How many questions from Sets, Relations and Functions appear in JEE?

Sets, Relations and Functions has appeared in 165 of the last 186 JEE Main and JEE Advanced papers — about 89% of them — contributing 313 questions in total across those papers.

Is Sets, Relations and Functions an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 89% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Sets, Relations and Functions questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Sets, Relations and Functions JEE Previous Year Questions with Answers

Q1·MathematicsIntegerJEE Advanced 2026

Let

S = \{1, 2, 3, \ldots, 10\}

. Consider the set

X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly 42 elements}\}

. Then the number of elements in

X

is ______.

Q2·MathematicsNumericalJEE Advanced 2026

Let

\mathbb{N}

denote the set of all positive integers. Consider the sets

A = \{1, 2, 3, 4, 5\}

and

B = \{1, 2, 3, 4, 5, 6, 7\}

. Let

S

be the set of all functions

f : A \to B

such that

f(2) \neq 2

and

f(4) \neq 4

. Consider the set

T = \{f \in S :

there exists a function

g : B \to \mathbb{N}

such that

g(f(x)) = 2^{x}

for all

x \in A\}

. Then the number of elements in the set

T

is _____.

Q3·MathematicsNumericalJEE Main 2026

Let

f

be a polynomial function such that

\log_2(f(x)) = \left(\log_2\left(2 + \dfrac{2}{3} + \dfrac{2}{9} + \ldots \infty\right)\right) \cdot \log_3\left(1 + \dfrac{f(x)}{f(1/x)}\right)

,

x > 0

and

f(6) = 37

. Then

\displaystyle\sum_{n=1}^{10} f(n)

is equal to ________.

Q4·MathematicsSingle correctJEE Main 2026

Consider the relation R on the set

\{-2, -1, 0, 1, 2\}

defined by

(a, b) \in R

if and only if

1 + ab > 0

. Then, among the statements: I. The number of elements in R is 17 II. R is an equivalence relation

(A)Only I is true
(B)Only II is true
(C)Both I and II are true
(D)Neither I nor II is true

Q5·MathematicsNumericalJEE Main 2026

Let

R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_{e}(x + y) \le 2\}

. Then the minimum number of elements, required to be added in

R

to make it a transitive relation, is __________.

Q6·MathematicsSingle correctJEE Main 2026

Let

f : \mathbb{R} \to \mathbb{R}

be defined as

f(x) = \frac{2x^{2} - 3x + 2}{3x^{2} + x + 3}

. Then

f

is :

(A)both one-one and onto
(B)one-one but not onto
(C)onto but not one-one
(D)neither one-one nor onto

Q7·MathematicsSingle correctJEE Main 2026

Let [·] denote the greatest integer function. If the domain of the function

f(x) = \sin^{-1}\left(\frac{x + [x]}{3}\right)

is [α, β), then α

^2

+ β

^2

is equal to:

(A)2
(B)5
(C)10
(D)13

Q8·MathematicsNumericalJEE Main 2026

Let A = {1, 4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation

R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + b_2 \text{ divides } a_2 + b_1\}

is _______.

Q9·MathematicsSingle correctJEE Main 2026

Let for some α ∈

\mathbb{R}

,

f : \mathbb{R} \to \mathbb{R}

be a function satisfying

f(x + y) = f(x) + 2y^{2} + y + \alpha xy

for all

x, y \in \mathbb{R}

. If

f(0) = -1

and

f(1) = 2

, then the value of

\sum_{n=1}^{5}(\alpha + f(n))

is:

(A)110
(B)140
(C)150
(D)170

Q10·MathematicsSingle correctJEE Main 2026

For the function

f : [1, \infty) \to [1, \infty)

defined by

f(x) = (x-1)^{4} + 1

, among the two statements: (I) The set

S = \{x \in [1, \infty) : f(x) = f^{-1}(x)\}

contains exactly two elements, and (II) The set

S = \{x \in [1, \infty) : f(x) = f^{-1}(x+1)\}

is an empty set,

(A)only (I) is TRUE
(B)only (II) is TRUE
(C)both (I) and (II) are TRUE
(D)neither (I) nor (II) is TRUE

Q11·MathematicsNumericalJEE Main 2026

If the domain of the function

f(x) = \sqrt{\log_{(0.6)}\left(\left|\frac{2x - 5}{x^{2} - 4}\right|\right)}

is

(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)

, then the value of

a + b + c + d + e

is ________.

Q12·MathematicsNumericalJEE Main 2026

Let

A = \{2, 3, 4, 5, 6\}

. Let R be a relation on the set

A \times A

given by

(x, y)R(z, w)

if and only if

x

divides

z

and

y \le w

. Then the number of elements in R is ________.

Q13·MathematicsSingle correctJEE Main 2026

If

g(x) = 3x^2 + 2x - 3

,

f(0) = -3

and

4g(f(x)) = 3x^2 - 32x + 72

, then

f(g(2))

is equal to:

(A) $\dfrac{25}{6}$
(B) $-\dfrac{25}{6}$
(C) $\dfrac{7}{2}$
(D) $-\dfrac{7}{2}$

Q14·MathematicsSingle correctJEE Main 2026

The sum of all the elements in the range of f(x)=Sgn(sinx)+Sgn(cosx)+Sgn(tanx)+Sgn(cotx),

x \neq \frac{n\pi}{2}

, n

\in

\mathbf{Z}

, where Sgn(t) =

\begin{cases} 1, & \text{if} \quad t > 0 \\ -1 & \text{if} \quad t < 0 \end{cases}

, is

(A)4
(B)2
(C)–2
(D)0

Q15·MathematicsSingle correctJEE Main 2026

Given below are two statements : Statement I : The function f : R → R defined by

f(x) = \dfrac{x}{1+|x|}

is one-one. Statement II : The function f : R → R defined by

f(x) = \dfrac{x^{2}+4x-30}{x^{2}-8x+18}

is many-one. In the light of the above statements, choose the correct answer from the options given below :

(A)Both Statement I and Statement II are false.
(B)Both Statement I and Statement II are true.
(C)Statement I is false but Statement II is true .
(D)Statement I is true but Statement II is false.

Q16·MathematicsSingle correctJEE Main 2026

Let R be a relation defined on the set

\{1,2,3,4\}\times\{1,2,3,4\}

by

R=\{((a,b),(c,d)):2a+3b=3c+4d\}

. Then the number of elements in R is

(A) $6$
(B) $18$
(C) $12$
(D) $15$

Q17·MathematicsSingle correctJEE Main 2026

If the domain of the function

f(x) = \sin^{-1}\left(\frac{1}{x^{2} - 2x - 2}\right)

, is

(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)

, then

\alpha + \beta + \gamma + \delta

is equal to

(A)2
(B)4
(C)3
(D)5

Q18·MathematicsSingle correctJEE Main 2026

If the domain of the function

f(x)=\log_{(10x^{2}-17x+7)}\left(18x^{2}-11x+1\right)

is

(-\infty,\ a)\cup(b,\ c)\cup(d,\ \infty)-\{e\}

, then

90(a+b+c+d+e)

equals:

(A) $170$
(B) $177$
(C) $307$
(D) $316$

Q19·MathematicsSingle correctJEE Main 2026

Let f be a function such that

3f(x) + 2f\left( \dfrac{m}{19x} \right) = 5x

,

x \ne 0

, where

m = \sum\limits_{i=1}^{9} (i)^{2}

. Then

f(5) - f(2)

is equal to

(A) $-9$
(B)36
(C)18
(D)9

Q20·MathematicsSingle correctJEE Main 2026

Consider two sets A =

\{x \in z : |(|x - 3| - 3)| \leq 1\}

and B=

\left\{ x \in \mathbb{R} - \{1,2\} : \dfrac{(x-2)(x-4)}{x-1}\log_e(|x-2|) = 0 \right\}

. Then the number of onto functions

f

: A→B is equal to :

(A)62
(B)79
(C)32
(D)81

Q21·MathematicsSingle correctJEE Main 2026

Let A={-2, -1, 0,1, 2, 3, 4}. Let R be a relation on A defined by xRy if and only if

2x + y \le 2

. Let

l

be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then

l

+ m + n is equal to :

(A)32
(B)34
(C)33
(D)35

Q22·MathematicsSingle correctJEE Main 2026

Let the relation R on the set M = {1, 2, 3,.......16} be given by

R = \{(x,y) : 4y = 5x - 3,\ x, y \in M\}

. Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to

(A)1
(B)2
(C)4
(D)3

Q23·MathematicsSingle correctJEE Main 2026

Let the domain of the function

f(x) = \log_3 \log_5 \left(7 - \log_2 (x^2 - 10x + 85)\right) + \sin^{-1}\left(\left|\frac{3x - 7}{17 - x}\right|\right)

be (α, β]. Then α + β is equal to :

(A)10
(B)12
(C)9
(D)8

Q24·MathematicsSingle correctJEE Main 2026

Let

f(x) = [x]^2 - [x + 3] - 3

,

x \in \mathbb{R}

where [•] is the greatest integer function. Then

(A) $f(x) > 0$ only for $x \in [4, \infty)$
(B) $f(x) < 0$ only for $x \in [-1, 3)$
(C) $\int_0^2 f(x)dx = -6$
(D) $f(x) = 0$ for finitely many values of x.

Q25·MathematicsSingle correctJEE Main 2026

Let

f

and g be functions satisfying

f(x+y) = f(x)\, f(y)

,

f(1) = 7

and

g(x+y) = g(xy)

,

g(1) = 1

, for all

x, y \in \mathbb{N}

.

\sum_{x=1}^{n}\left(\frac{f(x)}{g(x)}\right) = 19607

, then n is equal to :

(A)7
(B)5
(C)6
(D)4

Q26·MathematicsSingle correctJEE Main 2026

The number of elements in the relation

R = \{(x,y): 4x^{2} + y^{2} < 52,\ x,\ y \in Z\}

is

(A)77
(B)89
(C)67
(D)86

Q27·MathematicsSingle correctJEE Main 2026

The number of relations, defined on the set

\{a, b, c, d\}

, which are both reflexive and symmetric, is equal to:

(A)256
(B)16
(C)1024
(D)64

Q28·MathematicsSingle correctJEE Main 2026

Let

A = \{2, 3, 5, 7, 9\}

. Let R be the relation on A defined by

x

R

y

if and only if

2x \le 3y

. Let

\ell

be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then

\ell + m

is equal to:

(A)23
(B)25
(C)21
(D)27

Q29·MathematicsSingle correctJEE Main 2026

Let

A = \{x : |x^2 - 10| \le 6\}

and

B = \{x : |x - 2| > 1\}

. Then

(A) $A \cup B = (-\infty, 1] \cup (2, \infty)$
(B) $A - B = [2, 3)$
(C) $A \cap B = [-4, -2] \cup [3, 4]$
(D) $B - A = (-\infty, -4) \cup (-2, 1) \cup (4, \infty)$

Q30·MathematicsMultiple correctJEE Advanced 2025

Let N denote the set of all natural numbers, and Z denote the set of all integers. Consider the functions

f: N \to Z

and

g: Z \to N

defined by

f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd,} \\ (4-n)/2 & \text{if } n \text{ is even,} \end{cases}

and

g(n) = \begin{cases} 3 + 2n & \text{if } n \geq 0, \\ -2n & \text{if } n < 0. \end{cases}

Define

(g \circ f)(b) = g(f(n))

for all

n \in N

, and

(f \circ g(n)) = f(g(n))

for all

n \in Z

. Then which of the following statements is (are) TRUE?

(A) $g \circ f$ is NOT one-one and $g \circ f$ is NOT onto
(B) $f \circ g$ is NOT one-one but $f \circ g$ is onto
(C)g is one-one and g is onto
(D) $f$ is NOT one-one but $f$ is onto

Q31·MathematicsNumericalJEE Advanced 2025

Let the set of all relation R on the set

\{a, b, c, d, e, f\}

, such that R is reflexive and symmetric, and R contains exactly 10 elements be denoted by S. Then the number of elements is S is __________ .

Q32·MathematicsSingle correctJEE Main 2025

Let

A=\{0,1,2,3,4,5\}

. Let

R

be a relation on

A

defined by

(x,y)\in R

if and only if

\max\{x,y\}\in\{3,4\}

. Then among the statements

(S_1)

: The number of elements in

R

is 18, and

(S_2)

: The relation

R

is symmetric but neither reflexive nor transitive:

(A)both are true
(B)both are false
(C)only $(S_2)$ is true
(D)only $(S_1)$ is true

Q33·MathematicsIntegerJEE Main 2025

Let the domain of the function

f(x)=\cos^{-1}\left(\dfrac{4x+5}{3x-7}\right)

be

[\alpha,\beta]

and the domain of

g(x)=\log_2\big(2-6\log_{27}(2x+5)\big)

be

(\gamma,\delta)

. Then

\big|7(\alpha+\beta)+4(\gamma+\delta)\big|

is equal to ___

Q34·MathematicsSingle correctJEE Main 2025

Let

A=\{(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:|\alpha-1|\le 4 \text{ and } |\beta-5|\le 6\}

and

B=\{(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:16(\alpha-2)^2+9(\beta-6)^2\le 144\}

. Then

(A) $B\subset A$
(B) $A\cup B=\{(x,y):-4\le x\le 4,\ -1\le y\le 11\}$
(C)neither $A\subset B$ nor $B\subset A$
(D) $A\subset B$

Q35·MathematicsIntegerJEE Main 2025

The number of relations on the set

A=\{1,2,3\}

containing at most 6 elements including

(1,2)

, which are reflexive and transitive but not symmetric, is ______.

