2025年全国硕士研究生招生考试数学二

考试时间 180 分钟

一、选择题(本题共10小题,每小题5分,共50分.在每小题给出的四个选项中,只有一项符合题目要求,把所选项前的字母填在题后的括号内.)

1.
设函数z=z(x,y)z=z(x,y)z+lnzyxet2dt=0z+\ln z-\int_y^{x}e^{-t^{2}}dt=0确定,则zx+zy=()\frac\partial z{\partial x}+\frac{\partial z}{\partial y}=\left(\:\right)
A.
 
zz+1(ex2ey2)\frac z{z+1}(e^{-x^2}-e^{-y^2})
B.
 
zz+1(ex2+ey2)\frac z{z+ 1}( e^{- x^2}+ e^{- y^2})
C.
 
zz+1(ex2ey2)-\frac{z}{z+1}(e^{-x^{2}}-e^{-y^{2}})
D.
 
zz+1(ex2+ey2)-\frac{z}{z+1}(e^{-x^{2}}+e^{-y^{2}})

2.
已知函数f(x)=0xet2sintdtf(x)=\int_{0}^{x}e^{t^{2}}\sin tdtg(x)=0xet2dtsin2xg(x)=\int_{0}^{x}e^{t^{2}}dt\cdot\sin^{2}x,则()
A.
 
x=0是f(x)f(x)的极值点,也是g(x)g(x)的极值点
B.
 
x=0是f(x)f(x)的极值点,(0,0)是曲线y=g(x)的拐点
C.
 
x=0是f(x)f(x)的极值点,(0,0)是曲线y=f(x)的拐点
D.
 
(0,0)是曲线y=f(x)的拐点,也是曲线y=g(x)的拐点

3.
如果对微分方程y2ay+(a+2)y=0y^{\prime}-2ay^{\prime}+(a+2)y=0的任一解y(x)y(x),反常积分0+y(x)dx\int_0^+\infty y(x)dx均收敛,那么aa的取值范围是()
A.
 
(-2,-1]
B.
 
(-\infin,-1]
C.
 
(-2,0)
C.
 
(-\infin,0)

4.

设函数f(x),g(x)x=0设函数f(x),g(x)在x=0 的某去心邻域内有定义且恒不为零.若当x0x\to0时,f(x)f(x)g(x)g(x)的高阶无穷小,则当x0x\to0时,( )

A.
 
f(x)+g(x)=o(g(x))f( x) + g( x) = o( g( x) )
B.
 
f(x)g(x)=o(f2(x))f(x)g(x)=o(f^{2}(x))
C.
 
f(x)=o(eg(x)1)f(x)=o(e^{g(x)}-1)
D.
 
f(x)=o(g2(x))f(x)=o(g^{2}(x))

5.

设函数 f(x,y)连续,则22dx4x24f(x,y)dy=()\text{设函数 }f(x,y)\text{连续,则}\int_{-2}^{2}dx\int_{4-x^{2}}^{4}f(x,y)dy=\left(\quad\right)

A.
 

04[24yf(x,y)dx+4y2f(x,y)dx]dy\int_{0}^{4}\left[\int_{-2}^{-\sqrt{4-y}}f(x,y)dx+\int_{\sqrt{4-y}}^{2}f(x,y)dx\right]dy

B.
 

04[24yf(x,y)dx+4y2f(x,y)dx]dy\int_{0}^{4}\left[\int_{-2}^{\sqrt{4-y}}f(x,y)dx+\int_{\sqrt{4-y}}^{2}f(x,y)dx\right]dy

C.
 

04[24yf(x,y)dx+24yf(x,y)dx]dy\int_{0}^{4}\left[\int_{-2}^{-\sqrt{4-y}}f(x,y)dx+\int_{2}^{\sqrt{4-y}}f(x,y)dx\right]dy

D.
 

204dy4y2f(x,y)dx2\int_{0}^{4}dy\int_{\sqrt{4-y}}^{2}f(x,y)dx


6.
设单位质点P,QP,Q分别位于点(0,0)(0,0)(0,1)(0,1)处,PP从点(0,0)(0,0)出发沿xx轴正向移动,记GG为引力常量,则当质点PP移动到点(1,0)(1,0)时,克服质点QQ的引力所做的功为( )
A.
 
01G(x2+1)32dx\int_0^1 \frac{G}{(x^2+1)^{\frac{3}{2}}}dx
B.
 
01Gx(x2+1)32dx\int_0^1 \frac{Gx}{(x^2+1)^{\frac{3}{2}}}dx
C.
 
01G(x2+1)32dx\int_0^1 \frac{G}{(x^2+1)^{\frac{3}{2}}}dx
D.
 
01G(x+1)(x2+1)32dx\int_0^1 \frac{G(x+1)}{(x^2+1)^{\frac{3}{2}}}dx

7.

设函数f(x)连续,给出下列四个条件设函数f(x)连续,给出下列四个条件 limx0f(x)f(0)x①\lim_{x\to0}\frac{\left|f(x)\right|-f(0)}x存在;limx0f(x)f(0)x② \lim _x\to 0\frac {f( x) - \left | f( 0) \right | }x存在, limx0f(x)x③\lim_{x\to0}\frac{\left|f(x)\right|}x存在;limx0f(x)f(0)x④\lim_x\to0\frac{\left|f(x)\right|-\left|f(0)\right|}x存在,其中能得到“f(x)f(x)x=0x=0 处可导”的条件个数是()

A.
 
