AQA A Level Maths November 2021 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. Algebra and functions · 5 marks

   Express $\frac{x+2}{x^2-5x+6}-\frac{x-1}{x^2-4x+3}$ as a single fraction in its simplest form.

   Worked solution

   1. M1 for factorising $x^2-5x+6$ to $(x-2)(x-3)$.
   2. M1 for factorising $x^2-4x+3$ to $(x-1)(x-3)$.
   3. M1 for simplifying the second fraction: $\frac{x-1}{(x-1)(x-3)}$ cancels to $\frac{1}{x-3}$.
   4. M1 for creating a common denominator $(x-2)(x-3)$ and subtracting: $\frac{x+2}{(x-2)(x-3)}-\frac{x-2}{(x-2)(x-3)}$.
   5. A1 for reaching the final simplified answer $\frac{4}{(x-2)(x-3)}$.

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

   Three independent events $A$, $B$, and $C$ have probabilities $P(A)=0.5$, $P(B)=0.4$, and $P(C)=0.3$. Find the probability that exactly one of these events occurs.

   Worked solution

   1. M1 for identifying the three scenarios: $(A\cap B^{\prime }\cap C^{\prime })$, $(A^{\prime }\cap B\cap C^{\prime })$, and $(A^{\prime }\cap B^{\prime }\cap C)$.
   2. M1 for calculating the probabilities of these scenarios: $(0.5\times 0.6\times 0.7)+(0.5\times 0.4\times 0.7)+(0.5\times 0.6\times 0.3)$.
   3. A1 for evaluating and summing: $0.21+0.14+0.09=0.44$.

More on these topics: Algebra and functions · Integration · Probability

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 13 marks.
• Biggest question: Q5, 13 marks: flowerbeds as sectors: minimising edging.
• Marks by main topic: Statistical distributions 26, Integration 16, Trigonometry 14, Differentiation 10, Sampling and data 10, Probability 7, Hypothesis testing 7, Algebra and functions 5, Sequences and series 5.
• More Trigonometry in other sittings: June 2022 Q5 (3 marks), November 2020 Q9 (5 marks), June 2023 Q7 (14 marks).
• Against the other 5 sittings we mapped: more marks on Integration (16 marks here, 4 on average) and Trigonometry (14 marks here, 7 on average); fewer on Algebra and functions (5 marks here, 14 on average) and Exponentials and logarithms (0 marks here, 7 on average).
• Not on this paper: Proof, Coordinate geometry, Exponentials and logarithms, 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. Q1Arccos graph end point (multiple choice) (1 marks) · Trigonometry
2. Q2Simplifying an algebraic fraction (multiple choice) (1 marks) · Algebra and functions
3. Q3Limit from first principles (multiple choice) (1 marks) · Differentiation
4. Q4Binomial expansion in descending powers (5 marks) · Sequences and series
5. Q5Flowerbeds as sectors: minimising edging (13 marks) · Trigonometry, Differentiation
6. Q6Simplifying surd and index expressions (4 marks) · Algebra and functions
7. Q7Differential equation for a leaking roof (10 marks) · Integration
8. Q8Integration by parts with x cos x (6 marks) · Integration, Trigonometry
9. Q9Stationary points and a point of inflection (9 marks) · Differentiation
10. Q10Correlation in car data (multiple choice) (1 marks) · Sampling and data
11. Q11Binomial mean and variance (multiple choice) (1 marks) · Statistical distributions
12. Q12Systematic sample from an electoral register (3 marks) · Sampling and data
13. Q13Large data set: vehicle emissions extract (6 marks) · Sampling and data
14. Q14Two events and conditional probability (7 marks) · Probability
15. Q15Testing escape room times (7 marks) · Hypothesis testing
16. Q16Discrete distribution with a constant c (4 marks) · Statistical distributions
17. Q17Computer game wins: binomial probabilities (11 marks) · Statistical distributions, Probability
18. Q18Normal model for jam jar weights (10 marks) · 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.