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KVPY Sample Paper KVPY Stream-SX Model Paper-4

  • question_answer
    If the shaded portion represents the set of complex numbers then which of the following set of complex numbers satisfy the inequality \[{{\tan }^{-1}}({{\log }_{3}}\left| 2z-1 \right|>{{\tan }^{-1}}(\log 3\left| 2z+1 \right|)\]

    A)           

    B)

    C)

    D)

    Correct Answer: B

    Solution :

    [B]
    \[{{\log }_{3}}\left| 2z-1 \right|>{{\log }_{3}}\left| 2z+1 \right|\]
    \[\Rightarrow (2z-1)(2\bar{z}-1)>(2z+1)(2\bar{z}+1)\]
    \[\Rightarrow 4{{\left| \,z\, \right|}^{2}}-2\,(z+\bar{z})+1>4{{\left| \,z\, \right|}^{2}}+2\,(z+\bar{z})+1\]
    \[\Rightarrow 4.2\operatorname{Re}(z)<0\]
    \[\Rightarrow \operatorname{Re}(z)<0\]


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