If `a, b, c` are distinct odd integers and `omega` is non real cube root of unity, then minimum value of `| a omega^2 + b+ comega|`, is

Options

A. `0`
B. `3`
C. `1`
D. `2sqrt(3)`

Explanation

To solve the problem, we need to find the minimum value of \( | a \omega^2 + b + c \omega | \), where \( a, b, c \) are distinct odd integers and \( \omega \) is a non-real cube root of unity.

### Step-by-step Solution:

1. **Identify the values of \( \omega \) and \( \omega^2 \)**:
\[ \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i, \quad \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \]

2. **Substitute \( \omega \) and \( \omega^2 \) into the expression**:
\[ a \omega^2 + b + c \omega = a \left(-\frac{1}{2} - \frac{\sqrt{3}}{2} i\right) + b + c \left(-\frac{1}{2} + \frac{\sqrt{3}}{2} i\right) \]
This simplifies to:
\[ = -\frac{a}{2} - \frac{c}{2} + b + \left(-\frac{a \sqrt{3}}{2} + \frac{c \sqrt{3}}{2}\right)i \]

3. **Combine real and imaginary parts**:
The real part is:
\[ -\frac{a + c}{2} + b \]
The imaginary part is:
\[ \frac{\sqrt{3}}{2} (c - a) \]
Thus, we have:
\[ | a \omega^2 + b + c \omega | = \sqrt{\left(-\frac{a + c}{2} + b\right)^2 + \left(\frac{\sqrt{3}}{2} (c - a)\right)^2} \]

4. **Simplify the expression**:
\[ = \sqrt{\left(-\frac{a + c}{2} + b\right)^2 + \frac{3}{4} (c - a)^2} \]

5. **Set up for minimization**:
We need to find the minimum value of:
\[ \sqrt{\frac{1}{4} (a + c - 2b)^2 + \frac{3}{4} (c - a)^2} \]

6. **Choose distinct odd integers**:
To minimize the expression, we can choose the smallest distinct odd integers. Let's take:
\[ a = 1, \quad b = 3, \quad c = 5 \]

7. **Calculate the expression**:
Substitute \( a, b, c \) into the expression:
\[ = \sqrt{\frac{1}{4} (1 + 5 - 2 \cdot 3)^2 + \frac{3}{4} (5 - 1)^2} \]
\[ = \sqrt{\frac{1}{4} (6 - 6)^2 + \frac{3}{4} (4)^2} \]
\[ = \sqrt{0 + \frac{3}{4} \cdot 16} \]
\[ = \sqrt{12} = 2\sqrt{3} \]

### Final Answer:
The minimum value of \( | a \omega^2 + b + c \omega | \) is \( 2\sqrt{3} \).