UPSC CDS 1 2023 Math Paper

Concept:Determine if the given statements provide enough information to uniquely find the shortest side of a triangle, given the two longer sides are $24$ cm and $25$ cm.Explanation:Statement I:The angles are in the ratio $1:2:3$.Sum of angles $= 180^{\circ}$, so the angles are $30^{\circ}$, $60^{\circ}$, $90^{\circ}$.For a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, the sides opposite these angles are in the ratio $1:\sqrt{3}:2$.If the given sides $24$ and $25$ are the two longer sides, they would need to be in the ratio $\sqrt{3}:2$ to fit this pattern.$24:25$ is not equal to $\sqrt{3}:2$ (since $\frac{24}{25} = 0.96$, but $\frac{\sqrt{3}}{2} \approx 0.866$).Thus, the triangle cannot have these side lengths if the angles are $30^{\circ}$, $60^{\circ}$, $90^{\circ}$.Hence, Statement I alone is not sufficient to determine the shortest side.Statement II:The perpendicular from the vertex opposite the longest side ($25$ cm) has length $6.72$ cm.Let the shortest side be $x$ cm.Using area formula: area $= \frac{1}{2} \times \text{base} \times \text{height}$.Area taken with base $25$ cm: $\frac{1}{2} \times 25 \times 6.72 = 84$ cm${}^2$.Also, for a right triangle with sides $x$, $24$ cm, and $25$ cm, the area is $\frac{1}{2} \times x \times 24$.Equating: $\frac{1}{2} \times x \times 24 = 84$ gives $x = 7$ cm.Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, confirming a right triangle.Thus, Statement II alone gives $x = 7$ cm and is sufficient.Since Statement I alone is not sufficient but Statement II alone is sufficient, the question can be answered by Statement II alone.Answer:Option A is correct.