Question

The value of $\theta$ for which the system of equations $\left(\sin 3\theta \right)x-2y+3z=0,\left(\cos 2\theta \right)x+8y-7z=0$ and $2x+14y-11z=0$ has a non-trivial solution, is
($n\in Z)$

• A. $n\pi$
• B. $n\pi +(-1{)}^n\frac{\pi }{3}$
• C. $n\pi +(-1{)}^n\frac{\pi }{8}$
• D. $n\pi \pm \frac{\pi }{8}$

Answer 1

The correct option is A $n\pi$

$\left|\begin{array}{ccc}\sin 3\theta & -2& 3\\ \cos 2\theta & 8& -7\\ 2& 14& -11\end{array}\right|=0\Rightarrow 10\sin 3\theta +2(-11\cos 2\theta +14)+3(14\cos 2\theta -16)=0\Rightarrow 10\sin 3\theta +20\cos 2\theta -20=0\Rightarrow 3\sin \theta -4{\sin }^3\theta -2\left(2{\sin }^2\theta \right)=0\Rightarrow \sin \theta (3-4{\sin }^2\theta -4\sin \theta )=0\Rightarrow \sin \theta (2\sin \theta -1)(2\sin \theta +3)=0\Rightarrow \sin \theta =0\text{or}\sin \theta =\frac{1}{2}(\because \sin \theta \in [-1,1\left]\right)\Rightarrow \theta =n\pi \text{or}\theta =n\pi +(-1{)}^n\frac{\pi }{6}$