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. ( TEST coos 01234020FORM TP 2004101 MA Y/JUNE 2004

CARIBBEAN EXAMINATIONS COUNCILSECONDARY EDUCATION CERTIFICATE

EXAMINATIONMATHEMATICS

Paper 02 - General Proficiency

2 hours 40 minutes

INSTRUCTIONS TO CANDIDATES

1. Answer ALL questions in Section I, and ANY TWO in Section 11.

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)Geometry setMathematical tables (provided)Graph paper (provided)

DO NOT TURN TIDS PAGE UNTIL YOU ARE TOLD TO DO SO

Copyright © 2003 Caribbean Examinations Council.All rights reserved.

o12340201F 2004

Trigonometric ratios sin e opposi te side= hypotenuse

cos e adjacent side= hypotenuse

tan e opposite side= adjacent side

LIST OF FORMULAE

Volume of a prism

Volume of a right pyramid

Circumference

Area of a circle

Area of trapezium

" .Roots of quadratic equations

Area of triangle

Sine rule

Cosine rule

o1234020fF 2004

·.iPage 2

v = Ah where A is the area of a cross-section and h is the perpendicularlength.

v = 1 Ah where A is the area of the base and h is the perpendicular height.3

C = 2nr where r is the radius of the circle.

A = n? where r is the radius of the circle.

•A = ~ (a,+ b) h where a and b are the lengths of the parallel sides and h is

the perpendicular distance between the parallel sides.

If af + bx + c = 0,

-b ± ~b2 -4acthen x =2a

Opposite

Adjacent.

Area of /). = 1bh where b is the length of the base and h is the

perpendicular height~

Area of MBC = }ab sin C ~

( b >Area of MBC = .Is (s - a) (s - b) (s - c)

where s = a + b + c2

asin A

bsin B

csin C=

C~------~~--------~Aa2 = b2 + c2 - 2bc cos A

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r

;Pag~ j

SECTION I

Answer ALL the questions in this section.

All working must be clearly shown.

1. (a) Using a calculator, or otherwise, determine the exact value of

(i) 2 22.3 + 4.1

(ii) ~.I: _ 0.003

1 33- - 2-(iii) 3 5 (6 marks)

21.5

(b) (i) Write your answer in Part (a) (i) correct to one significant figure.

(ii) Write your answer in Part (a) (ii) in standard form.

(c) (i) Mr Mitchell deposited $40000 in a bank and earned simple interest at 7% perannum for two years.

Calculate the amount he will receive at the end of the two-year period.

(ii) Mr Williams bought a plot of land for $40 000. The value of the landappreciated by 7% each year.

Calculate the value of the land after a period of two years. (4 marks)

Total 12 marks

2. (a) Simplify:

(i)x2-Ix-I

(ii) 4ab2 + 2a2bab

(b) Express as a single fraction:3p + !L.""2 p

Cc) Solve for x, given

3x2 - 7x + 2 = 0

(4 marks)

(2 marks)

(4 marks)

Total 10 marks

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3. (a) A club has 160 members, some of whom play tennis (T) or cricket (C) or both. 97 playtennis, 86 play cricket and 10 play neither, x play both tennis and cricket.

(i) Draw a Venn diagram to represent this information.

(ii) How many members play both tennis and cricket? (5 marks)

(b) In a beauty contest, the scores awarded by eight judges were:

5.9 6.7 6.8 6.5 6.7 8.2 6.1 6.3

I

I !

I'I I:

(i) Using the eight scores, determine:

a) the mean

b) the median

c) the mode

(ii) . Only six scores are to be used. Which two scores may be omitted to leave thevalue of tlre median the same? (6 marks)

I

I

II

: I

" ."i Total 11marks

4. (a) (i) Using the formula

A~

t ='V 12n

(ii)

calculate the value of t when m = 20 and n = 48.

Express m as subject of the formula in (a) (i) above. (5 marks)

(b) In the diagram below, not drawn to scale, EFGH is a rectangle. The point D on HG is. A 0

such that ED = DG = 12 cm and GDF = 43 .

E r------------""7I F

H,.......------X----L--jr+-----'GD-+- 12cm ~

Calculate correct to one decimal place

(i) the length of GF

(ii) the length of HD

(iii) the size of the angle HDE. (7 marks)

Total 12marks

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-- •...,' tr...•..

Page S

5. An answer sheet is provided for this question.

(a) On the section of the answer sheet provided for 5 (a):

(i) write down the coordinates of the point P

(H) draw a line segment PQ through the point, P, such that the gradient of PQ is-3 . (3 marks)2

(b) On the section of the answer sheet provided for 5 (b):

(i) draw the reflection of quadrilateral A in the mirror line, labelled M1.

