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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 17

  • question_answer
    If complex number z satisfies the equation \[1+2|z{{|}^{2}}=\,\,|{{z}^{2}}+1{{|}^{2}}+2|z+1{{|}^{2}},\] then the value of \[|z(z+1)|\] is

    A) 1          

    B)        4                       

    C) 16                   

    D)        25

    Correct Answer: A

    Solution :

         [a] We have \[1+2|z{{|}^{2}}=|{{z}^{2}}+1{{|}^{2}}+2|z+1{{|}^{2}}\] \[\Rightarrow \,\,1+2z\bar{z}=({{z}^{2}}+1)({{\bar{z}}^{2}}+1)+2(z+1)(\bar{z}+1)\] \[\Rightarrow \,\,{{(z\bar{z})}^{2}}+2(z+\bar{z})+{{z}^{2}}+{{\bar{z}}^{2}}+2=0\] \[\Rightarrow \,\,{{(z\bar{z}-1)}^{2}}+{{(z+\bar{z}+1)}^{2}}=0\] \[\Rightarrow \,\,\,z\bar{z}=1\] and \[\Rightarrow \,\,\,z+\bar{z}+1=0\] \[\Rightarrow \,\,\,{{z}^{2}}+z=-1\Rightarrow |{{z}^{2}}+z|=1\]      


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