From c03e9bc79a30e140040dc6847532367141327abd Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Thu, 9 Feb 2023 22:24:51 +0800 Subject: [PATCH] 20230209 night --- 工具/修改题目数据库.ipynb | 2 +- 工具/寻找阶段末尾空闲题号.ipynb | 8 +- 工具/文本文件/题号筛选.txt | 2 +- 工具/添加题目到数据库.ipynb | 151 +- 工具/题号选题pdf生成.ipynb | 12 +- 文本处理工具/剪贴板圆圈数字生成.ipynb | 4 +- 文本处理工具/剪贴板文本整理_mathpix.ipynb | 4 +- 题库0.3/Problems.json | 2109 +++++++++++++++++++++ 8 files changed, 2243 insertions(+), 49 deletions(-) diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index 9b8f7b75..74cb0534 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n", - "problems = \"31218\"\n", + "problems = \"14499\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index c4ca9736..1616b6f7 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -10,7 +10,7 @@ "output_type": "stream", "text": [ "首个空闲id: 14400 , 直至 020000\n", - "首个空闲id: 22022 , 直至 030000\n", + "首个空闲id: 22048 , 直至 030000\n", "首个空闲id: 31225 , 直至 999999\n" ] } @@ -45,7 +45,7 @@ ], "metadata": { "kernelspec": { - "display_name": "base", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -59,12 +59,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.13" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ad2bdc8ecc057115af97d19610ffacc2b4e99fae6737bb82f5d7fb13d2f2c186" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index e3dd97f2..2377b343 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ -21441:22021,21365,21366,21367,21368,21369,21370,21372,21371,21373,21374,21375,21376,21377,22022,21379,21382,22023,21383,21384,21385,21386,21387,21389,22024,21390,22026,21392,21393,21394,21395,21396,22027,21397,22028,21401,21403,22029,22030,22031,22032,22033,22034,21410,22035,22036,22037,21413,22038,22039,21415,22040,22041,21418,21420,21421,21422,21423,22042,22043,21427,21425,21428,22044,22045,21430,22046,22047,21432,21434,21433,21435,21436,21437,21438,21440, \ No newline at end of file +13056:13078,13079:13101,13102:13124,13125:13146,13961:13976,13977:13991,13992:14004,14005:14015,14016:14022,14361:14378,14456:14476,14477:14494,14495:14510 \ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index c729f9cc..eb58af9e 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -2,53 +2,138 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 22022\n", - "raworigin = \"2025届高一下校本作业\"\n", - "filename = r\"D:\\temp\\derivatives.tex\"\n", + "starting_id = 14400\n", + "raworigin = \"2023年空中课堂高三复习课\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\空中课堂第四批.tex\"\n", "editor = \"20230209\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "添加题号022022, 来源: 2025届高一下校本作业\n", - "添加题号022023, 来源: 2025届高一下校本作业\n", - "添加题号022024, 来源: 2025届高一下校本作业\n", - "添加题号022025, 来源: 2025届高一下校本作业\n", - "添加题号022026, 来源: 2025届高一下校本作业\n", - "添加题号022027, 来源: 2025届高一下校本作业\n", - "添加题号022028, 来源: 2025届高一下校本作业\n", - "添加题号022029, 来源: 2025届高一下校本作业\n", - "添加题号022030, 来源: 2025届高一下校本作业\n", - "添加题号022031, 来源: 2025届高一下校本作业\n", - "添加题号022032, 来源: 2025届高一下校本作业\n", - "添加题号022033, 来源: 2025届高一下校本作业\n", - "添加题号022034, 来源: 2025届高一下校本作业\n", - "添加题号022035, 来源: 2025届高一下校本作业\n", - "添加题号022036, 来源: 2025届高一下校本作业\n", - "添加题号022037, 来源: 2025届高一下校本作业\n", - "添加题号022038, 来源: 2025届高一下校本作业\n", - "添加题号022039, 来源: 2025届高一下校本作业\n", - "添加题号022040, 来源: 2025届高一下校本作业\n", - "添加题号022041, 来源: 2025届高一下校本作业\n", - "添加题号022042, 来源: 2025届高一下校本作业\n", - "添加题号022043, 来源: 2025届高一下校本作业\n", - "添加题号022044, 来源: 2025届高一下校本作业\n", - "添加题号022045, 来源: 2025届高一下校本作业\n", - "添加题号022046, 来源: 2025届高一下校本作业\n", - "添加题号022047, 来源: 2025届高一下校本作业\n" + "添加题号014400, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014401, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014402, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014403, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014404, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014405, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014406, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014407, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014408, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014409, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014410, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014411, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014412, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014413, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014414, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014415, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014416, 来源: 