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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Ordinary Level
* 7 0 0 0 5 4 4 4 2 1 *
4037/12
ADDITIONAL MATHEMATICS
Paper 1
October/November 2013
2 hours
Candidates answer on the Question Paper.
No Additional Materials are required.
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 16 printed pages.
DC (SLM) 81102
© UCLES 2013
[Turn over
2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
Fortheequationax2+bx+c=0,
x=
Binomial Theorem
()
()
- b ! b 2 - 4ac .
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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3
1 Thediagramshowsthegraphof y = a sin (bx) + c for0 G x G 2r ,wherea,bandcarepositive
integers.
For
Examiner’s
Use
y
4
3
2
1
O
�
x
2�
–1
–2
–3
Statethevalueofa,ofbandofc.
a=
2 Findthesetofvaluesofkforwhichthecurve y = (k + 1) x 2 - 3x + (k + 1) liesbelowthe
x-axis.
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[3]
b=
c=
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[4]
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4
3 Showthat
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1 + sin i
cos i
+
= 2 sec i .
cos i
1 + sin i
[4]
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For
Examiner’s
Use
5
4 ThesetsAandBaresuchthat
For
Examiner’s
Use
1
A = $ x: cos x = , 0° G x G 620°. ,
2
B = " x: tan x = 3, 0° G x G 620°, .
(i) Findn(A).
[1]
(ii) Findn(B).
[1]
(iii) FindtheelementsofA,B.
[1]
(iv) FindtheelementsofA+B.
[1]
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6
5 (i) Find y (9 + sin 3x) dx .
[3]
(ii) Henceshowthat yr (9 + sin 3x) dx = ar + b ,whereaandbareconstantstobefound.
[3]
r
9
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For
Examiner’s
Use
7
6 Thefunction f (x) = ax 3 + 4x 2 + bx - 2 ,whereaandbareconstants,issuchthat 2x - 1isa
factor.Giventhattheremainderwhen f (x) isdividedby x - 2 istwicetheremainderwhen f (x) isdividedby x + 1,findthevalueofaandofb.
[6]
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For
Examiner’s
Use
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8
7 (a) (i) Findhowmanydifferent4-digitnumberscanbeformedfromthedigits
1,3,5,6,8and9ifeachdigitmaybeusedonlyonce.
[1]
[1]
(b) Ateamof6peopleistobeselectedfrom8menand4women.Findthenumberofdifferent
teamsthatcanbeselectedif
(i) therearenorestrictions,
[1]
(ii) theteamcontainsall4women,
[1]
(iii) theteamcontainsatleast4men.
[3]
(ii) Findhowmanyofthese4-digitnumbersareeven.
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For
Examiner’s
Use
9
8 (i) Onthegridbelow,sketchthegraphof y = (x - 2) (x + 3)
for- 5 G x G 4 ,and
statethecoordinatesofthepointswherethecurvemeetsthecoordinateaxes.
[4]
For
Examiner’s
Use
y
–5
–4
–3
–2
–1
O
1
2
3
(ii) Findthecoordinatesofthestationarypointonthecurve
(iii) Giventhatkisapositiveconstant,statethesetofvaluesofkforwhich
(x - 2) (x + 3) = k has2solutionsonly.
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4
y = (x - 2) (x + 3) .
x
[2]
[1]
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10
9 (a) Differentiate 4x 3 ln (2x + 1) withrespecttox.
(b) (i) Giventhat y =
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dy
2x
x+ 4
,showthat =
.
dx ^ x + 2h3
x+ 2
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[3]
[4]
For
Examiner’s
Use
11
c 5x + 20
dx .
3
e ^ x + 2h
(ii) Hencefind ydd
(iii) Henceevaluate ddy
2
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[2]
c 77 5x + 20
dx .
3
e2 ^ x + 2h
For
Examiner’s
Use
[2]
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12
10 Solutions to this question by accurate drawing will not be accepted.
Thepoints A (- 3, 2) and B (1, 4) areverticesofanisoscelestriangleABC,whereangle B = 90° .
(i) FindthelengthofthelineAB.
[1]
(ii) FindtheequationofthelineBC.
[3]
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For
Examiner’s
Use
13
(iii) FindthecoordinatesofeachofthetwopossiblepositionsofC.
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[6]
For
Examiner’s
Use
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14
J2 3N
OO .
11 (a) Itisgiventhatthematrix A = KK
4
1
L
P
(i) FindA+2I.
For
Examiner’s
Use
[1]
(ii) FindA2.
[2]
(iii) Usingyouranswertopart(ii)findthematrixBsuchthatA2B=I.
[2]
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15
J
-1 N
x
O ,showthatdet C ! 0 .
(b) GiventhatthematrixC = K 2
x
1
+
x
x
1
L
P
[4]
For
Examiner’s
Use
12 (a) Afunctionfissuchthat f (x) = 3x 2 - 1for- 10 G x G 8 .
(i) Findtherangeoff.
[3]
(ii) Writedownasuitabledomainforfforwhichf - 1 exists.
[1]
Question 12(b) is printed on the next page.
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16
(b) Functionsgandharedefinedby
For
Examiner’s
Use
g (x) = 4e - 2 for x d R ,
x
h (x) = ln 5x for x 2 0 .
(i) Find g - 1 (x) .
[2]
(ii) Solve gh (x) = 18 .
[3]
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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
Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
©UCLES2013
4037/12/O/N/13