2008-ASLM&S

HONG KONG EXAM]NATIONS AND ASSESSMENT AUTHORITY

HONG KONG ADVANCED LEVEL EXAMINATION 2OO8

II

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MATHEMATICS AND STATISTICS AS-LEVEL

8.30 am -

11.30 am (3 hours)

This paper must be answered in English

l. This paper consists of Section A and Section B.

2. Answer ALL questions in Section A, using the AL(E) answer book.

3. Answer any FOUR questions in Section B, using the AL(C) answer book.

4. Unless otherwise specified, all working must be clearly shown.

5. Unless otherwise specified, numerical answers should be either exact or given to 4 decimal places.

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4. A and B are two events. A' and B' are the complementary events of A and B respectively.l9rSuppose P(A)=i , P(lvB)=i, PQqIB)=i and P(B)=* ,where 0

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8. Abiologist studied the population of fruit fly I under limited food supply. Let r be the number ofdayssincethebeginningoftheexperimentand NO bethenumberoffruitfly I attime /. Thebiologist modelled the rate of change of the number of fruit tly A by

N'(') = 'o ,- ('> o) 'I + he-^'where h and k are positive constants.

,rl

$tIi+

(a) (i) Express ,r[jl -

r-] u, a linear tunction of r .' LN'(t) 'l

(ii) It is given that the intercepts on the vertical axis and the horizontal axis ofthe graph ofthe linear function in (i) are 1.5 and 7.6 respectively. Find the values of h ard k.

(4 marks)

(b) Take h = 4.5 , k = 0.2 and assume that N(0) = JQ .

(i) Let v= h+ekt,find I .'dt

Hence, or otherwise, find N(r) .

(iD The population of fruit fly B can be modelled by

M(r)= x(t*L"-k'\*t\"/ -'[- t- ) " 'where 6 is a constant. It is known that M(20) = N(20) .

(l) Find the value of 6.(2) Thebiologistclaimsthatthenumberoffruitfly I willbesmallerthanthatoffruit ;

[Hint: Consider the difference between the rates of change of the two populations.](1l marks)

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10. Assume that the number of visitors arriving at each corurter in an immigration hall is independentand follows a Plisson distrlbution with a rneanpl-3.9 visitols per minute. A counter is classified asbusy if at least 4 visitors arriving at it in one minute.

":'

(a) Find the probability that a counter is busy in a certain minute.(3 marks)

(b) An officer checks 4 counters in a certain minute.counter is found.

Find the probability that at least one busy

(2 marks)

If 10 counters are open, find the probability that more thanminute.

7 ofthem are busy in a certain

(3 marks)

(d) Suppose l0 counters are open and one of them is randomly selected. Find the probability thatmore than 7 of them are busy and the randomly selected counter is not busy in a certainminute.

(3 marks)

(e) The immigration hall is called congested if more than 90% of the open counters are busy in aminute. Suppose 15 counters in the hall are open. A senior officer checks the counters in acertain minute. It is given that more than 7 of the first 10 checked counters are busy. Findthe probability that the hall is congested.

(4 marks)

(c)

1

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12. offrcials of the Food Safety centre of a city inspect the imported "choy s*" !y selecting 40samples of "Choy i.,." to. each lorry and testing for an unregistered insecticide' A lorry of;Crr'"v Sum" is .iurrin.a as risky if more than 2 samples show positive results in the test'

Farm,4 supplies"ChoySum"tothecity. Pastdataindicatedthat l% oftheFarm l "ChoySum"showed positive ,"*ti, in the test. on a certain day, "choy Sum" supplied by Farm A istransportld by a number of lorries to the city'

(a) Find the probability that a lorry of "Choy Sum" is risky'

END OFPAPER

(3 marks)

(b) Find the probability that the 5ft lorry is the first lorry transportitg rislE "Choy tu-',, marks)

(c) If t lorries of "choy Sum" are-insp_ected, find the least value of t such that the probability offinding at least o.t. iotry of risky "Choy Sum" is greater than 0'05 (3 marks)

(d) Farm B also supplies "choy Sum" to the ciry' It is known that l'5% of the Farm B "choysum,, showed positive resulis in the test. on a certain day, "choy Sum" supplied by Farm

IandFarmBistransportedbySand12lorriesrespectivelytothecity.

