If $2{\sin ^2}\theta = 3\cos \theta ,$ where $0 \le \theta \le 2\pi $, then $\theta = $
$\frac{\pi }{6},\frac{{7\pi }}{6}$
$\frac{\pi }{3},\frac{{5\pi }}{3}$
$\frac{\pi }{3},\frac{{7\pi }}{3}$
None of these
In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length$(s)$ of the side$(s)$ of the triangle is (are)
$(A)$ $16$ $(B)$ $18$ $(C)$ $24$ $(D)$ $22$
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
One root of the equation $\cos x - x + \frac{1}{2} = 0$ lies in the interval
If $4{\sin ^4}x + {\cos ^4}x = 1,$ then $x =$
The number of elements in the set $S=$ $\left\{\theta \in[-4 \pi, 4 \pi]: 3 \cos ^{2} 2 \theta+6 \cos 2 \theta-\right.$ $\left.10 \cos ^{2} \theta+5=0\right\}$ is