Caribbean Examinations Council
Additional
Mathematics
CSEC® PAST PAPERS
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Cover photograph by Mrs Alberta Williams
With thanks to the students of the Sir Arthur Lewis Community College, St Lucia:
Akin Ogunlusi, Nechelle Joseph
CSEC® Additional Maths Past Papers
LIST OF CONTENTS
Paper 02 (03 May 2012)
3
Paper 032 (12 June 2012)
11
Paper 02 (07 May 2013)
13
Paper 032 (12 June 2013)
21
Paper 02 (06 May 2014)
23
Paper 032 (09 June 2014)
33
TEST CODE
FORM TP 2012037
CARIBBEAN
01254020
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
SECONDARY EDUCATION CERTIFICATE
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02 – General Proficiency
2 hours 40 minutes
03 MAY 2012 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
This paper consists of FOUR sections. Answer ALL questions in Section I,
Section II and Section III.
3.
Answer ONE question in Section IV.
4.
Write your solutions with full working in the booklet provided.
Required Examination Materials
Electronic Calculator (non programmable)
Geometry Set
Mathematical Tables (provided)
Graph Paper (provided)
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254020/F 2012
-2SECTION I
Answer BOTH questions.
1.
(a)
The functions f and g are defined by
f(x) = x3 + 1,
0<x<3
g(x) = x + 5,
x∈R
where R is the set of real numbers.
(i)
Determine the composition function g(f(x)).
(1 mark)
(ii)
State the range of g(f(x)). (1 mark)
(iii)
Determine the inverse of g(f(x)).
(2 marks)
(b)
If x + 2 is a factor of f(x) = 2x3 – 3x2 – 4x + a, find the value of a.
(2 marks)
(c)
Solve the equation 32x – 9(3–2x) = 8.
(5 marks)
(d)
(2 marks)
(i)
Express x3 = 10x–3 in the form log10x = ax + b.
(ii)
Hence, state the value of the gradient of a graph of log10x versus x.
(1 mark)
Total 14 marks
2.
(a)
(b)
The quadratic equation x2 – 4x + 6 = 0 has roots α and β.
Calculate the value of α2 + β2.
(5 marks)
Find the range of values of x for which
(c)
2x – 5
——— > 0.
3x + 1
(4 marks)
A customer repays a loan monthly by increasing the payment each month by $x. If the
customer repaid $50 in the 5th month and $70 in the 9th month, calculate the TOTAL amount
of money repaid at the end of the 24th month.
(5 marks)
Total 14 marks
GO ON TO THE NEXT PAGE
01254020/F 2012
-3SECTION II
Answer BOTH questions.
3.
(a)
The equation of a circle is given by x2 + y2 – 4x + 6y = 87.
(i)
A line has equation x + y + 1 = 0. Show that this line passes through the centre of
the circle.
(3 marks)
(ii)
Find the equation of the tangent to the circle at the point A (– 6, 3).
(b)
(4 marks)
→
→
→ 1 →
Given OA = a, OB = b, AP = — OA,
2
2
3
and b =
.
where a =
1
2
→
(i)
Write BP in terms of a and b.
(2 marks)
(ii)
Find |BP|.
(3 marks)
→
Total 12 marks
GO ON TO THE NEXT PAGE
01254020/F 2012
-44.
(a)
The diagram shows a sector of a circle centre O with an adjoining square. The radius of
the circle is 4 m.
�
If the sector AOC subtends an angle — at O, calculate, giving your answer in terms of
3
π
(i)
the area of the shape OACMN
(ii)
the perimeter of the shape OACMN.
(5 marks)
�
�
�
� √3
1
√2
Given that sin — = —–, cos — = — and sin — = cos — = —–, evaluate without using
4
3
4
3
2
2
2
7�
calculators, the exact value of cos —–.
(3 marks)
12
(b)
(c)
Prove the identity
1
1 – sin θ
——–——— ≡ ———— .
1 + tan θ
cos θ
cos θ
(4 marks)
Total 12 marks
GO ON TO THE NEXT PAGE
01254020/F 2012
-5SECTION III
Answer BOTH questions.
