Key Concepts and Formulas

• Cube Roots of Unity: The roots of the equation $x^3 = 1$ are $1, \omega, \omega^2$, where $\omega = e^{i2\pi/3}$ and $\omega^2 = e^{i4\pi/3}$. The non-real roots $\omega$ and $\omega^2$ are also the roots of $x^2 + x + 1 = 0$.
• Properties of Cube Roots of Unity: $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.
• Argument of a Complex Number: If $z = x + iy$, then $\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)$, considering the quadrant of $z$.

Step-by-Step Solution

1. Identify $z_0$ as a cube root of unity.

The problem states that $z_0$ is a root of $x^2 + x + 1 = 0$. This equation is well-known to have roots that are the non-real cube roots of unity, $\omega$ and $\omega^2$. Therefore, $z_0$ can be taken as $\omega$ (or $\omega^2$, the final answer would be the same). This identification simplifies calculations significantly.

2. Simplify the powers of $z_0$ using the property $\omega^3 = 1$.

We need to simplify $z_0^{81}$ and $z_0^{93}$. Since $z_0 = \omega$, we have:

• For $z_0^{81} = \omega^{81}$: Because $81$ is divisible by $3$, we have $81 = 3 \times 27$. Therefore, $\omega^{81} = (\omega^3)^{27} = (1)^{27} = 1$
• For $z_0^{93} = \omega^{93}$: Because $93$ is divisible by $3$, we have $93 = 3 \times 31$. Therefore, $\omega^{93} = (\omega^3)^{31} = (1)^{31} = 1$

This simplification is crucial, as it replaces complex numbers with real numbers.

3. Substitute the simplified powers of $z_0$ into the expression for $z$.

We are given $z = 3 + 6iz_0^{81} - 3iz_0^{93}$. Substituting $z_0^{81} = 1$ and $z_0^{93} = 1$, we get:

$z = 3 + 6i(1) - 3i(1) = 3 + 6i - 3i = 3 + 3i$

Now, $z$ is in the standard complex number form $a + bi$.

4. Calculate the argument of $z$.

We have $z = 3 + 3i$. The argument of $z$ is given by $\arg(z) = \tan^{-1}\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right)$. In this case, $\text{Re}(z) = 3$ and $\text{Im}(z) = 3$. Since both the real and imaginary parts are positive, $z$ lies in the first quadrant. Therefore,

$\arg(z) = \tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = \frac{\pi}{4}$

The argument of $z$ is $\frac{\pi}{4}$.

Common Mistakes & Tips

• Not Recognizing Cube Roots of Unity: Failing to recognize the equation $x^2 + x + 1 = 0$ as related to cube roots of unity leads to unnecessary calculations.
• Quadrant Errors: Always check the quadrant of the complex number when calculating the argument. The range of $\tan^{-1}(x)$ is $(-\pi/2, \pi/2)$, so adjustments might be needed based on the quadrant.
• Memorize Properties: Knowing the properties of cube roots of unity ($1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$) is essential for efficient simplification.

Summary

This problem tests the understanding of cube roots of unity and their properties. By recognizing $z_0$ as a cube root of unity, we can simplify the expression significantly. The powers of $z_0$ are simplified using $\omega^3 = 1$, and then the argument of the resulting complex number $z = 3 + 3i$ is calculated as $\frac{\pi}{4}$.

The final answer is \boxed{\frac{\pi}{4}}, which corresponds to option (A).