S6 Mock9 Paper1 S E

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HKDSE
MATH CP
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PAPER 1
HONG KONG EDUCATIONAL PUBLISHING COMPANY
HONG KONG DIPLOMA OF
SECONDARY EDUCATION EXAMINATION
Candidate Number
MATHEMATICS Compulsory Part
S6 MOCK EXAM 9 (2019)
PAPER 1
Question-Answer Book
Time allowed: 2¼ hours
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 1 and stick barcode labels in the spaces
provided on Pages 1, 3, 5, 7, 9 and 11.
2.
This paper consists of THREE sections, A(1), A(2) and B.
3.
Attempt ALL questions in this paper. Write your answers
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.
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 this book.
5.
Unless otherwise specified, all working must be clearly
shown.
6.
Unless otherwise specified, numerical answers should be
either exact or correct to 3 significant figures.
7.
The diagrams in this paper are not necessarily drawn to
scale.
8.
No extra time will be given to candidates for sticking on
the barcode labels or filling in the question number boxes
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after the ‘Time is up’ announcement.
CP S6 MOCK 9 PAPER 1-1
© 香港教育圖書公司
All Rights Reserved 2019
1
Section A(1) (35 marks)
1.
Simplify
(a 3b 2 )3
and express your answer with positive indices.
ab 4
(3 marks)
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2.
Factorize
(a) 16 x 2  8 xy  y 2 ,
(b) 16 x 2  8xy  y 2  9 .
(3 marks)
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CP S6 MOCK 9 PAPER 1-2
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3.
Make s the subject of the formula r  s 
1  7s
.
4
(3 marks)
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4.
Consider the compound inequality
2( x  5)  5 x  22 or x  2 (*)
(a) Solve (*).
(b) Write down the smallest positive integer satisfying (*).
(4 marks)
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CP S6 MOCK 9 PAPER 1-3
3
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5.
Sector OPQ is a thin metal sheet. The sheet PQRS is formed by cutting away sector OSR from
sector OPQ as shown in Figure 1.
Q
P
36 cm
S
a
O
R
12 cm
Figure 1
It is known that POQ = a, PS = QR = 36 cm, OS = OR = 12 cm and the area of
PQRS = 630 cm2.
(a) Find a.
(b) Find, in terms of , the perimeter of PQRS.
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CP S6 MOCK 9 PAPER 1-4
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(4 marks)
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6.
The marked price of a pair of shoes is $780. It is given that the marked price of the pair of
shoes is 30% higher than the cost.
(a) Find the cost of the pair of shoes.
(b) If the pair of shoes is sold at $570, find the percentage loss.
(4 marks)
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CP S6 MOCK 9 PAPER 1-5
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7.
The bar chart below shows the distribution of the ages of the students in a painting class for
children.
Number of students
Distribution of the ages of the students in the painting class
n
6
5
4
2
6
7
9
8
Age
10
If a student is randomly selected from the painting class, then the probability that the age of the
1
selected student is 6 is .
5
(b) Find the standard deviation of the above distribution.
(4 marks)
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CP S6 MOCK 9 PAPER 1-6
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(a) Find n.
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8.
In Figure 2, ABC = ADC, AB // DE, BAC = 58 and CDE = 116.
B
A
D
E
C
Figure 2
(a) Find ACD.
(5 marks)
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CP S6 MOCK 9 PAPER 1-7
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(b) Prove that ABC  CDA.
9.
A cake is termed standard if its weight is measured as 700 g correct to the nearest 10 g.
(a) Find the least possible weight of a standard cake.
(b) Someone claims that the total weight of 40 standard cakes can be measured as 27.7 kg
correct to the nearest 0.1 kg. Do you agree? Explain your answer.
(5 marks)
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CP S6 MOCK 9 PAPER 1-8
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Section A(2) (35 marks)
10. Figure 3 shows the graph for two cars, X and Y travelling on the straight road between town A
and town B during the period 9:00 to 10:30 in a morning. Y travels at a constant speed after it
leaves town B. It is given that town A and town B are 80 km apart.
B 80
X
36
Y
6
A
0
9:00
10:00
Time
10:30
Figure 3
(a) What is the speed of X during the period 10:00 to 10:30 in the morning?
(2 marks)
(b) When does Y leave town B?
(2 marks)
(c) Which car has the higher average speed during the period 9:00 to 10:30 in the morning?
Explain your answer.
(2 marks)
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CP S6 MOCK 9 PAPER 1-9
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Distance from town A (km)
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70
11. There are 23 members in a swimming club. The box-and-whisker diagram below shows the
distribution of the heights of the members in the swimming club. It is given that the mean of
this distribution is 173 cm.
Height (cm)
160
168
179
171
182
(a) Find the range and the inter-quartile range of the above distribution.
(3 marks)
(b) Three people of respective heights 164 cm, 172 cm and 175 cm now join the swimming
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CP S6 MOCK 9 PAPER 1-10
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club while a member of height 180 cm leaves the club. Find the mean and the median of
the heights of the members in the swimming club after the changes.
(3 marks)
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12. A solid metal right prism of base area 798 cm2 and height 20 cm is melted and recast into two
similar solid right pyramids. The bases of two pyramids are squares. The ratio of the base area
of the smaller pyramid to the base area of the larger pyramid is 4 : 25.
(a) Find the volume of the smaller pyramid.
(3 marks)
(b) If the length of the side of the base of the smaller pyramid is 24 cm, find the total surface
area of the larger pyramid.
(4 marks)
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CP S6 MOCK 9 PAPER 1-11
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13. It is given that y is the sum of two parts, one part varies as x and the other part varies as the
cube of x. When x = 2, y = 4 and when x = 1, y = 5.
(a) (i) Express y in terms of x.
(ii) Find y when x = 2.
(4 marks)
(b) Let f (x) = 2x  12x  9. Someone claims that the equation f (x) = 1 has 3 distinct
rational roots. Do you agree? Explain your answer.
(3 marks)
3
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CP S6 MOCK 9 PAPER 1-12
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CP S6 MOCK 9 PAPER 1-13
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14. It is given that A(2, 4) and B(5, 5) are two points on the rectangular coordinate plane. Denote
the straight line which passes through A and B by L.
(a) (i) Find the equation of L.
(ii) If L cut the y-axis at C, find the coordinates of C.
(3 marks)
(b) Let P be a moving point in the rectangular coordinate plane such that P is equidistant from
A and B. Denote the locus of P by .
(i) Find the equation of .
(ii) Let  cut the x-axis at D, find the coordinates of D.
(3 marks)
(c) Let O be the origin. Denote the point of intersection of L and  by E. Someone claims that
the area of OCED is less than 20. Do you agree? Explain your answer.
(3 marks)
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CP S6 MOCK 9 PAPER 1-14
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CP S6 MOCK 9 PAPER 1-15
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Section B (35 marks)
15. Let a and b be constants. Denote the graph of ax  log b y  1 by G. G passes through the
1

