Cartesian form: mixed practice Problem set B

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

1. (a) \(3z^* - z = 2 + 8i\)
2. (b) \(iz + 3z^* = -5 - 7i\)

Find the complex number \(z\) such that \(4z - 3 = 2iz + 7 - 10i\).

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

1. (a) \(x^2 + 28 = 0\)
2. (b) \(x^2 + 4x + 13 = 0\)
3. (c) \(2x^2 - 3x + 5 = 0\)

Solve \(4x^2 + 8x + 5 = 0\), and show the solutions on an Argand diagram.

Given that \(z = 3 - 7i + \dfrac{20}{2 - i}\), find \(z^*\).

\(a\) is a real number, \(z = 4 + ai\) and \(w = 2a - i\). Write each of these in terms of \(a\), in the form \(x + iy\).

1. (a) \(z - 3w\)
2. (b) \(zw\)

\(k\) is a real number. Write \(\dfrac{2k - 5i}{3k + i}\) in the form \(x + iy\).

\(a\) is a real number and \(z = \dfrac{a + 6i}{a - 4i}\). Find the values of \(a\) for which \(\text{Im}(z) = 1\).

\(a\) and \(b\) are real numbers, \(z = 3 + ai\) and \(w = b + i\). Given that \(zw = -3 + 9i\), find the possible values of \(a\) and \(b\).

\(z = 3 - 2i\) and \(w = -1 + 4i\). Show \(z\), \(z^*\), \(2z\), \(iz\), \(w\), \(z + w\) and \(z - w\) on one Argand diagram, and write down the coordinates of each point.

Let \(z = 0.8 - 0.6i\). 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 \(z^2 + zz^* = 32 + 24i\).

By writing \(z = x + iy\), show that \((z^2 + 2z)^* = (z^*)^2 + 2z^*\).

Solve the simultaneous equations \(3z - iw = 8 - 4i\) and \(z + 2iw = -2 + i\) for the complex numbers \(z\) and \(w\).