One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval

If sum of all the solutions of the equation $8\cos x \cdot \left( {\cos \left( {\frac{\pi }{6} + x} \right) \cdot \cos \left( {\frac{\pi }{6} - x} \right) - \frac{1}{2}} \right) = 1$ in $\left[ {0,\pi } \right]$ is $k\pi $then $k$ is equal to :

If $\sec x\cos 5x + 1 = 0$, where $0 < x < 2\pi $, then $x =$

If $sin^2x + sinx \,cosx -6cos^2x = 0$ and $-\frac{\pi}{2} < x < 0$, then the value of $cos2x$, is

The equation $\sqrt 3 \sin x + \cos x = 4$ has

The number of elements in the set $S =\left\{\theta \in[0,2 \pi]: 3 \cos ^4 \theta-5 \cos ^2 \theta-2 \sin ^2 \theta+2=0\right\}$ is $...........$.