Question

Consider the system of equations in $x,y,z$ as $x\sin 3\theta -y+z=0$, $x\cos 2\theta +4y+3z=0$ and $2x+7y+7z=0$. If the system has a non-trivial solution, then for any integer $n$, the values of $\theta$ are given by

• A. $[n+\frac{{(-1)}^n}{3}]\pi$
• B. $[n+\frac{{(-1)}^n}{4}]\pi$
• C. $[n+\frac{{(-1)}^n}{6}]\pi$
• D. $\frac{n\pi }{2}$

Answer 1

Answer: (C)

$[n+\frac{{(-1)}^n}{6}]\pi$

The given system of equations has a non-trivial solution.

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

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

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

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

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

$\Rightarrow \sin \theta =0\sin \theta =\left(\frac{1}{2}\right)$

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