中科院数学系统院高校招生考试试题

1 浙大考题

中国科学院大学

2016 年高校招生考试：数学(甲卷)

 满分100分,考试时间120分钟

1. (15分)求 $$\int_0^{ + \infty } {\frac{{{e^{ - ax}} - {e^{ - bx}}}}{x}dx} \quad \left( {b > a} \right).$$

2. (15分) $\sum_{i=1}^n a_n$发散, $a_n$为正项级数.求证:

(1) $\sum_{i=1}^\infty \frac{a_n}{S_n}$发散;

(2) $\sum_{i=1}^\infty \frac{a_{n+1}}{S_n}$发散.

3. (15分) 求

$$\int\limits_{{x^2} + {y^2} + {z^2} = {R^2}} {\frac{{dS}}{{\sqrt {{x^2} + {y^2} + {{\left( {z - h} \right)}^2}} }}} .$$

4. (15分) 设$A:V\to V$, $${H_{A,\alpha }}\left( t \right) = \left\{ {\varphi \left( t \right)\left| {\varphi \left( x \right) \in Q\left[ t \right],\varphi \left( x \right) \cdot \alpha = 0} \right.} \right\}$$ 中次数最小的一个.证: $\exists \alpha \in V$,使${H_{A,\alpha }}\left( t \right)$为$A$的极小多项式.

1.1 某同学面试问题

1. 求 $$\int_{ - \infty }^{ + \infty } {\frac{1}{{\left( {1 + {x^2}} \right)\left( {1 + {x^6}} \right)}}dx} .$$

2. 举一个无穷次可导却不解析的函数.

2 湖南大学考题

中国科学院大学

2016 年高校招生考试：数学(乙卷)

满分100分,考试时间120分钟

1. (15分)

(1) 求极限$\mathop {\lim }_{x \to - \infty } \left( {\sqrt {\left( {x + a} \right)\left( {x + b} \right)} + x} \right)$;

(2) 设$f(x)$满足$f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = \frac{3}{x}$,求$f(x)$的导数;

(3) 求$\mathop {\lim }_{n \to \infty } \sqrt[n]{{\frac{{{n^n}}}{{n!}}}}$.

2. (15分)设$r\geq 0$,求积分 $$\frac{1}{{2\pi }}\int_0^{2\pi } {\log \left( {1 - 2r\cos x + {r^2}} \right)dx} .$$

3. (10分)设$0<\mu <1,a>0$, $M_n$是$e^{-(x+ax^\mu )x^n}$在$(0,+\infty)$上的最大值.求 $$\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{M_n}}}{{n!}}} \right)^{{n^{ - \mu }}}}.$$

4. (10分)设函数$f(x)$在闭区间$[a,b]$上二次连续可微,并且$f(a)=f(b)=0$.证明不等式:

$${M^2} \le \frac{{{{\left( {b - a} \right)}^3}}}{2}\int_a^b {{{\left| {f''\left( x \right)} \right|}^2}dx} ,$$ 其中$M=\sup_{a\leq x\leq b}|f(x)|$.

5. (10分)设$A = \left( {\begin{array}{*{20}{c}}4&0\\0&1\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right)\left( {\begin{array}{*{20}{c}}3&0\\0&{ - 2}\end{array}} \right){\left( {\begin{array}{*{20}{c}}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right)^{ - 1}}$,求以下矩阵的特征根:

$$A + B,A \otimes \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right) \otimes B,A \otimes B.$$

注:对$A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}\\{{a_{21}}}&{{a_{22}}}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{{b_{11}}}&{{b_{12}}}\\{{b_{21}}}&{{b_{22}}}\end{array}} \right)$,张量积定义为$A \otimes B = \left( {\begin{array}{*{20}{c}}{{a_{11}}B}&{{a_{12}}B}\\{{a_{21}}B}&{{a_{22}}B}\end{array}} \right)$.

6. (10分)证明以下矩阵组成的集合是实数域上的线性空间,求其维数及一组基,并证明行列式$\det X$是二次型,写出其对应的双线性型.

$$M = \left\{ {X = \left( {\begin{array}{*{20}{c}}{{x_0} + {x_3}}&{{x_1} - i{x_2}}\\{{x_1} + i{x_2}}&{{x_0} - {x_3}}\end{array}} \right):{x_0},{x_1},{x_2},{x_3} \in \mathbb{R} } \right\}.$$

7. (20分)设可逆矩阵$A\in M_n(\mathbb C)$的特征值为$\lambda_1,\cdots,\lambda_n$.求线性变换

$$M_n(\mathbb C)\to M_n(\mathbb C),\quad X\mapsto AXA'$$

的全部特征值.

注：$M_n(\mathbb C)$表示定义在复数域$\mathbb C$上的$n$阶方阵.

