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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


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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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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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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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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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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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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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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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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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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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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)

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