返回特征解读

2022 年高考数学（新高考卷 1）

exams_raw/普通高考/2022/2022新高考1(山东,广东,湖南,湖北,河北,江苏,福建).pdf · HS-MATH-1024-v2.1-solution-aware

22 题0 个小问/题组
1

一、单选题 · 组合数学

若集合 M={x∣x<4}\htmlData{tutor-start=0,tutor-end=1}{M}\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 }\sqrt{\htmlData{tutor-start=17,tutor-end=18}{x}}\htmlData{tutor-start=19,tutor-end=20}{<}\htmlData{tutor-start=20,tutor-end=21}{4}\htmlData{tutor-start=21,tutor-end=23}{\}}, N={x∣3x⩾1}\htmlData{tutor-start=0,tutor-end=1}{N}\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}{3}\htmlData{tutor-start=12,tutor-end=13}{x} \htmlData{tutor-start=14,tutor-end=24}{\geqslant }\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=27}{\}}, 则 M∩N=\htmlData{tutor-start=0,tutor-end=1}{M} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{N}\htmlData{tutor-start=8,tutor-end=9}{=} ( ) (A) {x∣0⩽x<2}\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x} \htmlData{tutor-start=4,tutor-end=9}{\mid }\htmlData{tutor-start=9,tutor-end=10}{0} \htmlData{tutor-start=11,tutor-end=21}{\leqslant }\htmlData{tutor-start=21,tutor-end=22}{x}\htmlData{tutor-start=22,tutor-end=23}{<}\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=26}{\}} (B) {x∣13⩽x<2}\left\{\htmlData{tutor-start=7,tutor-end=8}{x} \htmlData{tutor-start=9,tutor-end=14}{\mid }\frac{\htmlData{tutor-start=20,tutor-end=21}{1}}{\htmlData{tutor-start=23,tutor-end=24}{3}} \htmlData{tutor-start=26,tutor-end=36}{\leqslant }\htmlData{tutor-start=36,tutor-end=37}{x}\htmlData{tutor-start=37,tutor-end=38}{<}\htmlData{tutor-start=38,tutor-end=39}{2}\right\} (C) {x∣3⩽x<16}\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x} \htmlData{tutor-start=4,tutor-end=9}{\mid }\htmlData{tutor-start=9,tutor-end=10}{3} \htmlData{tutor-start=11,tutor-end=21}{\leqslant }\htmlData{tutor-start=21,tutor-end=22}{x}\htmlData{tutor-start=22,tutor-end=23}{<}\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{6}\htmlData{tutor-start=25,tutor-end=27}{\}} (D) {x∣13⩽x<16}\left\{\htmlData{tutor-start=7,tutor-end=8}{x} \htmlData{tutor-start=9,tutor-end=14}{\mid }\frac{\htmlData{tutor-start=20,tutor-end=21}{1}}{\htmlData{tutor-start=23,tutor-end=24}{3}} \htmlData{tutor-start=26,tutor-end=36}{\leqslant }\htmlData{tutor-start=36,tutor-end=37}{x}\htmlData{tutor-start=37,tutor-end=38}{<}\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{6}\right\}

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。

2

一、单选题 · 数学竞赛/待细分

若 i(1−z)=1\mathrm{\htmlData{tutor-start=8,tutor-end=9}{i}}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{z}\htmlData{tutor-start=14,tutor-end=15}{)}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{1}, 则 z+zˉ=\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{+}\bar{\htmlData{tutor-start=7,tutor-end=8}{z}}\htmlData{tutor-start=9,tutor-end=10}{=} ( ) (A) −2\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{2} (B) −1\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{1} (C) 1\htmlData{tutor-start=0,tutor-end=1}{1} (D) 2\htmlData{tutor-start=0,tutor-end=1}{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 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。

