AQA Further Mathematics Paper 2 2025 Mark Scheme Pdf

Download the official AQA A-Level Further Mathematics Paper 2 mark scheme for June 2025 (7367/2). This **final version (v1.0)** provides full worked solutions, examiner guidance, and detailed mark allocation used in real marking.

---

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

The mark scheme outlines how examiners award marks:

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

👉 As stated on **page 3**, examiners:
- Award marks for **valid alternative methods**
- Use **standardisation** to ensure consistency
- Apply a **best-fit marking approach**

---

## **Key Answers & Methods**

---

### **Section A: Short Questions (Q1–Q4)** *(page 6)*
- Q1: \( b = -ac \)
- Q2: **30 cm²**
- Q3: Correct option: \( -\frac{3}{\sqrt{1-x^2}} \)
- Q4: \( 0 \le x \le 1 \)

---

### **Complex Numbers (Q5, page 6)**
- Uses real part condition:
\[
\text{Re}(z) = 15 - 4c
\]
- Final answer: **c = 2**

---

### **Numerical Methods (Q6, page 7)**
- Euler’s method applied
- Solves quadratic in k
- Final answer: **k ≈ 6.187**

---

### **Matrices (Q7, page 7)**
- Condition: \( AB = 0 \)
- General solution form:
\[
B =
\begin{bmatrix}
n & m \\
2n & 2m
\end{bmatrix}
\quad (n,m \neq 0)
\]

---

### **Graph Transformations (Q8, page 8)**
- Roots after transformation: **−8, 0, 6**
- Inequality solution:
\[
3 < x < 6
\]

---

### **Complex Numbers & Proof (Q9, pages 9–10)**
- \( \text{Re}(z) = -\frac{1}{2} \)
- Uses cube roots of unity
- Final result:
\[
\text{Re}(w) = -\frac{1}{2}
\]

---

### **Rational Functions (Q10, pages 11–12)**
- Asymptotes:
- \( x = -3 \)
- \( y = 2 \)
- Range:
\[
k \ge -\frac{8}{5}
\]
- Stationary point:
\[
\left(-\frac{4}{3}, -\frac{8}{5}\right)
\]

---

### **Vectors & Planes (Q11, pages 14–16)**
- Closest point: **(−3, 6, −1)**
- Plane equation:
\[
-12x + y + 5z = -43
\]
- Intersection point: **(13, −2, 23)**

---

### **Integration (Q12, page 17)**
- Uses completing square + substitution
- Final answer:
\[
\ln(8 - 4\sqrt{3} - 3\sqrt{5} + 2\sqrt{15})
\]

---

### **Matrices & Complex Transformations (Q13, pages 18–19)**
- Scale factor: **r = 2**
- Rotation: \( \theta = \frac{\pi}{3} \)
- Final values:
\[
x = \frac{\sqrt{3}}{256}, \quad y = -\frac{1}{256}
\]

---

### **Polar Coordinates (Q14, page 21)**
- Total area:
\[
\frac{67\pi}{2}
\]
- Volume:
\[
31.57 \text{ m}^3
\]

👉 The **diagram on page 20** shows the pool + island used in this calculation.

---

### **Improper Integrals (Q15, page 22)**
- Uses partial fractions + limits
- Final result:
\[
\ln\left(\frac{13}{12}\right)
\]

---

### **Area Problem (Q16, page 23)**
- Final answer:
\[
\frac{\pi\sqrt{3}}{5}
\]

---

### **Differential Equations (Q17, pages 24–26)**
- Model:
\[
\frac{dy}{dt} + 0.04y = 0.72 - 0.08t
\]
- Solution:
\[
y = 68 - 2t - 30e^{-0.04t}
\]
- Time result: **7 min 45 sec**
- Limitation: model assumes temperature decreases indefinitely

---

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

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

---

## **Why This Mark Scheme Matters**
- Shows **exact examiner expectations**
- Helps maximise **method marks**
- Teaches **full-solution structure**
- Essential for **A/A* performance**

---

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