Let the function $$f(\theta) = \begin{vmatrix} \sin \theta & \cos \theta & an \theta \\ \sin(\pi/6) & \cos(\pi/6) & an(\pi/6) \\ \sin(\pi/3) & \cos(\pi/3) & an(\pi/3) \end{vmatrix}$$ where $\theta \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$ and $f^{\prime}(\theta)$ denote the derivative of $f$ with respect to $\theta$. Which of the following statements is/are TRUE? (I) There exists $\theta \in \left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ such that $f^{\prime}(\theta) = 0$. (II) There exists $\theta \in \left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ such that $f^{\prime}(\theta) \neq 0$.

Evaluating $f(\theta)$ at the boundary points:

• For $\theta = \frac{\pi}{6}$, the first and second rows of the determinant are identical, so $f\left(\frac{\pi}{6}\right) = 0$.

• For $\theta = \frac{\pi}{3}$, the first and third rows of the determinant are identical, so $f\left(\frac{\pi}{3}\right) = 0$.

Since $f(\theta)$ is continuous on $\left[\frac{\pi}{6}, \frac{\pi}{3}\right]$ and differentiable on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$, by Rolle's Theorem, there exists at least one $\theta \in \left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ such that:

Thus, statement (I) is TRUE.

Expanding the determinant along the first row gives a non-constant function of the form $f(\theta) = c_1 \sin\theta + c_2 \cos\theta + c_3 an\theta$. Since $f(\theta)$ is not constant over the interval, its derivative $f^{\prime}(\theta)$ is not identically zero everywhere on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$. Therefore, there must exist some $\theta \in \left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ such that:

Thus, statement (II) is also TRUE.

Hence, both statements (I) and (II) are TRUE, matching option C.