Q36·MathematicsSingle correctJEE Main 2025

If the range of the function

f(x)=\dfrac{5-x}{x^2-3x+2}

,

x\ne 1,2

, is

(-\infty,\alpha]\cup[\beta,\infty)

, then

\alpha^2+\beta^2

is equal to:

(A)190
(B)192
(C)188
(D)194

Q37·MathematicsSingle correctJEE Main 2025

Let

f:\mathbb{R}\to\mathbb{R}

be a continuous function satisfying

f(0)=1

and

f(2x)-f(x)=x

for all

x\in\mathbb{R}

. If

\displaystyle\lim_{n\to\infty}\left\{f(x)-f\left(\dfrac{x}{2^n}\right)\right\}=G(x)

, then

\displaystyle\sum_{r=1}^{10}G(r^2)

is equal to:

(A)540
(B)385
(C)420
(D)215

Q38·MathematicsSingle correctJEE Main 2025

Let the domains of the functions

f(x)=\log_4\log_3\log_7\left(8-\log_2(x^2+4x+5)\right)

and

g(x)=\sin^{-1}\left(\dfrac{7x+10}{x-2}\right)

be

(\alpha,\beta)

and

[\gamma,\delta]

, respectively. Then

\alpha^2+\beta^2+\gamma^2+\delta^2

is equal to:

(A)15
(B)13
(C)16
(D)14

Q39·MathematicsSingle correctJEE Main 2025

Let

f, g:(1,\infty)\to\mathbb{R}

be defined as

f(x)=\dfrac{2x+3}{5x+2}

and

g(x)=\dfrac{2-3x}{1-x}

. If the range of the function

f\circ g:[2,4]\to\mathbb{R}

is

[\alpha,\beta]

, then

\dfrac{1}{\beta-\alpha}

is equal to

(A)68
(B)29
(C)2
(D)56

Q40·MathematicsSingle correctJEE Main 2025

Let

A=\{-3,-2,-1,0,1,2,3\}

and

R

be a relation on

A

defined by

xRy

if and only if

2x-y\in\{0,1\}

. Let

l

be the number of elements in

R

. Let

m

and

n

be the minimum number of elements required to be added in

R

to make it reflexive and symmetric relations, respectively. Then

l+m+n

is equal to:

(A)18
(B)17
(C)15
(D)16

Q41·MathematicsSingle correctJEE Main 2025

Let

A=\{-3,-2,-1,0,1,2,3\}

. Let

R

be a relation on

A

defined by

xRy

if and only if

0\le x^2+2y\le 4

. Let

l

be the number of elements in

R

and

m

be the minimum number of elements required to be added in

R

to make it a reflexive relation. Then

l+m

is equal to:

(A)19
(B)20
(C)17
(D)18

Q42·MathematicsSingle correctJEE Main 2025

Let

A=\{-2,-1,0,1,2,3\}

. Let

R

be a relation on

A

defined by

xRy

if and only if

y=\max\{x,1\}

. Let

l

be the number of elements in

R

. Let

m

and

n

be the minimum number of elements required to be added in

R

to make it reflexive and symmetric relations, respectively. Then

l+m+n

is equal to:

(A)12
(B)11
(C)13
(D)14

Q43·MathematicsSingle correctJEE Main 2025

If the domain of the function

f(x)=\log_e\!\left(\dfrac{2x-3}{5+4x}\right)+\sin^{-1}\!\left(\dfrac{4+3x}{2-x}\right)

is

[\alpha,\beta]

, then

\alpha^2+4\beta

is equal to:

(A)5
(B)4
(C)3
(D)7

Q44·MathematicsSingle correctJEE Main 2025

If the domain of the function

f(x)=\log_7\!\big(1-\log_4(x^2-9x+18)\big)

is

(\alpha,\beta)\cup(\gamma,\delta)

, then

\alpha+\beta+\gamma+\delta

is equal to:

(A)18
(B)16
(C)15
(D)17

Q45·MathematicsSingle correctJEE Main 2025

Let

f

be a function such that

f(x)+3f\!\left(\dfrac{24}{x}\right)=4x

,

x\ne 0

. Then

f(3)+f(8)

is equal to:

(A)11
(B)10
(C)12
(D)13

Q46·MathematicsSingle correctJEE Main 2025

Let

f:\mathbb{R}\to\mathbb{R}

be a function defined by

f(x)=\big|\,|x+2|-2|x|\,\big|

. If

m

is the number of points of local minima and

n

is the number of points of local maxima of

f

, then

m+n

is equal to:

(A)5
(B)3
(C)2
(D)4

Q47·MathematicsSingle correctJEE Main 2025

If the domain of the function

f(x)=\dfrac{1}{\sqrt{10+3x-x^2}}+\dfrac{1}{\sqrt{x+|x|}}

is

(a,b)

, then

(1+a)^2+b^2

is equal to:

(A)26
(B)29
(C)25
(D)30

Q48·MathematicsSingle correctJEE Main 2025

Let

A

be the set of all functions

f:Z\to Z

and

R

be a relation on

A

such that

R=\{(f,g):f(0)=g(1)\text{ and }f(1)=g(0)\}

. Then

R

is:

(A)Symmetric and transitive but not reflexive
(B)Symmetric but neither reflexive nor transitive
(C)Reflexive but neither symmetric nor transitive
(D)Transitive but neither reflexive nor symmetric

Q49·MathematicsSingle correctJEE Main 2025

Let

f:R\to R

be a twice differentiable function such that

(\sin x\cos y)(f(2x+2y)-f(2x-2y))=(\cos x\sin y)(f(2x+2y)+f(2x-2y))

, for all

x,y\in R

. If

f'(0)=\dfrac{1}{2}

, then the value of

24\,f''\left(\dfrac{5\pi}{3}\right)

is:

(A)2
(B) $-3$
(C)3
(D) $-2$

Q50·MathematicsSingle correctJEE Main 2025

Let

A=\{1,2,3,\dots,10\}

and

R

be a relation on

A

such that

R=\{(a,b):a=2b+1\}

. Let

(a_1,a_2),(a_2,a_3),(a_3,a_4),\dots,(a_k,a_{k+1})

be a sequence of

k

elements of

R

such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer

k

, for which such a sequence exists, is equal to:

(A)6
(B)7
(C)5
(D)8

Q51·MathematicsSingle correctJEE Main 2025

Let

S=N\cup\{0\}

. Define a relation R from S to R by:

R=\left\{(x,y):\log_e y=x\log_e\left(\dfrac{2}{5}\right),\ x\in S,\ y\in R\right\}

. Then, the sum of all the elements in the range of R is equal to

(A) $\dfrac{3}{2}$
(B) $\dfrac{5}{3}$
(C) $\dfrac{10}{9}$
(D) $\dfrac{5}{2}$

Q52·MathematicsSingle correctJEE Main 2025

If the domain of the function

\log_5(18x-x^2-77)

is

(\alpha,\beta)

and the domain of the function

\log_{(x-1)}\left(\dfrac{2x^2+3x-2}{x^2-3x-4}\right)

is

(\gamma,\delta)

, then

\alpha^2+\beta^2+\gamma^2

is equal to:

(A)195
(B)174
(C)186
(D)179

Q53·MathematicsSingle correctJEE Main 2025

Define a relation R on the interval

\left[0,\dfrac{\pi}{2}\right)

by x R y if and only if

\sec^2 x-\tan^2 y=1

. Then R is:

(A)an equivalence relation
(B)both reflexive and transitive but not symmetric
(C)both reflexive and symmetric but not transitive
(D)reflexive but neither symmetric nor transitive

Q54·MathematicsSingle correctJEE Main 2025

Let

f:R-\{0\}\to(-\infty,1)

be a polynomial of degree 2, satisfying

f(x)\,f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)

. If

f(K)=-2K

, then the sum of squares of all possible values of K is:

(A)1
(B)6
(C)7
(D)9

Q55·MathematicsSingle correctJEE Main 2025

Let

f:[0,3]\to A

be defined by

f(x)=2x^3-15x^2+36x+7

and

g:[0,\infty)\to B

be defined by

g(x)=\frac{x^{2025}}{x^{2025}+1}

. If both the functions are onto and

S=\{x\in Z:x\in A

or

x\in B\}

, then

n(S)

is equal to:

(A)30
(B)36
(C)29
(D)31

Q56·MathematicsSingle correctJEE Main 2025

If

f(x)=\frac{2^x}{2^x+\sqrt{2}}

,

x\in R

, then

\sum_{k=1}^{81} f\left(\frac{k}{82}\right)

is equal to:

(A)41
(B) $\frac{81}{2}$
(C)82
(D) $81\sqrt{2}$

Q57·MathematicsSingle correctJEE Main 2025

The relation

R=\{(x, y): x, y\in z

and

x+y

is even

\}

is:

(A)reflexive and transitive but not symmetric
(B)reflexive and symmetric but not transitive
(C)an equivalence relation
(D)symmetric and transitive but not reflexive

Q58·MathematicsSingle correctJEE Main 2025

Let

f:R\to R

be a function defined by

f(x)=(2+3a)x^2+\frac{a+2}{a-1}x+b

,

a\neq1

. If

f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy

, then the value of

28\sum_{i=1}^{5}|f(i)|

is:

(A)715
(B)735
(C)545
(D)675

Q59·MathematicsSingle correctJEE Main 2025

Let

f(x)=\dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}

. Then the value of

8\left(f\left(\dfrac{1}{15}\right)+f\left(\dfrac{2}{15}\right)+\ldots+f\left(\dfrac{59}{15}\right)\right)

is equal to

(A) $118$
(B) $92$
(C) $102$
(D) $108$

Q60·MathematicsSingle correctJEE Main 2025

The function

f:(-\infty,\infty)\to(-\infty,1)

, defined by

f(x)=\dfrac{2^x-2^{-x}}{2^x+2^{-x}}

is :

(A)One-one but not onto
(B)Onto but not one-one
(C)Both one-one and onto
(D)Neither one-one nor onto

Q61·MathematicsSingle correctJEE Main 2025

Let

A=\left\{x\in(0,\pi)-\left\{\tfrac{\pi}{2}\right\}:\log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2\right\}

and

B=\left\{x\ge 0:\sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\right\}

. Then

n(A\cup B)

is equal to:

(A)4
(B)2
(C)8
(D)6

Q62·MathematicsSingle correctJEE Main 2025

Let

X=\mathbb{R}\times\mathbb{R}

. Define a relation

R

on

X

as:

(a_1,b_1)\,R\,(a_2,b_2)\Leftrightarrow b_1=b_2

. Statement-I:

R

is an equivalence relation. Statement-II: For some

(a,b)\in X

, the set

S=\{(x,y)\in X:(x,y)\,R\,(a,b)\}

represents a line parallel to

y=x

. In the light of the above statements, choose the correct answer from the options given below :

(A)Both Statement-I and Statement-II are false.
(B)Statement-I is true but Statement-II is false.
(C)Both Statement-I and Statement-II are true.
(D)Statement-I is false but Statement-II is true.

Q63·MathematicsSingle correctJEE Main 2025

Let

A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:|x+y|\ge 3\}

and

B=\{(x,y)\in\mathbb{R}\times\mathbb{R}:|x|+|y|\le 3\}

. If

C=\{(x,y)\in A\cap B: x=0 \text{ or } y=0\}

, then

\displaystyle\sum_{(x,y)\in C}|x+y|

is :

(A)15
(B)18
(C)24
(D)12

Q64·MathematicsSingle correctJEE Main 2025

Let

f(x)=\log_e x

and

g(x)=\dfrac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}

. Then the domain of

f\circ g

is

(A) $\mathbb{R}$
(B) $(0,\infty)$
(C) $[0,\infty)$
(D) $[1,\infty)$

Q65·MathematicsSingle correctJEE Main 2025

Let

R=\{(1,2),(2,3),(3,3)\}

be a relation defined on the set

\{1,2,3,4\}

. Then the minimum number of elements, needed to be added in

R

so that

R

becomes an equivalence relation, is

(A) $10$
(B) $8$
(C) $9$
(D) $7$

Q66·MathematicsSingle correctJEE Main 2025

The number of non-empty equivalence relations on the set

\{1,2,3\}

is:

(A)6
(B)7
(C)5
(D)4

Q67·MathematicsSingle correctJEE Main 2025

Let

A=\{1,2,\dots,10\}

and

B=\{\tfrac{m}{n}:m,n\in A,\ m<n,\ \gcd(m,n)=1\}

. Then

n(B)

equals:

(A)31
(B)36
(C)37
(D)29

Q68·MathematicsIntegerJEE Main 2025

Let

A=\{1,2,3\}

. The number of relations on

A

, containing

(1,2)

and

(2,3)

, which are reflexive and transitive but not symmetric, is ______.

Q69·MathematicsSingle correctJEE Main 2025

Let

A=\{1,2,3,4\}

and

B=\{1,4,9,16\}

. Then the number of many-one functions

f:A\to B

such that

1\in f(A)

is equal to:

(A)127
(B)151
(C)163
(D)139

Q70·MathematicsSingle correctJEE Advanced 2024

Let

f : \mathbb{R} \to \mathbb{R}

be a function defined by

f(x) = \begin{cases} x^2 \sin\left(\frac{\pi}{x^2}\right) & ; \text{ if } x \neq 0 \\ 0 & ; \text{ if } x = 0 \end{cases}

Then which of the following statements is TRUE?

(A)f(x) = 0 has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$
(B)f(x) = 0 has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$
(C)The set of solutions of f(x) = 0 in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite
(D)f(x) = 0 has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$

Q71·MathematicsIntegerJEE Advanced 2024

Let

f : \mathbb{R} \to \mathbb{R}

be a function such that

f(x + y) = f(x) + f(y)

for all

x, y \in \mathbb{R}

, and

g : \mathbb{R} \to (0, \infty)

be a function such that

g(x + y) = g(x)g(y)

for all

x, y \in \mathbb{R}

. If

f\left(\frac{-3}{5}\right) = 12

and

g\left(\frac{-1}{3}\right) = 2

, then the value of

\left(f\left(\frac{1}{4}\right) + g(-2) - 8\right) g(0)

is ______

Q72·MathematicsIntegerJEE Advanced 2024

Let

a = 3\sqrt{2}

and

b = \frac{1}{5^{1/6}\sqrt{6}}

. If

x, y \in R

are such that

3x + 2y = \log_{a}(18)^{5/4}

and

2x - y = \log_{b}\left(\sqrt{1080}\right)

, then

4x + 5y

is equal to ______ .

Q73·MathematicsMultiple correctJEE Advanced 2024

Let

S = \left\{a + b\sqrt{2} : a, b \in Z\right\}

,

T_{1} = \left\{\left(-1 + \sqrt{2}\right)^{n} : n \in Z\right\}

, and

T_{2} = \left\{\left(1 + \sqrt{2}\right)^{n} : n \in N\right\}

. Then which of the following statements is(are) TRUE ?

(A) $Z \cup T_{1} \cup T_{2} \subset S$
(B) $T_{1} \cap \left(0, \frac{1}{2024}\right) = \phi$ , where $\phi$ denotes the empty set
(C) $T_{2} \cap (2024, \infty) \neq \phi$
(D)For any given $a, b \in Z$ , $\cos\left(\pi\left(a + b\sqrt{2}\right)\right) + i \sin\left(\pi\left(a + b\sqrt{2}\right)\right) \in Z$ if and only if $b = 0$ , where $i = \sqrt{-1}$

Q74·MathematicsNumericalJEE Main 2024

Let

A = \{2, 3, 6, 7\}

and

B = \{4, 5, 6, 8\}

. Let

R

be a relation defined on

A \times B

by

(a_1, b_1)\, R\, (a_2, b_2)

is and only if

a_1 + a_2 = b_1 + b_2

. Then the number of elements in

R

is ________.

Q75·MathematicsSingle correctJEE Main 2024

Let

f(x) = x^2 + 9

,

g(x) = \frac{x}{x-9}

and

a = f\circ g(10)

,

b = g\circ f(3)

. If

e

and

l

denote the eccentricity and the length of the latus rectum of the ellipse

\frac{x^2}{a} + \frac{y^2}{b} = 1

, then

8e^2 + l^2

is equal to.

(A)16
(B)8
(C)6
(D)12

Q76·MathematicsSingle correctJEE Main 2024

If the domain of the function

f(x) = \sin^{-1}\left(\frac{x - 1}{2x + 3}\right)

is

\mathbb{R} - (\alpha, \beta)

, then

12\alpha\beta

is equal to:

(A)36
(B)24
(C)40
(D)32

Q77·MathematicsSingle correctJEE Main 2024

Let

f(x)=\begin{cases}-a & \text{if } -a\le x\le 0\\ x+a & \text{if } 0<x\le a\end{cases}

where

a>0

and

g(x)=\dfrac{f(|x|)-|f(x)|}{2}

. Then the function

g:[-a,a]\to[-a,a]

is

(A)neither one-one nor onto.
(B)both one-one and onto.
(C)one-one.
(D)onto

Q78·MathematicsSingle correctJEE Main 2024

Let

A=\{2,3,6,8,9,11\}

and

B=\{1,4,5,10,15\}

. Let R be a relation on

A\times B

defined by

(a,b)\,R\,(c,d)

if and only if

3ad-7bc

is an even integer. Then the relation R is

(A)reflexive but not symmetric.
(B)transitive but not symmetric.
(C)reflexive and symmetric but not transitive.
(D)an equivalence relation.

