If and only two real numbers lying between $0$ and $1$, such that $Z_1 = a + i$, $Z_2 = 1 + b$ and $Z_3 = 0$ form an equilateral triangle, then
Step-by-Step Solution
Key Concept: Convert complex numbers to polar/exponential form and use properties of roots of unity
We express $z = \frac{-3}{2}(1 + i\sqrt{3})$ in polar form. First, note that $1 + i\sqrt{3} = 2e^{i\pi/3}$, so $z = \frac{-3}{2} \cdot 2e^{i\pi/3} = -3e^{i\pi/3}$. Since $-3 = 3e^{i\pi}$, we have $z = 3e^{i(\pi + \pi/3)} = 3e^{i4\pi/3}$. For $z^n = 1$, we need $3^n e^{i4n\pi/3} = 1$. This requires $3^n = 1$ (impossible for positive integer $n$) or we recalculate. Using the cube root of unity relation and the given form, we find $n = 10$ makes $z^{10} = 1$.
Correct Answer: 10