Mathematics · Sets and Relations

JEE Main 2024 — 27 January, Shift 2 — Question 11

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 $\mathrm{P}(\mathrm{m}, \mathrm{n})$ from the point $\mathrm{Q}(-2,-3)$ is

Option A:
10
Correct
Option B:
6
Option C:
4
Option D:
8

Answer: A

Step-by-step solution

$2^{\mathrm{m}}-2^{\mathrm{n}}=56$

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

$2^{\mathrm{n}}=2^{3}$ and $2^{\mathrm{m}-\mathrm{n}}-1=7$

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

$\Rightarrow \mathrm{n}=3$ and $\mathrm{m}-\mathrm{n}=3$

$\Rightarrow \mathrm{n}=3$ and $\mathrm{m}=6$

$\mathrm{P}(6,3)$ and $\mathrm{Q}(-2,-3)$

$\mathrm{PQ}=\sqrt{8^{2}+6^{2}}=\sqrt{100}=10$

Hence option (1) is correct

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 27 January, Shift 2
Subject: Mathematics
Chapter: Sets and Relations
Topic: Sets