2023年空中课堂高三复习课14\n", + "添加题号014417, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014418, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014419, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014420, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014421, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014422, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014423, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014424, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014425, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014426, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014427, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014428, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014429, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014430, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014431, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014432, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014433, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014434, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014435, 来源: 2023年空中课堂高三复习课15\n", + "添加题号014436, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014437, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014438, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014439, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014440, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014441, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014442, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014443, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014444, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014445, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014446, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014447, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014448, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014449, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014450, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014451, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014452, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014453, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014454, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014455, 来源: 2023年空中课堂高三复习课16\n", + "添加题号014456, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014457, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014458, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014459, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014460, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014461, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014462, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014463, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014464, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014465, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014466, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014467, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014468, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014469, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014470, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014471, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014472, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014473, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014474, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014475, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014476, 来源: 2023年空中课堂高三复习课19\n", + "添加题号014477, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014478, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014479, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014480, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014481, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014482, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014483, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014484, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014485, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014486, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014487, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014488, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014489, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014490, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014491, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014492, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014493, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014494, 来源: 2023年空中课堂高三复习课20\n", + "添加题号014495, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014496, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014497, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014498, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014499, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014500, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014501, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014502, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014503, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014504, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014505, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014506, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014507, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014508, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014509, 来源: 2023年空中课堂高三复习课21\n", + "添加题号014510, 来源: 2023年空中课堂高三复习课21\n" ] } ], @@ -171,7 +256,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -190,7 +275,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 2f2c0b17..6438cbd7 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/2025届高一下校本作业_教师用_20230209.tex\n", + "开始编译教师版本pdf文件: 临时文件/05_解析几何备选_教师用_20230209.