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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 \[\alpha \] is an imaginary cube root of unity, then for  \[n\in N\],  the value of \[{{\alpha }^{3n+1}}+{{\alpha }^{3n+3}}+{{\alpha }^{3n+5}}\] is [MP PET 1996; Pb. CET 2000]

    A) \[-1\]

    B) 0

    C) 1

    D) 3

    Correct Answer: B

    Solution :

    Since \[\alpha \] is an imaginary cube root of unity, let it be \[\omega \]then \[={{(\omega )}^{3n+1}}+{{(\omega )}^{3n+3}}+{{\omega }^{3n+5}}\] \[=\omega +1+{{\omega }^{5}},\,\,\,\,\,\{\because {{\omega }^{3n}}=1\]and \[{{\omega }^{3}}=1\}\]\[=\omega +1+{{\omega }^{2}}=0\]


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