spm trial 2009 addmath q&a (johor)

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    SULITNO. KAD PENGENALANANGKA GILTRAN

    3472/l

    JABATAN PELAJARAN NEGERI JOHOR

    PEPERIKSAAN PERCUBAAN SPM 2OA9ADDITIONAL MATIIEMATICSKertas ISept.2 Jam

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    Dua amJANGAN BUKA KBRTAS SOALAN1. Tulis nombor ka d pengenalan anangkagiliran andapadapetak angdisediakan.2. Kertassoalanni adalah alamdwibahasa.3. Soalandalam bahasa nggerismendahuluisoalan angsepadan alambahasaMelayu.4. Calon dibenarkanmenjawab keseluruhanatau sebahagian oalan sama ada dalambahasanggerisataubahasaMelayu.5. Calon dikehendakimembacamaklumatdihalaman elakang ertas oalan

    INI SEHTNGGA IBERITAHUUntukKegunaanemeriksa

    Soalan Markah Penuh Markah DioerolehiI 22 43 34 35 36 37 38 39 210 J2234 J5 26 47 A8 A9 A20 ,2l A22 J23 A+24 !25 A

    Jumlah 80Kertas oalanni

    347211 2oo9 akCiptapNJmengandungi0 halarnan ercetak

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    SULIT 3472t1T'he|ltov,ing -formulaemay be hetpfut n onsw-eringheqttestions The synbols giren are lhe onescommonly used.

    -b + rlh ' -4ac1 x= 2a2 r! ' x Ll '= Lt" 'n3 t! '+on- ai l 'n4 (u" ')n=a'" '5 log. mn = log am + laga n

    nl6 log" ' = lo g sn - log'nn7 log"m' : n logom

    ALGEBRA8 loc.blOe"D '-log. a

    fn:6r(n- l )d

    s"= lzr+(n- l )d l2'Tn = a"r n-t

    a(r ' - l \ u( l - r ' ) . .Sn= =--J, l . r+t)' r - l l - rS_ d l ' l .r" l -r "

    10l lt2

    CALCULUS

    . dy tly dudx du dx

    dv dv dul=tw, -;=t t ;+v;dx ax ax

    du dv.vtt clx dr dx, =- 1rdyv

    Areaundera curveI: lY. l r or

    f= lx tt VVotumegenerated

    tt= |/w" ctroroolt= l l r { - dy)-

    GEOMETRY

    1 Distancc =

    2 Midpoint

    5 A pointdividing segment f a l ine( ra,- mr, nv,+.y, )( r , .v)= i- ' i\ t l r n nt+t t /6 Areaof triangle:Jl,,r,r, +,r,-)/3r;/', ) - (t,.v,+r',y,+'t,y',)l

    (x.+x-(r , r )= l\ , .t , ;l r l={ . r -+.r , -

    ^ xt+y. ll) 7V-r-+.Y-

    -Yr+ .Y:'1 )

    341211 2oo9Iak Cipta PNJ SULIT

    - -'r, ' + (jr,

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    SULIT 34721r

    12marksl12markahlAnswerJawaPan:a) . . . . . .

    (b)

    Answer ll questfbnsDiagram shows negraph f he unct ionf x) =l x+ i i , for thedomain-2

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    Forexaminer's

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    3| ---ll3l

    4I r-lI t3 l

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    SULIT 3472t1

    3. aGivenhe unctionsf(x)=-:; andgf(x)= 5x 4.Findg(.r).-Diberiungsi"f x) = -:; dangf (x) = 5x 4 Carig(x) .x- l

    4.

    [3 marksj13maftahl

    AnswerlJawapan:

    Givenhat3 andk are he oots f hequadraticquation' +.r = p. Findhevalueof k andp.Diberibahawa dank ialahpunca-puncawrsamaan uadratik.ai nilaik dan p .13marksl

    [3markah]

    347211 2009 akCiptapNJ I Lihat sebelahSULTT

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    347211SULIT

    A Diagram5 shows he graphof /(x) = a(x - b)2+ c' State he valueol a 'b and c 'Rajah menunjukkan raf f (x) = alx - b)'1 c'Nyatakan ilai a 'b dan c

    DiagramRajah

    x-b)2 + c

    Answer/awaqan'.

    [3 marks][3markah]

    Forexaminer's

    use only

    6I r--1I l3 lt ,

    6. Givenhequadraticunction (x)= 2x' - 7' find he ange f r ifDiberiungsi uadratik-f x) = 2x' -7, cariulatbagi ika f (x)>

    Answer/ awapan

    347211 2ooeHahciptdJPNJ

    / (x) > I l .l l .

    [3 marks]13ma*ahl

    SULIT

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    I r--1I l r l

    9

    [JI t2 lr)

    SULIT7. Solve he equation '*'

    Se/esaikanpersamaan

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    [3 marks][3 markah]

    Answer/ awapan:

    Given hat og3 = loge , find hevalueof x.Diberi og3x = Iogg4, cari nilai x.[3marks][3 markah]

    Answer/awapan.