5.
(a)
Differentiate the following expression with respect to x, simplifying your answer.
3x + 4
———
x–2
(b)
(4 marks)
The point P (2, 10) lies on the curve y = 3x2 + 5x – 12. Find equations for
(i)
the tangent to the curve at P
(ii)
the normal to the curve at P.
(5 marks)
(c)
The length of the side of a square is increasing at a rate of 4 cms–1. Find the rate of increase
(5 marks)
of the area when the length of the side is 5 cm.
6.
(a)
Evaluate
∫
Total 14 marks
2
(4 marks)
(16 – 7x)3 dx.
1
(b)
dy
The point Q (4, 8) lies on a curve for which —– = 3x – 5. Determine the equation of the
dx
curve.
(3 marks)
(c)
�
Calculate the area between the curve y = 2 cos x + 3 sin x and the x-axis from x = 0 to —.
3
(d)
(3 marks)
Calculate the volume of the solid formed when the area enclosed by the curve y = x2 + 2
and the x-axis, from x = 0 to x = 3, is rotated through 360° about the x-axis.
[Leave your solution in terms of � ].
(4 marks)
Total 14 marks
GO ON TO THE NEXT PAGE
01254020/F 2012
-6SECTION IV
Answer only ONE question.
7.
(a)
A survey carried out in a town revealed that 25% of the households surveyed owned a
laptop computer and 70% owned a desktop computer. In addition, it was found that 12%
owned both a laptop and a desktop computer.
If a sample of households from the town is selected at random, determine the proportion
that own NEITHER a laptop NOR a desktop computer.
(4 marks)
(b)
A bag contains 4 red marbles, 3 black marbles and 3 blue marbles. Three marbles are
drawn at random without replacement from the bag.
Find the probability that the marbles
(i)
drawn are ALL of the SAME colour
(3 marks)
(ii)
contain EXACTLY 1 red marble.
(3 marks)
(c)
The probability of hiring a taxi from garage A, B or C is 0.3, 0.5 and 0.2 respectively. The
probability that the taxi ordered will be late from A is 0.07, from B is 0.1 and from C is
0.2.
(i)
Illustrate this information on a tree diagram showing the probability on all branches.
(3 marks)
(ii)
A garage is chosen at random, determine the probability that
a)
the taxi will arrive late
(3 marks)
b)
the taxi will come from garage C given that it is late.
(4 marks)
Total 20 marks
GO ON TO THE NEXT PAGE
01254020/F 2012
-78.
(a)
A car starting from rest at a point A, moves along a straight line reaching a velocity of
24 ms–1 by a constant acceleration of 6 ms–2. The car maintains this constant velocity
of 24 ms–1 for 5 seconds and is then brought to rest again by a constant acceleration
of –3 ms–2.
(i)
Using the graph sheet provided, draw a velocity-time graph to illustrate the
motion of the car.
(3 marks)
Determine the TOTAL distance travelled by the car.
(ii)
(3 marks)
(iii)
A second car, moving at a constant velocity of 32 ms–1 drives past point A, 3 seconds
after the first car left point A. Calculate the length of time after the first car started
that this second car meets it.
[Assume that the cars meet during the time when the first car is moving at a
constant velocity.]
(4 marks)
(b)
A particle moves in a straight line with acceleration given by a = (5t – 1) ms–2 at any time
t seconds. When t = 2 seconds, the particle has velocity 4 ms–1 and is 8 m from a fixed
point O. Determine
(i)
its velocity when t = 4
(5 marks)
(ii)
its displacement from O when t = 3.
(5 marks)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254020/F 2012
TEST CODE
FORM TP 2012037
01254020
MAY/JUNE 2012
CARIBBEAN
E XAM I NAT I O N S
COUNCIL
SECONDARY EDUCATION CERTIFICATE
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02 – General Proficiency
Answer Sheet for Question 8. (a) (i)
Candidate Number ............................................
ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET
01254020/F 2012
TEST CODE
FORM TP 2012038
CARIBBEAN
01254032
MAY/JUNE 2012
E XAM I NAT I O N S
COUNCIL
SECONDARY EDUCATION CERTIFICATE
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 032 – General Proficiency
ALTERNATIVE TO SBA
90 minutes
12 JUNE 2012 (p.m.)
Answer all parts of the given question.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254032/F 2012
-2Two sports clubs, P and Q, wish to use 600 m of fencing to enclose a court. They wish to determine
which design gives the maximum area.
(a)
Sports Club P uses the 600 m of fencing to make a rectangular court measuring 3x m by 2y m as
shown in the diagram below.
3x m
2y m
Sports Club Q uses the 600 m of fencing to make six equal-sized rectangular courts that are adjacent
to each other as shown in the diagram below. Each court measures x m by y m.
xm
xm
xm
ym
ym
For Sports Club P the mathematical problem is to maximize the area of enclosure to satisfy its
perimeter and the following conditions:
Maximize
Subject to
A = 6xy
6x + 4y = 600
(i)
Formulate the mathematical problem for Sports Club Q.
(2 marks)
(ii)
Determine the MAXIMUM area of the court for Sports Club Q.
(3 marks)
(iii)
Show that Sports Club P has the maximum area when a square enclosure is used
and determine the MAXIMUM possible area.
(4 marks)
(iv)
Suggest which sports club design should be used.
(b)
(1 mark )
The numbers log (a3 b7), log (a5 b12) and log (a8 b15) are the first three terms of an arithmetic series.
The 12th term of the series is log bn. Calculate the value of n.
(10 marks)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254032/F 2012
TEST CODE
FORM TP 2013037
01254020
MAY/JUNE 2013
CAR I B B EAN E XAM I NAT I O N S C O U N C I L
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02 – General Proficiency
2 hours 40 minutes
07 MAY 2013 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
DO NOT open this examination paper until instructed to do so.
2.
This paper consists of FOUR sections. Answer ALL questions in Section 1,
Section 2 and Section 3.
3.
Answer ONE question in Section 4.
4.
Write your solutions with full working in the booklet provided.
5.
A list of formulae is provided on page 2 of this booklet.
Required Examination Materials
Electronic calculator (non programmable)
Geometry Set
Mathematical Tables (provided)
Graph paper (provided)
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254020/F 2013
-2-
LIST OF FORMULAE
GO ON TO THE NEXT PAGE
01254020/F 2013
-3-
SECTION 1
Answer BOTH questions.
All working must be clearly shown.
1.
(a)
Let f(x) = x3 – x2 – 14x + 24.
(i)
Use the factor theorem to show that x + 4 is a factor of f(x).
(2 marks)
(ii)
Determine the other linear factors of f(x).
(3 marks)
(b)
A function f(x) is given by f(x) =
2x – 1
.
x+2
(3 marks)
(i)
Find an expression for the inverse function f –1(x).
(ii)
The function g is given by g(x) = x +1. Write an expression for the composite
function, fg(x). Simplify your answer.
(2 marks)
(c)
Given that 53x –2 = 7x + 2, show that
2(log5 + log7)
x =
.
(log125 – log7)
(4 marks)
Total 14 marks
2.
(a)
Let f(x) = 3x2 + 6x – 1.
(i)
Express f(x) in the form a(x + h)2 + k where h and k are constants.
(3 marks)
(ii)
State the minimum value of f(x).
(1 mark)
(iii)
Determine the value of x for which f(x) is a minimum.
(1 mark)
(b)
Find the set of values of x for which 2x2 + 3x – 5 ≥ 0.
(c)
Find the sum to infinity of the following series:
(4 marks)
1
2
1
2
– + –2 + –3 + –4 + ...
4
4
4
4
Note: This series can be rewritten as the sum of two geometric series.
(5 marks)
Total 14 marks
GO ON TO THE NEXT PAGE
01254020/F 2013
-4-
SECTION 2
Answer BOTH questions.
All working must be clearly shown.
(i)
A circle, C, has centre with coordinates A (2,1) and passes through the point
B (10,7).
Express the equation of the circle in the form x2 + y2 + hx + gy + k = 0, where
h, g and k are integers to be determined.
