# Edexcel Further Core Maths A-Level

It is advisable to check the official Edexcel Further Core Maths A-Level specification in case of any changes.

## Contents

## Complex Numbers

## Complex Numbers

- Real and imaginary numbers
- Addition, subtraction and multiplying complex numbers and simplifying powers of i
- Complex conjugates
- Division of a complex number by a complex number
- Argand diagrams
- Modulus and argument of a complex number
- Solving problems with complex numbers
- Square roots of a complex number
- Solving quadratic equations with complex roots
- Solving cubic equations
- Solving quartic equations
- Exam Questions - Complex numbers
- Geometrical effects of conjugating a complex number
- Exam Questions - Finding roots
- Modulus-argument form of a complex number
- Exponential Form (Euler's relation)
- Multiplication and division rules for mod and argument of two complex numbers
- Exam Questions - Further complex numbers

## De Moivre’s Theorem

## Loci in the Complex Plane

## Matrix Algebra

## Matrices

- Introduction and dimension of a matrix
- Addition and subtraction and multiplying a matrix by a scalar
- Matrix multiplication
- Identity and Inverse of a 2x2 matrix
- Exam Questions - Identity and inverse of a 2x2 matrix
- Transposed and symmetric matrices
- Finding the determinant of a 3x3 matrix
- Finding the inverse of a 3x3 matrix where it exists

## Simultaneous Equations by Matrix Methods

## Matrix Linear Transformations

- Linear transformations - rotations
- Linear transformations - reflections
- Linear transformations - enlargement
- How well do you know your transformations?
- Combinations of transformations
- Inverse matrices to reverse linear transformations
- Determinant as the area scale factor of a transformation
- Exam Questions - Matrix transformations

## Roots of Polynomial Equations

## Series

## Standard Summations

## Method of Differences

## Maclaurin’s Series

## Proof by Mathematical Induction

## Sum of Series

## Various Types of Proofs

## Further Calculus

## Applications of Integration – Volumes of revolution

- Volume of revolution about the x-axis
- Exam Questions - Volume of revolution about the x-axis
- Volume of revolution about the x-axis generated between curves
- Volume of Revolution about the y-axis
- Exam Questions - Volume of Revolution about the y-axis
- Volume of Revolution about the y-axis generated between curves
- Volume of revolution for a curve given in parametric form
- Exam Questions - Volume of revolution: parametric form

## Improper Integrals

## Mean Value of a Function

## Integrals involving Partial fractions

## Differentiating Inverse Trigonometric Functions

## Standard Integrals Involving Inverse Trigonometric Functions

## Vectors

## Scalar Product (Dot Product)

## Vector Equations of Lines

## Exam Questions – Vectors

## Planes

## Polar Coordinates and Curves

## Polar Coordinates

## Equations of Curves

## Area Bounded by a Polar Curve

## Tangents

## Hyperbolic Functions

## Hyperbolic Functions

- Definitions
- Graphs of sinh(x), cosh(x) and tanh(x)
- Graphs of sech(x), cosech(x) and coth(x)
- Solving equations using inverse and exponential functions
- Hyperbolic identities
- Osborn's rule
- Inverse hyperbolic functions and their graphs
- Expressing inverse hyperbolic functions as natural logarithms
- Solving hyperbolic equations using hyperbolic identities

## Differentiation of Hyperbolic Functions

## Standard Integrals Involving Hyperbolic Functions

## First Order Linear Differential Equations

## Exact Equations (Integrating Factors)

## Second Order Linear Differential Equations

## Equations of the form a d²y/dx² + b dy/dx + cy = 0

## Equations of the form a d²y/dx² + b dy/dx + cy = f(x)

- General solutions where f(x) = k (constant types)
- General solutions where f(x) = kx (linear types)
- General solutions where f(x) = kx
^{2}(quadratic types) - General solutions where f(x) = ke
^{px}(exponential types) - General solutions where f(x) = λ cosωx + µ sinωx (trig types)
- Special types of particular integrals
- Exam Questions - General solutions where f(x) = kx (linear types)
- Particular solutions using boundary conditions to solve differential equations
- Exam Questions - Exponential Type ke
^{px}(exponential types) - Exam Questions - Trig Type
- Exam Questions - Particular solutions using boundary conditions