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Mathematics Spec. Year 11 ATAR Course Textbook

Mathematics Spec. Year 11 ATAR Course Textbook
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Written by O. T. Lee

This book addresses the syllabus requirements of Units 1 and 2 of the Mathematics Specialist course of Western Australia.

The use of CAS/graphic calculators is seamlessly integrated into the teaching and learning process.  Questions have become more explicit in terms of the required methods and techniques.  Knowledge of CAS/graphic calculator techniques empower students to appreciate the relative efficiencies (and accuracies) of machine based techniques against traditional pencil and paper techniques.  However, the traditional pencil and paper techniques are the ones that convey the actual mathematical concepts and processes and form the backbone of  this book.  Machine based techniques are at best interpretative techniques.

The use of Hands-on-Tasks is continued in this book.  These tasks allow students to conceptualise mathematical concepts on their own without being explicitly “taught”.  This promotes relational understanding rather than factual knowledge of mathematical concepts and ideas.

A fully worked out Solution Manual is available as a resource for teachers.

Dr O.T. Lee

Mathematics Department

North Lake Senior Campus

Contents

  1      Combinatorics I

       1.1       The Factorial Notation

       1.2       The Multiplication Rule

       1.3       Arrangements/Permutations

   1.4       The Grouping Technique and the Complement Rule

   1.4.1   The Grouping Technique

      1.4.2   The Complement Rule

  2    Combinatorics II

    2.1       The Inclusion-Exclusion Principle for Two Sets/Events

   2.2       The Inclusion-Exclusion Principle for Three Sets/Events4

       2.3       The Pigeon Hole Principle (Dirichlet’s principle)    

  3    Combinatorics III

       3.1       The Combinatorial Notation

       3.2       Combinations or Selections

       3.3       Combinations and the inclusion-exclusion principle

       3.4       Combinations and Arrangements

    3.5       Pascal’s Triangle

  4    Introduction to Vectors

    4.1    Arrow Representation of Vectors

       4.1.1    Scalar multiple of a vector    

       4.1.2    Addition of Vectors (arrow representation)

       4.1.3    Subtraction of Vectors (arrow representation)

       4.1.4    Vectors and Triangle Trigonometry

   4.2    Component Representation of Vectors

   4.2.1    Unit vectors and Basis vectors

       4.2.2    Magnitude and direction of a vector

   4.2.3    Scalar Multiplication and Parallel Vectors

      4.2.4    Unit Vectors

   4.2.5    Alternative notation for vector in component form

      4.2.6    Addition and Subtraction of Vectors

   4.2.7    Working with Vectors in Component Form

  5      Position Vectors

5.1    Free Vectors and Position Vectors

5.2    Internal and External Division of a Line Segment

  6      Relative Vectors

      6.1       Relative Position/Displacement Vectors

6.2    Relative Velocity

  7    Vector Applications I

      7.1       Vectors using Arrow Representation

 7.2    Vectors in Component Form

 7.3    Position Vectors

  8      Vector Applications II

       8.1    Relative Displacement and Velocity

8.2    Interception & Collision Problems using Relative Vectors

 9        Scalar Product

       9.1       Scalar Product (Dot Product)

       9.1.1    Properties of the Scalar Product

    9.2    Geometric Interpretation of the Scalar Product

       9.3    Vector Projection

10      Vector Applications III

10.1    Closest Distance

10.2    Vector Components/Projections

11    Geometric Proofs & Circle Properties

11.1     Geometric Proofs

11.2       Properties of angles, parallel lines and triangles

11.3       Similar Triangles

11.4       Congruent Triangles

11.5       Circle Properties

12    Geometric Proofs using Vectors

12.1    Geometric Proofs involving Vectors I

12.2    Geometric Proofs involving Vectors II

13    Trigonometric Equations I

13.1    Review of Basic Concepts

13.2       Solving Trigonometric Equations

       13.2.1   General Solutions

13.3       Reciprocal Trigonometric Functions

 

14         Trigonometric Graphs130

14.1    Graphs of basic trigonometric functions

14.2    Graphs of Trigonometric functions of the form y = a f (bx + c) + d  

14.3       Graphs of reciprocal trigonometric functions

15       Trigonometric Identities

15.1    Pythagorean Identities

15.2       Proving Trigonometric Identities

15.3    Compound Angle Formulae

15.4    The Double and Half-Angle Formulae

15.5    Product to Sum and Sum to Product Formulae

15.6    Linear combinations of sine and cosine (Auxiliary Angles)

16    Matrix Algebra

16.1    Definitions

16.1.1    Equality of Matrices

16.2    Operations on Matrices

16.2.1    Matrix Addition and Subtraction

16.2.2    Scalar Multiplication

16.2.3    Basic Rules for Manipulation of Matrices

16.2.4    Matrix Multiplication

16.2.5    More Rules for Manipulating Matrices

16.2.6    Properties of Special Matrices

16.3    The Multiplicative Inverse of a Matrix

16.3.1    Determinant of a 2 ? 2 Matrix

16.3.2    Formula for the inverse of a 2 ? 2 matrix

16.4    Manipulating Matrix Equations

17    Systems of Linear Equations

17.1    Systems of Two Linear Equations

17.2       The Matrix Inversion Method

18    Applications using Matrices

19    Transformation Matrices

19.1       Linear Transformations in the x-y plane

19.2       Using Matrices to Represent Linear Transformations

19.2.1   Procedure for determining the matrix representation

             of a linear transformation

       19.2.2   Linear Transformations considered

19.3       Scale Factor for Area

19.4       Combining Transformations

20    Introduction to Complex Numbers

20.1       Introduction to Complex Numbers

20.2       Basic Properties of Complex Numbers (Cartesian Form)

20.3       Complex Numbers as Ordered Pairs

20.4       Quadratic Equations with Complex Roots

21    Methods of Proofs

21.1    Conjectures

21.2    Deductive Proofs

21.3       Counter Examples

21.4       Proof by Contradiction

21.5    Mathematical Induction

Answers

Index

©  O. T.  Lee


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