Label its image B.

(H) draw thereflection of quadrilateral B in the mirror line, labelled M2 •.

Label its image C.,-''.("

(4 marks)

(c) Complete the sentence in part (c) on your answer sheet, describing FULLY the singlegeometric transformation which maps quadrilateral A onto quadrilateral C.

(3 marks)

Total 10marks

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Page 6

6. The amount a plumber charges for services depends on the time taken to complete the repairsplus a fixed charge.

The graph below shows the charges in dollars (d) for repairs in terms of the number ofminutes (t) taken to complete the repairs.

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I!j'l~ i! I1 ;r·I'

(a)

(b)

(c)

(d)

(e)

(f)

oI2340201F 2004

··-~i+·t-f'-d+- -+r1j-t+l-!' I·+,-h- -t-t"r"" ++e--!-J+J ·ttt~l..ifh- i--t··P· -e-+-.,.+ ·-rt±+· ·4·t-H- H..t·.•..

10ij#.I;J;;t iiJii-;FiIit#tJUt;iit·o ro w ~ ~ ~ ~ m t

Number of minutes

What was the charge for a plumbing job which took 20 minutes? (1 mark )

How many minutes were spent completing repairs that cost:

(i) $38.00

(ii) $20.00? (2 marks)

What is the amount of the fixed charge? (1 mark)

Calculate the gradient of the line. (2 marks)

Write down the equation of the line in terms of d and t. (2 marks)

Determine the length of time taken to complete ajob for which the charge was $78_00.(3 marks)

Total 11 marks

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',Page 7

7. (a) A piece of wire is bent in the form of a circle and it encloses an area of 154 ern".

(i) Calculate:..'

a) the radius of the circle

b) the circumference of the circle.

22(Use rt = -)7

The same piece of wire is then bent in the form of a square.

(ii) Calculate the area enclosed by the square. (6 marks)

(b) The diagram below shows a map of Bay time drawn on a grid of 1 cm squares. The. scaleof the map is 1:100'000.

~ ~ "Sprin~

V~

flail

( Rose /Hall

\ /r-. 11

/South '- .-/Port ,

(i) Find to the nearest km, the shortest distance between Rose Hall and South Port.

(ii) Determine the bearing of South Port from Spring Hall.----------- -. - .-- -------

(6 marks)

Total 12 marks

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Page 8

8. Two recipes for making chocolate drinks are shown in the table below.

Cups of Milk Cups of chocolate

Recipe A 3 2

Recipe B 2 1

(a) What percent of the mixture using Recipe A is chocolate? (2 marks)

(b) By showing suitable calculations, determine which of the two recipes, A or B, is richerin chocolate. (2 marks)

(c) If the mixtures from Recipe A and Recipe B are combined, what is the percent ofchocolate in the new mixture? (2 marks)

(d) A vendor makes chocolate drink using Recipe A. 3 cups of milk and 2 cups of chocolatecan make 6 bottles of chocolate drink. A cup of milk costs $0.70 and a cup of chocolatecosts $1.15 .

. (i) What is the cost of making 150 bottles of chocolate drink?

(ii) What should be the selling price of each bottle of chocolate drink to make anoverall profit of 20%? (6 marks)

Total 12 marks

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-.:'Page 9

SECTION 11

Answer TWO questions in this section

ALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS ","' .

9. (a) The table below shows corresponding values for P and r.

P m

r 0.2

Given that P varies directly as r3, calculate the values of m and n. (6 marks)

(b) In the diagram below, not drawn to scale, AKLM and AST J are both rectangles,

Ar- ~3x~ ,S~~~3--~K

2x

Jr------------------4T

5

M-------:---------------------------fLGiven that AS = 3x cm, AJ = 2x cm, SK = 3 cm and JM = 5cm

(i) Obtain an expression, in terms of x, for the area of rectangle AKLM.

(ii) Given that the area ofrectangleAKLMis oOcm2, show that

2x2 + 7x -'- 15 = 0

(iii) Hence, calculate the value of x and state the length of AK and AM.(9 marks)

Total 15 marks

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Page 10,..

10. A vendor buys x kg of peanuts and y kg of cashew nuts.

(a) (i) To get a good bargain, she must buy a minimum of 10 kg of peanuts and aminimum of 5 kg of cashew nuts.

Write TWO inequalities which satisfy these conditions.

(ii) She buys no more than 60 kg of nuts. Peanuts cost $4.00 per kg and cashew nutscost $8.00 per kg and she spends at least $200. .

Write TWO in9ualities which satisfy these conditions.(5 marks)

(b) Using a scale of 2 cm to represent 10 kg on each axis, draw the graph of the FOURin~ualities in (a) (i) and (a) (ii),

On your graph, shade ONLY the region which satisfies all four inequalities.(6 marks)

(c) The profit on the sale of 1 kg of peanuts is $2.00 and on 1 kg of cashew nuts is $5.00.

