2019 Specialist Mathematics Written examination 2 · SPECIALIST MATHEMATICS Written examination 2...

SPECIALIST MATHEMATICSWritten examination 2

Wednesday 5 June 2019 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 6 6 60

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof23pages• Formulasheet• Answersheetformultiple-choicequestions

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• Youmaykeeptheformulasheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019

STUDENT NUMBER

Letter

SECTION A – continued

2019SPECMATHEXAM2(NHT) 2

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Question 1Thegraphofwhichoneofthefollowingrelationsdoesnothaveaverticalasymptote?

A. y xx

=−3 1

B. y xx

=++

5 21

2

2

C. y xx

=−4

23

D. yx x

=+1

42

E. y xx

=−+12

Question 2Thecurvegivenbyx =3 sec(t)+1andy =2 tan(t)–1canbeexpressedincartesianformas

A. ( ) ( )y x+

−−

=1

41

91

2 2

B. ( ) ( )x y+

−−

=1

31

21

2 2

C. ( ) ( )y x+

−−

=1

91

41

2 2

D. ( ) ( )x y−

++

=1

31

21

2 2

E. ( ) ( )x y−

−+

=1

91

41

2 2

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

SECTION A – continuedTURN OVER

3 2019 SPECMATH EXAM 2 (NHT)

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Question 3The maximal domain and range of the function f (x) = a cos–1(bx) + c, where a, b and c are real constants with a > 0, b < 0 and c > 0, are respectivelyA. [0, π] and [–a, a]

B. [0, π] and [–a + c, a + c]

C. −

1 1b b

, and [c, aπ + c]

D. 1 1b b

, −

and [c, aπ + c]

E. 1 1b b

, −

and [–aπ + c, aπ + c]

Question 4Which one of the following statements is false for z1, z2 ∈ C?

A. z zz

z− = ≠12 0,

B. z z z z1 2 1 2+ > +

C. zz

z zz

z1

2

1 2

22 2 0= ≠,

D. z z z z1 2 1 2=

E. zz

zz

z1

2

1

22 0= ≠,

Question 5The circle defined by z a z i+ = +3 , where a ∈ R, has a centre and radius respectively given by

A. a a8

98

38

12, ,−

+

B. a a8

98

9 964

2, ,−

+

C. a a8

98

18

153 7 2, ,−

−

D. −

+a a8

98

9 964

2, ,

E. −

+

a a8

98

38

12, ,



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Question 6P(z)isapolynomialofdegreenwithrealcoefficientswherez ∈ C.Threeoftherootsoftheequation P(z)=0arez=3–2i,z=4andz=–5i.ThesmallestpossiblevalueofnisA. 3B. 4C. 5D. 6E. 7

Question 7Thegradientofthelinethatisperpendicular tothegraphofarelationatanypointP(x,y)ishalfthegradientofthelinejoiningPandthepointQ(–1,1).Therelationsatisfiesthedifferentialequation

A. dydx

yx

=−+1

2 1( )

B. dydx

xy

=++

2 11

( )

C. dydx

xy

=−+

2 11

( )

D. dydx

xy

=+−

12 1( )

E. dydx

xy

=+−

2 11( )

Question 8Thetotalareaenclosedbetweenthex-axisandthegraphof f x x x x( ) = − −3 2 isclosesttoA. –2.015B. –1.008C. 1.008D. 2.015E. 2.824


5 2019SPECMATHEXAM2(NHT)

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Question 9

Withasuitablesubstitution, 5 11

2x dx−∫ canbeexpressedas

A. 51

2u du∫

B. 15 1

2u du∫

C. 54

9u du∫

D. 15 4

9u du∫

E. 5 5 14

9u du−∫

Question 10Euler’smethod,withastepsizeof0.1,isusedtoapproximatethesolutionofthedifferentialequation1ydydx

x= cos( ),withy =2whenx =0.

