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V.2

ZETAMATHSNational5Mathematics

RevisionChecklist

Contents:Expressions&Formulae Page

Rounding……………………………………………… 1 Surds……………………………………………………. 1Indices……………………………………..………….. 1

Algebra………………..………………………………. 1AlgebraicFractions.……………………………… 2Volumes………………………………………………. 2Gradient………………………………………………. 3Circles….………………………………………………. 3

Relationships

TheStraightLine………………………………….. 4SolvingEquations/Inequations…………….. 4SimultaneousEquations………………………. 4ChangetheSubject………………………….….. 5QuadraticFunctions.……………………………. 5PropertiesofShapes……………………………. 7SimilarShapes..……………………………………. 8Trigonometry(GraphsandEquations)…. 8

Applications

TriangleTrigonometry..………………………. 11Vectors….……………………………………………. 11Percentages………………………………………… 12Fractions……….……………………………………. 12Statistics…………………………………………….. 12

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ExpressionsandFormulae

Topic Skills ExtraStudy/Notes RoundingRoundtodecimalplaces e.g. 25.1241→25.1 to1

d.p.

34.676→34.68to2

d.p.

RoundtoSignificantFigures

e.g. 1276→1300 to2sig.figs.

0.06356→0.064 to2sig.figs.

37,684→37,700 to3sig.figs. 0.005832→0.00583 to3sig.figs.

Surds Simplifying LearnSquareNumbers:4,9,16,25,36,49,64,81,

100,121,144,169.Usesquarenumbersasfactors:

e.g. 50 = 25× 2 =5 2

Add/Subtract e.g.

50 + 8 = 25× 2 + 4 × 2 =5 2 +2 2 =7 2

Multiply/Divide e.g. 5× 15 = 5×15 = 75 = 25× 3 =5 3 48

3=

483= 16 = 4

RationaliseDenominator

Removesurdfromdenominator.

e.g. 1

3=

1× 3

3× 3=

3

9=

33

IndicesUseLawsofIndices 1. ax ×ay =ax+y e.g. a2 ×a3 =a2+3 =a5

2. ax ÷ay =ax−y a7 ÷a4 =a7−4 =a3 3. (ax )y =axy (a4)5 =a4×5 =a20

4.1ax=a−x 3

31 −=aa

5. a0 =1 a0 =1

ScientificNotation/StandardForm

Thefirstnumberisalwaysbetween1and10.e.g. 54,600=5.46×104 0.000978=9.78×10-4

(1.3×105)×(8×103)=10.4×108=1.04×109

Evaluateusingindices e.g. 932727 23 232

=== AlgebraExpandSingleBracket 12343 +=+ xx )( ExpandTwoBrackets UseFOIL(FirstsOutsidesInsidesLasts)oranother

suitablemethod(x+3)(x–2)=x2+3x–2x–6=x2+x-6

Knowthateveryterminthefirstbracketmustmultiplyeveryterminthesecond.e.g.(x+2)(x2–3x–4)=x3–3x2–4x+2x2–6x–8 =x3–x2–10x–8

SimplifyExpression Puttogetherthetermsthatarethesame:e.g. x2+4x+3–2x+8=x2+2x+11 a×a×a=a3

Factorise–CommonFactor

Takethefactorseachtermhasincommonoutsidethebracket:e.g. 4x2+8x=4x(x+2)NB:Alwayslookforacommonfactorfirst.


Topic Skills Notes Factorise–DifferenceofTwoSquares

Alwaystakesthesameform,onesquarenumbertakeawayanother.Easytofactorise:e.g. x2–9=(x+3)(x–3) 5x2–125=5(x2–25)(Commonfactorfirst) =5(x+5)(x–5)

Factorise–Trinomial(simple)

Useanyappropriatemethodtofactorise:e.g.OppositeofFOIL:• FactorsoffirsttermareFirstsinbrackets.• Lastsmultiplytogivelasttermandaddtogive

middleterm. x2–x–6=(x–3)(x+2)

Factorise–Trinomial(hard)

Thisismoredifficult.Usesuitablemethod.UsingoppositeofFOILabovewithtrialanderror.NB:TheOutsidesaddInsidesgiveacheckofthecorrectanswer:e.g. 3x2–13x–10 =(3x–5)(x+2)Check:3x×2+(-5)×x=6x-5x=-x ✗

=(3x+2)(x–5)Check:3x×(-5)+2×x=-15x+2x=-13x ✓

Iftheansweriswrong,scoreoutandtryalterativefactorsorpositions.Keepanoteofthefactorsyouhavetried.

