NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given p = 35^ , q = 25^ , and r = (-95^ ). Simplifying r: r = - (90^ + 5^ ) = - 5^ We need to find the value of p^2 + q^2 + r^2: p^2 + q^2 + r^2 = ^2 35^ + ^2 25^ + ^2 5^ Using the identities ^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 the sum-to-product formula C + D = 2 ( C+D 2 ) ( C-D 2 ): 70^ + 50^ = 2 ( 70^ + 50^ 2 ) ( 70^ - 50^ 2 ) = 2 60^ 10^ = 2 ( 1 2 ) 10^ = 10^ Substituting this back into the expression: p^2 + q^2 + r^2 = 3 2 - 1 2 ( 10^ - 10^ ) = 3 2 - 0 = 3 2 Answer: 3 2