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MATHEMATICS (SYLLABUS D)
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Cambridge International Examinations
Cambridge Ordinary Level
4024/21
May/June 2015
2 hours 30 minutes
Candidates answer on the Question Paper.
Additional Materials:
Geometrical instruments
Electronic calculator
READ THESE INSTRUCTIONS FIRST
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.
Section A
Answer all questions.
Section B
Answer any four questions.
If working is needed for any question it must be shown in the space below that question.
Omission of essential working will result in loss of marks.
You are expected to use an electronic calculator to evaluate explicit numerical expressions.
If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to
three significant figures. Give answers in degrees to one decimal place.
For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total of the marks for this paper is 100.
This document consists of 23 printed pages and 1 blank page.
DC (NH/CGW) 100477/2
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2
Section A [52marks]
Answerallquestionsinthissection.
1
(a) Afurnituresalesmanearned$36200lastyear.
(i) Hehadtopay22%ofthisamountastax.
Howmuchwasleftafterpayingtax?
Answer $.......................................... [2]
(ii) Hisearningsof$36200weremadeupof$25000basicsalaryplus8%ofthevalueofthe
furniturethathesold.
Calculatethevalueofthefurniturethathesold.
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Answer $.......................................... [3]
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3
(iii) Heboughtabookcasefromtheshopwhereheworked.
Itsmarkedpricewas$1080butbecauseheworkedthere,heonlypaid$756.
Calculatethepercentagediscountonthemarkedpricethathehadbeengiven.
Answer ........................................ %[2]
(b) George opened an account and invested a sum of money at 4.5% simple interest per year
for3years.Attheendofthe3yearsheclosedtheaccount,withdrawingatotalof$681.
CalculatetheamountthatGeorgeinvested.
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Answer $......................................... [3]
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2
Qisthepoint(–1,2),Risthepoint(3,10)andSisthepoint(–4,2).
(a) Calculatethelengthof QR.
Answer ....................................units[2]
t .
(b) Calculatethevalueof cosSQR
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Answer ............................................ [2]
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(c) ApointP(x,y)issuchthat PQ=PR.
(i) Showthat x+2y=13.
[2]
(ii) Pisontheline y=7.
FindthecoordinatesofP.
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Answer
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(...................,...................)[1]
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3
(a) (i)
D
C
A
B
IntrapeziumABCD,ABisparalleltoDC.DBandACarestraightlines.
Explainwhy theareaoftriangle ACB=theareaoftriangle ADB.
[1]
(ii)
G
H
E
K
F
Thediagramshowsthequadrilateral EHGK.
HFisparalleltoGKandEFKisastraightline.
(a) Nameatriangleequalinareatotriangle HFK.
Answer ............................................ [1]
(b) Henceshowthat theareaoftriangleHEK=theareaofquadrilateralHEFG.
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(b)
P
S
L
x°
y°
R
Q
M
TwocirclesintersectatLandM.
R and P are on the circumference of one circle. S and Q are on the circumference of the other
circle.
PLQandRLSarestraightlines.
t =x°andMLQ
t =y°.
PLR
t =x°.
(i) Completetheproofthat SMQ
Statement
Reason
t =SLQ
t x°= PLR
.............................................................................................
t =SMQ
t =x°
SLQ
............................................................................................. [2]
t =y°.
(ii) Provethat PRM
Statement
Reason
[2]
(iii) Completethefollowingstatement,givingyourreasons.
ThetrianglesPRMandQSMare.............................................
Reasons......................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
............................................................................................................................................... [3]
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4
[The volume of a cone =
1 2
4
πr h] [The volume of a sphere = πr3]
3
3
7.6
4.5
Asolidisformedbyjoiningaconeofradius4.5cmandheight7.6cmtoahemisphereofradius4.5cm
asshown.
(a) Calculatetheareaofthecirclewheretheyarejoined.
