UPSC NDA II 2018 Mathematics Solved Paper

Concept:The cosine rule (or law of cosines) relates the sides and an angle of a triangle.Explanation:For triangle $OPQ$ with points $O(0,0)$, $P(m,n)$, $Q(r,s)$ and angle $\alpha$ at $O$,$OP = \sqrt{m^2 + n^2}$, $OQ = \sqrt{r^2 + s^2}$, $PQ = \sqrt{(m-r)^2 + (n-s)^2}$.Using the cosine rule in $\triangle OPQ$: $\cos\alpha = \frac{OP^2 + OQ^2 - PQ^2}{2 \cdot OP \cdot OQ}$.Substituting the squares: $OP^2 = m^2+n^2$, $OQ^2 = r^2+s^2$, $PQ^2 = m^2+r^2-2mr + n^2+s^2-2ns$.Then numerator simplifies to $(m^2+n^2)+(r^2+s^2)-(m^2+r^2-2mr+n^2+s^2-2ns) = 2mr+2ns$.Thus $\cos\alpha = \frac{2(mr+ns)}{2\sqrt{(m^2+n^2)(r^2+s^2)}} = \frac{mr+ns}{\sqrt{(m^2+n^2)(r^2+s^2)}}$.Statement I gives $\cos\alpha = \frac{ms-nr}{\sqrt{(m^2+n^2)(r^2+s^2)}}$, which is different; hence Statement I is false.Statement II is the standard cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$, which is true for any triangle $ABC$.Therefore, Statement I is false, Statement II is true.Answer:Option D: Statement I is false, but Statement II is true.