2021 年高考数学（新高考卷 1）

exams_raw/普通高考/2021/2021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

一、单选题 · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x} \htmlData{tutor-start=6,tutor-end=11}{\mid }\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{<}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{<}\htmlData{tutor-start=16,tutor-end=17}{4}\htmlData{tutor-start=17,tutor-end=19}{\}}$ , $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{4}\htmlData{tutor-start=9,tutor-end=10}{,}\htmlData{tutor-start=10,tutor-end=11}{5}\htmlData{tutor-start=11,tutor-end=13}{\}}$ , 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) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=5}{\}}$ (B) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=7}{\}}$ (C) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{3}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{4}\htmlData{tutor-start=5,tutor-end=7}{\}}$ (D) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{4}\htmlData{tutor-start=7,tutor-end=9}{\}}$

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

已知 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{-}\mathrm{\htmlData{tutor-start=12,tutor-end=13}{i}}$ , 则 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{(}\bar{\htmlData{tutor-start=7,tutor-end=8}{z}}\htmlData{tutor-start=9,tutor-end=10}{+}\mathrm{\htmlData{tutor-start=18,tutor-end=19}{i}}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{=}$ ( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{6}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{2}\mathrm{\htmlData{tutor-start=11,tutor-end=12}{i}}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{2}\mathrm{\htmlData{tutor-start=11,tutor-end=12}{i}}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{6}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2}\mathrm{\htmlData{tutor-start=11,tutor-end=12}{i}}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2}\mathrm{\htmlData{tutor-start=11,tutor-end=12}{i}}$

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 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 · 平面几何

已知圆锥的底面半径为 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{2}}$ , 其侧面展开图为一个半圆, 则该圆锥的母线长为 ( ) (A) 2 (B) $\htmlData{tutor-start=0,tutor-end=1}{2}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{2}}$ (C) 4 (D) $\htmlData{tutor-start=0,tutor-end=1}{4}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{2}}$

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 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 · 代数

下列区间中, 函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{7}\sin\left(\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{-}\frac{\htmlData{tutor-start=24,tutor-end=27}{\pi}}{\htmlData{tutor-start=29,tutor-end=30}{6}}\right)$ 单调递增的区间是 ( ) (A) $\left(\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \frac{\htmlData{tutor-start=15,tutor-end=18}{\pi}}{\htmlData{tutor-start=20,tutor-end=21}{2}}\right)$ (B) $\left(\frac{\htmlData{tutor-start=12,tutor-end=15}{\pi}}{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=24}{\pi}\right)$ (C) $\left(\htmlData{tutor-start=6,tutor-end=9}{\pi}\htmlData{tutor-start=9,tutor-end=10}{,} \frac{\htmlData{tutor-start=17,tutor-end=18}{3}\htmlData{tutor-start=18,tutor-end=21}{\pi}}{\htmlData{tutor-start=23,tutor-end=24}{2}}\right)$ (D) $\left(\frac{\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=16}{\pi}}{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=26}{\pi}\right)$

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 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 · 平面几何

已知 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 是椭圆 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \frac{\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}{\htmlData{tutor-start=16,tutor-end=17}{9}}\htmlData{tutor-start=18,tutor-end=19}{+}\frac{\htmlData{tutor-start=25,tutor-end=26}{y}^{\htmlData{tutor-start=28,tutor-end=29}{2}}}{\htmlData{tutor-start=32,tutor-end=33}{4}}\htmlData{tutor-start=34,tutor-end=35}{=}\htmlData{tutor-start=35,tutor-end=36}{1}$ 的两个焦点, 点 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 上, 则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{M}\htmlData{tutor-start=2,tutor-end=3}{F}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{|} \htmlData{tutor-start=9,tutor-end=15}{\cdot }\htmlData{tutor-start=15,tutor-end=16}{|}\htmlData{tutor-start=16,tutor-end=17}{M}\htmlData{tutor-start=17,tutor-end=18}{F}_{\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{|}$ 的最大值为 ( ) (A) 13 (B) 12 (C) 9 (D) 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 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

