Contents  [See Pure: Vectors for sample notes & exercises]

 

 

Pure [A Level material is in green; STEP material is in orange]

 

Key to Exercises: E=easier; M=moderate; H=harder

 

 

General

Further Maths Exam Technique (14 pages; 10/12/19)

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

Reference (Pure) (7 pages; 4/1/21)

 

 

Algebra

Algebra [Important Ideas] (2 pages; 22/10/20)

Algebra - Introduction (3 pages; 25/8/20)

Partial Fractions (10 pages; 8/8/19)

Algebra Methods [STEP] (5 pages; 23/1/18)

Algebra Exs (M) (1 page; 16/12/20); Sol'ns (6 pages; 16/12/20)

 

 

Complex Numbers

Complex Numbers [Important Ideas] (2 pages; 22/10/20)

Complex Numbers - Part 1 (12 pages; 11/12/15)

Complex Numbers - Part 2 (11 pages; 18/9/19)

Complex Numbers - Part 3 (11 pages; 18/9/19)

Complex Number Methods [STEP] (9 pages; 28/2/15)

Complex Numbers Exs (E) (2 pages; 15/12/20) Sol'ns (3 pages; 15/12/20)

Complex Numbers Exs (M) (3 pages; 17/12/20) Sol'ns (13 pages; 17/12/20)

Complex Numbers Exs (H) (1 page; 15/12/20) Sol'ns (3 pages; 15/12/20)

 

 

Conics

Conics (4 pages; 17/8/19)

Parabolas (10 pages; 16/7/19)

Ellipses (7 pages; 18/8/19)

Hyperbolas (13 pages; 11/9/19)

Parabolas - Exs (H) (3 pages; 17/2/20); Sol'ns (7 pages; 29/12/19)

Ellipses - Exs (M/H) (1 page; 17/2/20); Sol'ns (3 pages; 30/12/19)

Hyperbolas - Exs (M/H) (2 pages; 17/2/20); Sol'ns (6 pages; 30/12/19)

 

 

Cubics

Cubics [Important Ideas] (3 pages; 24/10/20)

Cubics [STEP] (4 pages; 11/12/19)

Cubic Graphs Exs (3 pages; 29/7/16); Sol'ns (10 pages; 29/7/16)

 

 

Differential Equations (see also Mechanics: Oscillations)

Models (4 pages, 19/2/20)

1st order (3 pages, 23/2/19)

2nd order (linear, constant coeffs) (9 pages; 28/5/20)

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

Linear systems (7 pages; 14/9/19)

Differential Equations [STEP] (5 pages; 18/11/18)

 

 

Differentiation

Differentiation [Important Ideas] (1 page; 22/10/20)

Turning Points & Points of Inflexion [Important Ideas] (2 pages; 4/1/21)

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

Differentiation Exs (E/M) (2 pages; 18/12/20); Sol'ns (5 pages; 18/12/20)

Differentiation Exs (H) (1 page; 18/12/20); Sol'ns (3 pages; 18/12/20)

Turning points (3 pages; 7/4/20)

Points of inflexion (3 pages; 24/10/18)

 

 

Functions

Functions [Important Ideas] (2 pages; 22/10/20)

Modulus Function [Important Ideas] (2 pages; 22/10/20)

Curve Sketching Exs (E) (1 page; 29/12/20); Sol'ns (1 page; 29/12/20)

Curve Sketching Exs (M) (1 page; 29/12/20); Sol'ns (6 pages; 29/12/20)

Curve Sketching Exs (H) (2 pages; 29/12/20); Sol'ns (15 pages; 29/12/20)

Convexity & concavity (3 pages; 29/8/18)

Sketching rational functions (5 pages; 11/12/19)

y squared = f(x) (2 pages; 29/8/18)

Modulus function (3 pages; 4/9/18)

Functions - Misc (7 pages; 4/9/18)

 

 

Geometry

Circle Theorems (11 pages; 7/8/17)

Geometry Exs (2 pages; 1/1/21); Sol'ns (5 pages; 1/1/21)

 

 

Groups

Groups - Part 1 (10 pages; 22/10/16)

Groups - Part 2 (12 pages; 23/10/16)

Groups - Exs (H) (2 pages; 31/12/20); Sol'ns (8 pages; 31/12/20)

 

 

Hyperbolic Functions

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

Hyperbolic Functions Exs (E) (2 pages; 1/1/21); Sol'ns (5 pages; 1/1/21)

Hyperbolic Functions Exs (M) (1 page; 1/1/21); Sol'ns (3 pages; 1/1/21)

Hyperbolic Functions Exs (H) (2 pages; 1/1/21); Sol'ns (8 pages; 1/1/21)

 

 

Induction

Induction [Important Ideas] (4 pages; 22/10/20)

Induction (4 pages; 4/10/20)

Induction Exs (3 pages; 22/8/16); Sol'ns (23 pages; 3/10/20)

 

 

Inequalities

Inequalities [Important Ideas] (2 pages; 22/10/20)

Inequalities (7 pages; 5/9/18)

Inequalities Exs (E) (1 page; 2/1/21); Sol'ns (3 pages; 2/1/21)

Inequalities Exs (M) (1 page; 2/1/21); Sol'ns (3 pages; 2/1/21)

Inequalities Exs (H) (1 page; 2/1/21); Sol'ns (3 pages; 2/1/21)

 

 

Integration

Integration Methods [Important Ideas] (14 pages; 28/10/20)

Integration theory (5 pages; 28/9/18)

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

Improper integrals (3 pages; 20/9/18)

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

Integration - P2 Exs (Harder)(3 pages; 8/8/19); Sol'ns (22 pages; 8/8/19)

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

Integration - P4 Exs (Misc) (2 pages; 8/8/19); Sol'ns (6 pages; 8/8/19)

Integration Problems (1 page; 22/10/20); Sol'ns (3 pages; 22/10/20)

 

 

Logarithms

Logarithms [Important Ideas] (2 pages; 22/10/20)

Logarithms (4 pages; 13/2/17)

Logarithms - Exs (2 pages; 4/1/21); Sol'ns (8 pages; 4/1/21)

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

 

 

Matrices

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

3 simultaneous equations (8 pages; 24/3/20)

3 simultaneous equations - Exs [M](2 pages; 2/4/20); Sol'ns (8 pages; 2/4/20)

Transformations (9 pages; 1/10/19)

Transformations Exs (4 pages; 22/2/20); Sol'ns (14 pages; 22/2/20)

Transformation with zero determinant - Ex (7 pages; 5/3/21)

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

Invariant points & lines - conditions (12 pages; 16/4/20)

Invariant points and lines - Exs (1 page; 16/4/20); Sol'ns (3 pages; 16/4/20)

Shears (8 pages; 12/8/18)

Shears Exs [H] (1 page; 22/2/20); Sol'ns (4 pages; 22/2/20)

Eigenvectors (10 pages; 4/9/18)

Eigenvectors Exs [M/H] (3 pages; 20/3/20); Sol'ns (15 pages; 20/3/20)

Eigenvectors & Invariance (6 pages; 16/4/20)

General Exs [E/M/H] (3 pages; 6/1/21); Sol'ns (11 pages; 6/1/21)

 

 

Numbers

Integers [Important Ideas] (1 page; 22/10/20)

Integers - Exs (1 page; 22/10/20); Sol'ns (4 pages; 22/10/20)

Counting Problems (2 pages; 24/10/20); Sol'ns (8 pages; 22/10/20)

Number Theory (17 pages; 7/10/18)

 

 

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)

Num. Methods: Short Q'ns (3 pages; 12/6/20); Sol'ns (7 pages; 12/6/20)

 

 

Parametric Equations

Parametric equations (8 pages; 30/8/13)

 

 

Polar Curves

Polar Curves (21 pages; 2/11/20)

Polar Curves - Summary (11 pages; 2/11/20)

Polar Curves - Exs (1 page; 21/3/20); Sol'ns (6 pages; 14/1/20)

 

 

Polynomials

Polynomials [Important Ideas] (2 pages; 4/1/21)

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

Polynomials - Exs [E/M/H] (2 pages; 6/1/21); Sol'ns (7 pages; 7/1/21)

Roots of polynomials - Exs [M/H] (2 pages;13/1/21); Sol'ns (6 pages;13/1/21)

 

 

Proof (see also Induction)

Proof [Important Ideas] (3 pages; 22/10/20)

Proof - Problems (1 page; 24/10/20); Sol'ns (1 page; 24/10/20)

 

 

Quadratics

Quadratic Functions [Important Ideas] (2 pages; 22/10/20)

Quadratics - factorising (3 pages; 21/1/17)

