Sets, Relations & Functions · JEE Main previous year questions

387 questions asked across 19 exam years, newest first

Year-wise papers Topic-wise questions

Practise this topic

2026 47 questions

Q212026 · 2 Apr 2026 · Shift 2· Numerical answer
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 $y ⩽ w$ . Then the number of elements in R is ____.
Show answer
Answer: 120
Q72026 · 28 Jan 2026 · Shift 1
Let $S = {x^{3} + a x^{2} + b x + c : a, b, c \in N and a, b, c \leq 20}$ be a set of polynomials. Then the number of polynomials in $S$ , which are divisible by $x^{2} + 2$ , is
1. A.6
2. B.120
3. C.20
4. D.10
Show answer
Answer: D
Q102026 · 28 Jan 2026 · Shift 1
Let $f$ be a polynomial function such that $f (x^{2} + 1) = x^{4} + 5 x^{2} + 2$ , for all $x \in R$ . Then $0 \int 3 f (x) d x$ is equal to
1. A. $\frac{33}{2}$
2. B. $\frac{5}{3}$
3. C. $\frac{27}{2}$
4. D. $\frac{41}{3}$
Show answer
Answer: A
Q222026 · 28 Jan 2026 · Shift 1· Numerical answer
In a G.P., if the product of the first three terms is $27$ and the set of all possible values for the sum of its first three terms is $R - (a, b)$ , then $a^{2} + b^{2}$ is equal to ____ .
Show answer
Answer: 90
Q62026 · 28 Jan 2026 · Shift 2
Given below are two statements : Statement I : The function $f : R \to R$ defined by $f (x) = \frac{x}{1 + ∣ x ∣}$ is one-one. Statement II : The function $f : R \to R$ defined by $f (x) = \frac{x ^{2} + 4 x - 30}{x ^{2} - 8 x + 18}$ is many-one. In the light of the above statements, choose the correct answer from the options given below :
1. A.Statement I is true but Statement II is false
2. B.Both Statement I and Statement II are false
3. C.Both Statement I and Statement II are true
4. D.Statement I is false but Statement II is true
Show answer
Answer: C
Q252026 · 28 Jan 2026 · Shift 2· Numerical answer
Let $f$ be a differentiable function satisfying $f (x) = 1 - 2 x + \int_{0}^{x} e^{(x - t)} f (t) d t, x \in R$ and let $g (x) = \int_{0}^{x} (f (t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} d t, x \in R$ . If $p$ and $q$ are respectively the points of local minima and local maxima of $g$ , then the value of $∣ p + q ∣$ is equal to ________.
Show answer
Answer: 9
Q72026 · 24 Jan 2026 · Shift 1
If the domain of the function $f (x) = lo g_{(10 x^{2} - 17 x + 7)} (18 x^{2} - 11 x + 1)$ is $(- \infty, a) \cup (b, c) \cup (d, \infty) - {e}$ , then $90 (a + b + c + d + e)$ equals:
1. A.170
2. B.316
3. C.177
4. D.307
Show answer
Answer: B
Q112026 · 24 Jan 2026 · Shift 1
Let $R$ be a relation defined on the set ${1, 2, 3, 4} \times {1, 2, 3, 4}$ by $R = {((a, b), (c, d)) : 2 a + 3 b = 3 c + 4 d}$ . Then the number of elements in R is
1. A.6
2. B.15
3. C.12
4. D.18
Show answer
Answer: C
Q212026 · 24 Jan 2026 · Shift 1· Numerical answer
Let $(2 α, α)$ be the largest interval in which the function $f (t) = \frac{∣ t + 1∣}{t ^{2}}, t < 0$ , is strictly decreasing. Then the local maximum value of the function $g (x) = 2 lo g_{e} (x - 2) + α x^{2} + 4 x - α, x > 2$ , is ____
Show answer
Answer: 4
Q102026 · 23 Jan 2026 · Shift 1
Let $α$ and $β$ respectively be the maximum and the minimum values of the function $f (θ) = 4 (sin^{4} (\frac{7 π}{2} - θ) + sin^{4} (11 π + θ)) - 2 (sin^{6} (\frac{3 π}{2} - θ) + sin^{6} (9 π - θ)), θ \in R$ . Then $α + 2 β$ is equal to :
