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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers1 Mark
Question
Write the conjugate of $\frac{2-\text{i}}{(1-2\text{i})^2}.$
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Answer

$\frac{2-\text{i}}{(1-2\text{i})^2}=\frac{2-\text{i}}{1+4\text{i}^2-4\text{i}}$
$=\frac{2-\text{i}}{1-4-4\text{i}}$
$=\frac{2-\text{i}}{-3-4\text{i}}$
$=\frac{-2+\text{i}}{3+4\text{i}}$
$=\frac{\text{i}-2}{3+4\text{i}}\times\frac{3-4\text{i}}{3-4\text{i}}$
$=\frac{3\text{i}-4\text{i}^2-6+8\text{i}}{3^2-4^2\text{i}^2}$
$=\frac{11\text{i}+4-6}{9+16}$
$=\frac{-2}{25}+\frac{11}{25}\text{i}$
$\therefore$ Conjugate of $\frac{2-\text{i}}{(1-2\text{i})^2}=\Big(\overline{-\frac{2}{25}+\frac{11}{25}\text{i}}\Big)=-\frac{2}{25}-\frac{11}{25}\text{i}$
Hence, Conjugate of $\frac{2-\text{i}}{(1-2\text{i})^2}$ is $-\frac{2}{25}-\frac{11}{25}\text{i}.$

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