NDA 1 2025 Maths, question 75

Solution.

Answer: (d) is independent of both $p$ and $q$

1. Take logs: $(p + q)\log(x + y) = p\log x + q\log y$, then differentiate: $\frac{(p + q)(1 + y')}{x + y} = \frac{p}{x} + \frac{qy'}{y}$.
2. Rearranging gives $(py - qx)(xy' - y) = 0$, and the factor $py - qx$ cannot be zero on a real solution.
3. So $y' = y/x$, which contains neither $p$ nor $q$.

Trap: The curve itself depends on $p$ and $q$; the option means that the formula for $dy/dx$ has no $p$ or $q$ in it.