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

Maths sir · 3 steps

Write 2ᵐ - 2ⁿ = 56 and pull out 2ⁿ; the odd factor left over tells you m - n.

1. Set up. A set with k elements has 2ᵏ subsets. So 2ᵐ - 2ⁿ = 56, which factors as 2ⁿ(2ᵐ⁻ⁿ - 1) = 56.
2. Match the factors. 2ᵐ⁻ⁿ - 1 is odd and 2ⁿ is a power of 2. Since 56 = 8 × 7, the only match is 2ⁿ = 8 and 2ᵐ⁻ⁿ - 1 = 7. This gives n = 3 and m - n = 3, so m = 6.

   PQ = √((6-(-2))² + (3-(-3))²) = √(64 + 36) = √100

3. So. P(6, 3) is at distance 10 from Q(-2, -3), which is option (b).

The idea to keep. For 2ᵐ - 2ⁿ = N, factor out 2ⁿ. Split N into a power of 2 times an odd number; that odd number must be 2ᵐ⁻ⁿ - 1.

Answer from the NTA final answer key. Solution by PrepWiser.