Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

CHAPTER 12 REVIEW EXERCISES

CONCEPT CHECK EXERCISES

Determine each of the following as being either true or false. If it is false, explain why.

${(\sqrt{- 9})}^{2} = 9$ ${(\sqrt{- 9})}^{2} = 9$
$\frac{1 + j}{1 - j} = j$ $\frac{1 + j}{1 - j} = j$
$4/90 ° = 4 j$ $4/90 ° = 4 j$
$2 e^{π j} = - 2$ $2 e^{π j} = - 2$
${(2 ∠ \underline{120^{\circ}})}^{3} = 8$ ${(2 ∠ \underline{120^{\circ}})}^{3} = 8$
The phase angle $ϕ$ $ϕ$ for an impedance $Z = 4 - 4 j$ $Z = 4 - 4 j$ is $45 °$ $45 °$ .

PRACTICE AND APPLICATIONS

In Exercises 7–20, perform the indicated operations, expressing all answers in simplest rectangular form.

$(6 - 2 j) + (4 + j)$ $(6 - 2 j) + (4 + j)$
$(12 + 7 j) + (- 8 + 6 j)$ $(12 + 7 j) + (- 8 + 6 j)$
$(18 - 3 j) - (12 - 5 j)$ $(18 - 3 j) - (12 - 5 j)$
$(- 4 - 2 j) - \sqrt{- 49}$ $(- 4 - 2 j) - \sqrt{- 49}$
$5 j (6 - 5 j)$ $5 j (6 - 5 j)$
$- 3 j (4 - 7 j)$ $- 3 j (4 - 7 j)$
$(6 - 3 j) (4 + 3 j)$ $(6 - 3 j) (4 + 3 j)$
$(4 - 9 j) (1 + 2 j)$ $(4 - 9 j) (1 + 2 j)$
$\frac{3}{7 - 6 j}$ $\frac{3}{7 - 6 j}$
$\frac{48 j}{4 + 18 j}$ $\frac{48 j}{4 + 18 j}$
$\frac{6 - \sqrt{- 16}}{\sqrt{- 4}}$ $\frac{6 - \sqrt{- 16}}{\sqrt{- 4}}$
$\frac{3 + \sqrt{- 4}}{4 - j}$ $\frac{3 + \sqrt{- 4}}{4 - j}$
$\frac{5 j - (3 - j)}{4 - 2 j}$ $\frac{5 j - (3 - j)}{4 - 2 j}$
$\frac{2 - (6 - j)}{1 - 2 j}$ $\frac{2 - (6 - j)}{1 - 2 j}$

In Exercises 21–24, find the values of x and y for which the equations are valid.

$3 x - 2 j = y j - 9$ $3 x - 2 j = y j - 9$
$2 x j - 2 y = (y + 3) j - 3$ $2 x j - 2 y = (y + 3) j - 3$
$(3 + 2 j^{3}) (x + j y) = 4 + j^{9}$ $(3 + 2 j^{3}) (x + j y) = 4 + j^{9}$
$(x + j y) (7 j - 4) = j (x - 5)$ $(x + j y) (7 j - 4) = j (x - 5)$

In Exercises 25–28, perform the indicated operations graphically. Check them algebraically.

$(- 1 + 5 j) + (4 + 6 j)$ $(- 1 + 5 j) + (4 + 6 j)$
$(7 - 2 j) + (- 5 + 4 j)$ $(7 - 2 j) + (- 5 + 4 j)$
$(9 + 2 j) - (5 - 6 j)$ $(9 + 2 j) - (5 - 6 j)$
$(4 j + 8) - (11 - 3 j)$ $(4 j + 8) - (11 - 3 j)$

In Exercises 29–36, give the polar and exponential forms of each of the complex numbers.

$1 - j$ $1 - j$
$4 - 3 j^{3}$ $4 - 3 j^{3}$
$- 22 - 77 j$ $- 22 - 77 j$
$60 - 20 j$ $60 - 20 j$
$1.07 + 4.55 j$ $1.07 + 4.55 j$
$158 j - 327$ $158 j - 327$
5000
$- 4 j^{5}$ $- 4 j^{5}$

In Exercises 37–48, give the rectangular form of each number:

$2 (\cos 225 ° + j \sin 225 °)$ $2 (\cos 225 ° + j \sin 225 °)$
$48 (\cos 60 ° + j \sin 60 °)$ $48 (\cos 60 ° + j \sin 60 °)$
$5.011 (\cos 123.82 ° + j \sin 123.82 °)$ $5.011 (\cos 123.82 ° + j \sin 123.82 °)$
$2.417 (\cos 656.26 ° + j \sin 656.26 °)$ $2.417 (\cos 656.26 ° + j \sin 656.26 °)$
$0.62 ∠ \underline{- 72 °}$ $0.62 ∠ \underline{- 72 °}$
$20 ∠ \underline{160 °}$ $20 ∠ \underline{160 °}$
$27.08 ∠ \underline{346.27 °}$ $27.08 ∠ \underline{346.27 °}$
$1.689 ∠ \underline{194.36 °}$ $1.689 ∠ \underline{194.36 °}$
$2.00 e^{0.25 j}$ $2.00 e^{0.25 j}$
$e^{- 3.62 j}$ $e^{- 3.62 j}$
${(35.37 e^{1.096 j})}^{2}$ ${(35.37 e^{1.096 j})}^{2}$
$(13.6 e^{2.158 j}) (3.27 e^{3.888 j})$ $(13.6 e^{2.158 j}) (3.27 e^{3.888 j})$

In Exercises 49–64, perform the indicated operations. Leave the result in polar form.

$[3 (\cos 32 ° + j \sin 32 °)] [5 (\cos 52 ° + j \sin 52 °)]$ $[3 (\cos 32 ° + j \sin 32 °)] [5 (\cos 52 ° + j \sin 52 °)]$
$[2.5 (\cos 162 ° + j \sin 162 °)] [8 (\cos 115 ° + j \sin 115 °)]$ $[2.5 (\cos 162 ° + j \sin 162 °)] [8 (\cos 115 ° + j \sin 115 °)]$
$(40 ∠ \underline{18 °}) (0.5 ∠ \underline{245 °})$ $(40 ∠ \underline{18 °}) (0.5 ∠ \underline{245 °})$
$(0.1254 ∠ \underline{172.38 °}) (27.17 ∠ \underline{204.34 °})$ $(0.1254 ∠ \underline{172.38 °}) (27.17 ∠ \underline{204.34 °})$
$\frac{24 (\cos 165 ° + j \sin 165 °)}{{[3 (\cos 55 ° + j \sin 55 °)]}^{3}}$ $\frac{24 (\cos 165 ° + j \sin 165 °)}{{[3 (\cos 55 ° + j \sin 55 °)]}^{3}}$
$\frac{18 (\cos 403 ° + j \sin 403 °)}{{[2 (\cos 96 ° + j \sin 96 °)]}^{2}}$ $\frac{18 (\cos 403 ° + j \sin 403 °)}{{[2 (\cos 96 ° + j \sin 96 °)]}^{2}}$
$\frac{245.6 ∠ \underline{326.44 °}}{17.19 ∠ \underline{192.83 °}}$ $\frac{245.6 ∠ \underline{326.44 °}}{17.19 ∠ \underline{192.83 °}}$
$\frac{4 ∠ \underline{206 °}}{100 ∠ \underline{- 320 °}}$ $\frac{4 ∠ \underline{206 °}}{100 ∠ \underline{- 320 °}}$
$0.983 ∠ \underline{47.2 °} + 0.366 ∠ \underline{95.1 °}$ $0.983 ∠ \underline{47.2 °} + 0.366 ∠ \underline{95.1 °}$
$17.8 ∠ \underline{110.4 °} - 14.9 ∠ \underline{226.3 °}$ $17.8 ∠ \underline{110.4 °} - 14.9 ∠ \underline{226.3 °}$
$7644 ∠ \underline{294.36 °} - 6871 ∠ \underline{17.86 °}$ $7644 ∠ \underline{294.36 °} - 6871 ∠ \underline{17.86 °}$
$4.944 ∠ \underline{327.49 °} + 8.009 ∠ \underline{7.37 °}$ $4.944 ∠ \underline{327.49 °} + 8.009 ∠ \underline{7.37 °}$
${[2 (\cos 16 ° + j \sin 16 °)]}^{10}$ ${[2 (\cos 16 ° + j \sin 16 °)]}^{10}$
${[3 (\cos 36 ° + j \sin 36 °)]}^{6}$ ${[3 (\cos 36 ° + j \sin 36 °)]}^{6}$
${(7 ∠ \underline{110.5 °})}^{3}$ ${(7 ∠ \underline{110.5 °})}^{3}$
${(536 ∠ \underline{220.3 °})}^{4}$ ${(536 ∠ \underline{220.3 °})}^{4}$

In Exercises 65–68, change each number to polar form and then perform the indicated operations. Express the final result in rectangular and polar forms. Check by performing the same operation in rectangular form using a calcultor.

${(1 - j)}^{10}$ ${(1 - j)}^{10}$
${(\sqrt{3} + j)}^{8} {(1 + j)}^{5}$ ${(\sqrt{3} + j)}^{8} {(1 + j)}^{5}$
$\frac{{(5 + 5 j)}^{4}}{{(- 1 - j)}^{6}}$ $\frac{{(5 + 5 j)}^{4}}{{(- 1 - j)}^{6}}$
${(\sqrt{3} - j)}^{- 8}$ ${(\sqrt{3} - j)}^{- 8}$

In Exercises 69–72, find all the roots of the given equations.

$x^{3} + 8 = 0$ $x^{3} + 8 = 0$
$x^{3} - 1 = 0$ $x^{3} - 1 = 0$
$x^{4} + j = 0$ $x^{4} + j = 0$
$x^{5} - 32 j = 0$ $x^{5} - 32 j = 0$

In Exercises 73–76, determine the rectangular form and the polar form of the complex number for which the graphical representation is shown in the given figure.

In Exercises 77–88, solve the given problems.

