專升本數(shù)學(xué)電子題庫(kù)及答案

一、單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\frac{1}{\ln(x-1)}\)的定義域是(）A.\((1,+\infty)\)B.\((1,2)\cup(2,+\infty)\)C.\([1,+\infty)\)D.\([1,2)\cup(2,+\infty)\)2.當(dāng)\(x\to0\)時(shí),\(2x-\sin2x\)是\(x\)的()A.高階無(wú)窮小B.低階無(wú)窮小C.同階但不等價(jià)無(wú)窮小D.等價(jià)無(wú)窮小3.已知\(f(x)\)的一個(gè)原函數(shù)是\(e^{-x}\),則\(f^\prime(x)\)等于()A.\(e^{-x}\)B.\(-e^{-x}\)C.\(e^{-x}+C\)D.\(-e^{-x}+C\)4.定積分\(\int_{0}^{1}x^2dx\)的值為()A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.1D.25.設(shè)函數(shù)\(z=x^2y\),則\(\frac{\partialz}{\partialx}\)等于()A.\(2xy\)B.\(x^2\)C.\(y\)D.\(2x\)6.級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\)的和為()A.1B.\(\frac{1}{2}\)C.\(\frac{3}{2}\)D.27.向量\(\vec{a}=(1,2,3)\),\(\vec=(3,2,1)\),則\(\vec{a}\cdot\vec\)等于()A.10B.11C.12D.138.方程\(x^2+y^2-2x+4y+1=0\)表示的圓的圓心坐標(biāo)為()A.\((1,-2)\)B.\((-1,2)\)C.\((2,-1)\)D.\((-2,1)\)9.函數(shù)\(y=x^3-3x\)的單調(diào)遞增區(qū)間是()A.\((-\infty,-1)\cup(1,+\infty)\)B.\((-1,1)\)C.\((-\infty,0)\)D.\((0,+\infty)\)10.若\(A\)、\(B\)為兩個(gè)事件,且\(P(A)=0.4\),\(P(B)=0.3\),\(P(A\cupB)=0.6\),則\(P(A\capB)\)等于()A.0.1B.0.2C.0.3D.0.4答案:1.B2.C3.B4.A5.A6.A7.A8.A9.A10.A二、多項(xiàng)選擇題(每題2分,共10題)1.下列函數(shù)中,是奇函數(shù)的有（）A.\(y=x^3\sinx\)B.\(y=\ln(x+\sqrt{1+x^2})\)C.\(y=e^x-e^{-x}\)D.\(y=\frac{1}{x^2+1}\)2.下列極限存在的有()A.\(\lim_{x\to0}\frac{\sinx}{x}\)B.\(\lim_{x\to\infty}(1+\frac{1}{x})^x\)C.\(\lim_{x\to0}(1-x)^{\frac{1}{x}}\)D.\(\lim_{x\to0}\frac{e^x-1}{x}\)3.下列函數(shù)中,在其定義域內(nèi)連續(xù)的有()A.\(y=\sinx\)B.\(y=\lnx\)C.\(y=\frac{1}{x}\)D.\(y=\sqrt{x}\)4.下列求導(dǎo)運(yùn)算正確的有()A.\((x^n)^\prime=nx^{n-1}\)B.\((\sinx)^\prime=\cosx\)C.\((\lnx)^\prime=\frac{1}{x}\)D.\((e^x)^\prime=e^x\)5.下列積分計(jì)算正確的有()A.\(\int_{0}^{1}x^3dx=\frac{1}{4}\)B.\(\int_{0}^{\pi}\sinxdx=2\)C.\(\int_{-1}^{1}x^2dx=\frac{2}{3}\)D.\(\int_{0}^{e}\frac{1}{x}dx=1\)6.下列級(jí)數(shù)中,收斂的有()A.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n}\)C.\(\sum_{n=1}^{\infty}(-1)^n\frac{1}{n}\)D.