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Question

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

$[I] = 1$
$[I] = 0$
$I \leq \frac{π}{2}$
$I \leq \frac{π}{6}$

The correct answer is: $I less or equal than fraction numerator pi over denominator 6 end fraction$

Related Questions to study

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

$f left parenthesis x right parenthesis equals open square brackets fraction numerator left parenthesis x minus 2 right parenthesis to the power of 3 end exponent over denominator a end fraction close square brackets s i n invisible function application left parenthesis x minus 2 right parenthesis plus a c o s invisible function application left parenthesis x minus 2 right parenthesis comma open square brackets times close square brackets$ 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.

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

parallel

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 $stack l i m with x rightwards arrow 3 below blank fraction numerator l n left parenthesis 3 f left parenthesis x right parenthesis minus 2 right parenthesis over denominator 2 left parenthesis 1 minus f left parenthesis x right parenthesis right parenthesis end fraction text is equal end text$

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

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

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 )

parallel

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

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

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 $g left parenthesis x right parenthesis greater than 0 for all x greater than 0 text .If end text f left parenthesis 0 right parenthesis equals negative 3 text end text$ and $f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 4 text end text$ then
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 $g left parenthesis x right parenthesis greater than 0 for all x greater than 0 text .If end text f left parenthesis 0 right parenthesis equals negative 3 text end text$ and $f to the power of ´ end exponent left parenthesis 0 right parenthesis equals 4 text end text$ then
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 $I equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator sin invisible function application x over denominator square root of x end fraction d x text and end text J equals stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator cos invisible function application x over denominator square root of x end fraction d x text . end text$ 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 ?

parallel

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

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

If for a non-zero x, af $x plus b f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals fraction numerator 1 over denominator x end fraction minus 5 text , end text$ where $a to the power of not equal to end exponent b text , end text$ then $stretchy integral subscript 1 end subscript superscript 2 end superscript$ f(x)dx=

If for a non-zero x, af $x plus b f open parentheses fraction numerator 1 over denominator x end fraction close parentheses equals fraction numerator 1 over denominator x end fraction minus 5 text , end text$ where $a to the power of not equal to end exponent b text , end text$ then $stretchy integral subscript 1 end subscript superscript 2 end superscript$ f(x)dx=

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