數(shù)二考研真題及答案

一、單項選擇題1.當(dāng)\(x\to0\)時，若\(x-\sinx\)與\(ax^n\)是等價無窮小，則（）A.\(a=\frac{1}{6}\)，\(n=3\)B.\(a=\frac{1}{3}\)，\(n=3\)C.\(a=\frac{1}{6}\)，\(n=2\)D.\(a=\frac{1}{3}\)，\(n=2\)答案：A2.函數(shù)\(f(x)=\lim\limits_{t\to0}(1+\frac{\sint}{x})^{\frac{x^{2}}{t}}\)在\((-\infty,+\infty)\)內(nèi)（）A.連續(xù)B.有可去間斷點C.有跳躍間斷點D.有第二類間斷點答案：B3.設(shè)函數(shù)\(f(x)\)可導(dǎo)，且\(f(x)f^\prime(x)>0\)，則（）A.\(f(1)>f(-1)\)B.\(f(1)<f(-1)\)C.\(|f(1)|>|f(-1)|\)D.\(|f(1)|<|f(-1)|\)答案：C4.曲線\(y=\frac{1}{x}+\ln(1+e^{x})\)漸近線的條數(shù)為（）A.0B.1C.2D.3答案：D5.設(shè)函數(shù)\(z=z(x,y)\)由方程\(F(\frac{y}{x},\frac{z}{x})=0\)確定，其中\(zhòng)(F\)為可微函數(shù)，且\(F_2^\prime\neq0\)，則\(x\frac{\partialz}{\partialx}+y\frac{\partialz}{\partialy}=\)（）A.\(x\)B.\(z\)C.\(-x\)D.\(-z\)答案：B6.設(shè)\(M=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sinx}{1+x^{2}}\cos^{4}xdx\)，\(N=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin^{3}x+\cos^{4}x)dx\)，\(P=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(x^{2}\sin^{3}x-\cos^{4}x)dx\)，則有（）A.\(N<P<M\)B.\(M<P<N\)C.\(N<M<P\)D.\(P<M<N\)答案：D7.設(shè)\(A\)為三階矩陣，將\(A\)的第二列加到第一列得矩陣\(B\)，再交換\(B\)的第二行與第三行得單位矩陣，記\(P_1=\begin{pmatrix}1&0&0\\1&1&0\\0&0&1\end{pmatrix}\)，\(P_2=\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}\)，則\(A=\)（）A.\(P_1P_2\)B.\(P_1^{-1}P_2\)C.\(P_2P_1\)D.\(P_2P_1^{-1}\)答案：D8.設(shè)\(A\)為\(4\times3\)矩陣，\(\eta_1,\eta_2,\eta_3\)是非齊次線性方程組\(Ax=\beta\)的3個線性無關(guān)的解，\(k_1,k_2\)為任意常數(shù)，則\(Ax=\beta\)的通解為（）A.\(\frac{\eta_2+\eta_3}{2}+k_1(\eta_2-\eta_1)+k_2(\eta_3-\eta_1)\)B.\(\frac{\eta_2-\eta_3}{2}+k_1(\eta_2-\eta_1)+k_2(\eta_3-\eta_1)\)C.\(\frac{\eta_2+\eta_3}{2}+k_1(\eta_2+\eta_1)+k_2(\eta_3+\eta_1)\)D.\(\frac{\eta_2-\eta_3}{2}+k_1(\eta_2+\eta_1)+k_2(\eta_3+\eta_1)\)答案：A9.設(shè)\(f(x)\)為可導(dǎo)函數(shù)，且滿足\(\lim\limits_{x\to0}\frac{f(1)-f(1-x)}{2x}=-1\)，則曲線\(y=f(x)\)在點\((1,f(1))\)處的切線斜率為（）A.2B.-2C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：B10.