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Additional Mathematics Kertas 2 Sept2009 ^l /-lam 2'

JABATAN PELAJARAN NEGERI JOHOR PERCUBAAN SPM2OO9 PEPERIKSAAN ADDITIONAL MATHEMATICS KERTAS 2 Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1.

Kertqs soalan ini adalah dalan dt,ihahasa.

2.

Soalan dalam baha,saInggeris mendohuluisoalun yung,sepadundalcnnbahal;allfelayu.

3.

Calon dikehendakirnembacemaklumaldi halarnanbelakangkertas,soalanini.

4.

Culon dikehendakimenceraikanlwlamctn I9 dan ikat sehagaitnukahadapan hersama-sama de,nganj atvapan awla..

Kertassoalanini mengandungi t9 halamanbercetak dan I halarnan kosong SULIT

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Thefollowingformulae may be helpful in answeringthe questions. Thesymbolsgiven are the onescommonlywed. ALGEBRA x=

-b+ "[Y 4o;

8

2a

: fo8'b tog,a l Qgc a T,:a+ .(n-l )d

2

q ^ xa n=

3

a ^ +t{= a^' n

t0 s^:

4

(am)"= ann

ll

5

logo mn = log um * loga n

t2 ;="

6

lo g"m:bggn- logon n lo g a ttn : nlogum

7

e^* '

ltza+ ( n- r ) ell

T n = Q t'n'l

a(r"-l) =a(!-r") , r *1 r-l l-r o s-" = lrl.t " l-r

IJ

CALCULUS dv dv uvd\. L = u -+ v 'v&=dr

du

n

Areaundera curye =

du

dy dx

d), du

l vdx

dv

. . uqY dx--:_=-._*-;-' dr v dx u "

l

J'

or

b

=

lxdy

J'

Volume generated

du dr

-- llry' aY or Jb

--

l|d.- dv

GEOMETRY

'l Distance=

- r, )t + (.Y,- Y,)'

2 Midpoint (x.y)=

( x ,+ x , l-

t2 lrl={.r-+}ll

f

)

') -Y,+.Y:

')l

A pointdividinga segmentof a line ( tu, ,rv,, mv" \ -.* . lx, v) = frr+ m+ n ) n \ Areaof triangle= l,

;l (x,1,

^

+ x2y3+ x.!,,) - (xry, + xry, + x,yt)l

xi+ vi 17t

Vx- + -YSULIT

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SULIT STATISTICS

,-

&t/

; zI,'w,

zf,

,, _ n! " {n - r1''.

Lw,

Zr l_

n^ |

-r

l= ' .

3o=trt_-= v 1/ i I(-r-x )2

4

l) x-

o=

l0

P(A w B) --P (A) + P (B)- P (A a B)

11

P (X = r) = ' C ,P ' qn-' , P + q = |

12

Mean, p : np

[.!"-rl

5 m= r+l2

o =.[rW

lc

l f" l 6

nl

= _fu* r)!rr'

t4

I =9x100

__x-p o

O"

TRIGONOMETRY I A r c length ,s = r0 Areaof'sector. ,e = ! ,'e 2 2l | 3 s in * " o r' A : 2

9 sin (l + 8) : siM cos8 + cosl sin-B l 0 cos(l + B) : cosl cosB F sinl sinB fl

tnn l *1911$ tan(A +B ) = ' * " ' I + tanltan B

4 sec2A: | +tan2A 5 c os ecAz =1 + c o t2 A

t2

abc

sinA sinB sinC

6 s inM = 2 s i M c o s l 7 cos2A: cos'A - sin2A = 2c o s 2 A -r = l- 2 sin2A )tqn

