4.02r nth roots: of complex numbers

The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).

1. Sketch an Argand diagram to show these roots. [1]
2. Show that \(1 + \omega + \omega^2 = 0\). [2]
3. Hence evaluate
   1. \((2 + \omega)(2 + \omega^2)\), [2]
   2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]

Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]

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]

The Argand diagram shows the solutions to the equation \(z^5 = 1\) \includegraphics{figure_3}

1. Solve the equation $$z^5 = 1$$ giving your answers in the form \(z = \cos\theta + i\sin\theta\), where \(0 \leq \theta < 2\pi\) [2 marks]
2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon. [2 marks]
3. Show that if \(c = \cos\theta\), where \(z = \cos\theta + i\sin\theta\) is a solution to the equation \(z^5 = 1\), then \(c\) satisfies the equation $$16c^5 - 20c^3 + 5c - 1 = 0$$ [5 marks]
4. The Argand diagram on page 22 is repeated below. \includegraphics{figure_4} Explain, with reference to the Argand diagram, why the expression $$16c^5 - 20c^3 + 5c - 1$$ has a repeated quadratic factor. [3 marks]
5. \(O\) is the centre of a regular pentagon \(ABCDE\) such that \(OA = OB = OC = OD = OE = 1\) unit. The distance from \(O\) to \(AB\) is \(h\) By solving the equation \(16c^5 - 20c^3 + 5c - 1 = 0\), show that $$h = \frac{\sqrt{5} + 1}{4}$$ [5 marks]

Solve the equation \(z^3 = i\), giving your answers in the form \(e^{i\theta}\), where \(-\pi < \theta \leq \pi\) [4 marks]

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]

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]

Find the cube roots of the complex number \(z = 11 - 2i\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and correct to three decimal places. [7]

Find the three cube roots of the complex number \(2 + 3i\), giving your answers in Cartesian form. [9]

In this question you must show detailed reasoning. The complex number \(-4 + i\sqrt{48}\) is denoted by \(z\).

1. Determine the cube roots of \(z\), giving the roots in exponential form. [6]

The points which represent the cube roots of \(z\) are denoted by \(A\), \(B\) and \(C\) and these form a triangle in an Argand diagram.

1. Write down the angles that any lines of symmetry of triangle \(ABC\) make with the positive real axis, justifying your answer. [3]

In this question you must show detailed reasoning.

1. Show that \(e^{i\theta} - e^{-i\theta} = 2i\sin\theta\). [1]
2. Hence, show that \(\frac{2}{e^{2i\theta} - 1} = -(1 + i\cot\theta)\). [3]
3. Two series, \(C\) and \(S\), are defined as follows. $$C = 2 + 2\cos\frac{\pi}{10} + 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{10} + 2\cos\frac{2\pi}{5}$$ $$S = 2\sin\frac{\pi}{10} + 2\sin\frac{\pi}{5} + 2\sin\frac{3\pi}{10} + 2\sin\frac{2\pi}{5}$$ By considering \(C + iS\), find a simplified expression for \(C\) in terms of only integers and \(\cot\frac{\pi}{10}\). [8]
4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. [2]

11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\). \end{document}