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Unformatted text preview: 2000ASL I a S ’ HONG KdNG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION 2000 MATHEMATICS .AND STATISTICS ASLEVEL 830 am — 13.2.30 am (3 hours)
This paper must be answered in English I. I This paper consists of Secrion A and Section B.
2. Answer ALL questions in Section A, using the AME) answer Book.
3. Answer any FOUR questions in Section B, using the AL(C)2 answer book. I 4. , ' Unless otherwise speciﬁed, numerical answers should either be exact or given
to 4 decimal pIaees. zoooAsM & s—i ' 144 0 SW A (43 nanks) 1. Let 1n(xy)=1 mew». Show/that 3:3"?
y ' (II xy+x l
2. Let H<§. 1.. f 1  , ,
(3) Expand (l+2x)2 (14.39) 2 makings“ far I. . ,
ofxasfarasthe‘termin x3. , , , (b) Using (a) and the identity (I+2x)(l—2x+4:rz)a\l+8x3 , or
1 ' .
oﬂxerwisetexpand (1—2x+4x2) 2 inascendingpoweisofx asfm' / asthetermin x3 . 3. Figure 1 shows the graphs of 7 , ’
the two curves ‘ ‘ \ I eiand Cl: y l
l+x3 . C23 y Finds the area of the shaded
V region.
(5 marks) ZOOOASM & 8—2 145 A polynomial f(x) has the following properties (a) Find the range of values of x for which the graph of f(x) is concave
downward (i.e. convex upward). (b) Write down all the points of inﬂexion of the graph of ﬁx) . (c) Sketch the graph of t(x) .
(4 marks) A ﬁtness centre advertised a programme speciﬁcally designed for women
weighing 791g— or more, and claimed that their individual weights could be
reduced by at least 20kg on completion of the programme. Twentyone
women joined the programme and their weights in kg 'when they started are
shown below: Stemmmaﬂuniﬁ) \
7e\e\\2\355‘r«
s\1\r’4 5 ‘6.\6'7\‘8
90\2\\5\\8‘99 (a) Find the median and the interquartile range of these weights. (b) 0n completion‘of the programme, the median, lower quartile and upper
quartile of the weights of these women are 73 kg , 68 kg and 77 kg
respectively. The lightest and heaviest women weigh 60 kg and 82 kg
respectively. Draw two boxandwhisker diagrams in your answer book
comparing the weights of these women before and aﬁer the programme. (c) Referring to the box—andwhisker diagram in (b), someone claimed that
none of these women had reduced their individual weights by 20 kg or
more on completion of the programme. Determine whether this claim is correct or not. Explain your answer brieﬂy.
(6 marks) 2000ASM & 53 146 Mr. Chan has 6 cups of icecream in his refrigerator. There are 5 diﬁerent
ﬂavours as listed: cup of chocolate, cup of mango, cup of peach, cup of strawberry and
2 cups of vanilla. Mr. Chan randomly chooses 3 cups of the icecream. Find the probability that
(a) there is no vanilla ﬂavour icecream, (b)  there is exactly 1 cup of vanilla ﬂavour licecream.
(5 marks) 7. The number of people killed in a trafﬁc accident follows a Poisson distribution
K; with mean 0.1 . There are 5 trafﬁc accidents on a given day, ﬁnd the probability that there is at most 1 accident in which some people are killed.
(5 marks) A department store uses a machine to offer prizes for customers by playing
games A or B . The probability of a customer winning a prize in game A is
—5— and that in game B is g . Suppose each time the machine randomly
generates either game A or game B with probabilities 0.3 and 0.7
respectively. (a) Find the probability of a customer winning a prize in 1 trial. (b) The department store wants to adjust the probabilities of generating game A and game B so that the probability of a customer winning a
prize in 1 trial is % . Find the probabilities of generating game A and game B respectively.
(6 marks) 2000ASM & 8—4 147 * SECTION B (60 marks)  . 10.
Answer any FOUR questions in this section. Each question carries 15 marks.
Write your answers in the AL(C)2 answer book. A researcher studied the growth of a certain kind of bacteria. 100 000 such
bacteria were put into a beaker for cultivation. Let W’fﬁaﬁ elapsed after the cultivation has started and WWLdijC the A department store has two promotion plans, F and G , designed to inerease
its proﬁt, from which only one will be chosen. A marketing agent forecasts that
if x hundred thousand dollars is spent on a promotion plan, the respective rates
of change of its proﬁt with respect to x can be modelled by f(x) = l6+4xe'0'25x and got) = 16+ 6" Jl+8x . (a) Suppose that promotion plan F is adopted. (i) Show that f(x) 5 f(4) for x > 0 . (ii) If six hundred thousand dollars is spent on the plan, use the
trapezoidal rule with 6 subintervals to estimate the expected
increase in proﬁt to the nearest hundred thousand dollars. (6 marks) (b) Suppose that promotion plan G is adopted. (i) Show that g(x) is strictly increasing for x > 0 .
As x tends to inﬁnity, what value would g(x) tend to? '(ii) If six hundred thousand dollars is spent on the plan, use the substitution u = «)1 +83: , or otherwise, to ﬁnd the expected increase in proﬁt to the nearest hundred thousand dollars.
(7 marks) (c) The manager of the department store notices that if six hundred
. thousand dollars is spent on promotion, plan F will result in a bigger
proﬁt than, G . Determine which plan will eventually result in a bigger proﬁt if the amount spent on promotion increases indeﬁnitely. Explain your answer brieﬂy.
(2 marks) ZOOOASM & 8—5 148 l
l
; s grgeratg of the bacteria. The researcher obtained the following data: (a) The researcher suggested that r(t) can be modelled by r(t) = at” ,
where a and b are positive constants, (i) Express ln r(t) as a linear function of Int . (ii) Using the graph paper on Page 6, estimate graphically the value
of r(5) to 1 decimal place without ﬁnding the values of a and b .
(5 marks) (b) The researcher later observed that r(5) was 18.5 and considered the
model in (a) unsuitable. After reviewing some literature, he used the model r(t) =20— pe'q' , where p and q are positive constants.
(i) Express ln[20  r(t)] as a linear function of t . (ii) Using the graph paper on Page 7, estimate graphically the values
of p and q to 3 signiﬁcant ﬁgures. (iii) Estimate the total number of bacteria, to the nearest thousand, after 15 days of cultivation.
(10 marks) 2000ASM & 8—6 149 Page Total . . _
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1 151 2000ASM & 8—8 150 2000ASM & S—7 ll. A researcher studied the commercial ﬁshing situation in a certain ﬁshing zone.
Denoting the total catch of coral ﬁsh in that zone in 1 years time from January
1, 1992 by N(t) (in thousand tonnes). , he obtained the following data: The researcher modelled N(t) by In N(t)=a—eH“ where a and k are
constants. k —2k198 h th t "4 — +—ln—=0.
(a) S ow a e e e 55
Hence ﬁnd, to 2 decimal places, two sets of values of a and k. (4 marks) (b) The researcher later found out that N(7) = 170'. Determine which set of
values of a and k obtained in (a) will make the model ﬁt for the kno data. " Hence estimate, to the nearest thousand tonnes, the total possible catch of coral ﬁsh in that zone since January 1, 1992.
(4 marks) (c) The rate of change of the total catch of coral ﬁsh in that zone since . . . dN(t)
January 1, 1992 at time I IS given by dt . d N(t) (i) Show that = kN(t)eH".
‘
(ii) Using the values of a and k chosen in (b), determine in which year the maximum rate of change occurred. Hence ﬁnd, to the nearest integer, the volume of ﬁsh caught in that year.
(7 marks) ZOOO—ASM & 89 152 :12
f
I The milk produced by Farm A has been contaminated by dioxin. The amount
of dioxin presented in each bottle of milk follows a distribution with
mean 21113 (1 ng = 10’6 g) and standard deviation \,5_ng . Bbttles which
contain more than 12 ng of dioxin are classiﬁed as risky, and those which
contain more than 27 ng are hazardous. (a) Suppose a bottle of milk from Farm A is randomly chosen.
