Difference between revisions of "2002 AMC 12P Problems/Problem 23"

Revision as of 16:39, 1 July 2024

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$ , where $a$ and $b$ are positive real numbers. Find $a.$

$\text{(A) }\sqrt{118} \qquad \text{(B) }\sqrt{210} \qquad \text{(C) }2 \sqrt{210} \qquad \text{(D) }\sqrt{2002} \qquad \text{(E) }100 \sqrt{2}$

Note that $2002 = 11 \cdot 13 \cdot 14$ . With this observation, it becomes easy to note that $z = -14i$ is a root of the given equation.

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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 15: / Line 15: @@
 == Solution ==
-Note that <math>2002 = 11 \cdot 13 \cdot 14</math>.
+Note that <math>2002 = 11 \cdot 13 \cdot 14</math>. With this observation, it becomes easy to note that <math>z = -14i</math> is a root of the given equation.
-According to Wolfram-Alpha, the answer is <math>\boxed {\text{(A) }\sqrt{118}}</math>.
 == See also ==
 {{AMC12 box|year=2002|ab=P|num-b=22|num-a=24}}
 {{MAA Notice}}