Quick Answer: Grade 8–9 GCSE maths questions are hard because they combine two or three skills in one problem, such as algebra inside a probability question or surds inside an area question. Below are 10 original problems in the style of the hardest questions on AQA, Edexcel and Cambridge IGCSE Extended papers, each with a full worked answer and the mistake that costs students the most marks.
This post is for Year 10 and Year 11 students aiming for a grade 8 or 9 (or an A* at IGCSE), and for parents who want to see what the top of the paper actually looks like. If you want a gentler warm-up first, start with our post on hard maths questions that actually make you better at maths.
All ten questions are written for this post. They follow the style of exam questions but are not taken from past papers. Try each one on paper before you read the answer.
10 Hard GCSE Maths Questions With Answers
1. Algebraic proof: consecutive odd numbers
Prove that the difference between the squares of any two consecutive odd numbers is always a multiple of 8.
Answer:
Let the two consecutive odd numbers be 2n − 1 and 2n + 1, where n is an integer.
(2n + 1)² − (2n − 1)² = (4n² + 4n + 1) − (4n² − 4n + 1) = 8n
8n = 8 × n, and n is an integer, so the difference is always a multiple of 8. ∎
Common mistake: Writing the odd numbers as n and n + 2. That doesn't guarantee they're odd, so the proof gets few or no marks. Another mistake is testing a few numbers (3 and 5, 7 and 9) and calling that a proof. Examples are not a proof.
Typical of: AQA and Edexcel GCSE Higher. Algebraic proof often appears as a 3- or 4-mark question late on the paper.
2. Circle theorems: tangent and chord
A, B and C are points on a circle with centre O. TA is a tangent to the circle at A. Angle TAB = 64°. C lies on the major arc AB.
Find (a) angle OAB, (b) angle AOB and (c) angle ACB. Give a reason for each step.
Answer:
(a) A tangent meets a radius at 90°, so angle OAT = 90°. Angle OAB = 90° − 64° = 26°.
(b) OA = OB (both radii), so triangle OAB is isosceles and angle OBA = 26°. Angle AOB = 180° − 26° − 26° = 128°.
(c) The angle at the centre is twice the angle at the circumference, so angle ACB = 128° ÷ 2 = 64°. (This also follows directly from the alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.)
Common mistake: Getting the numbers right but writing weak reasons. "Because of the circle" earns nothing. You need the theorem in words, such as "the tangent is perpendicular to the radius". Also, if C were on the minor arc, angle ACB would be 180° − 64° = 116°, so read where C is.
Typical of: All three boards. Circle theorems with "give reasons" are standard on AQA, Edexcel and Cambridge IGCSE Extended papers.
3. Vectors: proving three points are collinear
OAB is a triangle with OA = a and OB = b. M is the midpoint of OA. N is the point on AB such that AN : NB = 2 : 1. The line OB is extended to X so that OB = BX.
Prove that M, N and X lie on a straight line.
Answer:
OM = ½a
ON = OA + AN = a + ⅔(b − a) = ⅓a + ⅔b
MN = ON − OM = ⅓a + ⅔b − ½a = −⅙a + ⅔b
OX = 2b, so MX = OX − OM = 2b − ½a
MX = 3(−⅙a + ⅔b) = 3 × MN
MX is a multiple of MN, so the lines are parallel. They share the point M, so M, N and X are collinear. ∎
Common mistake: Showing that one vector is a multiple of the other and stopping there. Parallel isn't the same as collinear. You must also say the vectors share a common point. Students also often write AB as a − b instead of b − a.
Typical of: Edexcel and AQA GCSE Higher, and Cambridge IGCSE Extended. Collinearity proofs are usually the final part of a multi-part vectors question.
4. Iteration: finding a root
(a) Show that the equation x³ − 5x − 1 = 0 has a root between x = 2 and x = 3.
(b) Show that the equation can be rearranged to x = ∛(5x + 1).
(c) Starting with x₀ = 2, use the iteration formula xₙ₊₁ = ∛(5xₙ + 1) to find the root correct to 2 decimal places.
Answer:
(a) When x = 2: 8 − 10 − 1 = −3. When x = 3: 27 − 15 − 1 = 11. There is a change of sign, and the function is continuous, so there's a root between 2 and 3.
(b) x³ − 5x − 1 = 0 → x³ = 5x + 1 → x = ∛(5x + 1).
(c) x₁ = 2.2240, x₂ = 2.2970, x₃ = 2.3199, x₄ = 2.3269, x₅ = 2.3291
x₄ and x₅ both round to 2.33. To confirm, substitute 2.325 and 2.335 into x³ − 5x − 1: you get about −0.057 and +0.056. The sign changes, so the root is x = 2.33 (2 d.p.).
Common mistake: Rounding each iterate before feeding it back in, which slowly drags the answer off. Use the full calculator value each time (the ANS key). Another mistake is stopping as soon as one value rounds to 2.33, without checking the next one or the bounds.
