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

The number of integral values of $k$, for which the equation $7 \cos x+5 \sin x=2 \mathrm{k}+1$ has a solution, is

The smallest positive value of $x$ in degrees satisfying the equation $\tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan (x) \tan \left(x-50^{\circ}\right)$ is

The number of solutions, of $2^{1+|\cos x|+|\cos x|^2+\ldots \ldots \cdots \cdots}=4$ in $(-\pi, \pi)$, is