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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2005

  • question_answer
    The amplitude of \[\frac{1+i\sqrt{3}}{\sqrt{3}+i}\] is :

    A)  \[\frac{\pi }{3}\]                                             

    B)  \[\frac{\pi }{4}\]

    C)  \[\frac{2\pi }{3}\]                                           

    D)  \[\frac{\pi }{6}\]

    Correct Answer: D

    Solution :

    Let          \[z=\frac{1+i\sqrt{3}}{\sqrt{3}\,+i}\]                 \[=\frac{1+i\sqrt{3}}{(\sqrt{3}+i)}\times \frac{(\sqrt{3}-i)}{(\sqrt{3}-i)}\]                 \[=\frac{\sqrt{3}\,-i+3i+\sqrt{3}}{3+1}=\frac{\sqrt{3}+i}{2}\]                 amp \[(z)={{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]                                 \[=\frac{\pi }{6}\]


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