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Worksheet: The Argument of a Complex Number

Q1:

A complex number is multiplied by another complex number 𝑧 , and then by the complex conjugate 𝑧 βˆ— . How is the argument of the original complex number affected?

  • AIt is increased by twice the argument of 𝑧 βˆ— .
  • BIt is increased by twice the argument of 𝑧 .
  • CIt is increased by πœ‹ .
  • DIt is unchanged.
  • EIt is increased by the argument of 𝑧 βˆ— .

Q2:

Consider the complex number 𝑧 = 7 + 7 𝑖 .

Find the argument of 𝑧 .

  • A 3 πœ‹ 4
  • B 7 √ 2
  • C7
  • D πœ‹ 4
  • E πœ‹ 2

Hence, find the argument of 𝑧 4 .

  • A πœ‹
  • B 2 πœ‹
  • C πœ‹ 1 6
  • D ο€» πœ‹ 4  4
  • E πœ‹ 4

Q3:

Given that principal argument ( 𝑍 ) = 5 πœ‹ 6 , determine principal argument ο€Ή 𝑍  2 .

  • A πœ‹ 6
  • B πœ‹ 3
  • C βˆ’ πœ‹ 6
  • D βˆ’ πœ‹ 3
  • E βˆ’ 2 πœ‹ 3

Q4:

Given that principal argument of 𝑍 = 1 3 πœ‹ 1 2 2 1 and principal argument of 𝑍 = 3 πœ‹ 4 2 2 , determine the principal argument of 𝑍 𝑍 4 1 2 2 .

  • A βˆ’ 1 1 πœ‹ 1 2
  • B 7 πœ‹ 1 2
  • C βˆ’ πœ‹ 1 2
  • D βˆ’ πœ‹ 6

Q5:

What is the argument of the complex number π‘Ž + 𝑏 𝑖 , where π‘Ž > 0 and 𝑏 > 0 ?

  • A t a n     ο€Ώ 𝑏 √ π‘Ž + 𝑏 
  • B t a n     ο€Ώ π‘Ž √ π‘Ž + 𝑏 
  • C t a n   ο€» π‘Ž 𝑏 
  • D t a n   ο€½ 𝑏 π‘Ž 
  • E t a n      √ π‘Ž + 𝑏 π‘Ž 

Q6:

Given that the principal argument of and the principal argument of , determine the principal argument of .

  • A
  • B
  • C
  • D
  • E

Q7:

Find the argument of the complex number 4 + 3 𝑖 in radians. Give your answer correct to two decimal places.

  • A0.67
  • B 0.54
  • C0.93
  • D 0.64
  • E0.89

Q8:

What does the argument of a complex number represent?

  • A the angle it makes with the positive imaginary axis
  • B its distance from the origin in the complex plane
  • C its real coordinate in the complex plane
  • D the angle it makes with the positive real axis
  • E its imaginary coordinate in the complex plane

Q9:

Given a complex number 𝑍 , where the principal argument of 𝑍 is πœƒ = 1 1 πœ‹ 1 2 , determine the principal argument of 1 0 𝑍 .

  • A 5 5 πœ‹ 6
  • B 1 7 πœ‹ 1 2
  • C 2 9 πœ‹ 3
  • D 1 1 πœ‹ 1 2

Q10:

Find the argument of the complex number 2 βˆ’ 7 𝑖 in radians. Give your answer correct to two decimal places.

  • A βˆ’ 0 . 7 7
  • B0.27
  • C βˆ’ 0 . 2 8
  • D βˆ’ 1 . 2 9
  • E1.3

Q11:

What is the argument of the complex number 𝑏 𝑖 , where 𝑏 < 0 ?

  • A βˆ’ πœ‹
  • B πœ‹
  • C πœ‹ 2
  • D βˆ’ πœ‹ 2
  • E πœ‹ 3

Q12:

What is the argument of the complex number 4 𝑖 ?

  • A βˆ’ πœ‹ 2
  • B πœ‹
  • C πœ‹ 3
  • D πœ‹ 2
  • E βˆ’ πœ‹