AMC12 Complex Numbers 2

Question 1

What is the sum of all cube roots of unity?

Accepted Answer

0

Answer

The cube roots of unity 1, ω, ω² are roots of z³ − 1 = 0, and by Vieta's formulas their sum equals 0.

Question 2

Using De Moivre's theorem, what is (cos(π/4) + i·sin(π/4))^8?

Accepted Answer

1

Answer

By De Moivre's theorem: cos(8·π/4) + i·sin(8·π/4) = cos(2π) + i·sin(2π) = 1.

Question 3

How many distinct solutions does z⁴ = 1 have in the complex numbers?

Accepted Answer

4

Answer

A degree-4 polynomial has exactly 4 complex roots; the four 4th roots of unity are 1, i, −1, −i.

Question 4

If z = 2(cos(π/3) + i·sin(π/3)), what is |z|?

Accepted Answer

2

Answer

In polar form z = r·(cos θ + i·sin θ), the modulus is r = 2.

Question 5

What is the principal argument of z = −1 + i√3?

Accepted Answer

2π/3

Answer

The point −1 + i√3 lies in the second quadrant; tan⁻¹(√3/1) = π/3, so the argument is π − π/3 = 2π/3.

AMC12 - American Mathematics Competition 12 Practice Test