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KVPY Sample Paper KVPY Stream-SX Model Paper-14

  • question_answer
    Let \[\alpha \] and \[\beta \] be the roots of equation \[p{{x}^{2}}+qx+r=0,\] \[P\ne 0.\] If P, q, r are in A.P. and \[\frac{1}{a}+\frac{1}{\beta }=4,\] then the value of \[\left| \alpha -\beta  \right|\]is:

    A) \[\frac{\sqrt{61}}{9}\]

    B) \[\frac{2\sqrt{17}}{9}\]

    C) \[\frac{\sqrt{34}}{9}\]

    D) \[\frac{2\sqrt{13}}{9}\]

    Correct Answer: D

    Solution :

    \[\frac{1}{\alpha }+\frac{1}{\beta }=4\]
    \[2q=P+r\]
    \[\Rightarrow \]   \[-2(\alpha +\beta )=1+\alpha \beta \]
    \[\Rightarrow \]   \[2-\left( \frac{1}{\alpha }+\frac{1}{\beta } \right)=\frac{1}{a\beta }+1\]
    \[\Rightarrow \]   \[\frac{1}{\alpha \beta }=-9\]
    Equation having roots \[\alpha ,\beta \]
    \[9{{x}^{2}}+4x-1=0\]
    \[\alpha ,\beta =\frac{-4\pm \sqrt{16+36}}{2\times 9}\]
    \[\left| \alpha -\beta  \right|=\frac{2\sqrt{3}}{9}.\]


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