A-Level Further Mathematics for Year 12 - Course 2: 3 x 3 Matrices, Mathematical Induction, Calculus Methods and Applications, Maclaurin Series, Complex Numbers and Polar Coordinates

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