2018 年高考数学（北京卷理科）

exams_raw/普通高考/2018/2018北京理.pdf · HS-MATH-1024-v2.1-solution-aware

20 题0 个小问/题组

一、选择题 · 组合数学

已知集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x} \htmlData{tutor-start=6,tutor-end=11}{\mid }\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{|}\htmlData{tutor-start=14,tutor-end=15}{<}\htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=18}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{0}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=17}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{=}$ （） (A) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=8}{\}}$ (B) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=12}{\}}$ (C) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=15}{\}}$ (D) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=15}{\}}$

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 1 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 数学竞赛/待细分

在复平面内，复数 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{-}\mathrm{\htmlData{tutor-start=19,tutor-end=20}{i}}}$ 的共轭复数对应的点位于（） (A) 第一象限 (B) 第二象限 (C) 第三象限 (D) 第四象限

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 2 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 数学竞赛/待细分

执行如图所示的程序框图，输出的 $\htmlData{tutor-start=0,tutor-end=1}{s}$ 值为（） (A) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}}$ (B) $\frac{\htmlData{tutor-start=6,tutor-end=7}{5}}{\htmlData{tutor-start=9,tutor-end=10}{6}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=7}{7}}{\htmlData{tutor-start=9,tutor-end=10}{6}}$ (D) $\frac{\htmlData{tutor-start=6,tutor-end=7}{7}}{\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{2}}$

**原卷图示 1**原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 3 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 数学竞赛/待细分

“十二平均律”是通用的音律体系，明代朱载堉最早用数学方法计算出半音比例，为这个理论的发展做出了重要贡献。十二平均律将一个纯八度音程分成十二份，依次得到十三个单音，从第二个单音起，每一个单音的频率与它的前一个单音的频率的比都等于 $\sqrt[\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{2}]{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 。若第一个单音的频率为 $\htmlData{tutor-start=0,tutor-end=1}{f}$ ，则第八个单音的频率为（） (A) $\sqrt[\htmlData{tutor-start=6,tutor-end=7}{3}]{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{f}$ (B) $\sqrt[\htmlData{tutor-start=6,tutor-end=7}{3}]{\htmlData{tutor-start=9,tutor-end=10}{2}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}\htmlData{tutor-start=15,tutor-end=16}{f}$ (C) $\sqrt[\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{2}]{\htmlData{tutor-start=10,tutor-end=11}{2}^{\htmlData{tutor-start=13,tutor-end=14}{5}}}\htmlData{tutor-start=16,tutor-end=17}{f}$ (D) $\sqrt[\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{2}]{\htmlData{tutor-start=10,tutor-end=11}{2}^{\htmlData{tutor-start=13,tutor-end=14}{7}}}\htmlData{tutor-start=16,tutor-end=17}{f}$

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 4 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 平面几何

某四棱锥的三视图如图所示，在此四棱锥的侧面中，直角三角形的个数为（） (A) 1 (B) 2 (C) 3 (D) 4

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 5 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 数学竞赛/待细分

设 $\boldsymbol{\htmlData{tutor-start=12,tutor-end=13}{a}}\htmlData{tutor-start=14,tutor-end=15}{,} \boldsymbol{\htmlData{tutor-start=28,tutor-end=29}{b}}$ 均为单位向量，则“ $\htmlData{tutor-start=0,tutor-end=1}{|}\boldsymbol{\htmlData{tutor-start=13,tutor-end=14}{a}}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{3}\boldsymbol{\htmlData{tutor-start=29,tutor-end=30}{b}}\htmlData{tutor-start=31,tutor-end=32}{|}\htmlData{tutor-start=32,tutor-end=33}{=}\htmlData{tutor-start=33,tutor-end=34}{|}\htmlData{tutor-start=34,tutor-end=35}{3}\boldsymbol{\htmlData{tutor-start=47,tutor-end=48}{a}}\htmlData{tutor-start=49,tutor-end=50}{+}\boldsymbol{\htmlData{tutor-start=62,tutor-end=63}{b}}\htmlData{tutor-start=64,tutor-end=65}{|}$ ”是“ $\boldsymbol{\htmlData{tutor-start=12,tutor-end=13}{a}} \htmlData{tutor-start=15,tutor-end=21}{\perp }\boldsymbol{\htmlData{tutor-start=33,tutor-end=34}{b}}$ ”的（） (A) 充分而不必要条件 (B) 必要而不充分条件 (C) 充分必要条件 (D) 既不充分也不必要条件

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 6 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 平面几何

在平面直角坐标系中，记 $\htmlData{tutor-start=0,tutor-end=1}{d}$ 为点 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\cos \htmlData{tutor-start=7,tutor-end=13}{\theta}\htmlData{tutor-start=13,tutor-end=14}{,} \sin \htmlData{tutor-start=20,tutor-end=26}{\theta}\htmlData{tutor-start=26,tutor-end=27}{)}$ 到直线 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{m}\htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{0}$ 的距离。当 $\htmlData{tutor-start=0,tutor-end=6}{\theta}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{m}$ 变化时， $\htmlData{tutor-start=0,tutor-end=1}{d}$ 的最大值为（） (A) 1 (B) 2 (C) 3 (D) 4

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 7 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一、选择题 · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=16}{\mid }\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{y} \htmlData{tutor-start=20,tutor-end=30}{\geqslant }\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{,} \htmlData{tutor-start=33,tutor-end=34}{a}\htmlData{tutor-start=34,tutor-end=35}{x}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{y}\htmlData{tutor-start=37,tutor-end=38}{>}\htmlData{tutor-start=38,tutor-end=39}{4}\htmlData{tutor-start=39,tutor-end=40}{,} \htmlData{tutor-start=41,tutor-end=42}{x}\htmlData{tutor-start=42,tutor-end=43}{-}\htmlData{tutor-start=43,tutor-end=44}{a}\htmlData{tutor-start=44,tutor-end=45}{y} \htmlData{tutor-start=46,tutor-end=56}{\leqslant }\htmlData{tutor-start=56,tutor-end=57}{2}\htmlData{tutor-start=57,tutor-end=59}{\}}$ ，则（） (A) 对任意实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ , $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=11}{\in }\htmlData{tutor-start=11,tutor-end=12}{A}$ (B) 对任意实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ , $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} \notin \htmlData{tutor-start=14,tutor-end=15}{A}$ (C) 当且仅当 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{0}$ 时, $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} \notin \htmlData{tutor-start=14,tutor-end=15}{A}$ (D) 当且仅当 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\frac{\htmlData{tutor-start=18,tutor-end=19}{3}}{\htmlData{tutor-start=21,tutor-end=22}{2}}$ 时, $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} \notin \htmlData{tutor-start=14,tutor-end=15}{A}$

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 8 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 代数

设 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 是等差数列，且 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{5}}\htmlData{tutor-start=20,tutor-end=21}{=}\htmlData{tutor-start=21,tutor-end=22}{3}\htmlData{tutor-start=22,tutor-end=23}{6}$ ，则 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式为 \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 9 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 平面几何

在极坐标系中，直线 $\htmlData{tutor-start=0,tutor-end=5}{\rho }\cos \htmlData{tutor-start=10,tutor-end=17}{\theta }\htmlData{tutor-start=17,tutor-end=18}{+} \htmlData{tutor-start=19,tutor-end=24}{\rho }\sin \htmlData{tutor-start=29,tutor-end=36}{\theta }\htmlData{tutor-start=36,tutor-end=37}{=} \htmlData{tutor-start=38,tutor-end=39}{a} \htmlData{tutor-start=40,tutor-end=42}{\ }\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{a}\htmlData{tutor-start=44,tutor-end=45}{>}\htmlData{tutor-start=45,tutor-end=46}{0}\htmlData{tutor-start=46,tutor-end=47}{)}$ 与圆 $\htmlData{tutor-start=0,tutor-end=4}{\rho}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{2} \cos \htmlData{tutor-start=12,tutor-end=18}{\theta}$ 相切，则 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}$ \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 10 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 代数

设函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\cos \left(\htmlData{tutor-start=16,tutor-end=23}{\omega }\htmlData{tutor-start=23,tutor-end=24}{x}\htmlData{tutor-start=24,tutor-end=25}{-}\frac{\htmlData{tutor-start=31,tutor-end=34}{\pi}}{\htmlData{tutor-start=36,tutor-end=37}{6}}\right) \htmlData{tutor-start=46,tutor-end=48}{\ }\htmlData{tutor-start=48,tutor-end=49}{(}\htmlData{tutor-start=49,tutor-end=55}{\omega}\htmlData{tutor-start=55,tutor-end=56}{>}\htmlData{tutor-start=56,tutor-end=57}{0}\htmlData{tutor-start=57,tutor-end=58}{)}$ 。若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=15}{\leqslant }\htmlData{tutor-start=15,tutor-end=16}{f}\left(\frac{\htmlData{tutor-start=28,tutor-end=31}{\pi}}{\htmlData{tutor-start=33,tutor-end=34}{4}}\right)$ 对任意的实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 都成立，则 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 的最小值为 \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 11 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 数学竞赛/待细分

