Further Maths - Preview

Links to Examples are in blue (links for other items appear in the password protected pages).

Recommended Reading

(1) “Calculus” (Geoffrey Matthews; publisher: John Murray)

(2) “Differential Equations” (Piaggio; publisher: G. Bell & Sons)

(3) "e: the story of a number" (Eli Maor; publisher: Princeton)

- can be read at the same time as studying A Level

(4) "Euler - The Master of Us All" (William Dunham; publisher: The Mathematical Association of America)

- assumes only A Level knowledge

- includes a section on solving cubic and quartic equations

(5) "Combinatorics - A very short introduction" (Robin Wilson; publisher: OUP)

- covers a wide range of interesting problems

- very little prior knowledge assumed

(6) "How to think like a mathematician" (Kevin Houston; publisher: CUP)

(7) “How to Make the World Add Up” (Tim Harford; publisher: The Bridge Street Press)

- How Statistics can be misleading

Pure 

General

Exam Technique (12 pages; 8/5/24) 

Pure Topics by Exam Board (22 pages; 7/8/19)

Reference (Pure) (7 pages; 13/1/23)

Clarifications (Pure) (3 pages; 2/9/23)

Prerequisite Pure Topics (for Further Maths and STEP) (4 pages; 6/12/21)

Algebra

  Introduction (3 pages; 25/8/20)

  Exercises (6 pages; 21/12/24)

Circle Theorems (11 pages; 7/8/17)

Complex Numbers - Part 1 (12 pages; 4/6/23)

Complex Numbers - Part 2 (12 pages; 4/6/23)

Complex Numbers - Part 3 (12 pages; 1/5/25)

Complex Numbers - Exs (45 pages; 22/3/25)

Complex Numbers - Further Exs (10 pages; 22/3/25)

Conics (4 pages; 17/8/19)

Parabolas (10 pages; 16/7/19)

Parabolas Exs (13 pages; 21/3/25)

Ellipses (7 pages; 18/8/19)

Ellipses Exs (6 pages; 21/3/25)

Hyperbolas (13 pages; 11/9/19)

Hyperbolas Exs (14 pages; 21/3/25)

Differential Equations (see also Mechanics: Oscillations)

Models (4 pages, 19/2/20)

1st order (3 pages, 23/2/19)

2nd order DEs - Ideas & Exs (17 pages; 4/3/25)

Approximate methods (9 pages; 27/2/21)

Linear systems (7 pages; 9/3/21)

Differentiation

Turning Points & Points of Inflexion (7 pages; 5/6/23)

Differentiation - Notes (4 pages; 3/9/18)

Differentiation Exs 1 2 3 4 5 6

Functions

Transformations of Functions: Ideas & Exs (8 pages; 28/12/24)

Transformations of Functions: Further Exs (10 pages; 28/12/24)

Curve Sketching Exs  1 2 3 4 5 6 7 8  9 10

Geometry Exs (12 pages; 6/3/25)

Hyperbolic Functions (10 pages; 26/7/16)

Hyperbolic Functions Exs (43 pages; 11/3/25)

Induction (3 pages; 2/6/23)

Induction Exs: Series (22 pages; 31/7/25)

Induction Exs: Sequences (14 pages; 31/7/25)

Induction Exs: Divisibility (18 pages; 31/7/25)

Induction Exs: Misc (12 pages; 31/7/25)

Inequalities

   Notes (3 pages; 29/12/24)

   Ideas & Exs (20 pages; 29/12/24)

   Further Exs (6 pages; 29/12/24)

Integration Methods (11 pages;15/6/24)

Integration theory (7 pages; 13/1/25)

Integration: Arc lengths & Surface areas (3 pages; 10/5/17)

Improper integrals (3 pages; 20/9/18)

Integration - P1 (E) Exs (1 page; 7/11/17); Sol'ns (8 pages; 7/11/17)

Integration - P2 (H) Exs (3 pages; 20/6/24); Sol'ns (24 pages; 20/6/24)

Integration - P3 (Hyperbolic) Exs (2 pages; 6/2/18); Sol'ns (12 pages; 17/4/20)

Integration Exs (Overview)

  General 2 5

  Improper Integrals 1 

  Arc Length 6

  Volume of Revolution / Surface Area 3 4 7 8

Logarithms (5 pages; 5/6/23)

Modelling Curves using Logs (4 pages; 7/8/17)

