MPM2D–ExamReviewUnits1&2–AnalyticGeometry

1. Solvethefollowinglinearsystemsalgebraicallybythemethodindicated:

a) elimination b) substitutioni) 2x–3y=-5 ii) 3y=-4x–1 i) 2x+y=10 ii) 2x+3y=0

-2x+7y=17 3x–2y=-22 3x+2y=9 x+y=2

2. Solvethefollowinglinearsystemgraphically: 1 53

y x= − and2 3 3x y+ =

3. Circlethepairoflineswhichareparallel.

i) 3 4 13x y+ = ii) 4 3 8x y− = iii) 3 54

y x= − + iv)4

83

y x= − +

4. Circlethepairoflineswhichareperpendicular.

i) 2 15x y+ = ii) 2 5x y+ = iii) 1 32

y x= − − iv)1

42

y x= +

5. Howmanysolutionsarethereineachofthefollowinglinearsystems? i) 4x–6y=7 ii) 3x–2y=5 iii) x+y=3 2x–3y=3 9x–6y=15 x–y=16. Thepoint(-3,5)isasolutiontowhichofthefollowinglinearsystems? i) x+y=2 ii) x–y=-7 iii) -2x+3y=21 2x–y=-10 3x+2y=1 4x+2y=-27. Determinethedistanceandslopeforeachpairofpoints: i) A(4,5)andB(-2,-3) ii) M(-1,7)andN(4,-2) iii) X(5,0)andY(0,-12)8. Describewhatismeantbythewordsscalene,isosceles,andequilateraltriangles.9. DeterminethelengthsofthesidesofthetrianglewithverticesA(0,0),B(4,3)andC(-4,3).WhattypeoftriangleisΔABC?10.ShowthatthetrianglewithverticesP(-2,-3),Q(4,1)andR(2,4)isarighttriangle.Whichoftheverticescontainstherightangle?11.Definethetermsquadrilateral,parallelogram,rhombus,rectangleandsquare.12.DeterminewhetherthequadrilateralwithverticesA(0,10),B(10,90),C(170,70)andD(160,-10)isaparallelogram,rhombus,rectangleorsquare.13.Determinethemidpointofthelinesegmentswiththegivenendpoints: i) S(10,-5)andT(-4,7) ii) J(3,0)andK(6,8) iii) M(4.6,-2.9)andN(-2.4,1.5)14.Determinetheequationofthelinewithaslopeof-2andwhichgoesthroughthepoint(3,-1).

15.Describethetermsmedian,altitudeandperpendicularbisector.16.AtrianglehasverticesatA(-1,3),B(7,5)andC(2,-3).DeterminetheequationofthemedianthroughvertexC.17.AtrianglehasverticesatP(-4,-3),Q(5,-6)andR(4,1).DeterminetheequationofthealtitudewhichgoesthroughvertexR.18.AtrianglehasverticesJ(-2,-5),K(6,-1)andL(0,5).DeterminetheequationoftheperpendicularbisectorofsideJK.19.Describethetermscentroid,orthocentreandcircumcentre.20.Definethetermscircle,radiusanddiameter.21.Acirclehasadiameterwithendpointsat(6,8)and(-6,-8).Determinethefollowing: i) Thecoordinatesofthecentreofthecircle iii) Thelengthofthediameter ii) Thelengthoftheradius iv) Theequationofthecircle22.Determinetheequationofacirclecenteredattheoriginandhavingaradiusof8units.23.Whatisthediameterofacirclewiththeequation 2 2 49x y+ = ?24.Thesumoftwonumbersis72.Theirdifferenceis48.Findthenumbers.25.Jacqueshasatotalof$155in$2and$5pizzacoupons.Ifhehas40couponsinall,howmanyofeachkinddoeshehave?26.Arectanglewithaperimeterof180cmisfourtimeslongerthanitiswide.Whatareitsdimensions?27.Fraser’sPlumbingcharges$50foraservicecall,plus$40/hforlabour.Gus’Plumbingcharges$30foraservicecall,plus$45/hforlabour. a) Whendobothcompanieschargethesame? b) Whichcompanywouldyouhireforarepairlasting5h?

