Complex numbers worksheet with answers

Free printable Complex numbers worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
ReIm11−1−122−2−233−3−3z₁z₂

For z₁ = 1 + j3 and z₂ = 1 + j3, find the real part of the quotient z₁/z₂.

z₁ = 1 + j3z₂ = 1 + j3
Re(z₁/z₂) =
2.Easy
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = 6 + j5 and z₂ = −4 − j4, find the modulus of the quotient, |z₁/z₂|.

z₁ = 6 + j5z₂ = −4 − j4
|z₁/z₂| =
3.Medium
ReIm22−2−244−4−4z₁z₂w

Given z₁ = 3∠40° and z₂ = 1.5∠−95°, find the imaginary part of w = z₁·z₂.

z₁ = 3∠40°z₂ = 1.5∠−95°
Im(w) =
4.Medium
ReIm22−2−244−4−4z₁z₂w

Given z₁ = 4∠70° and z₂ = 1.5∠−25°, find the imaginary part of w = z₁·z₂.

z₁ = 4∠70°z₂ = 1.5∠−25°
Im(w) =
5.Hard
ReIm11−1−122−2−233−3−3ABCD

Exactly one of the points A–D is a root of z³ = 27∠−90°. Click it.

w = 27∠−90°n = 3
  1. 1A (3∠150°)
  2. 2B (1.5∠−150°)
  3. 3C (3∠−150°)
  4. 4D (3∠−90°)
6.Hard
ReIm11−1−122−2−2z₀z₁z₂z₃

The roots of zⁿ = w with n = 4 and w = 16∠90° are zk = R1/n∠((φ + 360°·k)/n), k = 0, 1, 2, 3. Find the imaginary part of z₃.

w = 16∠90°n = 4k = 3
Im(z₃) =

Answer key

  1. 1. Re(z₁/z₂) = 1
  2. 2. |z₁/z₂| = 1.381
  3. 3. Im(w) = −3.686
  4. 4. Im(w) = 4.243
  5. 5. C (3∠−150°)
  6. 6. Im(z₃) = −1.848

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