The number of $\theta \in(0,4 \pi)$ for which the system of linear equations

$$ \begin{aligned} &3(\sin 3 \theta) x-y+z=2 \\\\ &3(\cos 2 \theta) x+4 y+3 z=3 \\\\ &6 x+7 y+7 z=9 \end{aligned} $$

has no solution, is :

<p>Given,</p> <p>$3(\sin 3\theta )x - y + z = 2$</p> <p>$3(\cos 2\theta )x + 4y + 3z = 3$</p> <p>$6x + 7y + 7z = 9$</p> <p>For no solutions determinant of coefficient will be = 0</p> <p>$\therefore$ $$D = \left| {\matrix{ {3\sin 3\theta } & { - 1} & 1 \cr {3\cos 2\theta } & 4 & 3 \cr 6 & 7 & 7 \cr } } \right| = 0$$</p> <p>$$ \Rightarrow 3\sin 3\theta (28 - 21) + 1(21\cos 2\theta - 18) + 1(21\cos 2\theta - 24) = 0$$</p> <p>$\Rightarrow 21\sin 3\theta + 42\cos 2\theta - 42 = 0$</p> <p>$\Rightarrow \sin 3\theta + 2\cos 2\theta - 2 = 0$</p> <p>$\Rightarrow 3\sin \theta - 4{\sin ^3}\theta + 2(1 - 2{\sin ^2}\theta ) - 2 = 0$</p> <p>$\Rightarrow 3\sin \theta - 4{\sin ^3}\theta - 4{\sin ^2}\theta = 0$</p> <p>$\Rightarrow 4{\sin ^3}\theta + 4{\sin ^2}\theta - 3\sin \theta = 0$</p> <p>$\Rightarrow \sin \theta (4{\sin ^2}\theta + 4\sin \theta - 3) = 0$</p> <p>$\therefore$ $\sin \theta = 0$</p> <p>$\Rightarrow \theta = \pi ,2\pi ,3\pi$ when $\theta \in (0,4\pi )$</p> <p>or,</p> <p>$4{\sin ^2}\theta + 4\sin \theta - 3 = 0$</p> <p>$\Rightarrow 4{\sin ^2}\theta + 6\sin \theta - 2\sin \theta - 3 = 0$</p> <p>$\Rightarrow 2\sin \theta (2\sin \theta + 3) - 1(2\sin \theta + 3) = 0$</p> <p>$\Rightarrow (2\sin \theta - 1)(2\sin \theta + 3) = 0$</p> <p>$\therefore$ $\sin \theta = {1 \over 2}$</p> <p>or,</p> <p>$\sin \theta = - {3 \over 2}$ [not possible as $\sin \in [ - 1,1]$]</p> <p>$\therefore$ $\sin \theta = {1 \over 2}$</p> <p>$$ \Rightarrow \theta = {\pi \over 6},{{5\pi } \over 6},{{13\pi } \over 6},{{17\pi } \over 6}$$</p> <p>$\therefore$ Possible values of $$\theta = \pi ,2\pi ,3\pi ,{\pi \over 6},{{5\pi } \over 6},{{13\pi } \over 6},{{17\pi } \over 6}$$</p> <p>$\therefore$ Total 7 values of $\theta$ possible.</p>