Difference between revisions of "2023 AMC 12B Problems/Problem 12"
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For complex number <math>u = a+bi</math> and <math>v = c+di</math> (where <math>i=\sqrt{-1}</math>), define the binary operation | 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 \ | + | <math>u \otimes v = ac + bdi</math> |
− | Suppose <math>z</math> is a complex number such that <math>z\ | + | 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> | <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> | ||
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let <math>z</math> = <math>a+bi</math>. | let <math>z</math> = <math>a+bi</math>. | ||
− | <math>z \ | + | <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> | This is equal to <math>z^{2} + 40 = a^{2}-b^{2}+40+2abi</math> |
Revision as of 17:03, 15 November 2023
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
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