3 西安交大考题

中国科学院大学

2016 年高校招生考试：数学(乙卷)

满分100分,考试时间120分钟

1. (15分)

(1) 求极限$\mathop {\lim }_{x \to - \infty } \left( {\sqrt {\left( {x + a} \right)\left( {x + b} \right)} + x} \right)$;

(2) 设$f(x)$满足$f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = \frac{3}{x}$,求$f(x)$的导数;

(3) 设$f:[0,1]\to R$连续,求$\mathop {\lim }_{n \to \infty } \int_0^1 {\int_0^1 \cdots } \int_0^1 {f\left( {\frac{{{x_1} \cdots {x_n}}}{n}} \right)d{x_1}d{x_2} \cdots d{x_n}}$.

解法一.设$|f|$最大值为$M$.对任何$\varepsilon>0$,存在$\delta>0$,使得当$|x-1/2|<\delta$时,有 $$\left|f(x)-f(\frac{1}{2})\right|<\varepsilon.$$

$$\begin{align*}&\int_{[0,1]^n}\left| f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)-f(\frac{1}{2})\right|dx_1dx_2\cdots dx_n\\\leq &\int_{\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|\geq\delta}\left| f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)-f(\frac{1}{2})\right|dx_1dx_2\cdots dx_n\\+&\int_{\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|<\delta}\left| f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)-f(\frac{1}{2})\right|dx_1dx_2\cdots dx_n\\\leq&2M\int_{\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|\geq\delta}dx_1dx_2\cdots dx_n+\varepsilon\\\leq&\frac{2M}{\delta^2}\int_{\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|\geq\delta}\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|^2dx_1dx_2\cdots dx_n+\varepsilon\\\leq&\frac{2M}{\delta^2}\int_{[0,1]^n}\left|\frac{x_1+x_2+\cdots+x_n}{n}-\frac{1}{2}\right|^2dx_1dx_2\cdots dx_n+\varepsilon\\=&\frac{M}{6n\delta^2}+\varepsilon.\end{align*}$$

因此 $$\limsup_{n\rightarrow\infty}\int_{[0,1]^n}\left| f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)-f(\frac{1}{2})\right|dx_1dx_2\cdots dx_n\leq \varepsilon.$$

令$\varepsilon\rightarrow0$即可.

解法二.由科尔莫格罗夫强大数定律得 $$\frac{{{X_1} + {X_2} + \cdots + {X_n}}}{n}\mathop \to \limits^{a.s.} E\left( {{X_i}} \right) = \frac{1}{2}\left( {n \to + \infty } \right).$$

又因为$f(x)$连续有界,由控制收敛定理可知

$$\mathop {\lim }\limits_{n \to \infty } E\left( {f\left( {\frac{{{X_1} + {X_2} + \cdots + {X_n}}}{n}} \right)} \right) = E\left( {\mathop {\lim }\limits_{n \to \infty } f\left( {\frac{{{X_1} + {X_2} + \cdots + {X_n}}}{n}} \right)} \right) = E\left( {f\left( {\mathop {\lim }\limits_{n \to \infty } \frac{{{X_1} + {X_2} + \cdots + {X_n}}}{n}} \right)} \right) = f\left( {\frac{1}{2}} \right).$$

2. (15分)设$r\geq 0$,求积分 $$\frac{1}{{2\pi }}\int_0^{2\pi } {\log \left( {1 - 2r\cos x + {r^2}} \right)dx} .$$

3. (20分)设$0<\mu <1,a>0$, $M_n$是$e^{-(x+ax^\mu )x^n}$在$(0,+\infty)$上的最大值.求 $$\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{M_n}}}{{n!}}} \right)^{{n^{ - \mu }}}}.$$

4. (10分)设$m$为正整数,方程$a\equiv b \mod m$定义为$m$能整除$a-b$.当$m$取何值时,以下线性方程组有整数解?

$$\left\{ \begin{array}{l}x + 2y - z \equiv 1\left( {\bmod m} \right),\\2x - 3y + z \equiv 4\left( {\bmod m} \right),\\4x + y - z \equiv 9\left( {\bmod m} \right).\end{array} \right.$$

5. (10分)设$A = \left( {\begin{array}{*{20}{c}}4&0\\0&1\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right)\left( {\begin{array}{*{20}{c}}3&0\\0&{ - 2}\end{array}} \right){\left( {\begin{array}{*{20}{c}}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right)^{ - 1}}$,求以下矩阵的特征根: $$A + B,A \otimes \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right) + \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right) \otimes B,A \otimes B.$$

注:对$A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}\\{{a_{21}}}&{{a_{22}}}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{{b_{11}}}&{{b_{12}}}\\{{b_{21}}}&{{b_{22}}}\end{array}} \right)$,张量积定义为$A \otimes B = \left( {\begin{array}{*{20}{c}}{{a_{11}}B}&{{a_{12}}B}\\{{a_{21}}B}&{{a_{22}}B}\end{array}} \right)$.