Q79·MathematicsSingle correctJEE Main 2024

Let

f(x)=\frac{1}{7-\sin 5x}

be a function defined on

R

. Then the range of the function

f(x)

is equal to:

(A) $\left[\frac{1}{8},\frac{1}{5}\right]$
(B) $\left[\frac{1}{7},\frac{1}{6}\right]$
(C) $\left[\frac{1}{7},\frac{1}{5}\right]$
(D) $\left[\frac{1}{8},\frac{1}{6}\right]$

Q80·MathematicsSingle correctJEE Main 2024

The function

f(x)=\dfrac{x^{2}+2x-15}{x^{2}-4x+9}

,

x\in\mathbb{R}

is

(A)both one-one and onto
(B)onto but not one-one
(C)neither one-one nor onto
(D)one-one but not onto

Q81·MathematicsSingle correctJEE Main 2024

Let the relations

R_{1}

and

R_{2}

on the set

X=\{1,2,3,\ldots,20\}

be given by

R_{1}=\{(x,y):2x-3y=2\}

and

R_{2}=\{(x,y):-5x+4y=0\}

. If

M

and

N

are the minimum number of elements required to be added in

R_{1}

and

R_{2}

, respectively, in order to make the relations symmetric, then

M+N

equals

(A) $8$
(B) $16$
(C) $12$
(D) $10$

Q82·MathematicsSingle correctJEE Main 2024

Let

A = \{1, 2, 3, 4, 5\}

. Let R be a relation on A defined by

xRy

if and only if

4x \le 5y

. Let m be the number of elements in R and n be the minimum number of elements from

A \times A

that are required to be added to R to make it a symmetric relation. Then

m + n

is equal to:

(A)24
(B)23
(C)25
(D)26

Q83·MathematicsSingle correctJEE Main 2024

Let

A=\{n\in[100,700]\cap\mathbb{N}:n\text{ is neither a multiple of 3 nor a multiple of 4}\}

. Then the number of elements in

A

is

(A) $300$
(B) $280$
(C) $310$
(D) $290$

Q84·MathematicsSingle correctJEE Main 2024

Let

f,g:\mathbb{R}\to\mathbb{R}

be defined as

f(x)=|x-1|

and

g(x)=e^x

for

x\ge 0

,

g(x)=x+1

for

x<0

. Then the function

f(g(x))

is:

(A)neither one-one nor onto
(B)one-one but not onto
(C)both one-one and onto
(D)onto but not one-one

Q85·MathematicsNumericalJEE Main 2024

If

S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}

, where

[t]

denotes the greatest integer less than or equal to

t

and

\{t\}

represents the fractional part of

t

, then

72\displaystyle\sum_{a \in S} a

is equal to ___.

Q86·MathematicsNumericalJEE Main 2024

Consider the function

f:\mathbb{R}\to\mathbb{R}

defined by

f(x)=\dfrac{2x}{\sqrt{1+9x^2}}

. If the composition of

f

,

\underbrace{(f\circ f\circ f\circ\ldots\circ f)}_{10\text{ times}}(x)=\dfrac{2^{10}x}{\sqrt{1+9\alpha x^2}}

, then the value of

\sqrt{3\alpha+1}

is equal to

Q87·MathematicsNumericalJEE Main 2024

In a survey of

220

students of a higher secondary school, it was found that at least

125

and at most

130

studied Mathematics; at least

85

and at most

95

studied Physics; at least

75

and at most

90

studied Chemistry;

30

studied both Physics and Chemistry;

50

studied both Chemistry and Mathematics;

40

studied both Mathematics and Physics and

10

studied none of these three subjects. Let

m

and

n

respectively be the least and the most number of students who studied all the three subjects. Then

m+n

is equal to ___

Q88·MathematicsSingle correctJEE Main 2024

Let a relation R on

\mathbb{N}\times\mathbb{N}

be defined as

(x_1,y_1)\,R\,(x_2,y_2)

if and only if

x_1\le x_2

or

y_1\le y_2

. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive. Then which one of the following is true?

(A)Only (II) is correct.
(B)Only (I) is correct.
(C)Both (I) and (II) are correct.
(D)Neither (I) nor (II) is correct.

Q89·MathematicsSingle correctJEE Main 2024

If the domain of the function

\sin^{-1}\left(\dfrac{3x-22}{2x-19}\right)+\log_e\left(\dfrac{3x^2-8x+5}{x^2-3x-10}\right)

is

(\alpha,\beta]

, then

3\alpha+10\beta

is equal to:

(A) $97$
(B) $100$
(C) $95$
(D) $98$

Q90·MathematicsSingle correctJEE Main 2024

Let

f:\mathbb{R}\to\mathbb{R}

and

g:\mathbb{R}\to\mathbb{R}

be defined as

f(x)=\begin{cases}\log_e x & ,\ x>0\\ e^{-x} & ,\ x\le0\end{cases}

and

g(x)=\begin{cases}x & ,\ x\ge0\\ e^x & ,\ x<0\end{cases}

. Then

g\circ f:\mathbb{R}\to\mathbb{R}

is:

(A)one-one but not onto
(B)neither one-one nor onto
(C)onto but not one-one
(D)both one-one and onto

Q91·MathematicsNumericalJEE Main 2024

Let

A=\{1,2,3,\ldots,20\}

. Let

R_1

and

R_2

be two relations on

A

such that

R_1=\{(a,b):b

is divisible by

a\}

,

R_2=\{(a,b):a

is an integral multiple of

b\}

. Then, number of elements in

R_1-R_2

is equal to ___

Q92·MathematicsSingle correctJEE Main 2024

Consider the relations

R_1

and

R_2

defined as

aR_1 b\Leftrightarrow a^2+b^2=1

for all

a,b\in\mathbb R

and

(a,b)R_2(c,d)\Leftrightarrow a+d=b+c

for all

(a,b),(c,d)\in\mathbb N\times\mathbb N

. Then

(A)Only $R_1$ is an equivalence relation
(B)Only $R_2$ is an equivalence relation
(C) $R_1$ and $R_2$ both are equivalence relations
(D)Neither $R_1$ nor $R_2$ is an equivalence relation

Q93·MathematicsSingle correctJEE Main 2024

If the domain of the function

f(x)=\dfrac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)

is

(-\infty,\alpha)\cup[\beta,\infty)

, then

\alpha^2+\beta^3

is equal to:

(A)140
(B)175
(C)150
(D)125

Q94·MathematicsNumericalJEE Main 2024

Let

A=\{1,2,3,4\}

and

R=\{(1,2),(2,3),(1,4)\}

be a relation on A. Let S be the equivalence relation on A such that

R\subset S

and the number of elements in S is n. Then, the minimum value of n is ______.

Q95·MathematicsNumericalJEE Main 2024

Let

A=\{1,2,3,\ldots\ldots100\}

. Let R be a relation on A defined by

(x,y)\in R

if and only if

2x=3y

. Let

R_1

be a symmetric relation on A such that

R\subset R_1

and the number of elements in

R_1

is

n

. Then the minimum value of

n

is ______.

Q96·MathematicsSingle correctJEE Main 2024

If

f(x)=\dfrac{4x+3}{6x-4}

,

x\ne\dfrac{2}{3}

and

(f\circ f)(x)=g(x)

, where

g:R-\left\{\dfrac{2}{3}\right\}\to R-\left\{\dfrac{2}{3}\right\}

, then

(g\circ g\circ g)(4)

is equal to

(A) $-\dfrac{19}{20}$
(B) $\dfrac{19}{20}$
(C) $-4$
(D) $4$

Q97·MathematicsNumericalJEE Main 2024

Let

A=\{1,2,3,\ldots,7\}

and let

P(A)

denote the power set of

A

. If the number of functions

f:A\to P(A)

such that

a\in f(a)

,

\forall a\in A

is

m^n

,

m,n\in\mathbb{N}

and

m

is least, then

m+n

is equal to ___

Q98·MathematicsNumericalJEE Main 2024

A group of

40

students appeared in an examination of

3

subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects,

20

students passed in Mathematics,

25

students passed in Physics,

16

students passed in Chemistry, at most

11

students passed in both Mathematics and Physics, at most

15

students passed in both Physics and Chemistry, at most

15

students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ___

Q99·MathematicsSingle correctJEE Main 2024

If the domain of the function

f(x)=\log_e\left(\dfrac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\dfrac{2x-1}{x+2}\right)

is

(\alpha,\beta]

, then the value of

5\beta-4\alpha

is equal to:

(A) $10$
(B) $12$
(C) $11$
(D) $9$

Q100·MathematicsNumericalJEE Main 2024

The number of symmetric relations defined on the set

\{1,2,3,4\}

which are not reflexive is ______.

Q101·MathematicsSingle correctJEE Main 2024

If the domain of the function

f(x)=\cos^{-1}\left(\dfrac{2-|x|}{4}\right)+\left(\log_e(3-x)\right)^{-1}

is

[-\alpha,\beta]-\{\gamma\}

, then

\alpha+\beta+\gamma

is equal to:

(A) $12$
(B) $9$
(C) $11$
(D) $8$

Q102·MathematicsSingle correctJEE Main 2024

If

f(x)=\begin{cases}2+2x, & -1\le x<0\\ 1-\dfrac{x}{3}, & 0\le x\le 3\end{cases}

;

g(x)=\begin{cases}-x, & -3\le x\le 0\\ x, & 0<x\le 1\end{cases}

, then range of

(f\circ g)(x)

is

(A) $(0,1]$
(B) $[0,3)$
(C) $[0,1]$
(D) $[0,1)$

Q103·MathematicsSingle correctJEE Main 2024

Let R be a relation on

Z\times Z

defined by

(a,b)R(c,d)

if and only if

ad-bc

is divisible by 5. Then R is

(A)Reflexive and symmetric but not transitive
(B)Reflexive but neither symmetric not transitive
(C)Reflexive, symmetric and transitive
(D)Reflexive and transitive but not symmetric

Q104·MathematicsNumericalJEE Main 2024

Let the set

C=\{(x,y)\mid x^2-2^y=2023,\ x,y\in\mathbb{N}\}

. Then

\displaystyle\sum_{(x,y)\in C}(x+y)

is equal to ___.

Q105·MathematicsSingle correctJEE Main 2024

If

R

is the smallest equivalence relation on the set

\{1,2,3,4\}

such that

\{(1,2),(1,3)\}\subset R

, then the number of elements in

R

is:

(A)10
(B)12
(C)8
(D)15

Q106·MathematicsSingle correctJEE Main 2024

Let

S=\{1,2,3,\ldots,10\}

. Suppose

M

is the set of all the subsets of

S

, then the relation

R=\{(A,B):A\cap B\neq\varphi;\,A,B\in M\}

is:

(A)symmetric and reflexive only
(B)reflexive only
(C)symmetric and transitive only
(D)symmetric only

Q107·MathematicsSingle correctJEE Main 2024

Let

f:\mathbb{R}-\left\{-\dfrac12\right\}\to\mathbb{R}

and

g:\mathbb{R}-\left\{-\dfrac52\right\}\to\mathbb{R}

be defined as

f(x)=\dfrac{2x+3}{2x+1}

and

g(x)=\dfrac{|x|+1}{2x+5}

. Then the domain of the function

f\circ g

is :

(A) $\mathbb{R}-\left\{-\dfrac52\right\}$
(B) $\mathbb{R}$
(C) $\mathbb{R}-\left\{-\dfrac74\right\}$
(D) $\mathbb{R}-\left\{-\dfrac52,-\dfrac74\right\}$

Q108·MathematicsSingle correctJEE Main 2024

The function

f:\mathbb N-\{1\}\to\mathbb N

defined by

f(n)=

the highest prime factor of

n

, is:

(A)both one-one and onto
(B)one-one only
(C)onto only
(D)neither one-one nor onto

Q109·MathematicsSingle correctJEE Main 2024

Let A and B be two finite sets with

m

and

n

elements respectively. The total number of subsets of the set A is 56 more than the total number of subsets of B. Then the distance of the point

P(m,n)

from the point

Q(-2,-3)

is

(A)10
(B)6
(C)4
(D)8

Q110·MathematicsMultiple correctJEE Advanced 2023

Let

S = (0, 1) \cup (1, 2) \cup (3, 4)

and

T = \{0, 1, 2, 3\}

. Then which of the following statements is(are) true?

(A)There are infinitely many functions from $S$ to $T$
(B)There are infinitely many strictly increasing functions from $S$ to $T$
(C)The number of continuous functions from $S$ to $T$ is at most 120
(D)Every continuous function from $S$ to $T$ is differentiable

Q111·MathematicsSingle correctJEE Main 2023

Negation of

p\wedge(q\wedge\sim(p\wedge q))

is

(A) $\sim(p\vee q)$
(B) $p\vee q$
(C) $(\sim(p\wedge q))\wedge q$
(D) $(\sim(p\wedge q))\vee p$

Q112·MathematicsSingle correctJEE Main 2023

If the domain of the function

f(x) = \log_e\left(4x^2 + 11x + 6\right) + \sin^{-1}(4x + 3) + \cos^{-1}\left(\frac{10x + 6}{3}\right)

is

(\alpha, \beta]

, then

36\,|\alpha + \beta|

is equal to:

(A) $63$
(B) $45$
(C) $72$
(D) $54$

Q113·MathematicsSingle correctJEE Main 2023

Let

[x]

denote the greatest integer function and

f(x)=\max\{1+x+[x],\,2+x,\,x+2[x]\}

,

0\le x\le 2

. Let

m

be the number of points in

[0,2]

, where

f

is not continuous and

n

be the number of points in

(0,2)

, where

f

is not differentiable. Then

(m+n)^2+2

is equal to:

(A)11
(B)2
(C)6
(D)3

Q114·MathematicsNumericalJEE Main 2023

Let

A = \{1, 2, 3, 4\}

and

R

be a relation on the set

A \times A

defined by

R = \{((a, b), (c, d)) : 2a + 3b = 4c + 5d\}

. Then the number of elements in

R

is:

Q115·MathematicsSingle correctJEE Main 2023

The negation of the statement

((A\wedge(B\vee C))\Rightarrow(A\vee B))\Rightarrow A

is:

(A)equivalent to $\sim A$
(B)equivalent to $\sim C$
(C)equivalent to $B\vee\sim C$
(D)a fallacy

Q116·MathematicsSingle correctJEE Main 2023

For

x\in\mathbb{R}

, two real valued functions

f(x)

and

g(x)

are such that

g(x)=\sqrt x+1

and

f(g(x))=x+3-\sqrt x

. Then

f(0)

is equal to:

(A) $1$
(B) $-3$
(C) $5$
(D) $0$

Q117·MathematicsNumericalJEE Main 2023

Let

A = \{-4, -3, -2, 0, 1, 3, 4\}

and

R = \{(a, b) \in A \times A : b = |a| \text{ or } b^2 = a + 1\}

be a relation on A. Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is

Q118·MathematicsSingle correctJEE Main 2023

The statement

(\sim(p \Leftrightarrow \sim q)) \wedge q

is equivalent to

(A) $(\sim p) \wedge (\sim q)$
(B) $p \wedge (\sim q)$
(C) $(\sim p) \vee q$
(D) $p \vee q$

Q119·MathematicsSingle correctJEE Main 2023

Among the two statements (S1):

(p\Rightarrow q)\wedge(q\wedge(\sim q))

is a contradiction and (S2):

(p\wedge q)\vee((\sim p)\wedge q)\vee(p\wedge(\sim q))\vee((\sim p)\wedge(\sim q))

is a tautology

(A)only (S2) is true
(B)only (S1) is true
(C)both are false
(D)both are true

Q120·MathematicsSingle correctJEE Main 2023

Let D be the domain of the function

f(x)=\sin^{-1}\left(\log_{3x}\left(\dfrac{6+2\log_3 x}{-5x}\right)\right)

. If the range of the function

g:D\to\mathbb{R}

defined by

g(x)=x-[x]

(

[x]

is the greatest integer function) is

(\alpha,\beta)

, then

\alpha^2+\dfrac{5}{\beta}

is equal to

(A)46
(B)135
(C)136
(D)45

Q121·MathematicsNumericalJEE Main 2023

The number of relations, on the set

\{1,2,3\}

containing

(1,2)

and

(2,3)

, which are reflexive and transitive but not symmetric, is _________.

