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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 11

  • question_answer
    If \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}}\] are any three roots of the equation \[{{z}^{6}}={{(z+1)}^{6}},\] then \[\arg \left( \frac{{{z}_{1}}-{{z}_{3}}}{{{z}_{2}}-{{z}_{3}}} \right)\] can be equal to

    A) \[\pi /6\]                  

    B) \[\pi /3\]                  

    C) \[\pi /4\]                  

    D) \[\pi \]

    Correct Answer: D

    Solution :

    [d] Equation \[{{z}^{6}}={{(z+1)}^{6}}\] \[\Rightarrow \,\,\,|{{z}^{6}}|\,\,=\,\,|{{(z+1)}^{6}}|\] \[\Rightarrow \,\,\,|z{{|}^{6}}\,\,=\,\,|z+1{{|}^{6}}\] \[\Rightarrow \,\,\,|z|\,\,=\,\,|z-\left( -1 \right)|\] So, z lies on the line segment joining the points 0 and \[-1\]. Hence, the roots of the equation are collinear. Given that \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}}\] are roots of the equation. \[\Rightarrow \,\,\,\,\arg \left( \frac{{{z}_{1}}-{{z}_{3}}}{{{z}_{2}}-{{z}_{3}}} \right)=0\]  or \[\pi \]


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