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

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

7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\frac { 1 } { 3 } \pi\). The point B represents \(\frac { 1 } { z }\).

1. Sketch an Argand diagram showing the origin O and the points A and B .
2. The point C is such that OACB is a parallelogram. C represents the complex number \(w\). Determine each of the following.

3 In this question you must show detailed reasoning.
The roots of the equation \(x ^ { 2 } - 2 x + 4 = 0\) are \(\alpha\) and \(\beta\).

1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.

7

1. 1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
   2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.

1

1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
   1. Write down \(z ^ { * }\).
   2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
   1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
   2. Convert \(w\) to modulus-argument form.

2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).

1. 1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
   2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
2. Express \(u\) in exact modulus-argument form.

3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = - 2 + 2 i\) and \(z _ { 2 } = 2 \left( \cos \frac { 1 } { 6 } \pi + i \sin \frac { 1 } { 6 } \pi \right)\).

1. Find the modulus and argument of \(z _ { 1 }\).
2. Hence express \(\frac { z _ { 1 } } { z _ { 2 } }\) in exact modulus-argument form.

3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.

1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
2. Find the complex number \(\frac { z } { w }\).

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).

1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).

1. The complex numbers \(z , v\) and \(w\) are related by the equation

$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and

$$w = \frac { z } { 1 - z }$$

1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).

1. Given that

$$\begin{aligned} z _ { 1 } & = 2 + 3 \\ \left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 } \\ \arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers,

1. write \(z _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) Give the exact value of \(r\) and give the value of \(\theta\) in radians to 4 significant figures.
2. Find \(z _ { 2 }\) giving your answer in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are integers.

1. (i) (a) Show that

$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$ where \(k\) is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.) Given that

• \(n\) is a positive integer
• \(\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }\) is a real number
  (b) use the answer to part (a) to write down the smallest possible value of \(n\).
  (ii) The complex number \(z = a + b \mathrm { i }\) where \(a\) and \(b\) are real constants.

Given that

• \(\left| z ^ { 10 } \right| = 59049\)
• \(\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }\) determine the value of \(a\) and the value of \(b\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$z _ { 1 } = - 4 + 4 i$$

1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
   1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
   2. \(\left( z _ { 2 } \right) ^ { 4 }\)
3. Show on a single Argand diagram
   1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
   2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)

1. Given that

$$\begin{aligned} z _ { 1 } & = 3 \left( \cos \left( \frac { \pi } { 3 } \right) + \mathrm { i } \sin \left( \frac { \pi } { 3 } \right) \right) \\ z _ { 2 } & = \sqrt { 2 } \left( \cos \left( \frac { \pi } { 12 } \right) - \mathrm { i } \sin \left( \frac { \pi } { 12 } \right) \right) \end{aligned}$$

1. write down the exact value of
   1. \(\left| Z _ { 1 } Z _ { 2 } \right|\)
   2. \(\arg \left( \mathrm { z } _ { 1 } \mathrm { z } _ { 2 } \right)\) Given that \(w = z _ { 1 } z _ { 2 }\) and that \(\arg \left( w ^ { n } \right) = 0\), where \(n \in \mathbb { Z } ^ { + }\)
2. determine
   1. the smallest positive value of \(n\)
   2. the corresponding value of \(\left| w ^ { n } \right|\)

1. A student was asked to answer the following:

For the complex numbers \(z _ { 1 } = 3 - 3 \mathrm { i }\) and \(z _ { 2 } = \sqrt { 3 } + \mathrm { i }\), find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) The student's attempt is shown below. \includegraphics[max width=\textwidth, alt={}, center]{33292670-3ad0-4125-a3bb-e4b7b21ed5f4-02_798_1109_534_338} The student made errors in line 1 and line 3
Correct the error that the student made in

1. 1. line 1
   2. line 3
2. Write down the correct value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)

2 In this question you must show detailed reasoning. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 3 - 7 \mathrm { i }\) and \(z _ { 2 } = 2 + 4 \mathrm { i }\).

1. Express each of the following as exact numbers in the form \(a + b \mathrm { i }\).
   1. \(3 z _ { 1 } + 4 z _ { 2 }\)
   2. \(z _ { 1 } z _ { 2 }\)
   3. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
2. Write \(z _ { 1 }\) in modulus-argument form giving the modulus in exact form and the argument correct to \(\mathbf { 3 }\) significant figures.

1

1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.