3

一、单选题 · 平面几何

在 △ABC\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, 点 D\htmlData{tutor-start=0,tutor-end=1}{D} 在边 AB\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} 上, BD=2DA\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{D}\htmlData{tutor-start=5,tutor-end=6}{A}. 记 CA→=m,CD→=n\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{C}\htmlData{tutor-start=17,tutor-end=18}{A}}\htmlData{tutor-start=19,tutor-end=20}{=}\boldsymbol{\htmlData{tutor-start=32,tutor-end=33}{m}}\htmlData{tutor-start=34,tutor-end=35}{,} \overrightarrow{\htmlData{tutor-start=52,tutor-end=53}{C}\htmlData{tutor-start=53,tutor-end=54}{D}}\htmlData{tutor-start=55,tutor-end=56}{=}\boldsymbol{\htmlData{tutor-start=68,tutor-end=69}{n}}, 则 CB→=\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{C}\htmlData{tutor-start=17,tutor-end=18}{B}}\htmlData{tutor-start=19,tutor-end=20}{=} ( ) (A) 3m−2n\htmlData{tutor-start=0,tutor-end=1}{3}\boldsymbol{\htmlData{tutor-start=13,tutor-end=14}{m}}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{2}\boldsymbol{\htmlData{tutor-start=29,tutor-end=30}{n}} (B) −2m+3n\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{2}\boldsymbol{\htmlData{tutor-start=14,tutor-end=15}{m}}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{3}\boldsymbol{\htmlData{tutor-start=30,tutor-end=31}{n}} (C) 3m+2n\htmlData{tutor-start=0,tutor-end=1}{3}\boldsymbol{\htmlData{tutor-start=13,tutor-end=14}{m}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{2}\boldsymbol{\htmlData{tutor-start=29,tutor-end=30}{n}} (D) 2m+3n\htmlData{tutor-start=0,tutor-end=1}{2}\boldsymbol{\htmlData{tutor-start=13,tutor-end=14}{m}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{3}\boldsymbol{\htmlData{tutor-start=29,tutor-end=30}{n}}

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。

4

一、单选题 · 数学竞赛/待细分

南水北调工程缓解了北方一些地区水资源短缺问题, 其中一部分水蓄入某水库. 已知该水库水位为海拔 148.5 m\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{4}\htmlData{tutor-start=2,tutor-end=3}{8}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{5} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{m}} 时, 相应水面的面积为 140.0 km2\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{4}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{0} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{m}}^{\htmlData{tutor-start=20,tutor-end=21}{2}}; 水位为海拔 157.5 m\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{5}\htmlData{tutor-start=2,tutor-end=3}{7}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{5} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{m}} 时, 相应水面的面积为 180.0 km2\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{8}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{0} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{k}\htmlData{tutor-start=16,tutor-end=17}{m}}^{\htmlData{tutor-start=20,tutor-end=21}{2}}. 将该水库在这两个水位间的形状看作一个棱台, 则该水库水位从海拔 148.5 m\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{4}\htmlData{tutor-start=2,tutor-end=3}{8}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{5} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{m}} 上升到 157.5 m\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{5}\htmlData{tutor-start=2,tutor-end=3}{7}\htmlData{tutor-start=3,tutor-end=4}{.}\htmlData{tutor-start=4,tutor-end=5}{5} \mathrm{~\htmlData{tutor-start=15,tutor-end=16}{m}} 时, 增加的水量约为 (7≈2.65)\htmlData{tutor-start=0,tutor-end=1}{(}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{7}} \htmlData{tutor-start=10,tutor-end=18}{\approx }\htmlData{tutor-start=18,tutor-end=19}{2}\htmlData{tutor-start=19,tutor-end=20}{.}\htmlData{tutor-start=20,tutor-end=21}{6}\htmlData{tutor-start=21,tutor-end=22}{5}\htmlData{tutor-start=22,tutor-end=23}{)} ( ) (A) 1.0×109 m3\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{.}\htmlData{tutor-start=2,tutor-end=3}{0} \htmlData{tutor-start=4,tutor-end=11}{\times }\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{0}^{\htmlData{tutor-start=15,tutor-end=16}{9}} \mathrm{~\htmlData{tutor-start=27,tutor-end=28}{m}}^{\htmlData{tutor-start=31,tutor-end=32}{3}} (B) 1.2×109 m3\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{.}\htmlData{tutor-start=2,tutor-end=3}{2} \htmlData{tutor-start=4,tutor-end=11}{\times }\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{0}^{\htmlData{tutor-start=15,tutor-end=16}{9}} \mathrm{~\htmlData{tutor-start=27,tutor-end=28}{m}}^{\htmlData{tutor-start=31,tutor-end=32}{3}} (C) 1.4×109 m3\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{.}\htmlData{tutor-start=2,tutor-end=3}{4} \htmlData{tutor-start=4,tutor-end=11}{\times }\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{0}^{\htmlData{tutor-start=15,tutor-end=16}{9}} \mathrm{~\htmlData{tutor-start=27,tutor-end=28}{m}}^{\htmlData{tutor-start=31,tutor-end=32}{3}} (D) 1.6×109 m3\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{.}\htmlData{tutor-start=2,tutor-end=3}{6} \htmlData{tutor-start=4,tutor-end=11}{\times }\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{0}^{\htmlData{tutor-start=15,tutor-end=16}{9}} \mathrm{~\htmlData{tutor-start=27,tutor-end=28}{m}}^{\htmlData{tutor-start=31,tutor-end=32}{3}}

