JEE Main 2022 (Online) 26th July Evening ShiftQuestion 1
If the sum of solutions of the system of equations $2\sin^2\theta-\cos2\theta=0$ and $2\cos^2\theta+3\sin\theta=0$ in the interval $[0,2\pi]$ is $k\pi$, then $k$ is equal to ________.
Trigonometric Ratio & Identities
Answer$3$
Solution
Concept. Write both equations in $\sin\theta$ and keep the common solutions. Formulas used. (1) $\cos2\theta=1-2\sin^2\theta$, $\cos^2\theta=1-\sin^2\theta$Step 1 - First equation. $2\sin^2\theta=1-2\sin^2\theta$, so $\sin^2\theta=\frac14$. Step 2 - Second equation. $2-2\sin^2\theta+3\sin\theta=0$, i.e. $(2\sin\theta+1)(\sin\theta-2)=0$, so $\sin\theta=-\frac12$. Step 3 - Solutions. $\theta=\frac{7\pi}6,\frac{11\pi}6$; sum $3\pi$, so $k=3$. Answer: 3