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Unformatted text preview: DOBASL
I 8: S HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG ADVANCED LEVEL EXAMINATION 2006 MATHEMATICS AND STATISTICS ASLEVEL 8.30 am — 11.30 am (3 hours)
This paper must be answered in English This paper consists of Section A and Section B. Answer ALL questions in Section A, using the AL(E) answer book. 2.
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.
©i§i%%i€ﬁﬁijl=i¥% R¥Iﬂii§ Hong Kong Examinations and Assessment Authority
All Rights Reserved 2006 2006—ASM & S—l SECTION A (40 marks)
Answer ALL questions in this section.
Write your answers in the AL(E) answer book. —1 (a) The coefﬁcients of x and x2 in the binomial expansion of [1+5] ’7
a
—l 1 . .
are ~— and  respectlvely, where a IS a nonzero constant and
18 24a 11 is a positive integer. (i) Find the values of a and n .
(ii) State the range of values of x for which the binomial expansion
—l
of [1 +1] ” is valid.
a (b) Using the results of (a) , or otherwise, (i) write down the binomial expansion of (9 +x) 2 in ascending powers of x as far as the term in x2 ; (ii) state the range of values of x for which the binomial expansion
41 of (9+x) 2 is valid.
(6 marks) After adding a chemical into a bottle of solution, the temperature S(t) of the
surface ofthe bottle can be modelled by S(t)=2(t+1)ze_’”+15, where S(t) is measured in °C , t (2 0) is the time measured in seconds after the chemical has been added and xi is a positive constant. It is given that
8(9) 2 S(19) . (a) Find the exact value of xi. . (b) Will the temperature of the surface of the bottle get higher than 90°C ?
Explain your answer. (6 marks) 2006—ASM & 8—2 V 2 2006ASM & S—3 3 The rate of change of the amount of water in litres ﬂowing into a tank can be
modelled by
500 f :_
(I) (r+2)2e" where t (2 0) is the time measured in minutes. (a) Using the trapezoidal rule with 5 sub—intervals, estimate the total
amount ofwater ﬂowing into the tank from i=1 to t=ll . d2f(t)
dt2 (b) Find . (c) Determine whether the estimate in (a) is an overestimate or
underestimate. (7 marks) The stem—and—leaf diagram below shows the distribution of the numbers of
books read by 24 students of a school in the ﬁrst term: Stem {tens} Leaf g units)
6 7 4
223567889
3455789 0 UJNV—‘O
oy—‘r—tUJ (a) Find the median and the interquartile range of the numbers of books
read. (b) The librarian of the school ran a reading award scheme in the second
term. The following table shows some statistics of the distribution of
the numbers of books read by these 24 students in the second term: Minimum Lower quartile Median Upper quartile Maximum
8 26 35 41 46 (i) Draw two box—andwhisker diagrams of the same scale to
compare the numbers ofbooks read by these students in the first
term and in the second term. (ii) The librarian claims that not less than 50% of these students read
at least 5 more books in the second term than that in the ﬁrst
term. Do you agree? Explain your answer. (7 marks) Go on to the next page 5. A and B are two events. Suppose that P(AﬂB):0.2 and P(AlB')=O.5 ,
where B’ is the complementary event of B . Let P(B) = b , where b <l . (a) Express P(AmB’) and P(A) in terms of b. (b) if A and B are independent events, ﬁnd the value(s) of b .
(7 marks) A teacher, Susan, built up a performance assessment scheme in which
her 100 students were awarded merit points according to their test scores. She
also used a normal distribution and a Poisson distribution to model the test scores and the merit points of the students respectively (see the following
table). Expected number of students *
Normal Poisson
distribution distribution Merit Observed number
points of students Test score (X) 50£X<150 0 20 14.65 l 24.66
150sX<250 41 44.00 34.52
250$X<350 28 33.45 24.17
350$X<450 9 6.38 11.28 X 2450 2 0.30 3.95 Sum = 100 Sum = 98.78 Sum : 98.58 hwai—A * Correct to 2 decimal places. (a) In the above table, why is the sum of the expected numbers of students
under each distribution less than 100 ? (b) The absolute values of the differences between the observed numbers of
students and the expected numbers of students are regarded as errors.
