Let s be the set of all complex numbers Z satisfying `|z^(2)+z+1|=1`. Then which of the following statements is/are TRUE?

Options

A. `|z+(1)/(2)| le (1)/(2)"for" a||z in S`
B. `|Z| le 2 "for all" z in S`
C. `|z+(1)/(2)| ge (1)/(2)"for all" z in S`
D. The set S has exactly four elements.

Answer

To solve the problem, we need to analyze the equation given by the condition \( |z^2 + z + 1| = 1 \).

### Step 1: Rewrite the equation
We start by letting \( w = z^2 + z + 1 \). The condition can be rewritten as:
\[ |w| = 1 \]
This implies that \( w \) lies on the unit circle in the complex plane.

### Step 2: Express \( w \) in terms of \( z \)
The expression for \( w \) is:
\[ w = z^2 + z + 1 \]

### Step 3: Set up the equation
We can express \( w \) in terms of its modulus:
\[ |z^2 + z + 1| = 1 \]

### Step 4: Analyze the roots
To find the values of \( z \), we can set \( w = e^{i\theta} \) for some \( \theta \) (since \( |w| = 1 \)). This gives us the equation:
\[ z^2 + z + (1 - e^{i\theta}) = 0 \]
This is a quadratic equation in \( z \).

### Step 5: Solve the quadratic equation
Using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 1, c = 1 - e^{i\theta} \):
\[ z = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (1 - e^{i\theta})}}{2 \cdot 1} \]
\[ z = \frac{-1 \pm \sqrt{1 - 4 + 4e^{i\theta}}}{2} \]
\[ z = \frac{-1 \pm \sqrt{4e^{i\theta} - 3}}{2} \]

### Step 6: Analyze the modulus of \( z \)
To find the modulus of \( z \), we need to analyze the expression:
\[ |z| = \left| \frac{-1 \pm \sqrt{4e^{i\theta} - 3}}{2} \right| \]

### Step 7: Determine the range of \( |z| \)
We can find the maximum and minimum values of \( |z| \) based on the values of \( e^{i\theta} \). The term \( 4e^{i\theta} - 3 \) will vary as \( \theta \) changes, affecting the modulus of \( z \).

### Step 8: Conclusion
After analyzing the modulus and the behavior of the quadratic equation, we can conclude the following statements about the set \( S \):
1. The values of \( |z| \) are bounded.
2. The maximum and minimum values can be derived from the analysis of the roots.

### Final Result
From the analysis, we can conclude that:
- The statements regarding the bounds of \( |z| \) can be verified against the options provided in the question.