Worksheet: Modulus of a Complex Number

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 3𝑖?

  • A3
  • B10
  • C2
  • D1
  • E10

Q3:

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

  • A10
  • B58
  • 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|=42
  • D|𝑍|=43
  • E|𝑍|=45

Q6:

Given that 𝑍=3𝑖, find |𝑍|.

Q7:

Given that 𝑍=33𝑖, find |𝑍|.

  • A|𝑍|=32
  • B|𝑍|=12
  • C|𝑍|=6
  • D|𝑍|=3
  • E|𝑍|=23

Q8:

Given that 𝑍=225𝑖, determine ||𝑍||.

  • A26
  • B26
  • C225𝑖
  • D2+25𝑖

Q9:

If 𝑟=5+2𝑖 and 𝑠=52𝑖, what is the modulus of 𝑟+𝑠?

Q10:

Given that 𝑍=39𝑖1+3𝑖, determine |𝑍|.

Q11:

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

  • A𝑎+𝑏
  • B𝑎+𝑏
  • C𝑎𝑏
  • D𝑎+𝑏
  • E𝑎+𝑏

Q12:

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

Q13:

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

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

Q14:

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

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

Q15:

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

  • A2𝑎+𝑏
  • B2𝑎+𝑏
  • C(𝑎𝑏)
  • D𝑎+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)(88𝑖)=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|=|𝑧𝑧| and 𝑧=12+5𝑖, which of the following is a possible value for 𝑧?

  • A1+5𝑖
  • B128𝑖
  • C258𝑖
  • D12+8𝑖
  • E12+5𝑖

Q19:

If 𝑧=49𝑖 and 𝑧=33𝑖, what is |𝑧𝑧|?

  • A13
  • B85
  • C85
  • D145
  • E145

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
  • B6
  • C3
  • D3

Q22:

What is |𝑧| given that 𝑧=28𝑖?

  • A36
  • B100
  • C217
  • D6
  • E215

Q23:

Consider the complex numbers 𝑧=34𝑖 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|=242, |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=53𝑖.

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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