TEST CODE
0592
FORM TP 9288 ruNE 1ee2 CARIBBEAN EXAMINATIONS COUNCIL SECONDARY EDUCATION CERTIFICATE
EXAMINATION
MATHEMATICS Paper 2 - General Proficiency
2
hours 40 minutes
1.
Answer ALL the questions in Section I, and any TWO in Section
2.
Begin the answer for each question on a new page.
3.
FulI marks may not be awarded unless full working or explanation is
II.
shown with the answer.
4.
Mathematical tables, formulae and graph paper are provided.
5.
Slide rules and silent electronic calcutators may be used for this paper.
5.
You are advised to use the first 10 minutes of the examination time to read through this paper. \trriting may begin during this I0-minute period.
DO NOT TURN THIS PAGE I.]NTIL YOU ARE TOLD TO DO SO Copyright @ Lggzcaribbean Examinations Council.
All rights reserved. 0592tF 92
\Page2
SECTION
I
Answer ALL the questions in this section.
l.
AII working must be clearly shown.
(a)
Calculate the exact value of
s'z,
+ lJ
4-2*
.
(
4 marks)
(b)
A piece of string 64 cm long, is divided in three pieces intheratio | :2: 5. 'Calculate the length of the longest piece. ( 3 marks)
(c)
A merchant sold a pen for $5.35, thereby making a proFrt of cost to
2.
i
(a)
(l)
the cost price of ttre pen to ttre merchant
(ii)
the selling price ttre merchant should request in order to make a
sotve
157aprofit
( Smarks)
k
(
i
1
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77o on the
him. Calculate
4 marks)
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PAGE
Page 3
(b)
a
The Venn diagram above illusrarcs some of the information given below. There arrc 100 members in a foreign language club. 48 members speak Spanish. 45 members speak French. 52 members speak German. 15 members speak Spanish and French. 18 members speak Spanish and German. 21 members speak German and French. Each member speaks ATI .EAST ONE of the three languages.
.
Let the number of members who speak all three languages be x.
(r)
Write an algebraic expression to represent the number of members in the shaded region.
(ir)
Describe the region shaded.
(ii|
Write an equation to show the toal number of members in the club.
(iv)
t i
r,
Hence, determine the number of members who speak all three
languages.
( Tmarks)
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Page 4
3. (a)
Solve the simultaneous equations:
4x-4Y= 2 lx+2Y=lJ (b)
(4marks)
Factorizecompletely:
(r) L -ef (ir) Z*-U-e (c)
The vectu
?
(3marks)
translates the point
Q,3) a
the point
(4,6).
4 ranslates the point Q,7) ta the point (- 4, 3). Write4 and 4 as column vectors.
The vector
(r) (ir) 4. (a) (r)
Hence,determinethevecror
900.
Measure and state the length
ofDC
the size of angle
O)
( Smarks)
Usingrulerandcompassesonly,constructaquadrilateralABCD in which AB = AD = 6 cm, BC - 4 cm, angle BAD = 600
andangle/8C =
(ii)
ig * g>.
ADC.
( 6 marks)
The images of^L (1,1) andN (2,3) under a single ransfumation Q are
L' (-1,1) andM (-3,2) respectively.
(r) (it)
Describe geometically the transformadon Q.
Determine the equation of the line
L'M.
( 6 marks)
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Page 5
U 5' (a)
O)
- -
Copy in your answer bmklet the table below for l(x) = 2-* x < x < 3, and calculate the missing values. for the domain
-2
3
Using a scale of 2 cm to represent 1 unit on the .r-axis and I cm !o represent I unit on the /(r) -axis, draw ttre graph of /(.r) for
-2
From your graph, determine
(r) (ir)
the values
of .r for which /(x) =
the minimum value
Q
of/(.r).
(ll
marks)
Notc: You slould atnchyow graph careful$ to tlv page onwhichyou the
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did
otlur workingfor this quesrton.
NE)ff
PAGE
Page 6
6.
-
The frequency distibution of the Mathematics marks obtained by 100 candidates is given
below.
0rt2t314151-
6t-
CI.]MULATIVE
NUMBER OF CANDIDATES
FREQUENCY
10
4
4
20
7
11
30
9
40
t2
50
l8
60
l3
70
t2
MARKS
7181-
80
11
90
9
91-
100
5
(a)
Copy the table above in your answer booklet and complete the cumulative frequency column.
O)
Using a scale of
(c)
From your cumulative frequency curve, estimate
1 cm to represent 10 marks on the.r-axis and I cm to represent 5 candidates on the y-axis, draw the cumulative frequency curve for the data-
(i) (ii)
marks chosen at random scored less
the number of candidates who scored at LEAST 45
i
the probability that a candidate
t
than 45
marks.
(1l marks)
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vl
PageT
\l
7.
