Sets-JEE-Mains-PYQ’s

Mathematics Questions

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

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

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

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

Click to View Answer

Correct Answer: [ ]

Let $A = \{ (\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6 \}$ and $B = \{ (\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144 \}$ . Then

(A) $A \subset B$

(B) $B \subset A$

(D) $A \cup B = \{ (x, y) : -4 \leq x \leq 4, -1 \leq y \leq 11 \}$

Click to View Answer

Correct Answer: [ ]

Let $A = \left\{ x \in (0, \pi) - \left\{\frac{\pi}{2}\right\} : \log_{\frac{2}{\pi}} |\sin x| + \log_{\frac{2}{\pi}} |\cos x| = 2 \right\}$ and $B = \{ x > 0 : \sqrt{x} (\sqrt{x} - 4) - 3\sqrt{x} - 2 + 6 = 0 \}$ . Then $n(A \cup B)$ is equal to:

(A) 4

(B) 8

(D) 2

Click to View Answer

Correct Answer: [ ]

Let $A = \{ (x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3 \}$ and $B = \{ (x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3 \}$ . If $C = \{ (x, y) \in A \cap B : x = 0 \text{ or } y = 0 \}$ , then $\sum_{(x,y) \in C} |x + y|$ is:

(A) 18

(B) 24

(D) 12

Click to View Answer

Correct Answer: [ ]

Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}$ . Then $n(B)$ is equal to:

(A) 29

(B) 31

(D) 36

Click to View Answer

Correct Answer: [ ]

Let $A=\{n \in[100,700] \cap \mathbb{N} : n \text{ is neither a multiple of 3 nor a multiple of 4}\}$ . Then the number of elements in $A$ is

(A) 300

(B) 310

(D) 280

Click to View Answer

Correct Answer: [ ]

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

(D) 6

Click to View Answer

Correct Answer: [ ]

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

(D) 9

Click to View Answer

Correct Answer: [ ]

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 $K$ cannot belong to the set:

(A) $\{80, 83, 86, 89\}$

(B) $\{84, 86, 88, 90\}$

(D) $\{84, 87, 90, 93\}$

Click to View Answer

Correct Answer: [ ]

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

(D) $P$ and $Q$

Click to View Answer

Correct Answer: [ ]

The number of elements in the set $\{x \in \mathbb{R} : (|x|-3)|x+4|=6\}$ is equal to :

(A) 4

(B) 2

(D) 1

Click to View Answer

Correct Answer: [ ]

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

(D) 38

Click to View Answer

Correct Answer: [ ]

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

(D) 45

Click to View Answer

Correct Answer: [ ]

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

(D) 55

Click to View Answer

Correct Answer: [ ]

Consider the two sets : $A = \{m \in \mathbb{R} : \text{both the roots of } x^{2}-(m+1)x+m+4=0 \text{ are real}\}$ and $B = [-3, 5)$ . Which of the following is not true?

(A) $A \cap B = \{-3\}$

(B) $B - A = (-3, 5)$

(D) $A - B = (-\infty, -3) \cup [5, \infty)$

Click to View Answer

Correct Answer: [ ]

If $A = \{x \in \mathbb{R} : |x| < 2\}$ and $B = \{x \in \mathbb{R} : |x-2| \geq 3\}$ ; then:

(A) $A - B = [-1, 2)$

(B) $A \cup B = \mathbb{R} - (2, 5)$

(D) $B - A = \mathbb{R} - (-2, 5)$

Click to View Answer

Correct Answer: [ ]

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) $B \cap C \neq \phi$

(D) If $(A-C) \subseteq B$ , then $A \subseteq B$

Click to View Answer

Correct Answer: [ ]

Two newspapers $A$ and $B$ are published in a city. It is known that $25\%$ of the city population 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

(D) 13.9

Click to View Answer

Correct Answer: [ ]

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) $2^{18}$

(D) $2^{15}$

Click to View Answer

Correct Answer: [ ]

Let $S=\{1,2,3, \ldots, 100\}$ . 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) $2^{50}(2^{50}-1)$

(D) $2^{50}+1$

Click to View Answer

Correct Answer: [ ]

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

(D) 38

Click to View Answer

Correct Answer: [ ]

Two sets $A$ and $B$ are as under:
$A=\{(a, b) \in R \times R: |a-5|<1 \text{ and } |b-5|<1\}$ ;
$B=\{(a, b) \in R \times R: 4(a-6)^{2}+9(b-5)^{2} \leq 36\}$ ;
Then:

(A) neither $A \subset B$ nor $B \subset A$

(B) $B \subset A$

(D) $A \cap B = \phi$ (an empty set)

Click to View Answer

Correct Answer: [ ]

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) $Q \not\subset P$

(D) $P = Q$

Click to View Answer

Correct Answer: [ ]

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 at least three elements are:

(A) 219

(B) 256

(D) 510

Click to View Answer

Correct Answer: [ ]

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: