1.12 Complex numbers questions for IB Maths AI

Write down $\mathrm{Re}(w)$.

Write down $z^{\ast }$.

Find $zw$, giving your answer in the form $a+b\mathrm{i}$, where $a,b\in \mathbb{R}$.

Write down $\mathrm{\mid }z\mathrm{\mid }$.

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.

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.

Without using a calculator, find ${\displaystyle \frac{z_1}{z_2}}$ in the form $a+b\mathrm{i}$, where $a,b\in \mathbb{Z}$.

Find the modulus and the argument of ${\displaystyle \frac{z_1}{z_2}}$.

Use technology to find ${\left({\displaystyle \frac{z_1}{z_2}}\right)}^4$ in the form $a+b\mathrm{i}$.

Write down the height of the nozzle above the floor.

Find the maximum height of the jet and the horizontal distance from the nozzle at which it occurs.

A decorative beam crosses the lobby at a height of $4.5$ metres. To check whether the water reaches the beam, the hotel engineer, Rosa, writes the equation $-0.35x^2+2.1x-3.7=0$.

Show that this equation has no real solutions.

Solve the equation $-0.35x^2+2.1x-3.7=0$, giving your answers in the form $a+b\mathrm{i}$.

State, with reference to your answer to part (b), how the real part of the solutions in part (d) relates to the shape of the jet.

Rosa adjusts the pump so that the jet is modelled by $h(x)=-0.35x^2+bx+0.8$, where $b>0$.

Find the least value of $b$ for which the jet reaches the beam.

Using the original model, find the horizontal distance from the nozzle at which the jet hits the floor.

By calculating the discriminant, show that this equation has no real solutions.

Find the two solutions of the equation, giving your answers in the form $a+b\mathrm{i}$.

State what your answer to part (b) tells Thabo about the ball.