sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions

Q 1 :

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? [2023]

10
9
21
15

1

2

3

4

(3)

$n (A) = 48, n (B) = 25, n (C) = 18$

$n (A \cup B \cup C) = 60 [Total]$

$n (A \cap B \cap C) = 5$

$(A \cup B \cup C) = \sum | A | - \sum | A \cap B | + | A \cap B \cap C |$

$\Rightarrow \sum | A \cap B | = 48 + 25 + 18 + 5 - 60 = 36$

Number of men who received exactly 2 medals

$= \sum | A \cap B | - 3 | A \cap B \cap C | = 36 - 15 = 21$

Q 2 :

Let $Ω$ be the sample space and $A \subseteq Ω$ be an event. Given below are two statements

(S1) : If P(A) = 0, then A = $ϕ$

(S2) : If P(A) = 1, then A = $Ω$

Then [2023]

both (S1) and (S2) are true
only (S1) is true
only (S2) is true
both (S1) and (S2) are false

1

2

3

4

(1)

$Ω$ be the sample space and A be an event. If $P (A) = 0 \Rightarrow A = ϕ$ If $P (A) = 1 \Rightarrow A = Ω$

Then both statements are true.

Q 3 :

The number of elements in the set ${n \in ℤ : | n^{2} - 10 n + 19 | < 6}$ is ___________ . [2023]

(6)

We have, $| n^{2} - 10 n + 19 |$ $< 6, n \in ℤ$

$\Rightarrow$ $| {(n - 5)}^{2} - 6 |$ $< 6$

$\Rightarrow 0 < {(n - 5)}^{2} < 12$

$\Rightarrow {(n - 5)}^{2} = 1, 4, 9$

$\Rightarrow n - 5 = \pm 1, \pm 2, \pm 3$

$∴ 6 values of n exist.$

Q 4 :

The number of elements in the set { $n \in ℕ : 10 \leq n \leq 100$ and $3^{n} - 3$ is a multiple of 7} is __________ . [2023]

(15)

$3^{n} - 3 = 7 k, k \in ℤ$

$\Rightarrow 3^{n} = 7 k + 3, k \in ℤ$ $...(i)$

$Now, 3 \equiv 3 mod 7$

$3^{3} \equiv -1 mod 7$

$3^{6} \equiv 1 mod 7$

$3^{7} \equiv 3 mod 7$

$3^{13} \equiv 3 mod 7$

$∴ n = 1, 7, 13, \dots will satisfy equation (i)$

This forms an A.P. with common difference 6.

Now, $N \in [10, 100]$

$∴ n = 13, 19, \dots$

Now, $13 + (n - 1) \cdot 6 \leq 100 \Rightarrow 13 + 6 n - 6 \leq 100$

$\Rightarrow 6 n \leq 100 - 7 \Rightarrow 6 n \leq 93 \Rightarrow n \leq 15.5 \Rightarrow n = 15$

So, there are 15 such numbers.

Q 5 :

Let $λ \in R$ and let the equation E be $| x^{2} | - 2 | x | + | λ - 3 | = 0 .$ Then the largest element in the set $S = {x + λ : x$ is an integer solution of E $}$ is _________ . [2023]

(5)

$| x |^{2}$ $- 2$ $| x |$ $+$ $| λ - 3 |$ $= 0$

${(| x | - 1)}^{2} +$ $| λ - 3 |$ $- 1 = 0$ $...(i)$

Given that $x$ is an integer $\Rightarrow$ $| x |$ $- 1$ is an integer

${(| x | - 1)}^{2} - 1$ is also an integer. Thus $| λ - 3 |$ is also an integer. Therefore $λ$ is also an integer, ${(| x | - 1)}^{2} \leq 1$ .

