Question

Which of the following are true?

• A. $\overline{\left(\overline{z}\right)}=z$
• B. $z\overline{z}\in R$
• C. $z+\overline{z}=0\Rightarrow z$ is purely real
• D. $z-\overline{z}=2Im\left(z\right)$

Answer 1

Answer: (A), (B)

$z\overline{z}\in R$

Let $z=a+bi$ we will check all the options one bye one

$\left(a\right)\overline{z}=a-bi$
$\Rightarrow \overline{\left(\overline{z}\right)}=\overline{a-bi}$
$\Rightarrow \overline{\left(\overline{z}\right)}=a+bi$

$\left(b\right)z\overline{z}=(a+bi)(a-bi)$
$\Rightarrow z\overline{z}=a^2+b^2$
$\Rightarrow z\overline{z}\in R$

$\left(c\right)z-\overline{z}=(a+bi)-(a-bi)$
$\Rightarrow z-\overline{z}=2bi$
$\Rightarrow z-\overline{z}=2Im\left(z\right)i$

$\left(c\right)z+\overline{z}=0$
$\Rightarrow (a+bi)+(a-bi)=0$
$\Rightarrow a=0\Rightarrow \text{purely imaginary}$

Hence, option $a,b$ are correct options.