Worksheet: Modulus of a Complex Number

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In this worksheet, we will practice using the general formula for calculating the modulus of a complex number.

Q1:

What is the modulus of the complex number 2 𝑖 ?

Q2:

What is the modulus of the complex number 3 − 𝑖 ?

  • A10
  • B3
  • C √ 2
  • D1
  • E √ 1 0

Q3:

What is the modulus of the complex number 3 + 7 𝑖 ?

  • A7
  • B3
  • C √ 5 8
  • D √ 1 0
  • E58

Q4:

What is the modulus of the complex number 3 + 4 𝑖 ?

Q5:

Given that 𝑍 = 8 + 4 𝑖 , find | 𝑍 | .

  • A | 𝑍 | = 4 √ 5
  • B | 𝑍 | = 3 2
  • C | 𝑍 | = 8 0
  • D | 𝑍 | = 4 √ 3
  • E | 𝑍 | = 4 √ 2

Q6:

Given that 𝑍 = 3 𝑖 , find | 𝑍 | .

  • A3
  • B 3 𝑖
  • C 3 + 3 𝑖
  • D − 3 𝑖
  • E − 3

Q7:

Given that 𝑍 = − 3 − √ 3 𝑖 , find | 𝑍 | .

  • A | 𝑍 | = √ 6
  • B | 𝑍 | = 1 2
  • C | 𝑍 | = 2 √ 3
  • D | 𝑍 | = 3
  • E | 𝑍 | = 3 √ 2

Q8:

Given that 𝑍 = 2 − 2 √ 5 𝑖 , determine | | 𝑍 | | .

  • A 2 + 2 √ 5 𝑖
  • B − 2 √ 6
  • C 2 √ 6
  • D − 2 − 2 √ 5 𝑖

Q9:

If 𝑟 = 5 + 2 𝑖 and 𝑠 = 5 − 2 𝑖 , what is the modulus of 𝑟 + 𝑠 ?

Q10:

Given that 𝑍 = 3 − 9 𝑖 1 + 3 𝑖 , determine | 𝑍 | .

Q11:

What is the modulus of the complex number 𝑎 + 𝑏 𝑖 , where 𝑎 and 𝑏 are real?

  • A 𝑎 + 𝑏
  • B √ 𝑎 + 𝑏
  • C √ 𝑎 + 𝑏
  • D 𝑎 + 𝑏
  • E √ 𝑎 − 𝑏

Q12:

If 𝑍 = 1 𝑍 , where 𝑍 is a complex number, what is | 𝑍 | ?

  • A0
  • B2
  • C 𝑖
  • D1

Q13:

Given that 𝑍 = ( 𝑎 + 𝑏 ) + 𝑖 ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑏 ) − 𝑖 ( 𝑎 + 𝑏 ) , where 𝑎 ∈ ℝ and 𝑏 ∈ ℝ , write 𝑍 in algebraic form and hence find | 𝑍 | .

  • A 𝑍 = 1 + 𝑖 , | 𝑍 | = √ 2
  • B 𝑍 = 1 − 𝑖 , | 𝑍 | = √ 2
  • C 𝑍 = 𝑖 , | 𝑍 | = 1
  • D 𝑍 = − 𝑖 , | 𝑍 | = 1

Q14:

Given that 𝑍 = ( 𝑎 + 𝑏 ) − 𝑖 ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑏 ) + 𝑖 ( 𝑎 + 𝑏 ) , where 𝑎 ∈ ℝ and 𝑏 ∈ ℝ , write 𝑍 in algebraic form and hence find | 𝑍 | .

  • A 𝑍 = 1 − 𝑖 , | 𝑍 | = √ 2
  • B 𝑍 = 1 + 𝑖 , | 𝑍 | = √ 2
  • C 𝑍 = − 𝑖 , | 𝑍 | = 1
  • D 𝑍 = 𝑖 , | 𝑍 | = 1

Q15:

Given the complex number 𝑧 = 𝑎 + 𝑏 𝑖 , what is the modulus of 𝑧 ?

  • A 𝑎 + 𝑏
  • B ( 𝑎 𝑏 )
  • C √ 𝑎 + 𝑏
  • D 2 𝑎 + 𝑏
  • E 2 √ 𝑎 + 𝑏

Q16:

Given that 𝑍 = 4 + 𝑖 , find | 𝑍 | .

  • A | 𝑍 | = √ 1 5
  • B | 𝑍 | = √ 1 7
  • C | 𝑍 | = 2
  • D | 𝑍 | = 4
  • E | 𝑍 | = 1 7

Q17:

Given that ( 𝑎 + 𝑏 𝑖 ) ( 8 − 8 𝑖 ) = 1 4 + 6 𝑖 , where 𝑎 and 𝑏 are real numbers, determine the value of 1 6 ( 𝑎 + 𝑏 ) .

