UPSC CDS I 2024 Math Paper

Concept:In a rhombus, diagonals are perpendicular bisectors. Using Pythagoras, if one diagonal or its relation to side is known, the other diagonal can be found, giving the ratio.Explanation:Let the side of the rhombus be $a$.Statement I: One diagonal equals the side, say $d_1 = a$.Half of $d_1$ is $\frac{a}{2}$. By Pythagoras, half of the other diagonal $d_2$ is $\sqrt{a^2 - \left(\frac{a}{2}\right)^2} = \frac{a\sqrt{3}}{2}$.Thus $d_2 = a\sqrt{3}$. The ratio $\frac{d_2}{d_1} = \sqrt{3}$ (or $\frac{d_1}{d_2} = \frac{1}{\sqrt{3}}$). So Statement I alone gives the ratio.Statement II: The longer diagonal equals $\sqrt{3}$ times the side, so $d_2 = a\sqrt{3}$.Half of $d_2$ is $\frac{a\sqrt{3}}{2}$. Half of the other diagonal $d_1$ is $\sqrt{a^2 - \left(\frac{a\sqrt{3}}{2}\right)^2} = \frac{a}{2}$.Thus $d_1 = a$. The ratio is the same: $\frac{d_2}{d_1} = \sqrt{3}$. So Statement II alone also gives the ratio.Each statement alone is sufficient to answer the question.Answer:Option B (If the Question can be answered by either Statement alone).