Worksheet: Argand Diagrams

In this worksheet, we will practice identifying complex numbers plotted on an Argand diagram and discovering their geometric properties.

Q1:

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

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

Q2:

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

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

Q3:

What does the modulus of a complex number represent?

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

Q4:

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

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

Q5:

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

  • AReflection in the line R e ( 𝑧 ) = I m ( 𝑧 )
  • BReflection in the imaginary axis
  • CReflection in the line R e ( 𝑧 ) = I m ( 𝑧 )
  • DReflection in the real axis
  • ERotation by 1 8 0 about the origin

Q6:

In what quadrant does 𝑧 lie?

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

Q7:

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

  • A ̄ 𝑍 = 3 3 𝑖
  • B ̄ 𝑍 = 3 + 3 𝑖
  • C ̄ 𝑍 = 3 + 3 𝑖
  • D ̄ 𝑍 = 3 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 ( 8 , 1 )
  • C ( 8 , 1 )
  • D ( 8 , 1 )

Q9:

Describe the geometric transformation that occurs when numbers in the complex plane are mapped to their sum with 3 2 𝑖 .

  • Aa translation by 3 2
  • Ba translation by 2 3
  • Ca translation by 2 3
  • Da translation by 3 2
  • Ea translation by 3 2

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 radians about the origin, what kind of number is 𝑧 ?

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

Q12:

Consider the complex number 𝑧 = 3 𝑖 .

Find the modulus of 𝑧 .

  • A1
  • B3
  • C 8
  • D 1 0
  • E 2

Hence, find the modulus of 𝑧 5 .

  • A 1 0 0 1 0
  • B10
  • C 1 0 1 0
  • D243
  • E 1 0

Q13:

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

  • A 1 1 + 2 𝑖
  • B 1 + 7 𝑖
  • C 7 𝑖
  • D 5 + 1 0 𝑖
  • E 5 + 1 3 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 6 + 7 𝑖 and the complex number 3 + 𝑏 𝑖 is 5.

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

Q16:

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

Q17:

Given that points 𝐴 and 𝐵 represent the complex numbers 𝑍 and 𝑍 on an Argand diagram, then 𝐵 is the image of 𝐴 under which transformation?

  • A reflection in the origin point
  • B reflection in the 𝑦 -axis
  • C reflection in the 𝑥 -axis

Q18:

Given that 𝑍 = 3 7 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q19:

Given that 𝑍 = 6 4 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q20:

Given that 𝑍 = 7 𝑖 , find the principal argument of 𝑍 .

  • A 𝜋
  • B 𝜋 2
  • C0
  • D 𝜋 2

Q21:

Given that 𝑍 = 1 2 + 3 2 𝑖 , find the principal argument of 𝑍 .

  • A 𝜋 3
  • B 5 𝜋 6
  • C 𝜋 3
  • D 2 𝜋 3

Q22:

What is the real part of the complex number shown?

Q23:

Let us consider a complex number, 𝑧 , with nonzero real and imaginary parts.

If the real and imaginary parts of 𝑧 are of the same sign, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A1st or 2nd
  • B2nd or 4th
  • C3rd or 4th
  • D1st or 3rd
  • E1st or 4th

If the real and imaginary parts of 𝑧 are of opposite signs, in which quadrant(s) of the Argand diagram could 𝑧 appear?

  • A2nd or 4th
  • B3rd or 4th
  • C1st or 2nd
  • D1st or 3rd
  • E1st or 4th

Q24:

What complex number lies at the midpoint of the complex numbers 𝑎 + 𝑏 𝑖 and 𝑥 + 𝑦 𝑖 , where 𝑎 , 𝑏 , 𝑥 , and 𝑦 are real, when they are represented on a complex plane?

  • A 𝑏 𝑦 2 + 𝑎 𝑥 2 𝑖
  • B ( 𝑎 + 𝑥 ) + ( 𝑏 + 𝑦 ) 𝑖
  • C 𝑎 𝑥 2 + 𝑏 𝑦 2 𝑖
  • D 𝑎 + 𝑥 2 + 𝑏 + 𝑦 2 𝑖
  • E ( 𝑎 𝑥 ) + ( 𝑏 𝑦 ) 𝑖

Q25:

In which quadrant of the Argand diagram does the complex number 3 2 𝑖 lie?

  • A second
  • B third
  • C first
  • D fourth

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