Complex number loci on Argand diagrams

A question is this type if and only if it asks to sketch or describe loci such as |z - a| = r, arg(z - a) = θ, or Re(z) = k on an Argand diagram, possibly finding intersections.

3 questions · Standard +0.3

4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
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CAIE P3 2015 November Q9
10 marks Standard +0.3
9
  1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
  2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE P3 2020 June Q10
12 marks Standard +0.3
10
  1. The complex number \(u\) is defined by \(u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }\), where \(a\) is real.
    1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
    2. Find the exact value of \(a\) for which \(\arg u ^ { * } = \frac { 1 } { 3 } \pi\).
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |\) and \(| z - 2 - \mathbf { i } | \leqslant 2\).
    2. Calculate the least value of \(\arg z\) for points in this region.
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OCR MEI FP1 2005 January Q8
12 marks Standard +0.3
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 \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).