AMC12 Complex Numbers 2

Question 1

What is the sum of all cube roots of unity?

Accepted Answer

0

Answer

3

Answer

1

Answer

−1

Question 2

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

Accepted Answer

1

Answer

i

Answer

−1

Answer

−i

Question 3

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

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 4

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

Accepted Answer

2

Answer

1

Answer

√3

Answer

4

Question 5

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

Accepted Answer

2π/3

Answer

π/3

Answer

π/6

Answer

5π/6

Question 6

What is i^100?

Accepted Answer

1

Answer

i

Answer

−1

Answer

−i

Question 7

What is (1 + i)² / (1 − i)?

Accepted Answer

−1 + i

Answer

1 + i

Answer

1 − i

Answer

−1 − i