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Math Line Questions by Experienced IB Maths Tutors

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Math Line Questions by Experienced IB Maths Tutors

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  1. 1.[Maximummark:8][withoutGDC] ConsiderthepointsA(3,9),B(6,13). FindthegradientofthelineAB.[2] WritedownthegradientofalineperpendiculartoAB.[1] LetMbethemidpointofthelinesegmentAB.FindthecoordinatesofM.[1] FindthedistancebetweenAandB(i.e.,thelengthAB).[2] FindthecoordinatesofthepointCifBisthemidpointofthelinesegmentAC.[2] 2.[Maximummark:8][withoutGDC] ConsiderthepointsA(4,6),B(b,2),andC(8,−4)whereb∈R.Findthevaluesofbineachofthe followingcases: (a)IfthegradientofthelineABis3.[2] 2 IfthemidpointofthelinesegmentABisM(6,4).[2] IfBisthemidpointofthelinesegmentAC.[1] IfthedistancebetweenAandBis6.[3] 3.[Maximummark:9][withoutGDC] ConsiderthelineLgivenby3x+4=y. (a)Writedown thegradientoftheline[1] they-intercept[1] thex-intercept[1] Drawthelineonthediagrambelow.[3] CheckifthepointsA(5,19)andB(6,20)lieontheline.[3] 4.[Maximummark:7][withoutGDC] FindtheequationofthelinepassingthroughA(2,5)andB(6,8). inthegradient-pointformy−y1=m(x−x1).[3] inthegradient-interceptformy=mx+c.[2] intheformax+by=d,wherea, b,anddareintegers.[2]

  2. 5.[Maximummark:6][withoutGDC] Thediagrambelowshowsthelinewithequation4x+3y=24.ThepointsAandBaretheyandx- interceptsrespectively.MisthemidpointofAB. Findthecoordinatesof thepointA;[2] thepointB;[2] thepointM.[2] 6.[Maximummark:5][withoutGDC] FindtheequationofthelinepassingthroughthepointsA(3,−4)andB(3,9).[2] FindtheequationofthelinepassingthroughthepointsC(5,7)andD(−2,7).[2] FindthepointofintersectionPbetweenthelinesL1andL2.[1] 7.[Maximummark:6] Findtheequationofthelinewhichisparalleltothex-axisandpassesthroughA(1,4).[2] Findtheequationofthelinewhichisparalleltothey-axisandpassesthroughA(1,4).[2] FindtheequationofthelinepassingthroughtheoriginandA(1,4).[2] 8.[Maximummark:9][withoutGDC] ConsiderthelineLonthediagrambelow. (a)Writedown thegradientoftheline[1] they-intercept[1] thex-intercept[1] Writedowntheequationofthelineinthegradient-interceptformy=mx+c.[2] GiventhatP(2,y)andQ(x,2)lieontheline,writedownthevaluesofxandy.[2] GiventhatA(a,−3)andB(−3,b)lieontheline,findthevaluesofaandb.[2] 9.[Maximummark:8][withoutGDC] ConsiderthelineLwithequationy+3x=5.ThelineL1isparalleltoLandpassesthroughthe point(7,−5). FindthegradientofL1.[1] FindtheequationofL1intheformy=mx+c.[3] Findthex-coordinateofthepointwherelineL1crossesthex-axis.[2] Drawthetwolinesonthediagrambelow.[2] 10.[Maximummark:9][withoutGDC] ConsiderthepointsA(3,6)andB(4,9).ThelineL1passesthroughAandB.

