题目

Is x<y?

(1) 3x<2y
(2) x>0 and y>0

选项

A.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.


B.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.


C.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.


D.

EACH statement ALONE is sufficient.


E.

Statements (1) and (2) TOGETHER are NOT sufficient.

解析

**问题:是否 \(x < y\)?** #### (1) \(3^x < 2^y\) 也就是说 \(x \neq y\)。 举例:若 \(x = y = 0\) \[ \Rightarrow 3^0 < 2^0 \Rightarrow 1 < 1 \] 这是不可能成立的。 再举例:\(x = 1, y = 2\) \[ 3^1 < 2^2 \quad \text{(成立,回答“是”)} \] 再举例:\(x = -2, y = -3\) \[ 3^{-2} < 2^{-3} \quad \text{(不成立,回答“否”)} \] **条件不充分。** #### (2) \(x > 0\) 且 \(y > 0\) 举例:\(x = 1, y = 0\)(不满足 \(y>0\),换例) \(x = 1, y = 0\) 不符合条件,换为 \(x=2, y=1\):回答“否” \(x = 1, y = 2\):回答“是” **条件不充分。** #### 同时考虑 (1) 和 (2) 此时只剩下 \(x\)、\(y\) 均为正数的情况,结合条件 (1),只有 \(x < y\) 才能满足 \(3^x < 2^y\)。 **答案:C。**
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