Let $S=\{\theta \in(-2 \pi, 2 \pi): \cos \theta+1=\sqrt{3} \sin \theta\}$.

Then $\sum\limits_{\theta \in \mathrm{S}} \theta$ is equal to :

The sum of all the integral values of $p$ such that the equation $3 \sin ^2 x+12 \cos x-3=p, x \in \mathbb{R}$, has at least one solution, is:

Let $S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a,\ a \in \mathbb{Z} \}$. Then $n(S)$ is equal to:

Number of solutions of $\sqrt{3} \cos 2 \theta+8 \cos \theta+3 \sqrt{3}=0, \theta \in[-3 \pi, 2 \pi]$ is :