The five questions
- Simplify 82/3 × 4−1/2, and then (√12 + √27) ÷ √3.
- Make x the subject of y = (2x + 3)/(x − 1).
- Solve the simultaneous equations y = x + 1 and x² + y² = 13.
- Write x² − 6x + 11 in completed-square form and give the coordinates of its turning point.
- Write 3/(x + 2) − 1/(x − 1) as a single fraction.
Do them on paper before reading on. What matters is not only whether you get the answers but whether you knew the method straight away. A question you get right after two false starts still counts as a gap, because in an A-level question it is one step out of several and you need your attention for the new part.
Worked answers, and what each one is testing
1. Indices and surds. 82/3 is the cube root of 8, squared, which is 4; 4−1/2 is 1/2; so the product is 2. For the surds, √12 = 2√3 and √27 = 3√3, so the top is 5√3 and dividing by √3 gives 5. A-level uses fractional and negative powers constantly, for instance to differentiate √x by writing it as x1/2. On our map: rational exponents and dividing radicals, which between them stand on 55 earlier concepts.
2. Rearranging with x on both sides. Multiply out: y(x − 1) = 2x + 3, so yx − y = 2x + 3. Collect the x terms: yx − 2x = y + 3. Factorise: x(y − 2) = y + 3, so x = (y + 3)/(y − 2). The factorising step is the one people miss, and it is the same move used to find an inverse function. On our map: solving a literal equation (formula) for a specified variable, which stands on 47 earlier concepts.
3. A linear and a quadratic together. Substitute y = x + 1 into the second equation: x² + (x + 1)² = 13, so 2x² + 2x − 12 = 0, or x² + x − 6 = 0. That factorises as (x + 3)(x − 2) = 0, so x = 2, y = 3 or x = −3, y = −2. Geometrically, this is where a line crosses a circle, which is how the same skill reappears in coordinate geometry. The question is substitution followed by solving a quadratic, so it draws on two concepts. On our map: solving by substitution and solving quadratics by factoring, which between them stand on 68 earlier concepts.
4. Completing the square. x² − 6x + 11 = (x − 3)² + 2, so the turning point is (3, 2), a minimum. A-level uses this form to find centres of circles and to sketch quadratics without a table of values. On our map: converting standard to vertex form (completing the square), which stands on 72 earlier concepts.
5. Algebraic fractions. Over the common denominator (x + 2)(x − 1), the top is 3(x − 1) − (x + 2) = 2x − 5, so the answer is (2x − 5)/((x + 2)(x − 1)). The usual slip is the sign on the 2 when subtracting the second bracket. This skill comes back in partial fractions and in integration. On our map: subtracting rational expressions (unlike denominators), which stands on 58 earlier concepts.
Those counts are a reminder of how far down each skill reaches. A slip in question 5 can come from the fraction arithmetic of primary school as easily as from the algebra on top of it, so when one of these goes wrong, try the plain-number version (1/3 − 1/4) before drilling the algebraic one.
What the two years contain
The content is set by the Department for Education and is the same for every exam board. Its subject content document says A-level maths must “build on the skills, knowledge and understanding set out in the whole GCSE subject content”, and that it “builds from GCSE level mathematics and introduces calculus and its applications”. The pure topics are proof, algebra and functions, coordinate geometry, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, numerical methods and vectors. The applied topics are statistics (sampling, data, probability, distributions and hypothesis testing) and mechanics (units, kinematics, forces and Newton’s laws, and moments).
The boards package it differently, but all three main ones set three two-hour papers of 100 marks each. Pearson Edexcel has two pure papers and a third with a statistics section and a mechanics section. AQA has one pure paper, then a paper that adds mechanics and one that adds statistics, both of which can also ask any of the pure content. OCR (Mathematics A) has a pure paper, a pure and statistics paper, and a pure and mechanics paper. Either way, pure maths runs through all three papers, and pure maths at this level is written in algebra. That is why the check above is all algebra: it is the part of GCSE every paper relies on. Specifications checked on 27 October 2026.
Summer bridging work: aiming it at your gaps
If your sixth form or college has set summer bridging work, the five questions above tell you which sections of it deserve most of the time. If it has not, build your own list from them: one topic at a time, starting with whichever of the five went worst, and starting lower than that topic if the plain-number version also went wrong.
Short sessions spread over the summer will leave more in place in September than the same hours crammed into the last week of August, and coming back to a topic a few days after you first fix it matters more than doing many questions on the day. The reasons, and a schedule you can run on paper, are in spaced repetition for maths. If GCSE itself needs redoing before any of this, the GCSE resit guide covers that route first.
