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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