Question

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

• A. $2$
• B. $8$
• C. $4$
• D. $6$

Answer 1

The correct option is D $6$

$5\cos x+4\sin x=2p+3$
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 $\{-4,-3,-2,-1,0,1\}$