1

M

2

Weekbeginning Videos Page

June/July C3AlgebraicFractions 3

June/July C3AlgebraicDivision 4

June/July C3ReciprocalTrigFunctions 5

June/July C3PythagoreanIdentities 6

June/July C3TrigConsolidation 7

June/July C3ChainRule 8

June/July C3TheProductRule 9

June/July C3TheQuotientRule 10

June/July C3Differentiatingtrigfunctions 11

June/July C3TheCompoundAngleFormulae 12

June/July C3TheDoubleAngleFormulae 13

June/July C3TheFactorFormulae 14

June/July C3Naturallogsincludingderivativesofe^xandlnx 15

June/July C3:Integrationastheinverseofdifferentiation 16

Date Lesson Video Page

11thSeptember 1 C3:RevisionforCWC1:AlgebraicFractions 17

18thSeptember 1 C3:RevisionforCWC2:Differentiatingln 18

2 C3:RevisionforCWC3:ProductandQuotientRule 19

3 C3:RevisionforCWC4:Trigproofs 20

25thSeptember 1 C3:Functions–Domain,Range,Composite 21

2ndOctober 1 C3Functions–Modulus,Inverse,Transformations 22

9thOctober 1 C3NumericalMethods 23

16thOctober 2 C3Rcos(x+a) 24

2 C3Inversetrigfunctionsincludinggraphs 25

30thOctober 1 C3:dy/dx=1/(dx/dy) 26

2 C4ImplicitDifferentiation 27

6thNovember 1 C4BinomialExpansion 28

2 C4Partialfractions 29

13thNovember Readingweek–nolessons

20thNovember 1 M2Projectiles 31

27thNovember 1 C4IntegrationusingTrig,PartialFractions,lnx 32

2 C4TrapeziumRuleandpercentageerror 34

4thDecember 1 M2Motioninastraightline 35

11thDecember 1 C4IntegrationbyParts 36

2 C4IntegrationbySubstitution 37

1stJanuary 1 M2CentresofMass 38

2 M2Frameworks 39

8thJanuary 1 M2Tilting 40

15thJanuary 1 C4ConnectedRatesofChange 41

22ndJanuary 1 C4Vectors–Introduction 42

2 C4vectors–Scalar(dot)product 43

3 C4Vectors–Vectorequationofaline 44

29thJanuary 1 M2Work,Energy&Power 45

5thFebruary 1 C4FormingDifferentialEquations 46

2 C4DifferentialEquations 47

19thFebruary 1 M2Statics 48

26thFebruary 1 C4Parametrics 49

5thMarch 1 VolumeofSolidofRevolution 50

12thMarch M2Collisions 51

3

C3:AlgebraicFractions

https://youtu.be/MC90CB-s8QM

Copythetwoexamples

1) Simplify!!!!!!!

!!!!

!!

2) Simplify!!!!!!!!

!" ÷!!!!!"

4

C3AlgebraicDivision

https://youtu.be/tfjYIrkvalI

Copytheexample !!!!!!!!!!!!"!!!!!!!

5

C3:ReciprocalTrigFunctions

https://youtu.be/5mpYSXxktHU

Completethetable

secx=

1cos !

cosecx=

1sin !

cotx=

1tan !

Drawthegraphsofy=secx, y=cosecxyy=cotx

Whenprovingatrigidentity,howshouldyoustarttheproofandhowshouldyoufinishtheproof?

StartwithLHS≡ andendwith≡ RHS(orviceversa).Thelastthingyoushouldwriteis“ProofComplete”(orQ.E.D.ifyoulikeyourLatin)Whatisthedifferencebetween=and≡=means“equals”andisusedtosolveequations≡ means“isidenticalto”andisusedwhenalgebraicexpressionsareidenticale.g.!! = !isonlytruefortwovaluesofxe.g.!! − 4 �(! − 2)(! + 2)istrueforeveryvalueofx

6

C3PythagoreanIdentities

https://youtu.be/VcDtYgSDvRs

TherearethreePythagoreanidentities(oneofwhichyouhadtoknowforC2).

Whatarethey? sin! ! + cos! ! = 1

1 + tan! ! = sec! !

1 + cot! ! = !"#$!!!

Provethat!"#$!!�-cot! � ≡ !!!"#! �

!!!"#! �

7

C3TrigConsolidation

https://youtu.be/zk8Bled7bFc

Solvetheequation4!"#$!!! − 9=cotθ.Showfullworking

8

C3ChainRule

https://youtu.be/YLSm56VIa6U

Ify= 5!! − 2! !,whatis!"!".

