AMC12 Complex Numbers 1 — Questions and Answers

Correct answer: 5

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

Correct answer: 8 − i

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

Correct answer: −i

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

Correct answer: −4

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

Correct answer: 11/5

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

Correct answer: 3 + i

Add real and imaginary parts separately: (1+2) + (2−1)i = 3 + i.

Correct answer: √3

z = √3/2 + i/2 and z̄ = √3/2 − i/2, so z + z̄ = √3 (the imaginary parts cancel).

AMC12 Complex Numbers 1

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

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