AMC12 Complex Numbers

Question 1

If z = 3 + 4i, what is |z|, the modulus of z?

Accepted Answer

5

Answer

The modulus is √(3² + 4²) = √(9 + 16) = √25 = 5.

Question 2

What is the product (2 + 3i)(1 − 2i)?

Accepted Answer

8 − i

Answer

Expanding: 2 − 4i + 3i − 6i² = 2 − i + 6 = 8 − i, since i² = −1.

Question 3

What is i^23?

Accepted Answer

−i

Answer

Since i repeats with period 4 and 23 ≡ 3 (mod 4), i^23 = i^3 = −i.

Question 4

If z = 1 + i, what is z^4?

Accepted Answer

−4

Answer

z² = (1+i)² = 2i, so z⁴ = (2i)² = 4i² = −4.

Question 5

What is the real part of (3 + 4i) / (1 + 2i)?

Accepted Answer

11/5

Answer

Multiply numerator and denominator by the conjugate (1−2i): (3+4i)(1−2i)/5 = (11−2i)/5, giving real part 11/5.

AMC12 - American Mathematics Competition 12 Practice Test