urlHost":"relative","child":null},{"key":"Decorative Arts","level":3,"type":"url","url":"/arts-fine-visual-performing/decorative-arts-chp","urlHost":"relative","child":null},{"key":"Film Making","level":3,"type":"url","url":"/arts-fine-visual-performing/film-making-chp","urlHost":"relative","child":null},{"key":"Graphics Art","level":3,"type":"url","url":"/arts-fine-visual-performing/graphics-art-chp","urlHost":"relative","child":null},{"key":"Muralist","level":3,"type":"url","url":"/arts-fine-visual-performing/muralist-chp","urlHost":"relative","child":null},{"key":"Music","level":3,"type":"url","url":"/arts-fine-visual-performing/music-chp","urlHost":"relative","child":null},{"key":"Painting & Drawing","level":3,"type":"url","url":"/arts-fine-visual-performing/painting-drawing-chp","urlHost":"relative","child":null},{"key":"Photography","level":3,"type":"url","url":"/arts-fine-visual-performing/photography-chp","urlHost":"relative","child":null},{"key":"Sculpture","level":3,"type":"url","url":"/arts-fine-visual-performing/sculpture-chp","urlHost":"relative","child":null},{"key":"Theatre","level":3,"type":"url","url":"/arts-fine-visual-performing/theatre-chp","urlHost":"relative","child":null},{"key":"All Arts Specializations","level":3,"type":"link","url":"/arts-fine-visual-performing/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/arts-fine-visual-performing-tdp-429912","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/arts-fine-visual-performing-tdp-429912?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/arts-humanities/articles-st-10","urlHost":"relative","child":null},{"key":"Trends in Arts ( Fine / Visual / Performing )","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/10","urlHost":"national","child":null}]]]},{"key":"Science","level":2,"type":"subItem","url":"/science-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Sc.","level":3,"type":"url","url":"/b-sc-chp","urlHost":"relative","child":null},{"key":"M.Sc.","level":3,"type":"url","url":"/m-sc-master-of-science-chp","urlHost":"relative","child":null},{"key":"Distance B.Sc.","level":3,"type":"url","url":"/distance-correspondence-b-sc-chp","urlHost":"relative","child":null},{"key":"Distance M.Sc.","level":3,"type":"url","url":"/distance-correspondence-m-sc-chp","urlHost":"relative","child":null},{"key":"All Science Courses","level":3,"type":"link","url":"/science/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Science Colleges in Mumbai","level":3,"type":"url","url":"/science/ranking/top-science-colleges-in-mumbai/101-2-0-151-0","urlHost":"relative","child":null},{"key":"Top Science Colleges in India","level":3,"type":"url","url":"/science/ranking/top-science-colleges-in-india/101-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Science Colleges in Pune","level":3,"type":"url","url":"/science/ranking/top-science-colleges-in-pune/101-2-0-174-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MCAER CET","level":3,"type":"url","url":"/science/mcaer-cet-exam","urlHost":"relative","child":null},{"key":"CUET UG","level":3,"type":"url","url":"/science/cuet-exam","urlHost":"relative","child":null},{"key":"CUET PG","level":3,"type":"url","url":"/science/cuet-pg-exam","urlHost":"relative","child":null},{"key":"NEST","level":3,"type":"url","url":"/college/niser-national-institute-of-science-education-and-research/nest-exam","urlHost":"relative","child":null},{"key":"IIT JAM","level":3,"type":"url","url":"/science/iit-jam-exam","urlHost":"relative","child":null},{"key":"JEST","level":3,"type":"url","url":"/science/jest-exam","urlHost":"relative","child":null},{"key":"All Science Exams","level":3,"type":"link","url":"/science/exams-st-11","urlHost":"relative","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Sc. in Computer Science colleges in India","level":3,"type":"url","url":"/science/colleges/b-sc-computer-science-colleges-india","urlHost":"relative","child":null},{"key":"B.Sc. in Biotechnology colleges in India","level":3,"type":"url","url":"/science/colleges/b-sc-biotechnology-colleges-india","urlHost":"relative","child":null},{"key":"B.Sc. in Agriculture colleges in India","level":3,"type":"url","url":"/science/colleges/b-sc-agriculture-colleges-india","urlHost":"relative","child":null},{"key":"Forensic Science colleges in India","level":3,"type":"url","url":"/science/colleges/forensic-science-colleges-india","urlHost":"relative","child":null},{"key":"Distance M.Sc. Colleges in India","level":3,"type":"url","url":"/science/colleges/distance-correspondence-m-sc-colleges-india","urlHost":"relative","child":null}]],[[],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/science-tdp-18","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/science-tdp-18?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/science/articles-st-11","urlHost":"relative","child":null},{"key":"Trends in Science","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/11","urlHost":"national","child":null},{"key":"Delhi University Cut-Offs 2020","level":3,"type":"url","url":"/university/university-of-delhi-24642/cutoff","urlHost":"national","child":null}]]]},{"key":"Architecture & Planning","level":2,"type":"subItem","url":"/architecture-planning-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Arch.","level":3,"type":"url","url":"/b-arch-bachelor-of-architecture-chp","urlHost":"relative","child":null},{"key":"M.Arch.","level":3,"type":"url","url":"/m-arch-master-of-architecture-chp","urlHost":"relative","child":null},{"key":"M.Plan","level":3,"type":"url","url":"/m-plan-master-of-planning-chp","urlHost":"relative","child":null},{"key":"All Architecture Courses","level":3,"type":"link","url":"/architecture-planning/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Architecture Colleges in India","level":3,"type":"url","url":"/architecture-planning/ranking/top-architecture-colleges-in-india/111-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Architecture Colleges in Bangalore","level":3,"type":"url","url":"/architecture-planning/ranking/top-architecture-colleges-in-bangalore/111-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Architecture Colleges in Mumbai","level":3,"type":"url","url":"/architecture-planning/ranking/top-architecture-colleges-in-mumbai/111-2-0-151-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"AAT","level":3,"type":"url","url":"/college/iit-kanpur-indian-institute-of-technology/aat-exam","urlHost":"relative","child":null},{"key":"NATA","level":3,"type":"url","url":"/architecture-planning/nata-exam","urlHost":"relative","child":null},{"key":"UPAT","level":3,"type":"url","url":"/university/university-of-petroleum-and-energy-studies/upat-exam","urlHost":"relative","child":null},{"key":"All Architecture & Planning Exams","level":3,"type":"link","url":"/architecture-planning/exams-st-12","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Environmental","level":3,"type":"url","url":"/architecture-planning/environmental-chp","urlHost":"relative","child":null},{"key":"Landscape","level":3,"type":"url","url":"/architecture-planning/landscape-chp","urlHost":"relative","child":null},{"key":"Rural (Regional)","level":3,"type":"url","url":"/architecture-planning/rural-regional-chp","urlHost":"relative","child":null},{"key":"Urban","level":3,"type":"url","url":"/architecture-planning/urban-chp","urlHost":"relative","child":null},{"key":"All Architecture Specializations","level":3,"type":"link","url":"/architecture-planning/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/architecture-tdp-21","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/architecture-tdp-21?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/architecture/articles-st-12","urlHost":"relative","child":null},{"key":"Trends in Architecture & Planning","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/12","urlHost":"national","child":null}]]]},{"key":"Accounting & Commerce","level":2,"type":"subItem","url":"/accounting-commerce-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Com.","level":3,"type":"url","url":"/b-com-chp","urlHost":"relative","child":null},{"key":"M.Com.","level":3,"type":"url","url":"/m-com-chp","urlHost":"relative","child":null},{"key":"CA","level":3,"type":"url","url":"/ca-chartered-accountant-chp","urlHost":"relative","child":null},{"key":"CS","level":3,"type":"url","url":"/cs-company-secretary-chp","urlHost":"relative","child":null},{"key":"Diploma in Accounting","level":3,"type":"url","url":"/accounting-commerce/diploma-in-accounting-chp","urlHost":"relative","child":null},{"key":"Diploma in Taxation","level":3,"type":"url","url":"/accounting-commerce/diploma-in-taxation-chp","urlHost":"relative","child":null},{"key":"All Accounting & Commerce Courses","level":3,"type":"link","url":"/accounting-commerce/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CA Colleges in India","level":3,"type":"url","url":"/accounting-commerce/colleges/ca-colleges-india","urlHost":"relative","child":null},{"key":"CS Colleges in India","level":3,"type":"url","url":"/accounting-commerce/colleges/cs-colleges-india","urlHost":"relative","child":null},{"key":"B.Com Colleges in India","level":3,"type":"url","url":"/accounting-commerce/colleges/b-com-colleges-india","urlHost":"relative","child":null},{"key":"M.Com Colleges in India","level":3,"type":"url","url":"/accounting-commerce/colleges/m-com-colleges-india","urlHost":"relative","child":null}],[{"key":"Top Ranked Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Commerce Colleges in India","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-india/97-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Commerce Colleges in Mumbai","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-mumbai-all/97-2-0-10224-0","urlHost":"relative","child":null},{"key":"Top Commerce Colleges in Delhi","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-delhi/97-2-0-74-0","urlHost":"relative","child":null},{"key":"Top Commerce Colleges in Pune","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-pune/97-2-0-174-0","urlHost":"relative","child":null},{"key":"Top Commerce Colleges in Bangalore","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-bangalore/97-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Commerce Colleges in Kolkata","level":3,"type":"url","url":"/accounting-commerce/ranking/top-commerce-colleges-in-kolkata/97-2-0-130-0","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"ICAI","level":3,"type":"url","url":"/accounting-commerce/accounting/icai-exam","urlHost":"relative","child":null},{"key":"ICSI","level":3,"type":"url","url":"/college/icsi-delhi-institute-of-company-secretaries-of-india/icsi-exam","urlHost":"relative","child":null},{"key":"CMA Exam","level":3,"type":"url","url":"/accounting-commerce/accounting/cma-exam","urlHost":"relative","child":null},{"key":"All Accounting & Commerce Exams","level":3,"type":"link","url":"/accounting-commerce/exams-st-13","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Accounting","level":3,"type":"url","url":"/accounting-commerce/accounting-chp","urlHost":"relative","child":null},{"key":"Taxation","level":3,"type":"url","url":"/accounting-commerce/taxation-chp","urlHost":"relative","child":null},{"key":"All Accounting & Commerce Specializations","level":3,"type":"link","url":"/accounting-commerce/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/accounting-tdp-55","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/accounting-tdp-55?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/accounting/articles-st-13","urlHost":"relative","child":null},{"key":"Trends in Accounting & Commerce","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/13","urlHost":"national","child":null},{"key":"Delhi University Cut-Offs 2020","level":3,"type":"url","url":"/university/university-of-delhi-24642/cutoff","urlHost":"national","child":null}]]]},{"key":"Banking, Finance & Insurance","level":2,"type":"subItem","url":"/banking-finance-insurance-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CFA","level":3,"type":"url","url":"/cfa-chartered-financial-analyst-level-1-2-3-chp","urlHost":"relative","child":null},{"key":"CFP","level":3,"type":"url","url":"/cfp-certified-financial-planner-chp","urlHost":"relative","child":null},{"key":"Diploma in Banking & Finance","level":3,"type":"url","url":"/diploma-in-banking-finance-insurance-chp","urlHost":"relative","child":null},{"key":"All Banking & Finance Courses","level":3,"type":"link","url":"/banking-finance-insurance/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Mutual Funds","level":3,"type":"url","url":"/banking-finance-insurance/mutual-funds-chp","urlHost":"relative","child":null},{"key":"CBS","level":3,"type":"url","url":"/cbs-chp","urlHost":"relative","child":null},{"key":"Corporate Banking","level":3,"type":"url","url":"/banking-finance-insurance/corporate-banking-chp","urlHost":"relative","child":null},{"key":"Health Insurance","level":3,"type":"url","url":"/banking-finance-insurance/health-insurance-chp","urlHost":"relative","child":null},{"key":"Investment Banking","level":3,"type":"url","url":"/banking-finance-insurance/investment-banking-chp","urlHost":"relative","child":null},{"key":"Capital Markets","level":3,"type":"url","url":"/banking-finance-insurance/capital-markets-chp","urlHost":"relative","child":null}],[{"key":"Top Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"AIMA Delhi","level":3,"type":"url","url":"/college/all-india-management-association-lajpat-nagar-delhi-2864","urlHost":"national","child":null},{"key":"XIME Bangalore","level":3,"type":"url","url":"/college/xavier-institute-of-management-and-entrepreneurship-bangalore-hosur-road-28230","urlHost":"national","child":null},{"key":"ISBF","level":3,"type":"url","url":"/college/isbf-lajpat-nagar-delhi-27907","urlHost":"national","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CFA","level":3,"type":"url","url":"/banking-finance-insurance/cfa-exam","urlHost":"relative","child":null}]],[[],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/banking-finance-insurance-tdp-430369","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/banking-finance-insurance-tdp-430369?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/banking-finance-insurance/articles-st-14","urlHost":"relative","child":null},{"key":"Trends in Banking, Finance & Insurance","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/14","urlHost":"national","child":null}]]]},{"key":"Aviation","level":2,"type":"subItem","url":"/aviation-chp","urlHost":"relative","child":[[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Cabin Crew / Air Hostess","level":3,"type":"url","url":"/aviation/cabin-crew-air-hostess-chp","urlHost":"relative","child":null},{"key":"Cargo Management","level":3,"type":"url","url":"/aviation/cargo-management-chp","urlHost":"relative","child":null},{"key":"Flying / Pilot Training","level":3,"type":"url","url":"/aviation/flying-pilot-training-chp","urlHost":"relative","child":null},{"key":"Ground Services","level":3,"type":"url","url":"/aviation/ground-services-chp","urlHost":"relative","child":null},{"key":"Aircraft Maintenance Engineering","level":3,"type":"url","url":"/aviation/aircraft-maintenance-engineering-chp","urlHost":"relative","child":null},{"key":"All Aviation Courses","level":3,"type":"link","url":"/aviation/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Indian Aviation Academy","level":3,"type":"url","url":"/college/indian-aviation-academy-andheri-west-mumbai-24815","urlHost":"national","child":null},{"key":"Rajiv Gandhi Aviation Academy","level":3,"type":"url","url":"/college/rajiv-gandhi-aviation-academy-begumpet-hyderabad-21635","urlHost":"national","child":null},{"key":"Hindustan Aviation Academy","level":3,"type":"url","url":"/college/hindustan-aviation-academy-marathahalli-bangalore-22133","urlHost":"national","child":null},{"key":"Frankfinn","level":3,"type":"url","url":"/college/frankfinn-institute-of-air-hostess-training-rajouri-garden-delhi-47223","urlHost":"national","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/aviation-tdp-24","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/aviation-tdp-24?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/aviation/articles-st-15","urlHost":"relative","child":null},{"key":"Trends in Aviation","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/15","urlHost":"national","child":null}]],[[],[],[]]]},{"key":"Teaching & Education","level":2,"type":"subItem","url":"/teaching-education/colleges/colleges-india","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Ed.","level":3,"type":"url","url":"/b-ed-chp","urlHost":"relative","child":null},{"key":"B.P.Ed.","level":3,"type":"url","url":"/b-p-ed-bachelor-of-physical-education-chp","urlHost":"relative","child":null},{"key":"B.Voc","level":3,"type":"url","url":"/b-voc-bachelor-of-vocational-education-chp","urlHost":"relative","child":null},{"key":"M.Ed.","level":3,"type":"url","url":"/m-ed-master-of-education-chp","urlHost":"relative","child":null},{"key":"M.P.Ed.","level":3,"type":"url","url":"/m-p-ed-master-of-physical-education-chp","urlHost":"relative","child":null},{"key":"D.Ed.","level":3,"type":"url","url":"/d-ed-diploma-in-education-chp","urlHost":"relative","child":null},{"key":"All Teaching & Education Courses","level":3,"type":"link","url":"/teaching-education/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Top Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"LSR Delhi","level":3,"type":"url","url":"/college/lady-shri-ram-college-for-women-university-of-delhi-lajpat-nagar-23895","urlHost":"national","child":null},{"key":"Lady Irwin College","level":3,"type":"url","url":"/college/lady-irwin-college-university-of-delhi-pragati-maidan-23897","urlHost":"national","child":null},{"key":"Loreto College Kolkata","level":3,"type":"url","url":"/college/loreto-college-park-street-kolkata-47646","urlHost":"national","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Ed Colleges in UP","level":3,"type":"url","url":"/teaching-education/colleges/b-ed-colleges-uttar-pradesh","urlHost":"relative","child":null},{"key":"B.Ed Colleges in Delhi NCR","level":3,"type":"url","url":"/teaching-education/colleges/b-ed-colleges-delhi-ncr","urlHost":"relative","child":null},{"key":"B.Ed Colleges in West Bengal","level":3,"type":"url","url":"/teaching-education/colleges/b-ed-colleges-west-bengal","urlHost":"relative","child":null},{"key":"B.Ed Colleges in Bihar","level":3,"type":"url","url":"/teaching-education/colleges/b-ed-colleges-bihar","urlHost":"relative","child":null},{"key":"B.Ed Colleges in Mumbai","level":3,"type":"url","url":"/teaching-education/colleges/b-ed-colleges-mumbai-all","urlHost":"relative","child":null}],[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"CTET","level":3,"type":"url","url":"/exams/ctet-exam","urlHost":"relative","child":null},{"key":"TSTET","level":3,"type":"url","url":"/exams/tstet-exam","urlHost":"relative","child":null},{"key":"UGC NET","level":3,"type":"url","url":"/exams/ugc-net-exam","urlHost":"relative","child":null},{"key":"RIE CEE","level":3,"type":"url","url":"/teaching-education/rie-cee-exam","urlHost":"relative","child":null},{"key":"PTET","level":3,"type":"url","url":"/teaching-education/ptet-exam","urlHost":"relative","child":null},{"key":"APEdCET","level":3,"type":"url","url":"/teaching-education/apedcet-exam","urlHost":"relative","child":null},{"key":"UP B.Ed JEE","level":3,"type":"url","url":"/teaching-education/up-b-ed-jee-exam","urlHost":"relative","child":null},{"key":"All Teaching & Education Exams","level":3,"type":"link","url":"/teaching-education/exams-st-16","urlHost":"relative","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Middle School","level":3,"type":"url","url":"/teaching-education/middle-school-chp","urlHost":"relative","child":null},{"key":"Physical Education","level":3,"type":"url","url":"/teaching-education/physical-education-chp","urlHost":"relative","child":null},{"key":"Pre Primary & Primary School","level":3,"type":"url","url":"/teaching-education/pre-primary-primary-school-chp","urlHost":"relative","child":null},{"key":"Secondary & Sr. Secondary School","level":3,"type":"url","url":"/teaching-education/secondary-sr-secondary-school-chp","urlHost":"relative","child":null},{"key":"Special Education","level":3,"type":"url","url":"/teaching-education/special-education-chp","urlHost":"relative","child":null},{"key":"Vocational Education","level":3,"type":"url","url":"/teaching-education/vocational-education-chp","urlHost":"relative","child":null},{"key":"All Teaching & Education Specializations","level":3,"type":"link","url":"/teaching-education/colleges/colleges-india","urlHost":"relative","child":null}],[],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/teaching-education-tdp-16","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/teaching-education-tdp-16?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/teaching-education/articles-st-16","urlHost":"relative","child":null},{"key":"Trends in Teaching & Education","level":3,"type":"url","url":"/teaching-education/articles-st-16","urlHost":"relative","child":null}]]]},{"key":"Nursing","level":2,"type":"subItem","url":"/nursing-chp","urlHost":"relative","child":[[[{"key":"Popular Courses","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"B.Sc. Nursing","level":3,"type":"url","url":"/nursing/b-sc-nursing-chp","urlHost":"relative","child":null},{"key":"M.Sc. in Nursing","level":3,"type":"url","url":"/nursing/m-sc-in-nursing-chp","urlHost":"relative","child":null},{"key":"All Nursing Courses","level":3,"type":"link","url":"/nursing/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/nursing-tdp-83","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/nursing-tdp-83?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/nursing/articles-st-17","urlHost":"relative","child":null},{"key":"Trends in Nursing","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/17","urlHost":"national","child":null}]],[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Nursing & Midwifery","level":3,"type":"url","url":"/nursing/nursing-midwifery-chp","urlHost":"relative","child":null},{"key":"All Nursing Specializations","level":3,"type":"link","url":"/nursing/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Colleges By Location","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Nursing Colleges in Uttar Pradesh","level":3,"type":"url","url":"/nursing/colleges/colleges-uttar-pradesh","urlHost":"relative","child":null},{"key":"Nursing Colleges in Bihar","level":3,"type":"url","url":"/nursing/colleges/colleges-bihar","urlHost":"relative","child":null},{"key":"Nursing Colleges in Bangalore","level":3,"type":"url","url":"/nursing/colleges/colleges-bangalore","urlHost":"relative","child":null},{"key":"Nursing Colleges in Kerala","level":3,"type":"url","url":"/nursing/colleges/colleges-kerala","urlHost":"relative","child":null},{"key":"Nursing Colleges in Kolkata","level":3,"type":"url","url":"/nursing/colleges/colleges-kolkata","urlHost":"relative","child":null},{"key":"Nursing Colleges in Delhi NCR","level":3,"type":"url","url":"/nursing/colleges/colleges-delhi-ncr","urlHost":"relative","child":null}],[]]]},{"key":"Beauty & Fitness","level":2,"type":"subItem","url":"/beauty-fitness-chp","urlHost":"relative","child":[[[{"key":"Popular Specializations","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Beauty Culture & Cosmetology","level":3,"type":"url","url":"/beauty-fitness/beauty-culture-cosmetology-chp","urlHost":"relative","child":null},{"key":"Massage & Spa Therapy","level":3,"type":"url","url":"/beauty-fitness/massage-spa-therapy-chp","urlHost":"relative","child":null},{"key":"Yoga","level":3,"type":"url","url":"/beauty-fitness/yoga-chp","urlHost":"relative","child":null},{"key":"All Beauty & Fitness Courses","level":3,"type":"link","url":"/beauty-fitness/colleges/colleges-india","urlHost":"relative","child":null}],[{"key":"Resources","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Ask a Question","level":3,"type":"url","url":"/tags/beauty-tdp-114","urlHost":"national","child":null},{"key":"Discussions","level":3,"type":"url","url":"/tags/beauty-tdp-114?type=discussion","urlHost":"national","child":null},{"key":"News and Articles","level":3,"type":"url","url":"/beauty/articles-st-19","urlHost":"relative","child":null},{"key":"Trends in Beauty & Fitness","level":3,"type":"url","url":"/analytics/ShikshaTrends/showETP/stream/19","urlHost":"national","child":null}],[]],[[],[],[]]]},{"key":"Universities and Colleges","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Central Universities","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"University of Delhi","level":3,"type":"url","url":"/university/delhi-university-du-24642","urlHost":"national","child":null},{"key":"JNU Delhi","level":3,"type":"url","url":"/university/jawaharlal-nehru-university-delhi-4225","urlHost":"national","child":null},{"key":"IGNOU Delhi","level":3,"type":"url","url":"/university/indira-gandhi-national-open-university-delhi-3030","urlHost":"national","child":null},{"key":"Banaras Hindu University","level":3,"type":"url","url":"/university/bhu-banaras-hindu-university-varanasi-23336","urlHost":"national","child":null}],[{"key":"Top State Universities","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"University of Mumbai","level":3,"type":"url","url":"/university/university-of-mumbai-856","urlHost":"national","child":null},{"key":"Anna University","level":3,"type":"url","url":"/university/anna-university-chennai-3084","urlHost":"national","child":null},{"key":"Gujarat University","level":3,"type":"url","url":"/university/gujarat-university-ahmedabad-29851","urlHost":"national","child":null},{"key":"CCS University","level":3,"type":"url","url":"/university/ccsu-chaudhary-charan-singh-university-meerut-19694","urlHost":"national","child":null}]],[[{"key":"Top Ranked Universities","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Universities in India","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-india/121-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Colleges in India","level":3,"type":"url","url":"/science/ranking/top-ranked-colleges-in-india/553-2-0-0-0","urlHost":"relative","child":null},{"key":"Top Universities in Bangalore","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-bangalore/121-2-0-278-0","urlHost":"relative","child":null},{"key":"Top Universities in Delhi","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-delhi/121-2-0-74-0","urlHost":"relative","child":null}],[{"key":"Top Ranked Universities","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Universities in Punjab","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-punjab/121-2-120-0-0","urlHost":"relative","child":null},{"key":"Top Universities in Chennai","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-chennai/121-2-0-64-0","urlHost":"relative","child":null},{"key":"Top Universities in Karnataka","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-karnataka/121-2-106-0-0","urlHost":"relative","child":null},{"key":"Top Universities in Maharashtra","level":3,"type":"url","url":"/science/ranking/top-university-colleges-in-maharashtra/121-2-114-0-0","urlHost":"relative","child":null}]],[[{"key":"Colleges by State","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Colleges in Maharashtra","level":3,"type":"url","url":"/colleges/maharashtra","urlHost":"relative","child":null},{"key":"Colleges in Karnatka","level":3,"type":"url","url":"/colleges/karnataka","urlHost":"relative","child":null},{"key":"Colleges in Uttar Pradesh","level":3,"type":"url","url":"/colleges/uttar-pradesh","urlHost":"relative","child":null},{"key":"Colleges in Kerala","level":3,"type":"url","url":"/colleges/kerala","urlHost":"relative","child":null}],[{"key":"Colleges by City","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Colleges in Delhi","level":3,"type":"url","url":"/colleges/delhi","urlHost":"relative","child":null},{"key":"Colleges in Bangalore","level":3,"type":"url","url":"/colleges/bangalore","urlHost":"relative","child":null},{"key":"Colleges in Mumbai","level":3,"type":"url","url":"/colleges/mumbai","urlHost":"relative","child":null},{"key":"Colleges in Hyderabad","level":3,"type":"url","url":"/colleges/hyderabad","urlHost":"relative","child":null}]]]}]]]},{"key":"Study Abroad","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Countries","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Countries","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"USA","level":3,"type":"url","url":"/usa","urlHost":"abroad","child":null},{"key":"UK","level":3,"type":"url","url":"/uk","urlHost":"abroad","child":null},{"key":"Canada","level":3,"type":"url","url":"/canada","urlHost":"abroad","child":null},{"key":"Australia","level":3,"type":"url","url":"/australia","urlHost":"abroad","child":null},{"key":"Germany","level":3,"type":"url","url":"/germany","urlHost":"abroad","child":null},{"key":"Ireland","level":3,"type":"url","url":"/ireland","urlHost":"abroad","child":null},{"key":"France","level":3,"type":"url","url":"/france","urlHost":"abroad","child":null},{"key":"Singapore","level":3,"type":"url","url":"/singapore","urlHost":"abroad","child":null},{"key":"New Zealand","level":3,"type":"url","url":"/new-zealand","urlHost":"abroad","child":null},{"key":"Japan","level":3,"type":"url","url":"/japan","urlHost":"abroad","child":null},{"key":"Italy","level":3,"type":"url","url":"/italy","urlHost":"abroad","child":null},{"key":"Finland","level":3,"type":"url","url":"/finland","urlHost":"abroad","child":null},{"key":"Netherlands","level":3,"type":"url","url":"/netherlands","urlHost":"abroad","child":null},{"key":"View All Countries","level":3,"type":"urlWithAnchor","url":"/abroad-countries-countryhome","urlHost":"abroad","child":null}],[{"key":"Universities/Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Top Universities in USA","level":3,"type":"url","url":"/usa/universities","urlHost":"abroad","child":null},{"key":"Top Universities in UK","level":3,"type":"url","url":"/uk/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Canada","level":3,"type":"url","url":"/canada/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Australia","level":3,"type":"url","url":"/australia/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Germany","level":3,"type":"url","url":"/germany/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Ireland","level":3,"type":"url","url":"/ireland/universities","urlHost":"abroad","child":null},{"key":"Top Universities in France","level":3,"type":"url","url":"/france/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Singapore","level":3,"type":"url","url":"/singapore/universities","urlHost":"abroad","child":null},{"key":"Top Universities in New Zealand","level":3,"type":"url","url":"/new-zealand/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Japan","level":3,"type":"url","url":"/japan/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Italy","level":3,"type":"url","url":"/italy/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Finland","level":3,"type":"url","url":"/finland/universities","urlHost":"abroad","child":null},{"key":"Top Universities in Netherlands","level":3,"type":"url","url":"/netherlands/universities","urlHost":"abroad","child":null},{"key":"Top Universities Abroad","level":3,"type":"urlWithAnchor","url":"/top-universities-in-abroad-abroadranking32","urlHost":"abroad","child":null}]]]},{"key":"Exams","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Exams","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Language Exams","level":3,"type":"sub-heading","url":"","urlHost":null,"child":null},{"key":"IELTS","level":3,"type":"url","url":"/exams/ielts","urlHost":"abroad","child":null},{"key":"TOEFL","level":3,"type":"url","url":"/exams/toefl","urlHost":"abroad","child":null},{"key":"PTE","level":3,"type":"url","url":"/exams/pte","urlHost":"abroad","child":null},{"key":"DET","level":3,"type":"url","url":"/exams/duolingo","urlHost":"abroad","child":null},{"key":"Aptitude Exams","level":3,"type":"sub-heading","url":"","urlHost":null,"child":null},{"key":"GRE","level":3,"type":"url","url":"/exams/gre","urlHost":"abroad","child":null},{"key":"GMAT","level":3,"type":"url","url":"/exams/gmat","urlHost":"abroad","child":null},{"key":"SAT","level":3,"type":"url","url":"/exams/sat","urlHost":"abroad","child":null}],[{"key":"Free IELTS Masterclass","level":3,"type":"url","url":"/shiksha-study-abroad-launches-free-shiksha-ielts-classes-articlepage-125137","urlHost":"abroad","child":null},{"key":"IELTS Preparation Resource","level":3,"type":"url","url":"/ielts/preparation","urlHost":"abroad","child":null},{"key":"IELTS Reading Test","level":3,"type":"url","url":"/exams/ielts/reading-ecp","urlHost":"abroad","child":null},{"key":"IELTS Writing Test","level":3,"type":"url","url":"/exams/ielts/writing-ecp","urlHost":"abroad","child":null},{"key":"IELTS Listening Test","level":3,"type":"url","url":"/exams/ielts/listening-ecp","urlHost":"abroad","child":null},{"key":"IELTS Speaking Test","level":3,"type":"url","url":"/exams/ielts/speaking-ecp","urlHost":"abroad","child":null}]],[[],[]]]},{"key":"Popular Programs","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top MS Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MS in USA","level":3,"type":"url","url":"/usa/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"MS in UK","level":3,"type":"url","url":"/uk/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"MS in Canada","level":3,"type":"url","url":"/canada/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"MS in Australia","level":3,"type":"url","url":"/australia/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"MS in Germany","level":3,"type":"url","url":"/germany/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"MS in Ireland","level":3,"type":"url","url":"/ireland/ms-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore MS Abroad","level":3,"type":"urlWithAnchor","url":"/ms-in-abroad-dc11509","urlHost":"abroad","child":null}],[{"key":"Top MBA Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MBA in USA","level":3,"type":"url","url":"/usa/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"MBA in UK","level":3,"type":"url","url":"/uk/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"MBA in Canada","level":3,"type":"url","url":"/canada/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"MBA in Australia","level":3,"type":"url","url":"/australia/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"MBA in Germany","level":3,"type":"url","url":"/germany/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"MBA in Ireland","level":3,"type":"url","url":"/ireland/mba-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore MBA Abroad","level":3,"type":"urlWithAnchor","url":"/mba-in-abroad-dc11508","urlHost":"abroad","child":null}],[{"key":"Top MEng Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MEng in USA","level":3,"type":"url","url":"/usa/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"MEng in UK","level":3,"type":"url","url":"/uk/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"MEng in Canada","level":3,"type":"url","url":"/canada/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"MEng in Australia","level":3,"type":"url","url":"/australia/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"MEng in Germany","level":3,"type":"url","url":"/germany/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"MEng in Ireland","level":3,"type":"url","url":"/ireland/meng-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore MEng Abroad","level":3,"type":"urlWithAnchor","url":"/meng-in-abroad-dc111842","urlHost":"abroad","child":null}]],[[{"key":"Top MIM Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MIM in USA","level":3,"type":"url","url":"/usa/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in UK","level":3,"type":"url","url":"/uk/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in Canada","level":3,"type":"url","url":"/canada/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in Australia","level":3,"type":"url","url":"/australia/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in Germany","level":3,"type":"url","url":"/germany/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in Ireland","level":3,"type":"url","url":"/ireland/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"MIM in France","level":3,"type":"url","url":"/france/mim-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore MIM Abroad","level":3,"type":"urlWithAnchor","url":"/mim-in-abroad-dc111849","urlHost":"abroad","child":null}],[{"key":"Top BE/BTech Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BE/B.Tech in USA","level":3,"type":"url","url":"/usa/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"BE/B.Tech in UK","level":3,"type":"url","url":"/uk/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Canada","level":3,"type":"url","url":"/canada/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Australia","level":3,"type":"url","url":"/australia/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Germany","level":3,"type":"url","url":"/germany/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Ireland","level":3,"type":"url","url":"/ireland/be-btech-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore BE/B.Tech Abroad","level":3,"type":"urlWithAnchor","url":"/be-btech-in-abroad-dc11510","urlHost":"abroad","child":null}],[{"key":"Top BBA Colleges","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BBA in USA","level":3,"type":"url","url":"/usa/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"BBA in UK","level":3,"type":"url","url":"/uk/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"BBA in Canada","level":3,"type":"url","url":"/canada/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"BBA in Australia","level":3,"type":"url","url":"/australia/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"BBA in Germany","level":3,"type":"url","url":"/germany/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"BBA in Ireland","level":3,"type":"url","url":"/ireland/bba-colleges-dc","urlHost":"abroad","child":null},{"key":"Explore BBA Abroad","level":3,"type":"urlWithAnchor","url":"/bba-in-abroad-dc111844","urlHost":"abroad","child":null}]]]},{"key":"Popular