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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank System of linear equations, Some special determinants, differentiation and integration of determinants

  • question_answer
    If \[a,b,c\] are respectively the \[{{p}^{th}},{{q}^{th}}{{r}^{th}}\]terms of an \[A.P.,\] the  \[\left| \,\begin{matrix}    a & p & 1  \\    b & q & 1  \\    c & r & 1  \\ \end{matrix}\, \right|=\] [Kerala (Engg.) 2002]

    A) 1

    B) -1

    C) 0

    D) pqr

    Correct Answer: C

    Solution :

    Let first term  = A and common difference = D \[\therefore a=A+(p-1)D\], \[b=A+(q-1)D\], \[c=A+(r-1)D\] \[\left| \begin{matrix}    \,a\,\,\, & p\,\,\, & 1\,  \\    \,b\,\,\, & q\,\,\, & 1  \\    \,c\,\,\, & r\,\,\, & 1  \\ \end{matrix} \right|=\left| \begin{matrix}    \,A+(p-1)D\,\,\, & p\,\,\, & 1\,  \\    A+(q-1)D\,\,\, & q\,\,\, & 1  \\    A+(r-1)D\,\,\, & r\,\,\, & 1  \\ \end{matrix} \right|\] Operate \[{{C}_{1}}\to {{C}_{1}}-D{{C}_{2}}+D{{C}_{3}}\] \[=\,\left| \begin{matrix}    \,A\,\, & p\,\, & 1\,  \\    \,A\,\, & q\,\, & 1  \\    \,A\,\, & r\,\, & 1  \\ \end{matrix} \right|=A\left| \begin{matrix}    \,\,1 & \,\,p & \,\,1\,  \\    \,1 & q & 1  \\    \,1 & r & 1  \\ \end{matrix} \right|=0\].


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