若 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{1} \htmlData{tutor-start=4,tutor-end=14}{\leqslant }\htmlData{tutor-start=14,tutor-end=15}{y} \htmlData{tutor-start=16,tutor-end=26}{\leqslant }\htmlData{tutor-start=26,tutor-end=27}{2}\htmlData{tutor-start=27,tutor-end=28}{x}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{y}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{x}$ 的最小值是 \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 12 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 代数

能说明“若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{>}\htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{)}$ 对任意的 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=5}{\in}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{]}$ 都成立，则 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{]}$ 上是增函数”为假命题的一个函数是 \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 13 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

二、填空题 · 平面几何

已知椭圆 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{:} \frac{\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}{\htmlData{tutor-start=16,tutor-end=17}{a}^{\htmlData{tutor-start=19,tutor-end=20}{2}}}\htmlData{tutor-start=22,tutor-end=23}{+}\frac{\htmlData{tutor-start=29,tutor-end=30}{y}^{\htmlData{tutor-start=32,tutor-end=33}{2}}}{\htmlData{tutor-start=36,tutor-end=37}{b}^{\htmlData{tutor-start=39,tutor-end=40}{2}}}\htmlData{tutor-start=42,tutor-end=43}{=}\htmlData{tutor-start=43,tutor-end=44}{1} \htmlData{tutor-start=45,tutor-end=47}{\ }\htmlData{tutor-start=47,tutor-end=48}{(}\htmlData{tutor-start=48,tutor-end=49}{a}\htmlData{tutor-start=49,tutor-end=50}{>}\htmlData{tutor-start=50,tutor-end=51}{b}\htmlData{tutor-start=51,tutor-end=52}{>}\htmlData{tutor-start=52,tutor-end=53}{0}\htmlData{tutor-start=53,tutor-end=54}{)}$ ，双曲线 $\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{:} \frac{\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}{\htmlData{tutor-start=16,tutor-end=17}{m}^{\htmlData{tutor-start=19,tutor-end=20}{2}}}\htmlData{tutor-start=22,tutor-end=23}{-}\frac{\htmlData{tutor-start=29,tutor-end=30}{y}^{\htmlData{tutor-start=32,tutor-end=33}{2}}}{\htmlData{tutor-start=36,tutor-end=37}{n}^{\htmlData{tutor-start=39,tutor-end=40}{2}}}\htmlData{tutor-start=42,tutor-end=43}{=}\htmlData{tutor-start=43,tutor-end=44}{1}$ 。若双曲线 $\htmlData{tutor-start=0,tutor-end=1}{N}$ 的两条渐近线与椭圆 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的四个交点及椭圆 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的两个焦点恰为一个正六边形的顶点，则椭圆 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的离心率为 \_\_\_\_\_\_；双曲线 $\htmlData{tutor-start=0,tutor-end=1}{N}$ 的离心率为 \_\_\_\_\_\_。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 14 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 平面几何

在 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{7}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{8}\htmlData{tutor-start=8,tutor-end=9}{,} \cos \htmlData{tutor-start=15,tutor-end=16}{B}\htmlData{tutor-start=16,tutor-end=17}{=}\htmlData{tutor-start=17,tutor-end=18}{-}\frac{\htmlData{tutor-start=24,tutor-end=25}{1}}{\htmlData{tutor-start=27,tutor-end=28}{7}}$ 。 (1) 求 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}$ ； (2) 求 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 边上的高。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 15 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 平面几何