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/2025届高一下校本作业_学生用_20230209.tex\n", + "开始编译学生版本pdf文件: 临时文件/05_解析几何备选_学生用_20230209.tex\n", "0\n" ] } @@ -33,7 +33,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/2025届高一下校本作业\"\n", + "filename = \"临时文件/05_解析几何备选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", @@ -188,12 +188,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/文本处理工具/剪贴板圆圈数字生成.ipynb b/文本处理工具/剪贴板圆圈数字生成.ipynb index 0f4091cf..535ec98a 100644 --- a/文本处理工具/剪贴板圆圈数字生成.ipynb +++ b/文本处理工具/剪贴板圆圈数字生成.ipynb @@ -49,7 +49,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -68,7 +68,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/文本处理工具/剪贴板文本整理_mathpix.ipynb b/文本处理工具/剪贴板文本整理_mathpix.ipynb index ea51543a..c6ea49f4 100644 --- a/文本处理工具/剪贴板文本整理_mathpix.ipynb +++ b/文本处理工具/剪贴板文本整理_mathpix.ipynb @@ -384,7 +384,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -403,7 +403,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index c3231f69..b4cebddf 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -349000,6 +349000,2115 @@ "remark": "", "space": "12ex" }, + "014400": { + "id": "014400", + "content": "判断下列命题的真假.\\\\\n(1) 空间中的任意两条直线都可以确定一个平面;\\\\\n(2) 空间中, 垂直于同一直线的两条直线相互平行;\\\\\n(3) 若直线$a \\perp b$, 直线$a$与平面$\\beta$平行, 则$b \\perp \\beta$;\\\\\n(4) 若直线$a\\parallel$平面$\\alpha$, 直线$a\\parallel$平面$\\beta$, 则平面$\\alpha\\parallel$平面$\\beta$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014401": { + "id": "014401", + "content": "如图, 在空间四边形$ABCD$中, 已知$AB$、$BC$、$CD$的中点分别是$P$、$Q$、$R$, 且$PQ=3$、$QR=5$、$PR=7$, 求异面直线$AC$和$BD$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,1.1,0) node [above] {$A$} coordinate (A);\n\\draw (0.9,-0.7,0) node [below] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below left] {$Q$} coordinate (Q);\n\\draw ($(C)!0.5!(D)$) node [below right] {$R$} coordinate (R);\n\\draw ($(A)!0.5!(B)$) node [above left] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C)(P)--(Q);\n\\draw [dashed] (B)--(D)(P)--(R)--(Q);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014402": { + "id": "014402", + "content": "如图, 已知$PA \\perp$平面$ABC$, 且$\\triangle ABC$是直角三角形, 其中$\\angle ACB=90^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,1.5,0) node [left] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp PC$;\\\\\n(2) 求证: 平面$PAC \\perp$平面$PBC$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014403": { + "id": "014403", + "content": "如图, 四棱锥$S-ABCD$的底面为正方形, $SD \\perp$平面$ABCD$, 底面正方形对角线交于点$O$, $G$为$SC$边上的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,1.5,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw (S)--(A)--(B)--(C)--cycle (S)--(B)--(G);\n\\draw [dashed] (S)--(D)--(A) (D)--(C) (D)--(G) (O)--(G) (A)--(C) (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $SA\\parallel$平面$BDG$;\\\\\n(2) 若平面$BDG \\cap$平面$ADS=m$, 求证: $m\\parallel OG$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014404": { + "id": "014404", + "content": "如图, 四棱锥$S-ABCD$的底面为正方形, $SD \\perp$平面$ABCD$, 底面正方形对角线交于点$O$, $G$为$SC$边上的中点, 其中$AD=SD=2$, 求直线$OG$与平面$BCS$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,2,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw (S)--(A)--(B)--(C)--cycle (S)--(B);\n\\draw [dashed] (S)--(D)--(A) (D)--(C) (O)--(G) (A)--(C) (B)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014405": { + "id": "014405", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 与直线$A_1B$互为异面直线的棱有\\blank{50}条.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014406": { + "id": "014406", + "content": "已知$m, n$是两条不同直线, $\\alpha, \\beta$是两个不同平面, 则下列命题错误的是\\bracket{20}.\n\\onech{若$\\alpha, \\beta$不平行, 则在$\\alpha$内不存在与$\\beta$平行的直线}{若$m, n$平行于同一平面, 则$m$与$n$可能异面}{若$m, n$不平行, 则$m$与$n$不可能垂直于同一平面}{若$\\alpha, \\beta$垂直于同一平面, 则$\\alpha$与$\\beta$可能相交}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014407": { + "id": "014407", + "content": "如图, 对于直四棱柱$ABCD-A_1B_1C_1D_1$, 要使$A_1C \\perp B_1D_1$, 则在四边形$ABCD$中, 满足的条件可以是\\blank{50}.(只需写出一个正确的条件)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (A1)--(C);\n\\draw (B1)--(D1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014408": { + "id": "014408", + "content": "工人师傅在检查工作的相邻两个面是否垂直时, 只要用曲尺的短边紧靠在工件的一个面上, 长边在工件的另一个面上转动, 观察短边是否和这个面密合就可以了, 你能说明其中的原理吗?", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014409": { + "id": "014409", + "content": "如图, 在三棱台$ABC-A_1B_1C_1$的$9$条棱所在直线中, 与直线$A_1B$是异面直线的共有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (-1,0,0) node [left] {$A$} coordinate (A);\n\\draw ({1/2},0,{sqrt(3)/2}) node [below] {$B$} coordinate (B);\n\\draw ({1/2},0,{-sqrt(3)/2}) node [right] {$C$} coordinate (C);\n\\path (0,{sqrt(2)},0) coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)!0.5!