Difference between revisions of "2023 AMC 12B Problems/Problem 12"

(Problem)
(Problem)
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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 \cdot v = ac + bdi</math>
+
<math>u \ding{39} v = ac + bdi</math>
  
Suppose <math>z</math> is a complex number such that <math>z\cdot z = z^{2}+40</math>. What is <math>|z|</math>?
+
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 16:50, 15 November 2023

Problem

For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation

$u \ding{39} v = ac + bdi$ (Error compiling LaTeX. Unknown error_msg)

Suppose $z$ is a complex number such that $z\din{39} z = z^{2}+40$ (Error compiling LaTeX. Unknown error_msg). What is $|z|$?

$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}$