  • A 1 3 2
  • B 2 9 1 6
  • C 29
  • D 1 3 1 0

Q18:

Consider the complex numbers 𝑧 and 𝑧 . Given that | 𝑧 | = | 𝑧 − 𝑧 | and 𝑧 = 1 2 + 5 𝑖 , which of the following is a possible value for 𝑧 ?

  • A 2 5 − 8 𝑖
  • B 1 2 − 8 𝑖
  • C − 1 2 + 5 𝑖
  • D 1 + 5 𝑖
  • E 1 2 + 8 𝑖

Q19:

If 𝑧 = − 4 − 9 𝑖 and 𝑧 = 3 − 3 𝑖 , what is | 𝑧 − 𝑧 | ?

  • A85
  • B13
  • C145
  • D √ 8 5
  • E √ 1 4 5

Q20:

What does the modulus of a complex number represent?

  • A its distance from the origin in the complex plane
  • B its real coordinate in the complex plane
  • C the angle it makes with the positive imaginary axis
  • D its imaginary coordinate in the complex plane
  • E the angle it makes with the positive real axis

Q21:

Given that | 𝑍 | = | 𝑍 + 6 | , determine the real part of the complex number 𝑍 .

  • A − 3
  • B − 6
  • C6
  • D3

Q22:

What is | 𝑧 | given that 𝑧 = 2 − 8 𝑖 ?

  • A 2 √ 1 5
  • B 2 √ 1 7
  • C100
  • D36
  • E6

Q23:

Consider the complex numbers 𝑧 = 3 − 4 𝑖 and 𝑤 = − 1 5 + 8 𝑖 .

Find | 𝑧 | and | 𝑤 | .

  • A | 𝑧 | = 5 , | 𝑤 | = 1 7
  • B | 𝑧 | = 2 5 , | 𝑤 | = 2 8 9
  • C | 𝑧 | = √ 7 , | 𝑤 | = √ 2 3
  • D | 𝑧 | = √ 1 7 , | 𝑤 | = √ 5
  • E | 𝑧 | = √ 7 , | 𝑤 | = √ 1 6 1

Calculate | 𝑧 𝑤 | . How does this compare with | 𝑧 | | 𝑤 | ?

  • A | 𝑧 𝑤 | = 2 2 , | 𝑧 | + | 𝑤 | = | 𝑧 𝑤 |
  • B | 𝑧 𝑤 | = 8 5 , | 𝑧 | | 𝑤 | = | 𝑧 𝑤 |
  • C | 𝑧 𝑤 | = 7 , 2 2 5 , √ | 𝑧 | | 𝑤 | = | 𝑧 𝑤 |
  • D | 𝑧 𝑤 | = √ 8 5 , | 𝑧 | | 𝑤 | ≥ | 𝑧 𝑤 |
  • E | 𝑧 𝑤 | = 2 √ 4 2 , | 𝑧 | | 𝑤 | ≤ | 𝑧 𝑤 |

Calculate | | 𝑧 𝑤 | | . How does this compare with | 𝑧 | | 𝑤 | ?

  • A | | 𝑧 𝑤 | | = 5 1 7 , | 𝑧 | | 𝑤 | = | | 𝑧 𝑤 | |
  • B | | 𝑧 𝑤 | | = 5 1 7 , | 𝑧 | | 𝑤 | = | | 𝑧 𝑤 | |
  • C | | 𝑧 𝑤 | | = 2 5 2 8 9 , | 𝑧 | | 𝑤 | ≥ | | 𝑧 𝑤 | |
  • D | | 𝑧 𝑤 | | = − 1 2 , | 𝑧 | − | 𝑤 | = | | 𝑧 𝑤 | |
  • E | | 𝑧 𝑤 | | = 1 7 5 , | 𝑧 | | 𝑤 | = | | 𝑧 𝑤 | |

Q24:

Consider the two complex numbers 𝑤 = − 1 + 7 𝑖 and 𝑧 = 5 − 3 𝑖 .

Calculate | 𝑤 | + | 𝑧 | to two decimal places.

Calculate | 𝑧 + 𝑤 | to two decimal places.

Which of the following relations do 𝑤 and 𝑧 satisfy?

  • A | 𝑤 | + | 𝑧 | ≤ | 𝑧 + 𝑤 |
  • B √ | 𝑤 | + | 𝑧 | = | 𝑧 + 𝑤 |
  • C | 𝑤 | + | 𝑧 | = 2 | 𝑧 + 𝑤 |
  • D | 𝑤 | + | 𝑧 | = | 𝑧 + 𝑤 |
  • E | 𝑤 | + | 𝑧 | ≥ | 𝑧 + 𝑤 |

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