Core Pure Mathematics — Papers 1 & 2 (Mandatory)
1. Proof
Core Pure
1a
Proof by Mathematical Induction
NotesPractice
1b
Induction for Series, Divisibility & Matrices
NotesPractice
2. Complex Numbers
Core Pure
2a
Polynomial Roots & Conjugate Pairs
NotesPractice
2b
Modulus-Argument Form & Argand Diagrams
NotesPractice
2c
Loci in Argand Diagrams
NotesPractice
2d
De Moivre’s Theorem & Multiple Angles
NotesPractice
2e
Exponential Form & nth Roots
NotesPractice
3. Matrices
Core Pure
3a
Matrix Operations & Determinants
NotesPractice
3b
Inverse Matrices (2×2 and 3×3)
NotesPractice
3c
Transformations & Invariant Points
NotesPractice
3d
Simultaneous Equations
NotesPractice
3e
Eigenvalues, Eigenvectors & Diagonalisation
NotesPractice
4. Further Algebra & Functions
Core Pure
4a
Roots of Polynomials & Transformations
NotesPractice
4b
Sums of Series & Method of Differences
NotesPractice
4c
Rational Functions & Inequalities
NotesPractice
4d
Conic Sections (Parabolas, Ellipses, Hyperbolas)
NotesPractice
5. Further Calculus
Core Pure
5a
Mean Value & Improper Integrals
NotesPractice
5b
Volumes & Surface Areas of Revolution
NotesPractice
5c
Integration with Partial Fractions
NotesPractice
5d
Maclaurin & Taylor Series
NotesPractice
5e
Limits, L’Hôpital’s Rule & Leibniz’s Theorem
NotesPractice
6. Further Vectors
Core Pure
6a
Vector & Cartesian Equations of Lines/Planes
NotesPractice
6b
Scalar & Vector (Cross) Products
NotesPractice
6c
Intersections & Distances
NotesPractice
7. Polar Coordinates
Core Pure
7a
Graphs of Polar Curves
NotesPractice
7b
Areas Enclosed by Polar Curves
NotesPractice
8. Hyperbolic Functions
Core Pure
8a
Definitions, Graphs & Identities
NotesPractice
8b
Calculus of Hyperbolic Functions
NotesPractice
9. Differential Equations
Core Pure
9a
First Order DEs & Integrating Factor
NotesPractice
9b
Second Order Differential Equations
NotesPractice
9c
Modelling Oscillations & Coupled DEs
NotesPractice
10. Numerical Methods
Core Pure
10a
Numerical Integration (Simpson’s Rule)
NotesPractice
10b
Numerical Solutions to DEs (Euler’s Method)
NotesPractice
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Mechanics — Paper 3 (Optional Application 1)
M1. Dimensional Analysis
Mechanics
M1a
Dimensions of Quantities
NotesPractice
M1b
Checking Formula Consistency
NotesPractice
M2. Momentum & Collisions
Mechanics
M2a
Conservation of Momentum (1D & 2D)
NotesPractice
M2b
Newton’s Law of Restitution
NotesPractice
M2c
Oblique Impacts
NotesPractice
M3. Work, Energy & Power
Mechanics
M3a
Work-Energy Principle
NotesPractice
M3b
Conservation of Energy
NotesPractice
M4. Circular Motion
Mechanics
M4a
Motion in a Horizontal Circle
NotesPractice
M4b
Motion in a Vertical Circle
NotesPractice
M5. Centres of Mass
Mechanics
M5a
Finding Centre of Mass (Shapes)
NotesPractice
M5b
Sliding and Toppling (Equilibrium)
NotesPractice
Statistics — Paper 3 (Optional Application 2)
S1. Discrete Distributions
Statistics
S1a
Expectation & Variance of E(X) and V(X)
NotesPractice
S1b
Probability Generating Functions (PGFs)
NotesPractice
S2. Poisson & Binomial Distributions
Statistics
S2a
Poisson Distribution & Additivity
NotesPractice
S2b
Approximations to the Binomial Distribution
NotesPractice
S3. Continuous Distributions
Statistics
S3a
PDFs, CDFs, Mean, Median & Mode
NotesPractice
S3b
Continuous Uniform (Rectangular) Distribution
NotesPractice
S4. Estimation & Confidence Intervals
Statistics
S4a
Central Limit Theorem
NotesPractice
S4b
Confidence Intervals for a Normal Mean
NotesPractice
S5. Hypothesis Testing
Statistics
S5a
Type I & Type II Errors
NotesPractice
S5b
Power of a Test
NotesPractice
S6. Chi-Squared Tests
Statistics
S6a
Goodness of Fit Tests
NotesPractice
S6b
Contingency Tables & Degrees of Freedom
NotesPractice
Discrete Mathematics — Paper 3 (Optional Application 3)
D1. Graphs and Networks
Discrete
D1a
Planarity & Eulerian Graphs
NotesPractice
D1b
Minimum Spanning Trees & Dijkstra’s
NotesPractice
D1c
Route Inspection (Postman) & TSP
NotesPractice
D2. Network Flows
Discrete
D2a
Finding Flows & Cuts
NotesPractice
D2b
Maximum Flow – Minimum Cut Theorem
NotesPractice
D3. Linear Programming
Discrete
D3a
Graphical Solution of Two Variables
NotesPractice
D3b
The Simplex Tableau
NotesPractice
D4. Critical Path Analysis
Discrete
D4a
Activity Networks & Float
NotesPractice
D4b
Gantt Charts & Resource Levelling
NotesPractice
D5. Game Theory
Discrete
D5a
Zero-Sum Games & Play-Safe Strategies
NotesPractice
D5b
Mixed Strategies & Dominance
NotesPractice
D6. Group Theory
Discrete
D6a
Binary Operations & Group Axioms
NotesPractice
D6b
Cayley Tables, Order & Cyclic Groups
NotesPractice
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