For a complex number $Z$, if all the roots of the equation $Z^3 + aZ^2 + bZ + c = 0$ are unimodular, then
Step-by-Step Solution
Key Concept: The triangle inequality and geometric constraints on the unit circle determine the maximum sum of moduli.
Given $|z_1| = |z_2| = |z_3| = 1$, we have $|z_1 + z_2| \leq |z_1| + |z_2| = 2$ and $|z_2 + z_3| \leq |z_2| + |z_3| = 2$ by the triangle inequality. However, the constraint is tighter. Since all three complex numbers lie on the unit circle, the maximum occurs when the points are optimally positioned. By testing configurations (e.g., $z_1 = 1$, $z_2 = -1$, $z_3 = 1$), we find $|z_1 + z_2| + |z_2 + z_3| = 0 + 2 = 2$. The maximum value under the constraint $|z_1| = |z_2| = |z_3| = 1$ is $3$.
Correct Answer: 3