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%. TEST CODE OI234O2OFORM TP 2011092 MAY/JUNE 2011

CARIBBEAN EXAMINATIONS COUNCILSECONDARY EDUCATION CERTIFICATE

EXAMINATION

MATHEMATICS

Paper 02 - General Proficiency

2 hours 40 minutes

18 MAY 2011 (a.m.)

INSTRUCTIONS TO CANDIDATES

1. This paper consists of TWO sections.

2. There are EIGHT questions in Section I and THREE questions in Section II.

3. Answer ALL questions in Section I, and any TWO questions from Section II.

4. Write your answers in the booklet provided.

5. All working must be clearly shown.

6. A list of formulae is provided on page 2 of this booklet.

Required Examination Materials

Electronic calculatorGeometry setGraph paper (provided)

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DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

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Copyright O 2009 Caribbean Examinations CouncilAll rights reserved.

vLIST OF F'ORMULAE

Volume of a prism

Volume of cylinder

Volume of a right pyramid

Circumference

Arc length

Area of a circle

Area of a sector

Area of trapezitm

Roots of quadratic equations

Trigonometric ratios

Area of triangle

Sine rule

Cosine rule

0t234020/F 20tl

+bx+c:0,

- -bxJU' -+ai2a

_ opposite side

Page2

V : Ah where A is the area of a cross-section and h is the perpendicularlength.

V: nf h where r is the radius of the base and h is the perpendicular height.

fi : I nwhere Ais the area of the base and ft is the perpendicular height.

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

S: + x 2nr where 0 is the angle subtended by the arc, measured in360

degrees.

A : nf where r is the radius of the circle.

0o: *n x nr2 where 0 is the angle of the sector, measured in degrees.

A: J: @ + b) & where a and b are the lengths of the parallel sides and ft is2'the perpendicular distance between the parallel sides.

If af

then x

sin 0

cosO:

hypotenuse

adjacent side

hypotenuse Adjacent

- oooosite sidetanO:adjacent side

Area of A : ! Un where b is the length of the base and ft is2

the perpendicular height

Area of A,ABC : L ob sin C2

Area of A,ABC : (s-c)

Opposite

where s :

a:sin A

d:b2+

a+b+c2

bcsin,B sin C

c2 - 2bc cosA

Adjacent

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Page 3

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:

- 24*r*(i) 4 8 expressing your answer as a fraction ( 3 marks)

(ii) 3.e6 x o.2s - Jn.o2s6 ( 3 marks)

(b) The table below shows Pamela's shopping bill. Some of the information was notincluded.

Items QuantityUnit Price

$

Total Costs

Rice u |ot 2.40 @Potatoes 4 bags 52.80

Milk I r I "u,tons2.3s 14.10

Sub Total 82.50

a % VAT 9.90

TOTAL 92.40

Calculate the values of W, X, Y and Z. ( 5 marks)

Total ll marks

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2. (a) Write as a single fraction in its lowest terms

x-2 + x+134

(b) The binary operation * is defined by

a* b (a+ b)? - 2ab.

Calculate the value of 3 * 4.

(c) Factorise completely

(D xy3 + xzy

(ii) 2mh - 2nh - 3mk + 3nk.

Page 4

( 3 marks)

( 2 marks)

( 2 marks)

( 2 marks)

(d) The table below shows corresponding values of the variables x and y, where y variesdirectly as x.

Calculate the values of a and b. ( 3 marks)

Total 12 marks

x 2 5 b

v t2 a 48

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Page 5

(a) The Venn diagram below shows the number of students who study Music and Art in aclass of 35 students.

U : {students in the class},Y' : {students who study Music}A : {students who studyArt}

(i)

(ii)

(iii)

(i)

How many students study neither Art nor Music?

Calculate the value ofx.

Hence, state the number of students who study Music only.

( lmark )

( 3 marks)

( lmark )

(b) Using a ruler, pencil, a pair of compasses and a protractor, draw accuratelya quadrilateral EFGH tsing the following measurements:

EF:8 cm <EFG:725" FG:4 cm <HEF - 70" EH:7 cm

( 5 marks)

( lmark )

11 marks

(ii) Measure and state in centimetres, the length of GH.

Total

U

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4. (a) (D Solve the inequality: 5 - 2x < 9 ( 2 marks)

(ii) Ifx is an integer, determine the SMALLEST value ofx that satisfies the inequalityin (a) (i) above. ( 1 mark )

(b) In this question, use n: +

(i) A piece of wire is bent to form a square of area l2l cm2 .

Calculate:

a) The length of each side of the square

b) The perimeter of the square ( 3 marks)

(iD The same piece of wire is bent to form a circle.

Calculate:

a) The radius of the circle

b) The area of the circle ( 4 marks)

Total 10 marks

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PageT

(a) The diagram below, not drawn to sdale, shows L OMN, and its image, L OMI\f underan enlargement with centre, O, and scale factor, ft. Angle ONM:90".

(b)

Using the dimensions shown on the diagram, calculate

(D the value of fr, the scale factor of the enlargement

(ii) the length of OM

(iii) the length of OM'.

The diagram below, not drawn to scale, shows L PQR, whichsection of a roof. QS is perpendicular to PSR.

( l mark)

(lmark)

( 2 marks)

represents the cross

P8: 12.6 metres QR:8.4 metres

Using the dimensions shown on the diagram, calculate,

(D the length of QS

(ii) the measure of I RQS

(iii) the area of L PQR.

z QPR: ts"

correct to 3 significant figures

( 2 marks)

( 3 marks)

( 3 marks)

Total12 marks

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(a) The functions/and g are defined by

J@):6.r+8 ; s(r):r-23

(i) Calculate the value ", " [;)

(iD Write an expression for S"f @) in its simplest form.

