Csec Mathematics May 2004

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FORM TP 2 4 TEST CODE 234 2MAY/JUNE 2004CARIBBEAN EXAMINATIONS COUNCIL

SECONDARY EDUCATION CERTIFICATEEXAMINATIONMATHEMATICS

Paper 02 - General Proficiency2 hours 40 minutes

( 1.7~Y2~i*.m.) )

INSTRUCTIONS TO CANDIDATES1. Answer ALL que st ions inSect ion I, a nd ANY TWO inSec tion II .2. Write your answers in the booklet provided.3. All working mus t be shown clear ly .4. A lis t of fonnulae is provided onpage 2 of this booklet .

Examination MaterialsElectronic calculator (non-programmable)Geometry setMathematical tables (provided)Graph paper (provided)

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOCopyright @ 2003 Caribbean Examinations Council.

All rights reserved.0 I234020/F 2004

LIST OF FORMULAEVolume of a prism

Volume of a right pyramidCircumferenceArea of a c ircle

Area of trapezium

Page 2

v =Ah where A is the area of a cross -sec tion and h is the perpendicularlength.v = 1 Ah where A is the area of the base and h i s t he perpend icul ar heigh t.C =2nr where r is the radius of the c ircle.A = nr where r is the radius of the c ircle.

A =i (a + b) h where a and b are the lengths of the paral le l s ides and h isthe perpendicular distance between the parallel sides.Roots ofquadraticequations IfaX + bx + c = 0,

th -b :t .Jb2 - 4acenx=- 2a

Trigonometr ic r at io s

Area of triangle

Sinerule

Cosine rule

0 I 2 34 02 0/ F 2 00 4

sina = opposite sidehypotenuse -~~~OPPOSiteAdjacentdjacent side

cos a = hypotenuse

tan a = oppositesideadjacentside

Area of ~ = 1 bh where b is the length of the base and h i s t he2

,ili~IMh.gh ~ h1 IArea ofMBC = J lbsinC < b )Area of MBC = .js(s - a) (s - b) (s - c)whe re s = a + b + c2 ~ b A--L_~-~sin A - sinB - sinCa2 = b2 + C2 - 2bc cos A

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5.

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An answer sheet i sprovided for thi s quest ion.(a) On the sec tion of the answer sheet provided for 5 (a) :

(i) write down the coordinates of the point P(ii) draw a line segmentPQ throughthe point,P, suchthatthe gradientof PQ is-3 . (3 marks)2

(b) On the sec tion of the answer sheet provided for 5 (b) :(i) draw the reflection of quadrilateral A inthe mirror line, labelled MI,

(ii)Label its image B.draw the reflection of quadrilateral B in the mirror line, labelled M2.

(4 marks)(c)

Label i ts image C.Complete the sentence inpart (c) on your answer sheet , descr ib ing FULLY the s inglegeometric transformation which maps quadrilateral A onto quadrilateral C. (3 marks)

Total 10marks


Page66. The amount a plumber charges for services depends on the t ime taken tocomple te the repai rsplus a fixed charge. .

The g raph be low shows the charges in doll ar s ti for repairs in terms of the number ofminutes t taken to complete the repairs.d

30

70

(a) What was the charge for a plumbing job which took 20 minutes? (1 mark)(b) How many minutes were spent completing repairs that cost:

(i) 38.00(ii) 20.oo? (2 marks)

(c) What i s the amount of the f ixed charge? (1 mark)(d) Calculate the gradient of the line. (2 marks)(e) Wri te down the equat ion of the l ine interms of d and t. (2 marks)(f) Detennine the length of t ime taken tocomple te a job for which the charge was 78.00.(3 marks)

Total 11 marks

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7. A pi ec e o fw ir e i s ben t in th e f onn of a c ir cl e and i t e nc lo ses ana re a o f 154 cm2 .a)(i) Calculate:

a) the radius of the circleb) the circumference of the circle.