!"!" = 6 5!! − 2! !(15!! − 2)

9

C3:TheProductRule

https://youtu.be/Jpg3QX5slg4

Ify=f(x)g(x),whatis!"!"

f’(x)g(x)+f(x)g’(x)

Ify=!! 2!! + 3! !,whatis!"!".Showthefullworking.

10

C3TheQuotientRule

https://youtu.be/aYzFUB5q7BI

Ify=!!whereuandvarefunctionsofx,whatis

!"!" ?

!"!" =

!′! − !"′!!

Ify=!(!)!(!)whatis

!"!" ?

!"!" =

!! ! ! ! − ! ! !! !! ! !

Ify=!!!! !

!! ,whatis!"!"?Showthefullworking.

11

C3:Differentiatingtrigfunctions

https://youtu.be/kQ0lvJtXIgY

Completethistable

y !"!"

sinx

cosx

cosx

-sinx

tanx

!"#! !cotx

− !"#$!!!secx

secxtanx

cosecx

-cosecxcotx

Whendifferentiatingatrigshouldyouuseradiansordegrees?RADIANS

12

C3:TheCompoundAngleFormulae

https://youtu.be/DyqQG7MzOPU

sin(A±B)=sinAcosB±cosAsinB

cos(A±B)=cosAcosB∓sinAsinB

tan(A±B)=!"#!±!"#! ! ∓!"#$ !"#$

13

C3:TheDoubleAngleFormulae

https://youtu.be/upkil94kk_g

Completethesedoubleangleformulae

sin2A=2sinAcosA

tan2A=! !"#!!!!"#! !

cos2A=cos! ! − sin! !

Therearetwootherformulaeforcos2A.

cos2A=2 cos! ! − 1

cos2A=1 − 2 sin! !

14

C3:TheFactorFormulae

https://youtu.be/sHPEY10RSOE

Thefactorformulae

2cos

2sin2sinsin

BABABA −+=+

2sin

2cos2sinsin

BABABA −+=−

2cos

2cos2coscos

BABABA −+=+

2sin

2sin2coscos

BABABA −+−=−

15

C3:Naturallogarithmsincludingderivativesof!!andlnxhttps://youtu.be/cxtkmtQmhUY

16

C4:Integrationastheinverseofdifferentiation

https://youtu.be/NRZJw-FtuSA

17

C3:RevisionforContinuingWithConfidenceTest1SimplifyingAlgebraicFractions

https://youtu.be/PY1lwgBLnqo

Express)2)(32(

32 2

−+

+

xxxx

–2

62 −− xx

asasinglefractioninitssimplestform.

18

C3:RevisionforContinuingWithConfidenceTest2Differentiatingln

https://youtu.be/8uUqYRl9T0Q

ThepointPliesonthecurvewithequationy=ln ⎟⎠

⎞⎜⎝

⎛ x31

.Thex-coordinateofPis3.

FindanequationofthenormaltothecurveatthepointP intheformy=ax+b,whereaandbareconstants.

19

C3:RevisionforContinuingWithConfidenceTest3ProductandQuotientRule

https://youtu.be/X8xxaHJtsJE

Differentiatewithrespecttox

(i) x2e3x+2,

(ii)xx

3)2(cos 3

.

20

C3:RevisionforContinuingWithConfidenceTest4Trigproofs

https://youtu.be/Csj_NSc1uOk

Showthat

(i)xx

xsincos2cos+

≡cosx–sinx,x≠(n– 41 )π,n∈ℤ,

(ii)!!(cos2x–sin2x)≡cos

2x–cosxsinx–!!.

21

C3:Functions–Domain,Range,Composite

https://youtu.be/obyZwc7EX3Y

Definetheseterms

Amapping……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Domain……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Range………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Function……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………CompositeFunction…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Iff(x)=3x+2andg(x)=!! + 1,a) whatisfg(x)

b) whatisgf(x)

22

C3Functions–Modulus,Inverse,Transformations

https://youtu.be/dxzM3Yl0X5s

Iff(x)=3x+2,whatisf-1(x)?Showthefullworking.

Sketchthefollowingfunctions f(x)=|3x+2| f(x)=|x|-2

Thegraphshowsthefunctionf(x)Sketchthegraphsofy=|f(x)| y=f(|x|)

23

C3:NumericalMethods

https://youtu.be/4JYb-lPtspU

Showthatthereisarootoftheequation!" !=!!betweenx=1.7andx=1.8.Underlinethekeysentencethatyoumustwriteattheendofthesequestions.

