All AQA Level 2 Further Maths topic questions

AQA Level 2 Further Maths · Topic C

Calculus questions

Practice questions covering calculus from this specification. Every question below is the real AQA question as it is printed on the paper, with its mark scheme.

Questions to attempt

6 questions on Calculus, each with its mark scheme and worked method. Reveal the answer when you have had a go.

C1 · Know that the gradient function dy/dx gives the gradient of the curve and measures the rate of change of y with respect to x
Q1
ORIGINAL
1
For $y=3x^2-5x$ , work out $\frac{dy}{dx}$ when $x=2$ .
[2 marks]
C2 · Know that the gradient of a function is the gradient of the tangent at that point
Q2
ORIGINAL
2
Work out the gradient of the tangent to $y=x^2-4x+1$ at the point where $x=1$ .
[2 marks]
C3 · Differentiation of kx^n where n is an integer, and the sum of such functions
Q3
ORIGINAL
3
Given $y=4x^3-3x^2+7x-9$ , work out $\frac{dy}{dx}$ .
[3 marks]
C4 · The equation of a tangent and normal at any point on a curve
Q4
ORIGINAL
4
Work out the equation of the tangent to $y=x^2+3x-1$ at the point where $x=1$ .
[3 marks]
C5 · Increasing and decreasing functions
Q5
ORIGINAL
5
Work out where $y=x^2-8x+1$ is decreasing and where it is increasing.
[3 marks]
C6 · Understand and use the notation d2y/dx2; know that it measures the rate of change of the gradient function
Q6
ORIGINAL
6
A curve has $\frac{dy}{dx}=6x^2-4x$ . Work out the value of $x$ at which the gradient is changing at a rate of $20$ , given $x>0$ .
[3 marks]

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Printable question pack — C CalculusAll 90 questions across 9 specification points — 84 more than the 6 above — with the full mark scheme. Guest accessible · browser print.

C Calculus revision notes

Real past-paper questions

18 questions shown as printed, 12 reference-only · newest series first

Question and mark-scheme extracts © AQA. Reproduced for educational use; the original paper is linked on every question. Contact hello@oscarsongtutoring.co.uk for removal.

