Further Complex Numbers (DP IB Applications & Interpretation (AI): 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.

Let $w=2-\mathrm{iz}$, where $w,z\in \mathrm{ℂ}$.

Find $w$ when

(i) $z=2e^{\frac{\mathrm{\pi }}{2}i}$

(ii) $z=\sqrt{2}{e}^{\frac{\mathrm{\pi }}{4}\mathrm{i}}$

On an Argand diagram the point $z$ can be transformed to the point $w$ by two transformations. Describe the two transformations and the order in which they are applied.

Hence, or otherwise, find the value of $z$ when $w=1+\mathrm{i}$.

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

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

It is given that $z_1=2\mathrm{cis}\left(\frac{\mathrm{\pi }}{4}\right)$ and $z_2=\sqrt{2}\mathrm{cis}\left(\frac{\mathrm{n\pi }}{12}\right),\mathrm{n}\in {\mathrm{\mathbb{Z}}}^{+}.$

Giving your answers in the form $r{e}^{i\theta },r\ge 0,-\pi <\theta \le \pi$, use technology to find the values of

(i) ${z_1}^3$

(ii) ${\left(z_1{z}_2\right)}^3$, for $n=2$.

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

Let $z=1+\mathrm{i}.$

Express $z$ in the form $z=a{e}^{\mathrm{i}b}$, where $a,b\in \mathrm{ℝ}$ , giving the exact values of $a$ and $b$.

Let $w_1=e^{ix}$ and $w_2=z{w}_1$.

i) Write $w_1+w_2$ in the form $e^{\mathrm{i}x}\left(c+id\right)$.

ii) Hence, find $Re\left(w_1+w_2\right)$  in the form $A\cos \left(x+a\right)$,  giving the exact value of $A$ ,where $A>0$ and $0<\alpha <\frac{\mathrm{\pi }}{2}$.

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

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

Use technology to find all the powers $z^n.$

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

Give your answer as an exact value.

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

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

Let $z_1=r_1{e}^{\mathrm{i}\theta }$ and $z_2={r_2}^{\mathrm{i}\left(\theta +\frac{\mathrm{\pi }}{2}\right)}$

Prove the results

(i) $\mathrm{Re}\left(z_1+z_2\right)=r_1\cos \theta -r_2\sin \theta$

(ii) $\mathrm{Im}\left(z_1+z_2\right)=r_1\sin \theta +r_2\cos \theta$ 

Using technology, or otherwise, show that

$\mathrm{Re}(2e^{\mathrm{i}5x}+6e^{\mathrm{i}\left(5x+1\right)}=7.28\cos \left(0.77+5x\right)$

The current, $I$, in an AC circuit can be modelled by the equation $I=a\cos \left(bt-c\right)$ where $b$ is the frequency and $c$ is the phase shift. 

Two AC voltage sources of the same frequency generate currents $I_A=12\cos \left(bt\right)$ and $I_B=15\cos \left(bt-\frac{\mathrm{\pi }}{4}\right)$.

Write down the maximum value and phase shift of the two currents $I_A$ and $I_B$ when they are each connected to the circuit alone.

The two AC voltage sources are connected to the circuit at the same time and the total current can be expressed as $I_A+I_B$.

Write down the maximum value and phase shift of $I_A+I_B$.

The height of a wave in metres, relative to a particular boat, can be modelled by the function $h\left(t\right)=0.5\sin \left(2t\right)$, where $t$ is the time in seconds. Observers on the boat are tracking a jumping dolphin. The height of the dolphin’s jumps can be modelled by the function $j\left(t\right)=2\sin \left(2t-0.5\right)$.

Find an expression for the height the dolphin can reach, at time $t$ seconds, when the height of the dolphin’s jump is affected by the height of the waves. Give your answer in the form $f\left(t\right)=A\sin \left(bt-c\right)$

Use technology to find the time when the dolphin first reaches its maximum height and write down the maximum height the dolphin reaches.

Find the time interval in the first two seconds when the height of dolphin will be above the wave

Let $z=2\mathrm{cis}\frac{3\mathrm{\pi }}{4}$.

