Question

Let
$\Delta =\left|\begin{array}{ccc}\frac{1}{\sin \theta \cos \varphi }& \frac{1}{\sin \theta \sin \varphi }& \frac{1}{\cos \theta }\\ \frac{-\cos \theta }{{\sin }^2\theta \cos \varphi }& \frac{-\cos \theta }{{\sin }^2\theta \sin \varphi }& \frac{\sin \theta }{{\cos }^2\theta }\\ \frac{\sin \varphi }{\sin \theta {\cos }^2\varphi }& \frac{-\cos \varphi }{\sin \theta {\sin }^2\varphi }& 0\end{array}\right|$ then

• A. $\Delta$ is dependent on $\theta$
• B. $\Delta$ is dependent on $\varphi$
• C. $\Delta$ is a constant
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $\Delta$ is dependent on $\theta$
B $\Delta$ is dependent on $\varphi$

Taking $\frac{1}{\sin \theta \cos \varphi },\frac{1}{\sin \theta \sin \varphi },\frac{1}{\cos \theta }$ common from $C_1,C_2,C_3$ respectively , we get

$\Delta =\frac{1}{{\sin }^2\theta \cos \theta \sin \varphi \cos \varphi }{\Delta }_1$

Where ${\Delta }_1=\left|\begin{array}{ccc}1& 1& 1\\ -\cot \theta & -\cot \theta & \tan \theta \\ \tan \varphi & -\cot \varphi & 0\end{array}\right|$

${\Delta }_1=\left|\begin{array}{ccc}0& 1& 1\\ 0& -\cot \theta & \tan \theta \\ \frac{1}{\sin \varphi \cos \varphi }& -\cot \varphi & 0\end{array}\right|$

Using $C_1\to C_1-C_2$ we get

Thus , ${\Delta }_1=\frac{1}{\sin \varphi \cos \varphi }[\tan \theta +\cot \theta ]$

$=\frac{1}{\sin \varphi \cos \varphi \sin \theta \cos \theta }$

Hence $\Delta =\frac{1}{{\sin }^3\theta {\cos }^2\theta {\sin }^2\varphi {\cos }^2\varphi }$