Question

If $\Delta =\left|\begin{array}{ccc}\arg z_1& \arg z_2& \arg z_3\\ \arg z_2& \arg z_3& \arg z_1\\ \arg z_3& \arg z_1& \arg z_2\end{array}\right|$
then $\Delta$ is divisible by

• A. $\arg (z_1+z_2+z_3)$
• B. $\arg z_1{z}_2{z}_3$
• C. $\arg z_1+\arg z_2+\arg z_3$
• D. $none$

Answer 1

Answer: (B), (C)

$\arg z_1{z}_2{z}_3$
$\arg z_1+\arg z_2+\arg z_3$

Ans. $\left(b\right).\left(c\right).$
Apply $C_1+C_2+C_3$ and take $\sum \arg z_1$ common.
$\therefore \Delta =\sum \left(\arg z_1\right)\left|\begin{array}{ccc}1& \arg z_2& \arg z_3\\ 1& \arg z_3& \arg z_1\\ 1& \arg z_1& \arg z_2\end{array}\right|$
Hence $\Delta$ is divisible by $\sum \arg z_1$
$\arg z_1+\arg z_2+\arg z_3=\arg \left(z_1{z}_2{z}_3\right).$