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8/10/2019 GCE O level A.math Specimen paper
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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 nonexact 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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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
xaxis. [4]
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Use
3 Show that1
1 2
cos
sin
sin
cossec
i
i
i
ii
++
+= . [4]
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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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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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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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7 (a) (i) Find how many different 4digit 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 4digit 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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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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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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(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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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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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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(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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(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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publisher will be pleased to make amends at the earliest possible opportunity.
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.