JEE MAIN - Mathematics (2024 - 27th January Evening Shift - No. 6)
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 :
8
10
4
6
Explanation
$$\begin{aligned} & 2^{\mathrm{m}}-2^{\mathrm{n}}=56 \\ & 2^{\mathrm{n}}\left(2^{\mathrm{m}-\mathrm{n}}-1\right)=2^3 \times 7 \\ & 2^{\mathrm{n}}=2^3 \text { and } 2^{\mathrm{m}-\mathrm{n}}-1=7 \\ & \Rightarrow \mathrm{n}=3 \text { and } 2^{\mathrm{m}-\mathrm{n}}=8 \\ & \Rightarrow \mathrm{n}=3 \text { and } \mathrm{m}-\mathrm{n}=3 \\ & \Rightarrow \mathrm{n}=3 \text { and } \mathrm{m}=6 \\ & \mathrm{P}(6,3) \text { and } \mathrm{Q}(-2,-3) \\ & \mathrm{PQ}=\sqrt{8^2+6^2}=\sqrt{100}=10 \end{aligned}$$
Hence option (1) is correct
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