In this question, take
*
=
3.14
In the figure above, not drawn to scale, the chord f/K subtends angle HOK ato,thecentreof thecircle. Angle HOK = 1200 and OH =l2cm. Calculate to three significant figures
(a) O) (c) (d)
the area of the circle the area of the minor the area
seolr';rr
OHK
of uiangleHOK
the length of
tlp minor arc I/r(.
(ll marks)
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u)
8.
In the figure above, not drawn to scale, ABCDE is apentagon inscribed in a
circle of centre O. The diameter AD is produced to
F.
Angle CDF = 1360
andangleBAD =72o.
(a)
O)
Calculate, giving reasons for your answer, the magniurde of angles
(r)
CDA'
(ii)
BcD
(iii)
AED
Given that
( 6marks)
OA = 15 cm and angle EAD =
calculate the length of AE.
350,
( 4marks)
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0592892
Page 9
V
SECTION
II
Answer TWO questions in this section.
RELATIONS AND FI.]NCTIONS
9.
A manufacturer produces two types of ball-point pens: Type L and Type M. There are at least 50 of Type L and at least 25 of Type M pens. The manufacturer, however, does notproduce moretlnn 80 ofTypelormore than CI of Type M or more than 1 20 of both Type,L and Type M taken together.
(a)
r
t
pens produced and y o pens produced, represent the number of Type M write THREE in> (not y including r 0 and equalities 0) which represent the above conditions.
Using
to represent the number of Type
)
'
(b)
Usinga scaleof I cm torepresent lOpens onEACHaxis, drawthegraph of the inequalities.
Identify the region which satisfies the inequalities.
(c)
The manufacturer makes oneachTypeM pen.
(i) (ii)
a
profit of
$
1
.50 on each TypeZ pen and $ I .10
Write an expression to represent his otal profit. Use the graph !o determine the values of x and y which give a maximum profit, and hence determine the maximum profiL
(15 marks)
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Page
10.
(a)
l0
3*
+ 5x -- 6, Solve the equation giving your answer correct to two decimal places.
( 5marks)
o) u)
frl
il
E ts 2 .lH
x
2 l-i
H U
z
tr
IA H
e
TIMEINMINUTES The graph above records the journeys of two cyclists travelling benveen towns A and 8. The cyclists begin their journeys at the same time.
Calculate
(r) (ir) (iii) (iv) (v)
the distance betrreen the two towns the time the cyclist from
I
takes for the joumey
the average speed of the cyclist from B, in metres per second the distance from town
I
where the cyclists met
the average speed, in metres per second, at
which the cyclist from would need to ravel after he met the cyclist from B, in order to complete the journey in the same time as the cyclist from B.
A
(10 marks)
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\l
Page 11
TRIGONOMETRY
V
/
11.
- |=I-
( 2marks)
(a)
hovethat 2cos2g
o)
The angles of depression from the top of a tower I to R and S are 320 and 220 respectively. The points R and S and the fmt of the tower are on the same horizontal plane. The height of the tower IX is 52 m. The bearings of R and S from X ue 2700 and 2200 respectively.
(i) (il)
Draw a sketch
o
2sin20.
represent the information given above.
Hence or otherwise, calculate
-
the distanceRS to one decimal place (13
the bearing of S from R.
12. In this qumtion take the radius of the earth to be 64()0 km and
marks)
r
to be
3.142.
(a)
The coordinates of the points P and Q on the earth's surface are (260 S, 250 w) and (600 N, 250 w) respectively.
Calculate
(r) (ir)
the shortestdistance fromP to Q the circumference df the circle of latitude Od w.
( Tmarks)
O)
X
I
and are both situated on latitude 6d N. Sration X is situated at (6d N, ld E) and station is situated west of X. The disrance between X and Y along the latitude 6d N is 1 800 km. ( 8 marks) Calculate the position of the racking station y. Two tracking stations
f
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Page 12
YECTORS ANDMATRICES
13. (a)
Points u,
v
and I{z have position vecrors
respectively. of
o)
Th"
V
and
W
arecollinear, determine the value (6 marks)
*ffii* ( ; '? ) *r*r"no
(i0
(a)
U,
1,.
(r)
la.
lf
a transformation
L
(ii)
-1) under
A,B,C
and
D ate a, b,
3g -b_
Prove that CD is parallelto AB.
Determine the ratio AB :
CD.
( 8 marks)
Thepoint E (3,2),undera shear.S, where theinvariantline is ther-axis, has image E'(7, 2).
Determine
(i)
the scale factor of the shear S
(ii)
the (2 x 2) matrix which represents S
(iii)
the coordinates of the image F ' of the poin t F (2, l) ( 7 marks) under the transformation S.
END OF TEST 0s92/p 92
( 1,
Derive ttre equation of tlre line onto which the line.r + y = 0 is ( 9 marks) mapped by the transformation L
Thepositionvectorsof thepoints
(r)
r.
Determine the coordinates of the image of the point the transformation
anda+[respectively.
O)
(:}), (3) ,,. (i)
