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7/28/2019 Paper 1 Que _eng
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HKDSE-MATH-CP 1-1 1 2012 Copyright by Vinci Mak
HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS Compulsory Part
PAPER 1
Question-Answer Book
(2 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 after
the Time is up announcement.
DSE-MOCK
MATH CP
PAPER 1Please stick the barcode label here.
Candidate Number
7/28/2019 Paper 1 Que _eng
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HKDSE-MATH-CP 1-2 2 2012 Copyright by Vinci Mak
SECTION A(1) (35 marks)
1. Simplify311
323 )(
ba
ba
and express your answer with positive indices.
(3 marks)
2. Consider the formula nnm 16)266(4 =+ .
(a) Make n the subject of the above formula.
(b) If the value ofm is decreased by 2, how will the value ofn be changed?
(3 marks)
3. Factorize
(a) 22 42336 yxyx + .
(b) yxyxyx 6342336 22 ++ .
(3 marks)
4. The marked price of a mobile phone is $888. It is given that the marked price ofthe mobile phone is 20% higher than the cost.
(a) Find the cost of the mobile phone.
(b) Find the maximum discount that can be given to the mobile phone such that
there will be no profit or loss.
(4 marks)
5. The ratio of the number of books owned by Vinci to the number of books owned
by Sophie is 3 : 4. If Sophie gives 5 of her own books to Vinci, both of them will
have the same number of books. Find the number of books owned by Vinci.
(4 marks)
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HKDSE-MATH-CP 1-3 3 2012 Copyright by Vinci Mak
6. In a polar coordinate system, the polar coordinates of the pointsA andB are
(4, 67 ) and (6, 247 ) respectively.
(a) Let O be the pole. AreA, O andB collinear? Explain your answer.
(b) The polar coordinates of the point Cis given by (r, ),
where rand are non-zero constants. It is known that COAO and
the area of AOC is 25 square units. Find the values ofrand .
(5 marks)
7. In Figure 7,ABCD is a semi-circle,BCF,ADF,AECandBED are straight lines.
If = 46BAC and = 32BDA , findx.
(4 marks)
8. The coordinates of the pointsA andB are (2, 1) and (7, 8) respectively.
A'is the reflection image ofA with respect to thex-axis.B is rotated
clockwise about the origin O through 270 toB'.
(a) Write down the coordinates ofA'andB'.
(b) Let P be a moving point in the rectangular coordinate plane such that
= 90''PBA . Find the equation of the locus ofP.
(5 marks)
9. For a set of six integers a, b, c, d, e and 16, where 166 < edcba ,
the mode is 8, the median is 9 and the mean is 10.
Find the two possible sets ofa, b, c, dand e.
(4 marks)
A
B
C
D
E
F
46o32o x
Figure 7
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7/28/2019 Paper 1 Que _eng
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HKDSE-MATH-CP 1-5 5 2012 Copyright by Vinci Mak
12. Figure 12 shows the graphs for Vinci and Sophie cycling on the same straight
road between townA and townB during the period 2:00 to 4:00 in an afternoon.
They rest for the same duration. It is given that town A and town B are 16km
apart.
(a) How long do they rest during the period?(1 mark)
(b) It is given that Vinci and Sophie meet at a place which is 8.5km from
townA after 3:00pm. When do they meet?
(3 marks)
(c) Vinci tells Sophie that they have the same average speed during the period.
Do you agree with him? Explain your answer.
(3 marks)
Distanc
efromt
ownA(
km)
0
2
4
6
10
16
2:00 2:21 3:00 4:00A
B
Time
Figure 12
Vinci
Sophie
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HKDSE-MATH-CP 1-6 6 2012 Copyright by Vinci Mak
13. Sector OCD is a thin metal sheet. The sheetABCD is formed by cutting away
sector OBA from sector OCD as shown in Figure 12(a).
It is known thatCOD =x,AD =BC= 20 cm, OA = OB = 40 cm
and)
CD = 80 cm.
(a) (i) Findx.
(ii) Find, in terms of, the area ofABCD.
(4 marks)
(b) The thin metal sheet ABCD described in (a) is divided into two parts,namely PQRS, with equal area. By joining AD and BC together, ABCD is
folded to form a hollow frustum X. Similarly, by joining PS and QR, PQRS
is folded to form another hollow frustum Y as shown in Figure 12(b). AreXand Ysimilar? Explain your answer.
(3 marks)
A
DC
B
O
80 cm
Figure 13(a)
x
40 cm
20 cm
A
DC
B
P
SR
Q
Figure 13(b)
X
Y
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HKDSE-MATH-CP 1-7 7 2012 Copyright by Vinci Mak
14. In Figure 14, Circle PQR is inscribed in OAB . It is given thatBQ = 3,
OP = 1 and AR = 2.
