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

In an Argand diagram, the points \(A\) and \(B\) are represented by the complex numbers \(- 3 + 2 \mathrm { i }\) and \(5 - 4 \mathrm { i }\) respectively. The points \(A\) and \(B\) are the end points of a diameter of a circle \(C\).

1. Find the equation of \(C\), giving your answer in the form $$| z - a | = b \quad a \in \mathbb { C } , \quad b \in \mathbb { R }$$ The circle \(D\), with equation \(| z - 2 - 3 i | = 2\), intersects \(C\) at the points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)
2. Find the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\)

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 \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\)

1. The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.

The centre of \(H\) represents the complex number \(\alpha\)

1. Show that \(\alpha = 3 + 7 \mathrm { i }\) Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
2. show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of \(H\) are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
3. 1. Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
   2. Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.

Given that a cubic equation has three distinct roots that all lie on the same straight line in the complex plane,

1. describe the possible lines the roots can lie on. $$f ( z ) = 8 z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(b , c\) and \(d\) are real constants.
   The roots of \(f ( z )\) are distinct and lie on a straight line in the complex plane.
   Given that one of the roots is \(\frac { 3 } { 2 } + \frac { 3 } { 2 } \mathrm { i }\)
2. state the other two roots of \(\mathrm { f } ( \mathrm { z } )\) $$g ( z ) = z ^ { 3 } + P z ^ { 2 } + Q z + 12$$ where \(P\) and \(Q\) are real constants, has 3 distinct roots.
   The roots of \(g ( z )\) lie on a different straight line in the complex plane than the roots of \(\mathrm { f } ( \mathrm { z } )\) Given that
   • \(f ( z )\) and \(g ( z )\) have one root in common
   • one of the roots of \(\mathrm { g } ( \mathrm { z } )\) is - 4
   • 1. write down the value of the common root,
     2. determine the value of the other root of \(\mathrm { g } ( \mathrm { z } )\)
   • Hence solve the equation \(\mathrm { f } ( \mathrm { z } ) = \mathrm { g } ( \mathrm { z } )\)

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.

9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation \(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).

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. \(\frac { z } { z ^ { * } }\)

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Write down, in terms of \(x\) and \(y\), an expression for \(z ^ { * }\), the complex conjugate of \(z\).
2. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
3. Find the complex number \(z\) such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$

1 It is given that \(z _ { 1 } = 2 + \mathrm { i }\) and that \(z _ { 1 } { } ^ { * }\) is the complex conjugate of \(z _ { 1 }\).
Find the real numbers \(x\) and \(y\) such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

2 The complex number \(z\) is defined by $$z = 1 + \mathrm { i }$$

1. Find the value of \(z ^ { 2 }\), giving your answer in its simplest form.
2. Hence show that \(z ^ { 8 } = 16\).
3. Show that \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).

6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).

1. 1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
   2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
   3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
2. Find in the form \(p + \mathrm { i } q\) the values of:
   1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
   2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).

6 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Write down, in terms of \(x\) and \(y\), an expression for $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).

3 The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$

1. Show that \(z _ { 1 } = \mathrm { i }\).
2. Show that \(\left| z _ { 1 } \right| = \left| z _ { 2 } \right|\).
3. Express both \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
4. Draw an Argand diagram to show the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 1 } + z _ { 2 }\).
5. Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$

3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. It is given that \(\alpha\) is of the form \(k \mathrm { i }\), where \(k\) is real. By substituting \(z = k \mathrm { i }\) into the equation, show that \(k = 4\).
2. Given that \(\beta = - 4\), find the value of \(\gamma\).

8

1. 1. Given that \(z ^ { 6 } - 4 z ^ { 3 } + 8 = 0\), show that \(z ^ { 3 } = 2 \pm 2 \mathrm { i }\).
   2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. Show that, for any real values of \(k\) and \(\theta\), $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
3. Express \(z ^ { 6 } - 4 z ^ { 3 } + 8\) as the product of three quadratic factors with real coefficients.

1 Given that \(z = 2 \mathrm { e } ^ { \frac { \pi \mathrm { i } } { 12 } }\) satisfies the equation $$z ^ { 4 } = a ( 1 + \sqrt { 3 } i )$$ where \(a\) is real:

1. find the value of \(a\);
2. find the other three roots of this equation, giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

7 The cubic equation \(27 z ^ { 3 } + k z ^ { 2 } + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. 1. In the case where \(\beta = \gamma\), find the roots of the equation.
   2. Find the value of \(k\) in this case.
3. 1. In the case where \(\alpha = 1 - \mathrm { i }\), find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\).
   2. Hence find the value of \(k\) in this case.
4. In the case where \(k = - 12\), find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1\) and \(\frac { 1 } { \gamma } + 1\).
   [0pt] [5 marks]