Question

If the equation ${\cos }^{2}x-k=\sin x(\sin x-2\cos x)$ has atleast one real solution. The number of possible integral values of $k$ is

• A. $1$
• B. $2$
• C. $3$
• D. $5$

Answer 1

The correct option is C $3$

Given, ${\cos }^{2}x-k=\sin x(\sin x-2\cos x)$
$\Rightarrow k=({\cos }^{2}x-{\sin }^{2}x)+2\sin x\cos x=\cos 2x+\sin 2x$
$\Rightarrow k\in [-\sqrt{2},\sqrt{2}]$
$\Rightarrow$ three integral values of $k=-1,0,1$