4.02k Argand diagrams: geometric interpretation

2 The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   1. \(| z - a | = | a |\),
   2. \(\arg ( z - 3 ) = \arg a\).

6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).

1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
   1. the real part of \(z\) is zero,
   2. \(\quad \arg z = \arg a\).

8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).

1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).

8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
2. Find the other two roots of the equation.
3. Represent the three roots on an Argand diagram.

2

1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
2. Represent these roots on an Argand diagram.

3

1. Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
2. Show the roots on an Argand diagram.

8

1. On a single Argand diagram, sketch the locus of points for which
   (A) \(| z - 3 \mathrm { j } | = 2\),
   (B) \(\quad \arg ( z + 1 ) = \frac { 1 } { 4 } \pi\).
2. Indicate clearly on your Argand diagram the set of points for which $$| z - 3 \mathrm { j } | \leqslant 2 \quad \text { and } \quad \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi .$$
3. (A) By drawing an appropriate line through the origin, indicate on your Argand diagram the point for which \(| z - 3 j | = 2\) and \(\arg z\) has its minimum possible value.
   (B) Calculate the value of \(\arg z\) at this point.

5

1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$

2 Indicate on a single Argand diagram

1. the set of points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\),
2. the set of points for which \(\arg ( z - 2 \mathrm { j } ) = \pi\),
3. the two points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\) and \(\arg ( z - 2 \mathrm { j } ) = \pi\).

9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\). \(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.

1. Write down the other two roots.
2. Represent these four roots on an Argand diagram.
3. Find the values of \(A , B , C\) and \(D\).

2 In this question, \(\theta\) is a real number with \(0 < \theta < \frac { 1 } { 6 } \pi\), and \(w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }\).

1. State the modulus and argument of each of the complex numbers $$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$ Illustrate these three complex numbers on an Argand diagram.
2. Show that \(( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta\). Infinite series \(C\) and \(S\) are defined by $$\begin{aligned} & C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots \\ & S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots \end{aligned}$$
3. Show that \(C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }\), and find a similar expression for \(S\).

2

1. Find the 4th roots of 16j, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate the 4th roots on an Argand diagram.
2. 1. Show that \(\left( 1 - 2 \mathrm { e } ^ { \mathrm { j } \theta } \right) \left( 1 - 2 \mathrm { e } ^ { - \mathrm { j } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 2 \cos \theta + 4 \cos 2 \theta + 8 \cos 3 \theta + \ldots + 2 ^ { n } \cos n \theta \\ & S = 2 \sin \theta + 4 \sin 2 \theta + 8 \sin 3 \theta + \ldots + 2 ^ { n } \sin n \theta \end{aligned}$$
   2. Show that \(C = \frac { 2 \cos \theta - 4 - 2 ^ { n + 1 } \cos ( n + 1 ) \theta + 2 ^ { n + 2 } \cos n \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).

5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.

1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}

4. A complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that $$| z + i | = 1$$

1. sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
2. Given that \(T\) maps \(| z + i | = 1\) to a circle \(C\) in the \(w\)-plane, find a cartesian equation of \(C\).

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\ S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{gathered}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that $$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$ Find a corresponding expression for \(C\).
2. P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
   1. Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
   2. The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.

6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 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.
3. State fully the geometrical relationship between \(O C\) and \(B A\).
4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
   

2

1. Solve the equation \(z ^ { 4 } = 2 ( 1 + \mathrm { i } \sqrt { 3 } )\), giving the roots exactly in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Sketch an Argand diagram to show the lines from the origin to the point representing \(2 ( 1 + i \sqrt { 3 } )\) and from the origin to the points which represent the roots of the equation in part (i).

4 The complex numbers 0,3 and \(3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\) are represented in an Argand diagram by the points \(O , A\) and \(B\) respectively.

1. Sketch the triangle \(O A B\) and show that it is equilateral.
2. Hence express \(3 - 3 e ^ { \frac { 1 } { 3 } \pi i }\) in polar form.
3. Hence find \(\left( 3 - 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } } \right) ^ { 5 }\), giving your answer in the form \(a + b \sqrt { 3 } \mathrm { i }\) where \(a\) and \(b\) are rational numbers.

3

1. Solve the equation \(z ^ { 6 } = 1\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), and sketch an Argand diagram showing the positions of the roots.
2. Show that \(( 1 + \mathrm { i } ) ^ { 6 } = - 8 \mathrm { i }\).
3. Hence, or otherwise, solve the equation \(z ^ { 6 } + 8 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

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 \(a + b \mathrm { i }\).

2

1. 1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   2. Beginning with an expression for \(\left( z + \frac { 1 } { z } \right) ^ { 4 }\), find the constants \(A , B , C\) in the identity $$\cos ^ { 4 } \theta \equiv A + B \cos 2 \theta + C \cos 4 \theta$$
   3. Use the identity in part (ii) to obtain an expression for \(\cos 4 \theta\) as a polynomial in \(\cos \theta\).
2. 1. Given that \(z = 4 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\) and that \(w ^ { 2 } = z\), write down the possible values of \(w\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\). Show \(z\) and the possible values of \(w\) in an Argand diagram.
   2. Find the least positive integer \(n\) for which \(z ^ { n }\) is real. Show that there is no positive integer \(n\) for which \(z ^ { n }\) is imaginary.
      For each possible value of \(w\), find the value of \(w ^ { 3 }\) in the form \(a + \mathrm { j } b\) where \(a\) and \(b\) are real.

1 In this question, give all non-real numbers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 < \theta < 2 \pi\).

1. Solve \(z ^ { 5 } = 1\).
2. Hence, or otherwise, solve \(z ^ { 5 } + 32 = 0\). Sketch an Argand diagram showing the roots.

6

1. Sketch on a single Argand diagram the loci given by
   1. \(\quad | z | = | z - 8 |\),
   2. \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. Indicate by shading the region of the Argand diagram for which $$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$

6 Sketch, on a single Argand diagram, the loci given by \(| z - \sqrt { 3 } - \mathrm { i } | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).

6 The complex number \(3 - 3 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   1. \(| z - a | = 3 \sqrt { 2 }\),
   2. \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
   3. Indicate, by shading, the region of the Argand diagram for which $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$