2021 AMC 12A Problems/Problem 13

Problem

Of the following complex numbers $z$ , which one has the property that $z^5$ has the greatest real part?

$\textbf{(A) }-2 \qquad \textbf{(B) }-\sqrt3+i \qquad \textbf{(C) }-\sqrt2+\sqrt2 i \qquad \textbf{(D) }-1+\sqrt3 i\qquad \textbf{(E) }2i$

Solution

Clearly $(-2)^5=-32$ and $(2i)^5=32i$ . B $= 2\text{cis}(150)$ , C $=2\text{cis}(135)$ , and D $=2\text{cis}(120)$ , so taking the real part of the 5th power of each we have B: $32\cos(650)=32\cos(30)=16\sqrt{3}$ . For C: $32\cos(675)=32\cos(-45)=16\sqrt{2}$ . For D: $32\cos(600)=32\cos(240)$ which is negative. Thus, the answer is $\boxed{\textbf{(B)}}$ . ~JHawk0224

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2021 AMC 12A Problems/Problem 13

Problem

Solution

See also