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MATHEMATICS (SYLLABUS D)
Paper 2
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Ordinary Level
October/November 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 (LEG/CGW) 84304/2 R
© UCLES 2013
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2
SectionA[52marks]
Answerallquestionsinthissection.
1
(a) Therateofexchangebetweendollars($)andpounds(£)is$1.56=£1.
Therateofexchangebetweeneuros(€)andpoundsis€1.10=£1.
(i) Amychanges£300intodollars.
CalculatehowmanydollarsAmyreceives.
Answer $............................................. [1]
(ii) Benchanges€770intopounds.
CalculatehowmanypoundsBenreceives.
Answer £............................................. [1]
(iii) Chrischanges$780intoeuros.
©UCLES2013
CalculatehowmanyeurosChrisreceives.
Answer €............................................ [2]
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For
Examiner’s
Use
3
(b) DebbiechangedsomedollarsintoJapaneseyen.
Therateofexchangewas81dollars=1yen.
Emmachangedthesamenumberofdollarsintoyen.
TherateofexchangeforEmmawas82dollars=1yen.
Emmareceived3feweryenthanDebbie.
Giventhatthenumberofdollarschangedeachtimeisx,findx.
©UCLES2013
For
Examiner’s
Use
Answer ............................................... [3]
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4
2
t =40˚andAC=8cm.
(a) ConstructthetriangleABCinwhich BAC
For
Examiner’s
Use
CisabovethelineAB,whichisdrawnforyou.
A
B
[2]
(b) Constructthelocusofallthepointsoutsidethetrianglethatare2cmfromtheperimeterof
thetriangle.
[2]
(c) FindandlabelthepointP,insidethetriangle,thatis6.5cmfromAand
equidistantfromBandC.
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[2]
5
3
ThelineABjoinsthepointA(–2,1)tothepointB(6,5).
(a) FindthecoordinatesofthemidpointofAB.
Answer (..............,..............)
[1]
(b) FindthegradientofAB.
Answer ............................................... [1]
(c) ABintersectsthey-axisatthepoint(0,c).
Findc.
For
Examiner’s
Use
Answer ............................................... [2]
(d) Express ABasacolumnvector.
Answer (e) Cisthepoint(5,2)andDisthepoint(h,k).
ThelinesABandCDareequalinlengthandparallel.
[1]
FindthecoordinatesofeachofthepossiblepointsD.
©UCLES2013
Answer (..............,..............)and(..............,..............)[3]
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6
4
Thetableshowsthedistributionofthemassesof100babiesatbirth.
Mass
(xkg)
1.5<xG2 2<xG2.5 2.5<xG3 3<xG3.5 3.5<xG4 4<xG4.5 4.5<xG5
Number
ofbabies
For
Examiner’s
Use
3
12
20
24
25
14
2
(a) Writedownthemodalclass.
Answer ............................................... [1]
(b) Forthispartofthequestionusethegridbelow.
Usingascaleof4cmtorepresent1kg,drawahorizontalx-axisfor1 G x G 5.
Usingascaleof2cmtorepresent5babies,drawaverticalaxisforfrequencyfrom0to30.
Usingyouraxes,drawafrequencypolygontorepresenttheseresults.
©UCLES2013
[2]
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7
(c) (i) Completethecumulativefrequencytablebelow.
Mass(xkg)
xG2
x G 2.5
Cumulativefrequency
3
15
xG3
x G 3.5
xG4
x G 4.5
For
Examiner’s
Use
xG5
100
[1]
(ii) Onthegridbelowdrawasmoothcumulativefrequencycurvetorepresenttheseresults.
100
90
80
70
60
Cumulative
50
frequency
40
30
20
10
0
1
1.5
(d) Useyourcurvetoestimate
(i) themedianmass,
(ii) the10thpercentile.
©UCLES2013
2
2.5
3
Mass (kg)
3.5
4
4.5
5 x
[2]
Answer .......................................... kg[1]
Answer .......................................... kg[1]
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8
5
(a) Solve
2
= 1.
3- x
Answer ............................................... [1]
(b) Factorise
(i) 5x + 5y ,
Answer ............................................... [1]
(ii) 9x 2 - 16 .
Answer ............................................... [1]
(c) (i) Factorise 2x 2 + 5x - 12 .
