Q85 / 160 · AP EAMCET 2014 · Mathematics

The least positive integer $n$ for which $(1+i)^n=(1-i)^n$, is

  • A
    $8$
  • B
    $2$
  • Correct answer
    C
    $4$
  • D
    $6$

Solution

Given, $(1+i)^n=(1-i)^n$ $$ \begin{aligned} & \Rightarrow \quad \frac{(1+i)^n}{(1-i)^n}=1 \\ & \Rightarrow \quad\left[\frac{(1+i) \times(1+i)}{(1-i) \times(1+i)}\right]^n=1 \\ & \Rightarrow \quad\left[\frac{1^2+i^2+2 i}{1^2-i^2}\right]^n=1 \\ & \Rightarrow \quad\left[\frac{1-1+2 i}{1+1}\right]^n=1 \\ & \Rightarrow \quad\left(\frac{2 i}{2}\right)^n=1 \\ & \Rightarrow \quad(i)^n=1 \\ & \therefore \quad n=4 \\ & \end{aligned} $$

Hence, the correct answer is option: 'C'.

Relevant Questions