AS & A Level Mathematics 9709 · Pure Mathematics 1 (for Paper 1) · 1.1

Quadratics: past paper questions

134 real questionsasked in 9 of the last 10 exam series2020–2026usually: Find, Solve, Show, State, Sketch

Real questions on quadratics

The newest 20 of 134 written questions. Open one to see it on the real paper.

9709/12 · May/June 2026 · Q1[3]
1 Solve the equation $3 x^{2} - 8 - \frac{16}{x ^{2}} = 0.$ See it on the paper →
9709/12 · May/June 2026 · Q10a[3]
A function f is defined by $f (x) = p x^{2} - 2 x + p + 4$ for $x \in R$ , where $p$ is a constant. (a) Find the exact values of $p$ for which the equation $f (x) = 0$ has repeated roots.See it on the paper →
9709/32 · May/June 2026 · Q1[3]
Find the quotient and remainder when $4 x^{3} - 4 x^{2} + 7 x + 2$ is divided by $2 x - 1$ .See it on the paper →
9709/32 · May/June 2026 · Q10a[4]
(a) Express $\frac{x + 3}{( x - 1 ) ( 2 x - 6 )}$ in partial fractions.See it on the paper →
9709/11 · October/November 2025 · Q1[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q1a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q1b[1]
1-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q1c[2]
2-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q2[3]
3-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · October/November 2025 · Q8a[3]
3-mark “Show” question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q10a(i)[3]
3-mark question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q10a(ii)[1]
1-mark “State” question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2025 · Q10b[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/23 · October/November 2025 · Q7a[2]
2-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/23 · October/November 2025 · Q7b[4]
4-mark “State” question — its maths and diagrams are best seen on the paper.
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9709/33 · October/November 2025 · Q1[4]
4-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/33 · October/November 2025 · Q2[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/33 · October/November 2025 · Q3[4]
4-mark “Solve” question — its maths and diagrams are best seen on the paper.
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9709/33 · October/November 2025 · Q10a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/11 · May/June 2025 · Q5a(i)[2]
2-mark question — its maths and diagrams are best seen on the paper.
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What examiners said

Most candidates looked at the final stage and used the equation v2 = u2 + 2as with v = 0, a = –2 and s = 16 to find V. Some tried to use the earlier stages of the motion but often failed to successfully find V.
— Examiner report, February/March 2025, 9709/42 Q2(a) (Cambridge)

Many candidates realised that the area under the pdf was equal to 1. The area could be found by finding the area of a trapezium, a rectangle plus a triangle, or a larger rectangle minus a triangle, or even by finding the equation of the line and using integration. All methods were seen, though the final method was not generally successful and was not an efficient method here. It was important that...
— Examiner report, February/March 2025, 9709/62 Q4(a)(i) (Cambridge)

Stronger candidates found the sum of the three products, (1 × 1) + (−7 × 18) + (1 × 135) = 10. Candidates should be aware that finding the full expansion with nine terms is unnecessary and time consuming.
— Examiner report, October/November 2023, 9709/11 Q1(b) (Cambridge)

Practise this topic

Answer these questions online and get each one marked against the official mark scheme, with what you missed and how to fix it.

Practise quadratics

In the syllabus

1.1 Quadratics

carry out the process of completing the square e.g. to locate the vertex of the graph of
y = ax2 + bx + c or to sketch the graph
find the discriminant of a quadratic polynomial e.g. to determine the number of real roots of the
equation ax2 + bx + c = 0. Knowledge of the term ‘repeated root’ is included.
solve quadratic equations, and quadratic By factorising, completing the square and using the
formula.
solve by substitution a pair of simultaneous e.g. x + y + 1 = 0 and x2 + y2 = 25,
2x + 3y = 7 and 3x2 = 4 + 4xy.
recognise and solve equations in x which are e.g. x4 – 5x2 + 4 = 0, 6x + x − 1 = 0,
tan x = 1 + tan x.

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