Q122·MathematicsSingle correctJEE Main 2023

Let

A=\{1,3,4,6,9\}

and

B=\{2,4,5,8,10\}

. Let

R

be a relation defined on

A\times B

such that

R=\{((a_1,b_1),(a_2,b_2)):a_1\le b_2 \text{ and } b_1\le a_2\}

. Then the number of elements in the set

R

is

(A)26
(B)160
(C)180
(D)52

Q123·MathematicsNumericalJEE Main 2023

The number of points, where the curve

f(x)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1

,

x\in\mathbb{R}

cuts the x-axis, is equal to

Q124·MathematicsNumericalJEE Main 2023

The number of ordered triplets of the truth values of

p, q

and

r

such that the truth value of the statement

(p \vee q) \wedge (p \vee r) \Rightarrow (q \vee r)

is True, is equal to _______ .

Q125·MathematicsSingle correctJEE Main 2023

The domain of the function

f(x)=\dfrac{1}{\sqrt{[x]^2-3[x]-10}}

is (where

[x]

denotes the greatest integer less than or equal to

x

)

(A) $(-\infty,-3)\cup(5,\infty)$
(B) $(-\infty,-2)\cup(6,\infty)$
(C) $(-\infty,-3]\cup(6,\infty)$
(D) $(-\infty,-3]\cup[6,\infty)$

Q126·MathematicsNumericalJEE Main 2023

Let

A=\{1,2,3,4,5\}

and

B=\{1,2,3,4,5,6\}

. Then the number of functions

f:A\to B

satisfying

f(1)+f(2)=f(4)-1

is equal to

Q127·MathematicsSingle correctJEE Main 2023

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

(A)10
(B)9
(C)21
(D)15

Q128·MathematicsSingle correctJEE Main 2023

The converse of the statement

((\sim p)\wedge q)\Rightarrow r

is

(A) $(\sim r)\Rightarrow p\wedge q$
(B) $(\sim r)\Rightarrow((\sim p)\wedge q)$
(C) $r\Rightarrow((\sim p)\wedge q)$
(D) $(p\vee(\sim q))\Rightarrow(\sim r)$

Q129·MathematicsSingle correctJEE Main 2023

The negation of the statement

(p\lor q)\land(q\lor(\sim r))

is

(A) $((\sim p)\lor r)\land(\sim q)$
(B) $((\sim p)\lor(\sim q))\land(\sim r)$
(C) $((\sim p)\lor(\sim q))\lor(\sim r)$
(D) $(p\lor r)\land(\sim q)$

Q130·MathematicsSingle correctJEE Main 2023

The statement

\sim[p\vee(\sim(p\wedge q))]

is equivalent to:

(A) $\sim(p\wedge q)\wedge q$
(B) $\sim(p\wedge q)$
(C) $\sim p\vee q$
(D) $(p\wedge q)\wedge(\sim p)$

Q131·MathematicsSingle correctJEE Main 2023

If

f(x)=\dfrac{(\tan 1°)x+\log_e(123)}{x\log_e(1234)-(\tan 1°)},\ x>0

, then the least value of

f(f(x))+f\left(f\left(\dfrac4x\right)\right)

is

(A) $8$
(B) $4$
(C) $2$
(D) $0$

Q132·MathematicsSingle correctJEE Main 2023

Let

A=\{2,3,4\}

and

B=\{8,9,12\}

. Then the number of elements in the relation

R=\{((a_1,b_1),(a_2,b_2))\in(A\times B,\ A\times B):a_1\text{ divides }b_2\text{ and }a_2\text{ divides }b_1\}

is:

(A) $36$
(B) $12$
(C) $18$
(D) $24$

Q133·MathematicsSingle correctJEE Main 2023

Negation of

p\Rightarrow q\Rightarrow q\Rightarrow p

is

(A) $(\sim p)\vee q$
(B) $(\sim q)\wedge p$
(C) $q\wedge(\sim p)$
(D) $p\vee(\sim q)$

Q134·MathematicsSingle correctJEE Main 2023

The negation of

p\wedge(\sim q)\vee(\sim p)

is equivalent to

(A) $p\wedge q$
(B) $p\wedge(\sim q)$
(C) $p\wedge(q\wedge(\sim p))$
(D) $p\vee(q\vee(\sim p))$

Q135·MathematicsSingle correctJEE Main 2023

Let

A=\{1,2,3,4,5,6,7\}

. Then the relation

R=\{(x,y)\in A\times A:x+y=7\}

is

(A)transitive but neither symmetric nor reflexive
(B)reflexive but neither symmetric nor transitive
(C)an equivalence relation
(D)symmetric but neither reflexive nor transitive

Q136·MathematicsNumericalJEE Main 2023

Let

A=\{0,3,4,6,7,8,9,10\}

and

R

be the relation defined on

A

such that

R=\{(x,y)\in A\times A: x-y \text{ is odd positive integer or } x-y=2\}

. The minimum number of elements that must be added to the relation

R

, so that it is a symmetric relation, is equal to

Q137·MathematicsSingle correctJEE Main 2023

Let the number of elements in sets

A

and

B

be five and two respectively. Then the number of subsets of

A\times B

each having at least 3 and at most 6 element is :

(A)792
(B)752
(C)782
(D)772

Q138·MathematicsNumericalJEE Main 2023

Let

A=\{1,2,3,4,\dots,10\}

and

B=\{0,1,2,3,4,5\}

. The number of elements in the relation

R=\{(a,b)\in A\times A:2(a-b)^{2}+3(a-b)\in B\}

is _____.

Q139·MathematicsSingle correctJEE Main 2023

The statement

(P\Rightarrow Q)\wedge(R\Rightarrow Q)

is logically equivalent to:

(A) $(P\vee R)\Rightarrow Q$
(B) $(P\Rightarrow R)\wedge(Q\Rightarrow R)$
(C) $(P\Rightarrow R)\vee(Q\Rightarrow R)$
(D) $(P\wedge R)\Rightarrow Q$

Q140·MathematicsSingle correctJEE Main 2023

Let

A=\{x\in\mathbb{R}:[x+3]+[x+4]\le 3\}

,

B=\left\{x\in\mathbb{R}:3^{x}\left(\displaystyle\sum_{r=1}^{\infty}\dfrac{3}{10^{r}}\right)^{x}<3^{-x}\right\}

, where

[t]

denotes greatest integer function. Then:

(A) $A=\varnothing,\,B\ne\varnothing$
(B) $A=B$
(C) $B\subset A,\,A\ne B$
(D) $A\subset B,\,A\ne B$

Q141·MathematicsSingle correctJEE Main 2023

Let the sets

A

and

B

denote the domain and range respectively of the function

f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}

, where

\lceil x\rceil

denotes the smallest integer greater than or equal to

x

. Then among the statements (S1):

A\cap B=(1,\infty)-\mathbb{N}

and (S2):

A\cup B=(1,\infty)

(A)only (S1) is true
(B)both (S1) and (S2) are true
(C)neither (S1) nor (S2) is true
(D)only (S2) is true

Q142·MathematicsNumericalJEE Main 2023

Let the point

(p,p+1)

lie inside the region

E=\{(x,y):3-x\le y\le\sqrt{9-x^{2}},\,0\le x\le 3\}

. If the set of all values of

p

is the interval

(a,b)

, then

b^{2}+b-a^{2}

is equal to _____.

Q143·MathematicsSingle correctJEE Main 2023

Among the statements: (S1):

(p\Rightarrow q)\lor(\sim p\land q)

is a tautology (S2):

(q\Rightarrow p)\Rightarrow(\sim p\land q)

is a contradiction

(A)neither (S1) nor (S2) is True
(B)only (S1) is True
(C)only (S2) is True
(D)both (S1) and (S2) are True

Q144·MathematicsSingle correctJEE Main 2023

Let

P(S)

denote the power set of

S=\{1,2,3,\ldots,10\}

. Define the relations

R_1

and

R_2

on

P(S)

as

AR_1B

if

(A\cap B^c)\cup(B\cap A^c)=\varnothing

and

AR_2B

if

A\cup B^c=B\cup A^c

,

\forall A,B\in P(S)

. Then:

(A)only $R_1$ is an equivalence relation
(B)only $R_2$ is an equivalence relation
(C)both $R_1$ and $R_2$ are equivalence relations
(D)both $R_1$ and $R_2$ are not equivalence relations

Q145·MathematicsSingle correctJEE Main 2023

Which of the following statements is a tautology?

(A) $p\vee(p\wedge q)$
(B) $(p\wedge(p\to q))\to\sim q$
(C) $(p\wedge q)\to(\sim(p)\to q)$
(D) $p\to(p\wedge(p\to q))$

Q146·MathematicsSingle correctJEE Main 2023

Let

R

be a relation on

\mathbb{R}

, given by

R=\{(a,b):3a-3b+\sqrt{7}\text{ is an irrational number}\}

. Then

R

is

(A)an equivalence relation
(B)reflexive and symmetric but not transitive
(C)reflexive but neither symmetric nor transitive
(D)reflexive and transitive but not symmetric

Q147·MathematicsSingle correctJEE Main 2023

Let

f:\mathbb{R}-\{0,1\}\to\mathbb{R}

be a function such that

f(x)+f\left(\dfrac{1}{1-x}\right)=1+x

. Then

f(2)

is equal to

(A) $\dfrac92$
(B) $\dfrac74$
(C) $\dfrac94$
(D) $\dfrac73$

Q148·MathematicsSingle correctJEE Main 2023

The negation of the expression

q\vee((\sim q)\wedge p)

is equivalent to

(A) $(\sim p)\vee(\sim q)$
(B) $p\wedge(\sim q)$
(C) $(\sim p)\vee q$
(D) $(\sim p)\wedge(\sim q)$

Q149·MathematicsSingle correctJEE Main 2023

Among the relations

S=\left\{(a,b):a,b\in\mathbb{R}-\{0\},\,2+\dfrac{a}{b}>0\right\}

and

T=\{(a,b):a,b\in\mathbb{R},\,a^2-b^2\in\mathbb{Z}\}

,

(A)neither $S$ nor $T$ is transitive
(B) $S$ is transitive but $T$ is not
(C) $T$ is symmetric but $S$ is not
(D)both $S$ and $T$ are symmetric

Q150·MathematicsSingle correctJEE Main 2023

(S1)

(p\Rightarrow q)\vee(p\wedge(\sim q))

is a tautology. (S2)

((\sim p)\Rightarrow(\sim q))\wedge((\sim p)\vee q)

is a contradiction. Then

(A)both (S1) and (S2) are correct
(B)only (S1) is correct
(C)only (S2) is correct
(D)both (S1) and (S2) are wrong

Q151·MathematicsSingle correctJEE Main 2023

Let

f:\mathbb{R}-\{2,6\}\to\mathbb{R}

be real valued function defined as

f(x)=\dfrac{x^2+2x+1}{x^2-8x+12}

. Then range of

f

is

(A) $\left(-\infty,-\dfrac{21}{4}\right]\cup(0,\infty)$
(B) $\left(-\infty,-\dfrac{21}{4}\right]\cup[1,\infty)$
(C) $\left(-\infty,-\dfrac{21}{4}\right]\cup\left[\dfrac{21}{4},\infty\right)$
(D) $\left(-\infty,-\dfrac{21}{4}\right]\cup[0,\infty)$

Q152·MathematicsSingle correctJEE Main 2023

Let R be a relation on

N\times N

defined by

(a,b)R(c,d)

if and only if

ad(b-c)=bc(a-d)

. Then R is

(A)transitive but neither reflexive nor symmetric
(B)symmetric but neither reflexive nor transitive
(C)symmetric and transitive but not reflexive
(D)reflexive and symmetric but not transitive

Q153·MathematicsSingle correctJEE Main 2023

If the domain of the function

f(x)=\frac{[x]}{1+x^{2}}

, where

[x]

is greatest integer

\le x

, is

[2,6)

, then its range is

(A) $\left(\frac{5}{26},\frac{2}{5}\right]$
(B) $\left(\frac{5}{37},\frac{2}{5}\right]-\left\{\frac{9}{29},\frac{27}{109},\frac{18}{89},\frac{9}{53}\right\}$
(C) $\left(\frac{5}{37},\frac{2}{5}\right]$
(D) $\left(\frac{5}{26},\frac{2}{5}\right]-\left\{\frac{9}{29},\frac{27}{109},\frac{18}{89},\frac{9}{53}\right\}$

Q154·MathematicsSingle correctJEE Main 2023

y=f(x)=\sin^{3}\left(\frac{\pi}{3}\left(\cos\left(\frac{\pi}{3\sqrt{2}}(-4x^{3}+5x^{2}+1)^{\frac{3}{2}}\right)\right)\right)

. Then, at

x=1

,

(A) $\sqrt{2}\,y'-3\pi^{2}y=0$
(B) $y'+3\pi^{2}y=0$
(C) $2y'+3\pi^{2}y=0$
(D) $2y'+\sqrt{3}\pi^{2}y=0$

Q155·MathematicsSingle correctJEE Main 2023

The number of values of

r\in\{p,q,\sim p,\sim q\}

for which

((p\wedge q)\Rightarrow(r\vee q))\wedge((p\wedge r)\Rightarrow q)

is a tautology, is:

(A) $3$
(B) $4$
(C) $1$
(D) $2$

Q156·MathematicsNumericalJEE Main 2023

Let

A=\{1, 2, 3, 5, 8, 9\}

. Then the number of possible functions

f: A \to A

such that

f(m \cdot n)=f(m) \cdot f(n)

for every

m, n \in A

with

m \cdot n \in A

is equal to _______.

