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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
www.zetamaths.com 1©ZetaMathsLimited
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.
www.zetamaths.com 2©ZetaMathsLimited
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
www.zetamaths.com 3©ZetaMathsLimited
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
www.zetamaths.com 4©ZetaMathsLimited
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
www.zetamaths.com 5©ZetaMathsLimited
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
www.zetamaths.com 6©ZetaMathsLimited
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
www.zetamaths.com 7©ZetaMathsLimited
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
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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
www.zetamaths.com 9©ZetaMathsLimited
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
www.zetamaths.com 10©ZetaMathsLimited
Topic Skills Notes
TrigIdentities Know:sin2x+cos2x=1
∴sin2x=1-cos2x andcos2x=1-sin2x
xcosxsinxtan =
Usetheabovefactstoshowonetrigfunctioncanbeanother.Startwiththelefthandsideoftheidentityandworkthroughuntilitisequaltotherighthandside.
and
www.zetamaths.com 11©ZetaMathsLimited
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
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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
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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.