TEST CODE 01254020
FORM TP 20220442
CARIBBEAN
MAY/JUNE 2022
EXAMINATIONS
COUNCIL
CARIBBEAN SECONDARY EDUCATION CERTIFICATE®
EXAMINATION
ADDITIONAL MATHEMATICS
Paper 02General Proficiency
2 hours 40 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This paper consists of SIX
2.
Write your answers in the spaces provided in this booklet.
3.
Do NOT write in the
4.
A list of formulae is provided on page 4 of this booklet.
5
If you need to rewrite any answer and there is not enough space to do so on the
questions.
Answer ALL
questions.
margins.
original page, you must use the extra page(s) provided at the back of this booklet.
Remember to draw
6.
a
line
through your original answer
If you use the extra page(s) you MUST write the question number clearly in
the box provided at the top of the extra page(s) and, where relevant, include
the question part beside the answer.
Required Examination Materias
Electronic calculator (non-programmable)
Geometry set
DO NOT TURN THIS PAGE UNTIL YOU ARE
TOLD TO DO SO.
Copyright© 2021 Caribbean Examinations Council
All rights reserved.
01254020/MJ/CSEC 2022
4
LIST OF FORMULAE
Arithmetic Series
T , a + n-1)d
Geometric Series
S,-[2a+(n-1nd]
S, " - 1
1, a r n - I
S
r-1
Circle
+2+2f* +2gy +c=0
Trigonometry
tan
Calculus
V
(At B)
v=
+y) where v =xi +yi
cos
A
cos Bsin A sin B
(Ab)1E tan Atan B
2=u+2as
u+ at
d
=
dx
=
tan At tan B
dr
d
Statistics
cos l a l b
sin (At B) = sin A cosB t cos A sin B
cos
Kinematics
x +f +(+g?=2
ab
V
Vectors
< r < l or d<1
an(ax + b)"-
COsax =-aSin ax
dx
dt
ut+at
sin ax = a cosax
dx
s=vdt
v=adt
S'=i
x=
--F)*
i=l
Probability
01254020/MJ/CSEC 2022
L
P(A
U
B)
=
P(A)
+
P(B)
-P(A
B)
GO ON TO THE NEXT PAGE
-5
SECTION I
ALGEBRA, SEQUENCES AND SERIES
ALL working must be clearly shown.
1.
(a)
Consider the quadratic equation qx? -
(4p)x + pq2 =
0, where p
and q are both
positive integers.
Express the sum AND product of the roots of the equation in terms ofp and q.
(3 marks)
i)
Determine the value for q such that the sum of the roots is equal to the product of
the roots.
(2 marks)
01254020/MJ/CSEC 2022
GO ON TO THE NEXT PAGE
-6
(iii)
If the
sum
of the roots
determine a value for p.
of the equation
is
20,
use
your answer from
(a) (ii)to
(1 mark)
Gv)
Hence, express the given quadratic equation
in terms of its numerical coeficients.
(1 mark)
b)
A series is given by
25-5+1-t5
Show that the series is
geometric.
(2 marks)
(ii)
Calculate the
sum
to
infinity of the series, giving the answer to 2 decimal places.
(2 marks)
GO ON TO THE NEXT PAGE
01254020/MJ/CSEC 2022
***
-7
()
A recent university graduate was offered a starting salary of $720 000 for the first year,
with increases of $5000 at the start of every year thereafter. Determine the number of
years (to the nearest whole number) that it would take for her annual salary to be 20%
greater than her salary in the first year.
(4 marks)
Total 15 marks
GO ON TO THE NEXT PAGE
01254020/MJ/CSEC 2022
-8
2.
(a)
When the polynomial expression 2x3
3x- cx + d is divided by (x + 1) and
-2), the same remainder of 64 is obatined.
Determine the value ofc and d.
(4 marks)
b)
Show that the expression
v25
45
is the same as -
3
2 marks)
01254020/MJ/CSEC 2022
GO ON TO THE NEXT PAGE
9
)
()
Given g ) = 6r + 12x
18, express gtr) in the form atx + H? + k.
3 marks)
(ii)
Using the expression derived in (c) (i), determine the roots of g().
(3 marks)
GO ON TO THE NEXT PAGE
01254020/MJ/CSEC 2022
10
(ii)
Hence, sketch
the
graph
of g{x)
on
the
following grid.
**
***
****7"
(**
******
::
**
.
..
....
...
**
****!***"
**********T********
***"{*
**.
*"**"**
""**"**":**"|*****"
:
*
***
.
.
**********
***
**
*****
**
*******
******
****|
********
**!**************
..
***********1**********
.
...
.