Answer .....................................cm2[2]
(b) Calculatethetotalvolumeofthesolid.
Answer .....................................cm3[2]
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(c) Another solid of the same type is made by joining a cone of radius 5cm and height h cm to a
hemisphereofradius5cm.
Theconeandhemispherehaveequalvolumes.
Calculatetheheightofthecone.
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Answer ...................................... cm[2]
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10
5
B
C
D
4 y°
A
x°
7
11
F
4
E
IntheframeworkABCDEF,BCDisastraightline,andCAisparalleltoDF.
t , BDE
t and DEF
t arerightangles.
ABD
AB=4m,DE=11mandEF=4m.
t =x°.
(a) FDE
Showthat x=20.0 correctto3significantfigures.
[2]
t = y°.
(b) BAC
Statingyourreasons,explainwhy y=x.
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(c) CalculateAC.
Answer ........................................ m[3]
(d) TheperpendiculardistancebetweentheparallellinesCAandDFis7m.
CalculatetheareaofACDF.
Answer .......................................m2[4]
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6
(a) Expandthebracketsandsimplify (x–1)(x2+x+1).
Answer ............................................ [2]
(b) Solvetheequation 3x
4
= 3.
x+2 x-2
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Answer ............................................ [3]
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(c) Solvethesesimultaneousequations.
4x–3y=4
4y–3x=–6.5
Answer x=......................................
y=...................................... [4]
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Section B[48marks]
Answerfourquestionsinthissection.
Eachquestioninthissectioncarries12marks.
7
(a) (i)
Evaluate
8 sin 54°
.
sin 18°
Answer ............................................ [1]
(ii) Evaluate 4.73 2 - 1.65 sin 43° .
Answer ............................................ [1]
(b)
B
x
A
60°
16
2x + 3
C
t =60°.
InthetriangleABC,BC=16cm and BAC
AB=xcm and AC=2x+3cm.
(i) Formanequationinxandshowthatitsimplifiesto 3x 2 + 9x - 247 = 0 .
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(ii) Solvetheequation 3x 2 + 9x - 247 = 0 , givingyouranswerscorrectto2decimalplaces.
Answer
x=.............................or.............................[3]
(iii) HencewritedownthelengthsofABandAC.
Answer
AB=....................cmAC=....................cm[1]
(iv) FindtheareaoftriangleABC.
Answer .....................................cm2[2]
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8
ThediagramshowsasectorAOBofacirclewithcentreOandradius9.3cm.
Theangleofthesectoris260°.
A
(a) (i) CalculatethelengthofthemajorarcAB.
9.3
B
O
260°
Answer ...................................... cm[2]
(ii) CalculatetheareaofthemajorsectorAOB.
Answer .....................................cm2[2]
(b) Asectorofradius0.8cm,centreO,isremovedfromthesectorAOBasshowninDiagramI.
Theshadedshapeisusedtomakepartofaconicalfunnel.
ADisjoinedtoBCasshowninDiagramII.
B
A
D
Diagram I
0.8
A,B
C
O
D,C
Diagram II
ThecircumferenceofthetopoftheconicalfunnelisthemajorarcAB,andthecircumferenceof
thebottomoftheconicalfunnelisthemajorarcCD.
(i) Calculatetheexternalsurfaceareaofthispartofthefunnel.
Answer .................................... cm2[2]
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(ii) Thefunneliscompletedbyattachinganopencylinderofheight5cm
tothebottomoftheconicalpart.
(a) Showthattheradiusofthecylinderis0.578cm,
correctto3significantfigures.
5
[2]
(b) Calculatetheexternalcurvedsurfaceareaofthiscylinder.
Answer .....................................cm2[2]
(c) Calculatethevolumeofthiscylinder.
Answer .....................................cm3[2]
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9
f(x)= x 3
(a) Completethefollowingtable.
x
–3
–2
–1
0
1
2
3
f(x)
[1]
(b) Usingascaleof2cmtorepresent1unit,drawahorizontalx-axisfor –3 G x G 3.
