4.02a Complex numbers: real/imaginary parts, modulus, argument

The cube roots of 1 are denoted by \(1\), \(\omega\) and \(\omega^2\), where the imaginary part of \(\omega\) is positive.

1. Show that \(1 + \omega + \omega^2 = 0\). [2]

\includegraphics{figure_1} In the diagram, \(ABC\) is an equilateral triangle, labelled anticlockwise. The points \(A\), \(B\) and \(C\) represent the complex numbers \(z_1\), \(z_2\) and \(z_3\) respectively.

1. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z_1 - z_3 = \omega(z_3 - z_2)\). [4]
2. Hence show that \(z_1 + \omega z_2 + \omega^2 z_3 = 0\). [2]

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]

It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).

1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]

\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)

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

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]

Given that \(z\) is a complex number and that \(z^*\) is the complex conjugate of \(z\) prove that \(zz^* - |z|^2 = 0\) [3 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]

In this question you must show detailed reasoning.

1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]

In this question, the argument of a complex number is defined as being in the range \([0, 2\pi)\). You are given that \(\omega_k\), where \(k = 0, 1, 2, ..., n-1\), are the \(n\) \(n\)th roots of unity for some integer \(n\), \(n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega_0 = 1\)).

1. With the help of a diagram explain why \(\omega_k = (\omega_1)^k\) for \(k = 2, ..., n-1\). [3]
2. Using the identity given in part (a), show that \(\sum_{k=0}^{n-1}\omega_k = 0\). [2]
3. Show that if \(z\) is a complex number then \(z + z^* = 2\text{Re}(z)\). [1]
4. Using the results from parts (b) and (c) show that \(\sum_{k=0}^{n-1}\text{Re}(\omega_k) = 0\). [1]
5. With the help of a diagram explain why \(\text{Re}(\omega_k) = \text{Re}(\omega_{n-k})\) for \(k = 1, 2, ..., n-1\). [1]

You should now consider the case when \(n = 5\).

1. 1. Use parts (d) and (e) to deduce that \(\cos\frac{4\pi}{5} = a + b\cos\frac{2\pi}{5}\), for some rational constants \(a\) and \(b\). [2]
   2. Hence determine the exact value of \(\cos\frac{2\pi}{5}\). [2]

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

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

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]

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]

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

1. Express \(z\) in the form \(z = re^{i\theta}\), where \(-\pi \leqslant \theta \leqslant \pi\). [3]
2. 1. Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of \(z\) when plotted in an Argand diagram. Give your answers correct to two decimal places.
   2. Write down the geometrical name of the triangle. [5]

When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3\sqrt{3}i\) lie on a circle. Find the equation of this circle. [4]

You are given that \(z = 3 - 4\mathrm{i}\).

1. Find
   [3] On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w^*\) is represented by the point \(B\).
2. Describe the geometrical relationship between the points \(A\) and \(B\). [2]

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]