Focus Smart Plus Maths Textbook M2 samplebook

BCB032038

BCB032038 978-616-541-313-8

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TPage Textbook Mathematics M2.indd 1

ISBN 978-616-541-313-8 First Published 2562

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Contents Chapter

1

Exponential Notation

1.1 Exponential Notation 1.2 Addition and Subtraction in Exponential Notation 1.3 Multiplication and Division in Exponential Notation 1.4 Combined Operations Using Exponential Notation Mastery Practice

2 4 7 10 14

Chapter

2

Real Numbers

15

Chapter

3

Polynomials

42

Chapter

4

Solving Quadratic Polynomial Equations

56

2.1 Rational Numbers, Irrational Numbers and Real Numbers 2.2 Squares 2.3 Square Roots 2.4 Cubes 2.5 Cube Roots Mastery Practice

3.1 Polynomials 3.2 Addition and Subtraction of Polynomials 3.3 Multiplication of Polynomials 3.4 Division of Polynomials by a Monomial Mastery Practice

4.1 Solving Quadratic Polynomial Equations in the Forms

ax 2 + bx = 0 and ax 2 + c = 0 4.2 Solving Quadratic Polynomial Equations in the Form

px 2 – q = 0 where p and q are Prefect Squares 4.3 Solving Quadratic Polynomial Equations in the Form

ax 2 + bx + c = 0 where a, b and c ≠ 0 Mastery Practice

Chapter

5

Pythagoras’ Theorem

5.1 Relationship between the Sides of a Right-angled Triangle 5.2 Converse of Pythagoras’ Theorem Mastery Practice

1

16 21 26 31 34 41

43 46 49 52 55

57 60 62 69

70

71 76 79

Chapter

6

Surface Areas and Volumes of Prisms and Cylinders

81

97

6.1 Surface Areas of Prisms and Cylinders 6.2 Volumes of Prisms and Cylinders Mastery Practice

82 87 95

Chapter

7

Parallel Lines and Angles

7.1 Angles Associated with Transversals and Parallel Lines Mastery Practice

98 107

Chapter

8

Transformations

110

Chapter

9

Congruence

147

10

Mean and Data Presentation

159

11

Geometrical Constructions

177

Chapter

Chapter

8.1 Transformation 8.2 Translation 8.3 Reflection 8.4 Rotation Mastery Practice

9.1 Congruence 9.2 Congruent Triangles Mastery Practice

10.1 Mean 10.2 Data Presentation Mastery Practice

11.1 Constructions Mastery Practice

111 113 121 131 144

148 151 158

160 162 175

178 204

Special Features

in This Book

Learning Outcomes States the learning objectives of each chapter.

Flashback

Test Yoursel f Evaluates the understanding of the students for every subtopic.

Lists the important mathematical terminologies of each chapter.

Helps students to recall the basic concepts for the chapter.

Math Online Provides direct access to useful websites by scanning the QR codes given.

Consists of brief and concise notes that summarise the concepts learnt in each chapter.

Mastery Practice Provides subjective questions covering the entire learning outcomes of each chapter.

Points out the important tips for students to take note.

Example

Points out the common mistakes that students make and the correct ways of answering questions.

Consists of sample questions with complete and comprehensive solutions.

Provides direct access to the interactive exercises by scanning the QR codes given.

1 Chapter

Exponential Notation

Flashback

By the end of this chapter, you should be able to

1. a2 × a3 =

• have a concept of real numbers expressed in exponential notation with integral indices.

2. x × x = 10

12

• explain results of expression in exponential notation in integral numbers, ratios and decimals.

3. 46 × 47 = 4 4. bn × bm =

• multiply and divide real numbers in the form of exponents with the same base and integral indices.

5. 102 × 105 = 6. a3 ÷ a = 7. a0 = 8. 1010 ÷ 102 = 10 9. 185 ÷ 183 = 18 10. b8 ÷ b3 =

Math Online Visit this website to know more about this chapter.

10. b5

9. 2

6. a2

5. 107

2. x22

1. a5

7. 1 3. 13 Answers:

Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

1

8. 8

4. b(n + m)

1.1 Exponential Notation A

Understanding exponential notation

Exponential notation are used when very large or small numbers such as 2,000,000,000,000, 390,000,000 and 0.0000000168 need to be written in a manner that is more easily understood.

The numbers in exponential notation is written based on the multiplication of a number with the powers of base number 10. Exponential notation uses a general format of A × 10n where 1  A  10 and n being the powers of base number 10.

