Mathematics CXC 2013

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TEST CO DE 1234 2

FO

R

TP 2 13 92 MAY/JUNE 2013

C A R I A N E X AM I N A T I

O N S

C O U N C I L

CARIBBEAN

SEC

ONDARY EDUCATION CERT

FICATE

EXAM

INATION

.

MATHEMATICS

Paper 02 - General Pro ficiency

2 hours 4 mi tes

( 22 MAY 2013 (a.m.) )

READ

TH

E FOL

LOW

ING INSTRUCT ONS CAREFULLY.

1. Thi s paper consists of TWO sect ions.

2. There are E IGHT questions in Section I and THREE questions in Section II.

3.

Answer ALL questions in Sec tion I, and any TWO ques tions from Section II .

4.

Write your answers in the booklet provided.

5. All working must be clearly shown.

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

Required Examination Materi als

Electron ic calculator

Geome

tr

y set

Graph

pa

per (provided)

DO NOT TURN THIS

PAGE UNTIL YOU A

RE

TOLD TO

DO

SO.

Copyright 2011 Caribbean Examinations Council

-

All

ri

ght

s reserved.

0123402 0/P 2013


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LIST OF FORMUL E

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

trapezium

Roots

of

quadratic equations

Trigonometric ratios

Area of triangle

Sine rule

Cosine rule

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Page 2

v = h where A is the area

of

a cross-section and h is the perpendicular

length.

v=

/ h

where

r

is the radius

of

the base and

h

is the perpendicular height.

v

=

t h

where A is the area

of

the base and h is the perpendicular height.

C = rrr

where

r

is the radius

of

the circle.

S

= X rrr

where

e

is the angle subtended by the arc , measured in

degrees.

A

= /

where

r

is the radius

of

the circle.

A

= xrr/ where e is the angle of the sector, measured in degrees.

A =

t

a b) h

where

a

and

b

are the lengths

of

the parallel sides and

h

is

the perpendicular distance between the parallel sides.

Ifax

2

bx c

=

0,

- 4ac

then

x =

2a

opposite side

sin

e

hypotenuse

Opposite

adjacent side

cos o

hypotenuse

opposite side

tan

e =

adjacent side

Area

of =

bh where b is the length

of

the base and h is

the perpendicular height.

Area

ofMBC

=

tab

sin

C

Area

ofMBC =

)s b ) s - c)

a

b c

where s

2

a

b c

sin

=

sin

B

=

sin C

GO ON TO THE

N XT

PAGE

djacent


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Page 3

SEC

TION

I

Answer

ALL

the questions in

th

is section.

All working

mus

t be clearly shown.

1.

(a)

Using a calculator, or otherwise, calculate the

EXACT

value of

1

- _

_1

5 3

(i)

(2 marks)

2

2

5

(2 marks)

(b) Smiley Orange Juice is sold in cartons

of

two different sizes at the prices s

hown

in the

table below.

(ii)

Carton Size

C os t

350 ml $

4.2

0

4

50ml

$5.1 3

Which size carton

of

orange ju ice is the BETTER buy? Justify your answer.

(3 mar ks)

(c) Faye borrowed $9 600 at 8% per annum compound interest.

(i) Calculate the interest on the loan for the first year.

(1

ma rk)

At the end of the first year, she repaid $4 368 .

(ii)

How

much did she still owe at the

beg

inn ing

of

the second year? (2 marks)

(iii) Calculate the interest on the remaining balance for the second year. (1 m

ar

k)

Total 11 ma rks

GO

ON

TO

TH

E N

EXT

PAGE

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Page 4

2. (a) Factorize completely :

(i) 2x

3

- 8x

2 marks)

(ii) 3i - 5x 2

2 marks)

(b) (i) Make C the of the formula F

=

C 32.

2 marks)

(ii) Given that F = 113, calculate the value of C.

1 mark)

(c) 5 tickets were sold for a concert . Of these x tickets were sold at 6 each, and the

remainder at 10 each.