    Findhesum o nfinig f hegeometriceries1 * I * -!- ..24Carihasiltambahingga etakterhinggaanagisirijanjangeometri*] *1*....), r

    12marksl12markah)

    Answer/Jawapan347211 2oo9 ok iDranNJ I Lihat sebelahSULIT

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    SULIT

    10 . Findhe east umberf ermsn hearithmeticrogressionsumexceeds 00.Caribilanganebufane*ecildalamaniangarithmetic5,tambahsebutanmelebihi00. [3 marks]13markahl

    Answer/Jawapan

    11 The irst hree onsecutiveositiveerms fa geometricrogressionre2 , y and18.Find hecommonatio f heprogression.Tiga sebutaneftamasuatuanianggeometrialah , y dan18'Carinisbah epunya agi aniang ersebut. 12ma*sl12maftahl

    Answer/awapan:....,347211 2oo9HahciptaJPNJ

    347211- 5, l, 3, .. so ha t ts

    - 1,3, ..supaya asil

    Forexaminer'suseonly

    l0| .--lI l r l

    l lr-= l2 lt ,

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    12.

    3472tt

    [3 marks[3 markah]

    Th e ariablesand , ar e elatedy heequatioy = +, wherem andn arenconstants. iagram1 snows he straightineobtained y plotting og ryagainst .Pembolehubah da ny dihubungkan lehpersamzotl = 4'0, manam dannn adalahpemalar. Raiah12 menuniukkangaris urusyang diperolehidenganmemplot log, y melawanx .

    Diagram 2Rajah12

    Find hevalueof m andof n.Cari nilaim dann.

    3472/1 2oo9HakciptaJPNJ

    Answer/ awaoan

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    SULIT 3472tLDiagram13 showsa straight ine PMQ.Rajah13 menunjukkangaris urus PMQ.

    P1-s ,-4t)

    QQ+7,s -21Diagram 3Rajah13

    Given hat a pointM(4,4)divideshe in esegmentoining he points (s + t'-4t)an d QQ+7,s - 2) in he ralio2: l Find hevalueof s and .Diberi titikM(4, 4) membahagi emberenggaris yang menyambungkan itikP(s + t,-4t) dan QQ + 7,s - 2) dalamnisbah2:1. Carinilais dan t. [ 4 marks ]14markahl

    Answer/JawapSl? s =t-

    347211 2oo9HakciDtd PNJ SULIT

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    SULIT 3472/l

    14 . Diagram4shows straightineABwhichntersectso hestraightineCDat hepointT(2,3)Raiah14menuniukkanaris urus B bersilangengan arisurusCD pada itikT(2,3)).

    *1"*q2Diagram4Rajah 4

    Theequationf hestraightl ineBisy=-1r+4 and heequationf hestraight2l ine CD is y=2y-1.Persamaan aris urus AB iatah = *1 , + 4 dan persamaan arisurus CD ialah

    LY=2x- lCalculatehe areaof he riangle 7D .Kirakan uas segitiga ATD .

    [3 marks]l3 markah

    Answer Jawapan

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    SULIT l2 347Zt l

    15. Diagram 5shows ectors and !. Draw he esul tantector f 2g- b.Rajah15 menunjukkan ektor a dan \. Lukiskan ektorpaduanbagi2a - b.12marks)[2 markah]

    Diagram 5Rajah'15Answer Jawapan

    3472/1 2oo9 Iok cir)to JPNJ ST]I , IT

    /,/ g

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    Forexaminer'suse only

    t6I r-iI l4 lr \

    PQ= 2?- 3! and

    -) -> ->VeKor PQ dan RS adalah selai. Diberi bahawa PQ =-tqRS= 2-FindCari(a) the valueof /rNilaibagi k(b) the ratioof PQ: RS

    Nisbahbagi PQ: RS

    SULIT

    347211 2oo9Hak cipta JPNJ

    -+ aThe vectorsPQ and Rf areparallel.t is given hat-+onS=(t +Ds-;b..

    3472t1l3

    16.

    2a -3b dan

    [4 marks]14ma*ahl

    Answer/Jawapan:(a'1(b) . . . . . . . . . . .

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    SULIT IAla ['orexaminer'suseonlv17. GivenhatcoS = p and0 is anacute ngle, xpressn erms f p .Diberibahawa kos0 = p, dan0 adalah uatu udut irus,ungkapkan alamsebutan bagi(a) tan d

    e(b) s in;z

    Answer/Jawapan: a)

    .AGiven y = :j , find he approximatealueof y , when ;r changes rom 2 to 2.02.x)4Diberiy =:, , cariperubahanilaibagiy , apabila berubah aripada kepadax2.02.

    14marksl14markahl

    Answer Jawaoan3472/1 2oo9Hak Cipta PNJ SULIT

    t7I r--I l4 l

    18.

    18

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    l5 3472/l

    Diagram 8shows seclorOPQwith entreO and adius cm.Rajah18menunjukkaneldorOPQ angberpusatO anejari5 cm.

    O Rlcm ODiagram 8Rajah 8

    GivenQR =1 cm and IPRO = 900, ind he area n cm2 of the shaded egion.Diberi QR=1 cm dan ./.PRO= 900 kirakan uasdatamcm2 bagi kawasan edorek.

    14marksl14matuahl

    Answer Jawapan

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    t6SULIT 3472t1 Forexaminer'suse only

    20 . IFind he coordinates f the turningpointson the curve / = 4x + - and determinehexmaximumDoint.ICari koordinatbagi titik-titikpusingan agi tengkungy = 4x + - dan tentukanx

    titik maksimum. 14marksl14markahl

    Answer Jawapan' . . . . . .20t--JI l4 l

    Given =# andx-!,[email protected])a'

    1- _1Diberi y=2-: dan

    Isi)axdy

    = 3g(r) where (r) sa functionf x. Findhevalue f

    = 3g(x) di mana g(x) ialah ungsibagi x . Carinilai

    347211 2oo9HakciptoJPNJAnswer Jawapan

    14marks)14markahl

    SULIT

    2l

    I r--I l4 lt l

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    SULIT 1l 3472t1

    Table22 shows he marksof a groupof studentsn an examination.Jadual 22 menunjukkanmarkah sekumpulanpelajaryang diperolehidalam suatupeperiksaan.

