Sets, Relations, and Functions

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2024 (Online) 6th April Morning Shift
#1018
Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, ..., 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ be 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) 16
(B) 12
(C) 8
(D) 10
HARD
2 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 22nd January Morning Shift
#1014
The number of non-empty equivalence relations on the set ${1, 2, 3}$ is :
(A) $7$
(B) $4$
(C) $5$
(D) $6$
3 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 22nd January Morning Shift
#1013
Let $A = {1, 2, 3, …, 10}$ and $B = {\frac{m}{n} : m, n \in A, m < n$ and $gcd(m, n) = 1}$. Then $n(B)$ is equal to :
(A) $29$
(B) $31$
(C) $37$
(D) $36$
4 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 23rd January Morning Shift
#1012
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) 9
(B) 8
(C) 7
(D) 10
5 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 23rd January Evening Shift
#1011
Let $A = {(x, y) ∈ R × R : |x + y| ⩾ 3}$ and $B = {(x, y) ∈ R × R : |x| + |y| ≤ 3}$. If $C = {(x, y) ∈ A ∩ B : x = 0$ or $y = 0}$, then $\sum_{(x, y) ∈ C} |x + y|$ is :
(A) 18
(B) 24
(C) 15
(D) 12
6 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 23rd January Evening Shift
#1010
Let $X = R \times R$. 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 true
(B) Statement I is true but Statement II is false
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
7 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 24th January Evening Shift
#1009
Let $A = {x \in (0, \pi) - {\frac{\pi}{2}} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2}$ and $B = {x \ge 0 : x(x-4) - 3|x-2| + 6 = 0}$. Then $n(A \cup B)$ is equal to :
(A) 4
(B) 8
(C) 6
(D) 2
8 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 28th January Morning Shift
#1008
The relation $R = {(x, y) : x, y ∈ Z$ 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
9 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 29th January Morning Shift
#1007
Define a relation R on the interval $[0, π/2)$ by $x$ R $y$ if and only if $\sec^2x - \tan^2y = 1$. Then R is :
(A) both reflexive and symmetric but not transitive
(B) both reflexive and transitive but not symmetric
(C) reflexive but neither symmetric not transitive
(D) an equivalence relation
10 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 29th January Evening Shift
#1006
Let $S = \mathbb{N} \cup \{0\}$. Define a relation R from S to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e (\frac{2}{5}), x \in S, y \in \mathbb{R}\}$. Then, the sum of all the elements in the range of $R$ is equal to:
(A) $\frac{3}{2}$
(B) $\frac{10}{9}$
(C) $\frac{5}{2}$
(D) $\frac{5}{3}$
11 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 2nd April Morning Shift
#1005
Let $A$ be the set of all functions $f: Z \rightarrow Z$ 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 reflective
(B) Symmetric but neither reflective nor transitive
(C) Transitive but neither reflexive nor symmetric
(D) Reflexive but neither symmetric nor transitive
MEDIUM
12 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 2nd April Evening Shift
#1004
Let $A = {1, 2, 3, ..., 100}$ and $R$ be a relation on $A$ such that $R = {(a, b) : a = 2b + 1}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), ..., (a_k, a_{k+1})$ 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) 8
(C) 7
(D) 5
MEDIUM
13 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 3rd April Morning Shift
#1003
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) 18
(B) 20
(C) 17
(D) 19
14 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 3rd April Evening Shift
#1002
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) 11
(B) 12
(C) 14
(D) 13
HARD
15 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 4th April Morning Shift
#1001
Consider the sets $A = \{(x, y) \in R \times R : x^2 + y^2 = 25\}$, $B = \{(x, y) \in R \times R : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in Z \times Z : x^2 + y^2 \le 4\}$ and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
(A) $15120$
(B) $18290$
(C) $17160$
(D) $19320$
16 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 4th April Evening Shift
#1000
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) 17
(B) 18
(C) 15
(D) 16
HARD
17 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Evening Shift
#999
Let $A = \{ (\alpha, \beta) \in R \times R : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6 \}$ and $B = \{ (\alpha, \beta) \in R \times R : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144 \}$. Then
(A) $A \subset B$
(B) $B \subset A$
(C) neither $A \subset B$ nor $B \subset A$
(D) $A \cup B = \{ (x, y) : -4 \leqslant x \leqslant 4, -1 \leqslant y \leqslant 11 \}$
18 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 8th April Evening Shift
#998
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
(S1): The number of elements in $R$ is $18$, and
(S2): The relation $R$ is symmetric but neither reflexive nor transitive
(A) both are false
(B) only (S1) is true
(C) only (S2) is true
(D) both are true
19 Competency UNDERSTAND MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 4th April Morning Shift
#997
Consider the sets $A = \{(x, y) \in R \times R : x^2 + y^2 = 25\}$, $B = \{(x, y) \in R \times R: x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in Z \times Z: x^2 + y^2 \leq 4\}$ and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
(A) 15120
(B) 18290
(C) 17160
(D) 19320
HARD
20 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 4th April Evening Shift
#996
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) 17
(B) 18
(C) 15
(D) 16
21 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Evening Shift
#995
Let $A = { (\alpha, \beta ) \in R \times R : |\alpha - 1| \leq 4$ and $|\beta - 5| \leq 6 }$

and $B = { (\alpha, \beta ) \in R \times R : 16(\alpha - 2)^{2}+ 9(\beta - 6)^{2} \leq 144 }$.