(i) Write an expression, in terms of x for

a)

the number of tickets sold at 10 each

1 mark)

b) the TOTAL amount of money coll ected for the sale of the 500 tickets.

1 mark)

(ii) The sum of 4 108 was collected for the sale of the 500 tickets.

Calculate the number of tickets sold at 6 eac h.

3 marks)

Total 12 marks

GO ON TO

TH

E

NE

XT PAGE

234

020lF 2013


5/13

Page5

(a)

Asurveyof the30 students inForm5 showedthat somestudentsused cam eras

(C)

or

mobilephonesM) totakephotographs.

20studentsusedmobilephones

4x students usedONLYcameras

x

studentsusedBOTHmobile phonesandcameras

2studentsdidnotuseeithercamerasorphones.

(i) CopytheVenndiagrambelowandcomplete ittoshow,intermsof

x

thenumber

ofstudentsineachregion . (3mar ks)

(ii) Writeanexpression,interms

of x,

whichrepresents theTOTALnumberofstudents

inthesurvey. (1

mark

(iii) Determinethenumber

of

studentsinForm5who usedONLYcameras.

(2

mar

ks)

(b) Inthediagrambelow,not dr awn toscale,AEC andADB arestra ight lines.


6/13

Page 6

4.

a) The diagram below shows an isosceles triangle

CD

G is the midpoint of

CD

c G

D

b)

i) Measure and state, in cent imetres, the length of

DE

ii) Measure and state, in degrees, the size of


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Page 7

5.

(a)

The incompletetable below showsone pair

of

values forA and R whereA isdirectly

proportionalto the squareofR.

A

36

R

:

5

(i)

(ii)

(ii i)

ExpressA intermsofR andaconstantk

Calculatethevalueof theconstantk.

Copy andcompletethe tab le.

(1

mark)

(2

ma

rks)

(2marks)

(b) Given

that e

x)

=

(i) /g(2)

(ii) /-1(3)

2x +

1

3 andg x)

=

4x

+

5,determinethevalues

of

:

(3

ma

rks)

(3marks)

Total mar ks

GO ON TO

THE

NEXT

PAGE

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Page 8

6.

(a) A car, travelling along a straight road at a constant speed of 54 kmJh, takes 20 seconds to

travel the distance between two sign posts.

Calculate

(i)

the speed of the car in

mls

(2 marks)

(ii)

the distance, in metres, between the two sign posts .

(2

mar ks)

(b) An answer sheet is p rovided for th is question.

The graph below shows triangle

LMN

and its image L'

M

N after undergoing a single

transformation.

7

y

5

3

1

o

1

2

3

4

5

6 7

8

9

10 11

12

13 x

(i) Describe fu lly the transformation that maps LMN onto

L' M ' N ' .

(2 mar ks)

(ii) n the answer she et provide d , draw triangle

L"

M" N" the image

of

triangle

LMN,

after a translation by the vector (

.

(2 marks)

(iii) Name and describe a combinat ion of

TW

transformations which

may

be used

to map

L" M"

N " onto

L' M '

N ' . (3

mar ks)

Total 11 m

ark

s

GO ON TO THE NEXT PAGE

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9/13

Page 9

7. The table below shows the amount, to the neare st dollar, spent by a group of 40 students at the

school canteen during a period of one week.

Num

ber of

Stud

en t

s C um ula

ti

ve F req uency

mount Spen t ($)

3

10

3

II - 20

7

10

I

2 1 - 30

9

19

I

3 1 - 40 11

41 - 50

8

51 - 60

2

(a) Copy and complete the table to show the cumulative frequency.

(2 mar ks)

(b) Using a scale of 1 em to represent $5 on the horizo nt al axis and 1 cm to rep resent

5 students on the

ver

t

ica

l axis , draw the cumulative frequency graph for the data.

(5 m

ark

s)

Mar ks will be awar ded for axes app ropr iately labelled, p

oints

cor r ectly

plo

tted, and

a smooth cu rve carefully d rawn.)

