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
ReIm11−1−122−2−233−3−3z₁ABCD

The Argand diagram shows z₁ = 1 − j2. Click the point that represents j·z₁.

z₁ = 1 − j2
  1. 1A (−2 − j)
  2. 2B (−1 + j2)
  3. 3C (2 − j)
  4. 4D (2 + j)
2.Easy
ReIm22−2−244−4−466−6−6z₁z₂

For z₁ = −1 + j and z₂ = −5 + j6, find the modulus of the product, |z₁z₂|.

z₁ = −1 + jz₂ = −5 + j6
|z₁z₂| =
3.Medium
ReIm11−1−122−2−2θz

The number z = 3·ej0.8 has its angle in radians. Find its imaginary part b.

r = 3θ = 0.8 rad
b =
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
ReIm22−2−244−4−4ABCD

Exactly one of the points A–D is a root of z⁴ = 81∠120°. Click it.

w = 81∠120°n = 4
  1. 1A (3∠−105°)
  2. 2B (3∠−150°)
  3. 3C (5.4∠−150°)
  4. 4D (3∠150°)
6.Hard
ReIm0.50.5−0.5−0.511−1−11.51.5−1.5−1.5z₁z₂

Solve z² + 3z + 4.5 = 0. The roots are a ± jb; find the real part a.

p = 3q = 4.5
a =

Answer key

  1. 1. D (2 + j)
  2. 2. |z₁z₂| = 11.05
  3. 3. b = 2.152
  4. 4. Im(w) = 4.243
  5. 5. B (3∠−150°)
  6. 6. a = −1.5

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