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

Solve the equation $z^3=27\mathrm{i}$, giving your answers in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

It is given that $z_1=2{\mathrm{e}}^{\frac{\pi }{3}\mathrm{i}}$ and $z_2=3 \text{cis}\left(\frac{n\pi }{12}\right)$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Find the value of $z_1{z}_2$ when $n=3$.

Find the least value of $n$ such that $z_1{z}_2\in {\mathrm{ℝ}}^{+}.$

Consider the complex number $w=\frac{z_1}{z_2}$, where $z_1=3-\sqrt{3}\mathrm{i}$ and $z_2=2 \text{cis}\left(\frac{2\pi }{3}\right)$.

Express $w$ in the form $r \text{cis} \theta$.

Plot $z_1$, $z_2$ and $w$ on the Argand diagram below.

Find the smallest positive integer value of $n$ such that $w^n$ is a real number. 

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

Express $z$ in the form $r \text{cis} \theta$, where $r>0$ and $-\pi <\theta \le \pi$.

Find the three roots of the equation $z^3=-1+\sqrt{3}\mathrm{i}$, giving your answers in the form $r \text{cis} \theta$, where $r>0$ and $-\pi <\theta \le \pi$.

Consider the equation $z^4-1=15$, where $z\in \mathrm{ℂ}$.

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

Plot the roots found in part (a) on an Argand diagram.

Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.

Consider the complex numbers $w=3\left(\cos \frac{\pi }{3}-\mathrm{i}\sin \frac{\pi }{3}\right)$ and $z=3-\sqrt{3}\mathrm{i}$.

Write $w$ and $z$ in the form $r \text{cis} \theta$, where $r>0$ and $-\pi <\theta \le \pi$.

Find the modulus and argument of $zw$.

Write down the value of $zw$.

Verify that $4+2\mathrm{i}$ and $-4-2\mathrm{i}$ are the square roots of $12+16\mathrm{i}$.

Hence, or otherwise, find the two roots of the equation $w^2+4w+\left(1-4\mathrm{i}\right)=0$, where $w\in \mathrm{ℂ}$. Give your answers in the form $a+b\mathrm{i}$, where $a,b\in \mathrm{ℝ}$.

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

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

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

(ii) Find the second root of the equation, giving your answer in the form ${\omega }_2=a+{\mathrm{e}}^{\mathrm{i}\theta }$, 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 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 $r \text{cis} \theta$, where $r>0$ and $-\pi <\theta \le \pi$.

Find the exact value of $w_1$.

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

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

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

Find the distinct values of $z^n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$, giving your answers in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

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

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

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

Plot ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ on an Argand diagram.

${\omega }_1$, ${\omega }_2$ 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 $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

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

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

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 $a,b\in \mathrm{ℝ}$.

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) $\text{arg}\left(\frac{z_1}{z_2}\right)=\text{arg}\left(z_1\right)-\text{arg}\left(z_2\right)$.

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

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 $f\left(z\right)=0$, plot 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 $z\in \mathrm{ℂ}$ and $p,q\in \mathrm{ℝ}$.

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

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

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

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

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

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 $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

By first expressing $1+\sqrt{3}\mathrm{i}$ and $-1+\mathrm{i}$ in the form $r \text{cis} \theta$, 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 $z$, where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

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

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 $n\in {\mathrm{\mathbb{Z}}}^{+}$, 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 also true for $n=0$ and for all negative integers $n$.

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

Show that

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

(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

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

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

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

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

Hence show that in general

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

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) $x=0,y=0$