8 t an2A :

t3 a2= b2+ c2 - 2bc cosl 14 Areaof triansle = f aDsinC -2

/

' * " :' l- Ia n ' A

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SectionA BahagianA 140marksl 140markahl Answerall questionsln thissecfion. Jawabsemuasoalan. equations: simultaneous 1 Solvethe following Giveyouranswerscorreclto threedecimalplaces. persamaanserentakberikut: Selesaikan Berikanjawapanandabetulkepada tiga titikperpuluhan. P+2q =l p' -3p+2q' =3 [5 marks] 15markahl The tangentto 2 A curvehas a gradientfunctionco.t-2x, wherea is a constant. to thestraightline 2y = *x +3 , the curveat the point (1,3)is perpendicular axt -2x , di mana a adalah pemalar. Satulengkungmempunyaifungsikecerunan (1,3) dengangarislurus pada adalah bersereniang titik Tangenkepadalengkung 2Y=-x+3' Find Cari [3 marks] 13markah)

(a) the valueof a, nilaibagi a ,

13marksl 13markah)

(b) the equationof the curve. persamaan lengkungtersebut.

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Ahmadhas 1000chickensin his poultryfarm.Everyweek,he willsell40of his chickens. Ahmadmempunyai1000ekor ayamdi ladangayamnya. Setiapminggu,dia akanmenjua!40 ekorayamnya. (a) Findthe totalnumberof chickenleftin hispoultryfarmafter21't week. Cari bilanganayamyang masihtinggaldi ladangayamnyase/epasmingguke-21. {4 marksl 14markahl (b) The costof feedingeachchickis RM2perweek. chickenfor thefirst Findthe totalamountof moneythathe spenton the remaining twelveweeks. PerbelanjaanatasmakananseekorayamadalahRM2seminggu. Kirakanjumlah perbelanjaanyang dibelanjakanatasjumlah ayamyang tinggal untukduabelasmingguyangpertama.. [3 marks] 13markahl

Height Tinggi(cm)

Numberof plants Bilangantanaman

20 -29

4

30-39 40-49 50-59 5

6 0 -6 9 I!.J_

1

IY

Table4 Jadual4 Table4 showsthe distribution of the heightsof plantsin a garden Giventhe medianis 47.5,find thevalueof a Jadual4 menunjukkantaburantinggitanamandi sebuahtaman. Diberi medianadalah47.5,carikannilai a . Hence,findthe varianceof thedistribution. Seterusnya,cari varianstaburantinggitanamantersebut.

13marksl 13markahl 13marksl 13markahl

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SULIT 5 Solutionsby scaledrawingwill not be accepted' Penyelesaiansecaralukisanberskalatidakditerima.

Diagram5 showsrhombusPQRS.Theequationof PS is 3y +7 x = 33 andthe equationof PR is .Y= -r + I I ' garislurusPS ialah 3y +7x =33 Rajah5 menunjukkanrombusPQRS.Persamaan dan persamaangais lurus PR ialah y = -'r + I I

Diagram5 Rajah5 (a) Find Cari

[3 marks] 13markahl

(i) the equationof QS, persamaangaris/urusQS,

[2 marksl 12markahl

(ii) the coordinates of S. koordinatbagi S.

(b) A pointf movessuchthatSf :IQ = 2'.1. Findtheequationof the locusof f, lokus T. SatutitikT bergerakdengankeadaan Sf :fQ = 2'.1 . Cari persamaan [3 marks] 13markahl

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6 (a) provethat

tos 2l = cos,4- sinI . cosl+sinl

t2 marksl

kos2A =kosA-sinl Buktikanbahawa kosA+sinA

t2markahl

(b) (i) Sketchthegraphof /=sin2r+2 for 0<x
untuky<x
[6marks] 16markahl

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SectionB BahagianB 140marksl 140markahl fromthissection. Answeranyfour questions Jawabmana-manaempat soalandaripadabahagianini.

liney=-x+8 ' 7 showsthecurvey=x2 +2 andthestraight 7 Diaoram

,=x2+2

Diagram7 Raiah7 Find Cari [3 marks] 13markahl

(a) the valueof k, nilaibagik,

14 marksl 14markahl

(b) the areaof the shadedregion, luasrantauberlorek,

Intermsof a, whenthe regionboundedby the curve' (c) the volumegenerated 360' aboutthey-axis' [3 marks] the y-axis andy = 6 is revolved janaan,dalam sebutann , apabilarantauyangdibatasiolehlengkung tsipadu itu, paski-ydany = 6 dikisarkanmelalui360' padapaksi-y, B markahl

