Let A = { z ∈ C : 1 ≤ | z − ( 1 + i ) ≤ 2 } and B = { z ∈ A : | z − ( 1 − i ) | = 1 } . Then, B :

A = { z ∈ c : 1 ≤ | z − ( 1 + i ) | ≤ 2 } | z − ( 1 + i ) | ≥ 1 and | z − ( 1 + i ) | ≤ 2 and also | z − ( 1 − i ) | = 1 hence B has infinite set.

The remainder when 32022 is divided by 5 is :

(3 * 3 * 3 * ………2022 times) ¸ 5 Remainder = (-1) (-1) (-1) ….1011 times) Remainder = -1 + 5 = 4

The number of distinct real roots of the equation x5 (x3−x2−x+1) + x(3x3−4x2−2x+4)−1=0 is …………..

x8−x7−x6+x5+3x4−4x3−2x2+4x−1=0 ⇒x7 (x−1)−x5 (x−1)+3x3 (x−1)−x (x2−1)+2x (1−x)+ (x−1)=0 ⇒ (x−1) (x2−1) (x5+3x−1)=0∴x=±1 are roots of above equation and x5 + 3x – 1 is a monotonic term hence vanishes at exactly one value of x other then 1 or 1. ∴ 3 real roots.

Let R be a relation defined on the set of natural numbers N as follows: R={(x,y):x∈N,y∈N,2x+y=41} . Find the domain and range of the relation R . Also, verify whether R is reflexive, symmetric, and transitive.

This is a Long Answer Type Question as classified in NCERT Exemplar Sol: Given?? that x∈ N, y∈N and 2x + y=41 ∴ Domain of R = {1, 2, 3, 4, 5, …, 20} and Range = {39, 37, 35, 33, 31, …, 1} Here, (3, 3) ∉ R as 2× 3 + 3 ≠ 41 So, R is not reflexive. R is not symmetric as (2, 37) ∈ R but (37, 2) ∉ R R is not transitive as (11, 19) ∈ R and (19, 3) ∈ R but (11, 3) ∉ R. Hence, R is neither reflexive, nor symmetric and nor transitive.

Maths NCERT Exemplar Solutions Class 12th Chapter One: Overview, Questions, Preparation

Q: The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Surface area, S = 4pr2 ? d s d t = 4 ? . 2 r d r d t = 8 ? d r d t = c o n s t a n t = k ( s a y ) ? d s d t = k ? s = k t + c ? 4 ? r 2 = k t + c Initially t = 0, r = 3 c = 36 p When t = 5, r = 7, k = 32p When t = 9, r = r, r = 9

Q: Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random are found to be 1 red and 1 black. If the probability that both balls come from Bag A is 6 1 1 , then n is equal to ________.

6 1 1 = Required probability After solving, we get n = 4

Q: Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to

Equation of tangent to the circle at (2, 4) is (4 + A) x + (8 + B) y + 2A + 2B + 2C = 0……. (i) Also equation of tangent to parabola y = x2 at (2, 4) is 4x – y = 4 ………. (ii) Comparing (i) and (ii) we get, A + C = 16

Q: Let f : R -> R be a continuous function such that f(3x) – f(x) =. If f(8) = 7, then f(14) is equal to:

f (3x)- f (x) = x Replace x→x3⇒f (x)−f (x3)=x3 Again replace x→x3⇒f (x3)−f (x32)−f (x32)=x32 ⇒f (3x)−f (0)=3x2putting x=83⇒f (8)−f (0)=4∴f (0)=3 Also putting x = 143 in f (3x) – 3 = 3x2⇒ F (14) – 3 = 7 f (14) = 10

Q: Let O be the origin and A be the point z1 = 1 + 2i. If B is the point z2, Re(z2) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuses, then which of the following is NOT true?

z2−0 (1+2i)−0=|OBOA|eiπ4⇒z2= (1+2i) (1+i)=−1+3i∴arg z2=π−tan−13 and |z2|=10 z1−2z2=3−4i∴arg (z1−2z2)=−tan−143⇒|z1−2z2|=|2+4i+1−3i|=10

Q: If the system of linear equations. 8x + y + 4z = -=2 x + y + z = 0 λx 3y = μ has infinitely many solutions, then the distance of the point (λ,μ−12) from the plane 8x + y + 4z + 2 = 0 is

