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Lesson: Argand Diagram

In this lesson, we will learn how to identify complex numbers plotted on an Argand diagram and discover their geometric properties.

Sample Question Videos

01:08

Worksheet • 16 Questions • 1 Video

Q1:

Find the value of 𝑍 given 𝑍 on the Argand diagram below.

  • A 𝑍 = βˆ’ 3 βˆ’ 5 𝑖
  • B 𝑍 = 3 + 5 𝑖
  • C 𝑍 = βˆ’ 3 + 5 𝑖
  • D 𝑍 = 3 βˆ’ 5 𝑖

Q2:

Given that the complex number 𝑍 is represented by the point ( βˆ’ 4 , βˆ’ 4 ) on the Argand diagram below, find | 𝑍 | .

  • A | 𝑍 | = 4 √ 2
  • B | 𝑍 | = 0
  • C | 𝑍 | = 3 2
  • D | 𝑍 | = 1 6
  • E | 𝑍 | = 4

Q3:

What does the modulus of a complex number represent?

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

Q4:

What complex number lies at the midpoint of 𝑧 1 and 𝑧 2 on the given complex plane?

  • A 2 + 2 𝑖
  • B 8 + 1 0 𝑖
  • C 4 + 4 𝑖
  • D 4 + 5 𝑖
  • E 5 + 4 𝑖

Q5:

Describe the geometric transformation that maps every complex number 𝑧 to its conjugate 𝑧 βˆ— .

  • Areflection in the real axis
  • Brotation by 1 8 0 ∘ about the origin
  • Creflection in the imaginary axis
  • Dreflection in the line R e I m ( 𝑧 ) = βˆ’ ( 𝑧 )
  • Ereflection in the line R e I m ( 𝑧 ) = ( 𝑧 )

Q6:

In what quadrant does 𝑧 βˆ— lie?

  • Athe third quadrant
  • Bthe first quadrant
  • Cthe fourth quadrant
  • Dthe second quadrant

Q7:

Find the value of Μ„ 𝑍 given 𝑍 on the Argand diagram below.

  • A Μ„ 𝑍 = βˆ’ 3 βˆ’ 3 𝑖
  • B Μ„ 𝑍 = 3 + 3 𝑖
  • C Μ„ 𝑍 = βˆ’ 3 + 3 𝑖
  • D Μ„ 𝑍 = 3 βˆ’ 3 𝑖

Q8:

If the number 𝑍 = 8 + 𝑖 is represented on Argand diagram by the point 𝐴 , determine the Cartesian coordinates of that point.

  • A ( 8 , 1 )
  • B ( βˆ’ 8 , βˆ’ 1 )
  • C ( 8 , βˆ’ 1 )
  • D ( βˆ’ 8 , 1 )

Q9:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with .

  • Aa translation by
  • Ba translation by
  • Ca translation by
  • Da translation by
  • Ea translation by

Q10:

Using the Argand diagram shown, find the value of 𝑧 + 𝑧   .

Q11:

The numbers in the complex plane are mapped to their product with a particular complex number 𝑧 . Given that this transforms the complex plane by a dilation with centre the origin followed by a rotation by πœ‹ radians about the origin, what kind of number is 𝑧 ?

  • Aa negative real number
  • Ba positive real number
  • Ca negative imaginary number
  • Da positive imaginary number

Q12:

Consider the complex number 𝑧 = 3 βˆ’ 𝑖 .

Find the modulus of 𝑧 .

  • A √ 1 0
  • B √ 2
  • C3
  • D √ 8
  • E1

Hence, find the modulus of 𝑧 5 .

  • A 1 0 0 √ 1 0
  • B √ 1 0
  • C243
  • D10
  • E 1 0 √ 1 0

Q13:

Find the complex number 𝑧 such that 4 + 3 𝑖 lies at the midpoint of 𝑧 and 3 βˆ’ 4 𝑖 when they are represented on a complex plane.

  • A 5 + 1 0 𝑖
  • B 5 + 1 3 𝑖
  • C 1 + 7 𝑖
  • D 7 βˆ’ 𝑖
  • E 1 1 + 2 𝑖

Q14:

Given that 𝑍 = 9 + 3 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q15:

Find the possible real values of 𝑏 such that the distance between the complex number βˆ’ 6 + 7 𝑖 and the complex number βˆ’ 3 + 𝑏 𝑖 is 5.

  • A 3 or 11
  • B 0.5 or 14.5
  • C 5
  • D 1.7 or 15.7
  • E 5 or 9

Q16:

Given that 𝑍 = βˆ’ 5 + 9 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.
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