2025年專升本數(shù)學(xué)真題及答案

一、單項(xiàng)選擇題1.函數(shù)\(y=\frac{1}{\sqrt{4-x^{2}}}\)的定義域是（）A.\((-2,2)\)B.\([-2,2]\)C.\((-\infty,-2)\cup(2,+\infty)\)D.\((-\infty,-2]\cup[2,+\infty)\)答案：A2.當(dāng)\(x\to0\)時(shí)，與\(x\)等價(jià)無(wú)窮小的是（）A.\(\sin2x\)B.\(\ln(1+x)\)C.\(1-\cosx\)D.\(2^{x}-1\)答案：B3.設(shè)函數(shù)\(f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)，且\(f^\prime(x_0)=2\)，則\(\lim\limits_{h\to0}\frac{f(x_0+h)-f(x_0)}{2h}\)的值為（）A.\(1\)B.\(2\)C.\(4\)D.\(\frac{1}{2}\)答案：A4.函數(shù)\(y=x^{3}-3x\)的單調(diào)遞增區(qū)間是（）A.\((-\infty,-1)\)和\((1,+\infty)\)B.\((-1,1)\)C.\((-\infty,0)\)D.\((0,+\infty)\)答案：A5.曲線\(y=\frac{1}{x}\)在點(diǎn)\((1,1)\)處的切線方程為（）A.\(x+y-2=0\)B.\(x-y-2=0\)C.\(x+y+2=0\)D.\(x-y+2=0\)答案：A6.\(\int\frac{1}{x^{2}}dx\)等于（）A.\(\frac{1}{x}+C\)B.\(-\frac{1}{x}+C\)C.\(\frac{2}{x^{3}}+C\)D.\(-\frac{2}{x^{3}}+C\)答案：B7.設(shè)\(f(x)\)為連續(xù)函數(shù)，則\((\int_{0}^{x}f(t)dt)^\prime\)等于（）A.\(f(x)\)B.\(f(t)\)C.\(\int_{0}^{x}f^\prime(t)dt\)D.\(f(x)-f(0)\)答案：A8.二元函數(shù)\(z=x^{2}+y^{2}\)在點(diǎn)\((1,1)\)處的全微分\(dz\)為（）A.\(2dx+2dy\)B.\(dx+dy\)C.\(4dx+4dy\)D.\(2dx+dy\)答案：A9.微分方程\(y^\prime-2y=0\)的通解是（）A.\(y=Ce^{2x}\)B.\(y=Ce^{-2x}\)C.\(y=Cxe^{2x}\)D.\(y=Cxe^{-2x}\)答案：A10.設(shè)矩陣\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)，則\(A\)的行列式\(\vertA\vert\)的值為（）A.\(-2\)B.\(2\)C.\(-10\)D.\(10\)答案：A二、多項(xiàng)選擇題1.下列函數(shù)中，在其定義域內(nèi)連續(xù)的函數(shù)有（）A.\(y=\frac{1}{x}\)B.\(y=\sinx\)C.\(y=\lnx\)D.\(y=\sqrt{x}\)答案：ABCD2.下列極限中，極限值為\(1\)的有（）A.\(\lim\limits_{x\to0}\frac{\sinx}{x}\)B.\(\lim\limits_{x\to0}(1+x)^{\frac{1}{x}}\)C.\(\lim\limits_{x\to\infty}(1+\frac{1}{x})^{x}\)D.\(\lim\limits_{x\to0}\frac{e^{x}-1}{x}\)答案：ABC3.設(shè)函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上可導(dǎo)，則下列說(shuō)法正確的有（）A.若\(f^\prime(x)>0\)在\((a,b)\)內(nèi)恒成立，則\(f(x)\)在\([a,b]\)上單調(diào)遞增B.若\(f^\prime(x)<0\)在\((a,b)\)內(nèi)恒成立，則\(f(x)\)在\([a,b]\)上單調(diào)遞減C.若\(f(x)\)在\([a,b]\)上有最大值，則最大值點(diǎn)一定是駐點(diǎn)D.