Sets, Relations & Functions

27 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q3·MCQ·Easy–moderate
Open
Reflexive and symmetric relations
The number of relations, defined on the set $\{a, b, c, d\}$ , which are both reflexive and symmetric, is equal to:
1. (1)256
2. (2)16
3. (3)1024
4. (4)64
MathsJEE Main 21 Jan 2026, Shift 2·Q13·MCQ·Easy–moderate
Open
Set operations on intervals
Let $A = \{x : \lvert x^2 - 10\rvert \le 6\}$ and $B = \{x : \lvert x - 2\rvert > 1\}$ . Then
1. (1) $A \cup B = (-\infty, 1] \cup (2, \infty)$
2. (2) $A - B = [2, 3)$
3. (3) $A \cap B = [-4, -2] \cup [3, 4]$
4. (4) $B - A = (-\infty, -4) \cup (-2, 1) \cup (4, \infty)$
MathsJEE Main 21 Jan 2026, Shift 2·Q17·MCQ·Easy–moderate
Open
Symmetric relations
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:
1. (1)23
2. (2)25
3. (3)21
4. (4)27
MathsJEE Main 22 Jan 2026, Shift 1·Q2·MCQ·Easy–moderate
Open
Symmetric relations
Let the relation $R$ on the set $M = \{1, 2, 3, \ldots, 16\}$ 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
1. (1)1
2. (2)2
3. (3)4
4. (4)3
MathsJEE Main 22 Jan 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
Counting elements of a relation
The number of elements in the relation $R = \{(x, y) : 4x^2 + y^2 < 52,\ x, y \in \mathbb{Z}\}$ is
1. (1)77
2. (2)89
3. (3)67
4. (4)86
MathsJEE Main 22 Jan 2026, Shift 2·Q14·MCQ·Moderate
Open
Domain of a function
Let the domain of the function $f(x) = \log_3\log_5\left(7 - \log_2(x^2 - 10x + 85)\right) + \sin^{-1}\left(\left\lvert\dfrac{3x - 7}{17 - x}\right\rvert\right)$ be $(\alpha, \beta]$ . Then $\alpha + \beta$ is equal to:
1. (1)10
2. (2)12
3. (3)9
4. (4)8
MathsJEE Main 22 Jan 2026, Shift 2·Q18·MCQ·Moderate
Open
Greatest integer function
Let $f(x) = [x]^2 - [x + 3] - 3$ , $x \in \mathbb{R}$ where $[\,\bullet\,]$ is the greatest integer function. Then
1. (1) $f(x) > 0$ only for $x \in [4, \infty)$
2. (2) $f(x) < 0$ only for $x \in [-1, 3)$
3. (3) $\displaystyle\int_0^2 f(x)\,dx = -6$
4. (4) $f(x) = 0$ for finitely many values of $x$ .
MathsJEE Main 22 Jan 2026, Shift 2·Q19·MCQ·Easy–moderate
Open
Functional equations
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 $x, y \in \mathbb{N}$ . If $\displaystyle\sum_{x=1}^{n}\left(\frac{f(x)}{g(x)}\right) = 19607$ , then $n$ is equal to:
1. (1)7
2. (2)5
3. (3)6
4. (4)4
We corrected a printing error in the original paper
Source printed “ $f(l) = 7$ ” and “ $g(l) = 1$ ” (letter l for the number 1); the missing “If” added before “ $\sum \ldots = 19607$ ”.
MathsJEE Main 23 Jan 2026, Shift 1·Q1·MCQ·Moderate
Open
Domain of logarithmic function
Let the domain of the function $f(x) = \log_3\log_5\log_7(9x - x^2 - 13)$ be the interval $(m, n)$ . Let the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ have eccentricity $\dfrac{n}{3}$ and the length of the latus rectum $\dfrac{8m}{3}$ . Then $b^2 - a^2$ is equal to:
1. (1)5
2. (2)11
3. (3)9
4. (4)7
MathsJEE Main 23 Jan 2026, Shift 1·Q6·MCQ·Moderate
Open
Reflexive and symmetric relations
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:
1. (1)32
2. (2)34
3. (3)33
4. (4)35
MathsJEE Main 23 Jan 2026, Shift 2·Q5·MCQ·Moderate
Open
Number of onto functions
Consider two sets $A = \{x \in \mathbb{Z} : \lvert(\lvert x - 3\rvert - 3)\rvert \le 1\}$ and $B = \left\{x \in \mathbb{R} - \{1, 2\} : \dfrac{(x-2)(x-4)}{x-1}\log_e(\lvert x - 2\rvert) = 0\right\}.$ Then the number of onto functions $f : A \to B$ is equal to:
1. (1)62
2. (2)79
3. (3)32
4. (4)81
We corrected a printing error in the original paper
Source printed “ $x \in z$ ” (lower-case z) for the set of integers.
MathsJEE Main 23 Jan 2026, Shift 2·Q16·MCQ·Easy–moderate
Open
Equivalence relations
Let $A = \{0, 1, 2, \ldots, 9\}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $\lvert x - y\rvert$ is a multiple of 3.