若 $\tan\htmlData{tutor-start=4,tutor-end=10}{\theta}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{2}$ , 则 $\frac{\sin\htmlData{tutor-start=10,tutor-end=16}{\theta}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=19}{+}\sin \htmlData{tutor-start=24,tutor-end=25}{2}\htmlData{tutor-start=25,tutor-end=31}{\theta}\htmlData{tutor-start=31,tutor-end=32}{)}}{\sin\htmlData{tutor-start=38,tutor-end=44}{\theta}\htmlData{tutor-start=44,tutor-end=45}{+}\cos\htmlData{tutor-start=49,tutor-end=55}{\theta}}\htmlData{tutor-start=56,tutor-end=57}{=}$ ( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{-}\frac{\htmlData{tutor-start=7,tutor-end=8}{6}}{\htmlData{tutor-start=10,tutor-end=11}{5}}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{-}\frac{\htmlData{tutor-start=7,tutor-end=8}{2}}{\htmlData{tutor-start=10,tutor-end=11}{5}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=7}{2}}{\htmlData{tutor-start=9,tutor-end=10}{5}}$ (D) $\frac{\htmlData{tutor-start=6,tutor-end=7}{6}}{\htmlData{tutor-start=9,tutor-end=10}{5}}$

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 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

若过点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{)}$ 可以作曲线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\mathrm{\htmlData{tutor-start=10,tutor-end=11}{e}}^{\htmlData{tutor-start=14,tutor-end=15}{x}}$ 的两条切线, 则 ( ) (A) $\mathrm{\htmlData{tutor-start=8,tutor-end=9}{e}}^{\htmlData{tutor-start=12,tutor-end=13}{b}} \htmlData{tutor-start=15,tutor-end=16}{<} \htmlData{tutor-start=17,tutor-end=18}{a}$ (B) $\mathrm{\htmlData{tutor-start=8,tutor-end=9}{e}}^{\htmlData{tutor-start=12,tutor-end=13}{a}} \htmlData{tutor-start=15,tutor-end=16}{<} \htmlData{tutor-start=17,tutor-end=18}{b}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{a} \htmlData{tutor-start=6,tutor-end=7}{<} \mathrm{\htmlData{tutor-start=16,tutor-end=17}{e}}^{\htmlData{tutor-start=20,tutor-end=21}{b}}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{b} \htmlData{tutor-start=6,tutor-end=7}{<} \mathrm{\htmlData{tutor-start=16,tutor-end=17}{e}}^{\htmlData{tutor-start=20,tutor-end=21}{a}}$

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 · 数学竞赛/待细分

有 6 个相同的球, 分别标有数字 1, 2, 3, 4, 5, 6, 从中有放回的随机取两次, 每次取 1 个球. 甲表示事件“第一次取出的球的数字是 1”, 乙表示事件“第二次取出的球的数字是 2”, 丙表示事件“两次取出的球的数字之和是 8”, 丁表示事件“两次取出的球的数字之和是 7”, 则 ( ) (A) 甲与丙相互独立 (B) 甲与丁相互独立 (C) 乙与丙相互独立 (D) 丙与丁相互独立

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

有一组样本数据 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ , 由这组数据得到新样本数据 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{y}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{y}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ , 其中 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{i}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{c}$ ( $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\cdots\htmlData{tutor-start=12,tutor-end=13}{,}\htmlData{tutor-start=13,tutor-end=14}{n}$ ), $\htmlData{tutor-start=0,tutor-end=1}{c}$ 为非零常数, 则 ( ) (A) 两组样本数据的样本平均数相同 (B) 两组样本数据的样本中位数相同 (C) 两组样本数据的样本标准差相同 (D) 两组样本数据的样本极差相同

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 · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为坐标原点, 点 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{(}\cos\htmlData{tutor-start=10,tutor-end=16}{\alpha}\htmlData{tutor-start=16,tutor-end=17}{,} \sin\htmlData{tutor-start=22,tutor-end=28}{\alpha}\htmlData{tutor-start=28,tutor-end=29}{)}$ , $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{(}\cos\htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{-}\sin\htmlData{tutor-start=22,tutor-end=27}{\beta}\htmlData{tutor-start=27,tutor-end=28}{)}$ , $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{3}}\htmlData{tutor-start=5,tutor-end=6}{(}\cos\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=17}{\alpha}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=23}{\beta}\htmlData{tutor-start=23,tutor-end=24}{)}\htmlData{tutor-start=24,tutor-end=25}{,} \sin\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=37}{\alpha}\htmlData{tutor-start=37,tutor-end=38}{+}\htmlData{tutor-start=38,tutor-end=43}{\beta}\htmlData{tutor-start=43,tutor-end=44}{)}\htmlData{tutor-start=44,tutor-end=45}{)}$ , $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{)}$ , 则 ( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{|}\overrightarrow{\htmlData{tutor-start=17,tutor-end=18}{O}\htmlData{tutor-start=18,tutor-end=19}{P}_{\htmlData{tutor-start=21,tutor-end=22}{1}}}\htmlData{tutor-start=24,tutor-end=25}{|} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=29}{|}\overrightarrow{\htmlData{tutor-start=45,tutor-end=46}{O}\htmlData{tutor-start=46,tutor-end=47}{P}_{\htmlData{tutor-start=49,tutor-end=50}{2}}}\htmlData{tutor-start=52,tutor-end=53}{|}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{|}\overrightarrow{\htmlData{tutor-start=17,tutor-end=18}{A}\htmlData{tutor-start=18,tutor-end=19}{P}_{\htmlData{tutor-start=21,tutor-end=22}{1}}}\htmlData{tutor-start=24,tutor-end=25}{|} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=29}{|}\overrightarrow{\htmlData{tutor-start=45,tutor-end=46}{A}\htmlData{tutor-start=46,tutor-end=47}{P}_{\htmlData{tutor-start=49,tutor-end=50}{2}}}\htmlData{tutor-start=52,tutor-end=53}{|}$ (C) $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{O}\htmlData{tutor-start=17,tutor-end=18}{A}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{O}\htmlData{tutor-start=43,tutor-end=44}{P}_{\htmlData{tutor-start=46,tutor-end=47}{3}}} \htmlData{tutor-start=50,tutor-end=51}{=} \overrightarrow{\htmlData{tutor-start=68,tutor-end=69}{O}\htmlData{tutor-start=69,tutor-end=70}{P}_{\htmlData{tutor-start=72,tutor-end=73}{1}}} \htmlData{tutor-start=76,tutor-end=82}{\cdot }\overrightarrow{\htmlData{tutor-start=98,tutor-end=99}{O}\htmlData{tutor-start=99,tutor-end=100}{P}_{\htmlData{tutor-start=102,tutor-end=103}{2}}}$ (D) $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{O}\htmlData{tutor-start=17,tutor-end=18}{A}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{O}\htmlData{tutor-start=43,tutor-end=44}{P}_{\htmlData{tutor-start=46,tutor-end=47}{1}}} \htmlData{tutor-start=50,tutor-end=51}{=} \overrightarrow{\htmlData{tutor-start=68,tutor-end=69}{O}\htmlData{tutor-start=69,tutor-end=70}{P}_{\htmlData{tutor-start=72,tutor-end=73}{2}}} \htmlData{tutor-start=76,tutor-end=82}{\cdot }\overrightarrow{\htmlData{tutor-start=98,tutor-end=99}{O}\htmlData{tutor-start=99,tutor-end=100}{P}_{\htmlData{tutor-start=102,tutor-end=103}{3}}}$

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 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 · 平面几何

已知点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 在圆 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{5}\htmlData{tutor-start=4,tutor-end=5}{)}^{\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{y}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{5}\htmlData{tutor-start=14,tutor-end=15}{)}^{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{6}$ 上, 点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{4}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{)}$ , $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{)}$ , 则 ( ) (A) 点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 到直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的距离小于 10 (B) 点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 到直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的距离大于 2 (C) 当 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{P}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{A}$ 最小时, $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{3}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{2}}$ (D) 当 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{P}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{A}$ 最大时, $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{3}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{2}}$