(a) Write down the perimeter of OAB .
(1 mark)
(b) A rectangular coordinate system with O as the origin is introduced in
Figure 14 so that the coordinates of A and B are (6, 0) and (0, 8)
respectively.
(i) Write down the coordinates of the orthocentre and circumcentre
of OAB .
(2 marks)
(ii) Find the coordinates of the incentre of OAB .
(2 marks)
(iii) Are the incentre, orthocentre and circumcentre of OAB collinear?
Explain your answer.
(2 marks)
O A
B
Q
P
R
Figure 14
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HKDSE-MATH-CP 1-8 8 2012 Copyright by Vinci Mak
SECTION B (35 marks)
15. (a) Simplify)1)(1(
3
1
1
1
1
++
+ xxxx.
(2 marks)
(b) nm xx )1()1( + , 33 )1()1( + xx and 42 )1()1( + xx , where m and n are
positive integers, are three polynomials of degree 6. It is given that the
H.C.F. and L.C.M. of the three polynomials are 3)1()1( + xx m andnxx )1()1( 3 + respectively. Write down all possible pairs ofm and n.
(2 marks)
16. There are 8 boys and 10 girls in a class. From the class, 7 students are randomly
selected to form the class committee.
(a) Find the probability that the class committee consists of 2 boys and 5 girls.
(2 marks)
(b) Find the probability that the class committee consists of at least 2 boys, and
there must be more girls than boys.(2 marks)
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HKDSE-MATH-CP 1-9 9 2012 Copyright by Vinci Mak
17. In Figure 17, the shaded region, including the boundary, is determined by three
inequalities.
(a) Write down the three inequalities.
(b) How many points (x,y), wherex andy are both integers, satisfy the three
inequalities in (a)?
(c) Find the maximum and minimum values of 123 + yx where (x,y) are
points satisfying (b).
(7 marks)
0
2
4
6
8
10
12
y
x2 4 6 8 10 12 14 16 18 20
Figure 17
3x + 5y = 60
2x y = 4
y = 1
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HKDSE-MATH-CP 1-10 10 2012 Copyright by Vinci Mak
18. In Figure 18,AB is a straight track 1000 m long on the horizontal ground.Eis a
small object moving alongAB. STis a vertical tower of height h m standing on
the horizontal ground. The angles of elevation of S from A and B are 30 and
20 respectively. = 40TAB .
(a) ExpressATandBTin terms ofh.Hence find h.
(5 marks)
(b) Let be the angle of elevation ofSfromE. Find the range of values of
asEmoves alongAB.
(3 marks)
30o
40o
20o
A B
S
T
h m
E
1000 m
Figure 18
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HKDSE-MATH-CP 1-11 11 2012 Copyright by Vinci Mak
19. Let f(x) x3log= .
(a) If the graph ofy = g(x) is obtained by translating the graph ofy = f(x) leftwards
by 4 units and upwards by 5 units, find g(x).
(b) A researcher performs an experiment to study the relationship between the
number of bacteriaA (u hundred million) and the temperature (s oC) under some
controlled conditions. From the data of u and s recorded in Table 19(a), the
researcher suggests using the formula u = f(s 4) 5 to describe the relationship.
s a1 a2 a3 a4 a5 a6 a7
u b1 b2 b3 b4 b5 b6 b7
Table 19(a)
(i) According to the formula suggested by the researcher, find the temperature
at which the number of the bacteria is zero.
(ii) The researcher then performs another experiment to study the relationship
between the number of bacteriaB (v hundred million) and the temperature
(t oC) under the same controlled conditions and the data of v and t are
recorded in Table 19(b).
t a1 4 a2 4 a3 4 a4 4 a5 4 a6 4 a7 4
v b1 + 5 b2 + 5 b3 + 5 b4 + 5 b5 + 5 b6 + 5 b7 + 5
Table 19(b)
Using the formula suggested by the researcher, propose a formula to express
v in terms oft.
(6 marks)
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HKDSE-MATH-CP 1-12 12 2012 Copyright by Vinci Mak
20. Vinci joins a saving plan by depositing in his bank account a sum of money at
the beginning of every year. At the beginning of the first year, he puts an initial
deposit of $P. Every year afterwards, he deposits 10% more than he does in the
previous year. The bank pays interest at a rate of 8% p.a., compounded yearly.
(a) Find, in terms ofP, an expression for the amount in his account at the end
of
(1) the first year,
(2) the second year,
(3) the third year.
(Note: You need not simplify your expressions)
(b) Using (a), or otherwise, show that the amount in his account at the end of
the nth year is )08.11.1(54$ nnP .
(7 marks)