Difference between revisions of "2021 AMC 12A Problems/Problem 13"
(→Solution) |
(→Solution) |
||
| Line 6: | Line 6: | ||
==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>. | ||
| − | \textbf{(B)} | + | <math>\textbf{(B)} = 2\text{cis}(150)</math>, |
| − | \textbf{(C)} | + | <math>\textbf{(C)} =2\text{cis}(135)</math> |
| − | \textbf{(D)} | + | <math>\textbf{(D)} =2\text{cis}(120)</math> |
| − | 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> | ||
| + | <math>\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}</math> | ||
| + | <math>\textbf{(D):} 32\cos(600)=32\cos(240)</math> which is negative. | ||
| + | Thus, the answer is <math>\boxed{\textbf{(B)}}</math>. | ||
~JHawk0224 | ~JHawk0224 | ||
Revision as of 16:41, 11 February 2021
Problem
Of the following complex numbers 
, which one has the property that 
has the greatest real part?
Solution
Clearly 
and 
.
,
Taking the real part of the 5th power of each we have
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. 