(Use 1t = 22)7The s ame pi ece o fw ir e i s t hen ben t in t he fonn o f a s quare .(ii) (6 marks)alculate the area enclosed by the square.

( b) The d iagr am below shows a map ofBay t ime d rawn onag rid of 1cm square s. The sc aleof the map is 1:100 000.

( i) F ind to the neare st l an , th e s hor te st di st ance between Rose Hall a nd Sout h Port.(ii) Determine the bearing of South Por t f rom Spr ing Hal l. (6 marks)

Total 12 marks

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Two recipes for making chocola te drinks are shown in the table below.

.What percent of the mixture using Recipe A ischocolate? (2 marks)By showing sui table calcula tions, determine which of the two recipes , A orB, i s r i cherin ch~colate. (2 marks)If the mixtures from Recipe A and Recipe B are combined, what is the percent ofchocolate in the new mixture? (2 marks)A vendor makes chocola te drink using Recipe A. 3 cups ofmilk and 2cups ofchocolatecan make 6 bot tl es ofchocolate drink. A cup ofmilk cos ts 0.70 and acup ofchocolatecosts 1.15.( i) What is t he cos t of making 150 bo ttle s o f chocola te d ri nk?(ii ) What s hou ld be the s el ling p ri ce of ea ch bot tle o f choco la te d rink to make an

overall profit of 20 ? (6 marks)Tota112 marks

J


'II

(a):1 (b). (c)I,iJ (d)';1IIII

i

CUpsof Milk Cups of chocolateRecipe A 3 2Recipe B 2 1


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Page9SECTION II

Answer TWO quest ions in thi s sec tionALGEBRA AND RELATIONS, FUNCTIONS AND GRAPHS

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

E8 4 62.50.2 2 nGiven that P varies directly as r3 calculate the values ofm and n 6marks)b) In the diagram below, not drawn to scale , AKLM and ASTJ are both rectangles.

A 3.r 3 K~J T5

M LGiven that AS = 3x cm, AJ = 2xcm, SK = 3cmandJM = 5cmi) Obtain anexpression, in terms of x for the area ofrectangle AKLMii) Given that the area of rectangle AKLM i s 60cm2, show that

2x2 + 7x - 15 = 0iii) Hence , calcula te the value of x and s ta te the length of AK and AM

9 marks)Total IS marks

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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 ofcashew nuts.

Write TWO inequalit ies which satisfy these conditions.ii) She buys no more than 60 kg of nuts. Peanuts cost 4 .00 per kg and cashew nuts

cost 8.00 per kg and she spends at least 200.Write TWO inequalit ies which satisfy these conditions.

Smarks)b) Using a scale of 2 cm to represent 10 kg on each axis, draw the graph of the FOURinequal it ies in a) i ) and a) i i) .

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

c) The p ro fi t on t he s al e of 1kg of peanut s is 2 .00 and on 1kg o f c ashew nuts i s 5 .00.i) Using your graph, determine the number of kilograms of each type of nut thevendor must sel l inorder tomake the maximum profi t.ii) Calculate the maximum profit. 4 marks)

Total IS marks

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a)

b)

iii)

P ag e 1 1

GEOMETRY AND TRIGONOMETRY

I n t he d ia gr am b el ow , V WZ a nd WXYZ ar e t wo c ir cl es i nt er ~t in g at W and - ; ; SVT is at an ge nt t o t he c ir cl e a t V VWX and vzy are straight lines, TVY = 7 8 a nd SVX = 51 .

i) C al cul at e t he si ze of E AC H o f t he f ol low in g a ng le s, g iv in g r ea so ns f or y ou ranswers.a) 1\VZWb) 1\Xyz 4marks)

i) D ra w a d ia gr am t o r ep re se nt t he i nf or ma ti on g iv en b el ow .Sh ow cl ea rl y t he n or th l in e i n y ou r di ag ram .

