17年數(shù)學(xué)一試題和答案

一、單項選擇題(每題2分,共10題)1.當(dāng)\(x\to0\)時,與\(x\)等價無窮小的是（）A.\(\sin2x\)B.\(x^2+x\)C.\(\tanx\)D.\(1-\cosx\)2.設(shè)函數(shù)\(f(x)\)在\(x=0\)處可導(dǎo),且\(f(0)=0\),\(f^\prime(0)=2\),則\(\lim_{x\to0}\frac{f(x)}{x}\)的值為()A.0B.1C.2D.不存在3.若函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處可微,則在該點處函數(shù)\(z=f(x,y)\)()A.必有極限B.一定連續(xù)C.偏導(dǎo)數(shù)一定連續(xù)D.以上都不對4.設(shè)\(A\)為\(n\)階方陣,且\(|A|=0\),則()A.\(A\)中必有兩行(列)元素對應(yīng)成比例B.\(A\)中至少有一行(列)向量是其余行(列)向量的線性組合C.\(A\)中必有一行(列）元素全為0D.\(A\)的秩\(r(A)=n-1\)5.設(shè)隨機變量\(X\)服從正態(tài)分布\(N(1,4)\),則\(P\{X\leq1\}\)的值為()A.0B.0.5C.1D.0.256.函數(shù)\(y=e^x\)的一個原函數(shù)是()A.\(e^x+1\)B.\(e^x-1\)C.\(e^x\)D.\(2e^x\)7.曲線\(y=x^3-3x^2+2x\)的拐點坐標為()A.\((1,0)\)B.\((-1,-6)\)C.\((0,0)\)D.\((2,-2)\)8.已知向量組\(\alpha_1=(1,2,3)\),\(\alpha_2=(2,4,k)\)線性相關(guān),則\(k\)的值為()A.3B.4C.5D.69.設(shè)\(D\)是由\(x=0\),\(y=0\),\(x+y=1\)所圍成的區(qū)域,則\(\iint_D2dxdy\)的值為()A.1B.2C.0.5D.410.冪級數(shù)\(\sum_{n=1}^{\infty}\frac{x^n}{n}\)的收斂半徑為()A.0B.1C.\(+\infty\)D.2二、多項選擇題(每題2分,共10題)1.下列函數(shù)中,在其定義域內(nèi)連續(xù)的有（）A.\(y=\frac{1}{x}\)B.\(y=\sinx\)C.\(y=\lnx\)D.\(y=\sqrt{x}\)2.設(shè)函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上可導(dǎo),則下列說法正確的是()A.\(f(x)\)在\([a,b]\)上一定連續(xù)B.\(f(x)\)在\((a,b)\)內(nèi)一定有最大值和最小值C.若\(f^\prime(x_0)=0\),則\(x_0\)為\(f(x)\)的極值點D.若\(f(x)\)在\((a,b)\)內(nèi)可導(dǎo),且\(f(a)=f(b)\),則至少存在一點\(\xi\in(a,b)\),使得\(f^\prime(\xi)=0\)3.對于\(n\)階方陣\(A\)和\(B\),下列結(jié)論正確的是（）A.\((AB)^T=B^TA^T\)B.\((A+B)^2=A^2+2AB+B^2\)C.若\(AB=AC\)且\(A\)可逆,則\(B=C\)D.若\(|AB|=0\),則\(|A|=0\)或\(|B|=0\)4.下列二次型中,是正定二次型的有（）A.\(f(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2\)B.\(f(x_1,x_2,x_3)=x_1^2+2x_2^2+3x_3^2-2x_1x_2\)C.\(f(x_1,x_2,x_3)=-x_1^2-x_2^2-x_3^2\)D.\(f(x_1,x_2,x_3)=x_1^2+x_2^2\)5.設(shè)隨機變量\(X\)與\(Y\)相互獨立,且\(X\simN(0,1)\),\(Y\simN(1,4)\),則()A.\(X+Y\simN(1,5)\)B.\(X-Y\simN(-1,5)\)C.\(2X+Y\simN(1,8)\)D.\(X+2Y\simN(2,17)\)6.下列積分中,值為0的有()A.\(\int_{-\pi}^{\pi}\sinxdx\)B.\(\int_{-1}^{1}x^3dx\)C.\(\int_{-\pi}^{\pi}\cosxdx\)D.\(\int_{-1}^{1}\frac{1}{x}dx\)7.曲線\(y=x^4-2x^2+3\)的單調(diào)區(qū)間為()A.在\((-\infty,-1)\)上單調(diào)遞增B.在\((-1,0)\)上單調(diào)遞減C.在\((0,1)\)上單調(diào)遞減D.在\((1,+\infty)\)上單調(diào)遞增8.設(shè)向量組\(\alpha_1,\alpha_2,\alpha_3\)線性無關(guān),則下列向量組中,線性無關(guān)的有()A.\(\alpha_1+\alpha_2,\alpha_2+\alpha_3,\alpha_3+\alpha_1\)B.\(\alpha_1-\alpha_2,\alpha_2-\alpha_3,\alpha_3-\alpha_1\)C.\(\alpha_1,\alpha_1+\alpha_2,\alpha_1+\alpha_2+\alpha_3\)D.\(\alpha_1-\alpha_2,\alpha_1+\alpha_2,2\alpha_1+3\alpha_2\)9.設(shè)\(D\)是由\(x^2+y^2\leq1\)所圍成的區(qū)域,則下列二重積分的值為\(\pi\)的有()A.\(\iint_D1dxdy\)B.\(\iint_D2dxdy\)C.