Question

The smallest positive integer n for which $(1 + i)^{2n} = (1 - i)^{2n}$ is :

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

Given: $\left[\frac{\left(1+i\right)}{\left(1-i\right)}\right]^{2n} = 1\quad...\left(i\right)$
Let $A = \frac{1+i}{1-i}\quad ...\left(ii\right)$
So $A = \frac{1+i}{1-i} \times\frac{1+i}{1+i} = \frac{2i}{1-i^{2}}$
Now $i^{2} = - 1$ and $i^{4} = 1$
$\therefore \quad A = \frac{2i}{2} = i$
So, from equations $\left(i\right)$ and $\left(ii\right)$,
$\left(i\right)^{2n} = 1 = i^{4}$
$\Rightarrow \quad2n = 4 \quad \therefore \quad n = 2.$