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A-Level Further Maths Paper 2 Past Papers & Drilling: Hyperbolic Functions, Complex Numbers, ODEs
TUTORZONE ORIGINAL DRILLING MOCK · CAMBRIDGE 9231 PAPER 2 (FURTHER PURE MATHEMATICS 2)
Written to the Cambridge 9231 syllabus (2026-2027): Paper 2 Further Pure 2 is 75 marks, 2 hours and 7-9 structured questions, worth 30% of A Level. The same course also has Paper 1 Further Pure 1 (75 marks, 2 hours, 60% AS / 30% A Level), 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 7-question, 75-mark Paper 2 drilling mock with answers and techniques.
What is A-Level Further Maths (9231) Paper 2 worth?
Direct answer: Paper 2 Further Pure 2 is 75 marks, 2 hours and 7-9 structured questions, worth 30% of the 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 2 technique: 2 hours for 7 questions, about 16 minutes each. The hyperbolic and complex-number questions are long, so secure them early; in the Maclaurin and ODE questions show the integrating factor and the constant clearly. Remember: cosh squared minus sinh squared = 1; characteristic equation (lambda squared) – (trace)(lambda) + det = 0; the reduction formula; and z = r cis theta.
How this paper was built
This is a TutorZone original drilling mock: the questions follow the Cambridge 9231 Further Mathematics structure (Paper 2 Further Pure 2: 75 marks, 2 hours, 7-9 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 seven questions test?
Direct answer: hyperbolic functions and their inverses, eigenvalues and Cayley-Hamilton, Maclaurin series, reduction formulae for powers of sine, complex numbers in modulus-argument form and first-order differential equations.
Paper 2: Further Pure 2 drilling (7 questions, 75 marks)
The seven questions cover the Further Pure 2 topics: hyperbolic functions, eigenvalues, Maclaurin series, reduction formulae, complex numbers and differential equations.
Question 1 (10 marks)
Given that sinh x = 3/4 with x > 0. (a) Find the exact values of cosh x and tanh x. (4) (b) Show that arsinh(3/4) = ln 2. (6)
Question 2 (12 marks)
A = [[4, 1], [2, 3]]. (a) Find the eigenvalues of A and a corresponding eigenvector for each. (7) (b) Verify that A squared = 7A – 10I. (5)
Question 3 (11 marks)
f(x) = ln(1 + 2x). (a) Find the first three non-zero terms of the Maclaurin expansion of f and state the range of convergence. (8) (b) Taking x = 0.1, compare the expansion with the true value. (3)
Question 4 (11 marks)
Let I(n) be the integral of (sin x)^n from 0 to pi/2, for n at least 0. (a) Show that I(n) = ((n – 1)/n) I(n – 2) for n at least 2. (7) (b) Hence find I(4) and I(5). (4)
Question 5 (11 marks)
z = 1 + i(root 3). (a) Write z in the form r cis theta and find z^6. (6) (b) Find the three roots of z^3 = 8i and verify that their sum is 0. (5)
Question 6 (10 marks)
Solve dy/dx + 2y = e^(-x) given that y(0) = 3. (a) Find the expression for y. (7) (b) Find y(1), correct to 3 significant figures. (3)
Question 7 (10 marks)
Given that cosh x = 5/3 with x > 0. (a) Find the exact values of sinh x and tanh x. (4) (b) Express x in terms of natural logarithms and give its value to 3 significant figures. (6)
If sinh x = 3/4, what is cosh x?
Direct answer: cosh squared minus sinh squared equals 1 and x is positive, so cosh x = 5/4 and tanh x = 3/5.
Paper 2 model answers
Question 1
(a) cosh squared minus sinh squared equals 1 and x is positive, so cosh x = 5/4 and tanh x = 3/5. (b) arsinh y = ln(y + root(y squared + 1)), so arsinh(3/4) = ln(3/4 + 5/4) = ln 2, about 0.693.
