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

Q 11 :

In a survey it is to be found that 70% of employees like bananas and 64% like apples. If $x$ % like both bananas and apples, then

$x \geq 34$
$x \leq 64$
$34 \leq x \leq 64$
All of these

1

2

3

4

(3)

$Let A and B denote the set of employees who like$ $bananas and apples respectively and total number of employees be 100 .$

$∴$ $n (A) = 70, n (B) = 64, n (U) = 100 and n (A \cap B) = x$

$Now, n (A \cup B) \leq 100$

$∴$ $n (A) + n (B) - n (A \cap B) \leq 100$

$\Rightarrow$ $70 + 64 - x \leq 100 \Rightarrow x \geq 34$ ...(i)

$Also, A \cap B \subseteq B$

$∴$ $n (A \cap B) \leq n (B) \Rightarrow x \leq 64$ ...(ii)

$By (i) and (ii), we have 34 \leq x \leq 64$

Q 12 :

If A and B are not disjoint sets, then $n (A \cup B)$ is equal to

$n (A) + n (B)$
$n (A) + n (B) - n (A \cap B)$
$n (A) + n (B) + n (A \cap B)$
$n (A) n (B)$

1

2

3

4

(2)

$Since A, B are not disjoint sets,$

$\Rightarrow$ $A \cap B \neq ϕ$

$So, n (A \cap B) \neq 0$

$So, n (A \cup B) = n (A) + n (B) - n (A \cap B)$

Q 13 :

Given $n (U) = 20$ , $n (A) = 12$ , $n (B) = 9$ , $n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to

17
9
11
3

1

2

3

4

(4)

$n [{(A \cup B)}^{C}] = n (U) - n (A \cup B)$

$= 20 - (n (A) + n (B) - n (A \cap B))$

$= 20 - (12 + 9 - 4) = 20 - 17 = 3$

Q 14 :

If $N_{a} = {a n : n \in N}$ , then $N_{5} \cap N_{7}$ = (Here $N$ is the set of natural numbers)

$N_{7}$
$N$
$N_{35}$
$N_{5}$

1

2

3

4

(3)

$N_{a} = {a n : n \in N}$

$N_{5} = {5, 10, 15, 20, 25, \dots}$

$N_{7} = {7, 14, 21, 28, 35, \dots}$

$N_{5} \cap N_{7} = {35, 70, 105, \dots} = N_{35}$

$∴$ $N_{5} \cap N_{7} = N_{35}$

Q 15 :

$A$ & $B$ are subsets of universal set $U$ such that $n (U) = 800$ , $n (A) = 300$ , $n (B) = 400$ & $n (A \cap B) = 100$ . The number of elements in the set $A^{c} \cap B^{c}$ is

100
200
300
400

1

2

3

4

(2)

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$ $= 300 + 400 - 100 = 600$

$n {(A \cup B)}^{c} = n (U) - n (A \cup B) = 800 - 600 = 200$

$∴$ $n (A^{c} \cap B^{c}) = n {(A \cup B)}^{c} = 200$

Q 16 :

In a certain town 25% families own a cell phone, 15% families own a scooter and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is

10,000
20,000
30,000
40,000

1

2

3

4

(3)

Q 17 :

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

$A = C$
$B = C$
$A \cap B = ϕ$
$A = B$

1

2

3

4

(2)

$Let x \in C and x \in A \Rightarrow x \in A \cap C$

$\Rightarrow x \in A \cap B . Thus x \in B$ $(∵ A \cap C = A \cap B)$

$Again suppose x \in C and x \notin A \Rightarrow x \in C \cup A$

$\Rightarrow x \in B \cup A \Rightarrow x \in B$

$Thus in both cases x \in C \Rightarrow x \in B$

$Hence C \subseteq B$ ...(1)

$Similarly we can show that B \subseteq C$ ...(2)

$Combining (1) and (2), we get B = C$

Q 18 :

For any two sets $A$ and $B$ , $A - (A - B)$ equals

$B$
$A - B$
$A \cap B$
$A^{c} \cap B^{c}$

1

2

3

4

(3)

$A - (A - B) = A \cap {(A \cap B^{c})}^{c} = A \cap (A^{c} \cup B)$

$= ϕ \cup (A \cap B) = A \cap B$

Q 19 :

Given $A = {x ∣ x is a root of x^{2} - 1 = 0}$ , $B = {x ∣ x is a root of x^{2} - 2 x + 1 = 0}$ . Then

