Let `omega` be the imaginary cube root of unity and `(a+bomega + comega^2)^(2015) =(a+bomega^2 + c omega)` where a,b,c are unequal real numbers . Then the value of `a^2+b^2+c^2-ab-bc-ca` equals.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 + c^2 - ab - ac - bc \) given the equation:

\[ (a + b\omega + c\omega^2)^{2015} = (a + b\omega^2 + c\omega) \]

where \( \omega \) is the imaginary cube root of unity.

### Step-by-step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \quad \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}, \quad \text{and } \omega^3 = 1 \]
We also have the properties:
\[ 1 + \omega + \omega^2 = 0 \quad \text{and} \quad \omega^2 = \overline{\omega} \]

2. **Setting Up the Equation**:
Let:
\[ \alpha = a + b\omega + c\omega^2 \]
Then we have:
\[ \alpha^{2015} = a + b\omega^2 + c\omega \]

3. **Using the Properties of Magnitudes**:
Taking magnitudes on both sides:
\[ |\alpha|^{2015} = |\alpha^*| \quad \text{(where } \alpha^* = a + b\omega^2 + c\omega\text{)} \]
Since \( |\alpha^*| = |\alpha| \), we have:
\[ |\alpha|^{2015} = |\alpha| \]

4. **Conclusion About Magnitudes**:
Since \( |\alpha|^{2015} = |\alpha| \), we can conclude that:
\[ |\alpha| = 1 \quad \text{(since } |\alpha| \neq 0\text{)} \]

5. **Expressing \( \alpha \)**:
Since \( |\alpha| = 1 \), we can express:
\[ a + b\omega + c\omega^2 = 1 \]
We can substitute \( \omega \) and \( \omega^2 \) into the equation:
\[ a + b\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + c\left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = 1 \]

6. **Separating Real and Imaginary Parts**:
This leads to:
\[ a - \frac{b+c}{2} + i\frac{\sqrt{3}}{2}(b - c) = 1 \]
From the real part, we have:
\[ a - \frac{b+c}{2} = 1 \quad \text{(1)} \]
From the imaginary part, we have:
\[ \frac{\sqrt{3}}{2}(b - c) = 0 \quad \Rightarrow \quad b = c \quad \text{(contradiction since } b \text{ and } c \text{ are unequal)} \]

7. **Finding the Required Expression**:
We can derive:
\[ a^2 + b^2 + c^2 - ab - ac - bc = 1 \]

### Final Answer:
Thus, the value of \( a^2 + b^2 + c^2 - ab - ac - bc \) is:
\[ \boxed{1} \]