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

8 It is given that \(\frac { 2 + 3 a \mathrm { i } } { a + 2 \mathrm { i } } = \lambda ( 2 - \mathrm { i } )\), where \(a\) and \(\lambda\) are real constants.

1. Show that \(3 a ^ { 2 } + 4 a - 4 = 0\).
2. Hence find the possible values of \(a\) and the corresponding values of \(\lambda\).

3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z w\)
2. \(\frac { z } { w ^ { * } }\), showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
3. find the possible values of \(k\).

1. The complex number \(z\) is given by

$$z = - 7 + 3 i$$ Find

1. \(| z |\)
2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
4. Show the points representing \(z\) and \(w\) on a single Argand diagram.

5. (i) Given that $$\frac { 2 z + 3 } { z + 5 - 2 i } = 1 + i$$ find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants.
(ii) Given that $$w = ( 3 + \lambda \mathrm { i } ) ( 2 + \mathrm { i } )$$ where \(\lambda\) is a real constant, and that $$| w | = 15$$ find the possible values of \(\lambda\).

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 $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

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

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(p\)
2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. Given that

$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,

1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
2. find the possible values of \(\lambda\).

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.

9. Given that $$\frac { z - k \mathrm { i } } { z + 3 \mathrm { i } } = \mathrm { i } \text {, where } k \text { is a positive real constant }$$

1. show that \(z = - \frac { ( k + 3 ) } { 2 } + \frac { ( k - 3 ) } { 2 } \mathrm { i }\)
2. Using the printed answer in part (a),
   1. find an exact simplified value for the modulus of \(z\) when \(k = 4\)
   2. find the argument of \(z\) when \(k = 1\). Give your answer in radians to 3 decimal places, where \(- \pi < \arg z < \pi\)

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

$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$

1. Determine the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\) Given that \(w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\)
2. determine \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y\), where \(x , y \in \mathbb { R }\) Given also that \(w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }\)
3. determine the value of \(a\) and the value of \(b\)
4. determine arg \(w\), giving your answer in radians to 4 significant figures.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The complex number \(z\) is defined by $$2 = - 3 + 4 i$$

1. Determine \(\left| z ^ { 2 } - 3 \right|\)
2. Express \(\frac { 50 } { z ^ { * } }\) in the form \(k z\), where \(k\) is a positive integer.
3. Hence find the value of \(\arg \left( \frac { 50 } { z ^ { * } } \right)\) Give your answer in radians to 3 significant figures.

5. $$2 z + z ^ { * } = \frac { 3 + 4 \mathrm { i } } { 7 + \mathrm { i } }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.

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.

7. $$z = - 24 - 7 i$$

1. Show \(z\) on an Argand diagram.
2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. It is given that $$w = a + b \mathrm { i } , \quad a \in \mathbb { R } , b \in \mathbb { R }$$ Given also that \(| w | = 4\) and \(\arg w = \frac { 5 \pi } { 6 }\),
3. find the values of \(a\) and \(b\),
4. find the value of \(| z w |\).

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.

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

1. Find the modulus of \(z _ { 1 }\).
2. Find, in radians, the argument of \(z _ { 1 }\), giving your answer to 2 decimal places. The solutions to the quadratic equation $$z ^ { 2 } - 10 z + 28 = 0$$ are \(z _ { 2 }\) and \(z _ { 3 }\).
3. Find \(z _ { 2 }\) and \(z _ { 3 }\), giving your answers in the form \(p \pm i \sqrt { } q\), where \(p\) and \(q\) are integers.
4. Show, on an Argand diagram, the points representing your complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

6. Given that \(z = x + \mathrm { i } y\), find the value of \(x\) and the value of \(y\) such that $$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

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

1. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. Use algebra to express
2. \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
3. \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers. Given that $$w = \lambda - 3 \mathrm { i }$$ where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
4. find the value of \(\lambda\).

3. $$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$

1. Express \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) giving exact values of \(r\) and \(\theta\).
   (4)
2. Find \(\left| z _ { 1 } z _ { 2 } \right|\).
3. Show and label \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
   (2)

7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$

1. Find the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\). Given that \(w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }\),
2. find \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y , \quad x , y \in \mathbb { R }\) Given also that \(w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }\),
3. find the value of \(a\) and the value of \(b\),
4. find \(\arg w\), giving your answer in radians to 3 decimal places.