$Case I:$ $| x |$ $- 1 = 0 \Rightarrow x = \pm 1$

$∴$ $| λ - 3 |$ $= 1 \Rightarrow λ = 4, 2$

$Possible values of λ + x = 5, 3, 1 .$

$Case II:$ $| x |$ $- 1 = 1 \Rightarrow x = \pm 2$

$∴$ $| λ - 3 |$ $= 0 \Rightarrow λ = 3$

$Possible values of x + λ = 5, 1 .$

$Considering all possible cases, the maximum value of λ + x = 5 .$

Q 6 :

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 [2024]

280
300
310
290

1

2

3

4

(2)

$Multiples of 3 are 102, 105, \dots, 699$

$a_{n} = 699 = 102 + (n - 1) \cdot 3$

$\Rightarrow 597 = 3 (n - 1) \Rightarrow n = 200$

$Now, multiples of 4 are 100, 104, \dots, 700$

$a_{m} = 700 = 100 + (m - 1) \cdot 4$

$\Rightarrow m = 151$

$Multiples of 3 and 4 are 108, 120, \dots, 696$

$a_{p} = 696 = 108 + (p - 1) \cdot 12$

$\Rightarrow p = 50$

$n (3 \cup 4) = n (3) + n (4) - n (3 \cap 4)$

$= n + m - p = 200 + 151 - 50 = 301$

$∴ The number of elements in A = 601 - 301 = 300$

Q 7 :

Let $S = {x \in R : {(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10}$ . Then the number of elements in S is [2024]

4
2
0
1

1

2

3

4

(2)

${(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10$

Let $\sqrt{3} + \sqrt{2} = t$

$t^{x} + \frac{1}{t^{x}} = 10$

Let $t^{x} = y, t h e n y + \frac{1}{y} = 10$

$\Rightarrow y^{2} - 10 y + 1 = 0$

$\Rightarrow y = \frac{10 \pm \sqrt{100 - 4}}{2} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4 \sqrt{6}}{2} = 5 \pm 2 \sqrt{6}$

${(\sqrt{3} + \sqrt{2})}^{x} = 5 \pm 2 \sqrt{6}$

$x \ln (\sqrt{3} + \sqrt{2}) = \ln (5 + 2 \sqrt{6}) or \ln (5 - 2 \sqrt{6})$

$x = \frac{\ln (5 + 2 \sqrt{6})}{\ln (\sqrt{3} + \sqrt{2})} or x = \frac{\ln (5 - 2 \sqrt{6})}{\ln (\sqrt{3} + \sqrt{2})}$

So, two real values of $x$ .

$∴$ The number of elements in S is 2.

Q 8 :

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 [2024]

6
8
10
4

1

2

3

4

(3)

$Total number of subsets of set A = 2^{m}$

$Total number of subsets of set B = 2^{n}$

According to question, $2^{m} - 2^{n} = 56$

$\Rightarrow 2^{n} (2^{m - n} - 1) = 2^{3} \times 7$

$\Rightarrow 2^{n} = 2^{3} and 2^{m - n} = 8 = 2^{3}$

$\Rightarrow n = 3 and m - n = 3 \Rightarrow m = 3 + n = 3 + 3 = 6$

$∴ m = 6 and n = 3$

$Now, distance between the points P (6, 3) and Q (- 2, - 3) is given by P Q = \sqrt{{(6 + 2)}^{2} + {(3 + 3)}^{2}} = \sqrt{64 + 36} = 10 units$

Q 9 :

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 m+n is equal to _______. [2024]

(45)

$Total number of students = n (S) = 220$

$n (M) \in [125, 130], n (P) \in [85, 95], n (C) \in [75, 90]$

$n (M \cup P \cup C) = 220 - 10 = 210$

$n (M \cap P) = 40, n (P \cap C) = 30, n (M \cap C) = 50$

$n (M \cup P \cup C) = \sum n (M) - \sum n (M \cap P) + n (M \cap P \cap C)$

$\Rightarrow n (M \cap P \cap C) = 210 + (40 + 30 + 50) - \sum n (M)$

$∵$ $(n {(M \cap P \cap C))}_{m a x} = n = \min {n (M \cap P), n (P \cap C), n (M \cap C)} = 30$

$∵$ $(\sum n {(M))}_{m a x} = 130 + 95 + 90 = 315$

$\Rightarrow (n {(M \cap P \cap C))}_{m i n} = m = 330 - 315 = 15$

$\Rightarrow n + m = 45$

Q 10 :

