JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Trigonometric Ratios and Identities

The expression $3\sin x+4\cos x$ takes every value in $[-5,5]$ and no value outside it, since its amplitude is $\sqrt{3^2+4^2}=5$.
A solution therefore exists exactly when

The integers in this range are $-6,-5,\dots ,4$, and there are $4-(-6)+1=11$ of them.
The number of integral values of $k$ is $11$.