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The value of the definite Integral text .  end text stretchy integral subscript negative 2008 end subscript superscript 2008 end superscript   fraction numerator f to the power of ´ end exponent left parenthesis x right parenthesis plus f to the power of ´ end exponent left parenthesis negative x right parenthesis over denominator left parenthesis 2008 right parenthesis to the power of x end exponent plus 1 end fraction d x text  equals end text

  1. f left parenthesis 2008 right parenthesis plus f left parenthesis negative 2008 right parenthesis    
  2. f left parenthesis 2008 right parenthesis minus f left parenthesis negative 2008 right parenthesis    
  3. 0    
  4. f left parenthesis negative 2008 right parenthesis minus f left parenthesis 2008 right parenthesis    

The correct answer is: f left parenthesis 2008 right parenthesis minus f left parenthesis negative 2008 right parenthesis

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The integral integral s i n invisible function application 2 x open parentheses 1 minus fraction numerator 3 over denominator 2 end fraction c o s invisible function application x close parentheses e to the power of s i n to the power of 2 end exponent invisible function application x plus c o s to the power of 3 end exponent invisible function application x end exponent d x text end textis equal to

The integral integral s i n invisible function application 2 x open parentheses 1 minus fraction numerator 3 over denominator 2 end fraction c o s invisible function application x close parentheses e to the power of s i n to the power of 2 end exponent invisible function application x plus c o s to the power of 3 end exponent invisible function application x end exponent d x text end textis equal to

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The value of stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator l o g subscript e end subscript invisible function application left parenthesis x plus 1 right parenthesis over denominator 1 plus x to the power of 2 end exponent end fraction d x text  is end text

The value of stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator l o g subscript e end subscript invisible function application left parenthesis x plus 1 right parenthesis over denominator 1 plus x to the power of 2 end exponent end fraction d x text  is end text

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If function f(x)equals c o s invisible function application x minus stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis x minus 1 right parenthesis f left parenthesis t right parenthesis d t comma text  then  end text f to the power of ´ ´ end exponent left parenthesis x right parenthesis plus f left parenthesis x right parenthesis text end textis equal to

If function f(x)equals c o s invisible function application x minus stretchy integral subscript 0 end subscript superscript x end superscript   left parenthesis x minus 1 right parenthesis f left parenthesis t right parenthesis d t comma text  then  end text f to the power of ´ ´ end exponent left parenthesis x right parenthesis plus f left parenthesis x right parenthesis text end textis equal to

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If I equals stretchy integral subscript 0 end subscript superscript 12 end superscript   fraction numerator d x over denominator square root of 1 minus x to the power of 2 n end exponent end root end fraction comma n element of N comma text end textthen

If I equals stretchy integral subscript 0 end subscript superscript 12 end superscript   fraction numerator d x over denominator square root of 1 minus x to the power of 2 n end exponent end root end fraction comma n element of N comma text end textthen

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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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If graph of the function y= f(x) is continuous and passes through point (3, 1) then stack l i m with x rightwards arrow 3 below   blank fraction numerator l n left parenthesis 3 f left parenthesis x right parenthesis minus 2 right parenthesis over denominator 2 left parenthesis 1 minus f left parenthesis x right parenthesis right parenthesis end fraction text  is equal end text

If graph of the function y= f(x) is continuous and passes through point (3, 1) then stack l i m with x rightwards arrow 3 below   blank fraction numerator l n left parenthesis 3 f left parenthesis x right parenthesis minus 2 right parenthesis over denominator 2 left parenthesis 1 minus f left parenthesis x right parenthesis right parenthesis end fraction text  is equal end text

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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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If graph of the function y= f(x) is continuous and passes through point (3, 1) then stack l i m with x rightwards arrow 3 below blank fraction numerator l n left parenthesis 3 f left parenthesis x right parenthesis minus 2 right parenthesis over denominator 2 left parenthesis 1 minus f left parenthesis x right parenthesis right parenthesis end fraction text end textis equal

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Let f be twice differentiable real valued function satisfying f to the power of ´ end exponent left parenthesis x right parenthesis not equal to 0 comma f left parenthesis x right parenthesis plus f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals negative x g left parenthesis x right parenthesis f to the power of ´ end exponent left parenthesis x right parenthesis where g left parenthesis x right parenthesis greater than 0 for all x greater than 0 text  .If  end text f left parenthesis 0 right parenthesis equals negative 3 text end textand f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 4 text end textthen
Statement -1:vertical line f left parenthesis x right parenthesis vertical line less or equal than 5 for all x greater than 0
Statement -2:left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of 2 end exponent plus open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent is decreasing for all x greater than 0.

Let f be twice differentiable real valued function satisfying f to the power of ´ end exponent left parenthesis x right parenthesis not equal to 0 comma f left parenthesis x right parenthesis plus f to the power of ´ ´ end exponent left parenthesis x right parenthesis equals negative x g left parenthesis x right parenthesis f to the power of ´ end exponent left parenthesis x right parenthesis where g left parenthesis x right parenthesis greater than 0 for all x greater than 0 text  .If  end text f left parenthesis 0 right parenthesis equals negative 3 text end textand f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 4 text end textthen
Statement -1:vertical line f left parenthesis x right parenthesis vertical line less or equal than 5 for all x greater than 0
Statement -2:left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of 2 end exponent plus open parentheses f to the power of ´ end exponent left parenthesis x right parenthesis close parentheses to the power of 2 end exponent is decreasing for all x greater than 0.

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Let I equals stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator sin invisible function application x over denominator square root of x end fraction d x text  and  end text J equals stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator cos invisible function application x over denominator square root of x end fraction d x text  . end text Then which one of the following is true ?

Let I equals stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator sin invisible function application x over denominator square root of x end fraction d x text  and  end text J equals stretchy integral subscript 0 end subscript superscript 1 end superscript   fraction numerator cos invisible function application x over denominator square root of x end fraction d x text  . end text Then which one of the following is true ?

maths-General
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text Iff end text text .  end text minus y minus e to the power of y end exponent comma g comma negative y minus y comma y greater than 0 text end textandE.. t minus stretchy integral subscript 0 end subscript superscript t end superscript   f left parenthesis t minus y right parenthesis g left parenthesis y right parenthesis d y text  , end textthen-

text Iff end text text .  end text minus y minus e to the power of y end exponent comma g comma negative y minus y comma y greater than 0 text end textandE.. t minus stretchy integral subscript 0 end subscript superscript t end superscript   f left parenthesis t minus y right parenthesis g left parenthesis y right parenthesis d y text  , end textthen-

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