(3 marks)
The line l is a tangent to the circle C at the point B. Find an equation for l.
(3 marks)
3.
(a)
(b)
(c)
(ii)
The position vectors of two points, P and Q, relative to a fixed origin, O, are 10i – 8j
and λi + 10j respectively, where λ is a constant.
→
→
Find the value of λ such that OP and OQ are perpendicular.
(3 marks)
The position vectors of A and B with respect to a fixed origin, O, are given by
→
→
OA = –2i +5j and OB = 3i – 7j respectively.
Find the unit vector in the direction of AB.
(3 marks)
Total 12 marks
4.
(a)
π
The diagram shows a sector cut from a circle of centre O. The angle at O is —. If the
6
5 (12 + π) cm, what is its area?
perimeter of the sector is —
(4 marks)
6
(b)
(c)
Solve the equation 2 cos2θ + 3 sin θ = 0 for 0 ≤ θ ≤ 360°.
(5 marks)
1
Given that tan (θ – α) = – and that tan θ = 3, use the appropriate compound angle
2
formula to find the value of the acute angle α.
(3 marks)
Total 12 marks
GO ON TO THE NEXT PAGE
01254020/F 2013
-5SECTION 3
Answer BOTH questions.
All working must be clearly shown.
5.
(a)
Given that y = x3 – 3x2 + 2. Find
(5 marks)
(i)
the coordinates of the stationary points of y
(ii)
the second derivative of y and hence determine the nature of EACH of the
stationary points.
(5 marks)
(b)
Differentiate y = (5x + 3)3sin x with respect to x, simplifying your result as far as
(4 marks)
possible.
Total 14 marks
6.
(a)
2
– 5 cos x) dx
(3sin
Find (5x
+4)xdx.
0
∫
2
(2 marks)
π
x) dxx) dx.
∫ ((33sinsinx x– 5–cos5 cos
(4 marks)
2
(b)
Evaluate
(c)
dy
A curve passes through the points P(0, 8) and Q(4, 0) and is such that — = 2 – 2x.
dx
0
Find the area of the finite region bounded by the curve in the first quadrant.
(8 marks)
Total 14 marks
GO ON TO THE NEXT PAGE
01254020/F 2013
-6SECTION 4
Answer ONLY ONE question.
All working must be clearly shown.
7.
(a)
Of the persons buying petrol at a service station, 40 per cent are females. Of the females,
30 per cent pay for their petrol with cash, and of the males, 65 per cent pay for their
petrol with cash.
(i)
Copy and complete the following tree diagram, by putting in all the missing
probabilities, to show this information.
(2 marks)
(ii)
What is the probability that a customer pays for petrol with cash?
(iii)
Determine which is the more likely event:
(b)
Event T:
Event V:
Customer is female, GIVEN that the petrol is paid WITH cash.
A male customer does NOT pay for petrol with cash.
(4 marks)
The marks obtained by 30 students on an English exam are given as
58
92
41
89
72
66
51
63
80
40
69
45
83
76
53
56
75
50
99
50
85
63
58
75
66
56
81
74
51
94
(3 marks)
(i)
State ONE advantage of using a stem and leaf diagram versus a box and whiskers
plot to display the data.
(1 mark)
(ii)
Construct a stem and leaf diagram to show the data.
(3 marks)
(iii)
Determine the median mark.
(2 marks)
(iv)
Calculate the semi inter-quartile range of the marks.
(3 marks)
(v)
Two students are chosen at random from the class.
Determine the probability that both scored less than 50 on the exam.
(2 marks)
Total 20 marks
GO ON TO THE NEXT PAGE
01254020/F 2013
-7-
8.
(a)
A particle starts from rest and accelerates uniformly to 20 m s–1 in 5 seconds. It continues
at this velocity for 10 seconds. It then accelerates again uniformly to a velocity of
60 m s–1 in 5 seconds. The particle then decelerates uniformly until it comes to rest,
15 seconds later.
(i)
On the graph paper provided, draw a velocity-time graph to illustrate the
motion of the particle.
