NDA 1 2025 Maths, question 64

Topic: Analytical geometry

Q64

If a line $\frac{x + 1}{p} = \frac{y - 1}{q} = \frac{z - 2}{r}$ , where $p = 2q = 3r$ , makes an angle $\theta$ with the positive direction of $y$ -axis, then what is $\cos 2\theta$ equal to?

(a) $- 31/49$ Correct answer
(b) $- 37/49$
(c) $31/49$
(d) $37/49$

Solution.

Answer: (a) $- 31/49$

The direction ratios are proportional to $(6,3,2)$ .
Thus $\cos^{2}\theta = 3^{2}/\left( 6^{2} + 3^{2} + 2^{2} \right) = 9/49$ .
Consequently $\cos 2\theta = 2\cos^{2}\theta - 1 = 18/49 - 1 = - 31/49$ .