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The number of distinct real roots of $open vertical bar table attributes columnalign center center center columnspacing 1em end attributes row cell sin invisible function application x end cell cell cos invisible function application x end cell cell cos invisible function application x end cell row cell cos invisible function application x end cell cell sin invisible function application x end cell cell cos invisible function application x end cell row cell cos invisible function application x end cell cell cos invisible function application x end cell cell sin invisible function application x end cell end table close vertical bar equals 0$ in the interval $negative pi over 4 less or equal than x less or equal than pi over 4$ is

0
2
1
3

The correct answer is: 1

Related Questions to study

If $0 less or equal than x less than 2 pi$ , then the number of real values of x, which satisfy the equation $cos invisible function application x plus cos invisible function application 2 x plus cos invisible function application 3 x plus cos invisible function application 4 x equals 0$ is :

If $0 less or equal than x less than 2 pi$ , then the number of real values of x, which satisfy the equation $cos invisible function application x plus cos invisible function application 2 x plus cos invisible function application 3 x plus cos invisible function application 4 x equals 0$ is :

The number of solutions of the equation $cos squared invisible function application open parentheses x plus pi over 6 close parentheses plus cos squared invisible function application x minus 2 cos invisible function application open parentheses x plus pi over 6 close parentheses cos invisible function application open parentheses pi over 6 close parentheses equals sin squared invisible function application pi over 6$ in the interval $left parenthesis negative pi divided by 2 comma pi divided by 2 right parenthesis$ is

The number of solutions of the equation $cos squared invisible function application open parentheses x plus pi over 6 close parentheses plus cos squared invisible function application x minus 2 cos invisible function application open parentheses x plus pi over 6 close parentheses cos invisible function application open parentheses pi over 6 close parentheses equals sin squared invisible function application pi over 6$ in the interval $left parenthesis negative pi divided by 2 comma pi divided by 2 right parenthesis$ is

The possible values of $theta element of left parenthesis 0 comma pi right parenthesis$ such that $sin invisible function application theta plus sin invisible function application 4 theta plus sin invisible function application 7 theta equals 0$ are :

The possible values of $theta element of left parenthesis 0 comma pi right parenthesis$ such that $sin invisible function application theta plus sin invisible function application 4 theta plus sin invisible function application 7 theta equals 0$ are :

parallel

The number of values of x in the interval [0, 3 $straight pi$ ] satisfying the equation , $2 sin squared invisible function application x plus 5 sin invisible function application x minus 3 equals 0$ is

The number of values of x in the interval [0, 3 $straight pi$ ] satisfying the equation , $2 sin squared invisible function application x plus 5 sin invisible function application x minus 3 equals 0$ is

The number of solutions of tan x + sec x = 2 cos x in [0, 2 $straight pi$ ], is

The number of solutions of tan x + sec x = 2 cos x in [0, 2 $straight pi$ ], is

Over $left square bracket negative pi comma pi right square bracket$ the solution of $2 sin squared invisible function application open parentheses x plus pi over 4 close parentheses plus square root of 3 cos invisible function application 2 x greater or equal than 0$ is

Over $left square bracket negative pi comma pi right square bracket$ the solution of $2 sin squared invisible function application open parentheses x plus pi over 4 close parentheses plus square root of 3 cos invisible function application 2 x greater or equal than 0$ is

parallel

Solution to inequality cos 2x + 5 cos x + 3 ≥ 0 over $left square bracket negative pi comma pi right square bracket$ is

Solution to inequality cos 2x + 5 cos x + 3 ≥ 0 over $left square bracket negative pi comma pi right square bracket$ is

Let $f left parenthesis x right parenthesis equals fraction numerator c o s e c to the power of 4 end exponent invisible function application x minus 2 c o s e c to the power of 2 end exponent invisible function application x plus 1 over denominator c o s e c invisible function application x left parenthesis c o s e c invisible function application x minus s i n invisible function application x right parenthesis plus fraction numerator s i n invisible function application x minus c o s invisible function application x over denominator sin invisible function application x end fraction plus c o t invisible function application x end fraction$ The sum of all the solutions of f (x) = 0 in [0, 100p] is

