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

  • question_answer
    Let \[x<1,\] then value of \[\left| \begin{matrix}    {{x}^{2}}+2 & 2x+1 & 1  \\    2x+1 & x+2 & 1  \\    3 & 3 & 1  \\ \end{matrix} \right|\] is

    A) Non-negative

    B) Non-positive

    C) Negative

    D) Positive

    Correct Answer: C

    Solution :

    [c] \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}},\] \[{{R}_{2}}\to {{R}_{2}}-{{R}_{3}},\] (reduces the determinant to:)             \[\left| \begin{matrix}    {{x}^{2}}-2x+1 & x-1 & 0  \\    2x-2 & x-1 & 0  \\    3 & 3 & 1  \\ \end{matrix} \right|\]             \[={{(x-1)}^{3}}-2{{(x-1)}^{2}}={{(x-1)}^{2}}(x-1-2)\] \[={{(x-1)}^{2}}(x-3),\] which is clearly negative for \[x<1\]


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