AS & A Level Mathematics 9709 · Pure Mathematics 3 (for Paper 3) · 3.1

Algebra: past paper questions

66 real questionsasked in 7 of the last 10 exam series2020–2026usually: Find, Show, Give, Determine, Solve

Real questions on algebra

The newest 20 of 66 written questions. Open one to see it on the real paper.

9709/12 · May/June 2026 · Q7a[4]
(a) The second, fourth and last terms of an arithmetic progression are $60$ , $52$ and $- 28$ respectively. Find the sum of all of the terms in the progression.See it on the paper →
9709/12 · May/June 2026 · Q7b(i)[3]
(b) The second, third and fourth terms of a geometric progression are $9$ , $k + 4$ and $k^{2}$ respectively, where $k$ is a negative constant. (i) Find the value of $k$ .See it on the paper →
9709/22 · October/November 2025 · Q4a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/22 · October/November 2025 · Q4b[3]
3-mark “Give” question — its maths and diagrams are best seen on the paper.
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9709/53 · October/November 2025 · Q2[4]
4-mark question — its maths and diagrams are best seen on the paper.
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9709/53 · October/November 2025 · Q3a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/53 · May/June 2025 · Q7a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/12 · February/March 2025 · Q3a[4]
4-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · February/March 2025 · Q3b[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/12 · February/March 2025 · Q5[6]
6-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · February/March 2025 · Q9a[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/32 · February/March 2025 · Q9b[3]
3-mark question — its maths and diagrams are best seen on the paper.
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9709/32 · February/March 2025 · Q9c[2]
2-mark question — its maths and diagrams are best seen on the paper.
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9709/13 · October/November 2024 · Q1[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/11 · May/June 2024 · Q8a[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/11 · May/June 2024 · Q8b[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/13 · May/June 2024 · Q1[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/13 · May/June 2024 · Q10a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/13 · May/June 2024 · Q10b[2]
2-mark “Find” question — its maths and diagrams are best seen on the paper.
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9709/13 · May/June 2024 · Q10c[3]
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What examiners said

Most candidates produced a fully correct expansion, but there was a minority who attempted to multiply out four brackets, usually unsuccessfully. Poorer attempts at this question did not apply the power of 4 correctly to the first and last terms or incorrectly simplified the other three terms. Not dealing with the minus sign correctly or ignoring it completely was a common error. Candidates are en...
— Examiner report, February/March 2025, 9709/12 Q3(a) (Cambridge)

Almost all candidates with an expansion selected the two terms needed to produce the x2 coefficient. A large majority scored both marks.
— Examiner report, February/March 2025, 9709/12 Q3(b) (Cambridge)

Many candidates scored no marks for this question because they started by evaluating S150 and S400. As highlighted in the General Comments, candidates are advised to read the question carefully before beginning a response. Candidates who attempted to find the correct summations that needed to be subtracted, or the first and last terms of the new progression, were able to score the majority of the...
— Examiner report, February/March 2025, 9709/12 Q5 (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 algebra

In the syllabus

3.1 Algebra

understand the meaning of |x|, sketch the Graphs of y = |f(x)| and y = f(|x|) for non-linear
functions f are not included.
e.g. |3x – 2| = |2x + 7|, 2x + 5 < |x + 1|.
divide a polynomial, of degree not exceeding 4,
use the factor theorem and the remainder e.g. to find factors and remainders, solve
polynomial equations or evaluate unknown coefficients
Including factors of the form (ax + b) in which the coefficient of x is not unity, and including calculation of remainders.
recall an appropriate form for expressing Excluding cases where the degree of the numerator
exceeds that of the denominator
(ax + b)(cx + d)(ex + f )
(ax + b)(cx + d)2
(ax + b)(cx2 + d)

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