For
Examiner’s
Use
Answer ............................................... [1]
(ii) Useyouranswertopart(c)(i)tosolvetheequation 2x 2 + 5x - 12 = 0 .
©UCLES2013
Answer x=.................or.................[1]
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9
(d) Asourceoflightisobservedfromadistanceofdmetres.
Theamountoflightreceived,Lunits,isinverselyproportionaltothesquareofthedistance.
For
Examiner’s
Use
GiventhatL=9whend=2,findthevalueofLwhend=3.
©UCLES2013
Answer ............................................... [2]
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10
6
(a)
For
Examiner’s
Use
A
D
C
50°
B
31
t = 50candBC=31m.
t = 90c, ACB
InthetriangleABC, ABC
t = 90c.
DisthepointonACsuchthat BDA
(i) ShowthatCD=19.93m,correctto2decimalplaces.
[2]
(ii) CalculateAD.
©UCLES2013
Answer ...........................................m[3]
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11
(b)
For
Examiner’s
Use
S
10°
55°
52
P
R
Q
TwoboatsareatthepointsPandQ.
RSisaverticalcliffofheight52m.
t = 10cand QStR = 55c.
PSQ
(i) StatetheangleofdepressionofPfromS.
Answer ............................................... [1]
(ii) Calculatethedistance,PQ,betweentheboats.
©UCLES2013
Answer ...........................................m[3]
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12
7
(a)
For
Examiner’s
Use
A
E
C
D
B
t and
IntriangleABC,DisthepointonBCsuchthatADbisects BAC
EisthepointonABsuchthatAE=AC.
(i) ShowthattrianglesAEDandACDarecongruent.
[3]
t = zc,
t = xc, EDB
t = ycand ACB
(ii) Giventhat ABD
findxintermsofyandz.
©UCLES2013
Answer x=......................................... [2]
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13
(b)
For
Examiner’s
Use
P
S
R
Q
t andRSbisects PRQ
t .
IntrianglePQR,QSbisects PQR
t
t
PQR = 42cand PRQ = 54c.
FindreflexangleQSR.
©UCLES2013
Answer ............................................... [2]
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14
SectionB[48marks]
Answerfourquestionsinthissection.
Eachquestioninthissectioncarries12marks.
8
(a)
For
Examiner’s
Use
B
14
10
A
O
Inthediagram,thecircleseachhavecentreO.
ABisachordofthelargercircleandalsoatangenttothesmallercircle.
AB=14cmandtheradiusofthelargercircleis10cm.
Findtheradiusofthesmallercircle.
Answer ......................................... cm[3]
(b)
S
Q
T
R
P
Inthediagram,PQandRSarechordsofacirclethatintersectatT.
(i) ShowthattrianglesPSTandRQTaresimilar.
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[3]
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15
(ii)
S
x
5
For
Examiner’s
Use
Q
T
11
R
P
ST=5cm,TR=11cmandTQ=xcm.
GiventhatPQ=18cm,showthatxsatisfiestheequation
x 2 - 18x + 55 = 0 .
[2]
(iii) Solvetheequation x 2 - 18x + 55 = 0 .
Giveeachsolutioncorrectto1decimalplace.
Answer x=.................or.................[3]
(iv) FindthedifferencebetweenthelengthsofPTandTQ.
©UCLES2013
Answer ......................................... cm[1]
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16
9
Thenumberofbacteriainacolonytrebleseveryhour.
Thecolonystartswith50bacteria.
Thetablebelowshowsthenumberofbacteria(y)inthecolonyafterthours.
Time
(thours)
0
1
2
2.5
3
3.5
Numberof
bacteria(y)
50
150
450
780
1350
2340
For
Examiner’s
Use
4
(a) Completethetable.
[1]
(b) Onthegridontheoppositepageplotthepointsinthetable,andjointhemwitha
smoothcurve.
[3]
(c) Useyourgraphtofindthenumberofbacteriainthecolonywhent=3.2.
Answer ............................................... [1]
(d) (i) Bydrawingatangent,estimatethegradientofthecurvewhent=2.5.
Answer ............................................... [2]
(ii) Whatdoesthisgradientrepresent?
Answer........................................................................................................................
..................................................................................................................................... [1]
(e) Giventhattheequationofthegraphis y = ka t ,findkanda.
