Question

If the determinant $\Delta =\left|\begin{array}{ccc}3& -2& \sin 3\theta \\ -7& 8& \cos 2\theta \\ -11& 14& 2\end{array}\right|=0$, then the value of $\sin \theta$ is

• A. $\frac{1}{3}$ or $1$
• B. $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{3}}{2}$
• C. $0$ or $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is C $0$ or $\frac{1}{2}$

Applying $R_2\to R_2+4R_1$ and $R_3\to R_3+7R_1$ we get
$\left|\begin{array}{ccc}3& -2& \sin 3\theta \\ 5& 0& \cos 2\theta +4\sin 3\theta \\ 10& 0& 2+7\sin 3\theta \end{array}\right|=0$
$\Rightarrow 2[5(2+7\sin 3\theta )-10(\cos 2\theta +4\sin 3\theta )]=0$
$\Rightarrow 2+7\sin 3\theta -2\cos 2\theta -8\sin 3\theta =0$
$\Rightarrow 2-2\cos 2\theta -\sin 3\theta =0$
$\Rightarrow \sin \theta (4{\sin }^2\theta +4\sin \theta -3)=0$
$\Rightarrow \sin \theta =0$ or $(2\sin \theta -1)=0$ or $(2\sin \theta +3)=0$
$\Rightarrow \sin \theta =0$ or $\sin \theta =\frac{1}{2}$