(i) Find the values of $z^2,z^3$ and $z^4$, giving your answers in the form  $a{e}^{\mathrm{i}\theta }$, where $a\in {\mathrm{ℝ}}^{+}$ and θ  is given as an exact value.

(ii) Plot $z,z^2,z^3$and  $z^4$ on an Argand diagram.

Find the exact value of $a$  such that successive integer powers of the complex number $w=\frac{z}{a+\mathrm{i}}$ lie on a unit circle.

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

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

Consider the complex numbers $z_1=2e^{\frac{\mathrm{\pi }}{3}\mathrm{i}},z_2=3-z_1$ and $z_3=\frac{z_1}{z_2}$.

Write $z_1,z_2$ and  $z_3$ in the form $a+b\mathrm{i}$ where $a,b\in \mathrm{ℝ}$, giving exact values of $a$ and $b$.

A scale model of a triangular patio is planned by representing the vertices of the triangle on an Argand diagram as the points $z_1,z_2$and $z_3$. Find the area of the triangle in the model.

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

(i) Plot the position of $z$ on an Argand diagram.

(ii) Express $z$ in the form $z=a{e}^{\mathrm{i}b}$, where $a,b\in \mathrm{ℝ}$, giving the exact value of $a$and the exact value of $b$.

Use technology to find the value of $\sqrt{z^3}$ . Give your answer in the form $z=a{e}^{\mathrm{i}b}$, where $a,b\in \mathrm{ℝ}$, giving the exact value of $a$ and the exact value of $b$.

Find the smallest positive integer $k$ such that $z^k$ is

(i) a positive real number,

(ii) a negative real number.

Explain why there is no possible integer value of $k$ such that $z^k$ is purely imaginary.

Given the points 1 and $z$ on an Argand diagram, where $z\ne 0$   is a complex number, explain how to find each of the following points by geometrical construction.  In each case provide a sketch to illustrate your answer.

$z^2$.

$\left(2-\mathrm{i}\right)z.$

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 $\tan \left(\frac{5\pi }{12}\right)=2+\sqrt{3}$.

Two power sources are connected to a single electrical circuit. At time $t$ seconds, the voltage, $V_1$ , provided by the first power source is modelled by $V_1=\mathrm{Im}\left(3{\mathrm{e}}^{12\mathrm{ti}}\right)$, and the voltage, $V_2$, from the second power source can be modelled by the function $V_2$.

The total voltage in the circuit, $V_T=V_1+V_2$,  is given by $V_T=10\sin \left(12t+20\right)$.

Find an expression for $V_2$ in the form $A\sin \left(Bt+C\right)$, where $A,B,C\in \mathrm{ℝ}.$

Write down the maximum voltage and the phase shift provided by the second power source, giving your answers correct to 2 decimal places.

A popular fast-food chain is considering opening a new restaurant in an up and coming part of New York.  They have looked at competition in the area and predict that the costs $C$, in thousands of USD, to run the new restaurant for the first 100 days after opening could be modelled by the function

$C\left(d\right)=30\sin \left(0.01d-4.5\right)+4,0<d\le 100,$ 

where $d$ is the number of days since opening.

The CEO of the company will only give permission for the new restaurant to open if the model predicts that the revenue will be greater than the costs by the 80^th day after opening.

The revenue $R$, in thousands of USD, for the first 100 days is predicted to follow the model

$R\left(d\right)=a\sin \left(0.01d+0.1\right)+4.5,0<d\le 100,a\in \mathrm{\mathbb{N}}.$

(i) Find the minimum value of  that will ensure the restaurant is making a profit by day 80.

(ii) For this value of $a$, show that the profits of the hotel can be modelled by the function $P\left(d\right)=A\sin \left(0.01d+b\right)+c$, giving the values of $A,b$and $c$ correct to four significant figures.

According to the model found in part (a) (ii), find

(i) the profit the restaurant is predicted to make on day 100, to the nearest thousand USD,

(ii) the first day for which the loss is less than $20 000 USD.

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) $x=0,y=0$