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 $x\in [\frac{-\pi }{4},\frac{\pi }{4}]$ is

• A. $0$
• B. $2$
• C. $1$
• D. $3$

Answer 1

The correct option is C $1$

$\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$

Applying $R_1\to R_1+R_2+R_3$ and taking common from $R_1$, we get
$(\sin x+2\cos x)\left|\begin{array}{ccc}1& 1& 1\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$

Applying $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$
$\Rightarrow (\sin x+2\cos x)\left|\begin{array}{ccc}1& 0& 0\\ \cos x& \sin x-\cos x& 0\\ \cos x& 0& \sin x-\cos x\end{array}\right|=0$
$\Rightarrow {(\sin x-\cos x)}^2(\sin x+2\cos x)=0$
$\Rightarrow x=\frac{\pi }{4}$, $x\in [-\frac{\pi }{4},\frac{\pi }{4}]$