Question

If $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$, the number of distinct real roots of $\Delta =0$ $\Delta =\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|\text{ is }$

• A. 0
• B. 2
• C. 1
• D. 3

Answer 1

Answer: (C)

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