考研數(shù)學(xué)一真題及答案

一、單項(xiàng)選擇題1.當(dāng)\(x\to0\)時(shí)，若\(x-\sinx\)與\(ax^n\)是等價(jià)無(wú)窮小，則（）A.\(a=\frac{1}{6}\)，\(n=3\)B.\(a=\frac{1}{6}\)，\(n=4\)C.\(a=\frac{1}{2}\)，\(n=3\)D.\(a=\frac{1}{2}\)，\(n=4\)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(diǎn)C.有跳躍間斷點(diǎn)D.有第二類間斷點(diǎn)答案：B3.設(shè)函數(shù)\(f(x)\)可導(dǎo)，且\(f(x)f'(x)>0\)，則（）A.\(f(1)>f(-1)\)B.\(f(1)<f(-1)\)C.\(|f(1)|>|f(-1)|\)D.\(|f(1)|<|f(-1)|\)答案：C4.設(shè)\(A\)為\(3\)階矩陣，將\(A\)的第\(2\)列加到第\(1\)列得矩陣\(B\)，再交換\(B\)的第\(2\)行與第\(3\)行得單位矩陣。記\(P_1=\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}\)，\(P_2=\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}\)，則\(A\)為（）A.\(P_1P_2\)B.\(P_1^{-1}P_2\)C.\(P_2P_1\)D.\(P_2P_1^{-1}\)答案：D5.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(gè)線性無(wú)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：A6.設(shè)二次型\(f(x_1,x_2,x_3)=a(x_1^2+x_2^2+x_3^2)+2x_1x_2+2x_2x_3+2x_3x_1\)的正、負(fù)慣性指數(shù)分別為\(1,2\)，則（）A.\(a>1\)B.\(a<-2\)C.\(-2<a<1\)D.\(a=1\)或\(a=-2\)答案：C7.設(shè)隨機(jī)變量\(X\)與\(Y\)相互獨(dú)立，且\(X\)服從標(biāo)準(zhǔn)正態(tài)分布\(N(0,1)\)，\(Y\)的概率分布為\(P(Y=0)=P(Y=1)=\frac{1}{2}\)，記\(F_Z(z)\)為隨機(jī)變量\(Z=XY\)的分布函數(shù)，則函數(shù)\(F_Z(z)\)的間斷點(diǎn)個(gè)數(shù)為（）A.0B.1C.2D.3答案：B8.設(shè)總體\(X\)的概率密度為\(f(x;\theta)=\begin{cases}\frac{1}{2\theta},&0<x<\theta\\\frac{1}{2(1-\theta)},&\theta\leqx<1\\0,&\text{其他}\end{cases}\)，其中參數(shù)\(\theta(0<\theta<1)\)未知，\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的簡(jiǎn)單隨機(jī)樣本，\(\overline{X}\)是樣本均值，則（）A.\(\frac{4}{n}\sum\limits_{i=1}^{n}X_i\)是\(\theta\)的無(wú)偏估計(jì)量B.\(2\overline{X}\)是\(\theta\)的無(wú)偏估計(jì)量C.\(\min\limits_{1\leqi\leqn}X_i\)是\(\theta\)的無(wú)偏估計(jì)量D.\(\max\limits_{1\leqi\leqn}X_i\)是\(\theta\)的無(wú)偏估計(jì)量答案：A二、多項(xiàng)選擇題1.設(shè)函數(shù)\(f(x)\)在\(x=0\)處連續(xù)，且\(\lim\limits_{h\to0}\frac{f(h^2)}{h^2}=1\)，則（）A.\(f(0)=0\)B.\(f_+'(0)=1\)C.\(f_-'(0)=1\)D.\(f'(0)=1\)答案：AB2.設(shè)函數(shù)\(f(x)\)具有二階連續(xù)導(dǎo)數(shù)，且\(f(x)>0\)，\(f'(0)=0\)，則函數(shù)\(z=f(x)\lnf(y)\)在點(diǎn)\((0,0)\)處取得極小值的一個(gè)充分條件是（）A.\(f(0)>1\)，\(f''(0)>0\)B.\(f(0)>1\)，\(f''(0)<0\)C.\(f(0)<1\)，\(f''(0)>0\)D.\(f(0)<1\)，\(f''(0)<0\)答案：A3.設(shè)函數(shù)\(y=y(x)\)由方程\(e^y+6xy+x^2-1=0\)確定，則\(y''(0)=\)（）A.\(-2\)B.\(-1\)C.\(1\)D.\(2\)答案：A4.下列曲線中有漸近線的是（）A.\(y=x+\sinx\)B.\(y=x^2+\sinx\)C.\(y=x+\sin\frac{1}{x}\)D.\(y=x^2+\sin\frac{1}{x}\)答案：C5.設(shè)向量組\(\alpha_1,\alpha_2,\alpha_3\)線性無(wú)關(guān)，則下列向量組線性相關(guān)的是（）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.\(\alpha_1-2\alpha_2,\alpha_2-2\alpha_3,\alpha_3-2\alpha_1\)D.\(\alpha_1+2\alpha_2,\alpha_2+2\alpha_3,\alpha_3+2\alpha_1\)答案：A6.設(shè)\(A\)為\(3\)階矩陣，\(P\)為\(3\)階可逆矩陣，且\(P^{-1}AP=\begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}\)。