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.Which of these equations represents the graph of y = 2ˣ translated by 3 units in the positive y-direction?
(a)y = 2ˣ + 3
(b)y = 2ˣ⁺³
(c)y = 2ˣ − 3
(d)y = 2(x + 3)
2.In a group of 50 people, 32 own a car, 20 own a bicycle and 9 own both a car and a bicycle. A person who owns a bicycle is picked at random. Work out the probability that this person does not own a car.
(a)11/50
(b)11/20
(c)9/20
(d)11/32
3.The point (6, 8) lies on the circle x² + y² = 100. Work out the gradient of the tangent to the circle at (6, 8).
(a)−3/4
(b)−4/3
(c)3/4
(d)4/3
4.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
5.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
6.A circle has equation x² + y² = 49. Work out the coordinates of the point(s) on the circle where the tangent is horizontal.
(a)(0, 7) and (0, −7)
(b)(7, 0) and (−7, 0)
(c)(7, 0) only
(d)(0, 7) only
7.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.
8.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.
(a)900
(b)50
(c)37.5
(d)1.25
9.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of OC, and N is the point on AC such that AN is twice NC. By finding the vectors MN and MB, show that M, N and B are collinear, and give the scalar k such that MN = k × MB.
(a)1/3
(b)2/3
(c)1/6
(d)3
10.A bead starts at position (2, −1) on a grid, in centimetres. It is moved by the column vector u, with top number 3 and bottom number 5, and then moved by the column vector v, with top number −7 and bottom number 2. Work out the coordinates of the bead's final position.
(a)(−5, 1)
(b)(5, 4)
(c)(−4, 7)
(d)(−2, 6)
11.240 people were asked whether they had been to the cinema in the last month. 100 of the people are under 30 years old and 140 are aged 30 or over. 65 of the under 30s and 35 of those aged 30 or over had been to the cinema. One of the people aged 30 or over is picked at random. Work out the probability that this person had been to the cinema.
(a)7/48
(b)1/4
(c)3/4
(d)7/20
12.The equation x³ − 5x − 3 = 0 can be rearranged to give an iterative formula of the form xₙ₊₁ = ∛(…). Work out which one of these is a correct rearrangement.
(a)xₙ₊₁ = ∛(5xₙ + 3)
(b)xₙ₊₁ = ∛(5xₙ − 3)
(c)xₙ₊₁ = (5xₙ + 3) ÷ 3
(d)xₙ₊₁ = ∛(3xₙ + 5)
13.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?
(a)40, 60, 30, 30
(b)4, 3, 2, 3
(c)4, 3, 0.5, 6
(d)4, 3, 2, 6
14.The point (9, 40) lies on the circle x² + y² = 1681, which has centre (0, 0). Work out the equation of the tangent to the circle at (9, 40), giving your answer in the form ax + by = c.
(a)9x − 40y = 1681
(b)9x + 40y = 1681
(c)40x + 9y = 1681
(d)9x + 40y = 41
15.Describe the single transformation that maps the graph of y = x² onto the graph of y = x² + 3.
y = x²y = x² + 3
(a)A translation by vector (0, 3)
(b)A translation by vector (0, −3)
(c)A translation by vector (3, 0)
(d)A stretch parallel to the y-axis, scale factor 3
16.The graph of y = f(x) passes through the point (0, 4). Work out the y-coordinate of the point where the graph of y = −f(x) + 5 crosses the y-axis.
(a)−4
(b)−9
(c)1
(d)9
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.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)

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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