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Maths-

General

Easy

Question

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

2550 $π$
2500 $π$
5000 $π$
5050 $π$

The correct answer is: 5000 $straight pi$

$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$

Related Questions to study

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

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.

parallel

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

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

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

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

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

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

parallel

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

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

Which is not a property of hydrophilic sols?

Which is not a property of hydrophilic sols?

chemistry-General

The path of a projectile in the absence of air drag is shown in the figure by dotted line. If the air resistance is not ignored then which one of the path is shown in the figure is appropriate for the projectile

If air resistance is taken into consideration then range and maximum height, both will decrease

The path of a projectile in the absence of air drag is shown in the figure by dotted line. If the air resistance is not ignored then which one of the path is shown in the figure is appropriate for the projectile

physics-General

If air resistance is taken into consideration then range and maximum height, both will decrease

parallel

Match Column - I with Column - II and select the correct option from the codes given below.

Match Column - I with Column - II and select the correct option from the codes given below.

Which graph represents auto catalysis?

Which graph represents auto catalysis?

chemistry-General

A) The frequency band of VHF is greater than UHF of TV transmission
B) Optical fiber transmission has frequency band of 1 THz to 1000 THz

A) The frequency band of VHF is greater than UHF of TV transmission
B) Optical fiber transmission has frequency band of 1 THz to 1000 THz

physics-General

parallel

A) Satellite communication uses different frequency bands for uplink and downlink
B) Bandwidth of video signals is 4.2 MHz

A) Satellite communication uses different frequency bands for uplink and downlink
B) Bandwidth of video signals is 4.2 MHz

physics-General

The TV broad casting bands are

The TV broad casting bands are

physics-General

The FM Radio broad casting band is

The FM Radio broad casting band is

physics-General

parallel

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