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。

二、多选题 · 平面几何

在正三棱柱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{C}_{\htmlData{tutor-start=17,tutor-end=18}{1}}$ 中, $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{1}$ , 点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 满足 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{B}\htmlData{tutor-start=17,tutor-end=18}{P}}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=27}{\lambda}\overrightarrow{\htmlData{tutor-start=43,tutor-end=44}{B}\htmlData{tutor-start=44,tutor-end=45}{C}}\htmlData{tutor-start=46,tutor-end=47}{+}\htmlData{tutor-start=47,tutor-end=50}{\mu}\overrightarrow{\htmlData{tutor-start=66,tutor-end=67}{B}\htmlData{tutor-start=67,tutor-end=68}{B}_{\htmlData{tutor-start=70,tutor-end=71}{1}}}$ , 其中 $\htmlData{tutor-start=0,tutor-end=8}{\lambda }\htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{[}\htmlData{tutor-start=13,tutor-end=14}{0}\htmlData{tutor-start=14,tutor-end=15}{,}\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{]}$ , $\htmlData{tutor-start=0,tutor-end=4}{\mu }\htmlData{tutor-start=4,tutor-end=8}{\in }\htmlData{tutor-start=8,tutor-end=9}{[}\htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{,}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{]}$ , 则 ( ) (A) 当 $\htmlData{tutor-start=0,tutor-end=7}{\lambda}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}$ 时, $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{P}$ 的周长为定值 (B) 当 $\htmlData{tutor-start=0,tutor-end=3}{\mu}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{1}$ 时, 三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{C}$ 的体积为定值 (C) 当 $\htmlData{tutor-start=0,tutor-end=7}{\lambda}\htmlData{tutor-start=7,tutor-end=8}{=}\frac{\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{2}}$ 时, 有且仅有一个点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ , 使得 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{P} \htmlData{tutor-start=7,tutor-end=13}{\perp }\htmlData{tutor-start=13,tutor-end=14}{B}\htmlData{tutor-start=14,tutor-end=15}{P}$ (D) 当 $\htmlData{tutor-start=0,tutor-end=3}{\mu}\htmlData{tutor-start=3,tutor-end=4}{=}\frac{\htmlData{tutor-start=10,tutor-end=11}{1}}{\htmlData{tutor-start=13,tutor-end=14}{2}}$ 时, 有且仅有一个点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ , 使得 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{B} \htmlData{tutor-start=7,tutor-end=12}{\perp}$ 平面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{P}$

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 · 代数

已知函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{x}^{\htmlData{tutor-start=8,tutor-end=9}{3}}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{a} \htmlData{tutor-start=13,tutor-end=19}{\cdot }\htmlData{tutor-start=19,tutor-end=20}{2}^{\htmlData{tutor-start=22,tutor-end=23}{x}} \htmlData{tutor-start=25,tutor-end=26}{-} \htmlData{tutor-start=27,tutor-end=28}{2}^{\htmlData{tutor-start=30,tutor-end=31}{-}\htmlData{tutor-start=31,tutor-end=32}{x}}\htmlData{tutor-start=33,tutor-end=34}{)}$ 是偶函数, 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：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 · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为坐标原点, 抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{y}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{p}\htmlData{tutor-start=11,tutor-end=12}{x}$ ( $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0}$ ) 的焦点为 $\htmlData{tutor-start=0,tutor-end=1}{F}$ , $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 上一点, $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{F}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴垂直, $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴上一点, 且 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{O}\htmlData{tutor-start=10,tutor-end=11}{P}$ . 若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{F}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{6}$ , 则 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的准线方程为 ______.

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 · 代数

函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{2}\ln \htmlData{tutor-start=17,tutor-end=18}{x}$ 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：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 · 数学竞赛/待细分

某校学生在研究民间剪纸艺术时, 发现剪纸时经常会沿纸的某条对称轴把纸对折. 规格为 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{0}\text{ \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{m}} \htmlData{tutor-start=13,tutor-end=20}{\times }\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{2}\text{ \htmlData{tutor-start=29,tutor-end=30}{d}\htmlData{tutor-start=30,tutor-end=31}{m}}$ 的长方形纸, 对折 1 次共可以得到 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{0}\text{ \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{m}} \htmlData{tutor-start=13,tutor-end=20}{\times }\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{2}\text{ \htmlData{tutor-start=29,tutor-end=30}{d}\htmlData{tutor-start=30,tutor-end=31}{m}}$ , $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{0}\text{ \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{m}} \htmlData{tutor-start=13,tutor-end=20}{\times }\htmlData{tutor-start=20,tutor-end=21}{6}\text{ \htmlData{tutor-start=28,tutor-end=29}{d}\htmlData{tutor-start=29,tutor-end=30}{m}}$ 两种规格的图形, 它们的面积之和 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{4}\htmlData{tutor-start=8,tutor-end=9}{0}\text{ \htmlData{tutor-start=16,tutor-end=17}{d}\htmlData{tutor-start=17,tutor-end=18}{m}}^{\htmlData{tutor-start=21,tutor-end=22}{2}}$ , 对折 2 次共可以得到 $\htmlData{tutor-start=0,tutor-end=1}{5}\text{ \htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{m}} \htmlData{tutor-start=12,tutor-end=19}{\times }\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{2}\text{ \htmlData{tutor-start=28,tutor-end=29}{d}\htmlData{tutor-start=29,tutor-end=30}{m}}$ , $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{0}\text{ \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{m}} \htmlData{tutor-start=13,tutor-end=20}{\times }\htmlData{tutor-start=20,tutor-end=21}{6}\text{ \htmlData{tutor-start=28,tutor-end=29}{d}\htmlData{tutor-start=29,tutor-end=30}{m}}$ , $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{0}\text{ \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{m}} \htmlData{tutor-start=13,tutor-end=20}{\times }\htmlData{tutor-start=20,tutor-end=21}{3}\text{ \htmlData{tutor-start=28,tutor-end=29}{d}\htmlData{tutor-start=29,tutor-end=30}{m}}$ 三种规格的图形, 它们的面积之和 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{8}\htmlData{tutor-start=8,tutor-end=9}{0}\text{ \htmlData{tutor-start=16,tutor-end=17}{d}\htmlData{tutor-start=17,tutor-end=18}{m}}^{\htmlData{tutor-start=21,tutor-end=22}{2}}$ , 以此类推. 则对折 4 次共可以得到不同规格图形的种数为 ______; 如果对折 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 次, 那么 $\sum_{\htmlData{tutor-start=6,tutor-end=7}{k}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{n}} \htmlData{tutor-start=15,tutor-end=16}{S}_{\htmlData{tutor-start=18,tutor-end=19}{k}}\htmlData{tutor-start=20,tutor-end=21}{=}$ ______ $\text{\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{m}}^{\htmlData{tutor-start=11,tutor-end=12}{2}}$ .

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 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 · 代数

已知数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}$ , $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\begin{cases} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{,} & \htmlData{tutor-start=33,tutor-end=34}{n} \text{ \htmlData{tutor-start=42,tutor-end=43}{为}\htmlData{tutor-start=43,tutor-end=44}{奇}\htmlData{tutor-start=44,tutor-end=45}{数}}\htmlData{tutor-start=46,tutor-end=47}{,} \\ \htmlData{tutor-start=51,tutor-end=52}{a}_{\htmlData{tutor-start=54,tutor-end=55}{n}}\htmlData{tutor-start=56,tutor-end=57}{+}\htmlData{tutor-start=57,tutor-end=58}{2}\htmlData{tutor-start=58,tutor-end=59}{,} & \htmlData{tutor-start=62,tutor-end=63}{n} \text{ \htmlData{tutor-start=71,tutor-end=72}{为}\htmlData{tutor-start=72,tutor-end=73}{偶}\htmlData{tutor-start=73,tutor-end=74}{数}}\htmlData{tutor-start=75,tutor-end=76}{.} \end{cases}$ (1) 记 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{n}}$ , 写出 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ , 并求数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式; (2) 求 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的前 20 项和.

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 · 数学竞赛/待细分

某学校组织“一带一路”知识竞赛, 有 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ 两类问题. 每位参加比赛的同学先在两类问题中选择一类并从中随机抽取一个问题回答, 若回答错误则该同学比赛结束; 若回答正确则从另一类问题中再随机抽取一个问题回答, 无论回答正确与否, 该同学比赛结束. $\htmlData{tutor-start=0,tutor-end=1}{A}$ 类问题中的每个问题回答正确得 20 分, 否则得 0 分; $\htmlData{tutor-start=0,tutor-end=1}{B}$ 类问题中的每个问题回答正确得 80 分, 否则得 0 分. 已知小明能正确回答 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 类问题的概率为 0.8, 能正确回答 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 类问题的概率为 0.6, 且能正确回答问题的概率与回答次序无关. (1) 若小明先回答 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 类问题, 记 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 为小明的累计得分, 求 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 的分布列; (2) 为使累计得分的期望最大, 小明应选择先回答哪类问题? 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：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。

四、解答题 · 平面几何

记 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的内角 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 的对边分别为 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ . 已知 $\htmlData{tutor-start=0,tutor-end=1}{b}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{c}$ , 点 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 在边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上, $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}\sin\htmlData{tutor-start=6,tutor-end=13}{\angle }\htmlData{tutor-start=13,tutor-end=14}{A}\htmlData{tutor-start=14,tutor-end=15}{B}\htmlData{tutor-start=15,tutor-end=16}{C} \htmlData{tutor-start=17,tutor-end=18}{=} \htmlData{tutor-start=19,tutor-end=20}{a}\sin \htmlData{tutor-start=25,tutor-end=26}{C}$ . (1) 证明: $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{b}$ ; (2) 若 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{D}\htmlData{tutor-start=5,tutor-end=6}{C}$ , 求 $\cos\htmlData{tutor-start=4,tutor-end=11}{\angle }\htmlData{tutor-start=11,tutor-end=12}{A}\htmlData{tutor-start=12,tutor-end=13}{B}\htmlData{tutor-start=13,tutor-end=14}{C}$ .

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。

四、解答题 · 平面几何

如图，在三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{D}$ 中，平面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{D} \htmlData{tutor-start=4,tutor-end=9}{\perp}$ 平面 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{D}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{D}$ ， $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}$ 的中点. (1) 证明： $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{A} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{D}$ ; (2) 若 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{O}\htmlData{tutor-start=11,tutor-end=12}{C}\htmlData{tutor-start=12,tutor-end=13}{D}$ 是边长为 $\htmlData{tutor-start=0,tutor-end=1}{1}$ 的等边三角形，点 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 在棱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{D}$ 上， $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{E}\htmlData{tutor-start=5,tutor-end=6}{A}$ ，且二面角 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{D}$ 的大小为 $45^\circ$ ，求三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{D}$ 的体积.

**原卷图示 1**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。

四、解答题 · 平面几何

在平面直角坐标系 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中，已知点 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{-}\sqrt{\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{7}}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{0}\htmlData{tutor-start=19,tutor-end=20}{)}$ ， $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{(}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{7}}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{0}\htmlData{tutor-start=18,tutor-end=19}{)}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{M}\htmlData{tutor-start=2,tutor-end=3}{F}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{|} \htmlData{tutor-start=9,tutor-end=10}{-} \htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{M}\htmlData{tutor-start=13,tutor-end=14}{F}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{|} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{2}$ . 记 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的轨迹为 $\htmlData{tutor-start=0,tutor-end=1}{C}$ . (1) 求 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的方程; (2) 设点 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 在直线 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{=} \frac{\htmlData{tutor-start=10,tutor-end=11}{1}}{\htmlData{tutor-start=13,tutor-end=14}{2}}$ 上，过 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的两条直线分别交 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ 两点和 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}$ 两点，且 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{T}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=11}{\cdot }\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{T}\htmlData{tutor-start=13,tutor-end=14}{B}\htmlData{tutor-start=14,tutor-end=15}{|} \htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{T}\htmlData{tutor-start=20,tutor-end=21}{P}\htmlData{tutor-start=21,tutor-end=22}{|} \htmlData{tutor-start=23,tutor-end=29}{\cdot }\htmlData{tutor-start=29,tutor-end=30}{|}\htmlData{tutor-start=30,tutor-end=31}{T}\htmlData{tutor-start=31,tutor-end=32}{Q}\htmlData{tutor-start=32,tutor-end=33}{|}$ ，求直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的斜率与直线 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 的斜率之和.

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。

四、解答题 · 代数

已知函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{1} \htmlData{tutor-start=11,tutor-end=12}{-} \ln \htmlData{tutor-start=17,tutor-end=18}{x}\htmlData{tutor-start=18,tutor-end=19}{)}$ . (1) 讨论 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 的单调性; (2) 设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}$ 为两个不相等的正数，且 $\htmlData{tutor-start=0,tutor-end=1}{b} \ln \htmlData{tutor-start=6,tutor-end=7}{a} \htmlData{tutor-start=8,tutor-end=9}{-} \htmlData{tutor-start=10,tutor-end=11}{a} \ln \htmlData{tutor-start=16,tutor-end=17}{b} \htmlData{tutor-start=18,tutor-end=19}{=} \htmlData{tutor-start=20,tutor-end=21}{a} \htmlData{tutor-start=22,tutor-end=23}{-} \htmlData{tutor-start=24,tutor-end=25}{b}$ ，证明： $\htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=3}{<} \frac{\htmlData{tutor-start=10,tutor-end=11}{1}}{\htmlData{tutor-start=13,tutor-end=14}{a}} \htmlData{tutor-start=16,tutor-end=17}{+} \frac{\htmlData{tutor-start=24,tutor-end=25}{1}}{\htmlData{tutor-start=27,tutor-end=28}{b}} \htmlData{tutor-start=30,tutor-end=31}{<} \text{\htmlData{tutor-start=38,tutor-end=39}{e}}$ .

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

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