Preface dditional aths IGCSE el aths IGCSE el dditional...

IGCSE O Level Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional

Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level

Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional Maths • IGCSE

O Level Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional























Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional Maths •

Preface

This book is a good companion for pupils preparing for grade 10 IGCSE examination for Additional Mathematics from the year 2014 onwards.

How to use the BookEach section of the book comprises concise key notes to explain important and relevant concepts pertinent to the topic covered in the section, follow by examples similar to examination questions. Pupils may use the example questions as a guide. If they are familiar with the topic in the section, they should try out the examples instead of referring to the solutions.

The last part of each section contains practice questions which can be used as further revision for weaker pupils or testing questions for those pupils who are more proficient in the topic.

Detail solutions for all practice questions are provided. However, pupils are advised to attempt the questions before referring to the solutions.

The book is best use for revision to prepare for examination. We believe the book presents ample examples and questions will benefit pupils in better understanding of each of topics hence gain more confidence in achieving good results.

Last but not least, we would like to thank Fairfield Book Publishers Pte Ltd for inviting Affinity Education Place to write this book.

Quek EM

Affinity Education Place

170_IGCSE O Level A Maths-3rd Pass- Print PDF 08/08/14

IGCSE O Level Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional


























Additional Maths • IGCSE O Level Additional Maths • IGCSE O Level Additional Maths •

Contents

1 Set Language and Notation 1

2 Functions 8

3 Quadratic Functions 21

4 Indices and Surds 29

5 Factors of Polynomials 39

6 Simultaneous Equations 54

7 Logarithmic and Exponential Functions 62

8 Straight Line Graphs – Cartesian Coordinates 74

9 Straight Line Graphs – Linear Law 85

10 Circular Measure 95

11 Trigonometry – Ratios and Graphs 103

12 Trigonometry – Equations and Identities 116

13 Permutations and Combinations 127

14 Binomial Expansions 134

15 Vectors in 2 Dimensions 141

16 Matrices 157

17 Differentiation – Techniques 171

18 Differentiation – Gradients, Tangents and Normals 182

19 Differentiation – Approximations 192

20 Differentiation – Rate of Change 200

21 Differentiation – Stationary Points, Maxima and Minima 206

22 Integration – Techniques 216

23 Integration – Plane Area 230

24 Kinematics – Moving Particles 241

Worked Solutions 251


IGCSE O Level Additional Maths Set Language and Notation 1

KEY NOTESSet Language and Notation

ϊϊ Set Language

1. Set is a collection or group of distinct objects. Some examples of Set:

A = {x : x is a positive even number less than 10}, then A = {2, 4, 6, 8}

B is the set of vowels in English alphabet, B = {a, e, i, o, u}.

C is the set of months in the year, C = {January, February, March, April, May, June, July, August, September, October, November, December}

2. The objects in a set are known as elements of the set. For example, D is the set of days in the week. The day Monday is an element of D, or Monday ∈D.

3. A set containing no element is known as empty set or null set. The notation to represent empty set is ∅ or { }.

4. Universal set, is the set which contains all possible elements. For example, set of real numbers is the Universal set of set of even numbers.

5. A subset is part of a set. For example,

The set of real numbers can be divided into rational and irrational numbers. Rational numbers and irrational numbers are subsets of set of real numbers.

If E is the set of all letters in English alphabet and B is the set of vowels in English alphabet, then B is a subset of E, or B E⊂ .

6. If a set S is exactly the same as T, then S is an improper subset of T or S T⊆ .

7. The number of elements in a set A is denoted as n (A).

8. A B∪ means the union of A and B, that is the set of elements from set A or set B.

9. A B∩ means the intersection of A and B, that is the set of elements from set A and set B.

10. The set containing all elements and of which all other sets are subsets is known as universal set, which is denoted as ε .

11. Complement of a set, denoted as A’, represents all elements from the universal set containing A but not from A.

1


2 IGCSE O Level Additional Maths Set Language and Notation

KEY N

OTES

Example 1:

Given ε = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

List the elements of the set

A = { x : x is a multiple of 4 }

B = { x : x is a multiple of 3 }

C = { x : x is a prime number }

Example 2:

Given ε = { x ∈ : −10 < x < 3}


A = { x : x < 0 }

B = { x : x > −5 }

C = { x : x is an even number }

Example 3:

ε = {x ∈ : 0 ≤ x ≤ 15}


A = { x : x is an even number }

B = { x : x is an odd number }

C = { x : x is a multiple of 4 }

D = { x : x is a multiple of 20 }

ϊŦ Solution:A = { 4, 8, 12 }

B = { 3, 6, 9, 12 }

C = {2, 3, 5, 7, 11 }

ϊŦ Solution:A = { −9, −8, −7, −6, −5, −4, −3, −2, −1 }

B = { −4, −3, −2, −1, 0, 1, 2 }

C = { −8, −6, −4, −2, 0, 2 }

ϊŦ Solution:A = { 0, 2, 4, 6, 8, 10, 12, 14 }

B = { 1, 3, 5, 7, 9, 11, 13, 15 }

C = { 4, 8, 12 }

D = { } or ∅



KEY NOTESϊϊ Venn Diagram

Suppose the universal set ε = {integers 0 to 10}, set A = { 2, 4, 6, 8 }, set B = {1, 3, 5, 7, 9}.