1
B.
 
2
C.
 
3
D.
 
4

8.
设矩阵(1202a000b)\begin{pmatrix}1&2&0\\2&a&0\\0&0&b\end{pmatrix}有一个正特征值和两个负特征值,则( )
A.
 
a>4,b>0a> 4, b> 0
B.
 
a<4,b>0a< 4, b> 0
C.
 
a>4,b<0a> 4, b< 0
D.
 
a<4,b<0a<4,b<0

9.

下列矩阵中,可以经过若干初等行变换得到矩阵(110100120000)\begin{pmatrix}1 & 1&0 & 1 \\0 & 0 & 1 & 2 \\0 & 0 & 0 & 0\end{pmatrix}的是()

A.
 

(110112132314)\begin{pmatrix}1&1&0&1\\1&2&1&3\\2&3&1&4\end{pmatrix}

B.
 

(110111251113)\begin{pmatrix}1&1&0&1\\1&1&2&5\\1&1&1&3\end{pmatrix}

C.
 

(100101030100)\begin{pmatrix}1&0&0&1\\0&1&0&3\\0&1&0&0\end{pmatrix}

D.
 

(112312232346)\begin{pmatrix}1&1&2&3\\1&2&2&3\\2&3&4&6\end{pmatrix}


10.
设3阶矩阵A,B满足r(AB)=r(BA)+1r(AB)=r(BA)+1,则()
A.
 
方程组(A+B)x=0(A+B)x=0只有零解
B.
 
方程组Ax=0Ax=0与方程组Bx=0Bx=0均只有零解
C.
 
方程组Ax=0Ax=0与方程组Bx=0Bx=0没有公共非零解
D.
 
方程组ABAx=0ABAx=0与方程组BABx=0BABx=0有公共非零解

二、填空题(本题共6小题,每小题5分,共30分,把答案填在题中横线上)

11.

1+ax(2x+a)dx=ln2\int_{1}^{+\infty} \frac{a}{x(2x+a)} dx = \ln 2,则 $a = $______.


12.

曲线 y=x33x2+13y = \sqrt[3]{x^3 - 3x^2 + 1} 的渐近线方程为 ______.


13.

limn1n2[ln1n+2ln2n++(n1)lnn1n]=\lim_{n \to \infty} \frac{1}{n^2} \left[ \ln \frac{1}{n} + 2 \ln \frac{2}{n} + \cdots + (n-1) \ln \frac{n-1}{n} \right] =______.


14.

已知函数 y=y(x)y = y(x){x=ln(1+2t)2t1y2teu2du=0\begin{cases} x = \ln (1 + 2t) \\ 2t - \int_{1}^{y^2t} e^{-u^2} du = 0 \end{cases} 确定,则 dydxt=0=\left. \frac{dy}{dx} \right|_{t=0}=______.


15.

微分方程 (2y3x)dx+(2x5y)dy=0(2y - 3x) dx + (2x - 5y) dy = 0 满足条件 y(1)=1y(1) = 1 的解为 ______.


16.

设矩阵 A=(α1,α2,α3,α4)A = (\alpha_1, \alpha_2, \alpha_3, \alpha_4),若 α1,α2,α3\alpha_1, \alpha_2, \alpha_3 线性无关,且 α1+α2=α3+α4\alpha_1 + \alpha_2 = \alpha_3 + \alpha_4,则方程组Ax=α1+4α4Ax=\alpha_{1}+4\alpha_{4}的通解为x=x=______.


三、解答题(本题共6小题,共70分,解答应写出文字说明、证明过程或演算步骤)

17.

计算011(x+1)(x22x+2)dx.\text{计算}\int_0^1\frac{1}{(x+1)(x^2-2x+2)}dx.


18.

设函数f(x)f(x)x=0x=0处连续,且limx0xf(x)e2sinx+1ln(1+x)+ln(1x)=3\lim_x\to0\frac{xf(x)-e^{2\sin x}+1}{\ln(1+x)+\ln(1-x)}=-3,证明f(x)f(x)x=0x=0处可导,并求f(0).f^\prime(0).


19.

设函数 f(x,y)f(x,y) 可微且满足 df(x,y)=2xeydx+ey(x2y1)dydf(x,y)=-2xe^{-y}dx+e^{-y}\left(x^{2}-y-1\right)dyf(0,0)=2f(0,0)=2,求 f(x,y)f(x,y),并求 f(x,y)f(x,y) 的极值.


20.

已知平面有界区域D={(x,y)x2+y24x,x2+y24y}D=\left\{(x,y)|x^{2}+y^{2}\leq4x,x^{2}+y^{2}\leq4y\right\},计算D(xy)2dxdy.\iint_{D}\left(x-y\right)^{2}dxdy.


21.

设函数f(x)f(x)在区间(a,b)(a,b)内可导,证明导函数f(x)f'(x)(a,b)(a,b)内严格单调增加的充分必要条件是:

(a,b)(a,b)内任意的x1,x2,x3x_{1},x_{2},x_{3},当x1x_{1}<x2x_{2}<x3x_{3}时,f(x3)f(x1)x2x1\frac{f(x_{3})-f(x_{1})}{x_{2}-x_{1}}<f(x3)f(x2)x3x2\frac{f(x_{3})-f(x_{2})}{x_{3}-x_{2}}.


22.

已知矩阵A=(41211121a)A=\begin{pmatrix}4&1&-2\\1&1&1\\-2&1&a\end{pmatrix}B=(k00060000)B=\begin{pmatrix}k&0&0\\0&6&0\\0&0&0\end{pmatrix}合同.

(1).

a_{a}的值及k_k的取值范围;

(2).

若存在正交矩阵ϱ\varrho,使得ϱTAQ=B\varrho^{T}AQ=B,求k_kϱ\varrho