2020 年高考数学（新高考卷 1）

exams_raw/普通高考/2020/2020新高考1(山东).pdf · HS-MATH-1024-v2.1-solution-aware

22 题0 个小问/题组

一、单选题 · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{1} \htmlData{tutor-start=8,tutor-end=18}{\leqslant }\htmlData{tutor-start=18,tutor-end=19}{x} \htmlData{tutor-start=20,tutor-end=30}{\leqslant }\htmlData{tutor-start=30,tutor-end=31}{3}\htmlData{tutor-start=31,tutor-end=33}{\}}$ , $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{2} \htmlData{tutor-start=8,tutor-end=9}{<} \htmlData{tutor-start=10,tutor-end=11}{x} \htmlData{tutor-start=12,tutor-end=13}{<} \htmlData{tutor-start=14,tutor-end=15}{4}\htmlData{tutor-start=15,tutor-end=17}{\}}$ , 则 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=7}{\cup }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{=}$ ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 数学竞赛/待细分

$\frac{\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{-}\mathrm{\htmlData{tutor-start=16,tutor-end=17}{i}}}{\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{2}\mathrm{\htmlData{tutor-start=31,tutor-end=32}{i}}}\htmlData{tutor-start=34,tutor-end=35}{=}$ ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 数学竞赛/待细分

6 名同学到甲、乙、丙三个场馆做志愿者, 每名同学只去 1 个场馆, 甲场馆安排 1 名, 乙场馆安排 2 名, 丙场馆安排 3 名, 则不同的安排方法共有 ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 数学竞赛/待细分

日晷是中国古代用来测定时间的仪器, 利用与晷面垂直的晷针投射到晷面的影子来测定时间. 把地球看成一个球(球心记为 $\htmlData{tutor-start=0,tutor-end=1}{O}$ ), 地球上一点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的纬度是指 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{A}$ 与地球赤道所在平面所成角, 点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 处的水平面是指过点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 且与 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{A}$ 垂直的平面. 在点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 处放置一个日晷, 若晷面与赤道所在平面平行, 点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 处的纬度为北纬 $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{0}^{\htmlData{tutor-start=4,tutor-end=9}{\circ}}$ , 则晷针与点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 处的水平面所成角为 ( )

**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：新高考卷 1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、单选题 · 数学竞赛/待细分

某中学的学生积极参加体育锻炼, 其中有 96% 的学生喜欢足球或游泳, 60% 的学生喜欢足球, 82% 的学生喜欢游泳, 则该中学既喜欢足球又喜欢游泳的学生数占该校学生总数的比例是 ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 数学竞赛/待细分

基本再生数 $\htmlData{tutor-start=0,tutor-end=1}{R}_{\htmlData{tutor-start=3,tutor-end=4}{0}}$ 与世代间隔 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 是新冠肺炎的流行病学基本参数. 基本再生数指一个感染者传染的平均人数, 世代间隔指相邻两代间传染所需的平均时间. 在新冠肺炎疫情初始阶段, 可以用指数模型: $\htmlData{tutor-start=0,tutor-end=1}{I}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\mathrm{\htmlData{tutor-start=13,tutor-end=14}{e}}^{\htmlData{tutor-start=17,tutor-end=18}{r}\htmlData{tutor-start=18,tutor-end=19}{t}}$ 描述累计感染病例数 $\htmlData{tutor-start=0,tutor-end=1}{I}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{)}$ 随时间 $\htmlData{tutor-start=0,tutor-end=1}{t}$ (单位: 天) 的变化规律, 指数增长率 $\htmlData{tutor-start=0,tutor-end=1}{r}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{R}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{T}$ 近似满足 $\htmlData{tutor-start=0,tutor-end=1}{R}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{r}\htmlData{tutor-start=9,tutor-end=10}{T}$ . 有学者基于已有数据估计出 $\htmlData{tutor-start=0,tutor-end=1}{R}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=8}{.}\htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{8}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{T}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{6}$ . 据此, 在新冠肺炎疫情初始阶段, 累计感染病例数增加 1 倍需要的时间约为 ( $\ln \htmlData{tutor-start=4,tutor-end=5}{2} \htmlData{tutor-start=6,tutor-end=14}{\approx }\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{.}\htmlData{tutor-start=16,tutor-end=17}{6}\htmlData{tutor-start=17,tutor-end=18}{9}$ ) ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是边长为 2 的正六边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{E}\htmlData{tutor-start=5,tutor-end=6}{F}$ 内的一点, 则 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{P}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{A}\htmlData{tutor-start=43,tutor-end=44}{B}}$ 的取值范围是 ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、单选题 · 代数

