Let `omega` be a complex number such that `2omega+1=z` where `z=sqrt(-3.)` `If|1 1 1 1-omega^2-1omega^2 1omega^2omega^7|=3k ,` then`k` is equal to : `-1` (2) `1` (3) `-z` (4) `z`

Options

A. 1
B. `z`
C. `-z`
D. `-1`

Explanation

To solve the problem, we need to find the value of \( k \) given the equation \( |1 \quad 1 \quad 1 \quad 1 - \omega^2 - 1 \omega^2 \quad 1 \omega^2 \omega^7| = 3k \) and the relationship \( 2\omega + 1 = z \) where \( z = \sqrt{-3} \).

### Step-by-Step Solution:

1. **Understanding the complex number \( \omega \)**:
We know that \( 2\omega + 1 = z \). Given that \( z = \sqrt{-3} \), we can express \( \omega \) in terms of \( z \):
\[ 2\omega = z - 1 \implies \omega = \frac{z - 1}{2} \]

2. **Using properties of \( \omega \)**:
We know that \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \). This means that \( \omega^2 = -1 - \omega \).

3. **Substituting in the determinant**:
The expression \( |1 \quad 1 \quad 1 \quad 1 - \omega^2 - 1 \omega^2 \quad 1 \omega^2 \omega^7| \) can be rewritten using \( \omega^7 = \omega \) (since \( \omega^3 = 1 \)):
\[ |1 \quad 1 \quad 1 \quad 1 - \omega^2 \quad 1 \omega^2 \omega| \]
This simplifies to:
\[ |1 \quad 1 \quad 1 \quad 1 - \omega^2 \quad \omega^2 \quad \omega| \]

4. **Calculating the determinant**:
We can calculate the determinant by performing row operations or using properties of determinants. However, we can also recognize that we can factor out common terms. The determinant can be simplified to:
\[ |1 \quad 1 \quad 1 \quad \omega^2 - \omega + 1| \]
Using the fact that \( \omega^2 = -1 - \omega \), we can substitute:
\[ |1 \quad 1 \quad 1 \quad -1 - \omega - \omega + 1| \]
This simplifies to:
\[ |1 \quad 1 \quad 1 \quad -2\omega| \]

5. **Final determinant calculation**:
The determinant can be computed as:
\[ |1 \quad 1 \quad 1 \quad -2\omega| = 3(-2\omega) = -6\omega \]
Therefore, we have:
\[ |-6\omega| = 6|\omega| \]

6. **Finding \( k \)**:
Since we know that \( |1 \quad 1 \quad 1 \quad 1 - \omega^2 - 1 \omega^2 \quad 1 \omega^2 \omega^7| = 3k \), we can equate:
\[ 6|\omega| = 3k \implies k = 2|\omega| \]

7. **Substituting \( \omega \)**:
From our earlier substitution, we have \( \omega = \frac{z - 1}{2} \). Thus:
\[ |\omega| = \left|\frac{z - 1}{2}\right| = \frac{|z - 1|}{2} \]
Therefore:
\[ k = 2 \cdot \frac{|z - 1|}{2} = |z - 1| \]

8. **Final value of \( k \)**:
Since \( z = \sqrt{-3} \), we can calculate \( |z - 1| \):
\[ |z - 1| = |\sqrt{-3} - 1| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
Thus, \( k = z \).

### Conclusion:
The value of \( k \) is equal to \( z \).