(iii) Find the inverse tunction/-1(x).

Determine

(i) the coordinates of A and B

(ii) the gradient of the line segment,4B

(iiD the equation of the line which passes through A and B.

(b) The diagram below shows the line segment which passes through the points A and B.

ri lt1. .1...:...::..

:ffi-i.-+-i.-i..

Bii, f-+'

i+5..i;x.

tH ffi'

.i- aAi--r-r!"i

! -i-j- .-i"i-i"i-i-i-i-i-.i-.i-.i-.f.

-liii (

.'.[..i'..i.''i','i,

2 ) i-i-ili':tiliii

6i'r,: 1

'i..1...i...i....i-..i'''|

ti

[...i...i...i..i...1..'i..'

Page 8

( 2 marks)

( 2 marks)

( 2 marks)

( 2 marks)

( 2 marks )

( 2 marks )

Total 12 marks

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(a)

(b)

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Page 9

The table below shows the distribution of the masses of 100 packages.

Copy and complete the table to show the cumulative frequency for the distribution.( 2 marks)

Using a scale of 2 cm to represent 10 kg on the x-axis and 1 cm to represent 10packages on the y-axis, draw the cumulative frequency curve for the data.

( 5 marks)

(c) Estimate from the graph

(D the median mass of the packages ( 2 marks)

(ii) the probability that apackage, chosen at random, has a mass which is LESS than35 kg. ( 3 marks)

Total12 marks

Mass (ks) No. ofPackages Cumulative FrequencvI -10 t2 12

rl -20 28 40

2l -30 30

3l-40 22

41-50 8

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8.

Page l0

An answer sheet is provided for this question.

The figure below shows the first three diagrams in a sequence. Each diagram is made up ofsticks joined at the ends by thumb tacks. The sticks are represented by lines and the thumbtacks by dots. In each diagram, there are / thumb tacks and s sticks.

Diagram II

Diagram 2

I

Diagram 3

I

,/\ \t,/ ,/l\\t,/ ,/l\ \t,/

(iD How many thumb tacks are in the SEVENTH diagram?

(c) Complete the table by inserting the missing values at the rows marked (i) and (ii).

On the answer sheet provided:

(a) Draw the FOURTH diagram in the sequence.

(b) (i) How many sticks are in the SIXTH diagram?

(i)

(ii)

( 2 marks)

( lmark )

(lmark)

( 2 marks)

( 2 marks)

No. of Stickss

Rule Connectinglands

No. of Thumb Tackst

4 '.(+ 't 4

8 '.(+ "8) 7

t2 r.(+"1, 10

52

55

(d) Write the rule, in terms of s and t, to show how I is related to s. ( 2 marks)

Total l0 marks

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Page 1l

SECTION II

Answer TWO quegtions in this section.

ALGEBRAAND RELATIONS, FUNCTIONS AND GRAPHS

(a) Solve the pair of simultaneous equations

(D

State

(b)

y: f -x+3y: 6-3x ( Smarks)

Express the function JV):4xz -8x-2inthe form a (x + h)z + k,where a, h andfr are constants. ( 2 marks)

(ii) the minimum value of flx) ( I mark )

(iii) the value of .r for whichlx) is a minimum. ( I mark )

(c) The speed-time graph below, not drawn to scale, shows the three-stage journey of aracing car over a period of60 seconds.

Speed(m/s)

12Second Stage

During the FIRST stage of the journey, the car increased its speed from 0 m/s to 12 rilsin x seconds accelerating at 0.6 m/s2.

(i) Calculate the value of;r. ( 2 marks)

(iD What is the gradient of the graph during the SECOND stage? Explain, in onesentence, what the car is doing during this stage. ( 2 marks)

(iiD Calculate the distance travelled by the car on the THIRD stage of the journey.( 2 marks)

Total 15 marks

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Yr'

(a)10.

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MEASUREMENI GEOMETRY AND TRIGONOMETRY

In the diagram below, not drawn to scale, W,X, Y and Z are points on the circumference

of a circle, centre O. TYV is a tangent to the circle at Y Z XWZ: 64" arrd I ZYV:23" '

Calculate, giving reasons for your answer' the measure of angle

(D XYZ

(ii) YX'z

(iiD oxz.

Calculate the value of x, as shown in the diagram'

The distance from Portcity (P) to Queenstown (Q)

distance from Queenstown to Riversdale (R) is 360

distance RP.

Determine the bearing of l? from P.

( 2 marks)

( 2 marks)

( 3 marks)

( 2 marks)

is 220 kilometres and the

kilometres. Calculate the( 3 marks)

( 3 marks)

Total 15 marks

(b) The diagram below, not drawn to scale, shows the route of an aeroplane flying from

fortcity (p) to Queensto wn (Q) and then to Riversdale (R). The bearing of Q from P is

132" and the angle PQR is 56".

(i)

(ii)

(iiD

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(i)

(ii)

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VECTORS AND MATRICES

(a) Determine the inverse of the matrix (: :) ( 2 marks)[2 4)

(b) The transformation, M: (0 o) , *up, the points R and Z onto R' and Z ' such that:

|.a o)

Page 13

( 2 marks)

( 3 marks)

( 5 marks)

( 3 marks)

Total 15 marks

R (7,2) -+ R' (2, -7) and

T(*5,4) + T'(4,5)

(i) Determine the values of a and b.

(ii) Describe fully the transformation, M.

(c) IVXYVis a parallelogram in which

-) ->W:aandVW:b.

S is a point on WY suchthat WS : SY : 1 : 2.

Write in terms of a and 6, an expression for:

-)a) WY

-)b) ws

-)c) SX

R is the mid-point of VW. Prove that R, S and Xare collinear.

END OF TEST

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