4.02m Geometrical effects: multiplication and division

10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).

1. Find the modulus and argument of \(w\).
2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.

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.

4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
   The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

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)

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

1. Given that \(| z | = 1\), sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { z + 7 \mathrm { i } } { z - 2 \mathrm { i } }$$
2. Show that \(T\) maps \(| z | = 1\) onto a circle in the \(w\)-plane.
3. Show that this circle has its centre at \(w = - 5\) and find its radius.

6. The transformation \(T\) maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation \(T\) is given by $$w = \frac { z } { i z + 1 } , \quad z \neq i$$ The transformation \(T\) maps the line \(l\) in the \(z\)-plane onto the line with equation \(v = - 1\) in the \(w\)-plane.

1. Find a cartesian equation of \(l\) in terms of \(x\) and \(y\). The transformation \(T\) maps the line with equation \(y = \frac { 1 } { 2 }\) in the \(z\)-plane onto the curve \(C\) in the \(w\)-plane.
2. 1. Show that \(C\) is a circle with centre the origin.
   2. Write down a cartesian equation of \(C\) in terms of \(u\) and \(v\).

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.

4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.

1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).

3. The complex numbers \(z\) and \(w\) are represented by the points \(Z\) and \(W\) in an Argand diagram. The complex number \(z\) is such that \(| z | = 6\) and \(\arg z = \frac { \pi } { 3 }\).
The point \(W\) is a \(90 ^ { \circ }\) clockwise rotation, about the origin, of the point \(Z\) in the Argand diagram.

1. Express \(z\) and \(w\) in the form \(x + \mathrm { i } y\).
2. Find the complex number \(\frac { z } { w }\).

5. \begin{figure}[h] \end{figure} The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)

1. Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
2. Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
3. Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\) A copy of Figure 1, labelled Diagram 1, is given on page 12.
4. Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
   \includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
   \section*{Diagram 1}

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

$$z _ { 1 } = - 4 + 4 i$$

1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
   1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
   2. \(\left( z _ { 2 } \right) ^ { 4 }\)
3. Show on a single Argand diagram
   1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
   2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)

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

$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$

1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
2. Sketch \(C\) and \(D\) on Argand diagrams.

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 \(T\).

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

$$w = z ^ { 2 }$$

1. Show that the line with equation \(\operatorname { Im } ( z ) = 1\) in the \(z\)-plane is mapped to a parabola in the \(w\)-plane, giving an equation for this parabola.
2. Sketch the parabola on an Argand diagram.

1

1. The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-02_321_592_276_347} On the copy of the diagram in the Resource Materials.
   • draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
   • indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
   • You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
   In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Resource Materials. You should join each point to the origin with a straight line.
   1. \(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
   2. \(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)

Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find

1. \(\left|\frac{z}{w}\right|\), [3]
2. \(\arg \left( \frac{z}{w} \right)\). [3]

1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
2. Show that \(\angle AOC = \angle DOB\). [2]
3. Find the area of triangle \(AOC\). [2]

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

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. [4]
2. Find the image under \(T\) in the \(w\)-plane of the circle \(|z| = 1\) in the \(z\)-plane. [6]
3. Sketch on separate diagrams the circle \(|z| = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane. [2]
4. Mark on your sketches the point \(P\), where \(z = i\), and its image \(Q\) under \(T\) in the \(w\)-plane. [2]

Given that \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{i\frac{\pi}{4}}\) state the value of \(\arg\left(\frac{z_1}{z_2}\right)\) Circle your answer. [1 mark] \(\frac{\pi}{12}\) \quad \(\frac{4}{3}\) \quad \(\frac{7\pi}{12}\) \quad \(2\)

In an Argand diagram, the points \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2i\).

1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + iy\), where \(x\) and \(y\) are real and exact. [6]

The points \(D\), \(E\) and \(F\) are the midpoints of the sides of triangle \(ABC\).

1. Find the exact area of triangle \(DEF\). [3]

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

1. Find the area of \(S\). [3]
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]

The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).

1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]