Further Complex Numbers (DP IB Analysis & Approaches (AA): HL): Exam Questions

Consider the equation $z^2+pz-2p-1=0,$ where $z\in \mathrm{ℂ},p\in \mathrm{ℝ}$.

Find the value of $p$ for which one of the two distinct roots is $z_1=2+\sqrt{3}\mathrm{i}$.

Find the range of values of $p$ for which the equation has two distinct, real roots.

Consider the complex number $\omega =-1+4\mathrm{i}$ .

Show that  $\omega$ is a root of the cubic equation

$z^3+5z^2+23z+51=0$

Find the other two roots of the cubic equation in part (a).

Consider $z=\mathrm{cis}\theta$ where $z\in \mathrm{ℂ},z\ne 1$.

Show that $\mathrm{Re}\left(\frac{1+\mathrm{z}}{1-\mathrm{z}}\right)=0$.

Consider the equation ${\left(z-2\right)}^2=\mathrm{i},\mathrm{z}\in \mathrm{ℂ}$.

(i) Verify that ${\omega }_1=2+e^{\mathrm{i}\frac{\mathrm{\pi }}{4}}$ is a root of this equation.

(ii) Find the second root of the equation, expressing your answer in the form ${\omega }_2=a+e^{\mathrm{i}\theta }$ where  $a\in \mathrm{ℝ}$ and  $\theta >0$.

The roots ${\omega }_1$ and ${\omega }_2$ are represented by the points A and B respectively on an Argand diagram.

Find AB.

Consider the equation $z^4+\left(1-4\mathrm{i}\right)z^2-4\mathrm{i}=0$, where $z\in \mathrm{ℂ}$.

Find the four distinct roots of the equation, giving your answers in the form $a+b\mathrm{i}$ where $a,b\in \mathrm{ℝ}$.

Consider the complex numbers $w_1=\frac{z_1}{z_2},z_1=\frac{\sqrt{2}{\mathrm{e}}^{-\frac{\pi }{3}\mathrm{i}}}{3}$ and  $z_2=2-2\sqrt{3}\mathrm{i}$.

Express

(i) $z_1$ in the form $a+b\mathrm{i}$

(ii) $z_2$ in the form $r\mathrm{cis}\theta$, where $r>0$ and $-\mathrm{\pi }<\theta <\mathrm{\pi }$.

Find the exact value of $w_1$.

Find $w_2=z_1{z}_2$, giving your answer in the form $r\mathrm{cis}\theta$, where $r>0$ and $-\pi <\theta <\pi$.

Without drawing an Argand diagram, describe the geometrical relationship between $z_1$ and $z_2$.

$z=\frac{\sqrt{3}}{2}\mathrm{i}-\frac{1}{2}$

Find all the powers $z^n$ .

Find the area of the shape made by the powers $z^n$ when plotted on an Argand diagram.

Let $z=\cos \theta +\mathrm{i}\sin \theta$.

Write down the value of $z{z}^{*}$.

Let $z_1=r_1\left(\cos {\theta }_1+\mathrm{i}\sin {\theta }_1\right)and{z}_2=r_2\left(\cos {\theta }_2+\mathrm{i}\sin {\theta }_2\right)$.

Prove the results

(i) $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

(ii) $\arg \frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}=\arg z_1-\arg z_2$

Using the results from part (b), describe fully the geometrical interpretation of dividing $z_1$ by $z_2$.

Consider the equation $z^3+27=0$ where $z\in \mathrm{ℂ}$. The complex numbers ${\omega }_1,{\omega }_2$ and ${\omega }_3$ are three distinct roots of the equation.

Find ${\omega }_1,{\omega }_2$ and ${\omega }_3$ giving your answers in the form $r\text{cis}\theta$.

Sketch ${\omega }_1,{\omega }_2\mathrm{and}{\omega }_3$ on the Argand diagram below.

${\omega }_1,{\omega }_2$and ${\omega }_3$ represent the vertices of a triangle.

Find the area of the triangle.

The complex numbers $z_1=a,z_2=3-2\mathrm{i}$ and $z_3$ are roots of the cubic equation $z^3+p{z}^2+qz-26=0,$ where $a,p,q\in \mathrm{ℝ}$. 

Find the values of $a,p$ and $q$.

Express $z_1,z_2$ and $z_3$ in the form $r{e}^{\mathrm{i}\theta }$, where $r>0$ and $0<\theta \le 2\mathrm{\pi }$.

Let  $f\left(z\right)=z^4+a{z}^3+6z^2+bz+65$,  where $a$ and $b$ are real constants.

Given that  $z=1+2\mathrm{i}$  is a root of the equation $f\left(z\right)=0$,  show the roots of $f\left(z\right)=0$ on the Argand diagram below.

Consider the equation  $p{z}^3+q{z}^2+8p^{3}z+5q=0$,  where $z\in \mathrm{ℂ},p,q\in \mathrm{ℝ}$. 

