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 44-20172017 and 44-20172017 respectively- Then F 1- F 2 lies between the numbersA- 0 and 0-45B- 0-45 and 0-9- 0-9 and 1-35D- 1-35 and 1-8

The correct option is C 0.9 and 1.35Given, F_1 and nbsp;F_2 be the fractional parts of nbsp; (44 √(2017))^2017 and (44+√(2017))^2017 respectively. Therefo ...

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Standard XII

Mathematics

Binomial Theorem for Any Index

The fractiona...

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 ${(44 - \sqrt{2017})}^{2017} and {(44 + \sqrt{2017})}^{2017}$ respectively. Then $F_{1} + F_{2}$ lies between the numbers

0 and 0.45

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0.45 and 0.9

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0.9 and 1.35

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1.35 and 1.8

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Solution

The correct option is C $0.9 and 1.35$
Given, $F_{1}$ and $F_{2}$ be the fractional parts of ${(44 - \sqrt{2017})}^{2017}$ and ${(44 + \sqrt{2017})}^{2017}$ respectively.

Therefore,
${(44 - \sqrt{2017})}^{2017} = 0 + F_{1} [∵ \sqrt{2017} = 44.91]$
and ${(44 + \sqrt{2017})}^{2017} = I_{1} + F_{2}$

Using binomial expansion,
${(44 - \sqrt{2017})}^{2017} =^{2017} C_{0} (44)^{2017} -^{2017} C_{1} (44)^{2016} (\sqrt{2017}) +^{2017} C_{2} (44)^{2015} (\sqrt{2017})^{2} - \dots - (\sqrt{2017})^{2017} = 0 + F_{1} \dots (i)$
${(44 + \sqrt{2017})}^{2017} =^{2017} C_{0} (44)^{2017} +^{2017} C_{1} (44)^{2016} (\sqrt{2017}) +^{2017} C_{2} (44)^{2015} (\sqrt{2017})^{2} + \dots + (\sqrt{2017})^{2017} = I_{1} + F_{2} \dots (i i)$

Adding equation $(i)$ and $(i i),$ we get
$I_{1} + F_{1} + F_{2} = 2 [^{2017} C_{0} (44)^{2017} +^{2017} C_{2} (44)^{2015} (\sqrt{2017})^{2} + \dots +^{2017} C_{2} (44)^{1} (\sqrt{2017})^{2016}]$
$\Rightarrow I_{1} + F_{1} + F_{2} = I_{2},$ where $I_{2}$ is integer.
$∴ F_{1} + F_{2}$ is an integer.

$0 < F_{1} < 1$
$0 < F_{2} < 1$
$∴ 0 < F_{1} + F_{2} < 2$
Between $0$ and $2,$ only integer is $1.$
$∴ F_{1} + F_{2} = 1$ which lies between $0.9 and 1.35$

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The fractional part of a real number x is x−[x], where [x] is the greatest integer less than or equal to x. Let F1 and F2 be the fractional parts of (44−√2017)2017 and (44+√2017)2017 respectively. Then F1+F2 lies between the numbers

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 ${(44 - \sqrt{2017})}^{2017} and {(44 + \sqrt{2017})}^{2017}$ respectively. Then $F_{1} + F_{2}$ lies between the numbers