Conjugates, division and equating parts Problem set A

Answer each part.

1. (a) Write down the conjugate of \(z = 4 - 9i\).
2. (b) \(z = 6 + 8i\). Find \(zz^*\).
3. (c) \(z = -7 + 2i\). Find \(z^*\) and \(zz^*\).

Let \(z = x + iy\), where \(x\) and \(y\) are real numbers.

1. (a) Show that \(z + z^*\) is real.
2. (b) Show that \(z^2 + (z^*)^2\) is real.

Write each quotient in the form \(x + iy\). Give exact fractions where needed.

1. (a) \(\dfrac{14 + 2i}{3 - i}\)
2. (b) \(\dfrac{4 + 7i}{1 + 2i}\)
3. (c) \(\dfrac{1}{2 - 3i}\)

Find the complex number \(z\) that satisfies each equation.

1. (a) \(z + 2z^* = 9 - 4i\)
2. (b) \(3z - z^* = -6 + 8i\)

\(a\) and \(b\) are real numbers. Find \(a\) and \(b\) in each equation.

1. (a) \((a + bi)(1 + 2i) = 7 + 4i\)
2. (b) \((a + bi)(2 - i) = 5 + 5i\)

In an alternating current circuit, the voltage is \(V = 30 + 10i\) volts and the impedance is \(Z = 4 + 2i\) ohms. The current is \(I = \dfrac{V}{Z}\).

1. (a) Find the current \(I\) in the form \(x + iy\).
2. (b) Show that \(ZZ^*\) is real and find its value.

Find the complex number \(z\) such that \(z(2 + i) + z^* = 5 + 3i\), where \(z^*\) is the conjugate of \(z\).

\(z = 1 + 2i\).

1. (a) Find \(zz^*\) and \(z + z^*\).
2. (b) Hence show that \(\dfrac{1}{z} + \dfrac{1}{z^*}\) is real, and find its value.