Chapter 10, Complex Numbers Video Solutions, Engineering Mathematics Through Applications

Problem 1

Let $u=2+j 3$ and $v=5+j 8$. Express the following in the form $a+j b$ where $a$ and $b$ are real:
a $u+v$
b $u-v$
c $u v$
d $v u$
e $u / v$
f $v / u$
g $2 u+v$
h $u^{2}$
i $v^{2}$
$\mathbf{j} u^{2}+v^{2}$

Chris Trentman

Numerade Educator

00:24

Problem 2

State the complex conjugate of the following:
$3-j 15,3+j 15, j 3, j, 0, \pi$ and $e$

Wendi Zhao

Numerade Educator

02:06

Problem 3

If $z=3-j$ express $z^{2}+7 z+13$ in the form $a+j b$ where $a$ and $b$ are real.

Lucas Finney

Numerade Educator

00:55

Problem 4

Simplify $j^{20}, j^{9}, j^{23}, j^{100}$ and $j^{1007}$.

Sherrie Fenner

Numerade Educator

04:27

Problem 5

i If $z=2+j$ find $z^{2}, z^{3}, z^{4}$ and $z^{4}-6 z^{2}+25$
ii Find one of the roots of $z^{4}-6 z^{2}+25=0$

Taylor Shimono

Numerade Educator

02:24

Problem 6

Determine the admittance, $Y=\frac{1}{Z}$, in a circuit, given that $Z=(75-j 40) \Omega$.
$(\Omega=\mathrm{ohm}$ is the SI unit of impedance.)

Thane Stiles

Numerade Educator

01:06

Problem 7

Assuming $a$ and $b$ are real, find them if a $a+j b=2+j 3$
b $a+j b=(2+j 3)-(2-j 3)$
c $a+j b=(1+j)^{2}$
d $a+j b=\frac{1}{3-j 4}$
e $2 a+j b=(1+2 j)^{2}$
f $1.8 a+j 3.4 b=\frac{7+j 5}{2-j}$

M Hassan Anwar

Numerade Educator

01:35

Problem 8

Express the following in the form $a+j b$ where $a$ and $b$ are real:
$\mathbf{a}-j 5(3+j 4)$
b $\frac{3+j 4}{j 5}$
c $(2-j 3)(1+j)$
$\mathbf{d}\left(\frac{3+j 4}{j 5}\right)-\left(\frac{2-j 3}{1-j}\right)$

M Hassan Anwar

Numerade Educator

02:49

Problem 9

If $Z_{1}=(1+j)$ ohm and $Z_{2}=(3+j 4)$ ohm are impedances connected in parallel in a circuit, find the total impedance, $Z_{\mathrm{t}}$, in the circuit given that
$$
\frac{1}{Z_{\mathrm{t}}}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}}
$$

Christina Shute

Numerade Educator

01:26

Problem 10

A current, $I$, in a magnetically coupled circuit satisfies
$$
20 I+j 100 I=200
$$
Find the current $I$.

Hast Aggarwal

Numerade Educator

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Problem 11

Find the exact values of $z$ which satisfy the following equations:
a $z^{2}+2 z+26=0$
b $z^{2}-2 z+3=0$
c $3 z^{2}-7 z+13=0$
By using a computer algebra system or a graphical calculator, plot on different axes the graphs $z^{2}+2 z+26$, $z^{2}-2 z+3$ and $3 z^{2}-7 z+13$
By looking at these graphs can you see why there are no real roots of these equations?

Victor Salazar

Numerade Educator

04:49

Problem 12

An a.c. voltage $V$ across a circuit is given by $V=(3+j 6)$ volts. Given that the impedance $Z$ of the circuit is $(8+j) \Omega$ find the current $I$ where
$$
I=\frac{V}{Z}
$$

Tanishq Gupta

Numerade Educator

02:55

Problem 13

Let $I_{\mathrm{p}}$ and $I_{\mathrm{s}}$ represent the primary and secondary currents through a mutually magnetically coupled circuit which satisfies the following equations:
$$
\begin{aligned}
10 &=4 I_{\mathrm{p}}+j 6 I_{\mathrm{p}}-j 4 I_{\mathrm{s}} \\
0 &=50 I_{\mathrm{s}}+j 12 I_{\mathrm{s}}+8 I_{\mathrm{s}}-j 4 I_{\mathrm{p}}
\end{aligned}
$$
Determine $I_{\mathrm{p}}$ and $I_{\mathrm{s}}$. (If you have difficulty with this question try it after Example 17.)

Erika Bustos

Numerade Educator

03:40

Problem 14

A resistance, $R$, and a capacitance, $C$, are connected in parallel. The impedance, $Z$, of the circuit is given by
$$
\frac{1}{Z}=\frac{1}{R}+\frac{1}{X_{c}}
$$
where $X_{c}=\frac{1}{j \omega C}$
$(\omega=$ angular frequency)
i Show that $Z=\frac{R}{1+j \omega C R}$.
ii Find the real and imaginary parts of $Z .(\operatorname{Re}(Z)$ and $\operatorname{Im}(Z)$ respectively.)

Khoobchandra Agrawal

Numerade Educator