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Worksheet: Finding the Modulus of a Complex Number

Q1:

What is the modulus of the complex number 2 𝑖 ?

Q2:

Given that 𝑍 = 3 𝑖 , find | 𝑍 | .

  • A βˆ’ 3
  • B 3 𝑖
  • C βˆ’ 3 𝑖
  • D3
  • E 3 + 3 𝑖

Q3:

What is the modulus of the complex number 3 βˆ’ 𝑖 ?

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

Q4:

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

  • A √ 1 0
  • B3
  • C7
  • D √ 5 8
  • E58

Q5:

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

Q6:

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

Q7:

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

Q8:

Given that 𝑍 = βˆ’ 3 βˆ’ √ 3 𝑖 , find | 𝑍 | .

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

Q9:

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

  • A βˆ’ 2 βˆ’ 2 √ 5 𝑖
  • B βˆ’ 2 √ 6
  • C 2 + 2 √ 5 𝑖
  • D 2 √ 6

Q10:

If π‘Ÿ = 5 + 2 𝑖 and 𝑠 = 5 βˆ’ 2 𝑖 , what is the modulus of π‘Ÿ + 𝑠 ?

Q11:

Given that 𝑍 = 6 ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   c o s s i n , find | 𝑍 | .

Q12:

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

Q13:

What is the modulus of the complex number π‘Ž + 𝑏 𝑖 , where π‘Ž and 𝑏 are real?

  • A π‘Ž + 𝑏 2 2
  • B π‘Ž + 𝑏
  • C √ π‘Ž + 𝑏
  • D √ π‘Ž + 𝑏 2 2
  • E √ π‘Ž βˆ’ 𝑏 2 2

Q14:

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

  • A 𝑖
  • B0
  • C2
  • D1

Q15:

Given that 𝑍 = ( π‘Ž + 𝑏 ) + 𝑖 ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑏 ) βˆ’ 𝑖 ( π‘Ž + 𝑏 ) , where π‘Ž ∈ ℝ and 𝑏 ∈ ℝ , write 𝑍 in algebraic form and hence find | 𝑍 | .

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

Q16:

Given that 𝑍 = ( π‘Ž + 𝑏 ) βˆ’ 𝑖 ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑏 ) + 𝑖 ( π‘Ž + 𝑏 ) , where π‘Ž ∈ 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 βˆ’ 𝑖 , | 𝑍 | = √ 2
  • D 𝑍 = βˆ’ 𝑖 , | 𝑍 | = 1

Q17:

What is the magnitude of the product of 𝑧 = π‘Ÿ ( πœƒ + 𝑖 πœƒ ) 1 c o s s i n and 𝑧 = 𝑠 ( πœ‘ + 𝑖 πœ‘ ) 2 c o s s i n ?

  • A π‘Ÿ 𝑠
  • B π‘Ÿ + 𝑠
  • C πœƒ + πœ‘
  • D π‘Ÿ 𝑠
  • E πœƒ Γ— πœ‘

Q18:

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

  • A | 𝑍 | = √ 4 1 , principle amplitude βˆ’ 1 5 0 ∘
  • B | 𝑍 | = 4 1 , principle amplitude 1 5 0 ∘
  • C | 𝑍 | = √ 4 1 , principle amplitude 1 5 0 ∘
  • D | 𝑍 | = 4 1 , principle amplitude βˆ’ 1 5 0 ∘

Q19:

Find the modulus and the principle amplitude of the number 𝑍 = βˆ’ 3 7 ο€Ό ο€Ό 5 πœ‹ 3  βˆ’ 𝑖 ο€Ό 5 πœ‹ 3   s i n c o s .

  • A | 𝑍 | = √ 3 7 , principle amplitude πœƒ = πœ‹ 6
  • B | 𝑍 | = 3 7 , principle amplitude πœƒ = βˆ’ πœ‹ 6
  • C | 𝑍 | = √ 3 7 , principle amplitude πœƒ = βˆ’ πœ‹ 6
  • D | 𝑍 | = 3 7 , principle amplitude πœƒ = πœ‹ 6

Q20:

Find the modulus and the principle 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 ∘ , principle amplitude πœƒ = 2 3 5 ∘
  • B | 𝑍 | = 1 6 3 0 5 c o s ∘ , principle amplitude πœƒ = βˆ’ 5 5 ∘
  • C | 𝑍 | = 1 6 3 0 5 c s c ∘ , principle amplitude πœƒ = 2 3 5 ∘
  • D | 𝑍 | = 1 6 3 0 5 s e c ∘ , principle amplitude πœƒ = βˆ’ 5 5 ∘

Q21:

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

  • A2, 6 0 ∘
  • B βˆ’ 2 , βˆ’ 3 0 ∘
  • C βˆ’ 2 , 6 0 ∘
  • D2, βˆ’ 3 0 ∘

Q22:

Find the modulus and the principal amplitude of the number 𝑍 = βˆ’ 2 3 ο€Ό ο€Ό 3 πœ‹ 2  βˆ’ 𝑖 ο€Ό 3 πœ‹ 2   c o s s i n .

  • A | 𝑍 | = √ 2 3 , principal amplitude πœƒ = βˆ’ πœ‹ 2
  • B | 𝑍 | = 2 3 , principal amplitude πœƒ = πœ‹ 2
  • C | 𝑍 | = √ 2 3 , principal amplitude πœƒ = πœ‹ 2
  • D | 𝑍 | = 2 3 , principal amplitude πœƒ = βˆ’ πœ‹ 2

Q23:

Given the complex number 𝑧 = π‘Ž + 𝑏 𝑖 , what is the modulus of 𝑧 2 ?

  • A 2 ο€» √ π‘Ž + 𝑏  2 2
  • B √ π‘Ž + 𝑏 2 2
  • C 2 ο€Ή π‘Ž + 𝑏  2 2
  • D π‘Ž + 𝑏 2 2
  • E ( π‘Ž 𝑏 ) 2