Δ=|81411λ−30|=12−3λ So for λ = 4, it is having infinitely many solutions. Δx=|−214011μ−30| = 6 3μ=0⇒−6−3μ=0 For μ=−2 distance of (4, −2, −12) from 8x + y + 4z + 2= 0 |32−2−2+264+1+16|=103 units

Q: Let A be a 2 × 2 matrix with det(A) = 1 and det ((A+I)(Adj(A)+I))=4. Then the sum of the diagonal elements of A can be:

| (A + I) (adj A + I)| = 4 |A adj A + A + Adj A + I| = 4 | (A)I + A + adj A + I|= 4|A| = 1 |A + adj A| = 4 A= [abcd]⇒adj A= [a−b−cd]⇒| (a+d)00 (a+d)|=4⇒a+d=±2

Updated on Jul 17, 2025 12:12 IST

By Vishal Baghel, Executive Content Operations

Table of content

Relations and Function Question and Answers
JEE Mains 2022
JEE Mains Solutions 2022,24th june , Maths, first shift

Relations and Function Question and Answers

Q.1. If $A = {1, 2, 3, 4}$ , define relations on $A$ which have properties of being:

(a) Reflexive, transitive but not symmetric

(b) Symmetric but neither reflexive nor transitive

(c) Reflexive, symmetric and transitive.

Sol:

$\begin{array}{l} G i v e n t h a t : A = {1, 2, 3} \\ a n d R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} \\ H e r e, 1 R 1, 2 R 2 a n d 3 R 3, S o, R i s r e f l e x i v e . \\ 1 R 2 b u t 2 R 1 o r 2 R 3 b u t 3 R 2, \\ S o, R i s n o t s y m m e t r i c . \\ 1 R 1 a n d 1 R 2 \Rightarrow 1 R 3, S o, R i s t r a n s i t i v e . \\ H e n c e, t h e c o r r e c t a n s w e r i s (a) . \end{array}$

Q.2. Let $R$ be a relation defined on the set of natural numbers N as follows:

$R = {(x, y) : x \in N, y \in N, 2 x + y = 41}$ . Find the domain and range of the relation $R$ . Also, verify whether $R$ is reflexive, symmetric, and transitive.

Sol:

$\begin{array}{l} G i v e n t h a t x \in N, y \in N a n d 2 x + y = 41 \\ ∴ D o m a i n o f R = {1, 2, 3, 4, 5, \dots, 20} \\ a n d R a n g e = {39, 37, 35, 33, 31, \dots, 1} \\ H e r e, (3, 3) \notin R \\ a s 2 \times 3 + 3 \neq 41 \\ S o, R i s n o t r e f l e x i v e . \\ R i s n o t s y m m e t r i c a s (2, 37) \in R b u t (37, 2) \notin R \\ R i s n o t t r a n s i t i v e a s (11, 19) \in R a n d (19, 3) \in R \\ b u t (11, 3) \notin R . \\ H e n c e, R i s n e i t h e r r e f l e x i v e, n o r s y m m e t r i c a n d n o r t r a n s i t i v e . \end{array}$

Q.3. Given $A = {2, 3, 4}, B = {2, 5, 6, 7}$ , construct an example of each of the following:

(a) An injective mapping from $A$ to $B$

(b) A mapping from $A$ to $B$ which is not injective

(c) A mapping from $B$ to $A$ .

Sol:

$\begin{array}{l} H e r e, A = {2, 3, 4} a n d B = {2, 5, 6, 7} \\ (i) L e t f : A \to B b e t h e m a p p i n g f r o m A t o B \\ f = {(x, y) : y = x + 3} \\ ∴ f = {(2, 5), (3, 6), (4, 7)} w h i c h i s a n i n j e c t i v e m a p p i n g . \\ (i i) L e t g : A \to B b e t h e m a p p i n g f r o m A \to B s u c h t h a t \\ ∴ g = {(2, 5), (3, 5), (4, 2)} w h i c h i s n o t a n i n j e c t i v e m a p p i n g . \\ (i i i) L e t h : B \to A b e t h e m a p p i n g f r o m B t o A \\ h = {(y, x) : x = y - 2} \\ ∴ h = {(5, 3), (6, 4), (7, 3)} w h i c h i s t h e m a p p i n g f r o m B t o A . \end{array}$

Q.4. Give an example of a map:

(i) Which is one-one but not onto

(ii) Which is not one-one but onto

(iii) Which is neither one-one nor onto.

