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Unformatted text preview: v‘ 2003-ASL SECTION A (40 marks)
& S ’ Answer ALL questions in this section.
Write your answers in the AL(E) answer book. HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG ADVANCED LEVEL EXAMINATION 2003 1. (a) Let I ax! <1 ‘ (i) Expand (1+ a x)'3 in ascending powers of x as far asthe term MATHEMATICS AND STATISTICS AS-LEVEL , . 3 _ lnX (ii) If the coefﬁcient of x3 » in the expansion of (lI—ax)‘3 is
8.30 am — 11.30 am (3 hours) , 30 . _ . t —- , find the value of a .
ThIs paper must be answered In Enghsh 27 (b) (i) Using the results in (a), or otherwise, expand (3—2):)‘3 in . ﬁr .
ascending powers of x as far as the term in x3 . I. This paper consists of Section A and Section B. . ‘.
(ii) State the range of values of x for which the expansion of
2. Answer ALL questions in Section A, using the AL(E) answer book. (3 _ 2x) -3 is valid.
3. Answer any FOUR questions in Section B, using the AL(C)2 answer book.. (7 marks)
4‘ Unless otherwise speciﬁed, all working must be clearly shown. 2.. After a ﬁxed amount of hot liquid is poured into a vessel, the rate of change of
, . . . the temperature 6 of the surface of the vessel can be modelled by
5. Unless otherWIse specrﬁed, numerical answers should be 'either exact or given
to 4 decimal places. 1
dt 1 ’
25(1+3te1°°) where‘ 6 is measured in °C and t (2 O) is the time measured in seconds.
Initially (t = O) , the temperature of the surface of the vessel is 16°C . - ;£.
(a) (i) Let u=1+3te10° , ﬁnd d—u. (ii) Using the result of (i), or otherwise, express 0 in terms of t . (b) Will the temperature of the surface of the vessel get higher than 95°C ? Explain your answer brieﬂy.
(7 marks) zoos-As-M & S—1 17 ' 2003-AS—M & 3—2 18 41‘ A chemical X is continuously added to a solution to form a substance Y . The
total amount of Y formed is given by 4x—9 V4):2 +3x+9 where x grams and y grams are the total amount of X added and the total
amount of Y formed respectively. y=3+ (a) Find % when 10 grams ofX is added to the Solution. (b) Estimate the total amount of Y formed if X is indeﬁnitely added to the
solution. (6 marks) A and B are two events. Suppose that P(AIB) = 0.5 , P(BIA) = 0.4 and
P(AUB) = 0.84 . Let P(A) = a , where a > O .
(a) Express P(AmB) and P(B), in terms of a. (b) Using the results of (a), or otherwise, ﬁnd the value of a . (c) Are A and B independent events? Explain your answer brieﬂy.
(7 marks) 2003—AS-M & S—3 19 A researcher conducted a study on the time (in minutes) spent on using the
Internet by university students. Thirty questionnaires were sent out and only
19 were returned. The results are as follows: 12 13 14 15 15 21 25 29
36 37 38 41 47 49 49 49
52 54 57 (a) Construct a stern and leaf diagram for these data. ' (b) Suppose that the researcher has received eight more questionnaires.
Three of them show that the time spent on using the Internet is one hour.
The others show that the time spent is more than one hour. (i) Find the revised median and the revised interquartile range of the
time spent. (ii) Describe brieﬂy the chahge in the mean and the change in the
range of the time spent.
(6 marks) The amount of money involved in a business transaction follows a normal
distribution with mean $215 and standard deviation $50 . Any transaction
with an amount more than $300 is classiﬁed as a Type A transaction. (a) Find the probability that a transaction will be classiﬁed as Type A. (b) Find the probability that in 7 randomly selected transactions, exactly 2
transactions will be classiﬁed as Type A. (c) Find the probability that the 8th randomly selected transaction is the 3rd
— transaction which is classiﬁed as Type A. (d) It is known that 64.8% of the transactions each exceeds $K . Find K .
(7 marks) 2003‘AS—M & 3—4 20 1 SECTION B (60 marks) . 8.
Answer any FOUR questions in this section. Each question carries 15 marks.
Write your answers in the AL(C)2 answer book. ' According to the past production record, an oil company manager modelled the
rate of change of the amount of oil production in thousand barrels by 20—4x 7 _ a+bx —_3
7—2x forall x¢2 and g(x)— 3+cx forall xi 0 , where a, b and c are positive constants. Deﬁne f (x) = Let C1 and C2 be the curves y=f(x) and y=g(x) respectively. It is given that the x-intercept and the y—intercept of C2 are —3 and 4 respectively. Also, it is known that C‘ and C2 have a common horizontal
asymptote. (a) Find the equations of the vertical asymptote(s) and horizontal
asymptote(s) to C] .
(2 marks) (b) Find the values of a, b and c .
(3 marks) (c) Sketch C1 and C2 on the same diagram and indicate the asymptote(s), intercept(s) and the point(s) of intersection of the two
curves. ‘ (5 marks) : (d) If the area enclosed by C1 , C2 and the straight line x = x1 , where 0 < )t < 1 , is 31n 3 square units, ﬁnd the exact value(s) of xi . 2
(5 marks) 2003-AS-M & S—5 21 m) = 5 + 24"” , where h and k are positive constants and t (Z O) is the time measured in
months. (a) Express ln(f (t) ~5) as a linear function of t .
