Question

If z is a complex number, then which of the following is not true

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

Answer 1

Answer: (C), (D)

The correct options are
C $argz=arg\overline{z}$
D $z^2={\overline{z}}^2$

Given,

$z$ is a complex no.

To find which statement is not true.

Statement$\left(1\right){\left|z\right|}^2={\left|z\right|}^2$

Let, $z=x+iy,\left|z\right|\sqrt{x^2+y^2}{\left|z\right|}^2=x^2+y^2$

Hence, it is true.

Statement $\left(2\right){\left|z\right|}^2={\left|\overline{z}\right|}^2{\left|z\right|}^2=\sqrt{{(x^2-y^2)}^2+{(2\times y)}^2}=\sqrt{x^4+y^4-2x^2{y}^2+2x^2{y}^2}{\left|z\right|}^2=\sqrt{x^4+y^4}\overline{z}=x-iy\left|\overline{z}\right|=\sqrt{x^2+y^2}$

Hence it is not true.

Statement $\left(3\right)argz=arg\overline{z}z=x+iy\overline{z}=x-iyargz={\tan }^{-1}\left(\frac{y}{x}\right)arg\overline{z}={\tan }^{-1}\left(\frac{-y}{x}\right)$
Hence,$argz\ne arg\overline{z}$

This statement is not true.

Statement$\left(4\right)z^2={\overline{z}}^2{z}^2=(x+iy{)}^2{\overline{z}}^2=(x-iy{)}^2{z}^2=x^2-y^2+2xyi{\overline{z}}^2=x^2-y^2-2xyiHence,z^2\ne {\overline{z}}^2$

This statement is not true.