Solving Equations with Complex Roots (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that is a root of the equation

show that

Use algebra to determine the other roots of , giving your answers in simplest form.

On an Argand diagram

• the root is represented by the point

• the other roots of are represented by the points and

• the number is represented by the point

Show on a single Argand diagram the positions of , , and

Determine the perimeter of the quadrilateral , giving your answer as a simplified surd.

where and are integers.

The complex number is a root of the equation

Write down another complex root of this equation.

Solve the equation completely.

Determine the value of and the value of .

When plotted on an Argand diagram, the points representing the roots of the equation form the vertices of a triangle.

Determine the area of this triangle.

where and are integers.

The complex number is a root of the equation

Write down another complex root of this equation.

Solve the equation completely.

Determine the value of and the value of .

When plotted on an Argand diagram, the points representing the roots of the equation form the vertices of a triangle.

Determine the area of this triangle.

In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.

Given that is a root of the equation

Write down another complex root of this equation.

Use algebra to determine the other 2 roots of the equation.

Show all 4 roots on a single Argand diagram.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

where is a constant.

Given that is a root of the equation

determine the value of

using algebra, solve completely, giving the roots in simplest form,

determine the modulus of the complex roots of

show the roots of on a single Argand diagram.

where and are constants.

The complex number is a root of the equation

Write down another complex root of this equation.

Solve the equation completely.

Determine the value of and the value of .

The equation

where and are real constants, has

• a complex root

• a repeated positive real root

Write down the other complex root of this equation.

Hence determine a quadratic factor of

Deduce the real root of the equation.

Hence determine the value of each of the constants , and

where and are integers.

The complex number is a root of the equation

Write down another complex root of this equation.

Determine the value of and the value of .

Show all the roots of the equation on a single Argand diagram.

where is a positive constant.

Find the four roots of giving your answers in terms of .

Given , express these four roots in the form , where

Show these four roots on a single Argand diagram.

Given that is a root of the equation

Write down the other complex root of the equation.

Given that is also a root of the equation,

find the value of and the value of .

where , , and are integers.

The complex numbers and are roots of the equation

Write down the other roots of this equation.

Show all the roots of the equation on a single Argand diagram.

Determine the values of , , and .