若定义在 $\mathbf{\htmlData{tutor-start=8,tutor-end=9}{R}}$ 的奇函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=8}{\infty}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{0}\htmlData{tutor-start=11,tutor-end=12}{)}$ 单调递减, 且 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{0}$ , 则满足 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{x}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)} \htmlData{tutor-start=8,tutor-end=18}{\geqslant }\htmlData{tutor-start=18,tutor-end=19}{0}$ 的 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的取值范围是 ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、多选题 · 数学竞赛/待细分

已知曲线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{m}\htmlData{tutor-start=4,tutor-end=5}{x}^{\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{n}\htmlData{tutor-start=11,tutor-end=12}{y}^{\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=17}{=}\htmlData{tutor-start=17,tutor-end=18}{1}$ ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、多选题 · 代数

下图是函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\sin\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=14}{\omega }\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=23}{\varphi}\htmlData{tutor-start=23,tutor-end=24}{)}$ 的部分图象, 则 $\sin\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=12}{\omega }\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=21}{\varphi}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{=}$ ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、多选题 · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{>} \htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b} \htmlData{tutor-start=9,tutor-end=10}{>} \htmlData{tutor-start=11,tutor-end=12}{0}$ , 且 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{1}$ , 则 ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、多选题 · 数学竞赛/待细分

信息熵是信息论中的一个重要概念. 设随机变量 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 所有可能的取值为 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \cdots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{n}$ , 且 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{X}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{p}_{\htmlData{tutor-start=10,tutor-end=11}{i}} \htmlData{tutor-start=13,tutor-end=14}{>} \htmlData{tutor-start=15,tutor-end=16}{0} \htmlData{tutor-start=17,tutor-end=19}{\ }\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{i}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=24}{,} \htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{,} \cdots\htmlData{tutor-start=34,tutor-end=35}{,} \htmlData{tutor-start=36,tutor-end=37}{n}\htmlData{tutor-start=37,tutor-end=38}{)}\htmlData{tutor-start=38,tutor-end=39}{,} \sum_{\htmlData{tutor-start=46,tutor-end=47}{i}\htmlData{tutor-start=47,tutor-end=48}{=}\htmlData{tutor-start=48,tutor-end=49}{1}}^{\htmlData{tutor-start=52,tutor-end=53}{n}} \htmlData{tutor-start=55,tutor-end=56}{p}_{\htmlData{tutor-start=58,tutor-end=59}{i}}\htmlData{tutor-start=60,tutor-end=61}{=}\htmlData{tutor-start=61,tutor-end=62}{1}$ , 定义 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 的信息熵 $\htmlData{tutor-start=0,tutor-end=1}{H}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{X}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{-}\sum_{\htmlData{tutor-start=12,tutor-end=13}{i}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{1}}^{\htmlData{tutor-start=18,tutor-end=19}{n}} \htmlData{tutor-start=21,tutor-end=22}{p}_{\htmlData{tutor-start=24,tutor-end=25}{i}} \log_{\htmlData{tutor-start=33,tutor-end=34}{2}} \htmlData{tutor-start=36,tutor-end=37}{p}_{\htmlData{tutor-start=39,tutor-end=40}{i}}$ . ( )

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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。

三、填空题 · 平面几何

斜率为 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{3}}$ 的直线过抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{y}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{4}\htmlData{tutor-start=10,tutor-end=11}{x}$ 的焦点, 且与 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ 两点, 则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}$ ______.

