4.02a Complex numbers: real/imaginary parts, modulus, argument

8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 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 \(B\).

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 \(\arg \left( z _ { 1 } z _ { 2 } \right) = \arg \left( z _ { 1 } \right) + \arg \left( z _ { 2 } \right)\).

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 \(k\).

6 The roots of the equation \(z ^ { 2 } - 6 z + 10 = 0\) are \(z _ { 1 }\) and \(z _ { 2 }\), where \(z _ { 1 } = 3 + \mathrm { i }\).

1. Write down the value of \(z _ { 2 }\).
2. Show \(z _ { 1 }\) and \(z _ { 2 }\) on an Argand diagram.
3. Show that \(z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }\).

11

1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
   1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
   2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).

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.

7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).

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 .

9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

5 The complex numbers \(u\) and \(v\) are given by \(u = 3 + 2 \mathrm { i }\) and \(v = 1 + 4 \mathrm { i }\).

1. Given that \(a u ^ { 2 } + b v ^ { * } = 7 + 36 \mathrm { i }\) find the values of the real constants \(a\) and \(b\).
2. Show the points representing \(u\) and \(v\) on an Argand diagram and hence sketch the locus given by \(| z - u | = | z - v |\). Find the point of intersection of this locus with the imaginary axis.

9 The complex number 3-4i is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

11

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.

9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

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 \(p\).

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]

The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 3 + 5\text{i} \quad \text{and} \quad z_2 = -2 + 6\text{i}$$

1. Show \(z_1\) and \(z_2\) on a single Argand diagram. [2]
2. Without using your calculator and showing all stages of your working,
   1. determine the value of \(|z_1|\) [1]
   2. express \(\frac{z_1}{z_2}\) in the form \(a + b\text{i}\), where \(a\) and \(b\) are fully simplified fractions. [3]
3. Hence determine the value of \(\arg \frac{z_1}{z_2}\) Give your answer in radians to 2 decimal places. [2]

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]

Given that \(z = 3 + 4i\) and \(w = -1 + 7i\).

1. find \(|w|\). [1]

The complex numbers \(z\) and \(w\) are represented by the points \(A\) and \(B\) on an Argand diagram.

1. Show points \(A\) and \(B\) on an Argand diagram. [1]
2. Prove that \(\triangle OAB\) is an isosceles right-angled triangle. [5]
3. Find the exact value of \(\arg \left( \frac{z}{w} \right)\). [3]

Given that \(z = 3 - 3i\) express, in the form \(a + ib\), where \(a\) and \(b\) are real numbers,

1. \(z^2\), [2]
2. \(\frac{1}{z}\), [2]
3. Find the exact value of each of \(|z|\), \(|z^2|\) and \(\left|\frac{1}{z}\right|\). [2]

The complex numbers \(z\), \(z^2\) and \(\frac{1}{z}\) are represented by the points \(A\), \(B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.

1. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram. [2]
2. Prove that \(\triangle OAB\) is similar to \(\triangle OCD\). [3]