Quadratics Exs [E/M] (1 page; 15/1/21); Sol'ns (3 pages; 15/1/21)

 

 

Sequences & Series

Series [Important Ideas] (3 pages; 16/12/20)

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

Arithmetic Seqs & Series Exs [E/M] (2 pp; 16/1/21) Sol'ns (6 pp; 16/1/21)

Seq & Series - Exs [E/M/H] (2 pages; 18/1/21); Sol'ns (7 pages; 18/1/21)

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

Method of Differences - Exs (E/M] (1 page; 6/10/20); Sol'ns (3 pp; 6/10/20)

Recurrence relations (16 pages; 5/9/18)

Recurrence relations - Ex (1 page; 13/7/20); Sol'ns (2 pages; 13/7/20)

Maclaurin Series (7 pages; 25/3/17)

Maclaurin Series Exs [E/H] (2 pages; 19/1/21); Sol'ns (8 pages; 19/1/21)

Taylor Series (5 pages; 27/3/18)

 

 

Transformations (of Functions)

Transformations [Important Ideas] (3 pages; 22/10/20)

Transformations (STEP) (6 pages; 12/12/19)

Transformations Exs (E/M) (1 page; 30/12/20); Sol'ns (2 pages; 30/12/20)

Transformations Exs (H) (1 page; 30/12/20); Sol'ns (7 pages; 30/12/20)

 

 

Trigonometry

Trigonometry [Important Ideas] (7 pages; 24/10/20)

Trigonometry - Part 1 (12 pages; 4/9/16)

Trigonometry - Part 2 (15 pages; 24/3/20)

t=tan(x/2) substitution (4 pages; 24/3/20)

Trigonometry Exs [E/M] (3 pages; 20/1/21); Sol'ns (12 pages; 20/1/21)

Trigonometry Exs [H] (1 page; 20/1/21); Sol'ns (7 pages; 20/1/21)

 

 

 

Vectors

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

Vector Methods [STEP] (4 pages; 14/3/15)

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)

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 [E] (1 page; 21/1/21); Sol'ns (4 pages; 21/1/21)

Vectors - Exs [M] (3 pages; 21/1/21); Sol'ns (15 pages; 21/1/21)

Vectors - Exs [H] (5 pages; 21/1/21); Sol'ns (21 pages; 21/1/21)

 

 

 

Miscellaneous

Miscellaneous [Important Ideas] (1 page; 22/10/20)

Linear Interpolation (5 pages; 7/2/16)

Misc. Exs [E/M](1 page; 19/1/21); Sol'ns (3 pages; 19/1/21)

 

 

 

 

Mechanics

 

General

Exam Board Topics - Mechanics (8 pages; 7/8/19)

Useful Results (2 pages; 5/5/20)

 

Centre of Mass

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 (3 pages; 25/3/20); Sol'ns (9 pages; 25/3/20) 

 

Circular Motion

Circular Motion [Important Ideas] (2 pages; 25/10/20): See I/I page

Circular Motion - Constant Speed (9 pages; 21/8/16)

Circular Motion - Variable Speed (13 pages; 30/9/15)

Circular Motion  Exs (1 page; 25/3/20); Sol'ns (2 pages; 21/1/20)

 

Energy

Energy (5 pages; 26/1/19)

Energy Exs (2 pages; 6/4/20); Sol'ns (8 pages; 22/1/20)

Gravitational Potential Energy (3 pages;15/2/19)

Hooke's Law (9 pages; 26/1/19)

 

Forces

Forces [Important Ideas] (16 pages; 12/11/20): See I/I page

Equilibrium [Important Ideas] (11 pages; 12/11/20): See I/I page

Forces - Exs (8 pages; 6/4/20); Sol'ns (30 pages; 6/4/20) 

Friction (3 pages; 24/10/18)

Frameworks (4 pages; 15/9/13)

Fictitious forces (5 pages; 9/7/15)

Systems of forces (5 pages; 14/11/18)

Toppling (2 pages; 13/11/20)

 

Impulse & Momentum

Collisions [Important Ideas] (3 pages; 25/10/20): See I/I page

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

Impulse & Momentum Exs (3 pages; 12/10/20); Sol'ns (11 pages; 12/10/20)

Collisions - reversal of direction (6 pages; 24/6/20)

Oblique impacts (4 pages; 23/8/18)

