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Complex Numbers

If 1 - in =...

Question

If $(1 - i)^{n} = 2^{n}$ , then $n =$

A

1

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B

0

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C

-1

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D

none of these

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Solution

The correct option is B 0
If $(1 - i)^{n} = 2^{n} + 0 i$ ......... (i)
We know that if two complex numbers are equal, their moduli must also be equal,
Therefore from (i),
we have $| (1 - i)^{n} | = | 2^{n} | \Rightarrow | 1 - i |^{n} = | 2 |^{n}$ ,
$\Rightarrow {[\sqrt{1^{2} + (- 1)^{2}}]}^{n} = 2^{n} \Rightarrow (\sqrt{2})^{n} = 2^{n}$
$\Rightarrow 2^{n / 2} = 2^{n}$
$\Rightarrow \frac{n}{2} = n$
$\Rightarrow n = 0$

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