**Module 1: Matrices - The determinant and inverse of a 3 x 3 matrix**

- Moving in to three dimensions
- Conventions for matrices in 3D
- The determinant of a 3 x 3 matrix and its geometrical interpretation
- Determinant properties
- Factorising a determinant
- Transformations using 3 x 3 matrices
- The inverse of a 3 x 3 matrix

**Module 2: Mathematical induction**

- The principle behind mathematical induction and the structure of proof by induction
- Mathematical induction and series
- Proving divisibility by induction
- Proving matrix results by induction

**Module 3: Further differentiation and integration**

- The chain rule
- The product rule and the quotient rule
- Differentiation of reciprocal and inverse trigonometric functions
- Integrating trigonometric functions
- Integrating functions that lead to inverse trigonometric integrals
- Integration by inspection
- Integration using trigonometric identities

**Module 4: Applications of Integration**

- Volumes of revolution
- The mean of a function

**Module 5: An Introduction to Maclaurin series**

- Expressing functions as polynomial series from first principles
- Maclaurin series
- Adapting standard Maclaurin series

**Module 6: Complex Numbers: De Moivre's Theorem and exponential form**

- De Moivre's theorem and it's proof
- Using de Moivre’s Theorem to establish trigonometrical results
- De Moivre’s Theorem and complex exponents

**Module 7: An introduction to polar coordinates**

- Defining position using polar coordinates
- Sketching polar curves
- Cartesian to polar form and polar to Cartesian form

**Module 8: Hyperbolic functions**

- Defining hyperbolic functions
- Graphs of hyperbolic functions
- Calculations with hyperbolic functions
- Inverse hyperbolic functions

* Differentiating and integrating hyperbolic functions