18 demanding questions: algebraic proof, functions, vectors, histograms, conditional probability and iteration.
🔺 Higher stretch — grade 8 and 9 topics
The top of a Higher paper is not the same content done faster — it is the statements that only appear at the top, and questions that hide two steps inside one instruction. This sheet gathers eighteen of them: algebraic proof and showing two expressions are equivalent, inverse and composite functions, transformations of graphs, the equation of a circle and its tangent, iteration, vectors used to construct a geometric argument, histograms with unequal class intervals, and conditional probability from a tree or a Venn diagram. It is deliberately hard and deliberately short. If you get half of it out, you are working where grades 8 and 9 are decided — and the worked answers are written to show the step that was actually being tested, not just the arithmetic around it.
- 1.A circle has centre (0, 0) and equation x² + y² = 36. Work out the coordinates of the two points where the circle crosses the y-axis.
- 2.The graph of y = f(x) passes through the point (2, 5). Write down the coordinates of the corresponding point on the graph of y = f(x − 4) + 1.
- 3.A photo printing service has two adverts for its price. Advert A: cost in pounds = 3(2n + 4) for n photos. Advert B: cost in pounds = 6n + 12. A customer says the two adverts always charge the same amount. Is the customer correct?
- 4.A circle has centre (0, 0) and equation x² + y² = 3721. The point (11, 60) lies on the circle. One of these is the gradient of the tangent to the circle at (11, 60). Work out which one.
- 5.Ben is asked to find the inverse of f(x) = 4 − 3x. He writes f⁻¹(x) = (4 − x)/3. Which statement about Ben's answer is correct?
- 6.The graph of y = f(x) has a minimum point at (4, −1). Write down the coordinates of the corresponding turning point on the graph of y = −f(x).
- 7.In a certain town, P(rain on Monday) = 0.3. If it rains on Monday, the probability that it also rains on Tuesday is 0.6. If it does not rain on Monday, the probability that it rains on Tuesday is 0.25. Work out the probability that it rains on Tuesday.
- 8.In triangle OAB, OA = a and OB = b. M is the midpoint of OA, and N is the midpoint of OB. Express the vector MN in terms of a and b.
- 9.Which expression is equivalent to 7x − 3(2x − 6)?
- 10.A cumulative frequency graph for the diameters, d mm, of 320 ball bearings is plotted from these points (upper class boundary, cumulative frequency): (10, 30), (20, 90), (30, 190), (40, 280), (50, 320). Estimate the diameter below which 90% of the ball bearings measure.
- 11.A student attempts to prove that the sum of any three consecutive integers is a multiple of 3. Line 1: Let the three consecutive integers be n, n + 1 and n + 2. Line 2: Their sum is n + (n + 1) + (n + 2) = 3n + 2. Line 3: 3n + 2 leaves a remainder of 2 when divided by 3, so it is not a multiple of 3. Line 4: So the sum of three consecutive integers is not always a multiple of 3. Which line contains the FIRST error?
- 12.A gym draws a histogram of the times, t minutes, that its members spend on one machine. The bar for 0 ≤ t < 10 has a frequency density of 1.8 per minute, the bar for 10 ≤ t < 25 has a frequency density of 3.2 per minute, and the bar for 25 ≤ t < 55 has a frequency density of 0.9 per minute. Members who spend 10 minutes or more on the machine pay an extra charge. Work out the number of members who pay the extra charge.
- 13.A rule multiplies the input by a fixed number and then adds a fixed number. An input of 1 gives an output of 5, and an input of 3 gives an output of 11. Work out the rule, writing the input as x and the output as y.
- 14.The iterative formula xₙ₊₁ = √(2xₙ + 3) is used repeatedly, starting from x₀ = 1. As n increases, the values of xₙ converge to a limit, L. Work out L.
- 15.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.
- 16.A surveyor marks two fixed points A and B, with position vectors OA = a and OB = b (in km) from a base station O. A relay mast P is to be placed on the line AB such that AP : PB = 3 : 2. Express the vector OP in terms of a and b.
- 17.A company's weekly profit, in £, is modelled by y = f(x), where x is the number of weeks since launch. The graph of y = f(x) has a maximum at (10, 45000). A rival company uses the same marketing strategy but starts trading 6 weeks later and has fixed costs £8000 higher every week, so its profit is modelled by y = f(x − 6) − 8000. In which week does the rival's maximum weekly profit occur, and what is it?
- 18.A circle has centre (0, 0) and equation x² + y² = 25. Work out the x-coordinates of the two points where the circle crosses the line y = 3.
What is on this worksheet?
The sheet holds 18 questions drawn from the MathsUK bank — the content areas covered: Algebra, Geometry and measures, Probability, Statistics (statements A6, A7, A13, A16, A20, G25, S3, P9). It is pitched at GCSE Higher and takes about 50 minutes to work through in full. It is built for independent practice, with full answers at the end for self-marking.
How to use the sheet well
- Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
- Do all 18 questions before checking — about 50 minutes is the guide, but there is no time pressure.
- Check the answers — press “Show answers” or print the answer page separately.
- Redo the questions you got wrong — twice as effective as doing 18 fresh ones.
- “New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.
Why this sheet helps
MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.
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