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。

5

一、单选题 · 数论

从 2 至 8 的 7 个整数中随机取 2 个不同的数, 则这 2 个数互质的概率为 ( ) (A) 16\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{6}} (B) 13\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}} (C) 12\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}} (D) 23\frac{\htmlData{tutor-start=6,tutor-end=7}{2}}{\htmlData{tutor-start=9,tutor-end=10}{3}}

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。

6

一、单选题 · 代数

记函数 f(x)=sin⁡(ωx+π4)+b(ω>0)\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}{=}\sin \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}{4}}\right)\htmlData{tutor-start=45,tutor-end=46}{+}\htmlData{tutor-start=46,tutor-end=47}{b}\htmlData{tutor-start=47,tutor-end=48}{(}\htmlData{tutor-start=48,tutor-end=54}{\omega}\htmlData{tutor-start=54,tutor-end=55}{>}\htmlData{tutor-start=55,tutor-end=56}{0}\htmlData{tutor-start=56,tutor-end=57}{)} 的最小正周期为 T\htmlData{tutor-start=0,tutor-end=1}{T}, 若 2π3<T<π\frac{\htmlData{tutor-start=6,tutor-end=7}{2} \htmlData{tutor-start=8,tutor-end=11}{\pi}}{\htmlData{tutor-start=13,tutor-end=14}{3}}\htmlData{tutor-start=15,tutor-end=16}{<}\htmlData{tutor-start=16,tutor-end=17}{T}\htmlData{tutor-start=17,tutor-end=18}{<}\htmlData{tutor-start=18,tutor-end=21}{\pi}, 且 y=f(x)\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}{)} 的图象关于点 (3π2,2)\left(\frac{\htmlData{tutor-start=12,tutor-end=13}{3} \htmlData{tutor-start=14,tutor-end=17}{\pi}}{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=24}{2}\right) 中心对称, 则 f(π2)=\htmlData{tutor-start=0,tutor-end=1}{f}\left(\frac{\htmlData{tutor-start=13,tutor-end=16}{\pi}}{\htmlData{tutor-start=18,tutor-end=19}{2}}\right)\htmlData{tutor-start=27,tutor-end=28}{=} ( ) (A) 1 (B) 32\frac{\htmlData{tutor-start=6,tutor-end=7}{3}}{\htmlData{tutor-start=9,tutor-end=10}{2}} (C) 52\frac{\htmlData{tutor-start=6,tutor-end=7}{5}}{\htmlData{tutor-start=9,tutor-end=10}{2}} (D) 3

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。

7

一、单选题 · 数学竞赛/待细分

设 a=0.1e0.1,b=19,c=−ln⁡0.9\htmlData{tutor-start=0,tutor-end=1}{a}\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=4,tutor-end=5}{1} \mathrm{\htmlData{tutor-start=14,tutor-end=15}{e}}^{\htmlData{tutor-start=18,tutor-end=19}{0}\htmlData{tutor-start=19,tutor-end=20}{.}\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}{=}\frac{\htmlData{tutor-start=32,tutor-end=33}{1}}{\htmlData{tutor-start=35,tutor-end=36}{9}}\htmlData{tutor-start=37,tutor-end=38}{,} \htmlData{tutor-start=39,tutor-end=40}{c}\htmlData{tutor-start=40,tutor-end=41}{=}\htmlData{tutor-start=41,tutor-end=42}{-}\ln \htmlData{tutor-start=46,tutor-end=47}{0}\htmlData{tutor-start=47,tutor-end=48}{.}\htmlData{tutor-start=48,tutor-end=49}{9}, 则 ( ) (A) a<b<c\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}{c} (B) c<b<a\htmlData{tutor-start=0,tutor-end=1}{c}\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}{a} (C) c<a<b\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{b} (D) a<c<b\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{c}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{b}

