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The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is
$0$
$2$
$1$
$3$
Solution
(c) Here, $(2\cos x + \sin x)\,\left| {\,\begin{array}{*{20}{c}}1&{\cos x}&{\cos x}\\1&{\sin x}&{\cos x}\\1&{\cos x}&{\sin x}\end{array}\,} \right|\, = 0$
or $(2\cos x + \sin x)\,\left| {\,\begin{array}{*{20}{c}}1&{\cos x}&{\cos x}\\0&{\sin x – \cos x}&0\\0&0&{\sin x – \cos x}\end{array}\,\,} \right|\, = 0$
or $(2\cos x + \sin x){(\sin x – \cos x)^2} = 0$
$\therefore $ $\tan x = – 2,\,1.\,$ But $\tan x \ne – 2$ in $\left[ { – \frac{\pi }{4},\,\,\,\,\frac{\pi }{4}} \right]$
$\therefore $$\tan x = 1$. So, $x = \frac{\pi }{4}$.