Answer k=..................a=.................[1]
(f) Thenumberofbacteriainanothercolonyisgivenbytheequation y = 500 + 500t .
(i) Onthesameaxes,drawagraphtorepresentthenumberofbacteriainthiscolony.
[2]
(ii) Statethevalueoftwhenthenumberofbacteriaineachcolonyisthesame.
©UCLES2013
Answer ............................................... [1]
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17
y
For
Examiner’s
Use
5000
4500
4000
3500
3000
Number
of bacteria
2500
2000
1500
1000
500
0
1
2
3
4
t
Time (t hours)
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18
10 Afueltankerdeliversfuelinacylindricalcontaineroflength9.5mandradius0.8m.
(a) Afterseveraldeliveries,thefuelremaininginthecontainerisshowninthediagram.
9.5
O
0.8
A
B
t = 90c.
ABishorizontal,Oisthecentreofthecircularcross-sectionand AOB
(i) Calculatethecurvedsurfaceareaofthecontainerthatisincontactwiththefuel.
Answer ......................................... m2[2]
(ii) Calculatethevolumeoffuelremaininginthecontainer.
Answer ......................................... m3[4]
(iii) Calculatethisvolumeremainingasapercentageofthevolumeofthewholecontainer.
©UCLES2013
Answer ...........................................%[2]
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For
Examiner’s
Use
19
(b) Thefuelispumpedthroughacylindricalpipeofradius4.5cmatarateof300cm/s.
(i) Calculatethevolumepumpedin1second.
Answer ....................................... cm3[1]
For
Examiner’s
Use
(ii) Calculatethetimetaken,inminutes,topump25000litresoffuel.
Giveyouranswercorrecttothenearestminute.
©UCLES2013
Answer ................................. minutes[3]
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20
11 ThediagramshowstrianglesAandB.
For
Examiner’s
Use
y
10
8
6
4
A
2
0
2
B
4
6
8
10
12
14
16
18
20
22
24
26
x
(a) (i) DescribefullythesingletransformationthatmapstriangleAontotriangleB.
Answer.........................................................................................................................
...................................................................................................................................... [2]
(ii) Findthematrixthatrepresentsthistransformation.
Answer f
p
[2]
(b) TriangleBismappedontotriangleCbythetransformationrepresentedbythe
2 0
m.
matrix c
0 1
(i) Onthegridabove,drawandlabeltriangleC.
(ii) Givethenameofthistransformation.
©UCLES2013
[2]
Answer ............................................... [1]
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21
(iii) Find the matrix that represents the inverse transformation that maps triangle C onto
triangleB.
Answer f
[2]
(iv) FindtheratioareaoftriangleC:areaoftriangleB.
p
For
Examiner’s
Use
Answer .....................:.....................[1]
(c) FindthematrixthatrepresentsthesingletransformationthatmapstriangleA
ontotriangleC.
Answer f
p
[2]
_________________________________________________________________________________
©UCLES2013
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22
12 (a)
For
Examiner’s
Use
A
65°
C
45°
B
t = 65c.
t = 45cand BAC
IntriangleABC, ABC
ACis5cmshorterthanBC.
(i) Showthat BC =
5 sin 65
.
sin 65 - sin 45
[3]
(ii) FindthelengthofBC.
©UCLES2013
Answer ......................................... cm[1]
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23
(b)
For
Examiner’s
Use
P
13
Q
6
10
R
S
IntrianglePQR,PQ=13cm,QR=6cmandRP=10cm.
QRisproducedtoS.
t ,givingyouranswerasafractioninitslowestterms.
(i) Findthevalueofcos PRQ
Answer ............................................... [3]
t .
(ii) Hencewritedownthevalueofcos PRS
Answer ............................................... [1]
TurnoverforThereSTofThiSqueSTion
©UCLES2013
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24
(c)
For
Examiner’s
Use
F
D
E
TriangleDEGhasthesameareaastriangleDEF,butisnotcongruenttotriangleDEF.
ThepointGislowerthanDEandGE=EF.
DrawthetriangleDEGinthediagramabove.
t = 30candML=2MN.
(d) IntriangleLMN, LMN
[1]
WhentheareaoftriangleLMNis18cm2,calculateMN.
Answer ......................................... cm[3]
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UniversityofCambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocal
ExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.
©UCLES2013
4024/21/O/N/13