NDA Mathematics Question (2026) — Solution

Step-by-step solution

Given f(x) = 2 ^2 x - 1 Using the double angle formula, we can write f(x) = 2x To find the intervals where the function is decreasing, we find its derivative and set it to be less than or equal to zero: f'(x) = -2 2x For f(x) to be decreasing, f'(x) 0 -2 2x 0 2x 0 The sine function is positive in the first and second quadrants, so: 2n 2x 2n + Dividing by 2, we get: n x n + 2 The length of the longest interval in which the function is decreasing is the difference between the upper and lower bounds of this interval: Length = (n + 2 ) - n = 2 Answer: 2