Question

Consider the system of linear equations in $x,y$ and $z$ given by $\left(\sin 3\theta \right)x-y+z=0,\left(\cos 2\theta \right)x+4y+3z=0,2x+7y+7z=0$. Find the values of $\theta$ for which the system has a non-trivial solution.

• A. $n\pi$
• B. $n\pi +{(-1)}^n\pi /6;n\in Z$
• C. $n\pi +{(-1)}^n\pi /3;n\in Z$
• D. $\frac{n\pi }{2}$

Answer 1

Answer: (A), (B)

The correct options are
A $n\pi$
B $n\pi +{(-1)}^n\pi /6;n\in Z$

Given system has non-trivial solution.
$\Rightarrow \left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$
$\Rightarrow 7\sin 3\theta +7\cos 2\theta -6+7\cos 2\theta -8=0$
$\Rightarrow \sin 3\theta +2\cos 2\theta =2$
$3\sin \theta -4{\sin }^3\theta +2(1-2{\sin }^2\theta )=2$
$-\sin \theta (4{\sin }^2\theta +4\sin \theta -3)=$
$\Rightarrow \sin \theta =0$ and $\sin \theta =\frac{1}{2}$
$\therefore \theta =n\pi$ and $\theta =n\pi +{(-1)}^n\pi /6;n\in Z$
Hence, options A and B.