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froman experiment. x and y , obtained 8 TableB showsthe valuesof twovariables, = Variables.rand y are relatedby theequation.y 2ox) + bx, whereaand b are constants. satu x dan y didapatidaripada nilaiduapembolehubah, Jadual 8 menunjukan denganpersamaany =2qx2 + bx, x dan y dihubungkan Pembolehubah eksperimen. dengankeadaana dan b adalahpemalar. A t

v

1 4 .5

1

q

o

7

25

39

tr A

Table 8 J a d u a lI

(a) Plot i

v

against.r, usinga scaleof 2 cm to 1 uniton bothaxes. 14marksi

Hence,drawthe lineof bestfit.

skala2 cm kepada'1unitpada graf ! lawanx , denganmenggunakan Ptotkan kedua-duapaksi. Seterusnya,lukiskangarisluruspenyuatanterbaik.

14markahJ

(b) Useyourgraphin 8(a),to findthevalueof Gunakangrafandadi 8(a), untukmencariniilai (i) a, (ii) b, (iii) y when x= 1.2. ), apab1ax = 1.2.

[6 marks] 16markahl

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OADEB,centreO andsectorfOFC withcentreL DiagramI showsa semicircle GiventhatOB = 5 cm and fO = 15cm. AADEBberpusata dan sebuahsektor Rajah9 menunjukkan sebuahsemibulatan TOFCberpusatT. DiberibahawaOB = 5 cm dan fO = 15 cm.

AOB 9 Dragram Rajah9 r =3.142) [Use/Guna Calculate Hitung (a) 4.TCO,

11markl

4TCO,

11markah)

(b) the perimeter, in cm,of theshadedregion, perimeter,dalamcm,kawasanberlorek. (c) the area,in cm2,of the shadedregion. luas,dalam cm2,kawasanberlorek.

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[5 marks] l5 markahl 14marksl 14markahl

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10 Diagram10 showstriangleoPQ.Thepointr lieson QP andthe points lieson op The straightlineOf intersects thestraightlineQS at the pointR. Rajah 10 menunjukkansegitigaOPQ.Titik T tertetakpada eP dan titik S tertetak pada OP. Garislurus OT bersilangdengangarislurus eS dl t/tk R.

Diagram 10 Rajah10 1r It is giventhatO"S=-OP, Qf =:QP, OP= p andOQ= q t.

= lrlr , gr =\e, , oF = p danOg= q Diberi bahawaos z) (a) Expressin termsof p and (1. Ungkapkandalamsebutanpdan q"

(r or, (il) gJ ,

(iii) .5.

14 marks) 14markah)

(b) GiventhatOn =n(fi andQF.=16 ,whererr and n are constants,findthe valueof m andof n . [6 marks] DiberiOR=rrOT aan QR=nQS , dengankeadaanm dann adatah pemalar, cari nilai m dan nilai rt. 16markahl

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S ULI ' I '

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11( a) ln a su rv e yc a rri e do u t i n a s c h o o l ,i t i s foundthat 2 out of 3 studentspassed their MathematicsTest. Dalam satu tinjauanyang dijalankanke atasmurid-muriddi sebuah sekolah didapati 2 daripada 3 orang murid lulusdalam ujian Matematik. (i) If 7 studentsfrom that schoolare chosenat random,calculatethe probability

passedtheirMathematics Test thatexactly6 students

[3 marks]

