If $2\,cos\,\theta  + sin\, \theta \, = 1$ $\left( {\theta  \ne \frac{\pi }{2}} \right)$ , then $7\, cos\,\theta + 6\, sin\, \theta $ is equal to

The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations

$ (y+z) \cos 3 \theta=(x y z) \sin 3 \theta $

$ x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z} $

$ (x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$ have a solution $\left(\mathrm{x}_0, \mathrm{y}_0, \mathrm{z}_0\right)$ with $\mathrm{y}_0 \mathrm{z}_0 \neq 0$, is

If $A + B + C = \pi$ & $sin\, \left( {A\,\, + \,\,\frac{C}{2}} \right) = k \,sin,\frac{C}{2}$ then $tan\, \frac{A}{2} \,tan \, \frac{B}{2}=$

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  = $ 

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

If  $a = \sin \frac{\pi }{{18}}\sin \frac{{5\pi }}{{18}}\sin \frac{{7\pi }}{{18}}$ and $x$ is the solution of the equatioin $y = 2\left[ x \right] + 2$ and $y = 3\left[ {x - 2} \right] ,$ where $\left[ x \right]$ denotes the integral part of $x,$ then $a$ is equal to :-