$A \cap B = A$
$A \cup B = ϕ$
$A \cup B = A$
$A \cap B = ϕ$

1

2

3

4

(3)

$We have A = {x ∣ x is a root of x^{2} - 1 = 0}$ $= {x ∣ x is a root of (x - 1) (x + 1) = 0} \Rightarrow A = {1, - 1}$

$B = {x ∣ x is a root of x^{2} - 2 x + 1 = 0}$ $= {x ∣ x is a root of {(x - 1)}^{2} = 0}$

$\Rightarrow B = {1}$

$\Rightarrow A \cup B = {1, - 1} = A$

Q 20 :

A set contains $n$ elements. The power set contains

$n$ elements
$2^{n}$ elements
$n^{2}$ elements
None of these

1

2

3

4

(2)

Total number of elements in power set of a set containing n elements = $2^{n}$ .

Q 21 :

In a group of 80 students, 50 play football, 45 play cricket and each student plays either football or cricket. Find the number of students who play both the games.

12
14
15
18

1

2

3

4

(3)

$Let F be the set of the students who play football$ $and C be the set of students who play cricket.$

$Then n (F) = 50 and n (C) = 45 .$

$Since each of the 80 students play atleast one of the two games,$

$we have n (F \cup C) = 80 .$

$n (F \cap C) = n (F) + n (C) - n (F \cup C) = 50 + 45 - 80 = 15 .$

Q 22 :

There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

is 12
is 4
is 2
cannot be determined from the given information

1

2

3

4

(3)

$Given, n (M) = 50, n (P) = 45, n (B) = 40 and$

$n (M \cap P) + n (P \cap B) + n (B \cap M) - 3 n (M \cap P \cap B) = 32$

$n (M \cup P \cup B) = 100 - 1 = 99$

$∴$ $n (M \cup P \cup B) = n (M) + n (P) + n (B) - n (M \cap P)$ $- n (P \cap B) - n (B \cap M) + n (M \cap P \cap B)$

$\Rightarrow$ $99 = 50 + 45 + 40 - 32 - 2 n (M \cap P \cap B)$

$\Rightarrow$ $- 4 = - 2 n (M \cap P \cap B) \Rightarrow n (M \cap P \cap B) = 2$

Q 23 :

Suppose $A_{1}, A_{2}, \dots, A_{30}$ are thirty sets each with five elements and $B_{1}, B_{2}, \dots, B_{n}$ are $' n'$ sets each with three elements.

Let $\cup_{i = 1}^{30} A_{i} = \cup_{j = 1}^{n} B_{j} = S$

Assume that each element of $S$ belongs to exactly ten of $A_{i}' s$ and exactly 9 of $B_{j}' s$ , then the value of $n$ is

90
15
9
45

1

2

3

4

(4)

$Since each A_{i} has 5 elements, and each element of$ $S belongs to exactly 10 of the A_{i}'s$

$∴$ $S = \cup_{i = 1}^{30} A_{i} \Rightarrow n (S) = \frac{1}{10} \sum_{i = 1}^{30} n (A_{i})$

$= \frac{1}{10} \times 5 \times 30 = 15$ ...(i)

$Again each B_{j} has 3 elements and each element of S belongs to$ $exactly 9 of B_{j}'s,$

$S = \cup_{j = 1}^{n} B_{j} \Rightarrow n (S) = \frac{1}{9} \sum_{j = 1}^{n} n (B_{j})$

$= \frac{1}{9} \times 3 n = \frac{n}{3}$ ...(ii)

$From (i) and (ii) we get$

$\frac{n}{3} = 15 \Rightarrow n = 45$

Q 24 :

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where $N$ is the set of natural numbers, then $X \cup Y$ is equal to

$Y - X$
$X$
$Y$
$N$

1

2

3

4

(3)

$X = {4^{n} - 3 n - 1},$ $Y = {9 (n - 1) : n \in N}$

$4^{n} - 3 n - 1$ $= {(1 + 3)}^{n} - 3 n - 1$

$= (1 + {}^{n}{C_{1}} 3 + {}^{n}{C_{2}} 3^{2} + \dots + 3^{n}) - 3 n - 1$

$= (1 + 3 n) + 9 ({}^{n}C_{2} + \dots + 3^{n - 2}) - 3 n - 1$

$= 9 ({}^{n}C_{2} + \dots + 3^{n - 2}),$ $n \geq 2 = 9 k$

Also for $n = 1$ , $4^{n} - 3 n - 1 = 0,$ a multiple of 9

Every element in $X$ is a multiple of 9. But $Y$ contains all multiples of 9. Hence $X \cup Y = Y .$

Q 25 :

In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is

45
0
25
35

1

2

3

4

(3)

Total number of students, $(N) = 60$

Let $n (C)$ be the number of students who play cricket and $n (T)$ be the number of students who play tennis.

