Question

If $\Delta =\left|\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \theta \sin \varphi & \sin \theta \cos \varphi & 0\end{array}\right|$, then

• A. $\Delta$ is independent of $\theta$
• B. $\Delta$ is independent of $\varphi$
• C. $\Delta$ is a constant
• D. $\frac{d\Delta }{d\theta }|{}_{\theta =\pi /2}{=0}^{}$

Answer 1

Answer: (B), (D)

The correct options are
B $\Delta$ is independent of $\varphi$
D $\frac{d\Delta }{d\theta }|{}_{\theta =\pi /2}{=0}^{}$

$\Delta =\left|\begin{array}{ccc}sin\theta cos\varphi & sin\theta sin\varphi & cos\theta \\ cos\theta cos\varphi & cos\theta sin\varphi & -sin\theta \\ -sin\theta sin\varphi & sin\theta cos\varphi & 0\end{array}\right|$
Expanding by $C_3$,
$\Delta =cos\theta \left(cos\theta sin\theta \right(co{s}^2\varphi +si{n}^2\varphi \left)\right)+sin\theta \left(si{n}^2\theta \right(co{s}^2\varphi +si{n}^2\varphi \left)\right)$
$=cos\theta \left(cos\theta sin\theta \right)+sin\theta \left(si{n}^2\theta \right)=sin\theta$
So, $\Delta$ is independent of $\varphi$
$\frac{d\Delta }{d\theta }=cos\theta {|}_{\frac{\pi }{2}}=0$