Question

If the sum of solutions of the system of equations 2sin2θ – cos2θ = 0 and 2cos2θ + 3sinθ = 0 in the interval [0, 2π] is kπ, then k is equal to _______.

Answer 1

The correct answer is 3
Equation (1)
$2\sin^2\theta=1–2\sin^2\theta$
$⇒\sin^2\theta=\frac{1}{4}$
$⇒\sin⁡\theta=±\frac{1}{2}$
$⇒ \theta= \frac{\pi }{6},\frac{5\pi}{6},\frac{7 \pi}{6},\frac{11\pi}{6}$
Equation(2)
$2\cos^2\theta+3\sin \theta=0$
$⇒2\sin^2\theta–3\sin \theta–2=0$
$⇒2\sin^2 \theta–4\sin \theta+\sin \theta–2=0$
$⇒(\sin \theta–2)(2\sin \theta+1)=0$
$⇒\sin⁡ \theta=\frac{−1}{2}$
$⇒ \theta =\frac{7 \pi}{6},\frac{11 \pi}{6}$
Hence, the Sum of solutions $=\frac{7 \pi +11 \pi}{6}$
$=\frac{18 \pi}{6}$
$=\frac{3}{\pi}$
$∴ k =3$