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FORM TP 2017085
TEST CODE
OI234O2O
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MAY/JUNE 2OI7
CARIBBEAN EXAMINATIONS COUNCIL
CARIBBEAN SECONDARY EDUCATION CERTIFICATE@
EXAMINATION
MATHEMATICS
Paper 02 - General Proficiency
2 hours 40 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
1.
This paper consists of TWO sections: I and II.
2.
Section I has EIGHT questions and Section
J.
Answer
4.
Write your answers in the spaces provided in this booklet.
5.
Do NOT write in the margins.
6.
All working MUST be clearly
7.
A list of formulae is provided on page 4 of this booklet.
8.
If you need to rewrite any answer and there is not enough space to do so on the original
page, you must use the extra page(s) provided at the back of this booklet. Remember
to draw a line through your original answer.
9
If you
II
has THREE questions.
ALL questions in Section I, and any TWO questions from Section II.
shown.
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 Materials
Electronic calculator
Geometry set
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
Copyright @ 2016 Caribbean Examinations Council
All rights reserved.
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LIST OF F'ORMULAE
I
V:
Volume of a prism
Ah where
length.
Volume of cylinder
V: nfh
Volume of a right pyramid
V
:;
Circumference
C
:2nr where r is the radius of the circle.
Arc
I
where r is the radius of the base and
h is the perpendicular
& is the
perpendicular height.
Ah where A is the area of the base and ft is the perpendicular height.
S: *0
length
and,
is the area of the cross-section
x 2nr where 0 is the angle subtended by the arc, measured in
degrees.
Area of a circle
A: nf where r is the radius of the circle.
Area of a sector
A: *-
Area of
trapezium
x
A: +
nl
where 0 is the angle of the sector, measured in degrees.
@ + b) ft where
a
and
b
are the lengths
of the parallel sides and ft
is the perpendicular distance between the parallel sides.
Roots of quadratic
equations If a* + bx I
then
c
-
.-----\'
1'ril
.s
... (
'.%
ti.
..E
"L;
:0,
rE
ilii
b+'[b'-4ac
.trt
-Sr
...\
-r'
fi
_,_-r*
Trigonometric ratios
Area of triangle
E
srnu:
opposite side
.tri-
nyporenuse
..Q]
cosO:
-Iypotenu-se
tane:
onoosite side
,=
Adjacent
aoJacent sloe
On
Area of A ABC
*n"r"b
is the length of the base and, h isthe perpendicular
:
)-ab sin C
s(s-a)(s-b)(s-c)
Area of A ABC :
ffi
a+b*c
s:
b
c
sin C
Cosine rule
d:b2+c2-2bccosA
sin
f
C
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b
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a
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B
a
Sine rule
H.f,I
,'E
e
Area ofA:
|height.
where
,o
Opposite
adjacent side
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n-
:=
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SECTION
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+
Answer ALL questions in this section.
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'-'-\-,
'ii
All working must be clearly shown.
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ti.
ts.,
rst:
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hr:
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1.
s
(a)
Using a calculator, or otherwise, calculate the EXACT value of
(i)
lqJ_ _ t2)*
\,,, L-T-'5J
E_-
fu.
A--1
E.
4
15
S:-,
. .
;;:
--.::
=
---\--1
re:,
gl
.-:-
2i.
Ea-|:i'-lla:
H-,
E.
!ii.
,ai-
|!{':
tr
.\.
riJ.
*
(2 marks)
t-,
,G}:,
a-
(ii)
.o.
-
(3.1 1.ls)'
0.00s
..s:
=
:.-=
-:8.
.+*t'
.!+l
--l
---\r
.rali
-kl
tri.
-rEL-j
-t-r-.
5-.
.q.
-Io.r,
:{!.:
'\r'
$
-tr.
It--
-S+,'.
-A--
-\r-
:+
:iq.r.
(2 marks)
:e=
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(b)
A store is promoting
a new
:r.
mobile phone under two plans: Plan A and Plan B.
-
t:.
-_.lla,
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The plans are advertised as shown in the table below.
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Plan A
Deposit
Monthly instalment
$400
s600
$65
$80
t2
6
0%
5%
Number of months to repay
Tax on ALL payments
(i)
+5
Plan B
iFi
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*
-'-h
,l-
:.8
Calculate the TOTAL cost of a phone under Plan A.
t:%
..-*
.-.*i
-
rli'
.Li
E
...lii
(2 marks)
(ii)
Determine which of the two plans, A or B, is the better deal. Justify your answer.
(2 marks)
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(c)
John's monthly electricity bill is based on the number ofkWh of electricity that he consumes
for that month. He is charged 55.10 per kWh of electricity consumed. For the month of
March 2016, two meter readings are displayed in the table below.
Meter Readings (kwh)
Beginning 01 March
Ending
3l March
03011
03307
H
(i)
Calculate the TOTAL amount that John pays for electricity consumption for the
month of March 2016.
E
(2 marks)
(iD
For the next month, April2016, John pays $2351.10 for electricity consumption.
Determine his meter reading at the end ofApril 2016.
(2 marks)
Total 12 marks
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(a)
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=
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actorize the following expres sions completely.
:v
xii
ri
(i) 6f -r&xy
-lt
'lrt
'r{
-Jrit
.trrl
:k
E
.I{
.F
E(
.E
fit
(!
,.ft
(2 marks)
(iD
4m2
-
7
.€5
."
(1 mark)
"Po
'-H
.:iE
FJ
(iii)
2P
rts
-3t -2
::+ii
fr}
.\Er
i(
-E-*
(2 marks)
(b)
Write as a single fraction and simpliff
5p+2 _ 3p-l
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(c)
A formula is given as d:
(D
li;)
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\i
Determine the value of dwhen
figures.
h:29.
Give your answer correct to 3 significant
(2 marks)
(ii)
Make ft the subject of the formula.
f+r
.+f-:
-.tlt--l
.*d.
-El
.E:
-t+.
.S:.
e.l
.(}:.
.+
r=
(2 marks)
Total
ll
marks
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3.
rue"
(a)
{
The universal set Uis defined as follows:
:=
:{Ei
i.r(
U: {x:x e lI, 2<x<12).
The sets
M
and l? are subsets
a
.-E
of U such that
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M: {oddnumbers}
R: {square numbers}
(i)
:-t\i
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.Ei
List the members of the subset M.
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.f
'-.S-
:-lrf
(1 mark)
(ii)
List the members of the subset R.
(1 mark)
(iiD
Draw a Venn diagram that represents the relationship among the defined subsets
of U.
:-I*,
.::G:
:E.
:.(!.
,:s.
(4 marks)
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(b)
I
Using a ruler, a pencil and a pair of compasses,
(D
construct accurately, the square ABCD, with sides 6 cm
(3 marks)
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:rs-::
:i+';
:t-ti-:l
s(r
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-*-
(ii)
construct, as an extension of your drawing in (b) (i), the trapezium DABQ so that
Z
ABQ:120"'
(2 marks)
[Note: Credit will be given for clearly drawn construction lines.]
(iiD
Hence, measure and state the length of BQ.
jE_-1
,.<t.
(1 mark)
-:ll[
rEt
_.Ei
Totall2 marks
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:l*
t\
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(a)
The
tunction/is defined asl(x)
(D
Find the value
:
{
!, -Z
ofl(3) + f (-3).
(2 marks)
(iD
Calculate the value of x for
which/(x)
:
5
!$
si
l-xri
ft.
A
,N.
.t--+.
-rG}
-+
-_fr-:
(2 marks)
(iiD
--v
i,s,
Determine the inverse function/-' (x).
:z
.tt
.-EI
.lii.
:'l+
E.
-ifl
i-ti.
(2 marks)
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The graph below shows two straight lines, 1., and (.r. Line (.,intercepts they-axis at (0,
Line 1., intercepts the x and y axes at (12,0) and (0, 6) respectively.
(b)
l)
v
z
:lsi-
/u,i
-l
:I+l
-H-.1
/ ,i,
t2
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a\l
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A:'-'l
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tti:
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8
6
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-+r.
/
\r
!(
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0
..G
.,
4
6
8
10
t2
t4
x
=
(i)
Calculate the gradient of the lines
/,
and (.,
fi.l
.!(.-
(2 marks)
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(ii)
Determine the equation of the line
ifl
1.,
(2 marks)
(iii)
.-E
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What is the relationship between [,, and (.r? Give a reason for your answer.
(2 marks)
Total12 marks
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ilit
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.-.t{.
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5.
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race
(a)
PTRS, not
drawn to scale, is a quadrilateral. Q is a point on PZ such that QT
:
QR
:
QP
Angle QRT:76".
