AQA A Level Maths June 2022 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 · 4 marks

   A cubic curve crosses the x-axis at $x=-2$, $x=1$, and $x=4$. It crosses the y-axis at the point $(0,16)$. Find the equation of the curve in the form $y=a{x}^3+b{x}^2+cx+d$.

   Worked solution

   1. M1 for writing the equation in the factored form $y=k(x+2)(x-1)(x-4)$.
   2. M1 for substituting $(0,16)$ to find $k$: $16=k(2)(-1)(-4)⟹16=8k$.
   3. A1 for finding $k=2$.
   4. A1 for expanding to get $y=2(x^3-3x^2-6x+8)=2x^3-6x^2-12x+16$.

2. Integration · 6 marks

   Given that $f^{″}(x)=12x-4$, $f^{\prime }(1)=2$ and $f(-1)=5$, find $f(x)$.

   Worked solution

   1. M1 for integrating $f^{″}(x)$ to find $f^{\prime }(x)=6x^2-4x+c_1$.
   2. M1 for substituting $f^{\prime }(1)=2$ to find $c_1$: $2=6(1{)}^2-4(1)+c_1⟹c_1=0$.
   3. M1 for substituting $c_1=0$ and integrating again to find $f(x)=2x^3-2x^2+c_2$.
   4. M1 for substituting $f(-1)=5$: $5=2(-1{)}^3-2(-1{)}^2+c_2$.
   5. M1 for solving for $c_2$: $5=-2-2+c_2⟹c_2=9$.
   6. A1 for the final correct function $f(x)=2x^3-2x^2+9$.

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: 19 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 13 marks.
• Biggest question: Q10, 13 marks: rational function: domain, range and stationary points.
• Marks by main topic: Statistical distributions 28, Algebra and functions 15, Integration 11, Probability 10, Exponentials and logarithms 7, Differentiation 7, Proof 6, Sampling and data 6, Hypothesis testing 6, Trigonometry 3, Sequences and series 1.
• More Algebra and functions in other sittings: November 2021 Q6 (4 marks), June 2023 Q6 (9 marks), November 2020 Q4 (7 marks).
• Against the other 5 sittings we mapped: more marks on Statistical distributions (28 marks here, 21 on average) and Integration (11 marks here, 5 on average); fewer on Sampling and data (6 marks here, 13 on average) and Trigonometry (3 marks here, 9 on average).
• Not on this paper: Coordinate geometry, 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. Q1Validity of a binomial expansion (multiple choice) (1 marks) · Sequences and series
2. Q2Region between a curve and a line (multiple choice) (1 marks) · Algebra and functions
3. Q3Solving f(x) = f inverse(x) (multiple choice) (1 marks) · Algebra and functions
4. Q4Integrating powers of x (2 marks) · Integration
5. Q5Graph of sin 2x and the number of solutions (3 marks) · Trigonometry
6. Q6Surfboard design: area from a parametric curve (9 marks) · Integration, Coordinate geometry
7. Q7Planet orbit data fitted with logs (7 marks) · Exponentials and logarithms
8. Q8Rates of change filling a cone (7 marks) · Differentiation
9. Q9Proof that an integer is even (6 marks) · Proof
10. Q10Rational function: domain, range and stationary points (13 marks) · Algebra and functions, Differentiation
11. Q11Normal standard deviation (multiple choice) (1 marks) · Statistical distributions
12. Q12Shape of a distribution (multiple choice) (1 marks) · Sampling and data
13. Q13Large data set: CO2 emissions (2 marks) · Sampling and data
14. Q14Call centre calls: binomial and a test (10 marks) · Statistical distributions, Hypothesis testing
15. Q15Sampling adults on an electoral register (3 marks) · Sampling and data
16. Q16Pet ownership: Venn diagram (10 marks) · Probability
17. Q17Normal model for working hours (6 marks) · Statistical distributions
18. Q18Normal model for athletes' heights (11 marks) · Statistical distributions
19. Q19Testing a bank campaign (6 marks) · Hypothesis testing

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.