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 ${\left(44+\sqrt{2017}\right)}^{2017}$
$\Rightarrow {\left(44+\sqrt{2017}\right)}^{2017}=I+F_2$
$\Rightarrow {\left(44-\sqrt{2017}\right)}^{2017}=F_1$
$\left[\because 0<44-\sqrt{2017}<1\right]$
$\therefore I+F_1+F_2$
$=2{(}^{2017}{C}_0{\left(44\right)}^{2017}{+}^{2017}{C}_2{\left(44\right)}^{2015}\left(2017\right)+\dots ]$
$\Rightarrow I+F_1+F_2=2$ (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 $1.35$