Maths-
General
Easy

Question

The value of integral subscript 0 superscript 100   left curly bracket square root of x right curly bracket d x (where {x} is the fractional part of x) is

  1. 50    
  2. 1    
  3. 100  
  4. None of these    

hintHint:

If a number is written as sum of a integer and fractional part, then x equals open square brackets x close square brackets plus open curly brackets x close curly brackets

The correct answer is: None of these


    integral subscript 0 superscript 100 open curly brackets square root of x close curly brackets d x
open parentheses open curly brackets x close curly brackets space i s space t h e space f r a c t i o n a l space p a r t space o f space x close parentheses
equals integral subscript 0 superscript 100 open curly brackets square root of x minus open vertical bar square root of x close vertical bar close curly brackets d x
equals integral subscript 0 superscript 100 open curly brackets square root of x close curly brackets d x minus integral subscript 0 superscript 100 open square brackets square root of x close square brackets d x
A s comma space integral subscript 0 superscript 100 open square brackets square root of x close square brackets d x rightwards double arrow 0 less or equal than x less or equal than 100
S o comma 0 less or equal than square root of x less or equal than 10
T h u s comma space i t space s h o u l d space b e space d i v i d e d space i n t o space 10 space p a r t s.
integral subscript 0 superscript 100 open curly brackets square root of x close curly brackets d x minus open square brackets integral subscript 0 superscript 10 open square brackets square root of x close square brackets d x plus integral subscript 10 superscript 20 open square brackets square root of x close square brackets d x plus integral subscript 20 superscript 30 plus integral subscript 30 superscript 40 open square brackets square root of x close square brackets d x integral subscript 40 superscript 50 open square brackets square root of x close square brackets d x plus integral subscript 50 superscript 60 open square brackets square root of x close square brackets d x plus integral subscript 60 superscript 70 open square brackets square root of x close square brackets d x plus integral subscript 70 superscript 80 open square brackets square root of x close square brackets d x plus integral subscript 80 superscript 90 open square brackets square root of x close square brackets d x plus integral subscript 90 superscript 100 open square brackets square root of x close square brackets d x close square brackets
equals open square brackets fraction numerator 2 open parentheses x close parentheses to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction close square brackets subscript 0 superscript 100 minus open square brackets integral subscript 0 superscript 10 0 d x plus integral subscript 10 superscript 20 1 d x plus integral subscript 20 superscript 30 2 d x plus integral subscript 30 superscript 40 3 d x integral subscript 40 superscript 50 4 d x plus integral subscript 50 superscript 60 5 d x plus integral subscript 60 superscript 70 6 d x plus integral subscript 70 superscript 80 7 d x plus integral subscript 80 superscript 90 8 d x plus integral subscript 90 superscript 100 9 d x close square brackets
equals open square brackets fraction numerator 2 open parentheses 100 close parentheses to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction close square brackets minus open square brackets open parentheses 0 close parentheses plus open parentheses 20 minus 10 close parentheses plus 2 open parentheses 30 minus 20 close parentheses plus 3 open parentheses 40 minus 30 close parentheses plus 4 open parentheses 50 minus 40 close parentheses plus 5 open parentheses 60 minus 50 close parentheses plus 6 open parentheses 70 minus 60 close parentheses plus 7 open parentheses 80 minus 70 close parentheses plus 8 open parentheses 90 minus 80 close parentheses plus 9 open parentheses 100 minus 90 close parentheses close square brackets
equals 2000 over 3 minus 450
equals 650 over 3The answer matches none of the above options.

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