The complex plane and the modulus Problem set B

The diagram shows the point that represents \(z = -5 + 4i\).

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, R and S, on an Argand diagram.

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

\(z = 6 - 8i\).

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) \(|15 + 8i|\)
2. (b) \(|-4 + 6i|\)
3. (c) \(|-7i|\)

Take the horizontal axis as real and the vertical axis as imaginary, with distances in millimetres. The tip of a robot arm moves from the origin to the point \(z = -15 + 8i\).

1. (a) Find the distance of the arm tip from the origin.
2. (b) A second arm tip is at \(-z\). Find the distance between the two arm tips.

\(z\) is a complex number with \(|z| = 13\) and \(\text{Re}(z) = 5\).

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 \(z = 4 + 7i\), \(z^*\), \(-z\) 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.