If $\bar z$ be the conjugate of the complex number $z$, then which of the following relations is false
$|z|\, = \,|\bar z|$
$z.\,\bar z = |\bar z{|^2}$
$\overline {{z_1} + {z_2}} = \overline {{z_1}} + \overline {{z_2}} $
$arg\,z = arg\,\bar z$
Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$
Let $S=\left\{Z \in C: \bar{z}=i\left(z^2+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z \in S}|z|^2$ is equal to
The amplitude of the complex number $z = \sin \alpha + i(1 - \cos \alpha )$ is
If $x+i y=\frac{a+i b}{a-i b},$ prove that $x^{2}+y^{2}=1$
$(z + a)(\bar z + a)$, where $a$ is real, is equivalent to