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The problem given involves vectors \( \mathbf{a} \) and \( \mathbf{b} \) with the following properties:

1. \( |\mathbf{a}| = 1 \)
2. \( |\mathbf{a} + 2\mathbf{b}| = 2 \)
3. \( (\mathbf{b} - 2\mathbf{a}) \perp \mathbf{b} \)

We need to find the magnitude \( |\mathbf{b}| \).

Let's solve this step by step.

### Step 1: Use the Magnitude Condition
From the first condition:
\[ |\mathbf{a}| = 1 \]

### Step 2: Use the Second Condition
\[ |\mathbf{a} + 2\mathbf{b}| = 2 \]
Squaring both sides:
\[ (\mathbf{a} + 2\mathbf{b}) \cdot (\mathbf{a} + 2\mathbf{b}) = 4 \]
\[ \mathbf{a} \cdot \mathbf{a} + 4 \mathbf{a} \cdot \mathbf{b} + 4 \mathbf{b} \cdot \mathbf{b} = 4 \]
Since \( |\mathbf{a}| = 1 \), we have \( \mathbf{a} \cdot \mathbf{a} = 1 \):
\[ 1 + 4 (\mathbf{a} \cdot \mathbf{b}) + 4 |\mathbf{b}|^2 = 4 \]
Rearranging:
\[ 4 |\mathbf{b}|^2 + 4 (\mathbf{a} \cdot \mathbf{b}) = 3 \]
\[ |\mathbf{b}|^2 + (\mathbf{a} \cdot \mathbf{b}) = \frac{3}{4} \]

### Step 3: Use the Perpendicular Condition
Since \( (\mathbf{b} - 2\mathbf{a}) \) is perpendicular to \( \mathbf{b} \):
\[ (\mathbf{b} - 2\mathbf{a}) \cdot \mathbf{b} = 0 \]
\[ \mathbf{b} \cdot \mathbf{b} - 2 \mathbf{a} \cdot \mathbf{b} = 0 \]
\[ |\mathbf{b}|^2 - 2 (\mathbf{a} \cdot \mathbf{b}) = 0 \]
\[ |\mathbf{b}|^2 = 2 (\mathbf{a} \cdot \mathbf{b}) \]

### Step 4: Solve the System of Equations
We have the following two equations:
1. \( |\mathbf{b}|^2 + (\mathbf{a} \cdot \mathbf{b}) = \frac{3}{4} \)
2. \( |\mathbf{b}|^2 = 2 (\mathbf{a} \cdot \mathbf{b}) \)

Substitute \( 2 (\mathbf{a} \cdot \mathbf{b}) \) for \( |\mathbf{b}|^2 \) in the first equation:
\[ 2 (\mathbf{a} \cdot \mathbf{b}) + (\mathbf{a} \cdot \mathbf{b}) = \frac{3}{4} \]
\[ 3 (\mathbf{a} \cdot \mathbf{b}) = \frac{3}{4} \]
\[ \mathbf{a} \cdot \mathbf{b} = \frac{1}{4} \]

Now, substitute \( \mathbf{a} \cdot \mathbf{b} = \frac{1}{4} \) into the second equation:
\[ |\mathbf{b}|^2 = 2 \left( \frac{1}{4} \right) \]
\[ |\mathbf{b}|^2 = \frac{1}{2} \]
\[ |\mathbf{b}| = \frac{\sqrt{2}}{2} \]

Therefore, the correct answer is:
\[ |\mathbf{b}| = \frac{\sqrt{2}}{2} \]

So, the correct option is B.

Do you want more details or have any questions?

Here are 5 more questions you might want to ask:
1. How do you determine if two vectors are perpendicular?
2. Can you explain the dot product in more detail?
3. What are some common vector operations and their properties?
4. How do you find the magnitude of a vector given its components?
5. What are some real-world applications of vector mathematics?

**Tip:** Always break down complex vector problems into smaller parts and use fundamental vector properties to simplify the solution process.

Explore the solution to a vector problem involving the magnitude of vectors and their perpendicular properties. This problem delves into advanced high school topics, including vector magnitudes, dot products, and vector perpendicularity theorems.

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