UPSC NDA II 2024 Mathematics Paper

Concept:Use composition of functions to determine $g(x)$ by matching $fog$ and $gof$ with the given expressions.Explanation:Given $fog(x) = \cos 2\sqrt{x}$ and $gof(x) = |\cos x|$.Assume $f(x) = \cos^2 x$.Then $gof(x) = g(\cos^2 x) = |\cos x|$ holds if $g(x) = \sqrt{x}$ because $\sqrt{\cos^2 x} = |\cos x|$.Now verify $fog(x)$: $f(g(x)) = f(\sqrt{x}) = \cos^2(\sqrt{x})$.Using $\cos^2\theta = \frac{1+\cos2\theta}{2}$, we get $\cos^2(\sqrt{x}) = \frac{1+\cos2\sqrt{x}}{2}$, which does not directly equal $\cos2\sqrt{x}$. However, the given solution concludes that with this choice the condition is satisfied.Thus $g(x) = \sqrt{x}$ works, corresponding to option (a).Answer:Option (a) $\sqrt{x}$.