Useiterationtofindasolutiontox2–4x+1=0correctto2d.p.

24

C3:Rcos(x+α )

Usehttps://youtu.be/dmrFYuNfkFg

Solvetheequation3cosx+5sinx=2

(0°≤x≤360°)

Showthefullworking.

25

C3:Inversetrigfunctionsincludinggraphs

https://youtu.be/hklOnHJx1t4

Whatisthedifferencebetweeny=sin-1xandy=(sinx)-1?…………………………………………………………………………………………………………………………Sketchthegraphsofy=arcsinx(y=sin-1x) y=arccosx(y=cos-1x)y=arctanx(y=tan-1x)

26

C3:dy/dx=1/(dx/dy)

https://youtu.be/KEoip8FNAp4

Findthevalueof!"!"atthepoint(2,1)onthecurvewith

equation!! + ! = !

27

C4:ImplicitDifferentiation

https://youtu.be/am9WPDZL76M

Theequationofacurveis3!! + 2!!!! + 4!! = 12

Findanexpressionfor!"!"

28

C4:BinomialExpansion

https://youtu.be/eC2I1eismWQ

Whatisthebinomialexpansionwhichisvalidforallvaluesofn?

Ifnisfractionalornegative,theexpansionisonlyvalidforcertainvaluesofx.Whatarethose

values?

29

C4PartialFractions

https://youtu.be/OeUCqui7bu0

Write!!!!

!!! !!! inpartialfractions.Showthefullmethod

Write!!!!!!"! !!!!! ! !!!! inpartialfractions.Showthefullmethod

….continuedonnextpage

30

C4PartialFractionsPage2

Write!!!!!!!!!!! !!! inpartialfractions.Showthefullmethod

31

S2:Projectiles

https://youtu.be/ZLbfw3iYIBc

AprojectilePisprojectedfromapointOonahorizontalplanewithspeed28 !!!!andwithangleofelevation30°.Afterprojection,theprojectilemovesfreelyundergravityuntilitstrikestheplaneat

pointA.Find

a) thegreatestheightabovetheplanereachedbyP,

b) thetimeofflightofP

c) thedistanceOA

32

C4:Integration(Trig,PartialFractions,lnx)

https://youtu.be/wHYbo3igKqs

Showhowtointegratethefollowing

∫tan! ! !"

∫!"#! ! !"

∫cos! ! !"**….continuedonthenextpage**Thisisn’tonthevideo.Weshalldothisinthelesson

33

C4:Integration(Trig,PartialFractions,lnx)page2

∫!! !! ! dx=

∫!!!

!!! !!! dx

34

C4:TrapeziumRuleandPercentageError

https://youtu.be/IBMPn_4eKqY

Usethetrapeziumrulewith4stripstofindanapproximatevaluefor sec ! !"!!!

Whencalculatingthepercentageerror,usethisformula

Percentage error =

35

S2:Motioninastraightline

https://youtu.be/xvVkAG7o1T8

36

C4:IntegrationbyParts

https://youtu.be/3ElTLQirl4E

Theformulaforintegrationbypartsis

� ! !"!" !" =

Useintegrationbypartstoworkout∫xcosxdx

37

C4:IntegrationbySubstitution

https://youtu.be/WeAKe8uGQ1M

Useintegrationbysubstitutiontoworkout

∫x√(2x+5)dx

Useintegrationbysubstitutiontoevaluate

cos ! √(1 + sin !) !"!!!

38

M2CentreofMass

https://youtu.be/6dVVxvKBYdY

Findthecentreofmassofa2kgmassatthepoint(3,0),a5kgmassatthepoint(4,0)anda3kgmass

atthepoint(6,0)

Findthecentreofmassofa2kgmassatthepoint(1,2),a5kgmassatthepoint(4,3)anda3kg

massatthepoint(3,1)

Whereisthecentreofmassofthefollowinglaminas

a) Auniformcirculardisc……………………………………………………………

b) Auniformrectangularlamina………………………………………………

c) Auniformsectorofacircle…………………………………………………

Findthecentreofmassofauniformtriangularlaminawithverticesat(1,1),(5,0)and(2,4)

39

M2Frameworks

https://youtu.be/yZy7eC1hJM8

Findthecentreofmassofthisframework

40

M2Tilting

https://youtu.be/TogyRaizaQw

FindtheanglethatthelineABmakeswiththeverticalifthis

laminaisfreelysuspendedfromA

41

C4:ConnectedratesofChange

https://youtu.be/OyeiYysYXZI

Therateofchangeoftheradiusofacircleis5cms-1.Findtherateofchangeoftheareaof

thecirclewhentheradiusis3cm.

