The Hardest Questions on a GCSE Maths Higher Paper, Walked Through Step by Step

On a GCSE Maths Higher paper, the last five questions are worth around 16 to 20 of the 80 marks. They are where grade 7 students run out of ideas and grade 9 students pick up the marks that separate them. They are also where most revision guides go quiet.

We have built step by step walkthroughs of the last five questions on two 2023 Edexcel Higher papers, free inside our Maths Skills Lab. This post goes through all ten, what makes each one hard, and the single idea that opens it up.

How the walkthroughs work

Each question opens with the wording exactly as it appeared on the paper. Your child presses the right arrow and the solution builds one step at a time, with the diagram drawn from the question's own words and everything not in use greyed out, so they always know which part of the question each step comes from.

Every question also has a New numbers button. It rewrites the question with fresh values in the same shape, so once your child has followed the paper version they can try a new one on their own and check each step. Show it all reveals the whole solution at once for checking.

Paper 1H, non-calculator, 19 May 2023

Question 20, inverse and composite functions (5 marks). Finding an inverse function and then working with two functions chained together. The idea that opens it up is the function machine. Write the steps of the function in order, then run them backwards with each step reversed. The inverse on the paper's numbers is (5x − 2)², and the final answer is x = 55.

Question 21, circle theorems (4 marks). An angle chase through a circle with a tangent, where the question asks for the theorems by name. Students often find the angle and lose marks for the reasons. The walkthrough writes a reason beside every line, and the answer is 71°.

Question 22, 3D trigonometry (2 marks). The angle between a line and a plane in a cuboid. Two marks, but most students do not know where to start. The idea is to find the right-angled triangle that contains the angle, then lift it out of the 3D shape and draw it flat.

Question 23, surds (4 marks). Rationalising a denominator with two terms. The key move is multiplying the top and bottom by the conjugate, the same bracket with the sign swapped, so the difference of two squares clears the surd from the bottom.

Question 24, quadratic inequalities (5 marks). Two quadratic inequalities that must both be true. Solve each by factorising, sketch both curves to see which side of each root is wanted, then keep only the overlap. The answer is −5/2 < x < 4/3. The second quadratic has a negative x² term, so its curve is n-shaped, and missing that flips the answer.

Paper 2H, calculator, 7 June 2023

Question 20, vectors (4 marks). Writing a vector across a triangle in terms of a and b, with a point dividing one side in a ratio. The idea is to plan a route along vectors you already know, and to reverse a vector's sign when you travel against it. The answer is (2/5)b + (3/5)a.

Question 21, transformations of graphs (2 marks). Where a set of points moves when a graph is translated and then reflected. Following a few key points through each change, one at a time, is more reliable than trying to picture the whole curve.

Question 22, probability (2 marks). A probability over n draws, with the answer as an expression rather than a number. The idea is to work out the probability of what you do not want and subtract it from 1. The answer is 1 − (1/5)ⁿ − (4/5)ⁿ.

Question 23, similar shapes (3 marks). Nested similar triangles and the ratio of areas. The trap is using the length ratio for areas. Lengths in a ratio of 1 : 2 give areas in a ratio of 1 : 4, so square the scale factor first, then subtract to find each band.

Question 24, regular octagons and surds (5 marks). The area of a shape enclosed by eight octagons, given in surd form. The idea is to split the shape into a square and four triangles, then work with the surds exactly. The area is 4(2 + √2)a².

What these ten questions have in common

None of them needs a topic that is off the syllabus. What makes them hard is that they combine two topics, or hide the method behind an unfamiliar diagram, or ask for an answer in algebra rather than a number. A student who knows every topic can still lose all five marks by not seeing the first step.

That is why the first line matters more than anything else on these questions. In the walkthroughs, look at how the first two or three presses always start by drawing or labelling something from the question. Getting something written down turns a blank page into a problem your child can work on.

How to use them

1. Try it on paper first. Give your child the question with no help for ten minutes, and ask them to write down anything useful, even if they cannot finish.
2. Walk through it. Open the walkthrough and step through at their own pace, stopping at the first step they did not think of.
3. Press New numbers. Try the same question with fresh values, without the walkthrough, then use Show it all to check.
4. Come back in a week. Try one more set of new numbers. If they can start it without help, that question type is secure.

For the grade 9 picture as a whole, read grade 7 vs grade 9 GCSE maths and how to get a grade 9. The full papers and mark schemes are linked from our free GCSE maths past papers guide.

If you want a teacher's eye on a full paper, our mock marking service marks it by hand and shows which questions are costing the most marks. Start with question 20 on Paper 1H.

Aadam, SHLC Tutors