The Cambridge AS & A Level Further Mathematics (9231) May/June 2026 papers and mark schemes are out, and they’re free. All four papers, 9231/12 to 9231/42, can be downloaded and printed on the Tutopiya Learning Portal. Below is a question-by-question breakdown of what each paper tested and how the mark scheme gave marks, taken from the papers themselves.

Key points from the 2026 9231 papers

• The pure papers rewarded long, multi-part questions. Paper 1 Question 7 on a polar curve was worth 17 marks, and Paper 2 Question 8 on matrices was worth 14.
• Calculus with hyperbolic and inverse functions ran through Paper 2: a Maclaurin series for sinh⁻¹(x + 1), an integral of sinh³ x and a differential equation with coth x.
• Differential equations came up in three places: a first-order equation with an integrating factor and a second-order equation on Paper 2, and v dv/dx and dv/dt equations on Paper 3.
• “Show that” results were common. Paper 1 Questions 3(a) and 7(a), Paper 2 Questions 2(a) and 5(a), and Paper 3 Question 6(a) all gave you the answer to prove.
• Paper 4 tested four different hypothesis tests: a sign test, a chi-squared test for independence, a paired t-test, and a two-sample confidence interval with a pooled variance.

All May/June 2026 9231 papers

Paper	Code	Duration	Marks	What it tests
Paper 1 Further Pure Mathematics 1	9231/12	2 hours	75	Further pure (AS and A Level)
Paper 2 Further Pure Mathematics 2	9231/22	2 hours	75	Further pure (A Level)
Paper 3 Further Mechanics	9231/32	1 hour 30 minutes	50	Further mechanics
Paper 4 Further Probability & Statistics	9231/42	1 hour 30 minutes	50	Further statistics

Timings and marks are taken from the front cover of each 2026 paper. All four papers are answered on the question paper and use the formula list (MF19). For the full syllabus content, see the Cambridge International AS & A Level Further Mathematics 9231 syllabus.

Which variant is this? Cambridge sets different versions of each paper (variants) for different time zones. The papers here are variant 2: 9231/12, /22, /32 and /42. All variants cover the same syllabus at the same difficulty, so variant 2 is just as good for practice as the one your school sat.

Download the 9231 May/June 2026 papers →

What came up in 9231 Paper 1 (Further Pure Mathematics 1), May/June 2026?

Paper 1 had seven questions worth 75 marks. The last two questions carried 31 of them.

Question	Topic	Marks
1	Series: Σ(4r − 3)(12r + 3) using standard results, the reciprocal series by the method of differences, then its sum to infinity	8
2	Proof by induction that an expression is divisible by 91	6
3	Matrices: a stretch followed by a shear, the line of invariant points, and an area condition using the determinant	9
4	Roots of 5x³ − 2x + 1 = 0: a cubic with roots 1/α², then sums of 1/α⁴, 1/α⁶ and 1/α⁸	10
5	Vectors: shortest distance between two lines, acute angle between two planes	11
6	Rational curve with a quadratic over a linear term: asymptotes, stationary points, sketch, modulus sketch and a modulus inequality in exact form	14
7	Polar curve r² = a/(1 + tan 2θ): point furthest from the initial line with a sign-change check, sketch, area by the substitution v = tan 2θ and partial fractions	17

What made it hard: Question 7(c) was worth 9 marks on its own. You had to set up the polar area, carry out the given substitution, split into partial fractions and combine logarithms to reach an exact answer.

What came up in 9231 Paper 2 (Further Pure Mathematics 2)?

Paper 2 had eight questions worth 75 marks.

