59985092 cxc mathematics may 2010 paper 2

FORM TP 2Dt00g7

CARIBBEAN EXAMINAT IONS COUNC

SECONDARY EDUCATION CERTIF ICA]-EEXAMINATION

MATHEMATICS

Paper 02 General Proficiency

2 hours 10 minutes

l 9 MAY 2010 (a .m. )

INSTRUC IONS TO CANDIDATES

This paper consis ts of TWO sect ions.

- l ' i rcre u lc [ : lGt l I 'c luest ions in Sect ion I anc i TI iRtr [1 quest ions in Sect iun IL

Answer ALL quest ions in Sect ion I . and any,TWO quest ions f iorn Sect ion I I "

Write your answers in the booklet provided.

. \ l l u o rk i r r r : r r . rus t bc c l cu r l v shou ,n .

A l is t o f lbrntu lae is pror ided o l l pase I o f th is booklet .

TIJST CODE OI234O2A

M A Y i J U N E 2 0 I O

I L

I

ffi]ffi

iI

I i .equired Examination Materials

E,lectron ic calculator( ieornetry set( i raph papcr (prov ided)

DO NOT TURN THIS PAGE UNTII , YOT] ARE TOLD TO DO SO.

-I-I

-E

-II

I

I

E

Copyright (t ' , 2008 Caribbean Exarninations Counci [email protected] r ights reserved.

0t231020/F 20 I 0

LIST OF FORMULAE

Volume of a prisrn

Volume of cyl inder

Volunre o1'a right pyramid

Circumference

Area of a circle

Area of trapeziurn

Roots of quadratic equations 17'a.r)

then .r

Trigonometric ratios

Area of t r iangle

V : Ah r,vhere I is the area of a cross-sectionlength.

V: nr:h r,vhere r is the radius of the base and h

I

V : = Ah where I is the area of the base and lt.'J

and h is the perpendicular

is the perpendicular height .

is the perpendicular height.

Oppos i te

L-:2Tr where l is the radius of the c i rc lc ' .

A: nrrwhere r is the radius of the c i rc le .

A:+Qr + l t) /r u,here o ancl b are the lengths of the paral lel sides and lr is

the perpendicular distance betrveert the par"al lel siclcs"

+ b - r ' + c - ' 0 .

-b t ,li: q*

s i n 0 =

2ct

opposi te s ide

hvpoterrr-rse

^ a d i a c e n t s i d cc o s H -

t a n O -

I l r 1 - r i r 1 . ' 1 1 1 1 - ;

o1-rposite srde

adjacent side

Areir 01'A-- ] b/, u,hcre /r is the length o1'the base and h is the

pcrpcndicuIar hei -uht

. , \ reu t r1 ' \ . 11 j ( - | , r i , s in ( '

S ine ru le

Cosine n-rle

Area of

where . r

Q :

s in I

. r ) : h2

^ A B C - v J ( i

t t * l ' , r c,)/-

h - : L ' *

sin B s in C

+ cr - 2bc cos ,4

Adjacent

01231020/F 20 1 0GO ON TO THE NEXT PACE,

( a )l . Deten-nine

( i )

( i i )

Mrs. Jack bought

( i ) C ' a l c u l a t c

to 2 s igni f rcant f igures

( 3 marks)

( 3 marks)

( I nrark )

SECTION I

Ansrver ALL the questions in this section.

Al l work ing must be c lear ly shown.

the EXACT value of :

I

l - ; - - '

a

')1 : x

)-)J

r < t 1 . 8 91 7

giv ing your answer correct

(b) 150 T-shirts fbr $ I 920 f i 'onr

the cost of ONE T-s l r i r t .

The T-shi r ts are so ld at $19.99 each.

C ' l l c L r la te

( i i 1

( i i i )

( i v )

thc an ' r t r r - ln t o f ' lnonc \ \ l l s . ,T r ick r -ccc i r c r i a t te r '

the TOTAL profit macie

the profit made as a percentagethe nearest rvhole number.

