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
ReIm55−5−51010−10−10z₁z₂ABCD

The Argand diagram shows z₁ = 3 + j and z₂ = −5 − j4. Click the point that represents z₁ − z₂.

z₁ = 3 + jz₂ = −5 − j4
  1. 1A (−2 − j3)
  2. 2B (8 + j5)
  3. 3C (8 − j5)
  4. 4D (5 + j8)
2.Easy
ReIm11−1−122−2−233−3−344−4−455−5−566−6−6z₁ABCD

The Argand diagram shows z₁ = 3 − j5. Click the point that represents z̄₁.

z₁ = 3 − j5
  1. 1A (−5 + j3)
  2. 2B (3 + j5)
  3. 3C (−3 − j5)
  4. 4D (−3 + j5)
3.Medium
ReIm11−1−122−2−2z₁z₂w

Given z₁ = 2∠40° and z₂ = 2∠110°, find the argument θ of w = z₁/z₂ in degrees, between −180° and 180°.

z₁ = 2∠40°z₂ = 2∠110°
θ = °
4.Medium
ReIm55−5−5θz

Find the argument θ of z = −7 − j in degrees, as the principal value between −180° and 180°.

z = −7 − j
θ = °
5.Hard
ReIm11−1−122−2−2θz

Find the imaginary part of (−2 − j2)⁵.

z = −2 − j2n = 5
Im(z⁵) =
6.Hard
ReIm11−1−122−2−233−3−3z₁z₂

The equation z² + 5z + 15.25 = 0 has two complex roots. Find their modulus |z|.

p = 5q = 15.25
|z| =

Answer key

  1. 1. B (8 + j5)
  2. 2. B (3 + j5)
  3. 3. θ = −70 °
  4. 4. θ = −171.9 °
  5. 5. Im(z⁵) = 128
  6. 6. |z| = 3.905

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