    Marks 1i ?Number f students 4 6 8 7Iable22Jadual22

    FindCari(a ) the maximum alueof x if he modemark s 8.nilai markah bagi x jika markahmod ialah8.(b ) the minimum alueof x if he meanof hemarks s greaterha n3nilai minimumbagi x jika markah min ebih daripada3.

    [3 marks]13markahJ

    Answer Jawapan

    Find he number f ways nwhich4 boysan d5 girls an be seated n a row fCari bilangan ara susunan4 budak elakidan 5 budakperempuan olehduduk.(a ) there s no restr ict iontiada syarat dikenakan(b ) the boysand hegir lsar eseated llernatelypelajar elakiduduk berselang selidenganpelajar perempuan. 14marks)14markahl

    Answer/Jawapan: a)

    317211 2007 lok Ciota JPNJ(b) . , . . .

    IL ihatsebelahSULIT

    22.

    22t--_lI r---I l r l

    )7tlI r-1I l4 l\ l

    (a)(b)

    23.

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    l8SULIT

    24. f abire 4 shows he numberof colouredmarbles n a bag.Jadual 24 menunjukkanbilanganguli bervvarna i dalam satu beg.

    3472t1

    Colour Number f marblesRed 4Green x+2Purple 2Table24Jadual 4

    A marble s drawnat random rom he bag.Given hat he probability f gettingaIgreenmarble s -. Find hevalueof -t .

    Sebijigulidikeluarkansscara rawak daripadabeg. Diberi kebarangkalianmendapatIsebiji ul ihi jau alah . Carini laix.J [2 marksl12markahl

    AnswerJawaDan . . . . . . . . .347211 2ao9HakciptaJpNJ

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    Forexonrineruseonlr-

    25

    l4 l( )

    l9SULIT 3472/l

    Diagram 5 showsa standard ormal istributionraphRajah25 menunjukkangraf taburannormalpiawai.

    The probabilityepresentedy he shaded egions 0.7514.Kebarangkalianawasanbedorek alah0.7514.(a ) Find hevalueof &

    Carinilaik.(b ) X is a contin uousandom ariables hich s normally istributed it ha mean80 and a variance f 9. Flnd hevalueof X when he z-scores t .

    X ialahpembolehubahawakselaniar angbeftabur ecaranormaldenganmin 80 dan varians . CarinilaiX iikaskor'z alah k .14marksl14markahl

    Answer/Jawapan: a)th ltv, , . . ' .

    END OF QUESTIONPAPER

    341211 2oo7Hukc'ipra I'NJ ILihat sebelahSULIT

    25 .

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    SULIT

    L Thisquestion aper onsistsf25 questionsKertassoakn ini mengandungi5 soalan2. Answer ll questions.Jawabsemuasoalan3. Writeyouranswersn thespacesrovidednthequestionaper.Tulisawapanandadalam uangyangdisediakan alsmkertassoalan.4. Showyourworking. t mayhelpyou o getmarks.Tunjukkanangkah-langkahentingdalamkerjamengiraanda. ni bolehmembantu ndauntukmendapatkan arkah.5. If youwish ochangeouranswer,ross ut heanswerhatyouhave one.Thenwritedown henewanswer.Sekiranyaanda hendakmenukarawapan,batalkanawapanyangtelah dibuat.Kemudian ttliscwapanyangbaru.6. Thediagramsn thequestionsrovided renot drawn o scale nless tated.Rajahyangmengiringisoalan fulak ilukis mengihttskalakecualidinyatakan.1. Themarksallocatedor each uestion reshownn brackes.Markahyangdiperuntukkan agi setiapsoalanditunjukkan alamkurungan-8. A listof formulaesprovidednpages to 5.Satusenarai umusdisediakan i halaman3 hingga5.9. A bookletof four-figuremathematicalablessprovided.Sebuah ukusifr matematik mpatangkadisediakan.10.You mayuseanon-prograrnmablecientific alculator.Anda dibenarkanmeng,gunakanalkulatorsaintifk yangtidakbolehdiprogram.I l. Hand n thisquestionapero he nvigilator t heendof theexamination.Serahkan ertassoalan ni kepada engawas eperilcsaani akhirpeperiksaan.

    3472/1 2oo9HakCipta PNJ

    20INFORMATIONFOR CANDIDATESMAKLUMAT UNTAK CALON

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    suLlr3472t1AdditionalMathematicsKertas 1September 0092 Jam

    347211

    JABATANPELAJARANNEGERI OHORPEPERIKSAANERCUBAAN PM 2OO9ADDITIONAL ATHEMATICSKertas1

    JANGANBTJKA UKUJAWAPANSEHINGGA IBERITAHU

    MARKINGSCHEME------J

    l(ertas oalanni mengandungihalaman ercetak

    [Lihat sebelahSULIT47211 lak CiptaJPNJ 2009

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    Submarks Totalmarks

    82B1

    2B1

    .x IJ =* - 9l8(3-) = 227(3 ' ,)g(3.1=23*.33 3*.32 2

    I

    I f-/ "l--:I '81

    I a " : ,l iI i

    rog,= os,qlu,g,i i log, =19"sJ-orany aseI I log19i tt-- T-_-I I l tl l "i l^ lI l )oo=-. . - . -i i r li l1t-*-f"---

