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 \[z=3-4i\], then  \[{{z}^{4}}-3{{z}^{3}}+3{{z}^{2}}+99z-95\]is equal to

    A) 5

    B) 6

    C) - 5

    D) - 4

    Correct Answer: A

    Solution :

    Given that \[z=3-4i\]Þ \[{{z}^{2}}=-7-24i\], \[{{z}^{4}}=-117-44i\]and \[{{z}^{4}}=-527+336i\] \\[{{z}^{4}}-3{{z}^{3}}+3{{z}^{2}}+99z-95=5\] Aliter: \[z=3-4i\]Þ \[{{(z-3)}^{2}}=-16\] Þ\[{{z}^{2}}-6z+25=0\] \[{{z}^{4}}-3{{z}^{3}}+3{{z}^{2}}+99z-95\] \[=({{z}^{2}}+3z-4)({{z}^{2}}-6z+25)+5\] \[=({{z}^{2}}+3z-4)(0)+5=5\]

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