The distribution with a smaller sum of errors will ﬁt the observed data
better. Which distribution, normal or Poisson, ﬁts the observed data
better? Explain your answer. (7 marks) 2006—ASM & S—4 4 2006—ASM & S—5 5 SECTEON B (60 marks)
Answer any FOUR questions in this section. Each question carries 15 marks.
Write your answers in the AL(C) answer book. Deﬁne f(x): a+bx 4 forall x¢4 . Let g(x)=f(—x) forall x¢—4 .
—x Let C1 and C2 be the curves y=f(x) and y=g(x) respectively. 1‘ It is given that the yintercept of C1 is 73 while the xintercept of C2
is —2 . (a) Find the values of a and b .
(2 marks) (b) (i) Find the equations of the vertical asymptote(s) and the horizontal
asymptote(s) to C1 . (ii) Sketch C1 and indicate its asymptote(s) and its intercept(s).
(5 marks) (c) On the diagram sketched in (b)(ii), sketch C2 and indicate its asymptote(s), its intercept(s) and the point(s) of intersection of the two
curves. (4 marks) (d) Find the area enclosed by C1 , C2 and the straight line y = 9 .
(4 marks) Go on to the next page An airline manager, Christine, notices that the weekly number of passengers of
the airline is declining, so she starts a promotion plan to boost the weekly
number of passengers. She models the rate of change of the weekly number of
passengers by dx 301 — 90 ~=— :20 ,
dt [2—6z‘+ll ( ) where x is the weekly number of passengers recorded at the end of a week in
thousands of passengers and t is the number of weeks elapsed since the start of
the plan. Christine ﬁnds that at the start ofthe plan ( i.e. t = O ) , the weekly number of
passengers is 40 thousand. (a) Let v=t2—6t+11,ﬁnd %. Hence, or otherwise, express x in terms of t .
(4 marks) (b) How many weeks after the start of the plan will the weekly number of passengers be the same as at the start of the plan?
(2 marks) (c) Find the least weekly number of passengers after the start of the plan.
Give your answer correct to the nearest thousand.
(3 marks) (d) The week when the weekly number of passengers drops to the least is
called the Recovery Week. (i) Find the change in the weekly number of passengers from the
Recovery Week to its following week. Give your answer correct
to the nearest thousand. (ii) Provethat(t+1)2—6(t+1)+11<3(12—6t+11) forall 1'. (iii) Christine’s assistant claims that after the Recovery Week, the
change in the weekly number of passengers from a certain week
to its following week will be greater than 25 thousand. Do you agree? Explain your answer.
(6 marks) 2006—AS—M & S—6 6 2006—ASM & S—7 7 After upgrading the production line of a cloth factory, two engineers, John and
Mary, model the rate of change of the amount of cloth production in thousand
metres respectively by f(z)=2522(t+10)3 and g(t)=28+keht2, where h and k are positive constants and t (2 0 ) is the time measured in
months since the upgrading of the production line. (a) Using the substitution u =t+10 , or otherwise, ﬁnd the total amount
of cloth production from t = 0 to t = 3 under John’s model.
(5 marks) (b) Express 1n (g0) —28) as a linear function of 22 .
(1 mark) (c) Given that the slope and the intercept on the vertical axis of the graph of
the linear function in (b) are measured to be 0.3 and 1.0 respectively,
estimate the values of h and k correct to 1 decimal place. (2 marks) (d) Using the estimated values of h and k obtained in (c) correct
to 1 decimal place, (i) expand g(t) in ascending powers of t as far as t6 , and hence estimate the total amount of cloth production from t=0 to
t = 3 under Mary‘s model; (ii) determine whether the estimate in (d)(i) is an over—estimate or an
underestimate; (iii) determine whether the total amount of cloth production from
t = 0 to t: 3 under Mary’s model is greater than that under
John’s model. (7 marks) (30 on to the next page 10. A researcher models the number of cars entering a roundabout in ﬁvesecond
time intervals (FSTIs) by a Poisson distribution with a mean of 4.7 cars per
FSTI, and the speed of a car entering the roundabout by a normal distribution
with a mean of 42.8 km/h and a standard deviation of 12 km/h . A car is
speeding if the speed of the car is over 50 km/h . (a) Find the probability that fewer than 6 cars enter the roundabout in a
certain FSTI.
