In this course, students will develop the mathematical understanding and skills to solve problems relating to:

- Proof by mathematical Induction
- Vectors and Matrices
- Complex Numbers 2

Proof by mathematical induction continues the developmental concept of proof from QCE Specialist Mathematics – Combinatorics, Vectors and Proof and QCE Specialist Mathematics – Complex Numbers, Trigonometry, Functions and Matrices. QCE Specialist Mathematics – Combinatorics, Vectors and Proof introduced a study of vectors with a focus on vectors in two-dimensional space. QCE Specialist Mathematics – Complex Numbers, Trigonometry, Functions and Matrices introduced complex numbers; this course extended the study of complex numbers to include complex arithmetic using the polar form.

In this course, students explore applications of matrices, study three-dimensional vectors, and are introduced to vector equations and vector calculus, with the latter extending students’ knowledge of calculus from Mathematical Methods. Cartesian and vector equations, together with equations of planes, enable students to solve geometric problems and problems involving motion in three-dimensional space.

These topics build on prior knowledge to enable a greater depth of analytical thinking and metacognition.

*Source – QCAA General Senior Syllabus 2019*

#### Complex Factors of Polynomials

- Topic - Factorise two-term quadratics over complex number field 1
- Topic - Factorise two-term quadratics over complex number field 2
- Topic - Factorise three-term quadratics over complex number field 1
- Topic - Factorise three-term quadratics over complex number field 2
- Topic - Factorise three-term quadratics over complex number field 3
- Topic - Factorise two-term quartics over complex number field
- Topic - Factorise three-term quartics over complex number field
- Topic - Find all factors of cubic polynomials over complex number field 1
- Topic - Find all factors of cubic polynomials over complex number field 2
- Topic - Find all factors of cubic polynomials over complex number field 3
- Topic - Find all factors of quartic polynomials over complex number field
- Topic - Complex roots and conjugate root theorem 1
- Topic - Complex roots and conjugate root theorem 2
- Topic - Complete Polynomials with Imaginary Coefficients using Factors 1
- Topic - Complete Polynomials with Imaginary Coefficients using Factors 2

#### Equations over Complex Number Field

- Video - Complex Roots of Quadratic Equations (4:06)
- Video - Complex Roots of Cubic Equations (2:35)
- Video - Complex Roots of Quadratic Equations using Quadratic Formula (4:19)
- Video - Quadratics in Complex Number Field using Quadratic Formula (2:41)
- Video - Quadratics in Complex Number Field using Completing Square (7:54)

#### Modulus-Argument (Polar) Form

- Video - Understanding Modulus-Argument Form (8:13)
- Topic - Horizontal Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Vertical Complex Numbers to Modulus-Argument (Polar) Form
- Topic - First Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Second Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Third Quadrant Complex Numbers to Modulus-Argument (Polar) Form
- Topic - Fourth Quadrant Complex Numbers to Modulus-Argument (Polar) Form

#### Conversion between Cartesian and Polar Forms

- Video - Complex Numbers in 1st Quarter to Mod-Arg Form (4:26)
- Video - Complex Numbers in 2nd Quarter to Mod-Arg Form (2:33)
- Video - Complex Numbers in 4th Quarter to Mod-Arg Form (4:29)
- Video - Complex Numbers in Purely Real to Mod-Arg Form (2:23)
- Video - Complex Numbers in Purely Imaginary to Mod-Arg Form (1:47)
- Video - Modulus-Argument Form to Cartesian Form (5:38)
- Video - Mod-Arg Form to Cartesian Form in Pure Real or Imaginary (4:23)
- Video - Simplifying Complex Numbers using Mod-Arg Form (3:45)
- Video - Exact Values of Sine & Cosine using Mod-Arg Form (4:57)

#### De Moivre's Theorem

- Video - Understanding De Moivre's Theorem (2:44)
- Video - Proof of De Moivre's Theorem Multiplication (3:42)
- Video - Proof of De Moivre's Theorem Division (2:20)
- Video - Proof of De Moivre's Theorem Subtraction (4:04)
- Video - Simplifying Complex Numbers using De Moivre's Theorem (7:42)
- Video - Purely Imaginary using De Moivre's Theorem (5:11)
- Video - Purely Real using De Moivre's Theorem (5:24)
- Video - Trigonometric Expression using De Moivre's Theorem (5:27)
- Video - De Moivre's Theorem involving Conjugate Pairs (6:58)
- Video - Trigonometric Properties & De Moivre's Theorem (6:04)

