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# 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

### Overview

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

• Fluency â€“ selecting and applying correct methods to answer with speed and efficiency
• Confidence â€“ critically assessing mathematical methods and investigating ways to apply them
• Problem solving â€“ analysing the â€˜unfamiliarâ€™ and identifying which skills and techniques you require to answer questions
• Constructing mathematical argument â€“ using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others
• Deep reasoning â€“ analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced:

• The determinant and inverse of a 3 x 3 matrix
• Mathematical induction
• Differentiation and integration methods and some of their applications
• Maclaurin series
• DeMoivreâ€™s Theorem for complex numbers and their applications
• Polar coordinates and sketching polar curves
• Hyperbolic functions

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A-level further mathematics course. Youâ€™ll also, be encouraged to consider how what you know fits into the wider mathematical world.

### Syllabus

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

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

• De Moivre's theorem and it's proof
• Using de Moivreâ€™s Theorem to establish trigonometrical results
• De Moivreâ€™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

### Taught by

Philip Ramsden and Phil Chaffe

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