Whenx =0.2,thevalueobtainedfor y,correcttofourdecimalplaces,isA. 2.2000B. 2.3089C. 2.3098D. 2.4189E. 2.4199

Question 11Thevectorresoluteof

a i j k= − +2 3 thatisperpendicular to

b i + j k= − is

A. − −( )23

i + j k

B. − −( )23

2i j + 3k

C. 13

8i j + 7k

−( )

D.

i j + 4k− 2

E.

i + j + 2k



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Question 12Giventhatθistheacuteanglebetweenthevectors

a i j k= − + −2 2 and

b i + 4j k,= − +4 7 thensin(2θ)isequalto

A. 2 2

B. 4 2

9

C. 2 2

9

D. 2 2

3

E. 4 2

3

Question 13Forthevectors

a i j k, b i 4j k and c i j k= + − = − − + = − − +3 2 6 λ tobelinearly dependent,thevalueofλ mustbeA. 0B. 1C. 2D. 3E. 4

Question 14

Thepositionvector

r( )t ofamassof3kgaftertseconds,wheret≥0,isgivenby

r( ) i j.t t t t= + −

10 16 4

32 3

Theforce,innewtons,actingonthemasswhent =2secondsisA. 16

j

B. 32

j

C. 48

j

D. 30 144

i j+

E. 16

Question 15Aliftacceleratesfromrestataconstantrateuntilitreachesaspeedof3ms–1.Itcontinuesatthisspeedfor10secondsandthendeceleratesataconstantratebeforecomingtorest.Thetotaltraveltimefortheliftis 30seconds.Thetotaldistance,inmetres,travelledbytheliftisA. 30B. 45C. 60D. 75E. 90



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Question 16Anobjectofmass2kgistravellinghorizontallyinastraightlineataconstantvelocityofmagnitude2ms–1.Theobjectishitinsuchawaythatitdeflects30°fromitsoriginalpath,continuingatthesamespeedinastraightline.Themagnitude,correcttotwodecimalplaces,ofthechangeofmomentum,inkgms–1,oftheobjectisA. 0.00B. 0.24C. 1.04D. 1.46E. 2.07

Question 17Aballisthrownverticallyupwardswithaninitialvelocityof7 6 ms–1,andissubjecttogravityandairresistance.Theaccelerationoftheballisgivenbyẍ=–(9.8+0.1v2),wherevms–1isitsvelocitywhenitisataheightofxmetresabovegroundlevel.Themaximumheight,inmetres,reachedbytheballisA. 5 loge(4)

B. loge 31( )

C. 5 2

21π

D. 5 loge(2)

E. 7 2

3π


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END OF SECTION A

Question 18ConsiderarandomvariableX withprobabilitydensityfunction

f xx x

x x( )

or =

≤ ≤< >

2 0 10 0 1

,,

Ifalargenumberofsamples,eachofsize100,istakenfromthisdistribution,thenthedistributionofthe

samplemeans,X ,willbeapproximatelynormalwithmeanE X( ) = 23andstandarddeviationsd X( )equal

to

A. 2

60

B. 2

6

C. 1

180

D. 1

18

E. 2

30

Question 19Bagsofpeanutsarepackedbyamachine.Themassesofthebagsarenormallydistributedwithastandarddeviationofthreegrams.Theminimumsizeofasamplerequiredtoensurethatthemanufacturercanbe98%confidentthatthesamplemeaniswithinonegramofthepopulationmeanisA. 37B. 38C. 48D. 49E. 60

Question 20Nitrogenoxideemissionsforacertaintypeofcarareknowntobenormallydistributedwithameanof0.875g/kmandastandarddeviationof0.188g/km.Fortworandomlyselectedcars,theprobabilitythattheirnitrogenoxideemissionsdifferbymorethan 0.5g/kmisclosesttoA. 0.030B. 0.060C. 0.960D. 0.970E. 0.977


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TURN OVER

CONTINUES OVER PAGE


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SECTION B – Question 1–continued

Question 1 (10marks)

Inthecomplexplane,Listhelinewithequation z z i+ = − −2 1 3 .

a. Verifythatthepoint(0,0)liesonL. 1mark

b. ShowthatthecartesianformoftheequationofLis 3 .y x= − 2marks

c. ThelineLcanalsobeexpressedintheform z z z− = −1 1 ,wherez1 ∈ C.