Then
(A) A $A \subset B$
(B) B $B \subset A$
(C) C neither $A \subset B$ nor $B \subset A$
(D) D $A \cup B = { (x, y) : -4 \leqslant x \leqslant 4, -1 \leqslant y \leqslant 11 }$
22 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 8th April Evening Shift
#994
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements

(S1): The number of elements in R is 18, and

(S2): The relation R is symmetric but neither reflexive nor transitive
(A) both are false
(B) only (S1) is true
(C) only (S2) is true
(D) both are true
HARD
23 Competency APPLY MCQ_SINGLE 4 Mark(s) 2025 JEE Main 2025 (Online) 3rd April Evening Shift
#987
If the probability that the random variable $X$ takes the value $x$ is given by

$P(X=x) = k(x+1)3^{-x}, x = 0, 1, 2, 3 \dots$, where $k$ is a constant, then $P(X \geq 3)$ is equal to
(A) $\frac{1}{9}$
(B) $\frac{8}{27}$
(C) $\frac{7}{27}$
(D) $\frac{4}{9}$
HARD
24 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 27th January Morning Shift
#1028
Let $S = {1, 2, 3, …, 10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = {(A, B) : A ∩ B ≠ 𝜙; A, B ∈ M}$ is :
(A) symmetric only
(B) reflexive only
(C) symmetric and reflexive only
(D) symmetric and transitive only
25 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 27th January Evening Shift
#1027
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) 8
(B) 10
(C) 4
(D) 6
26 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 29th January Morning Shift
#1026
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 transitive but not symmetric
(B) Reflexive and symmetric but not transitive
(C) Reflexive but neither symmetric nor transitive
(D) Reflexive, symmetric and transitive
27 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 29th January Evening Shift
#1025
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) $15$
(B) $10$
(C) $12$
(D) $8$
28 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 1st February Evening Shift
#1024
Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in R$ and $(a, b)R_2(c, d) \Leftrightarrow a+ d = b + c$ for all $(a, b), (c, d) \in N \times N$. Then:
(A) $R_1$ and $R_2$ both are equivalence relations
(B) Only $R_1$ is an equivalence relation
(C) Only $R_2$ is an equivalence relation
(D) Neither $R_1$ nor $R_2$ is an equivalence relation
29 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 4th April Evening Shift
#1023
Let a relation $R$ on $N \times N$ 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) Both (I) and (II) are correct.
(C) Neither (I) nor (II) is correct.
(D) Only (I) is correct.
30 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 29th January Morning Shift
#1022
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 transitive but not symmetric
(B) Reflexive and symmetric but not transitive
(C) Reflexive but neither symmetric nor transitive
(D) Reflexive, symmetric and transitive
31 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 29th January Evening Shift
#1021
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) $15$
(B) $10$
(C) $12$
(D) $8$
32 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 1st February Evening Shift
#1020
Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in R$ and $(a, b)R_2(c, d) \Leftrightarrow a+ d = b + c$ for all $(a, b), (c, d) \in N \times N$. Then:
(A) $R_1$ and $R_2$ both are equivalence relations
(B) Only $R_1$ is an equivalence relation
(C) Only $R_2$ is an equivalence relation
(D) Neither $R_1$ nor $R_2$ is an equivalence relation
33 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 4th April Evening Shift
#1019
Let a relation $R$ on $N \times N$ 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) Both (I) and (II) are correct.
