Question

Let $2{\sin }^{2}x+3\sin x-2>0$ and $x^2-x-2<0$ ( $x$ is measured in radians). Thcn $x$ lies in the interval

• A. $(\frac{\pi }{6}.\frac{5\pi }{6})$
• B. $(-1,\frac{5_{\pi }}{6})$
• C. $(-1,2)$
• D. $(\frac{\pi }{6},2)$

Answer 1

The correct option is D $(\frac{\pi }{6},2)$

first inequality gives
$(2\sin x-1)(\sin x+2)>0$
$(\sin x-\frac{1}{2})>0$
$\frac{\Pi }{6}<x<\frac{5\Pi }{6}$
second inequality gives
$(x+1)(x-2)<0$
$-1<x<2$
common part is $\frac{\Pi }{6}<x<2$