$∴$ $n (C) = 25, n (T) = 20, n (C \cap T) = 10$

$So, n (C \cup T) = n (C) + n (T) - n (T \cap C) = 25 + 20 - 10 = 35$

Now, number of students who play neither game

$= n (C \cup T)' = N - n (C \cup T) = 60 - 35 = 25$

Q 26 :

If $A = {5^{n} - 4 n - 1 : n \in N}$ and $B = {16 (n - 1) : n \in N}$ , then

$A = B$
$A \cap B = ϕ$
$A \subseteq B$
$B \subseteq A$

1

2

3

4

(3)

Q 27 :

If $A$ and $B$ are disjoint sets, then $B \cap A'$ , where $A'$ is complement of $A$ is equal to

$A$
$B$
$A'$
$B'$

1

2

3

4

(2)

$A$ and $B$ are disjoint sets.

$∴$ $A \cap B = ϕ$

$B \cap A' = B - A = B$ $[∵ B & A are disjoint sets]$

Q 28 :

Let $X = {n \in N : 1 \leq n \leq 50}$ . 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 ________.

(29)

Given, $A = {n \in X : n is a multiple of 2} = {2, 4, 6, \dots, 50}$

$B = {n \in X : n is a multiple of 7} = {7, 14, 21, 28, 35, 42, 49}$

Smallest subset of $X$ containing both $A$ and $B$ will be $A \cup B$ .

$∴$ $n (A \cup B) = n (A) + n (B) - n (A \cap B) = 25 + 7 - 3 = 29$

Q 29 :

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 ______.

(107)

Q 30 :

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

(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 31 :

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 __________.

(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 32 :

Number of integral values of $x$ , satisfying the equation ${[x]}^{2} - 5 [x] + 6 = 0,$ where $[\cdot]$ denotes the greatest integer function is _______.

(2)

$[x]$ $= \frac{5 \pm \sqrt{25 - 24}}{2 \cdot 1} = \frac{5 \pm \sqrt{1}}{2} = 3, 2$

Q 33 :

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

(5)

Let $X$ be the set of students who like to play cricket and $Y$ be the set of students who like to play football. Then, $X \cup Y$ is the set of students who like to play at least one game, and $X \cap Y$ is the set of students who like to play both games.

Given, $n (X) = 24,$ $n (Y) = 16,$ $n (X \cup Y) = 35$

Using $n (X \cup Y) = n (X) + n (Y) - n (X \cap Y),$

we get $35 = 24 + 16 - n (X \cap Y)$

Thus, $n (X \cap Y) = 5$

Q 34 :

If set A = {1, 3, 5}, then number of elements in $P {P (A)}$ is $2^{l}$ , where $l$ is ______.

(8)

$Given, A = {1, 3, 5}$

$∴$ $n {P (A)} = 2^{3} = 8$

$∴$ $n [P {P (A)}] = 2^{8}$

Q 35 :

Let $A = {n \in N : H . C . F . (n, 45) = 1}$ and Let $B = {2 k : k \in {1, 2, \dots, 100}}$ . Then the sum of all the elements of $A \cap B$ is ________.

(5264)

$We have A = {n \in N : H . C . F . (n, 45) = 1}$

$B = {2 k : k \in {1, 2, \dots, 100}}$

$B = {2, 4, 6, 8, 10, 12, 14, 16, 18, \dots, 200}$

Let the subset of $B$ , elements of which are divisible by 3.

$X = {6, 12, 18, \dots, 198} \Rightarrow n (X) = 33$

Let the subset of $B$ , elements of which are divisible by 5.

$Y = {10, 20, 30, \dots, 200} \Rightarrow n (Y) = 20$

Let the subset of $B$ , elements of which are divisible by both 3 and 5.

