AQA A Level Maths November 2020 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. Integration · 5 marks

   Find $\int \frac{x^2+1}{x(x-1)}dx$.

   Worked solution

   1. M1 for recognising the improper fraction and performing division to get $1+\frac{x+1}{x(x-1)}$.
   2. M1 for setting up partial fractions for the remainder: $\frac{x+1}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1}$.
   3. A1 for determining the constants $A=-1$ and $B=2$.
   4. M1 for formulating the complete integral: $\int (1-\frac{1}{x}+\frac{2}{x-1})dx$.
   5. A1 for integrating the terms correctly to $x-\ln |x|+2\ln |x-1|+c$.

2. Proof · 4 marks

   Prove by contradiction that there are infinitely many prime numbers.

   Worked solution

   1. M1 for assuming that there is a finite number of prime numbers, listing them as $p_1,p_2,\dots ,p_n$.
   2. M1 for constructing a new number $N=(p_1\times p_2\times \cdots \times p_n)+1$.
   3. M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
   4. A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

3. Sampling and data · 4 marks

   A dataset $x$ has mean $\overline{x}=5$ and standard deviation ${\sigma }_x=2$. A new dataset $y$ is formed using the linear transformation $y=ax+b$, where $a>0$. The new dataset has mean $\overline{y}=15$ and standard deviation ${\sigma }_y=6$. Find the values of $a$ and $b$.

   Worked solution

   1. M1 for linking the standard deviations: ${\sigma }_y=a{\sigma }_x$.
   2. A1 for solving for $a$: $6=a(2)⟹a=3$.
   3. M1 for linking the means: $\overline{y}=a\overline{x}+b$.
   4. A1 for substituting $a=3$ to find $b$: $15=3(5)+b⟹b=0$.

More on these topics: Integration · Proof · Sampling and data

What this paper covers

What this paper is like

• Shape: 18 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 14 marks.
• Biggest question: Q18, 14 marks: shirt defects: probability and binomial.
• Marks by main topic: Sampling and data 20, Sequences and series 19, Algebra and functions 15, Probability 15, Exponentials and logarithms 9, Statistical distributions 8, Hypothesis testing 7, Trigonometry 6, Integration 1.
• More Probability in other sittings: November 2021 Q14 (7 marks), June 2019 Q14 (7 marks), June 2022 Q16 (10 marks).
• Against the other 5 sittings we mapped: more marks on Sequences and series (19 marks here, 3 on average) and Sampling and data (20 marks here, 11 on average); fewer on Statistical distributions (8 marks here, 25 on average) and Differentiation (0 marks here, 10 on average).
• Not on this paper: Proof, Coordinate geometry, Differentiation, Numerical methods, Vectors.
• 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. Q1Properties of definite integrals (multiple choice) (1 marks) · Integration
2. Q2Value of R in R cos(x + a) (multiple choice) (1 marks) · Trigonometry
3. Q3Graphs that are not functions (tick one) (1 marks) · Algebra and functions
4. Q4Factor theorem and factorising a cubic (7 marks) · Algebra and functions
5. Q5Radioactive decay model (9 marks) · Exponentials and logarithms
6. Q6Transforming a graph (7 marks) · Algebra and functions
7. Q7Binomial coefficients and an equation in n (7 marks) · Sequences and series
8. Q8Geometric series with a sum to infinity of 96 (12 marks) · Sequences and series
9. Q9Proving a cosec and cot identity (5 marks) · Trigonometry
10. Q10Venn diagram with three events (multiple choice) (1 marks) · Probability
11. Q11Mount Everest temperatures (multiple choice) (1 marks) · Sampling and data
12. Q12Large data set: comparing two regions (4 marks) · Sampling and data
13. Q13Sampling in a school (6 marks) · Sampling and data
14. Q14Testing hospital waiting times (7 marks) · Hypothesis testing
15. Q15Sampling for a party election (5 marks) · Sampling and data
16. Q16Significance of a correlation coefficient (4 marks) · Sampling and data, Hypothesis testing
17. Q17Normal model for battery life (8 marks) · Statistical distributions
18. Q18Shirt defects: probability and binomial (14 marks) · Probability, Statistical distributions

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.