Specialization","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Masters","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"MS in Computer Science","level":3,"type":"url","url":"/ms-in-computer-science-engineering-from-abroad-ds11509277","urlHost":"abroad","child":null},{"key":"MS in IT/MIS","level":3,"type":"url","url":"/ms-in-it-networking-from-abroad-ds11509275","urlHost":"abroad","child":null},{"key":"MS in Mechanical","level":3,"type":"url","url":"/ms-in-mechanical-engineering-from-abroad-ds11509265","urlHost":"abroad","child":null},{"key":"MS in Electrical & Electronics","level":3,"type":"url","url":"/ms-in-electronics-electrical-engineering-from-abroad-ds11509263","urlHost":"abroad","child":null},{"key":"MBA in Finance","level":3,"type":"url","url":"/mba-in-finance-from-abroad-ds11508246","urlHost":"abroad","child":null},{"key":"MBA in Marketing","level":3,"type":"url","url":"/mba-in-marketing-from-abroad-ds11508249","urlHost":"abroad","child":null},{"key":"MBA in Human Resource","level":3,"type":"url","url":"/mba-in-human-resources-from-abroad-ds11508248","urlHost":"abroad","child":null},{"key":"MBA in International Business","level":3,"type":"url","url":"/mba-in-international-business-from-abroad-ds11508252","urlHost":"abroad","child":null},{"key":"Masters in Journalism","level":3,"type":"url","url":"/masters-in-journalism-courses-in-abroad-sl1328","urlHost":"abroad","child":null},{"key":"Masters of Humanities","level":3,"type":"url","url":"/masters-of-humanities-in-abroad-cl1244","urlHost":"abroad","child":null},{"key":"Masters of Law","level":3,"type":"url","url":"/masters-of-law-in-abroad-cl1245","urlHost":"abroad","child":null},{"key":"Masters of Medicine","level":3,"type":"url","url":"/masters-of-medicine-in-abroad-cl1243","urlHost":"abroad","child":null}],[{"key":"Bachelors","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"BE/B.Tech in Computer Science","level":3,"type":"url","url":"/be-btech-in-computer-science-engineering-from-abroad-ds11510277","urlHost":"abroad","child":null},{"key":"BE/B.Tech in IT/MIS","level":3,"type":"url","url":"/be-btech-in-it-networking-from-abroad-ds11510275","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Mechanical","level":3,"type":"url","url":"/be-btech-in-mechanical-engineering-from-abroad-ds11510265","urlHost":"abroad","child":null},{"key":"BE/B.Tech in Electrical & Electronics","level":3,"type":"url","url":"/be-btech-in-electronics-electrical-engineering-from-abroad-ds11510263","urlHost":"abroad","child":null},{"key":"Bachelors in Fashion","level":3,"type":"url","url":"/bachelors-in-fashion-design-courses-in-abroad-sl1331","urlHost":"abroad","child":null},{"key":"Bachelors in Media","level":3,"type":"url","url":"/bachelors-in-journalism-courses-in-abroad-sl1328","urlHost":"abroad","child":null},{"key":"Bachelors of Humanities","level":3,"type":"url","url":"/bachelors-of-humanities-in-abroad-cl1244","urlHost":"abroad","child":null},{"key":"Bachelors of Law","level":3,"type":"url","url":"/bachelors-of-law-in-abroad-cl1245","urlHost":"abroad","child":null},{"key":"Bachelors of Medicine","level":3,"type":"url","url":"/bachelors-of-medicine-in-abroad-cl1243","urlHost":"abroad","child":null},{"key":"B.Sc","level":3,"type":"url","url":"/bsc-in-abroad-dc111843","urlHost":"abroad","child":null}],[{"key":"Diploma/Certificate","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Computers Diploma","level":3,"type":"url","url":"/certificate-diploma-in-computers-in-abroad-cl1241","urlHost":"abroad","child":null},{"key":"Business Diploma","level":3,"type":"url","url":"/certificate-diploma-in-business-in-abroad-cl1239","urlHost":"abroad","child":null},{"key":"Engineering Diploma","level":3,"type":"url","url":"/certificate-diploma-in-engineering-in-abroad-cl1240","urlHost":"abroad","child":null}]]]},{"key":"Student Visas","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Student Visa Canada","level":3,"type":"url","url":"/canada-student-visa-guide-applycontent5772","urlHost":"abroad","child":null},{"key":"Student Visa USA","level":3,"type":"url","url":"/usa-student-visa-guide-applycontent5762","urlHost":"abroad","child":null},{"key":"Student Visa Australia","level":3,"type":"url","url":"/australia-student-visa-guide-applycontent5773","urlHost":"abroad","child":null},{"key":"Student Visa New Zealand","level":3,"type":"url","url":"/new-zealand-student-visa-guide-applycontent5776","urlHost":"abroad","child":null},{"key":"Student Visa Netherlands","level":3,"type":"url","url":"/netherlands-student-visa-guide-applycontent5996","urlHost":"abroad","child":null},{"key":"Student Visa Sweden","level":3,"type":"url","url":"/how-to-apply-for-sweden-student-visa-applycontent51016","urlHost":"abroad","child":null},{"key":"Student Visa UK","level":3,"type":"url","url":"/uk-student-visa-guide-applycontent5763","urlHost":"abroad","child":null},{"key":"Student Visa Germany","level":3,"type":"url","url":"/germany-student-visa-guide-applycontent5777","urlHost":"abroad","child":null},{"key":"Student Visa France","level":3,"type":"url","url":"/france-student-visa-guide-applycontent5774","urlHost":"abroad","child":null},{"key":"Student Visa Singapore","level":3,"type":"url","url":"/singapore-student-visa-guide-applycontent5775","urlHost":"abroad","child":null},{"key":"Student Visa Ireland","level":3,"type":"url","url":"/student-guide-to-ireland-guidepage-988","urlHost":"abroad","child":null},{"key":"Student Visa Finland","level":3,"type":"url","url":"/finland-student-visa-guide-applycontent5164235","urlHost":"abroad","child":null}]]]},{"key":"SOP/LOR","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"SOP","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"What is SOP?","level":3,"type":"url","url":"/sop-statement-of-purpose-applycontent1701","urlHost":"abroad","child":null},{"key":"Common Mistakes in SOP","level":3,"type":"url","url":"/common-mistakes-to-avoid-in-sop-applycontent11445","urlHost":"abroad","child":null},{"key":"Sample SOP for MBA","level":3,"type":"url","url":"/sample-sop-for-mba-applycontent1994","urlHost":"abroad","child":null},{"key":"Sample SOP for MS","level":3,"type":"url","url":"/sample-ms-sop-applycontent11421","urlHost":"abroad","child":null},{"key":"Sample SOP for Bachelors","level":3,"type":"url","url":"/sample-sop-for-ug-courses-applycontent1728","urlHost":"abroad","child":null},{"key":"Sample SOP for PhD","level":3,"type":"url","url":"/sop-for-phd-format-applycontent11417","urlHost":"abroad","child":null},{"key":"SOP for USA","level":3,"type":"url","url":"/sample-sop-for-usa-applycontent11437","urlHost":"abroad","child":null},{"key":"SOP for UK","level":3,"type":"url","url":"/sample-sop-for-uk-applycontent11439","urlHost":"abroad","child":null},{"key":"SOP for Canada","level":3,"type":"url","url":"/sample-sop-for-canada-applycontent11435","urlHost":"abroad","child":null},{"key":"SOP for Australia","level":3,"type":"url","url":"/sample-sop-for-australia-applycontent11449","urlHost":"abroad","child":null}],[{"key":"LOR","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"What is LOR?","level":3,"type":"url","url":"/lor-recommendation-letter-applycontent2702","urlHost":"abroad","child":null},{"key":"Common Mistakes in LOR","level":3,"type":"url","url":"/common-errors-in-a-recommendation-letter-applycontent2720","urlHost":"abroad","child":null},{"key":"Sample LOR for MBA","level":3,"type":"url","url":"/sample-recommendation-letter-for-mba-programs-applycontent2725","urlHost":"abroad","child":null},{"key":"Sample LOR for MS","level":3,"type":"url","url":"/sample-lor-for-ms-applycontent2723","urlHost":"abroad","child":null},{"key":"Sample LOR for Bachelors","level":3,"type":"url","url":"/sample-lor-for-ug-courses-applycontent2729","urlHost":"abroad","child":null},{"key":"Sample LOR for PhD","level":3,"type":"url","url":"/lors-for-phd-format-applycontent21403","urlHost":"abroad","child":null}]]]},{"key":"Scholarships","level":2,"type":"subItem","url":"/scholarships","urlHost":"abroad","child":[[[{"key":"By Course","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Scholarships for Bachelors","level":3,"type":"url","url":"/scholarships/bachelors-courses","urlHost":"abroad","child":null},{"key":"Scholarships for Masters","level":3,"type":"url","url":"/scholarships/masters-courses","urlHost":"abroad","child":null}],[{"key":"By Provider (for Masters)","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"College Specific Scholarships","level":3,"type":"url","url":"/scholarships/masters-courses?sc=internal","urlHost":"abroad","child":null},{"key":"Scholarships by External Agencies","level":3,"type":"url","url":"/scholarships/masters-courses?sc=external","urlHost":"abroad","child":null}],[{"key":"For Special Cases (for Masters) ","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"For Differently Abled","level":3,"type":"url","url":"/scholarships/masters-courses?sr=7","urlHost":"abroad","child":null},{"key":"For Women","level":3,"type":"url","url":"/scholarships/masters-courses?sr=1","urlHost":"abroad","child":null}]],[[{"key":"By Country","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Scholarships for USA","level":3,"type":"url","url":"/scholarships/usa-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Canada","level":3,"type":"url","url":"/scholarships/canada-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Australia","level":3,"type":"url","url":"/scholarships/australia-cp","urlHost":"abroad","child":null},{"key":"Scholarships for UK","level":3,"type":"url","url":"/scholarships/uk-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Germany","level":3,"type":"url","url":"/scholarships/germany-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Singapore","level":3,"type":"url","url":"/scholarships/singapore-cp","urlHost":"abroad","child":null},{"key":"Scholarships for New Zealand","level":3,"type":"url","url":"/scholarships/new-zealand-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Netherlands","level":3,"type":"url","url":"/scholarships/netherlands-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Ireland","level":3,"type":"url","url":"/scholarships/ireland-cp","urlHost":"abroad","child":null},{"key":"Scholarships for Sweden","level":3,"type":"url","url":"/scholarships/sweden-cp","urlHost":"abroad","child":null},{"key":"Scholarships for France","level":3,"type":"url","url":"/scholarships/france-cp","urlHost":"abroad","child":null}],[{"key":"By Type","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Need Based Scholarships","level":3,"type":"url","url":"/scholarships/masters-courses?st=need","urlHost":"abroad","child":null},{"key":"Merit Based Scholarships","level":3,"type":"url","url":"/scholarships/masters-courses?st=merit","urlHost":"abroad","child":null}]]]},{"key":"Education Loan","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Education Loan Guides","level":3,"type":"heading","url":"/education-loan-applycontent61319","urlHost":"abroad","child":null},{"key":"Step by Step Guide","level":3,"type":"url","url":"/step-by-step-guide-to-education-loans-applycontent61351","urlHost":"abroad","child":null},{"key":"Loan Eligibility Criteria","level":3,"type":"url","url":"/eligibility-criteria-for-education-loan-applycontent61405","urlHost":"abroad","child":null},{"key":"Documents Required for Loan","level":3,"type":"url","url":"/what-documents-are-required-for-an-education-loan-articlepage-1155","urlHost":"abroad","child":null},{"key":"Student Loan Vs Self Finance","level":3,"type":"url","url":"/education-loan-vs-self-funding-to-study-abroad-applycontent61357","urlHost":"abroad","child":null},{"key":"Best Education Loan Providers","level":3,"type":"url","url":"/top-education-loans-providers-in-india-applycontent61481","urlHost":"abroad","child":null},{"key":"Collateral for Education Loan","level":3,"type":"url","url":"/collateral-for-education-loan-everything-you-need-to-know-applycontent61359","urlHost":"abroad","child":null}],[{"key":"Guide for Countries","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Education Loan for USA","level":3,"type":"url","url":"/education-loan-for-usa-applycontent61343","urlHost":"abroad","child":null},{"key":"Education Loan for UK","level":3,"type":"url","url":"/education-loan-for-uk-applycontent61399","urlHost":"abroad","child":null},{"key":"Education Loan for Canada","level":3,"type":"url","url":"/education-loan-for-canada-applycontent61345","urlHost":"abroad","child":null},{"key":"Education Loan for Australia","level":3,"type":"url","url":"/education-loan-for-australia-applycontent61391","urlHost":"abroad","child":null},{"key":"Education Loan for Germany","level":3,"type":"url","url":"/education-loan-for-germany-applycontent61377","urlHost":"abroad","child":null},{"key":"Apply Education Loan for Abroad Studies","level":3,"type":"url","url":"/apply-education-loan","urlHost":"abroad","child":null}]]]},{"key":"Services","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Top Overseas Education Consultants in Cities","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Consultants in Delhi","level":3,"type":"url","url":"/apply/top-consultants-in-delhi","urlHost":"abroad","child":null},{"key":"Consultants in Mumbai","level":3,"type":"url","url":"/apply/top-consultants-in-mumbai","urlHost":"abroad","child":null},{"key":"Consultants in Pune","level":3,"type":"url","url":"/apply/top-consultants-in-pune","urlHost":"abroad","child":null},{"key":"Consultants in Kochi","level":3,"type":"url","url":"/apply/top-consultants-in-kochi","urlHost":"abroad","child":null},{"key":"Consultants in Bangalore","level":3,"type":"url","url":"/apply/top-consultants-in-bangalore","urlHost":"abroad","child":null},{"key":"Consultants in Kerala","level":3,"type":"url","url":"/apply/top-consultants-in-kerala","urlHost":"abroad","child":null},{"key":"Consultants in Chennai","level":3,"type":"url","url":"/apply/top-consultants-in-chennai","urlHost":"abroad","child":null},{"key":"Consultants in Hyderabad","level":3,"type":"url","url":"/apply/top-consultants-in-hyderabad","urlHost":"abroad","child":null},{"key":"Shiksha Counseling Service","level":3,"type":"url","url":"/apply","urlHost":"abroad","child":null},{"key":"Topper Counseling Service","level":3,"type":"url","url":"/topper-counseling-service","urlHost":"abroad","child":null}]]]}]]]},{"key":"Counseling","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Get Expert Guidance","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Ask a Question","level":3,"type":"url","url":"/","urlHost":"ask","child":null},{"key":"Discussions","level":3,"type":"url","url":"/discussions","urlHost":"ask","child":null}]]]},{"key":"Careers after 12th","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"By Stream","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Science","level":3,"type":"click","url":"/careers/opportunities","urlHost":"national","child":null},{"key":"Commerce","level":3,"type":"click","url":"/careers/opportunities","urlHost":"national","child":null},{"key":"Humanities","level":3,"type":"click","url":"/careers/opportunities","urlHost":"national","child":null}],[{"key":"Popular Careers","level":3,"type":"heading","url":"","urlHost":null,"child":null},{"key":"Aeronautical Engineer","level":3,"type":"url","url":"/careers/aeronautical-engineer-4","urlHost":"national","child":null},{"key":"Chartered Accountant","level":3,"type":"url","url":"/careers/chartered-accountant-23","urlHost":"national","child":null},{"key":"Computer Engineer","level":3,"type":"url","url":"/careers/computer-engineer-31","urlHost":"national","child":null},{"key":"Doctor","level":3,"type":"url","url":"/careers/medicine-76","urlHost":"national","child":null},{"key":"Hotel Manager","level":3,"type":"url","url":"/careers/hotel-manager-62","urlHost":"national","child":null},{"key":"Pilot","level":3,"type":"url","url":"/careers/pilot-91","urlHost":"national","child":null},{"key":"All other careers","level":3,"type":"urlWithAnchor","url":"/careers","urlHost":"national","child":null}]]]},{"key":"Courses after 12th","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"Science Stream","level":3,"type":"url","url":"/courses-after-12th/science","urlHost":"national","child":null},{"key":"Commerce Stream","level":3,"type":"url","url":"/courses-after-12th/commerce","urlHost":"national","child":null},{"key":"Arts Stream","level":3,"type":"url","url":"/courses-after-12th/arts","urlHost":"national","child":null},{"key":"All Class 12th Streams","level":3,"type":"urlWithAnchor","url":"/courses-after-12th","urlHost":"national","child":null}]]]},{"key":"Free Prep Material","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"NCERT Solutions","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"NCERT Solutions","level":3,"type":"url","url":"/ncert-solutions-preparation","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 12 Maths","level":3,"type":"url","url":"/preparation/maths-ncert-solutions-class-12th-ns","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 12 Physics","level":3,"type":"url","url":"/preparation/physics-ncert-solutions-class-12th-ns","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 12 Chemistry","level":3,"type":"url","url":"/preparation/chemistry-ncert-solutions-class-12th-ns","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 11 Maths","level":3,"type":"url","url":"/preparation/maths-ncert-solutions-class-11th-ns","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 11 Physics","level":3,"type":"url","url":"/preparation/physics-ncert-solutions-class-11th-ns","urlHost":"national","child":null},{"key":"NCERT Solutions for Class 11 Chemistry","level":3,"type":"url","url":"/preparation/chemistry-ncert-solutions-class-11th-ns","urlHost":"national","child":null}],[{"key":"NCERT Topics ","level":3,"type":"heading","url":null,"urlHost":null,"child":null},{"key":"NCERT Notes","level":3,"type":"url","url":"/preparation/ncert-notes-tp","urlHost":"national","child":null},{"key":"NCERT Class 12 Notes","level":3,"type":"url","url":"/preparation/ncert-class-12-notes-tp","urlHost":"national","child":null},{"key":"NCERT Class 11 Notes","level":3,"type":"url","url":"/preparation/ncert-class-11-notes-tp","urlHost":"national","child":null},{"key":"NCERT Class 12 Maths","level":3,"type":"url","url":"/preparation/ncert-class-12-maths-tp","urlHost":"national","child":null},{"key":"NCERT Class 12 Physics","level":3,"type":"url","url":"/preparation/ncert-class-12-physics-tp","urlHost":"national","child":null},{"key":"NCERT Class 12 Chemistry","level":3,"type":"url","url":"/preparation/ncert-class-12-chemistry-tp","urlHost":"national","child":null},{"key":"NCERT Class 11 Maths","level":3,"type":"url","url":"/preparation/ncert-class-11-maths-tp","urlHost":"national","child":null},{"key":"NCERT Class 11 Physics","level":3,"type":"url","url":"/preparation/ncert-class-11-physics-tp","urlHost":"national","child":null},{"key":"NCERT Class 11 Chemistry","level":3,"type":"url","url":"/preparation/ncert-class-11-chemistry-tp","urlHost":"national","child":null}]]]},{"key":"National Boards","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"CBSE","level":3,"type":"url","url":"/boards/cbse-board","urlHost":"national","child":null},{"key":"ICSE","level":3,"type":"url","url":"/boards/icse-board","urlHost":"national","child":null},{"key":"NIOS","level":3,"type":"url","url":"/boards/nios-board","urlHost":"national","child":null},{"key":"All Education Boards","level":3,"type":"urlWithAnchor","url":"/boards","urlHost":"national","child":null}]]]},{"key":"State Boards","level":2,"type":"subItem","url":"","urlHost":null,"child":[[[{"key":"UPMSP","level":3,"type":"url","url":"/boards/upmsp-board","urlHost":"national","child":null},{"key":"BSEB","level":3,"type":"url","url":"/boards/bseb-board","urlHost":"national","child":null},{"key":"PSEB","level":3,"type":"url","url":"/boards/pseb-board","urlHost":"national","child":null},{"key":"RBSE","level":3,"type":"url","url":"/boards/rbse-board","urlHost":"national","child":null},{"key":"JKBOSE","level":3,"type":"url","url":"/boards/jkbose-board","urlHost":"national","child":null}],[{"key":"GSEB","level":3,"type":"url","url":"/boards/gseb-board","urlHost":"national","child":null},{"key":"HPBOSE","level":3,"type":"url","url":"/boards/hpbose-board","urlHost":"national","child":null},{"key":"MPBSE","level":3,"type":"url","url":"/boards/mpbse-board","urlHost":"national","child":null},{"key":"BIEAP","level":3,"type":"url","url":"/boards/bieap-board","urlHost":"national","child":null}],[{"key":"BSEH","level":3,"type":"url","url":"/boards/bseh-board","urlHost":"national","child":null},{"key":"CGBSE","level":3,"type":"url","url":"/boards/cgbse-board","urlHost":"national","child":null},{"key":"WBBSE","level":3,"type":"url","url":"/boards/wbbse-board","urlHost":"national","child":null},{"key":"WBCHSE","level":3,"type":"url","url":"/boards/wbchse-board","urlHost":"national","child":null},{"key":"All Education Boards","level":3,"type":"urlWithAnchor","url":"/boards","urlHost":"national","child":null}]]]},{"key":"Abroad Counseling Service","level":2,"type":"subItem","url":"/apply","urlHost":"abroad","child":[[[]]]}]]]},{"key":"Shiksha Online","level":1,"type":"topItem","url":"","urlHost":null,"child":[[[{"key":"Technology","level":2,"type":"subItem","url":"/online-courses/technology-courses-certification-training-ct139","urlHost":"national","child":[[[{"key":"Courses In Technology","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Big Data","level":3,"type":"url","url":"/online-courses/big-data-courses-certification-training-st601","urlHost":"national","child":null},{"key":"Cloud Technologies","level":3,"type":"url","url":"/online-courses/cloud-technologies-courses-certification-training-st607","urlHost":"national","child":null},{"key":"Cybersecurity","level":3,"type":"url","url":"/online-courses/cybersecurity-courses-certification-training-st603","urlHost":"national","child":null},{"key":"Databases","level":3,"type":"url","url":"/online-courses/databases-courses-certification-training-st605","urlHost":"national","child":null},{"key":"IT Services","level":3,"type":"url","url":"/online-courses/masters-and-certificate-programs-courses-certification-training-st613","urlHost":"national","child":null},{"key":"Masters and Certificate Programs","level":3,"type":"url","url":"/online-courses/masters-and-certificate-programs-courses-certification-training-st613","urlHost":"national","child":null},{"key":"Networking and Hardware","level":3,"type":"url","url":"/online-courses/networking-and-hardware-courses-certification-training-st615","urlHost":"national","child":null},{"key":"Operating System","level":3,"type":"url","url":"/online-courses/operating-system-courses-certification-training-st617","urlHost":"national","child":null},{"key":"Programming","level":3,"type":"url","url":"/online-courses/programming-courses-certification-training-st619","urlHost":"national","child":null},{"key":"QA and Testing","level":3,"type":"url","url":"/online-courses/qa-and-testing-courses-certification-training-st621","urlHost":"national","child":null},{"key":"Web Development","level":3,"type":"url","url":"/online-courses/web-development-courses-certification-training-st644","urlHost":"national","child":null},{"key":"Software Tools","level":3,"type":"url","url":"/online-courses/software-tools-courses-certification-training-st293","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Learn Why Cybersecurity is essential","level":3,"type":"url","url":"/online-courses/what-is-cybersecurity-st603","urlHost":"national","child":null},{"key":"Learn about the 4Vs of Big Data","level":3,"type":"url","url":"/online-courses/what-is-big-data-analytics-st601-tg91","urlHost":"national","child":null},{"key":"An Ethical Hacker's Guide for getting from beginner to professional","level":3,"type":"url","url":"/online-courses/what-is-ethical-hacking-st603-tg161","urlHost":"national","child":null}],[{"key":"What's New","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"GPT 3.5 vs GPT 4: How Do They Differ?","level":3,"type":"url","url":"/online-courses/articles/gpt-3-point-5-vs-gpt-4-how-do-they-differ","urlHost":"national","child":null},{"key":"Exploring Python Pickle: The Ultimate Resource for Object Serialization and Storage","level":3,"type":"url","url":"/online-courses/articles/python-pickle/","urlHost":"national","child":null},{"key":"What is Web Service?","level":3,"type":"url","url":"/online-courses/articles/all-about-web-service/","urlHost":"national","child":null},{"key":"max() Function in Python","level":3,"type":"url","url":"/online-courses/articles/max-function-in-python/","urlHost":"national","child":null}]]]},{"key":"Data Science","level":2,"type":"subItem","url":"/online-courses/data-science-courses-certification-training-ct123","urlHost":"national","child":[[[{"key":"Courses In Data Science","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Data Science Basics","level":3,"type":"url","url":"/online-courses/data-science-basics-courses-certification-training-st549","urlHost":"national","child":null},{"key":"Data Science for HealthCare","level":3,"type":"url","url":"/online-courses/data-science-for-healthcare-courses-certification-training-st649","urlHost":"national","child":null},{"key":"Deep Learning","level":3,"type":"url","url":"/online-courses/deep-learning-courses-certification-training-st551","urlHost":"national","child":null},{"key":"Machine Learning","level":3,"type":"url","url":"/online-courses/machine-learning-courses-certification-training-st553","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Learn more about different versions of python","level":3,"type":"url","url":"/online-courses/what-is-python-st619-tg21","urlHost":"national","child":null},{"key":"How to get started with Data Science?","level":3,"type":"url","url":"/online-courses/what-is-data-science-ct123","urlHost":"national","child":null},{"key":"Learn about the difference between AI and ML","level":3,"type":"url","url":"/online-courses/what-is-machine-learning-st553","urlHost":"national","child":null}],[{"key":"What's New","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Activation Functions: With Real-life analogy and Python Code","level":3,"type":"url","url":"/online-courses/articles/activation-functions-with-real-life-analogy-and-python-code/","urlHost":"national","child":null},{"key":"Understanding Transformers: A Beginner's Guide to the Basics and Applications","level":3,"type":"url","url":"/online-courses/articles/transformer-nlp","urlHost":"national","child":null},{"key":"Difference Between Transportation Problem and Assignment Problem","level":3,"type":"url","url":"/online-courses/articles/difference-between-transportation-problem-and-assignment-problem/","urlHost":"national","child":null},{"key":"Midjourney AI Image Generator Pauses its Free Trials","level":3,"type":"url","url":"/online-courses/articles/midjourney-ai-image-generator-pauses-its-free-trials/","urlHost":"national","child":null}]]]},{"key":"Management","level":2,"type":"subItem","url":"/online-courses/management-courses-certification-training-ct135","urlHost":"national","child":[[[{"key":"Courses In Management","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Business Analytics","level":3,"type":"url","url":"/online-courses/business-analytics-courses-certification-training-st583","urlHost":"national","child":null},{"key":"Business Tools","level":3,"type":"url","url":"/online-courses/business-tools-courses-certification-training-st639","urlHost":"national","child":null},{"key":"Communication","level":3,"type":"url","url":"/online-courses/communication-courses-certification-training-st595","urlHost":"national","child":null},{"key":"CSR","level":3,"type":"url","url":"/online-courses/csr-courses-certification-training-st585","urlHost":"national","child":null},{"key":"Entrepreneurship","level":3,"type":"url","url":"/online-courses/entrepreneurship-courses-certification-training-st587","urlHost":"national","child":null},{"key":"Human Resources","level":3,"type":"url","url":"/online-courses/human-resources-courses-certification-training-st589","urlHost":"national","child":null},{"key":"Industry Programs","level":3,"type":"url","url":"/online-courses/industry-programs-courses-certification-training-st591","urlHost":"national","child":null},{"key":"Logistics and Supply Chain","level":3,"type":"url","url":"/online-courses/logistics-and-supply-chain-courses-certification-training-st627","urlHost":"national","child":null},{"key":"Marketing","level":3,"type":"url","url":"/online-courses/marketing-courses-certification-training-st593","urlHost":"national","child":null},{"key":"Masters and PG-Management","level":3,"type":"url","url":"/online-courses/masters-and-pg-management-courses-certification-training-st629","urlHost":"national","child":null},{"key":"Operations","level":3,"type":"url","url":"/online-courses/operations-courses-certification-training-st453","urlHost":"national","child":null},{"key":"Product Management","level":3,"type":"url","url":"/online-courses/product-management-courses-certification-training-st631","urlHost":"national","child":null},{"key":"Strategy and Leadership","level":3,"type":"url","url":"/online-courses/strategy-and-leadership-courses-certification-training-st633","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"What are the latest trends in Recruitment?","level":3,"type":"url","url":"/online-courses/what-is-recruitment-st589-tg1271","urlHost":"national","child":null},{"key":"What are the traits required to become a successful Entrepreneur?","level":3,"type":"url","url":"/online-courses/what-is-entrepreneurship-and-start-ups-st587-tg129","urlHost":"national","child":null},{"key":"Your guide to become a Business Analyst","level":3,"type":"url","url":"/online-courses/what-is-business-analytics-st583","urlHost":"national","child":null}],[{"key":"What's New","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Difference Between Authority and Responsibility","level":3,"type":"url","url":"/online-courses/articles/difference-between-authority-and-responsibility","urlHost":"national","child":null},{"key":"Basics of Google Display Network - Definition and Importance","level":3,"type":"url","url":"/online-courses/articles/google-display-network/","urlHost":"national","child":null},{"key":"Get Started with Google Ads Editor Today","level":3,"type":"url","url":"/online-courses/articles/google-ads-editor","urlHost":"national","child":null},{"key":"How can Google Ads Help You Advance Your Business Goals?","level":3,"type":"url","url":"/online-courses/articles/how-can-google-ads-help-business-goals","urlHost":"national","child":null}]]]},{"key":"Finance","level":2,"type":"subItem","url":"/online-courses/finance-courses-certification-training-ct133","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Accounting","level":3,"type":"url","url":"/online-courses/accounting-courses-certification-training-st577","urlHost":"national","child":null},{"key":"Banking","level":3,"type":"url","url":"/online-courses/banking-courses-certification-training-st579","urlHost":"national","child":null},{"key":"Investing","level":3,"type":"url","url":"/online-courses/investing-courses-certification-training-st581","urlHost":"national","child":null},{"key":"Law","level":3,"type":"url","url":"/online-courses/law-courses-certification-training-st351","urlHost":"national","child":null},{"key":"Insurance","level":3,"type":"url","url":"/online-courses/insurance-courses-certification-training-st373","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Learn about Corporate Finance concepts and tools","level":3,"type":"url","url":"/online-courses/what-is-corporate-finance-st577-tg207","urlHost":"national","child":null},{"key":"What you need to know about Commercial Banking","level":3,"type":"url","url":"/online-courses/what-is-commercial-banking-st579-tg491","urlHost":"national","child":null},{"key":"Tips, tools and techniques to get started with Risk Management","level":3,"type":"url","url":"/online-courses/what-is-risk-management-st579-tg79","urlHost":"national","child":null}]]]},{"key":"Creativity & Design","level":2,"type":"subItem","url":"/online-courses/creativity-and-design-courses-certification-training-ct125","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Architecture","level":3,"type":"url","url":"/online-courses/architecture-courses-certification-training-st555","urlHost":"national","child":null},{"key":"Fashion","level":3,"type":"url","url":"/online-courses/fashion-courses-certification-training-st557","urlHost":"national","child":null},{"key":"Web Design","level":3,"type":"url","url":"/online-courses/web-design-courses-certification-training-st305","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Why is a career in Graphic Design good?","level":3,"type":"url","url":"/online-courses/what-is-graphic-design-st305-tg297","urlHost":"national","child":null},{"key":"What is UI/UX Design all about?","level":3,"type":"url","url":"/online-courses/what-is-ui-ux-design-st305-tg1267","urlHost":"national","child":null},{"key":"Learn web design skills a designer should master","level":3,"type":"url","url":"/online-courses/what-is-web-designing-st305","urlHost":"national","child":null}]]]},{"key":"Emerging Technologies","level":2,"type":"subItem","url":"/online-courses/emerging-technologies-courses-certification-training-ct127","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"AR VR and Gaming","level":3,"type":"url","url":"/online-courses/ar-vr-and-gaming-courses-certification-training-st559","urlHost":"national","child":null},{"key":"BlockChain","level":3,"type":"url","url":"/online-courses/blockchain-courses-certification-training-st561","urlHost":"national","child":null},{"key":"Electric Vehicles","level":3,"type":"url","url":"/online-courses/internet-of-things-courses-certification-training-st563","urlHost":"national","child":null},{"key":"Internet of Things","level":3,"type":"url","url":"/online-courses/internet-of-things-courses-certification-training-st563","urlHost":"national","child":null},{"key":"Robotics","level":3,"type":"url","url":"/online-courses/robotics-courses-certification-training-st643","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Block chain 101: The complete guide","level":3,"type":"url","url":"/online-courses/what-is-blockchain-st561","urlHost":"national","child":null},{"key":"Learn how IOT is changing the world","level":3,"type":"url","url":"/online-courses/what-is-iot-and-connected-devices-st563-tg949","urlHost":"national","child":null},{"key":"A Beginner's guide to Augmented Reality","level":3,"type":"url","url":"/online-courses/what-is-augmented-reality-st559-tg1235","urlHost":"national","child":null}]]]},{"key":"Engineering-Non CS","level":2,"type":"subItem","url":"/online-courses/engineering-non-cs-courses-certification-training-ct131","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Aviation","level":3,"type":"url","url":"/online-courses/aviation-courses-certification-training-st569","urlHost":"national","child":null},{"key":"Electrical Engineering","level":3,"type":"url","url":"/online-courses/electrical-engineering-courses-certification-training-st571","urlHost":"national","child":null},{"key":"Masters and PG-Engineering","level":3,"type":"url","url":"/online-courses/masters-and-pg-engineering-courses-certification-training-st573","urlHost":"national","child":null},{"key":"Production and Industrial Engineering","level":3,"type":"url","url":"/online-courses/production-and-industrial-engineering-courses-certification-training-st575","urlHost":"national","child":null},{"key":"TeleCom","level":3,"type":"url","url":"/online-courses/telecom-courses-certification-training-st385","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Has technology helped manufacturing processes?","level":3,"type":"url","url":"/online-courses/what-is-manufacturing-technologies-st575-tg1039","urlHost":"national","child":null},{"key":"Learn about 5 steps of engineering design process","level":3,"type":"url","url":"/online-courses/what-is-engineering-design-st571-tg209","urlHost":"national","child":null},{"key":"How to start a career in Aviation Engineering","level":3,"type":"url","url":"/online-courses/what-is-aviation-engineering-st569-tg1193","urlHost":"national","child":null}]]]},{"key":"Healthcare","level":2,"type":"subItem","url":"/online-courses/healthcare-courses-certification-training-ct47","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Fitness and Nutrition","level":3,"type":"url","url":"/online-courses/fitness-and-nutrition-courses-certification-training-st623","urlHost":"national","child":null},{"key":"Healthcare Research","level":3,"type":"url","url":"/online-courses/healthcare-research-courses-certification-training-st625","urlHost":"national","child":null},{"key":"Healthcare Management","level":3,"type":"url","url":"/online-courses/healthcare-management-courses-certification-training-st353","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Your guide to become a qualified fitness instructor","level":3,"type":"url","url":"/online-courses/what-is-fitness-instruction-st623-tg453","urlHost":"national","child":null},{"key":"Clinical Research: Phases, Testing, Types & Trials","level":3,"type":"url","url":"/online-courses/what-is-clinical-research-st625-tg1211","urlHost":"national","child":null},{"key":"Learn about emerging trends in Biopharmaceutical Drug Discovery","level":3,"type":"url","url":"/online-courses/what-is-biopharmaceuticals-st353-tg465","urlHost":"national","child":null}]]]},{"key":"Energy And Environment","level":2,"type":"subItem","url":"/career-aptitude-test","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"/career-aptitude-test?action=viewCard","urlHost":"national","child":null},{"key":"Energy","level":3,"type":"url","url":"/online-courses/energy-courses-certification-training-st565","urlHost":"national","child":null},{"key":"Environment","level":3,"type":"url","url":"/online-courses/environment-courses-certification-training-st567","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Renewable Energy: Types, Forms & Sources","level":3,"type":"url","url":"/online-courses/what-is-renewable-energy-st565-tg265","urlHost":"national","child":null}]]]},{"key":"Social