\(\iint_D\frac{1}{2}dxdy\)D.\(\iint_D\pidxdy\)10.下列級數(shù)中,收斂的有()A.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n}\)C.\(\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\)D.\(\sum_{n=1}^{\infty}\frac{1}{2^n}\)三、判斷題(每題2分,共10題)1.若函數(shù)\(f(x)\)在點\(x_0\)處極限存在,則\(f(x)\)在點\(x_0\)處一定連續(xù)。()2.函數(shù)\(y=x^3\)在\(x=0\)處的導(dǎo)數(shù)為0,所以\(x=0\)是函數(shù)\(y=x^3\)的極值點。(）3.若\(A\)為\(n\)階方陣,且\(|A|\neq0\),則\(A\)可逆。（)4.二次型\(f(x_1,x_2,x_3)=x_1^2+x_2^2-x_3^2\)是正定二次型。()5.設(shè)隨機變量\(X\)與\(Y\)的協(xié)方差\(Cov(X,Y)=0\),則\(X\)與\(Y\)相互獨立。（)6.定積分\(\int_{a}^f(x)dx\)的值與積分變量\(x\)的符號無關(guān)。()7.若向量組\(\alpha_1,\alpha_2,\cdots,\alpha_m\)線性相關(guān),則其中必有一個向量可由其余向量線性表示。（)8.冪級數(shù)\(\sum_{n=0}^{\infty}a_nx^n\)在其收斂區(qū)間內(nèi)一定絕對收斂。()9.函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處的偏導(dǎo)數(shù)\(f_x(x_0,y_0)\)和\(f_y(x_0,y_0)\)都存在,則函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處可微。(）10.設(shè)\(A\)和\(B\)為兩個事件,若\(P(AB)=P(A)P(B)\),則\(A\)與\(B\)互不相容。（）四、簡答題(每題5分,共4題)1.求函數(shù)\(y=x^3-3x^2+5\)的單調(diào)區(qū)間和極值。-答案:先求導(dǎo)\(y^\prime=3x^2-6x=3x(x-2)\)。令\(y^\prime=0\),得\(x=0\)或\(x=2\)。當(dāng)\(x\lt0\)或\(x\gt2\)時,\(y^\prime\gt0\),函數(shù)單調(diào)遞增;當(dāng)\(0\ltx\lt2\)時,\(y^\prime\lt0\),函數(shù)單調(diào)遞減。極大值\(y(0)=5\),極小值\(y(2)=1\)。2.計算定積分\(\int_{0}^{1}xe^xdx\)。-答案:用分部積分法,設(shè)\(u=x\),\(dv=e^xdx\),則\(du=dx\),\(v=e^x\)。\(\int_{0}^{1}xe^xdx=[xe^x]_0^1-\int_{0}^{1}e^xdx=e-[e^x]_0^1=1\)。3.已知矩陣\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\),求\(A\)的逆矩陣\(A^{-1}\)。-答案:先求行列式\(|A|=1\times4-2\times3=-2\)。伴隨矩陣\(adj(A)=\begin{pmatrix}4&-2\\-3&1\end{pmatrix}\),則\(A^{-1}=-\frac{1}{2}\begin{pmatrix}4&-2\\-3&1\end{pmatrix}=\begin{pmatrix}-2&1\\\frac{3}{2}&-\frac{1}{2}\end{pmatrix}\)。4.設(shè)隨機變量\(X\)的概率密度函數(shù)為\(f(x)=\begin{cases}2x,&0\leqx\leq1\\0,&其他\end{cases}\),求\(E(X)\)。-答案:根據(jù)期望公式\(E(X)=\int_{-\infty}^{+\infty}xf(x)dx=\int_{0}^{1}x\cdot2xdx=\int_{0}^{1}2x^2dx=[\frac{2}{3}x^3]_0^1=\frac{2}{3}\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=\frac{1}{x-1}\)的漸近線情況。-答案:垂直漸近線:令分母\(x-1=0\),得\(x=1\),所以\(x=1\)是垂直漸近線。水平漸近線:\(\lim_{x\to\pm\infty}\frac{1}{x-1}=0\),所以\(y=0\)是水平漸近線。無斜漸近線。2.討論向量組\(\alpha_1=(1,1,1)\),\(\alpha_2=(1,2,3)\),\(\alpha_3=(1,3,k)\)的線性相關(guān)性。-答案：設(shè)\(k_1\alpha_1+k_2\alpha_2+k_3\alpha_3=0\),得到方程組\(\begin{cases}k_1+k_2+k_3=0\\k_1+2k_2+3k_3=0\\k_1+3k_2+kk_3=0\end{cases}\),其系數(shù)行列式\(D=\begin{vmatrix}1&1&1\\1&2&3\\1&3&k\end{vmatrix}=k-5\)。當(dāng)\(k=5\),\(D=0\),向量組線性相關(guān)；當(dāng)\(k\neq5\),\(D\neq0\),向量組線性無關(guān)。3.討論冪級數(shù)\(\sum_{n=1}^{\infty}\frac{(x-1)^n}{n}\)的收斂域。-答案:令\(t=x-1\),則冪級數(shù)變