(C) Neither (I) nor (II) is correct.
(D) Only (I) is correct.
34 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 6th April Morning Shift
#1017
Let $A = {n \in [100, 700] \cap N : n$ is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in $A$ is
(A) 300
(B) 310
(C) 290
(D) 280
35 Competency APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 6th April Evening Shift
#1016
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) $23$
(B) $26$
(C) $25$
(D) $24$
HARD
36 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2024 JEE Main 2024 (Online) 8th April Evening Shift
#1015
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) an equivalence relation.
(C) reflexive and symmetric but not transitive.
(D) transitive but not symmetric.
37 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 24th January Morning Shift
#1039
The relation $R = \{(a, b) : gcd(a, b) = 1, 2a \neq b, a, b \in Z\}$ is:
(A) reflexive but not symmetric
(B) transitive but not reflexive
(C) symmetric but not transitive
(D) neither symmetric nor transitive
38 Competency APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 29th January Evening Shift
#1038
Let R be a relation defined on $N$ as $aRb$ if $2a + 3b$ is a multiple of $5$, $a, b \in N$. Then R is
(A) an equivalence relation
(B) non reflexive
(C) symmetric but not transitive
(D) transitive but not symmetric
HARD
39 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 30th January Morning Shift
#1037
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) 7
(B) 3
(C) 4
(D) 5
40 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 31st January Morning Shift
#1036
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) symmetric and transitive but not reflexive
(B) reflexive and symmetric but not transitive
(C) transitive but neither reflexive nor symmetric
(D) symmetric but neither reflexive nor transitive
41 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 31st January Evening Shift
#1035
Among the relations
$S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^2 - b^2 \in Z}$,
(A) $S$ is transitive but $T$ is not
(B) both $S$ and $T$ are symmetric
(C) neither $S$ nor $T$ is transitive
(D) $T$ is symmetric but $S$ is not
42 Knowledge UNDERSTAND MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 8th April Evening Shift
#1034
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is :
(A) reflexive but neither symmetric nor transitive
(B) transitive but neither symmetric nor reflexive
(C) symmetric but neither reflexive nor transitive
(D) an equivalence relation
HARD
43 Competency APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 1st February Morning Shift
#1033
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 and transitive but not symmetric
(D) reflexive but neither symmetric nor transitive
EASY
44 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 1st February Evening Shift
#1032
Let $P(S)$ denote the power set of $S=${$1, 2, 3, …, 10$}. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if $(A \cap B^c) \cup (B \cap A^c) = \emptyset$ and $AR_2B$ if $A \cup B^c = B \cup A^c$, $\forall A, B \in P(S)$. Then :
(A) only $R_2$ is an equivalence relation
(B) both $R_1$ and $R_2$ are not equivalence relations
(C) both $R_1$ and $R_2$ are equivalence relations
(D) only $R_1$ is an equivalence relation
45 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 10th April Evening Shift
#1031
Let $A = {2, 3, 4}$ and $B = {8, 9, 12}$. Then the number of elements in the relation $R = {((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1$ divides $b_2$ and $a_2$ divides $b_1}$ is :
(A) 18
(B) 24
(C) 36
(D) 12
46 Competency APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 11th April Morning Shift
#1030
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) $15$
(C) $21$
(D) $9$
HARD
47 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2023 JEE Main 2023 (Online) 11th April Evening Shift
#1029
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) 180
(B) 26
(C) 52
(D) 160
48 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2022 JEE Main 2022 (Online) 28th June Evening Shift
#1044
Let $R_1 = \{(a, b) \in N \times N : |a - b| \le 13\}$ and $R_2 = \{(a, b) \in N \times N : |a - b| \ne 13\}$. 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
49 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2022 JEE Main 2022 (Online) 29th June Morning Shift
#1043
Let a set $A = A_1 \cup A_2 \cup ..... \cup A_k$, where $A_i \cap A_j = \phi$ for $i \neq j$, $1 \le j, j \le k$. 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.
