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Cambridge International Examinations
CambridgeOrdinaryLevel
* 8 1 2 6 0 6 2 8 4 7 *
4037/12
ADDITIONAL MATHEMATICS
Paper1
May/June 2014
2 hours
CandidatesanswerontheQuestionPaper.
Noadditionalmaterialsarerequired.
READ THESE INSTRUCTIONS FIRST
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Youareremindedoftheneedforclearpresentationinyouranswers.
Attheendoftheexamination,fastenallyourworksecurelytogether.
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion.
Thetotalnumberofmarksforthispaperis80.
Thisdocumentconsistsof15printedpagesand1blankpage.
DC(SJF/CGW)92626/1
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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 Show that
found.
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1 + sin A
cos A
+
cos A
1 + sin A
can be written in the form p sec A, where p is an integer to be
[4]
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2 (a) On the Venn diagrams below, draw sets A and B as indicated.
(i) (ii)
AB=
AB
[2]
(b) The universal set % and sets P and Q are such that n(%) = 20, n (P , Q) = 15, n (P) = 13 and
n (P + Q) = 4 . Find
(i) n (Q) ,
[1]
(ii) n ^(P , Q)lh,
[1]
(iii) n (P + Q l ) .
[1]
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3 (i) Sketch the graph of y = ^2x + 1h^x - 2h
for - 2 G x G 3, showing the coordinates of the
points where the curve meets the x- and y-axes.
[3]
(ii) Find the non-zero values of k for which the equation ^2x + 1h^x - 2h = k has two solutions only.
[2]
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4 the x-axis and the line x = a , where
1
0 1 a 1 1 radian, lies entirely above the x-axis. Given that the area of this region is square unit,
3
find the value of a.
[6]
The region enclosed by the curve
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y = 2 sin 3x ,
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5 1
(i) Given that 2 5x # 4 y = , show that 5x + 2y = –3.
8
(ii) Solve the simultaneous equations 2 5x # 4 y =
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1
and 7 x # 49 2y = 1.
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[3]
[4]
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1 3
3
m, Y = f 4 5 p and Z = ^1 2 3h. Write
2
6 7
down all the matrix products which are possible using any two of these matrices. Do not
evaluate these products.
6 2
(a) Matrices X, Y and Z are such that X = c
1
(b) Matrices A and B are such that A = c
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-4
-2
3
m and AB = c
1
-6
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9
m. Find the matrix B.
-3
[2]
[5]
9
7 The diagram shows a circle, centre O, radius 8 cm. Points P and Q lie on the circle such that the chord
PQ = 12 cm and angle POQ = i radians.
12 cm
Q
θ rad
P
8 cm
O
(i) Show that i = 1.696 , correct to 3 decimal places.
[2]
(ii) Find the perimeter of the shaded region.
[3]
(iii) Find the area of the shaded region.
[3]
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8 (a) (i) How many different 5-digit numbers can be formed using the digits 1, 2, 4, 5, 7 and 9 if no
digit is repeated?
[1]
(ii) How many of these numbers are even?
[1]
(iii) How many of these numbers are less than 60 000 and even?
[3]
(b) How many different groups of 6 children can be chosen from a class of 18 children if the class
contains one set of twins who must not be separated?
[3]
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9 A solid circular cylinder has a base radius of r cm and a volume of 4000 cm3.
(i) Show that the total surface area, A cm2, of the cylinder is given by A =
(ii) Given that r can vary, find the minimum total surface area of the cylinder, justifying that this area
is a minimum.
[6]
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8000
+ 2 rr 2 .
r
[3]
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10 In this question i is a unit vector due East and j is a unit vector due North.
At 12 00 hours, a ship leaves a port P and travels with a speed of 26 kmh–1 in the direction 5i + 12j.
(i) Show that the velocity of the ship is ^10i + 24jh kmh–1.
[2]
(ii) Write down the position vector of the ship, relative to P, at 16 00 hours.
[1]
(iii) Find the position vector of the ship, relative to P, t hours after 16 00 hours.
[2]
At 16 00 hours, a speedboat leaves a lighthouse which has position vector ^120i + 81jh km, relative to
P, to intercept the ship. The speedboat has a velocity of ^- 22i + 30jh kmh–1.
(iv) Find the position vector, relative to P, of the speedboat t hours after 16 00 hours.
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[1]
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(v) Find the time at which the speedboat intercepts the ship and the position vector, relative to P, of
the point of interception.
[4]
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11 (a) Solve
tan 2 x + 5 tan x = 0 for 0° G x G 180° .
[3]
2 cos 2 y - sin y - 1 = 0 for 0° G y G 360° .
[4]
(b) Solve
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(c) Solve
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r
sec `2z - j = 2 for 0 G z G r radians.
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[4]
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© UCLES 2014
4037/12/M/J/14