Question

The fractional part of a real number $x$ is $x-[x]$, where $[x]$ is the greatest integer less than or equal to $x$ Let $F_{1}$ and $F_{2}$ be the fractional parts of $\left(44-\sqrt{2017}\right)^{2017}$ and $\left(44+\sqrt{2017}\right)^{2017}$, respectively. Then, $F_{1}+F_{2}$ lies between the numbers

• A. 0 and 0.45
• B. 0.45 and 0.9
• C. 0.9 and 1.35
• D. 1.35 and 1.8

Answer 1

Answer: (C)

We have,
$F_{1}$ and $F_{2}$ be the fractional part of
$\left(44+\sqrt{2017}\right)^{2017}$ and $\left(44+\sqrt{2017}\right)^{2017} $
$\Rightarrow \left(44+\sqrt{2017}\right)^{2017}=I+F_{2}$
$\Rightarrow \left(44-\sqrt{2017}\right)^{2017}=F_{1}$
$\left[\because0<\,44-\sqrt{2017}<\,1\right]$
$\therefore I+F_{1}+F_{2}$
$=2 (^{2017}C_{0}\left(44\right)^{2017}+^{2017}C_{2}\left(44\right)^{2015} \left(2017\right)+\ldots]$
$\Rightarrow I+F_{1}+F_{2}=2$ (integer)
$\therefore I+F_{1}+F_{2}$ is an even integer.
$\therefore F_{1}+F_{2}$ is also integer
But, $0<\,F_{1}+F_{2}<\,2$
But, $F_{1}+F_{2}$ is an integer.
$F_{1}+F_{2}=1$
Hence, $F_{1}+F_{2}$ is lie between $0.9$ and $1.35$