50 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2022 JEE Main 2022 (Online) 27th July Morning Shift
#1042
Let $R_1$ and $R_2$ be two relations defined on $R$ by $aR_1b \Leftrightarrow ab \ge 0$ and $aR_2b \Leftrightarrow a \ge b$. 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
51 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2022 JEE Main 2022 (Online) 28th July Morning Shift
#1041
For $\alpha \in N$, consider a relation $R$ on $N$ given by $R = \{(x, y) : 3x + \alpha y$ 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 $7$
52 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2022 JEE Main 2022 (Online) 29th July Morning Shift
#1040
Let R be a relation from the set ${1, 2, 3, …, 60}$ to itself such that $R = {(a, b) : b = pq}$, where $p, q \geqslant 3$ are prime numbers}. Then, the number of elements in R is :
(A) $600$
(B) $660$
(C) $540$
(D) $720$
53 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 26th February Morning Shift
#1052
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) $S = {(x, y) | x^2 + y^2 = \sqrt{2}}$
(B) $S = {(x, y) | x^2 + y^2 = 2}$
(C) $S = {(x, y) | x^2 + y^2 = 1}$
(D) $S = {(x, y) | x^2 + y^2 = 4}$
54 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 16th March Morning Shift
#1051
The number of elements in the set {$x \in R : (|x| - 3) |x + 4| = 6$} is equal to :
(A) 4
(B) 2
(C) 3
(D) 1
55 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 16th March Evening Shift
#1050
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) $7$
56 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 17th March Morning Shift
#1049
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) Q and R
(B) None of these
(C) P and R
(D) P and Q
57 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 18th March Evening Shift
#1048
Define a relation $R$ over a class of $n imes n$ 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 reflexive, transitive but not symmetric
(B) $R$ is symmetric, transitive but not reflexive.
(C) $R$ is reflexive, symmetric but not transitive
(D) $R$ is an equivalence relation
58 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 27th July Evening Shift
#1047
Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = {(x, y) ∈ N × N: x^3 - 3x^2y - xy^2 + 3y^3 = 0}$. Then the relation $R$ is :
(A) symmetric but neither reflexive nor transitive
(B) reflexive but neither symmetric nor transitive
(C) reflexive and symmetric, but not transitive
(D) an equivalence relation
59 Competency APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 26th August Morning Shift
#1046
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}
EASY
60 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2021 JEE Main 2021 (Online) 31st August Morning Shift
#1045
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.
61 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 9th January Evening Slot
#1059
If $A = {x \in R : |x| < 2}$ and $B = {x \in R : |x – 2| \geq 3}$; then :
(A) $A – B = [–1, 2)$
(B) $A \cup B = R – (2, 5)$
(C) $A \cap B = (–2, –1)$
(D) $B – A = R – (–2, 5)$
62 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 2nd September Morning Slot
#1058
If $R = {(x, y) : x, y \in Z, x^2 + 3y^2 \le 8}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is :
(A) {0, 1}
(B) {-2, –1, 1, 2}
(C) {-1, 0, 1}
(D) {-2, –1, 0, 1, 2}
63 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 3rd September Morning Slot
#1057
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 ∩ B = {–$3$}
(B) B – A = (–$3$, $5$)
(C) A ∪ B = R
(D) A - B = ($-∝$, –$3$) ∪ ($5$, $∝$)
64 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 3rd September Evening Slot
#1056
Let $R_1$ and $R_2$ be two relation 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) $R_2$ is transitive but $R_1$ is not transitive.
(C) $R_1$ and $R_2$ are both transitive.
(D) $R_1$ is transitive but $R_2$ is not transitive.
