高考至今最難的數(shù)學(xué)試卷

高考至今最難的數(shù)學(xué)試卷一、選擇題（每題1分，共10分）

1.下列哪個(gè)函數(shù)不是初等函數(shù)？

A.$y=e^x$

B.$y=\lnx$

C.$y=\sqrt{x}$

D.$y=\sinx$

2.設(shè)函數(shù)$f(x)=\frac{1}{x}+\frac{1}{x-1}$，則$f'(x)$等于：

A.$-\frac{1}{x^2}$

B.$\frac{1}{x^2}$

C.$-\frac{1}{(x-1)^2}$

D.$\frac{1}{(x-1)^2}$

3.已知數(shù)列$\{a_n\}$滿足$a_1=1$，$a_{n+1}=a_n+\frac{1}{n}$，則$\lim_{n\rightarrow\infty}a_n$等于：

A.$\frac{1}{2}$

B.$\frac{3}{2}$

C.$1$

D.無極限

4.設(shè)$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$，$B=\begin{bmatrix}5&6\\7&8\end{bmatrix}$，則行列式$|AB|$等于：

A.$-72$

B.$72$

C.$-36$

D.$36$

5.已知等差數(shù)列$\{a_n\}$的前三項(xiàng)分別是$a_1=2$，$a_2=4$，$a_3=6$，則第$10$項(xiàng)$a_{10}$等于：

A.$18$

B.$20$

C.$22$

D.$24$

6.若$\triangleABC$的內(nèi)角$A$，$B$，$C$的正弦值分別為$\sinA=\frac{1}{2}$，$\sinB=\frac{3}{5}$，$\sinC=\frac{4}{5}$，則$\cosA$等于：

A.$\frac{3}{5}$

B.$\frac{4}{5}$

C.$-\frac{3}{5}$

D.$-\frac{4}{5}$

7.設(shè)$f(x)=\lnx+\sqrt{x}$，則$f'(1)$等于：

A.$1$

B.$\frac{1}{2}$

C.$\frac{1}{3}$

D.$\frac{1}{4}$

8.若$\log_2x=3$，則$x$等于：

A.$2$

B.$4$

C.$8$

D.$16$

9.設(shè)$A$是$n$階方陣，且$A^2=0$，則$A$必然是：

A.非零矩陣

B.零矩陣

C.單位矩陣

D.對(duì)角矩陣

10.設(shè)$f(x)=e^x+x^2$，則$f(x)$在$x=0$處的導(dǎo)數(shù)等于：

A.$1$

B.$2$

C.$3$

D.$e$

二、多項(xiàng)選擇題（每題4分，共20分）

1.下列哪些是實(shí)數(shù)域上的連續(xù)函數(shù)？

A.$f(x)=x^2$

B.$f(x)=\frac{1}{x}$

C.$f(x)=\sqrt{x}$

D.$f(x)=\lnx$

2.若$A$是$n$階可逆矩陣，則以下哪些性質(zhì)成立？

A.$A^T$是可逆矩陣

B.$A^{-1}$是可逆矩陣

C.$AA^{-1}=E$

D.$A^{-1}A=E$

3.下列哪些是數(shù)列$\{a_n\}$的收斂性質(zhì)？

A.$a_n$單調(diào)遞增且$\lim_{n\rightarrow\infty}a_n=L$

B.$a_n$單調(diào)遞減且$\lim_{n\rightarrow\infty}a_n=L$

C.$a_n$發(fā)散

D.$a_n$收斂但不一定單調(diào)

4.下列哪些是三角函數(shù)的性質(zhì)？

A.$\sin^2x+\cos^2x=1$

B.$\tanx=\frac{\sinx}{\cosx}$

C.$\sin(\pi-x)=\sinx$

D.$\cos(\pi-x)=-\cosx$

5.下列哪些是線性方程組的解的性質(zhì)？

A.如果方程組有唯一解，則系數(shù)矩陣和增廣矩陣的秩相等

B.如果方程組有無窮多解，則系數(shù)矩陣的秩小于增廣矩陣的秩

C.如果方程組無解，則系數(shù)矩陣的秩小于增廣矩陣的秩

D.如果方程組有唯一解，則系數(shù)矩陣的秩等于增廣矩陣的秩

三、填空題（每題4分，共20分）

1.設(shè)函數(shù)$f(x)=x^3-3x+2$，則$f(x)$的極小值點(diǎn)為$x=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\

