UPSC NDA II 2024 Mathematics Paper

Concept:The equation $\cos^{100} x - \sin^{100} x = 1$ can only be satisfied when $\cos^{100} x = 1$ and $\sin^{100} x = 0$ because both terms are non-negative and at most 1.Explanation:Since $\cos^2 x$ and $\sin^2 x$ are each in the interval $[0,1]$, their 100th powers also lie in $[0,1]$.Rewriting the equation as $\cos^{100} x = 1 + \sin^{100} x$, we note that the left side ≤ 1 and the right side ≥ 1.Equality occurs only when $\cos^{100} x = 1$ and $\sin^{100} x = 0$.This gives $\cos^2 x = 1$, so $\cos x = \pm 1$.The general solution for $\cos x = \pm 1$ is $x = n\pi$, where $n$ is any integer.Substituting $x = n\pi$ into the original equation: $\cos^{100}(n\pi) = (\pm1)^{100}=1$ and $\sin^{100}(n\pi)=0$, confirming the equality.Therefore the general solution is $x = n\pi$.Answer:Option A: $n\pi$