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A level Further Mathematics Content

England Wales Northern Ireland

A Level (Advanced Level) is usually studied over two years. The first year of an A Level is called AS (Advanced Subsidary) , the second year is called AS2.

DOE: AS and A-LevelFurther Mathematics Content 2016 Formulae A level maths

There are 9 components, labelled A to I :

(A ) Proof (B ) Complex numbers (C )Matrices (D ) Further algebra and functions (E )Further calculus (F ) Further vectors (G ) Polar coordinates (H ) Hyperbolic functions (I ) Differential equations


Scottish flag

There is overlap between AS Further Mathematics and the Scottish Higher Mathematics course and also between AS2 Further Mathematics and the Scottish Advanced Higher Mathematics and Advanced Higher Mathematics of Mechanics courses.

AS and A Level Content Overview

(A) Proof

A Proof

Mathematical proof Direct proof Proof by contradiction Proof by contrapositive Proof by induction

(B) Complex Numbers

cB Complex numbers



Complex numbers Complex conjugate Arithmetic operations Argand diagrams Loci Polar form De moivre's theorem Roots of a complex number

(C) Matrices

C Matrices

Matrices Matrix equality Matrix addition and subtraction Matrix notation Matrix elements Matrix Scalar multiplication Transpose of a matrix Matrix multiplication

Matrices - Determinant

2 x 2 matrix 3 x 3 matrix

Matrices - Inverse

Cofactor of a 3 x 3 matrix Adjoint of a 3 x 3 matrix Inverse of a square matrix Product of a square matrix and its inverse Using Gaussian elimination to find inverse of matrix

Special Matrices

Special matrices Null matrix Unit matrix Transformation matrices Reflections Rotations Scalings

Gaussian Elimination

Vedic way of solving simultaneous equations Augmented matrix
Elementary Row Operations Gaussian elimination Gaussian elimination : Excel solver Using Gaussian elimination to find inverse of matrix

(D) Further algebra and functions

D Further algebra and functions

Binomial Theorem

FactorialsPermutations Combinations Pascal’s Triangle. Binomial Co-efficients Binomial Co-efficient equations Binomial Expansions Binomial Theorem Probability and the Binomial Theorem Binomial Expansions and e

Functions

Refresher of functions Modulus function Inverse functions Polynomials Extrema Concavity and points of inflexion Odd and Even functions Asymptotes

Sequences and Series

Recurrence relations Fixed points Arithmetic sequences Arithmetic series nth term of arithmetic sequence Sum to n terms of an arithmetic sequence Geometric sequences Geometric series
Sum to n terms of a geometric sequence Infinite series Sum to infinity - geometric series Sum first n natural numbers
Sigma notation - Rules Common series - sigma notation

Power Series

Power series D'alembert's ratio test Absolute convergence Fibonacci series Centre of convergence Taylor's series Maclaurin's Series Maclaurin expansion

Application of summation formulae

Standard series Examples Iteration Iteration - graphical method First order process Second order process Rule of false position Newton-Raphson iteration

 

(E) Further calculus

E Further calculus

Integration refresher Areas under curves Area between curve and the x-axis Area between curve and the y-axis Standard integrals ex and 1/x Infinite integrals (discontinuities) sec2x Integrating Inverse trig functions Integration of rational functions
Volumes of revolution Integration by substitution. Integration by parts

Differentiating inverse functions Differentiating inverse trig functions Inverse trig functions:Recap
Implicit & explicit functions Logarithmic functions

(F) Further vectors

F Further vectors

Basic Vectors

Scalar and vector quantities Components Magnitude Position vectors Unit vector Zero vector Equal vectors Adding vectors Subtracting vectors Perpendicular vectors

Other Vectors

Collinearity Angle between vectors Scalar product Section formula Linear motion with bearings Vectors and bearings Relative Motion Vectors and forces

Vector Product

Direction Ratios and Cosines Vector product Right Hand Screw Rule Vector Product properties Components Scalar triple product

Vector equations of lines

Intersection of two lines Intersection of two planes The distance from a point to a plane The distance from a point to a line Equations of a line

Vector equation of planes

Planes Planes in space Vector equations The angle between two planes The distance between parallel planes Coplanar vectors Vector equation of a plane

(G) Polar coordinates

G Polar coordinates


Polar coordinates Polar form of complex numbers

(H) Hyperbolic functions

H Hyperbolic functions


(I) Differential equations

I Differential equations

 

Differential equations First-order differential equations Second-order differential equations Applications of differential equations

Applications of Calculus

Rectilinear motion Distance, velocity & acceleration Extrema of functions Approximating roots Uses of parametric equations Equations of motion Simple harmonic motion



Books

A selection of resources available at Amazon

AS Further Mathematics GCP: A-Level Maths Edexcel A-Level Further Mathematics
Understanding Mechanics Engineering Mathematics Engineering MathematicsAdvanced Higher Physics Theory Advanced Higher Physics Investigations Companion Advanced Higher Physics - Questions and Solutions

Maths Mutt at Amazon

New Advanced Higher Mathematics
Revision Notes Differentiation
By First Principles Quick Forty
Integration and Differential Equations

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© Alexander Forrest