NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given = -1 + -3 2 , which is the complex cube root of unity, . We know that ^3 = 1 and 1 + + ^2 = 0. Substituting = into the given expression, we simplify the powers of : ^ 19 = ^ 19 = ( ^3)^6 = ^ 35 = ^ 35 = ( ^3)^ 11 ^2 = ^2 ^ 25 = ^ 25 = ( ^3)^8 = ^ 38 = ^ 38 = ( ^3)^ 12 ^2 = ^2 The first term becomes: (1 + ^ 19 - ^ 35 )^ 100 = (1 + - ^2)^ 100 Since 1 + = - ^2, this reduces to: (- ^2 - ^2)^ 100 = (-2 ^2)^ 100 = 2^ 100 ^ 200 Since 200 = 3 66 + 2, ^ 200 = ^2. Thus, the first term is 2^ 100 ^2. The second term becomes: (1 - 3 ^ 25 + ^ 38 )^ 50 = (1 - 3 + ^2)^ 50 Since 1 + ^2 = - , this reduces to: (- - 3 )^ 50 = (-4 )^ 50 = 4^ 50 ^ 50 = (2^2)^ 50 ^ 50 = 2^ 100 ^ 50 Since 50 = 3 16 + 2, ^ 50 = ^2. Thus, the second term is 2^ 100 ^2. Subtracting the second term from the first term: 2^ 100 ^2 - 2^ 100 ^2 = 0 Answer: 0