4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

The function f is defined, for any complex number \(z\), by $$\text{f}(z) = \frac{iz - 1}{iz + 1}.$$ Suppose throughout that \(x\) is a real number.

1. Show that $$\text{Re f}(x) = \frac{x^2 - 1}{x^2 + 1} \quad \text{and} \quad \text{Im f}(x) = \frac{2x}{x^2 + 1}.$$ [2]
2. Show that \(\text{f}(x)\text{f}(x)^* = 1\), where \(\text{f}(x)^*\) is the complex conjugate of \(\text{f}(x)\). [2]

The complex numbers \(3 - 2i\) and \(2 + i\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + iy\) and showing clearly how you obtain these answers,

1. \(2z - 3w\), [2]
2. \((iz)^2\), [3]
3. \(\frac{z}{w}\). [3]

The complex number \(w\) is given by $$w = 10 - 5i$$

1. Find \(|w|\). [1]
2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [1]

The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + i)(z + 3i) = w$$

1. Use algebra to find \(z\), giving your answer in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

Given that $$\arg(\lambda + 9i + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,

1. find the value of \(\lambda\). [1]

The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number. Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]

Solve the equation \(2z - 5iz^* = 12\). [4]

The complex number \(z\) satisfies the equation \(z^2 - 4iz^* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

The complex number \(z\) satisfies the equation \(z + 2iz^* = 12 + 9i\). Find \(z\), giving your answer in the form \(x + iy\). [5]

The complex number \(z\) satisfies the equation \(z + 2iz^* + 1 - 4i = 0\). You are given that \(z = x + iy\), where \(x\) and \(y\) are real numbers. Determine the values of \(x\) and \(y\). [4]

In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.

1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]

In this question you must show detailed reasoning.

1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]

The complex number \(2 + i\) is denoted by \(z\).

1. Show that \(z^2 = 3 + 4i\). [2]
2. Plot the following on the Argand diagram in the Printed Answer Booklet.
   [1]
3. State the relationship between \(|z^2|\) and \(|z|\). [1]
4. State the relationship between \(\arg(z^2)\) and \(\arg(z)\). [1]

**In this question you must show detailed reasoning.** Given that \(z_1 = 3 + 2i\) and \(z_2 = -1 - i\), find the following, giving each in the form \(a + bi\).

1. \(z_1^* z_2\) [2]
2. \(\frac{z_1 + 2z_2}{z_2}\) [2]