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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 cube root of unity, then the value of \[{{(1-\omega +{{\omega }^{2}})}^{5}}+{{(1+\omega -{{\omega }^{2}})}^{5}}=\] [IIT 1965; MP PET 1997; RPET 1997]

    A) 16

    B) 32

    C) 48

    D) - 32

    Correct Answer: B

    Solution :

    \[{{(1-\omega +{{\omega }^{2}})}^{5}}+{{(1+\omega -{{\omega }^{2}})}^{5}}\] \[={{(-2\omega )}^{5}}+{{(-2{{\omega }^{2}})}^{5}}=-32{{\omega }^{3}}{{\omega }^{2}}-32{{({{\omega }^{3}})}^{3}}\omega \] \[=-32({{\omega }^{2}}+\omega )=-32(-1)=32\]


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