如图，在三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{C}_{\htmlData{tutor-start=17,tutor-end=18}{1}}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}} \htmlData{tutor-start=7,tutor-end=12}{\perp}$ 平面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ ， $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{E}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{F}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{G}$ 分别为 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{A}_{\htmlData{tutor-start=15,tutor-end=16}{1}}\htmlData{tutor-start=17,tutor-end=18}{C}_{\htmlData{tutor-start=20,tutor-end=21}{1}}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{B}\htmlData{tutor-start=25,tutor-end=26}{B}_{\htmlData{tutor-start=28,tutor-end=29}{1}}$ 的中点， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{=}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{5}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{C}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{A}\htmlData{tutor-start=20,tutor-end=21}{A}_{\htmlData{tutor-start=23,tutor-end=24}{1}}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{2}$ 。 (1) 求证： $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C} \htmlData{tutor-start=3,tutor-end=8}{\perp}$ 平面 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{E}\htmlData{tutor-start=2,tutor-end=3}{F}$ ； (2) 求二面角 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{C}_{\htmlData{tutor-start=8,tutor-end=9}{1}}$ 的余弦值； (3) 证明：直线 $\htmlData{tutor-start=0,tutor-end=1}{F}\htmlData{tutor-start=1,tutor-end=2}{G}$ 与平面 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{D}$ 相交。

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 16 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 数学竞赛/待细分

电影公司随机收集了电影的有关数据，经分类整理得到下表： | 电影类型 | 第一类 | 第二类 | 第三类 | 第四类 | 第五类 | 第六类 | | --- | --- | --- | --- | --- | --- | --- | | 电影部数 | 140 | 50 | 300 | 200 | 800 | 510 | | 好评率 | 0.4 | 0.2 | 0.15 | 0.25 | 0.2 | 0.1 | 好评率是指：一类电影中获得好评的部数与该类电影的部数的比值。假设所有电影是否获得好评相互独立。 (1) 从电影公司收集的电影中随机选取 1 部，求这部电影是获得好评的第四类电影的概率； (2) 从第四类电影和第五类电影中各随机选取 1 部，估计恰有 1 部获得好评的概率； (3) 假设每类电影得到人们喜欢的概率与表格中该类电影的好评率相等。用“ $\htmlData{tutor-start=0,tutor-end=3}{\xi}_{\htmlData{tutor-start=5,tutor-end=6}{k}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}$ ”表示第 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 类电影得到人们喜欢，“ $\htmlData{tutor-start=0,tutor-end=3}{\xi}_{\htmlData{tutor-start=5,tutor-end=6}{k}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{0}$ ”表示第 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 类电影没有得到人们喜欢 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{k}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{3}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{4}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{5}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{6}\htmlData{tutor-start=19,tutor-end=20}{)}$ 。写出方差 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=4}{\xi}_{\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{D}\htmlData{tutor-start=11,tutor-end=14}{\xi}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{D}\htmlData{tutor-start=21,tutor-end=24}{\xi}_{\htmlData{tutor-start=26,tutor-end=27}{3}}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{D}\htmlData{tutor-start=31,tutor-end=34}{\xi}_{\htmlData{tutor-start=36,tutor-end=37}{4}}\htmlData{tutor-start=38,tutor-end=39}{,} \htmlData{tutor-start=40,tutor-end=41}{D}\htmlData{tutor-start=41,tutor-end=44}{\xi}_{\htmlData{tutor-start=46,tutor-end=47}{5}}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{D}\htmlData{tutor-start=51,tutor-end=54}{\xi}_{\htmlData{tutor-start=56,tutor-end=57}{6}}$ 的大小关系。

**原卷题面及图示 1**原卷第 1 页 · question_region_fallback · 需复核

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 17 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 代数

设函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{[}\htmlData{tutor-start=8,tutor-end=9}{a}\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}} \htmlData{tutor-start=15,tutor-end=16}{-} \htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{4}\htmlData{tutor-start=19,tutor-end=20}{a}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{)}\htmlData{tutor-start=23,tutor-end=24}{x} \htmlData{tutor-start=25,tutor-end=26}{+} \htmlData{tutor-start=27,tutor-end=28}{4}\htmlData{tutor-start=28,tutor-end=29}{a} \htmlData{tutor-start=30,tutor-end=31}{+} \htmlData{tutor-start=32,tutor-end=33}{3}\htmlData{tutor-start=33,tutor-end=34}{]}\mathrm{\htmlData{tutor-start=42,tutor-end=43}{e}}^{\htmlData{tutor-start=46,tutor-end=47}{x}}$ . (1) 若曲线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}$ 在点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}$ 处的切线与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴平行，求 $\htmlData{tutor-start=0,tutor-end=1}{a}$ ； (2) 若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}$ 处取得极小值，求 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的取值范围.

