考研數(shù)學（一）試題及答案一、單項選擇題（每題1分，共20分）1.函數(shù)\(y=\frac{\ln(1-x)}{\sqrt{x+1}}\)的定義域是（）A.\((-1,1)\)B.\([-1,1)\)C.\((-1,0)\cup(0,1)\)D.\([-1,0)\cup(0,1)\)答案：A2.當\(x\to0\)時，\(f(x)=\frac{\sin2x}{x}\)是\(g(x)=\frac{1-\cosx}{x^2}\)的（）A.高階無窮小B.低階無窮小C.同階但非等價無窮小D.等價無窮小答案：C3.設函數(shù)\(f(x)\)在點\(x0\)處可導，則\(\lim\limits{h\to0}\frac{f(x0+h)-f(x0-h)}{h}\)等于（）A.\(f^\prime(x0)\)B.\(2f^\prime(x0)\)C.\(0\)D.\(f^\prime(2x0)\)答案：B4.曲線\(y=x^3-3x^2+1\)在點\((1,-1)\)處的切線方程為（）A.\(y=-3x+2\)B.\(y=3x-4\)C.\(y=-4x+3\)D.\(y=4x-5\)答案：A5.函數(shù)\(f(x)=x^3-3x\)在區(qū)間\([-2,2]\)上的最大值是（）A.\(2\)B.\(-2\)C.\(1\)D.\(-1\)答案：A6.設\(f(x)\)為連續(xù)函數(shù)，則\(\int{-a}^{a}f(x)dx\)等于（）A.\(0\)B.\(2\int{0}^{a}f(x)dx\)C.\(\int{0}^{a}[f(x)+f(-x)]dx\)D.\(\int{0}^{a}[f(x)-f(-x)]dx\)答案：C7.已知向量\(\vec{a}=(1,2,-1)\)，\(\vec=(2,-1,1)\)，則\(\vec{a}\cdot\vec\)等于（）A.\(-1\)B.\(0\)C.\(1\)D.\(2\)答案：B8.設\(A\)為\(n\)階方陣，且\(\vertA\vert=2\)，則\(\vert-2A\vert\)等于（）A.\((-2)^{n+1}\)B.\((-2)^n\)C.\(-2^{n+1}\)D.\(-2^n\)答案：A9.若線性方程組\(\begin{cases}x1+x2+x3=0\\x1+2x2+ax3=0\\x1+4x2+a^2x3=0\end{cases}\)有非零解，則\(a\)的值為（）A.\(1\)或\(2\)B.\(1\)或\(-2\)C.\(-1\)或\(2\)D.\(-1\)或\(-2\)答案：A10.設隨機變量\(X\)服從正態(tài)分布\(N(1,4)\)，則\(P\{-1<X\leq3\}\)等于（）（已知\(\varPhi(1)=0.8413\)）A.\(0.6826\)B.\(0.8413\)C.\(0.9544\)D.\(0.9974\)答案：A11.設函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)可導，\(x1,x2\in(a,b)\)，且\(x1<x2\)，則至少存在一點\(\xi\in(x1,x2)\)，使得（）A.\(f(x2)-f(x1)=f^\prime(\xi)(x2-x1)\)B.\(f(x2)-f(x1)=f(\xi)(x2-x1)\)C.\(f(x2)-f(x1)=f^{\prime\prime}(\xi)(x2-x1)\)D.\(f(x2)-f(x1)=f^{\prime\prime\prime}(\xi)(x2-x1)\)答案：A12.設\(y=e^{2x}\cos3x\)，則\(y^\prime\)等于（）A.\(2e^{2x}\cos3x-3e^{2x}\sin3x\)B.\(e^{2x}(2\cos3x+3\sin3x)\)C.\(2e^{2x}\cos3x+3e^{2x}\sin3x\)D.\(e^{2x}(2\cos3x-3\sin3x)\)答案：A13.曲線\(y=\lnx\)在點\((1,0)\)處的曲率為（）A.\(\frac{1}{2}\)B.\(\frac{1}{4}\)C.\(1\)D.\(2\)答案：A14.設\(D\)是由\(x=0\)，\(y=0\)，\(x+y=1\)所圍成的區(qū)域，則\(\iintDxdxdy\)等于（）A.\(\frac{1}{24}\)B.\(\frac{1}{12}\)C.\(\frac{1}{6}\)D.\(\frac{1}{3}\)答案：B15.已知冪級數(shù)\(\sum{n=0}^{\infty}anx^n\)的收斂半徑為\(R\)，則冪級數(shù)\(\sum{n=0}^{\infty}nanx^{n-1}\)的收斂半徑為（）A.\(R\)B.\(R^2\)C.\(\frac{R}{2}\)D.\(2R\)答案：A16.設\(f(x)\)是周期為\(2\pi\)的周期函數(shù)，它在\([-\pi,\pi]\)上的表達式為\(f(x)=\begin{cases}0,&-\pi\leqx<0\\x,&0\leqx<\pi\end{cases}\)，則\(f(x)\)的傅里葉級數(shù)在\(x=\pi\)處收斂于（）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(0\)D.\(\frac{3\pi}{2}\)答案：A17.設\(A\)是\(n\)階可逆矩陣，\(\lambda\)是\(A\)的一個特征值，則\(A^{-1}\)的一個特征值是（）A.\(\lambda\)B.\(\frac{1}{\lambda}\)C.\(\lambda^2\)D.