CompletetheSquare e.g.x2+8x-13=(x+4)2–13–16=(x+4)2-29 AlgebraicFractionsSimplifyingAlgebraicFractions

Step1:FactoriseexpressionStep2:Lookforcommonfactors.Step3:Cancelandsimplify6x2 −12xx2 + x −6

=6x(x −2)

(x+3)(x −2)=

6xx+3

AddandSubractFractions

Findacommondenominator.Thiscanbedoneeitherbyworkingoutthelowestcommondenominator,orbyusingSmileandKiss

bcbdac

bcbd

bcac

cd

ba

2310

23

210

235 +

=+=+

MultiplyFractions Multiplytopwithtop,bottomwithbottom:3a7c×4b5d

=12ab35cd

DivideFractions Invertsecondfractionandmultiply:

yxz

xyzx

xz

yx

zx

yx

149

2818

43

76

34

76 222

==×=÷

Volumes Volumeofaprism V=Areaofbasexheight

Volumeofacylinder V =πr2h

VolumeofaconeV = 1

3πr2h

VolumeofasphereV = 4

3πr3


Topic Skills Notes

Rearrangeeachoftheformulaetofindanunknown

e.g.Cylinderhasvolume400cm3andradius6cm,findtheheight

V =πr2h

h= 400π ×62

Vπr2

= h

Volumeofcompositeshapes

Thesearetwooftheabovecombined:LabelthemV1andV2

e.g. V1 =

43πr3 ÷2

V1=…

V2 =πr

2h V2=…

GradientFindthegradientofalinejoiningtwopoints

KnowthatgradientisrepresentedbythelettermStep1:SelecttwocoordinatesStep2:Labelthem(x1,y1)(x2,y2)Step3:Substitutethemintogradientformula

e.g. (-4,4),(12,-28)

m=y2 − y1x2 − x1

=(−28)−412−(−4)

=−−3216

=−2

CirclesLengthofArc Thisfindsthelengthofthearcofasectorofacircle:

LOA= angle360

×πd

LOAπd

=angle360

Forharderquestionsrearrangeformulatofindangle

AreaofSectorAOS = angle

360×πr2

AOSπr2

=angle360

Forharderquestionsrearrangeformulatofindangle

V1

V2

x1y1x2y2

or

or


Relationships

Topic Skills Notes StraightLine

Gradient • Representedbym• Measureofsteepnessofslope• Positivegradient–thelineisincreasing• Negativegradient–thelineisdecreasing

Y-intercept • Representedbyc• Showswherethelinecutsthey-axis• Findbymakingx=0

Findthegradientofalinejoiningtwopoints

KnowthatgradientisrepresentedbythelettermStep1:SelecttwocoordinatesStep2:Labelthem(x1,y1)(x2,y2)Step3:Substitutethemintogradientformula

e.g. (-4,4),(12,-28)

21632

)4(124)28(

xxyym12

12 −=−

=−−−−

=−−

=

Findequationofaline(fromgradientandy-intercept)

Step1:FindgradientmStep2:Findy-interceptcStep3:Substituteintoy=mx+c(seeabovefordefinitions)

Findequationofaline(fromtwopoints)

Usethiswhenthereareonlytwopoints(i.e.noy-intercept)Step1:FindgradientStep2:Substitueintoy–b=m(x–a)where(a,b)aretakenfromeitheroneofthepoints

Rearrangeequationtofindgradientandy-intercept

e.g. 3y+6x=12 3y=-6x+12 y=-2x+4 m=-2,c=4

Sketchlinesfromtheirequations

Step1:Rearrangeequationtotheformy=mx+c(seenoteabove)Step2:DrawatableofpointsStep3:Plotpointsoncoordinateaxes

SolvingEquations/Inequations SolvingEquations Usesuitablemethod:

e.g. 5(x+4)=2(x–5) 5x+20=2x–10 5x=2x–30 3x=-30 x=-10

Solvinginequations Solvethesamewayasequations.NB:Whendividingbyanegativechangethesign:e.g. -3x≤15 x≥-5

SimultaneousEquations

Solvebysketchinglines Step1:Rearrangelinestoformy=mx+cStep2:Sketchlinesusingtableofpoints(asabove)Step3:Findcoordinateofpointofintersection

Solvebysubstitution Thisworkswhenoneorbothequationsareoftheformy=ax+be.g.Solve 3x+2y=17 y=x+1Subequation2into1: 3x+2(x+1)=17 5x+2=17 x=3soy=3+1=4

x1y1x2y2

1

2

1


Topic Skills Notes

SimultaneousEquationsContd.

SolvebyElimination Step1:Scaleequationstomakeoneunknownequalwithoppositesign.Step2:AddEquationstoeliminateequaltermandsolve.Step3:Substitutenumbertofindsecondterm.e.g. 4a+3b=7 2a–2b=-14x2 8a+6b=14x3 6a–6b=-42+ 14a=-28 a=-2 substitutea=-2into 4(-2)+3b=7 3b=15 b=5Ans.a=-2,b=5

FormEquations Formequationsfromavarietyofcontextstosolveforunknowns

ChangetheSubject LinearEquations Rearrangeequationschangethesubject:

e.g.D=4C–3[C]D+3=4C

C43D=

+

43DC +

=

y=5(z+6)[z]

6z5y

+=

z65y

=−

65yz −=

Equationswithpowersorroots

e.g. hrV 2π= [r]

2rhV=

π

hVrπ

=

QuadraticFunctions

Quadraticsandtheirequations

y=x2 y=-x2 y=2x2 y=x2+5y=(x–3)2 y=(x+2)2-3

1

2

1

1

2

1

3

4

3

4

1


Topic Skills Notes Equationsofquadraticsy=kx2

Step1:IdentifycoordinatefromgraphStep2:Substituteintoy=kx2

Step3:Solvetofindke.g.Coordinate:(2,2)Substitution:2=k(2)2 2=4k k=0.5Quadratic:y=0.5x2

SketchingQuadraticsy=k(x+a)2+b

Step1:Identifyshape,ifk=1thengraphis+veorifk=-1thenthegraphis-veStep2:Identifyturningpoint(-a,b)Step3:Sketchaxisofsymmetryx=-aStep5:Findy-intercept(makex=0)Step4:Sketchinformation

SketchingQuadratics(Harder)y=(x+a)(x–b)

Step1:Identifyshape(+veor-ve)Step2:Identifyroots(x-intercepts)x=-a,x=bStep3:Findy-intercept(makex=0)Step4:Identifyturningpointe.g.y=(x+4)(x–2)+vegraph∴MinimumturningpointRoots:x=2,x=-4y-intercept:y=(0+4)(0–2)=-8TurningPoint(-1,-9)(seebelow)NB:Turningpointishalfwaybetweenroots.x-coord=(2+(-4))÷2=-1y-coord=(-1+4)(-1–2)=-9

SolvingQuadratics(findingroots)–Algebraically

Step1:EquatetozeroStep2:FactorisequadraticStep3:SeteachfactorequaltozeroStep4:Solveeachfactortofindrootse.g.y=x2+4x y=x2–5x–6 x(x+4)=0 (x–6)(x+1)=0 x=0orx+4=0 x–6=0orx+1=0 x=0orx=-4 x=6,x=-1

SolvingQuadratics(findingroots)–Graphically

Readrootsfromgraphx=2,x=-2

SolvingQuadratics–QuadraticFormula

Whenaskedtosolveaquadratictoanumberofdecimalplacesusethequadraticformula:

a2ac4bbx

2 −±−=

wherey=ax2+bx+c


Topic Skills Notes e.g.Solvey=x2–6x+2to1d.p.

