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

Revision as of 16:45, 15 November 2023

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

$u \cdot v = ac + bdi$

Suppose $z$ is a complex number such that $z\cdot z = z^{2}+40$ . What is $abs{z}$ ?

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

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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
-<math>u \earth v = ac \sum bdi</math>
+<math>u \cdot v = ac + bdi</math>
-<math>\textbf{(A) }\frac{9}{25}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{5}\qquad\textbf{(D) }\frac{25}{169}\qquad\textbf{(E) }\frac{4}{25}</math>
+Suppose <math>z</math> is a complex number such that <math>z\cdot z = z^{2}+40</math>. What is <math>abs{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>