Question 2
(a) The characteristic equation (lambda squared) – (trace)(lambda) + det = (lambda squared) – 7(lambda) + 10 = 0 gives lambda = 5 or 2; lambda = 5 gives (1, 1) and lambda = 2 gives (1, -2). (b) A squared = [[18, 7], [14, 11]] and 7A – 10I = [[18, 7], [14, 11]], so A squared = 7A – 10I.
Question 3
(a) The derivatives at 0 are 2, -4 and 16, so ln(1 + 2x) = 2x – 2x^2 + (8/3)x^3 – …, converging for |x| < 1/2. (b) At x = 0.1 the expansion gives 0.1827 while ln 1.2 = 0.1823, a difference of about 0.0003.
Question 4
(a) Integrating by parts and using (cos x) squared = 1 – (sin x) squared gives I(n) = (n – 1)(I(n – 2) – I(n)), so I(n) = ((n – 1)/n) I(n – 2). (b) I(4) = (3/4)(1/2)(pi/2) = 3pi/16, about 0.589; I(5) = (4/5)(2/3)(1) = 8/15, about 0.533.
Question 5
(a) The modulus is 2 and the argument is pi/3, so z = 2cis(pi/3) and z^6 = 2^6 cis(2pi) = 64. (b) The roots are 2cis(pi/6), 2cis(5pi/6) and 2cis(3pi/2), that is root 3 + i, -root 3 + i and -2i; their sum is 0.
Question 6
(a) The integrating factor is e^(2x), so the left-hand side is the derivative of y e^(2x); hence y e^(2x) = e^(x) + C and y = e^(-x) + C e^(-2x). With y(0) = 3, C = 2, so y = e^(-x) + 2 e^(-2x). (b) y(1) = 0.3679 + 2 x 0.1353 = 0.639.
Question 7
(a) sinh squared = cosh squared – 1 = 25/9 – 1 = 16/9 and x is positive, so sinh x = 4/3 and tanh x = 4/5. (b) x = arcosh(5/3) = ln(5/3 + 4/3) = ln 3, about 1.099.
How many questions and marks does this paper have?
Direct answer: 7 questions for 75 marks, matching Paper 2; allow the full 2 hours.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Hyperbolic identity wrong | Using cosh^2 + sinh^2 = 1 | It is cosh^2 – sinh^2 = 1 | cosh x = 5/4 |
| Eigenvector mismatch | Pairing (1, 1) with lambda = 2 | lambda = 2 pairs with (1, -2) | (1, -2) |
| Missing factorial | f”(0) x^2 without dividing by 2 | Divide each term by its factorial | (8/3)x^3 |
| Too few complex roots | Only one cube root of 8i | A cubic has three roots | three roots, sum 0 |
| Constant dropped in the ODE | Writing y = e^(-x) straight away | Keep C, then use the initial value | y = e^(-x) + 2e^(-2x) |
How to use this paper
First drill: the full 2 hours on all 7 questions, spending no more than 20 minutes on the hyperbolic and complex questions.
Mark it: check the identities, factorials and constants, and write down every mark lost.
Second drill: redo three to seven days later, aiming to finish inside 1 h 40 min.
What should I do after drilling Paper 2?
Direct answer: check every identity and constant, redo the paper in three to seven days, then drill Paper 3 Further Mechanics or Paper 4 Further Probability & Statistics for your route.
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 2?
Further Pure 2: hyperbolic functions, matrices (eigenvalues, Cayley-Hamilton), differentiation (Maclaurin series), integration (reduction formulae), complex numbers and differential equations; 75 marks, 2 hours, 7-9 structured questions.
Is Paper 2 needed for the AS Level?
No. Paper 2 counts for the A Level only (30%); the AS Level is Paper 1 plus Paper 3 or Paper 4.
Are the numerical answers reliable?
Yes. Every numeric answer was recomputed independently and both exact and approximate values are given.
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-1-mock-2027/ | A-Level Further Mathematics Paper 1 drilling mock |
Related: practice area · A-Level Further Mathematics Paper 1 drilling mock · A-Level Further Mathematics Paper 3 drilling mock · A-Level Further Mathematics Paper 4 drilling mock
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