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

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.

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.

9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),

1. find \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
4. the complex number represented by C,
5. the exact value of the radius of the circle.

9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),

1. find \(z _ { 2 }\) in the form \(a + i b\), where \(a\) and \(b\) are real.
2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
4. the complex number represented by C,
5. the exact value of the radius of the circle.

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. $$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z ^ { 2 }\),
2. \(\frac { z } { w }\).

2. $$z = \frac { 50 } { 3 + 4 \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\),

1. \(z\),
2. \(z ^ { 2 }\). Find
3. \(| z |\),
4. \(\arg z ^ { 2 }\), giving your answer in degrees to 1 decimal place.

5. $$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$

1. Find the value of \(| w |\). Find in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real constants,
2. \(z w\), showing clearly how you obtained your answer,
3. \(\frac { z } { w }\), showing clearly how you obtained your answer. Given that $$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
4. find the value of \(\lambda\).

9. Given that \(z = x + \mathrm { i } y\), where \(x \in \mathbb { R } , y \in \mathbb { R }\), find the value of \(x\) and the value of \(y\) such that $$( 3 - i ) z ^ { * } + 2 i z = 9 - i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

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.

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 \(\lambda\).

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.

4. The complex number \(z\) is given by $$z = \frac { p + 2 \mathrm { i } } { 3 + p \mathrm { i } }$$ where \(p\) is an integer.

1. Express \(z\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\).
2. Given that \(\arg ( z ) = \theta\), where \(\tan \theta = 1\) find the possible values of \(p\).

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

$$z _ { 1 } = p + 2 i \text { and } z _ { 2 } = 1 - 2 i$$ where \(p\) is an integer.

1. Find \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b\) i where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right| = 13\),
2. find the possible values of \(p\).

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.

4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)

1. \(Z\)
2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
3. find the value of \(p\) and the value of \(q\).

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

1. 1. Given that

$$\frac { 3 w + 7 } { 5 } = \frac { p - 4 \mathrm { i } } { 3 - \mathrm { i } } \quad \text { where } p \text { is a real constant }$$

1. express \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that arg \(w = - \frac { \pi } { 2 }\)
2. find the value of \(p\).
   (ii) Given that $$( z + 1 - 2 i ) ^ { * } = 4 i z$$ find \(z\), giving your answer in the form \(z = x + i y\), where \(x\) and \(y\) are real constants. \includegraphics[max width=\textwidth, alt={}, center]{89f82cd3-9afa-4431-bc74-a073909c903f-36_106_129_2469_1816}

3. \(z = 1 + \mathrm { i } \sqrt { 3 }\) Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.

1. \(z ^ { 2 } + z\),
2. \(\frac { z } { 3 - z }\),
   giving the exact values of \(a\) and \(b\) in each part.

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.

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 \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).

1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by

$$w = \frac { z + \mathrm { i } } { \mathrm { z } } , \quad z \neq 0 .$$

1. The transformation \(T\) maps the points on the line with equation \(y = x\) in the \(z\)-plane, other than \(( 0,0 )\), to points on a line \(l\) in the \(w\)-plane. Find a cartesian equation of \(l\).
2. Show that the image, under \(T\), of the line with equation \(x + y + 1 = 0\) in the \(z\)-plane is a circle \(C\) in the \(w\)-plane, where \(C\) has cartesian equation $$u ^ { 2 } + v ^ { 2 } - u + v = 0$$
3. On the same Argand diagram, sketch \(l\) and \(C\).