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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
    Find the value of \[{{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}=\] [UPSEAT 2002]

    A) 0

    B) 1

    C) \[\omega \]

    D) \[{{\omega }^{2}}\]

    Correct Answer: A

    Solution :

    \[{{[(1+\omega +{{\omega }^{2}})+\omega ]}^{3n}}-{{[(1+\omega +{{\omega }^{2}})+{{\omega }^{2}}]}^{3n}}\] \[={{\omega }^{3n}}-{{({{\omega }^{2}})}^{3n}}\]\[=\,{{({{\omega }^{3}})}^{n}}-{{({{\omega }^{3}})}^{2n}}\]\[={{1}^{n}}-{{1}^{2n}}=0\].


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