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S5AE Summer Assignment
HKCEE past paper – Mathematics Paper I(Section A) [2000/2002/2004]
Instructions:
All S5 students have to put your answer on a single line book or a few pieces of single
line paper with clear steps. Hand in your assignment on or before the first two days of
the next school year. Some of the chosen questions will be marked by your subject
teacher and the marks will be counted as the usual marks for the1st term in the next
school year.
2000 HKCEE MATHEMATICS PAPER 1A
SECTION A(1) (33 marks)
5
9
1.
Let C  ( F  32) . If C = 30, find F. (3 marks)
2.
Simplify
3.
Find the area of the sector in Figure 1.
(3 marks)
4.
In Figure 2, find a and x.
(4 marks)
5.
Solve
x 3 y
and express your answer with positive indices. (3 marks)
x2
11  2 x
 1 and represent the solution in Figure 3.
5
(4 marks)
6.
Let f ( x)  2 x 3  6 x 2  2 x  7 . Find the remainder when f(x) is divided by x  3 .
(3 marks)
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7.
In Figure 4, AD and BC are two parallel chords of the circle. AC and BD intersect at
E. Find x and y.
8.
9.
(4 marks)
On a map of scale 1:5 000, the area of the passenger terminal of the Hong Kong
International Airport is 220 cm2. What is the actual area, in m2, occupied by the
terminal on the ground? (4 marks)
Let L be the straight line passing through (-4, 4) and (6, 0).
(a) Find the slope of L.
(5 marks)
(b) Find the equation of L.
(c) If L intersects the y-axis at C, find the coordinates of C.
SECTION A(2) (33 marks)
10. (a) Solve 10 x 2  9 x  22  0 .
(2 marks)
th
(b) Mr. Tung deposited $10 000 in a bank on his 25 birthday and $9 000 on his
26th birthday. The interest was compounded yearly at r% p.a., and the total
amount he received on his 27th birthday was $22 000. Find r. (4 marks)
11. Figure 5 shows the cumulative frequency polygon of the distribution of the lengths
of 75 songs.
(a) Complete the tables below.
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(2 marks)
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Length
Cumulative
Length
(t seconds)
frequency
(t seconds)
t  220
3
200  t  220
3
t  240
16
220  t  240
13
t  260
46
240  t  260
30
t  280
t  300
Frequency
260  t  280
75
280  t  300
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(b) Find an estimate of the mean of the distribution. (2 marks)
(c) Estimate from the cumulative frequency polygon the median of the distribution.
(1 mark)
(d) What percentage of these songs have lengths greater than 220 seconds but not
greater than 260 seconds? (2 marks)
12. A box contains nine hundred cards, each marked with a different 3-digit number
from 100 to 999. A card is drawn randomly from the box.
(a) Find the probability that two of the digits of the number drawn are zero.
(2 marks)
(b) Find the probability that none of the digits of the number drawn is zero.
(2 marks)
(c) Find the probability that exactly one of the digits of the number drawn is zero.
(2 marks)
13. In Figure 6, ABCDE is a regular pentagon and CDFG is a square. BG produced
meets AE at P.
(a) Find BCG , ABP and APB .(5 marks)
(b) Using the fact that
AP
AB

, or otherwise, determine which line
sin ABP sin APB
segment, AP or PE, is longer. (3 marks)
14. An auditorium has 50 rows of seats. All seats are numbered in numerical order from
the first row to the last row, and from left to right, as shown in Figure 7. The first
row has 20 seats. The second row has 22 seats. Each succeeding row has 2 more
seats than the previous one.
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(a) How many seats are there in the last row?
(2 marks)
(b) Find the total number of seats in the first n rows.
Hence determine in which row the seat numbered 2000 is located. (4 marks)
2002 HKCEE MATHEMATICS PAPER 1A
SECTION A(1) (33 marks)
(ab2 ) 2
and express your answer with positive indices. (3 marks)
a5
1.
Simplify
2.
In Figure 1, the radius of the sector is 6 cm. Find the area of the sector in terms of
(3 marks)
.
3.
In Figure 2, OP and OQ are two perpendicular straight roads where OP = 100m and
OQ = 80m.
(a) Find the value of  .
(b) Find the bearing of P from Q.
(3 marks)
4.
Let f ( x)  x 3  2x 2  9x  18 .
(a) Find f(2)
(b) Factorize f(x) (3 marks)
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5.
For the set of data 4, 4, 5, 6, 8, 12, 13, 13, 13, 18, find
(a)
(b)
(c)
(d)
the mean,
the mode,
the median,
the standard deviation.(4 marks)
6.
The radius of a circle is 8cm. A new circle is formed by increasing the radius by
10%.
(a) Find the area of the new circle in terms of  .
(b) Find the percentage increase in the area of the circle. (4 marks)
7.
(a) Solve the inequality 3x  6  4  x .
(b) Find all integers satisfy both the inequalities 3x  6  4  x and 2 x  5  0 .
(4 marks)
In Figure 3, the straight line L : x  2 y  8  0 cuts the coordinates axe at A and B.
8.
(a) Find the coordinates of A and B.
(b) Find the coordinates of the mid-point of AB. (4 marks)
9.
In Figure 4, BD is a diameter of the circle ABCD. AB = AC and BDC  40 o . Find
(5 marks)
ABD .
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SECTION A(2) (33 marks)
10. In Figure 5, ABC is a triangle in which BAC  20 o and AB = AC. D, E are points
on AB and F is a point on AC such that BC = CE = EF = FD.