已知函數(shù)\(f(x)\)在\(x=0\)處可導(dǎo)，且\(f(0)=0\)，則\(\lim\limits_{x\to0}\frac{f(x^2)}{\sinx^2}=\)（）A.\(f^\prime(0)\)B.\(2f^\prime(0)\)C.\(\frac{1}{2}f^\prime(0)\)D.\(0\)答案：A二、多項選擇題1.下列函數(shù)中，在\(x=0\)處可導(dǎo)的有（）A.\(y=|x|^{\frac{1}{3}}\)B.\(y=\ln(1+x)\)C.\(y=e^{-|x|}\)D.\(y=\sqrt{1+x^2}\)答案：BD2.設(shè)函數(shù)\(f(x)\)連續(xù)，則下列函數(shù)中，必為偶函數(shù)的有（）A.\(F(x)=\int_{0}^{x}f(t^{2})dt\)B.\(F(x)=\int_{0}^{x}f^{2}(t)dt\)C.\(F(x)=\int_{0}^{x}t[f(t)-f(-t)]dt\)D.\(F(x)=\int_{0}^{x}t[f(t)+f(-t)]dt\)答案：AD3.下列關(guān)于函數(shù)極值的說法正確的是（）A.若\(f^\prime(x_0)=0\)，則\(x_0\)一定是\(f(x)\)的極值點B.若\(f(x)\)在\(x_0\)處取得極值，則\(f^\prime(x_0)=0\)C.函數(shù)\(f(x)\)在某區(qū)間內(nèi)的極大值不一定大于極小值D.函數(shù)\(f(x)\)的極值點一定是其駐點或不可導(dǎo)點答案：CD4.設(shè)函數(shù)\(z=f(x,y)\)具有二階連續(xù)偏導(dǎo)數(shù)，且\(\frac{\partial^{2}z}{\partialx\partialy}=0\)，\(\frac{\partial^{2}z}{\partialx^{2}}=a\)，\(\frac{\partial^{2}z}{\partialy^{2}}=b\)，則（）A.\(dz=\frac{\partialz}{\partialx}dx+\frac{\partialz}{\partialy}dy\)B.\(d^{2}z=a(dx)^{2}+2\frac{\partial^{2}z}{\partialx\partialy}dxdy+b(dy)^{2}\)C.當(dāng)\(a>0\)，\(b>0\)時，\(f(x,y)\)有極小值D.當(dāng)\(ab<0\)時，\(f(x,y)\)有鞍點答案：ABD5.設(shè)\(A\)為\(n\)階矩陣，下列命題中正確的是（）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\(r(A)=n-1\)，則\(r(A^)=1\)D.若\(r(A)<n-1\)，則\(A^=O\)答案：ACD6.設(shè)\(\alpha_1,\alpha_2,\alpha_3\)是三維向量，則對任意常數(shù)\(k,m\)，向量組\(\alpha_1+k\alpha_3,\alpha_2+m\alpha_3\)線性無關(guān)是向量組\(\alpha_1,\alpha_2,\alpha_3\)線性無關(guān)的（）A.必要條件B.充分條件C.充分必要條件D.既不充分也不必要條件答案：A7.已知函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上可積，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\([a,b]\)上一定有界C.\(\int_{a}^f(x)dx\)表示曲線\(y=f(x)\)與直線\(x=a\)，\(x=b\)，\(y=0\)所圍成圖形的面積D.若\(f(x)\)在\([a,b]\)上可積，則\(\int_{a}^f(x)dx=\lim\limits_{\lambda\to0}\sum_{i=1}^{n}f(\xi_i)\Deltax_i\)，其中\(zhòng)(\lambda=\max\{\Deltax_1,\Deltax_2,\cdots,\Deltax_n\}\)答案：BD8.