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The number of distinct real roots of the equaiton, $\left|\begin{array}{*{20}{c}}
{\cos \,\,x}&{\sin \,\,x}&{\sin \,\,x}\\
{\sin \,\,x}&{\cos \,\,x}&{\sin \,\,x}\\
{\sin \,\,x}&{\sin \,\,x}&{\cos \,\,x}
\end{array}\right|\,\, = \,\,0$ in the interval $\left[ { - \frac{\pi }{4},\frac{\pi }{4}} \right]$ is
$1$
$4$
$2$
$3$
Solution
$\begin{array}{*{20}{c}}
{\cos x}&{\sin x}&{\sin x}\\
{\sin x}&{\cos x}&{\sin x}\\
{\sin x}&{\sin x}&{\cos x}
\end{array} = 0$
${R_1} \to {R_1} – {R_2}$
${R_2} \to {R_2} – {R_3}$
$\begin{array}{*{20}{c}}
{\cos x – \sin x}&{\sin x – \cos x}&0\\
0&{\cos x – \sin x}&{\sin x – \cos x}\\
{\sin x}&{\sin x}&{\cos x}
\end{array} = 0$
${C_2} \to {C_2} + {C_3}$
$\begin{array}{*{20}{c}}
{\cos x – \sin x}&{\sin x – \cos x}&0\\
0&0&{\sin x – \cos x}\\
{\sin x}&{\sin x}&{\cos x}
\end{array} = 0$
Expanding using second row
$2\sin x{\left( {\sin x – \cos x} \right)^2} = 0$
$\sin x = 0$ or $\sin x = \cos x$
$x = 0$ or $x = \frac{\pi }{4}$