Oblique impact with plane - Exs (2 pages; 2/3/19)

Impulses & connected particles (13 pages; 28/8/18)

 

Kinematics

Kinematics (7 pages; 7/8/17)

Kinematics Exs (1 page; 6/4/20); Sol'ns (3 pages; 26/1/20)

Projectiles (9 pages; 5/5/20)

Suvat equations (5 pages; 24/1/17)

 

Oscillations

Differential equations: Oscillations (11 pages; 29/5/20)

Oscillations - Exs (2 pages; 6/4/20); Sol'ns (6 pages; 27/1/20)

 

Rotating Bodies

Rotating Bodies (5 pages; 19/8/19)

Rotating Bodies - Examples (6 pages; 16/3/18)

Moments of Inertia (7 pages; 23/3/18)

Rolling wheel - Point on circumference (5 pages; 1/2/21) 

Rolling wheel - Friction (6 pages; 1/2/21) 

 

 

 

Probability & Statistics

 

General

Probability & Statistics [Important Ideas] (9 pages; 4/10/20): See I/I page

Exam Board topics - Prob. & Stats (11 pages; 7/8/19)

Useful Results (4 pages; 26/5/20)

 

Continuous Distributions

Normal distribution (9 pages; 21/2/17)

 

Correlation & Regression

Correlation (12 pages; 23/8/19)

Rank Correlation (5 pages; 23/8/19)

Regression (8 pages; 23/8/19)

 

Discrete Distributions

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

Poisson distribution (8 pages; 30/5/20)

Approximations to Binomial & Poisson (4 pages; 30/1/17)

Geometric distribution (3 pages; 12/11/18)

Uniform distribution (2 pages; 20/3/20)

 

Probability

Counting (6 pages; 27/9/20)

Counting - Exs (2 pages 7/7/20); Sol'ns (5 pages 7/7/20)

Probability (2 pages; 25/3/18)

Probability Generating Functions (5 pages; 5/8/20)

 

Sampling

Sampling (4 pages; 7/8/17)

 

Tests

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

Hypothesis Tests - Types (8 pages; 24/1/20)

Confidence Intervals - Types (3 pages; 16/4/16)

Contingency Tables & Goodness of Fit (8 pages; 12/5/17)

Contingency Tables & Goodness of Fit - Important Ideas (3 pages; 24/3/20)

 

Miscellaneous

Statistics - Miscellaneous (3 pages; 21/8/16)

Statistics Exs (3 pages; 7/4/20); Sol'ns (9 pages; 7/4/20)

 

 

 

Discrete (aka Decision)

 

General

Short Q'ns (3 pages; 9/6/20); Sol'ns (7 pages; 9/6/20)

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

 

Algorithms (Misc.)

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

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

Algorithms - Exs (3 pages; 7/4/20); Sol'ns (5 pages; 3/2/20)

 

Allocation Problems

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

 

Critical Path Analysis

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

Critical Path Analysis - Part 2 (15 pages; 23/1/20)

Critical Path Analysis - Exs (3 pages; 7/4/20); Sol'ns (6 pages; 23/1/20)

 

Game Theory

Game Theory Exs (2 pages; 16/8/19); Sol'ns (7 pages; 16/8/19)

 

Graphs & Networks

Graphs Glossary (3 pages; 15/1/14)

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

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

Graphs & Networks - Exs (3 pages; 14/8/19); Sol'ns (3 pages; 14/8/19)

 

Linear Programming & Simplex method

Linear Programming (9 pages; 21/11/19)

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

Linear Programming Exs (4 pages; 16/8/19); Sol'ns (13 pages; 16/8/19)

Simplex method - Part 1 (15 pages; 31/1/18)

Simplex method - Part 2 (16 pages; 31/1/18)

Simplex method - Exercises (1 page; 14/8/19); Sol'ns (7 pages; 14/8/19)

 

Minimum Connector problem

Minimum Connector Problem (6 pages; 2/1/14)

Min. Connector Problem - Ex (1 page; 14/8/19); Sol'ns (2 pages; 14/8/19)

 

Route Inspection

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

Route Inspection Exs (2 pages; 14/8/19); Sol'ns (6 pages; 14/8/19)

 

Shortest Paths

Dijkstra's Algorithm (5 pages; 5/9/13)

Dijkstra's Algorithm - Ex (1 page; 14/8/19); Sol'ns (6 pages; 14/8/19)

Floyd's Algorithm (21 pages; 17/6/20)

Floyd's Algorithm Exs (2 pages; 14/8/19); Sol'ns (15 pages; 14/8/19)

 

Travelling Salesman

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

Travelling Salesman Ex (1 page; 14/8/19); Sol'ns (2 pages; 14/8/19)

 

 

STEP  

 

See also "Pure" page for notes relevant to STEP.

