A passage we wrote
Practise the comprehension section: a short passage and questions, ours
This is a passage we wrote in the style of a comprehension task, with our own questions and answers. It is not an OCR or MEI passage.
Our passage: square roots by averaging. To estimate √a for a positive number a, start with any positive guess x0. If x0 is too large, then a/x0 is too small, and the truth lies between them. A sensible next guess is their mean, x1 = ½(x0 + a/x0), and the process repeats as xn+1 = ½(xn + a/xn). After the first step every estimate is at least √a, by the inequality between the arithmetic and geometric means (1). The method improves very quickly: once an estimate is close, each step roughly doubles the number of correct decimal places (2).
Question 1, check
With a = 5 and x0 = 2, find x1 and x2. Answer: x1 = ½(2 + 2.5) = 2.25, and x2 = ½(2.25 + 5/2.25) = 2.236111..., already within 0.00005 of √5 = 2.236068...
Question 2, fill the gap at (1)
Show xn+1 ≥ √a. Answer: xn+1 − √a = (xn2 − 2xn√a + a) ÷ 2xn = (xn − √a)2 ÷ 2xn, which is never negative for xn > 0.
Question 3, explain (2)
The same identity shows the new error is the old error squared, divided by 2xn. For √5 the errors run 0.0139, then 0.0000431, then about 0.0000000004: squaring a small error roughly doubles its count of zeros.
Notice what the questions reward. Question 1 is routine technique. Question 2 asks the student to supply algebra the passage only asserted, which means reading the reference marker, finding the claim, and proving it in two lines. Question 3 asks for an explanation in words, backed by the algebra from Question 2. That is the pattern we drill: technique, then argument, then explanation, in that order, on a fresh passage roughly every fortnight in Year 13.
Our Paper 1 example. A 3 kg block on a smooth horizontal table is joined by a light inextensible string over a smooth pulley to a 2 kg mass hanging freely. Taking g = 9.8 m s−2, the equation for the whole system is 2g = 5a, so a = 3.92 m s−2, and for the block alone T = 3a = 11.76 N. The words smooth, light and inextensible are modelling assumptions, and naming them is part of the answer.
Six practice questions we wrote for H640 students (answers checked in Python)| # | Our question | Paper | Answer |
|---|
| 1 | Express (5x + 1)/((x − 1)(x + 2)) in partial fractions. | 1, 2 or 3 | 2/(x − 1) + 3/(x + 2) |
| 2 | Find the sum of 3r − 1 for r = 1 to 20. | Any | 610 |
| 3 | Solve cos 2x = 0.5 for 0° ≤ x ≤ 360°. | Any | 30°, 150°, 210°, 330° |
| 4 | A particle has velocity v = 6t − t2 m s−1. How far does it travel between t = 0 and t = 6? | 1 | 36 m |
| 5 | X ~ B(15, 0.35). Find P(X ≥ 8). | 2 | 0.113 |
| 6 | Using xn+1 = ½(xn + 7/xn) with x0 = 3, find x1. | 3 | 8/3, about 2.667 |
The passage, questions, examples and practice set were written by us on 31 October 2026. OCR publishes past papers, including comprehension papers, on its H640 qualification page.
