香 港 培 正 中 學
PUI CHING MIDDLE SCHOOL
二零零四年度 第二學段考試
科
目
數學及統計學
考試日期
答案紙頁數
4
別
******
考試時間
22 - 6 - 2005
試題紙頁數
1.
2.
3.
卷
******
8:30 ~ 11:30a.m.
姓 名
班 別
班 號
級別
中 六
時限
_三小時
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 (40 marks)
1.
The number of ants is given by
N (t ) 
A
,
1  4e  t
where A and are constants and t is the number of weeks which have elapsed since the population
was first examined. Find the values of A and given that initially there are 20 000 ants, and there
are 50 000 ants after 4 weeks.
(5 marks)
2.
(a) If x 
y
1 y
2
, find
(b) If x 2 y  e y  10 , find
3.
(a) If
dy
.
dx
(6 marks)
3x  5
A
B
C



, find the constants A, B and C.
2
x  x  x  1 x  1 x  1 ( x  1) 2
3
(b) Hence, find
4.
dy
.
dx
x
3
3x  5
dx .
 x2  x 1
(6 marks)
An environmental agency has determined that the rate of change of the quantity of pollutants
(measured in tons per year) that a certain company is pumping into a river is given by
dQ
 10e 0.03t ,
dt
where t is the number of years since the beginning of 2002.
(a) Find the total amount of pollutants pumped into the river from the beginning of 2002 to the
beginning of 2010. (Correct your answers to 4 significant figures.)
(b) On which year will the total amount of pollutants pumped into the river exceed 500 tons?
(6 marks)
Math&Stat2_M6_04
P.1
5.
In a game, five boys and two girls are selected from eight boys and four girls respectively to sit on
a row of seven chairs. In how many ways can be arranged if
(a) boys and girls sit on different sides of the seven chair?
(b) the girls sit next to each other?
(c) the chair in the middle must be occupied by a boy?
(8 marks)
6.
The coefficients of x and x2 in the expansion of
are 
(1  x) k
, where k and a are rational numbers,
e ax
5
23
and
respectively.
2
8
(a) Find the values of a and k.
(b) Hence, find the coefficient of x3 and also state the range of values of x for which the
expansion is valid.
(9 marks)
Math&Stat2_M6_04
P.2
Section B (60 marks)
7.
Given f ( x) 
(a)
3x
.
x 1
2
Find the equations of the asymptote(s) to the curve y = f(x).
(2 marks)
(b) Sketch the curve and indicate the asymptote(s), intercept(s), the turning point(s) and the
point(s) of inflection.
(8 marks)
(c)
8.
Find the area enclosed by the curve and the straight line y  1 .
(5 marks)
A textile factory has purchased a new weaving machine to increase its production of cloth. The
monthly output x (in km) of the machine, after t months, can be modeled by the function
x  100e 0.01t  65e 0.02t  35 , (0  t  62) .
(a) Suppose the cost of producing 1 km of cloth is US$300; the monthly maintenance fee of the
machine is US$300 and the selling price of 1 km of cloth is US$800.
(i)
Write down the profit function in terms of t.
(ii) Using differentiation, determine in which month will the greatest monthly profit be
obtained? Find also the profit, to the nearest US$, in that month.
(7 marks)
(b) The machine is regarded as “inefficient’ when the monthly profit falls below US$500 and it
should then be discarded.
(i) Determine the number of months the factory should keep the machine.
(ii) Estimate the total amount of cloth, to the nearest km, produced during the operation
period.
(iii) Find also the total profit, to the nearest US$, be obtained by the machine.
(8 marks)
Math&Stat2_M6_04
P.3
9.
Let f ( x) 
1
3
1 x2
where 0  x 
1
,
2
1
and I   2 f ( x) dx .
0
(a) (i)
Find the estimate I1 of I using the trapezoidal rule with 5 sub-intervals.
(ii) Find f ( x) and f ( x) .
(iii) Hence or otherwise, state whether I1 in (a)(i) is an over-estimate or under-estimate of I.
Explain your answer briefly.
(8 marks)
(b) (i) Use the binomial expansion to find a polynomial p(x) of degree 6 which approximates
f(x) for 0  x 
Let
1
.
2
1
2
0
I 2   p( x) dx . Find I 2 .
(ii) State whether I 2 in (b)(i) is an over-estimate or under-estimate of I. Explain your
answer briefly.
(7 marks)
10. A company has 10 supervisors and 20 junior staff. The mean and the standard deviation of the
weekly salaries of the supervisors are $5200 and $1200 respectively. The mean and the standard
deviation of the weekly salaries of the other staff are $3800 and $1000 respectively.
(a) Find the mean and the standard deviation of the weekly salaries of all the staff.
(7 marks)
(b) It is proposed to revised the weekly salaries of all the staff using the following formula:
yi  A 
B
( xi  4500) ,
1500
i  1, 2, 3, ,30.
where xi is the present weekly salary of the ith staff,
yi is the revised weekly salary of the ith staff, and
A and B are constants.
(i)
If, in the review, the highest weekly salary is increased from $9000 to $9950 and the
lowest weekly salary is increased from $1500 to $1700, find the values of A and B.
(ii) Find the mean and the standard deviation of the new weekly salaries of all the staff.
(6 marks)
(c) In another proposal, it is planned to increase the present weekly salary of every staff by $500.
Determine whether this proposal or the proposal in (b) would cost the company more.
Explain briefly.
(2 marks)
END OF PAPER
Math&Stat2_M6_04
P.4