UPSC NDA 2 2016 Mathematics Solved Paper

Concept:Direction ratios of a line joining two points are obtained by subtracting coordinates, and direction cosines satisfy $l^2+m^2+n^2=1$.Explanation:Step 1: For points $P(-2,1,-5)$ and $Q(4,-3,-1)$, compute direction ratios as $4-(-2)=6$, $-3-1=-4$, $-1-(-5)=4$.Simplify by dividing by 2 to get $<3,-2,2>$. So statement 1 is correct.Step 2: Direction cosines are $\frac{3}{\sqrt{3^2+(-2)^2+2^2}}$, $\frac{-2}{\sqrt{17}}$, $\frac{2}{\sqrt{17}}$ which are $\frac{3}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{2}{\sqrt{17}}$.Sum of squares: $\left(\frac{3}{\sqrt{17}}\right)^2+\left(\frac{-2}{\sqrt{17}}\right)^2+\left(\frac{2}{\sqrt{17}}\right)^2 = \frac{9}{17}+\frac{4}{17}+\frac{4}{17}=1$. So statement 2 is correct.Thus both statements are true.Answer:Both statements 1 and 2 are correct.