Focus SPM 2020 F4 KSSM - Additional Maths

CONTENTS

Mathematical Formulae iv

1Chapter

Functions 1

1.1 Functions 2
1.2 Composite Functions 14
1.3 Inverse Functions 19
SPM Practice 1 27

2Chapter Quadratic Functions 29
30
2.1 Quadratic Equations and Inequalities 37
2.2 Types of Roots of Quadratic Equations 39
2.3 Quadratic Functions 52
SPM Practice 2

3Chapter Systems of Equations 55

3.1 Systems of Linear Equations in Three Variables 56
3.2 Simultaneous Equations involving One Linear Equation and One Non-Linear
Equation 60
SPM Practice 3 65

4Chapter Indices, Surds and Logarithms 67
4.1 Laws of Indices 68
4.2 Laws of Surds 70
4.3 Laws of Logarithms 75
4.4 Applications of Indices, Surds and Logarithms 80
SPM Practice 4 85

5Chapter Progressions 87
88
5.1 Arithmetic Progressions 96
5.2 Geometric Progressions 104
SPM Practice 5

ii

6Chapter Linear Law 108
6.1 Linear and Non-Linear Relations 109
6.2 Linear Law and Non-Linear Relations 114
6.3 Applications of Linear Law 121
SPM Practice 6 124

7Chapter Coordinate Geometry 128

7.1 Divisor of a Line Segment 129
7.2 Parallel Lines and Perpendicular Lines 132
7.3 Areas of Polygons 137
7.4 Equations of Loci 142
SPM Practice 7 145

8Chapter Vectors 148

8.1 Vectors 149
8.2 Addition and Subtraction of Vectors 155
8.3 Vectors in a Cartesian Plane 166
SPM Practice 8 173

9Chapter Solution of Triangles 179
1 0Chapter
9.1 Sine Rule 180
9.2 Cosine Rule 184
9.3 Area of a Triangle 188
9.4 Application of Sine Rule, Cosine Rule and Area of a Triangle 191
SPM Practice 9 194

Index Numbers 196

10.1 Index Numbers 197
10.2 Composite Index 200
SPM Practice 10 211

Pre-SPM Model Paper 216
Answers 227



iii

1Chapter Learning Area : Algebra

Functions

Concept KEYWORDS
Map • Absolute value function – Fungsi
nilai mutlak
• Arrow diagram – Gambar rajah
anak panah
• Composite function – Fungsi
gubahan
• Domain – Domain
• Function – Fungsi
• Function notation – Tatatanda
fungsi
• Horizontal line test – Ujian garis
mengufuk
• Image – Imej
• Inverse function – Fungsi songsang
• Object – Objek
• Range – Julat
• Relation – Hubungan
• Vertical line test – Ujian garis
mencancang

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1

  Additional Mathematics  Form 4  Chapter 1  Functions

Chapter 1 1.1 Functions y

1. There are four different types of relations. 6 Each input value has
5 one output value.
One-to-one 4
3 x
2 123456
1

–2 –1–10
–2

y

Many-to-one 7 1234 Several input values
Types of relations 6 share one output
5 value.
4
3 x
2
1

–4 –3 –2 –1–10

One-to-many y

4

2 An input value may
0 2 4 6 8 10 12 14 16 x have two or more
–2 output values.

–4

y

Many-to-many 4 123 4 x For two or more
3 input values, there
2 are two or more
1 output values.

–4 –3 –2 –1–10
–2
–3
–4

2. Out of the four relations above, only two can be considered as functions:
• One-to-one relation.
• Many-to-one relation.

2

Additional Mathematics  Form 4  Chapter 1  Functions 

A Explaining functions using graphical 1 Chapter 1
representations and function
notations The following arrow diagram shows the relationship
between two sets X and Y.
1. Assuming the price of a movie ticket is RM10. (a) (b)
Hence, the amount paid depends on the number
of tickets purchased. Plus 1
(a) The relationship between the number of
tickets purchased and the total amount paid 1 2a 1
can be represented in a graphical form as 2 3b 4
follows. 3 4c 5

5

Set X Set Y Set X Set Y
(d)
(c) a
1 b
Squares 2 c
3
50 11 4
40 2
Set Y 30 –2 4
20
10 Set X Set Y Set X Set Y