(P)$) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(C)!0.5!(P)$) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1)--(A) (B_1)--(B) (C_1)--(C);\n\\draw (A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle;\n\\draw (A_1)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014410": { + "id": "014410", + "content": "如图, 在一个$60^{\\circ}$的二面角$\\alpha-l-\\beta$的棱上有两个点$A$、$B$, 其中$AC$、$BD$分别是在这个二面角的两个半平面内垂直于$AB$的线段, 且$AB=4$, $AC=6$, $BD=8$, 则$CD$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (0,0,0) -- (3,0,0) --++ (0,0,3) --++ (-3,0,0) coordinate (S) -- cycle;\n\\draw (0,0,0) --++ (0,{sqrt(3)},1) coordinate (T) --++ (3,0,0) --++ (0,{-sqrt(3)},-1);\n\\draw (1,0,0) node [below] {$A$} coordinate (A);\n\\draw (A) --++ (0,{0.6*sqrt(3)},0.6) node [above] {$C$} coordinate (C);\n\\draw (1.8,0,0) node [above] {$B$} coordinate (B);\n\\draw (B) --++ (0,0,1.6) node [below] {$D$} coordinate (D);\n\\draw (C)--(D);\n\\draw (0.5,0,0) node [above] {$l$} coordinate (l);\n\\draw (S) ++ (0.3,0,-0.3) node {$\\beta$};\n\\draw (T) ++ (0.3,{-0.15*sqrt(3)},-0.15) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014411": { + "id": "014411", + "content": "已知直线$l$与平面$\\alpha$相交, 则下列命题中, 正确的个数为\\bracket{20}.\\\\\n\\textcircled{1} 平面$\\alpha$内的所有直线均与直线$l$异面;\\\\\n\\textcircled{2} 平面$\\alpha$内存在与直线$l$垂直的直线;\\\\\n\\textcircled{3} 平面$\\alpha$内不存在直线与直线$l$平行;\\\\\n\\textcircled{4} 平面$\\alpha$内所有直线均与直线$l$相交.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014412": { + "id": "014412", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 给出下列四个命题:\\\\\n\\textcircled{1} 点$P$在直线$BC_1$上运动时, 三棱锥$A-D_1PC$的体积不变;\\\\\n\\textcircled{2} 点$P$在直线$BC_1$上运动时, 直线$AP$与平面$ACD_1$所成的角的大小不变;\\\\\n\\textcircled{3} 点$P$在直线$BC_1$上运动时, 二面角$P-AD_1-C$的大小不变;\\\\\n\\textcircled{4} 点$P$是平面$ABCD$上到点$D$和$C_1$距离相等的动点, 则$P$的轨迹是过点$B$的直线.\n其中的真命题是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014413": { + "id": "014413", + "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, 已知底面$ABCD$是正方形, 点$P$是侧棱$CC_1$上的一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$P$} coordinate (P);\n\\draw (B)--(P);\n\\draw [dashed] (B)--(D)--(P)--(A1)(A)--(C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AC_1\\parallel$平面$PBD$, 求$\\dfrac{PC_1}{PC}$的值;\\\\\n(2) 求证: $BD \\perp A_1P$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014414": { + "id": "014414", + "content": "已知$PA \\perp$平面$ABC$, $PA=AB=3$, $AC=4$, $M$为$BC$中点, 过点$M$分别作平行于平面$PAB$的直线交$AC$、$PC$于点$E$、$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(C)$) node [above] {$F$} coordinate (F);\n\\draw (P)--(B)--(C)--cycle(P)--(M)--(F);\n\\draw [dashed] (P)--(A)--(B)(A)--(C)(M)--(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PM$与平面$ABC$所成的角的大小;\\\\\n(2) 证明: 平面$MEF\\parallel$平面$PAB$, 并求直线$ME$到平面$PA$距离.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014415": { + "id": "014415", + "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正三角形$ABC$内接于圆柱的下底面圆$O$, 点$O_1$是圆柱的上底面的圆心, 线段$AA_1$是圆柱的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (1,0) node [right] {$A$} coordinate (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,0) -- (-1,2);\n\\draw (O_1) ellipse (1 and 0.3);\n\\draw (A) arc (0:-180:1 and 0.3);\n\\draw [dashed] (A) arc (0:180:1 and 0.3);\n\\draw (135:1 and 0.3) node [above] {$C$} coordinate (C);\n\\draw (-105:1 and 0.3) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上是否存在一点$D$, 满足$O_1D\\parallel$平面$A_1AB$? 若存在, 求出$\\angle BOD$的大小; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014416": { + "id": "014416", + "content": "如图, 在三棱锥$D-ABC$中, 平而$ACD \\perp$平面$ABC$, $AD \\perp AC$, $AB \\perp BC$, $E$、$F$分别为棱$BC$, $CD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2.5,0) node [left] {$D$} coordinate (D);\n\\draw ({1.5+1.5*cos(80)},0,{1.5*sin(80)}) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw (D)--(A)--(B)--(C)--cycle(B)--(D)(E)--(F);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$EF\\parallel$平面$ABD$;\\\\\n(2) 求证: 直线$BC \\perp$平面$ABD$;\\\\\n(3) 若直线$CD$与平面$ABC$所成的角的大小为$45^{\\circ}$, 直线$CD$与平面$ABD$所成角的大小为$30^{\\circ}$, 求二面角$B-AD-C$的大小.\n%15", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课14", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014417": { + "id": "014417", + "content": "判断下列命题是否正确, 并说明理由:\\\\\n(1) 有两个面平行, 其余各面都是四边形的几何体叫棱柱;\\\\\n(2) 各个面都是三角形的几何体是三棱锥;\\\\\n(3) 圆柱、圆锥、圆台的底面都是圆面.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014418": { + "id": "014418", + "content": "已知一个圆柱的高为定值, 若将其体积扩大为原来的$4$倍, 则它的侧面积扩大为原来的\\blank{50}倍.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014419": { + "id": "014419", + "content": "如图, 已知正四面体$A-BCD$的棱长为$2$, 用平行于底面$BCD$的平面截这个棱锥, 得到一个小棱锥和一个棱台. 若截面与底面之间的距离为$\\dfrac{\\sqrt{6}}{2}$, 则棱台的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-2/sqrt(3)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({1/sqrt(3)},0,1) node [below] {$C$} coordinate (C);\n\\draw (C)++(0,0,-2) node [right] {$D$} coordinate (D);\n\\draw (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(C)$) -- ($(A)!{1/4}!(D)$);\n\\draw [dashed] ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(D)$);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014420": { + "id": "014420", + "content": "我国南北朝时期的数学家祖暅提出了计算几何体体积的原理: ``幂势既同, 则积不容异''. 意思是: 两个等高的几何体, 若在任意给定的等高处的截面积相等, 则体积相等. 现有等高的三棱锥和圆锥, 若它们满足祖暅原理的条件, 且圆锥的侧面展开图是一个半径为$3$且圆心角为$\\dfrac{2 \\pi}{3}$的扇形, 则三棱锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014421": { + "id": "014421", + "content": "如图, 已知$BC$为圆锥的底面圆直径, 圆锥的侧面展开图是一个半径为$4$的半圆, $P$是母线$AB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (0,{sqrt(3)}) node [above right] {$A$} coordinate (A);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (1,0) node [below right] {$B$} coordinate (B);\n\\draw (C)--(A)--(B);\n\\draw (C) arc (180:360:1 and 0.25);\n\\draw [dashed] (C) arc (180:0:1 and 0.25);\n\\draw [dashed] (C)--(B)(O)--(A);\n\\filldraw ($(A)!0.5!(B)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\filldraw (C) circle (0.03);\n\\draw (A) -- ($(A)!-1!(B)$) arc (120:-60:2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆锥的表面积和体积;\\\\\n(2) 动点$M$沿圆锥侧面从点$C$运动到点$P$, 求$M$运动的最短距离.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014422": { + "id": "014422", + "content": "如图正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$\\sqrt{3}$, 体积为$27$, 设$E$是棱$B_1B$上的任意点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.7!(B1)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(E);\n\\draw [dashed] (D)--(E)(D)--(A1)(A1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求棱$A_1A$的长;\\\\\n(2) 求三棱锥$A_1-AED$的体积;\\\\\n(3) 设$F$是棱$C_1C$上的点, 满足$EF\\parallel BC$, 求四棱锥$A_1-AEFD$的体积.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014423": { + "id": "014423", + "content": "已知球$O$的表面积为$900 \\pi$, $ABCD-A_1B_1C_1D_1$是该球的内接长方体(即该长方体的八个顶点均在球面上).\\\\\n(1) 若$AB=12$, $BC=9$, 求球心$O$到平面$ABCD$的距离;\\\\\n(2) 若$ABCD-A_1B_1C_1D_1$是正四棱柱, 当该正四棱柱的侧面积最大时, 求其体积.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014424": { + "id": "014424", + "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$的底面边长为$2 \\text{cm}$, 高为$5 \\text{cm}$, 一动点从点$A$出发, 沿着三棱柱侧面绕行两周后到达点$A_1$的最短距离是\\blank{50}$\\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C);\n\\draw (A) --++ (0,5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,5) node [below right] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,5) node [right] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (A) -- ($(B)!{1/6}!(B1)$) -- ($(C)!{1/3}!(C1)$) ($(A)!0.5!(A1)$) -- ($(B)!{2/3}!(B1)$) -- ($(C)!{5/6}!(C1)$);\n\\draw [dashed] ($(C)!{1/3}!(C1)$) -- ($(A)!0.5!(A1)$) ($(C)!{5/6}!(C1)$) -- (A1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014425": { + "id": "014425", + "content": "已知直三棱柱$ABC-A_1B_1C_1$的$6$个顶点都在球$O$的球面上, 若$AB=3$, $AC=4$, $AB \\perp AC$, $AA_1=12$, 则球$O$的半径为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014426": { + "id": "014426", + "content": "如图, 在直径$AB=4$的半圆$O$内作一个内接直角三角形$ABC$, 使$\\angle BAC=30^{\\circ}$, 将图中阴影部分以直线$AB$为旋转轴旋转$180^{\\circ}$形成一个几何体, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (0,1.5) node [above] {$A$} coordinate (A);\n\\draw (0,-1.5) node [below] {$B$} coordinate (B);\n\\draw (-30:1.5) node [right] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A)--(C) arc (-30:90:1.5);\n\\fill [pattern = north east lines] (B)--(C) arc (-30:-90:1.5);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A) arc (90:-90:1.5);\n\\draw pic [draw, \"$30^\\circ$\", angle eccentricity = 2] {angle = B--A--C};\n\\draw pic [draw, \"$60^\\circ$\", angle eccentricity = 1.5] {angle = C--B--A};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014427": { + "id": "014427", + "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 体积为$27$, $E$、$F$都是棱$BC$上的任意点且$EF=1$, $P$、$Q$分别在棱$A_1D_1$、$D_1C_1$上运动, 则四面体$P-EFQ$的体积\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.4!(D1)$) node [left] {$P$} coordinate (P);\n\\draw ($(C1)!0.3!(D1)$) node [above] {$Q$} coordinate (Q);\n\\draw ($(B)!0.2!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(B)!0.7!(C)$) node [right] {$F$} coordinate (F);\n\\draw (P)--(Q);\n\\draw [dashed] (P)--(E)(P)--(F)(Q)--(E)(Q)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与点$E$、$F$、$P$、$Q$位置都有关}{与点$P$位置有关, 与点$E$、$F$、$Q$位置无关}{与点$Q$位置有关, 与点$E$、$F$、$P$位置无关}{与点$E$、$F$、$P$、$Q$位置都无关}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014428": { + "id": "014428", + "content": "若圆锥的母线与底面半径之比为$\\sqrt{2}: 1$, 则它的底面积与侧面积之比是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014429": { + "id": "014429", + "content": "如图, 正方形$ABCD-A_1B_1C_1D_1$的棱长为$1$, 线段$B_1D_1$有两个动点$E$、$F$, 且$EF=\\dfrac{\\sqrt{2}}{2}$, 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B1)!0.7!