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

If $\cos \theta + \cos 7\theta + \cos 3\theta + \cos 5\theta = 0$, then $\theta $

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

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

Number of solution $(s)$ of the equation ${\cos ^2}2x + {\cos ^2}\frac{{5x}}{4} = \cos 2x\,{\cos ^2}5x$ in $\left[ {0,\frac{\pi }{3}} \right]$ is

The sum of the solutions in $x \in (0,4\pi )$ of the equation $4\sin \frac{x}{3}\left( {\sin \left( {\frac{{\pi  + x}}{3}} \right)} \right)\sin \left( {\frac{{2\pi  + x}}{3}} \right) = 1$ is