Let's analyze the problem step-by-step:

Problem Summary:

We have an imaginary cubic root of unity given as: $\omega = \frac{-1 + \sqrt{3}i}{2}$ The task is to evaluate the given statements about $\omega$:

1. $\omega^{10} = \omega$
2. $|\omega| = 1$
3. $\arg(\omega) = \frac{2\pi}{3}$

We need to check which of these statements is correct.

Step 1: Understanding $\omega$

The cubic roots of unity are the solutions to the equation $x^3 = 1$. These roots are:

1. $1$ (the real root),
2. $\omega = \frac{-1 + \sqrt{3}i}{2}$, and
3. $\omega^2 = \frac{-1 - \sqrt{3}i}{2}$.

These roots satisfy: $\omega^3 = 1 \quad \text{and} \quad \omega^2 + \omega + 1 = 0.$

Step 2: Evaluate Each Statement

Statement (i): $\omega^{10} = \omega$

Since $\omega^3 = 1$, we can use this property to simplify powers of $\omega$: $\omega^{10} = \omega^{3 \times 3 + 1} = (\omega^3)^3 \cdot \omega = 1 \cdot \omega = \omega.$ So, Statement (i) is correct.

Statement (ii): $|\omega| = 1$

The modulus of $\omega = \frac{-1 + \sqrt{3}i}{2}$ is: $|\omega| = \sqrt{\left(\frac{-1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1.$ So, Statement (ii) is correct.

Statement (iii): $\arg(\omega) = \frac{2\pi}{3}$

To find the argument of $\omega$, we observe that $\omega$ is in the second quadrant (since it has a negative real part