z is a complex number such that `|Re(z)| + |Im (z)| = 4` then `|z|` can't be

Options

A. `sqrt(17/2)`
B. `sqrt(10)`
C. `sqrt(8)`
D. `sqrt(7)`

Answer

To solve the problem, we need to analyze the condition given for the complex number \( z \) such that \( |Re(z)| + |Im(z)| = 4 \). We will denote \( z = x + iy \), where \( x = Re(z) \) and \( y = Im(z) \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \( |Re(z)| + |Im(z)| = 4 \) can be rewritten as \( |x| + |y| = 4 \). This describes a diamond (or rhombus) shape in the coordinate plane with vertices at \( (4, 0) \), \( (0, 4) \), \( (-4, 0) \), and \( (0, -4) \).

2. **Finding the Range of \( |z| \)**:
The magnitude of the complex number \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
We need to find the minimum and maximum possible values of \( |z| \) given the constraint \( |x| + |y| = 4 \).

3. **Identifying the Maximum Value**:
The maximum distance from the origin to any point on the diamond occurs at the vertices. The maximum value of \( |z| \) is:
\[ |z|_{\text{max}} = |(4, 0)| = 4 \]
This occurs at the points \( (4, 0) \), \( (0, 4) \), \( (-4, 0) \), and \( (0, -4) \).

4. **Identifying the Minimum Value**:
To find the minimum value of \( |z| \), we can use the distance from the origin to the line \( |x| + |y| = 4 \). The shortest distance from the origin to the line can be calculated using the formula for the distance from a point to a line:
\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For the line \( x + y - 4 = 0 \) (which represents one of the edges of the diamond), we have \( A = 1, B = 1, C = -4 \), and the point \( (0, 0) \):
\[ \text{Distance} = \frac{|1(0) + 1(0) - 4|}{\sqrt{1^2 + 1^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

5. **Conclusion**:
Thus, the possible values for \( |z| \) lie in the range:
\[ 2\sqrt{2} \leq |z| \leq 4 \]
We need to identify which of the given options cannot be a value of \( |z| \).

6. **Checking the Options**:
- \( |z| \) can take values from \( 2\sqrt{2} \) to \( 4 \).
- Therefore, any value less than \( 2\sqrt{2} \) or greater than \( 4 \) cannot be \( |z| \).

Given that \( \sqrt{7} \) is approximately \( 2.64575 \), which is greater than \( 2\sqrt{2} \) (approximately \( 2.82843 \)), but less than \( 4 \), it is not in the range of \( |z| \).

### Final Answer:
Thus, the value that \( |z| \) cannot take is \( \sqrt{7} \).