A complex number z lies on the unit circle in the complex plane, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 3 - 2023 - Paper 1

To solve for the complex number z that lies on the unit circle, we first recognize that any complex number on the unit circle can be expressed as:

$z = e^{i heta}$

where $\theta$ is the angle formed with the positive x-axis. In this case, since it is given that z corresponds to an angle of $\frac{\pi}{3}$, we have:

$z = e^{i\frac{\pi}{3}} = \cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right) = \frac{1}{2} + i\frac{\sqrt{3}}{2}.$

Next, we evaluate the given complex numbers:

• A. -z:

$-z = -e^{i\frac{\pi}{3}} = e^{i\left(\frac{\pi}{3} + \pi\right)} = e^{i\frac{4\pi}{3}}$ (not equal to z)

• B. -z^2:

$-z^2 = -\left(e^{i\frac{\pi}{3}}\right)^2 = -e^{i\frac{2\pi}{3}}$ (not equal to z)

• C. -z^3:

$-z^3 = -\left(e^{i\frac{\pi}{3}}\right)^3 = -e^{i\pi} = 1$ (not equal to z)

• D. z^4:

$z^4 = \left(e^{i\frac{\pi}{3}}\right)^4 = e^{i\frac{4\pi}{3}}$ (not equal to z as well).

Thus, upon checking all options, only option B provides the correct transformation needed to obtain z, making it the correct answer.