Number of solutions of $5$ $cos^2 \theta  -3 sin^2 \theta  + 6 sin \theta  cos \theta  = 7$ in the interval $[0, 2 \pi] $ is :-

Number of solutions of equation $sgn(sin x) = sin^2x + 2sinx + sgn(sin^2x)$ in $\left[ { – \frac{{5\pi }}{2},\frac{{7\pi }}{2}} \right]$  is

(where $sgn(.)$ denotes signum function) –

If $\sin 2x + \sin 4x = 2\sin 3x,$ then $x =$

Let $\theta, \phi \in[0,2 \pi]$ be such that $2 \cos \theta(1-\sin \phi)=\sin ^2 \theta\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \phi-1, \tan (2 \pi-\theta)>0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy

$(A)$ $0 < \phi<\frac{\pi}{2}$ $(B)$ $\frac{\pi}{2} < \phi<\frac{4 \pi}{3}$

$(C)$ $\frac{4 \pi}{3} < \phi<\frac{3 \pi}{2}$ $(D)$ $\frac{3 \pi}{2} < \phi < 2 \pi$

The number of values of $x$ in the interval $[0, 5 \pi  ] $ satisfying the equation $3{\sin ^2}x – 7\sin x + 2 = 0$ is