Analysis on the HKCEE/HKDSE Questions HKDSE Examination Preparation Guide Points to Note in the Examination

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Content

Analysis on the HKCEE/HKDSE Questions HKDSE Examination Preparation Guide Points to Note in the Examination

Part 1 Key Stage 4 (S4 – S6)

1. Quadratic Equations 2

2. Functions and Graphs 20

3. Exponential and Logarithmic Functions 42

4. Equations of Straight Lines 60

5. More about Polynomials 80

6. Variations 98

7. More about Equations 118

8. More about Graphs of Functions 138

9. Basic Properties of Circles 164

Part 2 Key Stage 3 (S1 – S3)

10. Percentages 190

11. Polynomials, Indices and Surds 212

12. Rate and Ratio 233

13. Formulas and Equations 248

14. Basic Geometry 266

15. Coordinate Geometry 292

Mock Exam 313

Graph Papers 337

Sample

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Analysis on the HKCEE/HKDSE Questions

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

1. Quadratic

Equations ^Q11(b) ^Q5 ^Q10(b) ^Q12(c)

2. Functions and

Graphs ^Q4 ^Q10 Q11(a)(b) Q12, Q15 Q11(b) Q17

3. Exponential and

Logarithmic Functions

Q4 Q16(b)

4. Equations of Straight Lines

Q8,

Q17(a) Q12 Q13 Q13 Q12 Q13 Q12(c) Q12

5. More about

Polynomials ^Q4 ^Q10(b) ^Q14(a) ^Q13 ^Q12

6. Variations Q11(a) Q10(a) Q10(a) Q10(a) Q15(a) Q14(b) Q10(a) Q11(a) Q11 Q11

7. More about

Equations ^Q15(b)

8. More about Graphs of Functions

Q11(c) Q16

9. Basic Properties of Circles

Q9,

Q16(a) Q17 Q16(a)(b) Q17(a) Q16(a) Q17(a) Q8

10. Percentages ^Q6 ^Q5 ^Q3 ^Q6,

Q16(a) Q6 Q6 Q8,

Q16(b) Q7 Q7 Q7 Q4

11. Polynomials, Indices and Surds

Q1 Q3 Q1,Q6 Q2, Q3 Q1,Q3 Q2,Q3 Q1, Q3 Q2,Q3 Q1,Q3 Q2,Q3 Q1 Q1, Q3

12. Rate and Ratio ^Q5

13. Formulas and Equations

Q1, Q6,

Q16 Q2, Q7 Q1 Q1,Q7 Q6 Q1,Q6 Q5,Q6 Q1,Q6 Q2, Q3,

Q5 Q2, Q4

14. Basic

Geometry ^Q10 ^Q8 ^Q12 ^Q8 ^Q5 ^Q8 ^Q9 ^Q11 ^Q9

Q9

Q12(a)(b) Q7

15. Coordinate

Geometry ^Q16(b) ^Q7 Q17(b)(i) Q12(a)(b) Q8,Q9 Q17(a) Q8 Q6

Note: The numbers listed above refer to the question numbers in the HKCEE/HKDSE mathematics Paper 1 that year.

Q.1 represents questions in section A(1), Q.10 represents questions in section A(2), Q.15 represents questions in section B.

Chapter

Year

Sample Sample

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In order to prepare for the examination effectively. Candidates are advised to read the instruction of the assessment carefully.

A. Public Assessment Format

The HKDSE Mathematics curriculum consists of a Compulsory Part and an Extended Part (Module 1: Calculus and Statistics, Module 2: Algebra and Calculus). The following table lists out the details of the examination in the compulsory part:

Component Weighting Duration

Paper 1 Conventional questions 65% 135 minutes

Paper 2 Multiple-choice questions 35% 75 minutes

Paper 1 consists of two sections A and B in which all questions are to be attempted. Section A consists of questions on the foundation topics of the compulsory part and the foundation part of S1 – S3 curriculum. Section A is divided into two parts: Section A(1) (35 marks) consists of 8 to 11 elementary questions and Section A(2) (35 marks) will consist of 4 to 7 harder questions. On the other hand, Section B (35 marks) consists of 4 – 7 questions on the compulsory part and the whole curriculum of S1 – S3.

Paper 2 consists of two sections A and B in which all questions are to be attempted. Section A (about 66% of the paper mark) consists of questions on the foundation topics of the compulsory part and the foundation part of the S1 – S3 curriculum, whereas Section B (about 33% of the paper mark) consists of questions of the compulsory part and the whole S1 – S3 curriculum.

