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

  • question_answer If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are complex numbers such that \[|{{z}_{1}}|\,=\,|{{z}_{2}}|\,=\] \[\,|{{z}_{3}}|\,=\] \[\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1\,,\] then\[\text{ }|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|\] is [MP PET 2004; IIT Screening 2000]

    A) Equal to 1

    B) Less than 1

    C) Greater than 3

    D) Equal to 3

    Correct Answer: A

    Solution :

    \[1=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|\]\[=\left| \frac{{{z}_{1}}{{{\bar{z}}}_{1}}}{{{z}_{1}}}+\frac{{{z}_{2}}{{{\bar{z}}}_{2}}}{{{z}_{2}}}+\frac{{{z}_{3}}{{{\bar{z}}}_{3}}}{{{z}_{3}}} \right|\]\[(\because \,\,\,|{{z}_{1}}{{|}^{2}}=1={{z}_{1}}{{\overline{z}}_{1}},\text{etc})\] \[=\,|{{\bar{z}}_{1}}+{{\bar{z}}_{2}}+{{\bar{z}}_{3}}|\,=\,|\overline{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}|\,=\,|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|\] \[(\because \,\,\,|{{\bar{z}}_{1}}|=|{{z}_{1}}|)\]


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