Question

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

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (C)

2

Answer 2

$\begin{vmatrix}cos\,x&sin\,x&sin\,x\\\ sin\,x&cos\,x&sin\,x\\\ sin\,x&sin\,x&cos\,x\end{vmatrix}=0$
$R_{1} \rightarrow R_{1} -R_{2}$
$R_{2} \rightarrow R_{2}-R_{3}$
$\begin{vmatrix}cos\,x-sin\,x&sin\,x-cos\,x&0\\\ 0&cos\,x-sin\,x&sin\,x-cos\,x\\\ sin\,x&sin\,x&cos\,x\end{vmatrix}=0$
$C_{2} \rightarrow C_{2}+C_{1}$
$\left(cos\,x-sin\,x\right)\begin{vmatrix}1&0&0\\\ 0&cos\,x-sin\,x&sin\,x-cos\,x\\\ sin\,x&2\,sin\,x&cos\,x\end{vmatrix}=0$
Expanding using first row
$\left(2\,sin\,x+cos\,x\right)\left(sin\,x-cos\,x\right)^{2}=0$
$tan\,x=\cdot\frac{1}{2}$ or $tan\,x=1$
Hence two solutions are there in $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$