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.A student attempts to prove that the sum of any three consecutive integers is a multiple of 3. Line 1: Let the three consecutive integers be n, n + 1 and n + 2. Line 2: Their sum is n + (n + 1) + (n + 2) = 3n + 2. Line 3: 3n + 2 leaves a remainder of 2 when divided by 3, so it is not a multiple of 3. Line 4: So the sum of three consecutive integers is not always a multiple of 3. Which line contains the FIRST error?
(a)line 3
(b)line 4
(c)line 1
(d)line 2
2.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
3.A circle has centre (0, 0) and passes through the point (5, 12). Work out the equation of the circle.
(a)x² + y² = 119
(b)x² + y² = 169
(c)x² + y² = 17
(d)x² + y² = 13
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 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
6.The point (3, 4) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the gradient of the tangent to the circle at (3, 4).
(a)−4/3
(b)−3/4
(c)4/3
(d)3/4
7.A coastguard radar at the origin covers a circular region modelled by x² + y² = 400, where each unit represents 1 kilometre. A boat travels along the straight line that touches the boundary of the region at the point (12, 16). Work out the equation of the line the boat travels along.
(a)y = −3x/4 − 25
(b)y = −3x/4 + 25
(c)y = −4x/3 + 32
(d)y = 3x/4 + 7
8.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
9.A student is asked whether 3(x − 4) = 3x − 4 is an identity. Which statement gives the correct verdict and reason?
(a)It is an identity, because both sides begin with the term 3x, so they must be equivalent.
(b)It is not even an ordinary equation with a solution: expanding the left-hand side gives 3x − 12, and 3x − 12 = 3x − 4 would require −12 = −4, which is never true.
(c)It is not an identity, because the student forgot to multiply the 4 by 3 on both sides of the equation.
(d)It is not an identity, because the student needs to substitute a specific value of x before comparing the two sides.
10.The iterative formula xₙ₊₁ = √(2xₙ + 15) is used repeatedly, starting from x₀ = 1. Work out the value that xₙ approaches, correct to 2 decimal places.
(a)1.00
(b)5.00
(c)17.00
(d)4.99
11.f(x) = x³ − 5x − 6. Given that f(2.6) = −1.424 and f(2.7) = 0.183, work out what this shows about the equation x³ − 5x − 6 = 0.
y = x
(a)It has a solution between x = 2.6 and x = 2.7
(b)There is no root in this interval
(c)It has a solution between x = −2.6 and x = −2.7
(d)It shows that x = 2.6 is a solution of the equation
12.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
13.In a histogram of the masses, m grams, of some pebbles, the bar for the class 50 ≤ m < 80 has a frequency density of 2.4 per gram. Work out the number of pebbles in this class.
(a)12.5
(b)192
(c)72
(d)2.4
14.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
15.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
16.The equation x² − 3x − 7 = 0 can be solved using the iterative formula xₙ₊₁ = √(3xₙ + 7). The starting value is x₀ = 4, so x₁ is the value after the formula has been used once. Work out x₃ correct to 3 decimal places.
(a)4.535
(b)4.481
(c)13.149
(d)4.521
17.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
18.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

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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