Question

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

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

Answer 1

The correct option is C $\pi (n+\frac{{(-1)}^n}{6})$

Given system of equations
$x\sin 3\theta -y+z=0,x\cos 2\theta +4y+3z=0,2x+7y+7z=0$ are homogeneous system of linear equalities
since system has non trivial solution
$\therefore \left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0$
$\Rightarrow \sin 3\theta (28-21)+1(7\cos 2\theta -6)+(7\cos 2\theta -8)=0$
$\Rightarrow \sin \theta -4{\sin }^3\theta +2(1-2{\sin }^2\theta )-2=0$
either $\sin \theta =0or4{\sin }^2\theta +6\sin \theta -2\sin \theta =0$
$\Rightarrow (2\sin \theta -1)(2\sin \theta +3)=0$
$\therefore \sin \theta =\frac{1}{2},\sin \theta \ne \frac{3}{2}(\because \sin \theta >-1)$
$\therefore \theta =n\pi or\theta =n\pi +{(-1)}^n\frac{\pi }{6}\Rightarrow \theta =\pi [n+\frac{{(-1)}^n}{6}]$