Topic module: FP3 OCR

The order of an element is unchanged by an isomorphism

Isomorphism of groups

The order of a combination of cycles

Result that every permutation can be written as a composition of disjoint cycles

Cycle notation for permutations

Permutations under composition of permutations

Proof that all cyclic groups are Abelian

Generators

Identity and inverse elements

Binary operations

Definition

Exam Questions – Trig Type

Exam Questions – Exponential Type ke^px (exponential types)

Cyclic groups

Lagrange’s theorem

Subgroups

Order of a group

Commutative groups

Different types of groups

What defines a group

Shortest distance from a point to a plane

Modulus-argument form of a complex number

Shortest distance between two skew lines

Vector product form of a line

Applications : Area of a triangle and parallelogram

Vector product (cross product)

Exam Questions – General solutions where f(x) = kx (linear types)

Exam Questions – Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

Exam Questions – Particular solutions using boundary conditions

Exam Questions – Substitution types

Exam Questions – Exact equations (integrating factors)

Exam Questions – nth roots of a complex number

Exam Questions – Further complex numbers

The angle between two planes

The equation of the line of intersection between two non parallel planes

Finding the point of intersection between a line and a plane

Scalar product forms of a plane

Cartesian form of a plane

Parametric vector form of a plane

Cartesian form of a line

Using substitution to reduce a differential equation to a known form

Particular solutions using boundary conditions to solve differential equations

Special types of particular integrals

General solutions where f(x) = λ cosωx + µ sinωx (trig types)

General solutions where f(x) = ke^px (exponential types)

General solutions where f(x) = kx² (quadratic types)

General solutions where f(x) = kx (linear types)

General solutions where f(x) = k (constant types)

How to solve second order linear differential equations that equal zero

Using substitution to reduce a differential equation to a known form

Solving equations of the form dy/dx + Py = Q using an integrating factor

Exact equations where one side is the exact derivative of a product

Separating the variables and sketching a family of curves

Working with constants in log types

Differential Equations – Finding a general and a particular solution

nth roots of a complex number

Expressing sinⁿθ and cosⁿθ in terms of sin(kθ) and cos(kθ)

Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

De Moivre’s theorem

Multiplication and division rules for mod and argument of two complex numbers

Exponential Form (Euler’s relation)

Differential Equations – Exponential and trig type