大學(xué)數(shù)學(xué)競(jìng)賽數(shù)學(xué)類試題和答案

一、單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\sinx\)在\(x=0\)處的導(dǎo)數(shù)是(）A.0B.1C.-1D.22.極限\(\lim_{x\to0}\frac{\sinx}{x}\)的值為()A.0B.1C.-1D.不存在3.若\(f(x)\)在\([a,b]\)上連續(xù),在\((a,b)\)內(nèi)可導(dǎo),且\(f(a)=f(b)\),則至少存在一點(diǎn)\(\xi\in(a,b)\),使得()A.\(f'(\xi)>0\)B.\(f'(\xi)=0\)C.\(f'(\xi)<0\)D.\(f'(\xi)\)不存在4.設(shè)\(A\)為\(n\)階方陣,且\(\vertA\vert=0\),則\(A\)()A.必有一列元素全為0B.必有兩行元素對(duì)應(yīng)成比例C.必有一個(gè)列向量是其余列向量的線性組合D.任一列向量是其余列向量的線性組合5.級(jí)數(shù)\(\sum_{n=1}^{\infty}\frac{1}{n^p}\)當(dāng)()時(shí)收斂。A.\(p\leq1\)B.\(p>1\)C.\(p<1\)D.\(p\geq1\)6.向量\(\vec{a}=(1,2,3)\)與\(\vec=(-1,-2,-3)\)的夾角為()A.\(0\)B.\(\frac{\pi}{2}\)C.\(\pi\)D.\(\frac{3\pi}{2}\)7.函數(shù)\(z=x^2+y^2\)在點(diǎn)\((1,1)\)處沿向量\(\vec{l}=(1,1)\)的方向?qū)?shù)為()A.\(2\sqrt{2}\)B.\(2\)C.\(4\sqrt{2}\)D.\(4\)8.若\(f(x)\)的一個(gè)原函數(shù)是\(F(x)\),則\(\intf(2x)dx\)等于()A.\(F(2x)+C\)B.\(\frac{1}{2}F(2x)+C\)C.\(2F(2x)+C\)D.\(F(x)+C\)9.設(shè)\(A,B\)為\(n\)階方陣,且\(AB=O\),則下列結(jié)論正確的是()A.\(A=O\)或\(B=O\)B.\(\vertA\vert=0\)或\(\vertB\vert=0\)C.\(A+B=O\)D.\(A-B=O\)10.微分方程\(y''-2y'+y=0\)的通解是()A.\(y=C_1e^x+C_2e^{-x}\)B.\(y=(C_1+C_2x)e^x\)C.\(y=C_1+C_2e^x\)D.\(y=C_1e^x+C_2x\)二、多項(xiàng)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.\(y=\vertx\vert\)B.\(y=\frac{1}{x}\)C.\(y=\sinx\)D.\(y=\begin{cases}x+1,&x\geq0\\x-1,&x<0\end{cases}\)2.以下哪些是矩陣的初等行變換()A.交換兩行B.某一行乘以非零常數(shù)C.某一行乘以常數(shù)加到另一行D.交換兩列3.下列級(jí)數(shù)中,收斂的有()A.\(\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\)B.\(\sum_{n=1}^{\infty}\frac{1}{n}\)C.\(\sum_{n=1}^{\infty}\frac{1}{n^2}\)D.\(\sum_{n=1}^{\infty}(-1)^n\frac{1}{n}\)4.關(guān)于向量運(yùn)算,正確的有()A.\(\vec{a}\cdot\vec=\vec\cdot\vec{a}\)B.\(\vec{a}\times\vec=-\vec\times\vec{a}\)C.\((\vec{a}+\vec)\cdot\vec{c}=\vec{a}\cdot\vec{c}+\vec\cdot\vec{c}\)D.\((\vec{a}\times\vec)\times\vec{c}=\vec{a}\times(\vec\times\vec{c})\)5.