Hardest GCSE maths questions: 10 Higher tier types, worked through

The hardest GCSE maths questions are usually the multi-step ones near the end of each Higher paper: proofs, vector arguments, functions, iteration and probability that turns into a quadratic. In June 2026 a grade 9 on Pearson Edexcel's Higher tier took 208 out of 240 marks, about 87%, so you could drop around 30 marks and still get the top grade. This guide works through ten of these question types. Each one has an original exam-style question, a full solution and the step where marks usually go.

What are the hardest GCSE maths questions?

They're the questions that join two or three topics together and ask you to write a full argument as well as an answer. In the Summer 2023 examiner report on Edexcel Paper 3H, examiners listed the algebraic proof and vectors questions among those that proved more of a challenge. AQA's report on June 2025 Paper 2H listed the equation of a tangent to a circle and a proof with composite and inverse functions among the areas where students struggled.

Source: Medly summary of common errors. The algebraic proof, tangent and histogram rows draw on the June 2025 Edexcel Paper 1H report and the June 2025 AQA Paper 2H report. Marks are for the example questions in this guide. Last checked: 10 November 2026.

Which GCSE maths topics are grade 9 only?

None. Grades come from your total mark, so there's no list of grade 9 topics. What exists is Higher-only content. Pearson prints it in bold in its specification, as its tiering guidance explains, and AQA puts it in a "higher content" column. Most of the ten question types below draw on that content.

Grade 9 is about how few marks you drop. These were Pearson Edexcel's Higher boundaries in June 2026:

Source: Pearson grade boundaries, June 2026. Last checked: 10 November 2026.

AQA's Higher grade 9 was 219 out of 240 in its June 2026 grade boundaries. Both boards set three Higher papers of 80 marks each. OCR's J560 has three Higher papers of 100 marks each.

How to answer GCSE vector proof questions

Try this one:

Question (5 marks)

OAB is a triangle with OA = a and OB = b. M is the midpoint of OA. N is on AB with AN : NB = 2 : 1. OB is extended to R so that OB = BR. Prove that M, N and R lie on a straight line.

Answer

1. OM = ½a.
2. AB = b - a, so ON = a + ⅔(b - a) = ⅓a + ⅔b.
3. MN = ON - OM = -⅙a + ⅔b = ⅙(-a + 4b).
4. OR = 2b, so MR = 2b - ½a = ½(-a + 4b).
5. MR = 3MN, so MN and MR are parallel. They share the point M, so M, N and R lie on a straight line.

Most lost marks come at step 5. Showing one vector is a multiple of the other proves they're parallel. You also need to say they share a point, and mark schemes usually keep the final mark for that sentence.

How to do algebraic proof and circle theorem proofs

Try these ones:

Question (3 marks)

Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.

Answer

Write the odd numbers as 2n + 1 and 2n + 3, where n is an integer. Then (2n + 3)² - (2n + 1)² = (4n² + 12n + 9) - (4n² + 4n + 1) = 8n + 8 = 8(n + 1). That's 8 times an integer, so it's a multiple of 8.

Trying a few numbers proves nothing. Writing them as x and x + 2 only works if you also say that x is odd, and then you need an extra step to show that 4(x + 1) is a multiple of 8. Starting with 2n + 1 and 2n + 3 is simpler. End with a sentence that links your algebra back to the question.

Question (4 marks)

A, B and C are points on a circle with centre O, as shown in the diagram. Prove that angle AOB is twice angle ACB.

Answer

1. Draw a line from C through O and extend it to a point D.
2. OA = OC because they're radii, so triangle OAC is isosceles and angle OAC = angle OCA = x.
3. Angle AOD is an exterior angle of triangle OAC, so it equals x + x = 2x.
4. In the same way, OB = OC, so angle OBC = angle OCB = y and angle BOD = 2y.
5. Angle AOB = angle AOD + angle BOD = 2x + 2y = 2(x + y). Angle ACB = angle OCA + angle OCB = x + y, so angle AOB = 2 × angle ACB.

Marks here are for reasons. Each step needs one, and you can't use the theorem you're proving as a reason.

Tangents to circles and 3D trigonometry questions

Try these ones:

Question (3 marks)

P(3, -4) lies on the circle x² + y² = 25. Find the equation of the tangent at P.

Answer

The radius OP has gradient -4 ÷ 3 = -4/3. The tangent is perpendicular to the radius, so its gradient is 3/4. Using y - (-4) = ¾(x - 3) gives y = ¾x - 25/4, or 4y = 3x - 25. Check: when x = 3, 4y = -16, so y = -4.

The usual slip is using the radius gradient, or flipping the fraction without changing the sign.

Question (3 marks)

A cuboid is 8 cm long, 6 cm wide and 5 cm tall. A is a corner of the base and G is the opposite top corner, directly above C. Find the angle between AG and the base. Give your answer to 1 decimal place.

Answer

AC = √(8² + 6²) = 10 cm. Triangle ACG has a right angle at C, with CG = 5 cm. So tan θ = 5 ÷ 10 and θ = 26.6° to 1 decimal place.

The angle with a plane is measured to the line directly below, which here is AC. Using an edge of the base instead loses most of the marks. Check that your calculator is in degrees.