$X \cap Y = {30, 60, \dots, 180}, n (X \cap Y) = 6$

$Required sum = \sum B - [\sum X + \sum Y - \sum (X \cap Y)]$

$= 10100 - (3366 + 2100 - 630) = 5264$

Q 36 :

If $A$ is the void set $ϕ$ , then $P (A)$ has just one element $ϕ$ , i.e., $P (ϕ) = {ϕ}$ . So number of elements of $P [P (P (ϕ))]$ is _________.

(4)

$We have, P (ϕ) = {ϕ}$

$∴$ $P (P (ϕ)) = {ϕ, {ϕ}} \Rightarrow P [P (P (ϕ))] = {ϕ, {ϕ}, {{ϕ}}, {ϕ, {ϕ}}}$

Q 37 :

Given $n (U) = 20, n (A) = 12, n (B) = 9, n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to ______.

(3)

$n [{(A \cup B)}^{c}] = n (U) - n (A \cup B)$

$= 20 - (n (A) + n (B) - n (A \cap B)) = 20 - (12 + 9 - 4) = 20 - 17 = 3$

Q 38 :

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]

4
5
6
7

1

2

3

4

(2)

$A = \underset{101 terms}{\underset{⏟}{{1, 6, 11, 16, . . ., 501}}}$

$\Rightarrow T_{n} = 5 n - 4, where 1 \leq n \leq 101$

$B = \underset{71 terms}{\underset{⏟}{{9, 16, 23, . . ., 499}}}$

$\Rightarrow T_{m} = 7 m + 2, where 1 \leq m \leq 71$

For common difference of

$A \cap B = LCM {5, 7} = 35$

$A \cap B = {16, 51, . . .}$

$\Rightarrow T_{K} = 35 K - 19$

$\Rightarrow 16 \leq 35 K - 19 \leq 499$

$\Rightarrow 35 \leq 35 K \leq 518$

$\Rightarrow 1 \leq K \leq 14.8$

$∴$ K has 14 values

Now, terms divided by 3 in $A \cap B$ are {51, 156, 261, 366, 471} [ $∴$ LCM of 35 and 3 is 105]

Hence, the required number of terms is 5.

Q 39 :

In a survey, it was found out that 100 people don’t use any of laptop, mobile or wrist watches. 80 persons use all the three gadgets. There are 150 who use laptop and mobile, 200 who use mobile and wristwatch and 200 who use laptop and wristwatch. The number of people who use only laptop, only mobile and only wristwatch is equal. If this survey was conducted on 1000 persons, how many people use only wristwatch?

200
170
150
900

1

2

3

4

(2)

$3 x + 120 + 120 + 70 + 80 = 900$

$3 x = 510,$

$x = 170$

Q 40 :

Number of correct statements among the following is $k$ . then $2 k$ is _______

Statement I: Let $A = {1, 2, 3, 4, 5, 6, 7} .$ Define $B = {T \subseteq A : either 1 \notin T or 2 \in T}$ and $C = {T \subseteq A :$ the sum of all the elements of $' T'$ is a prime number $}$ . Then the number of elements in the set $B \cup C$ is 107.

Statement II: If sum of $n$ terms of any sequence is a quadratic in $' n'$ , then the sequence is an AP.

Statement III: Let $x = {(8 \sqrt{3} + 13)}^{13}$ and $y = {(7 \sqrt{2} + 9)}^{9} .$ If $[t]$ denotes the greatest integer $\leq t$ , then $[x] + [y]$ is even.

Statement IV: The expression $(x + {{(x^{3} - 1)}^{\frac{1}{2}})}^{5} + (x - {{(x^{3} - 1)}^{\frac{1}{2}})}^{5}$ is a polynomial of degree 7.

(6)

Topic Question Set

Q 11 :

In a survey it is to be found that 70% of employees like bananas and 64% like apples. If $x$ % like both bananas and apples, then

Q 12 :

If A and B are not disjoint sets, then $n (A \cup B)$ is equal to

Q 13 :

Given $n (U) = 20$ , $n (A) = 12$ , $n (B) = 9$ , $n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to

Q 14 :

If $N_{a} = {a n : n \in N}$ , then $N_{5} \cap N_{7}$ = (Here $N$ is the set of natural numbers)

Q 15 :

$A$ & $B$ are subsets of universal set $U$ such that $n (U) = 800$ , $n (A) = 300$ , $n (B) = 400$ & $n (A \cap B) = 100$ . The number of elements in the set $A^{c} \cap B^{c}$ is

Q 16 :

In a certain town 25% families own a cell phone, 15% families own a scooter and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is

Q 17 :

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

Q 18 :

For any two sets $A$ and $B$ , $A - (A - B)$ equals

Q 19 :

Given $A = {x ∣ x is a root of x^{2} - 1 = 0}$ , $B = {x ∣ x is a root of x^{2} - 2 x + 1 = 0}$ . Then

Q 20 :

A set contains $n$ elements. The power set contains

Q 21 :

In a group of 80 students, 50 play football, 45 play cricket and each student plays either football or cricket. Find the number of students who play both the games.

