UPSC CDS II 2025 Mathematics Paper

Concept:For two chords intersecting inside a circle, the product of the segments of one chord equals the product of the segments of the other chord: $AP \times PB = CP \times PD$.Explanation:We are given $AP \times PB = 48$. We need $CP \times PD$.Using Statement I alone: $AP = 8$, so $PB = 48 \div 8 = 6$. But $CP$ and $PD$ are unknown. So we cannot find $CP \times PD$.Using Statement II alone: $CP = 10$. But $PD$ is unknown. Also $AP \times PB$ is already known as 48, but without $CP$ we cannot find $PD$. So we cannot get $CP \times PD$.Using both statements together: From the chord theorem, $48 = 10 \times PD$. Thus $PD = 48 \div 10 = 4.8$. Then $CP \times PD = 10 \times 4.8 = 48$. So the answer is found.Answer:The question can be answered by using both the Statements together, but cannot be answered using either Statement alone. Hence option C is correct.