(3 marks)
(b) Find the probability that a car entering the roundabout is speeding.
(2 marks)
(c) Find the probability that the 6th car entering the roundabout is the
lst Speeding car. >_
(3 marks)
(d) The roundabout is hazardous in a certain FSTI if at least 4 cars enter
the roundabout in that FSTI and more than 2 of them are speeding.
(i) If exactly 4 cars enter the roundabout in a certain FSTI, ﬁnd the
probability that the roundabout is hazardous in that FSTI.
(ii) Given that fewer than 6 cars enter the roundabout in a certain
FSTI, ﬁnd the probability that the roundabout is hazardous in
that FSTI.
(7 marks)
2006—ASM & 8—8 8 ll. 2006—AS_M & S—9 9 A manufacturer of brand E grape juice starts a marketing campaign by issuing
points which can be exchanged for gifts. The number of points is shown on the
back of the lid of each can of brand E grape juice. The probabilities for a
customer to get a can of brand E grape juice with a 2—point lid and 5point lid
are 0.8 and 0.2 respectively. Atotal of 15 points or more can be exchanged
for a packet of potato chips while a total of 20 points or more can be exchanged
for a radio. (21) Find the probability that a customer can exchange for a packet of potato
chips in buying 5 cans of brand E grapejuice.
(3 marks) (b) A customer, Peter, buys 7 cans of brand E grapejuice. (i) Find the probability that only when the 7th can of brand E grape
juice has been opened, Peter gets a 5—point lid. (ii) Find the probability that only when the 7th can of brand E grape
juice has been opened, Peter can exchange for a radio. (iii) Given that Peter can exchange for a radio only when the 7th can
ofbrand E grapejuice has been opened, ﬁnd the probability that
the 7th can of brand E grapejuice has a 5—point lid. (iv) Given that Peter cannot get a packet of potato chips after opening 5 cans of brand E grapejuice, ﬁnd the probability that he can exchange for a radio only when the 7th can of brand E
grapejuice has been opened. (12 marks) Go on to the next page Table: Area under the Standard Normal Curve 12. There are many plants in a greenhouse and all ofthem are ofthe same species,
Assume that the numbers of infected leaves on the plants in the greenhouse are
independent and the number of infected leaves on each plant follows a Poisson
distribution with a mean of 2.6 . A plant with at least 4 infected leaves is
classiﬁed as unhealthy. (a) Find the probability that a certain plant in the greenhouse is unhealthy.
(3 marks) (b) A researcher, Teresa, inspects the plants one by one in the greenhouse.
She ﬁnds that the Mth inspected plant is the ﬁrst unhealthy plant. .0 (i) Find the probability that M is less than 5 . .1 .2 (ii) Given that M is less than 5 , ﬁnd the probability that M is :2 greater than 2 . I 5 l.6 (iii) If Teresa inspects In plants in the greenhouse, ﬁnd the least 1'7 value of In so that the probability of ﬁnding an unhealthy plant is greater than 0.95 . ' 7'0 (9 marks) 5:1 2.2 (c) It is given that there are 150 plants in the greenhouse. The number of 33 unhealthy plants in the greenhouse is recorded every Friday. Let N be "4
the number of unhealthy plants recorded on a Friday. Find the mean and the variance of N. g: (3 marks)
END OF PAPER
Note: An entry in the table is the proportion ofthe area under the entire curve which is between 2 = 0
and a positive value of z , Areas for negative values of z are obtained by symmetry,
Ar ) f 1 :iz'dx
z = e
OVZn
2006AS—M & 8—10 10 2006'AS'M & 5‘“ 11 ...
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