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KVPY Sample Paper KVPY Stream-SX Model Paper-28

  • question_answer
    The set of homoegeneous equations: \[tx+\left( t+1 \right)y+\left( t-1 \right)z=0;\left( t+1 \right)x+ty+\left( t+2 \right)\]\[z=0;\left( t-1 \right)x+\left( t+2 \right)y+tz=0\] has non-trivial solution for

    A) Three values of t

    B) two values of t

    C) One value of t

    D) no value of t

    Correct Answer: C

    Solution :

    Determinant of coefficients =
    \[\left| \begin{matrix}    t & t+1 & t-1  \\    t+1 & t & t+2  \\    t-1 & t+2 & t  \\ \end{matrix} \right|=\left| \begin{matrix}    t & 1 & -1  \\    t+1 & -1 & 1  \\    t-1 & 3 & 1  \\ \end{matrix} \right|\]
    \[=\left| \begin{matrix}    t & 1 & -1  \\    2t+1 & 0 & 0  \\    2t-1 & 4 & 0  \\ \end{matrix} \right|=-\,4(2t+1)\]
    For non-trivial solution \[t=-\frac{1}{2}.\]


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