2023 AMC 12B Problems/Problem 12: Difference between revisions

Latest revision as of 16:41, 24 October 2025

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

$u \otimes v = ac + bdi$

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

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

let $z$ = $a+bi$ .

$z \otimes z = a^{2}+b^{2}i$ .

This is equal to $z^{2} + 40 = a^{2}-b^{2}+40+2abi$

Since the real values have to be equal to each other, $a^{2}-b^{2}+40 = a^{2}$ . Simple algebra shows $b^{2} = 40$ , so $b = \pm 2\sqrt{10}$ .

The imaginary components must also equal each other, meaning $b^{2} = 2ab$ , or $b = 2a$ . This means $a = \frac{b}{2} = \pm \sqrt{10}$ .

Thus, the magnitude of $z$ is $\sqrt{a^{2}+b^{2}} = \sqrt{10+40} = \sqrt{50} = 5\sqrt{2}$ $=\text{\boxed{\textbf{(E) }5\sqrt{2}}}$

~Failure.net

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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

@@ Line 17: / Line 17: @@
 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>.
+Simple algebra shows <math>b^{2} = 40</math>, so <math>b = \pm 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>.
+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} = \pm \sqrt{10}</math>.
-Thus, the magnitude of z is <math> \sqrt{a^{2}+b^{2}} = \sqrt{50} = 5\sqrt{2}</math>
+Thus, the magnitude of <math>z</math> is <math> \sqrt{a^{2}+b^{2}} = \sqrt{10+40} = \sqrt{50} = 5\sqrt{2}</math>
 <math>=\text{\boxed{\textbf{(E) }5\sqrt{2}}}</math>