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Question

A function f is continuous on the interval left square bracket 0 comma pi right square bracket. text  If  end text I subscript 1 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript   f left parenthesis s i n invisible function application x right parenthesis d x text  and  end text I subscript 2 end subscript equals stretchy integral subscript 0 end subscript superscript pi end superscript   x f left parenthesis s i n invisible function application x right parenthesis d x then evaluate ratio fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction

  1. pi    
  2. fraction numerator pi over denominator 2 end fraction    
  3. p/1    
  4. fraction numerator pi over denominator 4 end fraction    

The correct answer is: fraction numerator pi over denominator 2 end fraction

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Statement - I : ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets ϕ left parenthesis x right parenthesis text end textattain its maximum value text  at  end text x equals fraction numerator pi over denominator 3 end fraction
Statement - 2:ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma ϕ left parenthesis x right parenthesis text end textincreasing function in open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets text  . end text

Statement - I : ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets ϕ left parenthesis x right parenthesis text end textattain its maximum value text  at  end text x equals fraction numerator pi over denominator 3 end fraction
Statement - 2:ϕ left parenthesis x right parenthesis equals stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis 3 s i n invisible function application t plus 4 c o s invisible function application t right parenthesis d t comma ϕ left parenthesis x right parenthesis text end textincreasing function in open square brackets fraction numerator pi over denominator 6 end fraction comma fraction numerator pi over denominator 3 end fraction close square brackets text  . end text

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f left parenthesis x right parenthesis equals open square brackets fraction numerator left parenthesis x minus 2 right parenthesis to the power of 3 end exponent over denominator a end fraction close square brackets s i n invisible function application left parenthesis x minus 2 right parenthesis plus a c o s invisible function application left parenthesis x minus 2 right parenthesis comma open square brackets times close square brackets denotes the greatest integer function, is continuous and differentiable in (4, 6) then.

f left parenthesis x right parenthesis equals open square brackets fraction numerator left parenthesis x minus 2 right parenthesis to the power of 3 end exponent over denominator a end fraction close square brackets s i n invisible function application left parenthesis x minus 2 right parenthesis plus a c o s invisible function application left parenthesis x minus 2 right parenthesis comma open square brackets times close square brackets denotes the greatest integer function, is continuous and differentiable in (4, 6) then.

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text If  end text f left parenthesis x right parenthesis equals open curly brackets table row cell s i n invisible function application open parentheses fraction numerator pi over denominator 2 end fraction left parenthesis x minus left square bracket x right square bracket right parenthesis close parentheses comma blank x less than 5 end cell row cell 5 left parenthesis b minus 1 right parenthesis comma blank x equals 5 text end text text i end text text s end text text end text text c end text text o end text text n end text text t end text text i end text text n end text text u end text text o end text text u end text text s end text text end text text a end text text t end text text end text x equals 5 comma left parenthesis a comma b element of R right parenthesis text end text text t end text text h end text text e end text text n end text text end text left parenthesis left square bracket. right square bracket end cell row cell a b to the power of 2 end exponent fraction numerator open vertical bar x to the power of 2 end exponent minus 11 x plus 24 close vertical bar over denominator x minus 3 end fraction comma blank x greater than 5 end cell end table close denotes greatest integer function)

text If  end text f left parenthesis x right parenthesis equals open curly brackets table row cell s i n invisible function application open parentheses fraction numerator pi over denominator 2 end fraction left parenthesis x minus left square bracket x right square bracket right parenthesis close parentheses comma blank x less than 5 end cell row cell 5 left parenthesis b minus 1 right parenthesis comma blank x equals 5 text end text text i end text text s end text text end text text c end text text o end text text n end text text t end text text i end text text n end text text u end text text o end text text u end text text s end text text end text text a end text text t end text text end text x equals 5 comma left parenthesis a comma b element of R right parenthesis text end text text t end text text h end text text e end text text n end text text end text left parenthesis left square bracket. right square bracket end cell row cell a b to the power of 2 end exponent fraction numerator open vertical bar x to the power of 2 end exponent minus 11 x plus 24 close vertical bar over denominator x minus 3 end fraction comma blank x greater than 5 end cell end table close denotes greatest integer function)

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A function f from integers to integers is defined as f left parenthesis x right parenthesis equals open curly brackets table row cell n plus 3 blank n element of o d d end cell row cell n divided by 2 blank n element of text end text text e end text text v end text text e end text text n end text text end text end cell end table text  suppose  end text k element of text  odd  end text close text  and end text f left parenthesis f left parenthesis f left parenthesis k right parenthesis right parenthesis right parenthesis equals 27 text end textthen the sum of digits k is

A function f from integers to integers is defined as f left parenthesis x right parenthesis equals open curly brackets table row cell n plus 3 blank n element of o d d end cell row cell n divided by 2 blank n element of text end text text e end text text v end text text e end text text n end text text end text end cell end table text  suppose  end text k element of text  odd  end text close text  and end text f left parenthesis f left parenthesis f left parenthesis k right parenthesis right parenthesis right parenthesis equals 27 text end textthen the sum of digits k is

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Let f(x)equals left square bracket n plus p s i n invisible function application x right square bracket cross times stack I with hat on top left parenthesis 0 comma p right parenthesis comma n stack I with hat on top Z to the power of ´ end exponent p to the power of ´ end exponent a prime number. The number of points at which f(x) is non-differentiable is ( [.] G.I.F )

Let f(x)equals left square bracket n plus p s i n invisible function application x right square bracket cross times stack I with hat on top left parenthesis 0 comma p right parenthesis comma n stack I with hat on top Z to the power of ´ end exponent p to the power of ´ end exponent a prime number. The number of points at which f(x) is non-differentiable is ( [.] G.I.F )

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