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KVPY Sample Paper KVPY Stream-SX Model Paper-20

  • question_answer
    The number of integral values of m for  "M quadratic expression, \[(1+2m){{x}^{2}}-2(1+3m)x+4(1+m),\] \[x\in R,\] is always positive, is:

    A) 3

    B) 8

    C) 7         

    D) 6

    Correct Answer: C

    Solution :

    Expression  is always positive if \[2m+1>0\]
    \[\Rightarrow \]   \[m>-\frac{1}{2}\]
    and       \[D<0\]
    \[\Rightarrow \]   \[{{m}^{2}}-6m-3<0\]
                \[3-\sqrt{12}<m<3+\sqrt{12}\]               ? (iii)
    \[\therefore \] Common interval is \[3-\sqrt{12}<m<3+\sqrt{12}\]
    \[\therefore \] Integral value of m \[\left\{ 0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6 \right\}.\]


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