数学试卷期末检测二答案

一、选择题（每题5分，共50分）
1. 下列各数中，属于有理数的是（）
A. $\sqrt{2}$
B. $\pi$
C. $\frac{1}{3}$
D. $\sqrt{3}$
答案：C
解析：有理数是可以表示为两个整数比的数，而$\frac{1}{3}$可以表示为
$\frac{1}{3} = \frac{1}{1} \times \frac{1}{3}$，因此$\frac{1}{3}$是有理数。

2. 若$a^2 - 3a + 2 = 0$，则$a$的值为（）
A. 1
B. 2
C. 1或2
D. 无法确定
答案：C
解析：由题意可得$a^2 - 3a + 2 = (a - 1)(a - 2) = 0$，所以$a = 1$或$a = 2$。

3. 在下列函数中，函数值域为$R$的是（）
A. $f(x) = x^2$
B. $f(x) = \sqrt{x}$
C. $f(x) = \frac{1}{x}$
D. $f(x) = |x|$
答案：C
解析：$f(x) = x^2$的函数值域为$[0, +\infty)$；$f(x) = \sqrt{x}$的函数值
域为$[0, +\infty)$；$f(x) = \frac{1}{x}$的函数值域为$\{y | y \neq 0\}$；$f(x) = |x|$的函数值域为$[0, +\infty)$。

因此，函数值域为$R$的是$f(x) =
\frac{1}{x}$。

4. 下列命题中，正确的是（）
A. 若$a > b$，则$a^2 > b^2$
B. 若$a > b$，则$a + c > b + c$
C. 若$a > b$，则$ac > bc$
D. 若$a > b$，则$\frac{a}{c} > \frac{b}{c}$
答案：B
解析：选项A中，当$a = 1$，$b = -2$时，$a^2 = 1$，$b^2 = 4$，不满足$a^2 > b^2$；选项C中，当$a = 1$，$b = -2$，$c = 2$时，$ac = 2$，$bc = -4$，不
满足$ac > bc$；选项D中，当$a = 1$，$b = -2$，$c = -2$时，$\frac{a}{c} = -\frac{1}{2}$，$\frac{b}{c} = 1$，不满足$\frac{a}{c} > \frac{b}{c}$。

因此，正确的命题是选项B。

5. 已知等差数列$\{a_n\}$中，$a_1 = 2$，$a_4 = 10$，求该数列的通项公式。

答案：$a_n = 3n - 1$
解析：由等差数列的性质，可得$a_4 = a_1 + 3d$，其中$d$为公差。

将已知条件
代入，得$10 = 2 + 3d$，解得$d = 2$。

因此，通项公式为$a_n = a_1 + (n -
1)d = 2 + (n - 1) \times 2 = 3n - 1$。

二、填空题（每题5分，共25分）
6. 若$\sin x + \cos x = \sqrt{2}$，则$\sin x \cos x =
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