UPSC NDA I 2025 Mathematics Paper

Concept:Implicit differentiation of the given equation, followed by algebraic simplification using the original relation, yields the derivative.Explanation:We have $(x+y)^{p+q}=x^{p}y^{q}$.Differentiate both sides with respect to $x$:Left: $(p+q)(x+y)^{p+q-1}\left(1+\frac{dy}{dx}\right)$.Right: $p x^{p-1}y^{q}+q x^{p}y^{q-1}\frac{dy}{dx}$.Collect $\frac{dy}{dx}$ terms:$\frac{dy}{dx}\left[(p+q)(x+y)^{p+q-1}-q x^{p}y^{q-1}\right]=p x^{p-1}y^{q}-(p+q)(x+y)^{p+q-1}$.Use $(x+y)^{p+q-1}=\frac{x^{p}y^{q}}{x+y}$ (from the original equation) to simplify both sides.After cancellation, we get $\frac{dy}{dx}=\frac{y}{x}$.This result does not contain $p$ or $q$.Answer:The derivative is independent of both $p$ and $q$. Option D.