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。

8

一、单选题 · 组合数学

已知正四棱锥的侧棱长为 l\htmlData{tutor-start=0,tutor-end=1}{l}, 其各顶点都在同一球面上. 若该球的体积为 36π\htmlData{tutor-start=0,tutor-end=1}{3}\htmlData{tutor-start=1,tutor-end=2}{6} \htmlData{tutor-start=3,tutor-end=6}{\pi}, 且 3⩽l⩽33\htmlData{tutor-start=0,tutor-end=1}{3} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{l} \htmlData{tutor-start=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{3} \sqrt{\htmlData{tutor-start=32,tutor-end=33}{3}}, 则该四棱锥体积的取值范围是 ( ) (A) [18,814]\left[\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{8}\htmlData{tutor-start=8,tutor-end=9}{,} \frac{\htmlData{tutor-start=16,tutor-end=17}{8}\htmlData{tutor-start=17,tutor-end=18}{1}}{\htmlData{tutor-start=20,tutor-end=21}{4}}\right] (B) [274,814]\left[\frac{\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{7}}{\htmlData{tutor-start=16,tutor-end=17}{4}}\htmlData{tutor-start=18,tutor-end=19}{,} \frac{\htmlData{tutor-start=26,tutor-end=27}{8}\htmlData{tutor-start=27,tutor-end=28}{1}}{\htmlData{tutor-start=30,tutor-end=31}{4}}\right] (C) [274,643]\left[\frac{\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{7}}{\htmlData{tutor-start=16,tutor-end=17}{4}}\htmlData{tutor-start=18,tutor-end=19}{,} \frac{\htmlData{tutor-start=26,tutor-end=27}{6}\htmlData{tutor-start=27,tutor-end=28}{4}}{\htmlData{tutor-start=30,tutor-end=31}{3}}\right] (D) [18,27]\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{8}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{7}\htmlData{tutor-start=7,tutor-end=8}{]}

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。

9

二、多选题 · 平面几何

已知正方体 ABCD−A1B1C1D1\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=11,tutor-end=12}{B}_{\htmlData{tutor-start=14,tutor-end=15}{1}} \htmlData{tutor-start=17,tutor-end=18}{C}_{\htmlData{tutor-start=20,tutor-end=21}{1}} \htmlData{tutor-start=23,tutor-end=24}{D}_{\htmlData{tutor-start=26,tutor-end=27}{1}}, 则 ( ) (A) 直线 BC1\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 与 DA1\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 所成的角为 90∘\htmlData{tutor-start=0,tutor-end=1}{9}\htmlData{tutor-start=1,tutor-end=2}{0}^{\htmlData{tutor-start=4,tutor-end=9}{\circ}} (B) 直线 BC1\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 与 CA1\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 所成的角为 90∘\htmlData{tutor-start=0,tutor-end=1}{9}\htmlData{tutor-start=1,tutor-end=2}{0}^{\htmlData{tutor-start=4,tutor-end=9}{\circ}} (C) 直线 BC1\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 与平面 BB1D1D\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{B}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{D}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{D} 所成的角为 45∘\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{5}^{\htmlData{tutor-start=4,tutor-end=9}{\circ}} (D) 直线 BC1\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}} 与平面 ABCD\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} 所成的角为 45∘\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{5}^{\htmlData{tutor-start=4,tutor-end=9}{\circ}}

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。

10

二、多选题 · 平面几何

已知函数 f(x)=x3−x+1\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}{x}^{\htmlData{tutor-start=8,tutor-end=9}{3}}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}, 则 ( ) (A) f(x)\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}{)} 有两个极值点 (B) f(x)\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}{)} 有三个零点 (C) 点 (0,1)\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=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{)} 是曲线 y=f(x)\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}{)} 的对称中心 (D) 直线 y=2x\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{x} 是曲线 y=f(x)\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}{)} 的切线

