Examples, ours
Three 9231 style problems, worked by us
One from Further Pure 1, one from Further Mechanics and one from Further Probability and Statistics. None is a Cambridge question.
Our Paper 1 problem, summation by differences. Find the sum of 1/(r(r + 1)) for r = 1 to n, and hence for n = 20. Write 1/(r(r + 1)) = 1/r − 1/(r + 1). Adding from r = 1 to n, every middle term cancels, leaving 1 − 1/(n + 1) = n/(n + 1). For n = 20 that is 20/21. As n grows the sum tends to 1, which is a typical follow-up part.
Our Paper 3 problem, Hooke's law. A light elastic string of natural length 0.8 m and modulus of elasticity 24 N hangs from a fixed point with a 1.2 kg particle at rest on its end. Take g = 10 m s−2 in this example. In equilibrium the tension equals the weight: 24x ÷ 0.8 = 12, so the extension is x = 0.4 m. The elastic potential energy stored is λx2 ÷ 2l = 24 × 0.16 ÷ 1.6 = 2.4 J.
Our Paper 4 problem, a probability generating function. A random variable X takes the values 0, 1 and 2 with probabilities 0.5, 0.3 and 0.2.
The generating function
G(t) = 0.5 + 0.3t + 0.2t2. Check G(1) = 1, as every probability generating function must.
G'(t) = 0.3 + 0.4t, so E(X) = G'(1) = 0.7.
The variance
G''(t) = 0.4, and Var(X) = G''(1) + G'(1) − [G'(1)]2 = 0.4 + 0.7 − 0.49 = 0.61.
Computing E(X2) directly, 0.3 + 0.8 = 1.1, gives 1.1 − 0.49 = 0.61 too.
Six practice questions we wrote for 9231 students (answers checked in Python)| No. | Question, ours | Paper | Answer |
|---|
| 1 | Find the sum of r3 for r = 1 to 10. | 1 | 3025 |
| 2 | Roots of x3 − 5x2 + 2x + 8 = 0 are α, β, γ. Find αβγ. | 1 | −8 |
| 3 | Use de Moivre's theorem to find (cos θ + i sin θ)4 when θ = π/8. | 2 | i |
| 4 | A particle on a light inextensible string moves in a vertical circle of radius 0.5 m. Find the least speed at the top for the string to stay taut, g = 10. | 3 | √5, about 2.24 m s−1 |
| 5 | X has density f(x) = 3x2 on [0, 1]. Find its median. | 4 | 0.51/3, about 0.794 |
| 6 | In a paired sign test, 8 of 10 differences are positive. Find P(X ≥ 8) under H0, one-tailed. | 4 | 0.0547 |
Our problems and practice set were written on 31 October 2026; none is a Cambridge question. The syllabus points teachers to Cambridge's School Support Hub for further 9231 support.
