Hardest GCSE Maths Questions

With the introduction of grade 9 at GCSE, which only 3.5% of students achieved in 2023 (Pearson), Maths exam papers have included some extremely challenging questions in recent years. These questions stretch the highest achieving students, and allow them to demonstrate their skills. 

Only 3.5% of students achieved a grade 9 in Maths in June 2023

At Save My Exams, we aim to make even the hardest questions easy to understand through our illuminating worked solutions and revision notes for GCSE Maths. Through my experience in the classroom as a maths teacher, I know that different students struggle with different topics, and that as a result there is no “hardest topic”. Therefore I have curated a selection of my personal favourite, hardest, GCSE Maths questions arranged by topic area, with links to our worked solutions.

Contents

Hardest Algebra Questions
Hardest Coordinate Geometry Questions
Hardest Number & Ratio Questions
Hardest Geometry Questions
Hardest Probability & Statistics Questions
Conclusion
FAQs

Hardest Algebra Questions

AQA, November 2021: 2H Q24

This question covers a great set of skills; algebraic fractions, forming an equation from a worded problem, and solving a quadratic. I often find that these topics are good indicators of a student’s confidence with algebra. The examiner report for this question stated “this question was poorly attempted with most students unable to progress. There were a significant number who made no attempt.” So this is a question where you could really gain some marks over other candidates to boost your grade. Our worked solution for this question can be found as Q11 in our Very Hard Algebraic Fractions topic question pack.

Edexcel, November 2020: 3H Q21b

There are two features that make this a tough question; a ratio involving algebra, and a quadratic in two variables. Both of these take a standard skill (ratio and quadratics) and give them an extra twist. Even if students manage to find the two possible ratios of x:y, they still have to decide which the correct solution is given the information in the question. Our worked solution is available as Q12 in our Very Hard Solving Quadratics topic question pack.

OCR, November 2017: H06 Q14b

Students struggle with proof questions due to the lack of structure or direct instructions

I commonly find that students struggle with proof questions due to the lack of structure or direct instructions. This question is no different, and requires students to set up their own expressions describing the values in the cross shape. The examiner’s report noted that “...almost all attempts at the proof merely consisted of further numerical examples”. While I would always recommend students test further numerical examples to help spot any patterns, making the leap to modelling with algebra helps distinguish top performers. A worked solution from one of our maths content creators is available as Q12b in our Very Hard Algebraic Proof topic question pack.

Hardest Coordinate Geometry Questions

Edexcel, November 2021: 2H Q19

I have taught A Level Maths extensively, and this question wouldn’t be out of place on a first year A Level paper. Firstly, students have to understand the link between simultaneous equations and the points of intersection. The simultaneous equations then have to be solved by substitution, which reduces to a quadratic. The question then finishes off by finding the midpoint; an essential coordinate geometry skill. You can view the full details of the solution in Q2 of our Very Hard Coordinate Geometry topic question pack. 

Edexcel, June 2019: 3H Q22

This is a fantastic example of a higher-level problem solving question. You are given a diagram, key facts, numerical values, algebraic values, and an angle. You must then piece these together to find the value of p. Spotting a route to a solution is not easy. I often advise students to start by working something out, which may then lead to finding some key information, unlocking the whole problem. The key to this problem is forming an equation for the circle in terms of p, and then finding the radius, which allows the equation to be solved for p. To see the full details of the path to the answer, see Q4 in our Very Hard Equation of a Circle topic question pack.

Hardest Number & Ratio Questions

Edexcel, June 2018: 2H Q21

The toughest questions combine the basics with another piece of knowledge

In my teaching experience, students are usually comfortable with the basic concept of upper and lower bounds and rounding. However, the toughest questions combine the basics with another piece of knowledge, in this case density, mass, and volume. Unusually, this question also asks for “the density of the wood” rather than the upper and lower bounds for the density, which is a more standard exam question. This means the final step in this question is to look for the highest degree of accuracy that both the upper and lower bounds can be rounded to, such that they are equal. To see the full solution written by one of our Maths content specialists, see Q3 in our Very Hard Rounding, Estimation & Bounds topic question pack.

AQA, June 2018: 1H Q26

I like this Paper 1 (non-calculator) question because it looks so simple on the outside, but the underlying maths can be quite tricky! There are only two short statements about the relative ratios and proportions of a, b, and c, and students have to then find a 3-part ratio. When I have taught lessons about ratio, 3-part ratios are always something students find difficult. The key to them is finding a part of the ratio that “overlaps” and can be a link between two 2-part ratios, which can then be joined together. See Q6 in our Very Hard Ratio Problem Solving topic question pack to view our full worked solution.