\(\sum_{n=1}^{\infty}\frac{1}{2^n}\)7.下列向量中,與向量\(\vec{a}=(1,1,0)\)垂直的有()A.\(\vec=(1,-1,0)\)B.\(\vec{c}=(0,0,1)\)C.\(\vecz3jilz61osys=(1,0,-1)\)D.\(\vec{e}=(0,1,-1)\)8.下列方程表示橢圓的有()A.\(\frac{x^2}{4}+\frac{y^2}{9}=1\)B.\(x^2+y^2=1\)C.\(\frac{x^2}{4}-\frac{y^2}{9}=1\)D.\(4x^2+9y^2=36\)9.下列函數(shù)中,有極值的有()A.\(y=x^2\)B.\(y=x^3\)C.\(y=\sinx\)D.\(y=e^x\)10.下列關(guān)于概率的說(shuō)法正確的有()A.\(0\leqP(A)\leq1\)B.\(P(\Omega)=1\)C.\(P(\varnothing)=0\)D.\(P(A\cupB)=P(A)+P(B)\)答案:1.BC2.ABD3.AD4.ABCD5.AC6.ACD7.AD8.AD9.AC10.ABC三、判斷題(每題2分,共10題)1.函數(shù)\(y=\sqrt{x-1}\)的定義域是\([1,+\infty)\)。()2.\(\lim_{x\to0}\frac{\sin2x}{x}=2\)。()3.若\(f(x)\)在\(x_0\)處可導(dǎo),則\(f(x)\)在\(x_0\)處連續(xù)。（)4.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)。()5.函數(shù)\(z=x^2+y^2\)在點(diǎn)\((0,0)\)處取得極小值。()6.級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n^3}\)收斂。()7.向量\(\vec{a}=(1,2,3)\)與向量\(\vec=(3,6,9)\)平行。()8.方程\(x^2+y^2+2x-4y+5=0\)表示一個(gè)點(diǎn)。()9.函數(shù)\(y=\frac{1}{x-1}\)在\((1,+\infty)\)上單調(diào)遞減。()10.若\(P(A)=0.5\),\(P(B)=0.4\),\(A\)與\(B\)互斥,則\(P(A\cupB)=0.9\)。（）答案:1.√2.√3.√4.√5.√6.√7.√8.√9.√10.√四、簡(jiǎn)答題(每題5分,共4題)1.簡(jiǎn)述函數(shù)單調(diào)性的判定方法。答案:對(duì)函數(shù)求導(dǎo),導(dǎo)數(shù)大于0則函數(shù)單調(diào)遞增,導(dǎo)數(shù)小于0則函數(shù)單調(diào)遞減。2.寫出牛頓-萊布尼茨公式。答案:若\(F^\prime(x)=f(x)\),則\(\int_{a}^f(x)dx=F(b)-F(a)\)。3.簡(jiǎn)述向量垂直的充要條件。答案:兩非零向量\(\vec{a}\)、\(\vec\)垂直的充要條件是\(\vec{a}\cdot\vec=0\)。4.簡(jiǎn)述古典概型的特點(diǎn)。答案:試驗(yàn)中所有可能出現(xiàn)的基本事件只有有限個(gè);每個(gè)基本事件出現(xiàn)的可能性相等。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^3-3x^2+1\)的單調(diào)性與極值。答案:求導(dǎo)得\(y^\prime=3x^2-6x\),令\(y^\prime=0\)得\(x=0\)或\(x=2\)。\(x\lt0\)或\(x\gt2\)時(shí)\(y^\prime\gt0\),函數(shù)遞增;\(0\ltx\lt2\)時(shí)\(y^\prime\lt0\),函數(shù)遞減。極大值\(y(0)=1\),極小值\(y(2)=-3\)。2.討論級(jí)數(shù)\(\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}\)的斂散性。答案:這是交錯(cuò)級(jí)數(shù),滿足萊布尼茨定理?xiàng)l件:\(\frac{1}{n}\)遞減且\(\lim_{n\to\infty}\frac{1}{n}=0\),所以該級(jí)數(shù)收斂。3.討論方程\(x^2+y^2+2x-2y+a