2025年考研數(shù)學(xué)二真題及答案

一、單項(xiàng)選擇題1.當(dāng)\(x\to0\)時(shí)，下列無(wú)窮小量中階數(shù)最高的是（）A.\(x^2-\sin^2x\)B.\(1-\cos\sqrt{x}\)C.\(\tanx-\sinx\)D.\(e^{x^2}-1-x^2\)答案：A2.函數(shù)\(f(x)=\frac{x^2-1}{x(x-1)}\)的間斷點(diǎn)個(gè)數(shù)為（）A.0B.1C.2D.3答案：C3.設(shè)函數(shù)\(y=y(x)\)由方程\(e^{x+y}-xy=1\)確定，則\(y^\prime(0)\)等于（）A.-1B.0C.1D.2答案：A4.已知函數(shù)\(f(x)\)具有二階導(dǎo)數(shù)，且\(\lim\limits_{x\to0}\frac{f(x)}{x}=0\)，\(f^{\prime\prime}(0)=4\)，則\(\lim\limits_{x\to0}\frac{f(x)}{x^2}\)的值為（）A.0B.1C.2D.4答案：C5.設(shè)函數(shù)\(f(x)\)在區(qū)間\([0,1]\)上連續(xù)，且\(\int_{0}^{1}f(x)dx=1\)，則\(\int_{0}^{1}\int_{x}^{1}f(x)f(y)dydx\)等于（）A.\(\frac{1}{2}\)B.1C.2D.4答案：A6.曲線\(y=\int_{0}^{x}(t-1)(t-2)dt\)在點(diǎn)\((0,0)\)處的切線方程是（）A.\(y=2x\)B.\(y=-2x\)C.\(y=x\)D.\(y=-x\)答案：A7.設(shè)函數(shù)\(z=f(x,y)\)由方程\(F(x+\frac{z}{y},y+\frac{z}{x})=0\)確定，其中\(zhòng)(F\)具有連續(xù)偏導(dǎo)數(shù)，且\(xF_1+yF_2\neq0\)，則\(x\frac{\partialz}{\partialx}+y\frac{\partialz}{\partialy}\)等于（）A.\(z-x-y\)B.\(x+y-z\)C.\(x+y+z\)D.\(-(x+y+z)\)答案：A8.設(shè)矩陣\(A=\begin{pmatrix}1&2&3\\0&1&2\\0&0&1\end{pmatrix}\)，則\(A^{-1}\)等于（）A.\(\begin{pmatrix}1&-2&1\\0&1&-2\\0&0&1\end{pmatrix}\)B.\(\begin{pmatrix}1&2&-1\\0&1&2\\0&0&1\end{pmatrix}\)C.\(\begin{pmatrix}1&-2&-1\\0&1&-2\\0&0&1\end{pmatrix}\)D.\(\begin{pmatrix}1&2&1\\0&1&2\\0&0&1\end{pmatrix}\)答案：A9.設(shè)向量組\(\alpha_1,\alpha_2,\alpha_3\)線性無(wú)關(guān)，向量\(\beta_1\)可由\(\alpha_1,\alpha_2,\alpha_3\)線性表示，向量\(\beta_2\)不能由\(\alpha_1,\alpha_2,\alpha_3\)線性表示，則必有（）A.\(\alpha_1,\alpha_2,\beta_1\)線性相關(guān)B.\(\alpha_1,\alpha_2,\beta_2\)線性相關(guān)C.\(\alpha_2,\alpha_3,\beta_1,\beta_2\)線性相關(guān)D.\(\alpha_1,\alpha_2,\alpha_3,\beta_1+\beta_2\)線性無(wú)關(guān)答案：D10.設(shè)\(A\)為\(n\)階實(shí)對(duì)稱矩陣，且\(A^2=A\)，\(r(A)=r\)，則\(\vertE+A\vert\)等于（）A.\(2^r\)B.\(2^{n-r}\)C.\(2^n\)D.\(2^{r(n-r)}\)答案：A二、多項(xiàng)選擇題1.下列關(guān)于函數(shù)極限的命題中，正確的是（）A.若\(\lim\limits_{x\tox_0}f(x)\)存在，\(\lim\limits_{x\tox_0}g(x)\)不存在，則\(\lim\limits_{x\tox_0}[f(x)+g(x)]\)不存在B.若\(\lim\limits_{x\tox_0}f(x)\)與\(\lim\limits_{x\tox_0}g(x)\)都不存在，則\(\lim\limits_{x\tox_0}[f(x)g(x)]\)不存在C.