JEE Mathematics Sets, Relations, and Functions Q #1009
KNOWLEDGE BASED
APPLY
4 Marks 2025 JEE Main 2025 (Online) 24th January Evening Shift MCQ SINGLE
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
Prev Next
Correct Answer: B
Explanation
For set A: $\log_{\frac{2}{\pi}} |\sin x| + \log_{\frac{2}{\pi}} |\cos x| = 2$ $\Rightarrow \log_{\frac{2}{\pi}} |\sin x \cdot \cos x| = 2$ $\Rightarrow |\sin x \cos x| = (\frac{2}{\pi})^2$ $\Rightarrow |\frac{1}{2} \sin 2x| = \frac{4}{\pi^2}$ $\Rightarrow |\sin 2x| = \frac{8}{\pi^2}$ Since $0 < \frac{8}{\pi^2} < 1$, there are 4 solutions for $x \in (0, \pi)$. For set B: Case 1: $x < 2$, $x(x-4) + 3(x-2) + 6 = 0$ $\Rightarrow x^2 - 4x + 3x - 6 + 6 = 0$ $\Rightarrow x^2 - x = 0$ $\Rightarrow x(x-1) = 0$, so $x = 0, 1$. Case 2: $x > 2$, $x(x-4) - 3(x-2) + 6 = 0$ $\Rightarrow x^2 - 4x - 3x + 6 + 6 = 0$ $\Rightarrow x^2 - 7x + 12 = 0$ $\Rightarrow (x-3)(x-4) = 0$, so $x = 3, 4$. Thus, $B = {0, 1, 3, 4}$, so $n(B) = 4$. Since $A$ and $B$ are disjoint, $n(A \cup B) = n(A) + n(B) = 4 + 4 = 8$.

More from this Chapter

MCQ_SINGLE
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:
NUMERICAL
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 _____________.
NUMERICAL
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 _________.
MCQ_SINGLE
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 :
NUMERICAL
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 ___________.
View All Questions