Conjugates, division and equating parts Problem set B

Answer each part.

1. (a) Write down the conjugate of \(z = -3 + 8i\).
2. (b) \(z = 9 - 12i\). Find \(zz^*\).
3. (c) \(z = 5 - 6i\). Find \(z^*\) and \(zz^*\).

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

1. (a) Show that \(z - z^*\) is purely imaginary.
2. (b) Show that \(z^2 - (z^*)^2\) is purely imaginary.

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

1. (a) \(\dfrac{11 - 10i}{4 + i}\)
2. (b) \(\dfrac{5 - 2i}{3 + 2i}\)
3. (c) \(\dfrac{2}{1 + 4i}\)

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

1. (a) \(2z + z^* = 12 - 5i\)
2. (b) \(z - 2z^* = -6 + 9i\)

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

1. (a) \((a + bi)(2 + i) = -6 + 7i\)
2. (b) \((a + bi)(3 - i) = 7 + i\)

In a control system the output signal is \(y = 5 + 10i\) and the input signal is \(u = 2 - i\), both in volts. The system ratio is \(R = \dfrac{y}{u}\).

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

Find the complex number \(z\) such that \(z(1 - 2i) + z^* = 4 - 6i\), where \(z^*\) is the conjugate of \(z\).

\(z = 3 - i\).

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