若\(\lim\limits_{x\tox_0}f(x)=0\)，且\(g(x)\)在\(x_0\)的某去心鄰域內(nèi)有界，則\(\lim\limits_{x\tox_0}[f(x)g(x)]=0\)D.若\(\lim\limits_{x\tox_0}f(x)=\infty\)，且\(\lim\limits_{x\tox_0}g(x)=\infty\)，則\(\lim\limits_{x\tox_0}[f(x)-g(x)]=\infty\)答案：AC2.設(shè)函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)可導(dǎo)，則下列結(jié)論正確的是（）A.若\(f(x)\)在\((a,b)\)內(nèi)單調(diào)增加，則\(f^\prime(x)\gt0\)在\((a,b)\)內(nèi)恒成立B.若\(f^\prime(x)\gt0\)在\((a,b)\)內(nèi)恒成立，則\(f(x)\)在\((a,b)\)內(nèi)單調(diào)增加C.若\(f(x)\)在\((a,b)\)內(nèi)有極值點(diǎn)\(x_0\)，則\(f^\prime(x_0)=0\)D.若\(f^\prime(x_0)=0\)，則\(x_0\)必為\(f(x)\)的極值點(diǎn)答案：BC3.下列積分值為零的是（）A.\(\int_{-1}^{1}x\sin^2xdx\)B.\(\int_{-\pi}^{\pi}x^2\cosxdx\)C.\(\int_{-1}^{1}(x+\sinx)dx\)D.\(\int_{-\pi}^{\pi}x\sinxdx\)答案：AC4.設(shè)函數(shù)\(z=f(x,y)\)具有二階連續(xù)偏導(dǎo)數(shù)，且\(\frac{\partial^2z}{\partialx\partialy}=0\)，則（）A.\(f(x,y)\)可表示為\(f(x,y)=g(x)+h(y)\)的形式B.\(z=f(x,y)\)的全微分\(dz=\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)C.\(\frac{\partial^2z}{\partialx^2}\)與\(\frac{\partial^2z}{\partialy^2}\)都恒為零D.函數(shù)\(z=f(x,y)\)所確定的曲面沒(méi)有鞍點(diǎn)答案：AB5.設(shè)\(A\)為\(n\)階矩陣，下列結(jié)論正確的是（）A.若\(A\)可逆，則\(A\)的伴隨矩陣\(A^\)也可逆B.若\(A\)不可逆，則\(A\)的伴隨矩陣\(A^\)的秩為0C.若\(A\)是正交矩陣，則\(A^TA=E\)D.若\(A\)是對(duì)稱矩陣，則\(A\)的特征值都是實(shí)數(shù)答案：ACD6.設(shè)向量組\(\alpha_1,\alpha_2,\cdots,\alpha_s\)線性相關(guān)，向量組\(\alpha_2,\alpha_3,\cdots,\alpha_s,\alpha_{s+1}\)線性無(wú)關(guān)，則（）A.\(\alpha_1\)可由\(\alpha_2,\alpha_3,\cdots,\alpha_s\)線性表示B.\(\alpha_{s+1}\)可由\(\alpha_1,\alpha_2,\cdots,\alpha_s\)線性表示C.向量組\(\alpha_1,\alpha_2,\cdots,\alpha_s\)的極大線性無(wú)關(guān)組與向量組\(\alpha_2,\alpha_3,\cdots,\alpha_s\)的極大線性無(wú)關(guān)組相同D.向量組\(\alpha_1,\alpha_2,\cdots,\alpha_s,\alpha_{s+1}\)的秩為\(s\)答案：AD7.設(shè)\(A\)為\(n\)階矩陣，且\(A\)的特征值為\(\lambda_1,\lambda_2,\cdots,\lambda_n\)，則（）A.\(\vertA\vert=\lambda_1\lambda_2\cdots\lambda_n\)B.\(A\)可逆的充要條件是\(\lambda_i\neq0\)，\(i=1,2,\cdots,n\)C.若\(A\)可相似對(duì)角化，則\(A\)有\(zhòng)(n\)個(gè)線性無(wú)關(guān)的特征向量D.若\(A\)為實(shí)對(duì)稱矩陣，則\(A\)一定可相似對(duì)角化答案：ABCD8.