Question

The number of solution of the equation $\mathrm{sgn}(\sin x)-\mathrm{sgn}\left({\sin }^{2}x\right)={\sin }^{2}x+2\sin x$ in $\left[\frac{-5\pi }{2},\frac{7\pi }{2}\right]$ is [Note: $\mathrm{sgn}(k)$ denotes signum function of $k$.]

• A. 10
• B. 6
• C. 13
• D. 9

Answer 1

Answer: (B)

$\text{ Case-I : }\sin x=0\Rightarrow x=-2\pi ,-\pi ,0,\pi ,2\pi ,3\pi$
$\mathrm{sgn}(\sin x)-\mathrm{sgn}\left({\sin }^{2}x\right)={\sin }^{2}x+2\sin x$
$\text{ has }6\text{ roots }$
$\text{ Case-II : }\sin x\ne 0$
$\mathrm{sgn}(\sin x)-1={\sin }^{2}x+2\sin x$
$\mathrm{sgn}(\sin x)=(1+\sin x{)}^2$
$\text{ no. real roots. }$