If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]

(18)

$We have, S = {a \in ℝ : | 2 a - 1 | = 3 [a] + 2 {a}}$

$Since for x \in ℝ, x = [x] + {x}$

$∴ 3 [a] + 2 {a} = 2 a + [a]$

$∴ | 2 a - 1 | = 2 a + [a]$

$For 2 a - 1 < 0, we have$

$- 2 a + 1 = 2 a + [a]$

$\Rightarrow 4 a = 1 - [a] \Rightarrow 4 a + [a] - 1 = 0$

$\Rightarrow a = \frac{1}{4}$

$Now, if 2 a - 1 > 0 \Rightarrow a > \frac{1}{2}$

$\Rightarrow 2 a - 1 = 2 a + [a]$

$\Rightarrow [a] = - 1 which is not possible as a > \frac{1}{2}$

$S o, a = \frac{1}{4} \in S$

$∴ 72 \sum_{a \in S} a = 72 \times \frac{1}{4} = 18$

Q 11 :

The number of elements in the set $S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z \geq 0}$ equals _______ . [2024]

(169)

$x + 2 y + 3 z = 42$

At $z = 0, x + 2 y = 42$

$y$ can take all value from 0 to 21

Hence 22 solutions.

Similarly,

at $z = 1, x + 2 y = 39$ $\to$ $20 solutions$

at $z = 2, x + 2 y = 36$ $\to$ $19 solutions$

at $z = 3, x + 2 y = 33$ $\to$ $17 solutions$

at $z = 4, x + 2 y = 30$ $\to$ $16 solutions$

at $z = 5, x + 2 y = 27$ $\to$ $14 solutions$

at $z = 6, x + 2 y = 24$ $\to$ $13 solutions$

at $z = 7, x + 2 y = 21$ $\to$ $11 solutions$

at $z = 8, x + 2 y = 18$ $\to$ $10 solutions$

at $z = 9, x + 2 y = 15$ $\to$ $8 solutions$

at $z = 10, x + 2 y = 12$ $\to$ $7 solutions$

at $z = 11, x + 2 y = 9$ $\to$ $5 solutions$

at $z = 12, x + 2 y = 6$ $\to$ $4 solutions$

at $z = 13, x + 2 y = 3$ $\to$ $2 solutions$

at $z = 14, x + 2 y = 0$ $\to$ $1 solutions$

Total cases = 169 solutions.

Q 12 :

Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to ________ . [2024]

(46)

$We have, x^{2} - 2^{y} = 2023 \Rightarrow 2^{y} = x^{2} - 2023$

$∴$ $Possible values of (x, y) are (45, 1)$

$S o, \sum_{(x, y) \in C} (x + y) = (45 + 1) = 46$

Q 13 :

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics, and 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, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is __________ . [2024]

(10)

Q 14 :

Let $A = {(α, β) \in R \times R : | α - 1 | \leq 4 and | β - 5 | \leq 6}$ and $B = {(α, β) \in R \times R : 16 {(α - 2)}^{2} + 9 {(β - 6)}^{2} \leq 144}$ . Then [2025]

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

1

2

3

4

(3)

$A : | α - 1 | \leq 4 and | β - 5 | \leq 6$

$\Rightarrow - 4 \leq α - 1 \leq 4 and - 6 \leq β - 5 \leq 6$

$\Rightarrow - 3 \leq α \leq 5 and - 1 \leq β \leq 11$

$B : 16 {(α - 2)}^{2} + 9 {(β - 6)}^{2} \leq 144$

$B : \frac{{(α - 2)}^{2}}{9} + \frac{{(β - 6)}^{2}}{16} \leq 1$

From Diagram, $B \subset A$ .

Q 15 :

Let A = {1, 2, 3, ..., 10} and $B = {\frac{m}{n} : m, n \in A, m < n and g c d (m, n) = 1}$ .

