4.02b Express complex numbers: cartesian and modulus-argument forms

1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.

1. \(z\)
2. \(z ^ { 2 }\)
3. \(\frac { z } { z ^ { * } }\)

4 The complex numbers \(w\) and \(z\) are defined as $$\begin{aligned} w & = 2 ( \cos \alpha + \mathrm { i } \sin \alpha ) \\ z & = 3 ( \cos \beta + \mathrm { i } \sin \beta ) \end{aligned}$$ Find the product \(w z\) Tick \(( \checkmark )\) one box. $$\begin{aligned} & 5 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) ) \\ & 6 ( \cos ( \alpha \beta ) + \mathrm { i } \sin ( \alpha \beta ) ) \\ & 5 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) ) \\ & 6 ( \cos ( \alpha + \beta ) + \mathrm { i } \sin ( \alpha + \beta ) ) \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_762_1206} \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_108_108_900_1206} \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-03_113_113_1032_1206}
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1. Sketch on the Argand diagram below, the locus of points satisfying the equation \(| z - 2 | = 2\) \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-14_1417_1475_790_399} 9
2. Given that \(| z - 2 | = 2\) and \(\arg ( z - 2 ) = - \frac { \pi } { 3 }\), express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
   [0pt] [3 marks]

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

1. Find
   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 \(B\).

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

3 In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval \([ 0,2 \pi )\).
Complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are defined by \(z _ { 1 } = 3 + 4 \mathrm { i }\) and \(z _ { 2 } = - 5 + 12 \mathrm { i }\).

1. Determine \(z _ { 1 } z _ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
2. Express \(z _ { 2 }\) in modulus-argument form.
3. Verify, by direct calculation, that \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).

2 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

2 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).

1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3 z - 4 z ^ { * }\)
   2. \(( z + 1 - 3 \mathrm { i } ) ^ { 2 }\)
   3. \(\frac { z + 1 } { z - 1 }\)
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.

2 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 \(z ^ { n }\).

11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).

1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
   [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
   (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
   (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
   (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).

7 The complex number \(z\) is given by \(- 20 + 21 \mathrm { i }\). Showing all your working,

1. find the value of \(| z |\),
2. calculate the value of \(\arg z\) correct to 3 significant figures,
3. express \(\frac { 1 } { z }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

13 The complex number \(w\) has modulus 1. It is given that $$w ^ { 2 } - \frac { 2 } { w } + k \mathrm { i } = 0$$ where \(k\) is a positive real constant.

1. Show that \(k = ( 3 - \sqrt { 3 } ) \sqrt { \frac { 1 } { 2 } \sqrt { 3 } }\).
2. Prove that at least one of the remaining two roots of the equation \(z ^ { 2 } - \frac { 2 } { z } + k i = 0\) has modulus greater than 1 .

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1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. 1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(( \mathrm { r } , \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).

10

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leq \pi\).
   (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.

5 The complex number \(z\) satisfies the equation \(2 z - \mathrm { i } = \mathrm { i } z + 2\).

1. Express \(z\) in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are rational numbers.
2. Find the exact value of \(| z |\) and of \(\arg ( z )\).
3. Express \(z ^ { 2 }\) in the form \(c + \mathrm { i } d\) where \(c\) and \(d\) are rational numbers.
4. Verify that \(\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)\) using an appropriate trigonometrical identity.

The complex number \(2 + \mathrm{i}\) is denoted by \(u\). Its complex conjugate is denoted by \(u^*\).

1. Show, on a sketch of an Argand diagram with origin \(O\), the points \(A\), \(B\) and \(C\) representing the complex numbers \(u\), \(u^*\) and \(u + u^*\) respectively. Describe in geometrical terms the relationship between the four points \(O\), \(A\), \(B\) and \(C\). [4]
2. Express \(\frac{u}{u^*}\) in the form \(x + \mathrm{i}y\), where \(x\) and \(y\) are real. [3]
3. By considering the argument of \(\frac{u}{u^*}\), or otherwise, prove that $$\tan^{-1}\left(\frac{4}{3}\right) = 2\tan^{-1}\left(\frac{1}{2}\right).$$ [2]

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

The complex number \(w\) is given by $$w = 10 - 5\text{i}$$

1. Find \(|w|\). [1]
2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [2]

The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + \text{i})(z + 3\text{i}) = w$$

1. Use algebra to find \(z\), giving your answer in the form \(a + b\text{i}\), where \(a\) and \(b\) are real numbers. [4]

Given that $$\arg(\lambda + 9\text{i} + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,

1. find the value of \(\lambda\). [2]

Given that \(z = 22 + 4i\) and \(\frac{z}{w} = 6 - 8i\), find

1. \(w\) in the form \(a + bi\), where \(a\) and \(b\) are real, [3]
2. the argument of \(z\), in radians to 2 decimal places. [2]

$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$

1. Given that \(a = 4\), find \(|z|\). [3]
2. Show that there is only one value of \(a\) for which \(\arg z = \frac{\pi}{4}\), and find this value. [6]

Given that \(z = 2 - 2i\) and \(w = -\sqrt{3} + i\),

1. find the modulus and argument of \(wz^2\). [6]
2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]

Given that \(z = 1 + \sqrt{3}i\) and that \(\frac{w}{z} = 2 + 2i\), find

1. \(w\) in the form \(a + ib\), where \(a, b \in \mathbb{R}\), [3]
2. the argument of \(w\), [2]
3. the exact value for the modulus of \(w\). [2]

On an Argand diagram, the point \(A\) represents \(z\) and the point \(B\) represents \(w\).

1. Draw the Argand diagram, showing the points \(A\) and \(B\). [2]
2. Find the distance \(AB\), giving your answer as a simplified surd. [2]

Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find

1. \(\left|\frac{z}{w}\right|\), [3]
2. \(\arg \left( \frac{z}{w} \right)\). [3]

1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
2. Show that \(\angle AOC = \angle DOB\). [2]
3. Find the area of triangle \(AOC\). [2]

$$z = -4 + 6i.$$

1. Calculate \(\arg z\), giving your answer in radians to 3 decimal places. [2]

The complex number \(w\) is given by \(w = \frac{A}{2 - i}\), where \(A\) is a positive constant. Given that \(|w| = \sqrt{20}\),

1. find \(w\) in the form \(a + ib\), where \(a\) and \(b\) are constants, [4]
2. calculate \(\arg \frac{w}{z}\). [3]