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Anonymous

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

Tags:
math homework
engineering aspects
quadratic equations
Mathematic Equations
EGN3420
Complex Numbers

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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Purchase answer to see full attachment

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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Anonymous

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Anonymous

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