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 $= 2\cis(150)$ (Error compiling LaTeX. Unknown error_msg), C $=2\cis(135)$ (Error compiling LaTeX. Unknown error_msg), and D $=2\cis(120)$ (Error compiling LaTeX. Unknown error_msg), 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
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All AMC 12 Problems and Solutions

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2021 AMC 12A Problems/Problem 13

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