Sol: $\begin{array}{l} \end{array}$

Commonly asked questions

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

$I. {(x, y) : xisaperson, y isthemonthofx}$ .

$II. {(a, b) : aisaperson, bisanancestorof a}$ .

If $A = {1, 2, 3, 4}$ , define relations on $A$ which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric and transitive.

Let $R$ be a relation defined on the set of natural numbers N as follows:

$R = {(x, y) : x \in N, y \in N, 2 x + y = 41}$ . Find the domain and range of the relation $R$ . Also, verify whether $R$ is reflexive, symmetric, and transitive.

Given $A = {2, 3, 4}, B = {2, 5, 6, 7}$ , construct an example of each of the following:

(a) An injective mapping from $A$ to $B$

(b) A mapping from $A$ to $B$ which is not injective

(c) A mapping from $B$ to $A$ .

Give an example of a map:

(i) which is one-one but not onto

(ii) which is not one-one but onto

(iii) which is neither one-one nor onto.

Let $A = R - {3}, B = R - {1}$ . Let $f : A \to B$ be defined by $f (x) = \frac{x - 2}{x - 3} \forall x \in A$ . Then Show that $f$ is bijective.

Let $A = [- 1, 1]$ , Then, discuss whether the following functions defined on $A$ are one-one, onto, or bijective:

(i) $f (x) = \frac{x}{2}$

(ii) $g (x) = | x |$

(iii) $h (x) = x | x |$

(iv) $k (x) = x^{2}$ .

Each of the following defines a relation on $N$ :

(i) $x$ is greater than $y, x, y \in N$

(ii) $x + y = 10, x, y \in N$

(iii) $x y, i s s q u a r e o f a n i n t e g e r x, y \in N$

(iv) $x + 4 y = 10, x, y \in N$ .

Determine which of the above relations are reflexive, symmetric, and transitive.

Let $A = {1, 2, 3, \dots, 9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d) i n A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .

Using the definition, prove that the function $f : A \to B$ is invertible if and only if $f$ is both one-one and onto.

Functions $f, g : R \to R$ are defined, respectively, by $f (x) = x^{2} + 3 x + 1$ , $g (x) = 2 x - 3$ , find

(i) $f \circ g$

(ii) $g \circ f$

(iii) $f \circ f$

(iv) $g \circ g$

Let $*$ be the binary operation defined on $Q$ . Find which of the following binary operations are commutative:

(i) $a * b = a - b \forall a, b \in Q$

(ii) $a * b = a^{2} + b^{2} \forall a, b \in Q$

(iii) $a * b = a + a b \forall a, b \in Q$

(iv) $a * b = {(a - b)}^{2} \forall a, b \in Q$ .

Let $*$ be binary operation defined on $R$ by $a * b = 1 + a b, \forall a, b \in R$ . Then the operation $*$ is:

(i) Commutative but not associative

(ii) Associative but not commutative

(iii) Neither commutative nor associative

(iv) Both commutative and associative.

Let $A = {a, b, c}$ and the relation $R$ be defined on $A$ as follows:

$R = {(a, a), (b, c), (a, b)}$ .

Then, write minimum number of ordered pairs to be added in $R$ to make $R$ reflexive and transitive.

Kindly consider the following

Let $f, g : R \to R$ be defined by $f (x) = 2 x + 1$ and $g (x) = x^{2} - 2$ , $\forall x \in R$ .

Respectively then, Find $g \circ f$ .

Let $f : R \to R$ be the function defined by $f (x) = 2 x - 3, \forall x \in R$ Write $f^{- 1}$ .

If $A = {a, b, c, d}$ and the function $f = {(a, b), (b, d), (c, a), (d, c)}$ , write $f^{- 1}$

If $f : R \to R$ is defined by $f (x) = x^{2} - 3 x + 2$ , write $f (f (x))$ .

Is $g = {(1, 1), (2, 3), (3, 5), (4, 7)}$ a function? If $g$ is described by $g (x) = α x + β$ , then what value should be assigned to $α$ and $β$ .

