UPSC NDA II 2015 Mathematics Paper

Concept:Direction ratios of a line are numbers proportional to its direction cosines.The angle $\theta$ between two lines with direction ratios $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ satisfies$\displaystyle \cos\theta = \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$.Explanation:Rewrite the first line $2x = 3y = -z$ in symmetric form.Divide each term by $6$: $\frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$.So its direction ratios are $(a_1,b_1,c_1) = (3,2,-6)$.Rewrite the second line $6x = -y = -4z$ similarly.Divide by $12$: $\frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$.Hence its direction ratios are $(a_2,b_2,c_2) = (2,-12,-3)$.Now compute the dot product:$a_1a_2+b_1b_2+c_1c_2 = 3\cdot2 + 2\cdot(-12) + (-6)\cdot(-3) = 6 - 24 + 18 = 0$.Since the numerator is zero, $\cos\theta = 0$.Therefore $\theta = 90^\circ$.Thus the two lines are perpendicular.Answer:Option A: are perpendicular.