 

Notes

STEP Exams - Tips (3 pages; 14/2/21)

STEP Exams - Preparation (3 pages; 4/6/20)

STEP Exams - Methods (8 pages; 4/6/20)

STEP Exams - Technique (7 pages; 4/6/20)

STEP - Important Ideas (26 pages; 15/2/21)

 

 

Index of q'ns, by topic

Pure (24 pages; 2/9/20)

Mechanics (9 pages; 2/9/20)

Probability & Statistics (7 pages; 2/9/20)

 

 

Exercises

STEP Exercises (5 pages; 2/12/20); Sol'ns (24 pages; 2/12/20)

 

 

Solutions for Past Papers

2005, Paper 1: 1  3  4  5  11  12  13  14 

2005, Paper 2: 1  4  6  10  11  

2005, Paper 3: 2  3  7 8  

2006, Paper 1: 1  2  3  4  5  6  7  8  9  10  11  12  13  14

2006, Paper 2: 2  4  6  12  

2006, Paper 3: 3  4  10  11  12  13  

2007, Paper 1: 1  2  3  4  5  7  12  

2007, Paper 2: 1  2  3  4 

2007, Paper 3: 1  2  3  7  

2008, Paper 1: 1  2  9  13  

2008, Paper 2: 1  3  4  5  6  7  10  

2008, Paper 3: 1 5  8  13  

2009, Paper 1: 1  2  9  12  

2009, Paper 2: 5  8  10  11  13

2009, Paper 3: 1  4  6  7  8  11  

2010, Paper 1: 1  2  3  5  9  

2010, Paper 2: 4  5  7  12  13

2010, Paper 3: 2  3  8  9 

2011, Paper 1: 1  2  6  9  

2011, Paper 2: 1  2  5  6  9  11  12

2011, Paper 3: 1  2  3  4  5  6  7  8  9  10  11  12  13

2012, Paper 1: 2  5  8  9  11  13

2012, Paper 2: 3  5  7  9  10  11  12  13

2012, Paper 3: 1  6  7  9  10  11  12  13

2013, Paper 1: 1 4 10 13

2013, Paper 2: 1 2 11 13

2013, Paper 3: 6 7 8 9

2014, Paper 1: 2 3 4 9 11

2014, Paper 2: 4 5 9 12

2014, Paper 3: 2 9 10 13

2015, Paper 1: 3 9 10 12

2015, Paper 2: 2 10

2015, Paper 3: 1 6 9 12

2016, Paper 1: 2 12 13

2016, Paper 2: 3 6 7 9 13

2016, Paper 3: 1 2 3 8

2017, Paper 1: 1 2 10 12 13

2017, Paper 2: 1 10 12 13

2017, Paper 3: 3 4 9 12 13

2018, Paper 1: 7 9 10 12 13

2018, Paper 2: 2 6 8 10 11 13

2018, Paper 3: 1 2 3 8 11

2019, Paper 1: 3 4 9 10 11

2019, Paper 2: 1 2 7 10 11

2019, Paper 3: 3 5 9 11 12

 

 

MAT

 

MAT - Index of Q'ns (14 pages; 4/11/20)

 

Notes

MAT Exam - Tips (3 pages; 14/2/21)

Notes on MAT paper (3 pages; 6/8/20)

MAT Exam Technique (12 pages; 4/9/20)

MAT - Important Ideas (17 pages; 15/2/20)

 

 

Exercises

MAT Exercises (3 pages; 2/12/20); Sol'ns (12 pages; 2/12/20)

 

 

Solutions for Past Papers

Specimen 1: MC 3 4 5

Specimen 2: MC 2 4

2007: MC 4 5

2008: MC 2 3 5

2009: MC 2

2010: MC 3

2011: MC 2

2012: MC 5

2013: MC 2 3 4 5

2014: MC 2 3 4 5

2015: MC 2 3 4 5

2016: MC

2017: MC 2

2018: MC

2019: MC 2 3

2020: MC