If the mappings $f$ and $g$ are given by

$f = {(1, 2), (3, 5), (4, 1)}$ and $g = {(2, 3), (5, 1), (1, 3)}$ , write $f ? g$ .

Let $C$ be the set of complex numbers. Prove that the mapping $f : C \to R$ given by $f (z) = ∣ z ∣, \forall z \in C$ , is neither one-one nor onto.

Let the function $f : R \to R$ be defined by $f (x) = c o s x, \forall x \in R .$ Show that $f$ is neither one-one nor onto.

Let $X = {1, 2, 3}$ and $Y = {4, 5}$ . Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not:

$I. f = {(1, 4), (1, 5), (2, 4), (3, 5)}$

$II. g = {(1, 4), (2, 4), (3, 4)}$

$III. h = {(1, 4), (2, 5), (3, 5)}$

$IV. k = {(1, 4), (2, 5)}$ .

If functions $f : A \to B$ and $g : B \to A$ satisfy $g ? f = I_{A}$ , then show that $f$ is one-one and $g$ is onto.

Let $f : R \to R$ be the function defined by $f (x) = \frac{1}{2 - c o s x}, \forall x \in R$ . Then, find the range of $f$ .

Let $n$ be a fixed positive integer. Define a relation $R$ in $Z$ as follows: $\forall a, b \in Z, a R b$ if and only if $a - b$ is divisible by $n$ . Show that $R$ is an equivalence relation.

Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b \forall a, b \in T$ . Then $R$ is:

(A) Reflexive but not transitive

(B) Transitive but not symmetric

(C) Equivalence

(D) None of these.

Consider the non-empty set consisting of children in a family and a relation $R$ defined as $a R b$ if $a$ is the brother of $b$ . Then $R$ is:

(A) Symmetric but not transitive

(B) Transitive but not symmetric

(C) Neither symmetric nor transitive

(D) Both symmetric and transitive.

The maximum number of equivalence relations on the set $A = {1, 2, 3}$ are

(A) 1

(B) 2

(C) 3

(D) 5

If a relation $R$ on the set ${1, 2, 3}$ be defined by $R = {(1, 2)}$ , then $R$ is

(A) Reflexive

(B) Transitive

(C) Symmetric

(D) None of these.

Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$ . Then $R$ is:

(A) An equivalence relation

(B) Reflexive, transitive but not symmetric

(C) Symmetric, transitive but not reflexive

(D) Neither transitive nor reflexive but symmetric

Let $A = {1, 2, 3}$ and consider the relation

$R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}$ .

Then $R$ is

(A) Reflexive but not symmetric

(B) Reflexive but not transitive

(C) Symmetric and transitive

(D) Neither symmetric, nor transitive

The identity element for the binary operation $*$ defined on $Q ~ {0}$ as $a * b = \frac{a b}{2} \forall a, b \in Q ~ {0}$ is

(A) 1

(B) 0

(C) 2

(D) None of these

If the set $A$ contains 5 elements and the set $B$ contains 6 elements, then the number of one-one and onto mappings from $A$ to $B$ is

(A) 720

(B) 120

(C) 0

(D) None of these

Let $A = {1, 2, 3, \dots, n}$ and $B = {a, b}$ . Then the number of surjections from $A$ into $B$ is

(A) ⁿp₂

(B) $2^{n} - 2$

(C) $2^{n} - 1$

(D) None of these

Let $f : R \to R$ be defined by $f (x) = \frac{1}{x} \forall x \in R$ . Then $f$ is

(A) One-one

(B) Onto

(C) Bijective

(D) $f$ is not defined

Let $f : R \to R$ be defined by $f (x) = 3 x^{2} - 5$ and $g : R \to R$ by $g (x) = \frac{X}{x^{2} + 1}$ . Then $g ? f$ is

(A) $\frac{3 x^{2} - 5}{9 x^{4} - 30 x^{2} + 26}$

(B) $\frac{3 x^{2} - 5}{9 x^{4} - 6 x^{2} + 26}$

(C) $\frac{3 x^{2}}{x^{4} + 2 x^{2} - 4}$

(D) $\frac{3 x^{2}}{9 x^{4} + 30 x^{2} - 2}$

Which of the following functions from $Z$ into $Z$ are bijections?

