Advanced practice coverage

The new questions check specific structures. None of these rows means that a whole specification clause is complete or that marks have been calibrated against an exam.

Mathematical checks include exact substitution and separate Python oracles. Written proofs, interpretation and examination demand still need a teacher’s review. Some difficulty settings change the formulation or number of steps; others currently change values within the same method.

Selected advanced structures
Question type	Structures	Official route references
Powers of a complex number Generator 0.5.0	rectangular power; power of a conjugate	9FM0 · core · 2 Complex numbers 7367 · core · B Complex numbers H245 · core · 2 Complex numbers H645 · core · Complex numbers
All arguments of complex roots Generator 0.5.0	roots of a positive real number; roots of a negative real number	9FM0 · core · 2 Complex numbers 7367 · core · B Complex numbers H245 · core · 2 Complex numbers H645 · core · Complex numbers
An Argand perpendicular bisector Generator 0.5.0	horizontal pair; oblique pair	9FM0 · core · 2 Complex numbers 7367 · core · B Complex numbers H245 · core · 2 Complex numbers H645 · core · Complex numbers
Intersecting an Argand circle and line Generator 0.5.0	horizontal chord; vertical chord	9FM0 · core · 2 Complex numbers 7367 · core · B Complex numbers H245 · core · 2 Complex numbers H645 · core · Complex numbers
Eigenvalues of a non-triangular matrix Generator 0.5.0	symmetric matrix; non-symmetric similar matrix	9FM0 · FP2 · 3.1 Further matrix algebra H245 · additional-pure · Matrices H645 · Y435 · Matrices 7367 · core · C10
An eigenvector of a non-triangular matrix Generator 0.5.0	first component fixed; second component fixed	9FM0 · FP2 · 3.1 Further matrix algebra H245 · additional-pure · Matrices H645 · Y435 · Matrices 7367 · core · C10
A matrix power acting on a vector Generator 0.5.0	sum of eigenvectors; difference of eigenvectors	9FM0 · FP2 · 3.1 Further matrix algebra H245 · additional-pure · Matrices H645 · Y435 · Matrices 7367 · core · C10
A parameter that makes a matrix singular Generator 0.5.0	parameter on diagonal; parameter off diagonal	9FM0 · core · 3 Matrices 7367 · core · C Matrices H245 · core · 3 Matrices H645 · core · Matrices
The algebra in a telescoping induction step Generator 0.5.0	square difference; cube difference	9FM0 · core · 1 Proof 7367 · core · A Proof H245 · core · 1 Proof H645 · core · Proof
A logarithm Maclaurin expansion Generator 0.5.0	logarithm of 1 + ax; logarithm of 1 − ax	9FM0 · core · 5 Further calculus 7367 · core · E Further calculus H245 · core · Further calculus H645 · core · Further calculus
A negative-power binomial expansion Generator 0.5.0	reciprocal linear expression; reciprocal square	9FM0 · core · 5 Further calculus 7367 · core · E Further calculus H245 · core · Further calculus H645 · core · Further calculus
A differential equation with a repeated root Generator 0.5.0	growing repeated root; decaying repeated root	9FM0 · core · 9 Differential equations 7367 · core · I Differential equations H245 · core · 9 Differential equations H645 · core · Differential equations
A polynomial particular integral Generator 0.5.0	constant forcing; linear forcing	9FM0 · core · 9 Differential equations 7367 · core · I Differential equations H245 · core · 9 Differential equations H645 · core · Differential equations
A first-order equation with a forcing term Generator 0.5.0	constant forcing; linear forcing	9FM0 · core · 9 Differential equations 7367 · core · I Differential equations H245 · core · 9 Differential equations H645 · core · Differential equations
Recovering e^x from a hyperbolic value Generator 0.5.0	given sinh x; given cosh x with x positive	9FM0 · core · 8 Hyperbolic functions 7367 · core · H Hyperbolic functions H245 · core · 5 Hyperbolic functions H645 · core · Hyperbolic functions
Hyperbolic double angles Generator 0.5.0	sinh of a double angle; cosh of a double angle	9FM0 · core · 8 Hyperbolic functions 7367 · core · H Hyperbolic functions H245 · core · 5 Hyperbolic functions H645 · core · Hyperbolic functions
Area of one polar rose petal Generator 0.5.0	cosine petal; sine petal	9FM0 · core · 7 Polar coordinates 7367 · core · G Polar coordinates H245 · core · 7 Polar coordinates H645 · core · Polar coordinates
Area swept by a polar spiral Generator 0.5.0	sector starting at zero; sector with nonzero lower limit	9FM0 · core · 7 Polar coordinates 7367 · core · G Polar coordinates H245 · core · 7 Polar coordinates H645 · core · Polar coordinates
An exact integral of an even sine power Generator 0.5.0	sin squared; sin to the fourth power; sin to the sixth power	9FM0 · core · 5 Further calculus 7367 · core · E Further calculus H245 · core · 6 Further calculus H645 · core · Further calculus
Intersecting a line and a plane Generator 0.5.0	find line parameter; find intersection point	9FM0 · core · 6 Further vectors 7367 · core · F Further vectors H245 · core · 8 Vectors H645 · core · Vectors
The closest point on a line Generator 0.5.0	origin to line; translated point to line	9FM0 · core · 6 Further vectors 7367 · core · F Further vectors H245 · core · 8 Vectors H645 · core · Vectors
A one-dimensional collision with restitution Generator 0.5.0	partially elastic collision; elastic collision	9FM0 · FM1 · Momentum and impulse 7367 · mechanics · Momentum and collisions H245 · mechanics · Momentum and impulse H645 · Y421 · Momentum and impulse H645 · Y431 · Momentum and impulse
A geometric waiting-time probability Generator 0.5.0	first success on trial k; wait longer than k trials	9FM0 · FS2 · 3.1 Geometric distributions H245 · statistics · 5.02f–g Geometric distributions H645 · Y422 · SR16–R17 Geometric distributions H645 · Y432 · SR16–R17 Geometric distributions H635 · Y412 · SR16–R17 Geometric distributions
A minimum spanning tree Generator 0.5.0	four-vertex complete network; tied edge weights	9FM0 · D1 · Decision 1: algorithms on graphs H645 · Y433 · Modelling with algorithms H245 · discrete · Algorithms on graphs H635 · Y413 · N5 Minimum connector problems
Fixed points of a reflection Generator 0.5.0	reflection in an axis; reflection in y = ±x	8365 · core · 5.3
A trigonometric equation using an identity Generator 0.5.0	sine substitution; cosine substitution; rearranged identity equation	8365 · core · 6.10

What still needs work

Three-dimensional matrix methods, resonant differential equations, full proof review, statistical inference and broad mechanics modelling remain incomplete. AS routes are mapped separately; an A-level topic is not automatically AS content. The full-course and complete-mock gates remain closed.

Coverage for every qualification · Read the worked methods