**Course Title **Advanced GCE in Further Mathematics

Further Mathematics is a facilitating subject enabling future study in a wide range of areas. It assumes all knowledge of Advanced GCE Mathematics and can only be taken in conjunction with this. Studying Further Mathematics is extremely challenging but it consolidates and reinforces standard Advanced Level Mathematics work, helping students to achieve their best possible grade in this. Students planning to take a highly mathematical degree such as Engineering, Sciences, Computing, Finance, Economics and Mathematics itself, will find they benefit enormously from taking Further Mathematics as it introduces new topics such as matrices and complex numbers that are vital in many STEM degrees. Students who have studied Further Mathematics often find the transition to such degrees far more straightforward. If students are passionate about maths then having a Further Mathematics qualification can bring benefit to students in any field as it clearly identifies them as having excellent analytical skills.

**Course Content**

**Further Pure Mathematics **

Further Mathematics broadens students’ knowledge through the introduction of two totally new but extremely important mathematical ideas that students will not have previously encountered:

- Complex Numbers: including their use in solving any quadratic, using the four operations: +, –, ., , knowing that polynomial equations having roots in conjugate pairs, understanding and drawing Argand diagrams, using the modulus – argument form and conversion, . and in modulus – argument form including the use of radians and the compound angle formulae and the ability to draw and interpret simple loci
- Matrices: including the operations: +, –, ., linear transformations in 2D and some in 3D, successive transformations in 2D, invariant points and lines, determinants and inverses of 2.2 matrices, considering the roots and coefficients of polynomial equations up to quartics

A Level Further Mathematics extends both this new content and the content found in A Level Mathematics to include:

- Proof by induction: including the use on series, divisibility tests and matrices
- More complex numbers: including the use of De Moivre’s Theorem, eiθ, complex roots and geometrical problems
- More matrices: including determinants and inverses of 3.3 matrices and solving 3 linear simultaneous equations with geometrical interpretations
- Standard formulae for sums of series
- Method of differences: including partial fractions
- Maclaurin series: including awareness of validity
- Vectors: including the equation of a line in 3D and a plane in vector and Cartesian forms, the scalar product and its applications, the intersection of lines and planes and the perpendicular distance between 2 lines, point to line and point to plane
- Calculus: including integration where the integrand extends to infinity, volumes of revolution, the mean value of a function, integration using partial fractions with quadratic factors, integration of inverse trigonometric functions, integration using trigonometric substitutions, the use of polar coordinates including considering the area enclosed by a polar curve
- Hyperbolic functions: including the definitions and graphs of sinhx, coshx and tanhx, the differentiation and Integration of hyperbolics, the use of inverse hyperbolic functions including logarithm forms, integration with hyperbolic substitutions
- Differential equations: including using an integrating factor, general and particular solutions, modelling, second order equations considering both auxiliary equation for homogeneous equations and using complementary function and particular integrals for non-homogeneous equations, considering the simple harmonic motion equation, the ability to model damped oscillations and the ability to model situations with 1 independent and 2 dependent variables as a pair of coupled 1st order simultaneous equations.

Synoptic assessment in Further Mathematics addresses students’ understanding of the connections between different elements of the subject. It involves the explicit drawing together of knowledge, understanding and skills learned in different parts of the Advanced GCE course through using and applying methods developed at earlier stages of study in solving problems. Making and understanding connections in this way is intrinsic to learning mathematics. It also looks at extending their knowledge of mathematics as well as deepening the understanding of ideas already touched on.

**Formal Assessment**

**All units are assessed at AS by 3 examinations in May 2020**

**The full GCE is assessed by 4 examinations in May and June 2021**

There is no requirement for assessed coursework for any of the units in Further Mathematics.

**Future Progression**

Further Mathematics qualifications are highly regarded and are warmly welcomed by universities. Students who take Further Mathematics are really demonstrating a strong commitment to their studies, as well as learning mathematics that is very useful for any mathematically rich degree. Some of the most prestigious universities require you to have a Further Mathematics qualification to study a mathematical degree whilst others often adjust their grade requirements more favourably to students with Further Mathematics. A Level Further Mathematics is a particularly good choice for students considering higher education in any Science or Maths-based course, ranging from: computer science, biochemical sciences, natural sciences, engineering, medical sciences and psychology to statistics, economics, accountancy, management and actuarial science.

**To succeed in Further Mathematics**

This is one of the most demanding A Level courses and although a candidate might have found progression through GCSE uncomplicated, A Level will require intensive study and reiteration of skills consistently throughout the course. Algebraic fluency is essential to succeed and should be coupled with the ability to process very abstract concepts and problems. It is recognised as one of the more intensive study programmes and requires a lot of self-resilience. A flair and passion for Maths is essential, as is being able to be independent.