NDA Mathematics Question (2025) — Solution

Step-by-step solution

The direction ratios of the line are proportional to p, q, r. Given p = 2q = 3r. Let p = 2q = 3r = 6k. Then p = 6k, q = 3k, r = 2k. The direction ratios of the line are proportional to 6, 3, 2. The direction cosines of the line are 6 6^2+3^2+2^2 , 3 6^2+3^2+2^2 , 2 6^2+3^2+2^2 , which simplifies to 6 7 , 3 7 , 2 7 . The line makes an angle with the positive direction of the y-axis, so = 3 7 . Using the double angle formula, 2 = 2 ^2 - 1. 2 = 2 ( 3 7 )^2 - 1 = 18 49 - 1 = - 31 49 . Answer: - 31 49