1. A.6
2. B.5
3. C.4
4. D.3
Show answer
Answer: B
Q192026 · 23 Jan 2026 · Shift 1
Let $A = {- 2, - 1, 0, 1, 2, 3, 4}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $2 x + y ⩽ 2$ . 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. A.34
2. B.32
3. C.33
4. D.35
Show answer
Answer: C
Q52026 · 23 Jan 2026 · Shift 2
Consider two sets $A = {x \in Z : ∣ (∣ x - 3∣ - 3) ∣ \leq 1}$ and $B = {x \in R - {1, 2} : \frac{( x - 2 ) ( x - 4 )}{x - 1} lo g_{e} (∣ x - 2∣) = 0}$ . Then the number of onto functions $f : A \to B$ is equal to :
1. A.32
2. B.81
3. C.79
4. D.62
Show answer
Answer: D
Q62026 · 23 Jan 2026 · Shift 2
Let $A = {0, 1, 2, \dots, 9}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $∣ x - y ∣$ 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. A.Statement I is correct but Statement II is incorrect
2. B.Both Statement I and Statement II are correct
3. C.Both Statement I and Statement II are incorrect
4. D.Statement I is incorrect but Statement II is correct
Show answer
Answer: D
Q52026 · 22 Jan 2026 · Shift 1
Let the relation R on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M}$ . Then the minimum number of elements required to be added in R , in order to make the relation symmetric, is equal to
1. A.4
2. B.3
3. C.1
4. D.2
Show answer
Answer: D
Q132026 · 22 Jan 2026 · Shift 1
Let $f (x) = x^{2025} - x^{2000}, x \in [0, 1]$ and the minimum value of the function $f (x)$ in the interval $[0, 1]$ be $(80)^{80} (n)^{- 81}$ . Then $n$ is equal to
1. A.-40
2. B.-41
3. C.-80
4. D.-81
Show answer
Answer: D
Q182026 · 22 Jan 2026 · Shift 1
If the domain of the function $f (x) = sin^{- 1} (\frac{5 - x}{3 + 2 x}) + \frac{1}{l o g _{e} ( 10 - x )}$ is $(- \infty, α] \cup [β, γ) - {δ}$ , then $6 (α + β + γ + δ)$ is equal to
1. A.66
2. B.68
3. C.70
4. D.67
Show answer
Answer: C
Q82026 · 21 Jan 2026 · Shift 1
The number of strictly increasing functions $f$ from the set ${1, 2, 3, 4, 5, 6}$ to the set ${1, 2, 3, \dots, 9}$ such that $f (i) \neq = i$ for $1 \leq i \leq 6$ , is equal to :
1. A.21
2. B.28
3. C.27
4. D.22
Show answer
Answer: B
Q152026 · 21 Jan 2026 · Shift 1
The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to:
1. A.16
2. B.64
3. C.256
4. D.1024
Show answer
Answer: B
Q172026 · 21 Jan 2026 · Shift 1
If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
1. A.5
2. B.2
3. C.3
4. D.1
Show answer
Answer: C
Q22026 · 21 Jan 2026 · Shift 2
Let $α$ and $β$ be the roots of the equation $x^{2} + 2 a x + (3 a + 10) = 0$ such that $α < 1 < β$ . Then the set of all possible values of $a$ is :
1. A. $(- \infty, \frac{- 11}{5}) \cup (5, \infty)$
2. B. $(- \infty, \frac{- 11}{5})$
3. C. $(- \infty, - 3)$
4. D. $(- \infty, - 2) \cup (5, \infty)$
Show answer
Answer: B
Q152026 · 21 Jan 2026 · Shift 2
Let $A = {2, 3, 5, 7, 9}$ . Let $R$ be the relation on $A$ defined by $x R y$ if and only if $2 x \leq 3 y$ . 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 symmetric relation. Then $l + m$ is equal to:
1. A.21
2. B.25
3. C.23
4. D.27
Show answer
Answer: B
Q212026 · 21 Jan 2026 · Shift 2· Numerical answer
Let $[\cdot]$ denote the greatest integer function and $f (x) = n \to \infty lim \frac{1}{n ^{3}} k = 1 \sum n [\frac{k ^{2}}{3 ^{x}}]$ . Then $12 j = 1 \sum \infty f (j)$ is equal to ____.
Show answer
Answer: 2