0 12345 Determine whether the relations are functions. Give
reasons for your answers.
Set X
Solution
(b) Note that each movie ticket purchased (a) Function. Each object has one image only even
(input value) in set X is mapped to one and
only one price (output value) in set Y. though element 5 does not have an object.
(b) Not a function. Object c has two images, which
For example, 1 → 10, 2 → 20, 3 → 30 and
so on. are 4 and 5.
(c) Function. Each object has only one image even
This type of relation is known as function.
(c) If f denotes a function which maps set X to though elements 2 and –2 have the same image.
(d) Not a function. Element 3 does not have an
set Y, then the function can be represented
by the following arrow diagram. image.

xfy SPM Tips

There are two types of relations which can be considered
as functions.

Capital

1 10 Pahang Kuantan
2 20 Perlis Kota Bharu
3 30 Kangar
4 40 Kelantan

Set X Set Y One-to-one relation
Material class

(d) Element 1 in set X is known as the object Steel Metal
while element 10 in set Y is known as the Copper Non-metal
image of 1. The same applies for 20, 30, and
40, which are the images of objects 2, 3, and Wood
4 respectively. Plastic

Many-to-one relation

Try Questions 1 – 2 in ‘Try This! 1.1’

3

Matematik Tambahan  Tingkatan 4  Jawapan 

ANSWERS

1Chapter (b) y
5
Functions
2
Try This! 1.1 1 Range: 0 ≤ f(x) ≤ 5
–3 –2 0 x
1. (a) 3 (b) 3 (c) 7
(c) y 3
2. (a) Function. Each object only has one image, even
though the element 6 does not have an object. 5 Range: 0 ≤ f(x) ≤ 5
3
(b) Not a function. Object 2 has three images, which are
2, 4, and 6. 0 x
58
(c) Not a function. Element s does not have an image. (d)
y
3. (a) This graph is not a graph of y as a function of x. There 7
are vertical lines which intersect the graph at two 5
different points. 3

(b) This graph is a graph of y as a function of x. Each
vertical line intersects the graph at one point only.

(c) This graph is a graph of y as a function of x. Each
vertical line intersects the graph at one point only.

(d) This graph is not a graph of y as a function of x. All
vertical lines intersect the graph at two different points,
except at the y-axis where the vertical line intersects
at one point only.

4. (a) (i) f : x → x3 or f(x) = x3
(ii) h : x → x + 1 or h(x) = x + 1
(b) A : r → πr 2 or A(r) = πr 2
(c) (i) f : x → 2x2 + 3x – 1 or f(x) = 2x2 + 3x – 1
(ii) g : x → sin x or g(x) = sin x

5. (a) 0 (b) –3 (c) —41 (d) – —32

6. (a) False (b) True (c) True

7. (a) Domain = {x, y, z}, range = {p, q} –4 – 2–3 0 x
(b) Domain = {–2, –1, 1, 2}, range = {1, 4} 2
(c) Domain = {1, 2, 3, 4, 5}, range = {12, 24, 36, 48, 50}
(d) Domain is x ∈ , range is y ∈ . 11. (a) –1 (b) 5 – t (c) 3 – 2t
(e) Domain is x ∈ , range is y > 3.
(f) Domain is x ∈ , x ≠ –5, range is y ∈ , y ≠ 0. 12. (a) 1 (b) 9x + 1 (c) 6z – 2
(g) Domain is x ∈ , range is y > 0.
(h) Domain is x . 0, range is L . 0.

8. (a) Domain 0 < x < 8, range 3 < y < 15. 13. (a) 2 (b) 4.5 (c) 5 – 2x
(b) Domain –6 < x < 1, range 0 < y < 9.
14. (a) 10 (b) –5, 3 (c) 5
9. (a) When the domain is , all values of y are possible.
The range is y ∈ . 15. 5 (b) – —32 , 3 (c) 1, —35

(b) When x is limited to negative values, all values of 16. (a) 7, 5
y will be at least 2. The range is y > 2.
17. (a) 3
(c) The range is {2, 3, 4, 5, 6}. (b) 1, 6

10. (a) y 18. (a) h = 3, k = 1
(b) 0 < f(x) < 9

–2 0 Range: 0 ≤ f(x) ≤ 2 19. (a) RM77 000
x (b) RM10 500
2 (c) 3
(d) V(n) is a function, Each input n will give rise to one
and only one output value of V(n).

227