18 demanding questions: algebraic proof, functions, vectors, histograms, conditional probability and iteration.

🔺 Higher stretch — grade 8 and 9 topics

Maths · GCSE Higher · 18 questions · ~50 min

MathsUKwww.geekhero.co.uk

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 box contains 9 red balls and 11 green balls. Two balls are taken out at random, one after the other, without being replaced. Given that both balls taken out are the same colour, work out the probability that both balls are red.
(a)55/91
(b)18/95
(c)9/20
(d)36/91
2.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?
(a)Wrong: sign kept, giving f⁻¹(x) = (−4 − x)/3
(b)Correct, but only because f is its own inverse
(c)Correct: 3y = 4 − x gives f⁻¹(x) = (4 − x)/3
(d)Wrong: correct inverse is f⁻¹(x) = (x − 4)/3
3.A rule turns each input x into an output y. The inputs are x = 0, 1, 2, 3 and the outputs are y = 4, 7, 10, 13. Work out the output when x = 5.
(a)19
(b)21
(c)22
(d)16
4.A histogram shows the ages, in years, of 250 members of a running club. The bar for the class 10 ≤ age < 20 has a frequency density of 4.5 members per year, the bar for 20 ≤ age < 35 has a frequency density of 6 members per year, and the bar for 50 ≤ age < 70 has a frequency density of 2.75 members per year. Work out the frequency of the remaining class, 35 ≤ age < 50.
(a)250
(b)115
(c)190
(d)60
5.The point (20, 21) lies on the circle x² + y² = 841, which has centre O(0, 0). The tangent to the circle at (20, 21) crosses the x-axis at P and the y-axis at Q. Work out the area of triangle OPQ, correct to 1 decimal place.
(a)842.0
(b)2.0
(c)1684.0
(d)580.7
6.The equation x³ − 3x − 4 = 0 has a root near x = 2. Four students each try a different iterative formula, all starting from x₀ = 2: xₙ₊₁ = ∛(3xₙ + 4); xₙ₊₁ = (xₙ³ − 4) ÷ 3; xₙ₊₁ = 4 ÷ (xₙ² − 3); xₙ₊₁ = xₙ³ − 2xₙ − 4. Only one of these formulas keeps producing values that settle near the root when it is repeated. Work out x₁, correct to 3 decimal places, for the formula that does this.
(a)2.154
(b)1.333
(c)0.000
(d)4.000
7.The graph of y = f(x) has a minimum turning point at (3, −5), crosses the x-axis at x = 1, and crosses the y-axis at (0, −2). Exactly one of these statements about the graph of y = −f(x + 2) is true. Which statement is true?
(a)y = −f(x + 2) crosses the x-axis at x = 3.
(b)y = −f(x + 2) has a maximum turning point at (5, 5).
(c)y = −f(x + 2) has a minimum turning point at (1, 5).
(d)y = −f(x + 2) has a maximum turning point at (1, 5).
8.A rectangular garden has width w metres and length (w + 3) metres. A gardener writes its perimeter as 2w + 3. Which statement corrects the gardener's mistake?
(a)It is 2(w + (w + 3)) = 4w + 6, not 2w + 3.
(b)It is 3w + 6, since the length doubles but not the width.
(c)It is w + (w + 3) = 2w + 3, matching the gardener.
(d)It is 4(w + 3) = 4w + 12, treating it as a square.
9.The graph of y = f(x) has x-intercepts at x = −2 and x = 6 and crosses the y-axis at (0, −12). Work out the x-intercepts and the y-intercept of y = −f(x).
(a)x = 2, x = −6; y-intercept (0, 12)
(b)x = 2, x = −6; y-intercept (0, −12)
(c)x = −2, x = 6; y-intercept (0, −12)
(d)x = −2, x = 6; y-intercept (0, 12)
10.A student says 4(2x − 3) is equivalent to 8x − 3. Which statement gives the correct verdict and reason?
(a)False — 4(2x − 3) = 8x − 12, not 8x − 3.
(b)True — 4(2x − 3) = 8x − 3 when the bracket is expanded.
(c)False — 4(2x − 3) = 2x − 12, only the −3 is multiplied.
(d)True — 4(2x − 3) = 8x − 3, since both are linear.
11.f(x) = x³ − 3x − 20, and the equation f(x) = 0 has exactly one solution. Work out the pair of consecutive integers between which that solution lies.
y = x
(a)2 and 3
(b)3 and 4
(c)4 and 5
(d)6 and 7
12.In a histogram of the masses, m grams, of some pebbles, the bar for the class 50 ≤ m < 80 has a frequency density of 2.4 per gram. Work out the number of pebbles in this class.
(a)12.5
(b)192
(c)72
(d)2.4
13.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.
(a)49
(b)79
(c)69
(d)39
14.e is the column vector with top number 5 and bottom number k. f is the column vector with top number 15 and bottom number 6. Given that f is 3 times e, work out the value of k.
(a)18
(b)9
(c)3
(d)2
15.A group of 80 pupils was asked whether they walk to school. 45 of the pupils are girls and 35 are boys. 30 of the girls walk to school and 14 of the boys walk to school. One of the girls is picked at random. Work out the probability that she walks to school. Give your answer in its simplest form.
(a)3/8
(b)11/20
(c)1/3
(d)2/3
16.A designer enlarges a drawing of a model car for a poster. She first enlarges the drawing by a scale factor of 1.5, and then enlarges that result by a further scale factor of 2. On the original drawing, the position of a wheel relative to the front bumper is given by the column vector with top number 4 and bottom number −3, in centimetres. What is the corresponding column vector on the poster, in centimetres?
(a)(7, 0)
(b)(12, −9)
(c)(12, −3)
(d)(4, −9)
17.A circle has centre (0, 0) and equation x² + y² = 100. A straight line is drawn from the origin through the point P(8, 9). Work out whether P lies inside, on, or outside the circle.
(a)P lies inside the circle
(b)P lies on the circle
(c)Cannot be determined from this information
(d)P lies outside the circle
18.A circle has centre (0, 0) and passes through the point (7, 24). Work out the equation of the circle.
(a)x² + y² = 49
(b)x² + y² = 625
(c)x² + y² = 31
(d)x² + y² = 576

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.

Similar worksheets worth a look

🧮 Paper 1 non-calculator warm-up — Higher · 20 questions · ~25 min
⚖️ Foundation to Higher crossover check · 20 questions · ~35 min
📈 Quadratics: factorise, complete the square, formula · 24 questions · ~45 min
⚗️ Ratio and proportion mastery — Higher · 24 questions · ~45 min