# Lesson Worksheet: Argand Diagram Mathematics

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

**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

**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).

**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.

**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 .

**Q21: **

True or False: Given that is a point on an Argand diagram representing a complex number, then the midpoint between and its conjugate lies on the real axis.

- ATrue
- BFalse

**Q25: **

Using the Argand diagram shown, find the value of .

- A
- B
- C
- D
- E