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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  \[\alpha \] and \[\beta \] are imaginary cube roots of unity, then the value of  \[{{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }\],is [MP PET 1998]

    A) 1

    B) \[-1\]

    C) 0

    D) None of these

    Correct Answer: C

    Solution :

    Since \[\alpha \] and \[\beta \] are complex roots of unity, we may write \[\alpha =\omega ,\beta ={{\omega }^{2}}\] Hence, \[{{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }={{\omega }^{4}}+{{({{\omega }^{2}})}^{28}}+\frac{1}{\omega .{{\omega }^{2}}}\] \[=\omega +{{\omega }^{56}}+1=\omega +{{\omega }^{2}}+1=0\]


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