JikaT orangmuriddaripadasekolahitu dipilihsecararawak,hitung bahawatepat6 orangmuridlulusdalamujianMatematik. kebarangkalian 13markahl (ii) lf thereare600students findthenumberof students whofailed in theschool, th e M a th e m a ti cTs e s t. 12marksl Jika terdapat 60A orang murid dalam sekolah itu, cari bilangan orang murrd yang gagal dalam ujian Matematik 12markahl ( b) A s ur v e yo n b o d y -m a s si s d o n e o n a groupof teachers.The mass of the teachers has a n o rma ld i s tri b u tl ow n i tha me anof 45 kg and standarddevi ati onof 10 kg Satu kajian terhadapberat badan sekumpulanguru telah drlakukan.Berat Badan guru- guru adalah mengikuttaburannormal dengan ntrn45 kg dan sisihan piawai 10 kg. ( i) A te a c h e ri s c h o s e na t ra n d o mfromthe group th a tth e b o dy-massof the teacheri s l essthan 42.5 kg. F i n dth e p ro b a b i l i ty Seorang guru dipilih secara rawak darrpadakumpulan tersebut. Cari kebarangkalianbahawa guru tersebutrnentpunyaiberat badan kurang dari 42 5 kg. ( ii) l l 1 2 .3 %o f th e te a c h e rsh a v ea body-massof morethan k kg, fi nd the v a l u eo f k .

beratbadanlebihdarik kg, carinilai Jika 12.3%gurutersebulmempunyai bagik. [5 marks] 15markahl

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SectionC BahagianC [20 marksl l2AmarkahJ fromthissection. Answertwo questions Jawabdua soalandaripadabahagianini. 12 A particleP movesalonga straightlineandpassesthrougha fixedpointO. vms-t,is givenby r'= 6t' -4t-2, wherer is thetime,in seconds, lts velocity, afterpassingthroughO. Suatuzarahbergerakdi sepanjangsuatugarisIurus dan melaluisatu titiktetapO t ialahmasa, Halajunya, u ms-t,diberioleh t'=6t2-4t-2,dengankeaclaan dalamsaaf,selepasmelalui O. (a) Find Cari

11markl 11markahJ

(i) the initialvelocityof the particleP, halajuawalzarahP,

12marks) 12ma*ahl

is zero, (ii) its velocityat the instantwhenthe acceleration halajunyapadaketika pecutanadalahsifar, (iii) the timeintervalduringwhichthe pafticlemovestowardsthe left, Julat masaapabilazarahitu bergerakarahke kiri,

12marksJ 12markahl

duringthefirstthreeseconds, ( b) the distance, in m, travelledby the particle jarak, dalam m, yang dilaluiolehzarahdalamtiga saatpertama, graphof the motionof the particlefor 0 < I < 3 Sketchthevelocitytime

[3 marks] 13markahl 12marks)

Lakargrafhalajumelawanmasabagipergerakanzarahitu untuk0 < I < 3. [2 markah]

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13 Table13 showsthe priceindicesintheyear2009 basedon theyear 2007of of a cake. fouritems,A, B, C, andD, usedin theproduction tahun2007 Jaclual13 menunjukkanindeksharga bagitahun 2009 berasaskan bagi empatitemA, B, C dan D, yangdigunakanuntukmembuatkek Item

PriceIndexin theyear2009 weightage basedon the year2007 130

B

144

(,

115

U

120

1

2n

Table13 Jadual13 its pricein2007. ( a) Giventhe priceof A is RM2.60in theyear2009,calculate

[2 marks] Diberiharga untukA adalahRM 2.60dalamtahun 2009,kirakanharganya 12markahl dalamtahun2Q07,

indexfor theyear2009basedon the year2007is (b) Giventhatthe composite

13marksl Diberi indeksgubahanbagi tahun2009berasaskantahun 2007 ialah 13markahl 125, carikan nilain 125, findthe valueof n.

pricein the Find the priceof the cakein theyear2007rt its corresponding year2009is RM46. 00. l2 marksl Cari hargakek dalam tahun 2007lika hargayangsepadandalamtahun 12markah) 20A9 iatahRM46. 00. (d)

