If `|omega|=1`, then the set of points `z=omega+1/omega` is contained in or equal to the set of points z satisfying.

Options

A. `"Re"(z) le 2` and `"Im"(z)=0`
B. `"Re"(z) le 1` and `"Im"(z)=0`
C. `|"Re"(z) le 2` and `"Im"(z)=0`
D. `|Re(z) le 1` and `"Im"(z)=0`

Explanation

To solve the problem, we need to analyze the expression \( z = \omega + \frac{1}{\omega} \) given that \( |\omega| = 1 \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
Since \( |\omega| = 1 \), we can express \( \omega \) in terms of trigonometric functions. We can write:
\[ \omega = e^{i\theta} = \cos\theta + i\sin\theta \]
where \( \theta \) is a real number.

2. **Calculating \( \frac{1}{\omega} \)**:
The reciprocal of \( \omega \) is:
\[ \frac{1}{\omega} = \frac{1}{\cos\theta + i\sin\theta} \]
To simplify this, we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{\omega} = \frac{\cos\theta - i\sin\theta}{\cos^2\theta + \sin^2\theta} = \cos\theta - i\sin\theta \]
(since \( \cos^2\theta + \sin^2\theta = 1 \)).

3. **Substituting into \( z \)**:
Now substitute \( \omega \) and \( \frac{1}{\omega} \) back into the expression for \( z \):
\[ z = \omega + \frac{1}{\omega} = (\cos\theta + i\sin\theta) + (\cos\theta - i\sin\theta) \]
Simplifying this gives:
\[ z = 2\cos\theta \]
The imaginary parts cancel out, leaving us with only the real part.

4. **Characterizing the Set of Points**:
Since \( z = 2\cos\theta \), the real part of \( z \) can vary depending on \( \theta \). The maximum value of \( \cos\theta \) is 1 and the minimum value is -1. Therefore, the range of \( z \) is:
\[ -2 \leq z \leq 2 \]
This means that the real part of \( z \) is constrained to the interval \([-2, 2]\), and the imaginary part is always zero.

5. **Conclusion**:
The set of points \( z = \omega + \frac{1}{\omega} \) is contained in the set of points satisfying:
\[ \text{Re}(z) \leq 2 \quad \text{and} \quad \text{Im}(z) = 0 \]

### Final Answer:
The set of points \( z = \omega + \frac{1}{\omega} \) is contained in or equal to the set of points satisfying:
\[ \text{Re}(z) \leq 2 \quad \text{and} \quad \text{Im}(z) = 0 \]