Question

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

• A. 0
• B. 2
• C. 1
• D. 3

Answer 1

Answer: (C)

1

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

$\sin x({\sin }^{2}x-{\cos }^{2}x)-\cos x(\cos x\sin x-{\cos }^{2}x)+\cos x({\cos }^{2}x-\cos x\sin x)=0$

${\sin }^{3}x-\sin x{\cos }^{2}x-{\cos }^{2}x\sin x+{\cos }^{3}x-{\cos }^{2}x\sin x=0$

$\sin x({\sin }^{2}x-{\cos }^{2}x)+2{\cos }^{2}x(\cos x-\sin x)=0$

$-\sin x(\sin x+\cos x)(\cos x-\sin x)+2{\cos }^{2}x(\cos x-\sin x)=0$

$(-\cos x-\sin x)[2{\cos }^{2}x-{\sin }^{2}x-\sin x\cos x]=0$

$\cos x-\sin x=0$

$\cos x=\sin x$

At $x=\frac{\pi }{4}$

$\therefore 1$ solution exists in $[\frac{-\pi }{4},\frac{\pi }{4}]$