Sciences","level":2,"type":"subItem","url":"/career-aptitude-test","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Education","level":3,"type":"url","url":"/online-courses/education-courses-certification-training-st599","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"TET 2020 :Notification, Syllabus, Eligibility, Dates, Exam Pattern","level":3,"type":"url","url":"/online-courses/articles/tet-exam-dates-syllabus-eligibility-notification/","urlHost":"national","child":null}]]]},{"key":"Personal Development","level":2,"type":"subItem","url":"/online-courses/personal-development-courses-certification-training-ct137","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Career Growth","level":3,"type":"url","url":"/online-courses/career-growth-courses-certification-training-st635","urlHost":"national","child":null},{"key":"Hobby And Passion","level":3,"type":"url","url":"/online-courses/hobby-and-passion-courses-certification-training-st597","urlHost":"national","child":null},{"key":"Languages","level":3,"type":"url","url":"/online-courses/languages-courses-certification-training-st371","urlHost":"national","child":null}],[{"key":"Career Guides","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Why Knowing English is important for today's world?","level":3,"type":"url","url":"/online-courses/what-is-ielts-spoken-english-st371-tg1071","urlHost":"national","child":null},{"key":"Tips and Tricks to communicate effectively","level":3,"type":"url","url":"/online-courses/articles/ways-to-effectively-speak-in-english/","urlHost":"national","child":null},{"key":"Your ultimate English Proficiency side","level":3,"type":"url","url":"/online-courses/articles/five-english-proficiency-tips-for-study-abroad","urlHost":"national","child":null}]]]},{"key":"Degree Programs","level":2,"type":"subItem","url":"/online-courses/degree-programs-courses-certification-training-ct148","urlHost":"national","child":[[[{"key":"Courses","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"Bachelors Program","level":3,"type":"url","url":"/online-courses/bachelors-program-courses-certification-training-st646","urlHost":"national","child":null},{"key":"Masters Program","level":3,"type":"url","url":"/online-courses/masters-program-courses-certification-training-st491","urlHost":"national","child":null}]]]},{"key":"Blog","level":2,"type":"subItem","url":"/online-courses/articles","urlHost":"national","child":[[[{"key":"What's New","level":3,"type":"heading","url":"","urlHost":"national","child":null},{"key":"10 Entrepreneurial Skills Every Professional Must Develop","level":3,"type":"url","url":"/online-courses/articles/entrepreneurial-skills","urlHost":"national","child":null},{"key":"Difference Between Gross Profit and Net Profit","level":3,"type":"url","url":"/online-courses/articles/difference-between-gross-profit-and-net-profit","urlHost":"national","child":null},{"key":"GPT 3.5 vs GPT 4: How Do They Differ?","level":3,"type":"url","url":"/online-courses/articles/gpt-3-point-5-vs-gpt-4-how-do-they-differ","urlHost":"national","child":null},{"key":"What is a Dividend: Benefits and Types","level":3,"type":"url","url":"/online-courses/articles/all-about-dividend/","urlHost":"national","child":null},{"key":"Exploring Python Pickle: The Ultimate Resource for Object Serialization and Storage","level":3,"type":"url","url":"/online-courses/articles/python-pickle/","urlHost":"national","child":null},{"key":"View All","level":3,"type":"urlWithAnchor","url":"/online-courses/articles/","urlHost":"national","child":null}]]]}]]]}]]]}},"saGnbLinks":{},"soGnbLinks":{},"childPageData":{},"examData":{},"examPageData":{"examBasicInfo":{"type":"preparation","conductedBy":"","url":"/preparation/maths-ncert-solutions-class-11th-ns","isRootUrl":"No","isConductingBody":"No","isExamPaid":0,"isAbroad":0,"groupMapping":[{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-binomial-theorem-4409-ns","primaryGroup":0,"rprpScore":113.51,"id":4409,"name":"Binomial Theorem"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-probability-4425-ns","primaryGroup":0,"rprpScore":237.04,"id":4425,"name":"Probability"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-statistics-4423-ns","primaryGroup":0,"rprpScore":349.45,"id":4423,"name":"Statistics"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-mathematical-reasoning-4421-ns","primaryGroup":0,"rprpScore":533.33,"id":4421,"name":"Mathematical Reasoning"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","primaryGroup":0,"rprpScore":117.24,"id":4419,"name":"Limits and Derivatives"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-introduction-to-three-dimensional-geometry-4417-ns","primaryGroup":0,"rprpScore":99200,"id":4417,"name":"Introduction to Three Dimensional Geometry"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-conic-sections-4415-ns","primaryGroup":0,"rprpScore":162.5,"id":4415,"name":"Conic Sections"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-straight-lines-4413-ns","primaryGroup":0,"rprpScore":251.28,"id":4413,"name":"Straight Lines"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-sequences-and-series-4411-ns","primaryGroup":0,"rprpScore":115.15,"id":4411,"name":"Sequences and Series"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-ns","primaryGroup":1,"rprpScore":100012.38,"id":4387,"name":"NCERT Maths 11th"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-permutations-and-combinations-4407-ns","primaryGroup":0,"rprpScore":455.94,"id":4407,"name":"Permutations and Combinations"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-linear-inequalities-4405-ns","primaryGroup":0,"rprpScore":0,"id":4405,"name":"Linear Inequalities"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-complex-numbers-and-quadratic-equations-4403-ns","primaryGroup":0,"rprpScore":210.99,"id":4403,"name":"Complex Numbers and Quadratic Equations"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-principle-of-mathematical-induction-4401-ns","primaryGroup":0,"rprpScore":0,"id":4401,"name":"Principle of Mathematical Induction"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-trigonometric-functions-4399-ns","primaryGroup":0,"rprpScore":216.67,"id":4399,"name":"Trigonometric Functions"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-relations-and-functions-4397-ns","primaryGroup":0,"rprpScore":337.19,"id":4397,"name":"Relations and Functions"},{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-sets-4395-ns","primaryGroup":0,"rprpScore":3346.14,"id":4395,"name":"Sets"}],"primaryGroup":{"groupUrl":"/preparation/maths-ncert-solutions-class-11th-ns","primaryGroup":1,"rprpScore":100012.38,"id":4387,"name":"NCERT Maths 11th"},"examPageId":4297,"examLogoUrl":{"url":"/public/images/defaultExamImage.jpeg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"classId":15071,"subjectId":15078,"className":"11","subjectName":"Maths","preparationEntityType":"chapter","preparationPageType":"solution","order":0,"highlightText":null,"isConductingAgency":0,"ampUrl":null,"id":14805,"name":"Maths Ncert Solutions class 11th","fullName":"NCERT Maths 11th"},"groupBasicInfo":{"groupId":4419,"groupName":"Limits and Derivatives","groupUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","examId":14805,"examPageId":4307,"isPrimary":0,"entitiesMappedToGroup":{"otherAttribute":[20,33,34,39],"year":[2023],"primaryHierarchy":[843],"course":[259]},"hierarchyData":{"843":{"stream":25,"substream":null,"specialization":null,"primary_hierarchy":1}},"attributeInfo":{"20":{"valueId":20,"valueName":"Full Time","attributeName":"Education Type","attributeId":5},"33":{"valueId":33,"valueName":"Classroom","attributeName":"Medium/Delivery Method","attributeId":7},"34":{"valueId":34,"valueName":"Distance/Correspondence","attributeName":"Medium/Delivery Method","attributeId":7},"39":{"valueId":39,"valueName":"Online","attributeName":"Medium/Delivery Method","attributeId":7}},"userId":32401605,"stateId":1,"countryId":2,"thumbnail":null,"created":"2023-05-02 21:00:39","topicId":7401219,"examConductingBody":"government","scoreAcceptanceLevel":"national","rprpScore":113.89,"blurb":null,"stateIds":null,"level":null,"order":0,"primaryStream":25},"contentInfo":{"homepage":{"wiki":null,"date":[],"file":{},"guideUrls":{"Summary":"/mediadata/examGuides/examSectionGuide4419_Overview_2025-09-13-03-37.pdf"},"sectionalData":{"sectionName":"homepage","entityType":"Summary","contentData":[{"id":890958,"h2Text":"","dataValue":"\u003cp>The subject matter experts at Shiksha created the \u003cstrong>Class 11 Limits and Derivatives\u003c/strong> NCERT solutions. The chapter introduces Calculus, a branch of mathematics that studies the change in the value of a function as the points in the domain change. \u003cstrong>Limits and Derivatives class 11\u003c/strong> Chapter 12 Maths includes the definitions of limits and derivatives and some algebra involving these concepts.\u003cbr>The step-by-step solutions help students to understand the concepts clearly and also provide them the confidence to appear in their school exam, CBSE Board exam, and other entrance tests later on. These solutions are accurate, and students can depend on them for their exam preparation. If they practiced only from these solutions, after going through the NCERT chapter, it would be more than sufficient for their examination preparation.\u003cbr>For comprehensive subject-wise NCERT notes of Class 11 and Class 12, \u003ca href=\"https://www.shiksha.com/preparation/ncert-notes-tp\">Check Here\u003c/a>. You will get the reliable notes for quick revision.\u003c/p> \n\u003ch3>Related Links\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%; height: 21px;\" border=\"1\"> \n \u003ctbody> \n \u003ctr style=\"height: 21px;\"> \n \u003ctd style=\"width: 33.3333%; height: 21px;\">\u003ca href=\"https://www.shiksha.com/preparation/maths-ncert-solutions-class-11th-ns\">NCERT Solutions for Class 11 Maths\u003c/a>\u003c/td> \n \u003ctd style=\"width: 33.3333%; height: 21px;\">\u003ca href=\"https://www.shiksha.com/preparation/ncert-class-11-notes-tp\">NCERT Class 11 Notes\u003c/a>\u003c/td> \n \u003ctd style=\"width: 33.3333%; height: 21px;\">\u003ca href=\"https://www.shiksha.com/preparation/ncert-solutions-ns\">NCERT Solutions Class 11 and 12 for Maths, Physics, Chemistry\u003c/a>\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table>","order":1,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"intro","wikiIdentifier":"237","status":"live","updateDateOn":false,"updatedOn":"2025-04-27 16:55:39","widgetData":[{"dataValue":"LIVE_BLOG_WIDGET","order":1,"dataCategoryType":"widget","updateDateOn":false}]},{"id":947470,"h2Text":"Class 11 Maths Chapter 12 Snapshot: Limits and Derivatives","dataValue":"\u003cp>Here is a quick summary of the \u003cstrong>Limit and Derivatives Class 11 Maths\u003c/strong>:\u003c/p> \n\u003cul> \n \u003cli>According to Chapter 12 of Class 11 Maths, \u003cspan class=\"mathml\" contenteditable=\"false\"> \n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"> \n \u003cmrow> \n \u003cmunder> \n \u003cmo>\n lim\n \u003c/mo> \n \u003cmrow> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n →\n \u003c/mo> \n \u003c!-- right arrow --> \n \u003cmsup> \n \u003cmi>\n a\n \u003c/mi> \n \u003cmo>\n -\n \u003c/mo> \n \u003c/msup> \n \u003c/mrow> \n \u003c/munder> \n \u003cmo>\n ⁡\n \u003c/mo> \n \u003c!-- function application --> \n \u003cmi>\n f\n \u003c/mi> \n \u003cmo>\n (\n \u003c/mo> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n )\n \u003c/mo> \n \u003c/mrow> \n \u003c/math> \u003c/span> is the expected value of f, which is called the left-hand limit of f at a, when x=a. Here, it is given that the values of f near x to the left of a.\u003c/li> \n \u003cli>To find the right-hand limit of f(x) at a, mathematically we show as - \u003cspan class=\"mathml\" contenteditable=\"false\"> \n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"> \n \u003cmrow> \n \u003cmunder> \n \u003cmo>\n lim\n \u003c/mo> \n \u003cmrow> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n →\n \u003c/mo> \n \u003c!-- right arrow --> \n \u003cmsup> \n \u003cmi>\n a\n \u003c/mi> \n \u003cmo>\n +\n \u003c/mo> \n \u003c/msup> \n \u003c/mrow> \n \u003c/munder> \n \u003cmo>\n ⁡\n \u003c/mo> \n \u003c!-- function application --> \n \u003cmi>\n f\n \u003c/mi> \n \u003cmo>\n (\n \u003c/mo> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n )\n \u003c/mo> \n \u003c/mrow> \n \u003c/math> \u003c/span>. It is the expected value of f at x = a, given the values of f near x to the right of a.\u003c/li> \n \u003cli>There is also the concept of common value when the right and left-hand limits coincide. It is denoted by \u003cspan class=\"mathml\" contenteditable=\"false\"> \n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"> \n \u003cmrow> \n \u003cmunder> \n \u003cmo>\n lim\n \u003c/mo> \n \u003cmrow> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n →\n \u003c/mo> \n \u003c!-- Unicode right arrow --> \n \u003cmi>\n a\n \u003c/mi> \n \u003c/mrow> \n \u003c/munder> \n \u003cmo>\n ⁡\n \u003c/mo> \n \u003c!-- Function application (invisible) --> \n \u003cmi>\n f\n \u003c/mi> \n \u003cmo>\n (\n \u003c/mo> \n \u003cmi>\n x\n \u003c/mi> \n \u003cmo>\n )\n \u003c/mo> \n \u003c/mrow> \n \u003c/math> \u003c/span>. \u003c/li> \n\u003c/ul> \n\u003cp>To get access to the chapter-wise Class 11 Maths NCERT Notes that offer solved examples and also provide the important topics, students must check \u003ca href=\"https://www.shiksha.com/preparation/ncert-class-11-maths-tp\">here\u003c/a>.\u003c/p>","order":2,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"209376","status":"live","updateDateOn":false,"updatedOn":"2025-07-28 14:46:56","widgetData":[{"dataValue":"POLLS_WIDGET","order":1,"dataCategoryType":"widget","updateDateOn":false}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":865224,"h2Text":"Class 11 Math Limits and Derivatives: Key Topics, Weightage ","dataValue":"\u003cp>\u003cstrong>Limits Class 11 Maths Chapter 12\u003c/strong> is the part of the broader \"Limits, Continuity & Differentiability\" topic in the JEE Main exam. Hence, the students must focus on the core concepts of the chapter. Before starting the preparation of the chapter, it is wise to know the topics of this chapter. See below the topics covered in this chapter:\u003c/p> \n\u003ctable style=\"border-collapse: collapse; width: 53.4434%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003cth style=\"width: 16.3912%;\">Exercise\u003c/th> \n \u003cth style=\"width: 37.0523%;\">Topics Covered\u003c/th> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 16.3912%;\">12.1\u003c/td> \n \u003ctd style=\"width: 37.0523%;\">Introduction\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 16.3912%;\">12.2\u003c/td> \n \u003ctd style=\"width: 37.0523%;\">Intuitive Idea of Derivatives\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 16.3912%;\">12.3\u003c/td> \n \u003ctd style=\"width: 37.0523%;\">Limits\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ch3>Class 11 Limits and Derivatives Weightage in JEE Main Exam\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003cth style=\"width: 33.3333%;\">Exam\u003c/th> \n \u003cth style=\"width: 33.3333%;\">Number of Questions\u003c/th> \n \u003cth style=\"width: 33.3333%;\">Weightage\u003c/th> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 33.3333%;\">\u003ca href=\"https://www.shiksha.com/engineering/jee-main-exam\">JEE Main\u003c/a>\u003c/td> \n \u003ctd style=\"width: 33.3333%;\">1 to 3 questions\u003c/td> \n \u003ctd style=\"width: 33.3333%;\">3.3% to 10%\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table>","order":3,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"188822","status":"live","updateDateOn":false,"updatedOn":"2025-03-05 15:41:04","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":947472,"h2Text":"Important Formulas of Class 11 Limits and Derivatives","dataValue":"\u003ch3>Important Formulae of Limits and Derivatives for CBSE and Competitive Exams\u003c/h3> \n\u003cul> \n \u003cli>\u003cstrong>First Principle of Differentiation\u003c/strong>\u003cspan class=\"katex-display\">\u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n lim\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 0\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003cmfrac>\n \u003cmrow>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n h\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmi>\n h\n \u003c/mi>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Power Functions:\u003c/strong> \n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n n\n \u003c/mi>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmsup>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n n\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (x^n) = n x^{n-1}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\u003cstrong>Trigonometric Functions\u003c/strong>: \n \u003cul> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n sin\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n cos\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\sin x) = \\cos x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n cos\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmi>\n sin\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\cos x) = -\\sin x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n tan\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n sec\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003c/mrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\tan x) = \\sec^2 x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n cot\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n cosec\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003c/mrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/msup>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\cot x) = -\\csc^2 x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n sec\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n sec\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n tan\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\sec x) = \\sec x \\tan x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n csc\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmi>\n cosec\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n cot\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\csc x) = -\\csc x \\cot x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003c/ul> \u003c/li> \n \u003cli>\u003cstrong>Exponential & Logarithmic Functions:\u003c/strong> \n \u003cul> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n e\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/msup>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (e^x) = e^x\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n a\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/msup>\n \u003cmi>\n ln\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (a^x) = a^x \\ln a\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003cspan class=\"katex-html\" aria-hidden=\"true\">\u003cspan class=\"base\">\u003cspan class=\"mord\">\u003cspan class=\"mfrac\">\u003cspan class=\"vlist-t vlist-t2\">\u003cspan class=\"vlist-r\">\u003cspan class=\"vlist-s\">\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/li> \n \u003cli>\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmsub>\n \u003cmrow>\n \u003cmi>\n log\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003c/mrow>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/msub>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmi>\n ln\n \u003c/mi>\n \u003cmo>\n ⁡\n \u003c/mo>\n \u003cmi>\n a\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} (\\log_a x) = \\frac{1}{x \\ln a}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003cspan class=\"katex-html\" aria-hidden=\"true\">\u003cspan class=\"base\">\u003cspan class=\"mord\">\u003cspan class=\"mfrac\">\u003cspan class=\"vlist-t vlist-t2\">\u003cspan class=\"vlist-r\">\u003cspan class=\"vlist-s\">\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/li> \n \u003c/ul> \u003c/li> \n \u003cli> \u003ch3>\u003cstrong>Rules of Differentiation\u003c/strong>\u003c/h3> \n \u003cul> \n \u003cli>\u003cstrong>Sum Rule:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmsup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n (f + g)' = f' + g'\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Difference Rule:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmsup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n (f - g)' = f' - g'\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Product Rule:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmsup>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmsup>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n (uv)' = u'v + uv'\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cbr>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Quotient Rule:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmsup>\n \u003cmrow>\n \u003cmo fence=\"true\">\n (\n \u003c/mo>\n \u003cmfrac>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmi>\n v\n \u003c/mi>\n \u003c/mfrac>\n \u003cmo fence=\"true\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmsup>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmi>\n u\n \u003c/mi>\n \u003cmsup>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003c/mrow>\n \u003cmsup>\n \u003cmi>\n v\n \u003c/mi>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/msup>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\left( \\frac{u}{v} \\right)' = \\frac{u'v - uv'}{v^2}\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003cspan class=\"katex-html\" aria-hidden=\"true\">\u003cspan class=\"base\">\u003cspan class=\"mord\">\u003cspan class=\"mfrac\">\u003cspan class=\"vlist-t vlist-t2\">\u003cspan class=\"vlist-r\">\u003cspan class=\"vlist-s\">\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/span>\u003c/li> \n \u003cli>\u003cstrong>Chain Rule:\u003c/strong> \u003cspan class=\"katex\">\u003cspan class=\"katex-mathml\">\n \u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\">\n \u003csemantics>\n \u003cmrow>\n \u003cmfrac>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmrow>\n \u003cmi>\n d\n \u003c/mi>\n \u003cmi>\n x\n \u003c/mi>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n f\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003cmo>\n ⋅\n \u003c/mo>\n \u003cmsup>\n \u003cmi>\n g\n \u003c/mi>\n \u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">\n ′\n \u003c/mo>\n \u003c/msup>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cannotation encoding=\"application/x-tex\">\n \\frac{d}{dx} f(g(x)) = f'(g(x)) \\cdot g'(x)\n \u003c/annotation>\n \u003c/semantics>\n \u003c/math>\u003c/span>\u003c/span>\u003c/li> \n \u003c/ul> \u003c/li> \n\u003c/ul>","order":4,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"209378","status":"live","updateDateOn":false,"updatedOn":"2025-07-28 14:46:56","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":866604,"h2Text":"Class 11 Maths Chapter 12 NCERT Solutions – Limits and Derivatives PDF Free","dataValue":"\u003cp>If students download the free \u003cstrong>Limits and Derivatives Class 11 PDF \u003c/strong>from the link given below, they will get the well-structured solutions of all the NCERT questions of chapter 12 Class 11 Maths. It will help them to do well in school exam, \u003ca href=\"https://www.shiksha.com/boards/cbse-board\">CBSE\u003c/a>, and the JEE Main exam.\u003c/p> \n\u003cp>\u003cspan style=\"color: #e03e2d; background-color: #fbeeb8;\">\u003cstrong>Chapter 5 Limits and Derivatives Class 11 NCERT Solutions:\u003cspan style=\"color: #000000;\"> \u003ca class=\"smce-docs smce-ftype-ico pdf smce-cta-link\" data-link=\"rd8uSHrTuyEzOSGklqFoV/HgfuSNdsbGyJZerPWBsCwsLucsIJ+84Fz+LylnFSnHYriwth6vFku/LnHyyyj0sAWSzhXw5wMhMTAh1ODlUIVdbat9cbpgrJ50O16StAeGITEzIXKT35ur\">Download PDF for Free\u003c/a> \u003c/span>\u003c/strong>\u003c/span>\u003c/p>","order":5,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"188836","status":"live","updateDateOn":false,"updatedOn":"2025-03-06 18:39:11","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":865226,"h2Text":"Class 11 Math Limits and Derivatives Exercise-wise Solution","dataValue":"\u003cp>\u003cstrong>Class 11th Math Limits and Derivatives\u003c/strong> helps students understand the fundamentals of calculus unit, including differentiation, application of derivatives, integral and more. There are various important topics in this chapter such as checking differentiability, continuity, derivatives of different functions, and more. Limits and Derivatives Class 11 Solutions will include solution of all the exercises in lucid language. \u003cstrong>Ex 13.1 Class 11\u003c/strong> will include problems related to the basic concepts of Limits and Derivatives such as using limit, finding continuity and differentiability. \u003cstrong>Class 11 Math Ex 13.2\u003c/strong> includes more concepts such as finding derivatives of various functions and using them as application. Miscellaneous exercise covers various topics covered throughout the chapter. Students can check all the solutions below;\u003c/p> \n\u003cp> \u003c/p> \n\u003cp> \u003c/p>","order":6,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"188824","status":"live","updateDateOn":false,"updatedOn":"2025-03-05 15:41:04","h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}},{"id":906274,"h2Text":"Class 11 Math Limits and Derivatives Exercise 13.1 Solution","dataValue":"\u003cp>\u003cstrong>Limits and Derivatives Exercise 13.1 focuses on fundamentals of the Limits. Student will be able to understand the application of limit to find first derivative of the given function for a given range. Ex 13.1 Class 11 Math in the Limits and Derivatives includes problems based on trigonometric functions, algebraic functions and exponential functions and finding their first derivative using the limits. Class 11 Math Ex 13.1 solution will contain solution of all 32 short and long answer type questions. Students can check the solutions below;\u003c/strong>\u003c/p> \n\u003ch3>Class 11 Math Limits and Derivatives Exercise 13.1 NCERT Solution\u003c/h3> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q1. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003c/mrow>\n \u003c/math>\u003c/span> x+ 3\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\">A.1.\u003cstrong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>x + 3 = 3 + 3 = 6.\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003cp> \u003c/p> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q2. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n lim\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/munder>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\">A.2.\u003cstrong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/munder>\n \u003cmrow>\n \u003cmo>\n (\n \u003c/mo>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003cmo>\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 7\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q3. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n lim\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\">A.3\u003cstrong>. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmsup>\n \u003cmrow>\n \u003cmi>\n r\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003cmo>\n ·\n \u003c/mo>\n \u003cmsup>\n \u003cmrow>\n \u003cmo stretchy=\"false\">\n (\n \u003c/mo>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmo stretchy=\"false\">\n )\n \u003c/mo>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/msup>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmi>\n π\n \u003c/mi>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table> \n\u003ctable style=\"border-collapse: collapse; width: 100%;\" border=\"1\"> \n \u003ctbody> \n \u003ctr> \n \u003ctd style=\"width: 100%; background-color: #ced4d9;\">\u003cstrong>Q4. \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n lim\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003ctr> \n \u003ctd style=\"width: 100%;\">A4.\u003cstrong> \u003cspan class=\"mathml\" title=\"Click to copy mathml\">\n \u003cmath>\n \u003cmrow>\n \u003cmunder>\n \u003cmrow>\n \u003cmi>\n l\n \u003c/mi>\n \u003cmi>\n i\n \u003c/mi>\n \u003cmi>\n m\n \u003c/mi>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n →\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003c/mrow>\n \u003c/munder>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmi>\n x\n \u003c/mi>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n ×\n \u003c/mo>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 4\n \u003c/mn>\n \u003cmo>\n −\n \u003c/mo>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 6\n \u003c/mn>\n \u003cmo>\n +\n \u003c/mo>\n \u003cmn>\n 3\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003cmo>\n =\n \u003c/mo>\n \u003cmfrac>\n \u003cmrow>\n \u003cmn>\n 1\n \u003c/mn>\n \u003cmn>\n 9\n \u003c/mn>\n \u003c/mrow>\n \u003cmrow>\n \u003cmn>\n 2\n \u003c/mn>\n \u003c/mrow>\n \u003c/mfrac>\n \u003c/mrow>\n \u003c/math>\u003c/span>\u003c/strong>\u003c/td> \n \u003c/tr> \n \u003c/tbody> \n\u003c/table>","order":7,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"content","wikiIdentifier":"190728","status":"live","updateDateOn":false,"updatedOn":"2025-05-26 11:18:14","widgetData":[{"id":904546,"dataValue":"ANA_FAQ","order":1,"subSectionId":"115","sectionName":"homepage","dataCategoryType":"widget","wikiIdentifier":"197266","status":"live","updateDateOn":false,"updatedOn":"2025-05-21 15:20:49"},{"dataValue":"EXAM_TIMELINE_WIDGET","order":2,"dataCategoryType":"widget","updateDateOn":false},{"dataValue":"COLLEGE_RECO_WIDGET_1","order":3,"dataCategoryType":"widget","updateDateOn":false},{"dataValue":"TopExam","dataCategoryType":"widget","updateDateOn":false}],"h2CategoryId":1,"h2CategoryText":"others","h2CategoryWidgetData":{"logoUrl":{"url":"/pwa/public/images/h2HeadingIcons/default.svg","urlHost":"https://images.shiksha.ws","absoluteUrl":null},"eventDates":null,"ctaInfo":null}}],"childPageDataConfig":{"homepage":{"categoryAllowed":{"intro":1,"content":30,"contactus":1},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null},"l4ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null}}},"tocList":[{"h2Text":"Class 11 Maths Chapter 12 Snapshot: Limits and Derivatives","h2Id":"content_toc_947470"},{"h2Text":"Class 11 Math Limits and Derivatives: Key Topics, Weightage ","h2Id":"content_toc_865224"},{"h2Text":"Important Formulas of Class 11 Limits and Derivatives","h2Id":"content_toc_947472"},{"h2Text":"Class 11 Maths Chapter 12 NCERT Solutions – Limits and Derivatives PDF Free","h2Id":"content_toc_866604"},{"h2Text":"Class 11 Math Limits and Derivatives Exercise-wise Solution","h2Id":"content_toc_865226"},{"h2Text":"Class 11 Math Limits and Derivatives Exercise 13.1 Solution","h2Id":"content_toc_906274"}],"pteTopExamWidget":{"widgetType":"TopExam","data":[{"label":"Engineering","viewMoreLink":"/engineering/exams-st-2","childPages":[{"label":"JEE Main","url":{"url":"/engineering/jee-main-exam","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/newLogo/113.jpg"},{"label":"BITSAT","url":{"url":"/engineering/bitsat-exam","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.com/mediadata/images/1684833088phpzOVgsB.png"},{"label":"COMEDK UGET","url":{"url":"/engineering/comedk-uget-exam","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.com/mediadata/images/1596168517phpr5wV8v.jpeg"},{"label":"SRMJEEE","url":{"url":"/engineering/srmjeee-exam","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.com/mediadata/images/1615444289phpFoCGx1.png"},{"label":"TNEA","url":{"url":"/b-tech/tnea-counselling","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/newLogo/110.jpg"},{"label":"VITEEE","url":{"url":"/engineering/viteee-exam","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/mobile5/images/CollegeShorlist/newLogo/170.jpeg"}]},{"label":"Boards","viewMoreLink":"/boards/exams-st-23","childPages":[{"label":"ISC","url":{"url":"/boards/isc-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"},{"label":"Maharashtra HSC","url":{"url":"/boards/maharashtra-hsc-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"},{"label":"CBSE 12th","url":{"url":"/boards/science-commerce-and-arts/cbse-12th-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"},{"label":"NIOS 12th","url":{"url":"/boards/nios-12th-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"},{"label":"UP 12th","url":{"url":"/boards/up-12th-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"},{"label":"MPBSE 12th","url":{"url":"/boards/mpbse-12th-board","urlHost":"relative","absoluteUrl":null},"img":"https://images.shiksha.ws/public/images/CollegeShorlist/default_logo.svg"}]}],"viewMoreCount":6},"updatedOn":"2025-07-28 14:46:56"}},"importantdates":{"wiki":null,"dates":null,"guideUrls":null,"sectionalData":{"sectionName":null,"entityType":null,"contentData":[],"childPageDataConfig":{"homepage":{"categoryAllowed":{"intro":1,"content":30,"contactus":1},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null},"l4ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null}}},"tocList":null,"pteTopExamWidget":null,"updatedOn":""}},"colleges":null,"registration":null,"mocktest":null,"books":null,"analysis":null,"preparation":null,"scorecard":null,"centre":null,"notification":null,"recruitment":null,"salary":null,"timetable":null,"compartment":null,"improvementexam":null,"revaluation":null,"marksheet":null,"selectionprocess":null,"eligibility":null,"hallticket":null,"form":null,"choicefilling":null,"sectionUrls":{"homepage":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns"},"comingDateInfo":null,"sectionNameMapping":{"mocktest":"Mock Test","class_12":"Class 12th","class_11":"Class 11th","class_10":"Class 10th","counselling":"Counselling","news":"News","rankpredictor":"Rank Predictor","homepage":"Overview","admitcard":"Admit Card","syllabus":"Syllabus","class_9":"Class 9th","timetable":"Time table","form":"Form","preptips":"Prep Tips","samplepapers":"Question Papers","analysis":"Analysis","hallticket":"Hall Ticket","colleges":"Colleges","cutoff":"Cut Off","salary":"Salary","books":"Books","callletter":"Call Letter","answerkey":"Answer Key","slotbooking":"Slot Booking","revaluation":"Revaluation","notification":"Notification","results":"Results","choicefilling":"Choice Filling","eligibility":"Eligibility","centre":"Centre","collegepredictor":"College Predictor","compartment":"Compartment","applicationform":"Application Form","scorecard":"Score Card","importantdates":"Dates","marksheet":"Marksheet","pattern":"Pattern","recruitment":"Recruitment","selectionprocess":"Selection Process","vacancies":"Vacancies","preparation":"Preparation","registration":"Registration","improvementexam":"Improvement Exam"},"updatedOn":"2025-07-28 14:46:56","defaultText":{"samplepapers":"Download previous year Maths Ncert Solutions class 11th question papers and sample papers. Use them for practice and improve your speed and accuracy. You can download Maths Ncert Solutions class 11th previous year question papers and the preparation guides for free.","importantdates":"Get all Maths Ncert Solutions class 11th 2023 important dates such as registration date, application date, admit card date, exam date, result date and counselling date. Sign up so that you don’t miss a single exam date or an update on Maths Ncert Solutions class 11th 2023 exam."},"dfpAdUnits":null,"ctpURL":{"url":"/search?q=Maths Ncert Solutions class 11th&s[]=25&ed[]=et_20::dm_39&ex[]=14805&pn=1&rf=examPage&apk=true&skr=false&rURL=false&pt=search_page","urlHost":"national","absoluteUrl":"https://www.shiksha.com/search?q=Maths Ncert Solutions class 11th&s[]=25&ed[]=et_20::dm_39&ex[]=14805&pn=1&rf=examPage&apk=true&skr=false&rURL=false&pt=search_page"},"ctpCount":0,"faq":{"@context":"https://schema.org","@type":"FAQPage","mainEntity":[{"@type":"Question","name":"40. Find the derivative of xn−anx−a for some constant a.","acceptedAnswer":{"@type":"Answer","text":"40. Given, f(x)= xn−anx−a. So, f′(x)=(x−a)ddx(xn−a3)−ddx(x−a)⋅(xn−an)(x−a)2=(x−a)⋅nxn−1−(xn−an)(x−a)2 =(x−a)⋅nxn−1−(xn−an)(x−a)2 =nxn−1⋅x−naxn−1−xn+an(x−a)2 =nxn−xx−naxn−1+an(x−a)2"}},{"@type":"Question","name":"42. Find the derivative of the following functions: (i)sin x cos x (ii) secx (iii)5 sec x + 4 cosx (iv) cosecx (v)3cot x + 5 cosec x (vi) 5sinx−6cosx+7 (vii) 2tanx−7secx","acceptedAnswer":{"@type":"Answer","text":"(i) f(x)=sin x cos x So, f?(x)=limh?0f(x+h)?f(x)h =limh?0sin(x+h)cos(x+h)?sinxcosxh =limh?012h×[2sin(x+h)cos(x+h)?2sinxcosx] =limh?012h[sin2(x+h)?sin2x] =limh?012h[2cos2(x+h)+2x2sin2(x+h)?2x2] =limh?01h[cos(2x+h)sinh] =limh?0cos(2x+h)×limh?0sinhh =cos(2x+0) =cos2x (ii) f(x)=secx So, f?(x)=limh?0f(x+h)?f(x)h =limh?01h[sec(x+h)?secx] =limh?01h[1cos(x+h)?1cosx] =limh?01h[cosx?cos(x+h)cos(x+h)cosx] =limh?01h[?2sin(x+x+h2)sin(x?(x+h)2)cos(x+h)cosx] =limh?01h[?2sin(2x+h2)sin(?h/2)cos(x+h)cosx] =limh?0(?1sin(2x+h2)cos(x+h)cosx×limh?0(?1)sinh/2h/2 =sinxcosx?cosx×1 =tanx?secx. (iii) Given f(x)=5 sec x+4 cosx. So, f?(x)=limh?0f(x+h)?f(x)h =limh?05sec(x+h)+4cos(x+h)?[5secx+4cosx]h. =limh?05h[sec(x+h)?secx]+limh?04h[cos(x+h)?cosx] =limh?05h[1cos(x+h)?1cosx]+limh?04h[?2sin(x+h+x2)sin(x+h?x2)] =limh?05h[cosx?cos(x+h)cos(x+h)(cosx)]+limh?04h[?2sin(2x+h2)sinh2] =limh?05h[?2sin(2x+h2)sin(?h/2)cos(x+h)cosx]?4limh?0sin(2x+h2)limh?0sinh/2h/2 =sin(2x+02)cos(x+0)cosx×1?4sin(2x2) =5sinxcosx?1cosx?4sinx =5tanx?secx?4sinx (iv) Given f(x)=cosecx f?(x)=limh?0f(x+h)?f(x)h =limh?01h[cosec(x+h)?cosecx] =limh?01h[1sin(x+h)?1sinx] =limh?01h[sinx?sin(x+h)sin(x+h)sinx] =limh?01h[2cos(x+x+h2)sin(x?(x+h)2)]sin(x+h)sinx] =limh?01h[2cos(x+x+h2)sin(x?(x+h)2)sin(x+h)sinx] =limh?01h[2cos(2x+h2)sin(?h/2)]sin(x+h)sinx] =limh?0cos(2x+h2)sin(x+h)sinx×(?1)sin(2)h/2) =cos(2x+02)sin(x+0)sinx×(?1) =?cosxsinx×1sinx =?cotx?cosecx (v) Given,f(x)=3 cot x+5cosecx. So, f?(x)=limh?0f(x+h)?f(x)h =2cos(x+0)cosx×1+7sin(2x+02)×(?1) =limh?03cot(x+h)+5cosec(x+h)?[3cotx+5cosx) h?03h[cot(x+h)?cotx]+limh?0 =?5h[cosec(x+h)?cosecx] =limh?03h[cos(x+h)sin(x+h)?cosxsinx]+limh?05h[1sin(x+h)?1sinx] =limh?03h[cos(x+h)sinx?cosxsin(x+h)sin(x+h)sinx]+limh?05h[sinx?sin(x+h)sin(x+h)sinx] =limh?03h[sin(x?(x+h))sin(x+h)sinx+limh?05h[2cos(x+x+h2)sin(x?