Findz1incartesianform. 2marks

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

11 2019 SPECMATH EXAM 2 (NHT)

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SECTION B – continuedTURN OVER

d. Find, in cartesian form, the point(s) of intersection of L and the graph of z = 4. 2 marks

e. Sketch L and the graph of z = 4 on the Argand diagram below. 2 marks

5

4

3

2

1

–1

–2

–3

–4

–5

O–5 –4 –3 –2 –1 1 2 3 4 5

Im(z)

Re(z)

f. Find the area of the sector defined by the part of L where Re(z) ≥ 0, the graph of z = 4 where Re(z) ≥ 0, and the imaginary axis where Im(z) > 0. 1 mark


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Question 2 (10marks)

Considerthefunction f withrule f xx xx

( ) .=+ +−

2

21

1

a. i. Statetheequationsoftheasymptotesofthegraphof f. 2marks

ii. Statethecoordinatesofthestationarypointsandthepointofinflection.Giveyouranswerscorrecttotwodecimalplaces. 2marks

iii. Sketchthegraphof f fromx =–6tox =6(endpointcoordinatesarenotrequired)onthesetofaxesbelow,labellingtheturningpointsandthepointofinflectionwiththeircoordinatescorrecttotwodecimalplaces.Labeltheasymptoteswiththeirequations. 3marks

y

x–6 –4 –2 2 4 6

2

1

O

–1

–2


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Considerthefunction fk withrule f x x x kxk ( ) ,=+ +−

2

2 1wherek ∈ R.

b. Forwhatvaluesofkwill fk haveno stationary points? 2marks

c. Forwhatvalueof k willthegraphof fk haveapointofinflectionlocatedonthey-axis? 1mark


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Question 3 (9marks)

h

60

40

20

O

–20

–30 30

y

x

Theverticalcross-sectionofabarrelisshownabove.Theradiusofthecircularbase (alongthex-axis)is30cmandtheradiusofthecirculartopis70cm.Thecurvedsidesofthe

cross-sectionshownarepartsoftheparabolawithrule y x= −

2

80454

. Theheightofthebarrelis 50cm.

a. i. Showthatthevolumeofthebarrelisgivenbyπ ( ) .900 800

50+∫ y dy 1mark

ii. Findthevolumeofthebarrelincubiccentimetres. 1mark


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Thebarrelisinitiallyfullofwater.WaterbeginstoleakfromthebottomofthebarrelsuchthatdVdt

hA

=−8000π cubiccentimetrespersecond,whereaftertsecondsthedepthofthewateris

hcentimetres,thevolumeofwaterremaininginthebarrelisVcubiccentimetresandtheuppermostsurfaceareaofthewaterisAsquarecentimetres.

b. Showthat dVdt

hh

=−

+400

4 45. 2marks

c. Finddhdt

intermsofh. Expressyouranswerintheform −+a hb chπ ( )

,2 wherea,bandcare

positiveintegers. 3marks

d. Usingadefiniteintegralintermsofh,findthetime,in hours,correcttoonedecimalplace,takenforthebarreltoempty. 2marks


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Question 4 (10marks)AsnowboarderattheWinterOlympicsleavesaskijumpatanangleofθdegreestothehorizontal,risesupintheair,performsvarioustricksandthenlandsatadistancedownastraightslopethatmakesanangleof45°tothehorizontal,asshownbelow.LettheoriginO ofacartesiancoordinate systembeatthepointwherethesnowboarderleavesthejump,withaunitvectorinthepositivexdirectionbeingrepresentedby

i andaunitvectorinthepositiveydirectionbeingrepresentedby

j.Distancesaremeasuredinmetresandtimeismeasuredinseconds.Thepositionvectorofthesnowboardertsecondsafterleavingthejumpisgivenby

r( ) i j, 0t t t t t t t= −( ) + − +( ) ≥6 0 01 6 3 4 9 0 013 2 3. . .

θ°

45°O

y

x

path of snowboarder

slope

a. Findtheangleθ °. 2marks

b. Findthespeed,inmetrespersecond,ofthesnowboarderwhensheleavesthejumpatO. 1mark


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c. FindthemaximumheightaboveOreachedbythesnowboarder.Giveyouranswerinmetres,correcttoonedecimalplace. 2marks

d. Showthatthetimespentintheairbythesnowboarderis60 3 1

49

+( )seconds. 3marks

e. Findthetotaldistancethesnowboardertravelswhileairborne.Giveyouranswerinmetres,correcttotwodecimalplaces. 2marks


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Question 5 (12marks)Apalletofbricksweighing500kgsitsonaroughplaneinclinedatanangleofα°tothehorizontal,

where tan( ) .α° =724

Thepalletisconnectedbyalightinextensiblecablethatpassesoverasmooth

pulleytoahangingcontainerofmassm kilogramsinwhichthereis10Lofwater.ThepalletofbricksisheldinequilibriumbythetensionTnewtonsinthecableandafrictionalresistanceforceof50g newtonsactingupandparalleltotheplane.Taketheweightforceexertedby1Lofwatertobegnewtons.

