Revise
The fastest marks in the paper: ideas that already repeated, and the shape the next paper is likely to take.
21 ideas asked in both papers
Each came up in 2025 and again in 2026, usually with new numbers. Answer it in your head, then tap to check.
Number of generators of a cyclic group of order n
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There are φ(n). For n = 8 that is 4.
Is R/I a field
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R/I is a field exactly when I is maximal. ℤₚ[x]/(f) is a field exactly when f is irreducible, and then it has p^(deg f) elements.
Rank and nullity of a given linear map
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rank + nullity = dimension of the domain.
Eigenvalues of a matrix, or from a polynomial that kills it
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Every eigenvalue is a root of any polynomial with p(A) = 0. A² = I gives ±1; A² = A gives 0 and 1.
Uniform convergence of nx/(1 + n²x²)
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Its maximum is 1/2, at x = 1/n, so it is not uniform on any interval containing 0. On [a, ∞) with a > 0 it is uniform.
Totally bounded, complete, compact metric spaces
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A metric space is compact exactly when it is complete and totally bounded.
Möbius map: image of a circle or the real axis
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Circles and lines go to circles and lines. Three points and their images fix the map.
∫₀^2π dθ / (a + b sin θ) by residues
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2π / √(a² − b²), for a > |b|.
Constant curvature and torsion
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The curve is a circular helix. τ = 0 gives a circle, κ = 0 a line.
Gaussian curvature of a surface
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A sphere of radius r has K = 1/r². For z = f(x, y), K = (f_xx f_yy − f_xy²) / (1 + f_x² + f_y²)², so z = x² + y² has K = 4 at the origin.
Singular solution of a Clairaut-type equation
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For y = px + f(p), eliminate p from y = px + f(p) and x + f′(p) = 0.
IVP: one, none or infinitely many solutions
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Lipschitz in y gives exactly one. y′ = y^(1/3), y(0) = 0 has infinitely many.
Classify a second-order PDE
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For Au_xx + 2Bu_xy + Cu_yy: B² − AC > 0 hyperbolic, = 0 parabolic, < 0 elliptic.
CF and PI of a constant-coefficient linear PDE
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Each factor (D − mD′) gives φ(y + mx) in the CF. For e^(ax + by), put D = a and D′ = b.
Euler–Lagrange equation and extremals
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∂F/∂y − d/dx (∂F/∂y′) = 0. If F has no x, then F − y′ ∂F/∂y′ is constant.
Solve a Fredholm or Volterra integral equation
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A separable kernel turns it into a small linear system. A Volterra convolution kernel is solved by Laplace transforms.
Poisson brackets and constants of motion
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dA/dt = [A, H] + ∂A/∂t, so A is conserved when that is 0. [qᵢ, pⱼ] = δᵢⱼ.
Small oscillations and normal modes
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Solve det(V − ω²T) = 0. The modes are orthogonal with respect to the mass matrix T.
Nature of an LPP solution
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Objective parallel to a binding edge: many optima. Region open in the improving direction: unbounded. Empty region: infeasible.
Interpolation: Lagrange, divided differences, Hermite
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n + 1 points fix one polynomial of degree at most n. Its n-th divided difference is the leading coefficient.
Divisor functions σ(n) and d(n)
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For n = ∏ pᵃ: d(n) = ∏ (a + 1) and σ(n) = ∏ (p^(a+1) − 1)/(p − 1).
What the 2019 paper adds
Half of the 2019 paper survives. These 8 of its ideas came back in 2025 or 2026, so the examiners have returned to them over seven years.
| Idea, first seen in 2019 | Asked again |
|---|---|
| Euler–Lagrange extremal of a given functional Methods | 2025 Q74, Q78 2026 Q76 |
| Möbius maps and cross-ratio Complex | 2025 Q62 2026 Q81 |
| Rouché: where the zeros of a polynomial lie Complex | 2026 Q77 |
| Rank and nullity of a given matrix or map Linear algebra | 2025 Q46, Q53 2026 Q75, Q90 |
| Centre of a non-abelian group of order p³ Algebra | 2026 Q56 |
| Degree of a field extension, such as [ℚ(√2, √3) : ℚ] Algebra | 2025 Q36 |
| ℓ² and Hilbert or Banach space facts Analysis | 2025 Q38, Q39 |
| Dual of a linear programme OR and numerical | 2026 Q65 |
Asked in 2019, not since
- Liouville’s theorem, and which functions are analytic everywhere (z̄, tan z, cosh z)
- Lebesgue measure of ℚ and of the irrationals; the Dirichlet function is Lebesgue- but not Riemann-integrable
- Tychonoff, the discrete topology, and which properties continuous maps preserve
- Operator norm, and why the null space of a bounded operator is closed
Likely shape of the next paper
A forecast from the 2025 and 2026 papers, checked against the syllabus. Use it to plan your time, not as a guarantee.
- Low
- High
Unit
Show the data
| Unit | Low | High |
|---|---|---|
| Algebra | 6 | 7 |
| Linear algebra | 8 | 10 |
| Analysis | 3 | 9 |
| Complex | 6 | 7 |
| Topology and DG | 7 | 8 |
| ODE and PDE | 7 | 7 |
| Methods | 5 | 8 |
| Mechanics | 7 | 7 |
| OR and numerical | 7 | 8 |
| Number theory | 6 | 7 |
In the syllabus, not asked yet
One pass over each before the exam. Topics marked 2019 have come up before.
| Unit | Not asked yet |
|---|---|
| Algebra | Sylow theorems, Cayley's theorem, PID, UFD and ED distinctions, Modules and Noetherian modules, Hilbert basis theorem |
| Linear algebra | Dual space and dual basis, Change of basis, Cayley–Hamilton, asked directly, Unitary and skew-Hermitian properties |
| Analysis | Riemann integrability criteria, Improper integrals, Lebesgue measure and integral (asked in 2019), Compactness and connectedness in metric spaces, Hahn–Banach, open mapping, closed graph, Operator norm (asked in 2019) |
| Complex | Liouville's theorem (asked in 2019), Taylor and Laurent expansions, Classifying singularities |
| Topology and DG | Product and quotient topology, Tychonoff theorem (asked in 2019), Urysohn lemma, Countability axioms, First and second fundamental forms, asked directly, Geodesics |
| ODE and PDE | Separation of variables for Laplace, heat and wave, Lagrange linear PDE (Pp + Qq = R) |
| Methods | Brachistochrone and catenary, Neumann series |
| Mechanics | Hamilton's variational principle, Sources, sinks and doublets, Complex potential, Sphere in a perfect fluid, Euler's equation of motion |
| OR and numerical | Simplex or Big-M by hand, Game theory, Simpson and trapezoidal rules, Picard's method, Gauss elimination, Spline interpolation |
| Number theory | Chinese remainder theorem, Möbius inversion and μ(n), Euler's theorem and φ(n) problems, Ramanujan's contributions, Vedic sutras by name |
Exam-day reminders
- No negative marking. Never leave a question blank.
- 100 questions in 120 minutes. Spend about 20 minutes on the general section, which leaves about 85 seconds per maths question.
- For “which is not true”, check every option before you choose.
- Two-step questions: write the first result down before you move on.
- A faulty question is usually given as bonus marks. If no option fits, pick the closest and move on.