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]

\[{{\log }_{3}}\left| 2z-1 \right|>{{\log }_{3}}\left| 2z+1 \right|\]

\[\Rightarrow (2z-1)(2\bar{z}-1)>(2z+1)(2\bar{z}+1)\]

\[\Rightarrow 4{{\left| \,z\, \right|}^{2}}-2\,(z+\bar{z})+1>4{{\left| \,z\, \right|}^{2}}+2\,(z+\bar{z})+1\]

\[\Rightarrow 4.2\operatorname{Re}(z)<0\]

\[\Rightarrow \operatorname{Re}(z)<0\]