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

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

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.

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.

1. A 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

$$w = \frac { z - 3 } { 2 \mathrm { i } - z } \quad z \neq 2 \mathrm { i }$$ The line in the \(z\)-plane with equation \(y = x + 3\) is mapped by \(T\) onto a circle \(C\) in the \(w\)-plane.

1. Determine
   1. the coordinates of the centre of \(C\)
   2. the exact radius of \(C\) The region \(y > x + 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. On a single Argand diagram
   1. sketch the circle \(C\)
   2. shade and label the region \(R\)

4. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z } { z + 3 } , \quad z \neq - 3$$ Under this transformation, the circle \(| z | = 2\) in the \(z\)-plane is mapped onto a circle \(C\) in the \(w\)-plane. Determine the centre and the radius of the circle \(C\).

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z - 1 } { z + 1 } , \quad z \neq - 1$$ The line in the \(z\)-plane with equation \(y = 2 x\) is mapped by \(T\) onto the curve \(C\) in the \(w\)-plane.

1. Show that \(C\) is a circle and find its centre and radius. The region \(y < 2 x\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Sketch circle \(C\) on an Argand diagram and shade and label region \(R\).

1. The complex number \(z = x + i y\) satisfies the equation

$$| z - 3 - 4 i | = | z + 1 + i |$$

1. Determine an equation for the locus of \(z\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Shade, on an Argand diagram, the region defined by $$| z - 3 - 4 i | \leqslant | z + 1 + i |$$ You do not need to determine the coordinates of any intercepts on the coordinate axes.

1. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z - \mathrm { i } } { z + 1 } \quad z \neq - 1$$ Given that \(T\) maps the imaginary axis in the \(z\)-plane to the circle \(C\) in the \(w\)-plane, determine (i) the coordinates of the centre of \(C\) (ii) the radius of \(C\)

9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$

1. find a cartesian equation for the locus of \(P\), simplifying your answer.
2. sketch the locus of \(P\).
   (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11\) i followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$

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 \(\triangle O C D\).

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 \(C\).

1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by

$$w = \frac { z } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. The region \(| z | < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
2. Shade the region \(R\) on an Argand diagram.

6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.

1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
3. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.

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 \(\pi\).

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 \(c\).
   

   END

2. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 1 - \mathrm { i } z } { z } , \quad z \neq 0$$ The transformation maps points on the real axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Find an equation of the line \(l\).

3. A transformation maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation is given by $$w = \frac { ( 2 + \mathrm { i } ) z + 4 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps the imaginary axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Determine a Cartesian equation of \(l\), giving your answer in the form \(a u + b v + c = 0\) where \(a , b\) and \(c\) are integers to be found.
(6)

3

1. Solve the equation \(z ^ { 2 } - 6 z + 36 = 0\), and give your answers in the form \(r ( \cos \theta \pm \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta \leqslant \pi\).
2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).

7

1. The complex number \(3 + 2 \mathrm { i }\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w ^ { * }\). Find
   1. the modulus of \(w\),
   2. the argument of \(w ^ { * }\), giving your answer in radians, correct to 2 decimal places.
2. Find the complex number \(u\) given that \(u + 2 u ^ { * } = 3 + 2 \mathrm { i }\).
3. Sketch, on an Argand diagram, the locus given by \(| z + 1 | = | z |\).

1 The complex number \(a + \mathrm { i } b\) is denoted by \(z\). Given that \(| z | = 4\) and \(\arg z = \frac { 1 } { 3 } \pi\), find \(a\) and \(b\).