審計專碩數(shù)學試題及答案

一、單項選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)在點\(x_0\)處可導，則\(\lim\limits_{\Deltax\rightarrow0}\frac{f(x_0+2\Deltax)-f(x_0)}{\Deltax}=()\)A.\(f'(x_0)\)B.\(2f'(x_0)\)C.\(\frac{1}{2}f'(x_0)\)D.不存在答案：B2.設\(f(x)=\ln(1+x)\)，則\(f^{(5)}(0)=()\)A.\(4!\)B.\(-4!\)C.\(\frac{1}{4!}\)D.\(-\frac{1}{4!}\)答案：A3.函數(shù)\(y=x^3-3x\)的單調(diào)遞減區(qū)間是()A.\((-\infty,-1)\)B.\((-1,1)\)C.\((1,+\infty)\)D.\((-\infty,+\infty)\)答案：B4.\(\int\frac{1}{x^2+1}dx=()\)A.\(\arctanx+C\)B.\(\ln(x^2+1)+C\)C.\(\frac{1}{2}\ln(x^2+1)+C\)D.\(\frac{1}{3}x^3-3x+C\)答案：A5.設\(z=x^2y\)，則\(\frac{\partialz}{\partialx}=()\)A.\(2xy\)B.\(x^2\)C.\(2x+y\)D.\(x^2+2y\)答案：A6.已知向量\(\vec{a}=(1,2,-1)\)，\(\vec=(2,-1,3)\)，則\(\vec{a}\cdot\vec=()\)A.\(-1\)B.\(1\)C.\(-3\)D.\(3\)答案：A7.下列矩陣中為對稱矩陣的是()A.\(\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}\)B.\(\begin{pmatrix}1&2&3\\2&1&4\\3&4&1\end{pmatrix}\)C.\(\begin{pmatrix}1&2&3\\-2&1&4\\-3&-4&1\end{pmatrix}\)D.\(\begin{pmatrix}1&0&0\\1&1&0\\1&1&1\end{pmatrix}\)答案：B8.若線性方程組\(\begin{cases}x_1+x_2+x_3=1\\2x_1+3x_2+4x_3=2\\3x_1+5x_2+7x_3=3\end{cases}\)的系數(shù)矩陣為\(A\)，增廣矩陣為\(\overline{A}\)，則\(r(A)\)與\(r(\overline{A})\)的關系是()A.\(r(A)>r(\overline{A})\)B.\(r(A)=r(\overline{A})\)C.\(r(A)<r(\overline{A})\)D.無法確定答案：B9.設隨機變量\(X\)服從正態(tài)分布\(N(1,4)\)，則\(P(X\leqslant1)=()\)A.\(0\)B.\(0.5\)C.\(1\)D.不確定答案：B10.從\(5\)個不同的元素中取出\(3\)個元素的組合數(shù)\(C_{5}^3=()\)A.\(10\)B.\(20\)C.\(30\)D.\(60\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數(shù)中，在\(x=0\)處連續(xù)的有()A.\(y=\sinx\)B.\(y=\frac{\sinx}{x}\)C.\(y=\begin{cases}x\sin\frac{1}{x}&x\neq0\\0&x=0\end{cases}\)D.\(y=\frac{1}{x}\)答案：AC2.設\(y=f(u)\)，\(u=g(x)\)，且\(y=f(g(x))\)可導，則\(y'\)等于()A.\(f'(u)g'(x)\)B.\(f'(g(x))g'(x)\)C.\(f(g'(x))\)D.\(g(f'(x))\)答案：AB3.函數(shù)\(y=e^x\)的特性有()A.單調(diào)遞增B.單調(diào)遞減C.過點\((0,1)\)D.是奇函數(shù)答案：AC4.以下積分正確的有()A.\(\intx^ndx=\frac{1}{n+1}x^{n+1}+C(n\neq-1)\)B.\(\int\sinxdx=\cosx+C\)C.\(\int\frac{1}{x}dx=\ln|x|+C\)D.\(\inte^xdx=e^x+C\)答案：ACD5.設\(z=f(x,y)\)，則全微分\(dz\)等于()A.\(\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(\frac{\partialz}{\partialy}dx+\frac{\partialz}{\partialx}dy\)C.\(z_xdx+z_ydy\)D.\(z_ydx+z_xdy\)答案：AC6.向量組\(\alpha_1,\alpha_2,\alpha_3\)線性相關的充要條件是()A.其中至少有一個向量可由其余向量線性表示B.齊次線性方程組\(x_1\alpha_1+x_2\alpha_2+x_3\alpha_3=0\)有非零解C.矩陣\(A=(\alpha_1,\alpha_2,\alpha_3)\)的秩\(r(A)<3\)D.\(\alpha_1,\alpha_2,\alpha_3\)的分量對應成比例答案：ABC7.對于\(n\)階方陣\(A\)，\(B\)，以下等式成立的有()A.\((AB)^T=B^TA^T\)B.\((A+B)^T=A^T+B^T\)C.\((kA)^T=kA^T(k\inR)\)D.\((A^T)^T=A\)答案：ABCD8.