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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。

三、填空题 · 代数

将数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=8}{\}}$ 与 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{3}\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=8}{\}}$ 的公共项从小到大排列得到数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ , 则 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的前 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 项和为 ______.

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、填空题 · 平面几何

某中学开展劳动实习, 学生加工制作零件, 零件的截面如图所示. $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为圆孔及轮廓圆弧 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 所在圆的圆心, $\htmlData{tutor-start=0,tutor-end=1}{A}$ 是圆弧 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 与直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{G}$ 的切点, $\htmlData{tutor-start=0,tutor-end=1}{B}$ 是圆弧 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 与直线 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 的切点, 四边形 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}\htmlData{tutor-start=2,tutor-end=3}{F}\htmlData{tutor-start=3,tutor-end=4}{G}$ 为矩形, $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{D}\htmlData{tutor-start=10,tutor-end=11}{G}$ , 垂足为 $\htmlData{tutor-start=0,tutor-end=1}{C}$ , $\tan \htmlData{tutor-start=5,tutor-end=12}{\angle }\htmlData{tutor-start=12,tutor-end=13}{O}\htmlData{tutor-start=13,tutor-end=14}{D}\htmlData{tutor-start=14,tutor-end=15}{C} \htmlData{tutor-start=16,tutor-end=17}{=} \frac{\htmlData{tutor-start=24,tutor-end=25}{3}}{\htmlData{tutor-start=27,tutor-end=28}{5}}$ , $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{H} \htmlData{tutor-start=3,tutor-end=13}{\parallel }\htmlData{tutor-start=13,tutor-end=14}{D}\htmlData{tutor-start=14,tutor-end=15}{G}$ , $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{F}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{2} \text{ \htmlData{tutor-start=13,tutor-end=14}{c}\htmlData{tutor-start=14,tutor-end=15}{m}}$ , $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2} \text{ \htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{m}}$ , $\htmlData{tutor-start=0,tutor-end=1}{A}$ 到直线 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{F}$ 的距离均为 $\htmlData{tutor-start=0,tutor-end=1}{7} \text{ \htmlData{tutor-start=9,tutor-end=10}{c}\htmlData{tutor-start=10,tutor-end=11}{m}}$ , 圆孔半径为 $\htmlData{tutor-start=0,tutor-end=1}{1} \text{ \htmlData{tutor-start=9,tutor-end=10}{c}\htmlData{tutor-start=10,tutor-end=11}{m}}$ , 则图中阴影部分的面积为 ______ $\text{\htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{m}}^{\htmlData{tutor-start=11,tutor-end=12}{2}}$ .

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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、填空题 · 平面几何

已知直四棱柱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{A}_{\htmlData{tutor-start=8,tutor-end=9}{1}}\htmlData{tutor-start=10,tutor-end=11}{B}_{\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{C}_{\htmlData{tutor-start=18,tutor-end=19}{1}}\htmlData{tutor-start=20,tutor-end=21}{D}_{\htmlData{tutor-start=23,tutor-end=24}{1}}$ 的棱长均为 2, $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{D}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{6}\htmlData{tutor-start=12,tutor-end=13}{0}^{\htmlData{tutor-start=15,tutor-end=20}{\circ}}$ . 以 $\htmlData{tutor-start=0,tutor-end=1}{D}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 为球心, $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{5}}$ 为半径的球面与侧面 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{C}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{B}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ 的交线长为 ______.

