2021 AMC 12A Problems/Problem 13

Revision as of 16:43, 11 February 2021 by Jhawk0224 (talk | contribs) (Solution)

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

\[\textbf{(B)} = 2\text{cis}(150)\] \[\textbf{(C)} =2\text{cis}(135)\] \[\textbf{(D)} =2\text{cis}(120)\] 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

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
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

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