AQA 8365 · June 2025 · Paper 1 · Q18 · 6 marks
6 marks
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Examiner report
This question was very well answered. Students should take extra care with their working when asked to ‘show that’ something is true. Question 18 (b) This question was more challenging and was a good discriminator. Most students awarded full marks used Alternative method 1. Some made errors when differentiating while others gave their answer as x = 15 rather than continuing to find the value of A. There were a relatively high number of non-attempts.
Text version
A 60 cm wire forms three sides of a rectangle and is attached to a fixed rod along the fourth side. Each of the two perpendicular wire sides has length x cm and the remaining wire side has length y cm. The rectangle has area A cm^2.
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AQA 8365 · June 2025 · Paper 1 · Q20 · 4 marks
4 marks
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Examiner report
This question was challenging for many students and was a good discriminator. After differentiation, most students tried to show that the gradient was always positive by either completing the square or using the discriminant. Some students used further differentiation and this approach did sometimes lead to a fully correct response.
Text version
y = x^3/3 - 5x^2 + 28x + 11.
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AQA 8365 · June 2025 · Paper 2 · Q5 · 3 marks
3 marks
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Examiner report
This question was well answered although some students made errors when substituting x = –1 after differentiating. Others made errors when solving the correct linear equation. Some did not attempt differentiation and substituted x = –1 into the equation of the curve.
Text version
The curve is y = x^5 + px^4, where p is a constant. When x = -1, the gradient is -3.
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AQA 8365 · June 2025 · Paper 2 · Q15 · 3 marks
3 marks
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Examiner report
This question was well answered. Most students substituted x = 3 into the given expression for the gradient and showed that it evaluated to zero. Question 15(b) This question was a good discriminator. The common error when attempting the second derivative was to write –18x–3. A significant number of students did not attempt a second derivative. Most who substituted x = 3 into the second derivative evaluated correctly with only a few concluding that the nature of the point was a maximum.
Text version
For a curve y = f(x), dy/dx = 1 - 9/x^2.
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AQA 8365 · June 2025 · Paper 2 · Q20 · 5 marks
5 marks
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Examiner report
This question was quite a good discriminator. Some students did not attempt differentiation but those that did usually gave the correct expression. Errors were quite often made when working out the gradient of the perpendicular line and some thought the gradient of the normal was – 4.
Text version
The curve is y = 4 + 2x - x^2. Its turning point is P(1, 5), so the tangent at P is horizontal. Q(3, 1) lies on the curve. The normal at Q meets the tangent at P at R.
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AQA 8365 · June 2024 · Paper 1 · Q5 · 3 marks
3 marks
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Examiner report
Lots of 3s and lots of 0s. Even after the comments from previous years, many students are still not splitting the fraction into 2 simplified terms before differentiating. Those that do usually score 3 marks.
Text version
The curve is y = (4x^3 + x^7)/x^4.
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AQA 8365 · June 2024 · Paper 1 · Q19 · 5 marks
5 marks
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Examiner report
A good spread of marks on this question. There is a very clear group of students who have been taught this who get all 5 marks in a clear manner testing the second differential to find the nature of the turning points. There are others who don’t look as though they’ve seen this topic at all. In between there are students who find the values of x correctly and then don’t test the nature of the turning points – just writing max and min or drawing a sketch is insufficient for a test which resulted in many students only being awarded 3 marks rather than the full 5 on this question. Errors in factorising or missing one of the values of x cost some students the first A mark but some of these went on to gain the third M mark for doing a correct second differential and then substituting in either x = 0 or x = 2 correctly to determine the nature of the point.
Text version
The equation of a curve is y = x^3 - 3x^2 + 5.
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AQA 8365 · June 2024 · Paper 2 · Q3 · 3 marks
3 marks
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Examiner report
Nearly all the students who attempted differentiation had at least one term correct. Some arithmetic errors were made when substituting x = 2 into their derivative and others made errors when rearranging to work out k. A significant number only substituted into the given expression for y and did not score any marks.
Text version
The equation of a curve is y = x^4 - 3kx^2, where k is constant. When x = 2, dy/dx = 23.
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AQA 8365 · June 2024 · Paper 2 · Q8 · 3 marks
3 marks
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Examiner report
This question was a good discriminator. The term x4 3 4 was often written as 12x–4 before an attempt at differentiation was made. Another common error was to add one to the negative power when differentiating. Nearly all gave an answer in the required form, but many only scored one mark for differentiating the first term correctly.
Text version
The curve is y = (1/2)x^2 + 3/(4x^4).
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AQA 8365 · June 2024 · Paper 2 · Q10 · 2 marks
2 marks
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Examiner report
A well answered question. The first and last statements were incorrect more often than the second one.
Text version
The curve y = f(x), where f is cubic, has a minimum at (a, 0) and a maximum at (1, b).
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AQA 8365 · June 2023 · Paper 1 · Q6 · 4 marks
4 marks
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Examiner report
Most candidates differentiated but then some didn’t know how to get the gradient when x = 1. We were generous about what we allowed to be used as a gradient for the third mark as we were examining them knowing how to find the equation of a straight line for this mark. A surprisingly high number of candidates found the equation of the normal rather than the tangent (and for the most part did it correctly gaining 3 marks).
Text version
The curve is y = x^6 + 4x^2 - 7 and contains the point (1, -2).
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AQA 8365 · June 2023 · Paper 1 · Q11 · 3 marks
3 marks
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Examiner report
Lots of full marks awarded on this question. 2 marks came from writing ±7 or from 55-6 not getting 49. Too many candidates made an error in their differentiation and couldn’t get more than 1 mark. 49 wasn’t enough to gain the A mark.
Text version
y = x^4/12 + 3x^2 + 4.
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AQA 8365 · June 2023 · Paper 2 · Q5 · 3 marks
3 marks
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Examiner report
Most differentiation was accurate but there were a significant proportion who did not realise that this was required. Those who correctly differentiated usually went on to equate to 6.75, although some took the cube root before dividing by 2 or took the square root instead of the cube root. The most common incorrect method was to solve y = 6.75 or to substitute x = 6.75 in the original equation.
Text version
y = 0.5x^4.
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AQA 8365 · June 2023 · Paper 2 · Q17 · 4 marks
4 marks
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Examiner report
It was rare to see a fully correct solution for this question. Most students did expand the brackets correctly even if they did not know how to proceed after that. However, a good number differentiated correctly and many of those did state that the result had to be > 0. Some thought that it was the second derivative that would be positive. Many who formed the correct inequality then either rearranged and gave an answer for a in terms of x or substituted x = 1 and gave the final answer a < 5 7 − .
Text version
h(x)=ax(3x^2-2)+5x, where a is a positive constant. The function is increasing for all real x.
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AQA 8365 · June 2022 · Paper 1 · Q6 · 3 marks
3 marks
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Examiner report
A very common error on this question was not expanding the brackets before differentiating. It was common to see the part on the outside of the brackets differentiated and then the part on the inside and at that point students attempted to multiply. For those that did expand the brackets first there were quite a few who were multiplying the powers rather than adding them. If they didn’t gain the M mark they couldn’t gain any A marks so whilst there were a few students who managed to gain the middle term in the answer this merited no marks. Students attempting the product rule were few and far between but those that did use this method usually gained full marks.
Text version
y=2x^4(x^3+2-3/x).
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AQA 8365 · November 2021 · Paper 1 · Q19 · 4 marks
4 marks
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Text version
y=f(x) is cubic. It is negative for x<5 and non-negative for x>=5. It is increasing for x<-1, decreasing for -1<x<2, and increasing for x>2.
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AQA 8365 · sample1 · Paper 2 · Q10 · 5 marks
5 marks
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Text version
y=(6x^9+x^8)/(2x^4).
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AQA 8365 · sample1 · Paper 2 · Q12 · 7 marks
7 marks
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Text version
An L-shaped region is made from two rectangles. Its labelled boundary lengths are 30x, 15x, 10x, 20x and y centimetres, and its perimeter is 252 cm. Its area is A cm^2.
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References open the specific AQA question paper when it is available. Missing, login-only or unverified question-paper sources are labelled on the affected row. Question content remains on the exam board's website.

June 20232 questions

Paper 1 · Q175 marks · Non-calculator · calculate · hardView question in paper (p.16)Mark scheme (p.23)
Paper 2 · Q97 marks · Calculator · multi · hardView question in paper (p.8)Mark scheme (p.15)

June 20223 questions

Paper 1: Q20 (p.18, 6 marks)

Paper 2: Q8 (p.6, 4 marks), Q12 (p.10, 5 marks)

November 20213 questions

Paper 1: Q2 (p.2, 3 marks)

Paper 2: Q13 (p.11, 4 marks), Q18 (p.16, 5 marks)

sample14 questions

Paper 1: Q5 (3 marks), Q8 (3 marks), Q15 (9 marks), Q19 (4 marks)

Question extracts are shown for study alongside a link to the original AQA paper. Reference-only rows carry series, paper, question number and marks.

Calculus questions

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