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.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
2.The graph of y = x² − 4x is translated by the vector (3, 0). Work out the equation of the image, giving your answer in the form y = x² + bx + c.
y = x² − 4xy = x²
(a)y = x² − 7x + 21
(b)y = x² + 2x − 3
(c)y = x² − 4x + 3
(d)y = x² − 10x + 21
3.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
4.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
5.OABC is a parallelogram, with OA = a and OC = c. X is the point on AC such that AX is a third of XC. Express the vector OX in terms of a and c.
(a)(1/2)a + (1/2)c
(b)(1/4)a + (3/4)c
(c)a + (1/4)c
(d)(3/4)a + (1/4)c
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.An equation has exactly one value of x that makes it true, but an identity is true for every value of x. Which of these best explains why 3x + 5 = 20 is an equation rather than an identity?
(a)It has an = sign, and equations always use =.
(b)3x + 5 cannot be simplified, so it must be an equation.
(c)Only x = 5 satisfies 3x + 5 = 20, not every value of x.
(d)A number on the right-hand side makes it an equation.
8.The 120 pupils in Year 11 at a school sat a maths test. Their marks m are grouped into classes of unequal width: 0 ≤ m < 40, 12 pupils; 40 ≤ m < 60, 24 pupils; 60 ≤ m < 70, 36 pupils; 70 ≤ m ≤ 100, 48 pupils. A histogram is drawn for these data. Work out the frequency density of the class 40 ≤ m < 60.
(a)2.4 pupils per mark
(b)24 pupils per mark
(c)0.2 pupils per mark
(d)1.2 pupils per mark
9.A circular running track is modelled on a grid whose centre is the origin, where each unit represents 1 metre. A floodlight at the point (30, 40) stands on the edge of the track. A second floodlight stands on the edge of the track at the point (0, k), where k is positive. Work out the value of k.
(a)40
(b)70
(c)50
(d)2500
10.Which line of algebra shows that the sum of two consecutive odd numbers is always a multiple of 4?
(a)(2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1)
(b)(2n + 1) + (2n + 1) = 4n + 2 = 2(2n + 1)
(c)(2n + 1) + (2n + 3) = 4n + 3
(d)n + (n + 2) = 2n + 2 = 2(n + 1)
11.A rectangular sheet of metal measures 20 cm by 12 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open box of volume 200 cm³. This gives x³ − 16x² + 60x − 50 = 0, which can be solved using the iterative formula xₙ₊₁ = (16xₙ² − xₙ³ + 50)/60. The starting value is x₀ = 1, so x₁ is the value after the formula has been used once. Work out x₃, and use it to find the longer side of the base of the box correct to 1 decimal place.
(a)1.1 cm
(b)9.7 cm
(c)17.7 cm
(d)18.9 cm
12.Describe a sequence of two transformations that maps the graph of y = x² onto the graph of y = −(x − 5)².
y = x²
(a)Reflect in the y-axis, then translate +5 in x.
(b)Translate +5 in x, then reflect in the x-axis.
(c)Translate −5 in x, then reflect in the x-axis.
(d)Translate +5 in y, then reflect in the x-axis.
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.
(a)y = 3x − 2
(b)y = x + 4
(c)y = 3x + 2
(d)y = 2x + 3
14.A closed cylinder has radius r cm and height (r + 5) cm. Its volume is 300 cm³, giving the equation πr²(r + 5) = 300, which can be solved using the iterative formula rₙ₊₁ = √(300 ÷ (π(rₙ + 5))). Taking r₀ = 3, work out r₃ correct to 2 decimal places.
(a)3.45
(b)3.38
(c)3.36
(d)4.82
15.The equation x³ + 4x − 9 = 0 is to be solved by iteration. Work out which one of these iterative formulas comes from a correct rearrangement of that equation.
(a)xₙ₊₁ = (9 + xₙ³)/4
(b)xₙ₊₁ = ∛(9 − 4xₙ³)
(c)xₙ₊₁ = ∛(9 − 4xₙ)
(d)xₙ₊₁ = ∛(9 + 4xₙ)
16.The distances, d km, cycled by 180 riders in a charity sportive are summarised by these cumulative frequencies: d < 30, 20 riders; d < 60, 60 riders; d < 80, 120 riders; d < 100, 160 riders; d < 130, 180 riders. Use interpolation to estimate the median distance cycled.
(a)130
(b)90
(c)70
(d)60
17.f(x) = x³ − 3x² − 4. Work out the pair of consecutive integers between which the solution of f(x) = 0 lies.
y = x
(a)4 and 5
(b)2 and 3
(c)0 and 1
(d)3 and 4
18.In a class of 30 pupils, 18 study French, 14 study German and 7 study both French and German. A pupil who studies French is picked at random. Work out the probability that this pupil also studies German.
(a)1/2
(b)7/25
(c)7/30
(d)7/18

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