Difference between revisions of "2023 AMC 12B Problems/Problem 12"
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==Problem== | ==Problem== | ||
− | For complex number <math>u = a+bi and < | + | For complex number <math>u = a+bi</math> and <math>v = c+di</math> (where <math>i=\sqrt{-1}</math>), define the binary operation |
− | < | + | <math>u \otimes v = ac + bdi</math> |
− | + | ||
− | </math>\textbf{(A) }\ | + | Suppose <math>z</math> is a complex number such that <math>z\otimes z = z^{2}+40</math>. What is <math>|z|</math>? |
+ | |||
+ | <math>\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}</math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | let <math>z</math> = <math>a+bi</math>. | ||
+ | |||
+ | <math>z \otimes z = a^{2}+b^{2}i</math>. | ||
+ | |||
+ | This is equal to <math>z^{2} + 40 = a^{2}-b^{2}+40+2abi</math> | ||
+ | |||
+ | Since the real values have to be equal to each other, <math>a^{2}-b^{2}+40 = a^{2}</math>. | ||
+ | Simple algebra shows <math>b^{2} = 40</math>, so <math>b</math> is <math>2\sqrt{10}</math>. | ||
+ | |||
+ | The imaginary components must also equal each other, meaning <math>b^{2} = 2ab</math>, or <math>b = 2a</math>. This means <math>a = \frac{b}{2} = \sqrt{10}</math>. | ||
+ | |||
+ | Thus, the magnitude of z is <math> \sqrt{a^{2}+b^{2}} = \sqrt{50} = 5\sqrt{2}</math> | ||
+ | <math>=\text{\boxed{\textbf{(E) }5\sqrt{2}}}</math> | ||
+ | |||
+ | ~Failure.net | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/Yw3W2ptPWYQ | ||
+ | |||
+ | |||
+ | |||
+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC12 box|year=2023|ab=B|num-b=11|num-a=13}} | ||
+ | {{MAA Notice}} |
Latest revision as of 01:52, 20 November 2023
Contents
Problem
For complex number and (where ), define the binary operation
Suppose is a complex number such that . What is ?
Solution 1
let = .
.
This is equal to
Since the real values have to be equal to each other, . Simple algebra shows , so is .
The imaginary components must also equal each other, meaning , or . This means .
Thus, the magnitude of z is
~Failure.net
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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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