Please stick the barcode label here.

HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY

HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 2012 I Candidate Number I I I I I I I I I I
MATHEMATICS Extended Part
Module 1 (Calculus and Statistics)
Question-Answer Book
8.30 am -11.00 am (2½hours)
This paper must be answered in English

INSTRUCTIONS
1. After the announcement of the start of the examination, you
should first write your Candidate Number in the space provided
on Page I and stick barcode labels in the spaces provided on
Pages 1, 3, 5 and 7.
2. This paper consists of Section A and Section B. Answer ALL
questions in this paper.

3. Write your answers for Section A in the spaces provided in this

Question-Answer Book. Do not write in the margins. Answers
written in the margins will not be marked.
4. Write your answers for Section B in the DSE(B) answer book.
Start each question (not part of a question) on a new page.

5. Graph paper and supplementary answer sheets will be supplied

on request. Write your Candidate Number, mark the question
number box and stick a barcode label on each sheet, and fasten
them with string INSIDE the book.
6. The Question-Answer book and the answer book will be
collected separately at the end of the examination.
7. Unless otherwise specified, all working must be clearly shown.
8. Unless otherwise specified, numerical answers should be either
exact or given to 4 decimal places.

9. The diagrams in this paper are not necessarily drawn to scale.

10. No extra time will be given to candidates for sticking on the
barcode labels or filling in the question number boxes after the
'Time is up'announcement.

©香港考試及評核局 保留版權
Hong Kong Examinations and Assessment Authority
All Rights Reserved 2012

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Section B (50 marks)
In this section, write your answers in the DSE(B) answer book.

10. Let I= f-}e dt . T

(a) (i) Use the trapezoidal rule with 6 sub-intervals to estimate I.

(ii) Is the estimate in (a)(i) an over-estimate or under-estimate? Justify your answer.

(7 marks)

J
2
2二
(b) Using a suitable substitution, show that I= 2 e 2 d.x .

(3 marks)

(c) Using the above results and the Standard Normal Distribution Table on page 14, show that冗< 3.25 .
(3 marks)

11. In a research of the radiation intensity of a city, an expert modelled the rate of change of the radiation intensity R (in
suitable units) by

dR a(30-t)+10
dt (t-35) 2 +b

where t (0�t�T) is the number of days elapsed since the start of the research, a , b and T are positive
constants.

It is known that the intensity increased to the greatest value of 6 units at t = 35 , and then decreased to the level as
61
at the start of the research at t = T . Moreover, the decrease of the intensity from t = 40 to t = 41 is ln —
50
units.

(a) Find the value of a .

(2 marks)

(b) Find the value of T.

(4 marks)

(c) Express R in terms of t .

(4 marks)

(d) For Ost s35 , when would the rate of change of the radiation intensity attain its greatest value?
(4 marks)

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12. A company provides cable-car service for tourists. Tourists complain that the waiting time for the cable-car is too
long. From past experience, the waiting time (in minutes) of a randomly selected tourist follows a normal
distribution with meanµ and standard deviation 9 .

(a) The customer service manager of the company conducts a survey on the waiting time to estimateµ.

(i) A random sample of 16 tourists is taken and their waiting times are recorded as below:
56 36 48 63 57 41 5O 43
56 55 62 46 55 69 38 50
Construct a 90% confidence interval forµ.

(ii) Find the least sample size to be taken such that the width of the 90% confidence interval forµ is less
than 6 minutes.
(7 marks)

(b) Suppose thatµ= 51.5 . The customer service manager of the company interviews tourists and will give a
coupon to a tourist whose waiting time is more than 65 minutes.

(i ) Find the probability that he gives less than 2 coupons to the first IO tourists interviewed.

(ii) Find the probability that the 5th coupon is given to the 20th tourist interviewed.
(6 marks)

13. Drunk driving is against the law in a city. The police set up an inspection block at the entrance of a certain highway
at night in order to arrest drunk drivers. From past experience, the number of drunk drivers arrested follows a
Poisson distribution with mean 2.3 per hour.

(a) Find the probability that at least 2 drunk drivers are arrested in a certain hour.
(2 marks)

(b) Given that at least 2 drunk drivers are arrested in a certain hour, find the probability that not more than 4
drunk drivers are arrested.
(3 marks)

(c) In a certain week, the police sets up an inspection block for three nights, all at the same period from 1:00 am to
2:00 am. It is known that the numbers of drunk drivers arrested in different nights are independent.

(i) Find the probability that the third night is the first night to have at least 2 drunk drivers arrested.

(ii) Find the probability that at least 2 drunk drivers are arrested in each of the 3 nights and there are totally
10 drunk drivers arrested.
(5 marks)

ENDOFPAPER

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 Standard Normal Distribution Table

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830
1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177
1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767
2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890
2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916
2.4 .4918 .4920 .492.2 .4925 .4927 .4929 .4931 .4932 .4934 .4936
2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952
2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964
2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974
2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981
2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986
3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993
3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995
3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4997
3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998
3.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998

Note : An entry in the table is the area under the standard normal curve between x = 0 and
x = z (z�0) . Areas for negative values of z can be obtained by symmetry.

－ __ A(z)
－－－

J。乙
-x2
z 1 —
A(z)= -e 2 dx

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