Evaluate $x^{2} - 2 x + 4$ $x^{2} - 2 x + 4$ for $x = 5 - 2 j .$ $x = 5 - 2 j .$
Evaluate $2 x^{2} + 5 x - 7$ $2 x^{2} + 5 x - 7$ for $x = - 8 + 7 j .$ $x = - 8 + 7 j .$
Solve for $x : x^{2} = 8 x - 41$ $x : x^{2} = 8 x - 41$ (Express the solutions in simplified form in terms of j.)
Solve for $x : 2 x^{2} = 6 x - 9$ $x : 2 x^{2} = 6 x - 9$ (Express the solutions in simplified form in terms of j.)
Are $1 - j$ $1 - j$ and $- 1 - j$ $- 1 - j$ solutions to the equation $x^{2} - 2 x + 2 = 0 ?$ $x^{2} - 2 x + 2 = 0 ?$
Show that $\frac{1}{2} (1 + j \sqrt{3})$ $\frac{1}{2} (1 + j \sqrt{3})$ is the reciprocal of its conjugate.
Solve for $x : {(1 + j x)}^{2} = 1 + j - x^{2}$ $x : {(1 + j x)}^{2} = 1 + j - x^{2}$
What is the argument for any negative imaginary number?
If $f (x) = 2 x - {(x - 1)}^{- 1},$ $f (x) = 2 x - {(x - 1)}^{- 1},$ find $f (1 + 2 j) .$ $f (1 + 2 j) .$
If $f (x) = x^{- 2} + 3 x^{- 1},$ $f (x) = x^{- 2} + 3 x^{- 1},$ find $f (4 + j) .$ $f (4 + j) .$
Using a calculator, express $5 - 3 j$ $5 - 3 j$ in polar form.
Using a calculator, express $25.0 e^{2.25 j}$ $25.0 e^{2.25 j}$ in rectangular form.

In Exercises 89–100, find the required quantities.

A 60-V ac voltage source is connected in series across a resistor, an inductor, and a capacitor. The voltage across the inductor is 60 V, and the voltage across the capacitor is 60 V. What is the voltage across the resistor?
In a series ac circuit with a resistor, an inductor, and a capacitor, $R = 6.50 Ω, X_{C} = 3.74 Ω,$ $R = 6.50 Ω, X_{C} = 3.74 Ω,$ and $Z = 7.50 Ω .$ $Z = 7.50 Ω .$ Find $X_{L} .$ $X_{L} .$
In a series ac circuit with a resistor, an inductor, and a capacitor, $R = 6250 Ω, Z = 6720 Ω,$ $R = 6250 Ω, Z = 6720 Ω,$ and $X_{L} = 1320 Ω .$ $X_{L} = 1320 Ω .$ Find the phase angle $ϕ .$ $ϕ .$
A coil of wire rotates at 120.0 r/s. If the coil generates a current in a circuit containing a resistance of $12.07 Ω,$ $12.07 Ω,$ an inductance of 0.1405 H, and an impedance of $22.35 Ω,$ $22.35 Ω,$ what must be the value of a capacitor (in F) in the circuit?
What is the frequency f for resonance in a circuit for which $L = 2.65$ $L = 2.65$ H and $C = 18.3 μ F ?$ $C = 18.3 μ F ?$
The displacement of an electromagnetic wave is given by $d = A (\cos ω t + j \sin ω t) + B (\cos ω t - j \sin ω t) .$ $d = A (\cos ω t + j \sin ω t) + B (\cos ω t - j \sin ω t) .$ Find the expressions for the magnitude and phase angle of d.
Two cables lift a crate. The tensions in the cables can be represented by $2100 - 1200 j N$ $2100 - 1200 j N$ and $1200 + 5600 j N .$ $1200 + 5600 j N .$ Express the resultant tension in polar form.
A boat is headed across a river with a velocity (relative to the water) that can be represented as $6.5 + 1.7 j mi / h .$ $6.5 + 1.7 j mi / h .$ The velocity of the river current can be represented as $- 1.1 - 4.3 j mi / h .$ $- 1.1 - 4.3 j mi / h .$ Express the resultant velocity of the boat in polar form.
In the study of shearing effects in the spinal column, the expression $\frac{1}{μ + j ω n}$ $\frac{1}{μ + j ω n}$ is found. Express this in rectangular form.
In the theory of light reflection on metals, the expression $\frac{μ (1 - k j) - 1}{μ (1 - k j) + 1}$ $\frac{μ (1 - k j) - 1}{μ (1 - k j) + 1}$ is encountered. Simplify this expression.
Show that $e^{j π} = - 1.$ $e^{j π} = - 1.$
Show that ${(e^{j π})}^{1/2} = j .$ ${(e^{j π})}^{1/2} = j .$
A computer programmer is writing a program to determine the n nth roots of a real number. Part of the program is to find the number of real roots and the number of pure imaginary roots. Write one or two paragraphs explaining how these numbers of roots can be determined without actually finding the roots.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for CHAPTER 12 REVIEW EXERCISES

Create new playlist

Sign In

Sign Up

Table of Contents for
CHAPTER 12 REVIEW EXERCISES