Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations
$\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$
$\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$
$\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$
has a non-trivial solution, then the value of $\theta$ is :
Correct Option: 2,
Case-I
$\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 3 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 3 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 3 \theta\end{array}\right|=0$
$\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}$
$\left|\begin{array}{ccc}2 & \sin ^{2} \theta & 4 \sin 3 \theta \\ 2 & 1+\sin ^{2} \theta & 4 \sin 3 \theta \\ 1 & \sin ^{2} \theta & 1+4 \sin 3 \theta\end{array}\right|=0$
$\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}, \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$
$\left|\begin{array}{ccc}0 & -1 & 0 \\ 1 & 1 & -1 \\ 1 & \sin ^{2} \theta & 1+4 \sin ^{3} \theta\end{array}\right|=0$
or $4 \sin 3 \theta=-2$
$\sin 3 \theta=-\frac{1}{2}$
$\theta=\frac{7 \pi}{18}$
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