Edexcel A Level Maths June 2023 Paper 3: question paper and mark scheme (9MA0/31, 9MA0/32)

3 A Level questions like these, with worked solutions

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

1. 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: $0.21+0.14+0.09=0.44$.

2. Sampling and data · 5 marks

   The data in a histogram is grouped into classes. A bar representing the class $100\le x<150$ has a width of $2.5\text{ cm}$ and a height of $4\text{ cm}$. The total area of all bars in the histogram is $90{\text{ cm}}^2$. If the bar for $100\le x<150$ represents a frequency of $20$, calculate the total frequency of the entire data set.

   Worked solution

   1. M1 for calculating the area of the specified bar: $2.5\times 4=10{\text{ cm}}^2$.
   2. M1 for establishing the relationship between area and frequency: $10{\text{ cm}}^2$ represents a frequency of $20$.
   3. A1 for calculating the scale factor: $1{\text{ cm}}^2$ represents $2$ items (since $20/10=2$).
   4. M1 for using the total area to find total frequency: $90{\text{ cm}}^2\times 2$.
   5. A1 for the final answer: $180$ items.

3. Kinematics · 8 marks

   Ship A starts at position $(0,5)$ relative to a port and moves with a constant velocity vector of $\left(\begin{array}{c}3\\ -1\end{array}\right){\text{ km h}}^{-1}$. Ship B starts at $(4,0)$ and moves with a constant velocity vector of $\left(\begin{array}{c}1\\ 2\end{array}\right){\text{ km h}}^{-1}$.

   1. Write down the position vectors of Ship A and Ship B at time $t$ hours.
   2. Find an expression for the square of the distance between the ships at time $t$, denoted $D^2(t)$.
   3. Find the exact time $t$ when the two ships are closest to each other.
   Worked solution

   1. M1 for writing the vector equations of motion.
      A1 for ${\mathbf{r}}_{\mathbf{A}}=\left(\begin{array}{c}3t\\ 5-t\end{array}\right)$ and ${\mathbf{r}}_{\mathbf{B}}=\left(\begin{array}{c}4+t\\ 2t\end{array}\right)$.
   2. M1 for finding the relative displacement vector ${\mathbf{r}}_{\mathbf{B}}-{\mathbf{r}}_{\mathbf{A}}=\left(\begin{array}{c}4-2t\\ 3t-5\end{array}\right)$.
      M1 for setting up the square of the magnitude: $D^2(t)=(4-2t{)}^2+(3t-5{)}^2$.
      A1 for expanding and simplifying to a quadratic: $D^2(t)=16-16t+4t^2+9t^2-30t+25=13t^2-46t+41$.
   3. M1 for recognizing that minimum distance occurs when the derivative of $D^2(t)$ is zero (or using the vertex formula $t=-b/2a$).
      M1 for solving $26t-46=0⟹26t=46$.
      A1 for finding exactly $t=\frac{23}{13}$ hours.

More on these topics: Probability · Sampling and data · Kinematics

What this paper covers

What this paper is like

• Shape: 6 questions, 50 marks in 1 h (about 1.2 minutes a mark); questions from 6 to 14 marks.
• Biggest question: Q6, 14 marks: normal model for hospital stays and a test.
• Marks by main topic: Statistical distributions 29, Probability 14, Sampling and data 7.
• More Statistical distributions in other sittings: June 2022 Q2 (12 marks), June 2024 Q1 (11 marks), November 2021 Q5 (11 marks).
• Against the other 6 sittings we mapped: more marks on Statistical distributions (29 marks here, 22 on average); fewer on Sampling and data (7 marks here, 16 on average).
• Not on this paper: Hypothesis testing.
• Calculator: allowed.

9MA0/31: question by question

Labels and topics are ours, read from the 9MA0/31 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Venn diagram with unknown probabilities (6 marks) · Probability
2. Q2Packets with prizes: binomial model and a test (9 marks) · Statistical distributions, Hypothesis testing
3. Q3Large data set: rainfall at Leeming (7 marks) · Sampling and data
4. Q4Normal model for men's heights (6 marks) · Statistical distributions
5. Q5A game with a spinner: probability distribution (8 marks) · Probability, Statistical distributions
6. Q6Normal model for hospital stays and a test (14 marks) · Statistical distributions, Hypothesis testing

What this paper is like

• Shape: 6 questions, 50 marks in 1 h (about 1.2 minutes a mark); questions from 3 to 13 marks.
• Biggest question: Q6, 13 marks: rod against a smooth wall, limiting equilibrium.
• Marks by main topic: Kinematics 33, Moments 13, Forces and Newton's laws 4.
• More Moments in other sittings: June 2022 Q4 (11 marks), June 2024 Q6 (9 marks), November 2021 Q3 (10 marks).
• Against the other 6 sittings we mapped: fewer on Forces and Newton's laws (4 marks here, 11 on average).
• Coverage: every topic on the list came up.
• Calculator: allowed.

9MA0/32: question by question

Labels and topics are ours, read from the 9MA0/32 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Car accelerating from rest (3 marks) · Kinematics
2. Q2Particle pulled on a rough horizontal plane (4 marks) · Forces and Newton's laws
3. Q3Velocity vector: speed and acceleration (9 marks) · Kinematics
4. Q4Particle moving with i and j vectors (10 marks) · Kinematics, Forces and Newton's laws
5. Q5Projectile clearing a wall (11 marks) · Kinematics
6. Q6Rod against a smooth wall, limiting equilibrium (13 marks) · Moments, Forces and Newton's laws

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