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This document consists of 16printed pages.
DC (SLM) 81102
UCLES 2013 [Turn over
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral
Certificate of Education Ordinary Level
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 NOTWRITE IN ANY
BARCODES.
Answer allthe questions.
Give non-exact numerical answers correct to 3 significant
figures, or 1 decimal place in the case of angles indegrees, 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.
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0
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ADDITIONAL MATHEMATICS 4037/12
Paper 1 October/November 2013
2 hours
Candidates answer on the Question Paper.No Additional Materials
are required.
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Mathematical Formulae
1. ALGEBRA
QuadraticEquation
For the equation ax2+ bx+ c= 0,
xa
b b ac
2
42
!=
- -.
Binomial Theorem
(a+ b)n= an+
(
n
1 )an1b+
(
n
2 )an2b2+ +
(
n
r)anrbr+ + bn,
where nis a positive integer and ( nr)= n!(n r)!r! .
2. TRIGONOMETRY
Identities
sin2A+ cos2A= 1
sec2A= 1 + tan2A
cosec2A= 1 + cot2A
Formulae for ABC
asinA
=b
sinB=
csin C
a2= b2+ c2 2bccosA
=1
2bcsinA
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For
Examiners
Use
1 The diagram shows the graph of ( )siny a bx c= + for 0 2xG G
r, where a, band care positiveintegers.
4
3
2
1
O
y
x
1
2
3
2
State the value of a, of band of c. [3]
a= b= c=
2 Find the set of values of kfor which the curve ( ) ( )y k x x
k1 3 12= + - + + lies below the
x-axis. [4]
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For
Examiners
Use
3 Show that1
1 2
cos
sin
sin
cossec
i
i
i
ii
++
+= . [4]
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For
Examiners
Use
4 The setsAandBare such that
: ,cosA x x x2
10 6 02 G G= =$ .,
: ,tanB x x x3 0 6 02 G G= =" ,.
(i) Find n(A). [1]
(ii) Find n(B). [1]
(iii) Find the elements ofA,B. [1]
(iv) Find the elements ofA+B. [1]
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For
Examiners
Use
5 (i) Find (9 3 )sin x xd+y . [3]
(ii) Hence show thatr
( )sin x x a b9 3 d9
r+ = +r
y , where aand bare constants to be found. [3]
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Examiners
Use
6 The function ( ) 4 2x ax x bxf 3 2= + + - , where aand bare
constants, is such that x2 1- is afactor. Given that the remainder
when ( )xf is divided by x 2- is twice the remainder when ( )xf is
divided by x 1+ , find the value of aand of b. [6]
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Examiners
Use
7 (a) (i) Find how many different 4-digit numbers can be formed
from the digits 1, 3, 5, 6, 8 and 9 if each digit may be used only
once. [1]
(ii) Find how many of these 4-digit numbers are even. [1]
(b) A team of 6 people is to be selected from 8 men and 4 women.
Find the number of different
teams that can be selected if
(i) there are no restrictions, [1]
(ii) the team contains all 4 women, [1]
(iii) the team contains at least 4 men. [3]
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For
Examiners
Use
8 (i) On the grid below, sketch the graph of ( ) ( )y x x2 3= -
+ for x5 4G G- , andstate the coordinates of the points where the
curve meets the coordinate axes. [4]
y
x
5 4 3 2 1 1 32 4O
(ii) Find the coordinates of the stationary point on the curve (
) ( )y x x2 3= - + . [2]
(iii) Given that kis a positive constant, state the set of
values of kfor which
( ) ( )x x k2 3- + = has 2 solutions only. [1]
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Examiners
Use
9 (a) Differentiate ( )lnx x4 2 13
+ with respect tox. [3]
(b) (i) Given that yx
x
2
2=
+, show that
x
y
x
x
2
4
d
d
3=
+
+
^ h. [4]
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For
Examiners
Use
(ii) Hence findyx
xx
2
5 20d
3+
+
^ h. [2]
(iii) Hence evaluate2
7yx
xx
2
5 20d
3+
+
^ h. [2]
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Examiners
Use
10 Solutions to this question by accurate drawing will not be
accepted.
The points ( , )A 3 2- and ( , )B 1 4 are vertices of an
isosceles triangleABC, where angle B 90= .
(i) Find the length of the lineAB. [1]
(ii) Find the equation of the lineBC. [3]
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(iii) Find the coordinates of each of the two possible positions
of C. [6]
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Examiners
Use
11 (a) It is given that the matrix2
4
3
1A =
J
LKK
N
POO.
(i) Find A+ 2I. [1]
(ii) Find A2. [2]
(iii) Using your answer to part (ii)find the matrix Bsuch that
A2B= I. [2]
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For
Examiners
Use
(b) Given that the matrix Cx x
x
1
12
=- +
-
x 1-
J
LKK
N
POO, show that det 0C ! . [4]
12 (a) A function f is such that ( ) 3 1x xf 2
= - for x10 8G G- .
(i) Find the range of f. [3]
(ii) Write down a suitable domain for f for which f 1- exists.
[1]
Question 12(b) is printed on the next page.
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Examiners
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(b) Functions g and h are defined by
( ) 4 2xg ex= - for x Rd ,
( ) 5lnx xh = for x 02 .
(i) Find ( )xg 1- . [2]
(ii) Solve ( ) 18xgh = . [3]
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a department of the University of Cambridge.
