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 2๐‘–?

Q2:

What is the modulus of the complex number 3โˆ’๐‘–?

  • A3
  • B10
  • Cโˆš2
  • D1
  • Eโˆš10

Q3:

What is the modulus of the complex number 3+7๐‘–?

  • Aโˆš10
  • Bโˆš58
  • C58
  • D3
  • E7

Q4:

What is the modulus of the complex number 3+4๐‘–?

Q5:

Given that ๐‘=8+4๐‘–, find |๐‘|.

  • A|๐‘|=32
  • B|๐‘|=80
  • C|๐‘|=4โˆš2
  • D|๐‘|=4โˆš3
  • E|๐‘|=4โˆš5

Q6:

Given that ๐‘=3๐‘–, find |๐‘|.

Q7:

Given that ๐‘=โˆ’3โˆ’โˆš3๐‘–, find |๐‘|.

  • A|๐‘|=3โˆš2
  • B|๐‘|=12
  • C|๐‘|=โˆš6
  • D|๐‘|=3
  • E|๐‘|=2โˆš3

Q8:

Given that ๐‘=2โˆ’2โˆš5๐‘–, determine ||๐‘||.

  • A2โˆš6
  • Bโˆ’2โˆš6
  • Cโˆ’2โˆ’2โˆš5๐‘–
  • D2+2โˆš5๐‘–

Q9:

If ๐‘Ÿ=5+2๐‘– and ๐‘ =5โˆ’2๐‘–, what is the modulus of ๐‘Ÿ+๐‘ ?

Q10:

Given that ๐‘=3โˆ’9๐‘–1+3๐‘–, determine |๐‘|.

Q11:

What is the modulus of the complex number ๐‘Ž+๐‘๐‘–, where ๐‘Ž and ๐‘ are real?

  • A๐‘Ž+๐‘๏Šจ๏Šจ
  • Bโˆš๐‘Ž+๐‘
  • Cโˆš๐‘Žโˆ’๐‘๏Šจ๏Šจ
  • D๐‘Ž+๐‘
  • Eโˆš๐‘Ž+๐‘๏Šจ๏Šจ

Q12:

If ๐‘=1๐‘, where ๐‘ is a complex number, what is |๐‘|?

Q13:

Given that ๐‘=(๐‘Ž+๐‘)+๐‘–(๐‘Žโˆ’๐‘)(๐‘Žโˆ’๐‘)โˆ’๐‘–(๐‘Ž+๐‘), where ๐‘Žโˆˆโ„ and ๐‘โˆˆโ„, write ๐‘ in algebraic form and hence find |๐‘|.

  • A๐‘=๐‘–, |๐‘|=1
  • B๐‘=1+๐‘–, |๐‘|=โˆš2
  • C๐‘=โˆ’๐‘–, |๐‘|=1
  • D๐‘=1โˆ’๐‘–, |๐‘|=โˆš2

Q14:

Given that ๐‘=(๐‘Ž+๐‘)โˆ’๐‘–(๐‘Žโˆ’๐‘)(๐‘Žโˆ’๐‘)+๐‘–(๐‘Ž+๐‘), where ๐‘Žโˆˆโ„ and ๐‘โˆˆโ„, write ๐‘ in algebraic form and hence find |๐‘|.

  • A๐‘=โˆ’๐‘–, |๐‘|=1
  • B๐‘=1+๐‘–, |๐‘|=โˆš2
  • C๐‘=1โˆ’๐‘–, |๐‘|=โˆš2
  • D๐‘=๐‘–, |๐‘|=1

Q15:

Given the complex number ๐‘ง=๐‘Ž+๐‘๐‘–, what is the modulus of ๐‘ง๏Šจ?

  • A2๏€ปโˆš๐‘Ž+๐‘๏‡๏Šจ๏Šจ
  • B2๏€น๐‘Ž+๐‘๏…๏Šจ๏Šจ
  • C(๐‘Ž๐‘)๏Šจ
  • Dโˆš๐‘Ž+๐‘๏Šจ๏Šจ
  • E๐‘Ž+๐‘๏Šจ๏Šจ

Q16:

Given that ๐‘=4+๐‘–, find |๐‘|.

  • A|๐‘|=4
  • B|๐‘|=17
  • C|๐‘|=2
  • D|๐‘|=โˆš15
  • E|๐‘|=โˆš17

Q17:

Consider the complex numbers ๐‘ง๏Šง and ๐‘ง๏Šจ. Given that |๐‘ง|=|๐‘งโˆ’๐‘ง|๏Šง๏Šจ๏Šง and ๐‘ง=12+5๐‘–๏Šง, which of the following is a possible value for ๐‘ง๏Šจ?