Town F i s 5 0 k In east of t own G.Town H i s o n a be ar ing o f 04 0 f rom t ow n FThe dis tance from F to H i s 6 5 k m.

ii) C al cu la te , t o t he n ea re st k il om et re , t he a ct ua l d is ta nc e GHC al cu lat e, t o t he n ear es t de gr ee , t he b ea ri ng of H f ro m G . 11marks)

T ot al 1 5 m a rk s


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P ag e 1 2a) G iv en t ha t s in e = -{3 ,0 ~ e ~ 9 0.2

i ) Express i n f ract ional or surd f orm t he value of cos e.S how t hat t he a re a of t ri a~gl e CDE i s 1 50 -{3 s qu ar e u ni ts , w he re CD = 3 0 u ni tsand DE = 2 0 u ni ts .

ii)

c

D

E

iii) C al cu la te t he l en gt h o f t h e s id e EC 7 marks)

b) In t hi s q ue st io n, u se 1 t = 3. 14 a nd a ssu me t he e ar th t o be a s ph er e of r adi us6 370km.Th e d ia gr am b el ow sh ow s a sk et ch o f t he ea rt h w it h t he G ree nw ic h M er id ia n an d t heEquator labelled.

N

GreenwichMeridianEquator

ST he t ow ns A and B ar e both on t he cir cl e of latit ude 24 N. The longi tude of A is1 08 E a nd t he l on gi tu de o f B i s 7 5 E .

Copy t he sketch above of the earth and i n~ er t t he poi nt s A and B o n y ou rdiagram.

i)



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P ag e 1 3 ii) Calculate, cor rect to the nearest kilometre,

a) t he r ad iu s o f t he c ir cl e o f l at it ud e 2 4 Nb) the shortest distance between A and B m eas ure d a lo ng t he c ir cl e of

latitude 24 N. 8 marks)T ot al I S m ar ks

VECTORS ANDMATRICES13.

The ver tices of a quadr ilater al, OABC a re 0 , 0 ), 4 , 2 ), 6 , 1 0) a nd 2 , 8 ) r e sp ec ti ve ly .U se a v ec to r m et ho d t o a ns we r t he q ue st io ns w hi ch f ol lo w.

a) Write as~olumn vector, in the form [;], the vector i) OA-7i i) CBCalculate loX/ the magnitude of oX

3 marks)b) 1 mar k) c) i ) State two geometr ical r elationships between the line segments OA and CB

ii) Explain why OABC is a parallelogram. 4 marks) d)

If M i s t he m id po in t o f t he d ia go na l DB and N i s t he m id po in t o f t he d ia go na l ACdeter mine the position vector-7 i), OM-7ONii)Hence, state one conclusion which can be made about thediagonals of the par allelogr amOABC 7 marks)

Total IS marks

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A n an sw er sh eet i s p ro vi de d f or t hi s qu est io n. a)

b)

On the answer sheet provided, per form the f ollowing transformations: i) Reflect triangle P in the y-axis.

Label its image Qii) D ra w the line y =x and reflect triangle Q in this line.

Label its image R S marks) i ii ) D es cr ib e, i n w or ds , t he s in gl e g eo me tr ic t ra ns fo rm at io n w hi ch m ap s t ri an gl e P

onto triangle R 3 marks) iv) Reflect triangle Q in the x-axis.

v)L ab el i ts i ma ge S .Wr it e d ow n t he 2 x 2 m at ri x f or t he t ran sfo rm at io n w hi ch m ap s t ri an gl e Ponto triangle S. 3 marks)

i ) Writ e down t he 2 x 2 matrices fora) a re fl ec ti on i n t he y -a xi sb) a r ef le ct io n i n t he l in e y = x

ii) U si ng t he t wo m at ri ce s i n b i ) a bo ve , o bt ai n a S IN GL E m at ri x f or a r ef le ct io ni n t he y -a xi s f ol lo we d b y a r ef le ct io n i n t h e l in e y =x 4 marks)

T ot al I S m ar ks

END OF TEST

234 2 F 4

Csec Mathematics May 2004

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