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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

  • question_answer
    Sum of common roots the equations \[{{z}^{3}}+2{{z}^{2}}+2z+1=0\]and \[{{z}^{1985}}+{{z}^{100}}+1=0\]

    A) -1       

    B) 1

    C) 0                    

    D) 1

    Correct Answer: A

    Solution :

    [a] we have, \[{{z}^{3}}+2{{z}^{2}}+2z+1=0\] \[\Rightarrow ({{z}^{3}}+1)+2z(z+1)=0\] \[\Rightarrow (z+1)({{z}^{2}}+z+1)=0\]\[\Rightarrow z=-1,\]\[\omega \],\[{{\omega }^{2}}\] Since z=-1 does not satisfy \[{{z}^{1985}}+{{z}^{100}}+1=0\]while z=\[\omega \],\[{{\omega }^{2}}\]satisfy it; hence, sum is \[\omega +{{\omega }^{2}}=-1\].          


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