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^s
I+
tr+
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s
:r-. .l
l
E'
i.. .l
-Er
.t
*
-A
l\t
POT
:=,
Determine, giving a reason for each step of your answer, the measure
(i)
of
angle RQT
(2 marks)
(ii)
angle PRT
r.!q.
(2 marks)
{rt.
.!(r
(iii)
.]q.
angle,SPT, given that angle
SRZ: l45o
and angle PSR
:
100"
(2 marks)
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(b)
{
The diagrambelow shows triangleABC anditsimage,A'B C', undera single transformation.
v
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-\
a
FI
'Ir(
.lri
\
c
\
I
&
*
\
B
t.
x
i
I
6
-)
1
I
I
-)
-4
(i)
E
..\
Describe completely the transformation that maps A ABC to L A'B'C'
(3 marks)
(iD
The translation vector
a: I i.l
[-),]
maps A
A'B'C'to
A,
AttBttctt
(2 marks)
On the diagram above, draw the L A"B"C".
Total
ll
marks
.R
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(a)
In this problem take n tobe!
7
I
.
The diagram below, not drawn to scale, shows a field in the shape of a sector of a circle,
with centre O and diameter 28 m. Angle POZ is 90".
Iq
S+
i\r
P
Eg
\
;-:
-Fr
E
o
Calculate
(i)
the area of the field
(2 marks)
(ii)
the perimeter of the fleld.
(3 marks)
..e
r\.
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(b)
{
The diagram below, not drawn to scale, shows a triangular prism ABCDEF. The crosssection is the right-angled triangle, ABC, where AB :6 cm and BC : l0 cm.
F
rE
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t)
rFr
s
.,.t=-
ri*
j
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DI
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10 cm
A
6cm
.'fr
--A-
E
B
'lq
Calculate
(D
.s.
.:S.
::%
the area of the triangle ABC
'I]
----txi
.t.i
,tr
lrt
f+
.l=a.
f;
..E'
:jt
.t
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.e.
..s.
.'.]{.'
r{.
,'.*-
..*
-:Vt
j-++
.!:i.
..E
f\I
.\.'
(2 marks)
F+
i:iX
:.&:,
--rA--
';E..
::'ql
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(ii)
:V:
the length of the prism,
{
if the volume is 540 cm3
I$
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::+.
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II}
H
li;
,s
l-.
.€
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(2 marks)
.=.
=..
.+.
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rg.
(iii)
the surf,ace area of the prism.
''84,
.{ii:
-.!t.i--:
-r{;r--'.
hi.
tr
]i+:frr
-+r--
\I-
-i(.
-NiN
'Fn.
.E
I\t
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,'-at'-'-
(2 marks)
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.tit
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marks
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.\
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{
The table below shows the speeds, to the nearest kmh-l, of 90 vehicles that pass a checkpoint.
ll
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:---.
if- -
I-;
t.-\
rq-
:
Speed (in
kmhi)
Frequency
Cumulative Frequency
-
i:.-
fil
{4
i.ca,
,*.'-|
ilri
0-19
5
5
20-39
ll
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40-59
26
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l:U
i|$
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(a)
(ii)
.{l
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5l
80-99
9
L<5.
100-l l9
2
fij
For the class interval20-39,
(i)
i-.til
60-79
as
The upper class limit is
l'.Q.
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written in the table above, complete the following sentences.
l'
i-...
i..t..-
...............
fi ;;;)
t_\
l.'{*I.
i.'f(
'..,=
:Ih
The class width is
(1 mark)
(iii)
-
'1-._
i rrril
:it!ii
-.Sl.
..8.
Sixteen vehicles passed a checkpoint at no more than
....kmh r.
(1 mark)
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(b)
(c)
Complete the table shown above by inserting the missing values for the cumulative
(2 marks)
frequency column.
t on the x-axis,
On the grid provided on page 21 , using a scale of 2 cm to represent 20 kmh
and 2 cm to represent l0 vehicles on the y-axis, draw the cumulative frequency curve to
(4 marks)
represent the information in the table.
:.*
i.tr=..
lA'.
l..sa
l<.
i.,Cr
i'Q
t
i-'--....
lri-t--
(d)
(i)
(ii)
l
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On your graph, draw reference lines to estimate the speed at which no more than
(1 mark)
50% of the vehicles drove as they passed the checkpoint.
t.....
t
- -'
What is the estimated speed?
;.
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f...-
r.E.
!.ii
'..1d
(1 mark)
Total 11 marks
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: lra
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;
The first four figures in a sequence are shown below. Figure 1 is a single black dot, while each
of the others consist of black dots arranged in an equilateral manner.
o
oo
ooo
o
oo
o
Figure
O
1
Figure 2
Figure 3
oo
ooo
oooo
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Figure 4
-l-\
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(a)
.t5:.
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Draw Figure 5 of the sequence in the space below
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tsi.
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--!-f_--
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&.
r+---
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(b)
(2 marks)
:c}I
(1 mark)
rl
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How many dots would be in Figure 6?
.-\Ji.
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;
The table below refers to the figures and the number of dots in each figure. Study the patterns
shown.
Figure, n
Number of Dots, d, in terms of n
1
I
Number of Dots Used, d
xlx(1+1)
1
2
]-"Zx(2+1)
2
3
2
ar:x(l+1)
J
6
2
11
n
(c)
complete the row which corresponds to Figure 11 in the table above.
(d)
Determine which figure in the sequence has 210 dots'
(2 marks)
(2 marks)
(e)
Figure z
Write a simplified algebraic expression for the number of dots, d, in the
(1 mark)
(0showthatthereisnodiagramthathasexactly1000dots.
(2 marks)
Total L0 marks
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SECTION
II
Answer TWO questions in this section.
ALGEBRA AND RELATIONS, F'UNCTIONS AND GRAPHS
9.
(a)
.k
.-\i..
The velocity-time graph below shows the motion of a cyclist over a period of 40 seconds
Velocity
(
ms')
10
05
(i)
15 20 25 30
Calculate the gradient
a)
3s
40
(s)
of
OA
(l
b)
mark)
AB
.'-ta:.'
----kl
----lr,i-'
++
'-.\'.
''E
.:\:l
(1 mark)
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(ii)
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E.
Complete the following statements
jSi+i
The cyclist started from rest, where his velocity was
ia.l
'-t!+.
{
ms-r, and steadily
li+-.'
.Li-:
k.\il
ms I each second during the first 25 seconds.
increased his velocity by
lr+
Li
.!iij.
Ff.-:
During the next 15 seconds, his velocity remained constant, that is, his acceleration
-r+
It-.
.€.
.ej
was ....................
ms-2.
(3 marks)
A-il
.-+.
(iii)
Determine the average speed of the cyclist over the 4O-second period'
.tr=.
(3 marks)
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012340201F 20t7
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0123402025
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(b)
Consider the following pair of simultaneous equations
x2
+
2xy:5
x+Y:3
(i)
WITHOUT solving, show that (1, 2) is a solution for the pair of simultaneous
equations.
..€i
=:
(2 marks)
(ii)
Solve the pair of simultaneous equations above to determine the other solution.
(5 marks)
Total 15 marks
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01234020tr. 2017
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MEASUREMENT, GEOMETRY AND TRIGONOMETRY
10.
(a)
P, Q, R and S are four points on the circumference of the circle shown below
Angle OR,S: 58'.
R
s
P
Using the geometrical properties of a circle to give reasons for each step of your answer,
determine the measure of
(D
tsPQ
ts-'l
5+..
lrttii:.
.t-.-.
s{'
r+.L='.
.e}'.
t.
.o.
+
(2 marks)
(iD
'*.
|It.
zoQS
.!(i
.=.
:t-,:
(3 marks)
-raxl
a
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(b)
AshipleavesPortAandsails52kmonabearing of 044" toPortB. Theshipthenchanges
course to sail to Port C,72km away, on a bearing of 105'.
(D
On the diagram below, not drawn to scale, label the known distances travelled and
the known angles.
N
B
C
A
(2 marks)
(ii)
Determine the measure of ZABC
j\=
(2 marks)
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0t234020[. 2017
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tilt
til
0123402028
ilil
flil
til
J
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-
aaaal
=-
r
Yaee:r
(iii)
Calculate, to the nearest km, the distance
,
lC.
J*['.,
lA-.i
!+
ri.,
lri:-i
\iil
1*1.,
Ha-j
,>.r:
.Ei.-
(3 marks)
(i")
Show that the bearing of
I
from C, to the nearest degree, is 260".
+at-!i.:
:ti---.
,-H-El
:l\r
rf-:r
..Q},.
.8.
(3 marks)
Total 15 marks
.Er,
.Ea-..
..t*,
i.ol-l
i.k-:':-q:l'.
:ift,
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0t234020/F 2017
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VECTORS AND MATRICES
:-=
.=
,=
.,$,
$
i-fi
l-rtt
11. (a)
Matrices
A andB
A_
(D
32
54
i.tj
Ir+
are such that
andB:
1_S;
'.-rr+
iti
0l
4
.t-k
::tiii
J -r.J
:"lEt
.\
Show by multiplying
A
.-Ec
',-_\--'
and B, that AB + BA.
r-j:-'---El
r6_
t
..4
:--*J-
:q
t...j
=
:::
-r-_-\-
.-{rt:
..S{,
-it!F-
..-'i
-
j-{iai
(2 marks)
(iD
FindA-1, the inverse
---!fai---_a{-_
-hi
.-t-'
i-Iii
-iraY'
-I*ti:tti+
.:1\:
ofl.
.:E--R
--E-
.:tt+,
rrG.
.E-.
.ls
,$l
=
ii=
(2 marks)
=
-.-.r*a-.
(iii)
:.i(
--g--
Write down the 2 x 2 matrix representing the matrix prodnct AA-l
i:tji-j
.-it-.
----FFr--
---E--i:-_':za:
..lii
r+j.
j-{{-i
--tii+-:
-rdi.
_-!a-:
(1 mark)
..Er:
--A---\t-
.:-+
,tG:
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0t234020/F 2017
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0123402030
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J
:.+:
:=
taaaata-
:i=
:.=
:jj=
r
raeel
(b)
:*ii.
(i)
Write the following pair of simultaneous equations as a matrix equation.
I+-,
E.
\+..
3x+2y:l
__!_
5x+4y:5
azr-.
r-1-:
$
tii.,
.l\i-:
=.:
F+:
!\:
S..
Fi.
O.
E:
_:q.
lrl,
€::
'e,i:
-'
(1 mark)
fi--
lE
S+-
(iD
E
Write the solution of your matrix equation in (b) (i) as a product of two matrices.
Iar
ti+j
\r
ft.
Nr-:
_E-
,ti,Q}::
e.
.G.
.Qll
:i,a'i,
ii{f-
.trt.
.!(-r
._*-.
t-i-:l
'hri:
+i
(2 marks)
E,
r\i
iti.:
ti.:
r*ij
E+
._s.:.
--lF-l
-Qi.
.R.:
€r::-_
.1{r'
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0t234020tF 2017
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ilil
0123402031
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(c)
The position vectors of the points P and Q relative to an origin , O, are
i jji-
.\E
:-1*J
-)
[+-l
OP
-+
and OQ
Ir.J
-
lsl
*
:__H
respectively.
:_-_-t
..i-
LoJ
The diagram below shows that
:5r
PR:3
OP and QS
:3
Fi
k
OQ.
.-.ha
.Ia
R
i--t\4-.-
-ti
.E
.s
A.
..Fi
.-.€i
E
'-.9
..4
OP
o
(i)
s
oQa
Express
i, tn" for- ['l
lv
vector
_)
1'E
-+
a
i:lii
.-f*
i:tri
..r=i
.-8.
OS
---E
,G
..S
.E
..ci
.$
(1 mark)
a
-)
PQ
..:s:
..I{i
..4+
--!i
::t
:
itr3
'.ts
.r-t
.*ii
.fril
i.-Jiii
.:!4
i:=-
(2 marks)
.:-€i
__r\'
..'-k
GO ON TO THE NEXT PAGE
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0t234020tr. 2017
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f------
f-----,'.
f- ---
t---[-----
-
-:
I ri-:
t-
rue"
{
-+
a
R^9.
(2 marks)
=
(ii)
:.:lr
State TWO geometrical relationships between PQ and RS
:.:ir
r+
rEi
s
'i/+
E
\i'
rr5-
E;
E
F+
!$
ha
\
E
*
h*
.G
>,
(2 marks)
:.O
e
Total 15 marks
.=
ajjjj
tli
E
.tt
=
-\
-ft)
END OF TEST
.Ei
T(
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IF'YOU FINISH BEFORE TIME IS CALLED, CHECI( YOUR WORI( ON THIS TEST.
t
.s
:A
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