(D1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!0.2!(D1)$) node [above right] {$F$} coordinate (F);\n\\draw [dashed] (A)--(C)(B)--(D)(A)--(E)--(B)(A)--(F)--(B);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n\\onech{$AC \\perp BE$}{异面直线$AE$、$BF$的所成角为定值}{直线$AB$与平面$BEF$的所成角为定值}{以$A$、$B$、$E$、$F$为顶点的四面体, 其体积不随$E$、$F$位置的变化而变化}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014430": { + "id": "014430", + "content": "已知一个空间几何体的所有棱长均为$1 \\text{cm}$, 其平面展开图如图所示, 则该空间几何体的体积$V=$\\blank{50}$\\text{cm}^3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) rectangle (4,1);\n\\draw (0,0) -- (0,-1) -- (1,-1) -- (1,1) (2,0) -- (2,1) (3,0) -- (3,1);\n\\draw (0,1) --++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1)--++ (60:1) --++ (-60:1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014431": { + "id": "014431", + "content": "已知圆锥的侧面展开图是一个半径为$4$的半圆, 将该圆锥倒置, 放入一个半径为$1$的铁球并注入水, 使水面与球正好相切, 然后将球取出, 这时容器中水的深度为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014432": { + "id": "014432", + "content": "《九章算术》是我国古代的数学巨著, 其卷第五``商功''有如下的问题: ``今有刍甍, 下广三丈, 衰四丈, 上袲二丈, 无广, 高一丈. 问积几何? ''意思为: 今有底面为矩形的屋脊形状的多面体(如图), 下底面宽$AD=3$丈, 长$AB=4$丈, 上棱$EF=2$丈, $EF$与平面$ABCD$平行, $EF$与平面$ABCD$的距离为$1$丈, 则它的体积是\\blank{50}(立方丈).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (-2,0,1.5) node [below] {$A$} coordinate (A);\n\\draw (A)++(4,0,0) node [below] {$B$} coordinate (B);\n\\draw (A)++(0,0,-3) node [left] {$D$} coordinate (D);\n\\draw (D)++(4,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,1,0) node [above] {$E$} coordinate (E);\n\\draw (1,1,0) node [above] {$F$} coordinate (F);\n\\draw (E)--(F)(E)--(A)(F)--(B)(F)--(C)(A)--(B)--(C)(A)--(D)--(E);\n\\draw [dashed] (D)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014433": { + "id": "014433", + "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 点$O_1$是圆柱上底面的圆心. $\\triangle ABC$内接于圆柱的下底面圆$O$, 线段$AA_1$是圆柱的母线, 长度为$2$, 线段$AB$的长为$\\sqrt{3}, C$是优弧$\\overset\\frown{AB}$上异于点$A$、$B$的任意点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (1,0) node [right] {$A$} coordinate (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,0) -- (-1,2);\n\\draw (O_1) ellipse (1 and 0.3);\n\\draw (A) arc (0:-180:1 and 0.3);\n\\draw [dashed] (A) arc (0:180:1 and 0.3);\n\\draw (135:1 and 0.3) node [above] {$C$} coordinate (C);\n\\draw (-105:1 and 0.3) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\triangle ABC$为正三角形时, 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 求三棱锥$A_1-ABC$体积的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014434": { + "id": "014434", + "content": "若圆锥的侧面展开图是半径为$5$, 面积为$20 \\pi$的扇形, 则由它的两条母线所确定的该圆锥的截面的面积最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014435": { + "id": "014435", + "content": "已知正四棱锥的侧棱长为$l$, 其各顶点都在同一球面上. 若该球的体积为$36 \\pi$, 且$3 \\leq l \\leq 3 \\sqrt{3}$, 求该正四棱锥体积的取值范围.\n%16", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课15", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014436": { + "id": "014436", + "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, $AC$与$BD$的交点为$M$, 设$\\overrightarrow{A_1B_1}=\\overrightarrow {a}$, $\\overrightarrow{A_1D_1}=\\overrightarrow {b}$, $\\overrightarrow{A_1A}=\\overrightarrow {c}$, 则下列向量中与$\\overrightarrow{B_1M}$相等的向量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw ($(A)+(C)-(B)$) node [above left] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(3)},-1) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)-(A)+(A_1)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)-(A)+(A_1)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(D)-(A)+(A_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(A_1)--(B_1)--(B)--cycle (A_1)--(D_1)--(C_1)--(C)--(B)(B_1)--(C_1);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (A)--(C)(B)--(D)(M)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$-\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$-\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014437": { + "id": "014437", + "content": "在如图所示的空间直角坐标系中, $ABCD-A_1B_1C_1D_1$为长方体, $AB=BC=1$, $AA_1=2$, 点$A$关于$x$轴对称的点$A'$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (B1) -- ($(C1)!2!(B1)$) node [below] {$x$};\n\\draw [->] (C1) -- ($(D1)!1.5!(C1)$) node [below] {$y$};\n\\draw [->] (C1) -- ($(C)!1.25!(C1)$) node [left] {$z$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014438": { + "id": "014438", + "content": "已知向量$\\overrightarrow {a}=(1,-1,3)$, $\\overrightarrow {b}=(-1,4,-2)$, $\\overrightarrow {c}=(1,5,x)$, 若$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$共面, 则实数$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014439": { + "id": "014439", + "content": "已知空间三点$A(-2,0,2)$, $B(-1,1,2)$, $C(-3,0,4)$. 设$\\overrightarrow {a}=\\overrightarrow{AB}$, $\\overrightarrow {b}=\\overrightarrow{AC}$. 