: c i l i ng , . \LL o f the T -sh i r t s( I m a r k )

( I m a r k )

of the cost price, giving your answer correct to( 2 marks)

Total I I rnarks

GO ON TO THE. NEXT PAGE01231020i F 201 0

*

( a )) Ci iven tha t r l - 1 . b :2 anc l

( i ) a t - b + c

( i i ) b1 ( '1

Wri tc the lb l lowing phrases

( i ) seven t i rnes the surn

( i i ) the product of l -WO

c' : --3. f ind the r, 'alue of-:

as a lgebr-a ic express ions:

o1- . r anc l . r '

consc'cutir e nurnbers rvhen the

Pagc 4

( l m a r k )

( I m a r k )

( l m a r k )

s r n u l l c r t t u r t t h c r i s . r '

( l m a r k )

( - t r r n l r r k . r

r I r r r r u ' k )

I I r r t r t t ' k r )

( 2 r n a r k s )

Total l2 marks

t l ie given in 1i : t 'n l t1 iot t

t

(b)

(cl)

( c ) So lve the pa i r o f s in rLr l taneous equat ions :

f-\ ' t .t '-: 7

.r f.t ' - I

Factori se conr 1'l I ete l rv :

( i ) 4,r ' r : l

( i i ) ?a .v 2ay - b r + l r r

( i i i t 3 , r r + l 0 - r 8

3 . (a) A survey u,as conducted among 40 tourists. f-hc rcsrr l ts u cr ' . . ' :

l f i r is i tcc i . \n t igua l . \ t30 v is i tcd Barbados ( B )3x visi ted both Antigua and Barbados.r visi ted neither Antigua nor Barbaclos

(i) Copy'and complete the Venn diagrerm below to representabor e.

-.-=-\- - - 1 - - -__ ._

I >I)

( i i )

( i i i )

01234020/F 20 1 0

Write an expression, in -rr,survey.

Calculate the value o{ ' .r .

( 2 marks)

to represent tl-re 1-OTAL number of tourists in thc( 2 marks)

( 2 marks)

GO ON TO T I IE NE,XT 'PAGF,

(b) The diagram below, not drawnwith cross section ABCDE. tr is

to scale, shows a wooden toy inthe midpoint of EC, and IBAE

the shape of a prism,: ZCBA : 90o.

-1.

Calculate

(i) the length of EF

( i i ) the l c -ng th o f DF

rr i i r the area of the face ABCDE.

5 c m

( I m a r k )

( 2 marks)

( 3 marks)

Total 12 marks

( 2 marks)

Total I I marks

\ \ i r . ' r r ' . r r r ic : J i rect l r :> the : t lLr r r r - ' r r i r . . thc r ar ia t ion eqr- rat ion is wr i t ten ! : kx2,u 'hereA r s ir c rrit: ti.tn t.

( i) Given that 1' : 50 when -r : 10, f ind the value of k. ( 2 marks)

( i i ) calculate the value ofy when x : 30. ( 2 marks)

(i) Using a ruler, a pencil and a pair of compasses, construct triangle EFG withE G : 6 c mlFEG: 60"and IEGF: 90o. ( 5 marks)

( i i ) Measure and state

(b)

the length of EF

the size of /-EtrG.

a)

b)

01234020/F 20 I 0GO ON TO THE NEXT PAGE

L

(a ) The functions / 'and g are defined as /(x):2t - 5 and g(.r) : xr + 3.

( i ) Calcu late the value of

/(4)

s/(4)

( i i ) F ind l ' r ( r ) .

The diagram below shows the graph of-y : x2 + 2x - 3 for the dornain - 4

graph above to deternr ine

the scale used on the -r-axis

the value ofy fbr which,r - - 1 .5

the values of r tor which y: 0

the range of values of-t,, giving your answerb are real numbers.

(

(

I mark )

2 marks)

marks)

< 2 .

( I m a r k )

( 2 marks )

( 2 marks )

in the fbrm a = t- < b, where a and( 2 marks )

Total l2 marks

GO ON TO THE NE,XT PAGE

a)

b )

(b )

( 2

< J

Use the

( i )

( i i )

( i i i )

( iv )

J , 1

I- t + 2.r

: t '

3

\ I I\ ' , ' Ii,\ )

I\i

! l

',1/,\

\

I N 2 \

\ f,\ .,

..:N\

-f

0t234020/F 2010

6.

An answer sheet is provided for this question.

(a) The diagram below, not drawn to scale, shows two straight lines, PQ andRS, intersectinga pair of parallel lines, TU and VW.

R P

Determine, giving a reason for E,ACH of your answers, the value of

( i ) r

( i i ) y .

( 2 marks)

( 2 marks)

(b) The diagram below shows triangle LMl,{,and its image, triangle L' M hr ,afterundergoinga rotat ion.

(i) Describe the rotation FULLY by stating

a ) the centre

b) the angle

c ) the direct ion. ( 3 marks)

( i i )

( i i i )

L,MN.

