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 $P(m, n)$ from the point $Q(-2,-3)$ is
(1) 10
(2) 6
(3) 4
(4) 8
Show Answer
Answer (1)
Solution
$$ \begin{aligned} & 2^{m}-2^{n}=56 \\ & 2^{n}\left(2^{m-n}-1\right)=2^{3} \times 7 \end{aligned} $$
$2^{n}=2^{3}$ and $2^{m-n}-1=7$
$\Rightarrow n=3$ and $2^{m-n}=8$
$\Rightarrow n=3$ and $m-n=3$
$\Rightarrow n=3$ and $m=6$
$P(6,3)$ and $Q(-2,-3)$
$PQ=\sqrt{8^{2}+6^{2}}=\sqrt{100}=10$
Hence option (1) is correct
