Question

If $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$, the number of distinct real roots of $\Delta=0$ $\Delta=\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} \text { is }$

• A. 0
• B. 2
• C. 1
• D. 3

Answer 1

Answer: (C)

Using $C_1 \rightarrow C_1+C_2+C_3$, we get
$\Delta=(\sin x+2 \cos x)\begin{vmatrix}1 & \cos x & \cos x \\1 & \sin x & \cos x \\1 & \cos x & \sin x\end{vmatrix}$
Applying $R_2 \rightarrow R_2-R_1, R_3 \rightarrow R_3-R_1$, we get
$\Delta=(\sin x+2 \cos x)\begin{vmatrix}1 & \cos x & \cos x \\0 & \sin -\cos x & 0 \\0 & 0 & \sin x-\cos x\end{vmatrix}$
$=(\sin x+2 \cos x)(\sin x-\cos x)^2$
$\Delta=0 \Rightarrow \tan x=-2, \tan x=1 .$
$\text { As }-\frac{\pi}{4} \leq x \leq \frac{\pi}{4},-1 \leq \tan x \leq 1$
Thus, $\tan x=1 \Rightarrow x=\pi / 4$.