α°

a. Labelallforcesactingonboththepalletofbricksandthehangingcontaineronthediagramabove,whenthepalletofbricksisinequilibriumasdescribed. 1mark

b. Showthatthevalueofmis80. 3marks


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SECTION B – Question 5–continuedTURN OVER

Suddenlythewateriscompletelyemptiedfromthecontainerandthepalletofbricksbeginstoslidedowntheplane.Thefrictionalresistanceforceof50gnewtonsactinguptheplanecontinuestoactonthepallet.

c. Findthedistance,inmetres,travelledbythepalletafter10seconds. 3marks


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SECTION B – continued

d. Whenthepalletreachesavelocityof3ms–1,waterispouredbackintothecontainerataconstantrateof2Lpersecond,whichinturnretardsthemotionofthepalletmovingdowntheplane.Let tbethetime,inseconds,afterthecontainerbeginstofill.

i. Writedown,intermsoft,anexpressionforthetotalmassofthehangingcontainerandthewateritcontainsaftertseconds.Giveyouranswerinkilograms. 1mark

ii. Showthattheaccelerationofthepalletdowntheplaneisgivenby g tt

( )5290

2−+

−ms for t∈[ )0 5, . 2marks

iii. Findthevelocityofthepalletwhent =4.Giveyouranswerinmetrespersecond,correcttoonedecimalplace. 2marks


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CONTINUES OVER PAGE


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Question 6 (9marks)Apaintcompanyclaimsthatthemeantimetakenforitspainttodrywhenmotorvehiclesarerepairedis3.55hours,withastandarddeviationof0.66hours.Assumethatthedryingtimeforthepaintfollowsanormaldistributionandthattheclaimedstandarddeviationvalueisaccurate.

a. LettherandomvariableX representthemeantimetakenforthepainttodryforarandomsampleof36motorvehicles.

WritedownthemeanandstandarddeviationofX . 2marks

Atacrashrepaircentre,itwasfoundthatthemeantimetakenforthepaintcompany’spainttodryon36randomlyselectedvehicleswas3.85hours.Themanagementofthiscrashrepaircentrewasnothappyandbelievedthattheclaimregardingthemeantimetakenforthepainttodrywastoolow.Totestthepaintcompany’sclaim,astatisticaltestwascarriedout.

b. WritedownsuitablenullandalternativehypothesesH0andH1respectivelytotestwhetherthemeantimetakenforthepainttodryislongerthanclaimed. 1mark

c. Writedownanexpressionforthepvalueofthestatisticaltestandevaluateitcorrecttothreedecimalplaces. 2marks

d. Usinga1%levelofsignificance,statewithareasonwhetherthecrashrepaircentreisjustifiedinbelievingthatthepaintcompany’sclaimofameantimetakenforitspainttodryof 3.55hoursistoolow. 1mark


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e. Atthe1%levelofsignificance,findthesetofsamplemeanvaluesthatwouldsupporttheconclusionthatthemeantimetakenforthepainttodryexceeded3.55hours.Giveyouranswerinhours,correcttothreedecimalplaces. 2marks

f. Ifthetrue meantimetakenforthepainttodryis3.83hours,findtheprobabilitythatthepaintcompany’sclaimisnotrejectedatthe1%levelofsignificance,assumingthestandarddeviationforthepainttodryisstill0.66hours.Giveyouranswercorrecttotwodecimalplaces. 1mark

END OF QUESTION AND ANSWER BOOK

SPECIALIST MATHEMATICS

Written examination 2

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

Victorian Certificate of Education 2019

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

SPECMATH EXAM 2

Specialist Mathematics formulas

Mensuration

area of a trapezium 12 a b h+( )

curved surface area of a cylinder 2π rh

volume of a cylinder π r2h

volume of a cone 13π r2h

volume of a pyramid 13 Ah

volume of a sphere 43π r3

area of a triangle 12 bc Asin ( )

sine ruleaA

bB

cCsin ( ) sin ( ) sin ( )

