malacca_stpm_trial_maths_t2_2011

8/3/2019 Malacca_STPM_Trial_Maths_T2_2011

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ffi-t

SULIT954/2Mathematics 220113 hours

954/2

MAJLTS PENGETUA SEKOLAH MALAYSIACAWANGANMELAKA

PEPERIKSAAN PERCUBAANSIJIL TINGGI PERSEKOLAtrAN MALAYSIA 2011

MATHEMATICS TPAPER2Threehoun

Instructions to candidates :DO NOT OPEN TEIS BOOKLET UNTILYOU ARE TOLD TO DO SO

Answer {l qu*tions. Answers may be written in either English or Malay.AII necessaryworking should beshown clearly.Non-exact numertcal answersmay begiven correct to three signifi.cant gures, or one decimalplace in the case of angles in degrees,unlessa different level of accuracy is specified in thequestion.Mathematical tables, a list of mathematicalfornulae and graph paper areprovided.

Arahan kepada calon:Jawab semtt soalan. Jawapan boleh ditulis dalam bahasa nggeris atau bahasa Melayu.Semuakerja yang perlu hendaklah ditunjukl


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CONFIDENTIAL* 2*This questionpaper s CONFIDENTIAL until the examinations over.

3

Find the generalsolutionofthe differential equationdv__ vi lx rz - 4'Given thaty = 1 when : 6, find theparticularsolutionby expressing intermsof x.

Find the valuesof S,where0" < 6 < 360o,which satisfu he equationws28 -- taza E.

In thefigure below,PE : 32 crnandz 5TR: z ?QR.R

(a) Statewo similar trianglesn the figure.O) Find the engthsof ?5 andFf.

At 12.00noon, a runawayhorse s at apositionof [2 i *.11]krn relative o afixed origin o and a hunter'sposition vectorrelative o the sameorigrn o is{-; - +;) tm. The horse s gallopingwith velocity {3 i *j} Ivah-a whenthe hunter s pursuing t with avelocity of (11 t + 3t Isnh-l(i) Find thepositionvector of thehunter elative o thehorseat time t.(ii) Calculatehe closestdistancebetween he horseand hehunter and

the time when heyare at this distancapart.

[3 marks]

[2 marks]

[5 marks]

[1 marks][5 marks]

[4marks]

[5marks]

srPM 95412*This questionpapers CONFIDENTIAL until the examination s over.

20 ctn


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CONFIDENTIALf

6

ln thefigureabove,?Q s thecommonchord or bothcircles. O s apoint onthe circle PGQ.ROSs the tangent o the circle POQat G. R*t and$N aretangentso thecircle FQNI{ at lvIandN respectively.aP andOQproducedmeet hecircle PqN$ at M and '{ respectively.Prove hat

(a) Rs s parallel o t{t'[,O) 4N$o:LMRo.

A storageankhas ahorizontalbaseof area5O0m2. At time t : 0, it isemptyand waterbegins o flow into it at a constantate of 3S ruxs-l. At thesameime,water begins o flow out ata rateproportional o f Ir, where r mis the depthof thewaterat time t s .

dh--: 0-04.dt(i) Show hatthe rateofir is givenby

dhdr :0.o2ts-{n).By using he substitutionr : 3 - r/8, show hat thoequationnpart i) becomes t" - **: o.o1r.

(ii) Giventhat: : 3 whent : O,solve his differentialequationandexpress in termsof r.

(ii| Findthe timewhen hedepthof waterreaches m'

[4 marks][8 marks]

[5 marks]

[3 marks]

[3 marks][2 marks]

When Ie: 1,

STPM 95412*This questionpapers CONFIDENTIAL until the exanination s over.


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CONFIDENTIAL*

7 For anymarriedcouplewho aremembersof a tennisclub,theprobabilitythatthehusband asa degrees

!and heprobabilitythatthe wife hasa

' -1*degrees | . Theprobability thatthe husband asa degreg giventhat thewife has adegree, s f .A marriedcouple s chosenat random.

