If $\sin \theta + 2\sin \phi + 3\sin \psi = 0$ and $\cos \theta + 2\cos \phi + 3\cos \psi = 0$ , then the value of $\cos 3\theta + 8\cos 3\phi + 27\cos 3\psi = $
$\cos (3\theta + 3\phi + 3\psi )$
$18\cos (\theta + \phi + \psi )$
$6\cos (\theta + \phi + \psi )$
$36\cos (\theta + \phi + \psi )$
Let $\theta, 0 < \theta < \pi / 2$, be an angle such that the equation $x ^2+4 x \cos \theta+\cot \theta=0$ has equal roots for $x$. Then $\theta$ in radians is
The general solution of $\tan 3x = 1$ is
The equation $\sqrt 3 \sin x + \cos x = 4$ has
The number of solutions of the equation $\sin (9 x)+\sin (3 x)=0$ in the closed interval $[0,2 \pi]$ is
If $tanA + cotA = 4$, then $tan^4A + cot^4A$ is equal to