Example 1 Write the following numbers using exponential ­notation. (a) 78,000,000 (b) 0.0000564

Solution (a) 78,000,000 = 7.8000000 × 107 Decimal point moves 7 places to the left. = 7.8 × 107 (b) 0.0000564 = 000005.64 × 10–5 Decimal point moves 5 places to the right. = 5.64 × 10–5 Try Questions 1 & 2 in Test Yourself 1.1

Example 2 Convert the following numbers written in the form of exponential notation into integer form. (a) 4.32 × 105 (b) 6.89 × 10–5 Mathematics

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2

The exponential notation is also widely known as the scientific notation or standard form.

Solution (a) 4.32 × 105 = 4.32000 = 432,000 (b) 6.89 × 10–5 =

6.89

= 0.0000689 Try Questions 3 & 4 in Test Yourself 1.1

B

Exponential notation for numbers between 1 and 10

The decimal point for a number between 1 and 10 (1  A  10) does not need to be moved when written using exponential notation.

When the power of base number 10 for any number is zero, it is equivalent to that particular number.

A × 100 = A

2.7 × 100 = 2.7

Example 3 a0 = 1

100 = 1

A × 100 = A × 1 = A

Write the following numbers using exponential notation. (a) 6.53 (b) 4.38

Solution (a) 6.53 = 6.53 × 100 (b) 4.38 = 4.38 × 100 Try Question 5 in Test Yourself 1.1

Test Yourself

1.1

1. Write the following numbers using exponential notation. (a) 4,600,000 (b) 530,000 (c) 409,700,000 (d) 700,000 2. Write the following decimals using exponential notation. (a) 0.00380 (b) 0.000004201 (c) 0.000063 (d) 0.000000000089

Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

3

3. Convert the following numbers written using ­exponential notation into integer form. (b) 3.12 × 106 (a) 4.75 × 104 7 (c) 4.08 × 10 (d) 9.81 × 103 4. Convert the following numbers written using ­exponential notation into decimals. (b) 2.34 × 10–5 (a) 5.81 × 10–3 –6 (c) 9.0 × 10 (d) 1.024 × 10–7 5. Write the following numbers using exponential notation. (a) 8.31 (b) 7.49 (c) 3.1 (d) 4.107

and Subtraction 1.2 Addition in Exponential Notation A

Addition in exponential notation

Addition is a process of finding the sum of two or more numbers using exponential notation. For addition of numbers in exponential notation that have the same power of base number 10: For example, the sum of A × 10n and B × 10n, equals to (A + B) × 10n.

Example 4 Find the sum of (a) 6.81 × 105 and 1.04 × 105 (b) 7.34 × 107 and 4.58 × 107

Solution (a) 6.81 × 105 and 1.04 × 105 = (6.81 + 1.04) × 105 = 7.85 × 105 (b) 7.34 × 107 and 4.58 × 107 = (7.34 + 4.58) × 107 = 11.92 × 107 = 1.192 × 108 Try Question 1 in Test Yourself 1.2

Mathematics

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A number written in exponential notation must only have one non-zero digit to the left of its decimal point.

Steps to add numbers in exponential notation that have different powers of base number 10 such as A × 10n + B × 10n + 1:  The power of base number 10 must be the same.

For example, A × 10n can be changed into

A

10

× 10n + 1.

 Find the sum of the numbers that have the same power of base number 10.

For example, A × 10n + B × 10n + 1

A

= × 10n + 1 + B × 10 n + 1 10 =(

A

10

+ B) × 10 n + 1

Example 5 Find the sum of (a) 6.54 × 105 and 2.93 × 104 (b) 9.49 × 105 and 8.9 × 104

Solution (a) 6.54 × 105 and 2.93 × 104 = 6.54 × 105 + 0.293 × 105 = 6.833 × 105 (b) 9.49 × 105 and 8.9 × 104 = 9.49 × 105 + 0.89 × 105 = 10.38 × 10 5 = 1.038 × 106 Try Question 2 in Test Yourself 1.2

B

Subtraction in exponential notation

Subtraction is the process of finding the ­difference between two numbers using exponential notation. For subtraction of numbers in exponential ­notation that have the same power of base ­number 10: For example, the difference between A × 10n and B × 10n, equals to (A – B) × 10n.

Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

5

Example 6 Find the difference between (a) 9.63 × 106 and 1.09 × 106 (b) 8.67 × 104 and 7.88 × 104

Solution (a) 9.63 × 106 and 1.09 × 106 = (9.63 – 1.09) × 106 = 8.54 × 106

(b) 8.67 × 104 and 7.88 × 104 = (8.67 – 7.88) × 104 = 0.79 × 10 4 = 7.9 × 103 Try Question 3 in Test Yourself 1.2

Steps to subtract numbers in exponential notation that have different powers of base number 10 such as A × 10n + 1 – B × 10n:



The power of base number 10 must be the same.

For example, B × 10n can be changed into



B × 10n + 1.

10

Find the difference between the numbers that have the same power of base number 10.

