TEST CODE
OOO592
FORM TP 9373 JUNE 1993 CARIBBEAN EXAMINATIONS COUNCIL SECONDARY EDUCATION CERTIFICATE EXAMINATION MATHEMATICS aper 02 - General Proficiency 2 hours 40 minutes
1.
Answer ALL questions in Section I, and any TWO in Section
2.
Begin the answer for each question on a new page.
3.
Full marks may not be awarded unless full working or explanation is shown with the answer.
4.
Mathematical tables, formulae and graph paper are provided.
5.
Slide rules and silent electronic calculators may be used for this paper.
6.
You are advised to use the first 10 minutes of the examination time to read through this paper. Writing may begin during this l0-minute
II.
period.
.t a\
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Copyright 0005921F 93
@.
tqg: Caribbean Examinarions Council. All rights reserved.
Page2
SECTION
I
Answer ALL the questions in this section. 1.
All working must be clearly shown.
(a)
Calculate the exact value of
$+ + (b)
Evaluate
t? + r+- /
to', givingyouranswerin {o.ooo+-*
( 4 marks) standardform.
( 3 marks)
(c)
A tourist exchanged US$200.00 for Jamaican currency at the rate of US$I.00 = J$18.81. She had o pay a govemment tax ofl?o ofthe amount exchanged. Calculate in Jamaican curency
(i) (ii)
2.
(a)
the tax paid the amount the tourist received.
( 4 marks)
There are 68 studens in Form V. 15 students study Mathematics only. 12 students study Physics only. 8 students study Physics and Chemistry only. 2 students study Physics and Mathematics only. 3 students study Mathematics, Physics and Chemisry. 4 students do not study any of these subjects.
(i)
Draw a carefully labelled Venn diagram to represent ( 2 marks) the information given above.
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Page 3
(ii)
Determine the number of students who surdy Physics. ( lmark )
(iii)
Given that
.r
students study Mathematics and Che'm-
istry only, and twice as many study Chemistry only, write an algebraic equation to represent *re information given and hence, calculate the value
of x.
(
3 marks)
o)
Given that a and b are unit vectors as shown in the diagram above,
(i)
-)
and
(ii)
-+
write the position vectors OP and OQ in terms of a
I -+
determine the length of. OP.
( Smarks)
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93
Page4 3.
(a) O) (c)
Solve
2P5^
( 4 marks)
T+ F=5'
y varies inversely as x2 and thaty =3when 2,calculate the value of y when r = 3. ( 3 marks)
Given that
x=
The cost of four chairs and a small table is $684' The cost of six chairs and a large table is $1196. The cost of the large table is TWICE the cost of the small table.
Given that a is the cost, in dollars, of in dollars, of a small table,
(r)
write
a
a
chairand D is the cost,
pair of simultaneous equations to represent the
information given
(ii) 4.
(a)
(l)
calculate the costof the large table.
5 marks)
Using ruler and compasses only, constructa triangle ABC wirh AB = 9.5cm, AC = 7.5cm and angle
BAC=
(ii)
(
6Ol.
Locatethepoint D such thu DB isperpendicularto AB and CD isparalleltoAB. Measure and state the length of BD in centimetres.
( 6 marks)
o)
Triangle PQR with P(1,1), 8(11,2), and R(1,9)ismapped onto triangle P'Q'R' wittl P' (3,3), 8'(8,3.5), and R'(3,7).
(i)
Using a scale of 2 cm to I unit on both axes, draw on graph paper triangles PQR and P'Q'R'.
Note:
Draw the .r-axis on the longgr side of the graph paper.
(ii)
Hence, describe fully the transformation which maps
triangle PQR ontoriangle
P'Q'R'.
( 6marks)
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Page 5' 5.
(a)
An aeroplane travels a distance of 3700 km in
O)
7| hours. Cal( 2 marks)
culate its average speed.
An aeroplane left Kingston at 21:00 hrs local time and ravelled for 9lhours arriving at l.os Angeles airport at 01:30 hrs local
time on the following day. Calculate ttre difference ifi time ( 4marks) between Kingston and Los Angeles. (c)
(i) (ii)
The angle of elevation from a point P on the ground to the top of a tower 20 m tall, is 650. Calculate the distance of P from the foot of the tower.
o the foot of the tower. Calculate the angle of depression of Y from the ( 6 marks) top of the tower. A point Y is 3.2m nearer than P
'
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Page 6
6.
A survey was taken among 100 customers to find out the time
spent
waiting in lines for service at ttrc bank. The following table shows the result of the survey..
Waiting time (in minutes)
1-
5
5
6-10
t2 l5
tl-15
t6-20 2r-25 26-30
31*35 (a)
No. of customers
19
2t 25 3
Construct a cumulative frequency able
above.
o
represent the data
( 2 marks)
o)
Using a scale of 2 cm to 5 minutes on the horizontal axis and 2 cm to l0 customers on the vertical axis, draw a cumulative frequency curve to illusfae the information. ( 4 marks)
(c)
Estimate the proportion of customers who waited more than 16 minutes.