    :i ,o i n=12i tI i ,,' t:*ol_:5rli?_ogi i , , - 2(2)i j ; [ : r -s)+( i?- l )4 j ,2ooi1-I is"200i--]*-i t r 1, .=:I i l , 18t ti i2 vi tf--T-_-i 12 j m=16andn=1.7411i ii l tos,m=4or- loez=-! or logt li I log, ,= logz t_"rlog,l ii lL_t_-

    J

    82B1

    ATlog, n

    L__,-.----_-L

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    S[b marks T- ; - | Totalmarksl=6ands=-20t=6 or s=--20s+ 3/= -2,2s 4t =16(Solve imultaneousquation)s+l+2t+14

    1'l .r ol-l - L-ol2' l ol4l

    ?

    821lo 2 o

    I-_* ' Ia l i a It l+ r - rA(0,4) rD(0,-1)

    2B1either aor-b

    (a) k=2a' ) * - 2 qk +7=2)@,: =3),"2

    g( f t+ )q ::h= l(2q-3[) @ equivalent2-nS = .1.pQ

    . - - ]

    /,l/\ \

    .( l t, \! \

    \

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    IIIIIIIt_

    t. 1r l l - O-(a) v ' -pl r - p '

    .e r : ;01 srn--=^l '?\ l )Ll^( g\o: l - 2sin ' l - |' \2)

    2.044 cmz1^1: (5) '(0.6435)_ aX3)2' ' 2l11$)' (0.643s) r 1{+X3)Shaded region = Apon AponIPOR = 36.870 or 0.6435 acl

    .V 1.44-0.06 -966y=--:x0.02- aa3Z) , ,? = - 4(21)x'5 or -4(24) 5dx

    ( -t \| -: .4 I\2 )d'y 2 . d2, ,-;* = -; ?r.rd=4 ,=-16dx ' x' ' dx '

    , l4-L- ' 2

    Q- = q- --1-o equivatentclx x'

    4B382

    4B382

    I

    IIB3

    82B1

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    4.;J

    ZA:i--J 2(--1) 3, - . ]l?r" -11't -" " ll r lIx- l

    t Izx-: l; t ? Ir l_ x- I B1(a)7(b) r=1x>0I*:31:J":!l , s25+x

    x t-28+x

    L2

    B1I

    B1

    Ia

    (a) k-0,679 2B1(Z>k;=A.2486

    I I (b)r=8203'i lI lo67e=tj lo(b) .r = 82..031

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    STJLIT

    AdditionalMathematicsKertas2Sept2009^ l- lam2'

    3472/2

    JABATAN PELAJARAN NEGERI JOHORPEPERIKSAANERCUBAAN PM2OO9ADDITIONAL MATHEMATICSKERTAS 2

    Du a am tigapuluh minit

    JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

    1. Kertqssoalan ni adalah dalan dt,ihahasa.2. Soalandalam baha,sanggeris mendohului oalunyung, epadundalcnnbahal;a lfelayu.3. Calon dikehendaki nembacemaklumaldi halarnanbelakangkertas soalanni.4. Culon dikehendakimenceraikanwlamctn 9 dan katsehagainukahadapanhersama- ama de, gan atvapan awla..

    Kertas oalanni mengandungi9 halaman ercetakan halarnanosongJ+ LLt ,1. SULIT

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    sulrT 2 347212The ollowingformulae may behelpful n answering hequestions. Thesymbolsgiven arethe onescommonlywed.

    -b+ "[Y 4o;x= 2a2 q^ xan= e^* '3 a^ +t{=a^'n4 (am)"= an n5 logo mn = log um * loga n6 log"m:bggn-logonn7 log attn : n logum

    tog,a fo8'blQgc aT,:a+.(n- l )ds^: l tza+(n-r)e l lTn = Qt'n' l

    ;= a(r" - l ) =a(! - r" ) , r *1" r- l l - rs- = o l r l . t l - r "

    ALGEBRA8

    t0l lt2IJ

    CALCULUSn

    " dy d), dudx du dr

    dv dv duv=uv. L=u-+v-'&drd\du dv

    ..uqYdxdr--:_=-._*-;-'vdxu-

    Areaundera curyel= lvdx orJ'

    b= lxdyJ'Volume generated-- llry' aY orJ-

    b-- l|d.- dv

    GEOMETRY

    'l Distance=2 Midpoint

    A pointdividing segment f a l ine( tu, -.*. , rv, mv"\lx, v) = \ frr+n m+n )Areaof trianglel,; l (x,1, + x2y3 x.! , , ) - (xry, + xry, + x,yt) l

    (x,+x,(x.y)= l-t2l l f )l r l={ . r -+}-^ xi+vi

    17 tVx-+-Y-3412t2

    -Y, .Y: )' ) l

    SULIT

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

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    SULITSTATISTICS

    3472/2

    l_i I ( - r -x)23o=tr t_-= l=' .l ) x-v1/

    ; zI,'w,Lw,, , _ n!" {n- r1''.n^ nl| =_--r fu* r)!rr 'P(Aw B)--P(A)+P(B) P(Aa B)P(X=r) ='C,P'qn- ' , P+ q = |Mean, p : npo =.[rW__x-po

