Complex Numbers Argand & Loci

4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).

2 On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \leqslant 3 , \operatorname { Re } z \geqslant - 2\) and \(\frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \pi\).

8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 marks]

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

In an Argand diagram the loci \(C_1\) and \(C_2\) are given by $$|z| = 2 \quad \text{and} \quad \arg z = \frac{1}{4}\pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
2. Hence find, in the form \(x + iy\), the complex number representing the point of intersection of \(C_1\) and \(C_2\). [2]

Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ where \(d\) is a real number.

1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C_1\) and \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + 2i}{iz}$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line. [4]

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 $$w = a z + b , \quad a , b \in \mathbb { C }$$

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

\(f(x) = \arcsin x\) State the maximum possible domain of \(f\) Tick \((\checkmark)\) one box. [1 mark] \(\{x \in \mathbb{R} : -1 \leq x \leq 1\}\) \(\left\{x \in \mathbb{R} : -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\right\}\) \(\{x \in \mathbb{R} : -\pi \leq x \leq \pi\}\) \(\{x \in \mathbb{R} : -90 \leq x \leq 90\}\)

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

1. find the modulus and argument of \(wz^2\). [6]
2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]

**In this question you must show detailed reasoning.** It is given that \(f(z) = z^3 - 13z^2 + 65z - 125\). The points representing the three roots of the equation \(f(z) = 0\) are plotted on an Argand diagram. Show that these points lie on the circle \(|z| = k\), where \(k\) is a real number to be determined. [9]

6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$

1. Find the modulus and argument of \(z\).
2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
   1. \(2 z + z ^ { * }\),
   2. \(\frac { \mathrm { i } z ^ { * } } { z }\).
   3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).

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.

1. $$z = 2 - 3 \mathrm { i }$$

1. Show that \(z ^ { 2 } = - 5 - 12 \mathrm { i }\). Find, showing your working,
2. the value of \(\left| z ^ { 2 } \right|\),
3. the value of \(\arg \left( z ^ { 2 } \right)\), giving your answer in radians to 2 decimal places.
4. Show \(z\) and \(z ^ { 2 }\) on a single Argand diagram.

1. The Argand diagram below shows the two points which represent two complex numbers, \(z_1\) and \(z_2\). \includegraphics{figure_1} On the copy of the diagram in the Printed Answer Booklet
   • 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 \(z_1 + z_2\).
   [2]
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 \(z_3z_4\) on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
   1. \(|z_3| = 1.5\) and \(|z_4| = 1.2\) [2]
   2. \(|z_3| = 0.7\) and \(|z_4| = 0.5\) [2]

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 10 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 2 \\ 4 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l1\) which is 12 metres away from \(P \cdot R\) is the point on \(l2\) such that \(QR\) is perpendicular to \(l1\).
3. Determine the length \(QR\).
   [0pt]
   5. (a) Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
   (b)
   Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
   (c) State the set of values for which the expansion in part (b) is valid.
   [0pt]
   6. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).
   (a) Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
   (b) Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
   (c) A triangle of area 4 square units is transformed by the matrix \(\mathbf { B }\). Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
   [0pt]
   7. (a) In this question you must show detailed reasoning. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
   (b) The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z | = | z - 2 i |\) and \(| z - 2 | = \sqrt { 5 }\) respectively.
   i. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\), showing any intercepts with the imaginary axis.
   ii. Indicate, by shading on your Argand diagram, the region \(\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}\).
   (c) i. Show that both of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfy \(| z - 2 | < \sqrt { 5 }\).
   ii. State, with a reason, which root of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfies \(| z | < | z - 2 i |\).
   (d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
   [0pt]
   [0pt]
   [0pt]
   [0pt]

Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}

Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}

2

1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
2. 1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
   2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]

3

1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z + 3 - 2 \mathrm { i } | = 2\).
2. Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Find the modulus and argument of \(u\).
3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.

1. (i) Given that

$$z _ { 1 } = 6 \mathrm { e } ^ { \frac { \pi } { 3 } \mathrm { i } } \text { and } z _ { 2 } = 6 \sqrt { 3 } \mathrm { e } ^ { \frac { 5 \pi } { 6 } \mathrm { i } }$$ show that $$z _ { 1 } + z _ { 2 } = 12 \mathrm { e } ^ { \frac { 2 \pi } { 3 } \mathrm { i } }$$ (ii) Given that $$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$ determine the least value of \(| z |\) as \(z\) varies.

\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]

7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).

1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
3. Describe the set of points geometrically.

1. A circle \(C\) in the complex plane has equation

$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$ where \(\alpha\) is a real constant with \(\alpha > 1\) Given that the imaginary axis is a tangent to \(C\)

1. sketch, on an Argand diagram, the circle \(C\)
2. explain why the value of \(\alpha\) is 3 The circle \(C\) is contained in the region $$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
3. Determine the maximum value of \(\beta\) Give your answer in radians to 3 significant figures.

Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(|z - 5 - 2i| \leq \sqrt{32}\) and Re (z) \(\geq\) 9. [6]

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]

1. (a) (i) Show on an Argand diagram the locus of points given by the values of \(z\) satisfying

$$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that \(\theta \in [ \alpha , \alpha + \pi ]\), where \(\alpha = - \arctan \left( \frac { 4 } { 3 } \right)\),
(ii) show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$ (b) (i) Show, by shading on your Argand diagram, the set of points \(A\).
(ii) Find the exact area of the region defined by \(A\), giving your answer in simplest form.

2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)

2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)

7

1. 1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
   2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.

6. The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that $$| z - 1 | = | z - 2 \mathrm { i } |$$ show that the locus of \(P\) is a straight line.

1. The complex number \(3 + 2i\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w^*\). Find
   1. the modulus of \(w\), [1]
   2. the argument of \(w^*\), giving your answer in radians, correct to 2 decimal places. [3]
2. Find the complex number \(u\) given that \(u + 2u^* = 3 + 2i\). [4]
3. Sketch, on an Argand diagram, the locus given by \(|z + 1| = |z|\). [2]

10

1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. 1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
   2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.

The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 + i\) and \(z_2 = 3 + 4i\).

1. Verify that \(|z_1| + |z_2| > |z_1 + z_2|\). [4]
2. Sketch on an Argand diagram the locus \(|z - z_1| = 2\). [2]

3

1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
2. Find the modulus and argument of each root.
3. Sketch an Argand diagram showing the points representing the roots.

2 In this question you must show detailed reasoning.

1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
2. Illustrate the number 25 i and its square roots on an Argand diagram.

9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\). Hence find all the roots of the equation $$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$ and deduce that none of them is real. Find the arguments of the two roots which have the smaller modulus.

Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]

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

Given that \(\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }\)

1. sketch the locus of \(P\) as \(z\) varies,
2. find the exact maximum possible value of \(| z |\)

It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).

1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]

4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).

4 Represent on an Argand diagram the region defined by \(2 < | z - ( 3 + 2 \mathrm { j } ) | \leqslant 3\).

11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).

1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
   [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
   (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
   (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
   (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).