65 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 4th September Morning Slot
#1055
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) $37$
(B) $65$
(C) $29$
(D) $55$
66 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 4th September Evening Slot
#1054
Let $\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$ where each $X_i$ contains $10$ elements and each $Y_i$ 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) $30$
(B) $50$
(C) $15$
(D) $45$
67 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2020 JEE Main 2020 (Online) 5th September Morning Slot
#1053
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) $36$
(C) $54$
(D) $38$
68 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2019 JEE Main 2019 (Online) 10th January Morning Slot
#1064
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) $42$
(B) $102$
(C) $1$
(D) $38$
69 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2019 JEE Main 2019 (Online) 12th January Morning Slot
#1063
Let $S = {1, 2, 3, … , 100}$. The number of non-empty subsets A of S such that the product of elements in A is even is :
(A) $2^{50} – 1$
(B) $2^{50} (2^{50} – 1)$
(C) $2^{100} – 1$
(D) $2^{50} + 1$
70 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2019 JEE Main 2019 (Online) 12th January Evening Slot
#1062
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) $2^{12}$
(B) $2^{18}$
(C) $2^{10}$
(D) $2^{15}$
71 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2019 JEE Main 2019 (Online) 9th April Evening Slot
#1061
Two newspapers A and B are published in a city. It is known that $25$% of the city populations reads A and $20$% reads B while $8$% reads both A and B. Further, $30$% of those who read A but not B look into advertisements and $40$% of those who read B but not A also look into advertisements, while $50$% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-
(A) $13.5$
(B) $13$
(C) $12.8$
(D) $13.9$
72 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2019 JEE Main 2019 (Online) 12th April Evening Slot
#1060
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 – B) $\subseteq$ C, then A $\subseteq$ C
(B) B $\cap$ C $\neq$ $\phi$
(C) (C $\cup$ A) $\cap$ (C $\cup$ B) = C
(D) If (A – C) $\subseteq$ B, then A $\subseteq$ B
73 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2018 JEE Main 2018 (Online) 15th April Morning Slot
#1067
Consider the following two binary relations on the set $A = {a, b, c}$:
$R_1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)}$ and
$R_2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}$.
Then:
(A) both $R_1$ and $R_2$ are not symmetric.
(B) $R_1$ is not symmetric but it is transitive.
(C) $R_2$ is symmetric but it is not transitive.
(D) both $R_1$ and $R_2$ are transitive.
74 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2018 JEE Main 2018 (Offline)
#1066
Two sets A and B are as under : A = {$(a, b) ∈ R × R : |a - 5| < 1$ and $|b - 5| < 1$}; B = {$(a, b) ∈ R × R : 4(a - 6)^2 + 9(b - 5)^2 ≤ 36$}; Then
(A) neither A ⊂ B nor B ⊂ A
(B) B ⊂ A
(C) A ⊂ B
(D) A ∩ B = 𝜙 ( an empty set )
75 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2018 JEE Main 2018 (Online) 16th April Morning Slot
#1065
Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$. Then :
(A) Range of $R_1$ is $\{2, 4, 8\}$.
(B) Range of $R_2$ is $\{1, 2, 3, 4\}$.
(C) Both $R_1$ and $R_2$ are symmetric relations.
(D) Both $R_1$ and $R_2$ are transitive relations.
76 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2016 JEE Main 2016 (Online) 10th April Morning Slot
#1068
Let $P = {\theta : sin\theta - cos\theta = \sqrt{2}cos\theta}$ and $Q = {\theta : sin\theta + cos\theta = \sqrt{2}sin\theta}$ be two sets. Then
(A) $P \subset Q$ and $Q - P \neq \phi$
(B) $Q \nsubseteq P$
(C) $P \nsubseteq Q$
(D) $P = Q$
77 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2015 JEE Main 2015 (Offline)
#1069
Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set $A \times B$, each having atleast three elements are
(A) 219
(B) 256
(C) 275
(D) 510
78 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2012 AIEEE 2012
#1070
Let $X = {1, 2, 3, 4, 5}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X$, $Z \subseteq X$ and $Y \cap Z$ is empty, is:
(A) $3^5$
(B) $2^5$
(C) $5^3$
(D) $5^2$
79 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2011 AIEEE
#1071
Let $R$ be the set of real numbers. Statement I: $A = \{(x, y) \in R \times R: y - x \text{ is an integer }\}$ is an equivalence relation on $R$. Statement II: $B = \{(x,y) \in R \times R: x = \alpha y \text{ for some rational number } \alpha\}$ is an equivalence relation on $R$.
(A) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
(B) Statement I is true, Statement II is false.
(C) Statement I is false, Statement II is true.
(D) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
80 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2010 AIEEE
#1072
Consider the following relations $R = \{(x, y) | x, y$ are real numbers and $x = wy$ for some rational number $w\}$; $S = \{(\frac{m}{n}, \frac{p}{q}) | m, n, p$ and $q$ are integers such that $n, q \neq 0$ and $qm = pn\}$. Then
(A) $R$ is an equivalence relation but $S$ is not an equivalence relation
(B) Neither $R$ nor $S$ is an equivalence relation
(C) $S$ is an equivalence relation but $R$ is not an equivalence relation
(D) $R$ and $S$ both are equivalence relations
81 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2009 AIEEE
#1073
If $A$, $B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then :
(A) $A = C$
(B) $B = C$
(C) $A \cap B = \phi$
(D) $A = B$
82 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2008 AIEEE
#1074
Let $R$ be the real line. Consider the following subsets of the plane $R \times R$: $S = {(x, y) : y=x+1 ext{ and } 0<x<2}$ $T = {(x, y) : x - y ext{ is an integer }}$, Which one of the following is true ?
