UPSC NDA II 2025 Mathematics Paper

Concept:To find the inverse, swap the variables x and y, then solve for y using algebraic manipulation and logarithms.Explanation:Start with the given function: y = (10^x - 10^{-x})/(10^x + 10^{-x}).Multiply numerator and denominator by 10^x to simplify: y = (10^(2x) - 1)/(10^(2x) + 1).Apply componendo and dividendo: (1+y)/(1-y) = 10^(2x).Take log base 10 on both sides: 2x = log10((1+y)/(1-y)).Solve for x: x = (1/2) log10((1+y)/(1-y)).Replace y with x to write the inverse function: f^{-1}(x) = (1/2) log10((1+x)/(1-x)).Answer:Option D: $\frac{1}{2}log_{10}(\frac{1+x}{1-x})$