Question	Topic	Marks
1	Maclaurin series for sinh⁻¹(x + 1) as far as x², exact constants	5
2	Reduction formula for an integral of (1 + t)ⁿ sin πt from 0 to 1, then I₂	7
3	Integral of sinh³ x by substitution u = cosh x; a first-order differential equation with coth x, solved with an integrating factor	11
4	Implicit differentiation of a curve with (x − y)⁵ and ln(xy): first and second derivatives at (−1, −2)	8
5	Upper and lower bounds for an integral using n rectangles, and the limit of their difference	10
6	Second-order differential equation with complex auxiliary roots and a quadratic right-hand side, with initial conditions	10
7	Sum of a geometric series in z, roots of 1 + z + … + z⁶ = 0, then de Moivre’s theorem to sum an infinite series	10
8	Systems of equations without a unique solution; using the characteristic equation to write A⁴ = pA² + qI; diagonalising for A⁷	14

Watch for: Question 6 was a full 10-mark differential equation. Marks were spread across the auxiliary equation, the complementary function, the particular integral, the general solution and the initial conditions, so one slip early on still leaves most marks available.

What came up in 9231 Paper 3 (Further Mechanics)?

Paper 3 had seven questions worth 50 marks.

Question	Topic	Marks
1	Centre of mass of a hemisphere (radius 3r) joined to a cylinder; angle when freely suspended from a point on the rim	7
2	A particle on two strings moving in a horizontal circle: find both tensions	5
3	Driving force and resistance 0.5v²: speed after 0.8 m (v dv/dx), then t in terms of v (dv/dt)	9
4	Oblique impact with a smooth barrier: e from a kinetic energy ratio, then a second barrier at 120°	8
5	Vertical circle on a string: tension in terms of θ, then the greatest height after the string goes slack	9
6	Elastic string: energy to show an expression for v², the maximum extension, the maximum speed	6
7	Projectile that strikes an inclined plane at right angles; find u and θ	6

What made it hard: Question 5(b) needed you to see that the particle leaves the circle when the tension is zero, then treat it as a projectile to find the extra height.

What came up in 9231 Paper 4 (Further Probability & Statistics)?

Paper 4 had six questions worth 50 marks.

Question	Topic	Marks
1	Why a Wilcoxon signed-rank test may not suit the data; a sign test for a median at 5%	7
2	Chi-squared test for independence of bus company and reliability, then which company seems least reliable	10
3	95% confidence interval for a difference in means, using a pooled variance	6
4	A piecewise cumulative distribution function: verify the median, find the constants, the PDF and E(1/X)	9
5	Paired t-test at 10% on a claim of more than 5 marks’ improvement, and the assumption it needs	8
6	Probability generating function: find the constants, Var(X), and P(Y = 0) for the sum of three observations	10

Watch for: in Question 2 some expected frequencies were below 5. The mark scheme expected you to combine rows or columns, and capped an answer with no cells combined at 5 out of 8.

How the 9231 May/June 2026 mark scheme gives marks

All four 2026 schemes use the same system as A Level Mathematics. Here is how it worked on this year’s questions.

1. M marks are for a method that is actually carried out. A method mark isn’t lost for slips, but it needs the method applied to this question. In Paper 1 Question 5(a), one M1 was for finding a common perpendicular by a cross product and another for the shortest-distance formula. The scheme said the scalar product and magnitudes had to be evaluated.

2. A marks depend on the method mark. An A mark can’t be given unless its M mark is earned. In Paper 2 Question 6, the final A1 for y needed the correct general solution and both constants. The scheme also said it “must have y =”, so leaving off “y =” costs a mark.

3. Proof by induction is marked step by step. Paper 1 Question 2 gave B1 for checking the base case and B1 for the inductive hypothesis. That B1 was lost if you assumed the result for all values of k. The final A1 needed the three previous marks and the result stated in full.

4. Starred and dependent marks. A DM mark depends on an earlier starred mark (M1*). In Paper 2 Question 3(b), the M1* was for integrating the right-hand side using your answer to part (a). The DM1 for substituting the initial condition depended on it, and was M0 if the expression had been wrongly manipulated first.

5. FT allows follow-through; CAO does not. In Paper 1 Question 4(b), B1FT let you use your own cubic from part (a), as long as it had positive powers and like terms collected. Question 4(c) then marked both sums CAO, so only the correct values scored. On Paper 1 Question 6(c), follow-through applied only to the asymptotes mark.

