Question

If the determinant $\Delta = \begin{vmatrix}3&-2&\sin\,3\theta\\ -7&8&\cos\,2\theta\\ -11&14&2\end{vmatrix} = 0$, 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

Answer: (C)

Applying $R_{2} \rightarrow R_{2}+4 R_{1}$
and $R_{3} \rightarrow R_{3}+7 R_{1}$ we get
$\begin{vmatrix} 3 & -2 & \sin 3 \theta \\ 5 & 0 & \cos 2 \theta+4 \sin 3 \theta \\ 10 & 0 & 2+7 \sin 3 \theta\end{vmatrix}=0$
$\Rightarrow 2[5(2+7 \sin 3 \theta)-10(\cos 2 \theta+4 \sin 3 \theta)]=0$
$\Rightarrow 2+7 \sin 3 \theta-2 \cos 2 \theta-8 \sin 3 \theta=0$
$\Rightarrow 2-2 \cos 2 \theta-\sin 3 \theta=0$
$\Rightarrow \sin \theta\left(4 \sin ^{2} \theta+4 \sin \theta-3\right)=0$
$\Rightarrow \sin \theta=0$ or $(2 \sin \theta-1)=0$ or $(2 \sin \theta+3)=0$
$\Rightarrow \sin \theta=0$ or $\sin \theta=\frac{1}{2}$