B. Standard Referencing

In the HKDSE, standards-referenced reporting will be adopted to report candidates’ results. Candidates’ levels of performance will be reported with reference to a set of standards as defined by cut scores on the variable or scale for a given subject (see the figure below).

There will be five cut scores in each subject to distinguish five levels of performance (1 – 5), with 5 being the highest.

A performance below the threshold cut score for Level 1 will be labelled as “Unclassified” (U). Level 5 candidates with the best performance will have their results annotated with the symbols ‘**’ and the next top group with the symbol ‘*’.

Cut scores

Variable/scale 1

U 2 3 4 5

HKDSE Examination Preparation Guide Sample

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A. Instructions Before the Start of the Examination

1. After you have taken your seat, put your essentials on your desk: a pencil, a ball-pen, a correction fluid, a ruler, a pair of compasses, a protractor, a calculator, your ID card and the examination permission. Put other belongings under your seat.

2. Listen carefully to the instructors. When the examination paper is delivered, DO NOT turn over or open the paper until you are told to do so.

3. When you are permitted to check the paper, read the instructions printed on the front page carefully. Then check if there are any missing pages or missing questions.

4. Make sure that your candidate number and seat number are written correctly on the answer book.

5. Put a watch on your desk to remind of the time used.

B. Guidelines in Answering Questions

1. When the examination starts, do not be eager to answer the questions. Scan the questions quickly first and start from the questions that you are most confident with.

2. Read the question carefully before answering it. Look for keywords such as “calculate”, “prove” and “explain”

etc. to find out what you are required to answer.

3. Use ball pens to answer questions. Show your working neatly.

4. When you are stuck in a question, just skip it and answer the next one. You may have time to come back to finish that question later, so do not panic.

5. Manage your time systematically on each question. Here are some suggestions:

Section Number of questions Time spent per question Total time spent

Section A(1) 8 to 11 questions ~ 4 minutes 35 minutes

Section A(2) 4 to 7 questions ~ 7 minutes 40 minutes

Section B 4 to 7 questions ~ 10 minutes 50 minutes

Time left for checking: 10 minutes 6. When applying a formula or geometrical properties, you are advised to show the formula or reasons instead of

substituting the values directly. If you do so, in case you give a wrong answer, you may still get some marks for correct methods used.

7. When drawing figures, always use pencils with appropriate tools such as ruler, pair of compasses and protractor.

8. In application problems, remember to check the units in each step and the answer.

9. Unless otherwise specified, you are advised to give the answers corrected to 3 significant figures. However, the values in the intermediate steps should be corrected to more than 3 significant figures, otherwise the answers obtained would not be accurate.

Points to Note in the Examination Sample

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x y

O

y = mx + c

x y

O y = c

x y

O

axis of symmetry

vertex

x y

O x

y

O

x y

O

x y

domain 1 • 2 • 3 •

• 5

• 10

• 15

co-domain

20 21

2 Functions and Graphs

Functions and Graphs

Graphs of Linear Functions

• The graph is a straight line.

• When m  0, the graph goes up from left to right. When m  0, the graph goes down from left to right.

• When m = 0, the function becomes a constant function and the graph is a horizontal line.

• x-intercept = mc

- , where m ≠ 0

• y-intercept = c

Definition of Functions

• If y is a function of x, then for every value of x, there is one and only one corresponding value of y.

• Notation: f(x), g(x), G(x), etc.

• Domain: Values of x such that f(x) is defined.

• Co-domain: All the possible values of f(x).

Linear Functions

In the form f(x) = mx + c, where m and c are constants.

Number of x-intercepts

• The number of x-intercepts is determined by

∆ = b² – 4ac.

(a) When ∆  0, there are two x-intercepts.

(b) When ∆ = 0, there is one x-intercept.

(c) When ∆  0, there is no x-intercept.

Graphs of y = ax

²

+ bx + c

• When a  0, the graph opens upwards.

When a  0, the graph opens downwards.

• The graph is symmetric about the axis of symmetry.

• The axis of symmetry must pass through the vertex.

• y-intercept = c

Graphs of y = a(x – h)

²

+ k

• Vertex = (h, k)

• If a  0, then (h, k) is the lowest point.

If a  0, then (h, k) is the highest point.

• The axis of symmetry is x = h.

Finding Optimum Values of y = ax

²

+ bx + c

Step 1: Rewrite the quadratic function y = ax² + bx + c into the form y = a(x – h)² + k by the method of completing the square, where h = a

b

-2 and k = a ac b

4 4 - ².

