UPSC NDA I 2017 Mathematics Solved Paper

Concept:For $f(x)=\cos x$ to be both one‑one and onto, the domain $X$ must be an interval where $\cos$ is strictly monotonic, and the codomain $Y$ must match the exact range of $f$ on that domain.Explanation:$\cos x$ is strictly decreasing on $[0,\pi]$, so it is one‑one on that interval.Its range on $[0,\pi]$ is $[-1,1]$, which must equal the codomain $Y$ for onto.Option A: $X=[0,\pi]$, $Y=[-1,1]$ satisfies both conditions.Option B: $X=\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ makes $\cos x$ not one‑one (symmetric about $0$), so it fails.Option C: $Y=(-1,1)$ excludes $-1$ and $1$, so not onto.Option D: $Y=[0,1]$ is a proper subset of the range, so not onto.Thus only option A meets all requirements.Answer:Option A: $X=[0,\pi]$ and $Y=[-1,1]$.