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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Conjugate, Modulus and Argument of complex number

  • question_answer
    Let \[z\] be a complex number, then the equation \[{{z}^{4}}+z+2=0\] cannot have a root, such that

    A) \[|z|\,<1\]

    B) \[|z|\,=1\]

    C) \[|z|\,>1\]

    D) None of these

    Correct Answer: A

    Solution :

    Suppose there exists a complex number \[z\] which satisfies the given equation and is such that  \[|z|\,<1\]. Then \[{{z}^{4}}+z+2=0\] Þ \[-2={{z}^{4}}+z\]Þ \[|-2|\,=\,|{{z}^{4}}+z|\] Þ \[2\le \,|{{z}^{4}}|+|z|\]Þ \[2<2,\] because\[|z|\,<1\] But \[2<2\] is not possible. Hence given equation cannot have a root \[z\] such that \[|z|<1\]


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