Question

Let $\Delta=\begin{vmatrix}\sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \\ \cos \theta \cos \phi & \cos \theta \sin \phi & -\sin \theta \\ -\sin \theta \sin \phi & \sin \theta \cos \phi & 0\end{vmatrix}$, then

• A. $\Delta$ is independent of $\theta$
• B. $\left.\frac{ d \Delta}{ d \theta}\right]_{\theta=\pi / 2}=0$
• C. $\Delta$ is a constant
• D. None of these

Answer 1

Answer: (B)

Applying $C _{1} \rightarrow C _{1}-(\cot \phi) C _{2}$, we get
$\Delta=\begin{vmatrix}0 & \sin \theta \sin \phi & \cos \theta \\ 0 & \cos \theta \sin \phi & -\sin \theta \\ -\sin \theta / \sin \phi & \sin \theta \cos \phi & 0\end{vmatrix}$
On expanding along $C _{1}$, we get
$\Delta=-\frac{\sin \theta}{\sin \phi}\left[\sin \phi \sin ^{2} \theta-\cos ^{2} \theta \sin \phi\right]$
$=\sin \theta$, This is independent of $\phi$.
Also, $\left.\frac{ d \Delta}{ d \theta}=\cos \theta \Rightarrow \frac{ d \Delta}{ d \theta}\right]_{\theta=\pi / 2}=\cos \left(\frac{\pi}{2}\right)=0$