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RAJASTHAN ­ PET Rajasthan PET Solved Paper-2011

  • question_answer
    Let\[{{z}_{1}}\]and\[{{z}_{2}}\]be complex numbers such that \[{{z}_{1}}\ne {{z}_{2}}\]and\[|{{z}_{1}}|=|{{z}_{2}}|.\]If\[{{z}_{1}}\]has positive real part and\[{{z}_{2}}\]has negative imaginary part, then\[\frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}}\]may be

    A)  zero

    B)  real and positive

    C)  real and negative

    D)  purely imaginary or zero

    Correct Answer: D

    Solution :

     \[\frac{{{z}_{1}}+{{z}_{2}}}{{{z}_{1}}-{{z}_{2}}}.\frac{{{\overline{z}}_{1}}+{{\overline{z}}_{2}}}{{{z}_{1}}-{{z}_{2}}}=\frac{2i\,\operatorname{Im}({{z}_{1}}.\,{{\overline{z}}_{2}})}{|{{z}_{1}}-{{z}_{2}}|}\] \[\Rightarrow \]Purely imaginary when\[\operatorname{Im}({{z}_{1}}.\,{{z}_{2}})\ne 0\]or zero.


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