6. AG means every step must be shown. Paper 1 Question 3(a) needed “some intermediate step”. Paper 2 Question 2(a), the reduction formula, needed limits and correct use of brackets. Paper 3 Question 6(a) gave A0 if any error was seen on the way to the printed result.

7. Accuracy and rejecting roots. Unless the question says otherwise, answers are accepted to 3 significant figures or 1 decimal place for degrees, and the scheme often says AWRT. Paper 3 Question 1(b) accepted AWRT 19.1° or 0.334 radians. Paper 3 Question 4(b) asked for θ to 3 significant figures. In Paper 3 Question 6(b), a negative root had to be explicitly rejected if you wrote it down.

8. Statistics conclusions must be in context and not definite. Paper 4 conclusions needed both variables named and uncertain language: “sufficient evidence to suggest”, not “prove”. The scheme also set caps for the wrong method. Using a biased variance in the Question 5 t-test capped you at 4 out of 7. A two-sample test instead of a paired one capped you at 2 out of 7.

Which papers should you sit?

The 2026 syllabus sets these routes:

• AS Level Further Mathematics: Paper 1 plus either Paper 3 (Further Mechanics) or Paper 4 (Further Probability & Statistics).
• Full A Level Further Mathematics: all four papers.

If you are taking AS, sit Paper 1 and the applied paper your school teaches. If you are taking the full A Level, practise all four. Paper 2 and Paper 1 together are 150 marks of pure maths.

How to use the 2026 papers as a mock

1. Keep them for last. Use older 9231 papers for topic practice and save May/June 2026 for a timed mock a few weeks before your exam.
2. Sit each paper in one go. Two hours for each pure paper, 1 hour 30 minutes for each applied paper.
3. Mark every M, A and B mark separately. Note where losing an M mark also cost you a dependent mark later in the question.
4. Rewrite your “show that” answers. Compare each line with the AG working in the scheme and add any step you skipped.
5. For Paper 4, write a conclusion template. Name both variables, use “evidence to suggest”, and check your test against the scheme’s caps.

More A Level Further Maths resources

• Cambridge A Level Further Maths past papers (9231, all years)
• A Level past papers 2026: all subjects

🔍 Looking for a specific past paper? Use our free Past Paper Finder to search by subject, year and exam board.

Need help with a 9231 topic? Book a free trial lesson with an A Level Further Maths tutor →

Frequently asked questions

Are the 9231 May/June 2026 past papers free? +

Yes. All four question papers (9231/12, 9231/22, 9231/32 and 9231/42) and their official mark schemes are free to download and print on the Tutopiya Learning Portal.

What topics came up in 9231 Paper 1 May/June 2026? +

Paper 1 had seven questions: summing a series with standard results and the method of differences, proof by induction of a divisibility result, matrix transformations, roots of a cubic, skew lines and planes, a rational curve with asymptotes and a modulus inequality, and a polar curve with an area found by substitution.

What came up in 9231 Paper 2 May/June 2026? +

Paper 2 covered a Maclaurin series for inverse sinh, a reduction formula, hyperbolic integration and a first-order differential equation, implicit differentiation, rectangle bounds for an integral, a second-order differential equation, complex numbers with de Moivre's theorem, and matrices including diagonalisation.

How is 9231 marked? +

The schemes use M marks for method, A marks for accuracy that need the method mark, and B marks that stand alone. Dependent marks (DM) are lost if the earlier starred mark is missing, and show-that results (AG) need every step shown.

Why can't I find 9231/11 or 9231/13 for 2026? +

Only variant 2 of the May/June 2026 papers is on the portal at the moment. Cambridge sets different variants for different time zones; variants 1 and 3 test the same syllabus in the same format, so variant 2 prepares you just as well.

When will the October/November 2026 9231 papers be available? +

Cambridge usually releases the October/November papers in the new year. We add them to the Learning Portal once they are published.

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