  3. (a)Find thegradientofthelineL1[1] theequationofthelineL1.[2] FindthelineL2whichisperpendiculartoL1andpassesthroughthepointA.[2] ExpressbothequationsofL1andL2intheformax+by=c,wherea, b,andc areintegers.[2] Writedownthesolutionofthetwosimultaneousequationsfoundin(c)andmakeacomment aboutthesolutionbysketchingthetwolinesL1,L2.[2] 11.[Maximummark:7][withoutGDC] ConsiderthepointsA(−3,7)andB(5,10). FindthegradientofthelineLpassingthroughAandB.[2] FindthecoordinatesofthemidpointMbetweenAandB.[1] FindtheequationofthelinewhichisperpendiculartoLandpassesthroughthepointM.(i.e.,the perpendicularbisectorofthelinesegmentAB)[2] FindthedistancebetweenthepointsAandM.[2] 12.[Maximummark:7][withoutGDC] ThepointsA(3,2)andB(7,4)areshowninthediagrambelow. LetLbetheperpendicularbisectorofthelinesegmentAB. FindtheequationofL.[5] Writedownthey-interceptofLanddrawanaccuratelineforLonthediagramabove.[2] [Maximummark:7][withGDC] FindtheperpendicularbisectorofthelinesegmentABwithA(9,14)andB(21,42),intheform ax+by=d,wherea,b,darepositiveintegers. [Maximummark:6][with/withoutGDC] FindthecoordinatesofapointPonL1:y=x+2giventhatthedistancebetweentheoriginandP is6. [Maximummark:6][withoutGDC] FindthecoordinatesofapointAonL1:y=x+2andapointBonL2:y=2x+1,suchthat M(6,9)isthemidpointofthelinesegmentAB. [Maximummark:6][withoutGDC] Thefollowingdiagramshowsthelines3x−5y=0,4x+y=7andthepointP(2,2).Alineis drawn from P to intersect with 3x − 5y= 0 at Q, and with 4x + y = 7 at R, so that P is the midpointofQR. FindtheexactcoordinatesofQandR.

  4. 17.[Maximummark:10][withoutGDC] LetA(4k,5k)beapointonthelineLwithequationx=y,wherekisaninteger. 45 VerifythatthepointAliesonthelineL.[2] FindthepossiblevaluesofkifthedistancebetweentheoriginandAequals12. [4] WritedownthecoordinatesofthetwopointsonthelineLwhosedistancefromtheoriginisequal to12.[2] Demonstrateonthediagrambelowtheresultofquestion(c).[2] 18.[Maximummark:13][with/withoutGDC] ConsiderthelineL1withequationy=3x−4.ThelineL2isparalleltoL1andpassesthroughthe pointA(2,10).ThelineL3isperpendiculartoL1andpassesthroughthepointA(2,10). (a)Findtheequation ofthelineL2[3] ofthelineL3.[3] ThelinesL1andL3intersectatpointB.FindthecoordinatesofpointB.[2] FindthedistancebetweenthepointsAandB.[2] SketchadiagramanddeducethedistancefromthepointAtothelineL1.[3] 19.[Maximummark:23] ThepointsA(4,3),B(8,3),andC(4,9)areshowninthediagrambelow. FindtheequationoftheperpendicularbisectorofthelinesegmentAB.[2] FindtheequationoftheperpendicularbisectorofthelinesegmentAC.[2] WritedownthecoordinatesofthepointofintersectionPofthetwobisectorsandshowthatPis themidpointofthelinesegmentBC.[3] Findtheareasofthetriangles ABC[2] ABP[2] ACP[2] FindtheequationoftheperpendicularbisectorLofthelinesegmentBCintheformax+by= d,wherea,b,dareintegers.[5] ShowthatthelineLdoesnotpassthroughA.[2] DrawthethreeperpendicularbisectorsofthesidesofABConthediagramabove.[3] 20.[Maximummark:10][with/withoutGDC] Thefollowingthreelinesl1,l2,andl3aredefinedwithequations: l1:x+y=6 l2:2x−y=8 l3:x=−3 FindthecoordinatesofthecommonpointAbetweenthelinesl1andl2.[2] Writedownthecoordinatesof (i)thecommonpointBbetweenthelinesl1andl3[2]

  5. (ii)thecommonpointCbetweenthelinesl2andl3.[2] (c)Hence,findtheareaofthetriangleABC.[4]

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