(x+h)2)sin(x+h)sinx =limh?03sin(x+h)sinx×limh?0(?1)sinhh+limh?05h[2cos(2x+h2)sin(?h/2)sin(x+h)sinx =3sinx?sinx×(?1)+5limh\u003c/"}},{"@type":"Question","name":"52. Kindly consider the following px2+qx+rax+b","acceptedAnswer":{"@type":"Answer","text":"52. Given, f (x) = px2+qx+rax+b f′(x)=(ax+b)ddx(px2+qx+r)−(px2+qx+r)ddx(ax+b)(ax+b)2 =(ax+b)(2xp+q)−(px2+qx+r)(a)(ax+b)2 =2apx2+2bpx+aqx+bq−apx2−aqx−ar(ax+b)2 =apx2+2bpx+bq−ar(ax+b)2"}},{"@type":"Question","name":"59. Kindly consider the following cosx1+sinx","acceptedAnswer":{"@type":"Answer","text":"59. Given, f (x) = cosx1+sinx So, f?(x) = (1+sinx)ddx(cosx)−cosxddx(1+sinx)(1+sinx)2 Putting (2) and (3) in (1) we get, f′(x)=(1+sinx)(−sinx)−cosx(cosx)(1+sinx)2 =sinx−sin2x−cos2x(1+sinx)2 =−sinx−(sin2x+cos2x)(1+sinx)2 =−(sinx+1)(1+sinx)2=−11+sinx."}},{"@type":"Question","name":"66. Kindly consider the following (x2 + 1) cos x","acceptedAnswer":{"@type":"Answer","text":"66. Given, f (x) = (x2 + 1) cos x f? (x) = (x2 + 1) ddxcosx+cosxddx (x2+1) = (x2+1) (−sinx)+cosx (2x+0) [? ddxcosx=−sinx] = x2 sin x sin x + 2x cos x."}}]},"prepFaq":null,"sectionsInfo":{"homepage":{"h1":"Limits and Derivatives Class 11 NCERT Solutions | Exercise-wise Answers","viewCount":958,"h2":"Maths Ncert Solutions class 11th 2023 Dates","h2Description":"Take a look at the table below for the complete schedule of Maths Ncert Solutions class 11th 2023 exam","guideUrl":"/mediadata/examGuides/examSectionGuide4419_Overview_2025-09-13-03-37.pdf","firstImageUrl":"","urlMapping":{"l2Url":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","l3Urls":null,"l3ToL4Urls":null}}},"officialWebsiteWithUTM":null,"firstImageInContent":null,"examPhase":"default_phase","prePostLabel":"Both","examMetaDataDTO":{"phoneNo":null,"officialWebsite":null,"isERWidget":false},"ctaInfoData":{"useEditorialCTA":true,"ctaInfoList":[],"examPhase":null},"class_9":null,"class_10":null,"class_11":null,"class_12":null,"questionPapersAvailable":false,"futureDateAvailable":false,"pattern":null,"syllabus":null,"results":null,"cutoff":null,"admitcard":null,"answerkey":null,"applicationform":null,"counselling":null,"samplepapers":{"wiki":null,"files":null,"guideUrls":null,"sectionalData":{"sectionName":null,"entityType":null,"contentData":[],"childPageDataConfig":{"homepage":{"categoryAllowed":{"intro":1,"content":30,"contactus":1},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null},"l4ChildPageConfig":{"categoryAllowed":{"intro":1,"content":30},"maxWikisAllowed":32,"entityType":"Summary","childPagesAssociated":["homepage","importantdates","samplepapers"],"defaultOrder":1,"l3ChildPageConfig":null,"l4ChildPageConfig":null}}},"tocList":null,"pteTopExamWidget":null,"updatedOn":""}},"slotbooking":null,"vacancies":null,"news":null,"callletter":null,"preptips":null,"sectionname":["homepage"],"l3Mapping":{},"l4Mapping":{}},"statusCode":200,"urlHash":"eyJ1cmwiOiIvcHJlcGFyYXRpb24vbWF0aHMtbmNlcnQtc29sdXRpb25zLWNsYXNzLTExdGgtbGltaXRzLWFuZC1kZXJpdmF0aXZlcy00NDE5LW5zIiwidXNlT2xkRm9ybWF0IjpmYWxzZSwic2hvd1NpbWlsYXJFeGFtcyI6ZmFsc2V9","url":"/preparation/maths-ncert-solutions-class-11th-ns","activeSection":"homepage","childPageHelper":{"id":null,"childPageName":"homepage","childPageType":"L2","l2Name":null,"l3Name":null,"uniqueChildPageName":"homepage","displayChildPageName":null,"displayParentPageName":null,"childPageUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns"},"breadCrumb":[{"name":"Home","url":"/","linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":true},{"name":"Preparation","url":null,"linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":false},{"name":"Maths Ncert Solutions class 11th","url":null,"linkText":null,"entityId":null,"entityType":null,"pwa":true,"isAbsoluteUrl":false}],"articleData":null,"featuredColleges":null,"similarExams":{"examTuple":null,"groupLevel":null,"pageNum":null,"totalPages":0},"examPhaseData":null,"epUpdates":null,"odnDetailsListResponseDTO":{"odnDetailsListResponse":{"odnRedAlert":null,"odnAlertList":null},"haveMoreTuples":false},"seoData":{"metaTitle":"Limits and Derivatives Class 11 NCERT Solutions | Exercise-wise Answers","metaDescription":"Explore complete NCERT Solutions for Class 11 Limits and Derivatives. Easy explanations, solved examples, and concept clarity.","h1Tags":"Limits and Derivatives Class 11 NCERT Solutions | Exercise-wise Answers","metaKeywords":"null","canonicalUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","seoUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","ampUrl":null,"noIndex":false,"noFollow":false},"applyOnline":null,"upcomingEvents":null,"acceptingWidget":{"requestData":null,"filters":null,"aggregateRatingConfig":null,"categoryInstituteTuple":[],"totalInstituteCount":0,"nameMapping":null,"aliasMapping":null,"defaultAppliedFilters":null,"filterOrder":null,"selectedFiltersCount":0,"paginationData":null,"categoryUrlHash":null,"loadMoreCourses":null,"examsUrl":null,"spellcheckDone":false,"keywordRelaxed":false,"pageUrl":null},"anaWidget":{"totalNumber":0,"listingType":null,"questionsDetail":null,"allQuestionURL":null,"questionOrder":null,"pageSize":10,"maxPagetoDisplay":1,"anaFaqData":null,"prepFaqData":{"fetchTypeToFaqData":{}},"anaWidgetPresent":true},"userInfo":{"userId":32401605,"username":"Pallavi112986","firstName":"Pallavi","lastName":"Pathak","designation":"Assistant Manager Content","imageUrl":"/mediadata/images/1641388025phprr2Sx1.jpeg","isAbroad":false},"videos":null,"examPageGuide":{"groupId":4419,"guideUrl":"/mediadata/examGuides/examGuide4419_2025-09-12-12-00.pdf","guideYear":2023},"groupsInfo":{"4387":{"url":"/preparation/maths-ncert-solutions-class-11th-ns","id":4387,"name":"NCERT Maths 11th"},"4395":{"url":"/preparation/maths-ncert-solutions-class-11th-sets-4395-ns","id":4395,"name":"Sets"},"4397":{"url":"/preparation/maths-ncert-solutions-class-11th-relations-and-functions-4397-ns","id":4397,"name":"Relations and Functions"},"4399":{"url":"/preparation/maths-ncert-solutions-class-11th-trigonometric-functions-4399-ns","id":4399,"name":"Trigonometric Functions"},"4401":{"url":"/preparation/maths-ncert-solutions-class-11th-principle-of-mathematical-induction-4401-ns","id":4401,"name":"Principle of Mathematical Induction"},"4403":{"url":"/preparation/maths-ncert-solutions-class-11th-complex-numbers-and-quadratic-equations-4403-ns","id":4403,"name":"Complex Numbers and Quadratic Equations"},"4405":{"url":"/preparation/maths-ncert-solutions-class-11th-linear-inequalities-4405-ns","id":4405,"name":"Linear Inequalities"},"4407":{"url":"/preparation/maths-ncert-solutions-class-11th-permutations-and-combinations-4407-ns","id":4407,"name":"Permutations and Combinations"},"4409":{"url":"/preparation/maths-ncert-solutions-class-11th-binomial-theorem-4409-ns","id":4409,"name":"Binomial Theorem"},"4411":{"url":"/preparation/maths-ncert-solutions-class-11th-sequences-and-series-4411-ns","id":4411,"name":"Sequences and Series"},"4413":{"url":"/preparation/maths-ncert-solutions-class-11th-straight-lines-4413-ns","id":4413,"name":"Straight Lines"},"4415":{"url":"/preparation/maths-ncert-solutions-class-11th-conic-sections-4415-ns","id":4415,"name":"Conic Sections"},"4417":{"url":"/preparation/maths-ncert-solutions-class-11th-introduction-to-three-dimensional-geometry-4417-ns","id":4417,"name":"Introduction to Three Dimensional Geometry"},"4419":{"url":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","id":4419,"name":"Limits and Derivatives"},"4421":{"url":"/preparation/maths-ncert-solutions-class-11th-mathematical-reasoning-4421-ns","id":4421,"name":"Mathematical Reasoning"},"4423":{"url":"/preparation/maths-ncert-solutions-class-11th-statistics-4423-ns","id":4423,"name":"Statistics"},"4425":{"url":"/preparation/maths-ncert-solutions-class-11th-probability-4425-ns","id":4425,"name":"Probability"}},"ctpLinksWidgetData":{"pageLinkDatas":[{"url":"/engineering/colleges/b-tech-colleges-india","urlText":"4068 Engineering Colleges in India"},{"url":"/bba/colleges/bba-colleges-india?rf=searchWidget&landing=ctp","urlText":"4495 BBA Colleges in India"},{"url":"/it-software/ai-robotics/colleges/colleges-india","urlText":"135 AI & Robotics colleges in India"},{"url":"/aviation/flying-pilot-training/colleges/colleges-india","urlText":"110 Flying / Pilot Training colleges in India"},{"url":"/accounting-commerce/colleges/b-com-colleges-india?rf=searchWidget&landing=ctp","urlText":"9804 B.Com Colleges in India"},{"url":"/accounting-commerce/colleges/ca-colleges-india","urlText":"35 CA Colleges in India"},{"url":"/design/colleges/colleges-india","urlText":"1673 Design colleges in India"},{"url":"/bba/colleges/bba-colleges-delhi-ncr","urlText":"445 BBA Colleges in Delhi/NCR"},{"url":"/teaching-education/colleges/b-ed-colleges-bihar","urlText":"196 B.Ed Colleges in Bihar"},{"url":"/teaching-education/colleges/b-ed-colleges-delhi-ncr","urlText":"382 B.Ed Colleges in Delhi/NCR"}]},"examNews":{"popular":[{"id":28413,"url":"/engineering/articles/jee-main-marks-vs-percentile-blogId-28413","title":"JEE Main Marks vs Percentile vs Rank 2025: Calculate Percentile and Rank Using Marks","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1704965393phpoShNWL_b.jpeg","summary":"Check your JEE Mains 2025 marks vs percentile vs Ranks through the comparative analysis done by shiksha.com for the students who appeared in JEE Main. Candidates can check how much percentile and rank they will get based on their JEE Main 2025 marks. Click here to know all the details on JEE Main Rank Vs Marks Vs Percentile.","lastModifiedDate":"2025-10-04 23:01:47","views":5351176,"userId":29781649,"comments":0,"shares":944,"discussionTopicId":5852191,"tagType":"exam","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":20003,"url":"/engineering/articles/nits-in-india-blogId-20003","title":"List of NIT Colleges in India 2025 - NIRF Ranking, Courses, Seats, Cutoff, Fees and Placement","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1579416701phpg320u6_b.jpeg","summary":"National Institute of Technology (NITs) are premier engineering colleges in India offering admission to degree courses at both undergraduate and postgraduate level. Check here the list of top NITs with their MHRD-NIRF rankings, courses offered, total seat intake and cut offs.","lastModifiedDate":"2025-09-04 14:50:43","views":3592259,"userId":29781649,"comments":205,"shares":313,"discussionTopicId":5232713,"tagType":"exam","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":117273,"url":"/boards/articles/cbse-10th-maths-question-paper-pdf-download-blogId-117273","title":"CBSE 10th Maths Question Paper 2025, 2024, 2023, 2022, 2020, 2019 PDF Download","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1709092170phpNR1yFa_b.jpeg","summary":"CBSE class 10 Maths question paper PDF 2025 is available here. Students can also download CBSE 10th Maths previous year question paper PDF to score well in class 10th board exam. Download CBSE 10th mathematics question paper 2025 along with PYQ of 2024, 2023, 2022, 2020 and 2019 here.","lastModifiedDate":"2025-09-13 14:22:03","views":3420312,"userId":65362629,"comments":0,"shares":4950,"discussionTopicId":7294495,"tagType":"multi","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":28509,"url":"/medicine-health-sciences/articles/neet-chapter-wise-weightage-blogId-28509","title":"NEET Chapter Wise Weightage 2026 PDF by NTA: Physics, Chemistry & Biology","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1743767698phpKlzK7b_b.jpeg","summary":"NEET Chapter Wise Weightage 2026 - Knowing the weightage of chapters in NEET has helped candidates excel in the examination. Check NEET chapter-wise weightage PDF for Physics, Chemistry, and Biology here. Knowing the chapter-wise weightage will help you prepare a rock solid preparation strategy for the exam day.","lastModifiedDate":"2025-10-08 18:32:53","views":3109315,"userId":18325315,"comments":115,"shares":4607,"discussionTopicId":5853393,"tagType":"exam","liveBlogStartDate":null,"liveBlogEndDate":null},{"id":12884,"url":"/engineering/articles/iits-in-india-blogId-12884","title":"List of Top IITs in India 2025: NIRF Ranking, Courses, Seats, Fees & Placement","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1568101678phpqodvPW_b.jpeg","summary":"Check out the ranking, seats and cut offs of all 23 IITs including IIT Kharagpur, IIT Bombay, IIT Madras, IIT Kanpur, IIT Delhi, IIT Guwahati, IIT Roorkee, IIT Ropar, IIT Bhubaneswar, IIT Gandhinagar, IIT Hyderabad etc. NIRF ranking 2025 for engineering has been released. Eight of the 23 IITs feature in the Top 10 of the NIRF engineering ranking.","lastModifiedDate":"2025-09-04 14:57:03","views":2559113,"userId":29781649,"comments":41,"shares":432,"discussionTopicId":3725168,"tagType":"multi","liveBlogStartDate":null,"liveBlogEndDate":null}],"latest":[{"id":211866,"url":"/news/engineering-gate-2026-last-date-to-apply-with-late-fee-live-link-steps-fees-status-and-exam-registration-details-blogId-211866","title":"GATE 2026 Last Date to Apply TODAY with Late Fee Live: Link, Steps, Fees, Status, and Exam Registration Details","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760268960phpUIys5L_b.jpeg","summary":"IIT Guwahati will close the GATE 2026 exam registration TODAY (October 13) with late fee. Students who haven’t registered yet must visit the website soon and apply. Registered students can check their GATE 2026 application status at GOAPS 2026. Read below to know about GATE 2026 last date of registration.","lastModifiedDate":"2025-10-13 01:37:00","views":0,"userId":2582250,"comments":0,"shares":0,"discussionTopicId":12417552,"tagType":"exam","liveBlogStartDate":"2025-10-13 00:17:00","liveBlogEndDate":"2025-10-14 20:17:00"},{"id":211856,"url":"/news/sarkari-exams-ibps-po-2025-mains-exam-analysis-live-updates-blogId-211856","title":"IBPS PO Mains Exam Analysis 2025 OUT LIVE Updates: Check Difficulty Level, Good Attempts & Review","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760214727phpkSFqBd_b.jpeg","summary":"The Institute of Banking Personnel Selection conducted the IBPS PO 2025 Mains exam on October 12, 2025. Candidates can check here the latest live updates about the IBPS PO 2025 Mains exam analysis including questions and answers, difficulty level & more. Check key guidelines and essential documents to carry.","lastModifiedDate":"2025-10-13 01:24:00","views":0,"userId":45077693,"comments":0,"shares":2,"discussionTopicId":12417294,"tagType":"exam","liveBlogStartDate":"2025-10-12 02:15:00","liveBlogEndDate":"2025-10-14 18:15:00"},{"id":211828,"url":"/news/engineering-jee-main-2026-registration-live-nta-form-date-syllabus-pattern-eligibility-jeemain-nta-nic-in-latest-news-blogId-211828","title":"JEE Main 2026 Registration Live: NTA Form, Date, Syllabus, Pattern, Eligibility, jeemain.nta.nic.in Latest News","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760097142phpB7sfYO_b.jpeg","summary":"NTA announces that the JEE Mains 2026 registration will start in October 2025. The JEE Main exam 2026 will be conducted in the usual bi-annual format- in January and April. Check the latest updates on JEE Main exam notification 2026, registration form, syllabus, pattern and more here.","lastModifiedDate":"2025-10-13 00:48:00","views":0,"userId":96365866,"comments":0,"shares":3,"discussionTopicId":12416734,"tagType":"exam","liveBlogStartDate":"2025-10-10 17:35:00","liveBlogEndDate":"2025-10-13 18:27:00"},{"id":211816,"url":"/news/boards-sslc-plus-two-time-table-2026-kerala-board-soon-live-updates-on-kerala-sslc-plus-two-exam-time-table-blogId-211816","title":"SSLC, Plus Two Time Table 2026 Kerala Board Soon; Live Updates on Kerala SSLC, Plus Two Exam Time Table","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1760091769php1bR6Vx_b.jpeg","summary":"Kerala SSLC, Plus Two time table 2026 is expected to be released any time soon. Check live updates on SSLC, Plus Two Time Table 2026 Kerala here. Students can download the SSLC Time table 2026 Kerala and DHSE Kerala time table 2026 from official website.","lastModifiedDate":"2025-10-13 00:38:00","views":0,"userId":48889573,"comments":0,"shares":0,"discussionTopicId":12416680,"tagType":"multi","liveBlogStartDate":"2025-10-10 16:01:00","liveBlogEndDate":"2025-10-17 18:59:00"},{"id":211570,"url":"/news/boards-up-board-exam-date-sheet-2026-class-10-12-live-updates-check-upmsp-high-school-inter-time-table-blogId-211570","title":"UP Board Exam Date Sheet 2026 Class 10, 12 (Soon) Live Updates; Check UPMSP High School, Inter Time Table","subTitle":null,"homepageImageUrl":"/mediadata/images/articles/1759830021phpM43SBz_b.jpeg","summary":"UPMSP is likely to announce the official UP Board 10th, 12th Time Table 2026 anytime soon. The board is likely to hold the final UPMSP Board Exams 2026 Class 10, 12 in February and March. Read on for the latest Updates on UP Board 10th, 12th Exam Date 2026 Time Table PDF.","lastModifiedDate":"2025-10-13 00:30:00","views":0,"userId":35358691,"comments":1,"shares":2,"discussionTopicId":12413844,"tagType":"multi","liveBlogStartDate":"2025-10-07 14:47:00","liveBlogEndDate":"2025-10-13 06:30:00"}]},"allExamPageGuide":null,"pageSchema":{"@context":"https://schema.org","@type":"Article","inLanguage":"en","headline":"Limits and Derivatives Class 11 NCERT Solutions | Exercise-wise Answers","description":"Explore complete NCERT Solutions for Class 11 Limits and Derivatives. Easy explanations, solved examples, and concept clarity.","keywords":"null","dateModified":"2025-07-28T14:46:56+05:30","datePublished":"2023-05-02T21:00:39+05:30","mainEntityOfPage":{"id":"https://www.shiksha.com/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","@type":"WebPage","name":"Maths Ncert Solutions class 11th"},"author":{"@type":"Person","name":"Pallavi Pathak","url":"https://www.shiksha.com/author/Pallavi112986"},"publisher":{"@type":"Organization","name":"Shiksha.com","logo":{"@type":"ImageObject","name":"Shiksha.com","url":"https://images.shiksha.ws/pwa/public/images/schema/shiksha_logo.jpg","height":"60","width":"600"}},"image":{"@type":"ImageObject","url":"https://images.shiksha.ws/pwa/public/images/schema/default_exam.jpg"}},"websiteSchema":{"@context":"https://schema.org","@type":"WebSite","name":"Shiksha.com","url":"https://www.shiksha.com"},"countryWidgetData":null,"admissionHelpWidgetTuples":null,"commentAndReplyCount":null,"commentCount":null,"comments":null,"userDetailsForComments":null,"abroadGtmParams":null,"abroadDFPparams":null,"abroadMISparams":null,"abroadExamDownloadCount":0,"abroadDownloadableRecommendedArticles":null,"abroadRelatedArticles":null,"abroadExamOldId":0,"lifeCycleTagId":0,"seoUrl":"/preparation/maths-ncert-solutions-class-11th-limits-and-derivatives-4419-ns","redirectUrl":null,"popularCoursesAfter12th":{"sections":{"arts":{"allCoursePageUrl":"/courses-after-12th/arts","courseMap":{"ba-humanities":{"label":"BA in Humanities & Social Sciences","childPages":{"SUAT":"/engineering/suat-exam/ba-hons-1016","JNUEE":"/university/jawaharlal-nehru-university/jnuee-exam","BHU UET":"/university/bhu-banaras-hindu-university/bhu-uet-exam/ba-hons-shastri-hons-413"},"chpurl":"/b-a-bachelor-of-arts-chp"},"bfa":{"label":"Bachelor of Fine Arts (BFA)","childPages":{"BHU UET":"/university/bhu-banaras-hindu-university/bhu-uet-exam/bpa-bfa-417","KUK Entrance Exam":"/university/kurukshetra-university/kuk-entrance-exam/bfa-780","JMI Entrance Exam":"/university/jamia-millia-islamia/jmi-entrance-exam/bfa-1159"},"chpurl":"/bfa-bachelor-of-fine-arts-chp"},"bdes":{"label":"Bachelor of Design in Animation (BDes)","childPages":{"UCEED":"/design/uceed-exam","NIFT Entrance Exam":"/design/nift-entrance-exam","NID Entrance Exam":"/design/communication-design/nid-entrance-exam"},"chpurl":"/bds-bachelor-of-dental-surgery-chp"},"ba-llb":{"label":"BA LLB (Bachelor of Arts + Bachelor of Laws)","childPages":{"CLAT":"/law/clat-exam","AILET":"/law/ailet-exam","LSAT India":"/law/lsat-india-exam"},"chpurl":"/bba-ll-b-chp"},"bjmc":{"label":"Bachelor of Journalism & Mass Communication (BJMC)","childPages":{"LUACMAT":"/bba/luacmat-exam/bjmc-776","SRMHCAT":"/university/srm-university-delhi-ncr-sonepat-haryana-srmuh/srmhcat-exam","GD Goenka Test":"/exams/gd-goenka-test-exam/bjmc-615"},"chpurl":"/b-j-m-c-bachelor-of-journalism-mass-communication-chp"}}},"commerce":{"allCoursePageUrl":"/courses-after-12th/commerce","courseMap":{"ca":{"label":"Chartered Accountancy (CA)","childPages":{"CA Foundation":"/accounting-commerce/accounting/ca-foundation-exam"},"chpurl":"/accounting-commerce/chartered-accountancy-chp"},"cs":{"label":"Company Secretary (CS)","childPages":{"ICSI Exam":"/college/icsi-delhi-institute-of-company-secretaries-of-india/icsi-exam"},"chpurl":"/accounting-commerce/company-secretary-chp"},"bcom":{"label":"BCom in Accounting & Commerce","childPages":{"BHU UET":"/university/bhu-banaras-hindu-university/bhu-uet-exam/bcom-hons-414","GLAET":"/university/gla-university/glaet-exam/bcom-936","GD Goenka Test":"/exams/gd-goenka-test-exam/bcom-598"},"chpurl":"/b-com-chp"},"bba-llb":{"label":"Bachelor of Business Administration & Bachelor of Law","childPages":{"CLAT":"/law/clat-exam","LSAT India":"/law/lsat-india-exam","AIBE":"/law/aibe-exam"},"chpurl":"/bba-ll-b-chp"},"bba-bms":{"label":"BBA/BMS","childPages":{"IPMAT":"/bba/ipmat-exam","NMIMS - NPAT":"/bba/nmims-npat-exam","SET":"/bba/set-exam"},"chpurl":"/bba-bachelor-of-business-administration-chp"}}},"science":{"allCoursePageUrl":"/courses-after-12th/science","courseMap":{"btech":{"label":"Bachelor of Technology (BE/BTech)","childPages":{"JEE Main":"/engineering/jee-main-exam","BITSAT":"/engineering/bitsat-exam","WBJEE":"/engineering/wbjee-exam"},"chpurl":"/b-tech-bachelor-of-technology-chp"},"barch":{"label":"Bachelor of Architecture (BArch)","childPages":{"NATA":"/architecture-planning/nata-exam","AAT":"/college/iit-kanpur-indian-institute-of-technology/aat-exam","ITM NEST":"/university/itm-university-gwalior/itm-nest-exam/barch-1058"},"chpurl":"/b-arch-bachelor-of-architecture-chp"},"bsc":{"label":"Bachelor of Science (BSc)","childPages":{"CUET":"/science/cuet-exam","UPCATET":"/science/upcatet-exam","IIT JAM":"/science/iit-jam-exam"},"chpurl":"/b-sc-chp"},"bca":{"label":"Bachelor of Computer Applications (BCA)","childPages":{"BUMAT":"/university/bharati-vidyapeeth/bumat-exam/bca-490","GSAT":"/science/gsat-exam/bca-827","SET":"/bba/set-exam/bca-1123"},"chpurl":"/bca-bachelor-of-computer-applications-chp"},"mbbs":{"label":"Bachelor of Medicine & Bachelor of Surgery (MBBS)","childPages":{"NEET":"/medicine-health-sciences/neet-exam","AYUSH":"/medicine-health-sciences/ayush-counselling","FMGE":"/medicine-health-sciences/fmge-exam"},"chpurl":"/mbbs-chp"}}}},"order":["science","commerce","arts"]},"beaconDataTracking":{"pageEntityId":14805,"extraData":{"countryId":[2],"stateId":[1],"groupId":[4419],"baseCourseId":[259],"deliveryMethod":[7],"hierarchy":[{"streamId":25,"substreamId":0,"specializationId":0,"primaryHierarchy":1}],"examIds":[14805]}},"gtmTracking":{"stream":25,"substream":0,"specialization":0,"baseCourseId":[259],"educationType":0,"deliveryMethod":7,"credential":0,"countryId":2,"stateId":1,"cityId":0},"faqAnaData":{"fetchTypeToFaqData":{"190728":[{"anaFaqId":39870268,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346548,"answerId":11346550,"questionText":"\u003cp>\u003cstrong>40. Find the derivative of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> for some constant a.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>40. \u003c/strong>Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>⋅\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>⋅\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>⋅\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>⋅\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":0,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-40-find-the-derivative-of-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-qna-11346548"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":45630154,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11586144,"answerId":11586160,"questionText":"\u003cp>\u003cstrong>42. Find the derivative of the following functions:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i)sin x cos x \u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> (ii)\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> (iii)5 sec x + 4 cosx\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>(iv) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> (v)3cot x + 5 cosec x\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(vi) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> (vii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>(i) f(x)=sin x cos x\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>(ii) f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(iii) Given f(x)=5 sec x+4 cosx.\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>?\u003c/mo>\u003cmn>4\u003c/mn>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>?\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>(iv) Given f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(v) Given,f(x)=3 cot x+5cosecx.\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmtext>?\u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/","order":1,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-42-find-the-derivative-of-the-following-functions-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-qna-11586144"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874278,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352960,"answerId":11352968,"questionText":"\u003cp>\u003cstrong>52.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>52. \u003c/strong>Given, f (x) = \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>p\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>q\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>p\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>q\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":2,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-52-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11352960"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874292,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353218,"answerId":11353220,"questionText":"\u003cp>\u003cstrong>59. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>59. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f?(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1731317257phprNOIm3_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731317257phprNOIm3.jpeg\" alt=\"\" width=\"370\" height=\"66\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>Putting (2) and (3) in (1) we get,\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003c/div>\u003c/div>","order":3,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-59-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11353218"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874306,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353296,"answerId":11353298,"questionText":"\u003cp>\u003cstrong>66. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 1) cos x\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>66. \u003c/strong>Given, f (x) = (\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 1) cos x\u003c/p>\u003cp>f? (x) = (\u003cem>x\u003c/em>\u003csup>2\u003c/sup> + 1) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmo>? \u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cem>x\u003c/em>\u003csup>2\u003c/sup> sin x sin x + 2x cos x.\u003c/p>","order":4,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-66-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353296"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":false,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874310,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356050,"answerId":11356056,"questionText":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>68. \u003c/strong>\u003cstrong>(x + cos x) (x tan x)\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>68. \u003c/strong>Given, f (x) = (x + cos x) (x tan x)\u003c/p>\u003cp>So, f?(x) = (x + cos x) \u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>Let g (x) = tan x.\u003c/p>\u003cp>So, g?(x) = \u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742469722phpjKVS9d.jpeg\" alt=\"\" width=\"189\" height=\"57\">\u003c/p>\u003cp> \u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003cstrong>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>= sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em> ______ (2)\u003c/p>\u003cp> Put (2) in (1) we get,\u003c/p>\u003cp>\u003cem> f\u003c/em>?(\u003cem>x\u003c/em>) = (\u003cem>x\u003c/em> + cos \u003cem>x\u003c/em>) (1 - sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>) +(1 - sin \u003cem>x\u003c/em>) (\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>)\u003c/p>\u003cp> We know that,\u003c/p>\u003cp> 1 + tan\u003csup>2\u003c/sup>\u003cem>x\u003c/em> = sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>\u003c/p>\u003cp> Þ 1 - sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em> = - tan\u003csup>2\u003c/sup>\u003cem>x\u003c/em>\u003c/p>\u003cp> So, \u003cem>f\u003c/em>?(\u003cem>x\u003c/em>) = - tan\u003csup>2\u003c/sup>\u003cem>x\u003c/em>(\u003cem>x\u003c/em> + cos \u003cem>x\u003c/em>) +(\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>) (1 - sin \u003cem>x\u003c/em>).\u003c/p>","order":5,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-68-lt-strong-gt-lt-qna-11356050"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":45630150,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11586012,"answerId":11586018,"questionText":"\u003cp>\u003cstrong>17. Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462933phpaZMuEe.jpeg\" alt=\"\" width=\"200\" height=\"62\">\u003c/picture>\u003c/p>","answerText":"\u003cp>\u003cstrong>\u003cpicture>17. Kindly go through the solution\u003c/picture>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462989phpO7uSFs.jpeg\" alt=\"\" width=\"518\" height=\"90\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>=\u003c/math>\u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= 4\u003c/p>","order":6,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-17-kindly-consider-the-following-lt-strong-gt-lt-p-gt-lt-p-gt-lt-picture-gt-lt-img-src-quot-qna-11586012"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870248,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345976,"answerId":11345982,"questionText":"\u003cp>\u003cstrong>30. If \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math> For what value (s) of a does \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> exists?\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>30.\u003c/strong> Given, \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmtext> \u003c/mtext>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>As | x | = \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>We can rewrite f (x) = \u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/p>\u003cp>Case 1: when a<0,\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> = exist such that a< 0\u003c/p>\u003cp>Case II when a> 0,\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> exist such that a>0.\u003c/p>\u003cp>Case III when a = 0.\u003c/p>\u003cp>L.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>R.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>Thus, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>≠\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> does not exist at a = 0.\u003c/p>","order":7,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-30-if-amp-nbsp-lt-math-gt-lt-mrow-gt-lt-mi-gt-f-lt-mi-gt-lt-mo-stretchy-quot-false-quot-gt-qna-11345976"},"anaFaqSource":"manual","generationSource":"manual","templateId":141,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869976,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345892,"answerId":11345896,"questionText":"\u003cp>\u003cstrong>22. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>22. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>Put y = x \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> . So that as y 0 cos\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731302984phpkNaKnH.jpeg\" alt=\"\" width=\"414\" height=\"339\">\u003c/picture>\u003c/p>","order":8,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-22-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345892"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869838,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345682,"answerId":11345686,"questionText":"\u003cp>\u003cstrong>12. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>Kindly go through the solution\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462682phpXmcqqU.jpeg\" alt=\"\" width=\"381\" height=\"68\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":9,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-12-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-gt-lt-qna-11345682"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869810,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345648,"answerId":11345652,"questionText":"\u003cp>\u003cstrong>10 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>z\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>10.\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>z\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>z\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>÷\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>z\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>z\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>z\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>z\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmrow> \u003cmo>[\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>z\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmfrac bevelled=\"true\"> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/msup> \u003cmo>−\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmfrac bevelled=\"true\"> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmi>z\u003c/mi> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>]\u003c/mo> \u003c/mrow> \u003cmo>÷\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>z\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmrow> \u003cmo>[\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmi>z\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmfrac bevelled=\"true\"> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>6\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/msup> \u003cmo>−\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmfrac bevelled=\"true\"> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>6\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmi>z\u003c/mi> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>]\u003c/mo> \u003c/mrow> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>We know that,\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003c/munder>\u003c/mrow>\u003c/math>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>So,\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742462588phpwGkj1s_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462588phpwGkj1s.jpeg\" alt=\"\" width=\"259\" height=\"63\">\u003c/picture>\u003c/div>\u003cdiv>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/mfrac> \u003cmo>×\u003c/mo> \u003cmn>6\u003c/mn> \u003cmo>=\u003c/mo> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/div>\u003c/div>","order":10,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-10-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345648"},"anaFaqSource":"manual","generationSource":"manual","templateId":105,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869802,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345540,"answerId":11345544,"questionText":"\u003cp>\u003cstrong>6. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>6.\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462078phplCeMlw.jpeg\" alt=\"\" width=\"338\" height=\"190\">\u003c/picture>\u003c/div>\u003cdiv>\u003ctable border=\"1\">\u003ctbody>\u003ctr>\u003ctd>\u003cstrong>Q6. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/td>\u003c/tr>\u003ctr>\u003ctd>\u003cp>\u003cstrong>6. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742462078phplCeMlw_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462078phplCeMlw.jpeg\" alt=\"\" width=\"1200\" height=\"675\">\u003c/picture>\u003c/div>\u003cdiv>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>3\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>1\u003c/mn> \u003cmn>0\u003c/mn> \u003cmsup> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>+\u003c/mo> \u003cmn>1\u003c/mn> \u003cmn>0\u003c/mn> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>5\u003c/mn> \u003cmo>.\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>\u003c/div>\u003cdiv>\u003cp>(0)4 + 5(0)3 + 10(0)2 + 10(0) + 5\u003c/p>\u003cp>= 5.\u003c/p>\u003c/div>\u003c/div>\u003c/td>\u003c/tr>\u003c/tbody>\u003c/table>\u003c/div>\u003c/div>","order":11,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-6-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345540"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874316,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356136,"answerId":11356148,"questionText":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>71.\u003c/strong> \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>71. \u003c/strong>Given, f (x) \u003cstrong>= \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmi>S\u003c/mi>\u003cmi>o\u003c/mi>\u003cmo>,\u003c/mo>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mrow>\u003cmo>⋅\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>⋅\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742470053phpIoPg6a_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742470053phpIoPg6a.jpeg\" alt=\"\" width=\"450\" height=\"94\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>Dividing numerator and denominator by cos2x we get,\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731318417phpZfJ4cO.jpeg\" alt=\"\" width=\"195\" height=\"76\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>·\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","order":12,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-71-lt-strong-gt-lt-qna-11356136"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869792,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345468,"answerId":11345470,"questionText":"\u003cp>\u003cstrong>1. \u003c/strong>\u003cspan style=\"font-weight: bold;\">l\u003c/span>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/munder>\u003c/mrow>\u003c/math>\u003c/span> x+ 3\u003c/strong>\u003c/p>","answerText":"\u003ch3>Exercise 13.1\u003c/h3>\u003cp>\u003cstrong>1. 1. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/munder>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>x + 3 = 3 + 3 = 6.\u003c/p>","order":13,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-1-amp-nbsp-lt-strong-gt-lt-span-style-quot-font-weight-bold-quot-gt-l-lt-span-gt-lt-strong-qna-11345468"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869804,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345582,"answerId":11345584,"questionText":"\u003cp>\u003cstrong>7. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>7. \u003c/strong>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742462163phpEpKisd.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462163phpEpKisd.jpeg\" alt=\"\" width=\"300\" height=\"56\">\u003c/picture>\u003c/p>\u003cdiv>\u003cdiv>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003c/div>\u003c/div>","order":14,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-7-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345582"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869808,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345624,"answerId":11345636,"questionText":"\u003cp>\u003cstrong>9. \u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow>Li\u003cmsub>\u003cmrow>m\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow> \u003c/math> \u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>Kindly go through the solution\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462419phpumlIvn.jpeg\" alt=\"\" width=\"271\" height=\"43\">\u003c/picture>\u003c/p>","order":15,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-9-lt-span-contenteditable-quot-false-quot-gt-lt-math-gt-lt-mrow-gt-li-lt-msub-gt-lt-mrow-qna-11345624"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869848,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345776,"answerId":11345784,"questionText":"\u003cp>\u003cstrong>18. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">18. \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi> x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731061609phpmS3cGH.jpeg\" alt=\"\" width=\"175\" height=\"94\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":16,"answerCount":2,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-18-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345776"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869846,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345744,"answerId":11345746,"questionText":"\u003cp>\u003cstrong>16. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>16. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":17,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-16-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345744"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869844,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345728,"answerId":11345736,"questionText":"\u003cp>\u003cstrong>15. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>15.\u003c/strong> \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":18,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-15-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345728"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869806,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345608,"answerId":11345616,"questionText":"\u003cp>\u003cstrong>8. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>8\u003c/mn>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>8. \u003c/strong>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742462272phpNRxprt_480x360.jpeg\" media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462272phpNRxprt_480x360.jpeg\"6 alt=\"\" width=\"402\" height=\"57\">\u003c/picture>\u003c/p>\u003cdiv>\u003cdiv>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742462272phpNRxprt_480x360.jpeg\"7 media=\" (max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462315phpbfVChJ.jpeg\" alt=\"\" width=\"239\" height=\"58\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003c/div>\u003c/div>\u003c/div>\u003c/div>","order":19,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-8-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345608"},"anaFaqSource":"manual","generationSource":"manual","templateId":105,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869842,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345716,"answerId":11345720,"questionText":"\u003cp>\u003cstrong>14. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>14. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731061432phpwzLB5i.jpeg\" alt=\"\" width=\"432\" height=\"153\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":20,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-14-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345716"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869850,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345792,"answerId":11345794,"questionText":"\u003cp>\u003cstrong>19. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>19. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= 0.\u003c/p>","order":21,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-19-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345792"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869798,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345518,"answerId":11345520,"questionText":"\u003cp>\u003cstrong>4. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>4. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":22,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-4-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345518"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869800,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345528,"answerId":11345532,"questionText":"\u003cp>\u003cstrong>5. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>5. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":23,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-5-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345528"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869854,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345808,"answerId":11345810,"questionText":"\u003cp>\u003cstrong>21. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>21. \u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003c/mtable>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003cmo>{\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731302877phpARxfZM.jpeg\" alt=\"\" width=\"130\" height=\"91\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":24,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-21-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-gt-lt-qna-11345808"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869794,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345496,"answerId":11345500,"questionText":"\u003cp>\u003cstrong>2. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>2. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":25,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-2-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345496"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869796,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345510,"answerId":11345514,"questionText":"\u003cp>\u003cstrong>3. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>3\u003cstrong>. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>r\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>π\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":26,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-3-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345510"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869812,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345660,"answerId":11345662,"questionText":"\u003cp>\u003cstrong>11. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo> \u003c/mo>\u003cmo>≠\u003c/mo>\u003cmo> \u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>11. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":27,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-11-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345660"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869840,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345698,"answerId":11345704,"questionText":"\u003cp>\u003cstrong>13. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>lim\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>13. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731061375phpx5yqQE.jpeg\" alt=\"\" width=\"128\" height=\"63\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":28,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-13-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-qna-11345698"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869852,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345802,"answerId":11345804,"questionText":"\u003cp>\u003cstrong>20. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>a\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>,\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>20. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= 1.\u003c/p>","order":29,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-20-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-gt-lt-qna-11345802"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869978,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345904,"answerId":11345914,"questionText":"\u003cp>\u003cstrong>23. Find \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/math>\u003c/span> and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/math>\u003c/span> , where f (x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>23.\u003c/strong> Given f (x) \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003cmo>}\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>for \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>left hand limit, L.H.S = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2 0 + 3 = 3.\u003c/p>\u003cp>Right hand limit, R.H.L = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= (0 + 1) = 3 1 = 3.\u003c/p>\u003cp>Thus, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>For \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>L.H.L = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>R.H.L = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Thus, \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>6\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":30,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-23-find-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-qna-11345904"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869980,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345922,"answerId":11345924,"questionText":"\u003cp>\u003cstrong>24. Find \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/math> , where \u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>24. \u003c/strong>Given \u003cstrong>\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> {\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>, \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmo>, \u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>? \u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>Now, L.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>1\u003csup>2\u003c/sup>- 1 = 0\u003c/p>\u003cp>And R.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math> (1)2 1 = 1 = 2.\u003c/p>\u003cp>Thus, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>≠\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> does not exist.\u003c/p>","order":31,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-24-find-lt-math-gt-lt-mrow-gt-lt-munder-gt-lt-mrow-gt-lt-mi-gt-l-lt-mi-gt-lt-mi-gt-i-lt-mi-qna-11345922"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869982,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345926,"answerId":11345928,"questionText":"\u003cp>\u003cstrong>25. Evaluate \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math> where f (x) = \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmfrac>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>25. \u003c/strong>Given f (x) = \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmfrac>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003cmo>,\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>?\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>We know that, \u003cmath>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo>≥\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>Now,\u003c/p>\u003cp>L.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>and R.H.L = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>Thus, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>≠\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>i e, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> does not exist.\u003c/p>","order":32,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-25-evaluate-lt-math-gt-lt-mrow-gt-lt-munder-gt-lt-mrow-gt-lt-mi-gt-l-lt-mi-gt-lt-mi-gt-i-qna-11345926"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869984,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345936,"answerId":11345938,"questionText":"\u003cp>\u003cstrong>26. Find \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> wheref (x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>0\u003c/mn>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>26. \u003c/strong>Given, f (x) \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo> {\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>, \u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>≠\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmn>0\u003c/mn>\u003cmo>, \u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731303189phpso1ZZf.jpeg\" alt=\"\" width=\"147\" height=\"70\">\u003c/picture>\u003c/p>\u003cp>L.H.S = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>R.H.L \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Thus, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>≠\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>i e, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> does not exist.\u003c/p>","order":33,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-26-find-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-munder-gt-qna-11345936"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869986,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345946,"answerId":11345948,"questionText":"\u003cp>\u003cstrong>27.Find \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>w\u003c/mi>\u003cmi>h\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>r\u003c/mi>\u003cmi>e\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> f (x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>27. \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo>|\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>|\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= | 5 |\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> 5\u003c/p>\u003cp>= 5 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>5\u003c/p>\u003cp>= 0\u003c/p>","order":34,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-27-find-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-gt-lt-qna-11345946"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39869988,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345952,"answerId":11345954,"questionText":"\u003cp>\u003cstrong>28. Supposef (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmn>4\u003c/mn>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> and if \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = f (1) what are possible values of a and b?\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>28. \u003c/strong>Given, f (x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmrow>\u003cmo> {\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>, \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmo>, \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>, \u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>Since we need \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we need, \u003c/p>\u003cp>LHL =\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = a + b × 1 = a + b\u003c/p>\u003cp>and RHL =\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> = b - a × 1 = b - a\u003c/p>\u003cp>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>:\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> we have the following equations\u003c/p>\u003cp>a + b = 4 ____ (1)\u003c/p>\u003cp>b - a = 4 ____ (2)\u003c/p>\u003cp>Adding (1) and (2) we get, \u003c/p>\u003cp>2b = 8\u003c/p>\u003cp>b = 4\u003c/p>\u003cp>Putting b = 4 in (1) we get\u003c/p>\u003cp>a + 4 = 4\u003c/p>\u003cp>a = 0\u003c/p>","order":35,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-28-supposef-x-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-gt-lt-mrow-qna-11345952"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870250,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11345990,"answerId":11345994,"questionText":"\u003cp>\u003cstrong>31. If the function f(x) satisfies \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> , evaluate \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>31\u003c/strong>. Given, \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> [\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmi>π\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":36,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-31-if-the-function-f-x-satisfies-amp-nbsp-lt-span-title-quot-click-to-copy-mathml-quot-gt-qna-11345990"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870266,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346140,"answerId":11346142,"questionText":"\u003cp>\u003cstrong>39. For some constants aand b, find the derivative of\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003c/strong>(x- a) + (x - b) \u003cstrong>(ii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> (iii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>(c) Given, f(x)=(x-a)(x-b)\u003c/p>\u003cp>where a and b are constants.\u003c/p>\u003cp>So,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=(x-a)+(x-b)\u003c/p>\u003cp>= 2x– a- b. \u003cmath>\u003c/math>\u003c/p>\u003cp>(ii) Given f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> where ab are constant\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>(iii) Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> where a and bare constants\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":37,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-39-for-some-constants-aand-b-find-the-derivative-of-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-qna-11346140"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870270,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346556,"answerId":11346564,"questionText":"\u003cp>\u003cstrong>41. Find the derivative of\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (ii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(iii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> (iv) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(v) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> (vi) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>41. (i) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>=2.\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>i\u003c/mi>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iii) \u003c/strong>Given, f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>⋅\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>⋅\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>⋅\u003c/mo>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>5\u003c/mn>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(v)\u003c/strong> Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>⋅\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>⋅\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>⋅\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>6\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(vi) \u003c/strong>Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":38,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-41-find-the-derivative-of-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-i-amp-nbsp-lt-span-title-qna-11346556"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870460,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346620,"answerId":11346636,"questionText":"\u003cp>\u003cstrong>44. Find the derivative of the following functions from first principle:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i) \u003c/strong>x\u003cstrong> (ii) \u003c/strong>(x)1 \u003cstrong>(iii) \u003c/strong>sin(x + 1) \u003cstrong>(iv) \u003c/strong>cos\u003cstrong> \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Find the derivative of the following functions (it is to be understood that a, b, c, d,\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>p, q, r and s are fixed non-zero constants and m and n are integers):\u003c/strong>\u003c/p>","answerText":"\u003cdiv>\u003cdiv>\u003cstrong>44.\u003c/strong>\u003c/div>\u003cdiv>\u003cstrong>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742465285phpQciQNb.jpeg\" alt=\"\" width=\"298\" height=\"151\">\u003c/picture>\u003c/strong>\u003c/div>\u003cdiv>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmfrac> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>−\u003c/mo> \u003cmi>h\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>x\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/div>\u003cdiv>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmfrac> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmi>h\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/div>\u003cdiv>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmo>=\u003c/mo> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/div>\u003cdiv>\u003cp>= - 1.\u003c/p>\u003cp>(ii) Given, \u003cem>f\u003c/em>(\u003cem>x\u003c/em>) = (-\u003cem>x\u003c/em>)\u003csup>-\u003c/sup>\u003csup>1\u003c/sup>\u003c/p>\u003cp>by first principle,\u003c/p>\u003cp>f(x)\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmunder> \u003cmrow> \u003cmi>l\u003c/mi> \u003cmi>i\u003c/mi> \u003cmi>m\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003cmo>→\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/munder> \u003cmfrac> \u003cmrow> \u003cmo stretchy=\"false\">[\u003c/mo> \u003cmo>−\u003c/mo> \u003cmo stretchy=\"false\">]\u003c/mo> \u003cmsup> \u003cmrow> \u003cmo stretchy=\"false\">(\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmi>h\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msup> \u003cmo>−\u003c/mo> \u003cmsup> \u003cmrow> \u003cmo stretchy=\"false\">(\u003c/mo> \u003cmo>−\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo stretchy=\"false\">)\u003c/mo> \u003c/mrow> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmrow> \u003cmi>h\u003c/mi> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003c/div>\u003c/div>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742465541phpd5zmnf_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742465541phpd5zmnf.jpeg\" alt=\"\" width=\"183\" height=\"146\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>(iii) Given, f(x) = sin(x + 1)\u003c/p>\u003cp>By first principle,\u003c/p>\u003cp>f’(x) = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>= cos (x + 1)\u003c/p>\u003cp>(iv) Given, f(x) = cos \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>By first principle,\u003c/p>\u003cp>f(x) = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742465751phpp3BZkD_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742465751phpp3BZkD.jpeg\" alt=\"\" width=\"420\" height=\"297\">\u003c/picture>\u003c/div>\u003c/div>\u003c/div>\u003c/div>\u003cp> \u003c/p>","order":39,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-44-find-the-derivative-of-the-following-functions-from-first-principle-lt-strong-gt-lt-p-qna-11346620"},"anaFaqSource":"manual","generationSource":"manual","templateId":119,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874274,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352928,"answerId":11352932,"questionText":"\u003cp>\u003cstrong>48. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>48. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f? (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>·\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":40,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-48-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11352928"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874282,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353088,"answerId":11353114,"questionText":"\u003cp>\u003cstrong>54.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cdiv class=\"photo-widget-full\">\n\u003cdiv class=\"figure\">\u003cpicture>\u003csource srcset=\"\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/1747917056phpkJ9BTH.jpeg\" alt=\"\" width=\"97\" height=\"31\" loading=\"lazy\">\u003c/picture>\u003c/div>\n\u003c/div>\n\u003cp> \u003c/p>","answerText":"\u003cdiv class=\"photo-widget-full\">\u003cdiv class=\"figure\">\u003cstrong>54. \u003c/strong>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/1747917155phpiaZ0iU.jpeg\" alt=\"\" width=\"251\" height=\"37\" loading=\"lazy\">\u003c/picture>\u003c/div>\u003cdiv class=\"figure\">\u003cp>So f? (x) \u003cstrong>= \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>=\u003c/mo>\u003cmn>4\u003c/mn>\u003cmo>×\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmn>2\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003c/div>\u003c/div>\u003cp> \u003c/p>","order":41,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-54-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353088"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874294,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353228,"answerId":11353230,"questionText":"\u003cp>\u003cstrong>60. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>60. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f?(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731317303phpOvz4n1.jpeg\" alt=\"\" width=\"535\" height=\"72\">\u003c/picture>\u003c/p>\u003cp>Let g(x) = cos x and p(x) = sin x.\u003c/p>\u003cp>{from so g’(x) A ) (upto equation 3)\u003c/p>\u003cp>Let g(x) = cos2 and p(x) = sin x.\u003c/p>\u003cp>So, g?(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731317359phpLiWI0l.jpeg\" alt=\"\" width=\"219\" height=\"78\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= -sin x ______ (2)\u003c/p>\u003cp>And p?(x) = \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>⋅\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= cos x _____ (3)\u003c/p>\u003cp>Putting (2) and (3) in (1) we get,\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":42,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-60-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11353228"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874296,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353238,"answerId":11353242,"questionText":"\u003cp>\u003cstrong>61.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>61. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f?(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>Let g(x) = cos x.\u003c/p>\u003cp>So, g?(x)\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= -sin x.\u003c/p>\u003cp>So, f?(x)\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>×\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":43,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-61-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353238"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874302,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353266,"answerId":11353268,"questionText":"\u003cp>\u003cstrong>64.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>64. \u003c/strong>Given, f (x) = \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>Let g?(x) = sin (x + a)\u003c/p>\u003cp>So, g?(x) = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003c/munder>\u003c/mrow>\u003c/math>\u003cstrong>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742469334phpzX3824_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742469334phpzX3824.jpeg\" alt=\"\" width=\"300\" height=\"208\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>\u003cstrong>= cos (x + a)\u003c/strong>\u003c/p>\u003cp>\u003cstrong>And P(x) = cos x\u003c/strong>\u003c/p>\u003cp>\u003cstrong>So, P?(x) = \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742469421phpxbliMG_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742469421phpxbliMG.jpeg\" alt=\"\" width=\"306\" height=\"104\">\u003c/picture>\u003c/div>\u003c/div>\u003c/div>\u003c/div>\u003cp>Thus, f?(x) = \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003cstrong>\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","order":44,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-64-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353266"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874304,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353284,"answerId":11353286,"questionText":"\u003cp>\u003cstrong>\u003cem>66. \u003c/em>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cem>Kindly consider the following\u003c/em>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cem>x\u003c/em>\u003csup>4\u003c/sup> (5 sin x 3 cos x)\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>\u003cem>65. \u003c/em>\u003c/strong>Given, f (x) = \u003cem>x\u003c/em>\u003csup>4\u003c/sup>. (5 sin x 3 cos x)\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>·\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>·\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>As \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>and \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Thus,\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>·\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":45,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-lt-em-gt-66-lt-em-gt-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-lt-em-gt-kindly-consider-the-qna-11353284"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874314,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356122,"answerId":11356124,"questionText":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>70. \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerText":"\u003cp> \u003cstrong>70. \u003c/strong>Given, f (x) \u003cstrong>= \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>So, f? (x) \u003cstrong>= \u003c/strong>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742469923phprlFy9G.jpeg\" alt=\"\" width=\"266\" height=\"120\">\u003c/p>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","order":46,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-70-lt-math-gt-lt-qna-11356122"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874318,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356162,"answerId":11356166,"questionText":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>72. (\u003cem>x\u003c/em> + sec \u003cem>x\u003c/em>) (\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>72. \u003c/strong>Given, f (x) = (x + sec x) (x tan x)\u003c/p>\u003cp>So, f?(x) = (x + see x)\u003cstrong> \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471139phpRI5F40_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471139phpRI5F40.jpeg\" alt=\"\" width=\"458\" height=\"46\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>Let g(x) = tan x.\u003c/p>\u003cp>So, g?(x)\u003cstrong> = \u003c/strong>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471188phpFPN8R5.jpeg\" alt=\"\" width=\"252\" height=\"42\">\u003c/p>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742470962php23HO2z_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742470962php23HO2z.jpeg\" alt=\"\" width=\"191\" height=\"66\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/strong> . _____ (2)\u003c/p>\u003cp>And P(x) = see x.\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471001phpwSwYwX_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471001phpwSwYwX.jpeg\" alt=\"\" width=\"246\" height=\"57\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471046phpjySO16_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471046phpjySO16.jpeg\" alt=\"\" width=\"322\" height=\"271\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/strong>= tan x sec x. ____ (3)\u003c/p>\u003cp> Putting (2) and (3) in (1) we get,\u003c/p>\u003cp>\u003cem> f\u003c/em>?(\u003cem>x\u003c/em>) = (\u003cem>x\u003c/em> + sec \u003cem>x\u003c/em>) (1 - sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>) + (\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>) (1 + tan \u003cem>x\u003c/em> sec \u003cem>x\u003c/em>)\u003c/p>","order":47,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-72-lt-em-gt-x-lt-qna-11356162"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874320,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356192,"answerId":11356194,"questionText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">Kindly consider the following\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">73. \u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">73. \u003c/span>\u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f?(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":48,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-lt-span-title-quot-click-to-copy-mathml-quot-gt-kindly-consider-the-following-lt-span-gt-qna-11356192"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870252,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346008,"answerId":11346010,"questionText":"\u003cp>\u003cstrong>32. If f (x) = \u003cmath>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmi>m\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmn>0\u003c/mn>\u003cmo>≤\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/math> . For what integers m and n does both \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> and \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> exist?\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>32. \u003c/strong>Given, f (x) = \u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmrow>\u003cmo>{\u003c/mo>\u003cmrow>\u003cmtable columnalign=\"left\">\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmi>m\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmi>x\u003c/mi>\u003cmo><\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mtd>\u003c/mtr>\u003cmtr columnalign=\"left\">\u003cmtd columnalign=\"left\">\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"left\">\u003cmn>0\u003c/mn>\u003cmo>≤\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>≤\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo>,\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>>\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/p>\u003cp>For \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>n = m\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> exist for n = m.\u003c/p>\u003cp>Again, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo>−\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>m\u003c/mi>\u003cmo>,\u003c/mo>\u003c/mrow>\u003c/math> Thus, \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math> exist for any integral value of m and n.\u003c/p>","order":49,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-32-if-f-x-amp-nbsp-lt-math-gt-lt-mrow-gt-lt-mrow-gt-lt-mo-gt-lt-mo-gt-lt-mtable-columnalign-qna-11346008"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870254,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346030,"answerId":11346038,"questionText":"\u003cp>\u003cstrong>33. Find the derivative\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>33. \u003c/strong>Given, f (x)=x\u003csup>2\u003c/sup>- 2 ., \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmo>? \u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>We have, \u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmsup>\u003cmrow>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>⇒\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>=20\u003c/p>","order":50,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-33-find-the-derivative-lt-strong-gt-lt-p-gt-qna-11346030"},"anaFaqSource":"manual","generationSource":"manual","templateId":97,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870256,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346052,"answerId":11346054,"questionText":"\u003cp>\u003cstrong>34. Find the derivative of x at x = 1.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>34. \u003c/strong>Given, f (x)=x, f (1)=? \u003c/p>\u003cp>We have, \u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003c/math> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>=1\u003c/p>","order":51,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-34-find-the-derivative-of-x-at-x-1-lt-strong-gt-lt-p-gt-qna-11346052"},"anaFaqSource":"manual","generationSource":"manual","templateId":99,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870258,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346060,"answerId":11346066,"questionText":"\u003cp>\u003cstrong>35. Find the derivative of 99x at x = l00\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>35. \u003c/strong>Given, f (x)= 99x, f (100)=? \u003c/p>\u003cp>So, f (100)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsub>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msub>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>×\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=99\u003c/p>","order":52,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-35-find-the-derivative-of-99x-at-x-l00-lt-strong-gt-lt-p-gt-qna-11346060"},"anaFaqSource":"manual","generationSource":"manual","templateId":165,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870260,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346078,"answerId":11346082,"questionText":"\u003cp>\u003cstrong>36. Find the derivative of the following functions from first principle.