2021 AMC 12A Problems/Problem 13: Difference between revisions

Revision as of 22:14, 11 February 2021

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

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

First, $\textbf{(B)} = 2\text{cis}(150), \textbf{(C)} =2\text{cis}(135)$ $, \textbf{(D)} =2\text{cis}(120)$ .

Taking the real part of the 5th power of each we have:

$\textbf{(A): }(-2)^5=-32$ ,

$\textbf{(B): }32\cos(650)=32\cos(30)=16\sqrt{3}$

$\textbf{(C): }32\cos(675)=32\cos(-45)=16\sqrt{2}$

$\textbf{(D): }32\cos(600)=32\cos(240)$ which is negative

$\textbf{(E): }(2i)^5$ which is imaginary

Thus, the answer is $\boxed{\textbf{(B)}}$ . ~JHawk0224

~ pi_is_3.14

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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 https://www.youtube.com/watch?v=AjQARBvdZ20
-== Video Solution by pi_is_3.14 (Using Polar Form and de Moivre's Theorem) ==
+== Video Solution by OmegaLearn (Using Polar Form and de Moivre's Theorem) ==
 https://youtu.be/2qXVQ5vBKWQ