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KVPY2017MathematicsFunctions

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- 2017 )²⁰¹⁷ and (44+ 2017 )²⁰¹⁷ respectively. Then F ₁+ F ₂ lies between the numbers

Options

  1. A0 and 0.45
  2. B0.45 and 0.9
  3. C0.9 and 1.35
  4. D1.35 and 1.8

Correct answer

C. 0.9 and 1.35

Step-by-step solution

I + F ₂=( 2017 +44)²⁰¹⁷ ~F ₂^ =( 2017 -44)²⁰¹⁷ ; 0 < F ₂^ < 1 I + F ₂- F ₂^ =2 [ ²⁰¹⁷ C ₁( 2017 )²⁰¹⁶(44)+ . ] F ₂= F ₂ ^ F ₂=(0.911)²⁰¹⁷ Now, F ₁=(44- 2017 )²⁰¹⁷=-(0.911)²⁰¹⁷ Fractional part can not - ve . So, F₁=1-(0.911)²⁰¹⁷ So, F₁+F₂=1 1 lie Between 0.9 & 1.35

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