For example, A × 10n + 1 – B × 10n

= A × 10n + 1 – = (A –

B

10

B

10

× 10n + 1

) × 10n + 1

Example 7 Find the difference between (a) 7.6 × 105 and 8.7 × 104 (b) 1.21 × 107 and 6.2 × 106

Solution (a) 7.6 × 105 and 8.7 × 104 = 7.6 × 105 – 0.87 × 105 = (7.6 – 0.87) × 105 = 6.73 × 105 (b) 1.21 × 107 and 6.2 × 106 = 1.21 × 107 – 0.62 × 107 = (1.21 – 0.62) × 107 = 0.59 × 107 = 5.9 × 106 Try Questions 4 & 5 in Test Yourself 1.2

Mathematics

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Test Yourself

1.2

1. Calculate each of the following.

(a) 4.7 × 105 + 3.3 × 105

(b) 6.2 × 107 + 3.6 × 107

(c) 8.12 × 104 + 2.29 × 104

(d) 4.84 × 109 + 7.4 × 109 2. Simplify each of the following.

(a) 6.3 × 104 + 7.2 × 105

(b) 4.8 × 109 + 8.4 × 108

(c) 3.11 × 107 + 4.22 × 106

(d) 9.47 × 106 + 5.44 × 105 3. Calculate each of the following.

(a) 3.9 × 107 – 2.7 × 107

(b) 4.19 × 106 – 3.43 × 106

(c) 8.95 × 105 – 7.26 × 105

(d) 9.81 × 108 – 8.94 × 108 4. Simplify each of the following.

(a) 7.3 × 109 – 8.2 × 108

(b) 1.9 × 108 – 3.4 × 107

(c) 2.38 × 104 – 4.82 × 103

(d) 1.74 × 107 – 8.9 × 106 5. Calculate each of the following.

(a) 6.3 × 10–3 + 3.2 × 10–4

(b) 4.3 × 10–7 + 6.8 × 10–7

(c) 5.2 × 10–6 – 3.4 × 10–6

(d) 1.2 × 10–5 – 2.3 × 10–4

and Division in 1.3 Multiplication Exponential Notation A

Multiplication in exponential notation

The general form to multiply two numbers using exponential notation is: (A × 10m)(B × 10n) = (A)(B) × 10m + n Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

7

Steps to multiply two numbers such as (A × 10m) and (B × 10n) using exponential notation.  Multiply A and B to get the first part of the answer.  Multiply the exponential parts by finding the sum of m and n which is the power of the base number 10.  Multiply both results from steps  and  to get the final answer.

Example 8 Calculate each of the following.

(a) (4 × 107)(9 × 102)

(b) (3.5 × 108)(–2 × 103)

(c) (–3 × 10–2)(–8 × 10–5)

Solution (a) (4 × 107)(9 × 102) = (4)(9) × 107 + 2

= 36 × 109

= 3.6 × 1010

(c) (–3 × 10–2)(–8 × 10–5) = (–3)(–8) × 10–2 + (–5)

= 24 × 10–7

= 2.4 × 10–6

(b) (3.5 × 108)(–2 × 103) = (3.5)(–2) × 108 + 3

= –7 × 1011

Try Questions 1 & 2 in Test Yourself 1.3

B

Division in exponential notation

The general form to divide two numbers using exponential notation is: (A × 10m) ÷ (B × 10n) = (

A ) × 10m – n B

Steps to divide two numbers such as (A × 10m) and (B × 10n) using exponential notation.

 Divide A by B to get the first part of the answer.  Divide the exponential parts by finding the difference between m and n which is the power of the base number 10.  Multiply both results from steps  and  to get the final answer.

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Example 9 Calculate each of the following. (a) (8 × 108) ÷ (2 × 105)

(b) (4.2 × 106) ÷ (–6 × 102)

(c) (–5.4 × 10 –6) ÷ (–9 × 10–9)

Solution (a)

(8 × 108) ÷ (2 × 105) 8 = × 108 – 5 2

= 4 × 103 (b)

(4.2 × 106) ÷ (– 6 × 102) 4.2 = × 106 – 2 –6

= – 0.7 × 104 = –7 × 103 (c)

(–5.4 × 10 –6) ÷ (–9 × 10–9) (–5.4) × 10–6 – (–9) = (–9)

= 0.6 × 103 = 6 × 102 Try Questions 3 & 4 in Test Yourself 1.3

Test Yourself

1.3

1. Calculate each of the following.

(a) (3 × 108)(4 × 1017)

(b) (4.7 × 103)(6 × 108)

(c) (5.2 × 108)(4.5 × 109)

(d) (7.1 × 103)(2.9 × 102) 2. Simplify each of the following.