若\(f(x)\)在\([a,b]\)上有最小值，則最小值點(diǎn)一定是導(dǎo)數(shù)不存在的點(diǎn)答案：AB4.下列積分中，計(jì)算正確的有（）A.\(\int_{0}^{1}x^{2}dx=\frac{1}{3}\)B.\(\int_{-1}^{1}x^{3}dx=0\)C.\(\int_{0}^{\pi}\sinxdx=2\)D.\(\int_{0}^{1}e^{x}dx=e-1\)答案：ABCD5.對(duì)于二元函數(shù)\(z=f(x,y)\)，下列說(shuō)法正確的有（）A.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處連續(xù)B.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處偏導(dǎo)數(shù)存在C.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處偏導(dǎo)數(shù)存在，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微D.若\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處偏導(dǎo)數(shù)連續(xù)，則\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處可微答案：ABD6.下列矩陣中，是方陣的有（）A.\(\begin{pmatrix}1&2\\3&4\end{pmatrix}\)B.\(\begin{pmatrix}1&2&3\end{pmatrix}\)C.\(\begin{pmatrix}1\\2\\3\end{pmatrix}\)D.\(\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\)答案：AD7.設(shè)\(A\)，\(B\)為\(n\)階方陣，下列等式成立的有（）A.\((A+B)^2=A^{2}+2AB+B^{2}\)B.\((AB)^T=B^TA^T\)C.\(\vertAB\vert=\vertA\vert\vertB\vert\)D.\((A+B)^{-1}=A^{-1}+B^{-1}\)答案：BC8.下列向量組中，線性相關(guān)的有（）A.\(\vec{a}=(1,0,0)\)，\(\vec=(0,1,0)\)，\(\vec{c}=(0,0,1)\)B.\(\vec{a}=(1,2,3)\)，\(\vec=(2,4,6)\)C.\(\vec{a}=(1,1,1)\)，\(\vec=(1,2,3)\)，\(\vec{c}=(2,3,4)\)D.\(\vec{a}=(1,0,0)\)，\(\vec=(0,1,0)\)，\(\vec{c}=(1,1,0)\)答案：BD9.設(shè)事件\(A\)，\(B\)相互獨(dú)立，且\(P(A)=0.5\)，\(P(B)=0.6\)，則下列說(shuō)法正確的有（）A.\(P(A\cupB)=0.8\)B.\(P(A\capB)=0.3\)C.\(P(A|B)=0.5\)D.\(P(B|A)=0.6\)答案：ABCD10.下列概率分布中，是離散型隨機(jī)變量概率分布的有（）A.\(P(X=k)=\frac{1}{2^{k}}\)，\(k=1,2,\cdots\)B.\(P(X=k)=\frac{C_{n}^{k}p^{k}(1-p)^{n-k}}{k!}\)，\(k=0,1,\cdots,n\)C.\(P(X=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}\)，\(k=0,1,\cdots\)D.\(P(X=x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\)，\(-\infty<x<+\infty\)答案：ABC三、判斷題1.函數(shù)\(y=\frac{1}{x-1}\)在\(x=1\)處沒(méi)有定義，所以\(x=1\)是該函數(shù)的間斷點(diǎn)。（√）2.若函數(shù)\(f(x)\)在點(diǎn)\(x_0\)處的左右導(dǎo)數(shù)都存在，則\(f(x)\)在點(diǎn)\(x_0\)處可導(dǎo)。