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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2010

  • question_answer
    If\[\left| \begin{matrix}    {{x}^{2}}+x & 3x-1 & -x+3  \\    2x+1 & 2+{{x}^{2}} & {{x}^{3}}-3  \\    x-3 & {{x}^{2}}+4 & 3x  \\ \end{matrix} \right|\] \[={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+.....+{{a}_{7}}{{x}^{7}},\] then the value of\[{{a}_{0}}\]is

    A)  25                                         

    B)  24

    C)  23                         

    D)         22

    E)  21

    Correct Answer: E

    Solution :

    We have \[\left| \begin{matrix}    {{x}^{2}}+x & 3x-1 & -x+3  \\    2x+1 & 2+{{x}^{2}} & {{x}^{3}}-3  \\    x-3 & {{x}^{2}}+4 & 3x  \\ \end{matrix} \right|\] \[={{x}^{2}}+x[(2+{{x}^{2}})(3x)-({{x}^{2}}+4)({{x}^{3}}-3)]\]                 \[-(3x-1)[(2x+1)3x-(x-3)({{x}^{3}}-3)]\]                                      \[+(-x+3)[(2x+1)({{x}^{2}}+4)\]                                                         \[-(x-3)(2+{{x}^{2}})\] \[=21+35x+12{{x}^{2}}-7{{x}^{3}}-9{{x}^{4}}\] \[+2{{x}^{5}}-{{x}^{6}}-{{x}^{7}}\] \[\Rightarrow \]               \[{{a}_{0}}=21\]


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