Step 2: If a  0, the function attains its minimum value k when x = h.

If a  0, the function attains its maximum value k when x = h.

Quadratic Functions

• In the form f(x) = ax² + bx + c, where a, b and c are constants with a ≠ 0.

• It can also be expressed in the form of f(x) = a(x – h)² + k, where a, h and k are constants with a ≠ 0.

Concept Map

x = h (h, k)

vertex

a  0

a  0 a  0

axis of symmetry

NF

(h, k) x = h

a  0

Sample

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4

HKDSE Exam Series — Mathematics Conventional Questions (Compulsory Part) (Upgraded Edition) Book A

As x – 5 = ±4, x = 1 is also a root.

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) × (c + di) = (ac – bd) + (ad + bc)i

(a + b) × (a – b) = a² – b²

Smart Review

A. Complex Numbers

1. Complex numbers（複數）are in the form of a + bi, where a and b are real numbers and i = - . a is called the real part（實部） 1 and b is called the imaginary part（虛部）. If a and b are real numbers, then a + bi and a – bi are a pair of conjugate complex numbers（共軛複數）.

For example, 4, –2i, 3 + 10i and 7 – 12i are complex numbers.

NF 2. If a + bi = c + di, then a = c and b = d.

NF 3. Operations of complex numbers (a) Addition and subtraction

For example, (i) (2 + 3i) + (7 + 3i) = (2 + 7) + (3 + 3)i

= 9 + 6i

(ii) (2 + 3i) – (7 + 3i) = (2 – 7) + (3 – 3)i

= –5 (b) Multiplication

For example, (5 + 3i) × (2 + 9i) = (10 – 27) + (45 + 6)i

= –17 + 51i (c) Division

For example, i

i 7

4 5

+ + =

i i

i i 7

4 5

7 :7 + +

- -

= i

7 1

28 5

7 1

35 4

2 2 2 2

+

+ + -

+

= 31i

50 33

+50

B. Quadratic Equations

1. An equation in the form ax² + bx + c = 0, where a ≠ 0, is called a quadratic equation in one unknown（一元二次方程）.

2. Solving quadratic equations by taking square root For example, solve (x – 5)² = 16.

x – 5 = 4 or x – 5 = –4 x = 9 or 1

c di a bi

+ + =

c d

ac bd

2 2

+ + +

c d

bc ad

2 2 i + -

Sample

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Concept Builder

1

Quadratic Equations

7 Determine whether each of the following is true or false.

1. -25 = 5i

2. 80 is not a complex number.

3. The square of a complex number must be a real number.

4. If bx² + ax + c = 0, then x =

a

b b ac

2

2 4

- ! - .

5. The equation x² – x – 4 = 0 has two distinct real roots.

6. The equation 2x² – x + 3 = 0 has no real roots.

7. If α and β are the roots of the equation 2x² + 5x + 1 = 0, then α + β =

25 and αβ = 2 - . 1

8. If 2 and 3 are the roots of a quadratic equation, the equation can be x² – 5x + 6 = 0.

NF

NF NF

Worked Examples

Section A(1)

1. (a) Solve the equation x² – 3x – 10 = 0.

(b) Hence solve the equation (x – 1)² – 3(x – 1) – 10 = 0.

(3 marks)

Solution Try Q.1 – 2.

(a) x² – 3x – 10 = 0

(x + 2)(x – 5) = 0 [1M]

x = –2 or 5 [1A]

(b) Let y = x – 1. The equation becomes y² – 3y – 10 = 0

y = –2 or 5 (By (a)) ` x – 1 = –2 or x – 1 = 5

x = –1 or 6 [1A]

Concept Builder Ans: 1. T 2. F 3. F 4. F 5. T 6. T 7. F 8. T

Sample

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1

Quadratic Equations

11

Point to Remember Point to Remember

Try to express the relationship between a and b by considering the real part and the imaginary part.

Point to Remember

To deal with fraction with complex number as denominator, we should multiply it with the conjugate of that complex number to change it to a real number.

Point to Remember Point to Remember

Write the expression in terms of the sum of roots and the product of roots.