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>36. (i) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> (ii) (x -1)(x-2)\u003c/strong>\u003c/p>\u003cp>\u003cstrong>(iii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> (iv) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>A.4.(i) \u003c/strong>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>7\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=0+3 x(x+ 0)\u003c/p>\u003cp>=3\u003cem>x\u003c/em>\u003csup>2\u003c/sup>\u003c/p>\u003cp>\u003cstrong>(ii)\u003c/strong> Given, f(x) =(x-1)(x-2)\u003c/p>\u003cp>=\u003cem>x\u003c/em>\u003csup>2\u003c/sup>- 3x+2\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= 2x – 3.\u003c/p>\u003cp>\u003cstrong>(iii) \u003c/strong>Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cmath>\u003c/math>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cstrong>(iv)\u003c/strong> Given, f(x)= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":53,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-36-find-the-derivative-of-the-following-functions-from-first-principle-lt-strong-gt-lt-p-qna-11346078"},"anaFaqSource":"manual","generationSource":"manual","templateId":115,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870262,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346102,"answerId":11346106,"questionText":"\u003cp>\u003cstrong>37. \u003c/strong>\u003cstrong>For the function\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmo>⋯\u003c/mo>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Prove that \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>37. \u003c/strong>\u003c/p>\u003cp>Given \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>⋅\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmo>⋅\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmo>…\u003c/mo>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmo>⋅\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>At x=0,\u003c/p>\u003cp>f(X) =1.\u003c/p>\u003cp>and at x=1,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>9\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>9\u003c/mn>\u003cmn>8\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>…\u003c/mo>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=100 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>×\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> 1\u003c/p>\u003cp>=100 \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>Hence, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>0\u003c/mn>\u003cmn>0\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":54,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-37-lt-strong-gt-lt-strong-gt-for-the-function-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-lt-qna-11346102"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870264,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346128,"answerId":11346130,"questionText":"\u003cp>\u003cstrong>38. Find the derivative of \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>⋯\u003c/mo>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span> for some fixed real number a.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>38. \u003c/strong>Given, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>We know that,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So,\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmo>?\u003c/mo>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/math>\u003cbr>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>?\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>a\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x \u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>and d\u003c/mi>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmn>0whereaisconstant\u003c/mn>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/span>\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmi>\u003c/mi>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/p>","order":55,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-38-amp-nbsp-find-the-derivative-of-lt-span-title-quot-click-to-copy-mathml-quot-gt-lt-math-qna-11346128"},"anaFaqSource":"manual","generationSource":"manual","templateId":117,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870272,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346568,"answerId":11346570,"questionText":"\u003cp>\u003cstrong>42. Find the derivative of cos x from first principle.\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>42. \u003c/strong>Given, f(x) = cos x\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>By first principle,\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":56,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-42-find-the-derivative-of-cos-x-from-first-principle-lt-strong-gt-lt-p-gt-qna-11346568"},"anaFaqSource":"manual","generationSource":"manual","templateId":95,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39870462,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11346660,"answerId":11346664,"questionText":"\u003cp>\u003cstrong>47. (\u003cem>ax\u003c/em> + \u003cem>b\u003c/em>) (\u003cem>cx\u003c/em> + \u003cem>d\u003c/em>)\u003csup>2\u003c/sup>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>47. \u003c/strong>Given, f (x) = (ax + b) (\u003cem>cx\u003c/em> + \u003cem>d\u003c/em>)\u003csup>2\u003c/sup>\u003c/p>\u003cp>So, f?(x) = (ax +b)\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":57,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-47-lt-em-gt-ax-lt-em-gt-lt-em-gt-b-lt-em-gt-lt-em-gt-cx-lt-em-gt-lt-em-gt-d-lt-em-gt-lt-sup-qna-11346660"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874270,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352304,"answerId":11352308,"questionText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">48. \u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">Kindly consider the following\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\"> \u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">48. \u003c/span>\u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f (x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":58,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-lt-span-title-quot-click-to-copy-mathml-quot-gt-48-amp-nbsp-lt-span-gt-lt-strong-gt-lt-p-qna-11352304"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874272,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352902,"answerId":11352906,"questionText":"\u003cp>\u003cstrong>49. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>49. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":59,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-49-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11352902"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874276,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352950,"answerId":11352952,"questionText":"\u003cp>\u003cstrong>51. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>51. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>p\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>9\u003c/mn>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmi>q\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>a\u003c/mi>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003cmi>p\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>r\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>q\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>r\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>","order":60,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-51-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11352950"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874280,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11352992,"answerId":11353004,"questionText":"\u003cp>\u003cstrong>53. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>53. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>So, f(x) = \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>4\u003c/mn>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>b\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":61,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-53-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11352992"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874284,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353134,"answerId":11353138,"questionText":"\u003cp>\u003cstrong>55.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(\u003cem>ax\u003c/em> + \u003cem>b\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em>\u003c/strong>\u003c/p>\n\u003cp> \u003c/p>","answerText":"\u003cp>\u003cstrong>55. \u003c/strong>Given, f (x) = (\u003cem>ax\u003c/em> + \u003cem>b\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em>\u003c/p>\u003cp>Chain rule, \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>u\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>u\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>u\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>u\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span> where\u003c/p>\u003cp>u (x) is a function of x.\u003c/p>\u003cp>So, f (x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":62,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-55-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353134"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874286,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353152,"answerId":11353156,"questionText":"\u003cp>\u003cstrong>56. (\u003cem>ax\u003c/em> + \u003cem>b\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em> (\u003cem>cx\u003c/em> + \u003cem>d\u003c/em>)\u003csup>m\u003c/sup>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>56. \u003c/strong>Given, f (x) = (\u003cem>ax\u003c/em> + \u003cem>b\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em> (\u003cem>cx\u003c/em> + \u003cem>d\u003c/em>)\u003csup>m\u003c/sup>\u003c/p>\u003cp>So, f’(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>c\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>m\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003cmi>m\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>m\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":63,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-56-lt-em-gt-ax-lt-em-gt-lt-em-gt-b-lt-em-gt-lt-em-gt-lt-sup-gt-n-lt-sup-gt-lt-em-gt-lt-em-qna-11353152"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874288,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353186,"answerId":11353190,"questionText":"\u003cp>\u003cstrong>57. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>sin(x + a)\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>57. \u003c/strong>Given, f (x) = sin (x + a)\u003c/p>\u003cp>So, f (x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>·\u003c/mo>\u003cmn>2\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= cos (x + a)\u003c/p>","order":64,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-57-amp-nbsp-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-qna-11353186"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874290,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353206,"answerId":11353212,"questionText":"\u003cp>\u003cstrong>58. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>cosec x. cot x\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>58. \u003c/strong>Given f (x) = cosec x. cot x.\u003c/p>\u003cp>By Leibnitz product rule,\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731315789php2UWC86.jpeg\" alt=\"\" width=\"387\" height=\"52\">\u003c/picture>\u003c/div>\u003cdiv>\u003cp>So, g(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>t\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmtable>\u003cmtr columnalign=\"center\">\u003cmtd columnalign=\"center\">\u003cmo>?\u003c/mo>csin\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>A\u003cmo>−\u003c/mo>B\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003c/mtd>\u003cmtd columnalign=\"center\">\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>A\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>B\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>A\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>B\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>.\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= -cosec\u003csup>2\u003c/sup>x.______(2)\u003c/p>\u003cp>And hx) = \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmrow>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731315901php5J9GY5.jpeg\" alt=\"\" width=\"333\" height=\"307\">\u003c/picture>\u003c/p>\u003c/div>\u003c/div>","order":65,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-58-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353206"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874300,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353254,"answerId":11353256,"questionText":"\u003cp>\u003cstrong>63.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>63. \u003c/strong>Given, f (x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>f?(x) = \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>·\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>·\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo>·\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmtext> \u003c/mtext>\u003c/mrow>\u003c/mfrac>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmo>_\u003c/mo>\u003cmo>_\u003c/mo>\u003cmo>_\u003c/mo>\u003cmo>_\u003c/mo>\u003cmo>_\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>{Copy (A)}\u003c/p>\u003cp>So, g?(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731317671phprYWklw.jpeg\" alt=\"\" width=\"216\" height=\"82\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= sin x ______ (2)\u003c/p>\u003cp>And p?(x) =\u003cstrong> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>⋅\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= cos x _____ (3)\u003c/p>\u003cp>So, put (2) and (3) in (1) we get,\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>f\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmo>′\u003c/mo>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>b\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>d\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>d\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>d\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>b\u003c/mi>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>d\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","order":66,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-63-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353254"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874298,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353246,"answerId":11353248,"questionText":"\u003cp>\u003cstrong>62.\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>sin\u003cem>\u003csup>n\u003c/sup>x\u003c/em>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>62. \u003c/strong>Given, f (x) =sin\u003cem>\u003csup>n\u003c/sup>x\u003c/em>\u003c/p>\u003cp>By chain rule, \u003c/p>\u003cp>f? (x) = n (sin \u003cem>x\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em>\u003csup>-1\u003c/sup> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> sin x\u003c/p>\u003cp>Let (gx) = sinx\u003c/p>\u003cp>So, g? (x) \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>g\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= cos x.\u003c/p>\u003cp>So, f? (x) = n (sin \u003cem>x\u003c/em>)\u003cem>\u003csup>n\u003c/sup>\u003c/em>\u003csup>-1\u003c/sup> cos x.\u003c/p>","order":67,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-62-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353246"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874308,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11353306,"answerId":11353308,"questionText":"\u003cp>\u003cstrong>67. \u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(\u003cem>ax\u003c/em>\u003csup>2\u003c/sup> + sin x) (p +q cos x)\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>67. \u003c/strong>Given, f (x) = (\u003cem>ax\u003c/em>\u003csup>2\u003c/sup> + sin x) (p +q cos x).\u003c/p>\u003cp>So, f? (x) = (\u003cem>ax\u003c/em>\u003csup>2\u003c/sup> + sin x) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span> \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmn>0\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>q\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>p\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>q\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>·\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>= q sin x (\u003cem>ax\u003c/em>\u003csup>2\u003c/sup> + sin x) + (p + q cos x) (2ax + cos x)\u003c/p>","order":68,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-67-lt-strong-gt-lt-p-gt-lt-p-gt-lt-strong-gt-kindly-consider-the-following-lt-strong-gt-lt-qna-11353306"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":39874312,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11356086,"answerId":11356092,"questionText":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">Kindly consider the following\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">69. \u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerText":"\u003cp>\u003cstrong>69. \u003c/strong>Given, f (x) = \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/p>\u003cp>So,\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmsup>\u003cmi>f\u003c/mi>\u003cmo>′\u003c/mo>\u003c/msup>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>4\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmo>−\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmtable columnalign=\"left\">\u003cmtr>\u003cmtd>\u003cmrow>\u003cmunder accentunder=\"true\">\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmsup>\u003cmrow>\u003c/mrow>\u003cmrow>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmn>5\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo stretchy=\"true\">_\u003c/mo>\u003c/munder>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003cmtr>\u003cmtd>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmtext> \u003c/mtext>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmtext> \u003c/mtext>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmtext> \u003c/mtext>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mtd>\u003c/mtr>\u003c/mtable>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>3\u003c/mn>\u003cmn>5\u003c/mn>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003cmn>5\u003c/mn>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>2\u003c/mn>\u003cmn>8\u003c/mn>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>3\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","order":69,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-lt-span-title-quot-click-to-copy-mathml-quot-gt-kindly-consider-the-following-lt-span-gt-qna-11356086"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false},{"anaFaqId":45630152,"entityId":4307,"entityType":"examPage","childPage":"homepage","wikiIdentifier":"190728","questionId":11586058,"answerId":11586064,"questionText":"\u003cp>\u003cstrong>\u003cpicture>29. Kindly consider the following\u003c/picture>\u003c/strong>\u003c/p>\n\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742463703phpBS7vhd.jpeg\" alt=\" width='1200' height=\" width=\"764\" height=\"71\">\u003c/picture>\u003c/p>","answerText":"\u003cp>\u003cstrong>29. \u003c/strong>\u003c/p>\u003cp>Given, \u003cem>f\u003c/em> (\u003cem>x\u003c/em>) = (\u003cem>x\u003c/em>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>x\u003c/em>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>)… (\u003cem>x\u003c/em>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp>So, \u003cstrong> \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>? \u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>= (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp>= 0 (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp> = 0\u003c/p>\u003cp>And\u003cstrong> \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>…\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>= (\u003cem>a\u003c/em>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>a\u003c/em>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>","order":70,"answerCount":1,"url":{"url":null,"urlHost":null,"absoluteUrl":"https://ask.shiksha.com/lt-p-gt-lt-strong-gt-lt-picture-gt-29-kindly-consider-the-following-lt-picture-gt-lt-strong-gt-lt-p-gt-lt-p-gt-qna-11586058"},"anaFaqSource":"manual","generationSource":"manual","templateId":93,"atfCtrIndex":0,"showViewOther":true,"quesTemplateId":null,"isCrawlable":true,"likesCount":0,"dislikesCount":0,"navigationUrl":null,"isQuestionHtml":1,"messageType":"QNA","prepQuestionDetails":null,"experiment":false,"crawlable":true,"level":0,"parentQuesId":0,"animated":false}]}},"prepAnaData":null,"recentLiveBlogs":[],"examResourcesWidget":null,"examCutoffDTO":{"gocCutoffData":[],"comboCutoffData":null,"filters":{},"aliasMapping":{},"nameMapping":{}},"examTimelineData":{"examName":null,"examConductingYear":null,"activePhase":null,"activeParentPhaseStatus":null,"heading":null,"timeline":null},"tagIds":[2864885],"pollsData":{"polls":null,"userResponse":null},"showGoogleAdSense":false,"preparationWidgetData":{"similarChapters":[{"id":15302,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Eight","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-eight-ns","order":0,"prepEntityType":"chapter"},{"id":15318,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Sixteen","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-sixteen-ns","order":0,"prepEntityType":"chapter"},{"id":15316,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Fifteen","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-fifteen-ns","order":0,"prepEntityType":"chapter"},{"id":15314,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Fourteen","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-fourteen-ns","order":0,"prepEntityType":"chapter"},{"id":15312,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-thirteen-ns","order":0,"prepEntityType":"chapter"},{"id":15310,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Twelve","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-twelve-ns","order":0,"prepEntityType":"chapter"},{"id":15308,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Eleven","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-eleven-ns","order":0,"prepEntityType":"chapter"},{"id":15306,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Ten","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-ten-ns","order":0,"prepEntityType":"chapter"},{"id":15304,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Nine","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-nine-ns","order":0,"prepEntityType":"chapter"},{"id":14805,"name":"Maths Ncert Solutions class 11th","url":"/preparation/maths-ncert-solutions-class-11th-ns","order":0,"prepEntityType":"chapter"},{"id":15300,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Seven","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-seven-ns","order":0,"prepEntityType":"chapter"},{"id":15298,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Six","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-six-ns","order":0,"prepEntityType":"chapter"},{"id":15296,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Five","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-five-ns","order":0,"prepEntityType":"chapter"},{"id":15294,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Four","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-four-ns","order":0,"prepEntityType":"chapter"},{"id":15292,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Three","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-three-ns","order":0,"prepEntityType":"chapter"},{"id":15290,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter Two","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-two-ns","order":0,"prepEntityType":"chapter"},{"id":15288,"name":"Maths NCERT Exemplar Solutions Class 11th Chapter One","url":"/preparation/maths-ncert-exemplar-solutions-class-11th-chapter-one-ns","order":0,"prepEntityType":"chapter"}]},"prepQuestions":null,"sectionalQuizData":{},"usersInfo":{"2582250":{"userId":2582250,"firstName":"UPASANA","lastName":"PRADHAN","name":"UPASANA PRADHAN","displayName":"UPASANA75632","city":64,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/UPASANA75632","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/UPASANA75632","isAbroad":false,"description":"Upasana has more than six years of experience in the education industry, especially in engineering and medical content articles. She’s an electronics and communication engineering graduate from DTU and covers engineering, IT news and articles at Shiksha. In her spare time, she has a soft spot for reading fictional novels and watching K-dramas.","twitterLink":"","facebookLink":"","linkedinLink":"https://www.linkedin.com/in/upasana-pradhan-785b42142/","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":78,"whatsappPermission":"yes"},"18325315":{"userId":18325315,"firstName":"Sreetama Datta","lastName":"Roy","name":"Sreetama Datta Roy","displayName":"SreetamaDattaRoy","city":74,"locality":null,"country":2,"imageURL":"/mediadata/images/1658923865php0Ah1UQ.jpeg","workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/SreetamaDattaRoy","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/SreetamaDattaRoy","isAbroad":false,"description":"Sreetama has over a decade of editorial experience in the higher education beat. A wanderlust by passion, she is also an avid reader and a music lover. Among friends, she is popular for her sense of humour.","twitterLink":"https://twitter.com/sreetama24","facebookLink":"","linkedinLink":"https://in.linkedin.com/in/sreetama-datta-bb418716","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":78,"whatsappPermission":"yes"},"29781649":{"userId":29781649,"firstName":"Satyendra","lastName":"Singh","name":"Satyendra Singh","displayName":"Satyendra118347","city":74,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/Satyendra118347","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/Satyendra118347","isAbroad":false,"description":"\u003cp style=\"text-align: justify;\">\u003cstrong>\u003cspan class=\"cf0\">Engineering Education & Entrance Exam Expert\u003c/span>\u003c/strong>\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cspan class=\"cf0\">\u003cstrong>Role\u003c/strong>:\u003c/span>\u003cspan class=\"cf1\"> Senior News Writer & Content Editor, Shiksha.com\u003c/span>\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>\u003cspan class=\"cf0\">Education:\u003c/span>\u003c/strong> \u003cspan class=\"cf1\">B.Tech\u003c/span>\u003cspan class=\"cf1\"> in Mechanical Engineering [IET], MA Education Philosophy\u003c/span>\u003c/p>\n\u003cp style=\"text-align: justify;\">Satyendra Singh is an Engineering graduate with a Master degree in Education Philosophy, working in Education journalism for the past 10 years. At Shiksha.com, he writes specifically in engineering entrance exams like JEE Main, GATE, MHT CET covering the admission process, syllabus & exam pattern updates and preparation modules. His engineering background helps him produce the right content required for engineering aspirants backed with factual accuracy. He has authored more than 500 articles, helping millions of aspirants make a decisive exam and career choice.\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>Highlights:\u003c/strong>\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>Domain Expertise:\u003c/strong> Engineering education background helps him developing content for preparation as well as admission process as per the demand of various exams and their aspirants.\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>Teaching Experience:\u003c/strong> Before shifting to education journalism, he has a brief stint as a Mathematics faculty guiding JEE and Olympiads aspirants. This helps him in understanding the dilemmas and tyranny of aspirants and write content accordingly.\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>Trusted Voice:\u003c/strong> Articles written by him has around 44 million page views, a testament of faith in his content put by aspirants.\u003c/p>\n\u003cp style=\"text-align: justify;\">Satyendra always believes that every student has the potential to shine, if guided with right and meaningful content.\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>Contact:\u003c/strong> \u003ca href=\"mailto:satyendra.singh@shiksha.com\" rel=\"nofollow\">satyendra.singh@shiksha.com\u003c/a>\u003c/p>\n\u003cp style=\"text-align: justify;\">\u003cstrong>LinkedIn:\u003c/strong> \u003ca href=\"https://www.linkedin.com/in/satyendra-singh-6764bb119/\" rel=\"nofollow\">Satyendra Singh\u003c/a>\u003c/p>","twitterLink":"","facebookLink":"","linkedinLink":"https://www.linkedin.com/in/satyendra-singh-6764bb119/","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":63,"whatsappPermission":"yes"},"35358691":{"userId":35358691,"firstName":"Md","lastName":"Shahzad","name":"Md Shahzad","displayName":"Md105063","city":74,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/Md105063","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/Md105063","isAbroad":false,"description":"Shahzad has over seven years of editorial experience in different domains of the education sector. He's been furnishing guidance and support to young students, offering valuable information to steer them on their educational journey. On the personal front, he is an avid reader and a music & movie lover. In his spare time, he loves to capture moments through lenses.","twitterLink":"","facebookLink":"","linkedinLink":"","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":63,"whatsappPermission":"no"},"45077693":{"userId":45077693,"firstName":"Shikha","lastName":"Goyal","name":"Shikha Goyal","displayName":"Shikha268477","city":74,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/Shikha268477","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/Shikha268477","isAbroad":false,"description":"Shikha Goyal is an author with expertise in various domains, especially engineering. She has postgraduate degrees in Mathematics as well as Mass Communication and Journalism. She has devoted her life to helping students find the right path to success and has been in the educational content field for more than 8 years in different roles, from writer to editor to subject matter expert. In her free time, she likes to read, travel, or bake.","twitterLink":"","facebookLink":"","linkedinLink":"","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":63,"whatsappPermission":"no"},"48889573":{"userId":48889573,"firstName":"Salviya","lastName":"Antony","name":"Salviya Antony","displayName":"SalviyaAntony29280","city":124,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/SalviyaAntony29280","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/SalviyaAntony29280","isAbroad":false,"description":"Salviya specialises in writing content on board exams. With a B.Tech degree from Kannur University,\nshe has 3+ years of work experience in the education industry. She loves travelling, cooking, and watching movies.","twitterLink":"","facebookLink":"","linkedinLink":"","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":77,"whatsappPermission":"no"},"65362629":{"userId":65362629,"firstName":"Mayank","lastName":"Uniyal","name":"Mayank Uniyal","displayName":"Mayank86659380","city":1616,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/Mayank86659380","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/Mayank86659380","isAbroad":false,"description":"Mayank Uniyal is a seasoned academic content creator at shiksha.com. Having a master's degree in International Business, he is creating content across several domains including CUET, Medicine and Universities. Apart from creating content, you may find him travelling to several destinations or trying authentic cuisines. ","twitterLink":"","facebookLink":"","linkedinLink":"","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":63,"whatsappPermission":"yes"},"96365866":{"userId":96365866,"firstName":"Samridhi","lastName":"Mishra","name":"Samridhi Mishra","displayName":"Samridhi2025818964","city":74,"locality":null,"country":2,"imageURL":null,"workExperience":null,"usergroup":"cms","profilePageURL":"/userprofile/Samridhi2025818964","abroadProfilePageURL":null,"passport":null,"anyProfileExists":true,"userContactInfo":null,"userPref":null,"userSCMProfile":null,"userEducation":null,"userPrefLocation":null,"userAdditionalInfo":null,"userSocialSignUpInfo":null,"userAuthorInfo":{"activeAuthor":true,"authorUrl":"/author/Samridhi2025818964","isAbroad":false,"description":"Samridhi Mishra is an experienced writer with a strong focus on engineering education. She creates engaging and informative content across various formats, including news articles, live blogs, exam pages, social media scripts and in-depth guides. With a deep understanding of competitive exams like JEE, GATE, and other engineering entrance tests, she aims to add real value to the education and exam-prep community. She can be contacted at samridhi.mishra@shiksha.com.","twitterLink":"","facebookLink":"","linkedinLink":"https://www.linkedin.com/in/samridhi-mishra-3793b9297/","abroad":false},"admissionTakenInfo":null,"allUserSCMProfile":null,"userConsentSection":null,"isSOUser":"NO","delete":"no","deactivate":"no","userCustomQuestion":null,"profileCompletionScore":70,"whatsappPermission":"yes"}},"isAbroad":false,"abroad":false,"domain":"national","isCriticalCSSRequired":true,"isOnline":false},"allExamData":{},"collegePredictorForm1Data":{},"collegePredictorForm2Data":{},"collegePredictorForm3Data":{},"collegePredictorResults":{},"rankPredictorForm3Data":{},"recommendationData":{},"NewPageFormData":{},"userCookieData":{"userCookieData":false},"articleData":{},"globalAPIErrorState":{"showErrorPage":false,"errType":"SNAP"},"articleDetailPage":{},"adpTagType":{},"comparePageData":{},"collegeReviewFormData":{},"OAFHomeData":{},"OAFClientPageData":{},"shortlistPageData":{},"careerTestPageData":{},"globalData":{"deviceType":"desktop","domain":"domestic","isAbroad":false,"isPdf":false,"mobileApp":false,"isRnApp":false,"appVersion":"","experimentName":"","isOnline":false,"recoWidgetType":null,"isGPTAccessEnabled":false},"abroadData":{"isAbroadEntity":false},"authorPageData":{},"reviewQuestPageData":{},"layout":{},"isUserInteracted":{"isUserInteracted":false},"currencyData":{"currency":[{"sourceCurrencyCode":"INR","destinationCurrencyCode":"INR","conversionFactor":1,"isSelected":true}]},"pclData":[],"editProfilePageData":{},"webinarDetailPage":{},"aboutUsPageData":{},"publicProfilePageData":{},"iulpTrackingConfig":null,"AnAPostingPage":{},"isGooglebot":{"isGooglebot":true},"otpSkipCase":{"otpSkipCase":false},"stickyData":{},"saMspPageData":{"pageNo":-1,"baseCourseInfo":[],"countryInfo":[],"recommendedExamsData":null,"baseCourseCountryCTPUrl":"","recommendedInstitutes":null,"institutesByLocation":null,"universityCollections":null,"currencyCodeRate":{"sourceCurrencyCode":null,"destinationCurrencyCode":null,"conversionFactor":0},"budgetFilter":[],"locationFilters":[],"scholarshipDetails":null,"specializationFilters":[],"examFilters":[],"applyCounsellingUrl":"","loadingData":false,"counsellors":null,"aliasMapping":null,"postCounselling":null,"inputParamString":"","rhsActivity":null,"partneredInstitutes":null,"currencyData":null,"articleWidget":[],"recommendedCourses":null,"mostpopularGuides":[],"bhstArticle":""},"rcpPageData":{},"atfCtaData":{},"upvoteData":[],"sectionToIndexMap":null,"rolPromptData":{},"sectionNameClickedOnTOC":"","instituteCompare":{},"isPDFCall":false,"inlineRegistrationFormData":null,"courseFinderResultPageData":{},"whatIsPageData":{},"cashbackState":{},"consentReducerData":{},"l2MenuNavigation":null,"uilpPageLoadEvent":null,"isCsrNavigation":false,"personalizedATFData":{"data":{},"loading":false,"error":null},"isExtendedBot":{"isExtendedBot":true},"PollPageData":{},"soMspPageData":{"allLoaders":{"topBannerWid":false,"categoryListWid":false,"horizontalCourseListWid":false,"soDiscoverTrendingWid":false,"topCareerOpportunitiesWid":false,"browseCoursesWid":false,"userReviewsWid":false,"articlesNewsListWid":false},"articleData":[],"topProviderData":[],"topCareerData":[],"popularCourses":[],"allCategories":[]}};
window.serviceWokrerFile = "service-worker.js";
window.COOKIEDOMAIN = ".shiksha.com";
window.sessionID = "";
window.fromWherePWA = "examPageData";
window.userCookieFlag = "false";
window.GA4Id = "G-K11YYW39J9";
window.isUserLoggedIn = false;
window.isHeaderFixed = false;
window.isCompareEnable = false;
window.shikshaProduct = 'default';
window.productPage = 'default';
window.deviceType = 'desktop';
window.preloadedUrls = [];
var jsFilesArr = []
setTimeout(() => {lazyLoadJs(document,"script", jsFilesArr)}, 900);window.onerror = function (msg, url, line, col, exception) { if (!url || url == '' || line == 0 || col == 0 || !exception) { return; } var img = new Image(); var srcUrl = 'https://apis.shiksha.com/appbeat/logJSError?msg=' + encodeURIComponent(msg) + '&line=' + encodeURIComponent(line) + '&url=' + encodeURIComponent(url) + '&col=' + encodeURIComponent(col) + '&exception=' + encodeURIComponent(exception) + '&currentUrl=' + encodeURIComponent(window.location.href); img.src = srcUrl; }
function getCookie(e){return document.cookie.length>0&&(c_start=document.cookie.indexOf(e+"="),-1!=c_start)?(c_start=c_start+e.length+1,c_end=document.cookie.indexOf(";",c_start),-1==c_end&&(c_end=document.cookie.length),unescape(document.cookie.substring(c_start,c_end))):""}
function setCookie(e,t,o,n){"undefined"==typeof COOKIEDOMAIN&&(COOKIEDOMAIN="");var c=new Date;c.setTime(c.getTime());var i=0;if(o="undefined"!=typeof o?o:0,n="undefined"!=typeof n?n:"days",0!=o){o="seconds"==n?1e3*o:1e3*o*60*60*24;var d=new Date(c.getTime()+o),i=d.toGMTString();"homepage"==n&&(i=getCookie("homepage_ticker_track")),document.cookie=e+"="+escape(t)+";path=/;domain="+COOKIEDOMAIN+(null==o?"":";expires="+i)}else document.cookie=e+"="+escape(t)+";path=/;domain="+COOKIEDOMAIN;return""==document.cookie?!1:!0}
function showShikshaAssistant(){
if(window.assistantInit === true){
return;
}
//if(window.location && window.location.href && window.location.href.indexOf("cbse") !==-1)
// return;

