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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 \[{{\alpha }^{4}}+{{\beta }^{4}}\] + \[\frac{1}{\alpha \beta }=\] [IIT 1977]

    A) 3

    B) 0

    C) 1

    D) 2

    Correct Answer: B

    Solution :

    Complex cube root of unity are \[1,\,\,\omega ,\,{{\omega }^{2}}\] Let\[\alpha =\omega ,\,\,\beta ={{\omega }^{2}}\]; Then  \[{{\alpha }^{4}}+{{\beta }^{4}}+{{\alpha }^{-1}}{{\beta }^{-1}}\]  \[={{\omega }^{4}}+{{({{\omega }^{2}})}^{4}}+({{\omega }^{-1}})\,\,{{({{\omega }^{2}})}^{-1}}=\omega +{{\omega }^{2}}+1=0\].


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