Q157·MathematicsSingle correctJEE Main 2023

The minimum number of elements that must be added to the relation

R=\{(a,b),(b,c)\}

on the set

\{a,b,c\}

so that it becomes symmetric and transitive is:

(A) $3$
(B) $4$
(C) $5$
(D) $7$

Q158·MathematicsSingle correctJEE Main 2023

The range of the function

f(x)=\sqrt{3 - x} + \sqrt{2 + x}

is:

(A) $[2\sqrt{2}, \sqrt{11}]$
(B) $[\sqrt{5}, \sqrt{13}]$
(C) $[\sqrt{2}, \sqrt{7}]$
(D) $[\sqrt{5}, \sqrt{10}]$

Q159·MathematicsNumericalJEE Main 2023

Let

f^1(x)=\dfrac{3x+2}{2x+3}

,

x\in\mathbb{R}-\left\{\dfrac{-3}{2}\right\}

. For

n\geq2

, define

f^n(x)=f^1\text{ of }f^{n-1}(x)

. If

f^5(x)=\dfrac{ax+b}{bx+a}

,

\gcd(a,b)=1

, then

a+b

is equal to

Q160·MathematicsSingle correctJEE Main 2023

Among the statements: (S1)

((p\vee q)\Rightarrow r)\Leftrightarrow(p\Rightarrow r)

and (S2)

((p\vee q)\Rightarrow r)\Leftrightarrow((p\Rightarrow r)\vee(q\Rightarrow r))

(A)only (S2) is a tautology
(B)only (S1) is a tautology
(C)neither (S1) nor (S2) is a tautology
(D)both (S1) and (S2) are tautologies

Q161·MathematicsSingle correctJEE Main 2023

Consider the following statements: P: I have fever; Q: I will not take medicine; R: I will take rest. The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

(A) $((\sim P) \vee \sim Q) \wedge ((\sim P) \vee R)$
(B) $(P \vee Q) \wedge ((\sim P) \vee R)$
(C) $((\sim P) \vee \sim Q) \wedge ((\sim P) \vee \sim R)$
(D) $(P \vee \sim Q) \wedge (P \vee \sim R)$

Q162·MathematicsSingle correctJEE Main 2023

The domain of

f(x) = \dfrac{\log_{(x+1)}(x - 2)}{e^{2\log_e x} - (2x + 3)}

,

x \in \mathbb{R}

is

(A) $\mathbb{R} - \{3\}$
(B) $(-1, \infty) - \{3\}$
(C) $(2, \infty) - \{3\}$
(D) $\mathbb{R} - \{-1, 3\}$

Q163·MathematicsSingle correctJEE Main 2023

If

p

,

q

and

r

three propositions, then which of the following combination of truth values of

p

,

q

and

r

makes the logical expression

\{(p \vee q) \wedge ((\sim p) \vee r)\} \to ((\sim q) \vee r)

false?

(A)p = T, q = T, r = F
(B)p = T, q = F, r = T
(C)p = F, q = T, r = F
(D)p = T, q = F, r = F

Q164·MathematicsSingle correctJEE Main 2023

Let

R

be a relation defined on

\mathbb{N}

as

a\,R\,b

if

2a+3b

is a multiple of

5,\,a,b\in\mathbb{N}

. Then

R

is:

(A)an equivalence relation
(B)transitive but not symmetric
(C)not reflexive
(D)symmetric but not transitive

Q165·MathematicsSingle correctJEE Main 2023

Let

f: \mathbb{R} \to \mathbb{R}

be a function such that

f(x) = \dfrac{x^2 + 2x + 1}{x^2 + 1}

. Then

(A) $f(x)$ is one-one in $[1, \infty)$ but not in $(-\infty, \infty)$
(B) $f(x)$ is one-one in $(-\infty, \infty)$
(C) $f(x)$ is many-one in $(-\infty, -1)$
(D) $f(x)$ is many-one in $(1, \infty)$

Q166·MathematicsNumericalJEE Main 2023

Let

f: \mathbb{R} \to \mathbb{R}

be a differentiable function that satisfies the relation

f(x + y) = f(x) + f(y) - 1

,

\forall x, y \in \mathbb{R}

. If

f'(0) = 2

, then

|f(-2)|

is equal to ________ .

Q167·MathematicsSingle correctJEE Main 2023

The statement

B\Rightarrow ((\sim A)\lor B)

is equivalent to:

(A) $A\Rightarrow (A\Leftrightarrow B)$
(B) $A\Rightarrow ((\sim A)\Rightarrow B)$
(C) $B\Rightarrow (A\Rightarrow B)$
(D) $B\Rightarrow ((\sim A)\Rightarrow B)$

Q168·MathematicsSingle correctJEE Main 2023

Let

f(x) = 2x^n + \lambda

,

\lambda \in \mathbb{R}

,

n \in \mathbb{N}

, and

f(4) = 133

,

f(5) = 255

. Then the sum of all the positive integer divisors of

(f(3) - f(2))

is

(A)60
(B)59
(C)61
(D)58

Q169·MathematicsSingle correctJEE Main 2023

Let

\Delta, \nabla \in \{\wedge, \vee\}

be such that

(p \to q)\,\Delta\,(p\,\nabla\,q)

is a tautology. Then

(A) $\Delta = \vee,\ \nabla = \vee$
(B) $\Delta = \vee,\ \nabla = \wedge$
(C) $\Delta = \wedge,\ \nabla = \vee$
(D) $\Delta = \wedge,\ \nabla = \wedge$

Q170·MathematicsSingle correctJEE Main 2023

The statement

(p\wedge(\sim q))\Rightarrow(p\Rightarrow(\sim q))

is:

(A)a tautology
(B)a contradiction
(C)equivalent to $p\vee q$
(D)equivalent to $(\sim p)\vee(\sim q)$

Q171·MathematicsSingle correctJEE Main 2023

The number of functions

f : \{1,2,3,4\} \to \{a \in \mathbb{Z} : |a| \le 8\}

satisfying

f(n) + \dfrac{1}{n}f(n+1) = 1

,

\forall n \in \{1,2,3\}

is

(A)1
(B)4
(C)2
(D)3

Q172·MathematicsSingle correctJEE Main 2023

Let

f:\mathbb{R}\to\mathbb{R}

be a function defined by

f(x)=\log_{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}

, for some

m

, such that the range of

f

is

[0,2]

. Then the value of

m

is

(A) $5$
(B) $4$
(C) $3$
(D) $2$

Q173·MathematicsNumericalJEE Main 2023

For some

a,b,c\in\mathbb{N}

, let

f(x)=ax-3

and

g(x)=x^b+c

,

x\in\mathbb{R}

. If

(f\circ g)^{-1}(x)=\left(\dfrac{x-7}{2}\right)^{1/3}

, then

(f\circ g)(ac)+(g\circ f)(b)

is equal to _______.

Q174·MathematicsNumericalJEE Main 2023

The minimum number of elements that must be added to the relation

R=\{(a,b),(b,c),(b,d)\}

on the set

\{a,b,c,d\}

so that it is an equivalence relation, is

Q175·MathematicsSingle correctJEE Main 2023

The equation

x^2-4x+[x]+3=x[x]

, where

[x]

denotes the greatest integer function, has:

(A)a unique solution in $(-\infty,1)$
(B)no solution
(C)exactly two solutions in $(-\infty,\infty)$
(D)a unique solution in $(-\infty,\infty)$

Q176·MathematicsSingle correctJEE Main 2023

Let

p

and

q

be two statements. Then

\sim(p\wedge(p\Rightarrow\sim q))

is equivalent to

(A) $p\vee(p\wedge q)$
(B) $p\vee(p\wedge(\sim q))$
(C) $(\sim p)\vee q$
(D) $p\vee((\sim p)\wedge q)$

Q177·MathematicsSingle correctJEE Main 2023

If

f(x)=\dfrac{2^{2x}}{2^{2x}+2},\ x\in\mathbb{R}

, then

f\left(\dfrac{1}{2023}\right)+f\left(\dfrac{2}{2023}\right)+\cdots+f\left(\dfrac{2022}{2023}\right)

is equal to

(A) $1011$
(B) $2010$
(C) $1010$
(D) $2011$

Q178·MathematicsSingle correctJEE Main 2023

The compound statement

(\sim(P\wedge Q))\vee((\sim P)\wedge Q)\Rightarrow((\sim P)\wedge(\sim Q))

is equivalent to

(A) $(\sim Q)\vee P$
(B) $((\sim P)\vee Q)\wedge(\sim Q)$
(C) $(\sim P)\vee Q$
(D) $((\sim P)\vee Q)\wedge((\sim Q)\vee P)$

Q179·MathematicsSingle correctJEE Main 2023

The relation

R=\{(a,b):\gcd(a,b)=1,\ 2a\ne b,\ a,b\in\mathbb{Z}\}

is:

(A)reflexive but not symmetric
(B)transitive but not reflexive
(C)symmetric but not transitive
(D)neither symmetric nor transitive

Q180·MathematicsNumericalJEE Advanced 2022

In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is __________.

Q181·MathematicsSingle correctJEE Main 2022

Let R be a relation from the set

\{1, 2, 3\ldots\ldots\ldots,60\}

to itself such that

R = \{(a, b) : b = pq,

where

p, q \geq 3

are prime numbers

\}

. Then, the number of elements in R is :

(A)600
(B)660
(C)540
(D)720

Q182·MathematicsSingle correctJEE Main 2022

The statement

\left(p \Rightarrow q\right) \vee \left(p \Rightarrow r\right)

is NOT equivalent to:

(A) $\left(p \wedge (\sim r)\right) \Rightarrow q$
(B) $(\sim q) \Rightarrow \left((\sim r) \vee p\right)$
(C) $p \Rightarrow (q \vee r)$
(D) $\left(p \wedge (\sim q)\right) \Rightarrow r$

Q183·MathematicsSingle correctJEE Main 2022

The statement

(p \wedge q) \Rightarrow (p \wedge r)

is equivalent to :

(A) $q \Rightarrow (p \wedge r)$
(B) $p \Rightarrow (p \wedge r)$
(C) $(p \wedge r) \Rightarrow (p \wedge q)$
(D) $(p \wedge q) \Rightarrow r$

Q184·MathematicsNumericalJEE Main 2022

Let

S = \{4,\,6,\,9\}

and

T = \{9,\,10,\,11,\,\ldots,\,1000\}

. If

A = \left\{a_{1}+a_{2}+\ldots+a_{k} : k \in N,\; a_{1},\,a_{2},\,a_{3},\,\ldots,a_{k} \in S\right\}

, then the sum of all the elements in the set

T - A

is equal to _______.

Q185·MathematicsSingle correctJEE Main 2022

The domain of the function

f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)

is :

(A) $[1, \infty)$
(B) $(-1, 2]$
(C) $[-1, \infty)$
(D) $(-\infty, 2]$

Q186·MathematicsSingle correctJEE Main 2022

Let p : Ramesh listens to music. q : Ramesh is out of his village r : It is Sunday s : It is Saturday Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as

(A) $\left((\sim q)\wedge(r \vee s)\right) \Rightarrow p$
(B) $\left(q \wedge (r \vee s)\right) \Rightarrow p$
(C) $p \Rightarrow \left(q \wedge (r \vee s)\right)$
(D) $p \Rightarrow \left((\sim q)\wedge(r \vee s)\right)$

Q187·MathematicsSingle correctJEE Main 2022

Let

\alpha

,

\beta

and

\gamma

be three positive real numbers. Let

f(x) = \alpha x^{5} + \beta x^{3} + \gamma x

,

x \in R

and

g : R \to R

be such that

g(f(x)) = x

for all

x \in R

. If

a_{1}

,

a_{2}

,

a_{3}

,...,

a_{n}

be in arithmetic progression with mean zero, then the value of

f\left( g\left( \frac{1}{n} \sum_{i=1}^{n} f\left(a_{i}\right) \right) \right)

is equal to :

(A)0
(B)3
(C)9
(D)27

Q188·MathematicsSingle correctJEE Main 2022

Let the operations

*, \odot \in \{\wedge, \vee\}

. If

(p * q) \odot (p \odot \sim q)

is a tautology, then the ordered pair

(*, \odot)

is :

(A) $(\vee, \wedge)$
(B) $(\vee, \vee)$
(C) $(\wedge, \wedge)$
(D) $(\wedge, \vee)$

Q189·MathematicsSingle correctJEE Main 2022

Let

S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\}

and

T=\left\{x \in Z: x^{2}-7|x|+9 \leq 0\right\}

. Then the number of elements in

S \cap T

is

(A)7
(B)5
(C)4
(D)3

Q190·MathematicsSingle correctJEE Main 2022

For

\alpha \in N

, consider a relation R on N given by

R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}

. The relation R is an equivalence relation if and only if :

(A) $\alpha = 14$
(B) $\alpha$ is a multiple of 4
(C)4 is the remainder when $\alpha$ is divided by 10
(D)4 is the remainder when $\alpha$ is divided by 7

Q191·MathematicsSingle correctJEE Main 2022

The domain of the function

f(x) = \sin^{-1}[2x^{2}-3] + \log_{2}\left(\log_{\frac{1}{2}}\left(x^{2}-5x+5\right)\right)

, where

[t]

is the greatest integer function, is :

(A) $\left(-\sqrt{\frac{5}{2}}, \frac{5-\sqrt{5}}{2}\right)$
(B) $\left(\frac{5-\sqrt{5}}{2}, \frac{5+\sqrt{5}}{2}\right)$
(C) $\left(1, \frac{5-\sqrt{5}}{2}\right)$
(D) $\left[1, \frac{5+\sqrt{5}}{2}\right)$

Q192·MathematicsSingle correctJEE Main 2022

\left(p\wedge r\right)\Leftrightarrow\left(p\wedge\left(\sim q\right)\right)

is equivalent to

\left(\sim p\right)

when r is

(A) $p$
(B) $\sim p$
(C) $q$
(D) $\sim q$

Q193·MathematicsSingle correctJEE Main 2022

Let

f, g : \mathbb{N}-\{1\} \to \mathbb{N}

be functions defined by

f(a) = \alpha

, where

\alpha

is the maximum of the powers of those primes p such that

p^{\alpha}

divides

a

, and

g(a) = a+1

, for all

a \in \mathbb{N}-\{1\}

. Then, the function

f + g

is

(A)one-one but not onto
(B)onto but not one-one
(C)both one-one and onto
(D)neither one-one nor onto

Q194·MathematicsSingle correctJEE Main 2022

Let

R_1

and

R_2

be two relations defined on

\mathbb{R}

by

a\,R_1\,b \Leftrightarrow ab \ge 0

and

a\,R_2\,b \Leftrightarrow a \ge b

, then

(A) $R_1$ is an equivalence relation but not $R_2$
(B) $R_2$ is an equivalence relation but not $R_1$
(C)both $R_1$ and $R_2$ are equivalence relations
(D)neither $R_1$ nor $R_2$ is an equivalence relation

Q195·MathematicsSingle correctJEE Main 2022

If the truth value of the statement

\left(P \wedge (\sim R)\right) \rightarrow \left((\sim R) \wedge Q\right)

is F, then the truth value of which of the following is F ?

(A) $P \vee Q \rightarrow \sim R$
(B) $R \vee Q \rightarrow \sim P$
(C) $\sim\left(P \vee Q\right) \rightarrow \sim R$
(D) $\sim\left(R \vee Q\right) \rightarrow \sim P$

Q196·MathematicsNumericalJEE Main 2022

Let

A=\{1,2,3,4,5,6,7\}

and

B=\{3,6,7,9\}

. Then the number of elements in the set

\{C\subseteq A:C\cap B\neq\phi\}

is________

Q197·MathematicsSingle correctJEE Main 2022

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

(A)4
(B)10
(C)11
(D)16

Q198·MathematicsSingle correctJEE Main 2022

The statement (~(p ⇔ ~q)) ∧ q is :

(A)a tautology
(B)a contradiction
(C)equivalent to (p ⇒ q) ∧ q
(D)equivalent to (p ⇒ q) ∧ p

Q199·MathematicsSingle correctJEE Main 2022

Negation of the Boolean expression

p\Leftrightarrow(q\Rightarrow p)

is

(A) $(\sim p)\wedge q$
(B) $p\wedge(\sim q)$
(C) $(\sim p)\vee(\sim q)$
(D) $(\sim p)\wedge(\sim q)$

Q200·MathematicsSingle correctJEE Main 2022

Which of the following statements is a tautology ?

