UPSC NDA 2 2017 Mathematics Solved Paper

Concept:The inverse of a function is found by swapping the variables and solving for the new dependent variable, using logarithmic properties.Explanation:Start with $y = 5^{\ln x}$. The domain is $x > 0$, and range is $y > 0$.Swap $x$ and $y$ to get $x = 5^{\ln y}$.Take natural logs on both sides: $\ln x = \ln(5^{\ln y}) = (\ln y)(\ln 5)$.Divide by $\ln 5$: $\frac{\ln x}{\ln 5} = \ln y$.Rewrite the left side using power rule: $\ln(x^{1/\ln 5}) = \ln y$.Since $\ln$ is one-to-one, $y = x^{1/\ln 5}$.For the inverse, we express $x$ in terms of $y$: $x = y^{1/\ln 5}$ with $y > 0$ (original range).This matches option A.Answer:A. $x = y^{\frac{1}{\ln 5}},\; y > 0$