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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 \[z=\frac{\sqrt{3}+i}{-2}\], then \[{{z}^{69}}\] is equal to [RPET 2001]

    A) 1

    B) - 1

    C) i

    D) - i

    Correct Answer: C

    Solution :

    \[z=\frac{\sqrt{3}+i}{-2}\] \[\Rightarrow \] \[iz=-\frac{-1+\sqrt{3i}}{2}=-\omega \] \[\Rightarrow \] \[z=\frac{-\omega }{i}=i\omega \]\[\Rightarrow \]\[{{z}^{69}}={{i}^{69}}.{{\omega }^{69}}=i\] \[(\because {{\omega }^{3n}}={{i}^{4n}}=1)\]


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