The number of integral values of $k$, for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution, is
$4$
$8$
$10$
$12$
One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval
$sin 3\theta = 4 sin\, \theta \,sin \,2\theta \,sin \,4\theta$ in $0\, \le \,\theta\, \le \, \pi$ has :
The variable $x$ satisfying the equation $\left| {\sin \,x\,\cos \,x} \right| + \sqrt {2 + {{\tan }^2}\,x + {{\cot }^2}\,x} = \sqrt 3$ belongs to the interval
Common roots of the equations $2{\sin ^2}x + {\sin ^2}2x = 2$ and $\sin 2x + \cos 2x = \tan x,$ are