4.02b Express complex numbers: cartesian and modulus-argument forms

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.

\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.

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 \(z _ { 3 }\).

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)

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.

7. A complex number \(z\) is given by $$z = a + 2 i$$ where \(a\) is a non-zero real number.

1. Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\). Given that \(z ^ { 2 } + 2 z\) is real,
2. find the value of \(a\). Using this value for \(a\),
3. find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
4. Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
5. Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).
   

4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.

1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
2. find the value of \(p\).
   (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
   Give your answers as exact values in their simplest form.
   II

7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).

1. Find each of these roots in the form \(a + b \mathrm { i }\).
2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
3. Illustrate the two roots on a single Argand diagram.
4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).

2. $$z = 6 - 6 \sqrt { 3 } i$$

1. 1. Determine the modulus of \(z\)
   2. Show that the argument of \(z\) is \(- \frac { \pi } { 3 }\) Using de Moivre's theorem, and making your method clear,
2. determine, in simplest form, \(z ^ { 4 }\)
3. Determine the values of \(w\) such that \(w ^ { 2 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.

4. (a) Express the complex number \(18 \sqrt { 3 } - 18 \mathrm { i }\) in the form $$r ( \cos \theta + \mathrm { i } \sin \theta ) \quad - \pi < \theta \leqslant \pi$$ (b) Solve the equation $$z ^ { 4 } = 18 \sqrt { 3 } - 18 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(- \pi < \theta \leqslant \pi\)

9. $$z = 4 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right) , \text { and } \boldsymbol { w } = 3 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$$ Express zw in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).

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

1. \(z ^ { 2 }\),
   (2)
2. \(\frac { 1 } { z }\).
   (2)
3. Find the exact value of each of \(| z | , \left| z ^ { 2 } \right|\) 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.
4. Show the points \(A , B , C\) and \(D\) on an Argand diagram.
5. Prove that \(\triangle O A B\) is similar to \(\triangle O C D\).

4. $$z = - 8 + ( 8 \sqrt { } 3 ) i$$

1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 3 }\),
3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

3. (a) Express the complex number \(- 2 + ( 2 \sqrt { 3 } ) \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).
(b) Solve the equation $$z ^ { 4 } = - 2 + ( 2 \sqrt { } 3 ) i$$ giving the roots in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , - \pi < \theta \leqslant \pi\).

6. Solve the equation $$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$ giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).

2. $$z = 5 \sqrt { } 3 - 5 i$$ Find

1. \(| z |\),
2. \(\arg ( z )\), in terms of \(\pi\). $$w = 2 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$ Find
3. \(\left| \frac { w } { z } \right|\),
4. \(\quad \arg \left( \frac { w } { z } \right)\), in terms of \(\pi\).

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

1. Find the modulus and the argument of \(z\). Using de Moivre's theorem,
2. find \(z ^ { 6 }\), simplifying your answer,
3. find the values of \(w\) such that \(w ^ { 4 } = z ^ { 3 }\), giving your answers in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\).

8. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 3 \mathrm { i } } { 1 + \mathrm { i } z } , \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 1\) in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.

1. Find a cartesian equation of the line \(l\). The circle \(| z - a - b \mathrm { i } | = c\) in the \(z\)-plane is mapped by \(T\) onto the circle \(| w | = 5\) in the \(w\)-plane.
2. Find the exact values of the real constants \(a\), \(b\) and \(c\).
   

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