Usingascaleof2cmtorepresent10units,drawaverticaly-axisfor –30 G y G 30 .
Usingyouraxes,plotthepointsinthetableandjointhemwithasmoothcurve.
Answer
[2]
(c) (i) Useyourgraphtosolve f(x)=–15.
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Answer ............................................ [1]
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(ii) Useyourgraphtofindasuchthat f - 1 ^ah = 1.7 .
Answer ............................................ [1]
(iii) Giventhat f - 1 ^ t h = u ,expresstintermsofu.
Answer t=....................................... [1]
(iv) Bydrawingatangentto y = f ^xh, estimatethegradientofthecurvewhenx=2.
Answer ............................................ [2]
(d) (i)
Usingthesameaxesdrawthelinethatrepresentsthefunction g ^xh = 5x + 3.
[2]
(ii) Hencefindthethreesolutionsoftheequation f ^xh = g ^xh.
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Answer
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x=............or............or............[2]
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20
10 Onedayafarmercollected300eggsfromhischickens.
Thetablebelowshowsthedistributionofthemassesoftheeggs.
Mass
(mgrams)
Frequency
42<mG46 46<mG48 48<mG50 50<mG54 54<mG58 58<mG66
60
40
48
72
56
24
(a) (i) Aneggischosenatrandom.
Calculatetheprobabilitythatthemassofthiseggisnotgreaterthan48grams.
Answer ........................................... [1]
(ii) Aneggischosenatrandomfromthe300eggs.
Anothereggischosenatrandomfromthosethatremain.
Calculatetheprobabilitythatthemassofoneeggisatmost46grams,andthemassofthe
otherismorethan58grams.
Answer ............................................ [2]
(b) Calculateanestimateofthemeanmassofanegg.
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Answer ......................................... g[3]
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(c) (i) Completethecumulativefrequencytable.
Mass
(mgrams)
mG42
mG46
Cumulative
Frequency
0
60
mG48
mG50
mG54
mG58
mG66
300
[1]
(ii) Onthegrid,drawasmoothcumulativefrequencycurvetoillustratethisinformation.
300
250
200
Cumulative
frequency
150
100
50
0
40
45
50
55
60
65
70
Mass (m grams)
[2]
(d) (i)
Useyourgraphtofindthemedianmassoftheeggs.
Answer ......................................... g[1]
(ii) Useyourgraphtofindtheinterquartilerange.
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Answer ......................................... g[2]
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11
(a)
ABCDEisapentagon.
AFB,AHEandBGCarestraightlines.
FisthemidpointofAB.
HisthemidpointofAE.
GdividesBCintheratio1:2.
AH = a, AF = a - b, BG = ED = c .
c
B
C
F
a-b
A
G
E
a
c
D
H
(i) Find FH .
Answer ............................................ [1]
(ii) Usingvectors,showthatGDisparalleltoFH.
[2]
(iii) Itisgiventhat c =
4
1
a + b .
5
5
(a) Express DC intermsofaandb.
Answer ............................................ [2]
(b) Find
AF : DC .
Answer ..................... :.....................[1]
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y
(b)
8
A
4
B
–4
C
A
8 x
B 4
0
C
–4
(i) ThetransformationTmapstriangleABContotriangle AlBlC l .
(a) DescribefullythetransformationT.
Answer........................................................................................................................... [2]
(b) ThematrixMrepresentsthetransformationT.
FindthematrixM.
J
K
Answer K
K
L
N
O
OO P
[2]
(ii) Triangle AlBlC l ismappedontotriangle AmB mC m byareflectioninthey-axis.
Drawandlabeltriangle AmB mC m .
[1]
(iii) TriangleABCismappedontotriangle AmB mC m byananticlockwiserotationabouttheorigin.
Statetheangleofrotation.
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Answer ............................................ [1]
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