(A) Neither S nor T is an equivalence relation on R
(B) Both S and T are equivalence relation on R
(C) S is an equivalence relation on R but T is not
(D) T is an equivalence relation on R but S is not
83 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2006 AIEEE
#1075
Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R = {(x, y) ∈ W × W |$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is
(A) reflexive, symmetric and not transitive
(B) reflexive, symmetric and transitive
(C) reflexive, not symmetric and transitive
(D) not reflexive, symmetric and transitive
84 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2005 AIEEE
#1076
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set $A = \{3, 6, 9, 12\}$. The relation is :
(A) reflexive and symmetric only
(B) an equivalence relation
(C) reflexive only
(D) reflexive and transitive only
85 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2004 AIEEE
#1077
Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ is:
(A) a function
(B) transitive
(C) not symmetric
(D) reflexive

NUMERICAL

1 Competency APPLY NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 22nd January Evening Shift
#1135
4 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 _________.
HARD
2 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 24th January Morning Shift
#1132
Let $S=\left\{p_1, p_2 \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \in A$, such that $x$ divides $y$, is ________ .
EASY
3 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Morning Shift
#1121
For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________
EASY
4 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Morning Shift
#1110
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 __________.
EASY
5 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 22nd January Evening Shift
#1104
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 _________.
EASY
6 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 24th January Morning Shift
#1101
Let $S=\left\{p_1, p_2 \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \in A$, such that $x$ divides $y$, is ________ .
EASY
7 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Morning Shift
#1090
For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________
EASY
8 Competency REMEMBER NUMERICAL 4 Mark(s) 2025 JEE Main 2025 (Online) 7th April Morning Shift
#1079
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 __________.
EASY
9 Competency APPLY NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 31st January Morning Shift
#1140
Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(2,3),(1,4)\}$ be a relation on $\mathrm{A}$. Let $\mathrm{S}$ be the equivalence relation on $\mathrm{A}$ such that $R \subset S$ and the number of elements in $\mathrm{S}$ is $\mathrm{n}$. Then, the minimum value of $n$ is __________.
HARD
10 Competency APPLY NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 31st January Evening Shift
#1139
Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.
HARD
11 Competency APPLY NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 1st February Morning Shift
#1138
Let $A=\{1,2,3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation 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 _____________.
HARD
12 Competency APPLY NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 4th April Morning Shift
#1137
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students 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 subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
HARD
13 Competency APPLY NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 9th April Morning Shift
#1136
5 Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A \times B$ by $(a_1, b_1) R(a_2, b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is __________.
HARD
14 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 30th January Evening Shift
#1111
The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is _________.
EASY
15 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 31st January Morning Shift
#1109
Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(2,3),(1,4)\}$ be a relation on $\mathrm{A}$. Let $\mathrm{S}$ be the equivalence relation on $\mathrm{A}$ such that $R \subset S$ and the number of elements in $\mathrm{S}$ is $\mathrm{n}$. Then, the minimum value of $n$ is __________.
EASY
16 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 31st January Evening Shift
#1108
Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.
EASY
17 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 1st February Morning Shift
#1107
Let $A=\{1,2,3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation 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 _____________.
EASY
18 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 4th April Morning Shift
#1106
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students 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 subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
EASY
19 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 9th April Morning Shift
#1105
Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A \times B$ by $(a_1, b_1) R(a_2, b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is __________.
EASY
20 Competency REMEMBER NUMERICAL 4 Mark(s) 2024 JEE Main 2024 (Online) 30th January Evening Shift
#1080
The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is _________.
EASY
21 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 24th January Evening Shift
#1120
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 __________.
EASY
22 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 25th January Morning Shift
#1119
Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.
EASY
23 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 6th April Morning Shift
#1118
Let $\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$ and $\mathrm{B}=\{0,1,2,3,4\}$. The number of elements in the relation $R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}$ is ___________.
