AQA A Level Further Mathematics Paper 1 2025 Mark Scheme Pdf

Description
Download the official AQA A-Level Further Mathematics Paper 1 mark scheme for June 2025 (7367/1). This **final version (v1.0)** includes full worked solutions, method marks, and examiner guidance used during marking. :contentReference[oaicite:0]{index=0}

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## **Mark Scheme Overview (pages 2–4)**

The mark scheme explains how examiners award marks:

- **M (Method):** Correct approach
- **A (Accuracy):** Correct answer from valid method
- **B (Independent):** Fact or result
- **R (Reasoning):** Logical justification
- **E (Explanation):** Clear explanation

👉 The **instructions on page 3** emphasise:
- Credit is given for **valid alternative methods**
- Marks follow a **“best fit” approach**
- Correct answers without working may still gain full marks (where appropriate)

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## **Section A: Short Questions (Q1–Q4)**

Quick 1-mark answers:

- **Q1:** \( c > 4 \)
- **Q2:** \( -z^* \) (reflection in imaginary axis)
- **Q3:** Expression involving \( \lim \frac{\sqrt{x}}{\cos x} \) cannot use L’Hôpital
- **Q4:** \( \text{sech}^2 x + \tanh^2 x = 1 \)

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## **Key Worked Answers & Methods**

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### **Complex Numbers (Q5, page 6)**
- Uses:
\[
\arg\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2
\]
- Final answer: **≈ 2.23 radians (128°)**

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### **Numerical Methods (Q6, page 7)**
- Simpson’s Rule with 5 ordinates
- Final answer: **1.360 (3 d.p.)**

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### **Algebraic Transformations (Q7, page 8)**
- Substitution: \( w = 2z - 1 \)
- Final equation:
\[
5w^3 + 23w^2 + 27w + 33 = 0
\]

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### **Geometry & Vectors (Q9, page 9)**
- Uses:
- Dot product
- Cross product
- Final magnitude: **|b| = √3 ≈ 1.73**

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### **Proof by Induction (Q10, page 10)**
- Common mistake: missing **base case (n = 1)**
- Required:
- Show true for n = 1
- Assume true for n = k
- Prove for n = k + 1

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### **Maclaurin Series (Q11, page 11)**
- Expansion:
\[
f(x) = \frac{1}{2} - \frac{1}{4}x + \frac{1}{48}x^3
\]

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### **Matrices & Eigenvalues (Q12, pages 12–13)**
- Eigenvalues: **1 and 0.5**
- Invariant line:
\[
y = \frac{1}{3}x
\]
- Matrix diagonalisation used to find limits of powers

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### **Complex Roots (Q13, page 14)**
- Roots:
- \( -4 + 3i \), \( -4 - 3i \), \( \frac{1}{4} \)
- Coefficients:
- \( r = 31 \), \( s = -25 \)

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### **Hyperbolic Functions (Q14, page 15)**
- Final solution:
\[
x = -\frac{13}{12}
\]

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### **3D Geometry (Q15, pages 16–17)**
- Parameter value: **t = 6**
- Vector equation derived using:
- Simultaneous planes
- Direction vectors

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### **Integration & Modelling (Q16, page 18)**
- Surface area of revolution:
- Final answer: **0.796 m²**
- Requires:
- Differentiation
- Substitution into surface area formula

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### **Differential Equations (Q17, pages 19–20)**
- Equation of motion:
\[
0.6\frac{d^2x}{dt^2} + \frac{4}{5}\frac{dx}{dt} + 8x = 0
\]
- Solution involves:
- Auxiliary equation
- Exponential + trig form

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### **Hyperbolic Graphs & Area (Q18, pages 21–22)**
- Inequality solution:
\[
\ln(\sqrt{5}-2) < x < 0 \quad \text{or} \quad x > \ln 4
\]
- Final area:
\[
\frac{9}{2} + \ln 2 + 15\ln\left(\frac{4}{3}\right)
\]

👉 The **diagram on page 30 of the question paper** corresponds directly to this area calculation.

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## **Assessment Objectives (page 5)**

- **AO1:** Procedures & accuracy
- **AO2:** Reasoning & proof
- **AO3:** Modelling & interpretation

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## **Why This Mark Scheme Matters**
- Shows **exact examiner expectations**
- Reveals **method marks (crucial for partial credit)**
- Helps you understand **how to reach full marks**
- Essential for mastering **Further Maths exam technique**

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Fully updated for the 2025/2026 academic year and optimised for instant download on markscheme.net.

This mark scheme is designed for Year 13 students studying AQA A-Level Further Mathematics.

It is ideal for exam revision, checking answers, understanding mark allocation, and improving problem-solving techniques for high-grade performance.