If $0<x<\pi$ and the matrix $\left[\begin{array}{cc}4 \sin x & -1 \\ -3 & \sin x\end{array}\right]$ is singular, then the values of x are :

$\frac{\pi}{3}, \frac{2 \pi}{3}$

Matrix is singular

⇒  $\left[\begin{array}{cc}4 \sin x & -1 \\ -3 & \sin x\end{array}\right] = 0$

⇒  4 sin²x - 3 = 0

⇒  $\sin^2x = \frac{3}{4}$

⇒  $\sin x = \frac{\sqrt{3}}{2}, \frac{-\sqrt{3}}{2} \Rightarrow x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}$  but as it is given $0<x<\pi$

so only first two values chosen

Option: 1