Triangle LMl,lis translated by the vecto , ( t l

t - ) 1\ "./

State TWO geometric relat ior-rships betrveen tr iangle Lh4l ' /and i ts image, tr iangle( 2 marks)

Determine the coordinates of the image of the point I under this transformation.( 2 marks)

Total 1l marks

GO ON TO THE NEXT PAGE

.)'L

\.1. . I

:X4 ' \

\

I \II

--1

IJ

0r234020/F 20 1 0

L

7.

Page [3

A class of 24 students threw the cricket ball at sports. The distance thrown by each student wasmeasured to the nearest metre. The results are shown below.

22

48

5 5

3 6

50

34

29

63

3 5

45

46

54

52.,)Z J

5 6,) ,\J Z

i n t c . n a l l r l l . )

represent 5histogram to

47

43

43

19

30

10

59

60

(

(

metres, and a scale of Iillustrate the data.

(

(a) Copy and complete the fiequency table for the data shown above.

Distance (m) Frequency

20 29 aJ

30 39 5

(b )

(c)

State the lou'er bor-rndan t i-rr the class

Using a scale of I cm on the r -ax is to

-)'-aris to represent I student. drar.v a

3 marks)

I rna rk )

cm on the

5 marks)

(d) Deternt ine

( i ) thc nunrber of st l ldents r i ho threw' the bal l a distance recorded as 50 rnetres orn r o r e ( l m a r k )

( i i ) the probabi l i ty that a stLldel t t . choscn i i t ratrdor-n. threuas 50 metres or more.

the bal l a distance recorded( I m a r k )

Total 11 marks

0r2340201F 20t0GO ON TO THE, NEXT PAGE

8.

An answer sheet is provided for this question.

The diagrar-n bclou,shows the first three figures in a sequence of figures. Each figure is made

up of squares of s ide I cm.

Fig. I Fig. 2

On your answer sheet, draw the FOURTH figure (Fig. 4) in the sequence.( 2 marks)

StLrcir rhc 1-rx11..1-n: in the table shown below, and on the answer sheet pror ided. cotttplr- ' tc

l l r c r i , r i : n r n r l - c l c . l r r t . t i i ) . ( i i i ) a n c J ( i r ) .

( a )

( b )

( i )

( i i )

( i i i )

( i v )

I Area ofFigure ,

t-J,T

Perimeterof Figure

(cm)

l x 6 2 : 1

2 4 2 x 6 - 2 : 1 0

aJ 9 3 x 6 - 2 : 1 6

4

5

l 5

l t

( 2 marks)

( 2 marks)

( 2 marks)

( 2 marks)

Tota l l0 marks

01234020/F 201 0GO ON TO THE, NE,XT PAGE

SECTTON II

Answer TWO questions in this section.

ALGEBRAAND RELATIONS, FUNCTTONS AND GRAPHS

g. (a) The diagram below shows the speed-time graph of the motion of an athlete durin-e a

race.

Speedin m/s

t 4

l 2

l 0

8

6

4

2

0 r 2 3 1 5 6 7 8 9 1 0 1 1 1 2 1 3Time in seconds

( i ) Using the graPh, determine

a) the MAXIMUM sPeed

b) t |e purnber of seconds fbr u,hich the speed vr 'as constant

c) the TOTAL distance col 'ered b1, the athlete durin,e the race.( 4 marks)

(ii) During which time-period of the race r'vas

a) the speed of the athlete increastng

b) the speed of the athlete decreasing

c) the acceleration of the athlete zero? ( 3 marks)

/ \

/ \

/ \

/ \

/ \

/ \

01231020/F 20 1 0

GO ON TO THE NE,XT PAGE

\

(b)

Page I I

A farmer supplies his neighbours with r pumpkins andy melons daily, using the follow-ing conditions:

First condition '. ),> 3

Secondcond i t i on : y<xThird condition : the total number of pumpkins and melons must not exceed 12.

( l m a r k )(i) Write an inequality to represent the THIRD condition.

( i i ) LJsing a scale of 1 cm to represent one pumpkin onto represent one melon on the r-axis, draw the graphs

the x-axis and I cmof the THREE lines

associated u. i th the THREE inequali t ies. ( 4 marks)

(iii) Identify, by shading, the region which satisfies the THREE inequalities.( l m a r k )

(iv) Determine, from your graph, the minimum values of x and y which satisfy the

conditions. ( 2 marks)

Total 15 marks

0t234020/F 20 1 0GO ON TO THE NEXT PAGE

(a )10.