函數(shù)\(f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)的充要條件有()A.左導(dǎo)數(shù)等于右導(dǎo)數(shù)B.極限\(\lim_{x\tox_0}\frac{f(x)-f(x_0)}{x-x_0}\)存在C.函數(shù)在\(x_0\)處連續(xù)D.函數(shù)在\(x_0\)處可微6.設(shè)\(A\)為\(n\)階方陣,下列哪些條件等價(jià)于\(A\)可逆()A.\(\vertA\vert\neq0\)B.\(A\)的秩為\(n\)C.\(A\)的列向量組線性無關(guān)D.存在\(n\)階方陣\(B\),使得\(AB=BA=E\)7.下列曲面中,屬于二次曲面的有（）A.球面B.柱面C.錐面D.拋物面8.對(duì)于多元函數(shù)\(z=f(x,y)\),下列說法正確的是()A.若偏導(dǎo)數(shù)存在,則函數(shù)一定連續(xù)B.若函數(shù)可微,則偏導(dǎo)數(shù)一定存在C.若偏導(dǎo)數(shù)連續(xù),則函數(shù)一定可微D.函數(shù)在某點(diǎn)連續(xù),則偏導(dǎo)數(shù)一定存在9.下列積分中,計(jì)算正確的有（）A.\(\int_{0}^{1}x^2dx=\frac{1}{3}\)B.\(\int_{-\pi}^{\pi}\sinxdx=0\)C.\(\int_{0}^{2\pi}\cosxdx=0\)D.\(\int_{1}^{e}\frac{1}{x}dx=1\)10.微分方程\(y'+p(x)y=q(x)\)的通解結(jié)構(gòu)為()A.它的一個(gè)特解B.對(duì)應(yīng)的齊次方程\(y'+p(x)y=0\)的通解C.它的一個(gè)特解加上對(duì)應(yīng)的齊次方程的通解D.兩個(gè)特解之差三、判斷題(每題2分,共10題)1.若\(f(x)\)在\(x_0\)處極限存在,則\(f(x)\)在\(x_0\)處一定連續(xù)。（)2.方陣\(A\)的行列式\(\vertA\vert\)與它的轉(zhuǎn)置矩陣\(A^T\)的行列式\(\vertA^T\vert\)相等。()3.級(jí)數(shù)\(\sum_{n=1}^{\infty}a_n\)收斂,則\(\lim_{n\to\infty}a_n=0\)。(）4.向量\(\vec{a}\)與\(\vec\)平行,則\(\vec{a}\times\vec=\vec{0}\)。（)5.函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處的偏導(dǎo)數(shù)\(f_x(x_0,y_0)\)就是函數(shù)\(z=f(x,y_0)\)在\(x=x_0\)處的導(dǎo)數(shù)。()6.若\(A\)為\(n\)階方陣,且\(A^2=A\),則\(A\)的特征值只能是0或1。（)7.定積分\(\int_{a}^f(x)dx\)的值與積分變量用什么字母表示無關(guān)。()8.二元函數(shù)\(z=f(x,y)\)的二階混合偏導(dǎo)數(shù)\(f_{xy}(x,y)\)與\(f_{yx}(x,y)\)一定相等。()9.微分方程\(y''+y=0\)的通解是\(y=C_1\cosx+C_2\sinx\)。()10.若\(A,B\)為\(n\)階方陣,且\(AB=BA\),則\((A+B)^2=A^2+2AB+B^2\)。（）四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=x^3-3x^2+1\)的極值點(diǎn)與極值。答:對(duì)\(y\)求導(dǎo)得\(y'=3x^2-6x=3x(x-2)\)。令\(y'=0\),解得\(x=0\)或\(x=2\)。當(dāng)\(x<0\),\(y'>0\);\(0<x<2\),\(y'<0\);\(x>2\),\(y'>0\)。所以\(x=0\)是極大值點(diǎn),極大值為\(y(0)=1\);\(x=2\)是極小值點(diǎn),極小值為\(y(2)=-3\)。2.