AQA A Level Maths June 2024 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. Algebra and functions · 5 marks

   When $f(x) = 2x^3 + px^2 + qx + 1$ is divided by $(2x-1)$ the remainder is 2. When $f(x)$ is divided by $(x+1)$ the remainder is -4. Find the values of $p$ and $q$.

   Worked answer

   1. M1 for applying the Remainder Theorem with $x = 0.5$: $f(0.5) = 2 \implies 2(\frac{1}{8}) + \frac{p}{4} + \frac{q}{2} + 1 = 2$.
   2. M1 for simplifying the first equation: $\frac{1}{4} + \frac{p}{4} + \frac{2q}{4} = 1 \implies p + 2q = 3$.
   3. M1 for applying the Remainder Theorem with $x = -1$: $f(-1) = -4 \implies -2 + p - q + 1 = -4 \implies p - q = -3$.
   4. M1 for solving the simultaneous equations (e.g., subtracting them gives $3q = 6$).
   5. A1 for stating $q = 2$ and $p = -1$.

2. Differentiation · 5 marks

   A curve has the equation $y = e^{-x}\sin x$. Find the exact x-coordinate of the first positive turning point of the curve.

   Worked answer

   1. M1 for using the product rule to differentiate: $\frac{dy}{dx} = -e^{-x}\sin x + e^{-x}\cos x$.
   2. M1 for setting the derivative equal to 0 to find turning points: $e^{-x}(\cos x - \sin x) = 0$.
   3. M1 for dividing by $e^{-x}$ (since $e^{-x} \ne 0$) to get $\cos x = \sin x$.
   4. M1 for dividing by $\cos x$ to obtain $\tan x = 1$.
   5. A1 for identifying the first positive solution as $x = \frac{\pi}{4}$.

3. Proof · 4 marks

   Prove by contradiction that if $a$ and $b$ are integers, then $a^2 - 4b \ne 2$.

   Worked answer

   1. M1 for assuming that there exist integers $a$ and $b$ such that $a^2 - 4b = 2$, which rearranges to $a^2 = 4b + 2 = 2(2b + 1)$.
   2. M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
   3. M1 for letting $a = 2k$, substituting to get $(2k)^2 = 4b + 2 \implies 4k^2 = 4b + 2$, and dividing by $2$ to get $2k^2 = 2b + 1 \implies 2(k^2 - b) = 1$.
   4. A1 for observing that the left-hand side $2(k^2 - b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2 - 4b \ne 2$.

More on these topics: Algebra and functions · Differentiation · Proof

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: "Explain" and "interpret" answers without context; "Show that" (Q7, Q18); Domain and range in set notation (Q17). 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.