FP3 Tutorials – OCR A Level Maths

Contents

First Order Differential Equations
Separating the Variables
Exact Equations (Integrating Factors)
Substitution Type
Second Order Differential Equations
Equations of the form ad²y/dx²+ bdy/dx+cy =f(x)
Using a Substitution
Complex Numbers
Further Complex Numbers
De Moivres Theorem
Vectors
Vector Product
Lines
Planes
Binary Operations
Binary Operations
Groups
Groups
Isomorphism

First Order Linear Differential Equations

Separating the Variables

  1. Differential Equations - Finding a general and a particular solution
  2. Working with constants in log types
  3. Differential Equations - Exponential and trig type
  4. Separating the variables and sketching a family of curves

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Exact Equations (Integrating Factors)

  1. Exact equations where one side is the exact derivative of a product
  2. Solving equations of the form dy/dx + Py = Q using an integrating factor
  3. Exam Questions - Exact equations (integrating factors)

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Substitution Types

  1. Using substitution to reduce a differential equation to a known form
  2. Exam Questions - Substitution types

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Second Order Linear Differential Equations

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

  1. How to solve second order linear differential equations that equal zero

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

  1. General solutions where f(x) = k (constant types)
  2. General solutions where f(x) = kx (linear types)
  3. General solutions where f(x) = kx2 (quadratic types)
  4. General solutions where f(x) = kepx (exponential types)
  5. General solutions where f(x) = λ cosωx + µ sinωx (trig types)
  6. Special types of particular integrals
  7. Exam Questions - General solutions where f(x) = kx (linear types)
  8. Particular solutions using boundary conditions to solve differential equations
  9. Exam Questions - Exponential Type kepx (exponential types)
  10. Exam Questions - Trig Type
  11. Exam Questions - Particular solutions using boundary conditions

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Using a Substitution

  1. Using substitution to reduce a differential equation to a known form

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Complex Numbers

Further Complex Numbers

  1. Modulus-argument form of a complex number
  2. Exponential Form (Euler's relation)
  3. Multiplication and division rules for mod and argument of two complex numbers
  4. Exam Questions - Further complex numbers

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De Moivre’s Theorem

  1. De Moivre's theorem
  2. Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
  3. Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
  4. Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)
  5. nth roots of a complex number
  6. Exam Questions - nth roots of a complex number

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Vectors

Vector Product

  1. Vector product (cross product)
  2. Applications : Area of a triangle and parallelogram

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Lines

  1. Cartesian form of a line
  2. Vector product form of a line
  3. Shortest distance between two skew lines

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Planes

  1. Parametric vector form of a plane
  2. Scalar product forms of a plane
  3. Cartesian form of a plane
  4. Finding the point of intersection between a line and a plane
  5. The equation of the line of intersection between two non parallel planes
  6. Shortest distance from a point to a plane
  7. The angle between two planes

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Binary Operations

Binary Operations

  1. Definition
  2. Binary operations
  3. Identity and inverse elements

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Groups

Groups

  1. What defines a group
  2. Different types of groups
  3. Commutative groups
  4. Order of a group
  5. Subgroups
  6. Lagrange’s theorem
  7. Cyclic groups
  8. Generators
  9. Proof that all cyclic groups are Abelian
  10. Permutations under composition of permutations
  11. Cycle notation for permutations
  12. Result that every permutation can be written as a composition of disjoint cycles
  13. The order of a combination of cycles

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Isomorphism

  1. Isomorphism of groups
  2. The order of an element is unchanged by an isomorphism

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