計(jì)算行列式\(\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}\)。答:第二行減去第一行的4倍,第三行減去第一行的7倍,得\(\begin{vmatrix}1&2&3\\0&-3&-6\\0&-6&-12\end{vmatrix}\)。再由第三行減去第二行的2倍,得\(\begin{vmatrix}1&2&3\\0&-3&-6\\0&0&0\end{vmatrix}=0\)。3.求\(\int\frac{1}{x^2-1}dx\)。答:將\(\frac{1}{x^2-1}\)分解為\(\frac{1}{2}(\frac{1}{x-1}-\frac{1}{x+1})\)。則\(\int\frac{1}{x^2-1}dx=\frac{1}{2}\int\frac{1}{x-1}dx-\frac{1}{2}\int\frac{1}{x+1}dx=\frac{1}{2}\ln\vertx-1\vert-\frac{1}{2}\ln\vertx+1\vert+C=\frac{1}{2}\ln\vert\frac{x-1}{x+1}\vert+C\)。4.已知向量\(\vec{a}=(1,2,-1)\),\(\vec=(2,-1,1)\),求\(\vec{a}\cdot\vec\)及\(\vec{a}\)與\(\vec\)夾角的余弦值。答：\(\vec{a}\cdot\vec=1×2+2×(-1)+(-1)×1=-1\)。\(\vert\vec{a}\vert=\sqrt{1^2+2^2+(-1)^2}=\sqrt{6}\),\(\vert\vec\vert=\sqrt{2^2+(-1)^2+1^2}=\sqrt{6}\)。夾角余弦值\(\cos\theta=\frac{\vec{a}\cdot\vec}{\vert\vec{a}\vert\vert\vec\vert}=\frac{-1}{\sqrt{6}×\sqrt{6}}=-\frac{1}{6}\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(f(x)=\begin{cases}x\sin\frac{1}{x},&x\neq0\\0,&x=0\end{cases}\)在\(x=0\)處的連續(xù)性與可導(dǎo)性。答:連續(xù)性:\(\lim_{x\to0}x\sin\frac{1}{x}=0=f(0)\),所以函數(shù)在\(x=0\)處連續(xù)?？蓪?dǎo)性:\(f'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to0}\sin\frac{1}{x}\),極限不存在,所以函數(shù)在\(x=0\)處不可導(dǎo)。2.討論方陣\(A\)可逆的判定方法及應(yīng)用場(chǎng)景。答:判定方法有:行列式\(\vertA\vert\neq0\);秩\(r(A)=n\);列向量組線性無關(guān)等。應(yīng)用場(chǎng)景如解線性方程組\(Ax=b\),當(dāng)\(A\)可逆時(shí),有唯一解\(x=A^{-1}b\);在矩陣運(yùn)算中簡(jiǎn)化計(jì)算等。3.討論級(jí)數(shù)收斂性判別法及其適用情況。答:比較判別法適用于與已知斂散性級(jí)數(shù)比較;比值判別法適用于通項(xiàng)含\(n!\)等形式;根值判別法適用于通項(xiàng)含\(n\)次冪形式。此外還有萊布尼茨判別法用于交錯(cuò)級(jí)數(shù)。不同判別法根據(jù)級(jí)數(shù)通項(xiàng)特點(diǎn)選擇使用。4.討論多元函數(shù)的偏導(dǎo)數(shù)、全微分與函數(shù)連續(xù)性之間的關(guān)系。答:函數(shù)可微則偏導(dǎo)數(shù)存在且函數(shù)連續(xù);偏導(dǎo)數(shù)連續(xù)則函數(shù)可微;但偏導(dǎo)數(shù)存在函數(shù)不一定連續(xù),函數(shù)連續(xù)偏導(dǎo)數(shù)也不一定存在。例如\(z=\sqrt{x^2+y^2}\)在\((0,0)\)連續(xù)但偏導(dǎo)數(shù)不存在;\(z=\begin{cases}\frac{xy}{x^2+y^