MEASUREMENT, GEOMETRY AND TRIGONOMETRY

not drawn to scal e" PO is a tansent to the circle PTSR, so that

marks)

marks)

marks)

its foot"f lagpole6 m a n d

In the diagram below,/.RPQ: 46"IRQP:32 "and TRQ is a straightl ine.

P

Calculate, giving a reason for EACH step of 1,'otrr ans\\er.

( i ) Z.PTR

( i i ) zrPR

( i i i ) tTSR.

( 2

(3

(2

(b) The diagram below, not drawn to scale. shows a vertical flagpolc. lrT. u'itlr

F. or"r thc horizontal plane LI '-G. LT and Of are *tt ' , - ' : r ihich 5Ll l lport thc

in i ts posit ion. The angle of eler at iott of f t iot l G is 5 5' . EF : 8 m, FG :

/_EFG : 120".

Calculate, giving your answer correct to

(i) the height, FT, of the flagpole

(ii) the length of trG

3 significant figures

(iii) the angle of elevation of T from E.

( 2 marks)

( 3 marks)

( 3 marks)

Total l5 marks

GO ON TO THE NEXT PAGE

01234020/F 2010

:

Pagc l 3

VECTORS AND MATRICt rS

t l . (a ) A and B nr -c tuo I r ? rna i r i ces such tha t

( l 1 ( s - r \A: I a r rd B : i

- " li l 5 l - 2 I i

( 2 marks)

( l m a r k )

/ r - \

write i I or the product of TWO matrices.\ 1 ) l

( 2 marks)

( iv) Hence, ca lcu late the values of r and_r , . ( 2 marks)

(b) -fhe diagrant bc' low. not drawn to scale" shows tr ianele . lKl, .

'''.-|--- ' ' / l

J . _

i\ ,

\ / r incl \ l r i 'c ntr ints on ,, /K ttncl JL re spectively. such t lrat

. / \ l = . . / A a n c l J , \ : ! . 1 1 . .

( i) C'opr thc dia-erarn in your answer booklet ancl shor,v the pcl ints Mand,V.

_ _ ) _ ) ( 2 m a r k s )

( i i ) Givcn thar . /A,1 : uarrd. t r l , , / :v ,rvr i te. irr terrns of z and y, alt crpression fbr

-)a) .IK

->b) i I \

--)c ) L L .

( i i i ) Us ing yoL l l ' f i r t c l ings in (b ) t i i ) . c iec lL rcc TWO geornet r i ca l re la t ionsh ips be twee lKL and ,ll,\'. ( 2 marks)

Total l5 marks

( i ) I ' i n c i . \ 8 .

l i i t I ) c t c r n i i r r e B r . t h e i l r v e r s e o f B .

t i t i r ( i i r e n t l l r t

5 - : \ i ' 2 \

: I r . ; =

l , j

K

( 4 marks)

END OF TEST

01234020/F 20 I 0

Answer Sheet for Question 6

( a ) R

TEST CODE OI234O2O

MAY/JUNE 201O

NCIL

Candidate Number .. . . . . . .

P

FORM TP 2010087CARIBBEAN EXAMINAT IONS COU

SECONDARY EDUCATION CERTIFICATEEXAMINATION

MATHEMATICS


( l )

( i i )

Y :

I

( b ) v

tI

X.'i',} I '

1\

//

\

:L :;z 1, \

I

N:

N' :

i -

1\l. i . . . i

: ' , - 4r r l

: i -3, . ' . [ .

,|

II

.v

I

( i ) a )

b )

c )

centre of rotation

angle of rotation

di rcct ion of ' ro tat ion

( i i )

( i i i )

LMIV and L' M N are and

lmase of L under the translation is (

ATTACH THIS ANSWER SHEET TO YOUR ANSWER BOOKLET

FORM TP 2010087TEST CODE OI234O2O

MAY/JUNE 20 1O

CARIBBEAN EXAMINAT IONS COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINATION

MATHEMATICS


Answer Sheet for Question 8 Candidate Number ........

( a )

( i i i )

( iv )

( b )

( i )

( i i )

FigureArea ofFigure(cm')

Perimeterof Figure

(cm)

l x 6 2 - 4

2 ,1- 2 x 6 - 2 : l 0

1J

c1 3 x 6 - 2 : 1 6

4

5

l 5

n

ATTACH THIS ANSWER SHEET TO YOUR ANSWtrR BOOKLET

59985092 cxc mathematics may 2010 paper 2

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