分式加減 測(cè)試題及答案

分式加減測(cè)試題及答案

一、單項(xiàng)選擇題（每題2分，共20分）1.計(jì)算\(\frac{1}{x}+\frac{2}{x}\)的結(jié)果是（）A.\(\frac{3}{x}\)B.\(\frac{3}{2x}\)C.\(\frac{1}{3x}\)D.\(\frac{2}{x}\)2.化簡(jiǎn)\(\frac{a}{a-b}-\frac{a-b}\)的結(jié)果是（）A.1B.\(\frac{a+b}{a-b}\)C.\(\frac{a-b}{a+b}\)D.a-b3.計(jì)算\(\frac{1}{x-1}-\frac{1}{x+1}\)，正確的結(jié)果是（）A.\(\frac{2}{(x-1)(x+1)}\)B.\(\frac{-2}{(x-1)(x+1)}\)C.\(\frac{2x}{(x-1)(x+1)}\)D.\(\frac{0}{(x-1)(x+1)}\)4.計(jì)算\(\frac{m}{m+3}-\frac{6}{9-m^{2}}+\frac{2}{m-3}\)的結(jié)果是（）A.1B.\(\frac{m-4}{m+3}\)C.\(\frac{m+4}{m-3}\)D.\(\frac{m+2}{m-3}\)5.化簡(jiǎn)\(\frac{x}{x-1}-\frac{1}{x^{2}-1}\)的結(jié)果為（）A.1B.\(\frac{x+1}{x^{2}-1}\)C.\(\frac{x-1}{x^{2}-1}\)D.\(\frac{x^{2}+1}{x^{2}-1}\)6.若\(\frac{1}{x}+\frac{1}{y}=3\)，則\(\frac{2x-xy+2y}{x+2xy+y}\)的值為（）A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.\(\frac{5}{7}\)D.\(\frac{1}{5}\)7.計(jì)算\(\frac{2}{x^{2}-4}+\frac{x}{x-2}\)的結(jié)果是（）A.\(\frac{x+2}{x-2}\)B.\(\frac{1}{x-2}\)C.\(\frac{x+2}{x^{2}-4}\)D.\(\frac{x}{x-2}\)8.化簡(jiǎn)\(\frac{a^{2}}{a-1}-a-1\)的結(jié)果是（）A.\(\frac{1}{a-1}\)B.\(\frac{a}{a-1}\)C.\(a\)D.19.計(jì)算\(\frac{a}{a+1}+\frac{1}{a+1}\)的結(jié)果是（）A.1B.\(\frac{a}{a+1}\)C.\(\frac{1}{a+1}\)D.\(\frac{a+1}{a}\)10.化簡(jiǎn)\(\frac{3}{x-1}-\frac{x+2}{x^{2}-1}\)的結(jié)果是（）A.\(\frac{2x+1}{x^{2}-1}\)B.\(\frac{2x+5}{x^{2}-1}\)C.\(\frac{3x+1}{x^{2}-1}\)D.\(\frac{3x+5}{x^{2}-1}\)二、多項(xiàng)選擇題（每題2分，共20分）1.下列計(jì)算正確的是（）A.\(\frac{1}{a}+\frac{1}=\frac{a+b}{ab}\)B.\(\frac{1}{a}-\frac{1}=\frac{b-a}{ab}\)C.\(\frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{a^{2}+b^{2}}{a^{2}b^{2}}\)D.\(\frac{1}{a^{2}}-\frac{1}{b^{2}}=\frac{b^{2}-a^{2}}{a^{2}b^{2}}\)2.化簡(jiǎn)\(\frac{x}{x-y}+\frac{y}{y-x}\)的結(jié)果可以是（）A.1B.