EASY
24 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 8th April Morning Shift
#1117
Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x, y) \in A \times A: x-y$ 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 ____________.
EASY
25 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 10th April Morning Shift
#1116
The number of elements in the set $\{ n \in Z:|{n^2} - 10n + 19| < 6\} $ is _________.
EASY
26 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 12th April Morning Shift
#1115
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 __________.
EASY
27 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 13th April Evening Shift
#1114
Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$ or $\left.b^{2}=a+1\right\}$ be a relation on $\mathrm{A}$. Then the minimum number of elements, that must be added to the relation $\mathrm{R}$ so that it becomes reflexive and symmetric, is __________
EASY
28 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 15th April Morning Shift
#1113
Let $A=\{1,2,3,4\}$ and $\mathrm{R}$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $\mathrm{R}$ is ____________.
EASY
29 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 15th April Morning Shift
#1112
The number of elements in the set $\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ is ___________.
EASY
30 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 24th January Evening Shift
#1089
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 __________.
EASY
31 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 25th January Morning Shift
#1088
Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.
EASY
32 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 6th April Morning Shift
#1087
Let $\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$ and $\mathrm{B}=\{0,1,2,3,4\}$. The number of elements in the relation $R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}$ is ___________.
EASY
33 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 8th April Morning Shift
#1086
Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x, y) \in A \times A: x-y$ 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 ____________.
EASY
34 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 10th April Morning Shift
#1085
The number of elements in the set $\{ n \in Z:|{n^2} - 10n + 19| < 6\} $ is _________.
EASY
35 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 12th April Morning Shift
#1084
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 __________.
EASY
36 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 13th April Evening Shift
#1083
Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$ or $\left.b^{2}=a+1\right\}$ be a relation on $\mathrm{A}$. Then the minimum number of elements, that must be added to the relation $\mathrm{R}$ so that it becomes reflexive and symmetric, is __________
EASY
37 Competency REMEMBER NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 15th April Morning Shift
#1082
Let $A=\{1,2,3,4\}$ and $\mathrm{R}$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $\mathrm{R}$ is ____________.
EASY
38 Knowledge APPLY NUMERICAL 4 Mark(s) 2023 JEE Main 2023 (Online) 15th April Morning Shift
#1081
The number of elements in the set $\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ is ___________.
HARD
39 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 24th June Evening Shift
#1128
The sum of all the elements of the set $\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\} $ is __________.
EASY
40 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th June Morning Shift
#1127
Let $A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $ and $B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $. Then A + B is equal to _____________.
EASY
41 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th June Morning Shift
#1126
Let A = {n $\in$ N : H.C.F. (n, 45) = 1} and Let B = {2k : k $\in$ {1, 2, ......., 100}}. Then the sum of all the elements of A $\cap$ B is ____________.
EASY
42 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 28th June Morning Shift
#1125
Let R1 and R2 be relations on the set {1, 2, ......., 50} such that R1 = {(p, pn) : p is a prime and n $\ge$ 0 is an integer} and R2 = {(p, pn) : p is a prime and n = 0 or 1}. Then, the number of elements in R1 $-$ R2 is _______________.
EASY
43 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 25th July Evening Shift
#1124
Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T \subseteq A$ : either $1 \notin T$ or $2 \in T\}$ and $C=\{T \subseteq A: T$ the sum of all the elements of $T$ is a prime number $\}$. Then the number of elements in the set $B \cup C$ is ________________.
EASY
44 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th July Evening Shift
#1123
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 ___________.
EASY
45 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 29th July Morning Shift
#1122
Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If $A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$ $\epsilon S\}$, then the sum of all the elements in the set $T-A$ is equal to __________.
EASY
46 Competency NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 24th June Evening Shift
#1097
26 The sum of all the elements of the set $$\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\} $$ is __________.
EASY
47 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th June Morning Shift
#1096
Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $$Then A + B is equal to _____________.
EASY
48 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th June Morning Shift
#1095
Let A = {n $\in$ N : H.C.F. (n, 45) = 1} and Let B = {2k : k $\in$ {1, 2, ......., 100}}. Then the sum of all the elements of A $\cap$ B is ____________.