6. (10分)证明以下矩阵组成的集合是实数域上的线性空间,求其维数及一组基,并证明行列式$\det X$是二次型,写出其对应的双线性型.

$$M = \left\{ {X = \left( {\begin{array}{*{20}{c}}{{x_0} + {x_3}}&{{x_1} - i{x_2}}\\{{x_1} + i{x_2}}&{{x_0} - {x_3}}\end{array}} \right):{x_0},{x_1},{x_2},{x_3} \in \mathbb{R} } \right\}.$$

7. (20分)设$\Omega$为含有$n$个元素的有限集合, $2^\Omega$为$\Omega$的幂集(即$\Omega$的所有子集构成的集合).对任意$A,B\in 2^\Omega$,定义数乘$0A=\emptyset$(空集), $1A=A$,加法$A+B=(A\cup B)\backslash (A\cap B)$(对称差).

(1) 证明$2^\Omega$关于以上数乘及加法为域$Z_2=\{0,1\}$ (注意在此域上$1+1=0$)上的线性空间,求其维数.

(2) 求$2^\Omega$的一维子空间个数.

(3) 取定非空$X\in 2^\Omega$,定义线性算子$T_X:2^\Omega\mapsto 2^\Omega$为$T_X A=A\cap X,A\in 2^\Omega$.求$T_X$的极小多项式,特征多项式,特征值和相应的特征子空间.

4 吉大考题

中国科学院大学

2016 年高校招生考试：数学(丙卷)

满分100分,考试时间120分钟

1. (15分)计算

(1) 求极限$\mathop {\lim }_{n \to \infty } \frac{{{1^{\alpha - 1}} + \cdots + {n^{\alpha - 1}}}}{{{n^\alpha }}} \quad {\alpha > 0}$.

(2) 已知$f'(a)$存在,$f(a)\neq0$,求$\mathop {\lim }_{n \to \infty } {\left( {\frac{{f\left( {a + \frac{1}{n}} \right)}}{{f\left( a \right)}}} \right)^n}$.

(3) 设$f:[0,1]\to \mathbb R$连续,求 $$\mathop {\lim }\limits_{n \to \infty } \int_0^1 {\int_0^1 \cdots } \int_0^1 {f\left( {{{\left( {{x_1} \cdots {x_n}} \right)}^{1/n}}} \right)d{x_1}d{x_2} \cdots d{x_n}} .$$

2. (15分)设$\phi (x)>0,f(x)>0$都是$[a,b]$上连续函数,求 $$\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\int_a^b {\phi \left( x \right){{\left( {f\left( x \right)} \right)}^n}dx} }}.$$

3. (20分)证明$\binom n1 - \frac{1}{2}\binom n2 + \frac{1}{3} \binom n3 - \cdots + (-1)^{n-1}\frac1n\binom nn = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$.

4. (10分)设$x,y$都是复数域上$n$阶方阵,定义$x^{(0)}=x,x^{(1)}=[x,y]\equiv xy-yx,x^{(j)}=[x^{(j-1)},y]$.证明

$$\sum\limits_{i = 0}^k {{y^i}x{y^{k - i}}} = \sum\limits_{j = 0}^k {\binom{k + 1}{j + 1} {y^{k - j}}{x^{\left( j \right)}}} .$$

5. (10分)给出平面中以下三条不同直线相交于一点的条件

$$ax+by+c=0,\quad bx+cy+a=0,\quad cx+ay+b=0.$$

求以下矩阵能对角化的条件:

$$\left( {\begin{array}{*{20}{c}}0&1&0\\0&0&1\\a&b&c\end{array}} \right).$$

6. (10分) 给出$M_2(\mathbb C)$中幂零矩阵所张成的线性空间的一组基.描述$M_n(\mathbb C)$中幂零矩阵所张成的线性空间.

7. (20分)证明$\cos x$是超越函数.

注：函数$f(x)$称为超越函数,如果不存在有限多个不全为零的$a_{pq},p,q=0,1,2,\cdots$,使得

$$\sum\limits_{p,q}a_{pq} x^p (f(x))^q=0,\quad \forall x\in \mathbb R.$$

5 大连理工考题

中国科学院大学

2016 年高校招生考试：数学(丁卷)

满分100分,考试时间120分钟

1. (15分)计算

(1) 求${\sqrt 2 ^{{{\sqrt 2 }^{{{\sqrt 2 }^ \cdots }}}}}$;

(2) 求$\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\frac{{{n^n}}}{{n!}}}}$;

(3) 求不定积分$\int {\frac{{x\ln x}}{{{{\left( {1 + {x^2}} \right)}^2}}}dx}$.

