香 港 培 正 中 學
PUI CHING MIDDLE SCHOOL
二零零二年度第一學段考試
科
目
考試日期
試題紙頁數
1.
2.
3.
數學及統計學
卷
22 - 1 - 2003
考試時間
4
答案紙頁數
別
******
8:30 ~ 10:30a.m.
姓 名
班 別
班 號
級別
中 六
時限
兩小時
備註
******
This paper consists of Section A and Section B.
Answer ALL questions in both Section A and Section B.
Unless otherwise specified, numerical answers should either be exact or given to 4 decimal
places.
SECTION A ( 60 marks)
2x  3
from first principles.
1 x
1.
Find the derivative of y 
2.
Find the constant term in the expansion of  3 x 
3.
A family of 5 invited 7 friends including Mr. Chan and his wife to join their social dinner at a
(4 marks)
9


1 
 .
x2 
(5 marks)
round table.
(a) Find the number of different ways in which all people can be seated at a round table.
(b) In how many ways can all people be seated so that the family of 5 and Chan's family must be
seated together respectively.
(6 marks)
4.
It is given that e xy 
x( x  1) 3
, where x  0 .
x2 1
(a)
Find the value of y when x  1 .
(b)
Find the value of
dy
when x  1 .
dx
(6 marks)
5.
At any time t (in hours), the relationship between the number N of tourists at a ski-resort and the
air temperature  C can be modelled by
N  2930  (  440)ln(   49)2 ,
where  45    40 .
(a) Express
dN
d
in terms of  and
.
dt
dt
(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)
Math&Stat1_M6_02
P.1
6.
(a)
If y 
e
x
for x  0 , find
x
dy
.
dx
(b) Use logarithmic differentiation to find f (x) of the function f ( x) 
x 1
,
x( x  2) 3
4
where x  2 .
(7 marks)
7.
In a game, four boys and three girls are selected from six boys and five girls respectively to sit on
a row of seven chairs.
(a) How many arrangements will be possible if boys and girls sit on different ends of the seven
chairs?
(b) How many arrangements will be possible if the chair in the middle must be occupied by a
girl?
(8 marks)
8.
(a) Write down the expansion of e x and e  x in ascending powers of x.
2
2
(b) Express e x  e  x in ascending powers of x.
2
(c) Expand
ex
(1 
2
2
1
2 x) 2
in ascending powers of x as far as the term in x3.
State the range of values of x for which the expansion is valid.
(9 marks)
9.
A new product is promoted to the market. A researcher suggests that the proportion of customers
who respond to the advertisement can be modelled by
P(t )  1  e 0.05t ,
where t is the number of days elapsed since the advertisement began to broadcast.
The marketing area contains 7 500 000 potential customers, and the revenue to the company is
$1.5 for each response. This revenue is exclusive of advertising cost. The fixed cost of producing
the advertisement is $800 000 and the variable cost is $12 000 for each day the advertisement
runs.
(a) Find the value of P(t) when t is very large and interpret the answer.
(b) What is the percentage of customer response after 28 days of advertising?
(c) After 28 days of advertising, what is the net profit, in the nearest dollar?
(9 marks)
Math&Stat1_M6_02
P.2
Section B(40 marks)
10. The government of a country enumerated the population at successive 5-year intervals from (the
beginning of ) 1970 and reported the following figures:
t
P(t)
5
6.75
10
9.11
15
12.30
20
16.60
25
22.41
It is suggested that the population P(t) (in millions) of the country in t years from 1970 can be
modelled by
P(t )  Ae kt .
(a) (i) Express ln P(t ) as a linear function of t and use the graph paper provided in Page 4 to
estimate the values of A and k. (Correct your answers to 1 decimal place.)
(ii) Find the population of the country at the beginning of 1970.
(iii) Find, correct to the nearest integer, the number of years that have elapsed
when the population first exceeded twice the population in (a)(ii).
(13 marks)
(b) Suppose that r(t) per cent of the total population of the country will become voter.
The function r(t) can be modelled by
r (t )  70(1  e 0.08t )  10 .
The number of voters N(t) (in millions) can be modelled by
N (t )  P(t )  r (t )% .
(i)
Determine the number of voters at the beginning of 1970.
(ii) Find the rate of increase of the number of voters at the beginning of 1970.
( 7 marks)
11. (a)
Find the equation(s) of the tangent(s) to the curve 3ay 2  x 2 ( x  a) at x = 2a.
(11 marks)
(b) (i) Find the coordinates of the point P at which this tangent with positive
gradient meets the curve again.
(ii) Show that the tangent in (b)(i) is the normal to the curve at P.
(9 marks)
Math&Stat1_M6_02
P.3
Please fill in the details in the boxes below. The completed graph should be torn off and submitted as
part of the solutions.
-----------------------------------------------------------------------------------------------------------------
Name
Class
Class Number
Graph paper for part (a)(i), Question 10
lnP(t)
3.5
3
2.5
2
1.5
1
0.5
t
0
5
15
10
END OF PAPER
Math&Stat1_M6_02
P.4
20
25