Question

Let $f(\theta)=\begin{vmatrix}-\sin \theta & 1 & 0 \\ 1 & 0 & \sin \theta \\ \sin \theta & 1 & 1\end{vmatrix}$. Number of solutions of the equation $f(\theta)=0$ in $(0,2 \pi)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

Clearly, $f(\theta)=2 \sin ^2 \theta-1=-\cos 2 \theta$
$\therefore f (\theta)=0 \Rightarrow \cos 2 \theta=0 \Rightarrow \theta=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$
So, number of solution are 4.