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1. 下列运算中正确的是(
A.$\frac{a + c}{b + c} = \frac{a}{b}$
B.$\frac{m}{a} - \frac{n}{b} = \frac{m - n}{a - b}$
C.$\frac{b}{a} - \frac{b + 1}{a} = \frac{1}{a}$
D.$\frac{1}{a + 3} + \frac{6}{a^2 - 9} = \frac{1}{a - 3}$
D
)A.$\frac{a + c}{b + c} = \frac{a}{b}$
B.$\frac{m}{a} - \frac{n}{b} = \frac{m - n}{a - b}$
C.$\frac{b}{a} - \frac{b + 1}{a} = \frac{1}{a}$
D.$\frac{1}{a + 3} + \frac{6}{a^2 - 9} = \frac{1}{a - 3}$
答案
D
解析
A. $\frac{a + c}{b + c} \neq \frac{a}{b}$
B. $\frac{m}{a} - \frac{n}{b} = \frac{mb - na}{ab} \neq \frac{m - n}{a - b}$
C. $\frac{b}{a} - \frac{b + 1}{a} = \frac{b - (b + 1)}{a} = \frac{-1}{a} \neq \frac{1}{a}$
D. $\frac{1}{a + 3} + \frac{6}{a^2 - 9} = \frac{a - 3}{(a + 3)(a - 3)} + \frac{6}{(a + 3)(a - 3)} = \frac{a + 3}{(a + 3)(a - 3)} = \frac{1}{a - 3}$
D
B. $\frac{m}{a} - \frac{n}{b} = \frac{mb - na}{ab} \neq \frac{m - n}{a - b}$
C. $\frac{b}{a} - \frac{b + 1}{a} = \frac{b - (b + 1)}{a} = \frac{-1}{a} \neq \frac{1}{a}$
D. $\frac{1}{a + 3} + \frac{6}{a^2 - 9} = \frac{a - 3}{(a + 3)(a - 3)} + \frac{6}{(a + 3)(a - 3)} = \frac{a + 3}{(a + 3)(a - 3)} = \frac{1}{a - 3}$
D
2. 化简$\frac{x^2 + y^2}{x - y} + \frac{2xy}{y - x}$的结果是(
A.$x + y$
B.$x - y$
C.$\frac{(x + y)^2}{x - y}$
D.$\frac{(x - y)^2}{x + y}$
B
)A.$x + y$
B.$x - y$
C.$\frac{(x + y)^2}{x - y}$
D.$\frac{(x - y)^2}{x + y}$
答案
B
解析
$\begin{aligned}\frac{x^2 + y^2}{x - y} + \frac{2xy}{y - x}&=\frac{x^2 + y^2}{x - y} - \frac{2xy}{x - y}\\&=\frac{x^2 + y^2 - 2xy}{x - y}\\&=\frac{(x - y)^2}{x - y}\\&=x - y\end{aligned}$
B
B
3. 计算：$\frac{3}{a - b} - \frac{2a + b}{a^2 - ab}$.
答案
解:原式=$\frac{3}{a-b}-\frac{2a+b}{a(a-b)}$=$\frac{3a-2a-b}{a(a-b)}$=$\frac{a-b}{a(a-b)}=\frac{1}{a}$.
1. 计算$\frac{x + 1}{x} - \frac{1}{x}$,结果正确的是(
A.$1$
B.$x$
C.$\frac{1}{x}$
D.$\frac{x + 2}{x}$
A
)A.$1$
B.$x$
C.$\frac{1}{x}$
D.$\frac{x + 2}{x}$
答案
A
解析
$\frac{x + 1}{x} - \frac{1}{x} = \frac{(x + 1) - 1}{x} = \frac{x}{x} = 1$，答案选A。
2. 化简$\frac{2}{a - 1} - \frac{1}{a + 1}$的结果是(
A.$\frac{a + 2}{(a - 1)(a + 1)}$
B.$\frac{a + 3}{(a - 1)(a + 1)}$
C.$\frac{a}{(a - 1)(a + 1)}$
D.$1$
B
)A.$\frac{a + 2}{(a - 1)(a + 1)}$
B.$\frac{a + 3}{(a - 1)(a + 1)}$
C.$\frac{a}{(a - 1)(a + 1)}$
D.$1$
答案
B
解析
$\begin{aligned}\frac{2}{a - 1} - \frac{1}{a + 1}&=\frac{2(a + 1)}{(a - 1)(a + 1)} - \frac{a - 1}{(a - 1)(a + 1)}\\&=\frac{2(a + 1)-(a - 1)}{(a - 1)(a + 1)}\\&=\frac{2a + 2 - a + 1}{(a - 1)(a + 1)}\\&=\frac{a + 3}{(a - 1)(a + 1)}\end{aligned}$
B
B
3. 计算：$x + \frac{1 - x^2}{x} = $
$\frac{1}{x}$
.答案
$\frac{1}{x}$
解析
$x+\frac{1-x^2}{x}$
$=\frac{x^2}{x}+\frac{1-x^2}{x}$
$=\frac{x^2 + 1 - x^2}{x}$
$=\frac{1}{x}$
$=\frac{x^2}{x}+\frac{1-x^2}{x}$
$=\frac{x^2 + 1 - x^2}{x}$
$=\frac{1}{x}$
4. 计算：$\frac{x^2}{x - 5} + \frac{25}{5 - x} = $
x+5
.答案
x+5
解析
$\frac{x^2}{x - 5} + \frac{25}{5 - x}$
$=\frac{x^2}{x - 5} - \frac{25}{x - 5}$
$=\frac{x^2 - 25}{x - 5}$
$=\frac{(x + 5)(x - 5)}{x - 5}$
$=x + 5$
$=\frac{x^2}{x - 5} - \frac{25}{x - 5}$
$=\frac{x^2 - 25}{x - 5}$
$=\frac{(x + 5)(x - 5)}{x - 5}$
$=x + 5$
5. 化简$\frac{b^2}{2a - b} + \frac{4a^2}{b - 2a}$的结果为
-2a-b
.答案
-2a-b
解析
$\frac{b^2}{2a - b} + \frac{4a^2}{b - 2a}$
$=\frac{b^2}{2a - b} - \frac{4a^2}{2a - b}$
$=\frac{b^2 - 4a^2}{2a - b}$
$=\frac{(b - 2a)(b + 2a)}{2a - b}$
$=-(b + 2a)$
$=-2a - b$
-2a - b
$=\frac{b^2}{2a - b} - \frac{4a^2}{2a - b}$
$=\frac{b^2 - 4a^2}{2a - b}$
$=\frac{(b - 2a)(b + 2a)}{2a - b}$
$=-(b + 2a)$
$=-2a - b$
-2a - b
6. 计算：$\frac{ac}{a - b} - \frac{bc}{a - b}$.
答案
解:$\frac{ac}{a-b}-\frac{bc}{a-b}=\frac{ac-bc}{a-b}$=$\frac{(a-b)c}{a-b}=c$.
7. 计算：$\frac{a^2}{a - 1} + \frac{1}{1 - a}$.
答案
解:原式=$\frac{a^{2}}{a-1}-\frac{1}{a-1}$=$\frac{a^{2}-1}{a-1}$=$\frac{(a+1)(a-1)}{a-1}$=a+1.
8. 计算：(1)$\frac{1}{2a} + \frac{c}{3ab^2}$；(2)$\frac{12}{m^2 - 9} + \frac{2}{3 - m}$.
答案
(1)$\frac{3b^{2}+2c}{6ab^{2}}$;
(2)$-\frac{2}{m+3}$.