a=1b=-6c=2

2286x

2286x

12214)6()6(

x2

+=

±=

×

××−−±−−=

2286x −

=

x=5.6 x=0.4

Discriminant b2–4acwherey=ax2+bx+cThediscriminantdescribesthenatureoftherootsb2–4ac>0tworealrootsb2–4ac=0equalroots(tangenttoaxis)b2–4ac<0

UsingtheDiscriminant Example1:Determinethenatureoftherootsofthequadraticy=x2+5x+4Solution:a=1,b=5,c=4b2–4ac=52–4×1×4=25–16=9Sinceb2–4ac>0thequadratichastworealroots.Example2:Determinep,wherex2+8x+phasequalrootsSolution:b2–4ac=082–4×1×p=0 64–4p=0 64=4p P=16

PropertiesofShapes

Circles

Pythagoras UsePythagorasTheoremtosolveproblemsinvolvingcirclesand3Dshapes.e.g.Findthedepthofwaterinapipeofradius10cm. ristheradius x2=102-92 x2=… x=4.4cm Depth=10–4.4=5.6cm

10cmx

18cm


Topic Skills Notes

SimilarShapes

LinearScaleFactorLength.Original

Length.NewFactor.Scale.Linear =

AreaScaleFactor 2

Length.OriginalLength.New

Factor.Scale.Area ⎟⎟⎠

⎞⎜⎜⎝

⎛=

VolumeScaleFactor 3

Length.OriginalLength.New

Factor.Scale.Volume ⎟⎟⎠

⎞⎜⎜⎝

⎛=

Trigonometry

TrigGraphs–SineCurve

y=asinbx+ca=maximaandminimaofgraphb=no.ofwavesbetween0and360⁰c=movementofgraphvertically

y=sinx maximaandminima1and-1,period=360⁰y=2sinxy=sin3xy=2sinx+2y=-sinx

2

-2

180⁰ 360⁰

y

x

2

-2

180⁰ 360⁰

y

x

4

1

-1

180⁰ 360⁰

y

x

1

-1

60⁰ 120⁰

y

x

1

-1

180⁰ 360⁰

y

x


Topic Skills Notes y=sin(x-30⁰)

TrigGraphs–CosineCurve

y=acosbx+ca=maximaandminimaofgraphb=no.ofwavesbetween0and360⁰c=movementofgraphvertically

y=cosxmaximaandminima1and-1,period=360⁰ThesametransformationsapplyforCosineasSine(above)

TrigGraphs–TanCurve

y=tanxnomaximaorminima,period=180⁰

SolvingTrigEquations KnowtheCASTdiagram

MemoryAid:AllStudentsTakeCare

Usethediagramabovetosolvetrigequations:Example1:Solve2sinx–1=0 2sinx=1 sinx=½ x=sin-1(½) x=30⁰,180⁰-30⁰

x=30⁰,150⁰

Example2:Solve4tanx+5=0 4tanx=-5 tanx=-5/4NB:tanxisnegativesotherewillbesolutionsinthesecondandfourthquadrant x==tan-1(5/4) xacute=51.3

⁰ x=180⁰-51.3⁰,360–51.3⁰

x=128.7⁰,308.7⁰

1

-1

210⁰ 360⁰

y

x30⁰

1

-1

180⁰ 360⁰

y

x

90⁰-90⁰

y

x


Topic Skills Notes

TrigIdentities Know:sin2x+cos2x=1

∴sin2x=1-cos2x andcos2x=1-sin2x

xcosxsinxtan =

Usetheabovefactstoshowonetrigfunctioncanbeanother.Startwiththelefthandsideoftheidentityandworkthroughuntilitisequaltotherighthandside.

and


National5LearningChecklist-Applications

Topic Skills ExtraStudy/Notes TriangleTrigonometry Triangle LabelTriangle

AreaofaTriangleCabA sin

21

=

SineRuleCc

Bb

Aa

sinsinsin==

UseSineRuletofindaside UseSineRuletofindanangle.NB: sinA=… A=sin-1(…)

