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Unformatted text preview: 96ASL
ﬁﬁfﬁﬁﬂﬂﬁTWIﬁﬂﬁ M&S HONG KONG EXAMINATIONS AUTHORITY
HONG KONG ADVANCED LEVEL EXAMINATION 1996 MATHEMATICS AND STATISTICS ASLEVEL 9.00 am—12.00 noon (3 hours)
This paper must be answered in English 1. This paper consists of Section A and Section B.
2. Answer ALL questions in Section A, using the AL(C1) answer book.
3. Answer any FOUR questions in Section B, using the AL(C2) answer book. 4‘ Unless othenvise speciﬁed, numerical answers should either be exact or
given to 4 decimal places. it 3 $§§Fﬁ§ﬂl§l$ ' ﬁﬂhﬁl—m z = 0 ,zéﬂifiﬁ z Zfix‘lﬁ‘ﬁﬁil‘z’lﬁlifaﬂﬁttﬁl ° are 2 Pﬁ‘éﬁﬁ’ﬂﬁfiﬁ
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96~ASM&S— 16 82 96ASM&S—1 83 SECTION A (40 marks)
Answer ALL questions in this section.
Write your answers in the AL(C1) answer hook. I. A stemandleaf diagram for the test scores of 30 students is shown below:
Stem (tens) Leafgunits)
l 0
2
3 0 2
4 4 5 8 9
5 0 l 2 6 8 8 9 9
6 2 3 3 5 5 8
7 l 2 2 4 4 4
8 2 5
9 l (a) Find the mean. mode and interquartile range of these scores. (b) If the score 71 is an incorrect record and the correct score is 11 ,
which of the statistics in (a) will have different values? Find the correct values ofthese statistics. (6 marks)
2. Let y=—IE (x>0). ﬁnd x dx
Hence or otherwise. ﬁnd I Inf (hr
x
(5 marks) x—l 3. Sketch the curve y = x _ 3 (x i 3) and indicate the asymptotes and intercepts.
(6 marks) 96ASM&S—2 84 4. Figure 1 shows a unit square target for shooting on the rectangular coordinate plane. The target is divided into three regions I, II and III by
the curves y=\/; and y=x3 . The scores for hitting the regions I, II and III are 10, 20 and 30
points respectively. y (a) Find the areas of the
three regions. (b) Suppose a child shoots
randomly at the target
twice and both shots hit
the target. Find the
probability that he will
score 40 points. (7 marks) Figure 1 5. At any time I (in hours), the relationship between the number N of tourists at a skiresort and the air temperature 0°C can be modelled by
N = 2930— (0+440) [ln(0+49)]z , where —45 S 0 S —40 . , I
(a) Express 93— in terms of 0 and ﬂ .
dt (11 (b) At a certain moment, the air temperature is —40 °C and it is falling at a rate of 0.5 °C per hour. Find, to the nearest integer, the rate of increase of the number of tourists at that moment.
(6 marks) 96ASM&S—3 35 'SECTION B (60 marks) I ’
Answer any FOUR questions in this section. Each question carries 15 marks. 6. A b E . . . _ .
company uys equ ll quantities of fuses, m 100 unit lots, from two Write your answers in the Aucz) answer book suppliers A and B. The company tests two fuses randomly drawn from each
lot, and accepts the lot if both fuses are nondefective. 1‘ is know“ lha‘ 4% 0f the {"505 from SUPPlicr A and 1% 0f the fuses from 8‘ There are several bags on a table each containing six cards numbered 0 , 1 ,
supplier B are defective. Assume that the quality of the fuses are 2 ’ 3 , 4 and 5 respectively.
independent of each other. (a) (i) Find the coefﬁcient of x5 in the expansion of . z . . 1 t. ,2
(a) What IS the prob tbtllty that a lot will be accepted (1+ x+x2 +x3 +x4 + x5)2 . (b) What is the probability that an acce ted lot came from su lier A?
