P1
Cambridge 9231 · Paper 1
7 Lessons · Steel Blue
Further Pure Mathematics 1
1
Lesson 1.1
Roots of Polynomial Equations
Relationships between roots and coefficients. Symmetric functions of roots for quadratics, cubics, and quartics.
2
Lesson 1.2Rational Funcations & Graph
Vertical, horizontal and oblique asymptotes. Sketching technique, range by discriminant, five curve transformations
3
Lesson 1.3Summation of Series
Method of differences, standard results for Σr, Σr², Σr³, and partial fraction decomposition of series.
4
Lesson 1.4Matrices & Linear Equations
Matrix algebra, determinants, inverses, and solving systems of linear equations including singular cases.
5
Lesson 1.5Polar Coordinates
Polar curves, conversion between Cartesian and polar forms, area under a polar curve, and arc length.
6
Lesson 1.6Vectors in 3D
Vector and Cartesian equations of lines and planes, angles, distances, and intersections in three-dimensional space.
7
Lesson 1.7Proof & Further Methods
Methods of proof including contradiction and counterexample, and further integration techniques.
P2
Cambridge 9231 · Paper 2
6 Lessons · Emerald Green
Further Pure Mathematics 2
1
Lesson 2.1Summation of Series
Standard results, method of differences, partial fractions route, and sums to infinity.
2
Lesson 2.2Mathematical Induction
Summation proofs, divisibility, matrix power proofs, and recurrence relation proofs — all four induction types.
3
Lesson 2.3Matrices
Operations, determinants, inverses (2×2 and 3×3), linear systems, and linear transformations with Argand diagrams.
4
Lesson 2.4Eigenvalues & Eigenvectors
Characteristic equation, finding eigenvectors, diagonalisation M = PDP⁻¹, and Cayley-Hamilton theorem.
5
Lesson 2.5Complex Numbers
Modulus-argument form, de Moivre's theorem, roots of unity, z+z⁻¹ method, and loci in the Argand diagram.
6
Lesson 2.6Further Calculus
Maclaurin series, improper integrals, reduction formulae, arc length, and surface of revolution.
P3
Cambridge 9231 · Paper 3
6 Lessons · Royal Purple
Further Mechanics
1
Lesson 3.1Projectile Motion
Range, maximum height, time of flight, trajectory equation, and projectile on inclined planes.
2
Lesson 3.2Equilibrium of a Rigid Body
Moments, couples, centre of mass, tilting and sliding conditions, and ladder problems.
3
Lesson 3.3Circular Motion
Horizontal and vertical circular motion, conical pendulum, conditions for complete circles, and loss of contact.
4
Lesson 3.4Hooke's Law & Elastic Strings
Natural length, extensions, elastic potential energy, energy conservation with strings and springs.
5
Lesson 3.5Motion under Variable Force
Newton's 2nd law with F as a function of time, velocity, or displacement — solving differential equations of motion.
6
Lesson 3.6Momentum & Impulse
Newton's experimental law, coefficient of restitution, direct and oblique impacts of smooth spheres and surfaces.
P4
Cambridge 9231 · Paper 4
5 Lessons · Amber
Further Probability & Statistics
1
Lesson 4.1Continuous Random Variables
PDF, CDF, expectation, variance, median, mode, and functions of a CRV including the Jacobian method.
2
Lesson 4.2Inference: Normal & t-distributions
Hypothesis tests and confidence intervals for means — z-test, t-test, two-sample tests, and paired tests.
3
Lesson 4.3χ²-Tests
Goodness-of-fit tests (uniform, Poisson, binomial, normal) and contingency table tests of association.
4
Lesson 4.4Non-Parametric Tests
Sign test, Wilcoxon signed-rank test, and Mann-Whitney U-test — ranking, tied values, and critical regions.
5
Lesson 4.5Probability Generating Functions
G(t) = E[tˣ], moments via derivatives, standard PGFs, sums of independent variables, compound distributions.
22
Lessons
100+
Worked Examples
80+
Practice Questions
22
Interactive Tools
4
Design Themes