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# A-level Further Mathematics for Year 13 - Course 1: Differential Equations, Further Integration, Curve Sketching, Complex Numbers, the Vector Product and Further Matrices

### 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, the 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 to

• Analytical and numerical methods for solving first-order differential equations
• The nth roots of unity, the nth roots of any complex number, geometrical applications of complex numbers.
• Coordinate systems and curve sketching.
• Improper integrals, integration using partial fractions and reduction formulae
• The area enclosed by a curve defined by parametric equations or polar equations, arc length and the surface area of revolution.
• Solving second-order differential equations
• The vector product and its applications
• Eigenvalues, eigenvectors, diagonalization and the Cayley-Hamilton Theorem.

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: First Order Differential Equations

• Solving first order differential equations by inspection
• Solving first order differential equations using an integrating factor
• Finding general and particular solutions of first-order differential equations
• Eulerâ€™s method for finding the numerical solution of a differential equation
• Improved Euler methods for solving differential equations.

Module 2: Further Complex Numbers

• The nth roots of unity and their geometrical representation
• The nth roots of a complex number and their geometrical representation
• Solving geometrical problems using complex numbers.

Module 3: Properties of Curves

• Cartesian and parametric equations for the parabola and rectangular hyperbola, ellipse and hyperbola.
• Graphs of rational functions
• Graphs of , , for given
• The focus-directrix properties of the parabola, ellipse and hyperbola, including the eccentricity.

Module 4: Further Integration Methods

• Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.
• Integrate using partial fractions including those with quadratic factors in the denominator
• Selecting the correct substitution to integrate by substitution.
• Deriving and using reduction formula

Module 5: Further Applications of Integration

• Finding areas enclosed by curves that are defined parametrically
• Finding the area enclosed by a polar curve
• Using integration methods to calculate the arc length
• Using integration methods to calculate the surface area of revolution

Module 6: Second Order Differential Equations

• Solving differential equations of form yâ€³ + ayâ€² + by = 0 where a and b are constants by using the auxiliary equation.
• Solving differential equations of form y â€³+ a y â€²+ b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to the complementary function

Module 7: The Vector (cross) Product

• The definition and properties of the vector product
• Using the vector product to calculate areas of triangles.
• The vector triple product.
• Using the vector triple product to calculate the volume of a tetrahedron and the volume of a parallelepiped
• The vector product form of the vector equation of a straight line
• Solving geometrical problems using the vector product

Module 8: Matrices - Eigenvalues and Eigenvectors

• Calculating eigenvalues and eigenvectors of 2 Ã— 2 and 3 Ã— 3 matrices.
• Reducing matrices to diagonal form.
• Using the Cayley-Hamilton Theorem

### Taught by

Philip Ramsden and Phil Chaffe

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