(A) $f (x) = x^{3}$

(B) $f (x) = x + 2$

(C) $f (x) = 2 x + 1$

(D) $f (x) = x^{2} + 1$

Let $f : R \to R$ be the functions defined by $f (x) = x^{3} + 5$ . Then $f^{- 1} (x)$ is

(A) ${(x + 5)}^{\frac{1}{3}}$

(B) ${(x - 5)}^{\frac{1}{3}}$

(C) ${(5 - x)}^{\frac{1}{3}}$

(D) $5 - x$

Let $f : A \to B$ and $g : B \to C$ be the bijective functions. Then ${(g \circ f)}^{- 1}$ is

(A) $f^{- 1} \circ g^{- 1}$

(B) $f \circ g$

(C) $g^{- 1} \circ f^{- 1}$

(D) $g \circ f$

Let $f : R - {\frac{3}{5}} \to R$ be defined by $f (x) = \frac{3 x + 2}{5 x - 3}$ . Then

(A) $f^{- 1} (x) = f (x)$

(B) $f^{- 1} (x) = - f (x)$

(C) ( $f \circ f$ )x=-x

(D) $f^{- 1} (x) = \frac{1}{19} f (x)$

Let $f : [0, 1] \to [0, 1]$ be defined by $f (x) = {\begin{matrix} x, & if x is rational \\ 1 - x, & if x is irrational \end{matrix}$

Then $(f \circ f) x i s$

(A) Constant

(B) $1 + x$

(C) $x$

(D) None of these

Let $f : [2, \infty) \to R$ be the function defined by $f (x) = x^{2} - 4 x + 5$ . Then the range of $f$ is

(A) $?$

(B) $[1, \infty)$

(C) $[4, \infty)$

(D) $[5, \infty)$

Let $f : N \to R$ be the function defined by $f (x) = \frac{2 x - 1}{2}$ and $g : Q \to R$ be another function defined by $g (x) = x + 2$ . Then $($ $(g \circ f)$
$\frac{3}{2}$ is

(A) 1

(B) 1

(C) $\frac{7}{2}$

(D) None of these

Let $f : R \to R$ be defined by:

Then $f (- 1) + f (2) + f (4)$ is

(A) 9

(B) 14

(C) 5

(D) None of these

Let $f : R \to R$ be given by $f (x) = t a n x$ . Then $f^{- 1} (1)$ is

(A) $\frac{π}{4}$

(B) ${n π + \frac{π}{4} : n \in ℤ}$

(C) Does not exist

(D) None of these

Let the relation $R$ be defined in $ℕ$ by $a R b$ if $2 a + 3 b = 30$ . Then $R =________.$

Let the relation $R$ be defined on the set $A = {1, 2, 3, 4, 5}$ by $R = {(a, b) : ∣ a^{2} - b^{2} ∣ < 8}$ . Then $R$ is given by ________

Let $f = {(1, 2), (3, 5), (4, 1)}$ and $g = {(2, 3), (5, 1), (1, 3)}$ . Then $g \circ f =______$ and $f \circ g =________.$

Kindly consider the following

If $f (x) = (4 - {(x - 7)}^{3}}$ , then $f^{- 1} (x) =________.$

State True or False for the statements in each of the Exercises 6 to 14:

Let $R = {(3, 1), (1, 3), (3, 3)}$ be a relation defined on the set $A = {1, 2, 3}$ . Then $R$ is symmetric, transitive but not reflexive.

Let $f : R \to R$ be the function defined by $f (x) = s i n (3 x + 2), \forall x \in R$ . Then $f$ is invertible.

Every relation which is symmetric and transitive is also reflexive.

An integer $m$ is said to be related to another integer $n$ if $m$ is a integral multiple of $n$ . This relation in $Z$ is reflexive, symmetric, and transitive.

Let $A = {0, 1}$ and $N$ be the set of natural numbers. Then the mapping $f : N \to A$ defined by $f (2 n - 1) = 0, f (2 n) = 1, \forall n \in N$ , is onto.

The relation $R$ on the set $A = {1, 2, 3}$ defined as $R = {(1, 1), (1, 2), (2, 1), (3, 3)}$ is reflexive, symmetric, and transitive.

The composition of functions is commutative.

The composition of functions is associative.

Every function is invertible.

A binary operation on a set has always the identity element.

