IIT-JEE 2001 ScreeningQuestion 1
The number of distinct real roots of $\begin{vmatrix}\sin x&\cos x&\cos x\\ \cos x&\sin x&\cos x\\ \cos x&\cos x&\sin x\end{vmatrix}=0$ in the interval $-\dfrac\pi4\le x\le\dfrac\pi4$ is
Determinant
Solution
The determinant is $(\sin x-\cos x)^2(\sin x+2\cos x)$. In $[-\pi/4,\pi/4]$, $\sin x=\cos x$ only at $x=\pi/4$, and $\tan x=-2$ is outside. So exactly one root.