Question

The number of solution of the equation $\operatorname{sgn}(\sin x)-\operatorname{sgn}\left(\sin ^2 x\right)=\sin ^2 x+2 \sin x$ in $\left[\frac{-5 \pi}{2}, \frac{7 \pi}{2}\right]$ is [Note: $\operatorname{sgn}( k )$ 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 $
$\operatorname{sgn}(\sin x)-\operatorname{sgn}\left(\sin ^2 x\right)=\sin ^2 x+2 \sin x$
$\text { has } 6 \text { roots }$
$\text { Case-II : } \sin x \neq 0 $
$ \operatorname{sgn}(\sin x)-1=\sin ^2 x+2 \sin x$
$ \operatorname{sgn}(\sin x)=(1+\sin x)^2$
$ \text { no. real roots. }$