AnswerstoUnits1&2Review

1.a)i)(2,3) ii)(-4,5) b)i)(11,-12) ii)(6,-4) 2.(6,-3) 3.i)andiii) 4.i)andiv)

5.i)none ii)infinite iii)one 6.Iii) 7.i)AB=10, 43

m = ii)MN 106= , 95

m = −

7.iii)XY=13, 125

m = 9.AB=5,BC=8,CA=5,isoscelestriangle 10. 90Q∠ = ° 12.Rectangle

13.i)(3,1) ii)(4.5,4) iii)(1.1,-0.7) 14.y=-2x+5 16.y=7x–17 17.y=3x–1118.y=-2x+1 21.i)(0,0) ii)10 iii)20 iv) 2 2 100x y+ = 22. 2 2 64x y+ = 23.1424.60,12 25.15($2)and25($5) 26.72cmX18cm 27.a)4hours b)Fraser’sPlumbing

MPM2D–ExamReviewUnits3,4&5–Quadratics

1. Aballisthrownupwardandthefollowingdataiscollected.

Time(s) 0 1 2 3 4 5 6Height(m) 15 39 53 57 51 35 9

i)Determinethevaluesofthefirstdifferencesandseconddifferences.ii)Whattypeofrelationshipexistsbetweenheightandtime?Why?

2. Identifywhichofthefollowingquadraticrelationsisexpressedinvertexform,standardformandfactoredform.

i) 22 6 9y x x= − + ii) ( )( )3 4 6y x x= − − + iii) ( )22 1 7y x=− − +

3. Giventhequadraticrelation ( )( )1 5 32

y x x= − − + ,determine

i) thezeros iv) theequationoftheaxisofsymmetry ii) thecoordinatesofthevertex v) directionofopening iii) theoptimalvalue vi) sketchtheparabola4. Determinethezerosofthefollowingquadraticrelations:

i) ( )( )3 2 1 4y x x= − + ii) ( )4 5y x x=− + iii) ( )( )5 2y x x= − + 5. Statethedirectionofopening,coordinatesofthevertex,numberofzeros,equationoftheaxisofsymmetryand

makessketchesofeachquadraticrelation: i) ( )22 4 3y x= − − ii) ( )23 2 1y x=− + − 6. Foreachquadraticrelationinquestion5,statetheoptimalvalue.Isitamaximumorminimum?

7. Describeinwordshowyouwouldobtainthegraphof ( )23 4 2y x= − − fromtransformationsofthegraphof2y x= .Makesketchesofbothparabolasonthesamegrid.

8. Reviewthemethodsoffactoring:commonfactor,differenceofsquares,perfectsquare,trinomials.

9. Factoreachofthefollowingexpressions: i) 22 6x x− ii) 29 64x − iii) 2 12 36x x+ + iv) 2 4 21x x− − v) 26 7 3x x− − vi) 22 12 16x x− +

10.Expressthequadraticrelation ( )( )4 2 3y x x= + − instandardform.

11.Expressthequadraticrelation 2 7 44y x x= + − infactoredform.12.Rewritethequadraticrelationsbelowinvertexformusingthemethodofcompletingthesquare,andthen

makeasketchofeachparabola. i) 22 12 19y x x= − + ii) 23 24 50y x x= − − −

13.Determinethezerosandvertexofthequadraticrelation 2 6 7y x x= − − .Makeasketch.

14.Solveeachquadraticequationbyfactoring.

i) 2 30 0x x− − = ii) 22 5 9x x= − iii) 26 2 7x x+ = 15.Solveeachquadraticequationinquestion14usingthequadraticformula.16.Thealtitudeofatriangleis2mlongerthanitsbase.Whatarethedimensionsofthealtitudeandthebaseiftheareaofthetriangleis40m2?17.Thesideofonesquareis3cmlongerthanthesideofanothersquare.Ifthesumoftheareasofthetwosquaresis65cm2,findthelengthsofthesidesofeachsquare.18.Boristhrowsaballverticallyupwardfromthetopofacliff.Theheightoftheballabovethebaseofthecliffis

approximatedbythemodel 265 10 5h t t= + − ,wherehistheheightinmetresandtisthetimeinseconds. i) Howhighisthecliff? ii) Howlongdoesittaketheballtoreachaheightof50mabovethebaseofthecliff? iii) Afterhowmanysecondsdoestheballhittheground?19.Arighttrianglehasaperimeterof36units.Ifthehypotenuseis15units,howlongaretheothertwosides?20.ThecityofOttawahasprovided300mofropetoenclosearectangularswimmingareaalongtheshoreofMooney’sBaybeach.Whatisthemaximumareathatcanbeenclosed,andthedimensionsoftherectangleofmaximumarea?[Note:thebeachisonesideoftherectangle.]

