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KVPY Sample Paper KVPY Stream-SX Model Paper-3

  • question_answer
    \[\text{Let}\,\,\text{M}\,=\,\left[ \begin{matrix}    0 & 1 & a  \\    1 & 2 & 3  \\    3 & b & 1  \\ \end{matrix} \right]\,\,\text{and}\,\,\,\text{adj}\,\,\text{M}\,=\,\left[ \begin{matrix}    -1 & 1 & -1  \\    8 & -6 & 2  \\    -5 & 3 & -1  \\ \end{matrix} \right]\] Where a and b are real numbers. Which of the following .options is/are correct?

    A) \[\text{det}\,(adj\,\,{{\text{M}}^{2}})\,=\,81\]

    B) \[a+b=3\]

    C) \[(adj\,{{\text{M}}^{-1}})\,+\,adj\,\,{{\text{M}}^{-1}}=\,-\text{M}\]

    D) \[\text{M}\left[ \begin{matrix}    \alpha   \\    \beta   \\    \gamma   \\ \end{matrix} \right]\,=\,\left[ \begin{matrix}    1  \\    2  \\    3  \\ \end{matrix} \right]\,,\,\text{the}\,\,\alpha \,-\,\beta +\,\gamma \,=\,3\]

    Correct Answer: B , C , D

    Solution :

    \[\text{M}=\left[ \begin{matrix}    0 & 1 & a  \\    1 & 2 & 3  \\    3 & b & 1  \\ \end{matrix} \right]\] and adj \[\text{M}=\left[ \begin{matrix}    -1 & 1 & -1  \\    8 & -6 & 2  \\    -5 & 3 & -1  \\ \end{matrix} \right]\]\[\Rightarrow \]adj\[\text{M}=\left[ \begin{matrix}    2-3b & ab-1 & -1  \\    8 & -6 & 2  \\    b-6 & 3 & -1  \\ \end{matrix} \right]\]\[=\left[ \begin{matrix}    -1 & 1 & -1  \\    8 & -6 & 2  \\    -5 & 3 & -1  \\ \end{matrix} \right]\]
    \[2-3b=-1;ab-1=1\]
    \[b-6=-\,5;a=2\]
    \[b=1\]
    Now,     \[\text{M}=\left[ \begin{matrix}    0 & 1 & 2  \\    1 & 2 & 3  \\    3 & 1 & 1  \\ \end{matrix} \right]\]
    \[\left| \text{M} \right|=8-10=-2\]
    \[\Rightarrow \]   \[a+b=3\]option [b]
                \[\left| \text{adj}\,({{\text{M}}^{2}}) \right|={{\left| {{\text{M}}^{2}} \right|}^{2}}\] \[={{\left| \text{M} \right|}^{4}}=16\]
    [c] \[{{\text{(adj}\,\text{M})}^{-1}}+\text{adj}\,({{\text{M}}^{-1}})\,\text{option}\,(c)\]\[=\,\text{adj}\,({{\text{M}}^{-1}})+\text{adj}\,({{\text{M}}^{-1}})\]\[=2\text{adj}\,({{\text{M}}^{-\,\,1}})\]\[=2(\left| {{\text{M}}^{-\,\,1}} \right|\text{M})=2\left( \frac{1}{-2}\text{M} \right)=\,-\,\text{M}\]
    [d] \[\text{M}\left[ \begin{matrix}    \alpha   \\    \beta   \\    \gamma   \\ \end{matrix} \right]\,=\,\left[ \begin{matrix}    1  \\    2  \\    3  \\ \end{matrix} \right]\]
    \[\left[ \begin{matrix}    0 & 1 & 2  \\    1 & 2 & 3  \\    3 & 1 & 1  \\ \end{matrix} \right]\left[ \begin{matrix}    \alpha   \\    \beta   \\    \gamma   \\ \end{matrix} \right]\,=\,\begin{matrix}    1  \\    2  \\    3  \\ \end{matrix}\]
    \[\beta +2\gamma =1\]
    \[\alpha +2\beta +3\gamma =2\]
    \[3\alpha +\beta +\gamma =1\]
    \[\alpha =1,\]\[\beta \]= \[-1,\]\[\gamma =1\]
    \[\alpha -\beta +\gamma =3\]Option [d]

    Solution :

    Same as Above

    Solution :

    Same as Above


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