Question

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

• A. $1$
• B. $4$
• C. $2$
• D. $3$

Answer 1

Answer: (C)

$2$

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

Applying $R1\to R1-R2andR3\to R3-R2$

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

$\to$ $(sinx-cosx{)}^2\left|\begin{array}{ccc}-1& 1& 0\\ \sin x& \cos x& \sin x\\ 0& 1& -1\end{array}\right|=0$

$\to (sinx-cosx{)}^2(cosx+sinx+sinx)=0$

Thus either, $cosx=sinxorcosx=-2sinxi.e.tanx=\frac{-1}{2}$

$\cos x=\sin x$ have one solution in $[\frac{-\pi }{4},\frac{\pi }{4}]whichis\left(\frac{\pi }{4}\right)$

And $tanx=\frac{-1}{2}$have one solution in $[\frac{-\pi }{4},\frac{\pi }{4}]whichis{\tan }^{-1}\left(\frac{-1}{2}\right)$