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

Correct Answer: 3

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 \(=\frac{7 \pi +11 \pi}{6}\)
\(=\frac{18 \pi}{6}\)
\(=\frac{3}{\pi}\)
\(∴ k =3\)