The number of elements in the set $S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}$ is$.....$

The number of solutions of $sin \,3x\, = cos\, 2x$ , in the interval $\left( {\frac{\pi }{2},\pi } \right)$ is

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is

Common roots of the equations $2{\sin ^2}x + {\sin ^2}2x = 2$ and $\sin 2x + \cos 2x = \tan x,$ are

If $\sin 5x + \sin 3x + \sin x = 0$, then the value of $x$ other than $0$ lying between $0 \le x \le \frac{\pi }{2}$ is

If $\cos \,\alpha  + \cos \,\beta  = \frac{3}{2}$ and $\sin \,\alpha  + \sin \,\beta  = \frac{1}{2}$ and $\theta $ is the the arithmetic mean of $\alpha $ and $\beta $ , then $\sin \,2\theta  + \cos \,2\theta $ is equal to