2024 年全国硕士研究生招生考试试题与答案 (数学一)

数学

本试卷共 4 页, 22 题. 全卷满分 150 分, 考试用时 120 分钟.

注意事项:

1. 答卷前, 考生务必将自己的姓名、准考证号填写在答题卡上.
2. 回答选择题时, 用铅笔把答题卡上对应题目的答案标号涂黑, 写在本试卷上无效.
3. 考试结束后, 将本试卷和答题卡一并交回.
4. 本试卷由 kmath.cn 自动生成.

一、单选题: 本题共 10 小题, 每小题 5 分, 共 40 分. 在每小题给出的四个选项中, 只有一项是符合题目要求的.

1. 已知函数 $f(x)\; =\; \backslash int\_\{0\}^\{x\}\; e^\{\backslash cos\; t\}\; dt$ , $g(x)\; =\; \backslash int\_\{0\}^\{\backslash sin\; x\}\; e^\{t^2\}\; dt$ , 则
   1. $f(x)$ 是奇函数, $g(x)$ 是偶函数
   2. $f(x)$ 是偶函数, $g(x)$ 是奇函数
   3. $f(x)$ 与 $g(x)$ 均为奇函数
   4. $f(x)$ 与 $g(x)$ 均为周期函数

[答案]: C [解析]: 【解析】由于 $e^\{\backslash cos\; t\}$ 是偶函数, 所以 $f(x)\; =\; \backslash int\_\{0\}^\{x\}\; e^\{\backslash cos\; t\}\; dt$ 是奇函数, 又 $g\text{'}(x)\; =\; e^\{(\backslash sin\; x)^2\}\; \backslash cos\; x$ 是偶函数, 所以 $g(x)$ 是奇函数. 故选 C.

1. 设 $P\; =\; P(x,\; y,\; z)$ 和 $Q\; =\; Q(x,\; y,\; z)$ 均为连续函数, $\backslash Sigma$ 为曲面 $Z\; =\; \backslash sqrt\{1\; -\; x^2\; -\; y^2\}$ ( $x\; \backslash le\; 0,\; y\; \backslash ge\; 0$ ) 的上侧, 则 $\backslash iint\_\{\backslash Sigma\}\; P\; dydz\; +\; Q\; dzdx\; =$
   1. $\backslash iint\_\{\backslash Sigma\}\; \backslash left(\backslash frac\{x\}\{z\}P\; +\; \backslash frac\{y\}\{z\}Q\backslash right)\; dxdy$
   2. $\backslash iint\_\{\backslash Sigma\}\; \backslash left(-\backslash frac\{x\}\{z\}P\; +\; \backslash frac\{y\}\{z\}Q\backslash right)\; dxdy$
   3. $\backslash iint\_\{\backslash Sigma\}\; \backslash left(\backslash frac\{x\}\{z\}P\; -\; \backslash frac\{y\}\{z\}Q\backslash right)\; dxdy$
   4. $\backslash iint\_\{\backslash Sigma\}\; \backslash left(-\backslash frac\{x\}\{z\}P\; -\; \backslash frac\{y\}\{z\}Q\backslash right)\; dxdy$

[答案]: A [解析]: 【解析】转换投影法, $z\; =\; \backslash sqrt\{1\; -\; x^2\; -\; y^2\}$ , $\backslash frac\{\backslash partial\; z\}\{\backslash partial\; x\}\; =\; -\backslash frac\{x\}\{z\}$ , $\backslash frac\{\backslash partial\; z\}\{\backslash partial\; y\}\; =\; -\backslash frac\{y\}\{z\}$

$$\backslash iint\_\{\backslash Sigma\}\; P\; dydz\; +\; Q\; dzdx\; =\; \backslash iint\_\{\backslash Sigma\}\; \backslash left(\backslash frac\{x\}\{z\}P\; +\; \backslash frac\{y\}\{z\}Q\backslash right)\; dxdy$$

故选 A.

1. 已知幂级数 $\backslash sum\_\{n=0\}^\{\backslash infty\}\; a\_n\; x^n$ 的和函数为 $\backslash ln(2+x)$ , 则 $\backslash sum\_\{n=0\}^\{\backslash infty\}\; n\; a\_\{2n\}\; =$

1. $-\backslash frac\{1\}\{6\}$
2. $-\backslash frac\{1\}\{3\}$
3. $\backslash frac\{1\}\{6\}$
4. $\backslash frac\{1\}\{3\}$

[答案]: A [解析]: 【解析】方法 1: $\backslash ln(2+x)\; =\; \backslash ln\; 2\; \backslash left(1\; +\; \backslash frac\{1\}\{2\}x\backslash right)\; =\; \backslash ln\; 2\; +\; \backslash ln\; \backslash left(1\; +\; \backslash frac\{1\}\{2\}x\backslash right)$

$$=\; \backslash ln\; 2\; +\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; (-1)^\{n-1\}\; \backslash frac\{\backslash left(\backslash frac\{1\}\{2\}x\backslash right)^n\}\{n\}$$