1 
points   1,  and  ,4  . Express y in terms of x.
2

2 
(4 marks)
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CP S6 MOCK 9 PAPER 1-16
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16. It is given that the coordinates of points A, B and C are (5, 0), (3, 0) and (4, 7) respectively.
Denote the circumcentre of ABC by G.
(a) Find the coordinates of G.
(3 marks)
(b) Someone claims that BGC is greater than 160. Do you agree? Explain your answer.
(3 marks)
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CP S6 MOCK 9 PAPER 1-17
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17. A box contains 5 red balls, 7 blue balls and 8 green balls.
(a) (i) If 3 balls are randomly drawn from the box one by one without replacement, find the
probability that exactly 2 blue balls are drawn.
(ii) If 3 balls are randomly drawn from the box one by one with replacement, find the
probability that exactly 2 blue balls are drawn.
(4 marks)
(b) 2 silver balls are added into the box. Ada repeats drawing one ball at a time randomly
from the box with replacement until a silver ball is drawn. Find the probability that she
gets a silver ball with at most 3 draws.
(3 marks)
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CP S6 MOCK 9 PAPER 1-18
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CP S6 MOCK 9 PAPER 1-19
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18. Figure 4(a) shows a solid pyramid ABCD with a triangular base, where AB = CB = 20 cm,
AC = 12 cm, AD = CD = 16 cm and ADB = 102.
D
C
C
N
D
M
B
A
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Figure 4(a)
(a) Find ABD.
Figure 4(b)
(2 marks)
(b) M and N are the mid-points of AD and CD respectively. A geometric model is made by
cutting off MBND from ABCD as shown in Figure 4(b). A craftsman claims that the area
of MBN is greater than 40 cm2. Do you agree? Explain your answer.
(5 marks)
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A
B
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CP S6 MOCK 9 PAPER 1-21
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19. John sets up an investment account in bank A. It is given that the total amount in the account at
the end of the first year is $20 000, and in subsequent years, the increased amount each year is
x % of the total amount at the end of the previous year, where x is a constant, and John deposits
$1200 in the account at the beginning of each year. It is found that the total amount in the
account at the end of the third year is $24 944.
(a) (i) Express, in terms of x, the total amount in the account at the end of the second year.
(ii) Find x.
(3 marks)
(b) (i) Express, in terms of n, the total amount in the account at the end of the nth year.
(ii) At the end of which year will the total amount in the account first exceed $60 000?
(5 marks)
n
The total amount in the account in Bank B at the end of the nth year ($)
1
22 500
2
24 663
John claims that the total amount in the account in bank A will be greater than that in
bank B at the end of a certain year. Do you agree? Explain your answer.
(3 marks)
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(c) John has another investment account in bank B. It is assumed that the total amount in the
account in bank B at the end of the nth year is $[p(1.1236)n + q], where p and q are
constants. The following information is given:
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END OF PAPER
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