JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer If \[z\ne 0\] is  a complex number, then

    A) \[\operatorname{Re}(z)=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\]

    B)   \[\operatorname{Re}({{z}^{2}})=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\]

    C) \[\operatorname{Re}(z)=0\Rightarrow \operatorname{Re}({{z}^{2}})=0\]

    D) None of these

    Correct Answer: A

    Solution :

    If \[z\ne 0\]. Let \[z=x+iy\] Þ \[{{z}^{2}}={{x}^{2}}-{{y}^{2}}+i(2xy)\] Re(z)= 0  Þ \[x=0\]. Therefore \[\operatorname{Im}({{z}^{2}})=2xy=0\] Thus \[\operatorname{Re}(z)=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\].

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