Contents

Contents

1 Number Systems ... 1

2 Quadratic Equations in One Unknown (I) ... 7

3 Quadratic Equations in One Unknown (II) ... 14

4 Functions and Graphs ... 21

5 Exponential Functions ... 30

6 Logarithmic Functions ... 38

7 Equations of Straight Lines ... 46

8 More about Polynomials ... 53

9 Basic Properties of Circles ... 60

10 More about Trigonometry ... 68

11 Applications of Trigonometric Functions ... 74

12 Variations ... 80

Revision Test ... 87

Preparing for the New School Year ... 92

... 95

Sample

Daily Life Mathematics

One day, Raymond asked Dr. Lau what the lowest temperature ever recorded on the surface of the Earth was. Dr. Lau said that the lowest temperature was around

108$ 75

- -

^ h_°C.

Can you help Raymond to find the lowest temperature?

Knowledge Reminder

1. Number Systems

2. Complex Numbers

(a) Complex numbers are numbers in the form a + bi, where a and b are real numbers, and i = -1.

(b) a is called the real part while b is called the imaginary part.

(c) Two complex numbers a + bi and c + di are equal if and only if a = c and b = d.

3. Operations of Complex Numbers

(a) Addition: (a + bi) + (c + di) = (a + c) + (b + d)i (b) Subtraction: (a + bi) – (c + di) = (a – c) + (b – d)i

(c) Multiplication: (a + bi) × (c + di) = (ac – bd) + (bc + ad)i i² = –1 (d) Division: c di

a bi +

+ = c di a bi

+

+ · c di i

c d

-

- = ac b

c d

d

2+ 2

+ + c d i

bc ad

2 2

- +

Let me try.

Can you find the answer?

Complex numbers

Real numbers Imaginary numbers

Irrational numbers Rational numbers

Integers Terminating decimals Recurring decimals

Negative integers Zero Positive integers

Unit 1

Number Systems

C oncept Builder

C oncept Builder

Determine whether each of the following statements is correct or not.

1. ‘5’ is a complex number. ^{Correct Incorrect}

2. The sum of two irrational numbers must be an irrational number. ^{Correct Incorrect} 3. The values of all trigonometric ratios are irrational. ^{Correct Incorrect} 4. If m and n are real numbers, (m + ni)(m – ni) must be a real number. Correct Incorrect

5. The square of a complex number must be a real number. ^{Correct Incorrect}

Exercise

Level 1

1. Determine whether the following numbers are rational or irrational.

(a) sin 45° (b) cos 60°

(c) tan 30° (d) 80

(e) 7

4 (f) 13

r2

Common Mistakes

Simplify 31 i

- and express the answer in the form a + bi.

Solution:

i 3

1

- = 3 i 1

- × i i 3 3

+ +

= 3 i i 3

2 2

+ -

= 3 i 8 +

= 3 8 + i

8

i Be aware that i² = –1 instead of 1, thus 3² – i² = 9 + 1 = 10. The answer should be 3

10 + i 10.

Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) S.4 – S.5

5

Sample

Convert the following decimals into fractions (2 – 3).

2. 0.6

.

_{3. 1.2}

.

₃

.

Simplify the following expressions and express the answers in the form a + bi (4 – 9). 4. (2 + 3i) + (4 – 5i) 5. (9 – 5i) – (11 + 4i) 6. i(5 – 6i)

7. (4 – 3i)(2i) 8. 4 5i

2

- 9. 5i

3

Level 2

Convert the following decimals into fractions (10 – 11).

10. –2.5

.

₁

.

_{11. 1.2}

.

₀₇

.

12. Calculate 0.2

.

₄

.

_{× 1.6}

.

₃

.

without using a calculator.

Simplify the following expressions and express the answers in the form a + bi (13 – 18). 13. (2 + 7i) + 3 2

- i

b l – 4 2

-7i

b l 14. (2 + 5i)²

Hint

Express the recurring decimals into fractions first.

15. 2 i 4

- 16. (7 – 2i)(9 + 4i)

17. i

i 4 3 3 2

+

- 18. i

i 3 2

4

- + 3 2i 5 +

19. Calculate the value of i ii 2

7 7

- ^- .

20. Is i + i² + i³ + … + i¹⁶ a real number? Explain your answer briefly.

21. Which of the following is/are a complex number?

A. 13 B. –7 C. 4 – 5i

D. All of the above

Multiple-choice Questions

22. Which of the following is an irrational number?

A. 3 × 12 B. 25 – 121 C. 12

2

D. ( 15 – 6)( 15 + 6)

Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) S.4 – S.5

5

Sample

23. Which of the following is NOT a real number?

A. π B. 13

4 C. 5 – i D. 133

24. W h i c h o f t h e f o l l ow i n g c a n b e transformed into a recurring decimal?

A. 27 B. 72 C. 8 5

D. 58

25. Which of the following is greater than 0.1

.

₂₃

.

_?

A. 0.12

.

₃

.

B. 0.123 C. 0.123 123 D. 0.1

.

₂

.

26. i⁵(ai + 2) = A. a + 2i B. a – 2i C. –a + 2i D. –a – 2i 27. ii

1 1

- + = A. 1 B. –1 C. i D. –i 28. i²⁰¹⁰ =

A. –1 B. –i C. 1 D. i

r Fo

HK

M ake Yourself Smarter

M ake Yourself Smarter

If a and b are real numbers and (a – 2i) – (4b + ai) + (3 + bi) = 0, find the values of a and b.

Enrichment Corner

Referring to the expression given in Daily Life Mathematics, the lowest temperature on the Earth was

108$ 75 cC

- -

^ h ₌ -1 108$ -1 75cC

= 8100i²cC = –90°C

Actually, the lowest temperature ever recorded on the surface of the Earth was −89.2°C in Antarctica on 21st July, 1983.

How about the highest temperature? You may find out more information from the following website:

http://www.universetoday.com/55043/earths-temperature/

Open-ended Question

Suppose x = p q and y = q

p, write down 3 sets of possible values of p and q such that x and y are rational numbers.

G ^lossary G ^lossary

complex number 複數 fraction 分數 imaginary number 虛數

imaginary part 虛部 integer 整數 irrational number 無理數

natural number 自然數 rational number 有理數 real number 實數

real part 實部 recurring decimal 循環小數 terminating decimal 有盡小數

Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) S.4 – S.5

5

Sample