(a) (6 × 109)(– 4 × 102)

(b) (–5 × 108)(–3.2 × 1011)

(c) (3 × 10–9)(4.7 × 103)

(d) (–7.2 × 10–3)(9.1 × 10–5)

Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

9

3. Calculate each of the following.

(a) (4 × 109) ÷ (1 × 107)

(b) (1.2 × 1012) ÷ (6 × 108)

(c) (2.56 × 107) ÷ (3.2 × 104)

(d) (1.196 × 1020) ÷ (2.3 × 106) 4. Simplify each of the following.

(a) (9.9 × 109) ÷ (–3 × 102)

(b) (–8 × 1021) ÷ (5 × 10–2)

(c) (– 4.5 × 1032) ÷ (–9 × 1020)

(d) (3.45 × 10–5) ÷ (–1.5 × 10–18)

Operations Using 1.4 Combined Exponential Notation To perform computations involving combined operations of addition and subtraction or multiplication and division using exponential notation, always work out from left to right.

Example 10 Solve each of the following. (a) 3.2 × 109 – 4.9 × 108 + 5.3 × 109 (b) (4.5 × 1012) × (2 × 106) ÷ (3 × 104)

Solution (a) 3.2 × 109 – 4.9 × 108 + 5.3 × 109

= 3.2 × 109 – 0.49 × 109 + 5.3 × 109 = 2.71 × 109 + 5.3 × 109 = 8.01 × 109

(b) (4.5 × 1012) × (2 × 106) ÷ (3 × 104) = 9 × 1018 ÷ 3 × 104

= 3 × 1014

Try Questions 1 & 2 in Test Yourself 1.4

Mathematics

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Work out from left to right.

Steps to perform computation involving any combination of the four basic operations and brackets:

 Work out the calculations within the ­brackets first.  Then perform the multiplication or ­division followed by the addition or subtraction, work out from left to right.

Example 11 Solve each of the following.

(a) (3.2 × 107) + (4.2 × 103) × (5 × 104)

(b) 6 × 109 × (5.7 × 105 – 6.2 × 104)

Solution (a) (3.2 × 107) + (4.2 × 103) × (5 × 104) = (3.2 × 107) + (2.1 × 108)

Perform multiplication first.

= 2.42 × 108

(b) 6 × 109 × (5.7 × 105 – 6.2 × 104) = 6 × 109 × (5.08 × 105) = 3.048 × 1015

Work out the calculations within the brackets first.

Try Questions 3 & 4 in Test Yourself 1.4

Test Yourself

1.4

1. Calculate each of the following.

(a) 8.6 × 106 + 9.8 × 105 – 6.5 × 105

(b) 3 × 107 – 7.05 × 107 + 2.15 × 106

(c) 9.46 × 1011 + 7.62 × 1011 – 5.26 × 1010

(d) 1.06 × 1015 – 4.51 × 1014 + 3.95 × 1014 2. Simplify each of the following.

(a) (6 × 107) × (4 × 108) ÷ (3 × 106)

(b) (2.1 × 106) × (7 × 108) ÷ (4.9 × 106)

(c) (– 6 × 1012) × (–5 × 10–3) ÷ (2 × 10–5)

(d) (2.5 × 10–12) × (–3 × 1010) ÷ (4 × 10–8)

Mathematics Focus Smart + MATHAYOM

Chapter 1 Exponential Notation

2

11

3. Solve each of the following.

(a) 4.1 × 1014 ÷ (5.2 × 107) × (9.6 × 106)

(b) (8.1 × 1012) ÷ (9 × 104) – (7.6 × 107)

(c) 8.08 × 109 + (9.9 × 1018) ÷ (1.1 × 108)

(d) (2.8 × 1013) × (1.2 × 106) – (7.8 × 1018) 4. Calculate each of the following.

(a) 1.38 × 1017 + (1.17 × 1017 – 3.62 × 1016)

(b) (3.46 × 1019 – 3.6 × 1019) × (9 × 108)

(c) –1.71 × 1013 + (5.5 × 1014 + 7.5 × 1013)

(d) 1.93 × 107 × (2 × 1014 ÷ 5 × 108)

Example 1 Write 0.00000542 using exponential notation.

0.00000542 = 5.42 × 106

0.00000542 = 5.42 × 10–6

Add a positive 6 to the exponent for the 6 places to the right that the decimal point moved.

Add a negative 6 to the exponent for the 6 places to the right that the decimal point moved.

Example 2 Calculate (2 × 108) × (4 × 103).

(2 × 108) × (4 × 103) = (2)(4) × 10(8)(3) = 8 × 1024

Mathematics

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2

The exponential is multiplied.

(2 × 108) × (4 × 103) = (2)(4) × 108 + 3 = 8 × 1011

The exponential is added.

BCB032038

BCB032038 978-616-541-313-8

,!7IG1G5-ebdbdi!

Cover F&B.indd 1

07/01/2019 18:17