p pp (6 marks) (ii) John takes two bags away from the table and randomly draws a card from each of them. Using (a)(i), or otherwise, find the
probability that the sum of the numbers on the two cards drawn 7. A reporter wished to find the mean number of children per family in a is 5 
village. She visited the only primary school in the village and asked all the
pupils there what the number of children, including themselves, was in each
of their families. The data obtained are presented in the following frequency (b) (i) Expand (1— x6)4 .
table: Number Ofd‘lldrcn (ii) Find the coefficient of x' , where r isa nonnegative
in the ram“ integer, in the expansion of (l—x)“4 for x< l . (4 marks) Number of pupils '3 (iii) Using (b)(i) and (b)(ii), or otherwise, find the coefficient of The reporter concludes that the mean number of children per family is 1_ x6 4
3 ‘ v . ' f r < 1 .
lxlO+2xll+3x6+4x9+5x4 x mummmns'o’mf 1x 0 M
10+ll+6+9+4 (7mark5)
ls her conclusion justiﬁed? Explain your answer.
(4 marks) (c) Joan takes four bags away from the table and randomly draws a card from each ofthem. Using (b)(iii), or otherwise, ﬁnd the probability that the sum of the numbers on the four cards drawn is 8 .
(4 marks) 96ASM&S—5 87
96~ASM&S—4 86 r The population size P of a species of reptiles living in a jungle increases at a rate of ‘1 dp Sew—2! (t20), E: where t is the time in month. It is known that P = 10 when I: O . (21) Use the trapezoidal rule with 6 subintervals to estimate I e ‘0 d! .
0 Hence estimate P, to the nearest integer, at I: 6 .
(7 marks) (b) Achemical plant was recently built near thcjungle. Pollution from
the plant atTects the growth of the population of the reptiles from
t: 6 onwards. An ecologist suggests that the population size of the
species of reptiles can then be approximated by P = kze'w’ 50 (t 2 6). (i) Using (a). ﬁnd the value of k correct to l decimal place. (ii) Determine the time at which the population size will attain its
maximum. Hence ﬁnd the maximum population size correct
to the nearest integer. (iii) Use the graph in Figure 2 to ﬁnd the value of t , correct to the
nearest integer, when the species of reptiles becomes extinct
due to pollution. (8 marks) 88 96ASM&S—6 The graph of y = e 0.0“ 96AS—M&S—7 Figure 2 89 a certain machine in a factory The monthly cost C(I) at time t of operating can be modelled by 10. Seat Number Centre Number a
b
m
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d
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a
C Page Total (0<ts36), ’—1 eb C(t) = a If you attempt Question 10, ﬁll in the details in the ﬁrst three
boxes above and tie this sheet INSIDE your answer book. 10(Cont’d) 3
1.70 2
1.44 Table 1 1.21 where t is in month and C(t) is in thousand dollars. Table 1 shows the values of C(t) when I = l, 2, 3, 4 . . 1 Lanai
a. ﬁg. III
IllIII I
1. 1+. .1. C0) + 1] as a linear function of t. l Express 1n (0 (a) Ham
an _ Ema ﬂan um. E
Em Use Table 1 and the graph paper on Page 8 to estimate graphically the values of 0 (ii) www.mmawﬁn and b correct to 1 decimal place. an “unﬁtwmmm. 2... a nﬁurmmm.
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Enﬂﬁﬂ
Eaaﬁ (8 marks) Using the values of a and b found in (a)(ii), estimate the monthly cost of operating this machine when I = 36 . (iii) income l’(!) generated by this machine at time t can The monthly (1}) mm ﬂan“
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I and 1’(() is in thousand dollars. The factory will stop using this machine when the monthly cost of operation exceeds the monthly income. where I isin month &
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a e the (ii) What is the total proﬁt generated by this machine? Giv answer correct to the ncarcst thousand dollars. (7 marks) 91 96ASM&S—9 90 96ASM&S—8 11. A machine discharges soda water once for each cup of soda water purchased.
