The number of all possible triplets $(a_1 , a_2 , a_3)$ such that $a_1+ a_2 \,cos \, 2x + a_3 \, sin^2 x = 0$ for all $x$ is

Number of solutions of the equation $2^x + x = 2^{sin \ x} +  \sin x$ in $[0,10\pi ]$ is -

If sum of all the solutions of the equation $8\cos x \cdot \left( {\cos \left( {\frac{\pi }{6} + x} \right) \cdot \cos \left( {\frac{\pi }{6} - x} \right) - \frac{1}{2}} \right) = 1$ in $\left[ {0,\pi } \right]$ is $k\pi $then $k$ is equal to :

The sides of a triangle are $\sin \alpha ,\,\cos \alpha $ and $\sqrt {1 + \sin \alpha \cos \alpha } $ for some $0 < \alpha < \frac{\pi }{2}$. Then the greatest angle of the triangle is.....$^o$

Let $A = \left\{ {\theta \,:\,\sin \,\left( \theta  \right) = \tan \,\left( \theta  \right)} \right\}$ and $B = \left\{ {\theta \,:\,\cos \,\left( \theta  \right) = 1} \right\}$ be two sets. Then

$sin^{2n}x + cos^{2n}x$ lies between