(A) $((\sim p) \vee q) \Rightarrow p$
(B) $p \Rightarrow ((\sim p) \vee q)$
(C) $((\sim p) \vee q) \Rightarrow q$
(D) $q \Rightarrow ((\sim p) \vee q)$

Q201·MathematicsSingle correctJEE Main 2022

Let a set

A = A_{1} \cup A_{2} \cup \ldots \cup A_{k}

, where

A_{i} \cap A_{j} = \phi

for

i \ne j

1 \le i, j \le k

. Define the relation R from A to A by

R = \{(x, y) : y \in A_{i}

if and only if

x \in A_{i}, 1 \le i \le k\}

. Then, R is :

(A)reflexive, symmetric but not transitive
(B)reflexive, transitive but not symmetric
(C)reflexive but not symmetric and transitive
(D)an equivalence relation

Q202·MathematicsNumericalJEE Main 2022

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If

f(g(x)) = 8x^2 - 2x

, and

g(f(x)) = 4x^2 + 6x + 1

, then the value of

f(2) + g(2)

is______.

Q203·MathematicsSingle correctJEE Main 2022

Negation of the Boolean statement

(p \vee q) \Rightarrow \left((\sim r) \vee p\right)

is equivalent to:

(A) $p \wedge (\sim q) \wedge r$
(B) $(\sim p) \wedge (\sim q) \wedge r$
(C) $(\sim p) \wedge q \wedge r$
(D) $p \wedge q \wedge (\sim r)$

Q204·MathematicsSingle correctJEE Main 2022

Let

\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}

be such that

(p \wedge q)\Delta((p \vee q) \Rightarrow q)

is a tautology. Then Δ is equal to :

(A) $\wedge$
(B) $\vee$
(C) $\Rightarrow$
(D) $\Leftrightarrow$

Q205·MathematicsNumericalJEE Main 2022

Let c, k ∈ R. If

f(x) = (c + 1)x^2 + (1 - c^2)x + 2k

and

f(x + y) = f(x) + f(y) - xy

, for all x, y ∈ R, then the value of

|2(f(1) + f(2) + f(3) + \ldots + f(20))|

is equal to ________.

Q206·MathematicsSingle correctJEE Main 2022

Let

R_1 = \{(a,b) \in N \times N : |a-b| \leq 13\}

and

R_2 = \{(a,b) \in N \times N : |a-b| \neq 13\}

. Then on N:

(A)Both $R_1$ and $R_2$ are equivalence relations
(B)Neither $R_1$ nor $R_2$ is an equivalence relation
(C) $R_1$ is an equivalence relation but $R_2$ is not
(D) $R_2$ is an equivalence relation but $R_1$ is not

Q207·MathematicsNumericalJEE Main 2022

Let

R_1

and

R_2

be relations on the set {1, 2, …, 50} such that

R_1 = \{(p,\, p^n) : p \text{ is a prime and } n \geq 0 \text{ is an integer}\}

and

R_2 = \{(p,\, p^n) : p \text{ is a prime and } n = 0 \text{ or } 1\}

. Then, the number of elements in

R_1 - R_2

is ______.

Q208·MathematicsSingle correctJEE Main 2022

Let p, q, r be three logical statements. Consider the compound statements

S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)

and

S_2 : p \rightarrow (q \vee r)

Then, which of the following is NOT true ?

(A)If $S_2$ is True, then $S_1$ is True
(B)If $S_2$ is False, then $S_1$ is False
(C)If $S_2$ is False, then $S_1$ is True
(D)If $S_1$ is False, then $S_2$ is False

Q209·MathematicsNumericalJEE Main 2022

The maximum number of compound propositions, out of

p \vee r \vee s

,

p \vee r \vee \sim s

,

p \vee \sim q \vee s

,

\sim p \vee \sim r \vee s

,

\sim p \vee \sim r \vee \sim s

,

\sim p \vee q \vee \sim s

,

q \vee r \vee \sim s

,

q \vee \sim r \vee \sim s

,

\sim p \vee \sim q \vee \sim s

that can be made simultaneously true by an assignment of the truth values to

p

,

q

,

r

and

s

, is equal to

Q210·MathematicsNumericalJEE Main 2022

Let

S = \{1, 2, 3, 4\}

. Then the number of elements in the set

\{f : S \times S \to S : f \text{ is onto and } f(a, b) = f(b, a) \geq a\ \forall\, (a, b) \in S \times S\}

is

Q211·MathematicsSingle correctJEE Main 2022

Let a function

\mathrm{f}:\mathbb{N} \to \mathbb{N}

be defined by

\mathrm{f(n)}=\begin{cases} 2\mathrm{n}, & \mathrm{n}=2,4,6,8,..... \\ \mathrm{n}-1, & \mathrm{n}=3,7,11,15,..... \\ \dfrac{\mathrm{n}+1}{2}, & \mathrm{n}=1,5,9,13,..... \end{cases}

then, f is

(A)one-one but not onto
(B)onto but not one-one
(C)neither one-one nor onto
(D)one-one and onto

Q212·MathematicsNumericalJEE Main 2022

Let

f : R \rightarrow R

be a function defined

f(x)=\frac{2e^{2x}}{e^{2x}+e}

. Then

f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+.....+f\left(\frac{99}{100}\right)

is equal to________.

Q213·MathematicsSingle correctJEE Main 2022

The Boolean expression

\left(\sim\left(p \wedge q\right)\right)\vee q

is equivalent to :

(A) $q \rightarrow \left(p \wedge q\right)$
(B) $p \rightarrow q$
(C) $p \rightarrow \left(p \rightarrow q\right)$
(D) $p \rightarrow \left(p \vee q\right)$

Q214·MathematicsNumericalJEE Main 2022

Let

S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

. Define

f:S\to S

as

f(n)=\begin{cases}2n, & \text{if } n=1,2,3,4,5\\ 2n-11 & \text{if } n=6,7,8,9,10\end{cases}

. Let

g:S\to S

be a function such that

\text{fog}(n)=\begin{cases}n+1 & \text{, if } n \text{ is odd}\\ n-1 & \text{, if } n \text{ is even}\end{cases}

, then

g(10)\ ((g(1)+g(2)+g(3)+g(4)+g(5)))

is equal to:

Q215·MathematicsSingle correctJEE Main 2022

Which of the following statement is a tautology?

(A) $((\sim q)\wedge p)\wedge q$
(B) $((\sim q)\wedge p)\wedge(p\wedge(\sim p))$
(C) $((\sim q)\wedge p)\vee(p\vee(\sim p))$
(D) $(p\wedge q)\wedge(\sim(p\wedge q))$

Q216·MathematicsSingle correctJEE Main 2022

Let

f(x) = \frac{x-1}{x+1}

,

x \in R - \{0, -1, 1\}

. If

f^{n+1}(x) = f(f^{n}(x))

for all

n \in N

, then

f^{6}(6) + f^{7}(7)

is equal to :

(A) $\frac{7}{6}$
(B) $-\frac{3}{2}$
(C) $\frac{7}{12}$
(D) $-\frac{11}{12}$

Q217·MathematicsSingle correctJEE Main 2022

Let f : ℝ → ℝ be defined as f(x) = x-1 and g : ℝ − {1, −1} → ℝ be defined as

g(x)=\frac{x^2}{x^2-1}

. Then the function fog is :

(A)one-one but not onto function
(B)onto but not one-one function
(C)both one-one and onto function
(D)neither one-one nor onto function

Q218·MathematicsSingle correctJEE Main 2022

Let r ∈ {p, q, ~p, ~q} be such that the logical statement

r\vee(\sim p)\Rightarrow(p\wedge q)\vee r

is a tautology. Then 'r' is equal to :

(A)p
(B)q
(C)~p
(D)~q

Q219·MathematicsSingle correctJEE Main 2022

Let

\Delta, \nabla \in \{\wedge, \vee\}

be such that

p \nabla q \Rightarrow ((p \Delta q) \nabla r)

is a tautology. Then

(p \nabla q) \Delta r

is logically equivalent to :

(A) $(p \Delta r) \vee q$
(B) $(p \Delta r) \wedge q$
(C) $(p \wedge r) \Delta q$
(D) $(p \nabla r) \wedge q$

Q220·MathematicsSingle correctJEE Main 2022

The negation of the Boolean expression

((\sim q) \wedge p) \Rightarrow ((\sim p) \vee q)

is logically equivalent to

(A) $p \Rightarrow q$
(B) $q \Rightarrow p$
(C) $\sim\left(p \Rightarrow q\right)$
(D) $\sim\left(q \Rightarrow p\right)$

Q221·MathematicsSingle correctJEE Main 2022

Consider the following two propositions: P1 : ~(p→ ~q) P2 : (p∧ ~ q)∧((~p)∨ q) If the proposition p → ((~p)∨ q) is evaluated as FALSE, then:

(A)P1 is TRUE and P2 is FALSE
(B)P1 is FALSE and P2 is TRUE
(C)Both P1 and P2 are FALSE
(D)Both P1 and P2 are TRUE

Q222·MathematicsNumericalJEE Main 2022

Let f : R → R be a function defined by

f(x) = \left(2\left(1 - \frac{x^{25}}{2}\right)\left(2 + x^{25}\right)\right)^{\frac{1}{50}}

. If the function

g(x) = f\left(f\left(f(x)\right)\right) + f\left(f(x)\right)

, the the greatest integer less than or equal to g (1) is ______

Q223·MathematicsSingle correctJEE Main 2022

Let

A = \{x \in R : |x+1| < 2\}

and

B = \{x \in R : |x-1| \ge 2\}

. Then which one of the following statements is NOT true ?

(A) $A - B = (-1,1)$
(B) $B - A = R - (-3,1)$
(C) $A \cap B = (-3,-1]$
(D) $A \cup B = R - [1,3)$

Q224·MathematicsSingle correctJEE Main 2022

The domain of the function

f(x)=\dfrac{\cos^{-1}\left(\dfrac{x^{2}-5x+6}{x^{2}-9}\right)}{\log_{e}(x^{2}-3x+2)}

is

(A) $(-\infty,1)\cup(2,\infty)$
(B) $(2,\infty)$
(C) $\left[-\dfrac{1}{2},1\right)\cup(2,\infty)$
(D) $\left[-\dfrac{1}{2},1\right)\cup(2,\infty)-\left\{\dfrac{3+\sqrt{5}}{2},\dfrac{3-\sqrt{5}}{2}\right\}$

Q225·MathematicsSingle correctJEE Main 2022

The number of choices of

\Delta\in\{\wedge,\vee,\Rightarrow,\Leftrightarrow\}

, such that

(p\Delta q)\Rightarrow((p\Delta\sim q)\vee((\sim p)\Delta q))

is a tautology, is

(A)1
(B)2
(C)3
(D)4

Q226·MathematicsSingle correctJEE Main 2022

Consider the following statements : A : Rishi is a judge. B : Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

(A) $B \to (A \vee C)$
(B) $(\sim B) \wedge (A \wedge C)$
(C) $B \to ((\sim A) \vee (\sim C))$
(D) $B \to (A \wedge C)$

Q227·MathematicsSingle correctJEE Main 2021

Which of the following is equivalent to the Boolean expression p ∧ ~q ?

(A)~ (q → p)
(B)~ p → ~q
(C)~ (p → ~q)
(D)~ (p → q)

Q228·MathematicsSingle correctJEE Main 2021

Negation of the statement

(p \vee r) \Rightarrow (q \vee r)

is :

(A) $p \wedge \sim q \wedge \sim r$
(B) $\sim p \wedge q \wedge \sim r$
(C) $\sim p \wedge q \wedge r$
(D) $p \wedge q \wedge r$

Q229·MathematicsSingle correctJEE Main 2021

Which of the following is not correct for relation R on the set of real numbers ?

(A) $(x, y) \in R \Leftrightarrow 0 < |x| - |y| \le 1$ is neither transitive nor symmetric.
(B) $(x, y) \in R \Leftrightarrow 0 < |x - y| \le 1$ is symmetric and transitive.
(C) $(x, y) \in R \Leftrightarrow |x| - |y| \le 1$ is reflexive but not symmetric.
(D) $(x, y) \in R \Leftrightarrow |x - y| \le 1$ is reflexive and symmetric.

Q230·MathematicsSingle correctJEE Main 2021

Let

*, \square \in \{\wedge, \vee\}

be such that the Boolean expression

(p * \sim q) \Rightarrow (p \square q)

is a tautology. Then :

(A) $* = \vee,\ \square = \vee$
(B) $* = \wedge,\ \square = \wedge$
(C) $* = \wedge,\ \square = \vee$
(D) $* = \vee,\ \square = \wedge$

Q231·MathematicsSingle correctJEE Main 2021

Let

f : \mathbf{N} \to \mathbf{N}

be a function such that

f(m + n) = f(m) + f(n)

for every

m, n \in \mathbf{N}

. If

f(6) = 18

, then

f(2) \cdot f(3)

is equal to :

(A)6
(B)54
(C)18
(D)36

Q232·MathematicsSingle correctJEE Main 2021

The statement

(p \wedge (p \to q) \wedge (q \to r)) \to r

is :

(A)a tautology
(B)equivalent to $p \to \sim r$
(C)a fallacy
(D)equivalent to $q \to \sim r$

Q233·MathematicsSingle correctJEE Main 2021

The Boolean expression

(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)

is equivalent to :

(A) $(p \wedge q) \Rightarrow (r \wedge q)$
(B) $(q \wedge r) \Rightarrow (p \wedge q)$
(C) $(p \wedge q) \Rightarrow (r \vee q)$
(D) $(p \wedge r) \Rightarrow (p \wedge q)$

Q234·MathematicsNumericalJEE Main 2021

If

A = \{x \in \mathbf{R} : |x - 2| > 1\}

,

B = \left\{x \in \mathbf{R} : \sqrt{x^{2} - 3} > 1\right\}

,

C = \left\{x \in \mathbf{R} : |x - 4| \geq 2\right\}

and

\mathbf{Z}

is the set of all integers, then the number of subsets of the set

(A \cap B \cap C)^{c} \cap \mathbf{Z}

is ___________.

Q235·MathematicsSingle correctJEE Main 2021

Let ℤ be the set of all integers, A = {(x, y) ∈ ℤ × ℤ :

(x - 2)^{2} + y^{2} \le 4

}, B = {(x, y) ∈ ℤ × ℤ :

x^{2} + y^{2} \le 4

} and C = {(x, y) ∈ ℤ × ℤ :

(x - 2)^{2} + (y - 2)^{2} \le 4

} If the total number of relation from A ∩ B to A ∩ C is

2^{p}

, then the value of p is :

(A)16
(B)25
(C)49
(D)9

Q236·MathematicsSingle correctJEE Main 2021

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

(A) $\{80, 83, 86, 89\}$
(B) $\{84, 86, 88, 90\}$
(C) $\{79, 81, 83, 85\}$
(D) $\{84, 87, 90, 93\}$

Q237·MathematicsSingle correctJEE Main 2021

Consider the two statements :

(S1) : (p \rightarrow q) \vee (\sim q \rightarrow p)

is a tautology.

(S2) : (p \wedge \sim q) \wedge (\sim p \vee q)

is a fallacy. Then :

(A)only (S1) is true.
(B)both (S1) and (S2) are false.
(C)both (S1) and (S2) are true.
(D)only (S2) is true.

