4.02l Geometrical effects: conjugate, addition, subtraction

5 The complex number 2 i is denoted by \(u\). The complex number with modulus 1 and argument \(\frac { 2 } { 3 } \pi\) is denoted by \(w\).

1. Find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(w , u w\) and \(\frac { u } { w }\).
2. Sketch an Argand diagram showing the points \(U , A\) and \(B\) representing the complex numbers \(u\), \(u w\) and \(\frac { u } { w }\) respectively.
3. Prove that triangle \(U A B\) is equilateral.

7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).

1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
   In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).

11 Throughout this question the use of a calculator is not permitted.

1. The complex numbers \(z\) and \(w\) satisfy the equations $$z + ( 1 + \mathrm { i } ) w = \mathrm { i } \quad \text { and } \quad ( 1 - \mathrm { i } ) z + \mathrm { i } w = 1$$ Solve the equations for \(z\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. The complex numbers \(u\) and \(v\) are given by \(u = 1 + ( 2 \sqrt { 3 } ) \mathrm { i }\) and \(v = 3 + 2 \mathrm { i }\). In an Argand diagram, \(u\) and \(v\) are represented by the points \(A\) and \(B\). A third point \(C\) lies in the first quadrant and is such that \(B C = 2 A B\) and angle \(A B C = 90 ^ { \circ }\). Find the complex number \(z\) represented by \(C\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.

8 Throughout this question the use of a calculator is not permitted.
The complex number \(u\) is defined by $$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. Find the exact modulus and argument of \(u\).
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | < 2\) and \(| z - u | < | z |\).

10

1. Find the complex number \(z\) satisfying the equation \(z ^ { * } + 1 = 2 \mathrm { i } z\), where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z + 1 - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
   2. Determine the difference between the greatest and least values of \(\arg z\) for points lying in this region.

7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find, in the form \(x + \mathrm { i } y\), the complex numbers
   1. \(u + v\),
   2. \(\frac { u } { v }\), showing all your working.
   3. State the argument of \(\frac { u } { v }\). In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
   4. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
   5. State fully the geometrical relationship between the line segments \(O A\) and \(B C\).

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).

1. On the Argand diagram below, sketch the locus of the points representing \(z\).
2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}

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

$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$ The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).

7. A transformation from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { ( 1 + \mathrm { i } ) z + 2 ( 1 - \mathrm { i } ) } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps points on the imaginary axis in the \(z\)-plane onto a line in the \(w\)-plane.

1. Find an equation for this line. The transformation maps points on the real axis in the \(z\)-plane onto a circle in the \(w\)-plane.
2. Find the centre and radius of this circle.

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

Given that $$| z - 2 i | = | z - 3 |$$

1. sketch the locus of \(P\). You do not need to find the coordinates of any intercepts. The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { \mathrm { i } z } { z - 2 \mathrm { i } } \quad z \neq 2 \mathrm { i }$$ Given that \(T\) maps \(| z - 2 i | = | z - 3 |\) to a circle \(C\) in the \(w\)-plane,
2. find the equation of \(C\), giving your answer in the form $$| w - ( p + q \mathrm { i } ) | = r$$ where \(p , q\) and \(r\) are real numbers to be determined.

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 \(R\)

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 \(R\).

2. 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 $$w = \frac { z + 2 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation \(T\) maps the circle \(| z | = 2\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane.
Find (i) the centre of \(C\),
(ii) the radius of \(C\).

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

$$w = \frac { z } { z + 4 \mathrm { i } } \quad z \neq - 4 \mathrm { i }$$ The circle with equation \(| z | = 3\) is mapped by \(T\) onto the circle \(C\) Determine

1. a Cartesian equation of \(C\)
2. the centre and radius of \(C\)

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

$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.

1. Show that \(C\) has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where \(k\) is a constant to be determined.
2. Hence determine
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\)

12. The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$

1. Show that \(T\) maps points on the half-line \(\arg ( z ) = \frac { \pi } { 4 }\) in the \(z\)-plane into points on the circle \(| w | = 1\) in the \(w\)-plane.
2. Find the image under \(T\) in the \(w\)-plane of the circle \(| Z | = 1\) in the \(z\)-plane.
3. Sketch on separate diagrams the circle \(| \mathbf { Z } | = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane.
4. Mark on your sketches the point \(P\), where \(z = \mathrm { i }\), and its image \(Q\) under \(T\) in the \(w\)-plane.

9. A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$| z - 3 \mathrm { i } | = 3$$

1. sketch the locus of \(P\).
2. Find the complex number \(z\) which satisfies both \(| z - 3 i | = 3\) and \(\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 2 \mathrm { i } } { z }$$
3. Show that \(T\) maps \(| z - 3 i | = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line.
   (5)(Total 11 marks)

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 \(C\).

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.

1. (a) Show that the transformation \(T\)

$$w = \frac { z - 1 } { z + 1 }$$ maps the circle \(| z | = 1\) in the \(z\)-plane to the line \(| w - 1 | = | w + \mathrm { i } |\) in the \(w\)-plane. The transformation \(T\) maps the region \(| z | \leq 1\) in the \(z\)-plane to the region \(R\) in the \(w\)-plane.
(b) Shade the region \(R\) on an Argand diagram.

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 \(R\).

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7

1. Use de Moivre's theorem to prove that $$\tan 3 \theta \equiv \frac { \tan \theta \left( 3 - \tan ^ { 2 } \theta \right) } { 1 - 3 \tan ^ { 2 } \theta } .$$
2. (a) By putting \(\theta = \frac { 1 } { 12 } \pi\) in the identity in part (i), show that \(\tan \frac { 1 } { 12 } \pi\) is a solution of the equation $$t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0 .$$ (b) Hence show that \(\tan \frac { 1 } { 12 } \pi = 2 - \sqrt { 3 }\).
3. Use the substitution \(t = \tan \theta\) to show that $$\int _ { 0 } ^ { 2 - \sqrt { 3 } } \frac { t \left( 3 - t ^ { 2 } \right) } { \left( 1 - 3 t ^ { 2 } \right) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t = a \ln b$$ where \(a\) and \(b\) are positive constants to be determined.

7 In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\).
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).

1. 1. A circle \(C\) in the complex plane is defined by the locus of points satisfying

$$| z - 3 i | = 2 | z |$$

1. Determine a Cartesian equation for \(C\), giving your answer in simplest form.
2. On an Argand diagram, shade the region defined by $$\{ z \in \mathbb { C } : | z - 3 \mathrm { i } | > 2 | z | \}$$ (ii) The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = z ^ { 3 }$$
3. Describe the geometric effect of \(T\). The region \(R\) in the \(z\)-plane is given by $$\left\{ z \in \mathbb { C } : 0 < \arg z < \frac { \pi } { 4 } \right\}$$
4. On a different Argand diagram, sketch the image of \(R\) under \(T\).