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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Self Evaluation Test - Determinats

  • question_answer
    If the value of the determinant \[\left| \begin{matrix}    a & 1 & 1  \\    1 & b & 1  \\    1 & 1 & c  \\ \end{matrix} \right|\] is positive, where \[a\ne b\ne c,\] then the value of abc

    A) Cannot be less than 1

    B) Is greater than \[-8\]

    C) Is less than \[-8\]

    D) Must be greater than 8

    Correct Answer: B

    Solution :

    [b]  \[\left| \begin{matrix}    a & 1 & 1  \\    1 & b & 1  \\    1 & 1 & c  \\ \end{matrix} \right|>0\] \[\Rightarrow \,\,\,\,a(bc-1)-1(c-1)+1(1-b)>0\] \[\Rightarrow \,\,\,\,abc-a-c+1+1-b>0\] \[\Rightarrow \,\,\,\,abc+2-(a+b+c)>0\] \[\Rightarrow \,\,\,\,abc>(a+b+c)-2\] Let; \[a=-1;\]  \[b=0\] & \[c=1\] Then;  \[0>-2\] [which is correct] Hence,  \[abc=0\Rightarrow abc>-8\]


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