NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given p = A + B and q = A + B. Squaring both equations: p^2 = ^2 A + ^2 B + 2 A B q^2 = ^2 A + ^2 B + 2 A B Adding the two equations: p^2 + q^2 = ( ^2 A + ^2 A) + ( ^2 B + ^2 B) + 2( A B + A B) p^2 + q^2 = 1 + 1 + 2 (A - B) = 2(1 + (A - B)) Subtracting q^2 from p^2: p^2 - q^2 = ( ^2 A - ^2 A) + ( ^2 B - ^2 B) - 2( A B - A B) p^2 - q^2 = - 2A - 2B - 2 (A + B) Using the sum-to-product formula 2A + 2B = 2 (A + B) (A - B): p^2 - q^2 = -2 (A + B) (A - B) - 2 (A + B) p^2 - q^2 = -2 (A + B)(1 + (A - B)) Dividing p^2 - q^2 by p^2 + q^2: p^2 - q^2 p^2 + q^2 = -2 (A + B)(1 + (A - B)) 2(1 + (A - B)) p^2 - q^2 p^2 + q^2 = - (A + B) Using the property ( - ) = - : - (A + B) = ( - (A + B)) = ( - A - B) Answer: ( - A - B)