Complex numbers 2

5 Use de Moivre's theorem to express \(\cos ^ { 4 } \theta\) in the form $$a \cos 4 \theta + b \cos 2 \theta + c$$ where \(a , b , c\) are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta d \theta$$ leaving your answer in terms of \(\pi\).

5 Use de Moivre's theorem to express \(\cos ^ { 4 } \theta\) in the form $$a \cos 4 \theta + b \cos 2 \theta + c$$ where \(a , b , c\) are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta d \theta$$ leaving your answer in terms of \(\pi\).

8

1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
   Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-18_2716_40_109_2009}
3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 Use De Moivre's Theorem to find the smallest positive angle \(\theta\) for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)

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 \(z ^ { n }\).

7

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 7 \theta = \frac { \operatorname { cosec } ^ { 7 } \theta } { 7 \operatorname { cosec } ^ { 6 } \theta - 56 \operatorname { cosec } ^ { 4 } \theta + 112 \operatorname { cosec } ^ { 2 } \theta - 64 }$$
2. Hence obtain the roots of the equation $$x ^ { 7 } - 14 x ^ { 6 } + 112 x ^ { 4 } - 224 x ^ { 2 } + 128 = 0$$ in the form \(\operatorname { cosec } q \pi\), where \(q\) is rational.

Use de Moivre's theorem to find the constants \(A\), \(B\) and \(C\) in the identity \(\sin^3 \theta \equiv A \sin \theta + B \sin 3\theta + C \sin 5\theta\). [4]

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Hence, given that $$z = \cos \theta + \mathrm { i } \sin \theta$$ show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
3. Given further that \(z + \frac { 1 } { z } = \sqrt { 2 }\), find the value of $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$

6

1. Use de Moivre's theorem to show that \(\sin ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )\).
2. Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + y \cot \theta = \sin ^ { 3 } \theta$$ for which \(y = 0\) when \(\theta = \frac { 1 } { 2 } \pi\).

Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots \\ & S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{aligned}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
2. Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
   Show \(z\) and its 4th roots in an Argand diagram.
   Find the product of the 4th roots and mark this as a point on your Argand diagram.

Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]

The complex number \(z = e^{\frac{i\pi}{3}}\) Which one of the following is a real number? Circle your answer. [1 mark] \(z^4\) \(z^5\) \(z^6\) \(z^7\)

The complex number \(z = e^{\frac{i\pi}{3}}\) Which one of the following is a real number? Circle your answer. [1 mark] \(z^4\) \(z^5\) \(z^6\) \(z^7\)

6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. 1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
   2. Verify that ( \(S , \bigcirc\) ) is a group.
   3. State the order of each element of \(( S , \bigcirc )\).
2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
3. 1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
   2. List all possible generators of \(( S , \bigcirc )\).

7

1. Use de Moivre's theorem to show that \(\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }\).
2. Hence find the exact roots of \(t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0\).

1. Use de Moivre's theorem to show that $$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$$ [3 marks]
2. Use de Moivre's theorem to express \(\sin 3\theta\) in terms of \(\sin \theta\) [2 marks]
3. Hence show that $$\cot 3\theta = \frac{\cot^3 \theta - 3\cot \theta}{3\cot^2 \theta - 1}$$ [4 marks]

6 Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\). Use the equation \(\cot 7 \theta = 0\) to show that the roots of the equation $$x ^ { 6 } - 21 x ^ { 4 } + 35 x ^ { 2 } - 7 = 0$$ are \(\cot \left( \frac { 1 } { 14 } k \pi \right)\) for \(k = 1,3,5,9,11,13\), and deduce that $$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$

Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]

Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.

2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z + 2 \mathrm { i } } { \mathrm { i } z } \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line.

5. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 z - 1 } { z + 3 } , \quad z \neq - 3$$ The circle in the \(z\)-plane with equation \(x ^ { 2 } + y ^ { 2 } = 1\), where \(z = x + \mathrm { i } y\), is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane. Find the centre and the radius of \(C\).

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]

3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that

1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\) (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)

1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
2. The transformation represented by the matrix \(\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points. [4 marks]

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

1. Express the complex number \(-2 + (2\sqrt{3})i\) in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\) [3]
2. Solve the equation $$z^3 = -2 + (2\sqrt{3})i$$ giving the roots in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\). [5]

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]

Given that \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{i\frac{\pi}{4}}\) state the value of \(\arg\left(\frac{z_1}{z_2}\right)\) Circle your answer. [1 mark] \(\frac{\pi}{12}\) \quad \(\frac{4}{3}\) \quad \(\frac{7\pi}{12}\) \quad \(2\)

Given that \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{i\frac{\pi}{4}}\) state the value of \(\arg\left(\frac{z_1}{z_2}\right)\) Circle your answer. [1 mark] \(\frac{\pi}{12}\) \quad \(\frac{4}{3}\) \quad \(\frac{7\pi}{12}\) \quad \(2\)

8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
2. Prove that the real part of \(\frac { 1 } { z }\) is constant.

3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.

3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.

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. Write down the fifth roots of unity.
2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).

3

1. Write down the fifth roots of unity.
2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).

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]

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 \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).

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}

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 \(\cos \frac { 1 } { 5 } \pi\).

In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]

$$f(z) = 3z^3 + pz^2 + 57z + q$$ where \(p\) and \(q\) are real constants. Given that \(3 - 2\sqrt{2}i\) is a root of the equation \(f(z) = 0\)

1. show all the roots of \(f(z) = 0\) on a single Argand diagram, [7]
2. find the value of \(p\) and the value of \(q\). [3]

5

1. Solve the equation \(z ^ { 2 } - 6 \mathrm { i } z - 12 = 0\), giving the answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
3. Find the exact modulus and argument of each root.
4. Hence show that the triangle \(O A B\) is equilateral.

1

1. Find \(a\) and \(b\) such that $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = \left( z ^ { 5 } - a \right) \left( z ^ { 3 } - b \right) .$$
2. Hence find the roots of $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).

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

1. Solve the equation

$$z ^ { 5 } - 32 i = 0$$ giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < 2 \pi\)

Express the complex number \(1 - i\sqrt{3}\) in modulus-argument form. Tick \((\checkmark)\) one box. [1 mark] \(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) \(2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\) \(2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\) \(2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)\)

1. The complex number \(z _ { 1 }\) is defined as

$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$

1. Without using your calculator show that $$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
2. Shade, on a single Argand diagram, the region \(R\) defined by $$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$ Given that the complex number \(z\) lies in \(R\)
3. determine the smallest possible positive value of \(\arg z\)

1

1. By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
2. Given that \(z z ^ { * } = 9\), describe the locus of \(z\).

Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5 = -16 + (16\sqrt{3})i,$$ giving each root in the form \(re^{i\theta}\). [4]

9

1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.

1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by

$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$ The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. [8]

The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.

1. Shade the region \(R\) on an Argand diagram. [2]

2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).