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

  • question_answer If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}},{{z}_{4}}\] are two pairs of conjugate complex numbers, then \[arg\left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+arg\left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\] equals

    A) 0

    B) \[\frac{\pi }{2}\]

    C) \[\frac{3\pi }{2}\]

    D) \[\pi \]

    Correct Answer: A

    Solution :

    We have \[{{z}_{2}}={{\overline{z}}_{1}}\]and \[{{z}_{4}}={{\overline{z}}_{3}}\] Therefore \[{{z}_{1}}{{z}_{2}}=|{{z}_{1}}{{|}^{2}}\]and  \[{{z}_{3}}{{z}_{4}}=|{{z}_{3}}{{|}^{2}}\] Now \[arg\left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+arg\left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)=arg\left( \frac{{{z}_{1}}{{z}_{2}}}{{{z}_{4}}{{z}_{3}}} \right)\] \[=arg\left( \frac{|{{z}_{1}}{{|}^{2}}}{|{{z}_{3}}{{|}^{2}}} \right)=arg\left( {{\left| \frac{{{z}_{1}}}{{{z}_{3}}} \right|}^{2}} \right)\]= 0 (\[\because \]Argument of positive real number is zero).


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