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 2𝑖?

Q2:

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

  • A3
  • B10
  • C√2
  • D1
  • E√10

Q3:

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

  • A√10
  • B√58
  • C58
  • D3
  • E7

Q4:

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

Q5:

Given that 𝑍=8+4𝑖, find |𝑍|.

  • A|𝑍|=32
  • B|𝑍|=80
  • C|𝑍|=4√2
  • D|𝑍|=4√3
  • E|𝑍|=4√5

Q6:

Given that 𝑍=3𝑖, find |𝑍|.

Q7:

Given that 𝑍=−3−√3𝑖, find |𝑍|.

  • A|𝑍|=3√2
  • B|𝑍|=12
  • C|𝑍|=√6
  • D|𝑍|=3
  • E|𝑍|=2√3

Q8:

Given that 𝑍=2−2√5𝑖, determine ||𝑍||.

  • A2√6
  • B−2√6
  • C−2−2√5𝑖
  • D2+2√5𝑖

Q9:

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

Q10:

Given that 𝑍=3−9𝑖1+3𝑖, determine |𝑍|.

Q11:

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

  • A𝑎+𝑏
  • B√𝑎+𝑏
  • C√𝑎−𝑏
  • D𝑎+𝑏
  • E√𝑎+𝑏

Q12:

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

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|.

  • A𝑍=𝑖, |𝑍|=1
  • B𝑍=1+𝑖, |𝑍|=√2
  • C𝑍=−𝑖, |𝑍|=1
  • D𝑍=1−𝑖, |𝑍|=√2

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|.

  • A𝑍=−𝑖, |𝑍|=1
  • B𝑍=1+𝑖, |𝑍|=√2
  • C𝑍=1−𝑖, |𝑍|=√2
  • D𝑍=𝑖, |𝑍|=1

Q15:

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

  • A2√𝑎+𝑏
  • B2𝑎+𝑏
  • C(𝑎𝑏)
  • D√𝑎+𝑏
  • E𝑎+𝑏

Q16:

Given that 𝑍=4+𝑖, find |𝑍|.

  • A|𝑍|=4
  • B|𝑍|=17
  • C|𝑍|=2
  • D|𝑍|=√15
  • E|𝑍|=√17

Q17:

Given that (𝑎+𝑏𝑖)(8−8𝑖)=14+6𝑖, where 𝑎 and 𝑏 are real numbers, determine the value of 16(𝑎+𝑏).

Q18:

Consider the complex numbers 𝑧 and 𝑧. Given that |𝑧|=|𝑧−𝑧| and 𝑧=12+5𝑖, which of the following is a possible value for 𝑧?

  • A1+5𝑖
  • B12−8𝑖
  • C25−8𝑖
  • D12+8𝑖
  • E−12+5𝑖

Q19:

If 𝑧=−4−9𝑖 and 𝑧=3−3𝑖, what is |𝑧−𝑧|?

  • A13
  • B85
  • C√85
  • D145
  • E√145

Q20:

What does the modulus of a complex number represent?

  • Aits real coordinate in the complex plane
  • Bthe angle it makes with the positive imaginary axis
  • Cits imaginary coordinate in the complex plane
  • Dthe angle it makes with the positive real axis
  • Eits distance from the origin in the complex plane

Q21:

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

  • A6
  • B−6
  • C3
  • D−3

Q22:

What is |𝑧| given that 𝑧=2−8𝑖?

  • A36
  • B100
  • C2√17
  • D6
  • E2√15

Q23:

Consider the complex numbers 𝑧=3−4𝑖 and 𝑤=−15+8𝑖.

Find |𝑧| and |𝑤|.

  • A|𝑧|=√17, |𝑤|=√5
  • B|𝑧|=25, |𝑤|=289
  • C|𝑧|=5, |𝑤|=17
  • D|𝑧|=√7, |𝑤|=√23
  • E|𝑧|=√7, |𝑤|=√161

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

  • A|𝑧𝑤|=85, |𝑧||𝑤|=|𝑧𝑤|
  • B|𝑧𝑤|=√85, |𝑧||𝑤|≥|𝑧𝑤|
  • C|𝑧𝑤|=2√42, |𝑧||𝑤|≤|𝑧𝑤|
  • D|𝑧𝑤|=22, |𝑧|+|𝑤|=|𝑧𝑤|
  • E|𝑧𝑤|=7,225, √|𝑧||𝑤|=|𝑧𝑤|

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

  • A||𝑧𝑤||=−12, |𝑧|−|𝑤|=||𝑧𝑤||
  • B||𝑧𝑤||=175, |𝑧||𝑤|=||𝑧𝑤||
  • C||𝑧𝑤||=25289, |𝑧||𝑤|≥||𝑧𝑤||
  • D||𝑧𝑤||=517, |𝑧||𝑤|=||𝑧𝑤||
  • E||𝑧𝑤||=517, |𝑧||𝑤|=||𝑧𝑤||

Q24:

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

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|

Q25:

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

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