  • A1+5๐‘–
  • B12โˆ’8๐‘–
  • C25โˆ’8๐‘–
  • D12+8๐‘–
  • Eโˆ’12+5๐‘–

Q18:

If ๐‘ง=โˆ’4โˆ’9๐‘–๏Šง and ๐‘ง=3โˆ’3๐‘–๏Šจ, what is |๐‘งโˆ’๐‘ง|๏Šจ๏Šง?

  • A13
  • B85
  • Cโˆš85
  • D145
  • Eโˆš145

Q19:

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

Q20:

Given that |๐‘|=|๐‘+6|, determine the real part of the complex number ๐‘.

  • A6
  • Bโˆ’6
  • C3
  • Dโˆ’3

Q21:

What is |๐‘ง| given that ๐‘ง=2โˆ’8๐‘–?

  • A36
  • B100
  • C2โˆš17
  • D6
  • E2โˆš15

Q22:

Consider the complex numbers ๐‘ง=3โˆ’4๐‘– and ๐‘ค=โˆ’15+8๐‘–.

Find |๐‘ง| and |๐‘ค|.

  • A|๐‘ง|=โˆš17, |๐‘ค|=โˆš5
  • B|๐‘ง|=25, |๐‘ค|=289
  • C|๐‘ง|=5, |๐‘ค|=17
  • D|๐‘ง|=โˆš7, |๐‘ค|=โˆš23
  • E|๐‘ง|=โˆš7, |๐‘ค|=โˆš161

Calculate |๐‘ง๐‘ค|. How does this compare with |๐‘ง||๐‘ค|?

  • A|๐‘ง๐‘ค|=85, |๐‘ง||๐‘ค|=|๐‘ง๐‘ค|
  • B|๐‘ง๐‘ค|=โˆš85, |๐‘ง||๐‘ค|โ‰ฅ|๐‘ง๐‘ค|
  • C|๐‘ง๐‘ค|=2โˆš42, |๐‘ง||๐‘ค|โ‰ค|๐‘ง๐‘ค|
  • D|๐‘ง๐‘ค|=22, |๐‘ง|+|๐‘ค|=|๐‘ง๐‘ค|
  • E|๐‘ง๐‘ค|=7,225, โˆš|๐‘ง||๐‘ค|=|๐‘ง๐‘ค|

Calculate ||๐‘ง๐‘ค||. How does this compare with |๐‘ง||๐‘ค|?

  • A||๐‘ง๐‘ค||=โˆ’12, |๐‘ง|โˆ’|๐‘ค|=||๐‘ง๐‘ค||
  • B||๐‘ง๐‘ค||=175, |๐‘ง||๐‘ค|=||๐‘ง๐‘ค||๏Šฑ๏Šง
  • C||๐‘ง๐‘ค||=25289, |๐‘ง||๐‘ค|โ‰ฅ||๐‘ง๐‘ค||
  • D||๐‘ง๐‘ค||=517, |๐‘ง||๐‘ค|=||๐‘ง๐‘ค||
  • E||๐‘ง๐‘ค||=๏„ž517, ๏„Ÿ|๐‘ง||๐‘ค|=||๐‘ง๐‘ค||

Q23:

Consider the two complex numbers ๐‘ค=โˆ’1+7๐‘– and ๐‘ง=5โˆ’3๐‘–.

Calculate |๐‘ค|+|๐‘ง| to two decimal places.

Calculate |๐‘ง+๐‘ค| to two decimal places.

Which of the following relations do ๐‘ค and ๐‘ง satisfy?

  • A|๐‘ค|+|๐‘ง|=|๐‘ง+๐‘ค|
  • Bโˆš|๐‘ค|+|๐‘ง|=|๐‘ง+๐‘ค|
  • C|๐‘ค|+|๐‘ง|=2|๐‘ง+๐‘ค|
  • D|๐‘ค|+|๐‘ง|โ‰ค|๐‘ง+๐‘ค|
  • E|๐‘ค|+|๐‘ง|โ‰ฅ|๐‘ง+๐‘ค|

Q24:

If ๐‘ง=1๐‘ง, where ๐‘ง is a complex number, what is |๐‘ง|?

Q25:

What is the general form for the modulus of a complex number ๐‘ง, where ๐‘ง=๐‘Ž+๐‘๐‘– and ๐‘Ž and ๐‘ are real numbers?

  • A|๐‘ง|=๐‘Ž+๐‘๏Šจ๏Šจ
  • B|๐‘ง|=โˆš๐‘Ž+๐‘๏Šจ๏Šจ
  • C|๐‘ง|=โˆš๐‘Ž+๐‘
  • D|๐‘ง|=โˆš(๐‘Žโˆ’1)+(๐‘โˆ’1)๏Šจ๏Šจ
  • E|๐‘ง|=โˆš๐‘Žโˆ’๐‘๏Šจ๏Šจ

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