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.

$\mathrm{f}\left(z\right)=z^3-13z^2+59z+pp\in \mathrm{\mathbb{Z}}$

Given that $z=3$ is a root of the equation $\mathrm{f}\left(z\right)=0$

show that $p=-87$

Use algebra to determine the other roots of $\mathrm{f}\left(z\right)=0$ , giving your answers in simplest form.

On an Argand diagram

• the root $z=3$ is represented by the point $P$

• the other roots of $\mathrm{f}\left(z\right)=0$ are represented by the points $Q$ and $R$

• the number $z=-9$ is represented by the point $S$

Show on a single Argand diagram the positions of $P$, $Q$ , $R$ and $S$

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

$\mathrm{f}\left(z\right)=2z^3+p{z}^2+qz-41$

where $p$ and $q$ are integers.

The complex number $5-4\mathrm{i}$ is a root of the equation $\mathrm{f}\left(\mathrm{z}\right)=0$

Write down another complex root of this equation.

Solve the equation $\mathrm{f}\left(\mathrm{z}\right)=0$ completely.

Determine the value of $p$ and the value of $q$.

When plotted on an Argand diagram, the points representing the roots of the equation $\mathrm{f}\left(z\right)=0$ form the vertices of a triangle.

Determine the area of this triangle.

$\mathrm{f}\left(z\right)=2z^3+p{z}^2+qz-41$

where $p$ and $q$ are integers.

The complex number $5-4\mathrm{i}$ is a root of the equation $\mathrm{f}\left(\mathrm{z}\right)=0$

Write down another complex root of this equation.

Solve the equation $\mathrm{f}\left(\mathrm{z}\right)=0$ completely.

Determine the value of $p$ and the value of $q$.

When plotted on an Argand diagram, the points representing the roots of the equation $\mathrm{f}\left(z\right)=0$ 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 $x=2+3i$ is a root of the equation

$2x^4-8x^3+29x^2-12x+39=0$

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.

$\mathrm{f}\left(z\right)=4z^3+p{z}^2-24z+108$

where $p$ is a constant.

Given that $-3$ is a root of the equation $\mathrm{f}\left(z\right)=0$

determine the value of $p$

using algebra, solve $\mathrm{f}\left(z\right)=0$ completely, giving the roots in simplest form,

determine the modulus of the complex roots of $\mathrm{f}\left(z\right)=0$

show the roots of $\mathrm{f}\left(z\right)=0$ on a single Argand diagram.

$\mathrm{f}\left(z\right)=2z^4-19z^3+A{z}^2+Bz-156$

where $A$and $B$are constants.

The complex number $5-\mathrm{i}$ is a root of the equation $\mathrm{f}\left(z\right)=0$

Write down another complex root of this equation.

Solve the equation $\mathrm{f}\left(z\right)=0$ completely.

Determine the value of $A$ and the value of $B$.

The equation

$x^4+A{x}^3+B{x}^2+Cx+225=0$

where $A,B$ and $C$ are real constants, has

• a complex root $4+3i$

• a repeated positive real root

Write down the other complex root of this equation.

Hence determine a quadratic factor of $x^4+A{x}^3+B{x}^2+Cx+225$

Deduce the real root of the equation.

Hence determine the value of each of the constants $A$, $B$ and $C$

$f\left(z\right)=2z^3–z^2+az+b$

where $a$ and $b$ are integers.

The complex number $-1-3i$ is a root of the equation $f\left(z\right)=0$

Write down another complex root of this equation.

Determine the value of $a$ and the value of $b$.

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

$f\left(x\right)=(9x^2+d)(x^2-8x+(10d+1\left)\right)$

where $d$ is a positive constant.

Find the four roots of $f\left(x\right)$ giving your answers in terms of $d$.

Given $d=4$, express these four roots in the form $a+ib$, where $a,b\in \mathrm{ℝ}$

Show these four roots on a single Argand diagram.

Given that $x=\frac{3}{8}+\frac{\sqrt{71}}{8} i$ is a root of the equation

$4x^3-19x^2+px+q=0$

Write down the other complex root of the equation.

Given that $x=4$ is also a root of the equation,

find the value of $p$ and the value of $q$.

$\mathrm{f}\left(z\right)=z^4+a{z}^3+b{z}^2+cz+d$

where $a$, $b$, $c$ and $d$ are integers.

The complex numbers $3+\mathrm{i}$ and $-1-2\mathrm{i}$ are roots of the equation $\mathrm{f}\left(z\right)=0$

Write down the other roots of this equation.

Show all the roots of the equation $\mathrm{f}\left(z\right)=0$ on a single Argand diagram.

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