Tags Class 12th

Class 12th

Get insights from 12k questions on Class 12th, answered by students, alumni, and experts. You may also ask and answer any question you like about Class 12th

Follow Ask Question

12k

Questions

Discussions

Active Users

Followers

New answer posted

a year ago

Class 12th Differential Equations

$S o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n t a n y s e c^{2} x d x + t a n x s e c^{2} y d y = 0$ is:

(A) $tanx + tany = k$

(B) $tanx -tan y = k$

(C) $\frac{tanx}{tany} = k$

(D) $tanx \cdot tany = k$

0 Follower 3 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} T h e g i v e n d i f f e r e n t i a l e q u a t i o n i s \\ t a n y s e c^{2} x d x + t a n x s e c^{2} y d y = 0 \\ \Rightarrow t a n x s e c^{2} y d y = - t a n y s e c^{2} x d x \\ \Rightarrow \frac{s e c^{2} y}{t a n y} . d y = \frac{- s e c^{2} x}{t a n x} . d x \\ I n t e g r a t i n g b o t h s i d e s, w e g e t \\ \Rightarrow \int \frac{s e c^{2} y}{t a n y} . d y = \int \frac{- s e c^{2} x}{t a n x} . d x \\ \Rightarrow l o g | t a n y | = - l o g | t a n x | + l o g c \\ \Rightarrow l o g | t a n y | + l o g | t a n x | = l o g c \\ \Rightarrow l o g | t a n x . t a n y | = l o g c \\ \Rightarrow t a n x . t a n y = k [? l o g c = k] \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

Integrating factor of the differential equation $c o s x \frac{d y}{d x} + y s i n x = 1$ is:

(A) $cosx$

(B) $tanx$

(C) $secx$

(D) $sinx$

0 Follower 2 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n d i f f e r e n t i a l e q u a t i o n i s \\ c o s x . \frac{d y}{d x} + y s i n x = 1 \\ \Rightarrow \frac{d y}{d x} + \frac{s i n x}{c o s x} y = \frac{1}{c o s x} \Rightarrow \frac{d y}{d x} + t a n x y = s e c x \\ H e r e, P = t a n x a n d Q = s e c x \\ I n t e g r a t i n g f a c t o r = e^{\int P d x} = e^{\int t a n x d x} = e^{l o g s e c x} = s e c x \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

Solution of the differential equation $x d y - y d x = 0$ represents:

(A) A rectangular hyperbola

(B) A parabola whose vertex is at the origin

(C) A straight line passing through the origin

(D) A circle whose center is at the origin

0 Follower 4 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n d i f f e r e n t i a l e q u a t i o n i s \\ x d y - y d x = 0 \\ \Rightarrow \frac{d y}{d x} = \frac{y}{x} \Rightarrow \frac{d y}{y} = \frac{d x}{x} \\ I n t e g r a t i n g b o t h s i d e s, w e g e t \\ \int \frac{d y}{y} = \int \frac{d x}{x} \\ \Rightarrow l o g y = l o g x + l o g c \\ \Rightarrow l o g y = l o g x c \\ \Rightarrow y = x c w h i c h i s a s t r a i g h t l i n e passing t h r o u g h t h e o r i g i n . \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

The differential equation for $y = A c o s α x + B s i n α x$ , where A and B are arbitrary constants, is

(A) $\frac{d^{2} y}{d x^{2}} - α^{2} y = 0$

(B) $\frac{d^{2} y}{d x^{2}} + α^{2} y = 0$

(C) $\frac{d^{2} y}{d x^{2}} + α y = 0$

(D) $\frac{d^{2} y}{d x^{2}} - α y = 0$

0 Follower 4 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n e q u a t i o n i s y = A c o s α x + B s i n α x \\ d i f f e r e n t i a t i n g b o t h s i d e s, w . r . t . x, w e g e t \\ \frac{d y}{d x} = - A s i n α x . α + B c o s α x . α \\ \frac{d y}{d x} = - A α s i n α x + B α c o s α x \\ A g a i n d i f f e r e n t i a t i n g b o t h s i d e s, w . r . t . x, w e g e t \\ \frac{d^{2} y}{d x^{2}} = - A α^{2} c o s α x - B α^{2} s i n α x \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - α^{2} (A c o s α x + B s i n α x) \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - α^{2} y \\ \Rightarrow \frac{d^{2} y}{d x^{2}} + α^{2} y = 0 \\ H e n c e, t h e c o r r e c t o p t i o n i s (b) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