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 18 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 平面几何

已知抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{y}^{\htmlData{tutor-start=6,tutor-end=7}{2}} \htmlData{tutor-start=9,tutor-end=10}{=} \htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{p}\htmlData{tutor-start=13,tutor-end=14}{x}$ 经过点 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{)}$ . 过点 $\htmlData{tutor-start=0,tutor-end=1}{Q}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)}$ 的直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 有两个不同的交点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ ，且直线 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{A}$ 交 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴于 $\htmlData{tutor-start=0,tutor-end=1}{M}$ ，直线 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{B}$ 交 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴于 $\htmlData{tutor-start=0,tutor-end=1}{N}$ . (1) 求直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的斜率的取值范围； (2) 设 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为原点， $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{Q}\htmlData{tutor-start=17,tutor-end=18}{M}} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=30}{\lambda }\overrightarrow{\htmlData{tutor-start=46,tutor-end=47}{Q}\htmlData{tutor-start=47,tutor-end=48}{O}}$ ， $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{Q}\htmlData{tutor-start=17,tutor-end=18}{N}} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=26}{\mu }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{Q}\htmlData{tutor-start=43,tutor-end=44}{O}}$ ，求证： $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=16}{\lambda}} \htmlData{tutor-start=18,tutor-end=19}{+} \frac{\htmlData{tutor-start=26,tutor-end=27}{1}}{\htmlData{tutor-start=29,tutor-end=32}{\mu}}$ 为定值.

