Further Pure Maths

AQA AS Level Further Pure Maths.

Further Pure Maths

Further Pure Maths topics

Proof by Mathematical Induction

Sum of Series

  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

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

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. Multiplication and division rules for mod and argument of two complex numbers
  17. Exam Questions - Further complex numbers

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 half-line
  3. Using complex numbers to represent regions on an Argand diagram

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

Up to Contents

Matrix Linear Transformations

  1. Linear transformations - rotations
  2. Linear transformations - reflections
  3. Linear transformations - enlargement
  4. Reflection in the line y=(tanθ)x
  5. How well do you know your transformations?
  6. Combinations of transformations
  7. 3D matrix transformations
  8. Invariant points

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

Series

Standard Summations

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

Up to Contents

Method of Differences

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

Up to Contents

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

Inequalities

Fractional Inequalities

  1. Fractional inequalities

Graphs of Rational Functions

Graphs of Rational Functions

  1. Graphs of rational functions

Conics

Conic Sections

  1. The conic sections of a circle, parabola, ellipse and hyperbola

Up to Contents

Parabola

  1. Cartsesian form
  2. Tangents and Normals using the discriminant
  3. Transformations of the parabola

Up to Contents

The Ellipse

  1. Cartesian form
  2. Tangents and Normals using the discriminant
  3. Transformations of the ellipse

Up to Contents

The Hyperbola

  1. Cartesian form and asymptotes
  2. Tangents and Normals using the discriminant
  3. Transformations of the hyperbola

Further Calculus

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

Up to Contents

Mean Value Theorem

  1. The mean value of a function

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

Lines

  1. Cartesian form of a line

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

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. Hyperbolic identities
  5. Expressing inverse hyperbolic functions as natural logarithms