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Trigonometrical Equations
hard
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.
A
$3$
B
$6$
C
$9$
D
$12$
(JEE MAIN-2022)
Solution
$2 \sin ^{2} \theta-\cos 2 \theta=0$
$2 \sin ^{2} \theta-\left(1-2 \sin ^{2} \theta\right)=0$
$\sin ^{2} \theta=\left(\frac{1}{2}\right)^{2}$
$\theta=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{11 \pi}{6}$
$2 \cos ^{2} \theta+3 \sin \theta=0$
$2 \sin ^{2} \theta-3 \sin \theta-2=0$
$\therefore \sin \theta=-\frac{1}{2}$
$\theta=\frac{7 \pi}{6}, \frac{11 \pi}{6}$
So, the common solution is
$\theta=\frac{7 \pi}{6}, \frac{11 \pi}{6}$
$\text { Sum }=\frac{7 \pi+11 \pi}{6}=3 \pi= k \pi$
$K =3$
Standard 11
Mathematics
