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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 \[n\] is a positive integer not a multiple of 3, then \[1+{{\omega }^{n}}+{{\omega }^{2n}}\] = [MP PET 2004]

    A) 3

    B) 1

    C) 0

    D) None of these

    Correct Answer: C

    Solution :

    Let \[n=3k+1\] \[{{\omega }^{n}}+{{\omega }^{2n}}={{\omega }^{3k+1}}+{{\omega }^{2(3k+1)}}={{\omega }^{3k}}\omega +{{\omega }^{6k}}{{\omega }^{2}}\]                      \[={{({{\omega }^{3}})}^{k}}.\omega +{{({{\omega }^{3}})}^{2k}}\].\[{{\omega }^{2}}=\omega +{{\omega }^{2}}=-1\] Hence \[1+{{\omega }^{n}}+{{\omega }^{2n}}=1-1=0\]


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