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Complex Numbers and Quadratic Equations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers and Quadratic Equations1 Mark
MCQ
Choose the correct answer. 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.
$|\text{z}^2|=|\text{z}|^2$
Let $z=x+y i$
$|z|=|x+y i|$ and $|z|^2=|x+y i|^2$
$\Rightarrow|z|^2=x^2+y^2 \ldots \text { (i) }$
Now, $z^2=x^2+y^2 i^2+2 x y i$
$z^2=x^2-y^2+2 x y i$
$|\text{z}^2|=\sqrt{(\text{x}^2-\text{y}^2)^2+(2\text{xy})^2}$
$=\sqrt{\text{x}^4+\text{y}^4-2\text{x}^2\text{y}^2+4\text{x}^2\text{y}^2}$
$=\sqrt{\text{x}^4+\text{y}^4+2\text{x}^2\text{y}^2}$
$=\sqrt{(\text{x}^2+\text{y}^2)^2}$
So, $|\text{z}|^2=\text{x}^2+\text{y}^2=|\text{z}|^2$
So, $|\text{z}|^2=|\text{z}^2|$

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