Complex Numbers & Argand Diagrams

1a4 marks

$f (z) = z^{4} + a z^{3} + b z^{2} + c z + d$

where $a, b, c$ and $d$ are real constants.

Given that $- 1 + 2 i$ and $3 - i$ are two roots of the equation $f (z) = 0$

Show all the roots of $f (z) = 0$ on a single Argand diagram.

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1b5 marks

Find the values of $a, b, c$ and $d .$

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3 marks

In an Argand diagram, the points $A$ and $B$ are represented by the complex numbers $- 3 + 2 i$ and $5 - 4 i$ respectively. The points $A$ and $B$ are the end points of a diameter of a circle $C$ .

(a) Find the equation of $C,$ giving your answer in the form

$| z - a | = b$ $a \in ℂ$ , $b \in ℝ$

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6 marks

The circle $D$ , with equation $| z - 2 - 3 i | = 2$ , intersects $C$ at the points representing the complex numbers $z_{1}$ and $z_{2}$

(b) Find the complex numbers $z_{1}$ and $z_{2}$

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Complex Numbers & Argand Diagrams (Edexcel A Level Further Maths: Core Pure): Exam Questions

Complex Numbers

Exponential Form & de Moivre's Theorem

Matrices

Properties of Matrices

Transformations using Matrices

Further Algebra & Functions

Roots of Polynomials

Series

Maclaurin Series

Hyperbolic Functions

Hyperbolic Functions

Further Calculus

Volumes of Revolution

Methods in Calculus

Further Vectors

Vector Lines

Vector Planes

Polar Coordinates

Polar Coordinates

Differential Equations

First Order Differential Equations

Second Order Differential Equations

Simple Harmonic Motion

Proof

Proof by Induction