= =

cosine rule c2 = a2 + b2 – 2ab cos (C )

Circular functions

cos2 (x) + sin2 (x) = 1

1 + tan2 (x) = sec2 (x) cot2 (x) + 1 = cosec2 (x)

sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x – y) = sin (x) cos (y) – cos (x) sin (y)

cos (x + y) = cos (x) cos (y) – sin (x) sin (y) cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

tan ( ) tan ( ) tan ( )tan ( ) tan ( )

x y x yx y

+ =+

−1tan ( ) tan ( ) tan ( )

tan ( ) tan ( )x y x y

x y− =

−+1

cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)

sin (2x) = 2 sin (x) cos (x) tan ( ) tan ( )tan ( )

2 21 2x x

x=

−

3 SPECMATH EXAM

TURN OVER

Circular functions – continued

Function sin–1 or arcsin cos–1 or arccos tan–1 or arctan

Domain [–1, 1] [–1, 1] R

Range −

π π2 2

, [0, �] −

π π2 2

,

Algebra (complex numbers)

z x iy r i r= + = +( ) =cos( ) sin ( ) ( )θ θ θcis

z x y r= + =2 2 –π < Arg(z) ≤ π

z1z2 = r1r2 cis (θ1 + θ2)zz

rr

1

2

1

21 2= −( )cis θ θ

zn = rn cis (nθ) (de Moivre’s theorem)

Probability and statistics

for random variables X and YE(aX + b) = aE(X) + bE(aX + bY ) = aE(X ) + bE(Y )var(aX + b) = a2var(X )

for independent random variables X and Y var(aX + bY ) = a2var(X ) + b2var(Y )

approximate confidence interval for μ x z snx z s

n− +

,

distribution of sample mean Xmean E X( ) = µvariance var X

n( ) = σ2

SPECMATH EXAM 4

END OF FORMULA SHEET

Calculus

ddx

x nxn n( ) = −1 x dxn

x c nn n=+

+ ≠ −+∫ 11

11 ,

ddxe aeax ax( ) = e dx

ae cax ax= +∫ 1

ddx

xxelog ( )( ) = 1 1

xdx x ce= +∫ log

ddx

ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dxa

ax c= − +∫ 1

ddx

ax a axcos( ) sin ( )( ) = − cos( ) sin ( )ax dxa

ax c= +∫ 1

ddx

ax a axtan ( ) sec ( )( ) = 2 sec ( ) tan ( )2 1ax dxa

ax c= +∫ddx

xx

sin−( ) =−

12

1

1( ) 1 0

2 21

a xdx x

a c a−

=

+ >−∫ sin ,

ddx

xx

cos−( ) = −

−

12

1

1( ) −

−=

+ >−∫ 1 0

2 21

a xdx x

a c acos ,

ddx

xx

tan−( ) =+

12

11

( ) aa x

dx xa c2 2

1

+=

+

−∫ tan

( )( )

( ) ,ax b dxa n

ax b c nn n+ =+

+ + ≠ −+∫ 11

11

( ) logax b dxa

ax b ce+ = + +−∫ 1 1

product rule ddxuv u dv

dxv dudx

( ) = +

quotient rule ddx

uv

v dudx

u dvdx

v

=

−

2

chain rule dydx

dydududx

=

Euler’s method If dydx

f x= ( ), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)

acceleration a d xdt

dvdt

v dvdx

ddx

v= = = =

2

221

2

arc length 1 2 2 2

1

2

1

2

+ ′( ) ′( ) + ′( )∫ ∫f x dx x t y t dtx

x

t

t( ) ( ) ( )or

Vectors in two and three dimensions

r = i + j + kx y z

r = + + =x y z r2 2 2

� � � � �ir r i j k= = + +ddt

dxdt

dydt

dzdt

r r1 2. cos( )= = + +r r x x y y z z1 2 1 2 1 2 1 2θ

Mechanics

momentum

p v= m

equation of motion

R a= m

2019 Specialist Mathematics Written examination 2 · SPECIALIST MATHEMATICS Written examination 2...

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Transcript of 2019 Specialist Mathematics Written examination 2 · SPECIALIST MATHEMATICS Written examination 2...