The complex plane and the modulus Problem set A

The diagram shows the point that represents \(z = 5 + 2i\).

1. (a) Write down \(-z\) and mark it on a copy of the diagram.
2. (b) Write down \(z^*\) and mark it on the diagram.
3. (c) Write down \(-z^*\) and mark it on the diagram. Describe the reflection that takes \(z\) to \(-z^*\).

The diagram shows two points, P and Q, on an Argand diagram.

1. (a) Write down the complex number represented by P.
2. (b) Find \(|P|\).
3. (c) Q represents \(w\). Find \(|w|\) to 3 significant figures.

\(z = -4 + 3i\).

1. (a) Write down \(-z\) and say how its point is related to the point for \(z\).
2. (b) Write down \(z^*\) and say how its point is related to the point for \(z\).
3. (c) Find \(|z|\), \(|z^*|\) and \(|-z|\).

Find each modulus. Give an exact value if it is an integer, and otherwise give 3 significant figures.

1. (a) \(|7 - 24i|\)
2. (b) \(|-5 - 3i|\)
3. (c) \(|12i|\)

Take east as the real axis and north as the imaginary axis, with distances in metres. A drone starts at the origin and flies to the point \(z = 9 - 12i\).

1. (a) Find the distance of the drone from the origin.
2. (b) A second drone is at \(z^*\). Find the distance between the two drones.

\(z\) is a complex number with \(|z| = 5\) and \(\text{Re}(z) = -3\).

1. (a) Find the two possible complex numbers \(z\).
2. (b) How are the two points related on the Argand diagram?

The points that represent \(z = 5 + 3i\), \(z^*\), \(-z\) and \(-z^*\) are the corners of a rectangle on an Argand diagram.

1. (a) Find the width and height of the rectangle.
2. (b) Find its perimeter and its area.