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Complex Numbers — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers2 Marks
Question
Express the following complex numbers in the standard form a + ib:
$\frac{1}{(2+\text{i})^2}$
✓

Answer

$\frac{1}{(2+\text{i})^2}=\frac{1}{2^2+(\text{i})^2+2\times2\times\text{i}}$
$=\frac{1}{4-1+4\text{i}}$
$=\frac{1}{3+4\text{i}}$
$\frac{1}{(3+4\text{i})}\times\frac{(3-4\text{i})}{(3-4\text{i})}$ [On rationalising the denominator]
$=\frac{3-4\text{i}}{3^2+4^2} \ \big[\because \ (\text{a}+\text{ib})(\text{a}-\text{ib})=\text{a}^2+\text{b}^2\big]$
$=\frac{3-4\text{i}}{25}$
$=\frac{3}{25}-\frac{4}{25}\text{i}$
$\therefore \ \frac{1}{(2+\text{i})^2}=\frac{3}{25}-\frac{4}{25}\text{i}$

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