JEE Advanced 2017 · Paper 1, Mathematics question 3

Official key: marks to all (a printing inconsistency in this question).

Simplify. For $x\in[n,n+1)$, $[x]=n$ and $\cos(\pi x+n\pi)=(-1)^n\cos\pi x$, so
$$f(x)=(-1)^nx\cos\pi x.$$

At an integer $n$:

• Right-hand value: $(-1)^nn\cos n\pi=n$.
• Left-hand limit (from $[n-1,n)$): $(-1)^{n-1}n\cos n\pi=-n$.

So $f$ jumps at every non-zero integer and is continuous at 0. As printed, the discontinuities are at $x=-1,1,2$.

Answer: (A), (C), (D) (marks to all)