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 ?

  • A √ 2
  • B3
  • C1
  • D √ 1 0
  • E10

Q3:

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

  • A √ 1 0
  • B3
  • C7
  • D √ 5 8
  • 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 √ 2
  • B | 𝑍 | = 8 0
  • C | 𝑍 | = 3 2
  • D | 𝑍 | = 4 √ 5
  • E | 𝑍 | = 4 √ 3

Q6:

Given that 𝑍 = 3 𝑖 , find | 𝑍 | .

  • A − 3
  • B 3 𝑖
  • C − 3 𝑖
  • D3
  • E 3 + 3 𝑖

Q7:

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

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

Q8:

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

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

Q9:

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

Q10:

Given that 𝑍 = 6 3 𝜋 4 + 𝑖 3 𝜋 4 c o s s i n , find | 𝑍 | .

Q11:

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

Q12:

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

  • A 𝑎 + 𝑏 2 2
  • B 𝑎 + 𝑏
  • C √ 𝑎 + 𝑏
  • D √ 𝑎 + 𝑏 2 2
  • E √ 𝑎 − 𝑏 2 2

Q13:

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

  • A 𝑖
  • B0
  • C2
  • D1

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

Q15:

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

Q16:

What is the magnitude of the product of 𝑧 = 𝑟 ( 𝜃 + 𝑖 𝜃 ) 1 c o s s i n and 𝑧 = 𝑠 ( 𝜑 + 𝑖 𝜑 ) 2 c o s s i n ?

  • A 𝑟 𝑠
  • B 𝑟 + 𝑠
  • C 𝜃 + 𝜑
  • D 𝑟 𝑠
  • E 𝜃 × 𝜑

Q17:

Find the modulus and the principal amplitude of the number 𝑍 = − 4 1 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ .

  • A | 𝑍 | = √ 4 1 , principal amplitude − 1 5 0 ∘
  • B | 𝑍 | = 4 1 , principal amplitude 1 5 0 ∘
  • C | 𝑍 | = √ 4 1 , principal amplitude 1 5 0 ∘
  • D | 𝑍 | = 4 1 , principal amplitude − 1 5 0 ∘

Q18:

Find the modulus and the principal amplitude of the number 𝑍 = − 3 7 5 𝜋 3 − 𝑖 5 𝜋 3 s i n c o s .

  • A | 𝑍 | = √ 3 7 , principal amplitude 𝜃 = 𝜋 6
  • B | 𝑍 | = 3 7 , principal amplitude 𝜃 = − 𝜋 6
  • C | 𝑍 | = √ 3 7 , principal amplitude 𝜃 = − 𝜋 6
  • D | 𝑍 | = 3 7 , principal amplitude 𝜃 = 𝜋 6

Q19:

Find the modulus and the principal amplitude of the number 𝑍 = 1 6 + 1 6 𝑖 3 0 5 t a n ∘ .

  • A | 𝑍 | = 1 6 3 0 5 s e c ∘ , principal amplitude 𝜃 = 2 3 5 ∘
  • B | 𝑍 | = 1 6 3 0 5 c o s ∘ , principal amplitude 𝜃 = − 5 5 ∘
  • C | 𝑍 | = 1 6 3 0 5 c s c ∘ , principal amplitude 𝜃 = 2 3 5 ∘
  • D | 𝑍 | = 1 6 3 0 5 s e c ∘ , principal amplitude 𝜃 = − 5 5 ∘

Q20:

Find the modulus and principal amplitude of the complex number − 2 ( 3 0 0 + 𝑖 3 0 0 ) s i n c o s ∘ ∘ .

  • A2, 6 0 ∘
  • B − 2 , − 3 0 ∘
  • C − 2 , 6 0 ∘
  • D2, − 3 0 ∘

Q21:

Find the modulus and the principal amplitude of the number 𝑍 = − 2 3 3 𝜋 2 − 𝑖 3 𝜋 2 c o s s i n .

  • A | 𝑍 | = √ 2 3 , principal amplitude 𝜃 = − 𝜋 2
  • B | 𝑍 | = 2 3 , principal amplitude 𝜃 = 𝜋 2
  • C | 𝑍 | = √ 2 3 , principal amplitude 𝜃 = 𝜋 2
  • D | 𝑍 | = 2 3 , principal amplitude 𝜃 = − 𝜋 2

Q22:

Given the complex number 𝑧 = 𝑎 + 𝑏 𝑖 , what is the modulus of 𝑧 2 ?

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

Q23:

Given that 𝑍 = 4 + 𝑖 , find | 𝑍 | .

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

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