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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Ordinary Level
* 0 5 1 0 3 4 8 7 8 5 *
4024/21
MATHEMATICS (SYLLABUS D)
Paper 2
May/June 2013
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, highlighters, 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 24 printed pages.
DC (KN/SW) 64205/3
© UCLES 2013
[Turn over
2
SectionA[52marks]
Answerallquestionsinthissection.
1
(a) Solve4(x–2)=7–x.
For
Examiner’s
Use
Answer x=......................................... [2]
(b)
Solvethesimultaneousequations.
2x+y=7
4x–3y=19
Answer x=.........................................
y=......................................... [3]
(c) (i) Writedowntheintegervaluesthatsatisfy–1<n <2.
Answer ............................................... [1]
(ii) Solve2–3y<8.
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Answer ............................................... [2]
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3
2
(a) Theinteriorangleofaregularpolygonis165o.
For
Examiner’s
Use
Howmanysideshasthepolygon?
Answer ............................................... [2]
(b)
B
D
p°
A
F
q°
E
C
G
FAECG andADBarestraightlines.DEisparalleltoBC.
t =p°and AED
t =q°.
(i) FAD
Findanexpressionintermsofpand/orqfor
t ,
(a) BCG
Answer ............................................... [1]
t .
(b) DBC
Answer ............................................... [1]
(ii) AE=7cm,EC=3cm,DE=5.6cmandDB=2.1cm.
(a) FindBC.
Answer ......................................... cm[1]
(b) FindAD.
Answer ......................................... cm[1]
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4
3
(a) Thediagramsshowparallelogramsmadefromsmalltriangles.
Parallelogram
1
2
3
For
Examiner’s
Use
4
5
(i) Completethetablebelow.
Parallelogramm
1
2
3
4
Numberofsmalltriangles
2
4
6
8
5
6
[1]
(ii) Find an expression, in terms of m, for the number of small triangles used to make
Parallelogramm.
Answer ............................................... [1]
(b) Thediagramsshowtrianglesmadefromthesamesmalltriangles.
Triangle
1
2
3
4
(i) Completethetablebelow.
Trianglen
1
2
3
4
Numberofsmalltriangles
1
4
9
16
5
5
6
[1]
(ii) Find an expression, in terms of n, for the number of small triangles used to make
Trianglen.
Answer ............................................... [1]
(iii) Triangleqismadefrom324smalltriangles.
Findq.
Answer ............................................... [1]
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5
(c) Thediagramsshowtrapeziumsmadefromthesamesmalltriangles.
Trapezium
1
Trapezium
4
2
For
Examiner’s
Use
3
5
(i) Bycomparingthediagramswiththoseinparts(a)and(b),findanexpression,interms
oft,forthenumberofsmalltrianglesusedtomakeTrapeziumt.
Answer ............................................... [1]
(ii) HowmanysmalltrianglesareusedtomakeTrapezium25?
Answer ............................................... [1]
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6
4
(a) Aboxofchocolatescontains10milkchocolatesand2plainchocolates.
Sachaeats3chocolateschosenatrandomfromthebox.
Thetreediagramshowsthepossibleoutcomesandtheirprobabilities.
First chocolate
Second chocolate
9
11
10
12
Third chocolate
milk
milk
2
11
plain
8
10
milk
2
10
plain
9
10
milk
1
10
plain
........
2
12
........
milk
milk
........
plain
........
For
Examiner’s
Use
plain
........
........
plain
milk
plain
(i) Completethetreediagram.
(ii) Expressingeachanswerasafractioninitslowestterms,findtheprobabilitythatSacha
[2]
(a) eats3milkchocolates,
Answer ............................................... [1]
(b) eats2milkchocolatesand1plainchocolateinanyorder.
Answer ............................................... [2]
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7
(b) Thefrequencydiagramshowsthedistributionofthenumberoflettersreceivedbyafamily
eachdayovera31dayperiod.
For
Examiner’s
Use
9
8
7
6
Number
of days
5
4
3
2
1
0
Forthisdistribution,find
(i) themode,
0
1
2
3
4
Number of letters
5
6
Answer ............................................... [1]
(ii) themedian.
Answer ............................................... [1]
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8
5
(a) (i)
For
Examiner’s
Use
Exchangerate
$1=€0.72
EddietravelsfromtheUSAtoGermany.
Hechanges$300intoeuros(€).
Howmanyeurosdoeshereceive?
Answer €............................................. [1]
(ii) WhenEddiereturnstotheUSAhehas€51leftthatheexchangesfor$75.
Whatexchangeratehasbeenusedinthiscase?