(a) Show hattheprobabilitythatbothof thernhavedegreess f. [2 marks](b) Find theprobabilitythat only oneofthem has adegree. [2 marks]

8 Ifthe discrete andomvariable is the numberofheadsobtainedn fourtosses fa fair coin. Find the probabilitydistributionofX .Hencg tabulateheprobabilitydistributionof

1Y:-- [4marks]a+ 2x9 A cafeteriahas wo coffeemakersd and . Thenumberof timesperweek

that.dbreaksdown hasaPoissondistributionwith mean0-3while thenumberof times hatE breaksdown n aweekhasapoissondistributionwith mean0.1.Fin4 correct o thee decimalplaces,heprobabilitythat n the nexttwoweeks,

(a) A will not breakdown at all,(b) eachmakerwill breakdownexactlyonce,(c) therewill be a total of4 breakdowns.

STPM954/2*This questionpaper s CONFIDENTIAL until theexaminations over.

[3 marks][3 marks][3 marks]


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CONFIDENTIAL*

10 The tablebelow shows he distance ravelled(in kilometer) rom their homesto the collegeby 10ff students.

Distance Numberof students0-4 25-9 rn10-L9 t320 -?9 1630-& 3tl45-59 2460-6S 11

(a) Draw a histogram o representhe abovedata. Hence,estimate hemedian. [4 marks]

O) Calculate he standard eviation or this dishibution andgive youranswercorect to one decimalplace. [3 marks]

(c) Calculate hepercentage f studentswho travelledat most58 knfrom their homes. [3 mmks]

11 Batteries or a transistor adiohave a mean ifespanundernormal usageof172hourswith a standard eviationof t houn,

Assuming hat the ifespansof the batteries ollow a normal disaibution,(a) find thevalueof } such hat 82!%ofthesebatterieshave a lifespan

of less han I hours, [3 marks](b) find the value of sif 60frrofthese batterieshave a lifespangreater

thanshours, [4 marks](c) calculate heprobabilitythattheradiowill operate or at least

162 hours f the radio uses our ofthese batteriesand equiresall ofthem o be working. t4 marksl

STPI0'{954/2*This questionpaper s CONFIDENTIAL until the examinatiql is 6ver.


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12 The continuous andomvariableX hasprobabilitydensity imction definedby

( lc , o1x12',f (x): l -z, la+z Ztx


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MARKING SCHEME - TRIAL EXAMINATION STPM MELAKA 2OI1MATHEMATICS T: PAPER2I r=-x2-4dydxdy

dxdy2)J(x - 2)(x+

dx=- x2-4 (separateariablescorrectly)dx(x-2)(x+2)

lny

4lnyy4

Y=7,x=6(1)4

K

1t 'J. I l= 4'LG4- 1,a21laxr f l 1 1 l= 4JlG-2) (r+Dld'1- ; Itn(r - 2) - tn(x+ 2)l + CwhereC = constanttx - 2't- tnlr]:rl + tnr wherenK = c-- tx-2\n\ r*z)

16-Zt=Kl6+z)'_. , lx-2\=21**z)4

lB 1llM 1l

lA 1ltM 1ltA 1ltsMl

2 cos202 cos20 - L

2cosa0-cos2e2 cosa0cos2o

= tan" 0sinz0cos20=!-cosz0

=l 7{2

cos0 = + 0.8409-- 32.77"147.23"2L2.77"327.23"= 32.8"147.2" 2L2.8"327.2"

lM 1ltM 1ltM 1l

tM 1l

IA I]t5Ml

Mathemotics72 20llTrial- Melafta - MARKINGSCHEME


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(a) A RIS andA RQParesimilar?s ,Rs/h'l\" , QP RPTS820 32:. TS = 5 cmRT TSRQ QPRT524 20:RT=6ctn:. PT = PR- RT