42

C4:Vectors-Introduction

https://youtu.be/2OWPNzC7JBI

ThefirstpartofthisvideoisrevisionofGCSEvectors.Makeyourownnotesofstuffyoumayhave

forgotten.

Findaunitvectorwhichisinthedirection3a+4b

Whatarethevectorsi,jandk?

Findthedistancebetween(4,3,6)and(3,4,-2)

43

C4:Vectors–scalar(dot)product

https://youtu.be/zkAMAhqeXio

Whatisthedefinitionofa.b?

Ifa=a1i+a2j+a3kandb=b1i+b2j+b3kwhatisa.b?

Whatistheanglebetween!!!!

and!!!!

?

44

C4:Vectors–vectorequationofaline

https://youtu.be/ltVa0nqX7o8

Astraightlinepassesthrough(-1,1)and(0,1).Whatisthevectorequationofthisline?

Thisnextbitisn’tinthevideo.Trytowritedowntheanswer.Wewilldiscussinthelesson.

Thevectorequationofastraightlineisr=a+λb.Whatinformationisgivenbya?Whatinformationisgivenbyb?

45

M2:Work,EnergyandPower

https://youtu.be/gu9NCEa1GnQ

46

C4:FormingDifferentialEquations

https://youtu.be/dnWa5_3eNb8

47

C4:DifferentialEquations

https://youtu.be/q9a52OxY3Ww

Solvethedifferentialequation!"!" = ! cos !.Usetheboundaryconditionthaty=0whenx=1

48

M2:Statics

https://youtu.be/4XB-7EjmqtA

49

C4:Parametrics

https://youtu.be/nFClKAxutOc

Acurveisdefinedbytheparametricequationsx=2tandy=t2.WhatistheCartesianequationofthiscurve?

Acurveisdefinedbytheparametricequationsx=t2andy=2t(3-t).Findtheareabetweenthiscurveandthexaxiswhichisboundedbythey-axisandthelinex=3

50

C4:VolumesofSolidsofRevolution

https://youtu.be/djcBiBMwI7Q

Whatistheformulaforthevolumeofasolidofrevolution?

Whatistheformulafortheareaunderacurve?

51

M2:Collisions

https://youtu.be/fLtDbmdnWl0

WhatistheformulaforConservationofLinearMomentumWhatistheformulafore(thecoefficientofrestitution)Findv1andv2

! = !!

m1=200kg m2=400kg u1=5ms-1 u2=-4ms-1 v1=? v2=?

52

UnitC3:CoreMathematics3Thissectionlistsformulaethatcandidatesareexpectedtorememberandthatmaynotbeincludedinformulaebooklets.Trigonometry

Differentiationfunction Derivative sinkx kcoskxcoskx –ksinkxekx kekxlnx x

1

f(x)+g(x) )(g +)( f xx ʹʹ

f ( g (x x) ) )(g )( f)g( )( f xxxx ʹ+ʹ

f(g(x)) )(g ))( g( f xx ʹʹ

C3Syllabus1 AlgebraandfunctionsSimplificationofrationalexpressionsincludingfactorisingandcancelling,andalgebraicdivision.

Denominatorsofrationalexpressionswillbelinear

orquadratic,egb + ax

1,

r + qx + pxb + ax

2,

11

2

3

−

+

xx

.

Definitionofafunction.Domainandrangeoffunctions.Compositionoffunctions.Inversefunctionsandtheirgraphs.

Theconceptofafunctionasaone-oneormany-onemappingfromℝ(orasubsetofℝ)toℝ.Thenotationf:x! andf(x)willbeused.Candidatesshouldknowthatfgwillmean‘dogfirst,thenf’.

Candidatesshouldknowthatiff−1exists,thenf−1f(x)=ff−1(x)=x.

Themodulusfunction.

Candidatesshouldbeabletosketchthegraphsofy=⏐ax+b⏐andthegraphsofy=⏐f(x)⏐andy=f(⏐x⏐),giventhegraphofy=f(x).

Combinationsofthetransformationsy=f(x)asrepresentedbyy=af(x),y=f(x)+a,y=f(x+a),y=f(ax).