設\(A\)是\(n\)階方陣，若\(Ax=0\)有非零解，則()A.\(|A|=0\)B.\(A\)的秩\(r(A)<n\)C.\(A\)的行向量組線性相關D.\(A\)的列向量組線性相關答案：ABCD9.下列關于概率的性質(zhì)正確的有()A.\(P(\varnothing)=0\)B.\(P(A\cupB)=P(A)+P(B)-P(A\capB)\)C.若\(A\subseteqB\)，則\(P(A)\leqslantP(B)\)D.\(P(A)+P(\overline{A})=1\)答案：ABCD10.以下關于排列數(shù)\(A_{n}^m\)與組合數(shù)\(C_{n}^m\)的關系正確的是()A.\(A_{n}^m=m!C_{n}^m\)B.\(C_{n}^m=\frac{A_{n}^m}{m!}\)C.\(A_{n}^m=\frac{n!}{(n-m)!}\)，\(C_{n}^m=\frac{n!}{m!(n-m)!}\)D.\(A_{n}^m>C_{n}^m\)答案：ABC三、判斷題（每題2分，共10題）1.若函數(shù)\(f(x)\)在\(x=a\)處不可導，則\(f(x)\)在\(x=a\)處一定不連續(xù)。(×)2.若\(y=f(x)\)在\([a,b]\)上連續(xù)，則\(y=f(x)\)在\((a,b)\)內(nèi)一定可導。(×)3.函數(shù)\(y=x^3\)的二階導數(shù)\(y''=6x\)。(√)4.\(\int\frac{1}{x}dx=\lnx+C\)（\(x>0\)），\(\int\frac{1}{x}dx=\ln(-x)+C\)（\(x<0\)），所以\(\int\frac{1}{x}dx=\ln|x|+C\)。(√)5.設\(z=f(x,y)\)，若\(\frac{\partialz}{\partialx}\)與\(\frac{\partialz}{\partialy}\)在點\((x_0,y_0)\)處都存在，則\(z=f(x,y)\)在點\((x_0,y_0)\)處可微。(×)6.若向量組\(\alpha_1,\alpha_2,\alpha_3\)線性無關，則向量組\(\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_3+\alpha_1\)也線性無關。(√)7.對于\(n\)階方陣\(A\)，\(B\)，若\(AB=E\)，則\(BA=E\)。(×)8.若\(n\)階方陣\(A\)可逆，則\(A\)的伴隨矩陣\(A^\)也可逆。(√)9.設隨機變量\(X\)服從參數(shù)為\(\lambda\)的泊松分布，\(E(X)=\lambda\)，\(D(X)=\lambda\)。(√)10.\(C_{n}^m=C_{n}^{n-m}\)。(√)四、簡答題（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\(y'=3x^2-6x-9=3(x^2-2x-3)=3(x-3)(x+1)\)。令\(y'=0\)，得\(x=-1\)或\(x=3\)。當\(x<-1\)時，\(y'>0\)；當\(-1<x<3\)時，\(y'<0\)；當\(x>3\)時，\(y'>0\)。所以極大值為\(y(-1)=10\)，極小值為\(y(3)=-22\)。2.計算定積分\(\int_{0}^{1}xe^xdx\)。答案：利用分部積分法，設\(u=x\)，\(dv=e^xdx\)，則\(du=dx\)，\(v=e^x\)。\(\int_{0}^{1}xe^xdx=[xe^x]_{0}^{1}-\int_{0}^{1}e^xdx=e-(e-1)=1\)。3.設向量\(\vec{a}=(1,2,3)\)，\(\vec=(2,-1,1)\)，求\(\vec{a}\times\vec\)。答案：\(\vec{a}\times\vec=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\1&2&3\\2&-1&1\end{vmatrix}=\vec{i}(2\times1-(-1)\times3)-\vec{j}(1\times1-2\times3)+\vec{k}(1\times(-1)-2\times2)=\vec{i}(2+3)-\vec{j}(1-6)+\vec{k}(-1-4)=5\vec{i}+5\vec{j}-5\vec{k}=(5,5,-5)\)。4.求齊次線性方程組\(\begin{cases}x_1+x_2+x_3=0\\x_1+2x_2+3x_3=0\\2x_1+3x_2+4x_3=0\end{cases}\)的通解。答案：對系數(shù)矩陣\(A=\begin{pmatrix}1&1&1\\1&2&3\\2&3&4\end{pmatrix}\)進行初等行變換為\(\begin{pmatrix}1&1&1\\0&1&2\\0&1&2\end{pmatrix}\)，進一步化為\(\begin{pmatrix}1&0&-1\\0&1&2\\0&0&0\end{pmatrix}\)。令\(x_3=t\)，則\(x_1=t\)，\(x_2=-2t\)，通解為\(\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=t\begin{pmatrix}1\\-2\\1\end{pmatrix}(t\inR)\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x-1}\)的單調(diào)性。答案：函數(shù)\(y=\frac{1}{x-1}\)的定義域為\(x\neq1\)。對\(y\)求導得\(y'=-\frac{1}{(x-1)^2}<0\)（\(x\neq1