Logarithms Exs 1 2 3 4 5

Matrices

Determinants & Inverses (14 pages; 3/6/19)

Transformations (9 pages; 1/10/19)

Invariant points and lines - introduction (4 pages; 16/4/20)

  Matrices - General (Exs) (7 pages; 29/3/25)

  Determinants (Exs) (8 pages; 29/3/25) 

  Invariant Points & Lines (Exs) (37 pages; 29/3/25)

  Inverses (Exs) (6 pages; 29/3/25) 

  Simultaneous Eq'ns (Exs) (17 pages; 29/3/25)

  Transformations (Exs) (14 pages; 29/3/25)

Numbers

Surds Exs 1 2 3

Numerical Methods

Calculator method (1 page; 5/10/20)

Errors (10 pages; 27/3/20)

Sol'n of Eq'ns: Bisection method (2 pages; 22/10/18)

Sol'n of Eq'ns: Method of False Position (4 pages; 22/10/18)

Sol'n of Eq'ns: Newton-Raphson (4 pages; 22/10/18)

Sol'n of Eq'ns: Secant method (2 pages; 12/6/20)

Sol'n of Eq'ns: Fixed Point (16 pages; 27/3/20)

Sol'n of Eq'ns: Comparison of methods (2 pages; 22/10/18)

Sol'n of Eq'ns: Convergence (3 pages; 12/6/20)

Numerical Integration (20 pages; 31/3/20)

Numerical Differentiation (8 pages; 31/3/20)

Approximating F'ns: Newton's Forward Difference (9 pages; 12/6/20)

Approximating F'ns: Lagrange's method (2 pages; 27/3/20)

Differential Eq'ns:Approximate methods (9 pages; 27/2/21)

Numerical Methods Exs

  Convergence (Exs) 2

  Integration (Exs) 3 4 5 6 7

  Sol'n of Eq'ns (Exs) 1

Partial Fractions (10 pages; 8/8/19)

Partial Fractions Exs 1

Polar Curves (30 pages; 25/2/25)

Polar Curves Exs 1 2 3 4 5

Polynomials

Quadratics (4 pages; 5/6/23)

Quadratics Exs 1 2 3 4 5

Cubics: See STEP/MAT/TMUA pages

Roots of polynomials (6 pages; 15/2/17)

Polynomials - Exs (18 pages; 5/4/25)

Sequences & Series

Series: Ideas & Exs (3 pages; 27/12/24)

Series: Further Exs (14 pages; 27/12/24)

Arithmetic Sequences & Series (2 pages; 23/1/17)

Arithmetic Series Exs 1 2 3 4

Method of Differences (3 pages; 6/10/20)

Maclaurin Series (7 pages; 23/7/21)

Maclaurin Series Exs (12 pages; 13/3/25)

Trigonometry

Basics (2 pages; 15/4/21)

Relations (3 pages; 15/4/21)

Sine & Cosine Rules (5 pages; 15/4/21)

Radians (4 pages; 15/4/21)

Sol'n of eq'ns (2 pages; 15/4/21)

Compound Angle Formulae (4 pages; 15/4/21)

Small Angle Approximations (2 pages; 15/4/21)

Inverse Functions (5 pages; 15/4/21)

Trigonometry - Exs  (29 pages; 7/4/25)

Vectors

Vectors [Important ideas] (18 pages; 21/1/21)

Vectors - Introduction (6 pages; 4/8/18)

Scalar Product (4 pages; 19/9/20)

Vectors - Lines (6 pages; 18/9/20)

Vectors - Planes (7 pages; 10/2/20)

Configuration of 3 planes (11 pages; 1/8/21)

Vectors - Angles (5 pages; 4/8/18)

Vectors - Intersections (3 pages; 4/8/18)

Vectors - Shortest distances (14 pages; 4/8/18)

Vector Product (8 pages; 4/8/18)

Triple Products (4 pages; 4/8/18)

Vectors: Misc (1 page; 4/4/18)

Vectors Exs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Mechanics

Exam Board Topics - Mechanics (8 pages; 9/7/21)

Useful Results (4 pages; 18/1/25)

Mechanics - Clarifications (11 pages; 24/4/23)

Centre of Mass - Contents (2 pages; 7/8/17)

Centre of Mass - Part 1 (13 pages; 7/8/17)

Centre of Mass - Part 2 (12 pages; 7/8/17)