JEE Mains 2022

Commonly asked questions

The number of distinct real roots of the equation x⁵ $(x^{3} - x^{2} - x + 1)$ + $x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is …………..

Let f : R -> R be a continuous function such that f(3x) – f(x) =. If f(8) = 7, then f(14) is equal to:

Let O be the origin and A be the point z₁ = 1 + 2i. If B is the point z₂, Re(z₂) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuses, then which of the following is NOT true?

If the system of linear equations.

8x + y + 4z = -=2

x + y + z = 0

$λ x$ 3y = $μ$

has infinitely many solutions, then the distance of the point $(λ, μ - \frac{1}{2})$ from the plane 8x + y + 4z + 2 = 0 is

Let A be a 2 × 2 matrix with det(A) = 1 and det $((A + I) (A d j (A) + I)) = 4 .$ Then the sum of the diagonal elements of A can be:

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a is $\frac{364}{3},$ is equal to:

Consider two G.P.’s. 2, 2², 2³, …… and 4, 4², 4³, …… of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is ${(2)}^{\frac{225}{8}}$ , then $\sum_{k = 1}^{n} k (n - k)$ is equal to:

If the function f(x) = ${\frac{l o g_{e} (1 - x + x^{2}) + l o g_{e} (1 + x + x^{2})}{\begin{matrix} s e c x - c o s x & k \end{matrix}}}, x \in (\frac{- π}{2}, \frac{π}{2}) - {0}$ is continuous at x = 0, then k is equal to:

If f(x) = ${\begin{matrix} x + a, x \leq 0 & | x - 4 |, x > 0 \end{matrix} a n d g (x) = {\begin{matrix} x + 1, x < 0 & {(x - 4)}^{2} + b, x \geq 0 \end{matrix}$ are continuous on R, then (gof)(2) + (fog)(2) is equal to:

Let f(x) = ${\begin{matrix} x^{3} - x^{2} + 10 x - 7, x \leq 1 & - 2 x + l o g_{2} (b^{2} - 4), x > 1 \end{matrix} .$ Then the set of all values of b, for which f(x) has maximum value at x = 1, is:

If a = $\underset{n \to \infty}{l i m} \sum_{k = 1}^{n} \frac{2 n}{n^{2} + k^{2}}$ and f(x) = $\sqrt[]{\frac{1 - c o s x}{1 + c o s x}}$ , x $\in$ (0, 1), then:

$I f \frac{d y}{d x}$ + 2y tan x = sin x, 0 < x < $\frac{π}{2}$ and y $(\frac{π}{3})$ = 0, then the maximum value of y(x) is:

A point P moves so that the sum of squares of its distances from the points (1, 2) and (2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A , B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to:

Let the tangent drawn to the parabola y²= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola $\frac{x^{2}}{α^{2}} - \frac{y^{2}}{β^{2}} = 1$ at the point ( α+ 4, β+ 4) does NOT pass through the point:

The length of the perpendicular from the point (1, 2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y – z = 0 = x – 2y + 3z – 5 is:

Let $\vec{a} = α \hat{i} + \hat{j} - \hat{k} a n d \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}, α > 0 .$ If the projection of $\vec{a} \times \vec{b}$ on the vector $- \hat{i} + 2 \hat{j} - 2 \hat{k}$ is 30, then equal to:

The mean and variance of a binomial distribution are and $\frac{α}{3}$ respectively. If P(X = 1) = $\frac{4}{243}$ then P(X = 4 or 5) is equal to:

Let E₁, E₂, E₃ be three mutually exclusive events such that P(E₁) = $\frac{2 + 3 p}{6}$ , P(E₃) = $\frac{2 - p}{8}$ and P(E3) = $\frac{1 - p}{2} .$ If the maximum and minimum values of p are p₁ and p₂, then (p₁ + p₂) is equal to:

Let S = ${θ \in [0, 2 π] : 8^{2 s i n^{2} θ} + 8^{2 c o s^{2} θ} = 16} .$ Then $n (S) + \sum_{θ \in S}^{} (s e c (\frac{π}{4} + 2 θ) c o s e c (\frac{π}{4} + 2 θ))$ is equal to:

$t a n (2 t a n^{- 1} \frac{1}{5} + s e c^{- 1} \frac{\sqrt[]{5}}{2} + 2 t a n^{- 1} \frac{1}{8})$ is equal to:

The statement $(\sim (p \Leftrightarrow \sim q)) \land q i s :$

If for some q, q, r $\in$ R, not at all have same sign, one of the roots of the equation $(p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0$ is also a root of the equation x² + 2x – 8 = 0, then $\frac{q^{2} + r^{2}}{p^{2}}$ is equal to………….

