EGN3420 University of South Florida Complex Numbers Homework Assignment 1

Anonymous

Question Description

This homework better to be done by computer not hand written so I can print it and hand it out

the assignment is attached

1. Let k be an integer. Solve the following: (a) i 4k (b) i −16k+3 (c) (i 5k−3 )(i −k+1)

2. Solve each of the following equations for z: (a) 21 − 5iz = 2zi (b) 49 + z 2 = 0 (c) z 2 = z(i − 5) (d) z = (1 − z)(1 − 4i) (e) z 2 + zi + 12 = 0

3. Solve the linear system of equations: z1 + iz2 = −1 z1 − z2 = i

4. Draw plots for the regions described by the following equations: (a) iIm(z) < 3 (b) |z| = 1 + Im(z) (c) Re(z) > |z| + 3

5. Show that: Re(z) ≤ |z|.

6. Simplify the following expressions: (a) 1+i 3−3i (b) 1 5i (c) 4 1−i (d) 1+i √ 7 (1−i) 3 1

7. Write the following numbers in polar form (r, θ) and express θ in both Arg(z) and arg0(z) forms: (a) 4 + 3i (b) i − 1 (c) −1 − i (d) −i

8. Using the answers from Question 7, express the following in polar form: (a) 4+3i i−1 (b) (4 + 3i)(i − 1)2 (c) (4 + 3i) √ i − 1

9. Find all the three cube roots of i.

10. Find (1 − i) 3 4 .

11. Solve the following quadratic equations: (a) z 2 + z + 1 = 0 (b) z 2 + zi + i = 0

12. Find the steady-state current, Is(t), in the following system. Given that, R = 10Ω, L = 10mH, C = 100µF and Vs(t) = 10cos(1000t). and see pic on the file attached

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EGN 3420 Homework Assignment 1 Complex Numbers Dr. Nasir Ghani September 2019 1. Let k be an integer. Solve the following: (a) i4k (b) i−16k+3 (c) (i5k−3 )(i−k+1 ) 2. Solve each of the following equations for z: (a) 21 − 5iz = 2zi (b) 49 + z 2 = 0 (c) z 2 = z(i − 5) (d) z = (1 − z)(1 − 4i) (e) z 2 + zi + 12 = 0 3. Solve the linear system of equations: z1 + iz2 = −1 z1 − z2 = i 4. Draw plots for the regions described by the following equations: (a) iIm(z) < 3 (b) |z| = 1 + Im(z) (c) Re(z) > |z| + 3 5. Show that: Re(z) ≤ |z|. 6. Simplify the following expressions: (a) (b) (c) (d) 1+i 3−3i 1 5i 4 1−i √ 1+i 7 (1−i)3 1 7. Write the following numbers in polar form (r, θ) and express θ in both Arg(z) and arg0 (z) forms: (a) 4 + 3i (b) i − 1 (c) −1 − i (d) −i 8. Using the answers from Question 7, express the following in polar form: (a) 4+3i i−1 (b) (4 + 3i)(i − 1)2 √ (c) (4 + 3i) i − 1 9. Find all the three cube roots of i. 3 10. Find (1 − i) 4 . 11. Solve the following quadratic equations: (a) z 2 + z + 1 = 0 (b) z 2 + zi + i = 0 12. Find the steady-state current, Is (t), in the following system. Given that, R = 10Ω, L = 10mH, C = 100µF and Vs (t) = 10cos(1000t). Figure 1: Figure for Question 12 2 ...
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moizeali1
School: Duke University

Find the attached solution

1. (a) Consider z = i 4k
z = i 4k
= (i 4 )
= (1)

k

k

=1

(b) Consider z = i −16 k +3

z = i −16 k +3
=
=

i3
i 16 k
i3

(i )

4 4k

−i
14 k
= −i
=

(c) Consider z = ( i 5 k −3 )( i − k +1 )

z = ( i 5 k −3 )( i − k +1 )
= ( i 5 k −3− k +1 )
= i 4k −2
=

i 4k
i2

(i )
=

4 k

−1

k

1
−1
= −1
=

2. Solving each of the following equations
(a).

21 − 5iz
−5iz
−5iz − 2zi
−7iz

= 2zi
= 2zi − 2i
= −21
= −21
−21
z =
−7i
−21 i
=

−7i i
−21i
=
−7i 2
−21i
=
7
z = −3i

(b).

49 + z 2 = 0
z 2 = −49
=  −49
= 7i
= 7i or − 7i
(c)

z 2 = z ( i − 5)
z 2 − z ( i − 5) = 0
z ( z − ( i − 5) ) = 0
z =0
or z = i − 5
(d)

z = (1 − z )(1 − 4i )
z = 1 − 4i − z − 4iz
2z = 1 − 41 + 4iz
2z − 4iz = 1 − 4i
2z (1 − 2i ) = 1 − 4i

Divide both side by 2(1-2i)
2z (1 − 2i )
1 − 4i
=
2 (1 − 2i ) 2 (1 − 2i )
z =

1 − 4i
2 (1 − 2i )

Multiplying numerator and denominator by (1+2i)
z =

(1 + 2i )
1 − 4i

2 (1 − 2i ) (1 + 2i )

=

1 + 2i − 4i + 8
2 (1 + 4 )

=

9 − 2i
10

(e). This is in a quadratic form so, basic formula of quadratic equation is used here

−i  i 2 − 4 1 12 −1  −1 − 48
z =
=
2
2
−1  −49 −1  7i
=
=
2
2
Therefore, roots are:
z =

−1 + 7i −1 − 7i
,
2
2

3. The given system of linear equations are:

z 1 + iz 2 = −1
z1 − z 2 = i
Subtr...

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