If $y = e^{- x} (A c o s x + B s i n x)$ , then $y$ is a solution of

(A) $\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} = 0$

(B) $\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0$

(C) $\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} + 2 y = 0$

(D) $\frac{d^{2} y}{d x^{2}} + 2 y = 0$

0 Follower 2 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n e q u a t i o n i s y = e^{- x} (A c o s x + B s i n x) \\ d i f f e r e n t i a t i n g b o t h s i d e s, w . r . t . x, w e g e t \\ \frac{d y}{d x} = e^{- x} (- A s i n x + B c o s x) - e^{- x} (A c o s x + B s i n x) \\ \frac{d y}{d x} = e^{- x} (- A s i n x + B c o s x) - y \\ A g a i n d i f f e r e n t i a t i n g b o t h s i d e s, w . r . t . x, w e g e t \\ \frac{d^{2} y}{d x^{2}} = e^{- x} (- A c o s x - B s i n x) - e^{- x} (- A s i n x + B c o s x) - \frac{d y}{d x} \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - e^{- x} (A c o s x + B s i n x) - [\frac{d y}{d x} + y] - \frac{d y}{d x} \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - y - \frac{d y}{d x} - y - \frac{d y}{d x} \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - 2 \frac{d y}{d x} - 2 y \Rightarrow \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} + 2 y = 0 \\ H e n c e, t h e c o r r e c t o p t i o n i s (c) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

The order and degree of the differential equation $\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{\frac{1}{4}} + x^{\frac{1}{5}} = 0$ respectively, are

(A) 2 and not defined

(B) 2 and 2

(C) 2 and 3

(D) 3 and 3

0 Follower 4 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n d i f f e r e n t i a l e q u a t i o n i s \\ \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{\frac{1}{4}} + x^{\frac{1}{5}} = 0 \\ \Rightarrow \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{\frac{1}{4}} = - x^{\frac{1}{5}} \\ Since, t h e o f \frac{d y}{d x} i s i n f r a c t i o n . \\ S o, t h e degree o f t h e g i v e n d i f f e r e n t i a l e q u a t i o n i s n o t d e f i n e d a s t h e o r d e r i s 2 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (a) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

The degree of the differential equation ${[1 + {(\frac{d y}{d x})}^{2}]}^{\frac{3}{2}} = \frac{d^{2} y}{d x^{2}}$ is

(A) 4

(B) $\frac{3}{2}$

(C) Not defined

(D) 2

0 Follower 9 Views

alok kumar singh

Contributor-Level 10

This is an Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} T h e g i v e n d i f f e r e n t i a l e q u a t i o n i s \\ {[1 + {(\frac{d y}{d x})}^{2}]}^{3 / 2} = (\frac{d^{2} y}{d x^{2}}) \\ S q u a r i n g b o t h s i d e s, w e h a v e \\ {[1 + {(\frac{d y}{d x})}^{2}]}^{3} = {(\frac{d^{2} y}{d x^{2}})}^{2} \\ S o, t h e degree o f t h e g i v e n d i f f e r e n t i a l e q u a t i o n i s 2 . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

0 0

New answer posted

a year ago

Class 12th Differential Equations

Choose the correct answer from the given four options in each of the Exercises from 1 to 42 (M.C.Q)

Q1.The degree of the differential equation ${(\frac{d^{2} y}{d x^{2}})}^{2} + {(\frac{d y}{d x})}^{2} = x s i n x (\frac{d y}{d x})$ is:

(A) 1

(B) 2

(C) 3

(D) Not defined

0 Follower 2 Views

alok kumar singh

Contributor-Level 10

This is a Objective Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} T h e degree o f g i v e n d i f f e r e n t i a l e q u a t i o n i s n o t d e f i n e d b e c a u s e t h e v a l u e o f \\ s i n (\frac{d y}{d x}) o n expansionwillbeincreasing p o w e r o f (\frac{d y}{d x}) . \\ H e n c e, t h e c o r r e c t o p t i o n i s (d) . \end{array}$

0 0

New question posted

a year ago

Class 12th Differential Equations

State True or False for the following:

(i) The integrating factor of a differential equation of the form $\frac{d y}{d x} + P_{1} x = Q_{1}$ is given by $e^{\int P_{1} d y} .$

(ii) solution of a differential equation of the type $\frac{d y}{d x} + P_{1} x = Q_{1}$ is given by $y \cdot I . F . = (I . F .) Q_{1} d y .$

(iii) Correct substitution for solving the differential equation of the type $\frac{d y}{d x} = f (x, y),$ where $f (x, y)$ is a homogeneous function of zero degree, is $y = v x$ .

(iv) Correct substitution for the solution of the differential equation of the type dx/dy = g(x, y) where g (x, y) is a homogeneous function of the degree zero is x = vy.

(v) Number of arbitrary constants in the particular solution of a differential equation of o

...more

State True or False for the following:

(i) The integrating factor of a differential equation of the form $\frac{d y}{d x} + P_{1} x = Q_{1}$ is given by $e^{\int P_{1} d y} .$

(ii) solution of a differential equation of the type $\frac{d y}{d x} + P_{1} x = Q_{1}$ is given by $y \cdot I . F . = (I . F .) Q_{1} d y .$

(iii) Correct substitution for solving the differential equation of the type $\frac{d y}{d x} = f (x, y),$ where $f (x, y)$ is a homogeneous function of zero degree, is $y = v x$ .

(iv) Correct substitution for the solution of the differential equation of the type dx/dy = g(x, y) where g (x, y) is a homogeneous function of the degree zero is x = vy.

(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two.

(vi) The differential equation representing the family of circles x² + (y – a)² = a² will be of order two.

(vii) The solution of dy/dx = (y/x)^1/3is y ^2/3-x^2/3= c.

(viii) Differential equation representing the family of curves

y e^x (A cos x + B sin x is d²y/dx² – 2dy/dx + 2y = 0

(ix) The solution of the differential equation dy/dx = x + 2y/x is x + y = kx².

(x) Solution of xdy/dx = y +x tan y/x is sin (y/x) =cx

(xi) The differential equation of all non- horizontal lines in a plane is d²y/dx²=0

less

0 Follower 3 Views

New answer posted

a year ago

Class 12th Differential Equations

Fill in the blanks for the following (I to xi)

(i) The degree of the differential equation $\frac{d^{2} y}{d x^{2}} + e^{\frac{d y}{d x^{}}} + 0$ is _________.

(ii) The degree of the differential equation is _________.

(iii) The number of arbitrary constants in the general solution of a differential equation of order three is _________.

(iv) $\frac{d y}{d x} + \frac{y}{x l o g x} = \frac{1}{x}$ is an equation is of the type _________.

(v) The general solution of a differential equation of the type $\frac{d y}{d x} + P_{1} x = Q_{1}$ is given by _________.

(vi) The solution of $\frac{x d y}{d x} + 2 y = x^{2}$ is _________.

(vii) The solution of $(1 + x^{2}) \frac{d y}{d x} + 2 x y - 4 x^{2} = 0$ is _________.

(viii) The solution of $y d x + (x + x y) d y = 0$ is _________.

(ix) The general solution of $\frac{d y}{d x} + y = s i n x$ is ____