The number of integral values of $k$, for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution, is

The number of values of $x$ in the interval $[0, 5\pi]$ satisfying the equation $3sin^2x\, \,-\,\, 7sinx + 2 = 0$ is

$\alpha=\sin 36^{\circ}$ is a root of which of the following equation

Let $S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}$ and $\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)$, then $\frac{1}{6}(\beta-14)^2$ is equal to

If $\tan \theta = - \frac{1}{{\sqrt 3 }}$ and $\sin \theta = \frac{1}{2}$, $\cos \theta = - \frac{{\sqrt 3 }}{2}$, then the principal value of $\theta $ will be

The solution of equation ${\cos ^2}\theta + \sin \theta + 1 = 0$ lies in the interval