If `|z_(1)-1|lt 1,|z_(2)-2|lt 2,|z_(3)-3|lt 3`, then `|z_(1)+z_(2)+z_(3)|`

Options

A. is less than 6
B. is more than 3
C. is less than 12
D. lies between 6 and 12

Answer

To solve the problem, we need to analyze the given conditions and find the modulus of the sum of the complex numbers \( z_1, z_2, \) and \( z_3 \).

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
We have three conditions:
- \( |z_1 - 1| < 1 \)
- \( |z_2 - 2| < 2 \)
- \( |z_3 - 3| < 3 \)

These inequalities imply that:
- \( z_1 \) lies within a circle of radius 1 centered at 1 on the complex plane.
- \( z_2 \) lies within a circle of radius 2 centered at 2.
- \( z_3 \) lies within a circle of radius 3 centered at 3.

2. **Finding the Range for Each \( z \)**:
- For \( z_1 \):
\[ |z_1 - 1| < 1 \implies 0 < z_1 < 2 \]
- For \( z_2 \):
\[ |z_2 - 2| < 2 \implies 0 < z_2 < 4 \]
- For \( z_3 \):
\[ |z_3 - 3| < 3 \implies 0 < z_3 < 6 \]

3. **Summing the Ranges**:
Now we need to find the range of \( z_1 + z_2 + z_3 \):
- The minimum value occurs when \( z_1, z_2, z_3 \) are all at their minimum:
\[ \text{Minimum} = 0 + 0 + 0 = 0 \]
- The maximum value occurs when \( z_1, z_2, z_3 \) are at their maximum:
\[ \text{Maximum} = 2 + 4 + 6 = 12 \]

4. **Finding the Modulus**:
Therefore, the modulus of the sum \( |z_1 + z_2 + z_3| \) can range from:
\[ |z_1 + z_2 + z_3| < 12 \]
and since \( z_1, z_2, z_3 \) can be zero, the lower bound is:
\[ |z_1 + z_2 + z_3| \geq 0 \]

5. **Conclusion**:
Thus, we conclude that:
\[ 0 < |z_1 + z_2 + z_3| < 12 \]

### Final Answer:
The modulus \( |z_1 + z_2 + z_3| \) can take any value in the range \( (0, 12) \).