設(shè)函數(shù)\(y=y(x)\)由參數(shù)方程\(\begin{cases}x=t^2+2t\\y=\ln(1+t)\end{cases}\)確定，則\(y=y(x)\)在\(x=3\)處的切線方程為（）A.當(dāng)\(x=3\)時，\(t=1\)B.\(\frac{dy}{dx}=\frac{1}{2(t+1)^2}\)C.切線斜率\(k=\frac{1}{8}\)D.切線方程為\(y-\ln2=\frac{1}{8}(x-3)\)答案：ACD9.設(shè)\(A\)為\(n\)階實對稱矩陣，下列結(jié)論正確的是（）A.\(A\)的特征值都是實數(shù)B.\(A\)必可相似對角化C.\(A\)有\(zhòng)(n\)個線性無關(guān)的特征向量D.存在正交矩陣\(Q\)，使得\(Q^{-1}AQ\)為對角矩陣答案：ABCD10.設(shè)函數(shù)\(f(x)\)在\(x=x_0\)處的某鄰域內(nèi)具有直到\(n\)階的連續(xù)導(dǎo)數(shù)，且\(f^\prime(x_0)=f^{\prime\prime}(x_0)=\cdots=f^{(n-1)}(x_0)=0\)，\(f^{(n)}(x_0)\neq0\)，則（）A.當(dāng)\(n\)為偶數(shù)且\(f^{(n)}(x_0)>0\)時，\(f(x)\)在\(x_0\)處取得極小值B.當(dāng)\(n\)為偶數(shù)且\(f^{(n)}(x_0)<0\)時，\(f(x)\)在\(x_0\)處取得極大值C.當(dāng)\(n\)為奇數(shù)時，\(f(x)\)在\(x_0\)處取得極值D.當(dāng)\(n\)為奇數(shù)時，\(f(x)\)在\(x_0\)處的切線與\(x\)軸平行答案：ABD三、判斷題1.若函數(shù)\(f(x)\)在\(x_0\)處極限存在，則\(f(x)\)在\(x_0\)處一定連續(xù)。（×）2.函數(shù)\(y=x^3\)在\(R\)上是單調(diào)遞增函數(shù)。（√）3.若\(f(x)\)在區(qū)間\([a,b]\)上可積，則\(|f(x)|\)在區(qū)間\([a,b]\)上也可積。（√）4.二元函數(shù)\(z=f(x,y)\)在點\((x_0,y_0)\)處的偏導(dǎo)數(shù)存在，則\(z=f(x,y)\)在點\((x_0,y_0)\)處一定可微。（×）5.若\(A\)，\(B\)為\(n\)階矩陣，且\(AB=O\)，則\(r(A)+r(B)\leqn\)。（√）6.向量組\(\alpha_1,\alpha_2,\cdots,\alpha_m\)線性相關(guān)的充要條件是其中至少有一個向量可由其余向量線性表示。（√）7.若\(A\)為\(n\)階可逆矩陣，則\(A\)的特征值都不為0。（√）8.函數(shù)\(y=\sinx\)的一個原函數(shù)是\(-\cosx+C\)（\(C\)為任意常數(shù)）。（√）9.若\(f(x)\)在\(x_0\)處二階可導(dǎo)，且\(f^\prime(x_0)=0\)，\(f^{\prime\prime}(x_0)>0\)，則\(x_0\)是\(f(x)\)的極小值點。（√）10.對于\(n\)階矩陣\(A\)，若存在可逆矩陣\(P\)，使得\(P^{-1}AP\)為對角矩陣，則\(A\)一定有\(zhòng)(n\)個不同的特征值。（×）四、簡答題1.求函數(shù)\(y=x^3-3x^2-9x+5\)的單調(diào)區(qū)間和極值。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)\(x<-1\)時，\(y^\prime>0\)，函數(shù)單調(diào)遞增；當(dāng)\(-1<x<3\)時，\(y^\prime<0\)，函數(shù)單調(diào)遞減；當(dāng)\(x>3\)時，\(y^\prime>0\)，函數(shù)單調(diào)遞增。所以極大值為\(y(-1)=10\)，極小值為\(y(3)=-22\)。單調(diào)遞增區(qū)間為\((-\infty,-1)\)和\((3,+\infty)\)，單調(diào)遞減區(qū)間為\((-1,3)\)。2.計算定積分\(\int_{0}