What is $\vert 2+i \vert$?
1. $1$
2. $\sqrt{3}$
3. $5$
4. $\sqrt{5}$
5. Something else!
Note:
* Correct Answer: D
* Use pythagoras in the complex plane
What is $(1+i)^2/(1-i)$?
1. $e^{i\pi/4}$
2. $\sqrt{2}e^{i\pi/4}$
3. $e^{i3\pi/4}$
4. $\sqrt{2}e^{i3\pi/4}$
5. Something else!
Note:
* Correct Answer: D
* Best to convert to euler with phase and just use the rules for adding and subtracting of exponents
Which point below best represents $4e^{i3\pi/4}$ on the complex plane?
<img src="./images/complex_numbers_graph.png" align="center" style="width: 600px";/>
Note:
* Correct Answer: D
What is $Re\left[\frac{e^{i\omega t}}{1+i}\right]$?
1. $\frac{1}{\sqrt{2}}\cos(\omega t + \pi/4)$
2. $\frac{1}{\sqrt{2}}\cos(\omega t - \pi/4)$
3. $\frac{1}2\cos(\omega t + \pi/4)$
4. $\frac{1}2\cos(\omega t - \pi/4)$
5. Something else
Note:
* Correct Answer: B
A resistor ($R$) and an inductor ($L$) are in parallel. What is the effective impedance, $Z_{eff}$ across these elements?
1. $R + L$
2. $R + i\omega L$
3. $1/(R+i\omega L)$
4. $\dfrac{1}{1/R -i/(\omega L)}$
5. Something else?
Note:
* Correct Answer: D
<img src="./images/RLC.png" align="right" style="width: 400px";/>
What is the total impedance of this circuit, $Z_{total}$?
1. $R + i\left(\omega L + \frac{1}{\omega C}\right)$
2. $R + i\left(\omega L - \frac{1}{\omega C}\right)$
3. $\frac{1}{R} + \frac{1}{i\omega L} + {i \omega C}$
4. $\dfrac{1}{\frac{1}{R} + \frac{1}{i\omega L} + {i \omega C}}$
5. None of these
Note:
* Correct Answer: B
AC voltage $V$ and current $I$ vs time $t$ are as shown:
<img src="./images/IV_graphs.png" align="center" style="width: 600px";/>
The graph shows that..
1. $I$ leads $V$ ( $I$ peaks before $V$ peaks )
2. $I$ lags $V$ ( $I$ peaks after $V$ peaks )
3. Neither
Note:
* Correct Answer: B