KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    If \[x+y,z\in R.x+y+z=4\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=6\], find the maximum possible value of z.

    A) \[\sqrt{5}-1\]

    B) \[\sqrt{3}\]

    C) 2

    D) none of these

    Correct Answer: C

    Solution :

    \[\therefore \]\[2xy={{(x+y)}^{2}}-({{x}^{2}}+{{y}^{2}})\]
    \[={{\left( 4-z \right)}^{2}}-\left( 6-{{z}^{2}} \right)=2{{z}^{2}}-8z+10\]
    The quadratic equation whose roots are x and y is \[{{\operatorname{t}}^{2}}-\left( x+y \right)t+xy=0\] or \[{{\operatorname{t}}^{2}}-\left( 4-z \right)t+{{z}^{2}}-4z+5=0\]
    Since x and y are real  \[{{\left( 4-z \right)}^{2}}-4\left( {{z}^{2}}-4z+5 \right)\ge 0\]\[\Leftrightarrow 3{{z}^{2}}-8z+4\le 0\Leftrightarrow \left( 3z-2 \right)\left( z-2 \right)\le 0\]\[\Leftrightarrow 2/3\le z\le 2.\]
    That z can take value 2 where\[x=y=1\].

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