Centre of Mass Exs 1 2 3 4 5 6

Forces

Forces [Important Ideas] (16 pages; 23/5/24)

Equilibrium [Important Ideas] (12 pages; 21/7/23)

Toppling (2 pages; 13/11/20)

Hooke's Law: Ideas & Exs (15 pages; 5/2/25)

Hooke's Law: Further Exs (10 pages; 5/2/25)

Forces - Exs (33 pages; 25/3/25)

Friction (5 pages; 1/6/24)

Friction Exs 1 2 3 4

Impulse & Momentum

Collisions - Important Ideas (4 pages; 22/4/21)

Impulse & Momentum (14 pages; 19/2/17)

Oblique impacts (4 pages; 23/8/18)

Oblique impact of ball with plane - Exs (10 pages; 5/5/25)

Impulse & Momentum Exs 1 2 3 4 

Kinematics

Kinematics (7 pages; 7/8/17)

Suvat equations (5 pages; 24/1/17)

Projectiles (11 pages; 10/3/21)

Kinematics Exs 1 2 3

Work & Energy

Work & Energy (4 pages; 21/11/20)

Work & Energy Exs 1 2 3 4

Probability & Statistics

Probability & Statistics [Important Ideas] (10 pages; 28/6/21)

Exam Board topics - Prob. & Stats (11 pages; 30/6/21)

Useful Results (4 pages; 26/4/24)

Binomial distribution (7 pages; 16/2/17)

Chi-Square Tests (12 pages; 19/4/24)

Chi-Square Exs

  Contingency Tables (Exs) 2a 2b

  Goodness of Fit (Ex) 1

Correlation & Regression

Correlation (13 pages; 27/2/23)

Rank Correlation (5 pages; 23/8/19)

Regression (8 pages; 23/8/19)
Correlation Exs 1 2 3

Discrete Random Variables (8 pages; 4/2/23)

Discrete Random Variables Exs 1 2 3 4

Geometric Distribution Exs 1 2

Hypothesis Tests - General (3 pages; 22/5/20)

Poisson distribution (7 pages; 10/2/21)

Poisson Distribution Exs 1 2 3 4

Decision (aka Discrete)

Topics by exam board (9 pages; 30/7/19)

Algorithms

Bin Packing Algorithms (3 pages; 7/8/18)

Maximum Matching algorithm (4 pages; 6/9/13)

Algorithms Exs (Overview)

   Purposes of algorithms (Exs) 1 7 8 9

   Bin Packing (Exs) 3 5 6

   Sorting (Exs) 2 4    

Critical Path Analysis - Part 1 (12 pages; 22/1/20)

Critical Path Analysis - Part 2 (14 pages; 7/11/20)

Critical Path Analysis Exs 1a 1b 2 3 4 5

Game Theory Exs 1 2 3

Graphs & Networks: Glossary (3 pages; 15/1/14)

Directed Edges (incl. Incidence & Distance matrices) (5 pages; 19/11/19)

Graphs & Networks Exs - 1 2

Planarity Algorithm (5 pages; 30/6/20)

Planarity algorithm Ex 1

Kuratowski's Thm (Ex) 1

Hungarian Algorithm  (9 pages; 28/8/17)

Linear Programming & Simplex method

Notes on Linear Programming (5 pages; 21/5/24)

Linear Programming [Introduction] (11 pages; 6/1/24)

Linear Programming Exs 1 2a 2b 

Notes on Simplex Method (5 pages; 21/5/24)

Simplex method - Part 1 [Introduction](15 pages; 9/1/24)

Simplex method - Part 2 [Introduction](18 pages; 21/5/24)

Simplex Exs 2c 8 9a 9b

Formulating network problems as LP problems (7 pages; 21/1/20)

Formulate as LP problem (Exs) 3 4 5 6 7

Minimum Connector Problem (6 pages; 21/11/20)

Minimum Connector Exs 1 2

Network Flows - Introduction (5 pages; 16/1/21)

Route Inspection Problem (5 pages; 12/9/17)

Route Inspection Exs (Overview)  1 2

Shortest Path

Dijkstra's Algorithm (5 pages; 19/11/20)

Dijkstra's algorithm Ex 1

Floyd's Algorithm (21 pages; 17/6/20)

Floyd's Algorithm Exs 1 2

Travelling Salesman problem (10 pages; 29/6/20)

Travelling Salesman problem Ex 1