NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given y = x ^ -1 x 1 - x^2 + 1 - x^2 We can rewrite the function as: y = x ^ -1 x 1 - x^2 + 1 2 (1 - x^2) Differentiating with respect to x using the quotient and chain rules: dy dx = ( ^ -1 x + x 1 - x^2 ) 1 - x^2 - x ^ -1 x ( -x 1 - x^2 ) 1 - x^2 + 1 2 ( -2x 1 - x^2 ) dy dx = 1 - x^2 ^ -1 x + x + x^2 ^ -1 x 1 - x^2 1 - x^2 - x 1 - x^2 dy dx = (1 - x^2) ^ -1 x + x 1 - x^2 + x^2 ^ -1 x (1 - x^2)^ 3/2 - x 1 - x^2 dy dx = ^ -1 x + x 1 - x^2 (1 - x^2)^ 3/2 - x 1 - x^2 dy dx = ^ -1 x (1 - x^2)^ 3/2 + x 1 - x^2 - x 1 - x^2 = ^ -1 x (1 - x^2)^ 3/2 Differentiating again with respect to x: d^2y dx^2 = 1 1 - x^2 (1 - x^2)^ 3/2 - ^ -1 x 3 2 (1 - x^2)^ 1/2 (-2x) (1 - x^2)^3 d^2y dx^2 = (1 - x^2) + 3x 1 - x^2 ^ -1 x (1 - x^2)^3 d^2y dx^2 = 1 (1 - x^2)^2 + 3x ^ -1 x (1 - x^2)^ 5/2 Substituting x = 0 into the second derivative: . d^2y dx^2 |_ x=0 = 1 (1 - 0)^2 + 0 = 1 Answer: 1