# Further Maths Contents

Further maths tutorials are listed here to help with your course of study

## Rational Expressions

## Graphs of Rational Functions

## Inequalities

## Fractional Inequalities

## Roots of Polynomial Equations

## Series

## Standard Summations

## Method of Differences

## Maclaurin’s Series

## Taylor’s Series

## Infinite sequences

## Infinite series

## Tests for convergence

## Conics

## Parabola

## Hyperbola (Rectangular)

## The Ellipse

## Matrix Algebra

## Matrices

## 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

## Further matrix algebra

- Transposed and symmetric matrices
- Finding the determinant of a 3x3 matrix
- Finding the inverse of a 3x3 matrix where it exists
- Linear transformations in 3 dimensions
- Using inverse matrices to reverse the effects of a linear transformation
- Eigenvalues and eigenvectors
- Reducing a symmetrical matrix to diagonal form

## Proof by mathematical induction

## Sum of Series

## 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

## Transformations of the Complex Plane

## Vectors

## Triple Scalar Product

## Systems of Linear Equations

## Solutions of systems of linear equations

## First Order Linear Differential Equations

## Exact Equations (Integrating Factors)

## Numerical Solution of Linear Differential Equations

## Solving differential equations of the form dy/dx = f(x)

## 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 - Particular solutions using boundary conditions

## Using a Substitution

## Polar Coordinates and Curves

## Polar Coordinates

## Equations of Curves

## Area Bounded by a Polar Curve

## 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

## Inverse Trigonometric Functions

## Hyperbolic Functions

## Differentiation – Continuity

## Rolle’s Theorem

## Mean Value Theorem

## L’Hopital’s Rule

## Integration

## Standard Integrals Involving Inverse Trigonometric Functions

## Standard Integrals Involving Hyperbolic Functions

## Deriving and using reduction formulae

## Applications of Integration – Arcs

## Applications of Integration – Surface area of revolution

## Integration as a limit of a sum

## Binary Operations

## Binary Operations

## Groups

## Groups

- What defines a group
- Different types of groups
- Commutative groups
- Order of a group
- Subgroups
- Lagrange’s theorem
- Cyclic groups
- Generators
- Proof that all cyclic groups are Abelian
- Permutations under composition of permutations
- Cycle notation for permutations
- Result that every permutation can be written as a composition of disjoint cycles
- The order of a combination of cycles