Then n(B) is equal to : [2025]

36
31
37
29

1

2

3

4

(2)

We have, A = {1, 2, 3, ..., 10}

$B = {\frac{m}{n} : m, n \in A, m < n and g c d (m, n) = 1}$

For m = 1, n = 2, 3, ..., 10 $\Rightarrow$ 9 cases

m = 2, n = 3, 5, 7, 9 $\Rightarrow$ 4 cases

m = 3, n = 4, 5, 7, 8, 10 $\Rightarrow$ 5 cases

m = 4, n = 5, 7, 9 $\Rightarrow$ 3 cases

m = 5, n = 6, 7, 8, 9 $\Rightarrow$ 4 cases

m = 6, n = 7 $\Rightarrow$ 1 cases

m = 7, n = 8, 9, 10 $\Rightarrow$ 3 cases

m = 8, n = 9 $\Rightarrow$ 1 cases

m = 9, n = 10 $\Rightarrow$ 1 cases

$∴$ n(B) = 31.

Q 16 :

Let $S = {(m, n) : m, n \in {1, 2, 3, . . . . ., 50}}$ . If the number of elements $(m, n)$ in S such that $6^{m} + 9^{n}$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to _______. [2026]

(1333)

$S = {1, 2, 3, \dots, 50}$

$p = (6^{m} + 9^{n}) is divisible by 5$

$No. of ways$

$6^{m} = {(5 λ + 1)}^{m} = 5 k + 1$

$9^{n} = {(10 - 1)}^{n} =$ $10 μ - 1$ if $n$ is odd

$\Rightarrow n must be odd$

$10 μ + 1$ if $n$ is even

$\Rightarrow No. of ways = 50 \times 25 = 1250$

$q \Rightarrow (m + n) is square of a prime$

	$m + n = 4$	$m + n = 9$	$m + n = 25$	$m + n = 49$
No. of	$↓$	$↓$	$↓$	$↓$
ways	3	8	24	48

$q = 3 + 8 + 24 + 48 = 83$

$= p + q = 1250 + 83 = 1333$

Q 17 :

The number of elements in the set

$S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}}$ is_____. [2026]

(16)

$\int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}$

$Use by parts$

$- x^{2} \cos x + x^{2} + 2 x^{2} \cos x - 2 x \sin x$

$- x^{2} \cos x + 2 x \sin x + 2 \cos x - 2 = x$

$\cos x = 1$

$x = 0, 2 π, 4 π, \dots, 30 π$

$Total Elements = 16$

Q 18 :

Let $A = {x : | x^{2} - 10 | \leq 6} and B = {x : | x - 2 | > 1} .$ Then [2026]

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

1

2

3

4

(4)

$| x^{2} - 10 |$ $\leq 6$

$- 6 \leq x^{2} - 10 \leq 6$

$4 \leq x^{2} \leq 16$

$A = [- 4, - 2] \cup [2, 4]$

$| x - 2 |$ $> 1$

$B = (- \infty, 1) \cup (3, \infty)$

$A \cup B = (- \infty, 1) \cup [2, \infty)$

$A \cap B = [- 4, - 2] \cup (3, 4]$

$A - B = [2, 3]$

$B - A = (- \infty, - 4) \cup (- 2, 1) \cup (4, \infty)$

Q 19 :

Let S be the set of the first 11 natural numbers. Then the number of elements in

$A = {B \subseteq S : n (B) \geq 2 and the product of all elements of B is even}$ is _______. [2026]

(1979)

$A = {1, 2, 3, \dots, 11}$

$∴$ $n (B) \geq 2 and product of all elements in B is even$

$Case (i):$

$n (B) = 2 \Rightarrow {}^{11}{C_{2}} - {}^{6}{C_{2}}$

$n (B) = 3 \Rightarrow {}^{11}{C_{3}} - {}^{6}{C_{3}}$

$n (B) = 4 \Rightarrow {}^{11}{C_{4}} - {}^{6}{C_{4}}$

$n (B) = 5 \Rightarrow {}^{11}{C_{5}} - {}^{6}{C_{5}}$

$n (B) = 6 \Rightarrow {}^{11}{C_{6}} - {}^{6}{C_{6}}$

$n (B) = 7 \Rightarrow {}^{11}{C_{7}}$

$⋮$

$n (B) = 11 \Rightarrow {}^{11}{C_{11}}$

$∴$ $number of set B = \sum_{r = 2}^{11} {}^{11}{C_{r}} - \sum_{r = 2}^{6} {}^{6}{C_{r}}$

