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. 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 $C_1\to C_1+C_2+C_3$, we get

$\left|\begin{array}{ccc}\sin x+2\cos x& \cos x& \cos x\\ \sin x+2\cos x& \sin x& \cos x\\ \sin x+2\cos x& \cos x& \sin x\end{array}\right|=0$

Taking $(\sin x+2\cos x)$ common from $C_1$, we get

$\Rightarrow (\sin x+2\cos x)\left|\begin{array}{ccc}1& \cos x& \cos x\\ 1& \sin x& \cos x\\ 1& \cos x& \sin x\end{array}\right|=0$

Applying $R_2\to R_2-R_1;R_3\to R_3-R_1$

$\Rightarrow (\sin x+2\cos x)\left|\begin{array}{ccc}1& \cos x& \cos x\\ 0& \sin x-\cos x& 0\\ 0& 0& \sin x-\cos x\end{array}\right|=0\Rightarrow (\sin x+2\cos x){(\sin x-\cos x)}^2=0\Rightarrow \sin x+2\cos x=0,\sin x-\cos x=0\Rightarrow \tan x=-2,\tan x=1$