## Topic 1: Proof by mathematical Induction

Summation notation or Sigma Notation

Why we need proof by mathematical induction

Proof by Mathematical Induction: Mathematical Dominoes

Proof by induction partial sums worked examples

Proof by Induction Divisibility proofs

## Topic 2: Vectors and Matrices

Dominance Matrices from start to finish

Dominance Matrices as a mathematical model

Markov Chains & Matrices: Part 1

Markov Chains Worked examples

Matrix Multiplication and Matrix Powers in Excel

Matrix Multiplication and Addition in Excel for Dominance Matrices

Leslie Matrices for population age distribution modelling

15 Vector Questions you should know how to do by the end of year 11

An Introduction to vectors in 3 dimensions

Converting 3D Vectors between Cartesian and Polar forms

Scalar or Dot Product in 3 Dimensions

Vector Functions Introduction and converting vector functions to cartesian functions

Converting Vector Equations to Cartesian Equations using trigonometric Identities

Vector equations and finding collision points

Vector Equations of Lines

Converting Vector equations to cartesian form and parametric equations

Parallel and perpendicular vector equations

Calculating distance from a point to a vector equation of a line

Vectors: Line of intersection & the angle between two planes

The Vector Product (Cross Product)

Vector Equation of a plane fully explained and shown graphically

Finding equations of planes: Three worked examples

Vectors: Calculating the distance from a point to a plane or a plane to a plane

Vectors Intersection points between lines and planes and the angle between a line and a plane

Vector Calculations & 3D Graphing on the Casio FX CG50AU

Intersections of 3D Lines, Skew Lines and angles between lines

Equation of spheres in cartesian and vector form

The geometric interpretation of systems of linear equations in 2 & 3 dimensions

Using Matrix Algebra to find an intersection point of 3 planes

Solving Simultaneous Equations on the Casio FX CG50AU

Gaussian elimination part 1

Guassian Elimination Part 2 Systems with lines as solutions

Gaussian Elimination on the Casio FX CG50AU

Gaussian Elimination part 3: systems resulting in No Solution

Gaussian Elimination part 4 Finding the equation of a plane resulting in an intersection

Vector Calculus Displacement, velocity Acceleration

Vector Calculus Anti differentiation worked example

Vector Calculus Differentiation worked examples

Vector Calculus Motion in a straight line

Vector Calculus Projectile Motion Calculating Launch Angle

Uniform Circular Motion Introduction

Uniform Circular Motion worked example

## Topic 3: Complex Numbers 2

Complex Numbers: A Very Fast Recap of i, Operations, Conjugates, Argand Diagrams & Modulus

Complex Numbers: Polar Form: Fast Recap including Operations on Polar Form.

Factorising & Solving quadratics over C

Factor & Remainder Theorems: Proof and Application

Conjugate root theorem: What it is and Proof

Factorising Polynomials over C worked examples

Calculating Roots of a complex number using De Moivre's Theorem

Roots of Unity

Sketching Subsets of the complex Plane