If `omega` is an imaginary cube roots of unity such that `(2+omega)^2=a+bomega,a,b in R` then value of `a+b` is

Options

A. 7
B. 6
C. 8
D. 5

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that \( (2 + \omega)^2 = a + b\omega \) where \( \omega \) is an imaginary cube root of unity.

### Step-by-step Solution:

1. **Identify the properties of \( \omega \)**:
- The cube roots of unity are \( 1, \omega, \omega^2 \) where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{-2\pi i / 3} \).
- We know that \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \).
- From this, we can express \( \omega^2 \) as \( \omega^2 = -1 - \omega \).

2. **Calculate \( (2 + \omega)^2 \)**:
\[ (2 + \omega)^2 = 2^2 + 2 \cdot 2 \cdot \omega + \omega^2 = 4 + 4\omega + \omega^2 \]

3. **Substituting \( \omega^2 \)**:
- Substitute \( \omega^2 \) with \( -1 - \omega \):
\[ (2 + \omega)^2 = 4 + 4\omega + (-1 - \omega) = 4 + 4\omega - 1 - \omega \]
- Simplifying this gives:
\[ = 3 + 3\omega \]

4. **Identify \( a \) and \( b \)**:
- From the equation \( (2 + \omega)^2 = a + b\omega \), we can compare coefficients:
\[ a = 3 \quad \text{and} \quad b = 3 \]

5. **Calculate \( a + b \)**:
\[ a + b = 3 + 3 = 6 \]

### Final Answer:
The value of \( a + b \) is \( 6 \).