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KVPY Sample Paper KVPY Stream-SX Model Paper-28

  • question_answer
    The number of positive integer solutions \[\left( x,y,z \right)\]for the system of simultaneous \[\operatorname{equations}\left\{ \begin{align}   & xy+xz=255, \\  & xy+yz=31 \\ \end{align} \right.\]is

    A) 3

    B) 2

    C) 1

    D) 0

    Correct Answer: B

    Solution :

    Consider the second equation
    \[xy+yz=y(x+z)=31=1\times 31\]
    It leads to\[y=1,x+z=31.\]. By substituting them into the first equation, it follows that
    \[x\left( 1+31-x \right)=255,\]
    \[{{x}^{2}}-32x+255=0,\]
    \[\left( x-15 \right)\left( x-17 \right)=0,\]
    \[\therefore {{x}_{1}}=15,{{x}_{2}}=17.\]
    Thus, the solutions are
    \[x=15,y=1,z=16\]
    and \[x=17,y=1,\,z=\text{ }14.\]


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