#### Roots of Unity in Complex Numbers

- Video - Nth Roots of Unity using Complex Numbers (8:04)
- Video - Complex Roots of Unity (6:03)
- Video - 4th Roots of Unity using Complex Numbers (3:09)
- Video - 5th Roots of Unity using Complex Numbers (1:57)
- Video - 6th Roots of Unity using Complex Numbers (2:30)
- Video - Cube Roots of Negative Unity using Complex Numbers (2:09)
- Video - Seventh Root of Unity (6:46)

#### Roots of Complex Numbers

- Video - 4th Roots of 4 using Complex Numbers (2:32)
- Video - 5th Roots of -32 using Complex Numbers (2:45)
- Video - Cube Roots of i using Complex Numbers (2:19)
- Video - 4th Roots of -16i using Complex Numbers (1:42)
- Video - 5th Roots of 1+i using Complex Numbers (2:07)
- Video - Cube Roots of Irrational Real Part & i using Complex Numbers (1:30)
- Video - Exact Values of Trigonometric Ratios involving Fifth Roots (6:37)

#### Basic Mathematical Induction

- Video - Introduction to Proof by Mathematical Induction (4:59)
- Video - Proof by Mathematical Induction 1 (9:09)
- Video - Proof by Mathematical Induction 2 (5:07)
- Video - Proof by Mathematical Induction 3 (5:50)
- Video - Proof by Mathematical Induction 4 (5:06)
- Video - Proof by Mathematical Induction 5 (4:05)
- Video - Proof by Mathematical Induction 6 (5:47)
- Topic - Basic Structure of Mathematical Induction
- Topic - Sum of Even Numbers by mathematical Induction: Assumption
- Topic - Sum of Even Numbers by Mathematical Induction: Proof
- Topic - Summation Proof by Mathematical Induction
- Topic - Production Formula by Mathematical Induction
- Topic - Sum of Fractions by Mathematical Induction
- Topic - Production of Fractions Proof by Mathematical Induction

#### Fundamental of Divisibility Proofs

- Video - Mathematical Induction Divisibility: Basic Divisibility 1 (5:10)
- Video - Mathematical Induction Divisibility: Basic Divisibility 2 (4:08)
- Video - Mathematical Induction Divisibility: Basic Divisibility 3 (8:39)
- Video - Mathematical Induction Divisibility: Multiple Indices (4:35)
- Topic - Basic Divisibility Proof by Mathematical Induction
- Topic - Basic Divisibility Proof by Mathematical Induction (Additional)
- Topic - Divisibility Proof of Non-Linear Indices
- Topic - Divisibility Proof of Non-Linear Indices (Additional)
- Topic - Divisibility Proof by Mathematical Induction using Substitutions
- Topic - Divisibility Proof by Mathematical Induction by Finding Initial Values

#### Divisibility Proofs of Multiples Indices

- Video - Mathematical Induction Divisibility: Two Indices 1 (4:43)
- Video - Mathematical Induction Divisibility: Two Indices 2 (6:27)
- Topic - Divisibility Proof with Two Indices
- Topic - Divisibility Proof with Two Indices (Additional)
- Topic - Divisibility Proof for Two Indices by Mathematical Induction
- Topic - Divisibility Proof for Three Consecutive Indices by Mathematical Induction

#### Lines in 2D

- Topic - Vector Equations with Direction Vectors & Fixed Points
- Topic - Vector Equations in Direction Vectors & Fixed Points by Unit Vectors
- Topic - Vector Equations with Direction Vectors and Intercepts
- Topic - Vector Equations Passing Two Points
- Topic - Parametric Equations of Vectors
- Topic - Parametric Equations of Vectors Passing Through two Points
- Topic - Cartesian Equations using Direction Vectors and Points

#### Applications of a Line in a Plane

- Topic - Initial Positions from Vector Equations
- Topic - Finding Speed of a Particle using its Vector Equation
- Topic - New Vector Equations for New Velocities
- Topic - Position of Vectors given Certain Times
- Topic - 2-Dimensional Velocity Vectors given New Speeds
- Topic - Time for Vector Equations
- Topic - Speed of Particles described as 2-Dimensional Vectors
- Topic - 3-Dimensional Velocity Vectors given New Speeds
- Topic - Shortest Distance using Vectors

#### Relationship between Lines

- Topic - Identifying Parallel or Not parallel for 2-Dimensional Lines
- Topic - Identifying Parallel or Not parallel for 3-Dimensional Lines
- Topic - Angles between Two Lines in 3-Dimensional Space
- Topic - Coordinates of Intersection of Lines in 3-Dimensional Space
- Topic - identify Positioning Two Lines in 3-Dimensional Space