11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer Let \[z,w\]be complex numbers such that \[\overline{z}+i\overline{w}=0\]and \[arg\,\,zw=\pi \]. Then arg z equals [AIEEE 2004]

    A) \[5\pi /4\]

    B) \[\pi /2\]

    C) \[3\pi /4\]

    D) \[\pi /4\]

    Correct Answer: C

    Solution :

    Given that arg  zw =\[\pi \]            .....(i) \[\bar{z}+i\bar{\omega }=0\Rightarrow \bar{z}=-i\bar{\omega }\]\[\Rightarrow z=i\omega \]\[\Rightarrow \omega =-iz\] From (i), arg\[(-i{{z}^{2}})=\pi \] \[arg\ (-i)+2arg(z)=\pi \] ; \[\frac{-\pi }{2}+2\ arg(z)=\pi \] \[2\,arg\,(z)=\frac{3\pi }{2}\]; \[a\,rg(z)=\frac{3\pi }{4}\]

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