Worked examples, ours
Core Pure and option examples we wrote
Two short Core Pure problems, an induction outline and a Further Mechanics 1 collision. None is a Pearson question.
Our Core Pure problem on complex numbers. Solve z2 − 4z + 13 = 0 and describe the roots on an Argand diagram.
Solving
The discriminant is 16 − 52 = −36, so z = (4 ± 6i) ÷ 2 = 2 ± 3i.
Real-coefficient quadratics always give conjugate pairs, which is a quick check.
Interpreting
The roots are reflections of each other in the real axis, at (2, 3) and (2, −3). Each has modulus √13.
Their sum 4 and product 13 match the coefficients, a second check worth one line.
Our Core Pure problem on matrices. For M = (2 1; 5 3), the determinant is 2 × 3 − 1 × 5 = 1, so M is invertible and M−1 = (3 −1; −5 2). Since det M = 1, M preserves area: a unit square maps to a parallelogram of area 1.
Our induction outline. Prove that the sum of r(r + 1) for r = 1 to n equals n(n + 1)(n + 2)/3. Base case n = 1: both sides are 2. Assume it for n = k. Adding (k + 1)(k + 2) gives k(k + 1)(k + 2)/3 + (k + 1)(k + 2) = (k + 1)(k + 2)(k + 3)/3, which is the formula for n = k + 1. Conclude in a sentence that the result holds for all positive integers n. Checked by us for n = 10: both sides are 440.
Our Further Mechanics 1 problem. A 3 kg sphere moving at 4 m s−1 hits a 1 kg sphere at rest, directly, with coefficient of restitution e = 0.5. Momentum: 3 × 4 = 3v1 + v2. Restitution: v2 − v1 = 0.5 × 4 = 2. Solving, v1 = 2.5 m s−1 and v2 = 4.5 m s−1, both in the original direction.
Six practice questions we wrote for 9FM0 students (answers checked in Python)| # | Our question | Paper | Answer |
|---|
| 1 | Find cosh(ln 3) exactly. | Core Pure | 5/3 |
| 2 | Find the Maclaurin series of ex sin x up to the x3 term. | Core Pure | x + x2 + x3/3 |
| 3 | The region under y = √x from x = 0 to 4 is rotated fully about the x-axis. Find the volume. | Core Pure | 8π |
| 4 | Find the area enclosed by the polar curve r = 2 cos θ. | Core Pure | π |
| 5 | X ~ Po(3). Find P(X = 2). | Further Statistics 1 | 0.224 |
| 6 | Apply one pass of quick sort to 9, 4, 7, 1, 8, choosing the middle item as pivot, as the specification directs. | Decision Mathematics 1 | Pivot 7; list becomes 4, 1, 7, 9, 8 |
All examples and practice questions were written by us on 31 October 2026. Pearson publishes past papers, mark schemes and examiners' reports for 9FM0 on its qualification page.
