Complex arithmetic operations

6. The complex number \(z\) is defined by $$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$ Given that the modulus of \(z\) is 5

1. write down the value of \(\lambda\)
2. determine the argument of \(z\), giving your answer in radians to one decimal place. In part (c) you must show detailed reasoning.
   Solutions relying on calculator technology are not acceptable.
3. Express in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are real,
   1. \(\frac { z + 3 i } { 2 - 4 i }\)
   2. \(\mathrm { Z } ^ { 2 }\)
4. Show on a single Argand diagram the points \(A\), \(B\), \(C\) and \(D\) that represent the complex numbers $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$

1. Find the modulus and the argument of \(z\), giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of \(k\).
2. Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and are given in terms of \(k\) where necessary,
   1. \(\frac { 4 } { z + 3 k }\)
   2. \(z ^ { 2 }\)
3. Given that \(k = 1\), plot the points \(A , B , C\) and \(D\) representing \(z , z ^ { * } , \frac { 4 } { z + 3 k }\) and \(z ^ { 2 }\) respectively on a single Argand diagram.

9. $$z = \frac { 1 } { 5 } - \frac { 2 } { 5 } \mathrm { i }$$

1. Find the modulus and the argument of \(z\), giving the modulus as an exact answer and giving the argument in radians to 2 decimal places. Given that $$\mathrm { zw } = \lambda \mathrm { i }$$ where \(\lambda\) is a real constant,
2. find \(w\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. Give your answer in terms of \(\lambda\).
3. Given that \(\lambda = \frac { 1 } { 10 }\)
   1. find \(\frac { 4 } { 3 } ( z + w )\),
   2. plot the points \(A , B , C\) and \(D\), representing \(z , z w , w\) and \(\frac { 4 } { 3 } ( z + w )\) respectively, on a single Argand diagram.

1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that \(\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }\)

1. determine \(\left| z _ { 2 } \right|\) Given also that \(p = - 4\)
2. determine the possible values of \(q\)
3. Show \(z _ { 1 }\) and the possible positions for \(z _ { 2 }\) on the same Argand diagram.

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = 2 + 8 \mathrm { i } \quad \text { and } \quad z _ { 2 } = 1 - \mathrm { i }$$ Find, showing your working,

1. \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
3. the value of \(\arg \frac { z _ { 1 } } { z _ { 2 } }\), giving your answer in radians to 2 decimal places.

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = 2 + 8 i \quad \text { and } \quad z _ { 2 } = 1 - i$$ Find, showing your working,

1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
3. the value of \(\arg \frac { Z _ { 1 } } { Z _ { 2 } }\), giving your answer in radians to 2 decimal places.

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$

1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram. Find, showing your working,
2. the value of \(\left| z _ { 1 } \right|\),
3. the value of \(\arg z _ { 1 }\), giving your answer in radians to 2 decimal places,
4. \(\frac { Z _ { 2 } } { Z _ { 1 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.

1. $$z = 2 - 3 \mathrm { i }$$

1. Show that \(z ^ { 2 } = - 5 - 12 \mathrm { i }\). Find, showing your working,
2. the value of \(\left| z ^ { 2 } \right|\),
3. the value of \(\arg \left( z ^ { 2 } \right)\), giving your answer in radians to 2 decimal places.
4. Show \(z\) and \(z ^ { 2 }\) on a single Argand diagram.

4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$

1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.

6. Given that \(z = - 3 + 4 \mathrm { i }\),

1. find the modulus of \(z\),
2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
4. Show the points \(A\) and \(B\) on an Argand diagram.

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

1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { id }\).
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\).

11. $$z = 4 \left( \cos \frac { \pi } { 4 } + \mathrm { i } \sin \frac { \pi } { 4 } \right) , \text { and } w = 3 \left( \cos \frac { 2 \pi } { 3 } + \mathrm { i } \sin \frac { 2 \pi } { 3 } \right)$$ Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).
[0pt] [P4 January 2003 Qn 1]

8 Given that \(z _ { 1 } = 2 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)\) and \(z _ { 2 } = 2 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)\) 8

1. Find the value of \(\left| z _ { 1 } z _ { 2 } \right|\) 8
2. Find the value of \(\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)\) 8
3. Sketch \(z _ { 1 }\) and \(z _ { 2 }\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-10_764_869_1546_587} 8
4. A third complex number \(w\) satisfies both \(| w | = 2\) and \(- \pi < \arg w < 0\) Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P \widehat { R } Q\). Fully justify your answer.