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.f(x) = x² − 1 and g(x) = 3x. Work out fg(4).
y = x² − 1
(a)36
(b)143
(c)11
(d)45
2.A drone flies from its base in three stages, each stage measured in metres east and north as a column vector. Stage 1 is the column vector with top number 30 and bottom number 40. Stage 2 is the column vector with top number −10 and bottom number 20. Stage 3 is the column vector with top number 15 and bottom number −5. Work out the column vector that would take the drone in a single straight flight back to its base from where it ends up.
(a)(−35, 55)
(b)(35, −55)
(c)(−35, −55)
(d)(35, 55)
3.Which of these is an identity?
(a)2(3x + 1) = 6x + 2
(b)5x + 1 = 5(x + 1)
(c)7 − x = x − 7
(d)4x − 3 = 3x + 5
4.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
5.A cuboid has a square base of side x metres and a height that is 3 m more than x. Its volume is 150 m³. This gives the equation x³ + 3x² − 150 = 0, which can be solved using the iterative formula xₙ₊₁ = ∛(150 − 3xₙ²). Taking x₀ = 4, work out x₂ correct to 2 decimal places.
(a)84.51
(b)4.39
(c)4.67
(d)6.32
6.In a survey, 120 adults were asked whether they have a driving licence. 70 of the adults are women and 50 are men. 45 of the women and 35 of the men have a driving licence. One of the adults who has a driving licence is picked at random. Work out the probability that this adult is a man.
(a)7/16
(b)5/12
(c)7/24
(d)7/10
7.A market stall's cost of hiring n tables is modelled by two formulas: Formula A: C = 3(2n + 5); Formula B: C = 6n + 15, where C is in pounds. A stallholder says the two formulas always give the same cost. Work out the cost given by each formula when n = 4, and use your results to decide whether the stallholder is correct.
(a)Formula A gives £39 and Formula B gives £39, but the stallholder is wrong because the two formulas use a different number of terms.
(b)Formula A gives £29 and Formula B gives £39, so the stallholder is wrong.
(c)Formula A gives £39 and Formula B gives £39, but this is only true when n = 4, so the stallholder is wrong.
(d)Formula A gives £39 and Formula B gives £39, and since 3(2n + 5) expands to 6n + 15 for every value of n, the stallholder is correct.
8.A student says that (x + 4)² is equivalent to x² + 16. For which value of x do the two expressions give the SAME result, making it look (misleadingly) like the student could be right?
(a)x = 4
(b)x = −4
(c)x = 0
(d)x = 8
9.A student is proving that (n + 1)² − n² is always an odd number. Which of these correctly completes the first line of algebra?
(a)(n + 1)² − n² = 1
(b)(n + 1)² − n² = 2n + 1
(c)(n + 1)² − n² = 2n
(d)(n + 1)² − n² = n² + 2n + 1
10.A circle has equation x² + y² = 25. Does the point (3, 4) lie on this circle?
(a)No, because (3 + 4)² = 49, not 25.
(b)Yes, because 3 + 4 = 7, and 7 is less than 25.
(c)No, because 3² + 4² = 17, not 25.
(d)Yes, because 3² + 4² = 25.
11.The masses, m kg, of 60 parcels are grouped like this: 0 ≤ m < 5, 22 parcels; 5 ≤ m < 10, 20 parcels; 10 ≤ m < 20, 9 parcels; 20 ≤ m < 30, 5 parcels; 30 ≤ m < 50, 4 parcels. Write down the class interval that contains the median mass.
(a)0 ≤ m < 5
(b)5 ≤ m < 10
(c)10 ≤ m < 20
(d)20 ≤ m < 30
12.A student claims: 'For every positive integer n, n² + n + 1 is a prime number.' Which value of n shows that this claim is false?
(a)n = 3
(b)n = 1
(c)n = 4
(d)n = 2
13.A bag contains 3 red counters and 5 blue counters. Three counters are taken out at random, one after another, without being replaced. Work out the probability that all three counters taken out are red.
(a)3/256
(b)9/112
(c)1/56
(d)27/512
14.A circle has centre (0, 0) and equation x² + y² = 64. Write down the radius of the circle.
(a)64
(b)8
(c)16
(d)32
15.In a histogram of the lengths, x cm, of some rods, the bar for 10 ≤ x < 30 has a frequency density of 3 per cm. The bar for 30 ≤ x < 45 is twice as tall as the bar for 10 ≤ x < 30. Work out the number of rods with a length in the class 30 ≤ x < 45.
(a)6
(b)45
(c)90
(d)120
16.The heights, h cm, of 80 plants are grouped like this: 0 ≤ h < 20, 14 plants; 20 ≤ h < 40, 22 plants; 40 ≤ h < 50, 16 plants; 50 ≤ h < 80, 28 plants. Write down the class interval that contains the lower quartile.
(a)40 ≤ h < 50
(b)50 ≤ h < 80
(c)20 ≤ h < 40
(d)0 ≤ h < 20
17.A circle has centre (0, 0) and equation x² + y² = 50. Work out the length of the diameter of the circle, correct to 1 decimal place.
(a)100.0
(b)25.0
(c)14.1
(d)7.1
18.y = 5 − 2x. Work out the value of x when y = 11.
(a)8
(b)3
(c)6
(d)−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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