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 graph of y = f(x) crosses the x-axis at x = −3 and x = 7, and crosses the y-axis at (0, 21). A second graph crosses the x-axis at x = −7 and x = 3, and crosses the y-axis at the same point, (0, 21). The second graph is y = g(x). Which of these could be the equation of g(x)?
(a)y = f(x) − 4
(b)y = f(−x)
(c)y = −f(x)
(d)y = −f(−x)
2.The graph of y = f(x) has a minimum turning point at (3, 2). Write down the coordinates of the minimum turning point of the graph of y = f(x) + 5.
(a)(3, 7)
(b)(−2, 2)
(c)(8, 2)
(d)(3, −3)
3.p is the column vector with top number 2 and bottom number 3. q is the column vector with top number −1 and bottom number 4. Work out 2p + q, giving your answer as a column vector in the form (top, bottom).
(a)(5, 2)
(b)(3, 10)
(c)(0, 11)
(d)(1, 7)
4.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)
5.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.
6.A histogram shows the times, t minutes, taken by 120 visitors to complete an escape room. The bar for 0 ≤ t < 10 has a frequency density of 5 visitors per minute, the bar for 10 ≤ t < 20 has a frequency density of 2 visitors per minute, the bar for 20 ≤ t < 40 has a frequency density of 1.5 visitors per minute, and the bar for 40 ≤ t < 60 has a frequency density of 1 visitor per minute. Work out which class contains the median time.
(a)The class with times from 0 up to 10
(b)The class with times from 20 up to 40
(c)The class with times from 10 up to 20
(d)The class with times from 40 up to 60
7.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
8.The function f(x) = x² for all real values of x has no inverse function, but g(x) = x² for x ≥ 0 does have one. Which statement correctly explains this?
y = x²
(a)Restricting any domain always creates an inverse
(b)Squares can never be reversed, under any conditions
(c)g's outputs are positive; f's could be negative
(d)g is one-to-one: f(3) = f(−3), removed by x ≥ 0
9.The equation x² = 5x − 3 is to be solved using iteration. Work out which of these iterative formulas comes from a correct rearrangement of the equation.
(a)xₙ₊₁ = 5(xₙ² + 3)
(b)xₙ₊₁ = (xₙ² + 3) ÷ 5
(c)xₙ₊₁ = (xₙ + 3) ÷ 5
(d)xₙ₊₁ = (xₙ² − 3) ÷ 5
10.Which expression is equivalent to 7x − 3(2x − 6)?
(a)13x − 18
(b)x − 18
(c)x + 18
(d)x + 6
11.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
12.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
13.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
14.In a histogram of the heights, h cm, of 90 seedlings, the class 12 ≤ h < 18 contains 36 seedlings. Work out the frequency density for this class.
(a)216
(b)6
(c)15
(d)3
15.A box holds 5 blue pens and 7 black pens. Two pens are taken at random, one at a time, and are not put back. The first pen taken is black. Work out the probability that the second pen taken is blue.
(a)5/12
(b)6/11
(c)4/11
(d)5/11
16.An allotment is in the shape of a rectangle. Its length is 5 m more than its width, x metres, and its area is 20 m². This gives x² + 5x − 20 = 0, which can be solved using the iterative formula xₙ₊₁ = 20 ÷ (xₙ + 5). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₂ correct to 2 decimal places.
(a)2.55
(b)−1.71
(c)2.00
(d)2.86
17.A scientist has grouped the lifetimes, in hours, of 300 batteries into classes of unequal width. She wants a diagram in which the number of batteries in a class is given by the area of its bar. Write down the type of diagram she should draw.
(a)A cumulative frequency diagram, with running totals plotted
(b)A bar chart, with frequency up the vertical axis
(c)A histogram, with frequency density up the vertical axis
(d)A pie chart, with one sector for each class interval
18.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

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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