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Unformatted text preview: SECTION A (40 marks) Answer ALL questions in this section.
HONG KONG EXAMINATIONS AUTHOR'TY Write your answers in the AL(C‘1) answer book. HONG KONG ADVANCED LEVEL EXAMINATION 1995 95—ASL
M818 MATH EMATICS AN D STATISTICS ASLEVEL 1. The numbers of hours spent by 25 students in studying for an examination are as follows:
1 l 8 25 21 18 25 7 32 29 18 1818,22 12 5 3o 19 15 20’ 50
25 10 26 23' 12 i 9.00 am12.00 noon (3 hours)
This paper must be answered in English (a) Copy and complete the following stemandleaf diagram for the above data:
' W Leaf tin 11
1. This paper consists of Section A and Section B.
' O 5 7 8
2. Answer ALL questions in Section A, using the AL(C1) answer book. 1
2
3. Answer any FOUR questions in Section B, using the AL(C2) answer book. 3
4
4. Unless otherwise speciﬁed, numerical answers should either be exact or 5 given to 4 decimal places.
(b) Find the mode, the median and the interquartile range 0f the numbers of hours spent by the 25 students.
(6 marks) ,_____
(b) If y = ——!—\J W where x>2 , use logarithmic
x + 1 x + 1 differentiation to ﬁnd 3% . (6 marks) 95ASM&S——1 52 95ASM&S—2 53 6. Figure 1 shows( graphs of the two curves 3. A teacher wants to divide a class of 18 students into 3 groups, each of C1 ; y = 22" +4
6 students, to do 3 different statistical projects. V
and C2: y = 5(2‘).
(a) In how many ways can the students be grouped? (b) If there are 3 girls in the class, ﬁnd the probability that there is
one girl in each group.
(6 marks) 4. The value M (in million dollars) of a house is modelled by the equation
M 1 l + d: 3t+4 Vt+25 where t is the number of years elapsed since the end of 1994 . The
value of the house was 3.1 million dollars at the end of 1994 . Figure 1 (a) Find, in terms of t , the value of the house 1 years after the end
Of 1994 ' (a) Find the coordinates of the points of intersection of C1 and C2 . (b) Find the rise in the value of the house between the end of 1994
and the end of 2000 . (b)
(7 marks) Hence, or otherwise, ﬁnd the area of the shaded region in Figure 1 bounded by C1 and C2 . If 2‘ = e“ forall 1, ﬁnd a. (8 marks) 5. An insurance company classiﬁes the aeroplanes it insures into class L (low
risk) and class H (high risk), and estimates the corresponding proportions
of the aeroplanes as 70% and 30% respectively. The company has also
found that 99% of class L and 88% of class H aeroplanes have no
accident withina year. If an aeroplane insured by the company has no 
accident within a year, what is the probability that it belongs to (a) class H ? (b) class L ?
(7 marks) 95AsMas4 55 95ASM&S3 54 SECTION B (60 marks) Answer any FOUR questions from this section. Each question carries 15 marks.
Write your answers in the AL(C2) answer book. 7. Let f(x) = (a) (1)
(ii)
(iii)
(1)) (i)
(ii)
95ASM&S5 1 where Osxgl,
\ll—x2 2 Find the estimate II of I using the trapezoidal rule with 5
subintervals. Find f/(x) and Wm. Using (a)(ii) or otherwise, state whether 11 in (a)(i) is an
overestimate or underestimate of I . Explain your answer
brieﬂy. ‘ (7 marks) Use the binomial expansion to ﬁnd a polynomial p(x) of degree 6 which approximates f(x) for 0 s x s :— . 1
Let I2 = fez p(x)dx. Find 12 . State whether I2 in (b)(i) is an overestimate or under— estimate of I . Explain your answer brieﬂy.
(8 marks) 56 A merchant sells compact discs (CD5). A market researcher suggests that
if each CD is sold for 3): , the number N(x) of CDs sold per week can be modelled by ’ N0) = ae“"‘ where a and b are constants. The merchant wants to determine the values of a and. b based on the
following results obtained from a survey: (a) (i) Express lnN(x) as a linear function of x. (ii) Use the graph paper on Page 6 to estimate graphically the
values of a and b correct to 2 decimal places.
(7 marks) (b) Suppose the merchant wishes to sell 400 CDs in the next Week.
Use the values of a and b estimated in (a) to determine the price of each CD. Give your answer correct to 1 decimal place.
(2 marks) (0) It is known that the merchant obtains CDs at a cost of $ 10 each.
Let C(x) dollars denote the weekly proﬁt. Using the values of a
and b estimated in (a), (i) express G(x) in terms of x ; (ii) , ﬁnd G,'(x) and hence determine the selling price for each CD in order to maximize the proﬁt.
(6 marks) 95ASM&S—6 57 Seat Number Centre Number r.
m
m
u
N
m
a
..m
d
m
C Page Total If you attempt Question 8, fill in the details in the ﬁrst three
boxes above and tie this sheet INSIDE your answer book. 8.(Cont‘d) This is a blank page 95ASM&58 . I uﬂI 11 ..
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7 6 5 2 .1 o 59 58 95AS~M&S7 9. A local engineering company estimates that if the loudness of the noise
released from the Hong Kong Stadium is to be reduced by x% , the cost Cl(x) , in thousand dollars, will be 800x
100 x C1(x)= Osx<100. 7 If the loudness of the noise released from the Stadium is to
be reduced by 80% , ﬁnd the cost required. (a) (i) (ii) If the Urban Council spends 2 million dollars for the work,
by what percentage can the loudness of the noise released from the Stadium be reduced?
(3 marks) (b) (i) Find C((x) and C,”(x). (ii) Determine the convexity and the vertical asymptote of the curve y = C,(x) . Sketch this curve.
' (7 marks) (c) An overseas engineering company estimates that if the loudness of
the noise released from the Hong Kong Stadium is to be reduced by 1% , the cost C2(x) , in thousand dollars, will be 400x
100 —x +1200, 0$x<100. C28) = (i) Find the value of x such that Cl(x) = Cz(x) . (ii) On the same graph sketched in (b)(ii), sketch the curve y = Cztx) .
Cltx) 2 ﬁrst. l [Hint You may sketch the curve y = (iii) If the Urban Council has a budget of 2 million dollars for
reducing the loudness of the noise released from the Stadium, use c(ii) to determine which company is more cost effective. '
(5 marks) 95ASM&S9 (,0 10. A shop special, ..g in bicycles recently opened up a new branch selling
bicycles only. Experience from other branches showed that the number of
bicycles sold in a day could be modelled by a Poisson distribution With
mean either 2 or 3. The branch manager recorded the number of bicycles
sold in a day for the ﬁrst 200 days as follows: Number of bicycles sold “I...
Observed frequency (dam E...” (a) (i) Using the Poisson distributions with means 2 and 3
respectively, ﬁll in the missing probabilities and expected
frequencies in Table l (Page 10). (ii) State which of the two Poisson distributions ﬁts the data
‘ better. (6 marks) (b) Adopting the model you have chosen in (a)(ii), what is the
probability that (i) no bicycle will be sold in a day? (ii) no bicycle will be sold in exactly 3 out of the next 7 days?
(5 marks) (c) The branch manager decides to sell tricycles as well. He knows
that the number of tricycles sold in a day can be modelled by a
Poisson distribution with mean 2. Adopting the model chosen in
(a)(ii) for the number of bicycles sold and assuming that the
numbers of bicycles and tricycles sold are independent of each
other, what is the probability that exactly three items are sold in a
day? i (4 marks) 95'AS‘M&S'1O 61 Candidate Ntuuber Centre Number Seat Number
' ' Page Total 10.(Cont’d) If you attempt Quwtion 10, fill 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 bicycles sold Observed
Bicycles Frequency *
Sold Expected Frequency Expected * Probability Frequency 0.0902
0 0527 * Correct to 1 decimal place. 95AS~M&s<11 62 95—ASM&S—1 2 This is a blank page 11. Madam Wong purchases cartons of oranges from a supplier every day.
Her buying policy is to randomly select ﬁve oranges from a carton and
accept the carton if all ﬁve are not rotten. Under usual circumstances.
2% of the oranges are rotten.
(a) Find the probability that a carton of oranges will be rejected by
‘ Madam Wong.
(3 marks)
(b) Every day, Madam Wong keeps on buying all the accepted cartons
of oranges and stops the buying exercise when she has to reject a
carton. What is the mean, correct to 1 decimal place, of the
number of cartons inspected by Madam Wong in a day?
(3 marks)
(c) Today, Madam Wong has a target of buying 2O acceptable cartons
of oranges from the supplier. Instead of applying the stopping rule
in (b), she will keep on inspecting the cartOns until her target is
achieved. Unfortunately, the supplier has a stock of 22 cartons
only.
(i) Find the probability that she can achieve her target.
(ii) Assuming she can achieve her target, ﬁnd the probability that
she needs to inspect 20 cartons only?
(7 marks)
((1) The supplier would like to import oranges of better quality so that
each carton will have at least a 95% probability of being accepted
by Madam Wong. If r% of these oranges are rotten, ﬁnd the
greatest acceptable value of r .
(2 marks)
95~AS—M&S13 64 12. A test is used to diagnose a disease. For people with the disease it is
known that the test scores follow a normal distribution with meari 70 and
standard deviation 5 . For people without the disease, the test Scores
follow another normal distribution with mean I u and the same standard deviation 5 . It is known that 33% of those people without the disease
wtll achieve a test score over 63.2 . (a) Find u . (3 marks) (b) It is estimated that 15% of the population of a city has the disease.
A doctor has proposed that a person be classiﬁed as having the disease if the person's test score exceeds 66 , otherwise the person
Will be classiﬁed as not having the disease. If a person is randomly selected from the population to take the test, (i) what is the probability that this person will be classiﬁed as
having the disease? (ii) ﬁnd the probability that this person will be misclassiﬁed.
‘ (12 marks) END OF PAPER 95AS—M&S‘l 4 65 ...
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