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。

11

二、多选题 · 平面几何

已知 O\htmlData{tutor-start=0,tutor-end=1}{O} 为坐标原点, 点 A(1,1)\htmlData{tutor-start=0,tutor-end=1}{A}\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=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} 在抛物线 C:x2=2py(p>0)\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{x}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{2} \htmlData{tutor-start=11,tutor-end=12}{p} \htmlData{tutor-start=13,tutor-end=14}{y}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{p}\htmlData{tutor-start=16,tutor-end=17}{>}\htmlData{tutor-start=17,tutor-end=18}{0}\htmlData{tutor-start=18,tutor-end=19}{)} 上, 过点 B(0,−1)\htmlData{tutor-start=0,tutor-end=1}{B}\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=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)} 的直线交 C\htmlData{tutor-start=0,tutor-end=1}{C} 于 P,Q\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} 两点, 则 ( ) (A) C\htmlData{tutor-start=0,tutor-end=1}{C} 的准线为 y=−1\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1} (B) 直线 AB\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} 与 C\htmlData{tutor-start=0,tutor-end=1}{C} 相切 (C) ∣OP∣⋅∣OQ∣>∣OA∣2\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=11}{\cdot }\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{O}\htmlData{tutor-start=13,tutor-end=14}{Q}\htmlData{tutor-start=14,tutor-end=15}{|} \htmlData{tutor-start=16,tutor-end=17}{>} \htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{O}\htmlData{tutor-start=20,tutor-end=21}{A}\htmlData{tutor-start=21,tutor-end=22}{|}^{\htmlData{tutor-start=24,tutor-end=25}{2}} (D) ∣BP∣⋅∣BQ∣>∣BA∣2\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=11}{\cdot }\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{B}\htmlData{tutor-start=13,tutor-end=14}{Q}\htmlData{tutor-start=14,tutor-end=15}{|} \htmlData{tutor-start=16,tutor-end=17}{>} \htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{B}\htmlData{tutor-start=20,tutor-end=21}{A}\htmlData{tutor-start=21,tutor-end=22}{|}^{\htmlData{tutor-start=24,tutor-end=25}{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 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。

12

二、多选题 · 代数

已知函数 f(x)\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}{)} 及其导函数 f′(x)\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=9}{\prime}}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{)} 的定义域均为 R\mathbf{\htmlData{tutor-start=8,tutor-end=9}{R}}, 记 g(x)=f′(x)\htmlData{tutor-start=0,tutor-end=1}{g}\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=8,tutor-end=14}{\prime}}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{)}. 若 f(32−2x),g(2+x)\htmlData{tutor-start=0,tutor-end=1}{f}\left(\frac{\htmlData{tutor-start=13,tutor-end=14}{3}}{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{2} \htmlData{tutor-start=21,tutor-end=22}{x}\right)\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=32}{g}\htmlData{tutor-start=32,tutor-end=33}{(}\htmlData{tutor-start=33,tutor-end=34}{2}\htmlData{tutor-start=34,tutor-end=35}{+}\htmlData{tutor-start=35,tutor-end=36}{x}\htmlData{tutor-start=36,tutor-end=37}{)} 均为偶函数, 则 ( ) (A) f(0)=0\htmlData{tutor-start=0,tutor-end=1}{f}\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=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{0} (B) g(−12)=0\htmlData{tutor-start=0,tutor-end=1}{g}\left(\htmlData{tutor-start=7,tutor-end=8}{-}\frac{\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{2}}\right)\htmlData{tutor-start=26,tutor-end=27}{=}\htmlData{tutor-start=27,tutor-end=28}{0} (C) f(−1)=f(4)\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,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=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{f}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{4}\htmlData{tutor-start=9,tutor-end=10}{)} (D) g(−1)=g(2)\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,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=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{g}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{)}

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。

13

三、填空题 · 数学竞赛/待细分

(1−yx)(x+y)8\left(\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{-}\frac{\htmlData{tutor-start=14,tutor-end=15}{y}}{\htmlData{tutor-start=17,tutor-end=18}{x}}\right)\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{y}\htmlData{tutor-start=30,tutor-end=31}{)}^{\htmlData{tutor-start=33,tutor-end=34}{8}} 的展开式中 x2y6\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{6}} 的系数为______. (用数字作答)

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。

14

三、填空题 · 平面几何

写出与圆 x2+y2=1\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{1} 和 (x−3)2+(y−4)2=16\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{3}\htmlData{tutor-start=4,tutor-end=5}{)}^{\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{y}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{4}\htmlData{tutor-start=14,tutor-end=15}{)}^{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{6} 都相切的一条直线的方程______.

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。

15

三、填空题 · 数学竞赛/待细分

若曲线 y=(x+a)ex\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\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}{a}\htmlData{tutor-start=6,tutor-end=7}{)} \mathrm{\htmlData{tutor-start=16,tutor-end=17}{e}}^{\htmlData{tutor-start=20,tutor-end=21}{x}} 有两条过坐标原点的切线, 则 a\htmlData{tutor-start=0,tutor-end=1}{a} 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 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。

16

三、填空题 · 平面几何

已知椭圆 C:x2a2+y2b2=1(a>b>0),C\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=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=44,tutor-end=45}{(}\htmlData{tutor-start=45,tutor-end=46}{a}\htmlData{tutor-start=46,tutor-end=47}{>}\htmlData{tutor-start=47,tutor-end=48}{b}\htmlData{tutor-start=48,tutor-end=49}{>}\htmlData{tutor-start=49,tutor-end=50}{0}\htmlData{tutor-start=50,tutor-end=51}{)}\htmlData{tutor-start=51,tutor-end=52}{,} \htmlData{tutor-start=53,tutor-end=54}{C} 的上顶点为 A\htmlData{tutor-start=0,tutor-end=1}{A}, 两个焦点为 F1,F2\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}, 离心率为 12\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}}. 过 F1\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}} 且垂直于 AF2\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{F}_{\htmlData{tutor-start=4,tutor-end=5}{2}} 的直线与 C\htmlData{tutor-start=0,tutor-end=1}{C} 交于 D,E\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{E} 两点, ∣DE∣=6\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{E}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{6}, 则 △ADE\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{D}\htmlData{tutor-start=12,tutor-end=13}{E} 的周长是______.

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 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。

17

四、解答题 · 代数

记 Sn\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{n}} 为数列 {an}\left\{\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{n}}\right\} 的前 n\htmlData{tutor-start=0,tutor-end=1}{n} 项和, 已知 a1=1,{Snan}\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}{1}\htmlData{tutor-start=7,tutor-end=8}{,}\left\{\frac{\htmlData{tutor-start=21,tutor-end=22}{S}_{\htmlData{tutor-start=24,tutor-end=25}{n}}}{\htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{n}}}\right\} 是公差为 13\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}} 的等差数列. (1) 求 {an}\left\{\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{n}}\right\} 的通项公式; (2) 证明: 1a1+1a2+⋯+1an<2\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}<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 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。

18

四、解答题 · 平面几何

记 △ABC\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} 的内角 A,B,C\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 a,b,c\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}, 已知 cos⁡A1+sin⁡A=sin⁡2B1+cos⁡2B\frac{\cos \htmlData{tutor-start=11,tutor-end=12}{A}}{\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{+}\sin \htmlData{tutor-start=21,tutor-end=22}{A}}\htmlData{tutor-start=23,tutor-end=24}{=}\frac{\sin \htmlData{tutor-start=35,tutor-end=36}{2} \htmlData{tutor-start=37,tutor-end=38}{B}}{\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{+}\cos \htmlData{tutor-start=47,tutor-end=48}{2} \htmlData{tutor-start=49,tutor-end=50}{B}}. (1) 若 C=2π3\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{=}\frac{\htmlData{tutor-start=8,tutor-end=9}{2} \htmlData{tutor-start=10,tutor-end=13}{\pi}}{\htmlData{tutor-start=15,tutor-end=16}{3}}, 求 B\htmlData{tutor-start=0,tutor-end=1}{B}; (2) 求 a2+b2c2\frac{\htmlData{tutor-start=6,tutor-end=7}{a}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{b}^{\htmlData{tutor-start=15,tutor-end=16}{2}}}{\htmlData{tutor-start=19,tutor-end=20}{c}^{\htmlData{tutor-start=22,tutor-end=23}{2}}} 的最小值.

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 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。

19

四、解答题 · 平面几何

如图, 直三棱柱 ABC−A1B1C1\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=10,tutor-end=11}{B}_{\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{C}_{\htmlData{tutor-start=19,tutor-end=20}{1}} 的体积为 4, △A1BC\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}_{\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{B} \htmlData{tutor-start=18,tutor-end=19}{C} 的面积为 22\htmlData{tutor-start=0,tutor-end=1}{2} \sqrt{\htmlData{tutor-start=8,tutor-end=9}{2}}. (1) 求 A\htmlData{tutor-start=0,tutor-end=1}{A} 到平面 A1BC\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{B} \htmlData{tutor-start=8,tutor-end=9}{C} 的距离; (2) 设 D\htmlData{tutor-start=0,tutor-end=1}{D} 为 A1C\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{C} 的中点, AA1=AB\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=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}, 平面 A1BC⊥\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{B} \htmlData{tutor-start=8,tutor-end=9}{C} \htmlData{tutor-start=10,tutor-end=15}{\perp} 平面 ABB1A1\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{B}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{1}}, 求二面角 A−BD−C\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}{D}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{C} 的正弦值.

原卷图示 1
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 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。

20

解答题 · 数学竞赛/待细分

一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯（卫生习惯分为良好和不够良好两类）的关系，在已患该疾病的病例中随机调查了 100 例（称为病例组），同时在未患该疾病的人群中随机调查了 100 人（称为对照组），得到如下数据： 不够良好良好病例组4060对照组1090\begin{array}{|c|c|c|} \hline & \text{\htmlData{tutor-start=39,tutor-end=40}{不}\htmlData{tutor-start=40,tutor-end=41}{够}\htmlData{tutor-start=41,tutor-end=42}{良}\htmlData{tutor-start=42,tutor-end=43}{好}} & \text{\htmlData{tutor-start=53,tutor-end=54}{良}\htmlData{tutor-start=54,tutor-end=55}{好}} \\ \hline \text{\htmlData{tutor-start=73,tutor-end=74}{病}\htmlData{tutor-start=74,tutor-end=75}{例}\htmlData{tutor-start=75,tutor-end=76}{组}} & \htmlData{tutor-start=80,tutor-end=81}{4}\htmlData{tutor-start=81,tutor-end=82}{0} & \htmlData{tutor-start=85,tutor-end=86}{6}\htmlData{tutor-start=86,tutor-end=87}{0} \\ \hline \text{\htmlData{tutor-start=104,tutor-end=105}{对}\htmlData{tutor-start=105,tutor-end=106}{照}\htmlData{tutor-start=106,tutor-end=107}{组}} & \htmlData{tutor-start=111,tutor-end=112}{1}\htmlData{tutor-start=112,tutor-end=113}{0} & \htmlData{tutor-start=116,tutor-end=117}{9}\htmlData{tutor-start=117,tutor-end=118}{0} \\ \hline \end{array} (1) 能否有 99% 的把握认为该疾病群体与未患该疾病群体的卫生习惯有差异？ (2) 从该地的人群中任选一人，A\htmlData{tutor-start=0,tutor-end=1}{A} 表示事件“选到的人卫生习惯不够良好”，B\htmlData{tutor-start=0,tutor-end=1}{B} 表示事件“选到的人患有该疾病”，P(B∣A)P(B‾∣A)\frac{\htmlData{tutor-start=6,tutor-end=7}{P}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{B} \htmlData{tutor-start=10,tutor-end=15}{\mid }\htmlData{tutor-start=15,tutor-end=16}{A}\htmlData{tutor-start=16,tutor-end=17}{)}}{\htmlData{tutor-start=19,tutor-end=20}{P}\htmlData{tutor-start=20,tutor-end=21}{(}\overline{\htmlData{tutor-start=31,tutor-end=32}{B}} \htmlData{tutor-start=34,tutor-end=39}{\mid }\htmlData{tutor-start=39,tutor-end=40}{A}\htmlData{tutor-start=40,tutor-end=41}{)}} 与 P(B∣A‾)P(B‾∣A‾)\frac{\htmlData{tutor-start=6,tutor-end=7}{P}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{B} \htmlData{tutor-start=10,tutor-end=15}{\mid }\overline{\htmlData{tutor-start=25,tutor-end=26}{A}}\htmlData{tutor-start=27,tutor-end=28}{)}}{\htmlData{tutor-start=30,tutor-end=31}{P}\htmlData{tutor-start=31,tutor-end=32}{(}\overline{\htmlData{tutor-start=42,tutor-end=43}{B}} \htmlData{tutor-start=45,tutor-end=50}{\mid }\overline{\htmlData{tutor-start=60,tutor-end=61}{A}}\htmlData{tutor-start=62,tutor-end=63}{)}} 的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标，记该指标为 R\htmlData{tutor-start=0,tutor-end=1}{R}. ① 证明：R=P(A∣B)P(A‾∣B)⋅P(A‾∣B‾)P(A∣B‾)\htmlData{tutor-start=0,tutor-end=1}{R} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{A} \htmlData{tutor-start=14,tutor-end=19}{\mid }\htmlData{tutor-start=19,tutor-end=20}{B}\htmlData{tutor-start=20,tutor-end=21}{)}}{\htmlData{tutor-start=23,tutor-end=24}{P}\htmlData{tutor-start=24,tutor-end=25}{(}\overline{\htmlData{tutor-start=35,tutor-end=36}{A}} \htmlData{tutor-start=38,tutor-end=43}{\mid }\htmlData{tutor-start=43,tutor-end=44}{B}\htmlData{tutor-start=44,tutor-end=45}{)}} \htmlData{tutor-start=47,tutor-end=53}{\cdot }\frac{\htmlData{tutor-start=59,tutor-end=60}{P}\htmlData{tutor-start=60,tutor-end=61}{(}\overline{\htmlData{tutor-start=71,tutor-end=72}{A}} \htmlData{tutor-start=74,tutor-end=79}{\mid }\overline{\htmlData{tutor-start=89,tutor-end=90}{B}}\htmlData{tutor-start=91,tutor-end=92}{)}}{\htmlData{tutor-start=94,tutor-end=95}{P}\htmlData{tutor-start=95,tutor-end=96}{(}\htmlData{tutor-start=96,tutor-end=97}{A} \htmlData{tutor-start=98,tutor-end=103}{\mid }\overline{\htmlData{tutor-start=113,tutor-end=114}{B}}\htmlData{tutor-start=115,tutor-end=116}{)}}； ② 利用该调查数据，给出 P(A∣B)\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=4,tutor-end=9}{\mid }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{)}，P(A∣B‾)\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=4,tutor-end=9}{\mid }\overline{\htmlData{tutor-start=19,tutor-end=20}{B}}\htmlData{tutor-start=21,tutor-end=22}{)} 的估计值，并利用①的结果给出 R\htmlData{tutor-start=0,tutor-end=1}{R} 的估计值. 附：K2=n(ad−bc)2(a+b)(c+d)(a+c)(b+d)\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}{)}}， P(K2⩾k)0.0500.0100.001k3.8416.63510.828\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 1
原卷图示 1原卷第 2 页 · qwen3.7-plus_layout_detection · 需复核
原卷图示 2
原卷图示 2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 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。

21

解答题 · 平面几何

已知点 A(2,1)\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=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)} 在双曲线 C:x2a2−y2a2−1=1 (a>1)\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}{a}^{\htmlData{tutor-start=41,tutor-end=42}{2}}\htmlData{tutor-start=43,tutor-end=44}{-}\htmlData{tutor-start=44,tutor-end=45}{1}} \htmlData{tutor-start=47,tutor-end=48}{=} \htmlData{tutor-start=49,tutor-end=50}{1} \htmlData{tutor-start=51,tutor-end=53}{\ }\htmlData{tutor-start=53,tutor-end=54}{(}\htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{>}\htmlData{tutor-start=56,tutor-end=57}{1}\htmlData{tutor-start=57,tutor-end=58}{)} 上，直线 l\htmlData{tutor-start=0,tutor-end=1}{l} 交 C\htmlData{tutor-start=0,tutor-end=1}{C} 于 P,Q\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q} 两点，直线 AP,AQ\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{A}\htmlData{tutor-start=5,tutor-end=6}{Q} 的斜率之和为 0. (1) 求 l\htmlData{tutor-start=0,tutor-end=1}{l} 的斜率； (2) 若 tan⁡∠PAQ=22\tan \htmlData{tutor-start=5,tutor-end=12}{\angle }\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{A}\htmlData{tutor-start=14,tutor-end=15}{Q} \htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{2}\sqrt{\htmlData{tutor-start=25,tutor-end=26}{2}}，求 △PAQ\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{A}\htmlData{tutor-start=12,tutor-end=13}{Q} 的面积.

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。

22

解答题 · 代数

已知函数 f(x)=ex−ax\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}{e}^{\htmlData{tutor-start=10,tutor-end=11}{x}} \htmlData{tutor-start=13,tutor-end=14}{-} \htmlData{tutor-start=15,tutor-end=16}{a}\htmlData{tutor-start=16,tutor-end=17}{x} 和 g(x)=ax−ln⁡x\htmlData{tutor-start=0,tutor-end=1}{g}\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}{x} \htmlData{tutor-start=10,tutor-end=11}{-} \ln \htmlData{tutor-start=16,tutor-end=17}{x} 有相同的最小值. (1) 求 a\htmlData{tutor-start=0,tutor-end=1}{a}； (2) 证明：存在直线 y=b\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{b}，其与两条曲线 y=f(x)\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}{)} 和 y=g(x)\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{g}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)} 共有三个不同的交点，并且从左到右的三个交点的横坐标成等差数列.

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。