Answer $1=€.................................... [1]
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9
(b) Gregbuys60gardenplantsatacostpriceof$2.00eachtosellinhisshop.
Hesells25ofthemataprofitof75%and18ofthemataprofitof35%.
Hesellstherestoftheplantsfor 45 ofthecostprice.
For
Examiner’s
Use
(i) Calculatetheprofitorlosshemakesfromsellingthese60plants,statingifitisaprofit
orloss.
Answer Gregmakesa...........................of$.....................[3]
(ii) Findthepercentageprofitorloss.
Answer ...........................................%[1]
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10
6
(a) ABCDisatrapeziumwithBCparalleltoAD.
9
B
For
Examiner’s
Use
C
18
A
55°
7
E
EisthepointonADsuchthatBEisperpendiculartoAD.
t =55°,AE=7cm,BE=18cmandBC =9cm.
BDA
Calculate
t ,
(i) BAE
D
Answer ............................................... [2]
(ii) theareaofthetrapeziumABCD.
Answer .........................................cm2[4]
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11
(b)
P
For
Examiner’s
Use
Q
112°
41°
S
PQRSisanothertrapezium.
t =112°and PRS
t =41°,eachmeasuredcorrecttothenearestdegree.
PQR
R
t .
Findthesmallestpossiblevalueof QRP
Answer ............................................... [2]
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12
7
(a) InanathleticsmatchBenwonthe100mracein9.98sandCalvinwonthe200mracein
19.94s.
For
Examiner’s
Use
Whatisthedifferenceintheiraveragespeeds?
Giveyouranswerinmetrespersecond,correcttotwodecimalplaces.
Answer ......................................... m/s[2]
(b)
Twocarseachcompleteajourneyof120km.
Thefirstcarisdrivenatanaveragespeedofx km/h.
Thesecondcarisdrivenatanaveragespeed3km/hfasterthanthefirstcar.
Thefirstcartakes6minuteslongertocompletethejourney.
(i) Writedownanequationinxandshowthatitsimplifiestox2+3x–3600=0.
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13
(ii) Solvetheequationx2+3x–3600=0,givingeachanswercorrectto
onedecimalplace.
For
Examiner’s
Use
Answer x=..................or..................[3]
(iii) Howmanyminutesdoesthefirstcartaketotravelthe120km?
Answer ................................. minutes[2]
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14
SectionB[48marks]
Answerfourquestionsinthissection.
Eachquestioninthissectioncarries12marks.
8
(a) Find
(i) f(–2),
f (x) =
4x - 3
2
Answer f(–2)=................................. [1]
(ii) f–1(x),
Answer f–1(x)=................................ [2]
(iii) thevalueofgsuchthatf(2g)=g.
Answer g=........................................ [2]
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For
Examiner’s
Use
15
(b)
For
Examiner’s
Use
B
C
A
E
D
BADandCAEarestraightlinesandBCisparalleltoED.
BA = c
(i) DescribefullythesingletransformationthatmapstriangleABContotriangleADE.
1
12
m, ED = c m and BA = 1 BD .
-2
-3
4
Answer..............................................................................................................................
...................................................................................................................................... [2]
(ii) Calculate BA .
Answer ............................................... [1]
(iii) FindCD.
Answer
 
[2]
Answer
 
[2]
(iv) FisthemidpointofBD.
Find EF .
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16
9
(a) ShapeIisacylinderwithradius4cmandheighthcm.
ThevolumeofShapeIis1500cm3.
For
Examiner’s
Use
4
(i) Findh.
h
Shape I
Answer ............................................... [2]
(ii) ShapeIismadebypouringliquidintoamouldatarateof0.9litresperminute.
Findthenumberofsecondsittakestopourthisliquidintothemould.
Answer ................................. seconds[1]
(b) ShapeIIisaprismoflength8cmwithatriangularcross-section,
shownshaded.
Twosidesoftheshadedtriangleareatrightanglestoeachotherand
havelengths5xcmand12xcm.
12x
GiventhatShapeIIalsohasavolumeof1500cm3,findx.
5x
8
Shape II
Answer ............................................... [2]
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17
(c) ShapeIIIisalsoaprismoflength8cmwithatriangularcross-section,
shownshaded.
Twosidesoftheshadedtriangleareatrightanglestoeachotherand
havelengths5ycmand12ycm.Thethirdsideisoflength13y cm.
12y
y satisfiestheequation4y2+16y–33=0.
For
Examiner’s
Use
13y
(i) Factorise4y2+16y–33.
8
5y
Shape III
Answer ............................................... [1]
(ii) Hencesolvetheequation4y2+16y–33=0.
