The number of values of $\theta\in(0,\pi)$  for which the system of linear equations $x + 3y + 7z = 0$$-x + 4y + 7z = 0$$(\sin3\theta)x + (\cos2\theta)y + 2z = 0$ has non-trival solution, is: Option 1)threeOption 2)fourOption 3)twoOption 4)one

Homogeneous system of linear equation -

$b=0$

- wherein

As we have learnt from the concept for non-trivial solution

$\Delta =\begin{vmatrix} 1 &3 &7 \\ -1 & 4 &7 \\ \sin 3\theta &\cos 2\theta & 2 \end{vmatrix}=0$

$\Delta =(8-7\cos 2\theta )-3(-2-7\sin 3\theta )+7(-\cos 2\theta -4\sin 3\theta )$

$=14-7\cos 2\theta+21\sin 3\theta-7\cos 2\theta -28\sin 3\theta$

$=14-14\cos 2\theta-7\sin 3\theta$

$=14-14(1-2\sin ^{2}\theta )-7(3\sin \theta -4\sin ^{3}\theta )$

$=-21\sin \theta+28\sin ^{3}\theta+28\sin ^{2}\theta$

$=7\sin \theta[-3+4\sin ^{2}\theta+4\sin \theta]$

$\sin \theta=0 \: or\: \sin \theta=\frac{1}{2}\: or\: \sin \theta=\frac{-3}{2}$

$for\: \: \theta \epsilon (0,\pi)$

$\theta =\frac{\pi}{6} \: \: and\: \: \frac{5\pi}{6}$

Option 1)

three

Option 2)

four

Option 3)

two

Option 4)

one

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