Edexcel A Level Maths June 2025 Paper 2: question paper and mark scheme (9MA0/02)

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1. Coordinate geometry · 8 marks

   Three points define the vertices of a triangle: \(A(-2, -1)\), \(B(4, 7)\), and \(C(8, 4)\).

   1. By calculating the gradients of \([AB]\) and \([BC]\), prove that triangle \(ABC\) is a right-angled triangle.

   2. Calculate the exact lengths of \([AB]\) and \([BC]\).

   3. Hence, find the exact area of the triangle \(ABC\).

   Worked answer

   1. Gradient \(m_{AB} = \frac{7 - (-1)}{4 - (-2)} = \frac{8}{6} = \frac{4}{3}\). A1
      Gradient \(m_{BC} = \frac{4 - 7}{8 - 4} = \frac{-3}{4}\). A1
      Since \(m_{AB} \times m_{BC} = \frac{4}{3} \times -\frac{3}{4} = -1\), \([AB]\) and \([BC]\) are perpendicular. M1
      Thus, triangle \(ABC\) has a right angle at \(B\). B1

   2. Length \(AB = \sqrt{(4 - (-2))^2 + (7 - (-1))^2} = \sqrt{36 + 64} = \sqrt{100} = 10\). A1
      Length \(BC = \sqrt{(8 - 4)^2 + (4 - 7)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\). A1

   3. Since the triangle is right-angled at \(B\), \(AB\) and \(BC\) act as the base and height. M1
      Area \(= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 5 = 25\) sq units. A1

2. Sequences and series · 5 marks

   Expand $(2-x)^{-2}$ up to the term in $x^2$. Use your expansion to find an approximate value for $\frac{1}{1.98^2}$.

   Worked answer

   1. M1 for rewriting as $2^{-2}(1 - \frac{x}{2})^{-2} = \frac{1}{4}(1 - \frac{x}{2})^{-2}$.
   2. M1 for expanding correctly: $\frac{1}{4}(1 + 2(\frac{x}{2}) + 3(\frac{x}{2})^2) = \frac{1}{4} + \frac{1}{4}x + \frac{3}{16}x^2$.
   3. M1 for deducing the appropriate value of $x$ to substitute is $x = 0.02$.
   4. M1 for substituting $x = 0.02$ into the expansion: $\frac{1}{4} + \frac{1}{4}(0.02) + \frac{3}{16}(0.0004)$.
   5. A1 for arriving at the correct approximation: $0.25 + 0.005 + 0.000075 = 0.255075$.

3. Vectors · 8 marks

   Consider the following two lines in 3D space: \[L_1: r = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} \quad \text{and} \quad L_2: r = \begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}\] Show algebraically that these two lines are skew (they are neither parallel nor do they intersect).

   Worked answer

   Set the components equal:
   \(1 + 2\lambda = 4 - \mu \implies 2\lambda + \mu = 3\) (Eq 1). M1
   \(-1 + \lambda = 2 + 2\mu \implies \lambda - 2\mu = 3\) (Eq 2). M1
   \(2 - 3\lambda = -1 + \mu \implies -3\lambda - \mu = -3\) (Eq 3). M1
   Solve Eq 1 and Eq 2: \(2(3 + 2\mu) + \mu = 3 \implies 6 + 5\mu = 3 \implies \mu = -0.6\). M1
   Substitute \(\mu = -0.6 \implies \lambda = 1.8\). A1
   Check in Eq 3: \(-3(1.8) - (-0.6) = -5.4 + 0.6 = -4.8\). M1
   However, the RHS of Eq 3 is \(-3\). Since \(-4.8 \neq -3\), there is no consistent solution. B1
   Therefore, the lines do not intersect. Since their direction vectors are not scalar multiples, they are skew. A1

More on these topics: Coordinate geometry · Sequences and series · Vectors

What this paper covers

Pure Mathematics 2: any of the pure content (Edexcel Pure topics 1 to 10).

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.

Pure

• Proof
• Algebra and functions
• Coordinate geometry
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods
• Vectors

Practise these topics (opens the practice engine)

• Pure I
• Pure II
• Pure III (calculus)

What the examiners said about this paper

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

• 9MA0/02: "Show that" (Q8); Inverse and composite functions (Q12); Modelling (Q9, Q11). 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)