Question

The number of values of $\theta \,\epsilon\, \left(0, \pi\right)$for which the system of linear equations
$x + 3y + 7z = 0$
$- x + 4y + 7z = 0$
$\left(sin 3\theta\right)x + \left(cos 2\theta\right)y +2z =0$
has a non-trivial solution, is

Answer 1

Since, the system of linear equations has, non-trivial solution then determinant of coefficient matrix $= 0$
$i. e.,\begin{vmatrix}sin 3\theta& cos 2\theta&2\\ 1&3&7\\ -1&4&7\end{vmatrix} = 0$
sin $3\theta \left(21- 28\right) - $cos $2\theta \left(7 + 7\right) + 2 \left(4 + 3\right) = 0$
sin $3\theta + 2$ cos $2\theta - 2 = 0$
$3$ sin $\theta - 4$ sin $^{3}\theta + 2 - 4$ sin $^{2}\theta - 2 = 0$
$4$ sin$^{3}\theta + 4$ sin $^{2}\theta - 3$ sin$\theta = 0$
sin$\theta \left(4 sin^{2}\theta + 4sin\theta - 3\right) = 0$
sin$\theta \left(4 sin^{2}\theta + 6 sin\theta - 2 sin\theta - 3\right) = 0$
sin $\theta \left[2 sin \theta \left(2 sin \theta -1\right) +3 \left(2 sin \theta -1\right)\right] =0$
sin $\theta\left(2 sin \theta-1\right)\left(2 sin \theta +3\right) =0$
sin $\theta = 0 sin \theta = \frac{1}{2}$ ($\because$ sin $\theta \ne -\frac{3}{2}$)
$\theta = \frac{\pi}{6}, \frac{5\pi}{6}$
Hence, for two values of $\theta$, system of equations has non-trivial solution