大三數(shù)學(xué)考試題目及答案

大三數(shù)學(xué)考試題目及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.設(shè)函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)，則\(\lim\limits_{\Deltax\to0}\frac{f(x_0-\Deltax)-f(x_0)}{\Deltax}=\)（）A.\(f'(x_0)\)B.\(-f'(x_0)\)C.\(0\)D.不存在答案：B2.若\(\intf(x)dx=F(x)+C\)，則\(\intf(ax+b)dx=\)（）（\(a\neq0\)）A.\(F(ax+b)+C\)B.\(\frac{1}{a}F(ax+b)+C\)C.\(aF(ax+b)+C\)D.\(\frac{1}{a}F(x)+C\)答案：B3.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.\((-\infty,+\infty)\)D.\((-1,+\infty)\)答案：A4.設(shè)向量\(\vec{a}=(1,2,-1)\)，\(\vec=(2,-1,3)\)，則\(\vec{a}\cdot\vec=\)（）A.\(1\)B.\(-1\)C.\(2-2-3=-3\)D.\(2+2-3=1\)答案：A5.二階線(xiàn)性常系數(shù)齊次微分方程\(y''+2y'+y=0\)的通解為（）A.\(y=(C_1+C_2x)e^{-x}\)B.\(y=C_1e^x+C_2e^{-x}\)C.\(y=C_1\cosx+C_2\sinx\)D.\(y=C_1e^{x}+C_2xe^{x}\)答案：A6.已知矩陣\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)，則\(\vertA\vert=\)（）A.\(1\times4-2\times3=-2\)B.\(1\times3-2\times4=-5\)C.\(1\times4+2\times3=10\)D.\(1\times3+2\times4=11\)答案：A7.設(shè)\(z=x^2y\)，則\(\frac{\partialz}{\partialx}=\)（）A.\(2xy\)B.\(x^2\)C.\(2x\)D.\(y\)答案：A8.無(wú)窮級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\)的和為（）A.\(1\)B.\(\frac{1}{2}\)C.\(2\)D.不存在答案：A9.曲線(xiàn)\(y=\lnx\)在點(diǎn)\((1,0)\)處的切線(xiàn)方程為（）A.\(y=x-1\)B.\(y=-x+1\)C.\(y=x\)D.\(y=-x\)答案：A10.若\(P(A)=0.6\)，\(P(B)=0.4\)，\(A\)與\(B\)相互獨(dú)立，則\(P(A\cupB)=\)（）A.\(0.76\)B.\(0.8\)C.\(0.6\)D.\(0.4\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在區(qū)間\([0,1]\)上滿(mǎn)足羅爾定理?xiàng)l件的是（）A.\(y=x^2-1\)B.\(y=\sinx\)C.\(y=e^x\)D.\(y=\ln(1+x)\)答案：AB2.下列積分中，收斂的是（）A.\(\int_{1}^{+\infty}\frac{1}{x^2}dx\)B.\(\int_{0}^{1}\frac{1}{\sqrt{x}}dx\)C.\(\int_{-\infty}^{+\infty}\sinxdx\)D.\(\int_{0}^{+\infty}e^{-x}dx\)答案：AD3.設(shè)\(A\)，\(B\)為\(n\)階方陣，則下列等式正確的是（）A.\((A+B)^2=A^2+2AB+B^2\)B.\((AB)^T=B^TA^T\)C.\(\vertAB\vert=\vertA\vert\vertB\vert\)D.若\(A\)可逆，則\((A^{-1})^{-1}=A\)答案：BCD4.對(duì)于向量組\(\alpha_1=(1,0,0)\)，\(\alpha_2=(0,1,0)\)，\(\alpha_3=(0,0,1)\)，\(\alpha_4=(1,1,1)\)，下列說(shuō)法正確的是（）A.向量組線(xiàn)性相關(guān)B.向量組線(xiàn)性無(wú)關(guān)C.\(\alpha_4=\alpha_1+\alpha_2+\alpha_3\)D.\(\alpha_1\)，\(\alpha_2\)，\(\alpha_3\)是向量組的一個(gè)極大線(xiàn)性無(wú)關(guān)組答案：BCD5.設(shè)\(z=f(x,y)\)是二元函數(shù)，則全微分\(dz=\)（）A.\(\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(\frac{\partialz}{\partialy}dx+\frac{\partialz}{\partialx}dy\)C.\(z_x'dx+z_y'dy\)D.\(z_y'dx+z_x'dy\)答案：AC6.下列函數(shù)是奇函數(shù)的有（）A.\(y=x^3\)B.\(y=\sinx\)C.\(y=e^x-e^{-x}\)D.\(y=\ln\frac{1-x}{1+x}\)答案：ABCD7.下列方程是一階線(xiàn)性微分方程的有（）A.\(y'+\frac{1}{x}y=\sinx\)B.\(y'+y^2=x\)C.\((x^2+1)y'-2xy=e^x\)D.\(y'=\frac{y}{x}+x\)答案：ACD8.設(shè)\(A=\begin{pmatrix}1&0&0\\0&2&0\\0&0&3\end{pmatrix}\)，則與\(A\)相似的矩陣有（）A.\(\begin{pmatrix}3&0&0\\0&1&0\\0&0&2\end{pmatrix}\)B.\(\begin{pmatrix}1&1&0\\0&2&0\\0&0&3\end{pmatrix}\)C.\(\begin{pmatrix}1&0&0\\0&3&0\\0&0&2\end{pmatrix}\)D.