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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
* 7 1 3 2 3 9 8 2 2 2 *
0606/23
ADDITIONAL MATHEMATICS
Paper 2
October/November 2013
2 hours
Candidates answer on the Question Paper.
Additional Materials:
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.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of
angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 80.
This document consists of 19 printed pages and 1 blank page.
DC (KN/SW) 67884/4
© UCLES 2013
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Mathematical Formulae
1. ALGEBRA
QuadraticEquation
Fortheequationax2+bx+c=0,
x=
Binomial Theorem
()
()
−b
b 2 − 4 ac
2a
()
()
n
n
n
(a+b)n=an+ 1 an–1b+ 2 an–2b2+…+ r an–rbr+…+bn,
n
n!
wherenisapositiveintegerand r =
(n–r)!r!
2. TRIGONOMETRY
Identities
sin2A+cos2A=1
sec2A=1+tan2A
cosec2A=1+cot2A
Formulae for ∆ABC
a
b
c
sinA = sinB = sinC
a2=b2+c2–2bccosA
1
2
∆= bcsinA
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1
Findthecoordinatesofthestationarypointsonthecurvey=x3–6x2–36x+16.
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(i) Findhowmanydifferentnumberscanbeformedusing4ofthedigits
1,2,3,4,5,6and7ifnodigitisrepeated.
Findhowmanyofthese4-digitnumbersare
(ii) odd,
[1]
(iii) oddandlessthan3000.
[3]
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[1]
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5
3
Findthesetofvaluesofkforwhichtheliney=3x–kdoesnotmeetthecurvey=kx2+11x–6.
[6]
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4
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y
r
c , 3m
2
(0, 3)
x
O
(a) (i) Thediagramshowsthegraphofy=A+Ctan(Bx)passingthroughthepoints(0,3)and
r
e , 3 o.FindthevalueofAandofB.
[2]
2
r
(ii) Giventhatthepoint e , 7 oalsoliesonthegraph,findthevalueofC.
8
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(b) Giventhat f (x) = 8 - 5 cos 3x ,statetheperiodandtheamplitudeoff.
[2]
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period...................................amplitude...................................
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5
(a) (i) IntheVenndiagrambelowshadetheregionthatrepresents^A j Bhl .
A
B
(ii) IntheVenndiagrambelowshadetheregionthatrepresents P k Q k Rl .
[1]
[1]
Q
P
R
(b) Express,insetnotation,thesetrepresentedbytheshadedregion.
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[1]
T
Answer...........................................................
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9
(c) TheuniversalsetandthesetsVandWaresuchthatn()=40,n(V )=18andn(W)=14.
Giventhatn (V k W ) = x and n ((V j W )l ) = 3x findthevalueofx.
YoumayusetheVenndiagrambelowtohelpyou.
For
Examiner’s
Use
[3]
V
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6
Theexpression2x3+ax2+bx+21hasafactor x+3andleavesaremainderof65when
dividedbyx–2.
(i) Findthevalueofaandofb.
[5]
(ii) Hencefindthevalueoftheremainderwhentheexpressionisdividedby2x+1.
[2]
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11
7
4 8
Theline4x+y=16intersectsthecurve - = 1atthepointsAandB.Thex-coordinateof
x y
A is less than the x-coordinate of B. Given that the point C lies on the line AB such that
AC :CB=1:2,findthecoordinatesofC.
[8]
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8
Solutionstothisquestionbyaccuratedrawingwillnotbeaccepted.
For
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y
X
D
A (–4, 6)
C (10, 4)
O
x
B (6, –4)
ThediagramshowsaquadrilateralABCD,withverticesA(−4,6),B(6,−4),C(10,4)andD.
TheangleADC=90°.ThelinesBCandADareextendedtointersectatthepointX.
(i) GiventhatCisthemidpointofBX,findthecoordinatesofD.
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(ii) HencecalculatetheareaofthequadrilateralABCD.
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9
Aparticletravelsinastraightlinesothat,t safterpassingthroughafixedpointO,itsvelocity,
vms–1,isgivenby v = 3 + 6 sin 2t .
(i) Findthevelocityoftheparticlewhent =
(ii) Findtheaccelerationoftheparticlewhent=2.
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.
4
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15
TheparticlefirstcomestoinstantaneousrestatthepointP.
(iii) FindthedistanceOP.
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10
For
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30 cm
Thevolumeofaconeof
heightHandradius Ris
120 cm
1 2
rR H
3
h cm
Thediagramshowsacontainerintheshapeofaconeofheight120cmandradius30cm.Wateris
pouredintothecontaineratarateof20r cm 3 s - 1 .
(i) Attheinstantwhenthedepthofwaterintheconeishcmthevolumeofwaterintheconeis
rh 3
V cm3.ShowthatV =
. [3]
48
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(ii) Findtherateatwhichhisincreasingwhenh=50. (iii) Findtherateatwhichthecircularareaofthewater’ssurfaceisincreasingwhenh=50. [4]
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11 Inthisquestioniisaunitvectordueeastandjisaunitvectorduenorth.
Attimet =0boatAleavestheoriginOandtravelswithvelocity(2i+4j)kmh–1.Alsoattimet=0
boat B leaves the point with position vector (–21i + 22j)km and travels with velocity
(5i+3j)kmh–1.
(i) WritedownthepositionvectorsofboatsAandBafterthours.
[2]
(ii) ShowthatAandBare25kmapartwhent=2.
[3]
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(iii) FindthelengthoftimeforwhichAandBarelessthan25kmapart.
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University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of
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©UCLES2013
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