AQA A Level Maths June 2025 Paper 1: question paper and mark scheme (7357/1)

Try 3 A Level questions like these

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

1. Coordinate geometry · 8 marks

   The line \(L_1\) has equation \(y = x + 2\) and the line \(L_2\) has equation \(y = -2x + 14\). The lines intersect at point \(A\).

   1. Find the coordinates of \(A\).

   2. Point \(B\) lies on \(L_1\) such that its \(x\)-coordinate is less than the \(x\)-coordinate of \(A\). The distance \(AB\) is exactly \(\sqrt{18}\) units. Find the coordinates of \(B\).

   3. The line \(L_3\) passes through point \(B\) and is perpendicular to \(L_1\). Find the equation of \(L_3\) and determine where it intersects \(L_2\).

   Worked answer

   1. Solve simultaneously: \(x + 2 = -2x + 14 \implies 3x = 12 \implies x = 4\). M1
      \(y = 4 + 2 = 6\). \(A(4, 6)\). A1

   2. Let \(B\) be \((x, x+2)\). Distance \(AB = \sqrt{18}\).
      \((x - 4)^2 + (x+2 - 6)^2 = 18 \implies (x - 4)^2 + (x - 4)^2 = 18\). M1
      \(2(x - 4)^2 = 18 \implies (x - 4)^2 = 9\). A1
      \(x - 4 = \pm 3 \implies x = 7\) or \(x = 1\). A1
      Since \(x\) for \(B\) is less than \(x\) for \(A\) (which is \(4\)), \(x = 1\).
      \(y = 1 + 2 = 3\). \(B(1, 3)\). A1

   3. \(L_3\) is perpendicular to \(L_1\) (\(m = 1\)), so \(m_3 = -1\). M1
      Passes through \(B(1, 3)\): \(y - 3 = -1(x - 1) \implies y = -x + 4\).
      Intersection with \(L_2\) (\(y = -2x + 14\)):
      \(-x + 4 = -2x + 14 \implies x = 10\). \(y = -10 + 4 = -6\).
      Intersection point is \((10, -6)\). A1

2. Trigonometry · 5 marks

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

   Worked answer

   1. M1 for using the identity $\text{cosec}^2 x = 1 + \cot^2 x$ to obtain $1 + 2\cot^2 x = 3$.
   2. M1 for rearranging to $\cot^2 x = 1$.
   3. M1 for taking the square root to find $\tan x = \pm 1$.
   4. A1 for finding the principal values $45^\circ$ and $135^\circ$.
   5. A1 for stating all four correct solutions: $45^\circ, 135^\circ, 225^\circ, 315^\circ$.

3. Exponentials and logarithms · 4 marks

   Given that $\log_2 x = p$ and $\log_2 y = q$, express $\log_2(8x^2 y^3)$ in terms of $p$ and $q$.

   Worked answer

   1. M1 for splitting the logarithm into a sum: $\log_2 8 + \log_2 x^2 + \log_2 y^3$.
   2. B1 for evaluating $\log_2 8 = 3$.
   3. M1 for applying the power law: $2\log_2 x + 3\log_2 y$.
   4. A1 for the final expression $3 + 2p + 3q$.

More on these topics: Coordinate geometry · Trigonometry · Exponentials and logarithms

What this paper covers

Pure mathematics only (AQA sections A to J).

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).

• 7357/1: "Show that" (Q10); Domain and range in set notation (Q12); Logarithms (Q10). 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.