Given that one of the distinct roots is $z_1=\frac{5}{2}$, find

(i) a pair of possible integer values for $p$ and $q$

(ii) the other two roots of the equation, $z_2$ and $z_3$, giving your answers in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

On an Argand diagram $z_1,z_2$ and $z_3$ are represented by the points A, B and C respectively.

Find the area of the triangle ABC.

Consider the complex numbers $z$ and $w$, where $z=\sqrt{3}-\mathrm{i}$, $\mathrm{Im}\left(\frac{{\mathrm{z}}^2}{w}\right)=0,\left|\frac{{\mathrm{z}}^2}{w}\right|=\frac{1}{2}\left|z\right|$. 

Use geometrical reasoning to find the two possibilities for $w$, giving your answers in exponential form.

Consider the equation $z^4-5a{z}^3+25a{z}^2-20abz+24ab=0$,  where $a,b\in \mathrm{\mathbb{Z}}$ and  $z\in \mathrm{ℂ}$. 

Given that one root is $a+a\mathrm{i}$ and another root is $b+b\mathrm{i}$, find the possible values of $a$and $b$.

Consider the complex number $z=1+\sqrt{3}\mathrm{i}$.

Use De Moivre’s theorem to find the value of $z^3$.

Use the principle of mathematical induction to prove, for all $n\in \mathrm{\mathbb{N}}$, that

${\left(\cos \theta +\mathrm{i}\sin \theta \right)}^n=\cos n\theta +\mathrm{i}\sin n\theta$

Show that the result in part (b) is true for all $n\in \mathrm{\mathbb{Z}}.$

Consider the equation ${\left(z+a\right)}^5+1=0,z\in \mathrm{ℂ}$.

Given that the product of the roots is 31, find the roots of the equation, expressing your answers in the form ${\omega }_n=b+e^{\mathrm{i}\theta }$, where $b\in \mathrm{ℝ}$ and $\theta >0$.

Let S be the sum of the roots found in part (a).

Show that $\mathrm{Im}\left(\mathrm{S}\right)=0$ and find the value of $\mathrm{Re}\left(\mathrm{S}\right)$ .

The roots ${\omega }_1,{\omega }_2,...,{\omega }_5$ are represented on an Argand diagram. 

Describe the geometrical shape made by the five roots.

Consider the equations $u^{*}+2v=2\mathrm{i}$ and $\mathrm{i}u+v^{*}=3$, where $u,v\in \mathrm{ℂ}$.  Find $\frac{u}{v}$ giving your answer in the form $r{e}^{\mathrm{i}\theta }$, where $r>0$ and $0<\theta <2\pi$.

By first expressing $1+\sqrt{3}\mathrm{i}$ and $-1+\mathrm{i}$ in the form $r\mathrm{cis}\theta$ where $r>0$ and $-\pi <\theta \le \pi$, show that $\tan \frac{5\pi }{12}=2+\sqrt{3}$.

Consider the complex number $z=1+\cos 2\theta +\mathrm{i}\sin 2\theta$.

Find the modulus and argument of z.

Solve $z=0$ for $-\pi <\theta \le \pi$.

Let $z={\mathrm{e}}^{\mathrm{i}\theta }$.

Show that

(i) $z+z^2+z^3+\cdots +z^n=\cos \theta +\cos 2\theta +\cdots +\cos n\theta +\mathrm{i}\left(\sin \theta +\sin 2\theta +\cdots +\sin n\theta \right)$,

(ii) $\left(2-z\right)\left(2-z^{*}\right)=5-4\cos \theta$.

Use the results found in part (a) to find the sum of the infinite series

$\frac{\sin \theta }{2}+\frac{\sin 2\theta }{2^2}+\frac{\sin 3\theta }{2^3}+\frac{\sin 4\theta }{2^4}+\cdots$

The primary square root of a complex number $z$ is defined as $\sqrt{z}=x+\mathrm{i}y$, where $x,y\in \mathrm{ℝ}$ and $x\ge 0$.  If $x=0$ then the value for $y$ is chosen such that $y\ge 0$.  Note that the other square root of $z$ will then be given by $-\sqrt{z}=-x-\mathrm{i}y$.

Show that

$x=\sqrt{\frac{\mathrm{Re}\left(z\right)+\sqrt{{\left(\mathrm{Re}\left(z\right)\right)}^2+{\left(\mathrm{Im}\left(z\right)\right)}^2}}{2}}$

Given that $x>0$, derive a formula for $y$ in terms of $x$and $\mathrm{Im}\left(z\right)$, and explain why $y$ in this case will always have the same sign (positive, negative, or zero) as $\mathrm{Im}\left(z\right)$.

Hence show that in general

$y=\pm \sqrt{\frac{-\mathrm{Re}\left(z\right)+\sqrt{{\left(\mathrm{Re}\left(z\right)\right)}^2+{\left(\mathrm{Im}\left(z\right)\right)}^2}}{2}}$

with the choice of the positive or negative value being dependent on the properties of $z$.

Explain what must be true of $z$ for each of the following to be true:

(i) $x=0,y\ne 0$

(ii) $x\ne 0,y=0$

(iii) $x=0,y=0$