The condition \( |Z - 1| \leq 2 \) means \( Z \) lies in a disk centered at 1 with radius 2.
Since \( \omega^3 = 1 \) and \( \omega \neq 1 \), \( \omega^3 - 1 = 0 \), which factors as:
\[ (\omega - 1)(\omega^2 + \omega + 1) = 0 \]
Since \( \omega \neq 1 \), \( \omega^2 + \omega + 1 = 0 \), so \( \omega^2 = -1 - \omega \).
Then,
\[ |\omega^2 Z - 1 - \omega| = |(-1 - \omega) Z - 1 - \omega| = |-(1 + \omega) Z - (1 + \omega)| = |1 + \omega| \cdot |Z + 1| = |Z + 1|. \]
By the Triangle Inequality,
\[ |Z + 1| = |Z - 1 + 2| \leq |Z - 1| + 2 \leq 2 + 2 = 4. \]
Equality occurs when \( Z = 3 \).
Also, by the Triangle Inequality,
\[ |Z + 1| = |Z - 1 + 2| \geq |2| - |Z - 1| \geq 2 - 2 = 0. \]
Equality occurs when \( Z = -1 \).
Hence, the set of possible values of \( a \) is:
\[ \boldsymbol{0 \leq a \leq 4.} \]
Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is: