Mathematics 9709 · syllabus 1.1
Cambridge A Level Mathematics 9709 — Quadratics topical past papers
Completing the square, the discriminant, quadratic inequalities and where a line meets a curve: the algebra that much of Paper 1 rests on.
- Questions
- 72
- Years
- 2009–2026
- With mark scheme
- 97%
- Questions 2022–2026
- 28
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What Quadratics covers
A quadratic ax² + bx + c can be written in completed-square form a(x + p)² + q, which shows the vertex (−p, q) and the line of symmetry at a glance. The discriminant b² − 4ac tells you how many real roots ax² + bx + c = 0 has: two if it is positive, one repeated root if it is zero, none if it is negative. You also solve quadratic inequalities, where a sketch of the parabola shows whether the answer is the region between the roots or outside them.
The topic reaches beyond single equations. A linear and a quadratic equation are solved together by substituting one into the other, and an equation such as x⁴ − 10x² + 9 = 0 is quadratic in x², so it is solved as one. When a line meets a curve, substitution gives a quadratic whose discriminant decides whether the line cuts, touches or misses the curve, which is how questions find the values of a constant k that make a line a tangent.
Quadratics is section 1.1 of Paper 1, Pure Mathematics 1, but its methods turn up across the course: completed squares give ranges and inverses in functions, a line meeting a circle in coordinate geometry produces a quadratic, trigonometric equations are often quadratic in sin x or tan x, and stationary points of cubics come from quadratic derivatives.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.1 Quadratics
What you should be able to do
- Complete the square on ax² + bx + c, including when a ≠ 1, and read off the vertex
- Use the discriminant to decide whether an equation has two, one or no real roots
- Solve quadratic inequalities and write the solution set correctly
- Solve a linear and a quadratic equation simultaneously by substitution
- Recognise and solve equations that are quadratic in a function of x, such as x² or √x
- Find the values of a constant for which a line meets, touches or misses a curve
Key ideas
- ax² + bx + c = a(x + b/(2a))² + c − b²/(4a), so the vertex is where x = −b/(2a)
- Discriminant b² − 4ac: > 0 two distinct real roots, = 0 one repeated root, < 0 no real roots
- x = (−b ± √(b² − 4ac)) / (2a)
- If α < β and a > 0, then a(x − α)(x − β) < 0 for α < x < β, and > 0 for x < α or x > β
- A line is a tangent to a curve when substituting gives a quadratic with discriminant 0
How Quadratics has been examined
From 2022 to 2026, Quadratics was on Paper 1 · Pure Mathematics 1 in 10 of 11 exam series, worth about 9% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 28 | 9% | 10 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 7.
Filed in Momentum under: Intersection Points.
A worked example
Find the set of values of k for which the line y = kx − 1 does not meet the curve y = x² − 3x + 3.
- Set the expressions equal: x² − 3x + 3 = kx − 1, so x² − (3 + k)x + 4 = 0.
- The line misses the curve when this equation has no real roots, so the discriminant is negative: (3 + k)² − 4 × 1 × 4 < 0.
- So (3 + k)² < 16, which means −4 < 3 + k < 4.
- Subtract 3 throughout: −7 < k < 1.
- Check the boundary k = 1: x² − 4x + 4 = (x − 2)² = 0, a repeated root, so the line just touches there, as expected.
Answer: −7 < k < 1
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Writing 'x < −3 and x > 2' for a region outside the roots: no number satisfies both, so the answer must be 'x < −3 or x > 2'.
- Dividing an inequality by a negative number, or by an expression whose sign is unknown, without reversing it; this changes the solution set, so collect terms on one side and factorise instead.
- Using b² − 4ac > 0 when a question says the line meets the curve: meeting includes touching, so the condition is b² − 4ac ≥ 0.
- Completing the square on 2x² − 12x + 5 without taking out the factor 2 first, which puts the vertex in the wrong place.
- Keeping every root of a disguised quadratic: if u = √x, a negative value of u gives no x and must be rejected.
Exam technique
- Sketch the parabola before writing the answer to an inequality; the sketch shows at once whether you need the region between the roots or outside them.
- For 'find the set of values of k', substitute, collect everything into one quadratic equal to zero, then write the discriminant condition in k and solve that inequality.
- Leave roots in surd form when the question asks for exact values; otherwise give them to 3 significant figures.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.