The number of 5-digit natural numbers, such that the product of their digits is 36, is………

The series of positive multiples of 3 is divided into sets : ${3}, {6, 9, 12}, {15, 18, 21, 24, 27}, . . . . . .$ Then the sum of the elements in the 11th set is equal to…………..

If the coefficients of x and x² in the expansion of (1 + x)^p (1 – x)^q, p, q 15, are -3 and -5 respectively, then the coefficient of x³ is equal to…………………

If $n (2 n + 1) \int_{0}^{1} {(1 - x^{n})}^{2 n} d x = 1177 \int_{0}^{1} {(1 - x^{n})}^{2 n + 1}$ dx, then n $\in$ N is equal to………….

Let a cure y = y(x) pass through the point (3, 3) and the area of the origin under this curve, above the x-axis and between the abscissae 3 and x (>3) be ${(\frac{y}{x})}^{3}$ . If the curve also passes through the point $(α, 6 \sqrt[]{10})$ in the first quadrant, then is equal to………….

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x – y = 3 respectively. If its orthocenter is (2, a), $- \frac{1}{2} < a < 2,$ then p is equal to…………

Let the function f(x) = 2x² – log_e x, x > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y² = 4ax at a point P on it passes through the point (8a, 8a – 1) but does not pass through the point $(- \frac{1}{a}, 0) .$ If the equation of the normal at P is $\frac{x}{α} + \frac{γ}{β} = 1,$ then a + b is equal to…………….

Let Q and R be two points on the line $\frac{x + 1}{2} = \frac{y + 2}{3} = \frac{z - 1}{2}$ at a distance $\sqrt[]{26}$ from the point P(4, 2, 7). Then the surface of the area of the triangle PQR is……………

JEE Mains Solutions 2022,24th june , Maths, first shift

Commonly asked questions

Let A = ${z \in C : 1 \leq | z - (1 + i) \leq 2}$ and B = ${z \in A : | z - (1 - i) | = 1}$ . Then, B :

The remainder when 3²⁰²²is divided by 5 is :

The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random are found to be 1 red and 1 black. If the probability that both balls come from Bag A is $\frac{6}{11},$ then n is equal to ________.

Let x²+ y² + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x² at (2, 4). Then A + C is equal to

The number of values of a for which the system of equations :

x + y + z = a

ax + 2ay + 3z = -1

x + 3ay + 5z = 4

is inconsistent, is

If the sum of the squares of the reciprocals of the roots a and b of the equation $3 x^{2} + λ x - 1 = 0$ is 15, then $6 {(α^{3} + β^{3})}^{2}$ is equal to :

The set of all values of k for which ${(t a n^{- 1} x)}^{3} + {(c o t^{- 1} x)}^{3} = k π^{3}, x \in R$ , is the interval :

Let S = ${\sqrt[]{n} : 1 \leq n \leq 50 a n d n i s o d d}$ .

Let $a \in S a n d A = [\begin{matrix} 1 & 0 & a \\ - 1 & 1 & 0 \\ - a & 0 & 1 \end{matrix}] .$ If $\sum_{a \in S}^{} d e t (a d j A)$ = 100l, then l is equal to :

For the function f(x) = 4 log_e(x – 1) – 2x² + 4x + 5, x > 1, which one of the following is NOT correct?

