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>. | Clearly <math>(-2)^5=-32</math> and <math>(2i)^5=32i</math>. | ||
| − | <math>\textbf{(B)} = 2\text{cis}(150)</ | + | <math>\textbf{(B)} = 2\text{cis}(150)\\$ |
| − | + | </math>\textbf{(C)} =2\text{cis}(135)\\$ | |
| − | <math>\textbf{(D)} =2\text{cis}(120) | + | <math>\textbf{(D)} =2\text{cis}(120)\\$ |
Taking the real part of the 5th power of each we have | Taking the real part of the 5th power of each we have | ||
| − | <math>\textbf{(B):} 32\cos(650)=32\cos(30)=16\sqrt{3}< | + | </math>\textbf{(B):} 32\cos(650)=32\cos(30)=16\sqrt{3}<math> |
| − | <math>\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}< | + | </math>\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}<math> |
| − | <math>\textbf{(D):} 32\cos(600)=32\cos(240)< | + | </math>\textbf{(D):} 32\cos(600)=32\cos(240)<math> which is negative. |
| − | Thus, the answer is <math>\boxed{\textbf{(B)}} | + | Thus, the answer is </math>\boxed{\textbf{(B)}}$. |
~JHawk0224 | ~JHawk0224 | ||
Revision as of 16:42, 11 February 2021
Problem
Of the following complex numbers 
, which one has the property that 
has the greatest real part?
Solution
Clearly 
and 
.
$\textbf{(B)} = 2\text{cis}(150)\$ (Error compiling LaTeX. Unknown error_msg)\textbf{(C)} =2\text{cis}(135)\$
$\textbf{(D)} =2\text{cis}(120)\$
Taking the real part of the 5th power of each we have$ (Error compiling LaTeX. Unknown error_msg)\textbf{(B):} 32\cos(650)=32\cos(30)=16\sqrt{3}$$ (Error compiling LaTeX. Unknown error_msg)\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}$$ (Error compiling LaTeX. Unknown error_msg)\textbf{(D):} 32\cos(600)=32\cos(240)
\boxed{\textbf{(B)}}$.
~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. 