CoosineRule Use Abccba cos2222 −+= tofindaside

Usebc

acbA

2cos

222 −+= tofindanangle

NB: cosA=… A=cos-1(…)

Bearings Useknowledgeofbearingstosolvetrigproblems.IncludingknowledgeofCorresponding,AlternateandSupplementaryangles.NB:ExtendrightnortharrowanduseZ-angles

Vectors 2DLineSegments Addorsubtract2DlineSegments

• Vectorsend-to-end• Arrowsinsamedirection

PositionVectors Thepositionvectorofacoordinateisthevectorfromtheorigintothecoordinate.E.g.A(4,-3)has

thepositionvector ⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

34

a

FindingaVectorfromTwoCoordinates

KnowthattofindavectorbetweentwopointsAand

Bthen abAB −= NB:Vectornotationforavectorbetweentwopoints

AandBis AB

3DVectors Determinecoordinatesofapointfromadiagramrepresentinga3DobjectLookatdifferenceinx,yandzaxesindividuallye.g.FindthecoordinatesofCC(15,9,0)

a b

a+b

BA

C

c

ab

BA

-ab

b-a

x

y

x

zy

B(15,9,6)

A(4,5,0)C


Topic Skills Notes VectorComponents AddandSubtract2Dand3Dvectorcomponents.

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

411

a and⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

523

b a+b=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+

+

+

542131

Multiplyvectorcomponentsbyascalar

2a=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

822

411

2

Findthemagnitudeofa2Dor3Dvector:

Forvectoru= ⎟⎟⎠

⎞⎜⎜⎝

⎛

yx

, u = 22 yx +

Forvectorv=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

zyx

, v = 222 zyx ++

PercentagesCompoundInterest Calculatemultiplierfrompercentage:

e.g.5%increase100%+5%=105%=1.05

Usemultipliertocalculatecompoundinterest/depreciation.e.g.£500with5%interestfor3years1.053x500

Percentageincrease/decrease %Increase/decrease= difference

original×100

ReversetheChange Findinitialamount.e.g.Watchreducedby30%to£42.70%=£42,1%=£0.60,100%=£60or42÷0.7=£60

FractionsAddandSubractFractions Findacommondenominator 2

3+45=1015+1215

AddandSubractMixedNumbers

Addorsubtractwholenumbers,ormakeanimproperfraction:

223+34

5=510

15+1215

or223+34

5=83+195

MultiplyFractions Multiplytopwithtop,bottomwithbottom:37×45=1235

MultiplyMixedNumbers

Maketopheavyfractionthenasabove:

337×45=237×45=9235

DivideFractions Invertsecondfractionandmultiply:67÷23=67×32=1810

=95

StatisticsComparingData

Calculatethemean: x = sum⋅of ⋅datanumber ⋅of ⋅terms

Findfivefiguresummary:L=lowestterm,Q1=lowerquartile,Q2=Median,Q3=upperquartile,h=highestterm

Interquartilerange:IQR=Q3–Q1middle50%ofdata


Topic Skills Notes

ComparingData(Contd.) Semi-Interquartilerange:SIQR= Q3−Q1

2

CalculateStandardDeviation:

s =Σx2 −

(Σx)2

nn−1

or s = Σ(x − x )2

n−1

KnowthatIQR,SIQRandstandarddeviationareameasureofthespreadofdata.Lowervaluemeansmoreconsistentdata.

Whencomparingdata,alwayscomparethemeasureofaverageandthemeasureofspreadi.e.comparethemediansorthemeansandthencomparetheSIQRorthestandarddeviation.Saywhateachcomparisonmeansinthecontext.e.g.OnaverageJohnexercisesmorebecausehismeanexercisetimeisgreater,butZahidismoreconsistentashisstandarddeviationissmaller.

LineofBestFit Useknowledgeofstraightlinetofindtheequationofalineofbestfit:y=mx+cory–b=m(x–a)

Useequationoflineofbestfittofindestimatefornewvalue.Usuallydosobysubstitutingvalueforxintoequation.

Drawbestfittingline:• Inlinewithdirectionofpoints• Roughlythesamenumberofpointsaboveandbelowline.

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