Given below are two statements:
Statement I: $n(R) = 36$
Statement II: $R$ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below
1. (1)Both Statement I and Statement II are correct
2. (2)Statement I is incorrect but Statement II is correct
3. (3)Statement I is correct but Statement II is incorrect
4. (4)Both Statement I and Statement II are incorrect
We corrected a printing error in the original paper
Source printed the set $A$ with a closing parenthesis, “2,....9)”, instead of a closing brace.
MathsJEE Main 24 Jan 2026, Shift 1·Q6·MCQ·Moderate
Open
Domain of logarithmic function
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:
1. (1)170
2. (2)177
3. (3)307
4. (4)316
MathsJEE Main 24 Jan 2026, Shift 1·Q12·MCQ·Easy–moderate
Open
Number of elements in relation
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
1. (1)6
2. (2)18
3. (3)12
4. (4)15
MathsJEE Main 24 Jan 2026, Shift 2·Q17·MCQ·Moderate
Open
Functional equations
Let $f$ be a function such that $3f(x) + 2f\left(\dfrac{m}{19x}\right) = 5x$ , $x \neq 0$ , where $m = \displaystyle\sum_{i=1}^{9}(i)^2$ . Then $f(5) - f(2)$ is equal to
1. (1) $-9$
2. (2)36
3. (3)18
4. (4)9
MathsJEE Main 28 Jan 2026, Shift 1·Q1·MCQ·Moderate
Open
Composite function equation
If $g(x) = 3x^2 + 2x - 3$ , $f(0) = -3$ and $4g(f(x)) = 3x^2 - 32x + 72$ , where $f$ is a differentiable function, then $f(g(2))$ is equal to:
1. (1) $\dfrac{25}{6}$
2. (2) $-\dfrac{25}{6}$
3. (3) $\dfrac{7}{2}$
4. (4) $-\dfrac{7}{2}$
We corrected a printing error in the original paper
Assumption “where $f$ is a differentiable function” added; the source leaves it unstated, and without it $f$ may change at $x = 16/3$ to the other root $(14 - 3x)/6$ of the equation, which gives $f(13) = -25/6$ (option 2).
MathsJEE Main 28 Jan 2026, Shift 2·Q8·MCQ·Easy–moderate
Open
One-one and many-one functions
Given below are two statements :
Statement I : The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{x}{1 + \lvert x\rvert}$ is one-one.
Statement II : The function $f : \mathbb{R} \to \mathbb{R}$ 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 :
1. (1)Both Statement I and Statement II are false.
2. (2)Both Statement I and Statement II are true.
3. (3)Statement I is false but Statement II is true.
4. (4)Statement I is true but Statement II is false.
MathsJEE Main 2 Apr 2026, Shift 1·Q21·Numerical·Moderate
Open
Domain of a function
If the domain of the function $f(x) = \sqrt{\log_{(0.6)}\left(\left\lvert\dfrac{2x-5}{x^2-4}\right\rvert\right)}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$ , then the value of $a + b + c + d + e$ is .
MathsJEE Main 2 Apr 2026, Shift 2·Q21·Numerical·Easy–moderate
Open
Number of elements in relation
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 .
MathsJEE Main 4 Apr 2026, Shift 1·Q4·MCQ·Easy–moderate
Open
Number of onto functions
The number of functions $f : \{1, 2, 3, 4\} \to \{a, b, c\}$ , which are not onto, is:
1. (1)48
2. (2)45
3. (3)51
4. (4)35
MathsJEE Main 4 Apr 2026, Shift 2·Q1·MCQ·Moderate
Open
Inverse of a function
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,
1. (1)only (I) is TRUE
2. (2)only (II) is TRUE
3. (3)both (I) and (II) are TRUE
4. (4)neither (I) nor (II) is TRUE
MathsJEE Main 4 Apr 2026, Shift 2·Q16·MCQ·Moderate
Open
Functional equations
Let for some $\alpha \in \mathbb{R}$ , $f : \mathbb{R} \to \mathbb{R}$ 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 $\displaystyle\sum_{n=1}^{5}\left(\alpha + f(n)\right)$ is:
1. (1)110
2. (2)140
3. (3)150
4. (4)170
MathsJEE Main 5 Apr 2026, Shift 2·Q21·Numerical·Moderate
Open
Counting elements of a relation
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 .
MathsJEE Main 6 Apr 2026, Shift 2·Q1·MCQ·Easy–moderate
Open
One-one and onto functions
Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}$ . Then $f$ is:
1. (1)both one-one and onto
2. (2)one-one but not onto
3. (3)onto but not one-one
4. (4)neither one-one nor onto
MathsJEE Main 6 Apr 2026, Shift 2·Q21·Numerical·Moderate
Open
Transitive closure of relation
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 .
MathsJEE Main 8 Apr 2026, Shift 2·Q1·MCQ·Easy–moderate
Open
Equivalence relations and counting
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
1. (1)Only I is true
2. (2)Only II is true
3. (3)Both I and II are true
4. (4)Neither I nor II is true
MathsJEE Main 8 Apr 2026, Shift 2·Q25·Numerical·Moderate
Open
Polynomial functional equation
Let $f$ be a polynomial function such that $\log_2(f(x)) = \left(\log_2\left(2 + \frac23 + \frac29 + \ldots\infty\right)\right)\cdot\log_3\left(1 + \frac{f(x)}{f\left(\frac1x\right)}\right),$ $x > 0$ and $f(6) = 37$ . Then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to .