Q238·MathematicsSingle correctJEE Main 2021

If the truth value of the Boolean expression

\left((p \vee q) \wedge (q \to r) \wedge (\sim r)\right) \to (p \wedge q)

is false, then the truth values of the statements p, q, r respectively can be:

(A)T F T
(B)F F T
(C)T F F
(D)F T F

Q239·MathematicsSingle correctJEE Main 2021

Let N be the set of natural numbers and a relation R on N be defined by R =

\left\{ (x, y) \in \mathbb{N} \times \mathbb{N} : x^3 - 3x^2 y - xy^2 + 3y^3 = 0 \right\}

. Then the relation R is :

(A)reflexive but neither symmetric nor transitive
(B)an equivalence relation
(C)symmetric but neither reflexive nor transitive
(D)reflexive and symmetric, but not transitive

Q240·MathematicsSingle correctJEE Main 2021

The compound statement

\left(P \vee Q\right) \wedge \left(\sim P\right) \Rightarrow Q

is equivalent to :

(A) $P \vee Q$
(B) $P \wedge \sim Q$
(C) $\sim\left(P \Rightarrow Q\right) \Leftrightarrow P \wedge \sim Q$
(D) $\sim\left(P \Rightarrow Q\right)$

Q241·MathematicsSingle correctJEE Main 2021

Which of the following is the negation of the statement "for all M > 0, there exists x ∈ S such that x ≥ M"?

(A)there exists M > 0, there exists x ∈ S such that x < M
(B)there exists M > 0, such that x ≥ M for all x ∈ S
(C)there exists M > 0, there exists x ∈ S such that x ≥ M
(D)there exists M > 0, such that x < M for all x ∈ S

Q242·MathematicsNumericalJEE Main 2021

Let A = {n ∈ N|n

^2

≤ n + 10,000}, B = {3k + 1|k ∈ N} and C = {2k|k ∈ N} , then the sum of all the elements of the set A ∩ (B − C) is equal to……

Q243·MathematicsNumericalJEE Main 2021

Let

S=\left\{1,2,3,4,5,6,7\right\}

. Then the number of possible function

f:S \to S

such that

f\left(m \cdot n\right)=f\left(m\right) \cdot f\left(n\right)

for every

m,n \in S

and

m \cdot n \in S

is equal to ____.

Q244·MathematicsSingle correctJEE Main 2021

Consider functions

f : A \to B

and

g : B \to C\left(A,B,C \subseteq R\right)

such that

\left(gof\right)^{-1}

exists, then :

(A)f and g both are onto
(B)f is onto and g is one-one
(C)f is one-one and g is onto
(D)f and g both are one-one

Q245·MathematicsSingle correctJEE Main 2021

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

(A)The match will not be played or weather is good and ground is not wet.
(B)The match will not be played or weather is not good and ground is wet.
(C)If the match will not be played, then either weather is not good or ground is wet.
(D)The match will be played and weather is not good or ground is wet.

Q246·MathematicsSingle correctJEE Main 2021

Let

g:N \rightarrow N

be defined as

g(3n+1)=3n+2,

g(3n+2)=3n+3,

g(3n+3)=3n+1

, for all n ≥ 0 Then which of the following statements is true?

(A)There exists a function $f:N \rightarrow N$ such that $gof=f$
(B)gogog = g
(C)There exists a one-one function $f:N \rightarrow N$ such that $fog=f$
(D)There exists on onto function $f:N \rightarrow N$ such that $fog=f$

Q247·MathematicsSingle correctJEE Main 2021

The Boolean expression

(p \Rightarrow q) \wedge (q \Rightarrow \sim p)

is equivalent to:

(A)q
(B)p
(C)~ p
(D)~ q

Q248·MathematicsSingle correctJEE Main 2021

Let

[x]

denote the greatest integer less than or equal to x. Then, the values of

x \in R

satisfying the equation

\left[e^x\right]^2 + \left[e^x + 1\right] - 3 = 0

lie in the interval :

(A) $[1, e)$
(B) $[\log_e 2, \log_e 3)$
(C) $[0, \log_e 2)$
(D) $[0, 1/\,e)$

Q249·MathematicsNumericalJEE Main 2021

The sum of all the elements in the set

\left\{n \in \{1,2,.......,100\} \middle| \text{H.C.F. of n and 2040 is 1}\right\}

is equal to...........

Q250·MathematicsSingle correctJEE Main 2021

Which of the following Boolean expressions is not a tautology?

(A) $(\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)$
(B) $(q \Rightarrow p) \vee (\sim q \Rightarrow p)$
(C) $(p \Rightarrow \sim q) \vee (\sim q \Rightarrow p)$
(D) $(p \Rightarrow q) \vee (\sim q \Rightarrow p)$

Q251·MathematicsSingle correctJEE Main 2021

Let

f:R-\left\{\frac{\alpha}{6}\right\}\to R

be defined by

f(x)=\frac{5x+3}{6x-\alpha}

. Then the value of

\alpha

for which

(fof)(x)=x,

for all

x\in R-\left\{\frac{\alpha}{6}\right\}

, is :

(A) $6$
(B) $8$
(C)No such $\alpha$ exists
(D) $5$

Q252·MathematicsSingle correctJEE Main 2021

Consider the following three statements : (A) if 3 + 3 = 7 then 4 + 3 = 8. (B) If 5 + 3 = 8 then earth is flat. (C) If both (A) and (B) are true then 5 + 6 = 17. Then, which of the following statements is correct?

(A)(A) and (C) are true while (B) is false
(B)(A) and (B) are false while (C) is true
(C)(A) is true while (B) and (C) are false
(D)(A) is false, but (B) and (C) are true

Q253·MathematicsSingle correctJEE Main 2021

Let [x] denote the greatest integer

\leq x

, where

x \in R

. If the domain of the real valued function

f(x) = \sqrt{\frac{\big|[x]\big|-2}{\big|[x]\big|-3}}

is

(-\infty, a) \cup [b, c) \cup [4, \infty)

,

a < b < c

, then the value of

a+b+c

is:

(A)-3
(B)8
(C)-2
(D)1

Q254·MathematicsSingle correctJEE Main 2021

The Boolean expression

(p \wedge \sim q) \Rightarrow (q \vee \sim p)

is equivalent to :

(A) $q \Rightarrow p$
(B) $\sim q \Rightarrow p$
(C) $p \Rightarrow q$
(D) $p \Rightarrow \sim q$

Q255·MathematicsSingle correctJEE Main 2021

The real valued function

f\left(x\right) = \dfrac{\mathrm{cosec}^{-1}x}{\sqrt{x - \left[x\right]}}

, where

[x]

denotes the greatest integer less than or equal to x, is defined for all x belonging to :

(A)all reals except integers
(B)all non-integers except the interval $[-1,1]$
(C)all integers except $0, -1, 1$
(D)all reals except the Interval $[-1,1]$

Q256·MathematicsSingle correctJEE Main 2021

If P and Q are two statements, then which of the following compound statement is a tautology ?

(A)((P ⇒ Q) ∧ ~ Q) ⇒ Q
(B)((P ⇒ Q) ∧ ~ Q) ⇒ ~ P
(C)((P ⇒ Q) ∧ ~ Q) ⇒ P
(D)((P ⇒ Q) ∧ ~ Q) ⇒ (P ∧ Q)

Q257·MathematicsSingle correctJEE Main 2021

Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that

PAP^{-1} = B

". Then which of the following is true ?

(A)R is symmetric, transitive but not reflexive,
(B)R is reflexive, symmetric but not transitive
(C)R is an equivalence relation
(D)R is reflexive, transitive but not symmetric

Q258·MathematicsSingle correctJEE Main 2021

Let f : R − {3} → R − {1} be defined by

f(x) = \frac{x-2}{x-3}

. Let g : R → R be given as g(x) = 2x − 3. Then, the sum of all the values of x for which

f^{-1}(x) + g^{-1}(x) = \frac{13}{2}

is equal to

(A)7
(B)2
(C)5
(D)3

Q259·MathematicsSingle correctJEE Main 2021

If the functions are defined as

f\left(x\right) = \sqrt{x}

and

g\left(x\right) = \sqrt{1 - x}

, then what is the common domain of the following functions :

f + g

,

f - g

,

f/g

,

g/f

,

g - f

where

(f \pm g)(x) = f(x) \pm g(x)

,

(f/g)(x) = \dfrac{f\left(x\right)}{g\left(x\right)}

(A) $0 \leq x \leq 1$
(B) $0 \leq x < 1$
(C) $0 < x < 1$
(D) $0 < x \leq 1$

Q260·MathematicsSingle correctJEE Main 2021

If the Boolean expression

(p \wedge q) \circledast (p \otimes q)

is a tautology, then

\circledast

and

\otimes

are respectively given by

(A) $\rightarrow, \rightarrow$
(B) $\wedge, \vee$
(C) $\vee, \rightarrow$
(D) $\wedge, \rightarrow$

Q261·MathematicsSingle correctJEE Main 2021

If the Boolean expression

(p \Rightarrow q) \Leftrightarrow (q * (\sim p))

is a tautology, then the Boolean expression

p * (\sim q)

is equivalent to :

(A) $q \Rightarrow p$
(B) $\sim q \Rightarrow p$
(C) $p \Rightarrow \sim q$
(D) $p \Rightarrow q$

Q262·MathematicsSingle correctJEE Main 2021

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

(A)P and Q
(B)P and R
(C)None of these
(D)Q and R

Q263·MathematicsSingle correctJEE Main 2021

Which of the following Boolean expression is a tautology ?

(A)(p ∧ q) ∨ (p ∨ q)
(B)(p ∧ q) ∨ (p → q)
(C)(p ∧ q) ∧ (p → q)
(D)(p ∧ q) → (p → q)

Q264·MathematicsSingle correctJEE Main 2021

Let

A = \{2, 3, 4, 5, ...., 30\}

and '

\simeq

' be an equivalence relation on

A \times A

, defined by

(a, b) \simeq (c, d)

, if and only if

ad = bc

. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

(A) $5$
(B) $6$
(C) $8$
(D) $7$

Q265·MathematicsSingle correctJEE Main 2021

Let

F_{1}

(A, B, C) = (A ∧ ~B) ∨ [~C ∧ (A ∨ B)] ∨ ~A and

F_{2}

(A, B) = (A ∨ B) ∨ (B → ~A) be two logical expressions. Then :

(A) $F_{1}$ is not a tautology but $F_{2}$ is a tautology
(B) $F_{1}$ is a tautology but $F_{2}$ is not a tautology
(C) $F_{1}$ and $F_{2}$ both area tautologies
(D)Both $F_{1}$ and $F_{2}$ are not tautologies

Q266·MathematicsSingle correctJEE Main 2021

Let A = {1,2,3……,10} and f: A→ A be defined as

f(k) = \begin{cases} k + 1 & \text{if k is odd} \\ k & \text{if k is even} \end{cases}

Then the number of possible functions g : A→A such that gof = f is :

(A) $10^{5}$
(B) $^{10}C_{5}$
(C) $5^{5}$
(D)5!

Q267·MathematicsSingle correctJEE Main 2021

Let

f(x) = \sin^{-1} x

and

g(x) = \frac{x^{2}-x-2}{2x^{2}-x-6}

. If

g(2) = \lim_{x \to 2} g(x)

, then the domain of the function fog is :

(A) $(-\infty, -2] \cup \left[-\frac{4}{3}, \infty\right)$
(B) $(-\infty, -1] \cup [2, \infty)$
(C) $(-\infty, -2] \cup [-1, \infty)$
(D) $(-\infty, -2] \cup \left[-\frac{3}{2}, \infty\right)$

Q268·MathematicsSingle correctJEE Main 2021

Let R = {P,Q)|P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1,−1) is the set:

(A) $S=\{(x,y)\,|\,x^{2}+y^{2}=1\}$
(B) $S=\{(x,y)\,|\,x^{2}+y^{2}=4\}$
(C) $S=\{(x,y)\,|\,x^{2}+y^{2}=\sqrt{2}\}$
(D) $S=\{(x,y)\,|\,x^{2}+y^{2}=2\}$

Q269·MathematicsSingle correctJEE Main 2021

Let f, g:

N \rightarrow N

such that

f(n + 1) = f(n) + f(1)

\forall n \in N

and g be any arbitrary function. Which of the following statements is NOT true ?

(A)f is one-one
(B)If fog is one-one, then g is one-one
(C)If g is onto, then fog is one-one
(D)If f is onto, then $f(n) = n\ \forall n \in N$

Q270·MathematicsSingle correctJEE Main 2021

The contrapositive of the statement “If you will work, you will earn money” is:

(A)If you will not earn money, you will not work
(B)You will earn money, if you will not work
(C)If you will earn money, you will work
(D)To earn money, you need to work

Q271·MathematicsSingle correctJEE Main 2021

A function f(x) is given by f(x) = 5

_{x}

5+

^{x}

5, then the sum of the series f (201 ) + f (202 ) + f (203 ) + ...... + f (3920 ) is equal to:

(A)192
(B)492
(C)392
(D)292

Q272·MathematicsSingle correctJEE Main 2021

The statement

A \rightarrow (B \rightarrow A)

is equivalent to:

(A) $A \rightarrow (A \wedge B)$
(B) $A \rightarrow (A \vee B)$
(C) $A \rightarrow (A \rightarrow B)$
(D) $A \rightarrow (A \leftrightarrow B)$

Q273·MathematicsSingle correctJEE Main 2021

Let

f : R \rightarrow R

be defined as

f(x) = 2x-1

and

g : R - \{1\} \rightarrow R

be defined as

g(x) = \frac{x - \frac{1}{2}}{x - 1}

. Then the composition function

f(g(x))

is :

(A)both one-one and onto
(B)onto but not one-one
(C)neither one-one nor onto
(D)one-one but not onto

Q274·MathematicsSingle correctJEE Main 2021

The statement among the following that is a tautology is:

(A) $A \wedge (A \vee B)$
(B) $B \rightarrow \left[A \wedge (A \rightarrow B)\right]$
(C) $A \vee (A \wedge B)$
(D) $\left[A \wedge (A \rightarrow B)\right] \rightarrow B$

Q275·MathematicsSingle correctJEE Main 2021

The negation of the statement

\sim p\wedge(p\vee q)

is :

(A) $\sim p\wedge q$
(B) $p\wedge\sim q$
(C) $\sim p\vee q$
(D) $p\vee\sim q$

Q276·MathematicsNumericalJEE Main 2021

If

a+\alpha=1, b+\beta=2

and

af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}, x\neq 0,

then the value of the expression

\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}

is__________.

Q277·MathematicsNumericalJEE Advanced 2020

Let the function

f : [0,1] \to \mathbb{R}

be defined by

f(x) = \frac{4^{x}}{4^{x} + 2}

Then the value of

f\left(\frac{1}{40}\right) + f\left(\frac{2}{40}\right) + f\left(\frac{3}{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)

is ________

Q278·MathematicsSingle correctJEE Advanced 2020

If the function

f : \mathbb{R} \to \mathbb{R}

is defined by

f(x) = |x|(x - \sin x)

, then which of the following statements is TRUE ?

(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto

Q279·MathematicsSingle correctJEE Main 2020

For a suitably chosen real constant a, let a function,

f : R - \{-a\} \to R

be defined by

f(x) = \frac{a - x}{a + x}

. Further suppose that for any real number

x \neq -a

and

f(x) \neq -a

,

(fof)(x) = x

. Then

f\left(-\frac{1}{2}\right)

is equal to:

(A) $\frac{1}{3}$
(B) $-\frac{1}{3}$
(C) $-3$
(D) $3$

Q280·MathematicsNumericalJEE Main 2020

Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m.n is ____.

Q281·MathematicsNumericalJEE Main 2020

Suppose that a function

f : R \to R

satisfies

f(x + y) = f(x)f(y)

for all

x, y \in R

and

f(1) = 3

. If

\sum_{i=1}^{n} f(i) = 363

, then n is equal to __________.

