Choose the correct answer. If ( 1+i 1-i )^ x =1, then: - Vidyadip

MCQ

Choose the correct answer. If $\Big(\frac{1+\text{i}}{1-\text{i}}\Big)^{\text{x}}=1,$ then:

A
$x = 2n + 1$
✓
$x = 4n$
C
$x = 2n$
D
$x = 4n + 1,$ where $n \in N$

✓

Answer

Correct option: B.

$x = 4n$

$\Rightarrow\Big[\frac{(1+\text{i})(1+\text{i})}{(1-\text{i})(1+\text{i})}\Big]^{\text{x}}=1$
$\Rightarrow\Big[\frac{1+2\text{i}+\text{i}^2}{1-\text{i}^2}\Big]^{\text{x}}=1$
$\Rightarrow\Big[\frac{2\text{i}}{1+1}\Big]^{\text{x}}=1$
$\Rightarrow\text{i}^{\text{x}}=1$
$\Rightarrow\text{x}=4\text{n},\text{n}\in\text{N}$

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