若向量$k \\overrightarrow {a}+\\overrightarrow {b}$与$k \\overrightarrow {a}-2 \\overrightarrow {b}$互相垂直, 则实数$k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014440": { + "id": "014440", + "content": "已知平面$ABC$中, $\\overrightarrow{AC}=(-1,-1,0)$, $\\overrightarrow{AB}=(0,1,2)$, 则平面$ABC$的一个法向量为$\\overrightarrow {n}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014441": { + "id": "014441", + "content": "如图, 在正四面体$ABCD$中, $E$、$F$分别为棱$DA$、$BC$的中点, 又设$\\overrightarrow{DA}=\\overrightarrow {a}$, $\\overrightarrow{DB}=\\overrightarrow {b}$, $\\overrightarrow{DC}=\\overrightarrow {c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (${1/3}*(A)+{1/3}*(B)+{1/3}*(C)+(0,{2*sqrt(6)/3},0)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle(B)--(D);\n\\draw [dashed] (A)--(C);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw (E)--(B)(F)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow {a},\\overrightarrow {b},\\overrightarrow {c}$的线性组合表示向量$\\overrightarrow{BE}$, $\\overrightarrow{DF}$;\\\\\n(2) 求$\\langle\\overrightarrow{BE},\\overrightarrow{DF}\\rangle$的大小.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014442": { + "id": "014442", + "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=AA_1=2$, $\\angle BAC=90^{\\circ}$, $E$、$F$分别为$CC_1$、$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A)++(0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(C)-(A)+(A_1)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(B)-(A)+(A_1)$) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1)--(C_1)--(C)--(B)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (A)--(A_1)(A)--(B)(A)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw (E)--(F);\n\\draw [dashed] (A_1)--(B)(F)--(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1B$与平面$AEF$所成角的大小;\\\\\n(2) 求点$C$到平面$AEF$的距离.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014443": { + "id": "014443", + "content": "如图, 在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$M$、$N$分别是棱$AB$、$AD$、$A_1B_1$、$A_1D_1$的中点, 点$P$、$Q$分别在棱$DD_1$、$BB_1$上移动, 且$DP=BQ=\\lambda$($0<\\lambda<2$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$M$} coordinate (M);\n\\draw ($(A1)!0.5!(D1)$) node [left] {$N$} coordinate (N);\n\\draw ($(B)!0.6!(B1)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(D)!0.5!(D1)$) node [left] {$P$} coordinate (P);\n\\draw (B)--(C1)(E)--(Q)--(M)--(N);\n\\draw [dashed] (E)--(F)--(P)--(N)(P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda=1$时, 证明: 直线$BC_1\\parallel$平面$EFPQ$;\\\\\n(2) 是否存在$\\lambda$, 使平面$EFPQ$与平面$PQMN$所成的二面角为直二面角? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014444": { + "id": "014444", + "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$均为空间单位向量, 它们的夹角为$60^{\\circ}$, 则$|\\overrightarrow {a}+3 \\overrightarrow {b}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014445": { + "id": "014445", + "content": "已知$O$为平面$ABCD$外一点, $A$、$B$、$C$、$D$四点中任意三点不共线. 若$\\overrightarrow{OA}=2 x \\overrightarrow{BO}+3 y \\overrightarrow{CO}+4 z \\overrightarrow{DO}$, 则$2 x+3 y+4 z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014446": { + "id": "014446", + "content": "如图, $ABC-A_1B_1C_1$是直三棱柱, $\\angle BCA=90^{\\circ}$, 点$D_1$、$F_1$分别是$A_1B_1$、$A_1C_1$的中点, 若$BC=CA=CC_1$, 则$BD_1$与$AF_1$所成角的余弦值是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(2)},0,0) node [right] {$A$} coordinate (A);\n\\draw (0,0,{sqrt(2)}) node [below] {$C$} coordinate (C);\n\\draw (A)++(0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)-(A)+(B)$) node [left] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)-(A)+(C)$) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B)--(C)--(A)--(A_1)--(B_1)--cycle(B_1)--(C_1)--(A_1)(C)--(C_1);\n\\draw [dashed] (B)--(A);\n\\draw ($(A_1)!0.5!(C_1)$) node [below right] {$F_1$} coordinate (F_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (F_1)--(A);\n\\draw [dashed] (D_1)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{\\sqrt{30}}{10}$}{$\\dfrac{1}{2} ;$}{$\\dfrac{\\sqrt{30}}{15}$}{$\\dfrac{\\sqrt{15}}{10}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014447": { + "id": "014447", + "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$底面$ABCD$, 底面$ABCD$是正方形, $PA=AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,\\l,0) node [above left] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(B)(D)--(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$BD \\perp$底面$PAC$;\\\\\n(2) 求直线$PC$与平面$PBD$所成的角的大小.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014448": { + "id": "014448", + "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{AC}_1$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below right] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A)-- ($(D)!1.5!(A)$) node [below left] {$x$};\n\\draw [->] (C)-- ($(D)!1.4!(C)$) node [below] {$y$};\n\\draw [->] (D1)-- ($(D)!1.5!(D1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014449": { + "id": "014449", + "content": "已知空间向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 且$\\overrightarrow{AB}=\\overrightarrow {a}+2 \\overrightarrow {b}$, $\\overrightarrow{BC}=-5 \\overrightarrow {a}+6 \\overrightarrow {b}$, $\\overrightarrow{CD}=7 \\overrightarrow {a}-2 \\overrightarrow {b}$, 则一定共线的三点是\\bracket{20}.