(1 mark) (b) Given that the slope and the intercept on the vertical axis of the graph of the linear function in (a) are —O.35 and 1.39 respectively, ﬁnd the
values of h and k correct to 1 decimal place. (2 marks) (c) The manager decides to start at production improvement plan and
predicts the rate of change of the amount of oil production in thousand barrels by .
g(t) = 5 +/ln(z + 1) + 2“k ”h , where h and k are the values obtained in (b) correct to 1 decimal place,
and t (2 O) is the time measured in months from the start of the plan. Using the trapezoidal rule with 5 sub-intervals, estimate the total
amount of oil production in thousand barrels from 122 to t=l2 . (2 marks) (d) It is known that g(t) in (c) satisﬁes d2g(t) 1
dtz =p(t)—q(t) , where q(t)=(t+1)2 (i) If 2' =2“ forall 120 , ﬁnd a.
(ii) Find p(t) . , (iii) It is known that there is no intersection between the curve
y=p(t) and the curve y=q(t) , , Where ZStSIZ .
Determine whether the estimate in (c) is an over-estimate or under-estimate.
( l 0 marks) 2003-AS-M & S—6 22 9. A researcher monitors the process of using micro-organisms to decompose
food waste to fertilizer. He records daily the pH value of the waste and models
its pH value by P(t)=a+%(t2 ~8t—8)e“’”, where t (Z 0) is the time measured in days, a and k are positive constants. When the decomposition process starts ( i.e. t = O) , the pH value of the waste
is 5.9 . Also, the researcher ﬁnds that P(8) —P(4) = 1.83 . (a) Find the values of a and k correct to 1 decimal place.
(5 marks) (b) Using the value of k obtained in (a) , (i) determine on which days the maximum pH value and the
minimum pH value occurred respectively ; d2P >0 forall [223.
dt2 (ii) prove that (8 marks) (c) Estimate the pH value of the waste after a very long time. 26-“ [ Note : Candidates may use 1im( t ) = 0 without proof. ] t—Mx: (2 marks) 2003-AS-M & S—7 23 10. A bank customer service centre records the number of incoming telephone calls
in ﬁve—minute time intervals (FMTIS) . The following table lists the number of
calls in a sample of 50 FMTls . _n umber of calls 1 Fre - uenc j (b) (C) Find the sample mean and the sample standard deviation of the data in
the table. '
~, (2 marks) The manager of the bank believes that the number of calls in a FMTI
follows a Poisson distribution and its mean can be estimated by the
sample mean obtained in (a). (1) Find the probability that there are fewer than 4 calls in a FMTI .
F (ii) Find the probability that there are fewer than 4 calls each
in exactly 2 FMTIs out of 6 consecutive FMTls.
(6 marks) Assume the model in (b) is adOpted and it is known that 55% of the
calls are from male customers and 45% of the calls are from female
customers. (i) If there are 3 calls in a FMTI , ﬁnd the probability that exactly
2 calls are from male customers. (ii) Find the probability that there are 2 calls in a FMTI and they
are both from male customers. - (iii) Given that there are fewer than 4 calls in a FMTI , ﬁnd the probability that there are at least 2 calls and all of these calls are
from male customers. (7 marks) 7001-AS-M X! 8—8 24 ll. ,1? In a game, two boxes A and 8 each contains n balls which are numbered 1 , 2 , , n . A player is asked to draw a ball randomly from each box. If the
number drawn from box A is greater than that from box VB , the player wins a
prize. (a) Find the probability that the two numbers drawn are the same.
V (1 mark) (b) Let p be the probability that a player wins the prize. (i) Find, in terms of p only, the probability that the number drawn
from box B is greater than that from box A . (ii) Using the result of(i), express p in terms of n: (iii) If the above game is designed so that at least 46% of the players
win the prize, ﬁnd the least value of n . (6 marks) (c) Two winners, John and Mary, are selected to play another game. They
take turns to throw a fair six-sided die. The ﬁrst player who gets a
number ‘6’ wins the game. John will throw the die ﬁrst. (i) Find the probability that John will win the game on his third
throw. (ii) Find the probability that John will win the game.
(iii) Given that Mary has won the game, ﬁnd‘the probability that Mary did not win the game before her third throw.
, (8 marks) 2003-AS—M & 5—9 25 12. A teacher randomly selected 7 students from a class of 13 boys and 17 girls
to form a group to take part in a ﬂag-selling activity. (a) Find the probability that the group consists of at least 1 boy and 1 girl.
(3 marks) (b) Given that the group consists of at least 1 boy and 1 girl, ﬁnd the probability that there are more than 3 girls in the group.
(3 marks) (c) A group of 3 boys and 4 girls is formed. It is known that the amount of
money collected by a boy and a girl in the activity can be modelled
respectively by normal distributions with the following means and standard deviations: Standard deviation .33.;_ $ 100
— $708 $100 Any student who collects more than $800 receives a certiﬁcate. (i) Find the probability that a particular boy in the group will receive
- a certiﬁcate. (ii) Find the probability that exactly 1 boy and 1 girl in the group
will receive certiﬁcates. (iii) Given that the group has received 2 certiﬁcates, ﬁnd the
probability that exactly 1 boy and 1 girl received the certiﬁcates.
(9 , marks) END OF PAPER 2003 —AS-M & S— 1 0 26 i... ...
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- Spring '11