# Worksheet: Argand Diagram

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

Q1:

Find the value of given on the Argand diagram below. • A
• B
• C
• D

Q2:

What is the real part of the complex number shown? Q3:

What is the imaginary part of the complex number shown? Q4:

If the number is represented on Argand diagram by the point , determine the Cartesian coordinates of that point.

• A
• B
• C
• D

Q5:

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

• Afourth
• Bsecond
• Cthird
• Dfirst

Q6:

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

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

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

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

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

Q7:

The Argand diagram shows the complex numbers , , , and . Which of these numbers is ? • A
• B
• C
• D

Q8:

What complex number lies at the midpoint of and on the given complex plane? • A
• B
• C
• D
• E

Q9: Q10:

Find the value of given on the Argand diagram below. • A
• B
• C
• D

Q11:

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

• AReflection in the line
• BReflection in the line
• CRotation by about the origin
• DReflection in the real axis
• EReflection in the imaginary axis

Q12:

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

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

Q13:

Seven complex numbers , , , , , , and are represented on the Argand diagram. Which of the complex numbers is ?

• A
• B
• C
• D
• E

What complex number is represented by ?

• A
• B
• C
• D
• E

Which complex number has equal real and imaginary parts?

• A
• B
• C
• D
• E

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

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

Q14:

Using the Argand diagram shown, find the value of . Q15:

Four complex numbers , , , and are shown on the Argand diagram. Find the image of the points , , , and under a transformation that maps to .

• A
• B
• C
• D
• E

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

• AThe transformation represents a reflection in the .
• BThe transformation represents a counterclockwise rotation by an angle of radians about the origin.
• CThe transformation represents a counterclockwise rotation by an angle of radians about the origin.
• DThe transformation represents a clockwise rotation by an angle of radians about the origin.
• EThe transformation represents a reflection in the .

Q16:

Three complex numbers , , and are represented on the Argand diagram. Find the image of points , , and under the transformation that maps to .

• A
• B
• C
• D
• E

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 with scale factor two centered at .
• BThe transformation represents a dilation with scale factor two centered at the origin.
• CThe transformation represents a dilation parallel to the with scale factor two centered at .
• DThe transformation represents a counterclockwise rotation by radians about the origin followed by a dilation with scale factor two centered at the origin.
• EThe transformation represents a dilation with scale factor centered at the origin.

Q17:

In which quadrant of the Argand plane does the complex number lie?

Q18:

Find all the solutions to .

• A
• B
• C
• D
• E

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

• 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 .
• 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 .
• EThe solutions form a regular hexagon centered at the origin with one vertex at .