AMC12 Complex Numbers 3

Question 1

For how many integers n with 1 ≤ n ≤ 100 is iⁿ + i^(−n) = 0?

Accepted Answer

50

Answer

25

Answer

75

Answer

100

Question 2

If z satisfies z + |z| = 2 + i, what is |z|?

Accepted Answer

5/4

Answer

3/4

Answer

√2

Answer

2

Question 3

What is the product of all primitive 6th roots of unity?

Accepted Answer

1

Answer

−1

Answer

0

Answer

i

Question 4

In the complex plane, what is the area of the triangle with vertices 0, z, and iz where z = 3 + 4i?

Accepted Answer

25/2

Answer

5

Answer

25

Answer

10

Question 5

If |z − 1| = |z + 1|, which best describes the locus of z in the complex plane?

Accepted Answer

The imaginary axis

Answer

The real axis

Answer

The unit circle

Answer

A parabola

Question 6

What is the value of (1 + i√3)^6?

Accepted Answer

64

Answer

−64

Answer

64i

Answer

8

Question 7

Let z₁ and z₂ be complex numbers with |z₁| = 2 and |z₂| = 3. What is the maximum possible value of |z₁ + z₂|?

Accepted Answer

5

Answer

1

Answer

6

Answer

√13