中文（繁體）
中文（简体）
English
A-Level Further Maths Paper 1 Past Papers & Drilling: Vectors, Matrices, Induction
TUTORZONE ORIGINAL DRILLING MOCK · CAMBRIDGE 9231 PAPER 1 (FURTHER PURE MATHEMATICS 1)
Written to the Cambridge 9231 syllabus (2026-2027): Paper 1 Further Pure 1 is 75 marks, 2 hours and 6-8 structured questions, worth 60% of AS and 30% of A Level. The same course also has Paper 2 Further Pure 2 (75 marks, 2 hours, 30%), Paper 3 Further Mechanics (50 marks, 1 h 30 min, 40% AS / 20%) and Paper 4 Further Probability & Statistics (50 marks, 1 h 30 min, 40% AS / 20%). This is a 6-question, 75-mark Paper 1 drilling mock with answers and techniques.
What is A-Level Further Maths (9231) Paper 1 worth?
Direct answer: Paper 1 Further Pure 1 is 75 marks, 2 hours and 6-8 structured questions, worth 60% of AS and 30% of A Level (Cambridge 9231, 2026-2027).
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Paper 1 (Further Pure Mathematics 1) | 6-8 structured questions, 75 marks | 60% AS / 30% A Level | 2 hours |
| Paper 2 (Further Pure Mathematics 2) | 7-9 structured questions, 75 marks | 30% A Level | 2 hours |
| Paper 3 (Further Mechanics) | 5-7 structured questions, 50 marks | 40% AS / 20% A Level | 1 h 30 min |
| Paper 4 (Further Probability & Statistics) | 5-7 structured questions, 50 marks | 40% AS / 20% A Level | 1 h 30 min |
Paper 1 technique: 2 hours for 6 questions, about 18 minutes each. Take roots of polynomials and series first, then matrices and polar coordinates, and leave vectors and induction until last. Remember: (sum of roots) = minus the coefficient of x^2; (sum of squares) = (sum)^2 – 2(pairwise sum); polar area = half the integral of r squared; invariant lines come from (2 + m)m = 1 + 2m.
How this paper was built
This is a TutorZone original drilling mock: the questions follow the Cambridge 9231 Further Mathematics structure (Paper 1 Further Pure 1: 75 marks, 2 hours, 6-8 structured questions, 30% of A Level) and the answers are our own, not Cambridge past papers or marking references.
Every numeric answer was recomputed programmatically, so you can use the model answers as a self-check after drilling.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Paper 1 (Further Pure Mathematics 1) | 60% AS / 30% A Level | about 18 min per question | roots, series, matrices, polar coordinates, vectors, induction |
| Paper 2 (Further Pure Mathematics 2) | 30% A Level | about 16 min per question | hyperbolic functions, eigenvalues, series, complex numbers, ODEs |
| Paper 3 (Further Mechanics) | 40% AS / 20% A Level | about 15 min per question | projectiles, rigid bodies, circular motion, Hooke’s law, momentum |
| Paper 4 (Further Probability & Statistics) | 40% AS / 20% A Level | about 15 min per question | continuous random variables, t-tests, chi-squared, non-parametric, PGFs |
What do the six questions test?
Direct answer: roots of polynomials, summation of series, matrices and invariant lines, polar area, shortest distance between skew lines, and proof by induction.
Paper 1: Further Pure 1 drilling (6 questions, 75 marks)
The six questions cover the Further Pure 1 topics: roots of polynomials, series, matrices, polar coordinates, vectors and induction.
Question 1 (12 marks)
The three roots of the cubic equation x^3 – 6x^2 + 11x – 6 = 0 are alpha, beta and gamma. (a) State the values of alpha + beta + gamma, alpha beta + beta gamma + gamma alpha and alpha beta gamma. (3) (b) Find alpha^2 + beta^2 + gamma^2. (3) (c) Find the cubic equation whose roots are 2alpha, 2beta and 2gamma. (6)
Question 2 (12 marks)
(a) Use partial fractions to show that 1/(r(r+1)) = 1/r – 1/(r+1), and find the sum from r = 1 to 9 of 1/(r(r+1)). (6) (b) Use the same method to find the sum from r = 1 to 8 of 1/(r(r+2)), giving your answer as a fraction. (6)
Question 3 (12 marks)
A = [[2, 1], [3, 4]]. (a) Find det A and A inverse. (6) (b) Solve 2x + y = 7 and 3x + 4y = 18. (3) (c) Find the invariant lines of B = [[2, 1], [1, 2]], that is lines y = mx mapped onto themselves. (3)
Question 4 (12 marks)
(a) Sketch the curve r = 2 + 2cos theta and state its axis of symmetry. (4) (b) Find the area enclosed by r = 2 + 2cos theta. (5) (c) State what curve r = 4cos theta is and find its area. (3)
Question 5 (13 marks)
Line l1: r = (1, 2, 3) + t(2, -1, 1); line l2: r = (3, 1, 5) + s(1, 1, 2). (a) Show that l1 and l2 are skew. (6) (b) Find the shortest distance between them. (7)
Question 6 (14 marks)
(a) Prove by induction that 4^n + 6n – 1 is divisible by 9 for every positive integer n. (8) (b) The sequence has u1 = 2 and u(n+1) = 3u(n) + 2; prove by induction that u(n) = 3^n – 1. (6)
How do you find the sum of the squares of the roots?
