If `0 lt a lt b lt c` and the roots `alpha,beta` of the equation `ax^2 + bx + c = 0` are non-real complex numbers, then

Options

A. `0lt|alpha|lt1/2`
B. `1/2lt|alpha|lt1`
C. `|alpha|gt1`
D. `|alpha|!=|beta|`

Explanation

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) under the conditions that \( 0 < a < b < c \) and that the roots \( \alpha \) and \( \beta \) are non-real complex numbers.

### Step-by-Step Solution:

1. **Understand the Condition for Non-Real Roots**:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are given by the formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the roots to be non-real, the discriminant must be negative:
\[ b^2 - 4ac < 0 \]

2. **Rearranging the Discriminant Condition**:
From the condition \( b^2 < 4ac \), we can rearrange it to express the relationship between \( a \), \( b \), and \( c \):
\[ \frac{b^2}{4ac} < 1 \]

3. **Magnitude of the Roots**:
The roots can be expressed in terms of their magnitudes. Since the roots are complex, we can express them as:
\[ \alpha = \frac{-b + i\sqrt{4ac - b^2}}{2a}, \quad \beta = \frac{-b - i\sqrt{4ac - b^2}}{2a} \]
The magnitude of the roots \( \alpha \) and \( \beta \) is given by:
\[ |\alpha| = |\beta| = \sqrt{\left(\frac{-b}{2a}\right)^2 + \left(\frac{\sqrt{4ac - b^2}}{2a}\right)^2} \]

4. **Calculating the Magnitude**:
Simplifying the expression for the magnitude:
\[ |\alpha| = |\beta| = \sqrt{\frac{b^2}{4a^2} + \frac{4ac - b^2}{4a^2}} = \sqrt{\frac{4ac}{4a^2}} = \frac{\sqrt{4ac}}{2a} = \frac{\sqrt{c}}{\sqrt{a}} \]

5. **Analyzing the Magnitude**:
Since \( 0 < a < b < c \), we know that \( c > a \). Therefore:
\[ \frac{\sqrt{c}}{\sqrt{a}} > 1 \]
This implies:
\[ |\alpha| > 1 \quad \text{and} \quad |\beta| > 1 \]

### Conclusion:
Thus, we conclude that both \( |\alpha| \) and \( |\beta| \) are greater than 1. Therefore, the correct options based on the conditions provided are options A and B.