EASY
49 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 28th June Morning Shift
#1094
Let R1 and R2 be relations on the set {1, 2, ......., 50} such that R1 = {(p, pn) : p is a prime and n $\ge$ 0 is an integer} and R2 = {(p, pn) : p is a prime and n = 0 or 1}. Then, the number of elements in R1 $-$ R2 is _______________.
EASY
50 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 25th July Evening Shift
#1093
Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T \subseteq A$ : either $1 \notin T$ or $2 \in T\}$ and $C=\{T \subseteq A: T$ the sum of all the elements of $T$ is a prime number $\}$. Then the number of elements in the set $B \cup C$ is ________________.
EASY
51 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 26th July Evening Shift
#1092
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 ___________.
EASY
52 Competency REMEMBER NUMERICAL 4 Mark(s) 2022 JEE Main 2022 (Online) 29th July Morning Shift
#1091
Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If $A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.$ $\epsilon S\}$, then the sum of all the elements in the set $T-A$ is equal to __________.
EASY
53 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 24th February Morning Shift
#1131
Let A = {n $ \in $ N: n is a 3-digit number} B = {9k + 2: k $ \in $ N} and C = {9k + $l$: k $ \in $ N} for some $l ( 0 < l < 9)$ If the sum of all the elements of the set A $ \cap $ (B $ \cup $ C) is 274 $ \times $ 400, then $l$ is equal to ________.
EASY
54 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 27th July Evening Shift
#1130
Let A = {n $\in$ N | n2 $\le$ n + 10,000}, B = {3k + 1 | k$\in$ N} an dC = {2k | k$\in$N}, then the sum of all the elements of the set A $\cap$(B $-$ C) is equal to _____________.
EASY
55 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 27th August Morning Shift
#1129
If A = {x $\in$ R : |x $-$ 2| > 1}, B = {x $\in$ R : $\sqrt {{x^2} - 3} $ > 1}, C = {x $\in$ R : |x $-$ 4| $\ge$ 2} and Z is the set of all integers, then the number of subsets of the set (A $\cap$ B $\cap$ C)c $\cap$ Z is ________________.
EASY
56 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 24th February Morning Shift
#1100
Let A = {n $ \in $ N: n is a 3-digit number} B = {9k + 2: k $ \in $ N} and C = {9k + $l$: k $ \in $ N} for some $l ( 0 < l < 9)$ If the sum of all the elements of the set A $ \cap $ (B $ \cup $ C) is 274 $ \times $ 400, then $l$ is equal to ________.
EASY
57 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 27th July Evening Shift
#1099
Let A = {n $\in$ N | n2 $\le$ n + 10,000}, B = {3k + 1 | k$\in$ N} an dC = {2k | k$\in$N}, then the sum of all the elements of the set A $\cap$(B $-$ C) is equal to _____________.
EASY
58 Competency REMEMBER NUMERICAL 4 Mark(s) 2021 JEE Main 2021 (Online) 27th August Morning Shift
#1098
If A = {x $\in$ R : |x $-$ 2| > 1}, B = {x $\in$ R : $\sqrt {{x^2} - 3} $ > 1}, C = {x $\in$ R : |x $-$ 4| $\ge$ 2} and Z is the set of all integers, then the number of subsets of the set (A $\cap$ B $\cap$ C)c $\cap$ Z is ________________.
EASY
59 Competency REMEMBER NUMERICAL 4 Mark(s) 2020 JEE Main 2020 (Online) 7th January Evening Slot
#1134
Let X = {n $ \in $ N : 1 $ \le $ n $ \le $ 50}. If A = {n $ \in $ X: n is a multiple of 2} and B = {n $ \in $ X: n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is ________.
EASY
60 Competency REMEMBER NUMERICAL 4 Mark(s) 2020 JEE Main 2020 (Online) 6th September Morning Slot
#1133
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 ______.
EASY
61 Competency REMEMBER NUMERICAL 4 Mark(s) 2020 JEE Main 2020 (Online) 7th January Evening Slot
#1103
Let X = {n $ \in $ N : 1 $ \le $ n $ \le $ 50}. If A = {n $ \in $ X: n is a multiple of 2} and B = {n $ \in $ X: n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is ________.
EASY
62 Competency REMEMBER NUMERICAL 4 Mark(s) 2020 JEE Main 2020 (Online) 6th September Morning Slot
#1102
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 ______.
EASY
63 Knowledge APPLY NUMERICAL 1 Mark(s)
#1078
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 ________.