if(!(document.getElementById("chat-container") == null)){
window.assistantInit = true;
var d = new Date();
var imported = document.createElement('script');
//imported.src = 'https://js.shiksha.ws/public/js/chatplugin.js?v=24920259ta';
imported.src = 'https://js.shiksha.ws/public/js/chatplugin.js?v='+d.getDate()+''+(d.getMonth()+1)+''+d.getFullYear()+'9THOct';
// imported.title = 'https://js.shiksha.ws/public/js/build/';
imported.async = 1;
// imported.defer = 1;
document.head.appendChild(imported);
}
}

window.isAbroad = false;

if(window.assistantInit === false || typeof(window.assistantInit) === "undefined"){
(window.isAbroad) ? showShikshaAssistant() : setTimeout(()=>{showShikshaAssistant()},3000);
}
window.hideGdpr = 0;
window.isInternationalUser = true;
window.ENV = 'production';
window.adSenseScriptStatus = 0;
window.fromWherePWA = "examPageData";
window.isPartytownEnabled = 'yes' == 'yes';

// window.showGoogleAdSense = 'no';

const scrollToElementById = (elementId, scrollOffset = 160) => {
const element = document.getElementById(elementId);
if(element){
window.scrollTo({
behavior: "smooth",
top: element.getBoundingClientRect().top - document.body.getBoundingClientRect().top - scrollOffset,
});
}
}