(3 marks)
From your graph determine
(ii)
a)
the total distance, in metres, travelled by the particle
(4 marks)
b)
the average velocity of the particle for the entire journey.
(2 marks)
(b)
A particle travels in a straight line in such a way that after t seconds its velocity, v, from
a fixed point, O, is given by the function v = 3t2 – 18t + 15.
Calculate
(3 marks)
(i)
the values of t when the particle is instantaneously at rest
(ii)
the distance travelled by the particle between 1 second and 3 seconds
(3 marks)
(iii)
the value of dv
— when
dt
a)
t = 2 seconds
(2 marks)
b)
t = 3 seconds.
(1 mark)
(iv)
Give an interpretation for the value in
a)
8 (b) (iii) a)
(1 mark)
b)
8 (b) (iii) b).
(1 mark)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254020/F 2013
TEST CODE
FORM TP 2013037
01254020
MAY/JUNE 2013
CAR I B B EAN E XAM I NAT I O N S C O U N C I L
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02 – General Proficiency
Answer Sheet for Question 8 (a)
Candidate Number .................................
ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET
01254020/F 2013
TEST CODE
FORM TP 2013038
01254032
MAY/JUNE 2013
CAR I B B EAN E XAM I NAT I O N S C O U N C I L
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 032 – General Proficiency
Alternative Paper
1 hour 30 minutes
12 JUNE 2013 (p.m.)
Answer the given questions.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254032/F 2013
-2Answer the given questions.
1.
(a)
A circle is drawn with centre at origin, O, and radius 6 cm. Find the coordinates of all
intersections of the circle with an origin centred square of side length 10 cm whose sides
are parallel to the coordinate axes as illustrated in Figure 1.
O
Figure 1
(10 marks)
(b)
The cuboid shown in Figure 2 has width x m and its length is twice its width. The
volume of the cuboid is 720 m3.
x
Figure 2
(i)
Find an expression for the height, h, of the cuboid in terms of x.
(ii)
Show that an expression for the surface area, A, of the cuboid is given by
2160
+ 4x2.
x
A=
Hence show that A has a stationary value when x = 3√10.
(iii)
(3 marks)
(3 marks)
3
(4 marks)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254032/F 2013
TEST CODE
FORM TP 2014037
CARIBBEAN
01254020
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02 – General Proficiency
2 hours 40 minutes
06 MAY 2014 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This paper consists of FOUR sections. Answer ALL questions in Section 1,
Section 2 and Section 3.
2.
Answer ONE question in Section 4.
3.
Write your solutions with full working in the booklet provided.
4.
A list of formulae is provided on page 2 of this booklet.
Required Examination Materials
Electronic Calculator (non programmable)
Geometry Set
Mathematical Tables (provided)
Graph Paper (provided)
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254020/F 2014
-2LIST OF FORMULAE
n
Sn = — [2a + (n – 1)d]
2
Arithmetic Series
Tn = a + (n – 1)d
Geometric Series
Tn = arn–1
Circle
x2 + y2 + 2fx + 2gy + c = 0
Vectors
v
v^ = —
|v|
Trigonometry
sin (A + B) ≡ sin A cos B + cos A sin B
a(rn – 1)
Sn = ————
r –1
(x + f)2 + (y + g)2 = r2
a•b
cos θ = ———
|a| × |b|
+
cos (A + B) ≡ cos A cos B
a
S∞ = ——, –1 < r < 1 or |r| < 1
1–r
|v| = √ (x2 + y2) where v = xi + yj
sin A sin B
tan A + tan B
tan (A + B) ≡ ——————–
1 tan A tan B
+
Differentiation
d
––– (ax + b)n = an(ax + b)n–1
dx
d
––– sin x = cos x
dx
d
––– cos x = –sin x
dx
n
Σx
Statistics
i=1
–x = ——
n
n
Σf x
i
=
i=1
i
i
———
,
n
Σf
n
Σ
n
Σ
fi xi2
(xi – –x)2
i=1
i=1
2
–2
S = ————— = ————
– (x)
n
n
i=1 i
Probability
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Kinematics
v = u + at
v2 = u2 + 2as
Σf
i=1 i
1
s = ut + — at2
2
GO ON TO THE NEXT PAGE
01254020/F 2014
-3SECTION 1
Answer BOTH questions.
