If $\sin \,\theta  + \sqrt 3 \cos \,\theta  = 6x - {x^2} - 11,x \in R$ , $0 \le \theta  \le 2\pi $ , then the equation has solution for

If $\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ is the solution of $4 \cos \theta+5 \sin \theta=1$, then the value of $\tan \alpha$ is

The number of real solutions $x$ of the equation $\cos ^2(x \sin (2 x))+\frac{1}{1+x^2}=\cos ^2 x+\sec ^2 x$ is

Statement $-1:$ The number of common solutions of the trigonometric equations $2\,sin^2\,\theta - cos\,2\theta  = 0$ and $2 \,cos^2\,\theta - 3\,sin\,\theta  = 0$ in the interval $[0, 2\pi ]$ is two.

Statement $-2:$ The number of solutions of the equation, $2\,cos^2\,\theta  - 3\,sin\,\theta  = 0$ in the interval $[0, \pi ]$ is two.

If $0\, \le \,x\, < \frac{\pi }{2},$ then the number of values of $x$ for which $sin\,x -sin\,2x + sin\,3x=0,$ is

The value of $\theta $ satisfying the given equation $\cos \theta + \sqrt 3 \sin \theta  = 2,$ is