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 :
Solution
<p>$$\begin{aligned}
& 2^{\mathrm{m}}-2^{\mathrm{n}}=56 \\
& 2^{\mathrm{n}}\left(2^{\mathrm{m}-\mathrm{n}}-1\right)=2^3 \times 7 \\
& 2^{\mathrm{n}}=2^3 \text { and } 2^{\mathrm{m}-\mathrm{n}}-1=7 \\
& \Rightarrow \mathrm{n}=3 \text { and } 2^{\mathrm{m}-\mathrm{n}}=8 \\
& \Rightarrow \mathrm{n}=3 \text { and } \mathrm{m}-\mathrm{n}=3 \\
& \Rightarrow \mathrm{n}=3 \text { and } \mathrm{m}=6 \\
& \mathrm{P}(6,3) \text { and } \mathrm{Q}(-2,-3) \\
& \mathrm{PQ}=\sqrt{8^2+6^2}=\sqrt{100}=10
\end{aligned}$$</p>
<p>Hence option (1) is correct</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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