(i) Find the probability that it is risky but not hazardous. (ii) If it is risky, ﬁnd the probability that it is hazardous.
(6 marks) (b) _ A distributor purchases bottles of milk from both Farm A and Farm B
and sells them under the same brand name ‘Healthy’. It is known that Wanna and the rest from Farm 8 . A bottle of
milk from Farm B has a probability of 0.058 of being risky and 0.004 of being hazardous. (i) If a randomly chosen bottle of ‘Healthy’ milk is risky, ﬁnd the
probability that it is from Farm B . (ii) If a randomly chosen bottle of ‘Healthy’ milk is risky, ﬁnd the
probability that it is a hazardous bottle from Farm B . (iii) The Health Department inspects 5 randomly chosen bottles of
‘Healthy’ milk. If 2 or more bottles of milk in the batch are
risky,the distributor’s license will be suspended immediately.
Find the probability that the license will be suspended. (9 marks) 2000ASM & 810 153 13. Boys B, , B2 and girls G, , G2 are students who have qualiﬁed to represent
their school in a singing contest. One, boy and one girl will form one team. The
team formed by B, and G, is denoted by B, G,, where i: l, 2 and j= 1,2.
A team can enter the second round of the contest if both team members do not
make any mistakes during their performance. Suppose that a student making
mistakes in a performance is an independent event, and the probabilities that B, , B2 , G, and G2 do not make any mistakes inaperformance are 0.9, 0.7,
0.8 and 0.6 respectively. (a) List all the possible teams that can be formed.
(1 mark) (b) Find the probability that B, G, can enter the second round of the
contest. '
(1 mark) (c) If a team is selected randomly to represent the school, ﬁnd the
probability that the team can enter the second round of the contest. «
‘ (2 marks) (d) If two teams B, G, and BI 02 are formed to represent the school, ﬁnd
the probability that (i) exactly one team can enter the second round of the contest, (ii) at least one team can enter the second round of the contest.
(5 marks) (e) Suppose that two teams are allowed to represent the school and each
student can 6th join one team. (i) If the two teams are formed randomly, ﬁnd the probability that
exactly one team can enter the second round of the contest. (ii) How should the teams be formed so that the school has a better
chance of having at least one team that can enter the second round of the contest?
(6 marks) 2000ASM & 8—11 154 14. The company which sells boxes of Energy Chips wanted to increase its sales
volume by issuing stamps which could be exchanged for toys. Stamps were put
into some of the boxes and a collection of 5 stamps could be exchanged for one
toy. Some students conducted a survey to study the chance of getting stamps.
They randomly selected 100 boxes of the Chips and counted the'number of
stamps in each box. The result is shown in the ﬁrst two columns of Table 1.
(a) The students tried to ﬁt the data by a binomial (n = 6) and a Poisson distribution. They took 1 as the mean of each of the distributions. Fill in the missing values in Table . ~
(4 marks)
(b) Suppose the absolute values of the differences between observed and expected frequencies are regarded as errors. The distribution with a smaller maximum error will ﬁt the data better. Which distribution in (a) is better? ( 1 mark)
(c) Assume the distribution chosen in (b) is adopted. (i) Find the probability of getting at least 1 stamp in a box of the
Chips. (ii) Mrs. Wong keeps on buying the Chips until she gets at least l
stamp. Find the probability that she needs not buy more than 3
boxes. (iii) Suppose Mr. Cheung buys 2 boxes of the Chips. If he gets
stamps in both boxes, ﬁnd the probability that he gets just
enough stamps to exchange for a toy. ( 10 marks) 2000ASM & 8—12 155 Candidate Number Page Total 14. (Cont’d) If you attempt Question 14, ll in the details in the ﬁrst three boxes
above and tie this sheet INSIDE your answer book. Table 1 Observed and expected frequencies of the number of boxes by the number of stamps in 100 boxes of Energy Chips This is a blank page. * Correct to 2 decimal places. END OF PAPER A _ _ 156
2000 s M & s 13 2000ASM & 3—14 157 ...
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