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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。

四、解答题 · 平面几何

在① $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{c}\htmlData{tutor-start=2,tutor-end=3}{=}\sqrt{\htmlData{tutor-start=9,tutor-end=10}{3}}$ , ② $\htmlData{tutor-start=0,tutor-end=1}{c} \sin \htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{3}$ , ③ $\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{=}\sqrt{\htmlData{tutor-start=8,tutor-end=9}{3}}\htmlData{tutor-start=10,tutor-end=11}{b}$ 这三个条件中任选一个, 补充在下面问题中, 若问题中的三角形存在, 求 $\htmlData{tutor-start=0,tutor-end=1}{c}$ 的值; 若问题中的三角形不存在, 说明理由. 问题: 是否存在 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ , 它的内角 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 的对边分别为 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ , 且 $\sin \htmlData{tutor-start=5,tutor-end=6}{A}\htmlData{tutor-start=6,tutor-end=7}{=}\sqrt{\htmlData{tutor-start=13,tutor-end=14}{3}} \sin \htmlData{tutor-start=21,tutor-end=22}{B}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{C}\htmlData{tutor-start=25,tutor-end=26}{=}\frac{\htmlData{tutor-start=32,tutor-end=35}{\pi}}{\htmlData{tutor-start=37,tutor-end=38}{6}}$ , ______?

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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。

四、解答题 · 代数

已知公比大于 1 的等比数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{4}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{0}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{3}}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{8}$ . (1) 求 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式; (2) 记 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{m}}$ 为 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 在区间 $(0, m] \ (m \in \mathbf{N}^*)$ 中的项的个数, 求数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{m}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的前 100 项和 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{0}}$ .

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1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。

