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\r
FORM
TP 2009092
CARIBBEAN
EXAMINATIONS COUNCIL
SECONDARY
EDUCATION CERTIFICATE
EXAMINATION
MATHEMATICS
Paper OZ
-
General
Proficiency
2
hours
40
minutes
20
MAY 2fi)9
(a.m.)
INSTRUCTIONS
TO
CANDIDATES
1.
Answer ALL
questions
in
Section
I,
and
ANY TWO in
Section
tr.
2.
Write
your
answers
in
the
booklet
provided.
3.
All working
must be shown
clearly.
4.
A list
of
formulae
is
provided
on
page
2
of
this
booklet.
Examination Materials
Electronic calculator
(non-programmable)
Geomeffy
set
Mathematical
tables
Graph
paper
(provided)
:
I
I
-
:
DO
NOT TURN THIS
PAGE TJNTIL
YOU ARE
TOLD
TO
DO
SO.
I
TEST CODE
OI234O2O
MAY/IUNE2OO9
:
Copyright @
2AO7
Caribbean
Examinations
Council@.
I
-
All
rights
reserved.
01234020tF 2009
I
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Y.,
LIST OF
FORMULAE
Volume of
a
prism
Volume of cylinder
Volume of
a right
pyramid
Circumference
Area of
a
circle
Area of trapezium
Rootsof
quadraticequations If
a*
+ bx
+ c
-b+
then x
=
Page2
V
=
Ah
where
A
is the area of
a
cross-section
and h
is the
perpendicular
length.
V
=nfhwhere
r is the radius
of the base and
h is
the
perpendicular height.
y
=
Anwhere
A is the area of
the base and h
is
the
perpendicular height.
C
=
2nr where
r is the radius of the
circle.
A
=
nf where r is the radius
of the circle.
A
=
|@
+ b) hwhere
a
and
b
arc thelengths of the
parallel
sides
and fu is
the
perpendicular distance between
the
parallel
sides.
-0,
Trigonometric ratios
Sine rule
Cosine
rule
2a
opposite
side
Epotenuf
adjacent side
hypotenuse
opposite side
afiacenGiae
sin0
=
cosO
=
tan0
=
0pposite
Area of triangle
Area of A
=
)nnwhere
D
is
the
length of the base
and
fo
is
the
perpendicular height
Area of LABC
=
loO
"in
C
AreaofMBC=W
a
+
b+
c
where s
=
bc
tirrB
=
sirrc
a
sin A
b2
-
4ac
Adjacent
0t2340248 2009
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l.
SECTION
I
Answer ALL
the
questions
in this
section.
All working
must be clearly shown.
(a)
Using
a calculator, or
otherwise,
calculate the
EXACT value of
2l
Page
3
(
3 marks)
(
3
marks)
(
lmark)
rate.
(
2
marks)
(
2 marks)
(
2 marks)
(
2
marks)
(D
2- + 1-
3,5
2
6-
5
giving your
answer
as
a cofllmon fraction
giving
your
answer
in standard
form.
The
basic
wage earned
by
a truck driver
for a 4O-hour week is
$560.00.
(D
Calculate
his hourly
rate.
For overtime work, the driver is
paid
one and
a
half times the basic hourly
(ii)
Calculate
his
overtime
wage
for 10 hours of overtime.
Factorise
completely:
(i)
2ax + 3ay
-
2bx
-
3W
(ii)
s*
-
20
(iii)
3*
+ 4x
-
t5
(ii)
(b)
(iii)
Calculate
the TOTAL wages earned by the truck driver for a 55-hour week.
(
3 marks)
Total
L2 marks
(a)
.
(b)
One
packet
of biscuits
costs
$x
and one cup of
ice
cream costs
$y.
One
packet
of biscuits and two cups of ice cream cost
$8.00,
while three
packets
of
biscuits
and one
cup
of
ice cream cost
$9.00.
(i)
Write a
pair
of
simultaneous equations in x and
y
to represent the
given
informationabove.
(
2marks)
(ii)
Solve
the equations obtained
in
(b) (i)
above to find the cost of one
packet
of biscuits
and
the cost
of one cup of
ice
cream.
(
4 marks)
Total
L2
marks
o.o2s6
)
,5)
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v
(a)
.
Page4
In a survey
of
50
students,
23 owned cellular
phones
l8 owned digital
cameras
x owned
cellular
phones
and
digital cameras
2x
owned neither.
Let
C
represent the set of
students
in the survey who owned cellular
phones,
and D the
set
of
students
who owned
digital
cameras.
(i)
Copy and complete
the
Venn
diagram below
to represent the information
obtained from
the survey.
(b)
(
2
marks)
(i1)
Write an expression
in.;r
for the TOTAL number of
students
in
the
iTfl"O ,
(iii)
Calculate the
value
of
x.
(
2
marks)
The diagram
below,
not drawn to scale, shows a rhonrbus, PgRs,with the
diagoni
PR
=
6
cm,
and
the
angle
RPQ
=
$Q".
