Worksheet: Argand Diagram

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

Q1:

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

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

Q2:

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

Q3:

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

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

Q4:

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

  • AReflection in the line ReIm(𝑧)=(𝑧)
  • BReflection in the line ReIm(𝑧)=−(𝑧)
  • CRotation by 180∘ about the origin
  • DReflection in the real axis
  • EReflection in the imaginary axis

Q5:

In what quadrant does 𝑧∗ lie?

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

Q6:

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

Q7:

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 )

Q8:

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

Q9:

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?

  • Areflection in the 𝑦-axis
  • Breflection in the origin point
  • Creflection in the 𝑥-axis

Q10:

What is the real part of the complex number shown?

Q11:

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?

  • A2nd or 4th
  • B3rd or 4th
  • C1st or 2nd
  • 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?

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

Q12:

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

  • Afourth
  • Bsecond
  • Cthird
  • Dfirst

Q13:

What is the imaginary part of the complex number shown?

  • A 2 √ 5
  • B4
  • C2
  • D − 4
  • E − 2

Q14:

The Argand diagram shows the complex numbers 𝑊, 𝑋, 𝑌, and 𝑍. Which of these numbers is 2−5𝑖?

  • A 𝑍
  • B 𝑌
  • C 𝑊
  • D 𝑋

Q15:

In which quadrant of the Argand diagram does the complex number √7−3𝜋4+𝑖−3𝜋4cossin lie?

  • Asecond
  • Bthird
  • Cfirst
  • Dfourth

Q16:

In which quadrant of the Argand plane does the complex number 7+9𝑖3−4𝑖 lie?

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

Q17:

Seven complex numbers 𝑧, 𝑧, 𝑧, 𝑧, 𝑧, 𝑧, and 𝑧 are represented on the Argand diagram.

Which of the complex numbers is −3+2𝑖?

  • A 𝑧
  • B 𝑧
  • C 𝑧
  • D 𝑧
  • E 𝑧

What complex number is represented by 𝑧?

  • A − 4 + 𝑖
  • B 1 − 4 𝑖
  • C 4 + 𝑖
  • D − 4 − 𝑖
  • E − 1 − 4 𝑖

Which complex number has equal real and imaginary parts?

  • A 𝑧
  • B 𝑧
  • C 𝑧
  • D 𝑧
  • E 𝑧

Which two complex numbers are a conjugate pair? What is their geometric relationship?

  • A 𝑧 and 𝑧 are a conjugate pair; they are related by a rotation about the origin of 𝜋 radians.
  • B 𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the real axis (𝑥-axis).
  • CNone of the complex numbers are a conjugate pair.
  • D 𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the imaginary axis (𝑦-axis).
  • E 𝑧 and 𝑧 are a conjugate pair; they are related by reflection in the real axis (𝑥-axis).

Q18:

Four complex numbers 𝑧, 𝑧, 𝑧, and 𝑧 are shown on the Argand diagram.

Find the image of the points 𝑧, 𝑧, 𝑧, and 𝑧 under a transformation that maps 𝑧 to 𝑖𝑧.

  • A 𝑧 = − 3 𝑧 = − 2 + 3 𝑖 𝑧 = 2 − 𝑖 𝑧 = − 𝑖
  • B 𝑧 = 3 𝑧 = 2 − 3 𝑖 𝑧 = − 2 + 𝑖 𝑧 = 𝑖
  • C 𝑧 = 3 𝑖 𝑧 = − 3 + 2 𝑖 𝑧 = 1 − 2 𝑖 𝑧 = 1
  • D 𝑧 = − 3 𝑖 𝑧 = 3 − 2 𝑖 𝑧 = 1 − 2 𝑖 𝑧 = − 1
  • E 𝑧 = − 3 𝑧 = − 2 − 3 𝑖 𝑧 = 2 + 𝑖 𝑧 = 𝑖

By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

  • AThe transformation represents a reflection in the 𝑦-axis.
  • BThe transformation represents a counterclockwise rotation by an angle of 𝜋2 radians about the origin.
  • CThe transformation represents a counterclockwise rotation by an angle of 𝜋 radians about the origin.
  • DThe transformation represents a clockwise rotation by an angle of 𝜋2 radians about the origin.
  • EThe transformation represents a reflection in the 𝑥-axis.

Q19:

Three complex numbers 𝑧, 𝑧, and 𝑧 are represented on the Argand diagram.

Find the image of points 𝑧, 𝑧, and 𝑧 under the transformation that maps 𝑧 to 2𝑧.

  • A 𝑧 = 4 𝑧 = − 𝑖 𝑧 = − 6 + 2 𝑖
  • B 𝑧 = 4 𝑧 = − 2 𝑖 𝑧 = − 6 + 4 𝑖
  • C 𝑧 = 2 𝑧 = − 2 𝑖 𝑧 = − 3 + 4 𝑖
  • D 𝑧 = 1 𝑧 = − 𝑖 2 𝑧 = − 3 2 + 𝑖
  • E 𝑧 = 4 𝑖 𝑧 = 2 𝑧 = − 4 − 6 𝑖

By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.

  • AThe transformation represents a dilation parallel to the 𝑦-axis with scale factor two centered at 𝑦=0.
  • BThe transformation represents a dilation with scale factor two centered at the origin.
  • CThe transformation represents a dilation parallel to the 𝑥-axis with scale factor two centered at 𝑥=0.
  • DThe transformation represents a counterclockwise rotation by 𝜋2 radians about the origin followed by a dilation with scale factor two centered at the origin.
  • EThe transformation represents a dilation with scale factor 12 centered at the origin.

Q20:

Find all the solutions to 𝑧=1.

  • A 𝑧 = 𝑖 , − 𝑖 , √ 3 2 + 1 2 𝑖 , − √ 3 2 + 1 2 𝑖 , √ 3 2 − 1 2 𝑖 , − √ 3 2 − 1 2 𝑖
  • B 𝑧 = 1 , − 1 , 𝑖 , − 𝑖 , √ 2 2 + √ 2 2 𝑖 , − √ 2 2 − √ 2 2 𝑖
  • C 𝑧 = 𝑖 , − 𝑖 , √ 3 2 + 𝑖 , − √ 3 2 + 𝑖 , √ 3 2 − 𝑖 , − √ 3 2 − 𝑖
  • D 𝑧 = 1 , − 1 , 1 + √ 3 2 𝑖 , 1 − √ 3 2 𝑖 , − 1 + √ 3 2 𝑖 , − 1 − √ 3 2 𝑖
  • E 𝑧 = 1 , − 1 , 1 2 + √ 3 2 𝑖 , 1 2 − √ 3 2 𝑖 , − 1 2 + √ 3 2 𝑖 , − 1 2 − √ 3 2 𝑖

By plotting the solutions on an Argand diagram, or otherwise, describe the geometric properties of the solutions of 𝑧=1.

  • AThe solutions are evenly spaced around the unit circle centered at the origin.
  • BThe solutions are evenly spaced around a unit semicircle centered at the origin in the half plane Im(𝑧)≥0.
  • CThe solutions are evenly spaced around the unit square centered at the origin.
  • DThe solutions form a regular hexagon inscribed within the unit circle centered at the origin with one vertex at 𝑧=𝑖.
  • EThe solutions form a regular hexagon centered at the origin with one vertex at 𝑧=√32+𝑖.

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