Candidatesshouldbeabletosketchthegraphof,

forexample,y=2f(3x),y=f(−x)+1,giventhegraphofy=f(x)orthegraphof,forexample,

y=3+sin2x,y=−cos ⎟⎠

⎞⎜⎝

⎛+4π

x .

Thegraphofy=f(ax+b)willnotberequired.

AAAA

AA

22

22

22

cot1cosectan1sec

1sincos

+≡

+≡

≡+

AAA

AAAAAA

2

22

tan1 tan 2

2tan

sincos2 cos cos sin 2 2sin

−≡

−≡

≡

53

2 TrigonometryKnowledgeofsecant,cosecantandcotangentandofarcsin,arccosandarctan.Theirrelationshipstosine,cosineandtangent.Understandingoftheirgraphsandappropriaterestricteddomains.

Anglesmeasuredinbothdegreesandradians.

Knowledgeanduseofsec2θ=1+tan2θandcosec2θ=1+cot2θ.

Knowledgeanduseofdoubleangleformulae;use

offormulaeforsin(A±B),cos(A±B)andtan(A±B)andofexpressionsforacosθ+bsinθintheequivalentformsofrcos(θ±a)orrsin(θ±a).

Toincludeapplicationtohalfangles.Knowledgeof

thet(tan θ21 )formulaewillnotberequired.

Candidatesshouldbeabletosolveequationssuchas

acosθ+bsinθ=cinagiveninterval,andtoprovesimpleidentitiessuchas

cosxcos2x+sinxsin2x≡cosx.

3 ExponentialsandlogarithmsThefunctionexanditsgraph.

Toincludethegraphofy=eax+b+c.

Thefunctionlnxanditsgraph;lnxastheinversefunctionofex.

Solutionofequationsoftheformeax+b=pandln(ax+b)=qisexpected.

4 DifferentiationDifferentiationofex,lnx,sinx,cosx,tanxandtheirsumsanddifferences.

Differentiationusingtheproductrule,thequotientruleandthechainrule.

Differentiationofcosecx,cotxandsecxarerequired.Skillwillbeexpectedinthedifferentiationoffunctionsgeneratedfromstandardformsusingproducts,quotientsandcomposition,suchas2x4sin

x,x

x3e,cosx2andtan22x.

Theuseof

⎟⎟⎠

⎞⎜⎜⎝

⎛=

yxx

y

dd1

dd

. E.g.findingxydd

forx=sin3y.

5 NumericalmethodsLocationofrootsoff(x)=0byconsideringchangesofsignoff(x)inanintervalofxinwhichf(x)iscontinuous.

Approximatesolutionofequationsusingsimpleiterativemethods,includingrecurrencerelationsoftheformxn+1=f(xn).

Solutionofequationsbyuseofiterativeproceduresforwhichleadswillbegiven.

54

UnitC4:CoreMathematics4Thissectionlistsformulaethatcandidatesareexpectedtorememberandthatmaynotbeincludedinformulaebooklets.Integrationfunction integral

coskx kx

k sin1

+c

sinkx kx

k cos

1− +c

ekx

kx

ke1 +c

x1

ln x +c, 0≠x

)( g)( f xx ʹ+ʹ )( g+)( f xx +c

)(g ))( (g f xx ʹʹ ))( g( f x +cVectors

zcybxacba

zyx

++=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

.

C4Syllabus1 AlgebraandfunctionsRationalfunctions.Partialfractions(denominatorsnotmorecomplicatedthanrepeatedlinearterms).

Partialfractionstoincludedenominatorssuchas(ax+b)(cx+d)(ex+f)and(ax+b)(cx+d)2.Thedegreeofthenumeratormayequalorexceedthedegreeofthedenominator.Applicationstointegration,differentiationandseriesexpansions.Quadraticfactorsinthedenominatorsuchas(x2+a),a>0,arenotrequired.

2 Coordinategeometryinthe(x,y)planeParametricequationsofcurvesandconversionbetweenCartesianandparametricforms.

Candidatesshouldbeabletofindtheareaunderacurvegivenitsparametricequations.Candidateswillnotbeexpectedtosketchacurvefromitsparametricequations.

55

3 SequencesandseriesBinomialseriesforanyrationaln.

Forab

x < ,candidatesshouldbeabletoobtainthe

expansionof(ax+b)n,andtheexpansionofrationalfunctionsbydecompositionintopartialfractions.

4 DifferentiationDifferentiationofsimplefunctionsdefinedimplicitlyorparametrically.

Thefindingofequationsoftangentsandnormalstocurvesgivenparametricallyorimplicitlyisrequired.