(d)
( 2 marks)
Calculate the probability that a customer chosen at random would have waited for more than27 minutes. ( 3 marks)
I \
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93
pageT Copy and complete the able below for the furrction
7.
)=5+x-?-f. x
-3
v
-16
a
-l
0
!' 2
I 4
2
2
3
-l
-10 ( 2marks)
Using a scale of 2 cm to I unit on the.r-axis and on they-aris, draw the graph of
)=5 +x-*tq-3<.rs3. (c)
I
cm to
I unit
( 4marks)
Using your graph or otherwise, determine the range of values of
.r
forwhich
x*
'tx-'>
-3.
( Smarks)
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Page 8
Notes for this question:
8.
T^ken =
+
Volume of cone
=
lnln
The diagram above, not dravyn to scale, shows the MAJOR sector, AO B, of acircle of radius 6 cm. It represents the net of a cone.
(a) O)
Show by calculation that the circumference of the base of the ( 2 marks) cone is 22cm,
Calculate
(r) (ii) (c)
the radius of the base of the cone
the height of the cone, grving your ilnswer correct !o ( 5 marks) one decimal place.
Calculate to two significant figures the volume of liqui( in ( 3 marks) litres, that the cone holds when filled.
i
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Page 9
SECTION
II
Answer TWO questions in this section.
RELATIONS AND FI.JNCTIONS 9.
(a)
Giventhat/(x) = x + 3 -'zx-',
(i)
derive/(-r) inttreform,f(x) = c + a(x+ b)2,where a,
(ii)
(iii) (b)
b
and
c
are constants
determine the value of .r at which the maximum value ot f(x) occurs state the maximum value
A composite function
of /("r).
( 6 marks)
/< is defined as
k(x)=(2x-l)'. (i) (ii) 10.
Express t(x) as gf(;r),where/(x) and g(.r) aretwo simpler functions. Show that
,t-t1x; = "f-tg-'(r).
(
9 marks)
A boy has $280. He wants to buy r records at $35 each and y tapes at $40 each. He must buy more than one ape but not more than four tapes. He must also buy at least three records.
(a)
Write THREE inequalities in .r and information.
(b)
(t) (ii)
(c)
y to represent
the above
( 4marks)
Usingascaleof2cmtorepresent I unitonEACHaxis, draw the graphs of the inequalities.
SHADE ttre region that satisfies the THREE in( Tmarks)
equalities.
Determine the maximum amount spentand statethe (x,y) value ( 4marks) that gives this amount.
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Page 10
TRIGONOMETRY AND GEOMETRY
n.
In the figure above, not drawn to scale, the quadrilateral PQRS isinscribed in the circle, centrc O. PR passes through O. The
tangents TP and,IS are drawn to the circle from
RSY
=
T.
Angle
200.
(i) (ii) (ii| (iv)
Calculate, giving reasons,
angle PQR
angle SPR
anglePST angle PTS.
( 7 marks)
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Page
ll
(b)
In niangle ABC above, not drawn to scale, AB AC = 7 cm and angle .BAC is o. Given that sin2 c[,
(i) (ii) (iii)
=
A point
K
K
to go ta
theglactrvalueofcos20,
thevalueofo, if90o < a < the length
1800
of BC, correct to one decimal
place.
is on a bearing of 0250
from another point
M.
8
Using a scale of
I
marks)
The boat leaves
M. The engine speed of the boat is 55 kmh-l. A wind
ing from the East at a speed of 25 kmtr
(a)
4 cm,
0.64, determine
(
t2.
=
is
blow-
1.
cm to represent 5 kmh-I, find by accurate
drawing
(D (ii) (b)
the course of the boat (12 marks)
the resultant speed of the boat.
If K is 12 km away from M,
and the boat leaves
K
at 07:00 hrs,
cdlculate the time at which the boat reaches M.
( 3 marks)
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93
Page12 VECTORS AND MATRICES
13.
(a)
Using a scale,of 2 cm to represent LPOR withP (lA), Q (3,1) and
o)
L
pQRis ransformed by the
(r) (ir)
matrix
Draw riangle
P/'R'
Determine the?
x
2 marix
LPQR.
t4.
(a)
(l ?)
Determine the coordinates of the image, L P'Q'R'.
fully. (c)
I unit on EACH axis, draw R(4,2). ( 3marks)
and describe the transformation ( 7 marks)
thatwill transform LP'Q'R'onto ( 5 marks)
P isthepoint(6,4)and Q isthepoint(8,2). M and N arethe mid points of OP and OQ respectively, where O is the origin.
(i) (ir) (iiD o)
l
Determine the vector PQ.
j
Determine trre vector
u?.
State ttre relationship between
MN
and PQ.
( E marks)
I
Given the equations
I
x - Y =-5 3x + 2y -*5, (l) (i0 (iii)
i
write ttre equations in matrix form
i
determine the inverse of the matrix hence, solve the
equations.
( 7 marks)
END OF TEST 0005921F 93
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