Pure Maths Notes & Worked Exercises 

 

A Level (ie non-FM) items are in green

Items relevant to STEP are in blue 

Background material is in orange

 

Miscellaneous

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

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

Miscellaneous Notes (1 page; 11/8/16)

Miscellaneous Exs (2 pages; 7/10/18); Solutions (7 pages; 7/10/18)

Useful results (6 pages; 22/1/17)

 

Algebra

Algebra methods (5 pages; 23/1/18)

Partial Fractions (10 pages; 8/8/19)

 

Complex Numbers

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

Complex Numbers - Part 2 (19 pages; 11/12/15)

Complex Numbers - methods (9 pages; 28/2/15)

Complex Nos - Part 1 Exs (3 pages; 14/8/19) Solutions (12 pages; 14/8/19)

Complex Nos - Part 2 Exs (1 page; 13/2/16); Solutions (5 pages; 13/2/16)

 

Conics

Conics (4 pages; 17/8/19)

Parabolas (10 pages; 16/7/19)

Ellipses (7 pages; 18/8/19)

Hyperbolas (13 pages; 12/8/19)

Parabolas - Exs (2 pages; 18/8/19); Sol'ns (7 pages; 18/8/19)

Ellipses - Exs (1 page; 18/8/19); Sol'ns (3 pages; 18/8/19)

Hyperbolas - Exs (1 page; 18/8/19); Sol'ns (4 pages; 18/8/19)

 

 

Differential Equations (see also Mechanics page)

1st order (3 pages, 23/2/19)

2nd order (linear, constant coeffs) (7 pages; 18/11/18)

Substitutions (5 pages; 18/11/18)

Approximate methods (4 pages; 18/11/18)

Linear systems (2 pages; 18/11/18)

 

 

Differentiation

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

Differentiation Exs (2 pages; 19/10/18); Solutions (6 pages; 19/10/18)

Turning points (1 page; 24/10/18)

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

 

Functions (See also Graphs)

Transformations of Functions (6 pages; 9/9/13)

Transformations  (STEP)(6 pages; 25/2/18)

Transformations Exs (1 page; 7/10/18); Solutions (5 pages; 7/10/18)

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; 7/10/18); Solutions (5 pages; 7/10/18)

 

Graphs (See also Functions)

Cubic Functions (3 pages; 25/2/18)

Cubic Graphs Exs (3 pages; 29/7/16); Solutions (10 pages; 29/7/16)

Curves: (Sketching) Rational Functions (5 pages; 25/2/18)

Curve Sketching Exs (2 pages; 12/8/19); Solutions (9 pages; 12/8/19)

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

 

Groups

Groups - Part 1 (10 pages; 22/10/16)

Groups - Part 2 (12 pages; 23/10/16)

Groups - Exs (2 pages; 16/8/19); Sol'ns (8 pages; 16/8/19)

 

Hyperbolic Functions

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

Hyperbolic Functions Exs (2 pages; 13/5/17); Solutions (9 pages; 13/5/17)

 

Inequalities

Inequalities (7 pages; 5/9/18)

Inequalities Exs (2 pages; 9/8/19); Solutions (6 pages; 9/8/19)

 

Integration

Integration methods (13 pages; 6/2/18)

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; 6/2/18)

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

 

Logarithms

Logarithms (4 pages; 13/2/17)

Logarithms Exs (1 page; 7/10/18); Solutions (5 pages; 7/10/18)

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

 

Matrices

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

Transformations (8 pages; 17/8/19)

Transformations Exs (4 pages; 14/8/19); Solutions (14 pages; 14/8/19)

Shears (8 pages; 12/8/18)

Shears Exs (1 page; 4/6/19); Solutions (4 pages; 4/6/19)

Invariant points and lines (6 pages; 25/10/18)

Eigenvectors (10 pages; 4/9/18)

Eigenvectors & Invariance (5 pages; 29/10/18) 

Eigenvectors Exs (3 pages; 13/8/19); Solutions (15 pages; 13/8/19)

General Exs (2 pages; 13/8/19); Solutions (10 pages; 13/8/19)

General - Advanced Exs (1 page; 30/10/18); Solutions (5 pages; 30/10/18)

 

Number Theory

Number Theory (17 pages; 7/10/18)

 

Numerical Methods

Calculator method (1 page; 22/10/18)

Errors (9 pages; 22/10/18)

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; 22/10/18)

Sol'n of Eq'ns: Fixed Point (15 pages; 22/10/18)

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

Approximating F'ns: Newton's Forward Difference (8 pages; 22/10/18)

Approximating F'ns: Lagrange's method (2 pages; 22/10/18)

Numerical Integration: Midpoint & Trapezium (7 pages; 22/10/18)

Numerical Integration: Simpson (7 pages; 22/10/18)

Convergence: Introduction (3 pages; 22/10/18)

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

Convergence: Numerical Integration (5 pages; 22/10/18)

Numerical Differentiation (3 pages; 22/10/18)

 

Parametric Equations

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

 

Polar Curves

Polar Curves (21 pages; 17/11/18)

Polar Curves - Exs (1 page; 12/8/19); Sol'ns (6 pages; 12/8/19)

 

Polynomials

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

Polynomials  Exs (1 page; 7/10/18); Solutions (4 pages; 7/10/18)

(Roots of) Polynomial eq'ns Exs (2 pages; 15/2/17); Solutions (6 pages; 15/2/17)

  

Proof

Induction (4 pages; 22/8/16)

Induction Exs (3 pages; 22/8/16); Solutions (23 pages; 30/1/18)

Method of Differences (2 pages; 28/1/19)

 

Quadratics

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

Quadratics Exs (1 page; 7/10/18); Solutions (3 pages; 7/10/18)

 

Sequences & Series

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

Arithmetic Seqs & Series Exs (2 pages; 23/1/17) Solutions (6 pages; 23/1/17)

Maclaurin Series (7 pages; 25/3/17)

Maclaurin Series Exs (2 pages; 8/3/17); Solutions (8 pages; 8/3/17)

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

Taylor Series (5 pages; 27/3/18)

 

Trigonometry

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

Trigonometry - Part 2 (15 pages; 4/9/16)

Trigonometry Exs (3 pages; 7/10/18); Solutions (12 pages; 7/10/18)

 

 

Vectors

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

Scalar Product (2 pages; 4/8/18)

Vector Product (8 pages; 4/8/18)

Triple Products (4 pages; 4/8/18)

Vectors - Lines (4 pages; 4/8/18)

Vectors - Planes (4 pages; 4/8/18)

Vectors - Angles (5 pages; 4/8/18)

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

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

Vectors - consistency of 3 eq'ns (8 pages; 4/7/16)

Vectors - Summary of methods (7 pages; 4/8/18)

Vectors (STEP)(4 pages; 14/3/15)

Vectors: Misc (1 page; 4/4/18)

Vectors - Lines & Planes: Exs (4 pages; 13/8/19); Sol'ns (21 pages; 13/8/19)

Vectors - Problems (2 pages; 13/8/19); Sol'ns (8 pages; 13/8/19)