Using Venn Diagram, the figure below is another way to express the sets.

Given set C contains all prime numbers in ε , hence C = {2, 3, 5, 7}.

Then,

A B∪ = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A C∪ = { }2 3 4 5 6 7 8, , , , , ,

B C∪ = {1, }2 3 5 7 9, , , ,

A B C∪ ∪ = { }1 2 3 4 5 6 7 8 9, , , , , , , ,

A B∩ = { }

A C∩ = {2}

B C∩ = {3,5,7}

A’ = { 0, 1, 3, 5, 7, 10 }

B’ = {0, 2, 4, 6, 8, 10}

C’ = {0, 1, 4, 6, 8, 9, 10}

Venn Diagram representation of universal setε , set A, set B and set C.

A B

1 3 5

7 9

2 4

6 8

0 10

ε

A B

14

6

8

0 10

2 3 5 7

9

ε



KEY N

OTES

Venn Diagram Representation

1. Union of sets A and B, A B∪ . 2. Intersection of sets A and B, A B∩ .

3. B is a proper subset of A, B A⊂ . 4. Complement of A, A’

Example 4:

Shade the region of the diagram that represents

(a) A ∩ B ∩ C (b) (A ∪ B) ∩ C (c) (A ∩ B) ∪ C

(d) A’∩ (B ∪ C) (e) A’ ∩ B ∩ C (f) A’ ∩ B’ ∩ C’

A B

ε

A B

ε

ε

A

B

ε

A

A'A'

ϊŦ Solution:

(a) A ∩ B ∩ C (b) (A ∪ B) ∩ C (c) (A ∩ B) ∪ C

(d) A’ ∩ (B ∪ C) (e) A’ ∩ B ∩ C (f) A’ ∩ B’ ∩ C’

εA

B C

εA

B C

εA

B C

εA

B C

εA

B C

εA

B C

A

B C



KEY NOTESExample 5:

Shade the region of the diagram that represents

(a) (A ∩ B) ∪ (A ∩ C) (b) A ∩ (B ∪ C)

(c) (A ∪ B) ∩ (A ∪ C) (d) A ∪ (B ∩ C)

ϊŦ Solution:(a) (A ∩ B) ∪ (A ∩ C) (b) A ∩ (B ∪ C)

(c) (A ∪ B) ∩ (A ∪ C) (d) A ∪ (B ∩ C)

εA

B C

A

B C

εA

B C

A

B C

εA

B C

A

B C

εA

B C

A

B C



PracticePractice Questions

1. ε = {a, b, c, d, e, f, g, h, i, j, k }

A = { a, d, f, j, k }

B = { b, c, f, g, h, k }

Find the followings.

(a) n(A)

(b) n(A’)

(c) n(B)

(d) n(B’)

(e) A ∪ B

(f) A ∩ B

(g) n(A ∪ B)

2. The whole numbers 1 to 20 are organised into two sets,

A = {x : x is an odd number }

B = {x : x is a multiple of 5 }

Complete the Venn diagram.

3. Two sets A and B are such that n(A) = 10 and n(B) = 6 with n( ε ) = 13. Find

(a) the smallest possible value of n (A ∩ B).

(b) the largest possible value of n (A ∩ B).

4. Define in terms of set A, B and C, the set representing the shaded regions in the following diagram

(a) _________________ (b) _________________ (c) _________________

εAA B

εA

B C

A

B C

εA

B C

A

B C

εA

B C

A

B C



5. On the Venn diagrams, shade the sets

(a) A B C∩( ) ∩’ ’ (b) A B C∩( ) ∩ ’

(c) ( ) ’A C B∪ ∩ (d) A C B∩( ) ∪

6. In a survey, 2800 students are interviewed. The number of students who have computers and mobile phone are recorded. Let C = {students who have one or more computers at home} and T = {students who have a mobile phone}. It was found that n(C) = 2150 and n(T) = 1500.

(a) Find the largest possible value of n C T∩( )’ ,

(b) What is the set notation representing “some students have neither a computer nor a mobile phone.

7. A mathematics quiz consists of 100 questions. Three pupils worked on these questions in following manner: first pupil worked on all odd-numbered questions, second pupil worked on question numbered 7, 14, 21, ..., third pupil worked on questions numbered 5, 10, 15, ... . How many questions are answered by two pupils?

8. The students in school study at least one of the subjects, namely Literature, Mathematics and Physics. Those students who study physics also study Mathematics.65 students study Mathematics only.12 students study Literature only.The number of students study physics and the number of students study Literature and Mathematics are the same.One-tenth of students who study Literature also study Physics.Altogether, there are 125 students.

(a) Draw a Venn diagram to illustrate the above information.

(b) Find the number of students taking Physics.

(c) Find the number of students taking both Mathematics and Literature.

