Difference between revisions of "2021 AMC 12A Problems/Problem 13"

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Thus, the answer is <math>\boxed{\textbf{(B)}}</math>.
 
Thus, the answer is <math>\boxed{\textbf{(B)}}</math>.
 
~JHawk0224
 
~JHawk0224
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==Video Solution by Hawk Math==
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https://www.youtube.com/watch?v=AjQARBvdZ20
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2021|ab=A|num-b=12|num-a=14}}
 
{{AMC12 box|year=2021|ab=A|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:54, 11 February 2021

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

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

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AMC 12 Problems and Solutions

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