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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 \[a\ne b\ne c,\] the value of x which satisfies the equation \[\left| \,\begin{matrix}    0 & x-a & x-b  \\    x+a & 0 & x-c  \\    x+b & x+c & 0  \\ \end{matrix}\, \right|=0\], is [EAMCET 1988; Karnataka CET 1991; MNR 1980;                                      MP PET 1988, 99, 2001; DCE 2001]

    A) \[x=0\]

    B) \[x=a\]

    C) \[x=b\]

    D) \[x=c\]

    Correct Answer: A

    Solution :

    Obviously, on putting \[x=0\], we observe that the determinant becomes \[{{\Delta }_{x=0}}=\left| \,\begin{matrix}    0 & -a & -b  \\    a & 0 & -c  \\    b & c & 0  \\ \end{matrix}\, \right|=a(bc)-b(ac)=0\] \[\therefore \] \[x=0\] is a root of the given equation. Aliter : Expanding \[\Delta \], we get \[\Delta \equiv -(x-a)\,[-(x+b)(x-c)]+(x-b)\,[(x+a)\,(x+c)]=0\] \[\Rightarrow \] \[2{{x}^{3}}-(2\,\Sigma ab)x=0\] \[\Rightarrow \] Either \[x=0\] or \[{{x}^{2}}=\sum{ab}\] (i.e.,  \[x=\pm \sum{ab})\] Again \[x=0\] satisfies the given equation.


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