    , - &t/zf,Zr

    [ . !"-rlr+l2 lcl f " l

    4 o=

    5 m=

    6 I =9x100O"

    l01112

    t4

    TRIGONOMETRYI Arclength,s=r02 Areaof'sector.e= ! ,'e3 sin2l * "or 'A: |

    2

    4 sec2A: +tan2A5 cosecz=1+cot2A

    6 sinM = 2 siM cosl7 cos2A: cos'A- sin2A=2cos2A-r= l- 2 sin2A

    ) tqn /8 tan2A: '* " : 'l - Ian'A

    3412t2

    sin (l + 8) : siM cos8 + cosl sin-Bcos l + B) : cosl cosB F sinl sinB

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

    9l0

    t2

    t314

    sinA sinB sinCa2= b2+ c2 2bccoslAreaof triansle = f aDsinC-2

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    SULIT )L+ILILSectionABahagianA140marksl140markahl

    Answer ll questionsn thissecfion.Jawab emua oalan.1 Solvehe ollowingimultaneousquations:Give ouranswersorreclo three ecimal laces.

    Selesaikanersamaan erentakerikutBerikanawapanandabetulkepada iga itikperpuluhan.P+2q =l

    p' -3p+2q' =3[5 marks]15markahl

    2 A curve asa gradientunction o.t2x, wherea is a constant.he tangentothe curve t hepoint(1,3)s perpendicularo hestraightine 2y = *x +3 ,Satuengkungmempunyaiungsi ecerunanxt -2x , di mana a adalahpemalar.Tangen epadaengkung ada itik 1,3)adalah ersereniangengan aris urus2Y=-x+3'FindCari(a) thevalue f a, [3 marks]

    nilai agi a , 13markah)(b) the equation f he curve. 13markslpersamaanengkung ersebut. 13markah)

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    SULIT 347212Ahmadhas 1000 hickensn hispoultryarm.Everyweek, ewillsell40of hischickens.Ahmadmempunyai1000ekor ayamdi ladangayamnya.Setiapminggu, iaakanmenjua! 0 ekorayamnya.(a) Find he otalnumber f chickeneft n hispoultryarmafter 1't week.

    Caribilanganayamyang masih inggaldi ladangayamnya e/epasmingguke-21{4 marksl14markahl

    (b) Thecostof eedin g ach hick s RM2perweek.Find he otalamount f moneyhathespent n he emaininghickenor he irsttwelveweeks.Perbelanjaan tasmakanan eekor yamadalahRM2seminggu.Kirakanumlah perbelanjaanangdibelanjakan tasumlahayamyangtinggaluntukduabelasminggu angpertama.. [3 marks]13markahl

    HeightTinggi(cm) Number fplantsBilangananaman20 -29 430-3940-4950-5960-69 5I ! .J_ IY 1

    TableJadual4Table4 shows hedistributionf heheights fplantsn a gardenGivenhemedian s47.5,ind hevalue f aJadual4 menunjukkanaburan inggi anaman i sebuah aman.Diberi medianadalah 7.5,carikan ilai a .Hence,ind hevariance f hedistribution.Seterusnya, ari variansaburaninggi anamanersebut.

    13marksl13markahl13marksl13markahl

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    SULIT 3472125 Solutions y scaledrawingwillnotbe accepted'Penyelesaian ecara ukisanberskalaidakditerima.

    Diagram shows hombus QRS. heequation f PS s 3y +7x = 33 and heequation f PR is .Y -r + I I 'Rajah menunjukkanombus QRS.Persamaanaris urusPS alah3y +7x

    =33dan persamaan ais lurusPR ialahy = -'r + I I

    DiagramRajah5(a) Find

    Cari(i ) the equation f QS ,

    persamaangaris urusQS,(i i) th ecoordinatesf S.

    koordinatbagiS.

    [3marks]13markahl[2marksl

    12markahl(b ) A point moves uch hatSf :I Q = 2'.1 Find heequation f he ocus f f,

    Satu itikT bergerakdengankeadaanSf :fQ = 2'.1 . CaripersamaanokusT.[3 marks]13markahl

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    SULIT 3472/2

    6 (a) provethat tos2l = cos 4 sin . t2 markslcosl+s in lBuktikan ahawa kos2A =kosA-sinl t2markahl

    kosA+sinA(b) ( i ) Sketchtheraph f /=s in2r+2 for0

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    SULIT

    SectionBBahagianB140marksl140markahl

    Answer ny ou r questionsrom hissection.Jawabmana-mana mpatsoalan aripada ahagianni.

    7 Diaoram showshecurve =x2 +2 and hestraightiney=-x+8 '

    DiagramRaiah7

    FindCari(a) th evalueof k,

    nilaibagik,(b) th eareaof heshadedegion,

    luas antauberlorek,

    3472t2SULIT

    34',7212

    [3 marks]13markahl14marksl

    14markahl(c) th evolume eneratedn erms f a, when he egion ounded y hecurve'

    they-axisandy = 6 is revolved60 'abouthey-axis' [3 marks]tsipaduanaan,dalamsebutan , apabilaantau angdibatasi leh engkungitu, paski-ydany = 6 dikisarkan elalui360' padapaksi-y, B markahl

    ,=x2+2

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    SULIT 3472t2

    Diagram shows semicircle ADEB, entreO andsector OFC withcentreGiven hatOB= 5 cmand O = 15cm.

