GATE CS 2014 Set 1 — Question 16

The function $f(\theta)$ is defined as a determinant. Notice the structure of the rows.
Row 1 depends on $\theta$.
Row 2 is the values of Row 1 evaluated at $\theta = \pi/6$.
Row 3 is the values of Row 1 evaluated at $\theta = \pi/3$.When $\theta = \pi/6$, Row 1 becomes identical to Row 2. A determinant with two identical rows is 0. Thus, $f(\pi/6) = 0$.
When $\theta = \pi/3$, Row 1 becomes identical to Row 3. Thus, $f(\pi/3) = 0$.Since $f(\theta)$ is a linear combination of differentiable functions ($\sin, \cos, \tan$) on the interval $[\pi/6, \pi/3]$, it is continuous and differentiable.
By Rolle's Theorem, since $f(\pi/6) = f(\pi/3) = 0$, there exists at least one $c \in (\pi/6, \pi/3)$ such that $f'(c) = 0$. So, statement (I) is TRUE.To check statement (II), we need to know if $f(\theta)$ is identically zero (constant). If it were constant zero, $f'(\theta)$ would always be 0.
Expanding the determinant along the first row:
$f(\theta) = \sin\theta(C_1) - \cos\theta(C_2) + \tan\theta(C_3)$, where $C_1, C_2, C_3$ are constants (minors).
Unless all coefficients are zero or the functions cancel out perfectly (which they don't for these specific values), the function is not a constant zero. For example, we can check a point like $\pi/4$. The rows would be linearly independent, so $f(\pi/4) \neq 0$.
Since $f(\theta)$ is not constant, its derivative is not identically zero. Thus, there exists some $\theta$ where $f'(\theta) \neq 0$. So, statement (II) is TRUE.Both statements are true.