O Level Additional Mathematics 4037 · Topic 4

Equations, inequalities and graphs: past paper questions

82 real questionsasked in 10 of the last 10 exam series2020–2025usually: Find, Solve, Show, Write, Give

Real questions on equations, inequalities and graphs

The newest 20 of 82 written questions. Open one to see it on the real paper.

4037/12 · October/November 2025 · Q1b[2]
2-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q1a[3]
3-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q1b[4]
4-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q8[7]
7-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q1a[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q1b[5]
5-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q2b(i)[2]
2-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q2b(ii)[3]
3-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q5a[1]
1-mark question — its maths and diagrams are best seen on the paper.
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4037/23 · October/November 2025 · Q6[9]
9-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/11 · May/June 2025 · Q2[4]
4-mark “Solve” question — its maths and diagrams are best seen on the paper.
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4037/12 · May/June 2025 · Q8a[5]
5-mark question — its maths and diagrams are best seen on the paper.
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4037/12 · May/June 2025 · Q8b[1]
1-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q1[4]
4-mark “Solve” question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q2a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q2b[2]
2-mark “Give” question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q2c[2]
2-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q2d[1]
1-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/22 · May/June 2025 · Q2e[3]
3-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2024 · Q1a[3]
3-mark “Find” question — its maths and diagrams are best seen on the paper.
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What examiners said

Most candidates could deal with 4 x − 3 < 2 x + 5 but obtaining the second limit was fraught with error. Many used 4 x − 3 < −2 x − 5 which demonstrated an incorrect understanding of the modulus principle. Frequently those that began this way later did not deal with the signs correctly and stumbled into a correct answer which was not accepted as it came from wrong working. An alternative approach...
— Examiner report, October/November 2023, 4037/13 Q1(b) (Cambridge)

This question was well answered. Many candidates found the correct value of x. Candidates were very successful at finding the gradient of AB and its midpoint. However, some were then unable to find the perpendicular gradient or make further progress. When attempting to find the gradient of the perpendicular line, some candidates only changed the sign of AB’s gradient instead of using the negative...
— Examiner report, May/June 2023, 4037/11 Q3(a) (Cambridge)

This question proved very challenging. The most common incorrect answer was (4, 9). Some candidates attempted to answer this question using vectors, but they were rarely successful.
— Examiner report, May/June 2023, 4037/11 Q3(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 equations, inequalities and graphs

In the syllabus

4.1 Equations, inequalities and graphs

Solve equations of the type For graphical solutions, an accurate graph is • |ax + b| = c (c ⩾ 0) expected. • |ax + b| = cx + d For algebraic methods, any valid method is acceptable. • |ax + b| = |cx + d| • |ax2 + bx + c| = d using algebraic or graphical methods.
Solve graphically or algebraically inequalities For graphical solutions, an accurate graph is of the type expected. • k|ax + b| > c (c ⩾ 0) For algebraic methods, any valid method is • k|ax + b| ⩽ c (c > 0) acceptable. • k|ax + b| ⩽ |cx + d| where k > 0 • |ax + b| ⩽ cx + d • |ax2 + bx + c| > d • |ax2 + bx + c| ⩽ d.
Use substitution to form and solve a quadratic For example: equation in order to solve a related equation. 4 2 • x 3 + x 3 - 12 = 0 2(1n 5x)2 + 1n 5x – 6 = 0 • 3ex = 12 – 5e –x.
Sketch the graphs of cubic polynomials and The points of intersection of the graph with the their moduli, when given as a product of three coordinate axes should be clearly labelled. linear factors.
Solve graphically cubic inequalities of the form • f(x) ⩾ d • f(x) > d • f(x) ⩽ d • f(x) < d where f(x) is a product of three linear factors and d is a constant.

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