MCQ
Choose the correct answer.
If z is a complex number, then:
If z is a complex number, then:
- A$|\text{z}^2|>|\text{z}|^2$
- B$|\text{z}^2|=|\text{z}|^2$
- C$|\text{z}^2|<|\text{z}|^2$
- D$|\text{z}^2|\geq|\text{z}|^2$
Solution:
Let z = x + yi
|z| = |x + yi| and |z|2 = |x + yi|2
⇒ |z|2 = x2 + y2 .....(i)
Now, z2 = x2 + y2i2 + 2xyi
z2 = x2 - y2 + 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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