AMC12 Complex Numbers 3

Question 1

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

Accepted Answer

50

Answer

iⁿ + i^(−n) = 0 exactly when n is odd: for odd n, iⁿ = i or −i and i^(−n) is its opposite, giving 50 values.

Question 2

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

Accepted Answer

5/4

Answer

Writing z = x + yi gives y = 1 and x + √(x²+1) = 2; solving yields x = 3/4 and |z| = 2 − 3/4 = 5/4.

Question 3

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

Accepted Answer

1

Answer

The primitive 6th roots are e^(iπ/3) and e^(5iπ/3); their product is e^(2πi) = 1.

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

Multiplication by i rotates z by 90°, so the triangle is a right isosceles triangle with legs |z| = 5; area = ½·5·5 = 25/2.

Question 5

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

Accepted Answer

The imaginary axis

Answer

z equidistant from 1 and −1 lies on the perpendicular bisector of the segment [−1, 1], which is the imaginary axis.

AMC12 - American Mathematics Competition 12 Practice Test