mathematics cxc 2009

8/11/2019 Mathematics CXC 2009

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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



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

a2=b2+c2-ZbccosA

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ON TO THE

NEXT

PAGE



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)

o1234020tF 2009

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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

GO ON TO THE NEXT PAGE

ot234020tF 2W9



\./

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.

(

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)

41234024/F 2009

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PAGE



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

01234020tF 2009

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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)

01234020ff

2AA9

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THE NEXT

PAGE



7.

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)

o1234020tF 2049

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ON TO

THE NEXT

PAGE



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

ot2340201F

2009

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NEXT

PAGE



(a)

.

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)

012340201F

2nO9

a

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TO

THE NEXT

PAGE



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

o723402A/F

2W9

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NE}ff PAGE



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)

01234A248

2009

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NE)ff

PAGE



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

GO ON TO

THE NEXT

PAGE

01234020tF

2009



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.

(

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)

0L234020tF

20A9

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