-(32-6)cm=26cm

lB 1l

lM 1llA 1l

tM l

lM lllA l lt6MI

4 | Let12.00hoursbet ime=0.h=horse.H=hunter.b. --G) rr'. = -i)Att=0, pVn

= :) (i)=-i)=fJ)-i) :) IM 1]u! -n. ! , .+tuln

= 0. '( :)/-3 + 8r\=t I\-5 + 2tl tM llM 1ltA1lDistanceapartbetween he horseand he hunter s the shorterwhen

afn ' f ln =0 (_;i l l l ='B(-3+86)+2(2t -s)=0

lB 1llM tl

Mathehaticsn 2011Trial- Melaka -- MARKINGSCHEME


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-34+6Bt=0.1 t = 0.5 hour

t = 0.5 Closest istance,a!'ldl= ,[tr +@ =,/v kn

_/1\- \-+)Timewhen heyareat thisdistance part= 12.30pm

tA1l

lA 1ltA 1lteMl

(a) zRoP - zOQPzOQP zNMPzROP_ ZNMP

RS / MN

(Angle in altematesegments)(Exteriorangleofa cyclic quadrilateral)

(Altemateangle)

tB1llB 1l[BUlB 1l

(b) Draw the ine OTl2 joinnglhe centerof two circlesIOIR.S (Tangent erpendicularo radii): .0P--OQ IIoM=oN (. :LPOT-zQoT) I

A OMT2 and A ONT2 are congmentf zOMTz- LONT|I(*) I zMOR - L0MT2 (Altemates'RS/lMN)IL zwOS= zoNTz (Al tematezs,RS//MN)

tB1ltBulB 1llB1llB llB tl

Martematics T2 2011Mal - Melaka - MARKING SCHEME


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-zMOR= zNOS

and LOMR = /- ONS:. zMRO = 1800-zMOR-zOMRandzNSO = 180" LNOS - LONSTherefore - MRO - L NSO

(From *)) lB 1l

tB l[12M]

6 l(D Iz-500hdVEdvdt

dll= 500--:-d.t-30-k lTd.ll

andl (*) lB1l500; =:O -

dh 30-K\NK!n

lM ll

dth- 1

dhdt

500)hIj = 0.04dt

2n -bo.o4 -"" ' -500k -L0

30 10\fitM 1llM 1l500

| /^ / i \=S0\r- ln)dhdt = 0.02(3- ,'lh) tA ll

Usinghesubstitutionx -3-tth, dll-: - = 0.02dtax -r: - -:--= | tM lI; - ^ t;-an zlndh dh dxdt dx dt

+Mathematicsn 2qllTnal- Melaka -- MARKING SCHEME


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-dx0.02 - -Zl h. dt__2(3_"#,, . f ' - 3) '# =o'01r

(ii) [' (, - 3\ a* = It o.o, o, (separateariablesJ3 \ x , Jo and imits)[ ' -: t" "]; = o'o1t

fx -3lnx - (3- 3ln3)l - 0.01

. . . r -1oo["-r*r ' "( i ) ]( i i i ) x=3-, ,14-tm

t = L00(3n3 - 2): . t=129.6s

tM 1l

tAutM1l

tM 1ltA1l

tM 1llA 1l[13M]

7 (a) P(Hlw)P(HnW)

=PJ.HJ\W)P(W)= P(Hlw) 'P(w)L7t=-x-t221,1Pl(H W') u (H'nW)l

., '= tP(H) P(H nw)l + lP(w)- P(Hnw)l3 1 r11r--r--? l- l ! t^, I5 Z \Z+/1150

IM 1]lA 1l

tMulA 1l[4 M]

MathematicsT2 2011Trial- Melaka - MARKINGSCEEME


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8 X is thenumber fheadsobtainedn four osses/ 1\x-B142)

P(X-0)=P(X-L)=D(V-)\ -r \ r \ - - ) -

P(x=3)=

P(X=4)-1Y=- L+zX

^r"(:)'G)"r"(:)'G)'",G)'(:)',,(:)'(i)'^r,(:)^ :)"

I1.6l r)6764T61

lo

(Y=y) I\ ti 1'7 LqD(V - ^, \ t-16

41A

6-

416

1-

iB l

[B 1] Any fourcorrect

[B 1] for valuev[B 1] conectas equired ntable orm[4 M]

9 | Let .4 = numberof times coffee makersA breaksdownperweek.