(a) Find CEF .(4 marks)
(b) Prove that AD = DF. (3 marks)
11. The area of a paper bookmark is A cm2 and its perimeter is P cm. A is a function of
P. It is known that A is the sum of two parts, one part varies as P and the other part
varies as the squares of P. When P = 24, A = 36 and when P = 18, A = 9.
(a) Express A in terms of P. (3 marks)
(b) (i) The best-selling paper bookmark has an area of 54 cm2. Find the
perimeter of this bookmark.
(ii) The manufacturer of the bookmarks wants to produce a gold miniature
similar in shape to the best-selling paper bookmark. If the gold miniature
has an area of 8 cm2, find its perimeter. (5 marks)
12. Two hundred students participated in a summer reading programme. Figure 6 shows
the cumulative frequency polygon of the distribution of the numbers of books read
by the participants.
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(a) The table below shows the frequency distribution of the numbers of books read
by the participants. Using the graph in Figure 6, complete the table.
Number of books read (x)
Number of participants
Award
0 x5
66
Certificate
5  x  15
15  x  25
Book coupon
64
25  x  35
35  x  50
Bronze medal
Silver medal
10
Gold medal
(b) Using the graph in Figure 6, find the inter-quartile range of the distribution.
(2 marks)
(c) Two participants were chosen randomly from those awarded with medals. Find
the probability that
(i)
they both won gold medals;
(ii)
they won different medals. (6 marks)
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13. A line segment AB of length 3 m is cut into three equal parts AC1, C1C2 and C2B as
shown in Figure 7(a).
On the middle part C1C2, an equilateral triangle C1C2C3 is drawn as shown in Figure
7(b).
(a) Find, in surd form, the area of triangle C1C2C3. (2 marks)
(b) Each of the line segments AC1, C1C3, C3C2 and C2B in Figure 7(b) is further
divided into three equal parts. Similar to the previous process, four smaller
equilateral triangles are drawn as shown in Figure 7(c). Find, in surd form, the
total area of all the equilateral triangles. (3 marks)
(c) Figure 7(d) shows all the equilateral triangles so generated when the previous
process is repeated again. What would the total area of all the equilateral
triangles become if this process is repeated indefinitely? Give your answer in
surd form. (4 marks)
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2004 HKCEE MATHEMATICS PAPER 1A
Section A(1)
(a 1b) 3
and express your answer with positive indices.
b2
1.
Simplify
2.
Make x the subject of the formula y 
3.
A sum of $5 000 is deposited at 2% p.a. for 3 years, compounded yearly. Find the
interest correct to the nearest dollar.
(3 marks)
In Figure 1, the graph of y   x 2  10x  25 touches the x-axis at A(a,0) and cuts
the y-axis at B(0, b) . Find a and b.
(3 marks)
4.
2
.
ax
5.
In Figure 2, find the bearing of B from A.
6.
Factorize
(a) a 2  ab  2a  2b
(b) 169 y 2  25
7.
(3 marks)
(3 marks)
(3 marks)
(4 marks)
The prices of an orange and an apple are $2 and $3 respectively. A sum of $46 is
spent buying some oranges and apples. If the total number of oranges and apples
bought is 20, find the number of oranges bought.
(4 marks)
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8.
A box contains nine cards numbered 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively.
9.
(a) If one card is randomly from the box, find the probability that the number
drawn is odd.
(b) If two cards are randomly drawn from the box and one by one with
replacement, find the probability that the product of the numbers drawn is
even.
In Figure 3, the area of the sector is 162 cm 2 .
(a) Find the radius of the sector.
(b) Find the perimeter of the sector in terms of  .
(5 marks)
Section A(2)
10. It is known that y is the sum of two parts, one part varies as x and the other part
varies as the square of x. When x = 3, y = 3 and when x = 4, y = 12.
(a) Express y in terms of x.
(4 marks)
(b) If x is an integer and y  42 , find all possible value(s) of x.
(4 marks)
11. A large group of students sat in a Mathematics test consisting of two papers, Paper I
and Paper II. The table below shows the mean, median, standard deviation and
range of the test marks of these students in each paper:
Test Paper
Mean
Median
Standard deviation
Range
Paper I
46.1 marks
46 marks
15.2 marks
91 marks
Paper II
60.3 marks
60 marks
11.6 marks
70 marks
A student, John scored 54 marks in Paper I and 66 marks in Paper II.
(a) Assume that the marks in each paper of the Mathematics test are normally
distributed. Relative to other students, did John perform better in Paper II than
in Paper I? Explain your answer.
(4 marks)
(b) In a mark adjustment, the Mathematics teacher added 4 marks to the test mark
of Paper I for each of these students. Write down the mean, the median and the
range of the test marks of Paper I after the mark adjustment. (3 marks)
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12. In Figure 4, AEC, AFB, BCD and DEF are straight lines. AB = AC, CD = CE and
CDE  36 o .
(a) Find
(i) AEF
(ii) BAC
(3 marks)
(b) Suppose AF = FB
(i)
Prove that AEB is a right angle.
(ii)
If AE = 10cm, find the area of ABC .
(6 marks)
13. In Figure 5, ABCD is a rhombus. The diagonals AC and BD cut at E.
(a) Find
(i)
the coordinates of E
(ii)
the equation of BD
(b) It is given that the equation of AD is x  7 y  65  0 . Find
(i)
(ii)
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the equation of BC,
the length of AB.
(4 marks)
(5 marks)