Cartesian form: mixed practice Problem set A

Find the complex number \(z\) in each case.

1. (a) \(z^* - 2z = 3 - 6i\)
2. (b) \(2iz + z^* = 4 + 5i\)

Find the complex number \(z\) such that \(5z - 2 = 3iz - 1 - 21i\).

Solve each equation. Give exact values where you can and otherwise 3 significant figures.

1. (a) \(x^2 + 20 = 0\)
2. (b) \(x^2 - 6x + 10 = 0\)
3. (c) \(3x^2 + 4x + 5 = 0\)

Solve \(4x^2 - 12x + 13 = 0\), and show the solutions on an Argand diagram.

Given that \(z = 2 + 5i - \dfrac{12}{1 + i}\), find \(z^*\).

\(a\) is a real number, \(z = 3a - 2i\) and \(w = 1 + ai\). Write each of these in terms of \(a\), in the form \(x + iy\).

1. (a) \(z + 2w\)
2. (b) \(zw\)

\(k\) is a real number. Write \(\dfrac{k + 4i}{2k - i}\) in the form \(x + iy\).

\(a\) is a real number and \(z = \dfrac{a + 3i}{a - 2i}\). Find the values of \(a\) for which \(\text{Im}(z) = 1\).

\(a\) and \(b\) are real numbers, \(z = 2 + ai\) and \(w = b + i\). Given that \(zw = -6 - 2i\), find the possible values of \(a\) and \(b\).

\(z = -2 + 3i\) and \(w = 4 + i\). Show \(z\), \(z^*\), \(2z\), \(iz\), \(w\), \(z + w\) and \(z - w\) on one Argand diagram, and write down the coordinates of each point.

Let \(z = 0.6 + 0.8i\). Use technology to find \(z^2\) and \(z^3\), and show \(z\), \(z^2\) and \(z^3\) on one Argand diagram. What do you notice about the distance of the three points from the origin?

Find the two complex numbers \(z\) such that \(z^2 + zz^* = 18 - 12i\).

By writing \(z = x + iy\), show that \((z^*)^3 = (z^3)^*\).

Solve the simultaneous equations \(2z + iw = 3 + 7i\) and \(z - 2iw = -1 - 4i\) for the complex numbers \(z\) and \(w\).