UPSC CDS II 2025 Mathematics Paper

Concept:Cross-multiply the given rational expression, then isolate terms and square both sides to remove square roots.Explanation:Start with $\frac{\sqrt{p+x}+\sqrt{p-x}}{\sqrt{p+x}-\sqrt{p-x}}=p$.Cross-multiply: $\sqrt{p+x}+\sqrt{p-x}=p(\sqrt{p+x}-\sqrt{p-x})$.Expand: $\sqrt{p+x}+\sqrt{p-x}=p\sqrt{p+x}-p\sqrt{p-x}$.Bring like terms together: $\sqrt{p+x}-p\sqrt{p-x}=-\sqrt{p-x}-p\sqrt{p-x}$ is messy; better: move $\sqrt{p-x}$ terms to left and $\sqrt{p+x}$ terms to right: $\sqrt{p+x}+p\sqrt{p-x}=p\sqrt{p+x}-\sqrt{p-x}$ is not correct. Actually reorganize: $\sqrt{p+x}+\sqrt{p-x}=p\sqrt{p+x}-p\sqrt{p-x}$ => $\sqrt{p+x}-p\sqrt{p+x}=-p\sqrt{p-x}-\sqrt{p-x}$ => $(1-p)\sqrt{p+x}=-(1+p)\sqrt{p-x}$.Square both sides: $(1-p)^2(p+x)=(1+p)^2(p-x)$.Expand: $(1-p)^2p+(1-p)^2x=(1+p)^2p-(1+p)^2x$.Bring $x$ terms to left, constants to right: $(1-p)^2x+(1+p)^2x=(1+p)^2p-(1-p)^2p$.Factor: $x[(1-p)^2+(1+p)^2]=p[(1+p)^2-(1-p)^2]$.Simplify squares: $(1-p)^2=1-2p+p^2$, $(1+p)^2=1+2p+p^2$. Their sum is $2(1+p^2)$, their difference is $4p$.Thus $x \cdot 2(1+p^2)=p \cdot 4p$ => $2(1+p^2)x=4p^2$ => $x=\frac{2p^2}{p^2+1}$.Answer:$x=\frac{2p^{2}}{p^{2}+1}$, which is option D.