Section B

9. Calculate i

i

4 3

2 11 +

- + (7 – 12i). (3 marks)

i i

4 3

2 11 +

- + (7 – 12i) = i

i

i i

4 3

2 11

4 3

+ # -

-

- + (7 – 12i) [1M]

= 8 33 6 44 i

4²+3²

- + - -

] g ] g + (7 – 12i)

= 25 i

25 50

- - + (7 – 12i) [1A]

= (–1 – 2i) + (7 – 12i)

= 6 – 14i [1A]

10. If (a + bi) × (1 – 2i) = 13 – i, where a and b are integers, find the values

of a and b. (5 marks)

L.H.S. = (a + bi) × (1 – 2i)

= (a + 2b) + (b – 2a)i [1M]

By comparing the real part and the imaginary part, ( 20a + 2b = 13 ... (1)

b – 2a = –1 ... (2) [1M]

2 × (1) : 2a + 4b = 26 ... (3)

(3) + (2) : 5b = 25 [1M]

b = 5 [1A]

Substituting b = 5 into (1), we have a + 2(5) = 13

a = 3 [1A]

11. If α and β are the roots of the equation x² – 4x + 2 = 0, find the values of

(a) α² – αβ + β², (3 marks)

(b) α³ + β³. (2 marks)

(a) α + β = 1

4 - -] g = 4,

αβ = 12 = 2 [1A]

α² – αβ + β² = (α + β)² – 3αβ α² + β² = (α + β)² – 2αβ [1M]

= 4² – 3(2)

= 10 [1A]

(b) α³ + β³ = (α + β)(α² – αβ + β²) [1M]

= (4)(10) (By (a))

= 40 [1A]

NF

Sample

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Warm Up Exercise

14

HKDSE Exam Series — Mathematics Conventional Questions (Compulsory Part) (Upgraded Edition) Book A

A B

Answer each of the following questions, each question carries 10 marks.

1. If a + 5i = 7 + (b – 3)i, find the values of a and b.

2. Find (2 + 5i) × (6i – 7).

3. Solve the equation 100 = (2x + 1)².

4. Solve 2x² + 5x + 2 = 0 by using the factor method.

5. Solve x² + 3x – 5 = 0 by using the quadratic formula.

6. By considering the discriminant of each of the following quadratic equations, determine the nature of its roots.

(a) x² + 4x + 4 = 0 (b) 2x² + 3x – 7 = 0 (c) 2x² – 5x + 12 = 0

7. If the equation kx² + 6x + 3 has two distinct real roots, find the range of values of k.

8. If α and β are the roots of the equation 2x² – 3x + 1 = 0, find the value of 1a + 1 b. 9. Form a quadratic equation whose roots are –2 and 4.

10. In the figure, the side of square B is 2 cm longer than that of square A. If the sum of the areas of the two squares is 130 cm², find the side length of square A.

/ 100

NF

NF NF

Sample

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1

Quadratic Equations

15

Mock Questions

Section A(1)

1. (a) Solve x² + 5x + 6 = 0.

(b) Hence solve (x + 2)² + 5(x + 2) + 6 = 0.

(3 marks)

2. (a) Solve 4x² – 81 = 0.

(b) Hence solve 4(x + 1)² – 81 = 0.

(3 marks)

3. Solve the equation (x + 3)(x – 5) = 9. (3 marks)

4. Solve the equation 3x² – 3x – 6 = x – 2. (3 marks)

5. Solve the equation 4(x + 3)(x – 1) = 21x. (3 marks)

6. Solve 3x² = 9x + 2. Give the answers in surd form if necessary. (3 marks)

7. Solve (x + 3)(1 – 2x) – 6 = 12x. (3 marks)

8. Solve (3x – 4)² = 6x. (3 marks)

9. Solve (2x – 1)(4x + 5) = (4x + 5)(12x + 5). (3 marks)

10. Suppose 7 is a root of the equation x² – kx – (2k + 4) = 0.

(a) Find the value of k.

(b) Hence find the other root.

(4 marks)

11. Suppose 3 is a root of the equation k = (x + 1)(2x – 1).

(a) Find the value of k.

(b) Hence find the other root.

(4 marks)

12. Find the number of real roots of the equation (3x + 1)(x + 3) = 2x(x – 8). Reference:HKCEE 07Q5 (3 marks)

13. Find the number of real roots of the equation 5x – 1 = x(3x + 2). (3 marks)

Sample Sample

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HKDSE Exam Series — Mathematics Conventional Questions (Compulsory Part) (Upgraded Edition) Book A Solution Guide

3 Exponential and

Logarithmic Functions

Warm Up Exercise

1.