\(\frac{x+y}{x-y}\)C.\(\frac{x-y}{x-y}\)D.\(\frac{x-y}{y-x}\)3.計(jì)算\(\frac{1}{x^{2}-1}+\frac{1}{x+1}\)，正確的步驟有（）A.先通分，\(\frac{1}{x^{2}-1}+\frac{1}{x+1}=\frac{1}{(x+1)(x-1)}+\frac{x-1}{(x+1)(x-1)}\)B.分子相加得\(\frac{1+x-1}{(x+1)(x-1)}\)C.化簡(jiǎn)得\(\frac{x}{(x+1)(x-1)}\)D.最終結(jié)果為\(\frac{x}{x^{2}-1}\)4.當(dāng)\(x\)取下列哪些值時(shí)，分式\(\frac{1}{x-2}+\frac{1}{x^{2}-4}\)有意義（）A.\(x\neq2\)B.\(x\neq-2\)C.\(x\neq0\)D.\(x\)為任意實(shí)數(shù)5.化簡(jiǎn)\(\frac{a}{a^{2}-4}-\frac{1}{2a-4}\)的結(jié)果可能是（）A.\(\frac{1}{2(a+2)}\)B.\(\frac{2a-2}{2(a^{2}-4)}\)C.\(\frac{2a-2}{2(a+2)(a-2)}\)D.\(\frac{a-1}{(a+2)(a-2)}\)6.計(jì)算\(\frac{m}{m-n}-\frac{n}{m+n}-\frac{2mn}{m^{2}-n^{2}}\)，正確的說(shuō)法有（）A.先通分，分母都化為\(m^{2}-n^{2}\)B.通分后式子變?yōu)閈(\frac{m(m+n)}{m^{2}-n^{2}}-\frac{n(m-n)}{m^{2}-n^{2}}-\frac{2mn}{m^{2}-n^{2}}\)C.分子運(yùn)算得\(m^{2}+mn-mn+n^{2}-2mn\)D.化簡(jiǎn)結(jié)果為\(\frac{(m-n)^{2}}{m^{2}-n^{2}}\)7.下列式子化簡(jiǎn)后與\(\frac{1}{x+1}\)相等的是（）A.\(\frac{x}{x(x+1)}\)B.\(\frac{x+1}{(x+1)^{2}}\)C.\(\frac{1}{x^{2}+2x+1}\times(x+1)\)D.\(\frac{1}{x^{2}-1}\div\frac{x-1}{x+1}\)8.若\(A=\frac{1}{x-1}\)，\(B=\frac{1}{x^{2}-1}\)，則\(A-B\)化簡(jiǎn)后為（）A.\(\frac{x}{x^{2}-1}\)B.\(\frac{x+1-1}{x^{2}-1}\)C.\(\frac{x}{(x+1)(x-1)}\)D.\(\frac{1}{x}\)9.計(jì)算\(\frac{1}{x+3}+\frac{6}{x^{2}-9}\)，正確的有（）A.先將\(\frac{6}{x^{2}-9}\)變形為\(\frac{6}{(x+3)(x-3)}\)B.通分后式子為\(\frac{x-3}{(x+3)(x-3)}+\frac{6}{(x+3)(x-3)}\)C.分子相加得\(x-3+6=x+3\)D.化簡(jiǎn)結(jié)果為\(\frac{1}{x-3}\)10.化簡(jiǎn)\(\frac{a^{2}}{a-2}-a-2\)，步驟正確的是（）A.變形為\(\frac{a^{2}}{a-2}-\frac{(a+2)(a-2)}{a-2}\)B.通分后分子為\(a^{2}-(a^{2}-4)\)C.計(jì)算得\(a^{2}-a^{2}+4=4\)D.結(jié)果為\(\frac{4}{a-2}\)三、判斷題（每題2分，共20分）1.\(\frac{1}{a}+\frac{2}{a}=\frac{3}{2a}\)（）2.\(\frac{a}{a-b}-\frac{a-b}=1\)（）3.