10% fromthe Giventhat the priceofitem D is estimatedtoincreaseby year2009to 2010,whiletheotheritemsremainunchanged. calculatethe compositeindexof thecakefor the year2010basedon the t3 marksl year2007. Diberi harga item D diiangkameningkatsebanyak10ok dari tahun 2009ke 2010, manakalaitem-itemlain tidak berubah. tahun2007. Kirakannomborindeksgubahanpada tahun2010berasaskan 13markahl

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14 Diagram14 showsa trianglePQR. Rajah14 menuniukkansebuahsegiigaPQR

14 Diagram Rajah14

[2 marksl

(a) Calculate/.PQR,

12markahl

Krakan {PQR,

(b) Sketchand labela trianglePQ'Rwhichhas a differentshapefromtrianglePQR suchthat lengthof PQ,PR and IPRQ remainunchanged. 12marksl State lPp'R, Lakardan labetkansebuahsegitigaPQ'Ryang berlainandaripadasegttlgaPQR dalam rajahdi atas,dengankeadaanpanjangPQ , PR danIPRQ dikekalkan' 12markahl Nyatakan /.PQ'R, (c) Hence,calculate Seferusnya,kirakan (i) the lengthof Q'Q,in cm, panjangQ'Q, dalamcm, (ii) the areaof thetrianglePRQ',in cm2. luassegitigaPRQ',dalam cm'.

[6 marks] 16 markahl

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15 Usegraphpaperto answerthisquestion. Gunakankerlasgrafuntukmeniawabsoalanini. A and B. typesoffurniture, Afactoryproducestwo Eachfurnitureneeds2 typesof rawmaterials , P and Q andthe numberof eachraw arepresentedin theTable'15 below. materials neededfor eachfurniture dua ienisperabottiaituA dan B. Sebttahkilangmengeluarkan Setiapperabotmemerlukanbahanmentah P dan Q dan bilanganbahanmentah dalamjadual 15 di bawah. yang diperlukanbagi setiapperabot ditunjukkan Numberof rawmaterials BilanqanBahanMentah

Table15 Jadual15 P leftin thefactoryrs30 and thenumberof raw Thenumberof rawmaterials materials Q leftis 24. is at mosttwicethe numberof A produced It is giventhatthe numberof furniture A andy unitsof x unltsoTfurniture furnitureI producedandthefactoryproduces furnitureB. BekalanbahanmentahP yangtinggaldalamkilangadalah 30 manakalabekalan bahanmentahQ yang tinggaldalamkilangadalah24. Diberibahawabilangan perabotA adalahselebih-lebihnya dua kali bilanganperabotB dan kilangitu x unit perabotA dany unitperabotB. mengeluarkan (a) Writethreeinequalities, otherthanx > 0 andy > 0, whichsatisfyall the constraints aoove. 13marksj selainx > A dan y > 0, yangmemenuhisemua Tulistiga ketaksamaan, kekangandi atas. l3 markah) (b) By usingthe scaleof 2 cm to 2 unitson thex-axisand2 cm to I uniton they-axis, theaboveconstraints. andshadetheregionRwhichsatisfies construct [3 marks] Denganmenggunakanskala2 cm kepada2 unitpada paksi-xdan 2 cm kepada 1 unit pada paksi-y,binadan lorekrantauR yangmemenuhisemLtakekangan di atas. 13markahl

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(c) Useyourgraphin '15(b), to find Gunakangrafandadi 15(b),untukmencari produced (i) the maximumnumberof unitsof furniture,B if 4 unitsof furnitureA are produced BitanganmaksimumunitperabotB yang dihasilkanjika bilanganunitperabotA yang dihasilkan adalah 4. (ii) the maximumprofitobtainedby thefactoryif the profitfromthe saleof a unit A /'sRM 200andthe profitfromthe saleof a unitof furnitureI is of furniture RM 2s0. Keuntunganmaksimumyang diperolehjika keuntungandaripadajualan seunitperabotA adalahRM 200 dankeuntungandaripadajualan seunit oerabotB adalah RM 254. 14marksl 14markahl

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