    Rajah menunjukkanebuah emibulatanADEBberpusat dansebuah ektorTOFCberpusatT. DiberibahawaOB = 5 cm dan O = 15 cm.

    10

    AOBDragramRajah9

    [Use/Guna =3.142)CalculateHitung(a) 4.TCO,

    4TCO,

    (b) th eperimeter,ncm ,of heshadedegion,perimeter, alamcm,kawasan erlorek.(c ) th earea, n cm2, f heshadedegion.

    luas, alamcm2, awasan erlorek.

    11markl11markah)

    [5 marks]l5 markahl14marksl

    14markahl

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    l211(a) ln a survey arr ied ut n a school,t s ound hat2 out of 3 students assedtheir Mathematics est.

    Dalam satu injauanyang dijalankan e atasmurid-muriddi sebuah sekolahdidapati2 daripada3 orangmurid lulusdalam ujian Matematik.(i ) If 7 students rom hat schoolare chosenat random, alculate he probability

    thatexactly studentsassedheirMathematicses t [3 marks]JikaT orangmuriddaripada ekolahtudipilih ecara awak,hitungkebarangkalianahawa epat6 orangmurid ulusdalamujianMatematik.13markahl( i i ) f here re600 tudentsn heschool,ind henumber f students ho ai ledthe Mathematicsest. 12marksl

    Jika terdapat60A orangmurid dalam sekolah tu, cari bilangan orang murrdyang gagal dalam ujian Matematik 12markahl(b) A surveyon body-masss doneon a groupof eachers. he massof he eachershas a normal ist r ibut loni tha meanof 45 kg andstandard eviat ion f 10 kg

    Satu kajian terhadapberat badan sekumpulanguru telah drlakukan.Berat Badanguru- guru adalah mengikut aburannormaldenganntrn45 kg dan sisihanpiawai10 kg.( i ) A teacher s chosen t random rom hegroupFind he probabi l i tyhat he body-mass f he eacher s ess han42.5kg.

    Seorangguru dipilihsecara awak darrpada umpulan ersebut.Cari kebarangkalian ahawaguru tersebut nentpunyai erat badan kurangdari 42 5 kg.( i i ) l l 12.3% f the eachers avea body-mass f more hank kg, ind hevalueof k.

    Jika12.3% uru ersebulmempunyaieratbadan ebihdarik kg, carinilaibagik. [5 marks]15markahl

    SULI ' I '

    3172/2SULI'T

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    t3SectionC

    BahagianC[20marksll2AmarkahJ

    Answerwo questionsrom hissection.Jawabdua soalan aripada ahagianni.

    12 A particle moves long straightineandpasseshrough fixedpointO.l tsveloci ty,ms-t ,sgiven y r'= 6t ' -4t-2, where is he ime, n seconds,afterpassing hroughO.Suatu arahbergerakdi sepanjang uatugaris urusdan melaluisatu titik etapOHalajunya, ms-t diberi leh t '=6t2-4t-2,dengankeaclaan ialahmasa,dalamsaaf,selepasmelaluiO.(a) Find

    Cari(i) th e nitial elocity f heparticle ,halaju walzarah ,(ii) it svelocity t he nstant hen heaccelerations zero,

    halajunya adaketikapecutan dalah ifar,(iii) he ime nterval uring hich hepafticle ovesowardshe eft,

    Julatmasaapabila arah tu bergerak rahke kiri,the distance,n m, travelled y heparticleuringhe irst hree econds,[3 marks]jarak,dalam m,yangdilalui lehzarahdalam iga saatpertama, 13markahlSketchhevelocitytime raph f hemotion f heparticleor0 < I < 3 12marks)Lakargrafhalajumelawanmasa agipergerakanarah tu untuk0 < I < 3.[2 markah]

    I Lihat setrelah

    sut_t'f

    3172/2SULIT

    3412t2

    11markl11markahJ12marks)

    12ma*ahl

    12marksJ12markahl(b)

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    SULIT14 Diagram 4 shows trianglePQR.

    Rajah 4menuniukkanebuah egiigaPQR

    Diagram 4Rajah 4

    (a) Calculate.PQR,Krakan {PQR,

    3472t2SULIT

    15 3472/2

    [2 marksl12markahl

    12marksl

    (c)

    (b) Sketchand abela trianglePQ'Rwhichhasa different hape rom rianglePQRsuch ha t ength fPQ,PR and PR Q remain nchanged.State Pp'R,Lakardan labetkan ebuahsegitigaPQ'RyangberlainandaripadasegttlgaPQRdalam rajahdi atas,dengankeadaan anjangPQ, PR dan PRQ dikekalkan'Nyatakan .PQ'R,Hence, alculateSeferusnya, irakan(i) the ength f Q'Q, n cm,panjangQ'Q, dalamcm,(i i) th eareaof he riangle RQ',ncm2.luassegitigaPRQ', alam cm'.

    12markahl

    [6 marks]16 markahl

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    SLTLIT l( ) 34'.12t215 Usegraph aperoanswerhis uestion.Gunakan erlas rafuntukmeniawaboalanni .

    Afactory roducestwoypes ffurni ture, and B.Each urniture eeds typesof ra wmaterialsP andQ and he number f each awmaterials eededor each urniturerepresentedn heTable 1 below.Sebttah ilangmengeluarkanua enisperabott aituA danB.Setiap erabotmemerlukan ahanmentahP danQ dan bilangan ahanmentahyangdiperlukanbagi setiap erabotditunjukkanalamadual15 di bawah.

