Worked examples, ours
One curve, three command words: worked examples we wrote
The same mathematics, asked three ways, needs three different amounts of writing. These problems are ours, not OCR's.
Our curve. Take y = x e−2x. We ask for its stationary point under three different instructions.
"Find the coordinates"
A graphical calculator's maximum function gives (0.5, 0.184). Under OCR's meaning of Find, that is a complete answer, though writing dy/dx = 0 first costs only a line.
"Determine the coordinates"
Justify it: dy/dx = e−2x − 2x e−2x = e−2x(1 − 2x). Since e−2x > 0, dy/dx = 0 only when x = 1/2, so y = 1/(2e), about 0.184.
"Exact coordinates"
The answer must be (1/2, 1/(2e)). Giving 0.184 here loses the final mark, because OCR defines exact as not rounded.
Our second problem, detailed reasoning. In this question you are told to show detailed reasoning. Solve 5x+1 = 32x, giving x to 3 significant figures.
- Take natural logarithms of both sides: (x + 1) ln 5 = 2x ln 3.
- Collect the x terms: x(2 ln 3 − ln 5) = ln 5.
- So x = ln 5 ÷ (2 ln 3 − ln 5) = 1.6094... ÷ 0.5878...
- x = 2.74 to 3 significant figures. The intermediate evaluation in step 3 is shown on purpose: a solver key alone would not count as a complete analytical method.
Our third problem, detailed reasoning with an integral. Evaluate the integral from 1 to 4 of (3√x − 2/x2) dx. Integrating gives [2x3/2 + 2/x] from 1 to 4. At 4 this is 16 + 0.5 = 16.5; at 1 it is 2 + 2 = 4. The value is 12.5. Under detailed reasoning the antiderivative and both substitutions must be on the page; the calculator's integral key may only be used to check.
Our fourth problem, Paper 3 style. A stone leaves flat ground at 25 m s−1, 35° up from the horizontal. With g = 9.8 m s−2, its upward speed starts at 25 sin 35° = 14.34 m s−1, so it lands after 2 × 14.34 ÷ 9.8 = 2.93 s, having travelled 25 cos 35° × 2.93 = 59.9 m sideways. The modelling sentence earns its own credit: the stone is a particle and air resistance is ignored, so a real stone would fall short.
Six practice questions we wrote for H240 students (answers checked in Python)| # | Our question | Paper | Answer |
|---|
| 1 | Solve 2 cos2 x + 3 sin x = 3 for 0° ≤ x ≤ 360°, showing detailed reasoning. | 1 or any pure section | x = 30°, 90°, 150° |
| 2 | Find the sum to infinity of 6 + 3 + 1.5 + ... | Pure | 12 |
| 3 | Differentiate x3 ln x and evaluate the derivative at x = e, exactly. | Pure | 4e2 |
| 4 | X ~ B(25, 0.2). Find P(X ≤ 2). | 2 | 0.0982 |
| 5 | T ~ N(64, 52). Find P(60 < T < 72). | 2 | 0.733 |
| 6 | A 4 kg block is pulled across a rough floor (μ = 0.4) by a horizontal 30 N force. Find its acceleration, g = 9.8. | 3 | 3.58 m s−2 |
All examples and practice questions above were written by us on 31 October 2026; none is an OCR question. OCR publishes sample assessment material and past papers on its H240 qualification page.
