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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Self Evaluation Test - Determinats

  • question_answer
    If \[a\ne p,\] \[b\ne q,\] \[c\ne r\]and \[\left| \begin{matrix}    p & b & c  \\    a & q & c  \\    a & b & r  \\ \end{matrix} \right|=0\] then the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is equal to

    A) \[-1\]

    B) \[1\]   

    C) \[-2\]

    D) \[2\]

    Correct Answer: D

    Solution :

    [d] \[Given\left| \begin{matrix}    p & b & c  \\    a & q & c  \\    a & b & r  \\ \end{matrix} \right|=0\] \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}},{{R}_{2}}\to {{R}_{2}}-{{R}_{3}}\] reduces the determinant to \[\left| \begin{matrix}    p-a & b-q & 0  \\    0 & q-b & c-r  \\    a & b & r  \\ \end{matrix} \right|=0\] \[\Rightarrow (p-a)(q-b)r+a(b-q)(c-r)-b(p-a)(c-r)=0\]\[\Rightarrow \,\] Dividing throughout by \[(p-a)(q-b)(r-c),\] we get \[\Rightarrow \frac{r}{r-c}+\frac{a}{p-a}+\frac{b}{q-b}=0\] \[\Rightarrow \frac{r}{r-c}+\frac{a}{p-a}+\frac{b}{q-b}=2\]


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