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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 $fraction numerator I subscript 2 end subscript over denominator I subscript 1 end subscript end fraction$

$π$
$\frac{π}{2}$
p/1
$\frac{π}{4}$

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

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Suppose A and B are two nonsingular matrices such that $A B equals B A to the power of 2 end exponent text and end text B to the power of 5 end exponent equals I comma text then end text$

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$stretchy integral subscript 0 end subscript superscript pi end superscript fraction numerator d x over denominator left parenthesis 5 plus 4 c o s invisible function application x right parenthesis to the power of 2 end exponent end fraction equals$

$stretchy integral subscript 0 end subscript superscript pi end superscript fraction numerator d x over denominator left parenthesis 5 plus 4 c o s invisible function application x right parenthesis to the power of 2 end exponent end fraction equals$

parallel

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

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

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

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

A point P moves in xy – plane in such a way that $left square bracket vertical line x vertical line right square bracket plus left square bracket vertical line y vertical line right square bracket equals 1 comma text where end text left square bracket. right square bracket$ denotes the greatest integer function. Area of the region representing all possible positions of the point P is equal to

A point P moves in xy – plane in such a way that $left square bracket vertical line x vertical line right square bracket plus left square bracket vertical line y vertical line right square bracket equals 1 comma text where end text left square bracket. right square bracket$ denotes the greatest integer function. Area of the region representing all possible positions of the point P is 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 $stretchy integral subscript 0 end subscript superscript 1 end superscript fraction numerator l o g subscript e end subscript invisible function application left parenthesis x plus 1 right parenthesis over denominator 1 plus x to the power of 2 end exponent end fraction d x text is end text$

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

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

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

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

parallel

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

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