Functions, iteration and quadratic inequalities worked examples

Try these ones:

Question (5 marks)

f(x) = 2x + 3 and g(x) = x². (a) Find f⁻¹(x). (b) Solve fg(x) = gf(x), giving your answers in the form a ± √b.

Answer

For (a), let y = 2x + 3, so x = (y - 3) ÷ 2 and f⁻¹(x) = (x - 3)/2. For (b), fg(x) means g first, so fg(x) = 2x² + 3 and gf(x) = (2x + 3)² = 4x² + 12x + 9. Setting them equal gives x² + 6x + 3 = 0. Completing the square, (x + 3)² = 6, so x = -3 ± √6.

Question (3 marks)

Solve 2x² + 5x ≤ 12.

Answer

Rearrange to 2x² + 5x - 12 ≤ 0 and factorise: (2x - 3)(x + 4) ≤ 0. The critical values are -4 and 3/2. The graph is U-shaped, so it's below the x-axis between them: -4 ≤ x ≤ 3/2. In set notation, that's {x : -4 ≤ x ≤ 3/2}. A quick sketch tells you which region you need.

Question (4 marks)

(a) Show that x³ + 4x - 1 = 0 has a solution between 0 and 1. (b) Using xₙ₊₁ = (1 - xₙ³) ÷ 4 with x₀ = 0, find x₃ to 4 decimal places.

Answer

For (a), the left side is -1 when x = 0 and 4 when x = 1. There's a change of sign, so there's a solution between 0 and 1. Write that sentence, because it carries a mark. For (b), x₁ = 0.25, x₂ = 0.24609375 and x₃ = 0.24627... = 0.2463. Use the ANS key so you never round in the middle.

Hard GCSE probability and histogram questions

Try these ones:

Question (5 marks)

A bag has n sweets. 6 are orange and the rest are yellow. Sam takes two sweets without replacing the first. The probability that both are orange is 1/3. Show that n² - n - 90 = 0 and find n.

Answer

1. For the first sweet, 6 of the n sweets are orange, so the probability it's orange is 6/n.
2. Sam doesn't put it back. That leaves 5 orange sweets out of n - 1, so the probability the second one is orange is 5/(n - 1).
3. Both picks need to be orange, so multiply: 6/n × 5/(n - 1) = 30 ÷ n(n - 1). On a tree diagram, this is the orange then orange branch.
4. Set this equal to 1/3. Cross-multiplying gives n(n - 1) = 90, so n² - n - 90 = 0.
5. Factorising, (n - 10)(n + 9) = 0. So n = 10, because you can't have a negative number of sweets.

Marks usually go on the second fraction. Without replacement, both the top and the bottom drop by one, because there's one fewer orange sweet and one fewer sweet in the bag.

Question (4 marks)

A histogram of journey times has three bars and no scale on the vertical axis. The bar for 0 < t ≤ 10 is 3 cm tall and represents 12 people. The bar for 10 < t ≤ 15 is 9 cm tall and the bar for 15 < t ≤ 30 is 2 cm tall. (a) How many people are there in total? (b) Estimate how many took more than 12 minutes.

Answer

Frequency is shown by area. The first bar has area 10 × 3 = 30 for 12 people, so each person is 2.5 units. The second bar is 5 × 9 = 45, or 18 people, and the third is 15 × 2 = 30, or 12 people. That's 42 in total. For (b), 12 to 15 minutes is 3/5 of the second bar, so 3/5 × 18 = 10.8. Add the 12 people in the last bar for about 23.

Pearson's June 2025 Paper 1H report said higher-attaining students were well prepared for histograms, so this is a topic you can secure with practice.

Where do students lose the most marks on hard GCSE maths questions?

Across these question types, the same habits cost marks:

• ending a proof without a final sentence
• rounding before the last step
• leaving the calculator in radians
• skipping the sketch on vectors, inequalities and 3D shapes
• leaving a part blank when the first line would earn a method mark

Proofs and long answers are hard to mark on your own. Medly AI covers GCSE Maths for AQA, Edexcel and OCR, and the free plan includes 5 marked questions, 20 AI tutor messages and 1 uploaded paper a day, with no card needed.

Frequently asked questions

What is the hardest topic in GCSE maths?

There's no official ranking. Proof questions, whether algebraic, vector or with functions, are among the questions that recent Pearson and AQA examiner reports list as challenging for students.

How many marks do you need for a grade 9 in GCSE maths?

In June 2026 it was 208 out of 240 on Pearson Edexcel's Higher tier and 219 out of 240 on AQA's. Boundaries change each year, so treat these as a guide.

Are the hardest questions always at the end of the paper?

Mostly, because papers tend to get harder as they go. A tricky question can still appear in the middle, so if you're stuck, move on and come back to it.

Is GCSE Further Maths harder than Higher GCSE maths?

There's no GCSE Further Maths in England (CCEA offers one in Northern Ireland), but AQA's Level 2 Certificate in Further Mathematics goes beyond Higher GCSE. It's a separate qualification that covers topics such as matrices and calculus.

How can I practise grade 9 maths questions?

A great idea would be to do the last five or six questions from real papers in our past papers directory, then mark them with the mark scheme. Topic question banks help with volume too, and our comparison of Medly AI and Physics & Maths Tutor shows how the two compare. The free Medly plan includes 5 marked questions, 20 AI tutor messages and 1 uploaded paper a day.

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