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.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
2.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
3.A rule turns each input x into an output y. An input of 1 gives an output of 1, an input of 2 gives an output of 4 and an input of 3 gives an output of 9. Work out the rule.
(a)y = 2x
(b)y = 3x
(c)y = x²
(d)y = 3x − 2
4.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
5.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
6.The masses, m kg, of 150 boxes are summarised by these cumulative frequencies: m < 5, 18 boxes; m < 10, 52 boxes; m < 20, 96 boxes; m < 35, 130 boxes; m < 60, 150 boxes. Work out the number of boxes with a mass in the class 10 ≤ m < 20.
(a)96
(b)54
(c)34
(d)44
7.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
8.The times, t seconds, taken by 142 competitors to complete a lap are grouped like this: 0 ≤ t < 10, 20 competitors; 10 ≤ t < 25, 12 competitors; 25 ≤ t < 45, 50 competitors; 45 ≤ t < 75, 60 competitors. A histogram is drawn. Write down the class whose bar is the tallest.
(a)10 ≤ t < 25
(b)45 ≤ t < 75
(c)0 ≤ t < 10
(d)25 ≤ t < 45
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 point (5, −12) lies on the circle x² + y² = 169, which has centre (0, 0). Work out the equation of the tangent to the circle at (5, −12), giving your answer in the form y = mx + c.
(a)y = −(12/5)x
(b)y = (5/12)x − 25/12
(c)y = (5/12)x − 169/12
(d)y = −(5/12)x − 119/12
11.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
12.The iterative formula xₙ₊₁ = xₙ³ − 2 is used repeatedly, starting from x₀ = 2. Which of these describes what happens to the sequence of values as n increases?
(a)It stays constant, since x₀ = 2 is a fixed point.
(b)It oscillates between two fixed values.
(c)It converges to a fixed value close to 2.
(d)It diverges, growing rapidly without limit.
13.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.
14.A market stall's cost of hiring n tables is modelled by two formulas: Formula A: C = 3(2n + 5); Formula B: C = 6n + 15, where C is in pounds. A stallholder says the two formulas always give the same cost. Work out the cost given by each formula when n = 4, and use your results to decide whether the stallholder is correct.
(a)Formula A gives £39 and Formula B gives £39, but the stallholder is wrong because the two formulas use a different number of terms.
(b)Formula A gives £29 and Formula B gives £39, so the stallholder is wrong.
(c)Formula A gives £39 and Formula B gives £39, but this is only true when n = 4, so the stallholder is wrong.
(d)Formula A gives £39 and Formula B gives £39, and since 3(2n + 5) expands to 6n + 15 for every value of n, the stallholder is correct.
15.A circle has centre (0, 0) and equation x² + y² = 36. Work out the coordinates of the two points where the circle crosses the y-axis.
(a)(0, 36) and (0, −36)
(b)(0, 6) and (0, −6)
(c)(0, 18) and (0, −18)
(d)(6, 0) and (−6, 0)
16.A circle has centre (0, 0) and radius 9. Work out the equation of the circle.
(a)x² + y² = 18
(b)x² + y² = 81
(c)x² − y² = 81
(d)x² + y² = 9
17.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
18.The graph of y = sin x is reflected in the x-axis to give the graph of y = g(x). Work out the value of g(90), where x is measured in degrees.
y = sin(x)
(a)−1
(b)1
(c)−0.5
(d)0

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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