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KVPY Sample Paper KVPY Stream-SX Model Paper-23

  • question_answer
    For a real number x let [x] denote the largest integer less or equal to x and \[\{x\}=x-[x].\]The smallest integer value of x for which \[\int\limits_{1}^{n}{[x]}\{x\}dx\] exceeds 2020 is

    A) 63

    B) 64      

    C) 90        

    D) 91

    Correct Answer: D

    Solution :

    We have,\[\int\limits_{1}^{n}{[x]\{x\}dx>2020}\] \[\Rightarrow \,\,\,\int\limits_{1}^{2}{\{x\}dx+2\int\limits_{2}^{3}{\{x\}}}\,dx+....(n-1)\]
    \[\int\limits_{n-1}^{n}{\{x\}dx>2020}\] \[\Rightarrow \,\,\left( 1+2+3....n-1 \right)\int\limits_{0}^{1}{x\,dx>2020}\]\[\Rightarrow \,\,\frac{\left( n-1 \right)\left( n \right)}{2}{{\left[ \frac{{{n}^{2}}}{2} \right]}^{1}}>2020\]\[\Rightarrow \,\,\frac{n\left( n-1 \right)\left( n \right)}{2\times 2}>2020\]
    \[\Rightarrow \,\,\,\,\,n\left( n-1 \right)>8080\]Maximum of x is 91.


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