(i) Find the probability that a lorry of "Choy Sum" supplied by Farm B is rislE'(ii) Find the probability that exactly 2 of these 20 lorries of "choy Sum" are risfty'(iii) It is given that exactly 2 of these 20 lorries of "choy Sum" are rrsty' Find the' proUaUility that these i lorries transport "Choy Sum" from Farm 'B ' (7 marks)

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Solution Marks Remarks

l. (a)t,-r)(-r-,')(i) #=,.[])t*t.V]#(*)2 +"

o 3a2 ')= I --x+-I- +"'28

l-o -3lT- 2

"11t-=uIs

.'.a=-3 ana t={(ii) The expansion is valid fo, l! . !

(b) ..+=,*1[l).!(L\'\u' m

"z[:o] s[:oJ/io 843l- s-

1l q 8ooJi _2s2e

800

IM

IA

IM

IA

IA

IA

1A

For binomial expansion-l

on (l+ ax) 2For any two terms correct.pp-l for omitting '...'

For both

For both correct

For RHS, accept 1.05375

Accept 3.16125 and 3.161

(7)

2. (a) y3 -rry=|

tu' 9 -( ro'* r.) = o

" du I d, ")dv- vdu 3y2

-u

IM

IA

Alternative Solution'tlu=l---

vdu^l--

t lra---v Idy'yt

dy=du

y22y3 +l

IM

IA

(b) 2^Xa=zlnu = x2 ln2I du

- 2xln'udxdu

-rxz .2xrn2dx

1M

1M

1A

Alternative Solution

u =2" - "*2k'2du

-

"x2tn2 .2xln2

d.r

=2"'.2xln2

IMIM

1A

26

LZ

VIn

I

I^II902 ---=t6

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I^II

YI

NI

YI

(or\ s (or \ l-l--t 'l:(e,lr Ie.l- (g wr)a- (S)a = (fl v ,V)d

!f = , .r.t

t'2-'*9 =oz "' ,l16

(g v V)a- (g)a + (p.)a = (g ny)4 :.

(c)

(q)

9

4(s)a.(s lv)a=@wv)a(e)

,n

zl+s'zult's+... = r

J + tro-aZl -tls + {u1- lz + {q1s'e ,o

'lt?ur tggg'91 : eruu 3uo1 e rege Snrp eqlgo uorle4uesuoc eql 'e'l998'91=

Tsss er+,r.o_rffl rr- fr *ffii,, ={eess'lr

+ tys-azt- [(s + r)ut - Q+ r;u11 E's]?i/

t9S8'9I + \.s-aZt-[(S + t)vt- Q+ rut]g'S - r'e'!998'9t ^,

Zl+ (Zul- Sq)e'S = ,J+Zt-(Sq-Zq)e's=0'.' .

0=r .0-/ ueqq

(6q7 acurs) C+ \.s_aZt-[(S+ t)q-(Z+r)q]g'9=I (s*t 7+r\ l. tPlno-'z'r.[

'

-

r Je sl ='(S+t Z+/\ ry \o_az.r*[

, _

, .Jr.r=*

(q)

(e)'e

(9)

VI

ntl

vt

hII

YI

WI

,Z- ,tEZV& yrxZ

'*Z- '{E Zi%Z=n - -,(e

Zulxz' ,*Z'-i- =rynp ry _.__r_=_ (c) np

^p ^p

\L)

YI

hII

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Solution Marks Remarks(d) P(A' A B') =l-P(Av B)

=t-a20

For P(,4'n B')*0Follow through

For P(l ') + P (B') * P(A' vFollow through

Alternative Solution IP(A' n B')=P(A')

-P(A' a B)-f,_1)_ft)_ [,_r./_[4.]