(c) Use your grap h to estimate

(i) the median amount of money spent

(2 mar ks)

(ii) the pro bability tha t a student chosen at random spent less than $23 during the

week. (2 marks)

Sho

w on yo

ur gra

ph,

using br

o

ken

lines, ho w t

hese

estimates were dete r mined .

Tot a l 11 mar ks

GO

ON

TO

THE NEXT

PAGE

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p ge -

SECT

ION II

Ans

wer

TW O q

uesti

ons in th is section.

AL

GEBRA

AND

REL

ATIONS, FUN

CT

IONS AND GRAPH S

9. (a) An answer she et is pr ovided for this quest ion.

Trish wishes to buy x oranges and

y

mangoes which she intends to carry in her bag.

H

er

bag has space for only 6 fruits.

(i)

Write an inequality to represent this information.

1

mar

-

To get a good bargain, she must buy AT LEA ST

2

man

goes.

(ii)

Write an inequality to repr esent this information. 1 mar k

More in formation about the number of oranges and mangoes associated with the good

bargain is represented by

y 2x.

(iii) Write the information represented by this inequality as a sentence in your 0

words. (2

marks

(iv) On the answer sh

eet

provided , draw the lines associated with the two inequali ies

obtained in

i )

and (ii) above.

(3

ma r ks

(v) Shad e on your graph the region which represents the solution set for the three

inequalities. 1

mark

(b) (i) Write 3x

2

-

12x

+

8 in the form a x

+

hi

+

k where a, hand k are constants .

(3 mar

(ii) Sketch the graph o y =

3x

2

-

12x

+ 8, showing on your sketch

a) the intercept on the y-axis

b) the coordinates

of

the minimu m point. (4

mar

Total 15

ma

r -

GO ON TO THE

NE

XT

P.

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Page 12

MEA

SURM

ENT, GEOMET

RY N

D TRIGONOMETRY

10.

(a) The diagram below, not drawn to scale,

shows a circle with centre O.

EBe

is a tangent

to the circle .


12/13

Page 13

b) The diagram below, not drawn to scale, shows three points R Sand F on the horizontal

ground. FT is a vertical tow er of height 25 m. The angle of elevation of the top of the

tower,

T

from

R

is 27.

R

is due east of

F

and S is due south of

F. S

43 .3 m.

T

~ R

-

-

i)

ii)

iii)

iv)

s

Sketch sepa rate diagram s of the triangles R T

TFS

and

S R.

Mark on EACH

diagram the given measures of sides and angles. 3 ma r ks)

Show, by calculation, that RF = 49.1 m.

Calculate the length of

SR

correct to 1 decimal place .

Calculate the angle of elevation of the top

of

the tower, T from S.

2 marks)

1 mar k )

2 marks)

Tota

l 15 mar ks

GO ON TO THE

NEXT

P

AGE

0 1234020lP 20 13


13/13

Page 14

VEC

TORS AN DM ATRICES

1

1

(a)

In the diagram below, not

dr

awn toscale,

P

and Q are the midpoints of

OA

and

AB

respectively.

OA

=

2aandOB

=

2b.

B

(i) Expressin termsofaand

b

thevectors

a)

AB

(2marks)

b)

PQ

.

(2

mar

ks)

(ii) State O geometricalrelationships thatex istbetweenOB andPQ.

G ive reasons for your answers. (2

mar

ks)

(b) GiventhatM

=

(i) Evaluate

M -

1

, the inverseof

M.

(2mar ks)

(ii) Showthatu: '

M = I.

(2mar ks)

(iii) Use amatrixmethodtosolvefor r,

S

t andu intheequation

(5marks)

Total15marks

ENDOF T EST

IF

YOUFINISHBEF

ORE

TIMEISCALLED CHECKYOU WORKONTHISTEST.

o

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2013

Documents

Mathematics CXC 2013