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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 \[(x-y)(x\omega -y)\] \[(x{{\omega }^{2}}-y)=\]

    A) \[{{x}^{2}}+{{y}^{2}}\]

    B) \[{{x}^{2}}-{{y}^{2}}\]

    C) \[{{x}^{3}}-{{y}^{3}}\]

    D) \[{{x}^{3}}+{{y}^{3}}\]

    Correct Answer: C

    Solution :

    \[(x-y)(x\omega -y)(x{{\omega }^{2}}-y)\] \[=({{x}^{2}}\omega -xy-xy\omega +{{y}^{2}})(x{{\omega }^{2}}-y)\] \[={{x}^{3}}-{{x}^{2}}y(1+\omega +{{\omega }^{2}})+x{{y}^{2}}(1+\omega +{{\omega }^{2}})-{{y}^{3}}\] \[={{x}^{3}}-{{y}^{3}}\] \[(\because 1+\omega +{{\omega }^{2}}=0)\]


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