1.13 Polar and Euler form questions for IB Maths AI

Write $z$ in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$. Give the exact values of $r$ and $\theta$.

Hence find $z^3$, giving your answer in the form $a+b\mathrm{i}$.

Find the time of sunrise on 1 January according to the model. Give your answer to the nearest minute.

The number of hours of daylight on day $t$ is $D(t)=S(t)-R(t)$.

Write $1.6\cos (0.0172t-2.9)$ and $1.4\cos (0.0172t+0.3)$ in the form $\mathrm{R}\mathrm{e}\left(p{\mathrm{e}}^{\mathrm{i}(0.0172t+q)}\right)$.

Hence, or otherwise, write $D(t)$ in the form $a\cos (0.0172t+b)+c$, where $a>0$ and $-\pi <b\le \pi$.

Find the day of the year with the most hours of daylight, and the number of hours of daylight on that day.

Find the number of days in the year on which there are more than 14 hours of daylight.

Write down $V_1$ and $V_2$ each as the imaginary part of a complex function of the form $a{\mathrm{e}}^{\mathrm{i}(100\pi t+c)}$.

Show that $V_1+V_2=\mathrm{I}\mathrm{m}\left(z{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $z=30+40{\mathrm{e}}^{\mathrm{i}\frac{\pi }{3}}$.

Find $z$ in the form $a+b\mathrm{i}$, giving the exact values of $a$ and $b$.

Hence write $V_1+V_2$ in the form $A\sin (100\pi t+\phi )$, where $A>0$ and $0<\phi <\frac{\pi }{2}$.

Find the first time $t>0$ at which the total voltage $V_1+V_2$ is a maximum. Give your answer in milliseconds.

The total voltage is applied to a circuit in which a resistor, an inductor and a capacitor are connected in series. Each component has a complex impedance:

• a resistor of $R$ ohms has impedance $R$

• an inductor of $L$ henrys has impedance $\mathrm{i}\omega L$

• a capacitor of $C$ farads has impedance $\frac{1}{\mathrm{i}\omega C}$

where $\omega$ is the angular frequency. For components in series, the total impedance $Z$ is the sum of their impedances. In Kwame's circuit $R=20$, $L=0.15$ and $C=100\times {10}^{-6}$.

Find $Z$ in the form $a+b\mathrm{i}$.

Find the modulus and the argument of $Z$.

The current in the circuit, $I$ amperes, is given by $I=\mathrm{I}\mathrm{m}\left(w{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $w=\frac{z}{Z}$.

Find $I$ in the form $B\sin (100\pi t+\psi )$, where $B>0$.

Find the time interval between a maximum of the voltage and the next maximum of the current. Give your answer in milliseconds.

The circuit is said to be at resonance when its total impedance is a real number.

Kwame can change the angular frequency of the supply. Find the frequency, $f=\frac{\omega }{2\pi }$ hertz, at which this circuit is at resonance.

Kwame now uses the original supply with only the resistor and the inductor, connected in parallel. For two components in parallel, the total impedance $Z_{\text{p}}$ satisfies

Find $Z_{\text{p}}$ in the form $a+b\mathrm{i}$.

Hence find the amplitude of the current in this parallel circuit.

State one reason why the current measured by Kwame might differ from the value predicted by this model.

Find $z_1{z}_2$, giving your answer in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

Find ${\displaystyle \frac{z_1}{z_2}}$, giving your answer in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

Write ${\displaystyle \frac{z_1}{z_2}}$ in the form $a+b\mathrm{i}$.

Write down the impedance, $Z_A$, of branch A and the impedance, $Z_B$, of branch B.

Find the modulus and argument of $Z_A$.

Branches A and B are connected in parallel. The total impedance, $Z$, of the circuit is given by

Show that $Z={\displaystyle \frac{50}{9+4\mathrm{i}}}$.

Write $Z$ in the form $a+b\mathrm{i}$, where $a,b\in \mathbb{R}$.

The voltage, $V$, the current, $I$, and the impedance, $Z$, are related by $V=IZ$. The supply voltage is $V=120$.

Find $I$, giving your answer in the form $a+b\mathrm{i}$.

Find $\mathrm{\mid }I\mathrm{\mid }$ and $\arg I$.

The current leads the voltage if $\arg I>\arg V$ and lags the voltage if $\arg I<\arg V$. State, giving a reason, whether the current leads or lags the voltage in Kwame's circuit.

Find the smallest positive integer $n$ for which $z^n$ is a real number.

For this value of $n$, find the value of $z^n$.

Find the smallest positive integer $m$ for which $z^m$ is purely imaginary, and write down the value of $z^m$.