2021 AMC 12A Spring Problem 13

Question

Of the following complex numbers

z,

which one has the property that

z^5

has the greatest real part?

-2

-\sqrt3 + i

-\sqrt2 + \sqrt2\, i

-1 + \sqrt3\, i

2i

Every choice has modulus

2,

so

z^5

has modulus

32;

only the angle matters

For angle

\theta,

the real part of

z^5

is

32\cos(5\theta);

find which

5\theta

is closest to a multiple of

360^\circ

Each listed number has modulus

2,

so

z^5

has modulus

32,

and its real part is

32\cos(5\theta),

where

\theta

is the argument of

z.

The arguments are

180^\circ,

150^\circ,

135^\circ,

120^\circ,

and

90^\circ.

Multiplying by

5

gives

900^\circ \equiv 180^\circ,

750^\circ \equiv 30^\circ,

675^\circ \equiv 315^\circ,

600^\circ \equiv 240^\circ,

and

450^\circ \equiv 90^\circ.

The largest cosine is

\cos 30^\circ,

from

z = -\sqrt3 + i,

giving real part

16\sqrt3.

Thus, the correct answer is B.

Problem 13 in other years