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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Expansion of determinants, Solution of equation in the form of determinants and properties of determinants

  • question_answer
    If \[p{{\lambda }^{4}}+q{{\lambda }^{3}}+r{{\lambda }^{2}}+s\lambda +t=\]\[\left| \,\begin{matrix}    {{\lambda }^{2}}+3\lambda  & \lambda -1 & \lambda +3  \\    \lambda +1 & 2-\lambda  & \lambda -4  \\    \lambda -3 & \lambda +4 & 3\lambda   \\ \end{matrix}\, \right|,\] the value of t is [IIT 1981]

    A) 16

    B) 18

    C) 17

    D) 19

    Correct Answer: B

    Solution :

    Since it is an identity in \[\left| \,\begin{matrix}    a(1+w) & b{{w}^{2}} & aw  \\    b(w+{{w}^{2}}) & c & b{{w}^{2}}  \\    c({{w}^{2}}+1) & aw & c  \\ \end{matrix}\, \right|\] so satisfied by every value of \[\lambda \]. Now put \[BC=[-b\,\,-a]\,\left[ \begin{align}   & \,\,\,a \\  & -a \\ \end{align} \right]=[{{a}^{2}}-ab]\] in the given equation, we have \[t=\left| \,\begin{matrix}    0 & -1 & 3  \\    1 & 2 & -4  \\    -3 & 4 & 0  \\ \end{matrix}\, \right|\,=-12+30=18\].


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