\(\begin{pmatrix}2&0&0\\0&1&0\\0&0&3\end{pmatrix}\)答案：ACD9.若\(\sum_{n=1}^{\infty}a_n\)收斂，則下列級(jí)數(shù)收斂的有（）A.\(\sum_{n=1}^{\infty}2a_n\)B.\(\sum_{n=1}^{\infty}(a_n+1)\)C.\(\sum_{n=1}^{\infty}a_n^2\)D.\(\sum_{n=1}^{\infty}\frac{a_n}{n}\)答案：ACD10.設(shè)\(X\)是隨機(jī)變量，其分布函數(shù)為\(F(x)\)，則下列性質(zhì)正確的是（）A.\(F(-\infty)=0\)B.\(F(+\infty)=1\)C.\(F(x)\)是單調(diào)遞減函數(shù)D.\(F(x)\)是右連續(xù)函數(shù)答案：ABD三、判斷題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)，則函數(shù)\(y=f(x)\)在點(diǎn)\(x_0\)處連續(xù)。（）答案：對(duì)2.\(\int_{a}^f(x)dx=-\int_^{a}f(x)dx\)。（）答案：對(duì)3.若\(A\)，\(B\)為\(n\)階方陣，且\(AB=0\)，則\(A=0\)或\(B=0\)。（）答案：錯(cuò)4.向量組\(\alpha_1=(1,2)\)，\(\alpha_2=(2,4)\)線(xiàn)性相關(guān)。（）答案：對(duì)5.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處的偏導(dǎo)數(shù)\(\frac{\partialz}{\partialx}\)，\(\frac{\partialz}{\partialy}\)存在，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微。（）答案：錯(cuò)6.函數(shù)\(y=\sinx\)的周期是\(2\pi\)。（）答案：對(duì)7.一階線(xiàn)性非齊次微分方程\(y'+p(x)y=q(x)\)的通解為\(y=e^{-\intp(x)dx}(\intq(x)e^{\intp(x)dx}dx+C)\)。（）答案：對(duì)8.若\(n\)階方陣\(A\)的行列式\(\vertA\vert\neq0\)，則\(A\)可逆。（）答案：對(duì)9.無(wú)窮級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n}\)收斂。（）答案：錯(cuò)10.若\(X\)服從正態(tài)分布\(N(\mu,\sigma^2)\)，則\(P(X\leqslant\mu)=0.5\)。（）答案：對(duì)四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=x^3-3x^2-9x+5\)的極值。答案：首先求導(dǎo)\(y'=3x^2-6x-9=3(x^2-2x-3)=3(x-3)(x+1)\)。令\(y'=0\)，得\(x=-1\)或\(x=3\)。當(dāng)\(x<-1\)時(shí)，\(y'>0\)；當(dāng)\(-1<x<3\)時(shí)，\(y'<0\)；當(dāng)\(x>3\)時(shí)，\(y'>0\)。所以\(x=-1\)時(shí)取極大值\(y(-1)=10\)，\(x=3\)時(shí)取極小值\(y(3)=-22\)。2.已知向量\(\vec{a}=(1,2,3)\)，\(\vec=(-1,0,1)\)，求\(\vec{a}\times\vec\)。答案：\(\vec{a}\times\vec=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\1&2&3\\-1&0&1\end{vmatrix}=\vec{i}(2\times1-0\times3)-\vec{j}(1\times1-(-1)\times3)+\vec{k}(1\times0-(-1)\times2)=\vec{i}(2-0)-\vec{j}(1+3)+\vec{k}(0+2)=2\vec{i}-4\vec{j}+2\vec{k}=(2,-4,2)\)。3.用拉格朗日乘數(shù)法求函數(shù)\(z=x^2+y^2\)在條件\(x+y-1=0\)下的極值。答案：設(shè)\(F(x,y,\lambda)=x^2+y^2+\lambda(x+y-1)\)。分別求偏導(dǎo)\(F_x=2x+\lambda=0\)，\(F_y=2y+\lambda=0\)，\(F_{\lambda}=x+y-1=0\)。由前兩式得\(2x+\lambda=2y+\lambda\)，即\(x=y\)，代入\(x+y-1=0\)得\(x=y=\frac{1}{2}\)。所以極值為\(z(\frac{1}{2},\frac{1}{2})=\frac{1}{2}\)。4.求矩陣\(A=\begin{pmatrix}1&2&3\\2&1&3\\3&3&6\end{pmatrix}\)的秩。答案：對(duì)矩陣\(A\)進(jìn)行初等行變換：\(A=\begin{pmatrix}1&2&3\\2&1&3\\3&3&6\end{pmatrix}\sim\begin{pmatrix}1&2&3\\0-3&-3\\0-3&-3\end{pmatrix}\sim\begin{pmatrix}1&2&3\\0&1&1\\0&0&0\end{pmatrix}\)，所以\(r(A)=2\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)在區(qū)間\((0,1)\)和\((1,+\infty)\)上的單調(diào)性。答案：對(duì)\(y=\frac{1}{x}\)求導(dǎo)得\(y'=-\frac{1}{x^2}\)。在區(qū)間\((0,1)\)和\((1,+\infty)\)上\(y'<0\)，所以函數(shù)\(y=\frac{1}{x}\)在\((0,1)\)和\((1,+\infty)\)上單調(diào)遞減。2.討論向量組\(\alpha_1=(1,1,1)\)，\(\alpha_2=(1,2,3)\)，\(\alpha_3=