2. (15分)设$\phi (x)>0,f(x)>0$都是$[a,b]$上连续函数,求

$$\mathop {\lim }\limits_{n \to \infty } \frac{{\int_a^b {\phi \left( x \right){{\left( {f\left( x \right)} \right)}^{n + 1}}dx} }}{{\int_a^b {\phi \left( x \right){{\left( {f\left( x \right)} \right)}^n}dx} }}.$$

3. (20分) 设$f(x)$为$[a,b]$上可微函数, $f(a)=f(b)=0$,但$f(x)$不恒等于零,则存在$\xi\in (a,b)$使得

$$\left| {f'\left( \xi \right)} \right| > \frac{4}{{{{\left( {b - a} \right)}^2}}}\int_a^b {f\left( x \right)dx} .$$

4. (10分)设$m$为正整数,方程$a\equiv b \mod m$定义为$m$能整除$a-b$.当$m$取何值时,以下线性方程组有整数解?

$$\left\{ \begin{array}{l}x \equiv 1\left( {\bmod \,2} \right),\\x \equiv 2\left( {\bmod \,3} \right),\\x\equiv 4\left( {\bmod \,5} \right).\end{array} \right.$$

5. (10分)证明代数数集合为可数集.

注：一个数称为代数数,如果它是某个系数为有理数的多项式的根.

6. (10分)设$n\geq2$,矩阵$A=(a_{ij})\in M_{n\times n}(\mathbb Z)$的每个元素要么是$-3$,要么是$4$,即$a_{ij}\in \{-3,4\}$. (1)设$S$是所有这些矩阵的和,求$S$及其秩$\mathrm{rank}\, S$; (2)证明行列式$|A^2|$是$7^{2n-2}$的倍数,即$7^{2n-2} |\, |A^2|$.

7. (20分)设$A = \left( {\begin{array}{*{20}{c}}a&1&0\\0&a&1\\0&0&a\end{array}} \right)\in M_{3\times 3}(\mathbb C)$,多项式$p(x)\in \mathbb C[x]$.

(1)证明: $p\left( A \right) = \left( {\begin{array}{*{20}{c}}{p\left( a \right)}&{p'\left( a \right)}&{p''\left( a \right)/2}\\0&{p\left( a \right)}&{p'\left( a \right)}\\0&0&{p\left( a \right)}\end{array}} \right)$. \quad (2)求$e^A$.

6 中科大考题

证明$AB$和$BA$有相同的特征多项式.

7 山大考题

中国科学院大学

2016 年高校招生考试：数学(X卷)

满分100分,考试时间120分钟

1. (15分)计算

(1) 求$\mathop {\lim }_{x \to 0} \frac{{\sqrt {1 + \tan x} - \sqrt {1 + \sin x} }}{{{x^3}}}$;

(2) 求$f(x)=x^{x^x}$的导数;

(3) 求 $$\mathop {\lim }\limits_{n \to \infty } \int_0^1 {\int_0^1 \cdots } \int_0^1 {\frac{{x_1^2 + x_1^2 + \cdots + x_n^2}}{{{x_1} + {x_2} + \cdots + {x_n}}}d{x_1}d{x_2}d{x_n}} .$$

2. (15分)已知$f\left( x \right) = \prod_{i = 1}^k {\left( {x - {a_i}} \right)}$,且 $$- \frac{{f'\left( x \right)}}{{f\left( x \right)}} = {c_0} + {c_1}x + {c_2}{x^2} + \cdots + {c_n}{x^n} + \cdots ,$$ 求$\mathop {\lim }\limits_{n \to \infty } \frac{{{c_n}}}{{{c_{n - 1}}}}$和$\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{c_n}}}$.

3. (20分) $a,b$为实数, $x^3+abx+b$在复数域上有重根,则$a,b$应满足什么条件?

4. (10分)求 $${\left( {\begin{array}{*{20}{c}}{{e^{i\theta }}}&{2i\sin \alpha }\\0&{{e^{i\theta }}}\end{array}} \right)^n}.$$

8 厦大考题

1. $A,B$特征值不同, $f_A,f_B$为其特征多项式.

(1) 存在$g(\lambda),h(\lambda)$使得 $$g(B)f_A(B)=I,h(A)g_B(A)=I.$$

(2) $AX-XB=0$只有零解;

(3) $AX-XB=C$有唯一解.

2. 设$f(x)=\frac1{1-x-x^2}$,证明$\sum_{n=1}^\infty\frac{n!}{f^{(n)}(0)}$收敛,其中$f^{(n)}(0)$表示$f(x)$在$0$点的$n$阶导数.