JEE Main 2021 · Mathematics · Complex Numbers · easy

The least positive integer n for which ((√3+i)/(√3-i))^n=1 is

Single correct option

The least positive integer

n

for which

\left(\dfrac{\sqrt3+i}{\sqrt3-i}\right)^{n}=1

Show answer and solution

Answer: (C) $6$

Write

\sqrt3+i=2e^{i\pi/6}

and

\sqrt3-i=2e^{-i\pi/6}

. Their ratio is

e^{i\pi/3}

, a primitive sixth root of unity. Then

e^{in\pi/3}=1

requires

n\pi/3=2\pi k

, i.e.

n

a multiple of

6

. The least positive value is

n=6

. Using

n=3

gives

-1

, and

n=12

works but is not the least.

Original question written in the JEE Main 2021 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake