Q. If $\left( \frac{1 + i}{1 - i} \right)^{x} = 1$, then
A) $x = 4n$, where $n\$ is any positive integer
B) $x = 2n$, where $n\$ is any positive integer
C) $x = 4n + 1$, where $n\$ is any positive integer
D) $x = 2n + 1$, where $n\$ is any positive integer
Explanation
(a)
$\left( \frac{1 + i}{1 - i} \right)^{x} = \left\lbrack \frac{(1 + i)(1 + i)}{(1 - i)(1 + i)} \right\rbrack^{x} = \left\lbrack \frac{(1 + i)}{1 - i^{2}} \right\rbrack^{x} = \left\lbrack \frac{1 - 1 + 2i}{2} \right\rbrack^{x}$
$\Rightarrow \ \ \ \ \ \left( \frac{1 + i}{1 - i} \right)^{x} = (i)^{x} = 1$ [given]
$\therefore\ \ \ \ \ x = 4n$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.