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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers1 Mark
MCQ
If $z$ is a non $-$ zero complex number, then $\Big|\frac{|\bar{\text{z}}}|^2{\text{z}}\bar{\text{z}}\Big|$ is equal to :
  • ✓
    $\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$
  • B
    $\big|{\text{z}}\big|$
  • C
    $\big|{\bar{\text{z}}}\big|$
  • D
    none of these
✓

Answer

Correct option: A.
$\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$
$\Big|\frac{|\bar{\text{z}}|^2}{\text{z}\bar{\text{z}}}\Big|=\Big|\frac{|\bar{\text{z}}|^2}{|\text{z}|^2}\Big| \ \Big(\because\text{z}\bar{\text{z}}=|\text{z}|^2\Big)$
Let $\text{z}=\text{a}+\text{ib}$
$\Rightarrow|\text{z}|=\sqrt{\text{a}^2+\text{b}^2}$
Let $\bar{\text{z}}=\text{a}-\text{ib}$
$\Rightarrow|\bar{\text{z}}|=\sqrt{\text{a}^2+\text{b}^2}$
$\therefore\Big|\frac{|\bar{\text{z}}|^2}{\text{z}\bar{\text{z}}}\Big|=\Big|\frac{|\bar{\text{z}}|^2}{|\text{z}|^2}\Big|$
$=\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$

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