    Table 5Jadual 5Thenumber f rawmaterials left n he actorys30and thenumber f rawmaterials left s 24.It sgiven hat henumber f urni ture producedsatmost wice henumber ffurniture producednd he actory roduces unlts Turniture and units ffurniture .Bekalan ahanmentahP yang inggal alam ilangadalah30 manakala ekalanbahanmentahQ yang inggal alam ilang dalah 4. Diberibahawa ilanganperabotA adalah elebih-lebihnyauakali bilangan erabotB dankilangtumengeluarkan unit perabotA dany unitperabotB.(a ) Write hree nequalities,ther hanx > 0 andy > 0, which atisfy l l he constraintsaoove. 13marksj

    Tulis igaketaksamaan,elain > A dany > 0, yangmemenuhisemuakekangan i atas. l3 markah)(b ) By using hescale f 2 cm o 2 units n hex-axisan d2 cm o I uniton hey-axis,constructndshadehe egion which atisf iesheabove onstraints.[3 marks]

    Denganmenggunakan kala cm kepada unitpada paksi-xdan2 cm kepada1 unit pada paksi-y, inadan orek antauR yangmemenuhi emLtaekangandi atas. 13markahl

    3472t2ST]I,IT

    Number f rawmaterialsBilanqan ahanMentah

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    SLILIT

    3472t2SUI,IT

    t7(c) Useyourgraphn 15(b),o in dGunakan rafandadi 15(b), ntukmencari

    (i ) th emaximum umber f units f urniture,Broducedf 4 units f urnitureareproducedBitanganmaksimum nitperabotByangdihasilkanika bilanganunitperabotAyangdihasilkan adalah 4.

    (ii) the maximum rofit btained y he actory f he profitrom hesaleof a unitof urniture /'sRM 20 0and heprofitrom hesaleof a unitof furniture isRM2s0.Keuntunganmaksimumyangdiperolehika keuntungan aripadaualanseunitperabotA adalahRM 200dankeuntungan aripadaualan seunitoerabotB adalah RM254. 14marksl14markahl

    END OF QUESTTON APERI Lihat sebelah

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    Jq/Zl!9Nama:Kelas:ArahanKepadaCalon

    I Tulisnamadankelas nda ada uang angdisediakan.2 Tandakan { ) untuk soalan angdijawab,3 Ceraikan eiaianni dan katsebagai ukahadapanersama-samaengan ukujawapan.

    I Lihat sebelah3472t2SULIT

    Bahagian Soalan SoalanDijawab MarkahPenuh MarkahDiperolehi(UntukKegunaanPemeriksa)A 5

    2 6J 74 65 86 I

    B 7 108 109 l0t0 t01i 10

    L 12 1013 t0T4 10l5 10

    Jumlah

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    SULIT 3472/2

    INFORMATION FOR CANDIDATESMAKLUMAT UNTUK CALON

    l. This questionpaperconsistsof threesections:Section A, Section B and Section C.Kertas soalan ini mengandungi iga bahagian: Bahagian A, Bahagian B danBahagian C.

    2. Answer al l questionsn SectionA, four queslions rom SectionB and two qucstionsfrom Secfion C.Jala,absemua soalqn dalam Bahagian A, empot soalan daripada Bahagian B dandua soalan daripada Bahogian C.

    3. Show your working. It may help you to get narks.Tttnjukkan langkah-langknh penting dalam kerja mengiro anda. Ini boleh membantuanda untuk mendapatkanmarkah .

    4. The diagrams n the questions rovidedare uot drawn to scaleunlessstated.Rajah yang mengiringi soalan tidak dilukis mengikutskala kecuali dinyatakan.

    5. The marksallocated or eachquestionand sub-partof a questionareshown nbrackets.Markah yang diperuntukkan bagi setiap soolctndttn kraian soalan ditunjukkandalam kurungan.

    6. A list of formulae s providedon pages2 to 3.Satu senarei rttmus disediakandi halarnan2 hinggo 3 .

    7. Graphpapersareprovided.Ke rt as gr af di sedi akan.

    8. You may use a non - progralnmable cientificcalculator.Anda dihenarkan menggunakankalkulutor saintifk yang, idak boleh diprogram.

    20

    3472t2SULIT

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    SULIT3472t2AdditionalMathematicsKertas2September 0092'h Jam

    3472t2

    JABATANPELAJARANNEGERI OHORPEPERIKSAAN ERCUBAAN PM 2OO9

    ADDITIONALMATHEMATICSKertas 2

    JANGANBUKA KERTASSOALAN N!SEHINGGADIBERITAHU

    RKINGSCHEUIE

    Kertas oalan nimengandungi7halaman ercetak

    [Lihat sebelahSULIT347212 Ha kCiptaJPNJ 2009

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    SULlT

    [ ]"*_

    2BAHAGIAN

    Solution

    3472t2

    Submarks roGimarks1- np=1-2q Orq=-- ' . lP l I2 l l

    Eliminate or q"(1 2q) ' - 3.(1*2q) + 2q2- 3 = OOrp2-3p +2"(\P )2-3*-0?_or eouivalent

    6q'+2q-5=03p'-8p-5=0

    Solve lrequadraticequation y using [email protected]@completinqhe sq-z r J+ "(oX-sj2(6)ri i&4:ll3&t2(3)

    q=0.761,-1.095orp=-0.523,3.189

    Ip: - 0.523,.189(9q = 0.761, 1.095

    if the workingcl fsolvingquadratic quations not shown

    ILihat sebelahSULIT

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    t_

    Jt*+"' 2x)rtx

    a=4

    Solution

    SULIT 347212

    2a)

    i, ,"/ - axt -zxdx

    2y=-y+JIIfl'ln = - -2

    *1r=2E]substitute= 1 n o axo2x = 2a (1) ' 2(1) '=

    Submarks

    [Lihat sebelahSULIT

    Totalmarks

    PFIIntegrate =

    4xo 2x'v ------.--'A. ) +L {-c

    Substitute= 1,y=3 into4xa 2x2y = :.i + c, to find he

    valueof c.y=x4-x2+3.