AnswerstoUnits3,4&5Review

1.i)firstdifferences:24,14,4,-6,-16,-26;seconddifferences:-10,-10,-10,-10,-10 ii)quadraticrelationshipsinceseconddifferencesareequal.2.i)standardform ii)factoredform iii)vertexform3.i)5,-3 ii)(1,8) iii)8 iv)x=1 v)downward4.i)½,-4 ii)0,-5 iii)5,-25.i)opensupward,V(4,-3),twozeros,axisofsymmetryx=4 ii)opensdownward,V(-2,-1),nozeros,x=-26.i)optimalvalue=-3,minimum ii)optimalvalue=-1,maximum7.verticalstretchbyafactorof3,horizontaltranslation4unitsright,verticaltranslation2unitsdown9.i)2x(x-3) ii)(3x-8)(3x+8) iii)(x=6)2 iv)(x-7)(x+3) v)(3x+1)(2x-3) vi)2(x-2)(x-4)

10. 24 4 24y x x= − − 11. ( )( )11 4y x x= + − 12.i) ( )22 3 1y x= − + ii) ( )23 4 2y x=− + − 13.zeros:7,-1;vertex(3,-16)

14.i)x=6,x=-5 ii)x= 12,x=-5 iii)x= 1

2,x= 2

3 15.sameasquestion14

16.base=8cm,altitude=10cm17.4cmX4cmand7cmX7cm18.i)65m ii)3seconds iii)4.74seconds19.9units,12units20.maximumarea=11250m2,length=150m,width=75m

MPM2D–ExamReview

Unit6–Trigonometry1. a) ExplainwhyΔABCissimilartoΔDEC. b) Writeaproportionstatementforthesetwotriangles.2. a) ExplainwhyΔPQR~ΔPTS. b) Writeaproportionstatementforthesetwotriangles.3. Solveforthelengthsxandy.(diagrambelow)

4. GiventhatΔABC~ΔPQRand 31

AB BC CAPQ QR RP

= = = ,determinehowmanytimes

biggertheareaofΔABCisthantheareaofΔPQR.5.GiventhatΔXYZ~ΔABCandthattheareaofΔXYZis25timestheareaofΔABC,determinethelengthof,determinethelengthofZifthelengthofBCis2cm.

6. Alinehasaslopeangleof63.5° andpassesthroughthepoint(2,3).Determine a) Theapproximateslopeoftheline b) Theequationoftheline

7. Findtheslopeangleoftheline1

52

y x= − .

8. Reviewthedefinitionsoftheprimarytrigratios:sine,cosine,tangent.9. Findthelengthoftheunknownsideorangle: i) ii) iii) 10.Reviewthesinelawandthecosinelawfortriangles.11.Find C∠ inΔABC,if 87A∠ = ° ,a=15cm,andc=8cm.Sketchthetriangle.12.SolveΔPQR,if 54 , 71P Q∠ = ° ∠ = ° ,andr=47cm.Sketchthetriangle.13.FindthemeasureofsidexinΔXYZify=7cm,z=9cm,and 93X∠ = ° .Sketchthetriangle.14.SolveΔSTRifs=11cm,t=9cmandr=8cm.Sketchthetriangle.

15.Fromthetopofan8mhouse,theangleofelevationtothetopoftheschool’sflagpoleacrossthestreetis9°.Theangleofdepressionis42°tothebottomofthepole.Howtallistheflagpole?Makeasketch.16.Twoplanesleaveanairportatthesametime.Onetravelsat355km/handtheotherat450km/h.Twohourslatertheyare800kmapart.Findtheanglebetweentheircourses.17.ShipAandshipBleaveaportatthesametime.ShipAtravels100kmatabearingof37°.ShipBtravels300kmatabearingof125°. a) Howfarapartarethetwoships? b) WhatbearingisshipBfromshipA’sposition?18.FrompointA,whichisduewestofamountain,theangleofelevationtothetopis29°.FrompointB,whichisdueeastofamountain,theangleofelevationtothetopis35°.IfpointsAandBare8.2kmapart,howhighisthemountain?19.Ajoggerruns3.40kmdirectlysouth,andthenturnsandruns5.80kmonabearingof300°.Whatdistanceandinwhatdirectionshouldthejoggerruntogobackdirectlytothestartingpoint?Showthestepsofyoursolution.

AnswerstoUnit6Review

1.a)AA~ b)AB BC CADE EC CD

= = 2.a)AA~ b)PT TS SPPQ QR RP

= =

3.x=18cm,y=5cm4.9times5.YZ=10cm6.a)m=2 b)y=2x–17.26.6°9.i)x=24.3cm ii)w=48.6° iii)h=28.3cm11.C=32.2°12.R=55°,p=46.4cm,q=54.3cm13.x=11.7cm14.S=80.4°,T=53.8°,R=45.8°15.9.4m16.58.2°17.313km,bearingof144°18.2.5km19.5.05km,bearingof84°