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Discovering Additional Mathematics is a textbooks designed for students in secondary schools. It follows the secondary Addtional Mathematics syllabus issued by the Ministry of Education, Singapore, which was implemented from 2007 onwards. This book covers the entire syllabus for the Singapore-Cambridge GCE O-level Additional Mathematics.

The emphasis of this books is on empowering students to learn mathematics both independently and effectively. Depending on the topic covered, different approaches have been used for the presentation of concepts to facilitate easy understanding by students. In some topics, an intuitive or an experimental approach is adopted to enable students to internalize concepts and instill in them an interest to explore the topics further.

The book includes clear and illustrative examples, class activities and diagrams to help students understand the concepts and apply them. IT skills and NE (National Education) values have also been incorporated where appropriate.

Key Features:

Contents

Chapter 1        Quadratic Equations
1.1                    Discriminant And The Nature Of Roots
1.2                    Discriminants And Quadratic Graphs
                          A.  Quadratic Graphs and the Nature of Roots
                          B.  Conditions for ax² + bx + c to be Always Positive (or Negative)
1.3                          Relationships Between The Roots And The Coefficients
                          A.  Forming a Quadratic Equation with Given Roots
                          B.  Sum and Product of the Roots

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Chapter 2        Indices And Surds
2.1                   Operations On Indices And Surds
                         A.  Further Examples on Indices
                         B.  Surds and Fractional Indices
                         C.  Addition and Subtraction of Surds
                         D.  Multiplication of Surds
                         E.  Division of Surds and Rationalising the Denominator

2.2                   Equations Involving Indices And Surds
                        A.  Equations Involving Indices
                        B.  Equations Involving Surds

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Chapter 3        Polynomials And Partial Fractions
3.1                   Multiplication Of Polynomials
                        A.  Definition and Notation of a Polynomial
                        B.  Multiplication of Polynomials
                        C.  Equality of Polynomials
3.2                   Division Of Polynomials
3.3                   Remainder Theorem
3.4                   Factor Theorem
                        A.  Finding Factors of a Polynomial
                        B.  Solving Cubic Equations
3.5                   Partial Fractions
                        A.  Proper Fractions with the Denominator Q(x) = (ax + b)(cx + d)
                        B.  Proper Fractions with the Denominator Q(x) = (ax + b)(cx + d)²
                        C.  Proper Fractions with the Denominator Q(x) = (ax + b)(x² + c²)
                        D.  Improper Fractions

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Chapter 4       Quadratic Inequalities And Modulus Functions
4.1                   Quadratic Inequalities
                        A.  Algebraic Method
                        B.  Graphical Method
4.2                   Modulus Functions
                        A.  Definitions
                        B.  Equations Involving Absolute Values
                        C.  Graphs of y =|f (x)|

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Chapter 5        Binomial Expansions
5.1                   Binomial Expansions
5.2                   The Binomial Theorem
                        A.  n! and   Notations
                        B.  General Term

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Chapter 6        Trigonometric Functions, Identities And Equations
6.1                   Definitions Of Trigonometric Functions
                        A.  Definitions
                        B.  Signs of Trigonometric Functions
                        C.  Special Angles
6.2                   Trigonometric Functions Of 180° ± θ, 360° ± θ, - θ and 90° - θ
                        A.  Trigonometric Functions for a General Angle
                        B.  Angles Greater than 360°
                        C.  Negative Angles
                        D.  Complementary Angles
6.3                   Graphs of Sine, Cosine, and Tangent Functions
                        A.  Graphs of y = sin x, y = cos x and y = tan x
                        B.  Graphs of y = a sin (bx) + c, y = a sin (x/b) + c, y = a cos (bx) + c, y = a cos (x/b)  + c, and y = a tan (bx)
6.4                   Simple Trigonometric Identities
                        A.  Fundamental Identities
                        B.  Proving of Identities
6.5                   Simple Trigonometric Equations
                        A.  Principal Values of sin^-1 x, cos^-1 x and tan^-1 x
                        B.  Basic Trigonometric Equations
                        C.  Further Trigonometric Equations

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Chapter 7        Further Trigonometric Identities
7.1                   Addition Formulae
                        A.  The Formulae for sin (A + B) and sin (A – B)
                        B.  The Formulae for cos (A + B) and cos (A – B)
                        C.  The Formulae for tan (A + B) and tan (A – B)
7.2                   Double Angle Formulae
                        A.  The Formulae for sin 2A, cos 2A and tan 2A
                        B.  The Formulae for sin²A and cos²A
7.3                   Factor Formulae
7.4                   The Expression a cos θ + b sin θ

