Matrices and Determinants 4 Question 28
29. Consider the system of linear equations in $x, y, z$
$(\sin 3 \theta) x-y+z=0,(\cos 2 \theta) x+4 y+3 z=0$ and $2 x+7 y+7 z=0$
Find the values of $\theta$ for which this system has non-trivial solution.
(1986, 5M)
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Answer:
Correct Answer: 29. $m<-\frac{15}{2}$ or $m>30$
Solution:
- The system of equations has non-trivial solution, if $\Delta=0$.
$\Rightarrow \quad\left|\begin{array}{ccc}\sin 3 \theta & -1 & 1 \\ \cos 2 \theta & 4 & 3 \\ 2 & 7 & 7\end{array}\right|=0$
Expanding along $C_{1}$, we get
$ \begin{array}{lr} \sin 3 \theta \cdot(28-21) & -\cos 2 \theta(-7-7)+2(-3-4)=0 \\ \Rightarrow & 7 \sin 3 \theta+14 \cos 2 \theta-14=0 \\ \Rightarrow & \sin 3 \theta+2 \cos 2 \theta-2=0 \\ \Rightarrow & 3 \sin \theta-4 \sin ^{3} \theta+2\left(1-2 \sin ^{2} \theta\right)-2=0 \\ \Rightarrow & \sin \theta\left(4 \sin ^{2} \theta+4 \sin \theta-3\right)=0 \\ \Rightarrow & \sin \theta(2 \sin \theta-1)(2 \sin \theta+3)=0 \\ \Rightarrow & \sin \theta=0, \sin \theta=\frac{1}{2} \\ \Rightarrow & {[\text { neglecting } \sin \theta=-3 / 2]} \\ & \theta=n \pi, n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z \end{array} $
