Edexcel A Level Maths June 2025 Paper 3: question paper and mark scheme (9MA0/31, 9MA0/32)

Try 3 A Level questions like these

Written and checked by us, not taken from the Pearson Edexcel paper. Tap "Worked answer" to see every step.

1. 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 answer

   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: $\frac{225}{512}$.

2. Sampling and data · 5 marks

   A histogram is drawn. The class $0 \le x < 20$ has frequency $10$, and the class $20 \le x < 30$ has frequency $30$. If the height of the bar for the first class is $2\text{ cm}$, find the height of the bar for the second class, assuming the width of the bars is proportional to the class width.

   Worked answer

   1. M1 for stating the class widths: $20$ and $10$.
   2. M1 for establishing the width ratio: if class width $20$ has a certain drawn width, class width $10$ has half that drawn width. Let drawn width of first bar be $2w$, then second bar is $w$.
   3. M1 for using Area $\propto$ Frequency for the first bar: Area $= 2w \times 2 = 4w$. So $4w$ represents frequency $10$, meaning $w$ represents frequency $2.5$.
   4. M1 for applying to the second bar: Area $= w \times h$. Frequency $= 30$. So $w \times h$ represents $30$. Since $w$ represents $2.5$, Area must be $12w$.
   5. A1 for solving for height: $w \times h = 12w \implies h = 12\text{ cm}$.

3. Kinematics · 8 marks

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

   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 answer

   1. M1 for writing the vector equations of motion.
      A1 for $\mathbf{r_A} = \begin{pmatrix} 3t \\ 5-t \end{pmatrix}$ and $\mathbf{r_B} = \begin{pmatrix} 4+t \\ 2t \end{pmatrix}$.
   2. M1 for finding the relative displacement vector $\mathbf{r_B} - \mathbf{r_A} = \begin{pmatrix} 4 - 2t \\ 3t - 5 \end{pmatrix}$.
      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 $26t - 46 = 0 \implies 26t = 46$.
      A1 for finding exactly $t = \frac{23}{13}$ hours.

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

What this paper covers

Statistics and Mechanics: Section A is statistics (topics 1 to 5), Section B is mechanics (topics 6 to 9). Pearson sometimes publishes the two sections as separate files (9MA0/31 and 9MA0/32).

We haven't mapped this paper question by question yet, so these are the topics it can test, from the specification's paper structure. Each links to a page where you can learn the topic and then practise questions on it.

Statistics

• Sampling and data
• Probability
• Statistical distributions
• Hypothesis testing

Mechanics

• Kinematics
• Forces and Newton's laws
• Moments

Practise these topics (opens the practice engine)

• Statistics

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.

What the examiners said about this paper

Summarised in our own words from the board's examiner report (Examiner Insights).

• 9MA0/31: Premature rounding and truncating (Q3, Q5); Binomial probabilities (Q1, Q3); Normal distribution (Q5). Official examiner report ↗
• 9MA0/32: "Show that" (Q2, Q6); Mechanics; Moments (Q6). Official examiner report ↗

Question-by-question topics, the top mistakes and what to focus on next →

Formula booklet

Mathematical Formulae and Statistical Tables (A Level Mathematics and Further Mathematics) · Pearson Edexcel · For 9MA0 / 9FM0, first certification 2018 (current)