const isTargetElementClicked = (target, targetCategory) => {
switch(targetCategory) {
case 'chat':
return target.closest('.chat--prompt-col, .prompt-chatboat, .primary-chat-icon');
case 'close':
return target.closest('.chat--prompt_close');
case 'exam':
return fromWherePWA != "applyPage" && target.closest('h1');
case 'noclicks':
return target.closest('.photo-widget-full, #college-search-input, #search-with-suggester-input, #filter-search-input, #flticon');
case 'tocAnchor':
return target.closest('.toc-anchor');
default:
return false;
}
}

// Interceptor listener — MUST be in capture phase before gtm
document.addEventListener('click', function (e) {
const target = e.target;

if (isTargetElementClicked(target, 'chat')) {
console.log('toggle chat');
e.stopImmediatePropagation();
setTimeout(toggleChatWindow, 50);
}
else if (isTargetElementClicked(target, 'close')) {
console.log('close chat');
e.stopImmediatePropagation();
setTimeout(() => {
closeChatPrompt();
showChatNotification();
}, 50);
}
else if(isTargetElementClicked(target, 'exam')){
e.stopImmediatePropagation()
}
else if(isTargetElementClicked(target, 'noclicks')){
e.stopImmediatePropagation();
}
else if(isTargetElementClicked(target, 'tocAnchor')){
e.stopImmediatePropagation();
e.preventDefault();
setTimeout(() => {
const liId = "li_" + target.href?.split("#")?.[1];
const liElement = document.getElementById(liId);
if(liElement){
liElement.click();
}
}, 50);
}
}, true); // capture = true

// Optimized GTM initialization for better INP score
let gtmInitialized = false;
let gtmInitScheduled = false;

// Modern scheduling utilities
function scheduleGTMInit() {
if (gtmInitScheduled) return;
gtmInitScheduled = true;

// Use scheduler API if available, fallback to requestIdleCallback
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
loadGTMScripts();
}, { priority: 'background' });
} else if ('requestIdleCallback' in window) {
requestIdleCallback(() => {
loadGTMScripts();
}, { timeout: 2000 });
} else {
// Fallback for older browsers
setTimeout(() => {
loadGTMScripts();
}, 10000);
}
}
//Important: moving this to App.js for experiment
if('yes' == 'yes'){
setTimeout(function(){lazyLoadJs(document,"script", ["https://www.googletagmanager.com/gtag/js?id=G-K11YYW39J9"],true);
window.dispatchEvent(new CustomEvent('ptupdate'));},0)
}

function loadGTMScripts() {
if (gtmInitialized) return;
gtmInitialized = true;

// Remove event listeners immediately to prevent re-triggering
document.body.removeEventListener('click', handleUserInteraction);
window.removeEventListener('scroll', handleUserInteraction, { passive: true });

// Load GTM with proper scheduling
if ('scheduler' in window && 'postTask' in window.scheduler) {
setTimeout(() => {
window.scheduler.postTask(() => {
initializeGTM();
}, { priority: 'background' });
}, 10000);
} else {
requestIdleCallback(() => {
initializeGTM();
}, { timeout: 8000 });
}
}

function initializeGTM() {
// GTM initialization code
(!function(e,t,n,o,a){
e[o]=e[o]||[];
e[o].push({"gtm.start":(new Date).getTime(),event:"gtm.js"});
var i=t.getElementsByTagName(n)[0],c=t.createElement(n),d="dataLayer"!=o?"&l="+o:"";
c.defer=!0;
c.src="https://www.googletagmanager.com/gtm.js?id="+a+d;
i.parentNode.insertBefore(c,i);
}(window,document,"script","dataLayer",false ? "GTM-WXXDPPBN" : "GTM-5FCGK6"));

// Load GPT script with delay
setTimeout(() => {
lazyLoadJs(document,"script",["https://www.googletagservices.com/tag/js/gpt.js"]);
}, 2000);
}

// Optimized user interaction handler
function handleUserInteraction() {
if (gtmInitialized) return;

// Use scheduler API for better performance
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
scheduleGTMInit();
}, { priority: 'user-visible' });
} else {
// Fallback with minimal delay
setTimeout(() => {
scheduleGTMInit();
}, 10);
}
}

// Attach optimized event listeners
document.body.addEventListener('click', handleUserInteraction, { passive: true });
window.addEventListener('scroll', handleUserInteraction, { passive: true });

window.addEventListener('message', receiveMessage, false);

function receiveMessage(event) {
if(event.data =='close_dfp'){
setTimeout(()=>{
closeBanner1();
},0)
}
}

if('0' == 1){
window.pushNotiConf = {
apiDomain : 'https://apis.shiksha.com',
deviceType : 'desktop',
fromPWA : true
};

function loadBNDependency(){if(null!=document.getElementById("pushNotificationContainer")){var e=new Date,t=e.getDate()+""+e.getMonth()+e.getFullYear()+"v10",o=Math.floor(100*Math.random()),n=function(){var e=document.createElement("script");e.src="https://js.shiksha.ws/pwa/public/unavoidable_js/shikshaPushNotification.js?"+t+o,e.async=!0,document.head.appendChild(e)},c=document.createElement("script");c.src="https://js.shiksha.ws/pwa/public/push-notification-helpers.js?"+t+o,c.async=!0,document.head.appendChild(c),"object"==typeof c.onload?c.onload=function(e){console.log("notification helper sdk loaded"),n()}:n()}}
}

window.disableInstant = true;

if (typeof history !== 'undefined' && 'scrollRestoration' in history) {
history.scrollRestoration = 'manual';
}

var spanForPDF = document.createElement('span');
spanForPDF.id = 'pageCSSPathForPDF';
spanForPDF.style = 'display:none;';
var att = document.createAttribute("csspath");
spanForPDF.setAttributeNode(att);
document.body.appendChild(spanForPDF);

function loadMultipleCSS(cssArrayList){
if(cssArrayList && cssArrayList.length>0){
cssArrayList.forEach(function(url) {
var script = document.createElement('link');
script.href = url;
script.rel = 'stylesheet';
script.type = 'text/css';
document.body.appendChild(script);
});
}
}

function loadMultipleScriptOneByOne(scriptArayList){
return new Promise( function( resolve, reject ) {
var listItem = scriptArayList.shift();
if(listItem){
var s = document.createElement('script');
var r = false;
s.type = 'text/javascript';
s.src = listItem;
s.async = true;
s.onerror = function(err) {
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
reject(err, s);
}
};
s.onload = s.onreadystatechange = function() {
// console.log(this.readyState); // uncomment this line to see which ready states are called.
if (!r && (!this.readyState || this.readyState == 'complete')) {
r = true;
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
resolve();
}
}
};
var t = document.getElementsByTagName('script')[0];
t.parentElement.insertBefore(s, t);
}else{
resolve();
}
});
}
function openChatBotRegistration(paramsObject,callBackFunction,onClose){
//versioning the files
var eVr=new Date();
var tVr= eVr.getDate()+""+eVr.getMonth()+eVr.getFullYear()+"98";
var cssArray = ['https://js.shiksha.ws/public/css/convMMP.css?v='+tVr];
var jsArray = ['https://js.shiksha.ws/public/js/jquery-3.5.1.min.js?v='+tVr,'https://js.shiksha.ws/public/js/vue.js?v='+tVr,'https://js.shiksha.ws/public/js/chatui.js?v='+tVr,'https://js.shiksha.ws/public/js/convMMP.js?v='+tVr];
// && paramsObject.entityId > 0
if(typeof(paramsObject) == 'object' && paramsObject !== null && (paramsObject.entityId > 0 || paramsObject.allowNullContId == true )){
if(typeof(loadConversationalMMP) === 'function'){
loadConversationalMMP(paramsObject,callBackFunction,onClose);
}else{
loadMultipleCSS(cssArray);
loadMultipleScriptOneByOne(jsArray).then(function (res) {
loadConversationalMMP(paramsObject,callBackFunction,onClose);
})
}
}

}
// window.blockDfpConfig = {'examPageMain': [21, 23], 'articlePage': []};
window.blockDfpConfig = {'examPageMain': [], 'articlePage': []};
window.shouldShowPEQ = true;
window.shouldShowDRC = true;
window.shouldEnableEventTracking = true;
window.shouldEnableExperimentalDFPEventTracking = false;<iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5FCGK6"
height="0" width="0" style="display:none;visibility:hidden"></iframe>!function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function(){n.callMethod?
n.callMethod.apply(n,arguments):n.queue.push(arguments)};if(!f._fbq)f._fbq=n;
n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0;
t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.appendChild(t,s)}(window,
document,'script','https://connect.facebook.net/en_US/fbevents.js');
fbq('init', '639671932819149');
fbq('track', "PageView");<img height="1" width="1" style="display:none"
src="https://www.facebook.com/tr?id=639671932819149&ev=PageView&noscript=1"
/>[118,51876,81059,26191,39658,89734,90205,55768,93281,75997,34246,17652,78029,93095,55232,82321,99774,54271,8424,90239,32215,75996,13891,97401,78305,25721,6429,16751,34680,57318,32714,78002,15303,4679,15724,64827,11898,32226,49277,68707,42397,874,93650]{"namedChunks":["desktop_header","ExamPageMain","ExamTopCard","PrepPageCTA","ExamApplyNow","ExamMenuCritical","ExamSectionalHomePage","RenderSectionHTML","SectionalECPToc","WikiLabelSectionalECP","AnATaggedFAQ","ANA","Icons","ExamSimilarWithTracking","PrepTypeWidget","AnACommon","ListWidget","CourseAfterWidget","MSPNewsComponent","desktop_footer","ShikshaCommonFooter"]}function stopHashScroll(event) {
event.preventDefault();
window.scrollTo(0, 0); // Scroll back to the top
}
// Function to re-enable scrolling
function enableHashScroll() {
window.removeEventListener("scroll", stopHashScroll);
}
const hash = window.location.hash;
if(hash) {
// Prevent the default scrolling behavior when the page has an anchor in the URL
window.addEventListener("scroll", stopHashScroll);
// Enable scroll when the page loads
window.addEventListener("load", enableHashScroll);
}window.dataLayer = window.dataLayer || [];
window.gtag = function (){dataLayer.push(arguments);}
window.gtag('js', new Date());
window.gtag('config', "G-K11YYW39J9");}{l i m}$