Further Pure Core Maths

OCR A Level Further Pure Core Maths.

Further Pure Core Maths

Further Pure Core Maths topics

Complex Numbers

Complex Numbers

  1. Real and imaginary numbers
  2. Addition, subtraction and multiplying complex numbers and simplifying powers of i
  3. Complex conjugates
  4. Division of a complex number by a complex number
  5. Argand diagrams
  6. Modulus and argument of a complex number
  7. Solving problems with complex numbers
  8. Square roots of a complex number
  9. Solving quadratic equations with complex roots
  10. Solving cubic equations
  11. Solving quartic equations
  12. Exam Questions - Complex numbers
  13. Geometrical effects of conjugating a complex number
  14. Exam Questions - Finding roots
  15. Modulus-argument form of a complex number
  16. Exponential Form (Euler's relation)
  17. Multiplication and division rules for mod and argument of two complex numbers
  18. Exam Questions - Further complex numbers
  19. Complex numbers specimen questions (mixed exam boards)

Up to Contents

Loci in the Complex Plane

  1. The locus of a point moving in a circle
  2. The locus of a point moving along a perpendicular bisector
  3. The locus of a point moving along a half-line
  4. Using complex numbers to represent regions on an Argand diagram

Up to Contents

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

Matrix Algebra

Matrices

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

Up to Contents

Simultaneous Equations by Matrix Methods

  1. Solving simultaneous linear equations in 2 unknowns
  2. Solving three linear simultaneous equations

Up to Contents

Matrix Linear Transformations

  1. Linear transformations - rotations
  2. Linear transformations - reflections
  3. Linear transformations - enlargement
  4. How well do you know your transformations?
  5. Combinations of transformations
  6. Inverse matrices to reverse linear transformations
  7. Determinant as the area scale factor of a transformation
  8. Exam Questions - Matrix transformations
  9. 3D matrix transformations
  10. Invariant points

Proof by Mathematical Induction

Various Types of Proofs

  1. Divisibility and multiple test proofs
  2. Exam Questions - Divisibility and multiple tests
  3. Matrix proofs
  4. Exam Questions - Matrix proofs

Vectors

Scalar Product (Dot Product)

  1. Scalar product
  2. Perpendicular vectors
  3. Exam Questions - Scalar product

Up to Contents

Vector Equations of Lines

  1. Vector equation of a line
  2. Angle between two lines
  3. Parallel lines
  4. Intersecting and skew lines
  5. Exam Questions - Parallel intersecting and skew lines
  6. Closest point to a line and shortest distance from the origin
  7. Shortest distance of a point to a line

Up to Contents

Exam Questions – Vectors

  1. Exam Questions - Vectors

Up to Contents

Vector Product

  1. Vector product (cross product)

Up to Contents

Lines: Vector to Cartesian form

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

Up to Contents

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. Shortest distance from a point to a plane
  6. Angle between a line and a plane
  7. The angle between two planes

Roots of Polynomial Equations

Roots of Polynomial Equations

  1. Relationships between the roots and coefficients of a quadratic equation
  2. Relationships between the roots and coefficients of a cubic equation
  3. Relationships between the roots and coefficients of a quartic equation

Further Partial Fractions

Partial Fractions

  1. Denominator contains 1 linear and 1 quadratic factor
  2. Improper types
  3. Exam Questions - Partial Fractions (Improper type)

Series

Standard Summations

  1. Sigma notation
  2. Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
  3. Sum of the squares of the first n natural numbers ∑r2
  4. Sum of the cubes of the first n natural numbers ∑r3
  5. Using known formulae to sum more complex series
  6. Exam Questions - Series

Up to Contents

Sum of Series – Proof by Induction

  1. Proof of the sum of the series ∑r
  2. Proof of the sum of the series ∑r²
  3. Proof of the sum of the series ∑r³
  4. Proof for other series
  5. Exam Questions - Sum of series

Up to Contents

Method of Differences

  1. Method of differences
  2. Exam Questions - Method of differences

Hyperbolic Functions

Hyperbolic Functions

  1. Definitions
  2. Graphs of sinh(x), cosh(x) and tanh(x)
  3. Solving equations using inverse and exponential functions
  4. The identity: cosh2x-sinh2x ≡ 1
  5. Expressing inverse hyperbolic functions as natural logarithms
  6. Solving hyperbolic equations using hyperbolic identities

Further Calculus

Maclaurin’s Series

  1. Maclaurin's series expansion
  2. Series expansion for ex
  3. Series expansion for sin(x) and cos(x)
  4. Series expansion for ln(1+x)
  5. Further series
  6. Exam Questions - Maclaurin’s series

Up to Contents

Improper Integrals

  1. Convergent and Divergent Integrals

Up to Contents

Applications of Integration – Volumes of revolution

  1. Volume of revolution about the x-axis
  2. Exam Questions - Volume of revolution about the x-axis
  3. Volume of revolution about the x-axis generated between curves
  4. Volume of Revolution about the y-axis
  5. Exam Questions - Volume of Revolution about the y-axis
  6. Volume of Revolution about the y-axis generated between curves
  7. Volume of revolution for a curve given in parametric form
  8. Exam Questions - Volume of revolution: parametric form

Up to Contents

Mean Value of a Function

  1. The mean value of a function

Up to Contents

Integrals involving Partial fractions

  1. Integrals involving partial fractions
  2. Exam Questions - Integrals involving partial fractions

Up to Contents

Differentiating Inverse Trigonometric Functions

  1. Derivatives of sin-1(x), cos-1(x) and tan-1(x)
  2. Mixed examples : Differentiating inverse trig functions

Up to Contents

Integrals Involving Inverse Trigonometric Functions

  1. Integrals of the form 1/(a2+x2) and 1/√(a2-x2)

Up to Contents

Differentiation: Hyperbolic Functions

  1. Differentiation of hyperbolic functions
  2. Differentiation of inverse hyperbolic functions

Up to Contents

Integrals Involving Hyperbolic Functions

  1. Integrating expressions involving hyperbolic functions

Polar Coordinates and Curves

Polar Coordinates

  1. Defining the position of a point
  2. Converting Cartesian coordinates to polar coordinates
  3. Converting polar coordinates to Cartesian coordinates

Up to Contents

Equations of Curves

  1. Converting the equation of a polar curve to Cartesian form
  2. Converting the equation of a Cartesian curve to polar form
  3. Sketching polar curves a spiral
  4. Sketching polar curves a half-line
  5. Sketching curves a circle and arc
  6. Sketching curves the cardioid r = a (1+cosθ)
  7. Sketching curves the curve r = asin 2θ

Up to Contents

Area Bounded by a Polar Curve

  1. Area bounded by a polar curve
  2. Area bounded by a cardioid and a loop
  3. Exam Questions - Area bounded by a polar curve

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

Up to Contents

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)

Up to Contents

Substitution Types

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

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

Up to Contents

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

Up to Contents

Using a Substitution

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

Up to Contents

Simple Harmonic Motion

  1. Solving the differential equation: ẍ = - ω2x
  2. Damped Oscillations

Up to Contents

Linear Systems

  1. Systems of differential equations