The number of distinct real roots of
Question:

The number of distinct real roots of

$\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0 \quad$ in the interval

$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:

 

  1. 4

  2. 1

  3. 2

  4. 3


Correct Option: , 2

Solution:

$\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0, \frac{-\pi}{4} \leq x \leq \frac{\pi}{4}$

Apply : $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \& \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$

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

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

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

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

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