The amount of soda water in each discharge is independently normally
distributed with mean 210 ml and standard deviation 15 ml . (a) Find the probability that the amount of a cup of soda water is between
200 m1 and 220 ml .
(2 marks) (b) Suppose cups of capacity 240 ml each are used.
(i) Find the probability that a discharge will overﬂow. (ii) What is the probability that there will be exactly 1 overﬂow
out of 30 discharges? (iii) ‘lf Sam buys a cup of soda water from the machine every day
starting on lst July, ﬁnd the probability that he will get the
second overﬂow on 3 lst July. (5 marks) (c) The vendor has decided to use cups of capacity 220 ml each and to
This is a blank lmgc repair the machine so that, on the average, 80 in 100 cups contain
more than 205 ml of soda water in each and only 1 in 100
discharges overﬂows. The amount of soda water in each discharge is
still independently normally distributed. (i) What will the new mean and standard deviation ofthe amount
of soda water in each discharge be? Give the answers correct
to 1 decimal place. (ii) If a discharge from the repaired machine overﬂows, ﬁnd the
probability that the amount of soda water in this discharge
exceeds 225 ml . Give the answer correct to 2 decimal places, (8 marks) 96ASM&S—l 0 92 96ASM&S—l 1 93 12. A factory supplies batches of lowquality electronic chips. Each batch
contains 6 chips, some of which may be defective. Table 2 (Page 12) shows
the frequency distribution of defective chips in each of 80 randomly selected
batches. (a) It is suggested that the number of defective chips in a batch could be
modelled by a binomial distribution with the probability that a chip is
defective being Oil . Fill in the missing expected frequencies, under
this distribution, in Table 2. (2 marks) (b) A buyer claims that the number of defective chips in a batch could be
approximated by a Poisson distribution with mean 1 . He has
calculated some expected frequencies as shown in Table 2. (i) Determine 1 correct to 1 decimal place. (ii) Fill in the missing expected frequencies, under this
distribution, in Table 2.
(3 marks) (c) A buyer compares the two distributions in (a) and (b) and adopts the
one which ﬁts the observed data better. He buys 4 batches of chips
" from the factory and classiﬁes a batch as good if all the chips in the
batch are nondefective. (i) Find the probability that at least 3 of these 4 batches are
good. ~ (ii) Suppose that at least 3 of these 4 batches are good 'and the
buyer buys 6 more batches Find the probability that exactly
8 of these 10 batches are good.
(10 marks) A 96AS~M&S— l 2 94 Candidate Number 12(Cont’d) D Page Total If you attempt Question 12, ﬁll in the details in the first three boxes above and tie this sheet INSIDE your answer book. Table 2 Observed and expected frequencies of defective chips
in each of 80 randomly selected batches Nombervof r v j . Observed ‘ * Correct to 1 decimal place. 96AS—M&S— l 3 . .Ex ectcd Frc ’uencvi“ 95 13. In city A, the occurrences of rainstorms are assumed to be independent. The
number of occurrences may be modelled by a Poisson distribution with mean
occurrence rate of 2 rainstorms per year. (a) Find the probability of having more than two rainstorms in a
particular year. (3 marks) (b) Last year, more than two rainstorms occurred. Estimate the number
of years which will elapse before the next occurrence of more than two
rainstorms in a year. Give the answer correct to the nearest integer. (3 marks) (c) Past experience suggests that the probability of having at least one
serious landslide in a year depends on the number of rainstorms in
that year as follows: Number of rainstorms
Probability of having at
least one serious landslide This is a blank page Find the probability that. in city A.
(i) there is no serious landslide in a particular year ; (ii) no rainstorm has occurred ifthcre is no serious landslide in a
particular year ; (iii) there is no serious landslide for at most 2 out of 5 years.
(9 marks) END OF PAPER 96—ASM&S—l4 96 96ASM&S—15 97 ...
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