Question

The number of values of $\theta ϵ\left(0,\pi \right)$for which the system of linear equations
$x+3y+7z=0$
$-x+4y+7z=0$
$\left(sin3\theta \right)x+\left(cos2\theta \right)y+2z=0$
has a non-trivial solution, is

Answer 1

Answer: 2

Since, the system of linear equations has, non-trivial solution then determinant of coefficient matrix $=0$
$i.e.,\left|\begin{array}{ccc}sin3\theta & cos2\theta & 2\\ 1& 3& 7\\ -1& 4& 7\end{array}\right|=0$
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$\theta \left(4si{n}^2\theta +4sin\theta -3\right)=0$
sin$\theta \left(4si{n}^2\theta +6sin\theta -2sin\theta -3\right)=0$
sin $\theta \left[2sin\theta \left(2sin\theta -1\right)+3\left(2sin\theta -1\right)\right]=0$
sin $\theta \left(2sin\theta -1\right)\left(2sin\theta +3\right)=0$
sin $\theta =0sin\theta =\frac{1}{2}$ ($\because$ 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