Hardest Geometry Questions 

Edexcel (International), January 2022: 1HR Q24

I have taught Maths to both GCSE and A Level students, and in both cases students quickly grasp the basics of vectors: finding routes between two points, and describing them algebraically. Often I find students at both levels struggle with questions involving ratios of lengths, and algebraic scalars (λ in this case). I picked out this GCSE question in particular because the strategy to solve it is used frequently in A Level textbook problems. If you enjoy tackling questions with problem solving like this, I would suggest that A Level Maths may be for you! I’ll give you a hint for this one: find two different paths from O to N, one involving λ, and set them equal to each other. To view the rest of our explanation, see Q9 in our Very Hard Vectors topic question pack.

Edexcel, June 2017: 1H Q22

Trigonometry questions, especially those using the cosine rule, usually appear in one of the calculator papers (Paper 2 and Paper 3). This question, however, is a great example of a non-calculator cosine rule question. Because it is a proof question, students have to be really particular and clear about their reasoning for the steps they use to receive full marks. When I mark students’ exam papers, this is a common reason for them to not receive full marks, and is something the examiners’ report mentions specifically.

Students need to ensure they use the correct mathematical language when giving their reasons and give all necessary detail. For example, "The angle between the tangent and the circle = 90" is not sufficient as the word radius is missing from their reason

See all the steps you need to include to receive full marks in Q5 of our Very Hard Sine, Cosine Rule & Area of Triangles topic question pack.

Hardest Probability & Statistics Questions

Edexcel, June 2015: 1H Q19

“Hannah’s Sweets” is possibly the most infamous GCSE Maths exam question of all time. When I consider the hardest questions I have seen in my many years as a maths teacher, it probably wouldn’t be in the top 10, but it caused an outcry online at the time. Hannah’s sweets even received coverage from news outlets like the BBC and The Guardian. 

“Hannah’s Sweets” is the most infamous GCSE Maths exam question of all time

What makes it appear as such a hard question is the way it builds up a description of a fairly standard conditional probability problem, but then asks to show that a quadratic equation is true! This overlap of topics had not been seen up to this point on a GCSE paper, which is why it caused such a stir. I think it makes an excellent question though, and really tests whether students know what they are doing with a conditional probability question. Have a go yourself first, and then check your answer with our fully worked solution. It is Q5 in our Very Hard Combined & Conditional Probability topic question pack.

Edexcel (International), January 2022: 2H Q20

When I teach histograms, I love using questions similar to this from a past Edexcel paper to test understanding of the topic, and practise problem solving skills. The specific feature that enables this is the blank y-axis on the graph. This makes the question a step harder than the usual “find the frequency” problems, and means you must work out the scale for the y-axis (frequency density) using information given in the problem. 

There is an alternate method too, where you can simply use the number of squares within the bars. This works as the areas in a histogram are proportional to the frequency. Both of these methods are shown in our worked solution of Q6 in our Very Hard Histograms topic question pack.

Conclusion

As a maths teacher, I know that it’s vital for students to build confidence answering questions that are difficult, or where the best method can’t be spotted straight away. This is because half of the exam paper is now aimed at grades 7 and above.

In a higher tier paper, half of the marks should be targeted at grades 9, 8 and 7 (Ofqual)

Once you have mastered the key skills of a topic, and are confident answering “standard” exam questions, the only way to improve at answering harder questions is by experiencing - and tackling - as many difficult questions as you can. There are no shortcuts to this unfortunately, but we do have plenty of challenging questions, all with worked solutions, in our Hard and Very Hard topic question packs. We also have GCSE Maths revision notes and past papers, for when you want to brush up on a particular topic, or practise a whole paper.

FAQs

What is the hardest GCSE Maths topic?

In my experience teaching GCSE Maths in English schools and as a GCSE Maths tutor, I would find it hard to pinpoint a hardest topic on any GCSE maths paper. Individual students will always vary in which topics they find hard compared to their peers. However, the following topics and skills are more common for students to find challenging:

• Algebraic Fractions
• Forming equations from words
• Algebraic Proof
• Coordinate Geometry & Circles
• Problem Solving with Ratios
• Vector Proof
• Trigonometry
• Circle Theorems & Geometric Proof
• Conditional Probability
• Histograms

What is the hardest Maths question?

There is certainly no “hardest maths question” as all students will vary in how difficult they find each topic. However, questions involving proof and/or including a high level of algebra are often the worst-answered on exam papers. The questions I have collated in this article are certainly a good starting point for seeing some of the hardest questions, but you can find plenty more in our Hard and Very Hard topic question packs.

Is Maths the hardest GCSE?

This is much more of a personal question. For some students Maths will be the subject they find hardest, and for others it will be the subject they find easiest. For most people it will fall somewhere in between. However hard or easy you find maths, practice is vital to improve. At Save My Exams we have resources - including practice papers - to help with your Maths revision, to test yourself, and to improve in GCSE Maths.