If z is a complex number, then: - Vidyadip

MCQ

If $z$ is $a$ complex number$,$ then:

A
$|\text{z}^2|>|\text{z}|^2$
✓
$|\text{z}^2|=|\text{z}|^2$
C
$|\text{z}^2|<|\text{z}|^2$
D
$|\text{z}^2|\geq|\text{z}|^2$

✓

Answer

Correct option: B.

$|\text{z}^2|=|\text{z}|^2$

$|z|^2 = |z^2|$
Let $z = x + yi$
$|z| = |x + yi|$ and $|z|^2 = |x + yi|^2$
$\Rightarrow |z|^2 = x^2 + y^2 .....(i)$
Now$, z^2 = x^2 + y^2i^2 + 2xyi$
$z^2 = x^2 - y^2+ 2xyi$
$|\text{z}^2|=\sqrt{(\text{x}^2-\text{y}^2)^2+(2\text{xy})^2}$
$=\sqrt{\text{x}^4+\text{y}^4-2\text{x}^2\text{y}^2+4\text{x}^2\text{y}^2}$
$=\sqrt{\text{x}^4+\text{y}^4+2\text{x}^2\text{y}^2}$
$=\sqrt{(\text{x}^2+\text{y}^2)^2}$
So$, |\text{z}|^2=\text{x}^2+\text{y}^2=|\text{z}|^2$
So$, |\text{z}|^2=|\text{z}^2|$

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