ε

A

CB

εA

CB

ε

A C

B

εA

C

B


8 IGCSE O Level Additional Maths Functions

KEY N

OTES

Functions

ϊϊ Relations and Functions

1. An association between the elements of two sets is called a relation. If a relation satisfies that each element in set A is mapped to exactly one element in set B, this relation is defined as a function or mapping.

2. Following diagrams illustrate mappings from set A to B.

3. Following relation is not a function. This is because two distinct elements, b and c from set A are mapped to same element, 3 in set B.

4. Checking a relation is a function

For a given graph y f x= ( ) , if each vertical line x = k where k is an element from domain of f intersects the graph exactly at one point, then f is a function.

For example, y = ± +x 4 , where x ∈ , x ≥ −4, is not a function. This is because any line x k k= > −, 4 intersects the graph at 2 points.

a 1

2

3

Set A Set B

b

c

a 1

2

3

Set A Set B

b

c

a 1

2

3

Set A Set B

b

c

2

−2

−4 0 x

y = ± + 4y x

2


IGCSE O Level Additional Maths Functions 9

KEY NOTESHowever, the graph of y = x2 – 4, x ∈ , is a function as any line x k k= ∈, intersects the graph at exactly one point.

5. In representing a function, f which mapped elements of set A to set B, the notation is

f A B:

6. If an element x in set A is mapped to an element y by function f in set B, then y = f (x).

The set A is known as Domain of f.

The set containing all images of elements from set A under f is known as Range of f.

7. For a graph of a function, elements of domain lies on the x-axis, and elements of range lies on the y-axis.

8. Composite Functions

A mapping followed by another mapping is a composite function. Function f followed by g is expressed as gf x( ) , that is g f x( )( ) .

In general, gf x fg x( ) ≠ ( )

Note: ff (x) = f 2(x) and fff (x) = f 3(x).

−2 20

−4

x

y

= −2 4y x

x

gf

f (x) ( )g f x

( )f g xx

fg

g(x)



KEY N

OTES

For example, suppose f xx

: →2

, where x > 0 and g x x: → + 1 , where x > −1 , then

gf x gx x

( ) =

= +2 2

1

fg x f xx

( ) = +( ) =+

12

1

ff x fx

x( ) =

=2

f x ffx

f xx

3 2 2( ) =

= ( ) =

g x g x x x2 1 1 1 2( ) = +( ) = + + = +

9. One-to-One Functions

For a given function y f x= ( ) , if any horizontal line y = k where k is an element of range of f, intersects the graph exactly at one point, then f is a one to one function.

Some examples of one-to-one function are

f xx

: →3

3 for x ∈

g: xx

→6

, x x∈ ≠, 0

10. Inverse Functions

Let f x y: → be a function, then its inverse function is to map y to x. It is written asf y x− →1 : .

Note that

(a) Domain of f −1 = Range of f .

(b) Range of f −1 = Domain of f .

(c) ff x− ( )1 = f f x− ( )1 = x .

(d) fg x f g x( ) ( ) ≠ ( )− − −1 1 1 but fg x g f x( ) ( ) = ( )− − −1 1 1 .

11. The inverse function f −1 of f exists only if f is one-to-one function.

x

Domainof f

Rangeof f

f

y

−1f


IGCSE O Level Additional Maths Functions 11

KEY NOTES 12. Geometrical representation of Inverse Functions.

On the Cartesian plan, the graph of y f x= ( )−1 is the reflection of y f x= ( ) along the line y x= .

Some examples of function which has inverse function:

(a) f x ex( ) = , f x x− ( ) =1 ln

(b) g x x( ) = − ( )ln , g x e x− −( ) =1

(c) h x x( ) = +2 1 , h xx− ( ) =

−1 12

(0, 2)

(−1, 0)

(0, −1)

0 (2, 0)

y

x

y = f(x)

y = x

y = f−1(x)



KEY N

OTES

Example 1:

Functions f and g are defined on the set of real numbers by f xx

: →−3

2, x ≠ 2 , and

g x xx

: → −27

, x > 0

Write in similar form (a) f−1 and (b) g−1. Hence determine the domain of f−1 and g−1.

13. The modulus or absolute of a function f is defined as

f xf x

f x

f x

f x( ) =

( )− ( )

( ) ≥( ) <

,.

for , for

00

14. Graphs of Modulus function

To draw the graph of y f x= ( ) , first determine x-intercept of the graph of y f x= ( ) . The portion of graph above x-axis remains unchanged, and reflects the portion wheref x( ) < 0 about x-axis.

ϊŦ Solution:

(a) Let y f x= ( )−1 , then f y x( )= .

f yy

( ) =−3

2

3

2 −=

yx

23

− =yx

yx

= −23

∴ f xx

− → −1 23

:

(b) Let y g x= ( )−1 , then g y yy

( ) = −27

27

yy

x− =

2 7 02y xy− − =

yx x

=± − ( ) −( )2 4 2 7

4

Since y > 0 , yx x

=+ +2 56

4. ∴ g x

x x− →+ +1

2 564

:

Domain of f −1 are all real values except x = 0.

Domain of g−1 are all real values.


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