Q282·MathematicsSingle correctJEE Main 2020

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be:

(A)63
(B)54
(C)38
(D)36

Q283·MathematicsNumericalJEE Main 2020

Let

A = (a, b, c)

and

B = (1, 2, 3, 4)

. Then the number of elements in the set

C = \{f : A \to B \mid 2 \in f(A) \text{ and } f \text{ is not one-one}\}

is __________.

Q284·MathematicsSingle correctJEE Main 2020

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. if x% of the people read both the newspapers, then a possible value of x can be:

(A)29
(B)55
(C)37
(D)65

Q285·MathematicsSingle correctJEE Main 2020

Let [t] denote the greatest integer

\le t

. Then the equation in x,

[x]^{2} + 2[x+2] - 7 = 0

has:

(A)infinitely many solutions
(B)exactly two solutions
(C)no integral solution
(D)exactly four integral solutions

Q286·MathematicsSingle correctJEE Main 2020

Let

\bigcup_{i=1}^{50} X_{i} = \bigcup_{i=1}^{n} Y_{i} = T

, where each

X_{i}

contains 10 elements and each Yi contains 5 elements. If each element of the set T is an element of exactly 20 of sets

X_{i}

's and exactly 6 of sets

Y_{i}

's then n is equal to:

(A)15
(B)30
(C)45
(D)50

Q287·MathematicsSingle correctJEE Main 2020

Consider the two sets: A = {m ∈ R : both the roots of

x^{2}-(m+1)x+m+4=0

are real} and B = [−3, 5). Which of the following is not true?

(A) $A\cap B=\{-3\}$
(B) $B-A=(-3,\ 5)$
(C) $A-B=(-\infty,\ -3)\cup(5,\ \infty)$
(D) $A\cup B=R$

Q288·MathematicsSingle correctJEE Main 2020

Let

R_1

and

R_2

be two relations defined as follows:

R_1 = \{(a,b) \in R^2 : a^2 + b^2 \in Q\}

and

R_2 = \{(a,b) \in R^2 : a^2 + b^2 \notin Q\}

, where

Q

is the set of all rational numbers. Then:

(A)Neither $R_1$ nor $R_2$ is transitive.
(B) $R_1$ is transitive but $R_2$ is not transitive.
(C) $R_1$ and $R_2$ are both transitive
(D) $R_2$ is transitive but $R_1$ is not transitive.

Q289·MathematicsSingle correctJEE Main 2020

If

A = \{x \in R : |x| < 2\}

and

B = \{x \in R : |x - 2| \ge 3\}

; then:

(A) $B - A = R - (-2, 5)$
(B) $A \cap B = (-2, -1)$
(C) $A - B = [-1, 2)$
(D) $A \cup B = R - (2, 5)$

Q290·MathematicsSingle correctJEE Main 2020

Let

f:(1,3)\to R

be a function defined by

f(x)=\dfrac{x[x]}{1+x^{2}}

, where [x] denotes the greatest integer

\le x

. Then the range of f is:

(A) $\left(\dfrac{2}{5},\dfrac{4}{5}\right]$
(B) $\left(\dfrac{2}{5},\dfrac{1}{2}\right)\cup\left(\dfrac{3}{5},\dfrac{4}{5}\right]$
(C) $\left(\dfrac{3}{5},\dfrac{4}{5}\right)$
(D) $\left(\dfrac{2}{5},\dfrac{3}{5}\right]\cup\left(\dfrac{3}{4},\dfrac{4}{5}\right)$

Q291·MathematicsSingle correctJEE Main 2020

The inverse function of

f(x)=\dfrac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}}, x\in(-1,1)

, is

(A) $\dfrac{1}{4}(\log_{8}e)\log_{e}\left(\dfrac{1-x}{1+x}\right)$
(B) $\dfrac{1}{4}\log_{e}\left(\dfrac{1-x}{1+x}\right)$
(C) $\dfrac{1}{4}\log_{e}\left(\dfrac{1+x}{1-x}\right)$
(D) $\dfrac{1}{4}(\log_{8}e)\log_{e}\left(\dfrac{1+x}{1-x}\right)$

Q292·MathematicsNumericalJEE Main 2020

Let

X=\{n \in N : 1 \le n \le 50\}

. If

A=\{n \in X : n \text{ is a multiple of } 2\}

and

B=\{n \in X, n \text{ is a multiple of } 7\}

, then the number of elements in the smallest subset of X containing both A and B is __________.

Q293·MathematicsSingle correctJEE Main 2020

If

g(x)=x^{2}+x-1

and

(gof)(x)=4x^{2}-10x+5

, then

f\left(\frac{5}{4}\right)

is equal to

(A) $\frac{3}{2}$
(B) $\frac{1}{2}$
(C) $-\frac{3}{2}$
(D) $-\frac{1}{2}$

Q294·MathematicsSingle correctJEE Main 2019

For

x \in \left(0, \dfrac{3}{2}\right)

, let

f(x) = \sqrt{x}

,

g(x) = \tan x

and

h(x) = \dfrac{1-x^{2}}{1+x^{2}}

. If

\phi(x) = ((h \circ f) \circ g)(x)

, then

\phi\left(\dfrac{\pi}{3}\right)

is equal to :

(A) $\tan \dfrac{11\pi}{12}$
(B) $\tan \dfrac{\pi}{12}$
(C) $\tan \dfrac{5\pi}{12}$
(D) $\tan \dfrac{7\pi}{12}$

Q295·MathematicsSingle correctJEE Main 2019

Let A, B and C be sets such that

\phi \neq A \cap B \subseteq C

. Then which of the following statements is not true?

(A)If $(A-C) \subseteq B$ , then $A \subseteq B$
(B)If $(A-B) \subseteq C$ , then $A \subseteq C$
(C) $(C \cup A) \cap (C \cup B) = C$
(D) $B \cap C \neq \phi$

Q296·MathematicsSingle correctJEE Main 2019

For

x \in R

, let

[x]

denote the greatest integer

\le x

, then the sum of the series

\left[-\dfrac{1}{3}\right] + \left[-\dfrac{1}{3} - \dfrac{1}{100}\right] + \left[-\dfrac{1}{3} - \dfrac{2}{100}\right] + \cdots + \left[-\dfrac{1}{3} - \dfrac{99}{100}\right]

(A) $-135$
(B) $-153$
(C) $-133$
(D) $-131$

Q297·MathematicsSingle correctJEE Main 2019

Let f(x) = x

^{2}

, x ∈ R. For any A ⊆ R, define g(A) = {x ∈ R : f(x) ∈ A}. If S = [0, 4], then which one of the following statements is not true?

(A)f(g(S)) ≠ f(S)
(B)f(g(S)) = S
(C)g(f(S)) ≠ S
(D)g(f(S)) = g(S)

Q298·MathematicsSingle correctJEE Main 2019

The domain of the definition of the function

f\left(x\right)=\dfrac{1}{4-x^{2}}+\log\left(x^{3}-x\right)

is

(A) $\left(1,2\right)\cup\left(2,\infty\right)$
(B) $\left(-1,0\right)\cup\left(1,2\right)\cup\left(3,\infty\right)$
(C) $\left(-1,0\right)\cup\left(1,2\right)\cup\left(2,\infty\right)$
(D) $\left(-2,-1\right)\cup\left(-1,0\right)\cup\left(2,\infty\right)$

Q299·MathematicsSingle correctJEE Main 2019

If the function

f: R-\{1, -1\} \to A

defined by

f(x) = \dfrac{x^{2}}{1-x^{2}}

, is surjective, then A is equal to:

(A) $R - [-1, 0)$
(B) $R - (-1, 0)$
(C) $R - \{-1\}$
(D) $[0, \infty]$

Q300·MathematicsSingle correctJEE Main 2019

If

f(x)=\log_{e}\left(\frac{1-x}{1+x}\right)

,

|x|<1

, then

f\left(\frac{2x}{1+x^{2}}\right)

is equal to:

(A) $2f(x)$
(B) $\left(f(x)\right)^{2}$
(C) $2f(x^{2})$
(D) $-2f(x)$

Q301·MathematicsSingle correctJEE Main 2019

Let

f(x)=a^{x}

(a>0)

be written as

f(x)=f_{1}(x)+f_{2}(x)

, where

f_{1}(x)

is an even function and

f_{2}(x)

is an odd function. Then

f_{1}(x+y)+f_{1}(x-y)

equals

(A) $2f_{1}(x)f_{2}(y)$
(B) $2f_{1}(x)f_{1}(y)$
(C) $2f_{1}(x+y)f_{2}(x-y)$
(D) $2f_{1}(x+y)f_{1}(x-y)$

Q302·MathematicsSingle correctJEE Main 2019

Let Z be the set of integers. If

A = \{x \in Z : 2^{(x+2)(x^{2}-5x+6)} = 1\}

and

B = \{x \in Z : -3 < 2x - 1 < 9\}

, then the number of subsets of the set A

\times

B, is :

(A) $2^{15}$
(B) $2^{18}$
(C) $2^{12}$
(D) $2^{10}$

Q303·MathematicsSingle correctJEE Main 2019

Let

f:R\rightarrow R

be defined by

f(x)=\dfrac{x}{1+x^{2}}

,

x\in R

. Then the range of f is:

(A) $\left[-\dfrac{1}{2},\dfrac{1}{2}\right]$
(B) $R-[-1,1]$
(C) $R-\left[-\dfrac{1}{2},\dfrac{1}{2}\right]$
(D) $(-1,1)-\{0\}$

Q304·MathematicsSingle correctJEE Main 2019

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

(A)102
(B)42
(C)1
(D)38

Q305·MathematicsSingle correctJEE Main 2019

If the Boolean expression

(p\oplus q)\wedge(\sim p\ominus q)

is equivalent to

p\wedge q

, where

\oplus,\ominus\in\{\wedge,\vee\}

, then the ordered pair

(\oplus,\ominus)

is :

(A) $(\vee,\wedge)$
(B) $(\vee,\vee)$
(C) $(\wedge,\vee)$
(D) $(\wedge,\wedge)$

Q306·MathematicsSingle correctJEE Main 2019

For

x \in R - [0,1]

, let

f_1(x) = \dfrac{1}{x}

,

f_2(x) = 1-x

and

f_3(x) = \dfrac{1}{1-x}

be three given functions. If a function,

J(x)

satisfies

(f_2 \circ J \circ f_1)(x) = f_3(x)

then

J(x)

is equal to:

(A) $f_3(x)$
(B) $\dfrac{1}{x}f_3(x)$
(C) $f_2(x)$
(D) $f_1(x)$

Q307·MathematicsSingle correctJEE Main 2019

Let

A = \{x \in R : x \text{ is not a positive integer}\}

. Define a function

f : A \rightarrow R

as

f(x) = \frac{2x}{x - 1}

then f is

(A)injective but nor surjective
(B)not injective
(C)surjective but not injective
(D)neither injective nor surjective

Q308·MathematicsSingle correctJEE Advanced 2018

Let

E_{1} = \left\{x \in R : x \neq 1 \text{ and } \frac{x}{x-1} > 0\right\}

and

E_{2} = \left\{x \in E_{1} : \sin^{-1}\left(\log_{e}\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}

\left(\text{Here, the inverse trigonometric function } \sin^{-1}x \text{ assumes values in } \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].\right)

Let

f : E_{1} \to R

be the function defined by

f(x) = \log_{e}\left(\frac{x}{x-1}\right)

and

g : E_{2} \to R

be the function defined by

g(x) = \sin^{-1}\left(\log_{e}\left(\frac{x}{x-1}\right)\right)

. The correct option is :

LIST-I	LIST-II
P.The range of $f$ is	1. $\left(-\infty, \frac{1}{1-e}\right] \cup \left[\frac{e}{e-1}, \infty\right)$
Q.The range of $g$ contains	2. $(0, 1)$
R.The domain of $f$ contains	3. $\left[-\frac{1}{2}, \frac{1}{2}\right]$
S.The domain of $g$ is	4. $(-\infty, 0) \cup (0, \infty)$
	5. $\left(-\infty, \frac{e}{e-1}\right]$
	6. $(-\infty, 0) \cup \left(\frac{1}{2}, \frac{e}{e-1}\right]$

(A)P $\to$ 4; Q $\to$ 2; R $\to$ 1; S $\to$ 1
(B)P $\to$ 3; Q $\to$ 3; R $\to$ 6; S $\to$ 5
(C)P $\to$ 4; Q $\to$ 2; R $\to$ 1; S $\to$ 6
(D)P $\to$ 4; Q $\to$ 3; R $\to$ 6; S $\to$ 5

Q309·MathematicsNumericalJEE Advanced 2018

The value of

\left(\left(\log_{2} 9\right)^{2}\right)^{\frac{1}{\log_{2}\left(\log_{2} 9\right)}} \times \left(\sqrt{7}\right)^{\frac{1}{\log_{4} 7}}

is ______ .

Q310·MathematicsMultiple correctJEE Advanced 2015

Let

f(x) = \sin\left(\dfrac{\pi}{6}\sin\left(\dfrac{\pi}{2}\sin x\right)\right)

for all

x \in \mathbb{R}

and

g(x) = \dfrac{\pi}{2}\sin x

for all

x \in \mathbb{R}

. Let

(f \circ g)(x)

denote

f(g(x))

and

(g \circ f)(x)

denote

g(f(x))

. Then which of the following is (are) true ?

(A)Range of $f$ is $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$
(B)Range of $f \circ g$ is $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$
(C) $\displaystyle\lim_{x \to 0} \dfrac{f(x)}{g(x)} = \dfrac{\pi}{6}$
(D)There is an $x \in \mathbb{R}$ such that $(g \circ f)(x) = 1$

Q311·MathematicsSingle correctJEE Advanced 2014

Let

f_{1} : \mathbb{R} \to \mathbb{R}

,

f_{2} : [0, \infty) \to \mathbb{R}

,

f_{3} : \mathbb{R} \to \mathbb{R}

and

f_{4} : \mathbb{R} \to [0, \infty)

be defined by

f_{1}(x) = \begin{cases} |x| & \text{if } x < 0 \\ e^{x} & \text{if } x \ge 0 \end{cases}

;

f_{2}(x) = x^{2}

;

f_{3}(x) = \begin{cases} \sin x & \text{if } x < 0 \\ x & \text{if } x \ge 0 \end{cases}

and

f_{4}(x) = \begin{cases} f_{2}(f_{1}(x)) & \text{if } x < 0 \\ f_{2}(f_{1}(x)) - 1 & \text{if } x \ge 0 \end{cases}

List – I	List – II
P. $f_{4}$ is	1.onto but not one-one
Q. $f_{3}$ is	2.neither continuous nor one-one
R. $f_{2} \circ f_{1}$ is	3.differentiable but not one-one
S. $f_{2}$ is	4.continuous and one-one

(A)P-3, Q-1, R-4, S-2
(B)P-1, Q-3, R-4, S-2
(C)P-3, Q-1, R-2, S-4
(D)P-1, Q-3, R-2, S-4

Q312·MathematicsMultiple correctJEE Advanced 2014

Let

f: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}

be given by

f(x) = (\log(\sec x + \tan x))^3

. Then

(A) $f(x)$ is an odd function
(B) $f(x)$ is a one-one function
(C) $f(x)$ is an onto function
(D) $f(x)$ is an even function

Q313·MathematicsMultiple correctJEE Advanced 2013

If

3^{x} = 4^{x-1}

, then

x =

(A) $\frac{2\log_{3} 2}{2\log_{3} 2 - 1}$
(B) $\frac{2}{2 - \log_{2} 3}$
(C) $\frac{1}{1 - \log_{4} 3}$
(D) $\frac{2\log_{2} 3}{2\log_{2} 3 - 1}$

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