Complex numbers 2

4 Solve the equation \(z ^ { 3 } = i\), giving your answers in the form \(e ^ { i \theta }\), where \(- \pi < \theta \leq \pi\)
[0pt] [4 marks]

1. Solve the equation

$$z ^ { 5 } = 32$$ Give your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\)

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 \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).

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.

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

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.

7

1. State the sum of the series \(1 + \mathrm { w } + \mathrm { w } ^ { 2 } + \mathrm { w } ^ { 3 } + \ldots + \mathrm { w } ^ { \mathrm { n } - 1 }\), for \(w \neq 1\).
2. Show that \(( 1 + i \tan \theta ) ^ { k } = \sec ^ { k } \theta ( \cos k \theta + i \sin k \theta )\), where \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
3. By considering \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { \mathrm { k } }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \sec ^ { k } \theta \sin k \theta = \cot \theta \left( 1 - \sec ^ { n } \theta \cos n \theta \right)$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
4. Hence find \(\sum _ { k = 0 } ^ { 6 m - 1 } 2 ^ { k } \sin \left( \frac { 1 } { 3 } k \pi \right)\) in terms of \(m\).

5 By expanding \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }\), where \(z = e ^ { \mathrm { i } \theta }\), show that \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).

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 \(\theta = \frac { 1 } { 2 } \pi\).

2 The complex number \(z = e ^ { \frac { i \pi } { 3 } }\)
Which one of the following is a real number?
Circle your answer.
[0pt] [1 mark]
\(z ^ { 4 }\)
\(z ^ { 5 }\)
\(z ^ { 6 }\)
\(z ^ { 7 }\)

2 The complex number \(z = e ^ { \frac { i \pi } { 3 } }\)
Which one of the following is a real number?
Circle your answer.
[0pt] [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 \(( S , \bigcirc )\).

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)

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

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.

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 \(t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0\).

13

1. Use de Moivre's theorem to show that $$\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ 13
2. Use de Moivre's theorem to express \(\sin 3 \theta\) in terms of \(\sin \theta\)
   13
3. Hence show that $$\cot 3 \theta = \frac { \cot ^ { 3 } \theta - 3 \cot \theta } { 3 \cot ^ { 2 } \theta - 1 }$$

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 $$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$

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 \(C\).

2. (a) Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$ (b) Hence find 3 distinct solutions of the equation \(16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + 1 = 0\), giving your answers to 3 decimal places where appropriate.

6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.

1. Use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ where \(n\) is a positive integer.
2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
3. Hence find all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)

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.

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 In this question, \(w\) denotes the complex number \(\cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\).

1. Express \(w ^ { 2 } , w ^ { 3 }\) and \(w ^ { * }\) in polar form, with arguments in the interval \(0 \leqslant \theta < 2 \pi\).
2. The points in an Argand diagram which represent the numbers $$1 , \quad 1 + w , \quad 1 + w + w ^ { 2 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 }$$ are denoted by \(A , B , C , D , E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.)
3. Write down a polynomial equation of degree 5 which is satisfied by \(w\).

9 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 \(k = 0,1,2 , \ldots , n - 1\), are the \(n n ^ { \text {th } }\) 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 \(\omega _ { k } = \left( \omega _ { 1 } \right) ^ { k }\) for \(k = 2 , \ldots , n - 1\).
2. Using the identity given in part (a), show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \omega _ { \mathrm { k } } = 0\).
3. Show that if \(z\) is a complex number then \(z + z ^ { * } = 2 \operatorname { Re } ( z )\).
4. Using the results from parts (b) and (c) show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = 0\).
5. With the help of a diagram explain why \(\operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = \operatorname { Re } \left( \omega _ { \mathrm { n } - \mathrm { k } } \right)\) for \(k = 1,2 , \ldots , n - 1\). You should now consider the case where \(n = 5\).
6. 1. Use parts (d) and (e) to deduce that \(\cos \frac { 4 \pi } { 5 } = \mathrm { a } + \mathrm { b } \cos \frac { 2 \pi } { 5 }\), for some rational constants \(a\) and \(b\).
   2. Hence determine the exact value of \(\cos \frac { 2 \pi } { 5 }\).

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. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\).

2. $$z = 5 \sqrt { } 3 - 5 i$$ Find

1. \(| z |\),
2. \(\arg ( z )\), in terms of \(\pi\). $$w = 2 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$ Find
3. \(\left| \frac { w } { z } \right|\),
4. \(\quad \arg \left( \frac { w } { z } \right)\), in terms of \(\pi\).

4 Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)

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.

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.

4 （a）Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients．
［4 marks］
4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
堛的 增
4 (b) Find the value of \(p\) and the value of \(r\).

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 \(0 \leqslant \theta < 2 \pi\).

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 \(z\).

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 \(\arg z\)

5 The complex number \(z\) is defined by \(z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }\). Find, showing all your working,

1. an expression for \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\),
2. the two square roots of \(z\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

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 \(- \pi < \theta \leqslant \pi\).

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 \(p\).

7

1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. State the modulus and argument of each root.
3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$