Question

The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is:

• A. 4
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0,-\frac{\pi }{4}\le x\le \frac{\pi }{4}$
Apply $:R_1\to R_1-R_2\&R_2\to R_2-R_3$
$\left|\begin{array}{ccc}\sin x-\cos x& \cos x-\sin x& 0\\ 0& \sin x-\cos x& \cos x-\sin x\\ \cos x& \cos x& \sin x\end{array}\right|=0$
$(\sin x-\cos x{)}^2\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ \cos x& \cos x& \sin x\end{array}\right|=0$
$(\sin x-\cos x{)}^2(\sin x+2\cos x)=0$
$\therefore x=\frac{\pi }{4}$