$= 2^{11} - (1 + 11) - (2^{6} - 7)$

$= 2048 - 64 - 5$

$= 1979$

$Alternate Solution:$

$Total subsets = 2^{11}$

$No. of subsets having odd terms only = 2^{6}$

$No. of subsets having one term only & also having even terms = 5$

$Required ways = 2^{11} - 2^{6} - 5 = 1979$

Topic Question Set

Q 1 :

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? [2023]

Q 2 :

Let $Ω$ be the sample space and $A \subseteq Ω$ be an event. Given below are two statements

(S1) : If P(A) = 0, then A = $ϕ$

(S2) : If P(A) = 1, then A = $Ω$

Then [2023]

Q 3 :

The number of elements in the set ${n \in ℤ : | n^{2} - 10 n + 19 | < 6}$ is ___________ . [2023]

Q 4 :

The number of elements in the set { $n \in ℕ : 10 \leq n \leq 100$ and $3^{n} - 3$ is a multiple of 7} is __________ . [2023]

Q 5 :

Let $λ \in R$ and let the equation E be $| x^{2} | - 2 | x | + | λ - 3 | = 0 .$ Then the largest element in the set $S = {x + λ : x$ is an integer solution of E $}$ is _________ . [2023]

Q 6 :

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 [2024]

Q 7 :

Let $S = {x \in R : {(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10}$ . Then the number of elements in S is [2024]

Q 8 :

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 [2024]

Q 10 :

If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]

Q 11 :

The number of elements in the set $S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z \geq 0}$ equals _______ . [2024]

Q 12 :

Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to ________ . [2024]

(10)

Q 14 :

Let $A = {(α, β) \in R \times R : | α - 1 | \leq 4 and | β - 5 | \leq 6}$ and $B = {(α, β) \in R \times R : 16 {(α - 2)}^{2} + 9 {(β - 6)}^{2} \leq 144}$ . Then [2025]

Q 15 :

Let A = {1, 2, 3, ..., 10} and $B = {\frac{m}{n} : m, n \in A, m < n and g c d (m, n) = 1}$ .

Then n(B) is equal to : [2025]

Q 16 :

Let $S = {(m, n) : m, n \in {1, 2, 3, . . . . ., 50}}$ . If the number of elements $(m, n)$ in S such that $6^{m} + 9^{n}$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to _______. [2026]

Q 17 :

The number of elements in the set

$S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}}$ is_____. [2026]

Q 18 :

Let $A = {x : | x^{2} - 10 | \leq 6} and B = {x : | x - 2 | > 1} .$ Then [2026]

Q 19 :

Let S be the set of the first 11 natural numbers. Then the number of elements in

$A = {B \subseteq S : n (B) \geq 2 and the product of all elements of B is even}$ is _______. [2026]

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Topic Question Set

Q 1 : 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? [2023]

Q 2 : Let Ω be the sample space and A⊆Ω be an event. Given below are two statements (S1) : If P(A) = 0, then A = ϕ (S2) : If P(A) = 1, then A = Ω Then [2023]

Q 3 : The number of elements in the set {n∈ℤ:|n2-10n+19|<6} is ___________ . [2023]

Q 4 : The number of elements in the set {n∈ℕ:10≤n≤100 and 3n-3 is a multiple of 7} is __________ . [2023]

Q 5 : Let λ∈R and let the equation E be |x2|-2|x|+|λ-3|=0. Then the largest element in the set S={x+λ:x is an integer solution of E} is _________ . [2023]

Q 6 : Let A={n∈[100,700]∩N:n is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A is [2024]

Q 7 : Let S={x∈R:(3+2)x+(3-2)x=10}. Then the number of elements in S is [2024]

Q 8 : 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 [2024]

Q 10 : If S={a∈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∑a∈Sa is equal to ________ [2024]

