Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $\left(\frac{1+ i \sqrt{3}}{1- i \sqrt{3}}\right)^{ n }$ is an integer, then n is
A
1
B
2
✓
3
D
4
✓
Answer
Correct option: C.
3
(C) $\frac{1+i \sqrt{3}}{1-i \sqrt{3}}=\left(\frac{1+i \sqrt{3}}{1-i \sqrt{3}}\right)\left(\frac{1+i \sqrt{3}}{1+i \sqrt{3}}\right)$ $=\frac{-2+ i 2 \sqrt{3}}{4}$ $=\frac{-1+i \sqrt{3}}{2}=\omega$ $\therefore \quad\left(\frac{1+ i \sqrt{3}}{1- i \sqrt{3}}\right)^{ n }=\omega^{ n }=\omega^3=1 \Rightarrow n =3$
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