(r)
(ii)
Using
a ruler, a
pencil,
and a
pair
of compasses, csnstruct
the
rhombus
P@iRS
accurately.
(
4marts)
Join
OS.
Measure
and
state,
in
centimetres; the
length
of
QS.
(
2 marks)
Total l.L marks
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\./
4.
(a)
The
table below
shows two
readings
taken from an aircraft's
flight record.
Time Distance
Travelled
(km)
08:55
09:O7
957
1083
For the
period
of time between
the
two readings, calculate
(i)
the
distance
travelled
in kilometres
(ii)
the
average speed
of the aircraft in km/tr.
(b)
The map shown below
is
drawn
to
a
scale of
1:50 000.
straight
line.
Calculate the actual
distance, in kilometres,
from Lto
M.
Page 5
(
lmark)
(
3 marks)
(
2 marks)
(
2
marks)
Total
Ll
marks
(1)
(ii)
(iii)
Measure and
state,
in
centimetres,
the distance on
the
map
from Lto
M
alonga
The actual distanco
between
two
points
is 4J km, Calculate
the
number of
centimetres that
shguld be used to,represent
th,is distance on,the map.
'
i
'
(3marks)
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5.
(a)
Given
thatflx)
=
2x
-
5
and
g(x)
=
*
-
31,
calculate the value
of
(i)
fr-z)
(i1)
ffi1)
(iii)
"f-I(3).
Giventhaty
=*+2x-3
(i)
Copy
and complete thetable
below.
Page
6
(
lmark)
(
2
marks)
(
2
narks)
(b)
(ii)
--....
(
2 marks)
\--
Using a
scale of
2 cm to reprecnt I
unit
o
tfue.r-uis and-l-im ts
represont
1
unit
on
the
y-axis,
draw the
graph
of
y
=
f
a
2x
-
3 fq
-
4
<
x
3
2.
(
smarks)
Total
12
marks
x 4 -3 -2
-1
0
I
2
v
5
-3
4
-3
5
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6.
The diagram below
shows triangles
A, B andD.
The line
y
=
x is
also
shown.
(a)
Describe,
FULLY,
the single
transformation
which
(i)
triangle
D
(ii)
triangle
B.
State
the
coordinates
of
the
vertices
of triangle
C,
reflection
in the
line
y
=
x.
PageT
maps
triangleA
onto
(
3 marks)
(
3 marks)
the
image
of triangle
A after
a
(
4
marks)
Total
L0 marks
(b)
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7.
Page 8
An answer
sheet is
provided
for
this
question.
The table
below shows
ttre
time
to the
nearest
minfie, that
80
students
waited
to
be served
at
a
school's canteen.
Waiting
Time
(minutes) No. of Students
Cumulative
Frequericy
1-5
6-10
11-15
t6-20
2t
-25
26-30
31-35
36-40
4
7
11
18
22
l0
5
3
4
t1
22
Copy and
complete
the table,
&e
cumulative frequeney.
-
\
(
2
marks)
On
the answer sheet
provided,
use
the
values from
your
table to
cbmplete
the
cumulative
frequency
curve.
(
4
marks)
Use
your
graph
frqm
(b)
above
to
estimate
(i)
the median
for the
data
(
2
marks)
(ii)
the
number of studeffs
who
waited for
nomore
than 29 minutes
(
2marks)
(iii)
the
probability
ttrat
a student, chosen
at random from the
Soup,
waited
for
no
more
than
17
minutes.
(
2
marks)
Total
L2
marks
(a)
(b)
(c)
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Page 9
An answer
sheet
is
provided for this
question.
The
drawings
below
show
the
first
three
diagfams
in a sequence.
Each diagram
in
the sequence
is
obtained
by drawing
a
l-unit
square
on each
side
that
forms
the
perimeter
of
the
previous
For
example,
Diagram
2
is
obtained
by
drawing
a l-unit
square
on
each of the
four
sides
of
Diagram
1.
On
the
ansYver
sheet
provided:
(a)
Draw Diagram4
in the
sequence.
(
2
marks)
(b)
Complete
the table
by
inserting
the
appropriate
values at
the rows
marked
(i),
(ii)
and
(
8
marks)
Total l0
marks
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(a)
.
Page 10
SECIION
II
Answer
TWO
questions
in this
section.
RELATIONS, FIINCTIONS AND
GRAPHS
Solve
the
pair
of
simultaneous
equations
y=4
-
2x
y=2*-3x+7.
(4marks)
Express
2*
-
3x
+
L
in
the
fonna(x
+
h)2
+
&,
where a,harrdkareteal
numbers.
(
3
marks)
(b)
(c)
(e)
Using your
answer
from
(b)
above, or otherwise,
calculate
(D
ttre
minimum
vahrc
of
?-*
-3x+
I
(ii)
the value of x
for
which the
minimum
occurs,
Sketch
the
graph
of
y
=
*
-
3x
+
l, clearly showing
the
coordinates
of
the
minimum
point
the value
of
the
y-intercept
the
values
of x
where
the
graph
cuts-the x-axis.
