If bar z be conjugate of complex number z , then which of following relations is false

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4-1.Complex numbers

normal

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}} $

$arg\,z = arg\,\bar z$

Solution

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

Standard 11

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