a a b

4 2 3

3 = ( )

a a b

4 3 1 2 3 2 1

#

= a ab

3 4 2 3

= a¹^-³⁴b²³

= a b^-³¹ ²³

= a b

3 1 2 3

2. log log

8 16 =

log log 2 2

3 4

= log log

3 2

4 2

= 3 4

3. log a² + log a³ = 2 log a + 3 log a

= 5 log a 4. (a) log 12 = log (2² × 3) = log 2² + log 3 = 2 log 2 + log 3 = 2a + b

(b) log 15 = log (3 × 10 ÷ 2) = log 3 + log 10 – log 2 = b + 1 – a

5. 4^2x = 8 (2²)^2x = 2³ 2^4x = 2³ 4x = 3

x = 4 3

6. 2^x – 5 · 2^x + 8 = 0 (1 – 5) · 2^x + 8 = 0 –4 · 2^x + 8 = 0 2^x = 2 x = 1

7. log (x + 3) = log (2x + 4) x + 3 = 2x + 4

x = –1

8. log₃ (x – 1) = 2.3 x – 1 = 3^2.3

x = 3^2.3 + 1

= 13.5 (cor. to 3 sig. fig.) 9. (a) & (b)

x y

0 1 2 3

–1

–2

1

–1 2 3

–2

y = log2 x y = x

y = 2^x y = 2^–x

10. Sound intensity level = 10 log 10

10

12 8 -

d - n dB

= 10 log 10⁴dB

= 40 dB

Mock Questions

Section B 1.

p q p q

1 4 3

3 - =

p q p q

1 4 3 1 3

_ - i

[1M]

= p q

p q

3 1

3 4 3

-

= p³ 3 q

1 1 3

- -d n -4 [1M]

= p q¹⁰³ ^-³¹

= q p

3 1 3 10

[1A]

Common mistakes

The answer should be expressed with positive indices as required.

Sample

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23 23

3

Exponential and Logarithmic Functions

2. m

m n n

2

2 2 3 4

- = m²⁴m^{- -}^] ²^gn³^-² [1M]

= m m n²^ ² h⁴¹ [1M]

= m²⁺²¹n⁴¹

= m n²⁵ ⁴¹ [1A]

3. ³ab³ a b² ^{1 2}

3

# ^-

^ h ^_ ⁱ ^{= ab}^{^} ³^h³¹^#^{^}^{a b}² ^-¹^h²¹^#²³ ^[1M]

= a b3 (a b )

1 2 1

4 3

# ^-

= a b a b ⁴

3 3

1 3

# 2 ^- [1M]

= a3 b

1 2 3 1

4 + -3

= a b¹¹⁶ ⁴¹ [1A]

4.

log

log log x

x x

3 1

2+

= 1

1 log

log log

x

x x

3

2 +2

[1M]

= 1

2 2

1 log

log x

x

3

c + m

[1M]

= 15 2 [1A]

5.

1

log log

x y

x y x y

6 3

2

4 2

+ -

= log log

log log log log

x y

6 3

1 1

2

4 2

+

+ - -

[1M]

= log log

log log log log

x y

x y x y

6 3

2 4 2

+

+ + +

[1M]

= 6log 3log

log log

x y

6 3

+ +

= 1 [1A]

6. log log log

16

1 243 3264

2 - 3 #

c m

= log 2 log

log log

32 3 64

2 4

3 2 5

#

- -

a k [1M]

= log

4 log 2 5

2 2

5 6

# c- - m

= 13

log log

5 2

6 2

2 #

c- m [1M]

= 5

-39 [1A]

Guidelines

Use the change of base formula: loga b = log log a b.

7. log 1 x²-y²

d n

= log 1

x+y x-y

^ h^ h

= G

= log x y

1

c + m + log x y 1

c - m [1M]

= –2log x+y– log (x – y) [1M]

= –2a – b [1A]

Guidelines

log (x + y) = 2 log x+y

8. a² + b² = 3ab a² + b² – 2ab = ab

(a – b)² = ab [1A]

log (a – b)² = log ab [1M]

2 log (a – b) = log a + log b log (a – b) =

21 (log a + log b) [1A]

9. (a) log 21 = log (3 × 7)

= log 3 + log 7 [1M]

= r + s [1A]

(b) log 27

343 = log 27 – log 343

=

21 log 3³ –

21 log 7³ [1M]

= 3 2log 3 –

23 log 7 = (r s)

2

3 -

[1A]

10. (a) Let y = 0.694444 ...

10y = 6.944444 ...

10y – y = 6.944444 ... – 0.694444 ... [1M]

9y = 6.25 y = .

9 6 25

= 3625 [1A]

Guidelines

To express a recurring decimal into fraction, try to remove the recurring part.