若\(P=(\alpha_1,\alpha_2,\alpha_3)\)，\(Q=(\alpha_1+\alpha_2,\alpha_2,\alpha_3)\)，則\(Q^{-1}AQ\)為（）A.\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&1\end{pmatrix}\)B.\(\begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}\)C.\(\begin{pmatrix}2&0&0\\0&1&0\\0&0&1\end{pmatrix}\)D.\(\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}\)答案：B7.設(shè)\(X_1,X_2,\cdots,X_n\)為來自總體\(N(\mu,\sigma^2)\)的簡(jiǎn)單隨機(jī)樣本，樣本均值\(\overline{X}=9.5\)，參數(shù)\(\mu\)的置信度為\(0.95\)的雙側(cè)置信區(qū)間的置信上限為\(10.8\)，則\(\mu\)的置信度為\(0.95\)的雙側(cè)置信區(qū)間為（）A.\((8.2,10.8)\)B.\((9.5,10.8)\)C.\((8.8,10.2)\)D.\((9.0,10.0)\)答案：A8.設(shè)隨機(jī)變量\(X\)服從正態(tài)分布\(N(\mu_1,\sigma_1^2)\)，隨機(jī)變量\(Y\)服從正態(tài)分布\(N(\mu_2,\sigma_2^2)\)，且\(P(|X-\mu_1|<1)>P(|Y-\mu_2|<1)\)，則必有（）A.\(\sigma_1<\sigma_2\)B.\(\sigma_1>\sigma_2\)C.\(\mu_1<\mu_2\)D.\(\mu_1>\mu_2\)答案：A9.設(shè)隨機(jī)變量\(X\)與\(Y\)相互獨(dú)立，且\(X\)服從標(biāo)準(zhǔn)正態(tài)分布\(N(0,1)\)，\(Y\)的概率分布為\(P(Y=0)=P(Y=1)=\frac{1}{2}\)，記\(F_Z(z)\)為隨機(jī)變量\(Z=XY\)的分布函數(shù)，則（）A.\(F_Z(z)\)是連續(xù)函數(shù)B.\(F_Z(z)\)有一個(gè)間斷點(diǎn)C.\(P(Z=0)=\frac{1}{2}\)D.\(P(Z=1)=\frac{1}{2}\)答案：BC10.設(shè)總體\(X\)的概率密度為\(f(x;\theta)=\begin{cases}\frac{2x}{3\theta^2},&\theta<x<2\theta\\0,&\text{其他}\end{cases}\)，其中\(zhòng)(\theta\)是未知參數(shù)，\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的簡(jiǎn)單隨機(jī)樣本，若\(E(c\sum\limits_{i=1}^{n}X_i^2)=\theta^2\)，則\(c\)的值可以為（）A.\(\frac{2}{5n}\)B.\(\frac{1}{3n}\)C.\(\frac{2}{3n}\)D.\(\frac{1}{5n}\)答案：A三、判斷題1.若函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)可導(dǎo)，且\(f'(x)>0\)，則\(f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增。（）答案：對(duì)2.若級(jí)數(shù)\(\sum\limits_{n=1}^{\infty}a_n\)收斂，則級(jí)數(shù)\(\sum\limits_{n=1}^{\infty}a_n^2\)也收斂。（）答案：錯(cuò)3.已知\(A,B\)為\(n\)階矩陣，若\(AB=0\)，則\(r(A)+r(B)\leqn\)。（）答案：對(duì)4.設(shè)\(A\)為\(n\)階矩陣，若\(A\)與對(duì)角矩陣相似，則\(A\)有\(zhòng)(n\)個(gè)線性無(wú)關(guān)的特征向量。（）答案：對(duì)5.設(shè)隨機(jī)變量\(X\)服從參數(shù)為\(\lambda\)的泊松分布，則\(E(X^2)=\lambda^2+\lambda\)。（）答案：對(duì)6.若\(X\)與\(Y\)相互獨(dú)立，且\(X\simN(0,1)\)，\(Y\simN(0,1)\)，則\(X+Y\simN(0,2)\)。（）答案：對(duì)7.設(shè)總體\(X\)的均值\(\mu\)未知，\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的樣本，則\(\frac{1}{n}\sum\limits_{i=1}^{n}(X_i-\overline{X})^2\)是總體方差\(\sigma^2\)的無(wú)偏估計(jì)量。（）答案：對(duì)8.函數(shù)\(y=e^x\)與\(y=\lnx\)的圖像關(guān)于直線\(y=x\)對(duì)稱。（）答案：對(duì)9.若\(f(x)\)是周期為\(T\)的連續(xù)函數(shù)，則\(\int_{a}^{a+T}f(x)dx=\int_{0}^{T}f(x)dx\)對(duì)任意實(shí)數(shù)\(a\)都成立。（）答案：對(duì)10.設(shè)\(A\)為\(m\timesn\)矩陣，若\(Ax=0\)只有零解，則\(A^TA\)是正定矩陣。（）答案：對(duì)四、簡(jiǎn)答題1.求函數(shù)\(f(x)=(x-1)^2(x+1)^3\)的單調(diào)區(qū)間與極值。答案：先求\(f(x)\)的導(dǎo)數(shù)\(f'(x)=2(x-1)(x+1)^3+3(x-1)^2(x+1)^2=(x-1)(x+1)^2(5x+1)\)。令\(f'(x)=0\)，得\(x=1,x=-1,x=-\frac{1}{5}\)。當(dāng)\(x<-\frac{1}{5}\)時(shí)，\(f'(x)>0\)