解答题 · 数学竞赛/待细分

为加强环境保护，治理空气污染，环境监测部门对某市空气质量进行调研，随机抽查了 100 天空气中的 PM2.5 和 SO $_{\htmlData{tutor-start=2,tutor-end=3}{2}}$ 浓度（单位： $\htmlData{tutor-start=0,tutor-end=3}{\mu}$ g/m $^{\htmlData{tutor-start=2,tutor-end=3}{3}}$ ），得下表： $\begin{array}{|c|c|c|c|} \hline \text{\htmlData{tutor-start=38,tutor-end=39}{S}\htmlData{tutor-start=39,tutor-end=40}{O}}_{\htmlData{tutor-start=43,tutor-end=44}{2}} \htmlData{tutor-start=46,tutor-end=56}{\setminus }\text{\htmlData{tutor-start=62,tutor-end=63}{P}\htmlData{tutor-start=63,tutor-end=64}{M}\htmlData{tutor-start=64,tutor-end=65}{2}\htmlData{tutor-start=65,tutor-end=66}{.}\htmlData{tutor-start=66,tutor-end=67}{5}} & \htmlData{tutor-start=71,tutor-end=72}{[}\htmlData{tutor-start=72,tutor-end=73}{0}\htmlData{tutor-start=73,tutor-end=74}{,} \htmlData{tutor-start=75,tutor-end=76}{5}\htmlData{tutor-start=76,tutor-end=77}{0}\htmlData{tutor-start=77,tutor-end=78}{]} & \htmlData{tutor-start=81,tutor-end=82}{(}\htmlData{tutor-start=82,tutor-end=83}{5}\htmlData{tutor-start=83,tutor-end=84}{0}\htmlData{tutor-start=84,tutor-end=85}{,} \htmlData{tutor-start=86,tutor-end=87}{1}\htmlData{tutor-start=87,tutor-end=88}{5}\htmlData{tutor-start=88,tutor-end=89}{0}\htmlData{tutor-start=89,tutor-end=90}{]} & \htmlData{tutor-start=93,tutor-end=94}{(}\htmlData{tutor-start=94,tutor-end=95}{1}\htmlData{tutor-start=95,tutor-end=96}{5}\htmlData{tutor-start=96,tutor-end=97}{0}\htmlData{tutor-start=97,tutor-end=98}{,} \htmlData{tutor-start=99,tutor-end=100}{4}\htmlData{tutor-start=100,tutor-end=101}{7}\htmlData{tutor-start=101,tutor-end=102}{5}\htmlData{tutor-start=102,tutor-end=103}{]} \\ \hline \htmlData{tutor-start=114,tutor-end=115}{[}\htmlData{tutor-start=115,tutor-end=116}{0}\htmlData{tutor-start=116,tutor-end=117}{,} \htmlData{tutor-start=118,tutor-end=119}{3}\htmlData{tutor-start=119,tutor-end=120}{5}\htmlData{tutor-start=120,tutor-end=121}{]} & \htmlData{tutor-start=124,tutor-end=125}{3}\htmlData{tutor-start=125,tutor-end=126}{2} & \htmlData{tutor-start=129,tutor-end=130}{1}\htmlData{tutor-start=130,tutor-end=131}{8} & \htmlData{tutor-start=134,tutor-end=135}{4} \\ \hline \htmlData{tutor-start=146,tutor-end=147}{(}\htmlData{tutor-start=147,tutor-end=148}{3}\htmlData{tutor-start=148,tutor-end=149}{5}\htmlData{tutor-start=149,tutor-end=150}{,} \htmlData{tutor-start=151,tutor-end=152}{7}\htmlData{tutor-start=152,tutor-end=153}{5}\htmlData{tutor-start=153,tutor-end=154}{]} & \htmlData{tutor-start=157,tutor-end=158}{6} & \htmlData{tutor-start=161,tutor-end=162}{8} & \htmlData{tutor-start=165,tutor-end=166}{1}\htmlData{tutor-start=166,tutor-end=167}{2} \\ \hline \htmlData{tutor-start=178,tutor-end=179}{(}\htmlData{tutor-start=179,tutor-end=180}{7}\htmlData{tutor-start=180,tutor-end=181}{5}\htmlData{tutor-start=181,tutor-end=182}{,} \htmlData{tutor-start=183,tutor-end=184}{1}\htmlData{tutor-start=184,tutor-end=185}{1}\htmlData{tutor-start=185,tutor-end=186}{5}\htmlData{tutor-start=186,tutor-end=187}{]} & \htmlData{tutor-start=190,tutor-end=191}{3} & \htmlData{tutor-start=194,tutor-end=195}{7} & \htmlData{tutor-start=198,tutor-end=199}{1}\htmlData{tutor-start=199,tutor-end=200}{0} \\ \hline \end{array}$ (1) 估计事件“该市一天空气中 PM2.5 浓度不超过 75，且 SO $_{\htmlData{tutor-start=2,tutor-end=3}{2}}$ 浓度不超过 150”的概率； (2) 根据所给数据，完成下面的 $\htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{2}$ 列联表： $\begin{array}{|c|c|c|} \hline \text{\htmlData{tutor-start=36,tutor-end=37}{S}\htmlData{tutor-start=37,tutor-end=38}{O}}_{\htmlData{tutor-start=41,tutor-end=42}{2}} \htmlData{tutor-start=44,tutor-end=54}{\setminus }\text{\htmlData{tutor-start=60,tutor-end=61}{P}\htmlData{tutor-start=61,tutor-end=62}{M}\htmlData{tutor-start=62,tutor-end=63}{2}\htmlData{tutor-start=63,tutor-end=64}{.