KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    If \[a\in \left[ -\text{ }20,0 \right]\], then probability that the graph of the function \[y=16{{x}^{2}}+8\left( a+5 \right)\text{ }x-7a-5\] is strictly above the \[x-\]axis is

    A) \[\frac{7}{20}\]

    B) \[\frac{13}{20}\]

    C) \[\frac{17}{20}\]

    D) \[\frac{3}{20}\]

    Correct Answer: B

    Solution :

    the graph of \[y=16{{x}^{2}}+8(a+5)x-7a-5\]is strictly above the x-axis
    \[\Rightarrow \]\[y>0\,\forall \,x\in \mathbf{R}\]\[\Rightarrow \]\[16{{x}^{2}}+8(a+5)x-7a-5>0\,\forall \,x\in R\]
    The above inequality holds if discriminant <0
    [\[\because \]coefficient \[{{x}^{2}}>0\]]
    \[\Rightarrow \]\[64{{\left( a+5 \right)}^{2}}-4.16\left( -7a-50 \right)<0\]\[\Rightarrow \]\[{{a}^{2}}+17a+30<0\]\[\Rightarrow \]\[\left( a+2 \right)\left( a+15 \right)<0\]\[\Rightarrow \]\[-15<a<-2\]
    Given \[-20\le a\le 0\] and favourable cases \[-15<a<-2\]
    \[\therefore \] Required probability \[=\frac{length\,of\,interval\left( -15,-2 \right)}{length\,of\,interval\left( -20,0 \right)}\]
    \[=\frac{-2-\left( -15 \right)}{0-\left( -20 \right)}=\frac{13}{20}\]

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