Question

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

• A. $4$
• B. $3$
• C. $1$
• D. $2$

Answer 1

The correct option is C $1$

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

Taking $\cos x$ common from each row

$\left|\begin{array}{ccc}\tan x& 1& 1\\ 1& \tan x& 1\\ 1& 1& \tan x\end{array}\right|=0,\Rightarrow {\tan }^{3}x-3\tan x+2=0\Rightarrow (\tan x-1)({\tan }^{2}x+\tan x-2)=0\Rightarrow (\tan x-1)(\tan x+2)(\tan x-1)=0\Rightarrow \tan x=1,2$

$\tan x=2$ is not possible $\because (-\frac{\pi }{4}\le x\le \frac{\pi }{4})$