\(\frac{1}{\lambda^2}\)答案：B18.設\(A\)為\(n\)階正交矩陣，則下列結論不正確的是（）A.\(\vertA\vert=\pm1\)B.\(A^TA=E\)C.\(A\)的列向量組是單位正交向量組D.\(A\)的行向量組線性相關答案：D19.設隨機變量\(X\)和\(Y\)相互獨立，且\(X\simN(0,1)\)，\(Y\simN(1,1)\)，則（）A.\(P\{X+Y\leq0\}=\frac{1}{2}\)B.\(P\{X+Y\leq1\}=\frac{1}{2}\)C.\(P\{X-Y\leq0\}=\frac{1}{2}\)D.\(P\{X-Y\leq1\}=\frac{1}{2}\)答案：B20.設總體\(X\)服從正態(tài)分布\(N(\mu,\sigma^2)\)，\(X1,X2,\cdots,Xn\)是來自總體\(X\)的樣本，\(\bar{X}\)是樣本均值，則\(\frac{n(\bar{X}-\mu)^2}{\sigma^2}\)服從（）A.\(N(0,1)\)B.\(N(\mu,\frac{\sigma^2}{n})\)C.\(t(n-1)\)D.\(\chi^2(1)\)答案：D二、多項選擇題（每題2分，共20分）1.下列函數(shù)中，在其定義域內(nèi)連續(xù)的有（）A.\(y=\frac{1}{x-1}\)B.\(y=\ln(1+x^2)\)C.\(y=\sqrt{x+1}\)D.\(y=\frac{\sinx}{x}\)答案：BCD2.設函數(shù)\(f(x)\)在點\(x0\)處可導，則\(f(x)\)在點\(x0\)處（）A.連續(xù)B.可微C.有極限D.有定義答案：ABCD3.下列積分中，值為零的有（）A.\(\int{-1}^{1}x\cosxdx\)B.\(\int{-1}^{1}x\sinxdx\)C.\(\int{-1}^{1}x^2\cosxdx\)D.\(\int{-1}^{1}x^2\sinxdx\)答案：BD4.設向量\(\vec{a}=(1,0,-1)\)，\(\vec=(0,1,1)\)，則（）A.\(\vec{a}\cdot\vec=0\)B.\(\vec{a}\times\vec=(1,-1,1)\)C.\(\vec{a}\)與\(\vec\)垂直D.\(\vec{a}\)與\(\vec\)平行答案：AC5.設\(A\)為\(n\)階方陣，且\(\vertA\vert=0\)，則（）A.\(A\)的列向量組線性相關B.\(A\)的行向量組線性相關C.\(A\)不可逆D.\(A\)的秩小于\(n\)答案：ABCD6.設隨機變量\(X\)服從參數(shù)為\(\lambda\)的泊松分布，則\(P\{X=k\}\)等于（）A.\(\frac{e^{-\lambda}\lambda^k}{k!}\)B.\(Cn^kp^k(1-p)^{n-k}\)（這里\(n\)與本題無關，只是列出二項分布公式對比）C.\(\frac{\lambda^k}{k!}\sum{n=k}^{\infty}\frac{e^{-\lambda}\lambda^n}{n!}\)D.\(1-\sum{k=0}^{\infty}\frac{e^{-\lambda}\lambda^k}{k!}\)答案：A7.設函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上連續(xù)，則下列說法正確的有（）A.\(\int{a}^f(x)dx\)是一個常數(shù)B.\(\int{a}^{x}f(t)dt\)是\(x\)的函數(shù)C.\(\int{a}^f(x)dx\)與積分變量的記法無關D.\(\int{a}^f(x)dx=\int{a}^f(u)du\)答案：ABCD8.已知冪級數(shù)\(\sum{n=0}^{\infty}an(x-x0)^n\)在\(x=2\)處收斂，則該冪級數(shù)在\(x=0\)處（）A.絕對收斂B.條件收斂C.發(fā)散D.斂散性不確定答案：A9.設\(A\)為\(n\)階方陣，且\(A^2=A\)，則\(A\)的特征值可能是（）A.\(0\)B.\(1\)C.\(2\)D.\(-1\)答案：AB10.設總體\(X\)服從正態(tài)分布\(N(\mu,\sigma^2)\)，\(X1,X2,\cdots,Xn\)是來自總體\(X\)的樣本，\(S^2\)是樣本方差，則（）A.\(\frac{(n-1)S^2}{\sigma^2}\)服從\(\chi^2(n-1)\)分布B.\(\frac{\bar{X}-\mu}{S/\sqrt{n}}\)服從\(t(n-1)\)分布C.\(\sum{i=1}^{n}\frac{(Xi-\bar{X})^2}{\sigma^2}\)服從\(\chi^2(n-1)\)分布D.\(\frac{\sum{i=1}^{n}(Xi-\mu)^2}{\sigma^2}\)服從\(\chi^2(n)\)分布答案：AD三、判斷題（每題1分，共10分）1.若函數(shù)\(f(x)\)在點\(x0\)處可導，則\(f(x)\)在點\(x0\)處一定連續(xù)。（）答案：√2.函數(shù)\(y=\frac{1}{x^2+1}\)在\((-\infty,+\infty)\)上單調(diào)遞增。（）答案：×3.若\(\int{a}^f(x)dx=0\)，則\(f(x)\)在\([a,b]\)上恒為零。（）答案：×4.向量組\(\vec{a}1=(1,2,3)\)，\(\vec{a}2=(4,5,6)\)，\(\vec{a}3=(7,8,9)\)線性無關。（）答案：×5.若\