= Solution
<Complex conjugation> reverses the <argument>, reciprocation also inverts the <modulus>, and multiplication by $-1$ adds $\pi$ to the <argument>. With all arguments chosen in $[0,2\pi)$ and $0<\phi<\pi/2$,
$$
\boxed{z^*=2e^{i(2\pi-\phi)},\qquad z^{-1}=\frac12e^{i(2\pi-\phi)},\qquad -z=2e^{i(\pi+\phi)}.}
$$
Thus $z$ is in the first quadrant, $z^*$ and $z^{-1}$ lie on the same fourth-quadrant ray at radii two and one-half, and $-z$ is diametrically opposite $z$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-1-complex-transformations.png]
{title=Conjugation, reciprocation and negation on the Argand plane}
{description=Original diagram using $\phi=\pi/6$ to illustrate the general locations and moduli of the four points.}
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