table row cell table row blank cell f left parenthesis x right parenthesis equals fraction numerator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses to the power of 2 end exponent over denominator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses plus 1 minus c o t invisible function application x plus c o t invisible function application x end fraction end cell row blank cell text  defined for  end text R minus n pi comma n element of I end cell row blank cell f left parenthesis x right parenthesis equals fraction numerator left parenthesis c o t invisible function application x right parenthesis to the power of 4 end exponent over denominator 1 plus c o t to the power of 2 end exponent invisible function application x end fraction end cell row blank cell f left parenthesis x right parenthesis equals 0 rightwards double arrow c o t invisible function application x equals 0 rightwards double arrow x equals left parenthesis 2 n minus 1 right parenthesis fraction numerator pi over denominator 2 end fraction end cell row blank cell therefore x equals fraction numerator pi over denominator 2 end fraction comma fraction numerator 3 pi over denominator 2 end fraction comma fraction numerator 5 pi over denominator 2 end fraction comma horizontal ellipsis horizontal ellipsis comma fraction numerator 199 pi over denominator 2 end fraction end cell row blank cell s u m equals fraction numerator pi over denominator 2 end fraction left square bracket 1 plus 3 plus 5 plus horizontal ellipsis horizontal ellipsis plus 199 right square bracket end cell end table end cell row cell text end text text ( end text text 100 end text text end text text s end text text o end text text l end text text u end text text t end text text i end text text o end text text n end text text s end text text ) end text text end text end cell row cell equals fraction numerator pi over denominator 2 end fraction times fraction numerator 100 over denominator 2 end fraction times 200 equals 5000 pi end cell end table

Let $f left parenthesis x right parenthesis equals fraction numerator c o s e c to the power of 4 end exponent invisible function application x minus 2 c o s e c to the power of 2 end exponent invisible function application x plus 1 over denominator c o s e c invisible function application x left parenthesis c o s e c invisible function application x minus s i n invisible function application x right parenthesis plus fraction numerator s i n invisible function application x minus c o s invisible function application x over denominator sin invisible function application x end fraction plus c o t invisible function application x end fraction$ The sum of all the solutions of f (x) = 0 in [0, 100p] is

table row cell table row blank cell f left parenthesis x right parenthesis equals fraction numerator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses to the power of 2 end exponent over denominator open parentheses c o s e c to the power of 2 end exponent invisible function application x minus 1 close parentheses plus 1 minus c o t invisible function application x plus c o t invisible function application x end fraction end cell row blank cell text  defined for  end text R minus n pi comma n element of I end cell row blank cell f left parenthesis x right parenthesis equals fraction numerator left parenthesis c o t invisible function application x right parenthesis to the power of 4 end exponent over denominator 1 plus c o t to the power of 2 end exponent invisible function application x end fraction end cell row blank cell f left parenthesis x right parenthesis equals 0 rightwards double arrow c o t invisible function application x equals 0 rightwards double arrow x equals left parenthesis 2 n minus 1 right parenthesis fraction numerator pi over denominator 2 end fraction end cell row blank cell therefore x equals fraction numerator pi over denominator 2 end fraction comma fraction numerator 3 pi over denominator 2 end fraction comma fraction numerator 5 pi over denominator 2 end fraction comma horizontal ellipsis horizontal ellipsis comma fraction numerator 199 pi over denominator 2 end fraction end cell row blank cell s u m equals fraction numerator pi over denominator 2 end fraction left square bracket 1 plus 3 plus 5 plus horizontal ellipsis horizontal ellipsis plus 199 right square bracket end cell end table end cell row cell text end text text ( end text text 100 end text text end text text s end text text o end text text l end text text u end text text t end text text i end text text o end text text n end text text s end text text ) end text text end text end cell row cell equals fraction numerator pi over denominator 2 end fraction times fraction numerator 100 over denominator 2 end fraction times 200 equals 5000 pi end cell end table

The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x tan 3x is
where n $element of$ I

We know that
tan x · tan 2x · tan 3x = tan 3x – tan 2x – tan x
hence tan x + tan 2x + tan 3x = tan 3x – tan 2x – tan x
Þ tan x + tan 2x = 0
\ tan 2x = tan (– x)
2x = np – x

x equals fraction numerator n pi over denominator 3 end fraction comma n element of I

The general solution of the trigonometric equation tan x + tan 2x + tan 3x = tan x · tan 2x tan 3x is
where n $element of$ I

We know that
tan x · tan 2x · tan 3x = tan 3x – tan 2x – tan x
hence tan x + tan 2x + tan 3x = tan 3x – tan 2x – tan x
Þ tan x + tan 2x = 0
\ tan 2x = tan (– x)
2x = np – x

x equals fraction numerator n pi over denominator 3 end fraction comma n element of I

parallel

The sum of all the solution of cotθ= sin 2θ(θ ≠ n $straight pi$ , n integer), 0 , $less or equal than theta less or equal than pi$ is

For the given relation cos q = (2 sin q cos q) sin q, sin q¹ 0

table row cell text end text text o end text text r end text text end text s i n invisible function application theta equals plus-or-minus fraction numerator 1 over denominator square root of 2 end fraction text end text text o end text text r end text text end text c o s invisible function application theta equals 0 end cell row cell text end text text o end text text r end text text end text theta equals fraction numerator pi over denominator 4 end fraction comma fraction numerator 3 pi over denominator 4 end fraction comma fraction numerator pi over denominator 2 end fraction end cell end table

Then the sum of roots is 3p/2

The sum of all the solution of cotθ= sin 2θ(θ ≠ n $straight pi$ , n integer), 0 , $less or equal than theta less or equal than pi$ is

For the given relation cos q = (2 sin q cos q) sin q, sin q¹ 0

table row cell text end text text o end text text r end text text end text s i n invisible function application theta equals plus-or-minus fraction numerator 1 over denominator square root of 2 end fraction text end text text o end text text r end text text end text c o s invisible function application theta equals 0 end cell row cell text end text text o end text text r end text text end text theta equals fraction numerator pi over denominator 4 end fraction comma fraction numerator 3 pi over denominator 4 end fraction comma fraction numerator pi over denominator 2 end fraction end cell end table

Then the sum of roots is 3p/2

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction$

space space space space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction U sin g space L apostrophe space H o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent minus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent open parentheses negative 1 close parentheses equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent equals 1 fifth cross times open parentheses 1 plus 0 close parentheses to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus 0 right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent equals 1 fifth plus 1 half equals fraction numerator 2 plus 5 over denominator 10 end fraction equals 7 over 10.

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction$

space space space space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction U sin g space L apostrophe space H o s p i t a l space t h e o r e m comma space equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent minus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent open parentheses negative 1 close parentheses equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent equals 1 fifth cross times open parentheses 1 plus 0 close parentheses to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus 0 right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent equals 1 fifth plus 1 half equals fraction numerator 2 plus 5 over denominator 10 end fraction equals 7 over 10.

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses pi cos squared space x close parentheses over denominator x squared end fraction$

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space equals straight pi cross times 1 cross times 1 equals straight pi

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses pi cos squared space x close parentheses over denominator x squared end fraction$

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction equals L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin space open parentheses straight pi open parentheses 1 minus sin squared straight x close parentheses close parentheses over denominator x squared end fraction equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses straight pi minus πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space open square brackets sin left parenthesis straight pi minus straight a right parenthesis equals sina close square brackets equals straight pi open curly brackets straight L subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin open parentheses πsin squared straight x close parentheses over denominator πsin squared straight x end fraction cross times fraction numerator sin squared straight x over denominator straight x squared end fraction close curly brackets space space space space space space space space space space space space equals straight pi cross times 1 cross times 1 equals straight pi

parallel

$L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of x minus e cubed over denominator x not stretchy rightwards arrow 0 end fraction$

$L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of x minus e cubed over denominator x not stretchy rightwards arrow 0 end fraction$

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of 3 x end exponent minus e cubed over denominator x end fraction$

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of 3 x end exponent minus e cubed over denominator x end fraction$

$Lt subscript x not stretchy rightwards arrow 0 end subscript invisible function application fraction numerator e to the power of 7 x end exponent minus 1 over denominator 7 x end fraction$

Lt subscript x not stretchy rightwards arrow 0 end subscript invisible function application fraction numerator e to the power of 7 x end exponent minus 1 over denominator 7 x end fraction

rightwards arrow y

x rightwards arrow 0 y rightwards arrow 7 cross times 0 y rightwards arrow 0

limit as y rightwards arrow 0 of fraction numerator e to the power of y minus 1 over denominator y end fraction equals 1

$Lt subscript x not stretchy rightwards arrow 0 end subscript invisible function application fraction numerator e to the power of 7 x end exponent minus 1 over denominator 7 x end fraction$

Lt subscript x not stretchy rightwards arrow 0 end subscript invisible function application fraction numerator e to the power of 7 x end exponent minus 1 over denominator 7 x end fraction

rightwards arrow y

x rightwards arrow 0 y rightwards arrow 7 cross times 0 y rightwards arrow 0

limit as y rightwards arrow 0 of fraction numerator e to the power of y minus 1 over denominator y end fraction equals 1

parallel

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