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Algebra of Complex Numbers

If Re 1+iy ...

Question

If $R e {(1 + i y)}^{3} = - 26$ , where $y$ is a real number, then the value of $| y |$ is

A

2

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B

3

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C

4

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D

6

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E

9

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Solution

The correct option is C $3$
Let $z = {(1 + i y)}^{3}$
$= 1 + 3 i y + 3 i^{2} y^{2} + i^{3} y^{3}$
$= 1 + 3 i y - 3 y^{2} - i y^{3}$
$= (1 - 3 y^{2}) + (3 y - y^{3}) i$
Now, $Re (z) = - 26$
$\Rightarrow 1 - 3 y^{2} = - 26$
$\Rightarrow 27 = 3 y^{2} \Rightarrow y^{2} = 9$
$\Rightarrow y = \pm 3$ or $| y | = 3$

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