(i) Using your graph, determine the number ofkilograms of each type of nut thevendor must sell in order to make the maximum profit.

(ii) Calculate the maximum profit. (4 marks)

ITotal 15 marks

!ii

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r!

11.

(b)

-Page 11

GEOMETRY AND TRIGONOMETRY

Ca) In the diagram below, VWZ and WXYZ are two circles intersecting at Wand Z. svr is a. A "

tangent to the circle at V, VWX and VZYare straight lines, TVY = 78° and SVX = 51°.

y

(i) Calculate the size of EACH of the following angles, giving reasons for youranswers.

."a) VZW

b) "Xyz (4 marks)

(i) Draw a diagram to represent the information given below.-,

Show clearly the north line in your diagram.

Town F is 50 km east of town G.

Town H is on a bearing of 040° from town F.

The distance from F to H is 65 km.

(ii) Calculate, to the nearest kilometre, the actual distance GH.

(iii) Calculate, to the nearest degree, the bearing of H from G. (11 marks)

Total 15 marks

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....- ...

Page 12

12. (a) Given that sin e = -{3 ,0° ~ 0 ~ 90°.2

(i) Express in fractional or surd form the value of cos o.

(ii) Show that the area of triangle CDE is 150 -.r3 square units, where CD = 30 unitsand DE = 20 units.

I

'I, II,

!i

D

E

i1 .! (iii) Calculate the length of the side EC. (7 marks)

(b) In this question, use n = 3.14 and assume the earth to bea sphere of radius6370 km.

The diagram below shows a sketch of the earth with the Greenwich Meridian and theEquator labelled.

N

GreenwichMeridian ----'lr--~

~~~~--Equator

!

S

The towns A and B are both on the circle of latitude 24° N. The longitude of A isios' E and the longitude of B is 75° E.

(i) Copy the sketch above of the earth and insert the points A and B on yourdiagram.

! I!

i

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(ii) Calculate, correct to the Dearest kilometre,

a) the radius of the circle of latitude 24° N

b) the shortest distance between A and B, measured along the circle oflatitude 24° N. (8 marks)

Total 15 marks

VECTORS AND MATRICES

13. The vertices of a quadrilateral, OABC, are (0, 0), (4" 2), (6, 10) and (2,8) respectively.

Use a vector method to answer the questions which follow.

(a) Write as a column vector, in the form [;], the vector

(i) at .~

(ii) CB

Calculate Iat I' the magnitude of at.(3 marks)

(b) (1 mark)

(c)\

State two geometrical relationships between the line segments OA and CB.(i)

(ii) Explain why OARC is a parallelogram. (4 marks)

(d) If M is the midpoint of the diagonal OB, and N is the midpoint of the diagonal AC,determine the position vector

-)(i) OM

-)

(ii) ON

Hence, state one conclusion which can be made about the diagonals of the parallelogramOABC. (7 marks)

TatallSmarks

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· ~~;'..

Page 14

14. An answer sheet is provided for this question.

(a) F On the answer sheet provided, perform the following transformations:

(i) Reflect triangle P in the y-axis.

Label its image Q.

(ii) Draw the line y = x and reflect triangle Q in this line.

Label its image R. (5 marks)

(iii) Describe, in words, the single geometric transformation which maps triangle Ponto triangle R. (3 marks)

(iv) Reflect triangle Q in the x-axis.

Label its image S.

(v) Write down the 2 x 2 matrix for the transformation which maps triangle Ponto triangle S. (3 marks)

Cb) (i) Write down the 2 x 2 matrices for

a) a reflection in the y-axis

b) a reflection in the line y = x.

(ii) Using the two matrices in b (i) above, obtain a SINGLE matrix for a reflectionin the y-axis followed by a reflection in the line y = x. (4 marks)~.

:![Total 15 marks

ENoOF'TEST,.,

o12340201F 2004

TEST CODE 01234U2UFORM TP 2004101

CARIBBEAN EXAMINATIONSMA YIJUNE 2004

COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINATION

MATHEMATICS

Paper 02 General Proficiency

Answer Sheet for Question 5

Ca)

Cb)

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Cc) The single geometric transformation which maps quadrilateral A onto quadrilateral C is

o 1234020/F 2004

ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

TEST CODE 01234020FORM TP 2004101

CARIBBEAN EXAMINATIONS COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINA TIONMATHEMATICS

Paper 02 General Proficiency

MAY/JUNE 2004

Candidate Number .Answer Sheet for Question 14

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ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLETo12340201F 2004