Consider the system of equations in $x, y, z$ as
$x \sin 3θ-y+z=0$
$x \cos 2θ+ 4y + 3z = 0$
$2x + 7y+7z=0$
If this system has a non-trivial solution, then for any integer n, values of θ are given
Answer & explanation
Correct answer: option 3
The given system of equations has a non-trivial solution.
$∴\begin{vmatrix}\sin 3θ&-1&1\\\cos 2θ&4&3\\2&7&7\end{vmatrix}=0$
$⇒7 \sin 3θ+ 14 \cos 2θ-14=0$
$⇒3 \sin θ-4 \sin^3 θ+2-4 \sin^2θ-2=0$
$⇒\sin θ (4 \sin^2 θ + 4 \sin θ - 3) = 0$
$⇒\sin θ (2 \sin θ + 3) (2 \sin θ-1)=0$
$⇒\sin θ =0$ or, $\sin θ =\frac{1}{2}⇒ θ=nπ$ or, $θ=nπ+(-1)^n\frac{π}{6}$