ALL working must be clearly shown.
1.
(a) (i)
The function f is defined by f : x → 1 – x2, x ∈
.
(1 mark)
Show that f is NOT one-to-one.
1
(ii)
The function g is defined by g : x → — x – 3, x ∈
2
.
(2 marks)
a)
Find fg(x), and clearly state its domain.
Determine the inverse, g–1, of g and sketch on the same pair of axes, the
(3 marks)
graphs of g and g–1.
(b)
(c)
b)
When the expression 2x3 + ax2 – 5x – 2 is divided by 2x – 1, the remainder is –3.5.
Determine the value of the constant a.
(3 marks)
The length of a regtangular kitchen is y m and the width is x m. If the length of the kitchen
is half the square of its width and its perimeter is 48 m, find the values of x and y (the
dimensions of the kitchen). (5 marks)
2.
(a)
Total 14 marks
Given that f(x) = –2x2 – 12x – 9.
Express f(x) in the form k + a (x + h)2, where a, h and k are integers to be determined.
(i)
(3 marks)
(ii)
State the maximum value of f(x).
(1 mark)
(iii)
Determine the value of x for which f(x) is a maximum.
(1 mark)
(b)
Find the set of values of x for which 3 + 5x – 2x2 < 0.
(4 marks)
(c)
A series is given by 0.2 + 0.02 + 0.002 + 0.0002 + ...
(i)
Show that this series is geometric.
(3 marks)
(ii)
Find the sum to infinity of this series, giving your answer as an exact fraction.
(2 marks)
Total 14 marks
GO ON TO THE NEXT PAGE
01254020/F 2014
-4SECTION 2
Answer BOTH questions.
ALL working must be clearly shown.
3.
(a)
(b)
(i)
Determine the value of k such that the lines x + 3y = 6 and kx + 2y = 12 are
perpendicular to each other.
(3 marks)
(ii)
A circle of radius 5 cm has as its centre the point of intersection of the two
perpendicular lines in (i). Determine the equation for this circle.
(3 marks)
RST is a triangle in the coordinate plane. Position vectors R, S, and T relative to an
origin, O, are
1
3
,
and
1
–1
4
respectively.
4
(i)
^ = 90°.
Show that TRS
(4 marks)
(ii)
Determine the length of the hypotenuse.
(2 marks)
[Hint: A rough drawing of RST might help].
Total 12 marks
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01254020/F 2014
-54.
(a)
Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians.
A
9 cm
H
0.7 rad
O
B
C
Figure 1.
(i)
Find the area of the sector OAB.
(2 marks)
(ii)
Hence, find the area of the shaded region, H.
(b)
(4 marks)
1 and cos —
� =—
� = —–,
√ 3 show that
Given that sin —
2
6
6
2
� = —
1 ( √ 3 cos x – sin x), where x is acute.
cos x + —
6
2
(2 marks)
1
tan θ sin θ
Prove the identity ——–——– ≡ 1 + ——– .
cos θ
1 – cos θ
(4 marks)
Total 12 marks
(c)
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01254020/F 2014
-6SECTION 3
Answer BOTH questions.
ALL working must be clearly shown.
5.
(a)
The equation of a curve is y = 3 + 4x – x2. The point P (3, 6) lies on the curve.
Find the equation of the tangent to the curve at P, giving your answer in the form
ax + by + c = 0, where a, b, c, ∈
(b)
(4 marks)
.
Given that f(x) = 2x3 – 9x2 – 24x + 7.
(i)
Find ALL the stationary points of f(x).(5 marks)
(ii)
Determine the nature of EACH of the stationary points of f(x).
6.
(a)
Evaluate
(b)
(c)
Evaluate
∫
∫
4
(5 marks)
Total 14 marks
(4 marks)
x (x2 – 2) dx.
2
–�3
(4 cos x + 2 sin x) dx, leaving your answer in surd form.
