AQA A Level Maths Specimen Paper 3: question paper and mark scheme (7357/3)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the AQA paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Differentiation · 5 marks

   A curve has parametric equations $x=\ln t$ and $y=t^2$, $t>0$. Find the equation of the normal to the curve at the point where $t=1$. Give your answer in the form $ax+by+c=0$.

   Worked solution

   1. M1 for finding the coordinates at $t=1$: $x=0$, $y=1$.
   2. M1 for calculating $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=2t$, giving $\frac{dy}{dx}=2t^2$.
   3. M1 for evaluating the tangent gradient at $t=1$ to get $m_t=2$.
   4. M1 for determining the normal gradient $m_n=-\frac{1}{2}$ and using $(0,1)$ to form the equation $y-1=-\frac{1}{2}(x-0)$.
   5. A1 for rearranging to the required integer format: $x+2y-2=0$.

2. Integration · 4 marks

   Find $\int (\sin x+\cos x{)}^{2}dx$.

   Worked solution

   1. M1 for expanding the bracket to ${\sin }^{2}x+2\sin x\cos x+{\cos }^{2}x$.
   2. M1 for simplifying using ${\sin }^{2}x+{\cos }^{2}x\equiv 1$ and $2\sin x\cos x\equiv \sin 2x$.
   3. M1 for integrating $1+\sin 2x$.
   4. A1 for $x-\frac{1}{2}\cos 2x+c$.

3. Probability · 3 marks

   A bag contains 5 green counters and 3 yellow counters. A counter is drawn at random, its colour is noted, and it is replaced. This is done 3 times in total. Find the probability that exactly 2 green counters are drawn.

   Worked solution

   1. M1 for identifying the 3 ways to draw exactly 2 green counters: GGY, GYG, YGG.
   2. M1 for calculating the probability of one such arrangement, e.g., $(\frac{5}{8})\times (\frac{5}{8})\times (\frac{3}{8})=\frac{75}{512}$.
   3. A1 for multiplying by 3 to find the total probability: $\frac{225}{512}$.

More on these topics: Differentiation · Integration · Probability

What this paper covers

What this paper is like

• Shape: 15 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 13 marks.
• Biggest question: Q3, 13 marks: differential equation for weed growth.
• Marks by main topic: Sampling and data 23, Statistical distributions 21, Integration 19, Differentiation 12, Algebra and functions 8, Trigonometry 6, Probability 6, Sequences and series 5.
• More Integration in other sittings: November 2020 Q1 (1 marks), November 2021 Q7 (10 marks), June 2022 Q6 (9 marks).
• Against the other 6 sittings we mapped: more marks on Integration (19 marks here, 6 on average) and Sampling and data (23 marks here, 12 on average); fewer on Hypothesis testing (0 marks here, 6 on average) and Exponentials and logarithms (0 marks here, 6 on average).
• Not on this paper: Proof, Coordinate geometry, Exponentials and logarithms, Numerical methods, Vectors, Hypothesis testing.
• Calculator: allowed.

7357/3: question by question

Labels and topics are ours, read from the 7357/3 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Area under y = x^2 - 9 (multiple choice) (1 marks) · Integration
2. Q2Decking frame shaped as a sector (6 marks) · Trigonometry
3. Q3Differential equation for weed growth (13 marks) · Integration, Exponentials and logarithms
4. Q4Integration by parts with ln 2x (5 marks) · Integration, Exponentials and logarithms
5. Q5Binomial expansion and an approximation (5 marks) · Sequences and series
6. Q6Partial fractions and an exact integral (8 marks) · Algebra and functions, Integration
7. Q7Concave section of a graph and an area (12 marks) · Differentiation, Integration
8. Q8Sampling households who eat out (2 marks) · Sampling and data
9. Q9Households that grow their own food (3 marks) · Probability
10. Q10Large data set: correlation coefficients (7 marks) · Sampling and data
11. Q11Staffed checkouts: probability (3 marks) · Probability
12. Q12Teacher illness: binomial test (10 marks) · Statistical distributions, Hypothesis testing
13. Q13Household data: summary statistics (8 marks) · Sampling and data
14. Q14Mean spending: normal test (11 marks) · Statistical distributions, Hypothesis testing
15. Q15Questionnaire sample from a small town (6 marks) · Sampling and data

Some Paper 3 statistics questions use AQA's large data set. We don't write large data set questions, so use your class materials for those.