P(A, 91 e-0.60.60 I tu rl-0! |, i 0.5488 0.549 I to tto) BfPo(o.z) | ts rt

,4r = numberof timescoffee makersA breaksdown n next two weeks. IB = numberof times coffee makersB breaksdownperweek. I81 = numberof times coffee makersB breaksdown Iin next wo weeks. IA-Po(0.3)

I(a) Aa-Po(0.6) ltBrl

P(A1= landf,, - 1) - P(Ar= 1) x P(81- 1) I= (s-06x 0.6) e-0.2 0.2) | ttut t- 0.0539 | tA l l

Mathenmticsn 2qllTrial- Melaka - MARKING SCHEME


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(c) T = total number ofbreak downT = Ar* B1-Po(0.8)

P(T = 4\ _ e-o Bo.ga4t- 0.00767

lB lllM 1l[A 1][eM]

(a)

Frequency ensity

1. 6

l. l1. 00. E0.4

Median,m69.5Distancex

median mx/_1oo_(n2 - 2 -"":'m = 37 k:trt

D1scale,axislabel,allpointsjoinedDIThe sizeof thehistogramD1CorrectBoundaries

lA 1l

9.5 19.579.5- 29.5

44.5 59.559.5 69.5

MathematicsT2 2011Trial- Melaka -- MARnNG SCHEME


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(b)

Standard eviation o '':

= ^1288.46- 17.0(c) Let number f students ho travelled t most58 km - x .

/ 58-44.5 \x=65*[rn=-rr=, ; 'nx - 86.6Percentage f studentswho travelied at most 58 km

86.6_ _ x 1000/0- 86.6o/o

X/"\'tf ,lu3680r-( t* /

2733.25.5 19.579.5 - 29.541.0709.5 44.5

44.5 59.559.5 69.5 +a t oL.I J

f=I00= 164270

Mathematicsn 2qllTtial- Melaka - MARKING SCHEME


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l1 Let

(a)

X is the ifespan n hours of a batteryx-N(t72,92)

P(X < h)r h- t72tP\2. g )

P(Z< z)z

= 0.82= 0.82= 0.82= 0.915

lL L729

h-172 = 0.915 1720(b)

hP(x > s)

r s-172rP\2, s )P(Z > z)

aQ)z

s-L72s

P(x > L62)

- 180.235 180.2= 0.6- 0.6+ 0.6- f ta- -0.253- -0.253

(c)- 169.723 - 169.7

r t62 - 172t,-Plz> , ) _P(z>-1.11.1)= R(-1.111)= 0.8667

P al l 4 batteries wt for at least !62 hours) = 0.8667a-- 0.564

[M l] forstandardization

IM I]

tA1llM 1l

lB lltMutA 1l

tM ltA1ltM1l[A 1][1]M]

Mathematics72 2011Trial - Meldka -- MARKING SCHEME


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tM 1ltM 1l

lA 1llM 1l

lA 1IlM 1l

tA 1l

[B 1] (forr3)

=l iaxJo 3f* '1 '=l- lL6J,t26

v2Ftx)= 7=F(z)I,' !, * z) ," [ -z -l '- ' -+ l -L+zxl'13 ),= -'-5.,r--i.,)]

,x


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F(x)

0.8u.o

y=_T+2x_2.

v- x26.

11.522.s33.54

(c) P(x>1) =L-P(x

malacca_stpm_trial_maths_t2_2011

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