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

\(z = 5\sqrt{3} - 5i\) Find

1. \(|z|\), [1]
2. \(\arg(z)\), in terms of \(\pi\). [2]

$$w = 2\left[\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right]$$ Find

1. \(\left|\frac{w}{z}\right|\), [1]
2. \(\arg\left(\frac{w}{z}\right)\), in terms of \(\pi\). [2]

$$z = 4\left(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right) \text{ and } w = 3\left(\cos \frac{2\pi}{3} + i\sin \frac{2\pi}{3}\right).$$ Express \(zw\) in the form \(r(\cos \theta + i \sin \theta)\), \(r > 0\), \(-\pi < \theta < \pi\). [3]

Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]

The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.

1. \(z + 5w\), [2]
2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
3. \(\frac{1}{w}\). [2]

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

1. Find \(|z|\) and \(\arg z\). [2]
2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]

The complex numbers \(2 + 3\text{i}\) and \(4 - \text{i}\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \text{i}y\), showing clearly how you obtain your answers.

1. \(z + 5w\), [2]
2. \(z^*w\), where \(z^*\) is the complex conjugate of \(z\), [3]
3. \(\frac{1}{w}\). [2]

The complex numbers \(a\) and \(b\) are given by \(a = 7 + 6\text{i}\) and \(b = 1 - 3\text{i}\). Showing clearly how you obtain your answers, find

1. \(|a - 2b|\) and \(\arg(a - 2b)\), [4]
2. \(\frac{b}{a}\), giving your answer in the form \(x + \text{i}y\). [3]

Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2\mathrm{j}\) and \(\beta = -2 - \mathrm{j}\).

1. Represent \(\beta\) and its complex conjugate \(\beta^*\) on an Argand diagram. [2]
2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]

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

Simplify $$\frac{\cos\left(\frac{6\pi}{13}\right) + i\sin\left(\frac{6\pi}{13}\right)}{\cos\left(\frac{2\pi}{13}\right) - i\sin\left(\frac{2\pi}{13}\right)}$$ Tick (\(\checkmark\)) one box. [1 mark] \(\cos\left(\frac{8\pi}{13}\right) + i\sin\left(\frac{8\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{8\pi}{13}\right) - i\sin\left(\frac{8\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{4\pi}{13}\right) + i\sin\left(\frac{4\pi}{13}\right)\) \(\square\) \(\cos\left(\frac{4\pi}{13}\right) - i\sin\left(\frac{4\pi}{13}\right)\) \(\square\)

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$z + w^* = 5$$ $$3z^* - w = 6 + 4i$$ Find \(z\) and \(w\) [5 marks]

The matrix \(\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a\) and \(b\) are positive real numbers, and \(\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\) Use \(\mathbf{C}\) to show that \(\cos \frac{\pi}{12}\) can be written in the form \(\frac{\sqrt{m + n}}{2}\), where \(m\) and \(n\) are integers. [7 marks]

The complex number \(z\) is such that $$z = \frac{1 + \text{i}}{1 - k\text{i}}$$ where \(k\) is a real number.

1. Find the real part of \(z\) and the imaginary part of \(z\), giving your answers in terms of \(k\) [2 marks]
2. In the case where \(k = \sqrt{3}\), use part (a) to show that $$\cos \frac{7\pi}{12} = \frac{\sqrt{2} - \sqrt{6}}{4}$$ [5 marks]

In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).

1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3z - 4z^*\) [2]
   2. \((z + 1 - 3\text{i})^2\) [2]
   3. \(\frac{z + 1}{z - 1}\) [2]
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
   [3]

Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]

The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).

1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]

A complex number is defined by \(z = -3 + 4\mathrm{i}\).

1. 1. Express \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi \leqslant \theta \leqslant \pi\).
   2. Express \(\bar{z}\), the complex conjugate of \(z\), in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [4]

Another complex number is defined as \(w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)\).

1. Express \(zw\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [3]

Solve the equation \(2z + iz = \frac{-1 + 7i}{2 + i}\).

1. Give your answer in Cartesian form [7]
2. Give your answer in modulus-argument form. [4]

The complex numbers \(z\) and \(w\) are represented, respectively, by points \(P(x, y)\) and \(Q(u,v)\) in Argand diagrams and $$w = z(1 + z)$$

1. Show that $$v = y(1 + 2x)$$ and find an expression for \(u\) in terms of \(x\) and \(y\). [4]
2. The point \(P\) moves along the line \(y = x + 1\). Find the Cartesian equation of the locus of \(Q\), giving your answer in the form \(v = au^2 + bu\), where \(a\) and \(b\) are constants whose values are to be determined. [5]

Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]

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]