Track / Stage 4 / 230 of 340 #490 of 2000

Problem 490

AMC 12 late, AIME early

Algebra Difficulty 4.9 Find the answer

2. Set $A=\left\{x \mid x^{2}-3 x+2=0\right\}, C=\left\{x \mid x^{2}-m x+2=0\right\}, A \cap C=C$ , then the value of the real number $m$ is
A. 3
B. 3 or $-2 \sqrt{2}<m<2 \sqrt{2}$
C. $-2 \sqrt{2}<m<2 \sqrt{2}$
D. None of the above

Your answer

Official solution

2. B From the problem, we know that $A=\{1,2\}$ . Since $A \cap C=C$ , i.e., $C \subseteq A$ , $C$ can contain at most the elements 1 and 2. When $C$ contains 1 or 2, $m=3$ in both cases, at which point $A=C$ ; when $C=\varnothing$ , the equation $x^{2}-m x+2=0$ has no real roots, then $\Delta=m^{2}-8<0$ . That is, $-2 \sqrt{2}<$ $m<2 \sqrt{2}$ . In summary, $m=3$ or $-2 \sqrt{2}<m<2 \sqrt{2}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.