Difference between revisions of "2021 AMC 12A Problems/Problem 13"
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==Solution== | ==Solution== | ||
| − | + | <cmath>\textbf{(B)} = 2\text{cis}(150)</cmath> | |
| − | + | <cmath>\textbf{(C)} =2\text{cis}(135)</cmath> | |
| − | < | + | <cmath>\textbf{(D)} =2\text{cis}(120)</cmath> |
| − | < | ||
Taking the real part of the 5th power of each we have | Taking the real part of the 5th power of each we have | ||
| − | </ | + | <cmath>\textbf{(A):} (-2)^5=-32</cmath> |
| − | < | + | <cmath>\textbf{(B):} 32\cos(650)=32\cos(30)=16\sqrt{3}</cmath> |
| − | < | + | <cmath>\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}</cmath> |
| − | Thus, the answer is < | + | <cmath>\textbf{(D):} 32\cos(600)=32\cos(240)</cmath> which is negative. |
| + | <cmath>\textbf{(E):} (2i)^5</cmath> which is imaginary. | ||
| + | Thus, the answer is <math>\boxed{\textbf{(B)}}</math>. | ||
~JHawk0224 | ~JHawk0224 | ||
Revision as of 15:43, 11 February 2021
Problem
Of the following complex numbers 
, which one has the property that 
has the greatest real part?
Solution
![\[\textbf{(B)} = 2\text{cis}(150)\]](http://latex.artofproblemsolving.com/2/7/2/272822faf3172a60e6d6d26a5d1aec1b48783da4.png)
![\[\textbf{(C)} =2\text{cis}(135)\]](http://latex.artofproblemsolving.com/4/a/5/4a53f8db420043fe44fa8111fc55b691e775eb0a.png)
![\[\textbf{(D)} =2\text{cis}(120)\]](http://latex.artofproblemsolving.com/2/5/e/25ef8f98eb2de4ebbc20b95d2b6e7e2bfc44b23d.png)
Taking the real part of the 5th power of each we have
![\[\textbf{(A):} (-2)^5=-32\]](http://latex.artofproblemsolving.com/c/8/8/c88764279d696ec72fff747e640e0868093404c0.png)
![\[\textbf{(B):} 32\cos(650)=32\cos(30)=16\sqrt{3}\]](http://latex.artofproblemsolving.com/4/3/0/430b4fec7668f2b0bde47a5e7125a0b84e0b8a6c.png)
![\[\textbf{(C):} 32\cos(675)=32\cos(-45)=16\sqrt{2}\]](http://latex.artofproblemsolving.com/3/7/e/37effff5fca8c3a660ba53d7b8ed6c7fced02d4c.png)
![\[\textbf{(D):} 32\cos(600)=32\cos(240)\]](http://latex.artofproblemsolving.com/8/3/5/835aa9ae201547ffa94688b88507103ae7b64292.png)
which is negative.
![\[\textbf{(E):} (2i)^5\]](http://latex.artofproblemsolving.com/b/b/5/bb5221228e1ece98303492f7aaa6e419075aec4c.png)
which is imaginary.
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. 
