egn 3420 homework assignment 1 complex numbers

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 âˆ’16k+3 (c) (i 5kâˆ’3 )(i âˆ’k+1)

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

3. Solve the linear system of equations: z1 + iz2 = âˆ’1 z1 âˆ’ 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) â‰¤ |z|.

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

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

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

9. Find all the three cube roots of i.

10. Find (1 âˆ’ 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 = 10â„¦, L = 10mH, C = 100ÂµF and Vs(t) = 10cos(1000t). and see pic on the file attached