4.02r nth roots: of complex numbers

9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\). Hence find all the roots of the equation $$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$ and deduce that none of them is real. Find the arguments of the two roots which have the smaller modulus.

Let \(\omega = \cos \frac { 1 } { 5 } \pi + \mathrm { i } \sin \frac { 1 } { 5 } \pi\). Show that \(\omega ^ { 5 } + 1 = 0\) and deduce that $$\omega ^ { 4 } - \omega ^ { 3 } + \omega ^ { 2 } - \omega = - 1$$ Show further that $$\omega - \omega ^ { 4 } = 2 \cos \frac { 1 } { 5 } \pi \quad \text { and } \quad \omega ^ { 3 } - \omega ^ { 2 } = 2 \cos \frac { 3 } { 5 } \pi$$ Hence find the values of $$\cos \frac { 1 } { 5 } \pi + \cos \frac { 3 } { 5 } \pi \quad \text { and } \quad \cos \frac { 1 } { 5 } \pi \cos \frac { 3 } { 5 } \pi$$ Find a quadratic equation having roots \(\cos \frac { 1 } { 5 } \pi\) and \(\cos \frac { 3 } { 5 } \pi\) and deduce the exact value of \(\cos \frac { 1 } { 5 } \pi\).

9 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).

1. Show that \(\omega\) is a root of the equation.
2. Write down the other four roots of the equation.
3. Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
4. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
5. Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found.

9 The cube roots of unity are represented on the Argand diagram below by the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244} The points \(L , M\) and \(N\) are the midpoints of the line segments \(A B , B C\) and \(C A\) respectively. Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points \(A , B , C , L , M\) and \(N\). \section*{END OF QUESTION PAPER}

3

1. Show that \(\frac { - 3 + \sqrt { 3 } \mathrm { i } } { 2 } = \sqrt { 3 } \mathrm { e } ^ { \frac { 5 } { 6 } \pi \mathrm { i } }\).
2. Hence determine the exact roots of the equation \(z ^ { 5 } = \frac { 9 ( - 3 + \sqrt { 3 } \mathrm { i } ) } { 2 }\), giving the roots in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

1. Justify each of the following statements.
   1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   2. \(A B C D E\) is a regular pentagon.
   3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   4. \(b ^ { * } = e\)
   5. \(a + b + c + d + e = 0\)
   6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

8

1. 1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
   2. Write down the five other non-real roots in terms of \(\omega\).
2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
3. Show that:
   1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
   2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\). There are no questions printed on this page There are no questions printed on this page \section*{There are no questions printed on this page} \end{document}

8

1. Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
   1. \(\quad 4 ( 1 + i \sqrt { 3 } )\);
   2. \(4 ( 1 - i \sqrt { 3 } )\).
2. The complex number \(z\) satisfies the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
3. 1. Solve the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Illustrate the roots on an Argand diagram.
4. 1. Explain why the sum of the roots of the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ is zero.
   2. Deduce that \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).

8

1. Write down the five roots of the equation \(z ^ { 5 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Hence find the four linear factors of $$z ^ { 4 } + z ^ { 3 } + z ^ { 2 } + z + 1$$
3. Deduce that $$z ^ { 2 } + z + 1 + z ^ { - 1 } + z ^ { - 2 } = \left( z - 2 \cos \frac { 2 \pi } { 5 } + z ^ { - 1 } \right) \left( z - 2 \cos \frac { 4 \pi } { 5 } + z ^ { - 1 } \right)$$
4. Use the substitution \(z + z ^ { - 1 } = w\) to show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

7

1. 1. Express each of the numbers \(1 + \sqrt { 3 } \mathrm { i }\) and \(1 - \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
   2. Hence express $$( 1 + \sqrt { 3 } i ) ^ { 8 } ( 1 - i ) ^ { 5 }$$ in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
2. Solve the equation $$z ^ { 3 } = ( 1 + \sqrt { 3 } \mathrm { i } ) ^ { 8 } ( 1 - \mathrm { i } ) ^ { 5 }$$ giving your answers in the form \(a \sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \theta }\), where \(a\) is a positive integer and \(- \pi < \theta \leqslant \pi\).

