Straight Lines
Subsets Condition — Distance Between Points
nta_pyq_2024_jan
Grade 11
Question:
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
Step-by-Step Solution
Key Concept: $2^m-2^n=56=2^3\times7$. $2^n(2^{m-n}-1)=2^3\times7\Rightarrow n=3$, $2^{m-n}-1=7\Rightarrow m-n=3\Rightarrow m=6$. $P=(6,3)$, $Q=(-2,-3)$.
$m=6,n=3$. $PQ=\sqrt{64+36}=10$.
Correct Answer: 1