Practice section 6 of 9

Consider the equation $2z^3+pz^2+qz-65=0$.

(i) If $p$ and $q$ are real numbers, what are the possibilities for real and complex roots for this equation?

It is given that $z=3+2i$ is a root of the equation $2z^3+pz^2+qz-65=0$, where $p$ and $q$ are real numbers.

(ii) Find the values of $p$ and $q$.

Do not use a calculator in answering this question.

Find the roots of the equation $w^2(1-i)+4w+(10+10i)=0$, giving your answers in cartesian form $a+ib$.

One of the roots of the equation $z^4-z^3+4z^2+3z+p=0$ is $1-2i$, where $p$ is a constant.

(i) A student says that “One of the other roots must be $1+2i$.” Explain why this statement is not entirely correct.

(ii) Determine the value of $p$.

(iii) Find the exact values of all the other roots of the equation.

Write down the real root of $w^3+8=0$, where $w$ is a complex number.

Hence, find the other roots of $w^3+8=0$ in exact form.

Deduce the roots of $w^4+w^3+8w+8=0$.

Determine the roots of $z$ such that $z^4-iz^3+8iz+8=0$.

The point $A$ represents a fixed complex number $a$ where $\pi/4<\arg(a)<\pi/2$. The complex numbers $-ia$ and $-a$ are represented by the points $B$ and $C$ respectively.

On a single clearly labelled Argand diagram, mark the points $A$, $B$ and $C$.

Describe the geometrical relationship between

(a) $A$ and $B$, and

(b) $A$ and $C$.

Let $z=-1-\sqrt3i$.

(i) State the value of $|-1-\sqrt3i|$ and $\arg(-1-\sqrt3i)$.

(ii) Give the geometrical meanings of the statements “$|-1-\sqrt3i|$” and “$\arg(-1-\sqrt3i)$”.