Alternative Solution 2P(A' r: B') = P(A') + P(B) - P(A' w B')

= P(A') + P(B) - P((A r: B)')

=[,-1).(,- 3 )-[,-1 1)t s/1. lo/1.610)

Hence A,

il20

*0and B' are not mutually exclusive.

IMI

Alternative Solution

;,"*++).(,_*)=3

2+P(A'v B') since P(l'u.B') < I

Hence A' and B' are not mutually exclusive.IMI

(7\

5. (a) Let $X be the amount of money spent by a randomly selected customer.r(x > :oooo)= o.zqz,( , rtoooo_ tt) = o.ro,I oooo ).'.

p(o . , .3oooo- P) = o.rr*[ 6000 )

...3oooo_t =o.z6000

i'e' # = 25800

(b) The required probability_p0osoo

6Z

Surssnu IIe oJe^{ sexeJoslaqel pue ur8rro:og (1-dd)

edeqs rogsa1o1dru,(sz rog

sldecrslut rog

()

1,_/ | |:+,-l : t/y,,Ai

e s t E z t /lo I- ;, t- t- s- s- L- B- trrrrr-l':'r't"

-"'1t' _--..' I Z_=xt: -l' -- -: t=t II

t,II

t' ' (x "=nrt-' lz+e/ l/

xj/(q)

z- = x sl J ol eloldruxse Issrue^ eqlJo uoqenbe erll '.'Z+x +Z-

Solution Remarks

For numerator

Follow through

OR

oR 1.0063

(pp-l) if *

*^not rejected

5

=... * f?G -2x)dx

"8x+2

(c) (i) f (r\ =3x -2x+2f '(x) = (x+2)(3)-(3x-2)(l)(x+2)2

8a(x+2)'

So the equation of Z, is .y - k = f'(h)(x - h)

3h-2 8v-ffi= rn*;(*-n1

(h+2)2 y -(3h-2)(h+2) =916-91,

8x -

(h +2)2 y +3h2 -

4h -

4 = o

(iD (1) If Z1 passes through the origin, then8(o)

-

(ft +z)2 (o) +3h2 -

4h -

4 = o

h=2 or ] (rejectedsince P liesinthefirstquadrant)J

...p=3(2)-2 =1(2) +2SloPe of t, =@)-t-=-2

-8Hence the equation of Z, is y

-l = -2(x -2)i.e.2x+Y-5=0

(2) The x-intercept of Z2 is IISo the required area

-

(r3x-2 *.!(1-z)rurz x*2 2\2 )"J

=fr('-#)*.i(i)u,3

= [3x-8hlx+ ,ii.;

=!*glo2

For

t,

Itl{Iq

ttl

.

III

Io

1A+IA

IM

30

I

qSnorql /Y\olloJ

rrrral $I eql Jod

euo reqlrg

'lreJJoJ sr rurelc s,1sr8o1otq eql ecueH'

OZ < I roJ (rN < (r)y,,t os' OZ < t ueqaa, Surseercu sr (7)p - (l)ltt puu (02)N = (97)61 ecutg

LtEt'st=#

Solution Marks Remarksrro I lr*7 ate. (a) (i) Jo +0,=

l0 ' (,F* *zrlt*zs' *zrlt*s2 *zr[t*t.s2.il.'o? )

- 2(4) 40 [' )=

1.305182044=1.3052So the increase oftemperature is about 1.3052"C '

d(t -\(u) -r-Jr +i l=--dt\40 ) qorlt*t2

a'-( t.1u,')= ' 'o,z(+0, )

"+40(t + tt )2>0

Hence it is an over-estimate.

