If `f(z)=(7-z)/(1-z^2)` , where `z=1+2i ,` then `|f(z)|` is

Options

A. `(|z|)/(2)`
B. `|z|`
C. `2|z|`
D. None of these

Explanation

To find the modulus of the function \( f(z) = \frac{7 - z}{1 - z^2} \) where \( z = 1 + 2i \), we will follow these steps:

### Step 1: Calculate \( z^2 \)
First, we need to compute \( z^2 \):
\[ z = 1 + 2i \]
\[ z^2 = (1 + 2i)^2 = 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2 = 1 + 4i - 4 = -3 + 4i \]

### Step 2: Substitute \( z \) and \( z^2 \) into \( f(z) \)
Now substitute \( z \) and \( z^2 \) into the function:
\[ f(z) = \frac{7 - (1 + 2i)}{1 - (-3 + 4i)} \]
\[ = \frac{7 - 1 - 2i}{1 + 3 - 4i} = \frac{6 - 2i}{4 - 4i} \]

### Step 3: Simplify \( f(z) \)
Next, we simplify \( f(z) \):
\[ f(z) = \frac{6 - 2i}{4 - 4i} \]
To simplify, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ f(z) = \frac{(6 - 2i)(4 + 4i)}{(4 - 4i)(4 + 4i)} \]

### Step 4: Calculate the denominator
Calculate the denominator:
\[ (4 - 4i)(4 + 4i) = 4^2 - (4i)^2 = 16 - (-16) = 32 \]

### Step 5: Calculate the numerator
Now calculate the numerator:
\[ (6 - 2i)(4 + 4i) = 24 + 24i - 8i - 8i^2 = 24 + 16i + 8 = 32 + 16i \]

### Step 6: Combine results
Now we can combine the results:
\[ f(z) = \frac{32 + 16i}{32} = 1 + \frac{1}{2}i \]

### Step 7: Find the modulus of \( f(z) \)
Now we need to find the modulus of \( f(z) \):
\[ |f(z)| = \sqrt{\left(1\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{1 + \frac{1}{4}} = \sqrt{\frac{4}{4} + \frac{1}{4}} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \]

### Final Answer
Thus, the modulus \( |f(z)| \) is:
\[ |f(z)| = \frac{\sqrt{5}}{2} \]