（×）3.函數(shù)\(y=x^{2}\)在\((-\infty,0)\)上是凸函數(shù)。（√）4.\(\int_{0}^{2\pi}\cosxdx=0\)。（√）5.若二元函數(shù)\(z=f(x,y)\)在點(diǎn)\((x_0,y_0)\)處的偏導(dǎo)數(shù)\(f_x(x_0,y_0)=0\)且\(f_y(x_0,y_0)=0\)，則點(diǎn)\((x_0,y_0)\)一定是函數(shù)\(z=f(x,y)\)的極值點(diǎn)。（×）6.設(shè)\(A\)，\(B\)為\(n\)階方陣，若\(AB=0\)，則\(A=0\)或\(B=0\)。（×）7.向量組\(\vec{a}=(1,2,3)\)，\(\vec=(2,4,6)\)線性相關(guān)。（√）8.若事件\(A\)與\(B\)互斥，則\(P(A\cupB)=P(A)+P(B)\)。（√）9.設(shè)\(X\)服從正態(tài)分布\(N(\mu,\sigma^{2})\)，則\(E(X)=\mu\)，\(D(X)=\sigma^{2}\)。（√）10.線性方程組\(Ax=b\)有解的充要條件是\(r(A)=r(A|b)\)。（√）四、簡(jiǎn)答題1.求函數(shù)\(y=x^{3}-3x^{2}+5\)的極值。先求導(dǎo)數(shù)\(y^\prime=3x^{2}-6x=3x(x-2)\)。令\(y^\prime=0\)，解得\(x=0\)和\(x=2\)。當(dāng)\(x<0\)時(shí)，\(y^\prime>0\)，函數(shù)遞增；當(dāng)\(0<x<2\)時(shí)，\(y^\prime<0\)，函數(shù)遞減；當(dāng)\(x>2\)時(shí)，\(y^\prime>0\)，函數(shù)遞增。所以\(x=0\)時(shí)取極大值\(y(0)=5\)，\(x=2\)時(shí)取極小值\(y(2)=1\)。2.計(jì)算定積分\(\int_{1}^{e}\frac{1}{x}(1+\lnx)dx\)。令\(u=1+\lnx\)，則\(du=\frac{1}{x}dx\)。當(dāng)\(x=1\)時(shí)，\(u=1+\ln1=1\)；當(dāng)\(x=e\)時(shí)，\(u=1+\lne=2\)。原積分化為\(\int_{1}^{2}udu\)，根據(jù)積分公式\(\intudu=\frac{1}{2}u^{2}+C\)，可得\([\frac{1}{2}u^{2}]_{1}^{2}=\frac{1}{2}(4-1)=\frac{3}{2}\)。3.求二元函數(shù)\(z=x^{2}+2xy-y^{2}\)的偏導(dǎo)數(shù)\(z_x\)和\(z_y\)。求\(z_x\)時(shí)，把\(y\)看作常數(shù)，對(duì)\(x\)求導(dǎo)，\(z_x=2x+2y\)。求\(z_y\)時(shí)，把\(x\)看作常數(shù)，對(duì)\(y\)求導(dǎo)，\(z_y=2x-2y\)。4.已知矩陣\(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\)，求\(A\)的逆矩陣\(A^{-1}\)。先求行列式\(\vertA\vert=1\times4-2\times3=-2\)。伴隨矩陣\(A^{}=\begin{pmatrix}4&-2\\-3&1\end{pmatrix}\)。根據(jù)逆矩陣公式\(A^{-1}=\frac{1}{\vertA\vert}A^{}\)，可得\(A^{-1}=-\frac{1}{2}\begin{pmatrix}4&-2\\-3&1\end{pmatrix}=\begin{pmatrix}-2&1\\\frac{3}{2}&-\frac{1}{2}\end{pmatrix}\)。五、討論題1.討論函數(shù)\(y=\frac{x^{2}-1}{x-1}\)的連續(xù)性，并說(shuō)明間斷點(diǎn)類型。首先化簡(jiǎn)函數(shù)\(y=\frac{x^{2}-1}{x-1}=x+1(x\neq1)\)。函數(shù)在\(x\neq1\)處是一次函數(shù)，是連續(xù)的。在\(x=1\)處，函數(shù)無(wú)定義。