AQA Further Mathematics Paper 3d 2025 Mark Scheme Pdf

Download the official AQA A-Level Further Mathematics Paper 3 (Discrete) mark scheme for June 2025 (7367/3D). This **final version (v1.0)** contains full worked solutions, examiner guidance, and detailed mark allocation used during standardisation.

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

The mark scheme explains how examiners award marks:

- **M (Method):** Correct process
- **A (Accuracy):** Correct answer
- **B (Independent):** Standalone marks
- **R (Reasoning):** Logical justification
- **E (Explanation):** Clear explanation

👉 As stated on **page 3**, examiners:
- Credit **valid alternative methods**
- Apply **best-fit marking**
- Use standardisation to ensure consistency

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## **Section A: Multiple Choice Answers (Q1–Q3)** *(page 6)*

- **Q1:** 3
- **Q2:** \( P = 25,\ x = 0,\ y = 5 \)
- **Q3:** Bottom-right Cayley table

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## **Key Worked Answers & Methods**

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### **Minimum Spanning Tree (Q4, page 7)**
- Correct arcs identified using Prim’s algorithm
- Total length:
\[
2.7 + 1.4 + 1.3 + 3.4 = 8.8 \text{ miles}
\]

- **Q4(b):**
- Hoole–Upton link **not part of MST**
- Therefore, total length **unchanged**

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### **Critical Path Analysis (Q5, pages 8–9)**

- Correct earliest start & latest finish times shown on network diagram
- **Float of activity F = 4 days**

👉 The **completed network diagram on page 8** shows timings like:
- A: 0 → 8
- C: 6 → 24
- J finishes at **35 days**

- Gantt chart must:
- Include all activities
- Show **critical path clearly**

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### **Network Flows (Q6, page 10)**

- Supersource: **C and H**
- Supersink: **D and G**

- Cut value:
\[
79 \text{ litres per second}
\]

- Conclusion:
- Max flow = **79** (matches cut)
- Andrew’s claim is **correct** (max-flow min-cut theorem)
- Adding supersource/sink **does NOT increase flow**

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### **Linear Programming (Q7, pages 11–12)**

- Optimal solution:
- Malt: **30 loaves**
- Rye: **60 loaves**

- Maximum profit:
\[
£240
\]

- Limitation:
- Assumes **all loaves are sold**

👉 The **graph on page 11** shows feasible region and optimal vertex (30, 60)

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### **Planar Graphs (Q8, pages 13–15)**

- Using Euler’s formula:
\[
n = 2
\]

- Graph properties:
- Vertices: **4**
- Edges: **4**
- Faces: **2**

- **Eulerian case:**
- Graph is **bipartite (K₂,₂)** → claim valid

- **Semi-Eulerian case:**
- Graph contains a **cycle**
- Not a tree → claim invalid

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### **Game Theory (Q9, pages 16–17)**

- **Q9(a)(i):**
- Strategy **C is dominated by B**

- **Q9(a)(ii): Mixed strategy:**
- \( p = \frac{5}{11} \) for A
- \( \frac{6}{11} \) for B
- 0 for C

- **Q9(b): LP inequalities**
\[
6p_1 + 3p_2 + 7p_3 \ge v
\]
\[
2p_1 + 4p_2 + p_3 \ge v
\]
\[
4p_1 + 3p_2 + 8p_3 \ge v
\]

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### **Group Theory (Q10, page 18)**

- **Identity element:**
\[
(0, 1)
\]

- **Inverse of (a, b):**
\[
\left(\frac{a}{a^2 + b^2},\ \frac{-b}{a^2 + b^2}\right)
\]

- Requires:
- Left and right operation proof
- Algebraic manipulation

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

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

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## **Why This Mark Scheme Matters**
- Shows **exact answers + examiner expectations**
- Highlights **method marks (crucial for partial credit)**
- Explains **how to structure full solutions**
- Essential for mastering **Discrete Maths Paper 3**

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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, particularly the Discrete option.

It is ideal for revision, checking answers, understanding mark allocation, and improving exam technique for top grades.