4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

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

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$

4

1. It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
   1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
   2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
2. It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q\) i is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).

3

1. Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
2. It is given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where \(p\) and \(q\) are real numbers.
   1. Show that \(p = - 11\) and find the value of \(q\).
   2. Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
   3. Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).
      \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}

3

1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
   It is given that \(\alpha = 1 + \mathrm { i }\).
   1. Find the value of \(k\).
   2. Show that \(\beta + \gamma = 4\).
   3. Find the values of \(\beta\) and \(\gamma\).

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 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$ where \(p\) and \(q\) are constants, has three complex roots, \(\alpha , \beta\) and \(\gamma\). It is given that \(\beta = - 2 + 3 \mathrm { i }\) and \(\gamma = 1 + 2 \mathrm { i }\).

1. 1. Write down the value of \(\alpha \beta \gamma\).
   2. Hence show that \(( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }\).
   3. Hence find \(\alpha\), giving your answer in the form \(m + n \mathrm { i }\), where \(m\) and \(n\) are integers.
2. Find the value of \(p\).
3. Find the value of the complex number \(q\).

3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.

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.

5 The Argand diagram below shows the points representing 1 and \(z\), where \(| z | = 2\). \includegraphics[max width=\textwidth, alt={}, center]{26cec6f9-78a7-4f0b-969a-13ad02510c25-3_577_595_312_242} Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.

• \(\mathrm { Z } ^ { * }\)
• \(\frac { 1 } { z }\)
• \(1 + z\)
• iz

2 Fig. 2 shows two complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) represented on an Argand diagram. \begin{figure}[h] \end{figure}

1. On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.
   In the case where \(z _ { 1 } = 1 + 2 \mathrm { i }\) and \(z _ { 2 } = 3 + \mathrm { i }\), find \(\frac { z _ { 2 } - z _ { 1 } } { z _ { 1 } ^ { * } }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.

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 \(\frac { z } { w }\).

The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).

1. 1. Find the modulus and argument of \(z w\).
   2. Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
2. The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
3. The complex conjugate of \(v\) is denoted by \(\bar { v }\). Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.
   

1. The complex number \(z\) is given by \(z = 3 + \lambda \mathrm { i }\), where \(\lambda\) is a positive constant. The complex conjugate of \(z\) is denoted by \(\bar { z }\).

Given that \(z ^ { 2 } + \bar { z } ^ { 2 } = 2\), find the value of \(\lambda\).

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.

8 Throughout this question the use of a calculator is not permitted.

1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.

5 Throughout this question the use of a calculator is not permitted. The complex numbers \(w\) and \(z\) satisfy the relation $$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$

1. Given that \(z = 1 + \mathrm { i }\), find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Given instead that \(w = z\) and the real part of \(z\) is negative, find \(z\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

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.