UPSC CDS II 2025 Mathematics Paper

Concept:In a rhombus, diagonals are perpendicular bisectors. The side length is given by $\sqrt{(\frac{d_1}{2})^2 + (\frac{d_2}{2})^2}$, where $d_1$ and $d_2$ are the diagonals.Explanation:The diagonals are in ratio 5 : 12. Let them be $5k$ and $12k$.Using Statement I: Sum of diagonals = $34$ cm. So $5k + 12k = 34$ ⇒ $17k = 34$ ⇒ $k = 2$. Thus $d_1 = 10$ cm, $d_2 = 24$ cm. Side = $\sqrt{(10/2)^2 + (24/2)^2} = \sqrt{5^2 + 12^2} = \sqrt{169} = 13$ cm. Neither diagonal equals 13 cm. Hence the question is answered (answer "No").Using Statement II: Side length is given as 13 cm. Using the same side formula: $13 = \sqrt{(\frac{5k}{2})^2 + (\frac{12k}{2})^2} = \sqrt{(\frac{25k^2}{4} + \frac{144k^2}{4})} = \sqrt{\frac{169k^2}{4}} = \frac{13k}{2}$. Solving gives $k = 2$, so diagonals are 10 cm and 24 cm. Again, no diagonal equals 13 cm. The question is answered.Thus either statement alone is sufficient to determine the answer.Answer:B. The Question can be answered by using either Statement alone