所以, 当 $n>0$ , $a\_\{2n\}\; =\; -\backslash frac\{1\}\{2n\; \backslash cdot\; 2^\{2n\}\}$ , 所以, $\backslash sum\_\{n=0\}^\{\backslash infty\}\; n\; a\_\{2n\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; n\; a\_\{2n\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; n\; \backslash cdot\; \backslash left(-\backslash frac\{1\}\{2n\; \backslash cdot\; 2^\{2n\}\}\backslash right)\; =\; -\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2^\{2n+1\}\}\; =\; -\backslash frac\{\backslash left(\backslash frac\{1\}\{2\}\backslash right)^3\}\{1-\backslash frac\{1\}\{4\}\}\; =\; -\backslash frac\{1\}\{6\}$ 故选 A. 方法 2:

$$[\backslash ln(2+x)]\text{'}\; =\; \backslash frac\{1\}\{2+x\}\; =\; \backslash frac\{1\}\{2\backslash left(1+\backslash frac\{x\}\{2\}\backslash right)\}\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; (-1)^n\; \backslash left(\backslash frac\{x\}\{2\}\backslash right)^n$$

$$\backslash ln(2+x)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; (-1)^n\; \backslash frac\{\backslash left(\backslash frac\{1\}\{2\}x\backslash right)^\{n+1\}\}\{n+1\}\; +\; C\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; (-1)^\{n-1\}\; \backslash frac\{\backslash left(\backslash frac\{1\}\{2\}x\backslash right)^n\}\{n\}\; +\; C$$

$$S(0)\; =\; C\; =\; \backslash ln(2+0)\; =\; \backslash ln\; 2$$

所以, $\backslash sum\_\{n=0\}^\{\backslash infty\}\; n\; a\_\{2n\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; n\; a\_\{2n\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; n\; \backslash cdot\; \backslash left(-\backslash frac\{1\}\{2n\; \backslash cdot\; 2^\{2n\}\}\backslash right)\; =\; -\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{2^\{2n+1\}\}\; =\; -\backslash frac\{\backslash left(\backslash frac\{1\}\{2\}\backslash right)^3\}\{1-\backslash frac\{1\}\{4\}\}\; =\; -\backslash frac\{1\}\{6\}$ 故选 A.

4. 设函数 $f(x)$ 在区间 $(-1,\; 1)$ 上有定义, 且 $\backslash lim\_\{x\; \backslash to\; 0\}\; f(x)\; =\; 0$ , 则

1. 当 $\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{f(x)\}\{x\}\; =\; m$ 时, $f\text{'}(0)\; =\; m$
2. 当 $f\text{'}(0)\; =\; m$ 时, $\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{f(x)\}\{x\}\; =\; m$
3. 当 $\backslash lim\_\{x\; \backslash to\; 0\}\; f\text{'}(x)\; =\; m$ 时, $f\text{'}(0)\; =\; m$
4. 当 $f\text{'}(0)\; =\; m$ 时, $\backslash lim\_\{x\; \backslash to\; 0\}\; f\text{'}(x)\; =\; m$

[答案]: B [解析]: 【解析】因为 $f\text{'}(0)\; =\; m$ , 所以 $f(x)$ 在 $x=0$ 处连续, 从而 $\backslash lim\_\{x\; \backslash to\; 0\}\; f(x)\; =\; f(0)\; =\; 0$ , 所以

$$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{f(x)\}\{x\}\; =\; \backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{f(x)\; -\; f(0)\}\{x\; -\; 0\}\; =\; m,\; \backslash text\{\; 故选\; B.\}$$

对于 A 选项, $\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{f(x)\}\{x\}\; =\; m$ , 推不出来 $f\text{'}(0)\; =\; m$ ; 对于 C 选项, $f\text{'}(x)$ 在 $x=0$ 处不一定连续; 对于

D 选项, $f\text{'}(x)$ 在 $x=0$ 处极限未必存在.

5. 在空间直角坐标系 $O-xyz$ 中, 三张平面 $\backslash pi\_i:\; a\_i\; x\; +\; b\_i\; y\; +\; c\_i\; z\; =\; d\_i$ ( $i=1,\; 2,\; 3$ ) 的位置关系如图所示,

记 $\backslash alpha\_i\; =\; (a\_i,\; b\_i,\; c\_i,\; d\_i)$ , 若 $r\; \backslash begin\{pmatrix\}\; \backslash alpha\_1\; \backslash \backslash \; \backslash alpha\_2\; \backslash \backslash \; \backslash alpha\_3\; \backslash end\{pmatrix\}\; =\; m$ , $r\; \backslash begin\{pmatrix\}\; \backslash beta\_1\; \backslash \backslash \; \backslash beta\_2\; \backslash \backslash \; \backslash beta\_3\; \backslash end\{pmatrix\}\; =\; n$ , 则

1. $m=1,\; n=2$
2. $m=n=2$
3. $m=2,\; n=3$
4. $m=n=3$

数学试题第 1 页 (共 10 页)

数学试题第 2 页 (共 10 页)