Q 11 : The number of elements in the set S={(x,y,z):x,y,z∈Z, x+2y+3z=42, x,y,z≥0} equals _______ . [2024]

Q 12 : Let the set C={(x,y)∣x2-2y=2023, x,y∈N}. Then ∑(x,y)∈C(x+y) is equal to ________ . [2024]

(10)

Q 14 : Let A={(α,β)∈R×R:|α–1|≤4 and |β–5|≤6} and B={(α,β)∈R×R:16(α–2)2+9(β–6)2≤144}. Then [2025]

Q 15 : Let A = {1, 2, 3, ..., 10} and B={mn:m,n∈A, m<n and gcd(m,n)=1}. Then n(B) is equal to : [2025]

Q 16 : Let S={(m,n):m,n∈{1,2,3,.....,50}}. If the number of elements (m,n) in S such that 6m+9n is a multiple of 5 is p and the number of elements (m,n) in S such that m+n is a square of a prime number is q, then p+q is equal to _______. [2026]

Q 17 : The number of elements in the set S={x:x∈[0,100] and ∫0xt2sin(x-t)dt=x2} is_____. [2026]

Q 18 : Let A={x:|x2-10|≤6} and B={x:|x-2|>1}. Then [2026]

Q 19 : Let S be the set of the first 11 natural numbers. Then the number of elements in A={B⊆S:n(B)≥2 and the product of all elements of B is even} is _______. [2026]

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Q 1 :

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? [2023]

Q 2 :

Let $Ω$ be the sample space and $A \subseteq Ω$ be an event. Given below are two statements

(S1) : If P(A) = 0, then A = $ϕ$

(S2) : If P(A) = 1, then A = $Ω$

Then [2023]

Q 3 :

The number of elements in the set ${n \in ℤ : | n^{2} - 10 n + 19 | < 6}$ is ___________ . [2023]

Q 4 :

The number of elements in the set { $n \in ℕ : 10 \leq n \leq 100$ and $3^{n} - 3$ is a multiple of 7} is __________ . [2023]

Q 5 :

Let $λ \in R$ and let the equation E be $| x^{2} | - 2 | x | + | λ - 3 | = 0 .$ Then the largest element in the set $S = {x + λ : x$ is an integer solution of E $}$ is _________ . [2023]

Q 6 :

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 [2024]

Q 7 :

Let $S = {x \in R : {(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10}$ . Then the number of elements in S is [2024]

Q 8 :

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 [2024]

Q 10 :

If $S = {a \in R : | 2 a - 1 | = 3 [a] + 2 {a}}$ , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then $72 \sum_{a \in S} a$ is equal to ________ [2024]

Q 11 :

The number of elements in the set $S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z \geq 0}$ equals _______ . [2024]

Q 12 :

Let the set $C = {(x, y) ∣ x^{2} - 2^{y} = 2023, x, y \in N}$ . Then $\sum_{(x, y) \in C} (x + y)$ is equal to ________ . [2024]

Q 14 :

Let $A = {(α, β) \in R \times R : | α - 1 | \leq 4 and | β - 5 | \leq 6}$ and $B = {(α, β) \in R \times R : 16 {(α - 2)}^{2} + 9 {(β - 6)}^{2} \leq 144}$ . Then [2025]

Q 15 :

Let A = {1, 2, 3, ..., 10} and $B = {\frac{m}{n} : m, n \in A, m < n and g c d (m, n) = 1}$ .

Then n(B) is equal to : [2025]

Q 16 :

Let $S = {(m, n) : m, n \in {1, 2, 3, . . . . ., 50}}$ . If the number of elements $(m, n)$ in S such that $6^{m} + 9^{n}$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to _______. [2026]

Q 17 :

The number of elements in the set

$S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} \sin (x - t) d t = x^{2}}$ is_____. [2026]

Q 18 :

Let $A = {x : | x^{2} - 10 | \leq 6} and B = {x : | x - 2 | > 1} .$ Then [2026]

Q 19 :

Let S be the set of the first 11 natural numbers. Then the number of elements in

$A = {B \subseteq S : n (B) \geq 2 and the product of all elements of B is even}$ is _______. [2026]