Question

The number of distinct real roots of $\begin{vmatrix}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{vmatrix}=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:

• A. 4
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$\begin{vmatrix}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{vmatrix}=0,-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$
Apply $: R_{1} \rightarrow R_{1}-R_{2} \& R_{2} \rightarrow R_{2}-R_{3}$
$\begin{vmatrix}\sin x-\cos x & \cos x-\sin x & 0 \\ 0 & \sin x-\cos x & \cos x-\sin x \\ \cos x & \cos x & \sin x\end{vmatrix}=0$
$(\sin x-\cos x)^{2}\begin{vmatrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ \cos x & \cos x & \sin x\end{vmatrix}=0$
$(\sin x-\cos x)^{2}(\sin x+2 \cos x)=0$
$\therefore x=\frac{\pi}{4}$