Complex Numbers

  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).

Sample Solution

ACED ESSAYS