}\htmlData{tutor-start=64,tutor-end=65}{5}} & \htmlData{tutor-start=69,tutor-end=70}{[}\htmlData{tutor-start=70,tutor-end=71}{0}\htmlData{tutor-start=71,tutor-end=72}{,} \htmlData{tutor-start=73,tutor-end=74}{1}\htmlData{tutor-start=74,tutor-end=75}{5}\htmlData{tutor-start=75,tutor-end=76}{0}\htmlData{tutor-start=76,tutor-end=77}{]} & \htmlData{tutor-start=80,tutor-end=81}{(}\htmlData{tutor-start=81,tutor-end=82}{1}\htmlData{tutor-start=82,tutor-end=83}{5}\htmlData{tutor-start=83,tutor-end=84}{0}\htmlData{tutor-start=84,tutor-end=85}{,} \htmlData{tutor-start=86,tutor-end=87}{4}\htmlData{tutor-start=87,tutor-end=88}{7}\htmlData{tutor-start=88,tutor-end=89}{5}\htmlData{tutor-start=89,tutor-end=90}{]} \\ \hline \htmlData{tutor-start=101,tutor-end=102}{[}\htmlData{tutor-start=102,tutor-end=103}{0}\htmlData{tutor-start=103,tutor-end=104}{,} \htmlData{tutor-start=105,tutor-end=106}{7}\htmlData{tutor-start=106,tutor-end=107}{5}\htmlData{tutor-start=107,tutor-end=108}{]} & & \\ \hline \htmlData{tutor-start=123,tutor-end=124}{(}\htmlData{tutor-start=124,tutor-end=125}{7}\htmlData{tutor-start=125,tutor-end=126}{5}\htmlData{tutor-start=126,tutor-end=127}{,} \htmlData{tutor-start=128,tutor-end=129}{1}\htmlData{tutor-start=129,tutor-end=130}{1}\htmlData{tutor-start=130,tutor-end=131}{5}\htmlData{tutor-start=131,tutor-end=132}{]} & & \\ \hline \end{array}$ (3) 根据 (2) 中的列联表，判断是否有 99% 的把握认为该市一天空气中 PM2.5 浓度与 SO $_{\htmlData{tutor-start=2,tutor-end=3}{2}}$ 浓度有关？附： $\htmlData{tutor-start=0,tutor-end=1}{K}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{=} \frac{\htmlData{tutor-start=14,tutor-end=15}{n}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{a}\htmlData{tutor-start=17,tutor-end=18}{d}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{b}\htmlData{tutor-start=20,tutor-end=21}{c}\htmlData{tutor-start=21,tutor-end=22}{)}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}{\htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{a}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{b}\htmlData{tutor-start=32,tutor-end=33}{)}\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{c}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{)}\htmlData{tutor-start=38,tutor-end=39}{(}\htmlData{tutor-start=39,tutor-end=40}{a}\htmlData{tutor-start=40,tutor-end=41}{+}\htmlData{tutor-start=41,tutor-end=42}{c}\htmlData{tutor-start=42,tutor-end=43}{)}\htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{b}\htmlData{tutor-start=45,tutor-end=46}{+}\htmlData{tutor-start=46,tutor-end=47}{d}\htmlData{tutor-start=47,tutor-end=48}{)}}$ ， $\begin{array}{|c|c|c|c|} \hline \htmlData{tutor-start=32,tutor-end=33}{P}\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{K}^{\htmlData{tutor-start=37,tutor-end=38}{2}} \htmlData{tutor-start=40,tutor-end=50}{\geqslant }\htmlData{tutor-start=50,tutor-end=51}{k}\htmlData{tutor-start=51,tutor-end=52}{)} & \htmlData{tutor-start=55,tutor-end=56}{0}\htmlData{tutor-start=56,tutor-end=57}{.}\htmlData{tutor-start=57,tutor-end=58}{0}\htmlData{tutor-start=58,tutor-end=59}{5}\htmlData{tutor-start=59,tutor-end=60}{0} & \htmlData{tutor-start=63,tutor-end=64}{0}\htmlData{tutor-start=64,tutor-end=65}{.}\htmlData{tutor-start=65,tutor-end=66}{0}\htmlData{tutor-start=66,tutor-end=67}{1}\htmlData{tutor-start=67,tutor-end=68}{0} & \htmlData{tutor-start=71,tutor-end=72}{0}\htmlData{tutor-start=72,tutor-end=73}{.}\htmlData{tutor-start=73,tutor-end=74}{0}\htmlData{tutor-start=74,tutor-end=75}{0}\htmlData{tutor-start=75,tutor-end=76}{1} \\ \hline \htmlData{tutor-start=87,tutor-end=88}{k} & \htmlData{tutor-start=91,tutor-end=92}{3}\htmlData{tutor-start=92,tutor-end=93}{.}\htmlData{tutor-start=93,tutor-end=94}{8}\htmlData{tutor-start=94,tutor-end=95}{4}\htmlData{tutor-start=95,tutor-end=96}{1} & \htmlData{tutor-start=99,tutor-end=100}{6}\htmlData{tutor-start=100,tutor-end=101}{.}\htmlData{tutor-start=101,tutor-end=102}{6}\htmlData{tutor-start=102,tutor-end=103}{3}\htmlData{tutor-start=103,tutor-end=104}{5} & \htmlData{tutor-start=107,tutor-end=108}{1}\htmlData{tutor-start=108,tutor-end=109}{0}\htmlData{tutor-start=109,tutor-end=110}{.}\htmlData{tutor-start=110,tutor-end=111}{8}\htmlData{tutor-start=111,tutor-end=112}{2}\htmlData{tutor-start=112,tutor-end=113}{8} \\ \hline \end{array}$

