Difference between revisions of "2021 AMC 12A Problems/Problem 13"
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==Solution== | ==Solution== | ||
| − | {{ | + | Clearly <math>(-2)^5=-32</math> and <math>(2i)^5=32i</math>. B <math>= 2\cis(150)</math>, C <math>=2\cis(135)</math>, and D <math>=2\cis(120)</math>, so taking the real part of the 5th power of each we have B: <math>32\cos(650)=32\cos(30)=16\sqrt{3}</math>. For C: <math>32\cos(675)=32\cos(-45)=16\sqrt{2}</math>. For D: <math>32\cos(600)=32\cos(240)</math> which is negative. Thus, the answer is <math>\boxed{\textbf{(B)}}</math>. |
| + | ~JHawk0224 | ||
==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 16:38, 11 February 2021
Problem
Of the following complex numbers 
, which one has the property that 
has the greatest real part?
Solution
Clearly 
and 
. 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: 
. For C: 
. For D: 
which is negative. Thus, the answer is 
.
~JHawk0224
See also
| 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. 