設(shè)函數(shù)\(f(x)\)在\([a,b]\)上連續(xù)，在\((a,b)\)內(nèi)可導(dǎo)，則下列說(shuō)法正確的是（）A.若\(f(a)=f(b)\)，則至少存在一點(diǎn)\(\xi\in(a,b)\)，使得\(f^\prime(\xi)=0\)B.若\(f(a)\neqf(b)\)，則至少存在一點(diǎn)\(\xi\in(a,b)\)，使得\(\frac{f(b)-f(a)}{b-a}=f^\prime(\xi)\)C.若\(f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增，則\(f^\prime(x)\geq0\)在\((a,b)\)內(nèi)恒成立D.若\(f^\prime(x)\geq0\)在\((a,b)\)內(nèi)恒成立，則\(f(x)\)在\([a,b]\)上單調(diào)遞增答案：ABCD9.設(shè)\(A\)和\(B\)為\(n\)階矩陣，且\(AB=BA\)，則（）A.\(A\)和\(B\)有相同的特征值B.\(A\)和\(B\)可同時(shí)相似對(duì)角化C.若\(\lambda\)是\(A\)的特征值，則對(duì)應(yīng)于\(\lambda\)的\(A\)的特征向量也是\(B\)的特征向量D.\(r(AB)\leq\min\{r(A),r(B)\}\)答案：BD10.設(shè)函數(shù)\(y=y(x)\)是二階常系數(shù)非齊次線性微分方程\(y^{\prime\prime}+py^\prime+qy=f(x)\)的一個(gè)特解，\(y_1(x)\)和\(y_2(x)\)是對(duì)應(yīng)的齊次方程\(y^{\prime\prime}+py^\prime+qy=0\)的兩個(gè)線性無(wú)關(guān)的解，則該非齊次方程的通解為（）A.\(y=C_1y_1(x)+C_2y_2(x)+y(x)\)B.\(y=C_1y_1(x)+C_2[y_1(x)+y_2(x)]+y(x)\)C.\(y=C_1[y_1(x)-y_2(x)]+C_2y_2(x)+y(x)\)D.\(y=C_1y(x)+C_2y_1(x)+y_2(x)\)答案：ABC三、判斷題1.若函數(shù)\(f(x)\)在\(x_0\)處可導(dǎo)，則\(f(x)\)在\(x_0\)處一定連續(xù)。（√）2.函數(shù)\(y=\vertx\vert\)在\(x=0\)處可導(dǎo)。（×）3.若\(\int_{a}^f(x)dx=0\)，則\(f(x)\)在\([a,b]\)上恒為零。（×）4.設(shè)\(z=f(x,y)\)具有二階連續(xù)偏導(dǎo)數(shù)，且\(A=\frac{\partial^2z}{\partialx^2}\)，\(B=\frac{\partial^2z}{\partialx\partialy}\)，\(C=\frac{\partial^2z}{\partialy^2}\)，當(dāng)\(AC-B^2\lt0\)時(shí)，點(diǎn)\((x_0,y_0)\)是\(z=f(x,y)\)的鞍點(diǎn)。（√）5.若矩陣\(A\)與\(B\)相似，則\(A\)與\(B\)有相同的特征多項(xiàng)式。（√）6.向量組\(\alpha_1,\alpha_2,\cdots,\alpha_s\)線性相關(guān)的充要條件是其中至少有一個(gè)向量可由其余向量線性表示。（√）7.若\(A\)為\(n\)階正交矩陣，則\(\vertA\vert=1\)。（×）8.函數(shù)\(y=e^x\)是微分方程\(y^{\prime\prime}-y=0\)的一個(gè)解。（√）9.若\(A\)是\(n\)階方陣，且\(r(A)\ltn\)，則\(A\)的列向量組線性相關(guān)。（√）10.定積分\(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)對(duì)任何函數(shù)\(f(x)\)都成立。（×）四、簡(jiǎn)答題1.求函數(shù)\(y=x^3-3x^2-9x+5\)的單調(diào)區(qū)間與極值。答案：先求導(dǎo)\(y^\prime=3x^2-6x-9=3(x^2-2x-3)=3(x-3)(x+1)\)。令\(y^\prime=0\)，得\(x=-1\)或\(x=3\)。當(dāng)\(x\lt-1\)時(shí)，\(y^\prime\gt0\)，函數(shù)單調(diào)遞增；當(dāng)\(-1