********?*******
*****
**i***
***********!*"
*****:*****:*************
*****2*************1*****]***
:
:
:
***
*******
***
******
***
*******1
(3 marks)
Total 15 marks
01254020/MJ/CSEC 2022
GO ON TO THE NEXT PAGE
- 11-
SECTION II
coORDINATE GEOMETRY, VECTORS AND TRIGONOMETRYY
ALL working must be clearly shown.
3
(a)
The equation of a circle is r2 +y2+ 4x-8y+10
0.
Determine the coordinates of its centre AND the length of the radius of the circle.
(3marks)
(i)
Determine the equation ofthe tangent to the circle at the point P(-5, 5).
(3 marks)
GO ON TO THE NEXT PAGE
01254020/MJ/CSEC 2022
(b)
(c)
are
- 12-
such that OX
OX and OY are perpendicular.
The vectors OX and OY
=4i+j
and
OY=i-4j.
Show that the vectors
(5 marks)
of
AB which subtends an angle
The diagram below, not drawn to scale, shows a chord
0.5° (0.5 radians) at the centre, 0, ofa circle of radius 10 cm. Given that the area oftriangle
10 em
Q.5
(3 marks)
GO ON TO THE NEXT PAGE
AOB r 2 sin 0, calculate the area of the shaded region.
01254020/MJ/CSEC 2022
PONOT
WRITENTHIS AREA
G0ANOTMRITNIRAAKSARE
T
A
4.
(a)
(b)
Given that f«)
=
.14 SECTION II
INTRODUCTORY CALCULUS
ALL working must be clearly shown.
x{5-x}, determine f"().
3 marks)
GO ON TO THE NEXT PAGE
(3 marks)
3 marks)
Differentiate EACH of the following expressions with respect to x, simplifying your answer
(1+2x) (x +2)
2 sin 3x+ cos x
where possible.
)
(ii)
01254020/MJ/CSEC 2022
tiiiiti:
DO NOT WRITE IN THIS
AREA
RRRRRRRRRRRRRRRRRRRRRRRRIRRRRRRRRRRRRR
DOKRRRRKIARSRRRKRARIRRRRRRRRRiiR
NOTRIN 7HIS AR
R
DO NO
RITE IN THK AR
5.
(b)
(a)
Determine (4 cos -6 sin 0) de.
Evaluate 3 - x d .
01254020/MJ/CSEC 2022
- 16-
(2 marks)
(3 marks)
GO ON TO THE NEXT PAGE
DO NOT WRITE IN THIS AREA
Do NOT WRITEN THIS AREA
DoNOT WRITEIN
THIS
ARE
ii
i
ii
RRERRRRERR
N
3
B
6.
(a)
- 18 -
SECTION Iv
PROBABILITY AND STATISTICS
ALL working must be clearly shown.
Ata school canteen, 80% ofthe students (S) purchase chips (C) and 55% purchase chicken
s
Complete the following Venn diagram to illustrate this information.
(K). Of the students who purchase chicken, 11% do not purchase chips.
)
K
(4 marks)
GO ON TO THE NEXT PAGE
(1 mark)
Determine the probability that a student chosen at random purchases ONLY chicken
or ONLY chips.
01254020/MJ/CSEC 2022
- 19 -
(b)
The
PCA)
i)
probabilities of the occurrence of two events, A and B, are given by
3
4. P(B) = and P(A UB)= 10
Determine
P(AOB)
(3 marks)
(ii)
PAB).
(3 marks)
(c)
State, with a reason, whether Events A and B are independent.
*******°.**.
***°
*************
(1 mark)
GO ON TO THE NEXT PAGGE
01254020/MJ/CSEC 2022
20
(d)
The
)
following table shows the
marks obtained
by 27 students
in
a
Mathematics
30
35
39
42
45
45
52
59
61
61
65
69
70
71
75
77
79
81
83
83
85
87
89
90
Construct
a
stem-and-leaf diagram
to
test.
909598
display this data.
(4 marks)
i)
State ONE
advantage of using a stem-and-leaf diagram to display
the data.
.
*o*****.***
*****
(1 mark)
01254020/MJ/CSEC 2022
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- 21
(ii)
State the range of values of the marks obtained
by
the students.
******.********
*****°°*******
(1 mark)
(iv)
The data
displayed in the following box-and-whisker plot.
are
H
30
52
71
85
98
State TWO distinct observations about the data as seen in the box-and-whisker
plot.
*°°*°**.
***°*°°**
*************
2 marks)
Total 20 marks
END OF TEST
IF YOU FINISH BEFORE TIME IS
CALLED, CHECK YOUR WORK
ON THIS TEST.
01254020/MJ/CSEC 2022