KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    The positive, integer k for which \[\frac{{{k}^{2}}}{3{{k}^{3}}+500}\] is a maximum is

    A) 5                                 

    B) 6      

    C) 7         

    D) 8

    Correct Answer: C

    Solution :

    Let \[f(x)=\frac{{{x}^{2}}}{3{{x}^{3}}+500}\] and \[g(x)=3x+\frac{500}{{{x}^{2}}}\]  
    If \[g(x)\]is minimum, then \[f(x)\]is maximum \[g'(x)=3-\frac{1000}{{{x}^{3}}}\]\[\Rightarrow \]\[g''(x)\frac{3000}{{{x}^{4}}}>0\]
    Minimum value of \[g(x)\]at \[x={{\left( \frac{1000}{3} \right)}^{1/3}}\]\[={{(333.33)}^{1/3}}\]
    Where \[6<x<7\]
    So greatest term of the sequence \[\left( \frac{{{k}^{2}}}{3{{k}^{3}}+500} \right)\] is either \[k=6\] or \[k=7\]
    \[\therefore \]\[f(6)=\frac{36}{1148}\]
    \[\therefore \]\[f(7)>f(6)\]
    \[\therefore \] A \[k=7\]is maximum.

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