(b) (i) 100(ln16)2 -

630lnxs + 1960 = 968

50(lnxs)2 -3l5lnx6 + 496 = 0

31 t6lrl -r,r =

" l0 5xo *22.1980 or 24.5325

. 2001n x 630(ii) w'(x) =.'.w'(x)

cc

lCerlOC(c) 3u

ICaIIOC(c) 8ur

,Qt

lcsrroc roleJerunu JoJ vlrJ.loc uuoJ Jol8Jelunu JoJ I trl) Sursn Jopuruouep JoJ htrl

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lceuoc urroJ roJ I II

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hIIhil

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n

VI9I0'0 ^v

/I+WI+I^trtnnn00960'0

(exes np; o - i ,rGtzts LenrD IP + slGeus Lsns' 0),Qrlrqeqord pennber eq1

mIF

r'l+hil+hlfirn00960'0=

lGtzts tg v s' o - i rGtztg L9? 9' 0 [ ) + s(6Ez,ts Lsrs' 0)] x o rGtzt s ts v s' o),$rpqeqord pannbsr eq1 (e)

0l 9t0'0 =

0 * i

* Gtzts tgvs'o - t) uGtzts t9?9'o) ofr +OI

7 " rGtzts Lens' 0 - i sGEZEs L9?e'0) ofc =,Qrpquqord pennber eq1 (p)

,rrrooor1loo:,Gtztstsvs'0

- r) 8(6zrsrgrs'0) 0lJ +(e&esDv;o - i 6GEZtsLe?s'O oll + 0{6tztsLsv;0) =

&lgqeqord partnber eq1 (c)

(r)VI

VI+WI

(e)YI

htrI+WI

lz)VIhII

gLs6'0 !(eeustgvs'o),Gtztstsvs'o - I) + GtztsLgns'O zGtztsLgrs'o - r) +

Gtzestgvs'o)GrctsDvs'o - r) + Gtztstgvs'il *&lllqeqord pertnber eq1

z uo-iiqo-S-s IIEmIIV-

VIhII

8ls6'0 =,Gtzts ts v s' O + (erc* t9 t 9' o - 0 rG tzts ts v9' o) rtc +

) zGtT,tsLsn;OP + ,(a*tster;o - DGtztsLg?9'o) ]c =,$y11qeqord pa:rnbar eq1

f -86-i1fr

ios-sIiiEu5itV-

zGtzESLgng'o-

YIWI

8LS6'0 =

,G*tsnvs'o-t)-r= (pelceqc sr Jermoc qlt eql JoUe prmoJ ore srelunoc ,tsnq ou)4-1=,tlyrquqord pelnber eq1(q)

(e)89rS'0 0

6tZtSL9v9'0 =(ir.izirio) t- T-t-t-( o c-ac6'e o'r'z6't o't-'r6't ot-'o6't ) ,{Urqeqord pennbar

eq1e)

VI

IAil+htrl

$lJeruau$FBIAIuormlos

Solution(a) The requl.g0e-1.8 l.gle-1.8 l.g2e-I.8 l.g3e-1.8 1.g4e-I.8+-+-+-+-0!|2!3!4l

* 0.963593339=

0.9636

Remarks

lM for P(X s 4)lM for Poisson probabilitywith correct 2

For standardization

For any one correct

For all correct

lM for form correct

lM for form correct

lM for any 2 caseslM for denominator using (a)lA for all correct

(b) oo =r(rri#) =p(Z >2)=0.5-0.47i2=0.0228o. =p( z< 2 -3\[ 0 8 )= '(' < -l '25) = 0'5 - 0'3944 = 0' 1056Pt =l- Po - Pz =l-0.0228-0.1056 = 0.8716

(c) (D The required probability= cl pzpt2 + C? pz2 po= 3(0. l0s6x0 .87 rc)z + 3(0. 1056)2 (0.0228)x 0.241431455x 0.2414

(ii) The required probability= ctpz2 poz *fi.rrn'po * pro= 6(0. I 056)2 (0.022q2 + I 2(0. I 056)(0. 87 I 6)2 (0. 0228) + (0.S7 I 6)4x 0.599107436- 0.5991

(d) The required probabilityt's'i.-' * (o.l0s6)2 . r#, 0.241431455) + t'soi.-" (0. ssstoi436)

- l.lg635%ng=

0.0883

IM

1A

IA

34

s

) Sursn Joleurruouep loJ htrllceJ.loc urroJ JoJ wl

lsexoc sews II3 JoJ htrIlcerros esec 1 ,(ue JoJ hll

((t)(p) q pepre^\e eq UBJ)rpqeqord lerruourg roJ hll

lcauoc seses JoJ IAII

\L)

