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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank De Moivre's theorem and Roots of unity

  • question_answer
    If \[{{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{n}}\] is an integer, then n is  [UPSEAT 2002]

    A) 1

    B) 2

    C) 3

    D) 4

    Correct Answer: C

    Solution :

     \[\frac{1+i\sqrt{3}}{1-i\sqrt{3}}=\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\,\left( \frac{1+i\sqrt{3}}{1+i\sqrt{3}} \right)=\frac{-2+i2\sqrt{3}}{4}\]  \[=\,\frac{-1+i\sqrt{3}}{2}=\omega \] \ \[{{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{n}}={{\omega }^{n}}={{\omega }^{3}}=1\Rightarrow n=3\].


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