Typical of: AQA and Edexcel GCSE Higher. Iteration isn't on every IGCSE syllabus, so check your specification.
5. Functions: composite and inverse
f(x) = 2x + 3 and g(x) = x².
(a) Find fg(x). (b) Find gf(x). (c) Find f⁻¹(x). (d) Solve fg(x) = gf(x), giving your answers in surd form.
Answer:
(a) fg(x) = f(x²) = 2x² + 3
(b) gf(x) = g(2x + 3) = (2x + 3)² = 4x² + 12x + 9
(c) Let y = 2x + 3. Then x = (y − 3) ÷ 2, so f⁻¹(x) = (x − 3)/2
(d) 2x² + 3 = 4x² + 12x + 9 → 0 = 2x² + 12x + 6 → x² + 6x + 3 = 0
Using the quadratic formula (or completing the square: (x + 3)² = 6), x = −3 + √6 or x = −3 − √6 (about −0.55 and −5.45).
Common mistake: Doing the functions in the wrong order. fg(x) means apply g first, then f. Students also expand (2x + 3)² as 4x² + 9, missing the 12x.
Typical of: Edexcel GCSE Higher and Cambridge IGCSE Extended, where composite and inverse functions are tested often. AQA includes them too.
6. Probability: forming and solving a quadratic
A bag contains n sweets. 6 of the sweets are orange and the rest are green. Two sweets are taken at random without replacement. The probability that both sweets are orange is ⅓.
(a) Show that n² − n − 90 = 0. (b) Find n. (c) Given that the two sweets are the same colour, find the probability that they are both orange.
Answer:
(a) P(both orange) = (6/n) × (5/(n − 1)) = 30/(n(n − 1)) = ⅓
So n(n − 1) = 90, which gives n² − n − 90 = 0.
(b) (n − 10)(n + 9) = 0, so n = 10 or n = −9. A number of sweets can't be negative, so n = 10.
(c) With 6 orange and 4 green:
P(both orange) = 6/10 × 5/9 = 30/90
P(both green) = 4/10 × 3/9 = 12/90
P(same colour) = 42/90
P(both orange | same colour) = 30/42 = 5/7
Common mistake: In (a), writing 6/n × 6/n, as if the first sweet were put back. In (c), giving 30/90 = ⅓ as the answer. "Given that" means you only look at the outcomes where the colours match, so you divide by 42/90.
Typical of: All three boards. The "show that" quadratic is a common grade 9 probability question. Conditional probability is on the AQA and Edexcel Higher and Cambridge IGCSE Extended specifications.
7. Surds: dividing by a surd
A rectangle has area (13 + 7√3) cm². Its width is (2 + √3) cm.
(a) Find the length of the rectangle in the form a + b√3, where a and b are integers. (b) Find the perimeter in the same form.
Answer:
(a) Length = (13 + 7√3) ÷ (2 + √3). Multiply the top and bottom by (2 − √3):
Numerator: (13 + 7√3)(2 − √3) = 26 − 13√3 + 14√3 − 21 = 5 + √3
Denominator: (2 + √3)(2 − √3) = 4 − 3 = 1
Length = 5 + √3 cm
Check: (5 + √3)(2 + √3) = 10 + 5√3 + 2√3 + 3 = 13 + 7√3 ✓
(b) Perimeter = 2[(5 + √3) + (2 + √3)] = 14 + 4√3 cm
Common mistake: Multiplying by (2 + √3) instead of the conjugate (2 − √3). Another common slip is √3 × √3 = 9 instead of 3, which gives −63 instead of −21 in the numerator.
Typical of: AQA and Edexcel GCSE Higher, and Edexcel International GCSE. Rationalising a denominator also appears in Cambridge IGCSE Extended.
8. Quadratic sequences: is it a term?
The first five terms of a sequence are 4, 9, 18, 31, 48.
(a) Find an expression for the nth term. (b) Show that 1,000 is not a term in the sequence. (c) Find the first term that is greater than 1,000.
Answer:
(a) First differences: 5, 9, 13, 17. Second difference: 4. So the sequence starts with 2n².
Subtract 2n² (2, 8, 18, 32, 50) from the sequence: 2, 1, 0, −1, −2. This is −n + 3.
nth term = 2n² − n + 3
(b) Set 2n² − n + 3 = 1,000, so 2n² − n − 997 = 0. The discriminant is 1 + 4 × 2 × 997 = 7,977. This isn't a square number (89² = 7,921 and 90² = 8,100), so n can't be a whole number. 1,000 is not a term.
(c) n = 22 gives 2(484) − 22 + 3 = 949. n = 23 gives 2(529) − 23 + 3 = 1,038. The first term over 1,000 is the 23rd term, 1,038.
Common mistake: Using half the second difference wrongly. The coefficient of n² is the second difference divided by 2 (here 4 ÷ 2 = 2), not the second difference itself.