Direct answer: use (sum)^2 – 2(pairwise sum). For x^3 – 6x^2 + 11x – 6 = 0: 36 – 22 = 14.
Paper 1 model answers
Question 1
(a) Comparing coefficients: sigmas of the roots give 6, pairwise sum 11 and product 6. (b) (sum)^2 – 2(pairwise sum) = 36 – 22 = 14. (c) Replace x by x/2 and multiply by 8: x^3 – 12x^2 + 44x – 48 = 0 (roots 2, 4, 6).
Question 2
(a) All middle terms cancel: 1 – 1/10 = 9/10. (b) 1/(r(r+2)) = (1/2)(1/r – 1/(r+2)), leaving (1/2)(1 + 1/2 – 1/9 – 1/10) = (1/2)(58/45) = 29/45, about 0.644.
Question 3
(a) det A = 2 x 4 – 1 x 3 = 5, A inverse = (1/5)[[4, -1], [-3, 2]]. (b) x = (4 x 7 – 1 x 18)/5 = 2, y = (-3 x 7 + 2 x 18)/5 = 3. (c) B(1, m) = (2 + m, 1 + 2m); the condition (2 + m)m = 1 + 2m gives m^2 = 1, so the invariant lines are y = x (lambda = 3) and y = -x (lambda = 1).
Question 4
(a) A cardioid, symmetric about the initial line theta = 0. (b) Area = (1/2) times the integral of (2 + 2cos theta)^2 from 0 to 2pi = (1/2)(12pi) = 6pi, about 18.85. (c) r = 4cos theta is a circle with centre (2, 0) and radius 2, area 4pi, about 12.57.
Question 5
(a) The first two component equations give t = 1 and s = 0; the third then reads 4 against 5, so there is no common point. The direction vectors (2, -1, 1) and (1, 1, 2) are not multiples, so the lines are not parallel: they are skew. (b) n = d1 x d2 = (-3, -3, 3), |n| = 3 x root 3; (Q – P) dot n = 3; the shortest distance is 3/(3 x root 3) = 1/root 3, about 0.577.
Question 6
(a) For n = 1, 4 + 6 – 1 = 9. Assume 4^k + 6k – 1 = 9M; then 4^(k+1) + 6(k + 1) – 1 = 4(4^k + 6k – 1) – 18k + 9 = 9(4M – 2k + 1), a multiple of 9, so the result holds for every positive integer n. (b) u1 = 3 – 1 = 2; assuming u(k) = 3^k – 1 gives u(k+1) = 3(3^k – 1) + 2 = 3^(k+1) – 1, so u(n) = 3^n – 1.
How many questions and marks does this paper have?
Direct answer: 6 questions for 75 marks, matching Paper 1; allow the full 2 hours.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Vieta signs reversed | Reading sum of roots as -6 | Sum of roots = minus the x^2 coefficient | 6 |
| No remaining terms | Stopping at 1/r – 1/(r+1) | Write the surviving terms 1 – 1/(n+1) | 9/10 |
| Missing half in polar area | Using the integral of r squared | The area is half the integral of r squared | 6pi |
| Cross-product slips | A wrong sign in d1 x d2 | Check each component of the determinant | 1/root 3 |
| No induction conclusion | Stopping after the hypothesis step | State that the result holds for all positive integers n | multiple of 9 |
How to use this paper
First drill: the full 2 hours on all 6 questions, with nothing but the formula list beside you.
Mark it: check each step, especially signs and the final conclusion.
Second drill: redo three to seven days later, aiming to finish inside 1 h 40 min.
What should I do after drilling Paper 1?
Direct answer: log what you lost, redo the paper in three to seven days, then move on to Paper 2 Further Pure 2 and, if your route needs it, Paper 3 or Paper 4.
FAQ
Is this a Cambridge past paper?
No. This is not a Cambridge past paper: the questions and answers are TutorZone originals, not marking references.
What is in A-Level Further Maths 9231 Paper 1?
Further Pure 1: roots of polynomials, rational functions, series, matrices, polar coordinates, vectors and induction; 75 marks, 2 hours, 6-8 structured questions.
How much is Paper 1 worth?
Paper 1 is 60% of the AS Level and 30% of the A Level; the AS Level needs Paper 1 plus Paper 3 or Paper 4, while the A Level needs all four papers.
Can I just memorise the answers?
No. The marks are for the working; log whatever you lost and redo the same question three days later.
Official references
| Resource | What to check |
|---|---|
| Cambridge A-Level Further Mathematics 9231 syllabus (2026-2027) | Confirm the marks, times and weightings for Papers 1-4 |
| https://www.tutorzone.com.hk/en-alevel-further-mathematics-9231-paper-2-mock-2027/ | A-Level Further Mathematics Paper 2 drilling mock |
Related: practice area · A-Level Further Mathematics Paper 2 drilling mock · A-Level Further Mathematics Paper 3 drilling mock · A-Level Further Mathematics Paper 4 drilling mock
Next step: find a tutor / register as a tutor
After drilling, the hard part is telling a conceptual error from a writing error. Tell us the year group, the school and your available times, and TutorZone will match you with a tutor who knows the HKEAA assessment requirements.
Demand for tutors in this subject is steady, and those who can explain the marking scheme and where marks are won are especially sought after. Register free; you pay only when a match succeeds.
