NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given p = 35^ , q = 25^ , and r = (-95^ ) = - (90^ + 5^ ) = - 5^ . p + q + r = 35^ + 25^ - 5^ Using A + B = 2 ( A+B 2 ) ( A-B 2 ): p + q + r = 2 30^ 5^ - 5^ = 2 ( 1 2 ) 5^ - 5^ = 0 We know that (p + q + r)^2 = p^2 + q^2 + r^2 + 2(pq + qr + rp) = 0 pq + qr + rp = - 1 2 (p^2 + q^2 + r^2) Now, p^2 + q^2 + r^2 = ^2 35^ + ^2 25^ + ^2 5^ Using ^2 = 1 - 2 2 and ^2 = 1 + 2 2 : p^2 + q^2 + r^2 = 1 - 70^ 2 + 1 - 50^ 2 + 1 + 10^ 2 = 3 2 - 1 2 ( 70^ + 50^ - 10^ ) Using C + D = 2 ( C+D 2 ) ( C-D 2 ): 70^ + 50^ = 2 60^ 10^ = 2 ( 1 2 ) 10^ = 10^ So, p^2 + q^2 + r^2 = 3 2 - 1 2 ( 10^ - 10^ ) = 3 2 Therefore, pq + qr + rp = - 1 2 ( 3 2 ) = - 3 4 Answer: - 3 4