JEE Main & Advanced Sample Paper JEE Main Sample Paper-4

  • question_answer
    Area of geometric figure formed by the roots of\[{{\left( 1+\frac{1}{z} \right)}^{4}}=1\] (where \[z\in \]complex number) is

    A)  2                            

    B) \[\frac{\sqrt{3}}{4}\]

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

    D)  None

    Correct Answer: D

    Solution :

    \[{{\left( 1+\frac{1}{z} \right)}^{4}}=1\]\[\Rightarrow \]\[{{(z+1)}^{4}}={{z}^{4}}\]\[\Rightarrow \]\[|z+1|+|z|\] \[\Rightarrow \]\[(z+1)(\overline{z}+1)=z.\overline{z}\Rightarrow z+\overline{z}=-1\]\[\Rightarrow \]\[2\operatorname{Re}(z)=-1\] \[\therefore \]roots of given equation lie on line and so all points collinear.

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