Question

If the sum of solutions of the system of equations $2 \sin ^2 \theta-\cos 2 \theta=0 $ and $2 \cos ^2 \theta+3 \sin \theta=0$ in the interval $[0,2 \pi]$ is $k \pi$, then $k$ is equal to _____

Answer 1

$2 \sin ^2 \theta-\cos 2 \theta=0 $
$2 \sin ^2 \theta-\left(1-2 \sin ^2 \theta\right)=0$
$ \Rightarrow \sin ^2 \theta=\left(\frac{1}{2}\right)^2$
$ \theta=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{11 \pi}{6}$
$ 2 \cos ^2 \theta+3 \sin \theta=0$
$ \Rightarrow 2 \sin ^2 \theta-3 \sin \theta-2=0$
$ \therefore \sin \theta=-\frac{1}{2} $
$ \theta=\frac{7 \pi}{6}, \frac{11 \pi}{6}$
So, the common solution is
$\theta=\frac{7 \pi}{6}, \frac{11 \pi}{6} $
Sum $=\frac{7 \pi+11 \pi}{6}=3 \pi=k \pi$
$K = 3$