Answer y=..................or..................[1]
(iii) Findtheareaoftheshadedtriangle.
Answer ........................................cm2[1]
(iv) FindthetotalsurfaceareaofShapeIII.
Answer ........................................cm2[3]
(d) Find
Volume of Shape III
asafractioninitssimplestform.
Volume of Shape II
Answer ............................................... [1]
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18
10 (a) Thetableshowssomevaluesofxandthecorrespondingvaluesofyfor y =
x
–3
y
–2.5
–2
–1.5
–1
1
1.5
2
2.5
0.96
1.5
2.67
6
6
2.67
1.5
0.96
(i) Completethetable.
(ii) Onthegriddrawthegraphof y =
6.
x2
For
Examiner’s
Use
3
[1]
6
for–3GxG3.
x2
y
8
7
6
5
4
3
2
1
–3
–2
–1
0
1
2
3
x
[2]
(iii) Useyourgraphtofindthevaluesofxwheny=2.
Answer x=..................or..................[1]
(iv) Bydrawingatangent,findthegradientofthecurvewhenx=1.5.
Answer ............................................... [2]
(v) Bydrawingasuitablelineonthegrid,solvetheequation
6
= 2 - x.
x2
Answer x=......................................... [2]
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19
(b) Thegraphshowsasketchofy=5a x.
For
Examiner’s
Use
y
(4, b)
(2, 45)
P
x
O
Twopointsonthecurveare(2,45)and(4,b).
(i) Findthevaluesofaandb.
Answer a=.........................................
b=......................................... [2]
(ii) Findthecoordinatesofthepoint,P,wherethegraphcrossesthey-axis.
Answer (.....................,.....................) [1]
(iii) FindthegradientofthestraightlinejoiningthepointsPand(2,45).
Answer ............................................... [1]
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20
11 (a) Thescalediagramshowsthepositions,AandB,oftwoboats.
For
Examiner’s
Use
North
A
Scale: 1 cm to 50 m
B
(i) Findtheactualdistancebetweenthetwoboats.
Answer ............................................m[1]
(ii) AthirdboatispositionedatC,suchthatAC=350mandBC=300m.
CiseastofthelineAB.
UserulerandcompassestofindC.
[2]
(iii) MeasurethebearingofCfromA.
Answer ............................................... [1]
(iv) AfourthboatispositionedatD,suchthatACisthelineofsymmetryofthe
quadrilateralABCD.
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CompletethequadrilateralABCD.
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[2]
21
(b) Thediagramshowsthepositions,P,QandR,ofthreebuoys.
ThebearingofQfromPis054o,PQ=250m,QR=340mandPR=160m.
For
Examiner’s
Use
Q
North
250
54°
P
340
160
R
(i) CalculatethebearingofRfromP.
Answer ............................................... [4]
(ii) CalculatetheareaoftrianglePQR.
Answer .......................................... m2[2]
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22
12 (a) Thedistributionoftheweightsofluggagefor140passengersisshowninthetable.
Weightof
luggage
(wkg)
01wG6 61wG10 101wG14 141wG16 161wG18 181wG22 221wG30
Frequency
For
Examiner’s
Use
15
14
20
24
31
24
12
(i) Calculateanestimateofthemeanweightofluggage.
Answer ........................................... kg[3]
(ii) Onthegridopposite,drawahistogramtorepresentthisdata.
(iii) Estimatetheprobabilitythatapassenger,chosenatrandom,hasluggageweighingless
than13kg.
[3]
Answer ............................................... [2]
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23
For
Examiner’s
Use
0
2
4
6
8
10
12
14
16
18
Weight of luggage (kg)
20
22
24
26
28
30 w
TURNOVERFORTHERESTOFTHEQUESTION
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24
(b) Thepiechartrepresentsthedistributionofthebirthplacesofagroupof60students.
Do not
write in this
margin
Singapore
South Africa
48°
126°
Pakistan
54° 42°
Australia
United
Kingdom
(i) FindthenumberofstudentsinthegroupwhowereborninAustralia.
Answer ............................................... [1]
(ii) CalculatethepercentageofstudentsinthegroupwhowereborninSouthAfrica.
Answer ............................................%[1]
(iii) Fourmorestudentsjointhegroup.
Ofthese,twostudentswereborninPakistan,oneinSingaporeandoneinChina.
Anewpiechartistobedrawnusingtheinformationaboutthewholegroupofstudents.
Forthenewpiechart,calculatetheangleofthesectorthatrepresentsthestudentsborn
inPakistan.
Giveyouranswercorrecttothenearestdegree.
Answer ............................................... [2]
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