Single locus sketching

A question is this type if and only if it asks to sketch a single locus defined by a modulus equation (circle) or argument equation (half-line/ray) on an Argand diagram.

8 questions · Moderate -0.4

4.02k Argand diagrams: geometric interpretation
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CAIE P3 2012 June Q4
7 marks Moderate -0.3
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 \(| z - u | = | u |\).
CAIE P3 2005 November Q7
8 marks Moderate -0.3
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 $$| z | = | z - 1 - 2 \mathrm { i } |$$
CAIE P3 2024 November Q1
4 marks Moderate -0.3
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}
OCR FP1 2006 January Q7
10 marks Moderate -0.8
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 \(| z + 1 | = | z |\).
OCR FP1 2008 June Q2
7 marks Standard +0.3
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. \(\arg ( z - 3 ) = \arg a\).
OCR MEI FP1 2007 January Q8
11 marks Moderate -0.3
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) \(0 < \arg ( z - m ) < \frac { \pi } { 4 }\)
AQA Further AS Paper 1 Specimen Q9
3 marks Moderate -0.5
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]
Pre-U Pre-U 9794/2 2016 June Q4
6 marks Moderate -0.8
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]