Complex Numbers And Quadratic Equations question 286
Question: If $ {{( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} )}^{n}} $ is an integer, then n is [UPSEAT 2002]
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: C
Solution:
$ \frac{1+i\sqrt{3}}{1-i\sqrt{3}}=( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} )( \frac{1+i\sqrt{3}}{1+i\sqrt{3}} )=\frac{-2+i2\sqrt{3}}{4} $ $ =\frac{-1+i\sqrt{3}}{2}=\omega $ \ $ {{( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} )}^{n}}={{\omega }^{n}}={{\omega }^{3}}=1\Rightarrow n=3 $ .
BETA