Q 22 :

There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

Q 24 :

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where $N$ is the set of natural numbers, then $X \cup Y$ is equal to

Q 25 :

In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is

Q 26 :

If $A = {5^{n} - 4 n - 1 : n \in N}$ and $B = {16 (n - 1) : n \in N}$ , then

Q 27 :

If $A$ and $B$ are disjoint sets, then $B \cap A'$ , where $A'$ is complement of $A$ is equal to

Q 28 :

Let $X = {n \in N : 1 \leq n \leq 50}$ . 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 ________.

Q 29 :

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 ______.

Q 30 :

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

Q 31 :

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 __________.

Q 32 :

Number of integral values of $x$ , satisfying the equation ${[x]}^{2} - 5 [x] + 6 = 0,$ where $[\cdot]$ denotes the greatest integer function is _______.

Q 33 :

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

Q 34 :

If set A = {1, 3, 5}, then number of elements in $P {P (A)}$ is $2^{l}$ , where $l$ is ______.

Q 35 :

Let $A = {n \in N : H . C . F . (n, 45) = 1}$ and Let $B = {2 k : k \in {1, 2, \dots, 100}}$ . Then the sum of all the elements of $A \cap B$ is ________.

Q 36 :

If $A$ is the void set $ϕ$ , then $P (A)$ has just one element $ϕ$ , i.e., $P (ϕ) = {ϕ}$ . So number of elements of $P [P (P (ϕ))]$ is _________.

Q 37 :

Given $n (U) = 20, n (A) = 12, n (B) = 9, n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to ______.

Q 38 :

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]

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Topic Question Set

Q 11 : In a survey it is to be found that 70% of employees like bananas and 64% like apples. If x% like both bananas and apples, then

Q 12 : If A and B are not disjoint sets, then n(A∪B) is equal to

Q 13 : Given n(U)=20, n(A)=12, n(B)=9, n(A∩B)=4, where U is the universal set, A and B are subsets of U, then n[(A∪B)c] equals to

Q 14 : If Na={an:n∈N}, then N5∩N7 = (Here N is the set of natural numbers)

Q 15 : A & B are subsets of universal set U such that n(U)=800, n(A)=300, n(B)=400 & n(A∩B)=100. The number of elements in the set Ac∩Bc is

Q 16 : In a certain town 25% families own a cell phone, 15% families own a scooter and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is

Q 17 : If A, B and C are three sets such that A∩B=A∩C and A∪B=A∪C, then

Q 18 : For any two sets A and B, A−(A−B) equals

Q 19 : Given A={x∣x is a root of x2-1=0}, B={x∣x is a root of x2-2x+1=0}. Then

Q 20 : A set contains n elements. The power set contains

Q 21 : In a group of 80 students, 50 play football, 45 play cricket and each student plays either football or cricket. Find the number of students who play both the games.

Q 22 : There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

Q 23 : Suppose A1,A2,…,A30 are thirty sets each with five elements and B1,B2,…,Bn are 'n' sets each with three elements. Let ∪i=130Ai=∪j=1nBj=S Assume that each element of S belongs to exactly ten of Ai's and exactly 9 of Bj's, then the value of n is

Q 24 : If X={4n-3n-1: n∈N} and Y={9(n-1):n∈N}, where N is the set of natural numbers, then X∪Y is equal to

Q 25 : In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is

Q 26 : If A={5n-4n-1:n∈N} and B={16(n-1):n∈N}, then

Q 27 : If A and B are disjoint sets, then B∩A′, where A′ is complement of A is equal to

Q 28 : Let X={n∈N:1≤n≤50}. If A={n∈X:n is a multiple of 2} and B={n∈X:n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is ________.

Q 29 : Let A={1,2,3,4,5,6,7}. Define B={T⊆A: either 1∉T or 2∈T} and C={T⊆A:T the sum of all the elements of T is a prime number}. Then the number of elements in the set B∪C is ______.

Q 30 : The number of elements in the set {n∈ℤ:|n2-10n+19|<6} is ________.

Q 31 : The number of elements in the set {n∈ℕ:10≤n≤100 and 3n-3 is a multiple of 7} is __________.

Q 32 : Number of integral values of x, satisfying the equation [x]2-5[x]+6=0, where [·] denotes the greatest integer function is _______.

Q 33 : In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

Q 34 : If set A = {1, 3, 5}, then number of elements in P{P(A)} is 2l, where l is ______.

Q 35 : Let A={n∈N:H.C.F.(n,45)=1} and Let B={2k:k∈{1,2,…,100}}. Then the sum of all the elements of A∩B is ________.

Q 36 : If A is the void set ϕ, then P(A) has just one element ϕ, i.e., P(ϕ)={ϕ}. So number of elements of P[P(P(ϕ))] is _________.

Q 37 : Given n(U)=20, n(A)=12, n(B)=9, n(A∩B)=4, where U is the universal set, A and B are subsets of U, then n[(A∪B)c] equals to ______.

Q 38 : Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A∩B, which are divisible by 3, is: [2026]

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Q 11 :

In a survey it is to be found that 70% of employees like bananas and 64% like apples. If $x$ % like both bananas and apples, then

Q 12 :

If A and B are not disjoint sets, then $n (A \cup B)$ is equal to

Q 13 :

Given $n (U) = 20$ , $n (A) = 12$ , $n (B) = 9$ , $n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to

Q 14 :

If $N_{a} = {a n : n \in N}$ , then $N_{5} \cap N_{7}$ = (Here $N$ is the set of natural numbers)

Q 15 :

$A$ & $B$ are subsets of universal set $U$ such that $n (U) = 800$ , $n (A) = 300$ , $n (B) = 400$ & $n (A \cap B) = 100$ . The number of elements in the set $A^{c} \cap B^{c}$ is

Q 16 :

In a certain town 25% families own a cell phone, 15% families own a scooter and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is

Q 17 :

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

Q 18 :

For any two sets $A$ and $B$ , $A - (A - B)$ equals

Q 19 :

Given $A = {x ∣ x is a root of x^{2} - 1 = 0}$ , $B = {x ∣ x is a root of x^{2} - 2 x + 1 = 0}$ . Then

Q 20 :

A set contains $n$ elements. The power set contains

Q 21 :

In a group of 80 students, 50 play football, 45 play cricket and each student plays either football or cricket. Find the number of students who play both the games.

Q 22 :

There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of three subjects. Only one student passed in all the subjects. Then the number of students failing in all the three subjects

Q 24 :

If $X = {4^{n} - 3 n - 1 : n \in N}$ and $Y = {9 (n - 1) : n \in N}$ , where $N$ is the set of natural numbers, then $X \cup Y$ is equal to

Q 25 :

In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is

Q 26 :

If $A = {5^{n} - 4 n - 1 : n \in N}$ and $B = {16 (n - 1) : n \in N}$ , then

Q 27 :

If $A$ and $B$ are disjoint sets, then $B \cap A'$ , where $A'$ is complement of $A$ is equal to

Q 28 :

Let $X = {n \in N : 1 \leq n \leq 50}$ . 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 ________.

Q 29 :

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 ______.

Q 30 :

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

Q 31 :

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 __________.

Q 32 :

Number of integral values of $x$ , satisfying the equation ${[x]}^{2} - 5 [x] + 6 = 0,$ where $[\cdot]$ denotes the greatest integer function is _______.

Q 33 :

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?

Q 34 :

If set A = {1, 3, 5}, then number of elements in $P {P (A)}$ is $2^{l}$ , where $l$ is ______.

Q 35 :

Let $A = {n \in N : H . C . F . (n, 45) = 1}$ and Let $B = {2 k : k \in {1, 2, \dots, 100}}$ . Then the sum of all the elements of $A \cap B$ is ________.

Q 36 :

If $A$ is the void set $ϕ$ , then $P (A)$ has just one element $ϕ$ , i.e., $P (ϕ) = {ϕ}$ . So number of elements of $P [P (P (ϕ))]$ is _________.

Q 37 :

Given $n (U) = 20, n (A) = 12, n (B) = 9, n (A \cap B) = 4$ , where $U$ is the universal set, $A$ and $B$ are subsets of $U$ , then $n [{(A \cup B)}^{c}]$ equals to ______.

Q 38 :

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]