Question

The number of integral values of $p$, for which the equation $5\cos x+4\sin x=2p+3$ has real solutions is

Answer 1

$5\cos x+4\sin x=2p+3$
For the equation to have solution, $2p+3$ should lie in the range of $5\cos x+4\sin x$

Now,
$-\sqrt{5^2+4^2}\le 5\cos x+4\sin x\le \sqrt{5^2+4^2}\Rightarrow -\sqrt{5^2+4^2}\le 2p+3\le \sqrt{5^2+4^2}\Rightarrow -\sqrt{41}\le 2p+3\le \sqrt{41}\Rightarrow -\sqrt{41}-3\le 2p\le \sqrt{41}-3\Rightarrow \frac{-\sqrt{41}-3}{2}\le p\le \frac{\sqrt{41}-3}{2}$

Hence, the integral values of $p$ are $\{-4,-3,-2,-1,0,1\}$