KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    The set of equations: \[\lambda x-y+\left( \cos \theta  \right)z=0;3x+y+2z=0;\] \[\left( cos\theta  \right)x+y+2z=0; 0\le \theta <2\pi ,\] has non -trivial solutions

    A) For no value of \[\lambda \operatorname{and}\theta \]

    B) For all values of \[\lambda \]and\[\theta \]

    C) For all values of\[\lambda \], and only two values of\[\theta \]

    D) For only one value of \[\lambda \] and all values of \[\theta \]

    Correct Answer: A

    Solution :

    determinant of coefficients \[=\left| \begin{matrix}    \lambda  & -1 & \cos \theta   \\    3 & 1 & 2  \\    \cos \theta  & 1 & 2  \\ \end{matrix} \right|=\cos \theta -{{\cos }^{2}}\theta +6\] and this is positive for all \[\theta \] since \[\left| \cos \theta  \right|\le 1.\]the only solution is therefore the trivial solution.

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