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 \ding{39} 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\din{39} 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> | ||
Revision as of 17:50, 15 November 2023
Problem
For complex number 
and 
(where 
), define the binary operation
$u \ding{39} v = ac + bdi$ (Error compiling LaTeX. Unknown error_msg)
Suppose 
is a complex number such that $z\din{39} z = z^{2}+40$ (Error compiling LaTeX. Unknown error_msg). What is 
?
