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 \(\frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \pi\).

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1. The complex number \(w\) is such that $$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$ It is given that \(w = x + \mathrm { i } y\), where \(x\) and \(y\) are real and \(x > 0\) Find an equation for \(y\) in terms of \(x\) 8
2. The complex number \(z\) satisfies both $$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$ The region \(R\) is the locus of \(z\) Sketch the region \(R\) on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511} 8
3. \(\quad z _ { 1 }\) is the point in \(R\) at which \(| z |\) is minimum. 8
4. 1. Calculate the exact value of \(\left| z _ { 1 } \right|\) 8
5. (ii) Express \(z _ { 1 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.

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 In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).

6. The complex number \(z\) on an Argand diagram is represented by the point \(P\) where $$| z + 1 - 13 i | = 3 | z - 7 - 5 i |$$ Given that the locus of \(P\) is a circle,

1. determine the centre and radius of this circle. The complex number \(w\), on the same Argand diagram, is represented by the point \(Q\), where $$\arg ( w - 8 - 6 \mathrm { i } ) = - \frac { 3 \pi } { 4 }$$ Given that the locus of \(P\) intersects the locus of \(Q\) at the point \(R\),
2. determine the complex number representing \(R\).

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

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.

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.

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$

1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).

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 \(C\)

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 \(S\).
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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,
   (a) \(2 z + z ^ { * }\),
   (b) \(\frac { \mathrm { i } z ^ { * } } { z }\).
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 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 \(b\).

1 The complex number \(\omega\) is shown below on the Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-02_597_650_632_689} Which of the following complex numbers could be \(\omega\) ?
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \cos ( - 2 ) + i \sin ( - 2 ) \\ & \cos ( - 1 ) + i \sin ( - 1 ) \\ & \cos ( 1 ) + i \sin ( 1 ) \\ & \cos ( 2 ) + i \sin ( 2 ) \end{aligned}$$ □
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15. Given that \(z = 2 - 2 \mathrm { i }\) and \(w = - \sqrt { 3 } + \mathrm { i }\),

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

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.

7. The complex number \(z\) is represented by the point \(P ( x , y )\) in the Argand diagram and $$| z - 4 - \mathrm { i } | = | z + 2 |$$

1. Find the equation of the locus of \(P\).
2. Give a geometric interpretation of the locus of \(P\).

1. A curve has equation

$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$

1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
3. find the possible values of \(\tan \theta\)

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. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that \(\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }\)

1. determine \(\left| z _ { 2 } \right|\) Given also that \(p = - 4\)
2. determine the possible values of \(q\)
3. Show \(z _ { 1 }\) and the possible positions for \(z _ { 2 }\) on the same Argand diagram.

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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 \(/ _ { 1 }\) and \(/ _ { 2 }\). \(Q\) is a point on \(/ 1\) which is 12 metres away from \(P \cdot R\) is the point on \(/ 2\) such that \(Q R\) is perpendicular to \(/ 1\).
3. Determine the length \(Q R\).
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   5. (a) Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
4. 
   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 \(x\).
5. State the set of values for which the expansion in part (b) is valid.
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   6. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(= \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).
6. Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
7. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
8. 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.
9. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
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   7. (a) In this question you must show detailed reasoning. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
10. 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 \(\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}\).
11. 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 \(| z | < | z - 2 i |\).
12. On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
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1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z ^ { 5 } = - 16 + ( 16 \sqrt { } 3 ) i$$ giving each root in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

1. Write down the modulus and argument of the complex number \(\mathrm { e } ^ { \mathrm { j } \pi / 3 }\).
2. The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number \(a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }\) corresponds to the complex number \(b\). Show A and the two possible positions for B in a sketch. Express \(a\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the two possibilities for \(b\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
3. Given that \(z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }\), show that \(z _ { 1 } ^ { 6 } = 8\). Write down, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), the other five complex numbers \(z\) such that \(z ^ { 6 } = 8\). Sketch all six complex numbers in a new Argand diagram. Let \(w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }\).
4. Find \(w\) in the form \(x + \mathrm { j } y\), and mark this complex number on your Argand diagram.
5. Find \(w ^ { 6 }\), expressing your answer in as simple a form as possible.

1. Show that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(z \neq - 1\) then $$\frac { z - 1 } { z + 1 } = \mathrm { i } \tan \frac { 1 } { 2 } \theta$$
2. Hence, or otherwise, show that if \(z\) is a cube root of unity then $$\frac { z ^ { 3 } - 1 } { z ^ { 3 } + 1 } + \frac { z ^ { 2 } - 1 } { z ^ { 2 } + 1 } + \frac { z - 1 } { z + 1 } = 0$$
3. Hence write down three roots of the equation $$\left( z ^ { 3 } - 1 \right) \left( z ^ { 2 } + 1 \right) ( z + 1 ) + \left( z ^ { 2 } - 1 \right) \left( z ^ { 3 } + 1 \right) ( z + 1 ) + ( z - 1 ) \left( z ^ { 3 } + 1 \right) \left( z ^ { 2 } + 1 \right) = 0$$ and find the other three roots. Give your answers in an exact form.

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
   (a) \(| z - a | = | a |\),
   (b) \(\arg ( z - 3 ) = \arg a\).

2

1. On an Argand diagram draw the locus of points which satisfy \(\arg ( z - 4 \mathrm { i } ) = \frac { \pi } { 4 }\).
2. Give, in complex form, the equation of the circle which has centre at \(6 + 4 \mathrm { i }\) and touches the locus in part (i).

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.

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

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.

4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).

1. The locus of points \(z = x + \mathrm { i } y\) that satisfy

$$\arg \left( \frac { z - 8 - 5 i } { z - 2 - 5 i } \right) = \frac { \pi } { 3 }$$ is an arc of a circle \(C\).

1. On an Argand diagram sketch the locus of \(z\).
2. Explain why the centre of \(C\) has \(x\) coordinate 5
3. Determine the radius of \(C\).
4. Determine the \(y\) coordinate of the centre of \(C\).

5 The variable complex number \(z\) is given by $$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$ where \(\theta\) takes all values in the interval \(- \pi < \theta \leqslant \pi\).

1. Show that \(| z - \mathrm { i } | = 2\), for all values of \(\theta\). Hence sketch, in an Argand diagram, the locus of the point representing \(z\).
2. Prove that the real part of \(\frac { 1 } { z + 2 - \mathrm { i } }\) is constant for \(- \pi < \theta < \pi\).

9. Given that $$\frac { z - k \mathrm { i } } { z + 3 \mathrm { i } } = \mathrm { i } \text {, where } k \text { is a positive real constant }$$

1. show that \(z = - \frac { ( k + 3 ) } { 2 } + \frac { ( k - 3 ) } { 2 } \mathrm { i }\)
2. Using the printed answer in part (a),
   1. find an exact simplified value for the modulus of \(z\) when \(k = 4\)
   2. find the argument of \(z\) when \(k = 1\). Give your answer in radians to 3 decimal places, where \(- \pi < \arg z < \pi\)