If the tangent at the point (x₁, y₁) on the curve y = x³ + 3x² + 5 passes through the origin, then (x₁, y₁) does NOT lie on the curve:

The sum of absolute maximum and absolute minimum values of the function f(x) = |2x² + 3x + 2| + sin x cos x in the interval [0, 1] is :

If ${a_{i}}_{i = 1}^{n},$ where n is an even integer, is an arithmetic progression with common difference 1, and $\sum_{i = 1}^{n} a_{i} = 192,$ then n is equal to :

If x = x(y) is the solution of the differential equation $y \frac{d x}{d y} = 2 x + y^{3} (y + 1) e^{y}, x (1) = 0$ ; then x(e) is equal to :

Let lx – 2y = µ be a tangent to the hyperbola $a^{2} x^{2} - y^{2} = b^{2} .$ Then ${(\frac{λ}{a})}^{2} - {(\frac{μ}{b})}^{2}$ is equal to

Let $\hat{a}, \hat{b}$ and be unit vectors. If $\hat{c}$ be vector such that the angle between $\hat{a} a n d \vec{c} i s \frac{π}{12},$ and $\hat{b} = \vec{c} + 2 (\vec{c} \times \hat{a}),$ then ${| 6 \vec{c} |}^{2}$ is equal to :

If a random variable X follows the Binomial distribution B (33, p) such that 3P (X = 0) = P (X = 1) then the value of $\frac{P (X = 15)}{P (X = 18)} - \frac{P (X = 16)}{P (X = 17)}$ is equal to :L

The domain of the function f(x) = $\frac{c o s^{- 1} (\frac{x^{2} - 5 x + 6}{x^{2} - 9})}{l o g_{e} (x^{2} - 3 x + 2)}$ is :

Let S = ${θ \in [- π, π] - {\pm \frac{π}{2}} : s i n θ t a n θ + t a n θ = s i n 2 θ}$ . If $\sum_{θ \in S}^{} c o s 2 θ,$ then T = n(S) is equal to

The number of choices for $Δ \in {\land, \lor, \Rightarrow, \Leftrightarrow}$ , such that $(p Δ q) \Rightarrow ((p Δ ~ q) \lor (~ p) Δ q)$ is a tautology, is :

The number of one-one functions f:{a, b, c, d} ® {0, 1, 2, ……., 10} such that 2f (a) – f(b) + 3f(c) + f(d) = 0 is ___________.

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, -2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is __________

Let $A (\frac{3}{\sqrt[]{a}}, \sqrt[]{a}), a > 0$ , be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If D (3 cos q, a sin q) is a point in the fourth quadrant such that the maximum area of $Δ A C D$ is 12 square units, then a is equal to _________.

Let a line having direction ratios 1, -4, 2 intersect the lines $\frac{x - 7}{3} = \frac{y - 1}{- 1} = \frac{z + 2}{1}$ and $\frac{x}{2} = \frac{y - 7}{3} = \frac{z}{1}$ at the points A and B. Then (AB)² is equal to _________.

The number of points where the function f(x) = ${\begin{cases} | 2 x^{2} - 3 x - 7 | i f x \leq - 1 \\ [4 x^{2} - 1] i f - 1 < x < 1 \\ | x + 1 | + | x - 2 | i f x \geq 1 \end{cases}$ [t] denotes the greatest integer $\leq$ t is discontinuous is __________-.

Let f (θ) = $s i n θ + \int_{- \frac{π}{2}}^{\frac{π}{2}} (s i n θ + t c o s θ) f (t) d t .$ Then the value of $| \int_{0}^{\frac{π}{2}} f (θ) d θ |$ is

Let $\underset{0 \leq x \leq 2}{M a x} {\frac{9 - x^{2}}{5 - x}} = α a n d \underset{0 \leq x \leq 2}{M i n} = {\frac{9 - x^{2}}{5 - x}} = β$

If $\int_{β - \frac{8}{3}}^{2 α - 1} M a x {\frac{9 - x^{2}}{5 - x}, x} d x = α_{1} + α_{2} l o g_{e} (\frac{8}{15})$ then $α_{1} + α_{2}$ is equal to __________.

If two tangents drawn from a point (a, b) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of ${(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2}$ equals __________.

Let S be the region bonded by the curves y = x³ and y² = x. The curve y = 2|x| divides S into two regions of areas R₁, and R₂ _________.

If max {R₁, R₂} = R₂, then $\frac{R_{2}}{R_{1}}$ is equal to _________.

If the shortest distance between the lines $\vec{r} = (- \hat{i} + 3 \hat{k}) + λ (\hat{i} - a \hat{j}) a n d \vec{r} = (- \hat{j} + 2 \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})$ is $\sqrt[]{\frac{2}{3}}$ , then the integral value of ‘a’ is equal to __________.

Maths NCERT Exemplar Solutions Class 12th Chapter One Exam

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