Q. If $\overline{\mathcal{z}}$ be the conjugate of the complex number $\mathcal{z}$, then which of the following relations is false?
A) $\left| \mathcal{z} \right| = |\overline{\mathcal{z}}|$
B) $\mathcal{z.\ }\overline{\mathcal{z}} = \left| \overline{\mathcal{z}} \right|^{2}$
C) $\overline{\mathcal{z}_{1} + \mathcal{z}_{2}} = {\overline{\mathcal{z}}}_{1} + {\overline{\mathcal{z}}}_{2}$
D) $arg\ \left( \mathcal{z} \right) = \arg{(\overline{\mathcal{z}})}$
Explanation
(d)
Let $\ \mathcal{z =}x + iy\ \Rightarrow \ \overline{\mathcal{z}} = x - iy$
Since, $\arg{\left( \mathcal{z} \right) = \tan^{- 1}\frac{y}{x}}$
and $\arg{\left( \overline{\mathcal{z}} \right) = \tan^{- 1}\left( \frac{- y}{x} \right)}$
$\Rightarrow \ \arg{\left( \mathcal{z} \right) \neq \arg{(\overline{\mathcal{z}})}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.