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 rule turns each input x into an output y. The inputs are x = −1, 0, 1, 2 and the matching outputs are y = 5, 3, 1, −1. Work out the rule.
- 2.The point (18, 24) lies on the circle x² + y² = 900, which has centre (0, 0). The tangent to the circle at (18, 24) crosses the y-axis at the point Q. Work out the y-coordinate of Q.
- 3.The masses, m kg, of 160 fish caught by a trawler in one day are grouped into classes of unequal width: 0 ≤ m < 10, 40 fish; 10 ≤ m < 30, 60 fish; 30 ≤ m < 45, 30 fish; 45 ≤ m < 50, 30 fish. A histogram is to be drawn from this table. Which set of frequency densities, listed in the same order as the classes above, is correct?
- 4.u is the column vector with top number 5 and bottom number 1. v is the column vector with top number 2 and bottom number 4. Work out u − v, giving your answer as a column vector in the form (top, bottom).
- 5.A circle has centre (0, 0) and passes through the point (12, 35). Work out the equation of the circle.
- 6.The graph of y = f(x) crosses the x-axis at x = −3 and x = 7, and crosses the y-axis at (0, 21). A second graph crosses the x-axis at x = −7 and x = 3, and crosses the y-axis at the same point, (0, 21). The second graph is y = g(x). Which of these could be the equation of g(x)?
- 7.f(x) = x + 3 and g(x) = 2x. Work out fg(x).y = x + 3
- 8.The point (6, 8) lies on the circle x² + y² = 100. Work out the gradient of the tangent to the circle at (6, 8).
- 9.The equation x² + 2x − 5 = 0 can be solved using the iterative formula xₙ₊₁ = 5/(xₙ + 2). The starting value is x₀ = 1, so x₁ is the value after the formula has been used once. Work out x₂ correct to 2 decimal places.
- 10.A call centre records the length, t seconds, of 100 calls: 0 ≤ t < 20, 15 calls; 20 ≤ t < 30, 24 calls; 30 ≤ t < 50, 40 calls; 50 ≤ t < 80, 21 calls. The manager's target is for a call to be finished in under 35 seconds. Estimate the number of calls that met the target.
- 11.Two ordinary fair dice are rolled and the two scores are added together. Given that the total is an even number, work out the probability that the total is 8.
- 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.y = 5 − 2x. Work out the value of x when y = 11.
- 14.A number machine multiplies its input by 3 and then adds 7. The output is 1. Work out the input.
- 15.A circle has centre (0, 0) and passes through the point (7, 24). Work out the equation of the circle.
- 16.Which of these equations represents the graph of y = 2ˣ translated by 3 units in the positive y-direction?
- 17.The point (−4, 3) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the equation of the tangent to the circle at (−4, 3), giving your answer in the form y = mx + c.
- 18.The graph of y = f(x) has a minimum turning point at (3, 2). Write down the coordinates of the minimum turning point of the graph of y = f(x) + 5.
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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