UPSC NDA I 2015 Mathematics Paper

Concept:The direction cosines $l, m, n$ of a line are calculated from its direction ratios $(a, b, c)$ using the formula:$l = \frac{\pm a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{\pm b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{\pm c}{\sqrt{a^2 + b^2 + c^2}}$Explanation:Given direction ratios: $a = 12$, $b = 4$, $c = 3$.First, compute the magnitude:$\sqrt{a^2 + b^2 + c^2} = \sqrt{12^2 + 4^2 + 3^2} = \sqrt{144 + 16 + 9} = \sqrt{169} = 13$Now substitute into the formula for direction cosines (taking the positive sign, as the question likely expects the principal direction):$l = \frac{12}{13}, \quad m = \frac{4}{13}, \quad n = \frac{3}{13}$Thus, the direction cosines are $(12/13, 4/13, 3/13)$.Answer:$\boxed{\left(\frac{12}{13}, \frac{4}{13}, \frac{3}{13}\right)}$ which corresponds to option A.