UPSC NDA I 2023 Mathematics Solved Paper

Concept:The angle between two lines is given by the cosine formula using their direction ratios.Explanation:The direction ratios of the two lines are $(2, -1, 2)$ and $(k, 3, 5)$.The angle between them is $\frac{\pi}{4}$.Using $\cos\theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$, we get:$\cos\frac{\pi}{4} = \frac{2k + (-1)(3) + 2\cdot5}{\sqrt{2^2+(-1)^2+2^2}\,\sqrt{k^2+3^2+5^2}}$.Simplify: $\frac{1}{\sqrt{2}} = \frac{2k+7}{3\sqrt{k^2+34}}$.Square both sides and cross-multiply: $9(k^2+34) = 2(4k^2+28k+49)$.Expand: $9k^2 + 306 = 8k^2 + 56k + 98$.Rearrange: $k^2 - 56k + 208 = 0$.Factorise: $(k-4)(k-52)=0$.Thus $k=4$ or $k=52$.Since only $k=4$ is among the given options, the required value is $4$.Answer:$4$