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Chapter 8        Coordinate Geometry Of Lines And Rectilinear Figures
8.1                   Parallel Lines And Perpendicular Lines
                        A.  Parallel Lines
                        B.  Perpendicular Lines
8.2                   Midpoint Of Line Segment
8.3                   Areas Of Rectilinear Figures
                        A.  Area of Triangle
                        B.  Area of Polygon

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Chapter 9        Simultaneous Equations In Two Unknowns
9.1                   The Identity Matrix And The Inverse Matrix
9.2                   Solving Simultaneous Linear Equations By The Inverse Matrix Method
                        A.  Equations in Matrix Form
                        B.  Conditions for the Number of Solutions
9.3                   Simultaneous Equations: One Linear And One Non-linear
                        A.  Substitution Method
                        B.  Number of Points of Intersection between a Quadratic Curve and a Straight Line

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Chapter 10      Coordinate Geometry Of Curves And Circles
10.1                 Graphs of y = axⁿ And y² = kx
                        A.  Graphs of y = axⁿ
                        B.  Graphs of y² = kx
10.2                 Equations Of Circles
                        A.  Standard Form of a Circle
                        B.  General Form of a Circle
10.3                 Intersection Of A Circle And A Straight Line

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Chapter 11      Proofs In Plane Geometry
11.1                 Basic Proofs Involving Triangles, Quadrilaterals And Circles
                        A.  Building Blocks of Geometry
                        B.  Problems Involving Triangles and Quadrilaterals
                        C.  Problems Involving Circles
11.2                 Midpoint Theorem And Intercept Theorem For Triangles
                        A.  Midpoint Theorem
                        B.  The Intercept Theorem for Triangles
11.3                 Tangents, Chords And Secants Of A Circle
                        A.  Tangent-chord Theorem (Alternate Segment Theorem)
                        B.  Intersecting Chords Theorem
                        C.  Tangent-secant Theorem

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Chapter 12      Derivatives Of Functions And Rules Of Differentiation
12.1                 The Idea Of Derivative
12.2                 Derivatives Of Powers Of x
                        A.  Derivative of xⁿ
                        B.  Constant Multiple Rule
                        C.  Sum Rule and Difference Rule
12.3                 Product Rule And Quotient Rule
                        A.  The Product Rule
                        B.  The Quotient Rule
12.4                 Chain Rule
12.5                 Derivatives Of Trigonometric Functions
12.6                 Second Derivatives

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Chapter 13      Tangents, Normals, And Rates Of Change
13.1                 Tangents And Normals
                        A.  Tangents
                        B.  Normals
13.2                 Rates Of Change
13.3                 Connected Rates Of Change

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Chapter 14      Maxima And Minima Problems
14.1                 Increasing And Decreasing Functions
14.2                 Stationary Points
                        A.  Types of Stationary Points
                        B.  First Derivative Test
                        C.  Second Derivative Test for Maxima and Minima
14.3                 Maxima And Minima Problems

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Chapter 15      Exponential And Logarithmic Functions
15.1                 Exponential Functions
                        A.  Definition of an Exponential Function
                        B.  Graphs of Exponential Functions
                        C.  The Natural Exponential Function
15.2                 Logarithmic Functions
                        A.  Definition of Logarithm
                        B.  Laws of Logarithms
                        C.  Logarithmic Functions and Their Graphs
15.3                 Equations Involving Exponential And Logarithmic Functions
                        A.  Equations Involving Exponential Functions
                        B.  Equations Involving Logarithmic Functions
                        C.  Applications
15.4                 Transformation Of Relationships To Linear Form
                        A.  Idea of Transformation to Linear Form
                        B.  Determination of Relationships
15.5                 Derivatives Of e^x And ln x
                        A.  Derivative of ex
                        B.  Derivative of ln x

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Chapter 16      Indefinite Integrals
16.1                 Idea of Indefinite Integral
16.2                 Rule Of Integration, Integration Of xⁿ And Integration Of (ax + b)ⁿ
                        A.  Integration of xⁿ
                        B.  Rules of Integration
                        C.  Integration of (ax + b)ⁿ
16.3                 Applications Of Indefinite Integrals
16.4                 Integration Of Trigonometric Functions
16.5                 Integration Of Exponential Functions And 1/ax + b
                        A.  Integration of e^x and e^ax+b
                        B.  Integration of 1/ ax + b

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Chapter 17      Definite Integrals
17.1                 Idea Of Definite Integral And Its Evaluation
                        A.  Idea of Definite Integral
                        B.  Evaluation of Definite Integrals
17.2                 Area Bounded By A Curve And Lines Parallel To The Axes
                        A.  Area between a Curve and the x-axis
                        B.  Area Between a Curve and the y-axis
17.3                 Areas Below The X-Axis

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Chapter 18      Kinematics Problems
18.1               Applications Of Differentiation
18.2               Applications Of Integration

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Answers

Samples