\(\frac{1}{x-1}-\frac{1}{x+1}=\frac{2}{x^{2}-1}\)（）4.分式\(\frac{1}{x}+\frac{1}{y}\)的最簡(jiǎn)公分母是\(xy\)（）5.計(jì)算\(\frac{2}{x^{2}-4}+\frac{x}{x-2}=\frac{x+2}{x-2}\)（）6.化簡(jiǎn)\(\frac{a^{2}}{a-1}-a-1=\frac{1}{a-1}\)（）7.\(\frac{1}{x^{2}-1}+\frac{1}{x+1}=\frac{x}{x^{2}-1}\)（）8.當(dāng)\(x=2\)時(shí)，分式\(\frac{1}{x-2}+\frac{1}{x^{2}-4}\)有意義（）9.化簡(jiǎn)\(\frac{a}{a^{2}-4}-\frac{1}{2a-4}=\frac{1}{2(a+2)}\)（）10.計(jì)算\(\frac{m}{m-n}-\frac{n}{m+n}-\frac{2mn}{m^{2}-n^{2}}=\frac{(m-n)^{2}}{m^{2}-n^{2}}\)（）四、簡(jiǎn)答題（每題5分，共20分）1.計(jì)算\(\frac{3}{x-2}+\frac{1}{2-x}\)。-答案：\(\frac{3}{x-2}+\frac{1}{2-x}=\frac{3}{x-2}-\frac{1}{x-2}=\frac{3-1}{x-2}=\frac{2}{x-2}\)。2.化簡(jiǎn)\(\frac{x^{2}}{x-1}-x-1\)。-答案：\(\frac{x^{2}}{x-1}-x-1=\frac{x^{2}}{x-1}-\frac{(x+1)(x-1)}{x-1}=\frac{x^{2}-(x^{2}-1)}{x-1}=\frac{1}{x-1}\)。3.計(jì)算\(\frac{1}{x^{2}-9}+\frac{1}{x+3}\)。-答案：先對(duì)\(x^{2}-9\)因式分解為\((x+3)(x-3)\)，則原式\(=\frac{1}{(x+3)(x-3)}+\frac{x-3}{(x+3)(x-3)}=\frac{1+x-3}{(x+3)(x-3)}=\frac{x-2}{(x+3)(x-3)}\)。4.化簡(jiǎn)\(\frac{a}{a-b}+\frac{b-a}\)。-答案：\(\frac{a}{a-b}+\frac{b-a}=\frac{a}{a-b}-\frac{a-b}=\frac{a-b}{a-b}=1\)。五、討論題（每題5分，共20分）1.討論在分式加減運(yùn)算中，如何確定最簡(jiǎn)公分母？-答案：先對(duì)各分母因式分解，取各分母所有因式的最高次冪的乘積作為最簡(jiǎn)公分母。比如分母\(x^{2}-4=(x+2)(x-2)\)與\(x-2\)，最簡(jiǎn)公分母就是\((x+2)(x-2)\)。2.分式加減運(yùn)算中容易出現(xiàn)哪些錯(cuò)誤？如何避免？-答案：常見(jiàn)錯(cuò)誤有通分錯(cuò)誤、符號(hào)錯(cuò)誤、去括號(hào)錯(cuò)誤等。避免方法是準(zhǔn)確因式分解找最簡(jiǎn)公分母，注意符號(hào)變化，去括號(hào)遵循法則，計(jì)算后要檢查。3.舉例說(shuō)明分式加減運(yùn)算在實(shí)際生活中的應(yīng)用。-答案：比如工程問(wèn)題，一項(xiàng)工程甲單獨(dú)做\(x\)天完成，乙單獨(dú)做\(y\)天完成，兩人合作一天完成的工作量就是\(\frac{1}{x}+\frac{1}{y}\)，通過(guò)分式運(yùn)算可求解相關(guān)工程問(wèn)題。4.當(dāng)分式加減運(yùn)算結(jié)果為0時(shí)，說(shuō)明了什么？-答案：說(shuō)明分子為0且分母不為0。例如\(\frac{x-1}{x+1}-\frac{x-1}{x+1}=0\)，