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 19 题

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

三、解答题 · 组合数学

设 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 为正整数，集合 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=13}{\alpha }\htmlData{tutor-start=13,tutor-end=14}{|} \htmlData{tutor-start=15,tutor-end=22}{\alpha }\htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{t}_{\htmlData{tutor-start=28,tutor-end=29}{1}}\htmlData{tutor-start=30,tutor-end=31}{,} \htmlData{tutor-start=32,tutor-end=33}{t}_{\htmlData{tutor-start=35,tutor-end=36}{2}}\htmlData{tutor-start=37,tutor-end=38}{,} \cdots\htmlData{tutor-start=45,tutor-end=46}{,} \htmlData{tutor-start=47,tutor-end=48}{t}_{\htmlData{tutor-start=50,tutor-end=51}{n}}\htmlData{tutor-start=52,tutor-end=53}{)}\htmlData{tutor-start=53,tutor-end=54}{,} \htmlData{tutor-start=55,tutor-end=56}{t}_{\htmlData{tutor-start=58,tutor-end=59}{k}} \htmlData{tutor-start=61,tutor-end=65}{\in }\htmlData{tutor-start=65,tutor-end=67}{\{}\htmlData{tutor-start=67,tutor-end=68}{0}\htmlData{tutor-start=68,tutor-end=69}{,} \htmlData{tutor-start=70,tutor-end=71}{1}\htmlData{tutor-start=71,tutor-end=73}{\}}\htmlData{tutor-start=73,tutor-end=74}{,} \htmlData{tutor-start=75,tutor-end=76}{k} \htmlData{tutor-start=77,tutor-end=78}{=} \htmlData{tutor-start=79,tutor-end=80}{1}\htmlData{tutor-start=80,tutor-end=81}{,} \htmlData{tutor-start=82,tutor-end=83}{2}\htmlData{tutor-start=83,tutor-end=84}{,} \cdots\htmlData{tutor-start=91,tutor-end=92}{,} \htmlData{tutor-start=93,tutor-end=94}{n}\htmlData{tutor-start=94,tutor-end=96}{\}}$ . 对于集合 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 中的任意元素 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{x}_{\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{,} \cdots\htmlData{tutor-start=30,tutor-end=31}{,} \htmlData{tutor-start=32,tutor-end=33}{x}_{\htmlData{tutor-start=35,tutor-end=36}{n}}\htmlData{tutor-start=37,tutor-end=38}{)}$ 和 $\htmlData{tutor-start=0,tutor-end=6}{\beta }\htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{y}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{y}_{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \cdots\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=32}{y}_{\htmlData{tutor-start=34,tutor-end=35}{n}}\htmlData{tutor-start=36,tutor-end=37}{)}$ ，记 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{)} \htmlData{tutor-start=17,tutor-end=18}{=} \frac{\htmlData{tutor-start=25,tutor-end=26}{1}}{\htmlData{tutor-start=28,tutor-end=29}{2}}\htmlData{tutor-start=30,tutor-end=31}{[}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{x}_{\htmlData{tutor-start=35,tutor-end=36}{1}} \htmlData{tutor-start=38,tutor-end=39}{+} \htmlData{tutor-start=40,tutor-end=41}{y}_{\htmlData{tutor-start=43,tutor-end=44}{1}} \htmlData{tutor-start=46,tutor-end=47}{-} \htmlData{tutor-start=48,tutor-end=49}{|}\htmlData{tutor-start=49,tutor-end=50}{x}_{\htmlData{tutor-start=52,tutor-end=53}{1}} \htmlData{tutor-start=55,tutor-end=56}{-} \htmlData{tutor-start=57,tutor-end=58}{y}_{\htmlData{tutor-start=60,tutor-end=61}{1}}\htmlData{tutor-start=62,tutor-end=63}{|}\htmlData{tutor-start=63,tutor-end=64}{)} \htmlData{tutor-start=65,tutor-end=66}{+} \htmlData{tutor-start=67,tutor-end=68}{(}\htmlData{tutor-start=68,tutor-end=69}{x}_{\htmlData{tutor-start=71,tutor-end=72}{2}} \htmlData{tutor-start=74,tutor-end=75}{+} \htmlData{tutor-start=76,tutor-end=77}{y}_{\htmlData{tutor-start=79,tutor-end=80}{2}} \htmlData{tutor-start=82,tutor-end=83}{-} \htmlData{tutor-start=84,tutor-end=85}{|}\htmlData{tutor-start=85,tutor-end=86}{x}_{\htmlData{tutor-start=88,tutor-end=89}{2}} \htmlData{tutor-start=91,tutor-end=92}{-} \htmlData{tutor-start=93,tutor-end=94}{y}_{\htmlData{tutor-start=96,tutor-end=97}{2}}\htmlData{tutor-start=98,tutor-end=99}{|}\htmlData{tutor-start=99,tutor-end=100}{)} \htmlData{tutor-start=101,tutor-end=102}{+} \cdots \htmlData{tutor-start=110,tutor-end=111}{+} \htmlData{tutor-start=112,tutor-end=113}{(}\htmlData{tutor-start=113,tutor-end=114}{x}_{\htmlData{tutor-start=116,tutor-end=117}{n}} \htmlData{tutor-start=119,tutor-end=120}{+} \htmlData{tutor-start=121,tutor-end=122}{y}_{\htmlData{tutor-start=124,tutor-end=125}{n}} \htmlData{tutor-start=127,tutor-end=128}{-} \htmlData{tutor-start=129,tutor-end=130}{|}\htmlData{tutor-start=130,tutor-end=131}{x}_{\htmlData{tutor-start=133,tutor-end=134}{n}} \htmlData{tutor-start=136,tutor-end=137}{-} \htmlData{tutor-start=138,tutor-end=139}{y}_{\htmlData{tutor-start=141,tutor-end=142}{n}}\htmlData{tutor-start=143,tutor-end=144}{|}\htmlData{tutor-start=144,tutor-end=145}{)}\htmlData{tutor-start=145,tutor-end=146}{]}$ . (1) 当 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{3}$ 时，若 $\htmlData{tutor-start=0,tutor-end=7}{\alpha }\htmlData{tutor-start=7,tutor-end=8}{=} \htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{0}\htmlData{tutor-start=17,tutor-end=18}{)}$ ， $\htmlData{tutor-start=0,tutor-end=6}{\beta }\htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{)}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=16}{\alpha}\htmlData{tutor-start=16,tutor-end=17}{)}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{)}$ 的值； (2) 当 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{4}$ 时，设 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的子集，且满足：对于 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 中的任意元素 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ ，当 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ 相同时， $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{)}$ 是奇数；当 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ 不同时， $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{)}$ 是偶数. 求集合 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 中元素个数的最大值； (3) 给定不小于 2 的 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，设 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的子集，且满足：对于 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 中的任意两个不同的元素 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ ， $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{)} \htmlData{tutor-start=17,tutor-end=18}{=} \htmlData{tutor-start=19,tutor-end=20}{0}$ . 写出一个集合 $\htmlData{tutor-start=0,tutor-end=1}{B}$ ，使其元素个数最多，并说明理由.

题解状态：标准答案与规范题解待补充

题目标签：北京卷理科第 20 题

解题过程

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