Typical of: All three boards, including Cambridge IGCSE Extended.
9. Histograms: estimating from frequency density
The times taken by 100 people to finish a puzzle were recorded. A histogram was drawn with these classes and bar heights:
| Time, t (minutes) | Frequency density |
| 0 < t ≤ 10 | 1.2 |
| 10 < t ≤ 15 | 4 |
| 15 < t ≤ 20 | 5.6 |
| 20 < t ≤ 30 | 2.4 |
| 30 < t ≤ 50 | 0.8 |
(a) Estimate how many people took between 12 and 25 minutes. (b) Estimate the median time.
Answer:
Frequency = frequency density × class width, so the frequencies are 12, 20, 28, 24 and 16 (total 100).
(a) 12 to 15 minutes: 3 × 4 = 12. 15 to 20 minutes: 28. 20 to 25 minutes: 5 × 2.4 = 12. Estimate = 12 + 28 + 12 = 52 people.
(b) The median is the 50th value. The cumulative frequencies are 12, 32, 60, so the median lies in the 15 < t ≤ 20 class. We need 50 − 32 = 18 of the 28 people in that class.
Median ≈ 15 + (18 ÷ 28) × 5 ≈ 18.2 minutes
Common mistake: Reading the bar height as the frequency. On a histogram with unequal class widths, the area of the bar is the frequency, not the height.
Typical of: AQA and Edexcel GCSE Higher, and Cambridge IGCSE Extended.
10. Similar shapes: area and volume ratios
Two bottles are mathematically similar. Their surface areas are 250 cm² and 360 cm². The larger bottle has a volume of 540 cm³. Find the volume of the smaller bottle.
Answer:
Area ratio = 250 : 360 = 25 : 36
Length ratio = √25 : √36 = 5 : 6
Volume ratio = 5³ : 6³ = 125 : 216
Smaller volume = 540 × 125/216 = 312.5 cm³
Common mistake: Using the area ratio straight on the volume (540 × 250/360 = 375 cm³), or cubing the area ratio. Always go back to the length ratio first, then square it for area or cube it for volume.
Typical of: All three boards. Similar volumes is a regular grade 8–9 question on AQA, Edexcel and Cambridge IGCSE Extended papers.
How to Get Better at Grade 8–9 Questions
- Learn the reasons as well as the methods. Circle theorems, proofs and vectors all lose marks for missing or vague reasons. Practise writing the theorem in full.
- Spot the hidden topic. Question 6 looks like probability but is really a quadratic. Question 7 looks like area but is really surds. Ask yourself which topic is doing the work.
- Keep exact values. Leave answers in surd form when asked, and use the full calculator value during iteration.
- Check your answer. Multiply back (Question 7), substitute back (Question 4) or sense-check the size (Question 10: the smaller bottle should hold less).
- Mark your work against a method. When you get a question wrong, find the exact line where your working went wrong and redo it from there. Timed mock exam practice helps you do this under exam conditions.
How Improve ME Institute Helps
Improve ME Institute teaches GCSE and IGCSE Maths in small groups of a maximum of six, matched to each student's exam board, so practice questions follow the style of the papers they will actually sit. Every student starts with a free Diagnostic Test, which shows which grade 8–9 topics need work before teaching begins.
Frequently Asked Questions
Q: What makes a GCSE maths question grade 9 level?
A: Grade 9 questions usually combine more than one topic, give less guidance and need clear reasoning, such as proofs, "show that" questions or problems where you have to choose the method yourself. The maths itself is still on the specification. What makes it hard is putting the pieces together.
Q: Which GCSE maths topics are the hardest?
A: Students most often find algebraic proof, vectors, circle theorem reasons, conditional probability, iteration and functions difficult. These topics are tested in different ways from year to year, so memorising one type of question isn't enough.
Q: Are these questions the same for GCSE and IGCSE?
A: Most of them overlap. Circle theorems, vectors, functions, surds, sequences, histograms and similar shapes are on AQA, Edexcel and Cambridge IGCSE Extended. Some topics, such as iteration, are not on every IGCSE syllabus, so check your board's specification.
Q: How many hard questions should I practise each week?
A: Quality matters more than quantity. Two or three hard questions a week, attempted properly and then reviewed line by line, usually do more than a long list of questions skimmed with the answers open.
Q: Can I use these questions as a mock exam?
A: They are good for targeted practice, but a real paper mixes easier and harder questions and has a time limit. Use these to strengthen weak topics, then sit full timed papers from your board.
Ready for the Top of the Paper?
Pick the question above that you found hardest and try it again tomorrow without looking at the answer. If the same topics keep tripping your child up, a free Diagnostic Test at Improve ME Institute in Gold & Diamond Park, Al Quoz, will show exactly where the marks are being lost. You can also see how we teach Maths in Dubai.
