NDA Mathematics Question (2026) — Solution

Step-by-step solution

Given g(x) - f(x) = f(4-x) g(x) = f(x) + f(4-x) Replacing x with 4-x, we get: g(4-x) = f(4-x) + f(x) = g(x) Let I = _0^4 f(4-x) g(4-x) \,dx Using the property _0^a h(x)\,dx = _0^a h(a-x)\,dx, we have: I = _0^4 f(x) g(x) \,dx Adding the two expressions for I: 2I = _0^4 f(4-x) g(4-x) \,dx + _0^4 f(x) g(x) \,dx Since g(4-x) = g(x), this becomes: 2I = _0^4 f(4-x) + f(x) g(x) \,dx Substituting f(x) + f(4-x) = g(x): 2I = _0^4 g(x) g(x) \,dx = _0^4 1\,dx 2I = [x]_0^4 = 4 I = 2 Answer: 2