Single locus sketching

4 The complex number \(u\) is defined by \(u = \frac { ( 1 + 2 \mathrm { i } ) ^ { 2 } } { 2 + \mathrm { i } }\).

1. Without using a calculator and showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Sketch an Argand diagram showing the locus of the complex number \(z\) such that \(| z - u | = | u |\).

7 The equation \(2 x ^ { 3 } + x ^ { 2 } + 25 = 0\) has one real root and two complex roots.

1. Verify that \(1 + 2 \mathrm { i }\) is one of the complex roots.
2. Write down the other complex root of the equation.
3. Sketch an Argand diagram showing the point representing the complex number \(1 + 2 \mathrm { i }\). Show on the same diagram the set of points representing the complex numbers \(z\) which satisfy $$| z | = | z - 1 - 2 \mathrm { i } |$$

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}

7

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

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) \(\arg ( z - 3 ) = \arg a\).

8 It is given that \(m = - 4 + 2 \mathrm { j }\).

1. Express \(\frac { 1 } { m }\) in the form \(a + b \mathrm { j }\).
2. Express \(m\) in modulus-argument form.
3. Represent the following loci on separate Argand diagrams.
   (A) \(\arg ( z - m ) = \frac { \pi } { 4 }\) (B) \(0 < \arg ( z - m ) < \frac { \pi } { 4 }\)

9

1. Sketch on the Argand diagram below, the locus of points satisfying the equation \(| z - 2 | = 2\) \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-14_1417_1475_790_399} 9
2. Given that \(| z - 2 | = 2\) and \(\arg ( z - 2 ) = - \frac { \pi } { 3 }\), express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
   [0pt] [3 marks]

2
3 2 The diagram below shows a locus on an Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-02_855_962_1085_539} Which of the equations below represents the locus shown above?
Circle your answer.
[0pt] [1 mark] \(| z - 2 + 3 \mathrm { i } | = 2\) \(| z + 2 - 3 \mathrm { i } | = 2\) \(| z - 2 + 3 \mathrm { i } | = 4\) \(| z + 2 - 3 \mathrm { i } | = 4\)