(4 marks)
0
dy
A curve passes through the point P (2, –5) and is such that —– = 6x2 – 1.
dx
(3 marks)
(i)
Determine the equation of the curve.
(ii)
Find the area of the finite region bounded by the curve, the x-axis, the line x = 3
and the line x = 4.
(3 marks)
Total 14 marks
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01254020/F 2014
-7SECTION 4
Answer ONLY ONE question.
ALL working must be clearly shown.
7.
(a)
There are 60 students in the sixth form of a certain school. Mathematics is studied by 27
of them, Biology by 20 of them and 22 students study neither Mathematics nor Biology.
If a student is selected at random, what is the probability that the student is studying
(i)
both Mathematics and Biology?
(3 marks)
(ii)
Biology only?
(2 marks)
(b)
Two ordinary six-sided dice are thrown together. The random variable S represents the
sum of the scores of their faces landing uppermost.
(i)
Copy and complete the sample space diagram below.
69
57
4
10
38
26
1
2
1
5
2
3
4
Sample space diagram of S
(ii)
6
(1 mark)
Find
a)
P (S > 9)
(2 marks)
b)
P (S < 4).
(1 mark)
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01254020/F 2014
-8(iii)
d
0
1
2
3
4
5
P (D = d)
1
—
6
a
2
—
9
b
1
—
9
c
Find the values of a, b and c.
(3 marks)
(c)
Let D be the difference between the scores of the faces landing uppermost. The
table below gives the probability of each possible value of d.
The aptitude scores obtained by 51 applicants for a supervisory job are summarized in the
following stem and leaf diagram.
5|1 means 51
3
1
5
9
4
2
4
6
8
9
5
1
3
3
5
6
7
9
6
0
1
3
3
3
5
6
8
8
9
7
1
2
2
2
4
5
5
5
6
8
8
0
1
2
3
5
8
8
9
9
0
1
2
6
8
8
9
9
(4 marks)
(i)
Find the median and quartiles for the data given.
(ii)
Construct a box-and-whisker plot to illustrate the data given and comment on the
distribution of the data.
(4 marks)
Total 20 marks
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01254020/F 2014
-98.
(a)
Figure 2 below, not drawn to scale, shows the motion of a car with velocity, V, as it moves
along a straight road from Point A to Point B. The time, t, taken to travel from Point A to
Point B is 90 seconds and the distance from Point A to Point B is 1410 m.
Figure 2.
(i)
What distance did the car travel from Point A towards Point B before starting to
decelerate?
(2 marks)
(ii)
Calculate the deceleration of the car as it goes from 25 m s–1 to 10 m s–1.
(5 marks)
(1 mark)
(iii)
For how long did the car maintain the speed of 10 m s–1?
(iv)
From Point B, the car decelerates uniformly, coming to rest at a Point C and
covering a further distance of 30 m. Determine the average velocity of the car
over the journey from Point A to Point C. (2 marks)
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01254020/F 2014
- 10 (b)
A particle travels along a straight line. It starts from rest at a point, P, on the line and after
10 seconds, it comes to rest at another point, Q, on the line. The velocity v m s–1 at time
t seconds after leaving P is
v = 0.72t2 – 0.096t3
for 0 < t < 5
v = 2.4t – 0.24t2
for 5 < t < 10
At maximum velocity the particle has no acceleration.
(i)
Find the time when the velocity is at its maximum.
(3 marks)
(ii)
Determine the maximum velocity.
(2 marks)
(iii)
Find the distance moved by the particle from P to the point where the particle
attains its maximum velocity.
(5 marks)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254020/F 2014
TEST CODE
FORM TP 2014038
CARIBBEAN
01254032
MAY/JUNE 2014
E XAM I NAT I O N S
COUNCIL
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 032 – General Proficiency
ALTERNATIVE
1 hour 30 minutes
09 JUNE 2014 (p.m.)
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This paper consists of ONE question. Answer the given question.
2.
Write your solutions with full working in the booklet provided.
3.
A list of formulae is provided on page 2 of this booklet.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright © 2011 Caribbean Examinations Council
All rights reserved.
