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

**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 and on the given complex plane?

- A
- B
- C
- D
- E

**Q5: **

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

- AReflection in the line
- BReflection in the imaginary axis
- CReflection in the line
- DReflection in the real axis
- ERotation by about the origin

**Q6: **

In what quadrant does lie?

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

**Q7: **

Find the value of given on the Argand diagram below.

- A
- B
- C
- D

**Q8: **

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

- A
- B
- C
- D

**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 .

**Q11: **

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

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

**Q12: **

Consider the complex number .

Find the modulus of .

- A1
- B3
- C
- D
- E

Hence, find the modulus of .

- A
- B10
- C
- D243
- E

**Q13: **

Find the complex number such that lies at the midpoint of and when they are represented on a complex plane.

- A
- B
- C
- D
- E

**Q14: **

Given that , find the principal argument of rounded to the nearest two decimal places.

**Q15: **

Find the possible real values of such that the distance between the complex number and the complex number 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 , find the principal argument of rounded to the nearest two decimal places.

**Q17: **

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

- A reflection in the origin point
- B reflection in the -axis
- C reflection in the -axis

**Q18: **

Given that , find the principal argument of rounded to the nearest two decimal places.

**Q19: **

Given that , find the principal argument of rounded to the nearest two decimal places.

**Q20: **

Given that , find the principal argument of .

- A
- B
- C0
- D

**Q21: **

Given that , find the principal argument of .

- A
- B
- C
- D

**Q22: **

What is the real part of the complex number shown?

**Q23: **

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?

- A1st or 2nd
- B2nd or 4th
- C3rd or 4th
- 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?

- 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 and , where , , , and are real, when they are represented on a complex plane?

- A
- B
- C
- D
- E

**Q25: **

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

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