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1

1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [2 marks]
2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [5 marks]

11 An Argand diagram with the point A representing a complex number \(z _ { 1 }\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-4_716_778_932_239} The complex numbers \(z _ { 2 }\) and \(z _ { 3 }\) are \(z _ { 1 } \mathrm { e } ^ { \frac { 2 } { 3 } \mathrm { i } \pi }\) and \(z _ { 1 } \mathrm { e } ^ { \frac { 4 } { 3 } \mathrm { i } \pi }\) respectively.

1. 1. On the copy of the Argand diagram in the Printed Answer Booklet, mark the points B and C representing the complex numbers \(z _ { 2 }\) and \(z _ { 3 }\).
   2. Show that \(z _ { 1 } + z _ { 2 } + z _ { 3 } = 0\).
2. Given now that \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are roots of the equation \(z ^ { 3 } = 8 \mathrm { i }\), find these three roots, giving your answers in the form \(\mathrm { a } + \mathrm { ib }\), where \(a\) and \(b\) are real and exact.

5

1. In this question you must show detailed reasoning.
   Determine the sixth roots of - 64 , expressed in \(r \mathrm { e } ^ { \mathrm { i } \theta }\) form.
2. Represent the roots on an Argand diagram.

10

1. Show on an Argand diagram the points representing the three cube roots of unity.
2. 1. Find the exact roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\), expressing them in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta < \pi\).
   2. The points representing the cube roots of unity form a triangle \(\Delta _ { 1 }\). The points representing the roots of the equation \(z ^ { 3 } - 1 = \sqrt { 3 } \mathrm { i }\) form a triangle \(\Delta _ { 2 }\). State a sequence of two transformations that maps \(\Delta _ { 1 }\) onto \(\Delta _ { 2 }\).
   3. The three roots in part (b)(i) are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\). By simplifying \(z _ { 1 } + z _ { 2 } + z _ { 3 }\), verify that the sum of these roots is zero.
   4. Hence show that \(\sin 20 ^ { \circ } + \sin 140 ^ { \circ } = \sin 100 ^ { \circ }\).

1. (a) Use de Moivre's theorem to prove that

$$\sin 7 \theta = 7 \sin \theta - 56 \sin ^ { 3 } \theta + 112 \sin ^ { 5 } \theta - 64 \sin ^ { 7 } \theta$$ (b) Hence find the distinct roots of the equation $$1 + 7 x - 56 x ^ { 3 } + 112 x ^ { 5 } - 64 x ^ { 7 } = 0$$ giving your answer to 3 decimal places where appropriate.

1. The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.

The centre of \(H\) represents the complex number \(\alpha\)

1. Show that \(\alpha = 3 + 7 \mathrm { i }\) Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
2. show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of \(H\) are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
3. 1. Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
   2. Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.

1. (a) Determine the roots of the equation

$$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\) (b) Show the roots of the equation in part (a) on a single Argand diagram.
(c) Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$ (d) Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)

10 In this question you must show detailed reasoning.

1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).

11 In this question you must show detailed reasoning. In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively. \begin{figure}[h] \end{figure}

1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).

8 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. 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.
2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.

4 In this question you must show detailed reasoning.
You are given that \(z = \sqrt { 3 } + \mathrm { i }\). \(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.

1. Determine the value of \(n\).
2. Find the value of \(z ^ { n }\).

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}

1. Justify each of the following statements.
   1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   2. \(A B C D E\) is a regular pentagon.
   3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   4. \(b ^ { * } = e\)
   5. \(a + b + c + d + e = 0\)
   6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

6

1. Find the three roots of \(z ^ { 3 } = 1\), giving the non-real roots in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Given that \(\omega\) is one of the non-real roots of \(z ^ { 3 } = 1\), show that $$1 + \omega + \omega ^ { 2 } = 0$$
3. By using the result in part (b), or otherwise, show that:
   1. \(\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }\);
   2. \(\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega\);
   3. \(\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi\), where \(k\) is an integer.