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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2011

  • question_answer
        If\[f(x)=\left| \begin{matrix}    a & -1 & 0  \\    ax & a & -1  \\  a{{x}^{2}} & ax & a  \\ \end{matrix} \right|,\]then\[f(2x)-f(x)\]is not divisible by

    A)  \[x\]                                    

    B)  \[a\]

    C)  \[2a+3x\]          

    D)  \[d{{x}^{2}}\]

    Correct Answer: D

    Solution :

                     Applying\[{{R}_{3}}\to {{R}_{3}}-x{{R}_{2}}\]and \[{{R}_{2}}\to {{R}_{2}}-x{{R}_{1}}\] \[\therefore \] \[f(x)=\left| \begin{matrix}    a & -1 & 0  \\    0 & a+x & -1  \\    0 & 0 & a+x  \\ \end{matrix} \right|=a{{(a+x)}^{2}}\] \[\therefore \] \[f(2x)-f(x)=a[{{(a+2x)}^{2}}-{{(a+x)}^{2}}]\] \[=a(a+2x-a-x)(a+2x+a+x)\]                 \[=ax(2a+3x)\]


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