UPSC CDS II 2025 Mathematics Paper

Concept:We use cross-multiplication and algebraic factorisation to convert the given proportion into a product form equal to zero.Explanation:Start with $\frac{p+q}{q+r}=\frac{r+s}{s+p}$. Cross-multiply: $(p+q)(s+p) = (r+s)(q+r)$.Expand both sides: $ps + p^2 + qs + pq = rq + r^2 + sq + sr$.Bring all terms to one side: $p^2 - r^2 + ps - sr + pq - rq = 0$.Factor groups: $(p-r)(p+r) + s(p-r) + q(p-r) = 0$.Factor out $(p-r)$: $(p-r)(p+r+s+q) = 0$.Hence, either $p-r=0$ or $p+q+r+s=0$.So $p = r$ or the sum of all four variables is zero.Answer:Either $p+q+r+s=0$ or $p=r$. Therefore, option C is correct.