The number of values of θ ∈ (0, π) for which the system of linear...

For non-trivial solutions

\left|\begin{cases}1 & 3 & 7 \\ - 1 & 4 & 7 \\ \sin 3\theta & \cos 2\theta & 2\end{cases}\right| = 0

$⇒ 4\sin^{3}\theta + 4\sin^{2}\theta - 3sin\theta = 0$

$⇒ sin\theta = 0 or$sin\theta = \dfrac{1}{2} or sin\theta = - \dfrac{3}{2}$<br /><br />Since$\theta \in \left(0, \pi\right)

$⇒ \theta =$\dfrac{\pi}{6}$and$\dfrac{5\pi}{6} will satisfy the equation$</p><p>because$sin\theta = - \dfrac{3}{2} i$s not possible as range of$sin\theta$is \left[- 1, 1\right]$

$⇒$ Number of $\theta$ ' $s = 2$