01254032/F 2014
-2LIST OF FORMULAE
n
Sn = — [2a + (n – 1)d]
2
Arithmetic Series
Tn = a + (n – 1)d
Geometric Series
Tn = arn–1
Circle
x2 + y2 + 2fx + 2gy + c = 0
Vectors
v
v^ = —
|v|
Trigonometry
sin (A + B) ≡ sin A cos B + cos A sin B
a(rn – 1)
Sn = ————
r–1
(x + f)2 + (y + g)2 = r2
a•b
cos θ = ———
|a| × |b|
+
cos (A + B) ≡ cos A cos B
a
S∞ = ——, –1 < r < 1 or |r| < 1
1–r
|v| = √ (x2 + y2) where v = xi + yj
sin A sin B
tan A + tan B
tan (A + B) ≡ ——————–
1 tan A tan B
+
Differentiation
d
––– (ax + b)n = an(ax + b)n–1
dx
d
––– sin x = cos x
dx
d
––– cos x = –sin x
dx
n
Σx
Statistics
i=1
–x = ——
n
n
Σf x
i
=
i=1
i
i
———
,
n
Σf
n
Σ
n
Σ
fi xi2
(xi – –x)2
i=1
i=1
2
–2
S = ————— = ————
– (x)
n
n
i=1 i
Probability
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Kinematics
v = u + at
v2 = u2 + 2as
Σf
i=1 i
1
s = ut + — at2
2
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01254032/F 2014
-31.
(a)
A student has to compute the area under the graph of a function. He reasons that he can do
so by subdividing the area into an infinitely large number of rectangles. To help himself,
he investigates by finding the area under the graph of the function f(x) = x over the interval
[0,1], using the method of circumscribed rectangles as shown in Figure 1.
f(x) = x
f(x)
0
x1 x2 x3 ... xn-1 1
x
Figure 1. Circumscribed Rectangles
(i)
The student subdivides the interval [0, 1] into n equal subintervals. Calculate the
(1 mark)
width, ∆ x, of each subinterval.
(ii)
Let the points of subdivision be x0 = 0, x1, x2, x3, ..., xn–1, xn = 1 as shown in
Figure 1.
Find the values of x1, x2, x3, ..., xn–1 in terms of n.
Determine the heights h1, h2, h3, ..., hn of the circumscribed rectangles over each
of the respective n subintervals.
(2 marks)
(iii)
(1 mark)
(iv)
Determine the area A1, A2, A3, ..., An of the respective circumscribed rectangles.
(2 marks)
(v)
Show that the sum, Sn, of the areas of these circumscribed rectangles is given by
n+1
Sn = ——– .
2n
(3 marks)
(Hint: You will need to evaluate the sum of a series. State any theorem used.)
(vi)
a)
Compute S(n) for n = 10, 20, 50 and 100, giving your answers to three decimal
places.
(2 marks)
b)
What number does S(n) approach as n gets larger?
(1 mark)
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01254032/F 2014
-4(b)
The variables x and y are related by a law of the form y = axn, where a and n are integers.
The approximate values for y, corresponding to the given values of x are shown in Table 1.
Table 1
x
2
3
4
5
6
7
y
50
250
775
1875
3900
7200
(i)
Use logarithms to reduce this relation to a linear form, giving your values of lg x
and lg y correct to two decimal places where appropriate.
(2 marks)
(ii)
Using the graph paper provided and a scale of 2 cm to represent 0.1 units on the
x-axis, and 1 cm to represent 0.2 units on the y-axis, plot a suitable straight line
graph of lg y against lg x. (2 marks)
(iii)
Use your straight line graph to estimate the value of the constant a and the value
(4 marks)
of the constant n.
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
01254032/F 2014
TEST CODE
FORM TP 2014038
01254032
MAY/JUNE 2014
CARIBBEAN
E XAM I NAT I O N S
COUNCIL
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 032 – General Proficiency
ALTERNATIVE
Graph Sheet for Question 1 (b) (ii)
Candidate Number .............................................
ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET
01254032/F 2014