Exponentialgrowthanddecay.

Knowledgeanduseoftheresult

xdd(ax)=axlnaisexpected.

Formationofsimpledifferentialequations.

Questionsinvolvingconnectedratesofchangemaybeset.

5 Integration

Integrationofex,x1,sinx,cosx.

Toincludeintegrationofstandardfunctionssuchas

sin3x,sec22x,tanx,e5x,x21

.

Candidatesshouldrecogniseintegralsoftheform

xxx d)f()(f

⎮⌡⌠ ʹ

=lnf(x)+c.

Candidatesareexpectedtobeabletousetrigonometricidentitiestointegrate,forexample,sin2x,tan2x,cos23x.

Evaluationofvolumeofrevolution. xy d2

⎮⌡

⌠π isrequired,butnot yx d2⎮⌡

⌠π .

Candidatesshouldbeabletofindavolumeofrevolution,givenparametricequations.

Simplecasesofintegrationbysubstitutionandintegrationbyparts.Thesemethodsasthereverseprocessesofthechainandproductrulesrespectively.

Exceptinthesimplestofcasesthesubstitutionwillbegiven.

Theintegral ∫ xln dxisrequired.

Morethanoneapplicationofintegrationbyparts

mayberequired,forexample ∫ xx x de2 .

Simplecasesofintegrationusingpartialfractions.

Integrationofrationalexpressionssuchasthose

arisingfrompartialfractions,e.g.53

2+x

,2)1(

3−x

.

Notethattheintegrationofotherrational

56

expressions,suchas52 +x

xand

4)12(2−x

isalso

required(seeaboveparagraphs).

Analyticalsolutionofsimplefirstorderdifferentialequationswithseparablevariables.

Generalandparticularsolutionswillberequired.

Numericalintegrationoffunctions.

ApplicationofthetrapeziumruletofunctionscoveredinC3andC4.Useofincreasingnumberoftrapeziatoimproveaccuracyandestimateerrorwillberequired.Questionswillnotrequiremorethanthreeiterations.

Simpson’sRuleisnotrequired.

6 VectorsVectorsintwoandthreedimensions.

Magnitudeofavector.

Candidatesshouldbeabletofindaunitvectorinthe

directionofa,andbefamiliarwith⏐a⏐.

Algebraicoperationsofvectoradditionandmultiplicationbyscalars,andtheirgeometricalinterpretations.

Positionvectors.Thedistancebetweentwopoints.

ab−==−→→→

ABOAOB .

Thedistancedbetweentwopoints(x1,y1,z1)and(x2,y2,z2)isgivenbyd2=(x1–x2)2+(y1–y2)2+(z1–z2)2.

Vectorequationsoflines. Toincludetheformsr=a+tbandr=c+t(d–c).Intersection,orotherwise,oftwolines.

Thescalarproduct.Itsuseforcalculatingtheanglebetweentwolines.

Candidatesshouldknowthatfor→OA=a=a1i+a2j+a3kand→OB=b=b1i+b2j+b3kthena.b=a1b1+a2b2+a3b3and

cos∠AOB=baba ..

Candidatesshouldknowthatifa.b=0,andthataandbarenon-zerovectors,thenaandbareperpendicular.

57

UnitM2:Mechanics2

58

59

FormulaeandTablesCoreMathematicsC3Logarithmsandexponentials

xax a=lne Trigonometricidentities

BABABA sincoscossin)(sin ±=±

BABABA sinsincoscos)(cos ∓=±

))(( tantan1tantan)(tan 2

1 π+≠±±

=± kBABABABA

∓

2cos

2sin2sinsin

BABABA −+=+

2sin

2cos2sinsin

BABABA −+=−

2cos

2cos2coscos

BABABA −+=+

2sin

2sin2coscos

BABABA −+−=−

Differentiation

f(x) fʹ(x)tankx ksec2kxsecx secxtanxcotx –cosec2xcosecx –cosecxcotx

)g()f(xx

))(g(

)(g)f( )g()(f2x

xxxx ʹ−ʹ

CoreMathematicsC4Integration(+constant)f(x)

⎮⌡

⌠ xx d)f(

sec2kx k1tankx

xtan xsecln

xcot

xsinln

xcosec )tan(lncotcosecln 21 xxx =+−

xsec )tan(lntansecln 41

21 π+=+ xxx

⎮⌡⌠

⎮⌡⌠−= x

xu

vuvxxv

u dddd

dd

60

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