Llsg'0 *08rr9r tt0'0

"

&gllqeqord parnber eq1

zLto'0 x08LtgILtl'0 x

o(rct esozzt o) r r(rct as ozzt o - D .?rl [ . (e s t L6n Loo' il,Gs t 6t too o - r) f cl +l(lrososozzo'o) r,(rctanzzo'o - r) zlrl[(r9 t L6nLoo'O LGst L6n Loo'o - r),'JJ +Ir(utawzzud ot(L6l6gozzo'o - r) rfrJI o(tgt L6nLoo'il sGgtrctnco- r) fe] =

firpquqord perrnber eq1

tzzU'0 x168690220'0 =

[,(sr o'o) rr(s I 0'0 - il olc + (s t o'o) ur(st 0'0 - I) .rIJ + or(s t o'o - D] - r =,firpqeqord pelnber eq1

(rs)

(lr)

(l)(p)

' L sl 1 Jo enle^ lsuel eql ecuaH

ezzztssts.s =yffi!.,96'0q > 9$Z0SZ66',out1

96'0> ft9z0sz66'0

'i::,':"*'::i: _l _:

VIhtrt+I^trI

VI

WI+I^It

VI

(r)YI

NI

hII

Fs0'0 < (t9et6rt00'0)r-t(tgtL6vLoo'o- r) + "'+

aL6v Lol'O z(t9tL6n Loo'o - i + G9e L6v too'dcsttur too'o__ ! . ,*t* rn ouorlnlos e^rlBrusllv

IIIIIIII

s0' 0 < ?(89 L6v Loo' il + Gw tav n0'0 - r) r-1([9 uev nu Or-1, + . . .

+

,Gw tev nu o - D rGs e.t6n Loo' il $ c + r- rGs e L6v Loo' o - :D3.s t L6n Loo' o) f c

IIIII

i r"l

tr00'0 =Gst6vtoo'o) rGstrcvtoo'o - r) =

,$rlrqeqord pernber eq1(q)

sr00'0 !t9tL6tL00'0 *

[.(ro'o)rr(ro'o - i olJ+ (r0'0)6r(10'0 - r),fJ + or(ro'o - r)] - r =.Z(e)

\Z)

VIhII

()

YI

I^trI+WI

s{J?lueu$lrBIuormlos

,11

6t

'lurq ue^IE eqlJo osn e{BIuplnoc peldtua11e oI{1( esoql pu peldue[B seleplpuec ,{ueu ool loN 'rood

'(f) (q) eleldruoc o1 3utpe3 ,{q pereputq ere.,'t seleplpuBc atuos 'rleJ

'uorler8alut op 01 uot1rultsqns,(1dde ]ou plnoc seleplpuecJo requnu V 'llJ

'punoJ eJel* so{elsttu sselorcc euros q8noql'poo8 ,ften

'poo3,ften0s

(z)

(r) (l)(r) (q)

(l) (e)(r) (e)s

'(Z) ttud qcer lou plnoc scueq puu .,ut8ttoq8norql sessed r7,, Jo uolllpuoc aql esn 01 pelleJ seleprpuec .(ue61 'rood

'an-lnc e Jo lerruou pue lue8uelaql uoo,^ loq uotlBleJ eql qll,^A IEIIIlueJ lou ele.!\ sel8plpusc Jo requnu v 'rle{

'pooD

'saloldur.,(se oql pug plnoc seleprpuec lsoIAI 'poo8 ,{ren

0ql

L9

(r)

(l) (c)

(q)

(e) t

IBJoUAC uI eCueuuoJJod0lr) ,Qpepdo4r0qunNuorlsen)

(suorlsanb 9 Jo lno t Jo ooroqc y) g uoqcag

',(russeceu lou sI qclqa ',(lyectueqceu: uoIlBIAopptpuuls aql elelnclec ol palJl euros q8noqlle uotlsenb sql elpuuq plnoc seleppuec ,(ueyq 'poog9