Sketch on
your
graph
of
y
-
Z*
-
Zx+ 1, the
line
which
values
of
x
and
y
calculated
in
(a)
above.
L
mark
)
l
mark
)
(
4
marks)
intersects the curve
at the
(
2
marks)
Total
15
marks
(
(
(d)
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v
Page
11
(a)
T1re
owner of
a shop wishes
to buy x
guitars
and
y
violins.
To
satisfy
the demands of
his customers,
fhe
nurnber
of
violins
must be less than m
equal to the
number
of
guitars.
(i)
Write
an
inequality
to represent
this informatioa.
(
lmark)
The
cost
of orp
guitar
is
$150
and
the cost of
one
violin
is
$300-
He
has
$4
500
to
spend on the
purchase
of
these
instruments.
(ii)
Write
an
inequality
to
represent
this infoilnation.
(
2
marks)
To
get
a
good
bargain,
the
owner
of
the shop
rnust
buy
at
least 5
violins.
(iii)
Wiite
an
inequality to
represent
this
information.
(
1
mark
)
(i)
Using
a
scale
of
2
cmon
the
horizontal
axis
to
represont
5
guitars,
and 2
cm
on the vertical
axis to represent
5 violins,
draw
the
graphs
of
the
lines
associated
with
the
THREE inequalities written in
(a)
(i),
(ii)
and
(iii)
above.
(
4
marks)
(ii)
Shade
the
region
on
your
graph
that satisfies
all
THREE
inequalities.
(iii)
State
the
coudinates of
the
vertices
of
the
shaded
region.
(
Lmark)
(
2
marks)
The
owner
of
the shop sells the
instruments
to make
a
profit
of
$60
on each
guitar
and
$10O
on each
violin.
(b)
(c)
(D
Express
the
TOTAL
profit
in
terms
of * and
y.
(ii)
Calculate the
maximum
profit.
(
I"
mark
)
(
3
marks)
Total
L5
marks
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11.
Page
72
GEOMETRY
AND
TRIGONOMETRY
The
diagram
beloW,
not
drawn
to
scale,
shows
a
circle,
centre
O.
The line
DCE
is
a
tangentio
the
circle.
Angle
ACE
=
48o
and
angle
OCB
=
26o
'
(a)
Calculate:
(i)
ZABC
(ii)
4o,
(iii)
zBcD
(iv)
ZBAC
(v)
toAC
(vD
IOAB
(
Lmark)
(
Lmark)
(
lmark)
(
l mark)
(
l
mark)
(
L
mark)
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V
12.
Page
14
In this
question,
use 7T
-
,
ana assume
that the
earth is a sphere
of radius
6370
km.
7'
The diagram
below,
not drawn
to scale,
shows
a sketch
of
the earth
with the
North
and
South
Poles
labelled
N
and
S
respectively.
Arcs
representing
circles of
longitude 35"8
and
15"W,
and
circles
of latitude
0"
and
60oN
are
drawn
but
not labelled.
(a)
Copy
the
sketch
and
(i)
label the
arc
which
represents:
a)
60"N
b)
35"E
(ii)
insert the
points:
a)
J
(60"N,
35"E)
(
2
marks)
b)
K(60oN,
15"W)
(2marks)
(b)
Calculate,
to the nearest
kilometre,
the
SHORTEST
distance
from
(D
J
to
the
North
Polb
measured
along the
common
circle of
longitude
(ii)
J
to K
measured
along the
cornmon
circle of latitude.
(
3
marks)
(
a
marks)
(c)
A
point 11
is located
2002
km
due
Calculate
the
latitude
of
11.
south
of K
along the
common
circle of
longitude-
(
4
marks)
Total
15
marks
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01234020tF
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tP
(a)
3.
In the diagram
below,
not drawn to scale,
midpoint of
AB,
and D is such that OD
=
ZDA.
-+ -+
OA
=
3a and
OB
=b.
Page 15
(
4marks)
(
2marks)
B is the midpoint
of
OX,
C
is
the
The vectors a and b are such that
The
points
A
origin
(0,0).
Express
(i)
(ii)
the vectors AB and
the
position
vector
VECTORS
AND MATRICES
-+
(-z\
-)
(+\
and B have
position
vectors
Oo
=
[
,.,J
*O
O,
=lr)where
O
is
the
The
point
G
lies on the
line AB such thatAG
=
LrAn.
(
/,\
in trre rorm
[r.J
-+
-)
AG
-+
oG.
(b)
(i)
Write
in
terms
of
a and b:
-+
a) AB
-+
b)
AC
-+
c)
DC
-+
d) DX
(6marks)
State TWO
geometrical
relationships between
DX and, DC.
(
2 marks)
State
ONE
geometrical
relationship between
the
points
D,
C,
andX.
(
lmark)
Total
15 marks
(ii)
(iii)
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