**原卷图示 2**原卷第 2 页 · qwen3.7-plus_layout_detection · 需复核

**原卷图示 3**原卷第 2 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：新高考卷 1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。

解答题 · 数学竞赛/待细分

如图，四棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}$ 的底面为正方形， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{D} \htmlData{tutor-start=3,tutor-end=8}{\perp}$ 底面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ . 设平面 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{D}$ 与平面 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的交线为 $\htmlData{tutor-start=0,tutor-end=1}{l}$ . (1) 证明： $\htmlData{tutor-start=0,tutor-end=1}{l} \htmlData{tutor-start=2,tutor-end=7}{\perp}$ 平面 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{C}$ ； (2) 已知 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{D}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 上的点，求 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{B}$ 与平面 $\htmlData{tutor-start=0,tutor-end=1}{Q}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{D}$ 所成角的正弦值的最大值.

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1第 20 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。

解答题 · 平面几何

已知函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{a}\htmlData{tutor-start=8,tutor-end=9}{e}^{\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{-} \ln \htmlData{tutor-start=22,tutor-end=23}{x} \htmlData{tutor-start=24,tutor-end=25}{+} \ln \htmlData{tutor-start=30,tutor-end=31}{a}$ . (1) 当 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{e}$ 时，求曲线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}$ 在点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}$ 处的切线与两坐标轴围成的三角形的面积； (2) 若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=15}{\geqslant }\htmlData{tutor-start=15,tutor-end=16}{1}$ ，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：新高考卷 1第 21 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。

解答题 · 平面几何

已知椭圆 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \frac{\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}{\htmlData{tutor-start=16,tutor-end=17}{a}^{\htmlData{tutor-start=19,tutor-end=20}{2}}} \htmlData{tutor-start=23,tutor-end=24}{+} \frac{\htmlData{tutor-start=31,tutor-end=32}{y}^{\htmlData{tutor-start=34,tutor-end=35}{2}}}{\htmlData{tutor-start=38,tutor-end=39}{b}^{\htmlData{tutor-start=41,tutor-end=42}{2}}} \htmlData{tutor-start=45,tutor-end=46}{=} \htmlData{tutor-start=47,tutor-end=48}{1} \htmlData{tutor-start=49,tutor-end=51}{\ }\htmlData{tutor-start=51,tutor-end=52}{(}\htmlData{tutor-start=52,tutor-end=53}{a}\htmlData{tutor-start=53,tutor-end=54}{>}\htmlData{tutor-start=54,tutor-end=55}{b}\htmlData{tutor-start=55,tutor-end=56}{>}\htmlData{tutor-start=56,tutor-end=57}{0}\htmlData{tutor-start=57,tutor-end=58}{)}$ 的离心率为 $\frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{2}}}{\htmlData{tutor-start=16,tutor-end=17}{2}}$ ，且过点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)}$ . (1) 求 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的方程； (2) 点 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{N}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 上，且 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{M} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{N}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{D} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{M}\htmlData{tutor-start=10,tutor-end=11}{N}$ ， $\htmlData{tutor-start=0,tutor-end=1}{D}$ 为垂足. 证明：存在定点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{|}$ 为定值.

题解状态：标准答案与规范题解待补充

题目标签：新高考卷 1第 22 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。