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Cambridge International Examinations
CambridgeOrdinaryLevel
* 4 2 9 6 3 4 5 2 6 7 *
4037/13
ADDITIONAL MATHEMATICS
Paper1
October/November 2014
2 hours
CandidatesanswerontheQuestionPaper.
Noadditionalmaterialsarerequired.
READ THESE INSTRUCTIONS FIRST
WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.
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Answerallthequestions.
Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseof
anglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.
Theuseofanelectroniccalculatorisexpected,whereappropriate.
Youareremindedoftheneedforclearpresentationinyouranswers.
Attheendoftheexamination,fastenallyourworksecurelytogether.
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion.
Thetotalnumberofmarksforthispaperis80.
Thisdocumentconsistsof15printedpagesand1blankpage.
DC(SJF/CGW)97557
©UCLES2014
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2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
Binomial Theorem
()
()
b 2 − 4 ac .
2a
−b
()
()
n
n
n
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
.
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
sin A = sin B = sin C a2 = b2 + c2 – 2bc cos A
∆=
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1
bc sin A
2
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3
1 The diagram shows the graph of
positive integers.
y = a cos bx + c for 0° G x G 360° , where a, b and c are
y
7
1
O
180°
State the value of each of a, b and c.
360°
x
[3]
a=
b=
c=
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2 The line
of AB.
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4y = x + 8
cuts the curve
xy = 4 + 2x
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at the points A and B. Find the exact length
[5]
5
3 The universal set  is the set of real numbers. Sets A, B and C are such that
A = " x : x 2 + 5x + 6 = 0, ,
B = " x: ^x - 3h^x + 2h^x + 1h = 0, ,
C = " x : x 2 + x + 3 = 0, .
(i) State the value of each of n(A), n(B) and n(C).
n ^Ah =
n ^Bh =
n ^Ch =
[3]
(ii) List the elements in the set A , B .
[1]
(iii) List the elements in the set A + B .
[1]
(iv) Describe the set C l .
[1]
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4 (a) Solve
3 sin x + 5 cos x = 0 for 0° G x G 360° .
[3]
(b) Solve
J
rN
cosec K3y + O = 2 for 0 G y G r radians.
4P
L
[5]
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7
5 (a) A drinks machine sells coffee, tea and cola. Coffee costs $0.50, tea costs $0.40 and cola costs
$0.45. The table below shows the numbers of drinks sold over a 4-day period.
Coffee
Tea
Cola
Tuesday
12
2
1
Wednesday
9
3
0
Thursday
8
5
1
Friday
11
2
0
(i) Write down 2 matrices whose product will give the amount of money the drinks machine
took each day and evaluate this product.
[4]
(ii) Hence write down the total amount of money taken by the machine for this 4-day period. [1]
J 2 4N
O and XY = I, where I is the identity matrix. Find the
(b) Matrices X and Y are such that X = K
5
1
L
P
matrix Y.
[3]
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6 The diagram shows a sector, AOB, of a circle centre O, radius 12 cm. Angle AOB = 0.9 radians. The
point C lies on OA such that OC = CB.
A
C
O
0.9 rad
12 cm
B
(i) Show that OC = 9.65 cm correct to 3 significant figures.
[2]
(ii) Find the perimeter of the shaded region.
[3]
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7 (iii) Find the area of the shaded region.
Solve the equation
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[3]
1 + 2 log 5 x = log5 ^18x - 9h.
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[5]
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Given that f ^xh = x ln x 3 , show that f l^xh = 3 ^1 + ln xh.
8 (i)
(ii) Hence find y ^1 + lnxh d x .
(iii) Hence find
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[3]
[2]
2
y1 lnx d x in the form p + ln q , where p and q are integers.
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[3]
11
9 (a) Given that the first 3 terms in the expansion of ^5 - qxh p are 625 - 1500x + rx 2 , find the value of
each of the integers p, q and r.
[5]
(b) Find the value of the term that is independent of x in the expansion of
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J
1 N
K2x + 3O .
4x P
L
[3]
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10 (a) Solve the following simultaneous equations.
5x
=1
25 3y - 2
3x
= 81
27 y - 1
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[5]
13
(b) The diagram shows a triangle ABC such that AB = ^2 + 3h cm, BC = ^1 + 2 3h cm
and AC = 2 cm.
A
^2 +
B
3h cm
2 cm
^1 + 2 3h cm
C
Without using a calculator, find the value of cos A in the form a + b 3 , where a and b are
constants to be found.
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[4]
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11 The diagram shows part of the curve y = ^x + 5h^x - 1h2 .
y
y = (x + 5)(x – 1)2
O
(i) Find the x-coordinates of the stationary points of the curve.
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x
[5]
15
(ii) Find y ^x + 5h^x - 1h2 d x .
[3]
(iii) Hence find the area enclosed by the curve and the x-axis.
[2]
(iv) Find the set of positive values of k for which the equation ^x + 5h^x - 1h2 = k has only one real
solution.
[2]
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BLANk pAGE
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© UCLES 2014
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