四、計(jì)算題（每題10分，共50分）

1.計(jì)算定積分$\int_0^1(x^2-2x+1)\,dx$。

2.設(shè)函數(shù)$f(x)=\frac{x}{x^2+1}$，求$f'(x)$。

3.解線性方程組$\begin{cases}2x+3y-z=5\\x-y+2z=-1\\3x+2y-z=4\end{cases}$。

4.已知數(shù)列$\{a_n\}$滿足$a_1=1$，$a_{n+1}=2a_n+3$，求$\lim_{n\rightarrow\infty}\frac{a_n}{3^n}$。

5.設(shè)$A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$，求$A$的特征值和特征向量。

6.計(jì)算三重積分$\iiint_D(x^2+y^2)\,dV$，其中$D$是由$x^2+y^2\leq1$和$z\geq0$所圍成的區(qū)域。

7.設(shè)$f(x)=e^x\sinx$，求$f''(x)$。

8.解微分方程$\frac{dy}{dx}=3xy^2$，初始條件為$y(0)=1$。

9.設(shè)$A$是$3\times3$的實(shí)對(duì)稱矩陣，且$A^2=-A$，求$A$的特征值。

10.計(jì)算級(jí)數(shù)$\sum_{n=1}^{\infty}\frac{n^2}{n!}$的和。

本專業(yè)課理論基礎(chǔ)試卷答案及知識(shí)點(diǎn)總結(jié)如下：

一、選擇題答案及知識(shí)點(diǎn)詳解：

1.B.$y=\lnx$，因?yàn)閷?duì)數(shù)函數(shù)的定義域是正實(shí)數(shù)。

2.D.$\frac{1}{x^2}$，因?yàn)閷?dǎo)數(shù)的定義是$\frac{df}{dx}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$。

3.C.$1$，因?yàn)?a_n=1+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}$是調(diào)和級(jí)數(shù)，其極限為$1$。

4.B.$72$，因?yàn)樾辛惺降挠?jì)算是按照第一行展開，得到$1\cdot8-2\cdot7=8-14=-6$，第二行展開得到$3\cdot8-4\cdot7=24-28=-4$，所以行列式為$-6+(-4)=-10$。

5.A.$18$，因?yàn)榈炔顢?shù)列的通項(xiàng)公式是$a_n=a_1+(n-1)d$，其中$d$是公差，所以$a_{10}=2+9\cdot2=20$。

6.B.$\frac{4}{5}$，因?yàn)?\cosA=\sqrt{1-\sin^2A}=\sqrt{1-\left(\frac{1}{2}\right)^2}=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$。

7.C.$\frac{1}{3}$，因?yàn)?f'(x)=\fracrad0w9g{dx}(\lnx+\sqrt{x})=\frac{1}{x}+\frac{1}{2\sqrt{x}}$，在$x=1$處，$f'(1)=\frac{1}{1}+\frac{1}{2\sqrt{1}}=1+\frac{1}{2}=\frac{3}{2}$。

8.C.$8$，因?yàn)?\log_2x=3$意味著$2^3=x$，所以$x=8$。

9.B.零矩陣，因?yàn)槿绻?A^2=0$，則$A$的每一個(gè)元素平方后都為$0$，所以$A$必須是零矩陣。

10.D.$e$，因?yàn)?f'(x)=e^x+2x$，在$x=0$處，$f'(0)=e^0+2\cdot0=1$。

二、多項(xiàng)選擇題答案及知識(shí)點(diǎn)詳解：

1.A,C,D。$f(x)=x^2$和$f(x)=\sqrt{x}$在實(shí)數(shù)域上連續(xù)，$f(x)=\frac{1}{x}$和$f(x)=\lnx$在定義域內(nèi)連續(xù)。

2.A,B,C,D。這些都是可逆矩陣的性質(zhì)。

3.A,B,D。收斂性質(zhì)包括單調(diào)有界和極限存在。

4.A,B,C,D。這些都是三角函數(shù)的基本性質(zhì)。

5.A,B,C。這些是線性方程組的解的性質(zhì)。

三、填空題答案及知識(shí)點(diǎn)詳解：

1.$x=-1$，因?yàn)?f'(x)=3x^2-3$，令$f'(x)=0$得到$x=-1$。

2.$x=0$，因?yàn)?f''(x)=6x$，在$x=0$處，$f''(x)=0$。

3.$x=-1$，因?yàn)?A$的行列式為$0$，所以$A$是奇異的。

4.$x