    Nqte:.y: J(*4x'' - Zx)h must have at least1 ter"m ith power ncreased yAt.

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    A"t

    - -Solulion347212

    tr_lUseTn-a+(n_1)d721=40+20"(40)

    1000-*840=160

    b)

    OR otherval idmethod

    l.JseSn= !; tz^+ (n- 1 )dlI rrrooo) 11x-0)l

    l , lfl-ihat sebelahrUILI

    Sutrmarks Totalmarks

    3472t2 HakCipta.JPNJ 009

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    *L=39.5or*F=7or "fr=d lPl IUse median ormula47.5=.39.5 + (

    al' \ ,^I r ' -Y--1l-* (7\2\ 1 '

    Zf*= 1425rZl*- 72917smFfr, 'tGt" f 4- | ---st - t -

    Submarks

    Using he formuiadistance ,,n7- - - - . -

    / t r t \Jtr- : ) ' +(y-4) '=2JG-7) ' r ' -8) ' \ :_l\-_,9r1rqv1-qql_-PvJ?7--q--]=Y

    SULIT {- --olution ---

    347212

    [Lihat sebelahSULIT

    *aWith "f , and F correspondingo "L

    /+-"1Zf - t-J

    72917.5 ,1425,2-30-- ' -30- ' '

    i i ) 3y+7x=33y=x+1y=x+1

    s (3,4)ST = 2TQCan be implied s ormula. Pl I

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    9UUr[_i-- m347212Solution

    Use dentityCos2A-Sin2A=Cos2A

    LHS= RHSNs mistake allowed

    b) i )

    Graph inl per ioCin sxs nAmplitude

    Drawing f thestraightine rom he equationnvolvingX and y, eithergradientOR y intercept f straightl-inemust be correct.

    Fr*ltir*lE]

    [Lihat sebelahgUL[

    ( rr

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    SULIT 7BAHAGIAN

    3472t2

    Submarks

    [Lihat sebelahSULIT

    Totalmarks

    c)

    Solvehesimultaneous

    lntegrate2+2

    lntegratex'+2)+ areaof riangieT +rc3

    lntegratez x2n t tt'- 2),1t,

    x?'+2=*x+8x2+x-6x=2,x=-3

    F2 u '1Use imit I ' in o [ \x) - +2xJJOr find he area riangle% (6)(6)

    in . , rQ)| -2yl

    IiL

    -n

    (24.667)

    l imi tse/ \\ Kl /\*__r/

    \I\

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    SULIT

    Frcmgraplr ): againtb=- 2.45[-2.S0

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    3472t2vxI

    7

    6

    5

    4

    3

    2

    :ir^"r,t 'I.4

    iir{r i. .^t; !1i,l

    7212Hak" i$t5J]rNru''20f.rs

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    SULIT 3472t2t

    Solution

    10a)i )i i )i i i )

    Use he riangleawfor Of or g..t or .V

    1)or =lo. io

    2;q)J

    I;0

    t2- -p+;QOJ N1

    f f i=6q1p*IQR=n1-c+1n)

    Use he triangle aw o findoQ=on*nO171j tPo Jn,q-ntp*nq=q

    1arJn=--- .m=--1A.4- a{Both arecorrectl

    ,.-*\ Comparehecoefficient/ Kt )\*-/ of p andq.l 1rn. n=0JL

    2-m+n;1_)af-(-n)+n=1J1 -

    Jm= --n2

    [Lihat sebelahLULlT

    347212 Ha kCiptaJPNJ 2009

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    SULIT t2 347212

    [Lihat sebelahSULII

    Solution11a)i) 2 ----1I lpr I

    0.2048UsenqI600(: )") .

    J

    /^\\ Kr )

    b)i) r-\\Kr/- \ oaorsl_rurI

    k-45 = 1.16l0

    Submarks Totalmarks

    -\-____,zoo !!1 |

    UseZP(2.#",P(Xtk)=0.123p(z> L:45 )=0.123 i0

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    SULIT 13BAHAGIAN

    3472t2

    [Lihat sebelahSULIT

    Totalmarks$olution Subrng$s12a) )i i ) ,1, ,Useal = 0. o finddt

    1t= laI- | , ,y=tr( - ) -J

    6=.---J

    1I*4G) -2a_)

    i i )

    b)

    v

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    Ztw = (130)1 (140)3 (115)2n (120)nt-t , I- \ tw r--rUse = T, = 125 \.KIJ

    t\1_Nln=2

    ,,=-13,:t l100t32I {tl r!$;) +'l:():!_?g)l0

    sur=rT

    2'60 x I00= 130X \_,Ll!]_] x=RM2.0o

    347212

    [Lihat sebelahsuLlI

    14

    Solution13a) User 9- x loo ( l )

    eon rvv

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