Question

Consider the system of linear equation in x, y, z.
$\left(sin3\theta \right)x-y+z=0$
$\left(cos2\theta \right)x+4y+3z=0$
$2x+7y+7z=0$
Find the values of $\theta$ for which this system has non-trivial solution

• A. $\theta =n\pi +(-1{)}^n\frac{\pi }{6}\forall n\in I$
• B. $\theta =\left(m\right)\pi ,m\in I$
• C. $\theta =(m+1/2)\pi ,m\in I$
• D. $\theta =n\pi +(-1{)}^n\frac{\pi }{3}\forall n\in I$

Answer 1

Answer: (A), (B)

$\theta =n\pi +(-1{)}^n\frac{\pi }{6}\forall n\in I$
$\theta =\left(m\right)\pi ,m\in I$

The system of the equation has a non-trivial solution if $△=0$

$\Rightarrow \left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$

Expanding along $C_1$, we get

$\sin 3\theta .(28-21)-\cos 2\theta (-7-7)+2(-3-4)=0$

$\Rightarrow 7\sin 3\theta +14\cos 2\theta -14=0$

$\Rightarrow \sin 3\theta +2\cos 2\theta -1=0$

$\Rightarrow 3\sin \theta -4{\sin }^3\theta +2(1-2{\sin }^2\theta )-2=0$

$\Rightarrow \sin \theta ({\sin }^2\theta +4\sin \theta -3)=0$

$\Rightarrow \sin \theta (2\sin \theta -1)(2\sin \theta +3)=0$

$\Rightarrow \sin \theta =0,\sin \theta =\frac{1}{2}$ (neglecting $\sin \theta =-\frac{3}{2}$)

$\Rightarrow \theta =n\pi ,n\pi +{(-1)}^n\frac{\pi }{6}n\in Z$