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Complex Numbers — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers3 Marks
Question
Solve the system of equations $\text{Re}(\text{z}^2)=0,|\text{z}|=2.$
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Answer

$\text{Re}(\text{z}^2)=0,|\text{z}|=2$
Let $\text{z}=\text{x}+\text{iy}$
$\text{z}^2=0$
$\Rightarrow(\text{x}+\text{iy})^2=0$
$\Rightarrow\text{x}^2+2\text{ixy}-\text{y}^2=0$
$\Rightarrow\text{x}^2-\text{y}^2=0 \ ...(\text{i}),$ which is the real part of $(\text{x}+\text{iy})^2.$
$|\text{z}|=2$
$\Rightarrow\sqrt{\text{x}^2+\text{y}^2}=2$
$\Rightarrow\text{x}^2+\text{y}^2=4 \ ...(\text{ii})$
Adding (i) and (ii), we get
$2\text{x}^2=4$
$\Rightarrow\text{x}^2=2$
$\Rightarrow\text{x}=\pm\sqrt{2},\text{y}=\pm\sqrt{2}$
$\text{x}+\text{iy}=\sqrt{2}+\text{i}\sqrt{2}$
$=\sqrt{2}-\text{i}\sqrt{2}$
$=\sqrt{2}+\text{i}\sqrt{2}$

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