'pelnber sr ftrlrqeqord luolllpuoc e leql oJE,^ B lou ele,t\ soleplpuec oluos 'rIE{s

'sluela .,1uepuedeput,, Jo leql qlIA pesnJuoc pue sluala ((ellsnlcxe,(11eqnru,,;o uorlrugop eqlJo elns Jou ere,^^ SelBpIpLIEo ,(ueu areqrrl (p) Ued ldecxe'poo3,ftenv

',(pedo.rd ,( Jo lltutl eqlJo uollelou IEcIItuaqlBu aql lueserd lou ppoc salsplpuc eulos 'poogc

ur uollcuru cruqlueSol ol{l Jo esn eql ql! ^ JeIIIIUBJ lou oJe,^'uoItrBIlueroJJIp

seleprpuec oluos 'poocZ

eql

JOJ

(e) ur uorsuedxa eqt ,(1dde 1ou

Jo senlu^ 3o e8ue: eql lno llo,{\

'pelnber uolleulxorddeplnoc solprpuec ,(uery '(q) ued ur rood reqld

'prlA sr uorsuedxe lelr.uourq eql qclq,^tou plnoc soteprpuec euog '(t)(e) ued ur pooE ,ften

I

IEJeueD uI ocusuuoJlodJequnNuoqsen|

(,tos1nduo3) y uo;1ceg

aJuEruJoJJod 6solBplpuBJ

QuestionNumber

Popularity(%) Performance in General

e (a) (i)(iD

(b) (i) (ii)(iii)

(iv)

5l Very.good.

Poor. Many candidates were not aware that the second derivative of the givend3 x d,2,

equatron ,r *,

rather than fI .

Good.

that dY should be found and somedtPoor. Many candidates did not realisefailed to apply chain rule to find dI .

dt

very poor. Most candidates could not interpret their own mathematical f,rndingsand hence failed to make use of the results to make judgement.

l0(a)

(b)

(c)

(d)

(e)

97 Very good.

Very good.

Good.

Poor. Many candidates overlooked that a joint probability should be considered.Fair. Many candidates were able to handle conditional probabilities but somewere careless.

11(a)

(b)

(c) (i) (ii)

(d)

83 Very good.

Good.

Fair. some candidates did not do the counting right and missed some of theeligible events.

Poor. Many candidates had difficulty in identi$ing the joint probabilitiesrequired for the numerator of the conditional probability.

40

tv

'8ur>1urqt c1leure1s.ds e,r.ordrut pue s1decuocgo Surpuelsropun reueq ur dloq ue+o ucc uotluluese:d reelc pue sloqru(s IEcIleuoI{}Btu Jo esn redot4 't

'Je/v\suE Ieuu eql ro3 ,(curncceSoeer8ep pelnbar er{l saerqce 01 Japro ur sdels alerpeuretut aq1 ur seceld lelutcep sJoIU esn plnoqs sepplpueJ 'Z

'ruelqord e o^los ol lro,$u^lo rreqt Jo sllnssr ornJo esn e1ur ppoc feql 1sql it\oqs q pallq lnq sllDls rleql polsrlsuouop seleplpuec,{ueyrtr 'senbruqcal pcrltuer{1cru oql pqqeq Eunreeru eql 01 suolluraplsuoc orour a,rt8 ppot{s s4eplpue3 'l

:suorl"pueuuocor pue slueuluoc Isjeuac

JoJeqr.unueqtr

'slua.{e eql ?uqunoc ut .{1yncg3tp p"q ssleplpuec euos 'JIed

'slue^e lue^eleJEurlunoc ur ,Qpcgglp peJslunocue so1eplpuc euos 'poog

'serllenbeur Euttpueq q fUIHs ssel ars^{ saPplpLIBJ 'JIBg

'pooE &sn

'poo8 {:enZ9

(lrXu)

(r)(p)

(c)

(q)

(e)zt

IBJau3C ur ecuuuuoJJed(%)f,rue1ndod

JeqlunNuo4sen$