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 $\mathrm{k} \pi$, then k is equal to ______.

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Trigonometric Equations

If the sum of solutions of the system of equations

2 sin^{2} θ - cos 2 θ = 0

and

2 cos^{2} θ + 3 sin θ = 0

in the interval

[0, 2 π]

k π

, then k is equal to ______.

Answer: 3

From the first equation

2 sin^{2} θ - cos 2 θ = 0

, substitute

cos 2 θ = 1 - 2 sin^{2} θ

to get

4 sin^{2} θ = 1

, so

sin θ = \pm \frac{1}{2}

. This gives

θ = \frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}

.

From the second equation

2 cos^{2} θ + 3 sin θ = 0

, use

cos^{2} θ = 1 - sin^{2} θ

to get

2 sin^{2} θ - 3 sin θ - 2 = 0

. Solving:

(2 sin θ + 1) (sin θ - 2) = 0

, so

sin θ = - \frac{1}{2}

(since

sin θ = 2

is impossible). This gives

θ = \frac{7 π}{6}, \frac{11 π}{6}

.

The common solutions are

θ = \frac{7 π}{6}, \frac{11 π}{6}

. Their sum is

\frac{7 π + 11 π}{6} = \frac{18 π}{6} = 3 π

. Therefore

k = 3

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer