Straight Lines Question 4
Question 4 - 2024 (27 Jan Shift 2)
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
(1) 10
(2) 6
(3) 4
(4) 8
Show Answer
Answer (1)
Solution
$$ \begin{aligned} & 2^{m}-2^{n}=56 \ & 2^{n}\left(2^{m-n}-1\right)=2^{3} \times 7 \end{aligned} $$
$2^{\mathrm{n}}=2^{3}$ 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
