Question

If $\overline{z}$ be the conjugate of the complex number z, then which of the following relations is false
[MP PET 1987]

• A. $|z|=|\overline{z}|$
• B. $z.\overline{z}|\overline{z}{|}^2$
• C. $\overline{z_1}+\overline{z_2}=\overline{z_1}+\overline{z_2}$
• D. $argz=arg\overline{z}$

Answer 1

The correct option is D $argz=arg\overline{z}$

Let z = x +iy, z = x - iy
Since $arg\left(z\right)=\theta =ta{n}^{-1}\frac{y}{x}$
$arg\left(\overline{z}\right)=\theta =ta{n}^{-1}\left(\frac{-y}{x}\right)$
Thus $arg\left(z\right)\ne arg\left(\overline{z}\right).$