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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  \[\omega \] is a complex cube root of unity, then \[(1+\omega )(1+{{\omega }^{2}})\] \[(1+{{\omega }^{4}})(1+{{\omega }^{8}})...\]to \[2n\] factors = [AMU 2000]

    A)  0

    B) 1

    C) \[-1\]

    D) None of these

    Correct Answer: B

    Solution :

    \[(1+\omega )(1+{{\omega }^{2}})(1+{{\omega }^{4}})(1+{{\omega }^{8}})......\]upto \[2n\] factors \[=(-{{\omega }^{2}})(-\omega )(1+\omega )(1+{{\omega }^{2}}).....\]upto \[2n\] factors \[=1.1.1.....\]upto \[n\] factors = 1


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