Solving Equations with Complex Roots (Edexcel IAL Further Maths): Exam Questions

$f(z)=z^{4}+6z^{3}+76z^{2}+az+b$

where $a$ and $b$ are real constants.

Given that $-3+8i$ is a complex root of the equation $f(z)=0$

write down another complex root of this equation.

Hence, or otherwise, find the other roots of the equation $f(z)=0$

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

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

Solutions relying entirely on calculator technology are not acceptable.

$f(z)=z^{3}-13z^{2}+59z+pp\inℤ$

Given that $z=3$ is a root of the equation $f(z)=0$

show that $p=-87$

Use algebra to determine the other roots of $f(z)=0$ , giving your answers in simplest form.

On an Argand diagram

• the root $z=3$ is represented by the point $P$
• the other roots of $f(z)=0$ are represented by the points $Q$ and $R$
• the number equation$z=-9$ 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.

$f(z)=2z^{3}+pz^{2}+qz-41$

where $p$ and $q$ are integers.

The complex number $5-4i$ is a root of the equation equation$f(z)=0$

Write down another complex root of this equation.

Solve the equation $f(z)=0$ 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 $f(z)=0$ form the vertices of a triangle.

Determine the area of this triangle.

$f(z)=2z^{3}+pz^{2}+qz-41$

where $p$ and $q$ are integers.

The complex number $5-4i$ is a root of the equation equation$f(z)=0$

Write down another complex root of this equation.

Solve the equation $f(z)=0$ 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 $f(z)=0$ 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.

$f(z)=4z^{3}+pz^{2}-24z+108$

where $p$ is a constant.

Given that $-3$ is a root of the equation $f(z)=0$

determine the value of $p$

using algebra, solve equation$f(z)=0$ completely, giving the roots in simplest form,

determine the modulus of the complex roots of $f(z)=0$

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

$f(z)=2z^{4}-19z^{3}+Az^{2}+Bz-156$

where $A$and $B$are constants.

The complex number $5-i$ is a root of the equation $f(z)=0$

Write down another complex root of this equation.

Solve the equation $f(z)=0$ completely.

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

The equation

$x^{4}+Ax^{3}+Bx^{2}+Cx+225=0$

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 $x^{4}+Ax^{3}+Bx^{2}+Cx+225$

Deduce the real root of the equation.

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

$f(z)=2z^{3}--z^{2}+az+b$

where $a$ and $b$ are integers.

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

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 $f(z)=0$ on a single Argand diagram.

$f(x)=(9x^{2}+d)(x^{2}-8x+(10d+1))$

where $d$ is a positive constant.

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

Given $d=4$, express these four roots in the form $a+ib$, where $a,b\inℝ$

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 $q$.

$f(z)=z^{4}+az^{3}+bz^{2}+cz+d$

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

The complex numbers $3+i$ and $-1-2i$ are roots of the equation $f(z)=0$

Write down the other roots of this equation.

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

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