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

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 – selecting and applying correct methods to answer with speed and efficiency * Confidence – critically assessing mathematical methods and investigating ways to apply them * Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions * Constructing mathematical argument – using mathematical tools such as diagrams, graphs, the logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others * Deep reasoning – 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. You’ll also be encouraged to consider how what you know fits into the wider mathematical world. 3b:T528, How t

• How to find the general or particular solution to a first-order differential equation by inspection or by using an integrating factor.How to find a numerical solution to a differential equation using the Euler method or an improved Euler method..How to find the nth roots of unityHow to find the nth roots of a complex number in the form How to use complex roots of unity to solve geometrical problems.How to identify the features of parabolas, rectangular hyperbolae, ellipses and hyperbolae defined by Cartesian and parametric equations.How to identify features of graphs defined by rational functions.How to define a parabola, ellipse or hyperbola using focus-directrix properties and eccentricity.How to evaluate improper integrals.How to integrate using partial fractionsHow to derive and use reduction formulaeHow to find areas enclosed by curves that are defined parametrically.How to find the area enclosed by a polar curve.How to calculate arc length.How to calculate the surface area of revolution.How to find the auxiliary equation for a second order differential equation.

• An understanding of the content of the course A-Level Further Mathematics for Year 12: Course 1 and Course 2 is required