O Level Additional Mathematics 4037 · Topic 14

Calculus: past paper questions

207 real questionsasked in 10 of the last 10 exam series2020–2025usually: Find, Use, Show, Determine, Give

Real questions on calculus

The newest 20 of 207 written questions. Open one to see it on the real paper.

4037/12 · October/November 2025 · Q9b[1]
1-mark “Determine” question — its maths and diagrams are best seen on the paper.
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4037/12 · October/November 2025 · Q12a[4]
4-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/12 · October/November 2025 · Q12b[2]
2-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q2[3]
3-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q3[4]
4-mark question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q4a[4]
4-mark “Calculate” question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q4b[2]
2-mark “Calculate” question — its maths and diagrams are best seen on the paper.
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4037/13 · October/November 2025 · Q5[8]
8-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q7a[2]
2-mark question — its maths and diagrams are best seen on the paper.
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4037/22 · October/November 2025 · Q7b[4]
4-mark question — its maths and diagrams are best seen on the paper.
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4037/11 · May/June 2025 · Q6a[5]
5-mark question — its maths and diagrams are best seen on the paper.
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4037/11 · May/June 2025 · Q6b(i)[1]
1-mark “Show” question — its maths and diagrams are best seen on the paper.
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4037/11 · May/June 2025 · Q6b(ii)[2]
2-mark “Use” question — its maths and diagrams are best seen on the paper.
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4037/11 · May/June 2025 · Q9[10]
10-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/12 · May/June 2025 · Q12[6]
6-mark question — its maths and diagrams are best seen on the paper.
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4037/21 · May/June 2025 · Q1a[3]
3-mark “Use” question — its maths and diagrams are best seen on the paper.
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4037/21 · May/June 2025 · Q5[4]
4-mark question — its maths and diagrams are best seen on the paper.
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4037/21 · May/June 2025 · Q9d(ii)[5]
5-mark “Find” question — its maths and diagrams are best seen on the paper.
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4037/21 · May/June 2025 · Q10a(i)[3]
3-mark question — its maths and diagrams are best seen on the paper.
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4037/21 · May/June 2025 · Q10a(ii)[3]
3-mark question — its maths and diagrams are best seen on the paper.
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What examiners said

Weaker candidates did not score because they did not attempt the differentiation. Some candidates expanded brackets first, but then differentiated successfully and found two roots from their quadratic, gaining at least two marks. Those who used the product and chain rules sometimes slipped up differentiating the first term: (2x ‒ 1) ‒1 was seen more frequently than expected. Most errors occurred i...
— Examiner report, October/November 2023, 4037/13 Q6(a) (Cambridge)

Many candidates started again rather than use their graph, often setting the original expression equal to k and being unable to deal with the algebraic manipulation. Some candidates wrote down k < 0 and no more. As many had not differentiated in part (a), they struggled to find the maximum and therefore the second inequality in k. A few found the y-coordinate of the maximum point and then wrote k...
— Examiner report, October/November 2023, 4037/13 Q6(c) (Cambridge)

A good proportion of fully correct answers were seen to this part of the question. The majority of candidates who attempted to differentiate did so correctly. Some candidates may have improved if they had reread the question as they found the equation of the tangent and not the equation of the normal. Some candidates did not seem to be aware that by substituting x = 1 into their derivative, they w...
— Examiner report, October/November 2023, 4037/22 Q5(a) (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 calculus

In the syllabus

14.1 Calculus

Understand the idea of a derived function. Only an informal understanding of the idea of a limit is expected, and the technique of differentiation from first principles is not required.
Use the notations R V dy d2 y S d JKdyNOW f′(x), f″(x), , dx dx2 SS dx KdxOWW = T L PX dy δx, δx → 0, . dx.
Know and use the derivatives of the standard Includes constant multiples, sums and composite functions xn (for any rational n), sin x, cos x, functions (use of chain rule). tan x, ex, ln x. For example, differentiate `3x 2 + 4j .
Differentiate products and quotients of functions.
Use differentiation to find gradients, tangents and normals.
Use differentiation to find stationary points. Points of inflexion are not included.
Apply differentiation to connected rates of change, small increments and approximations.
Apply differentiation to practical problems involving maxima and minima.
Use the first and second derivative tests to Points of inflexion are not included. discriminate between maxima and minima. Full justification of conclusions is expected. An explanation of how to distinguish between a maximum point and a minimum point may be required. Unless specified otherwise, any valid method is allowed.
Understand integration as the reverse process Solutions for indefinite integrals should include an of differentiation. arbitrary constant.
Integrate sums of terms in powers of x, Solutions for indefinite integrals should include an 1 1 arbitrary constant. including and . x ax + b.
Integrate functions of the form: Includes the case where n = –1. n • (ax + b) for any rational n For trigonometric functions angles will always be in • sin (ax + b) radians. • cos (ax + b) Solutions for indefinite integrals should include an arbitrary constant. • sec2 (ax + b) • eax+b.

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