AQA A Level Maths June 2025 Paper 2: question paper and mark scheme (7357/2)

Try 3 A Level questions like these

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1. Trigonometry · 6 marks

   Solve $3 \cot^2 \theta + 5 \text{cosec} \theta + 1 = 0$ for $0^\circ \le \theta \le 360^\circ$.

   Worked answer

   1. M1 for substituting $\cot^2 \theta = \text{cosec}^2 \theta - 1$ to form $3(\text{cosec}^2 \theta - 1) + 5\text{cosec} \theta + 1 = 0$.
   2. M1 for simplifying to the quadratic $3\text{cosec}^2 \theta + 5\text{cosec} \theta - 2 = 0$.
   3. M1 for factorising to $(3\text{cosec} \theta - 1)(\text{cosec} \theta + 2) = 0$.
   4. M1 for showing $\text{cosec} \theta = \frac{1}{3} \implies \sin \theta = 3$ yields no real solutions.
   5. M1 for solving $\text{cosec} \theta = -2 \implies \sin \theta = -0.5$.
   6. A1 for finding the two correct angles: $210^\circ$ and $330^\circ$.

2. Sequences and series · 4 marks

   The first three terms of the expansion $(a+bx)^6$ are $64 - 192x + cx^2$. Find the values of the constants $a$, $b$, and $c$.

   Worked answer

   1. M1 for equating the first term: $a^6 = 64 \implies a=2$.
   2. M1 for equating the $x$ coefficient to find $b$: $\binom{6}{1}a^5 b = -192 \implies 6(32)b = -192 \implies 192b = -192$.
   3. A1 for obtaining $b = -1$.
   4. A1 for finding the $x^2$ coefficient: $c = \binom{6}{2}a^4 b^2 = 15(16)(-1)^2 = 240$.

3. Kinematics · 6 marks

   A drone is programmed to fly in a straight line. Its position vector \(r\) (in km) at time \(t\) (in hours) is \(r = \begin{pmatrix} -5 \\ 12 \end{pmatrix} + t \begin{pmatrix} 8 \\ -6 \end{pmatrix}\).
   Calculate the exact speed of the drone in km h\(^{-1}\), and find its distance from the origin at \(t=2\).

   Worked answer

   Velocity vector \(v = \begin{pmatrix} 8 \\ -6 \end{pmatrix}\). Speed \(= |v| = \sqrt{8^2 + (-6)^2} = \sqrt{64+36} = 10\) km h\(^{-1}\). M1A1
   Position at \(t=2\): \(r(2) = \begin{pmatrix} -5 \\ 12 \end{pmatrix} + 2 \begin{pmatrix} 8 \\ -6 \end{pmatrix} = \begin{pmatrix} 11 \\ 0 \end{pmatrix}\). M1A1
   Distance from origin \(= \sqrt{11^2 + 0^2} = 11\) km. M1A1

More on these topics: Trigonometry · Sequences and series · Kinematics

What this paper covers

Section A is pure mathematics; Section B is mechanics (AQA sections P to S).

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

Mechanics

• Kinematics
• Forces and Newton's laws
• Moments

Practise these topics (opens the practice engine)

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

What the examiners said about this paper

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

• 7357/2: "Show that" (Q7); Mechanics; Moments (Q18). Official examiner report ↗

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

Formula booklet

Formulae for A-level Mathematics · AQA · Version 1.2 (first issued August 2017), for 7356 / 7357 (current)

Checked against AQA’s own booklet on 11 October 2026.