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Let's go over each exercise from your document in detail. I'll walk you through solving them step by step.

### **Exercice 1:**

We are given:
- \( z = 5i \) 
- \( z' = 10 + 5i \)

You are asked to **prove that \( z \times z' = \overline{z'} \)** by calculating it.

1. **First, calculate \( z \times z' \):**
   \[
   z \times z' = 5i \times (10 + 5i)
   \]
   Apply the distributive property:
   \[
   = 5i \times 10 + 5i \times 5i = 50i + 25i^2
   \]
   Since \( i^2 = -1 \), this becomes:
   \[
   = 50i + 25(-1) = 50i - 25 = -25 + 50i
   \]

2. **Now, calculate the conjugate \( \overline{z'} \):**
   \[
   z' = 10 + 5i \quad \Rightarrow \quad \overline{z'} = 10 - 5i
   \]
   This means you need to verify if:
   \[
   -25 + 50i = 10 - 5i
   \]
   Clearly, this is **false**. Therefore, it seems there might be a mistake or typo in the problem statement. Usually, for a product like this, the result could be expected to be something related to \( z \times z' \) matching another form.

### **Exercice 2:**

We are asked to examine the function \( f(z) = z^2 - 14z + 74 \) for any complex number \( z \).

- **Part a)** asks you to show that \( z = \overline{z'} \).  
  To solve this, you'll need to calculate \( f(z) \) and \( f(\overline{z}) \) to see if they are equal.

- **Part b)** involves dividing \( z \) by \( z' \), but the exact details are obscured in the image. Please clarify the specific details for this part, so I can help better.

### **Exercice 3:**

We are given:
- \( z_1 = 5 + i \)
- \( z_2 = -2 + 3i \)

You are asked to **show that \( z_1 + z_2 \) is a real number, and that \( z_1 - z_2 \) is purely imaginary.**

1. **First, compute \( z_1 + z_2 \):**
   \[
   z_1 + z_2 = (5 + i) + (-2 + 3i) = 5 - 2 + i + 3i = 3 + 4i
   \]
   This is **not a real number** but a complex number with real part 3 and imaginary part 4.

2. **Now, compute \( z_1 - z_2 \):**
   \[
   z_1 - z_2 = (5 + i) - (-2 + 3i) = 5 + i + 2 - 3i = 7 - 2i
   \]
   This is **not purely imaginary** either; it has both real and imaginary parts.

It seems the problem might be testing a condition or equation for these specific numbers. Please verify the statement again or provide any additional context.

---

### Let me know if you'd like more details or if you have further questions. Here are some related questions to deepen your understanding:

1. What is the geometric interpretation of the conjugate \( \overline{z} \) in the complex plane?
2. How do the properties of complex multiplication and conjugation relate to symmetry in the complex plane?
3. What happens when you multiply a complex number by its conjugate?
4. How would the nature of \( z_1 + z_2 \) and \( z_1 - z_2 \) change if one of the numbers were purely imaginary or purely real?
5. Can you think of a scenario where \( z_1 \times z_2 \) equals \( \overline{z_2} \)? How would that affect the interpretation of the problem?

---

### Tip:
When working with complex numbers, always simplify using \( i^2 = -1 \) early to avoid mistakes later on.

Learn how to solve exercises involving complex numbers, including multiplication, conjugation, and quadratic functions. This tutorial provides step-by-step solutions to exercises requiring the use of complex number properties and function evaluation.

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