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 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.
(a)1.36
(b)1.49
(c)1.37
(d)1.67
2.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
3.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.
4.240 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
5.f(x) = 2x² + 1 and g(x) = x − 1. Work out fg(x), giving your answer in expanded form.
y = 2x² + 1
(a)2x² − 4x + 2
(b)2x² − 1
(c)2x²
(d)2x² − 4x + 3
6.For two events A and B, P(A) = 0.6 and P(A and B) = 0.15. Work out P(B | A).
(a)0.15
(b)0.09
(c)0.25
(d)0.45
7.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
8.A garage services 200 cars in one week. 120 of the cars are petrol cars and the rest are diesel cars. 30 of the petrol cars and 24 of the diesel cars fail the service. One of the cars that failed is picked at random. Work out the probability that it is a diesel car.
(a)4/9
(b)3/25
(c)3/10
(d)2/5
9.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) − 6 crosses the y-axis.
(a)10
(b)−6
(c)4
(d)−2
10.In a year group of 60 pupils, 25 play football, 18 play tennis and 8 play both football and tennis. A pupil who plays football or tennis or both is picked at random. Work out the probability that this pupil plays both sports.
(a)8/43
(b)8/35
(c)8/25
(d)2/15
11.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
12.A plumber charges a call-out fee of £30 plus £25 per hour worked. Work out the total charge for a job that takes 3 hours.
(a)£165
(b)£105
(c)£75
(d)£90
13.Forty pupils in class P and forty pupils in class Q each solved a puzzle. The times, in seconds, were summarised using cumulative frequency. For class P the lower quartile is 24, the median is 38 and the upper quartile is 46. For class Q the lower quartile is 30, the median is 35 and the upper quartile is 44. Write down the statement that correctly compares the two classes.
(a)Q was slower on average and less consistent
(b)Q was faster on average and more consistent
(c)Q was faster on average and less consistent
(d)Q was slower on average and more consistent
14.A proof that (n + 3)² − (n − 3)² is always a multiple of a certain number begins: Line 1: (n + 3)² − (n − 3)² = (n² + 6n + 9) − (n² − 6n + 9). Which expression correctly completes Line 2?
(a)18
(b)12n
(c)6n
(d)2n² + 18
15.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
16.The point (4, 2) lies on the circle x² + y² = 20. Work out the equation of the tangent to the circle at (4, 2).
(a)y = −0.5x + 4
(b)y = −2x − 10
(c)y = 2x − 6
(d)y = −2x + 10
17.Triangle ABC is translated by the column vector with top number 3 and bottom number −5 to form triangle A′B′C′. Triangle A′B′C′ is then translated by the column vector with top number −7 and bottom number 2 to form triangle A″B″C″. Work out the single column vector that translates triangle ABC directly to triangle A″B″C″.
(a)(10, −7)
(b)(4, −3)
(c)(−4, 3)
(d)(−4, −3)
18.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

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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