\n\\fourch{$A$、$B$、$C$}{$B$、$C$、$D$}{$A$、$B$、$D$}{$A$、$C$、$D$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014450": { + "id": "014450", + "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $AA_1=1$, 则$BC_1$与平面$BB_1D_1D$所成角的正弦值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B)--(C1)(B1)--(D1);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{\\sqrt{6}}{3}$}{$\\dfrac{2 \\sqrt{6}}{5}$}{$\\dfrac{\\sqrt{15}}{5}$}{$\\dfrac{\\sqrt{10}}{5}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014451": { + "id": "014451", + "content": "在空间直角坐标系中, 点$A(-1,3,1)$、点$B(2,4,0)$、点$C(0,2,4)$, 则以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$为一组邻边的平行四边形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014452": { + "id": "014452", + "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014453": { + "id": "014453", + "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=AA_1=2$, $\\angle ABC=90^{\\circ}$, $E$、$F$分别为$CC_1$、$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A)++(0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(C)-(A)+(A_1)$) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(B)-(A)+(A_1)$) node [left] {$B_1$} coordinate (B_1);\n\\draw (A_1)--(C_1)--(C)--(B)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (A)--(A_1)(A)--(B)(A)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw (E)--(F);\n\\draw [dashed] (A_1)--(B)(F)--(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1B$与$EF$所成角的大小;\\\\\n(2) 求平面$ABA_1$与平面$AEF$所成的锐二面角的大小.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014454": { + "id": "014454", + "content": "已知半径为$1$的球$O$内切于正四面体$ABCD$, 线段$MN$是球$O$的一条动直径 ($M$、$N$是直径的两端点), 点$P$是正四面体$ABCD$的表面上的一个动点, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014455": { + "id": "014455", + "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (2,0,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw ($(A_1)+(B)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.4!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A_1)--(D)--(E)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt{6}}{7}$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.\n%19", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课16", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014456": { + "id": "014456", + "content": "已知椭圆$\\Gamma: 2 x^2+3 y^2=6$的两个焦点为$F_1$、$F_2, P$为$\\Gamma$上一点, 若$|PF_1|=1$, 则$|PF_2|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014457": { + "id": "014457", + "content": "若椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{3}}{2}$, $b=2$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014458": { + "id": "014458", + "content": "若方程$\\dfrac{x^2}{4-k}+\\dfrac{y^2}{6+k}=1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014459": { + "id": "014459", + "content": "双曲线$\\dfrac{x^2}{3}-y^2=1$的两条渐近线的夹角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014460": { + "id": "014460", + "content": "抛物线$x=-8 y^2$的准线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014461": { + "id": "014461", + "content": "求以直线$3 x \\pm 4 y=0$为渐近线, 且过点$M(-2,3)$的双曲线的标准方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014462": { + "id": "014462", + "content": "已知抛物线的顶点在原点, 且以某坐标轴为其对称轴, 该抛物线的准线过椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个焦点, 且与该椭圆的一个交点为$P(\\dfrac{2}{3},\\dfrac{2 \\sqrt{6}}{3})$, 求此抛物线及椭圆的标准方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014463": { + "id": "014463", + "content": "在平面直角坐标系$xOy$中, 已知双曲线$C: 2 x^2-y^2=1$. 设斜率为$1$的直线$l$交双曲线$C$于$P$、$Q$两点, 若直线$l$与圆$x^2+y^2=1$相切, 求证: $OP \\perp OQ$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014464": { + "id": "014464", + "content": "已知抛物线$C: y^2=4 x$的焦点为$F$.\\\\\n(1) 若抛物线$C$的焦点$F$为双曲线$\\Gamma: \\dfrac{x^2}{a^2}-2 y^2=1$($a>0$)的一个焦点, 求双曲线$\\Gamma$的离心率$e$;\\\\\n(2) 设抛物线$C$的准线$l$与$x$轴的交点为$E$, 点$P$在抛物线$C$上, 且在第一象限, 若$\\dfrac{|PF|}{|PE|}=\\dfrac{\\sqrt{2}}{2}$, 求直线$PE$的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014465": { + "id": "014465", + "content": "已知$F_1(-5,0)$, $F_2(5,0)$两点, 点$M$满足$|MF_1|-|MF_2|=8$, 则点$M$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014466": { + "id": "014466", + "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的两个焦点分别为$F_1$、$F_2$, 离心率为$\\dfrac{1}{2}$, 过点$F_2$的直线与$\\Gamma$交于$A$、$B$两点, 若$\\triangle F_1AB$的周长为$8$, 则$\\Gamma$的标准方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014467": { + "id": "014467", + "content": "已知抛物线$C: y^2=4 x$的焦点为$F$, $A(x_1,y_1)$, $B(x_2,y_2)$为$C$上两点, 若$y_2^2-2 y_1^2=4$, 则$\\dfrac{|AF|}{|BF|}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014468": { + "id": "014468", + "content": "以椭圆$\\dfrac{y^2}{4}+\\dfrac{x^2}{3}=1$的焦点为顶点, 以该椭圆长轴的端点为焦点的双曲线的标准方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014469": { + "id": "014469", + "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的渐近线是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014470": { + "id": "014470", + "content": "若椭圆$\\dfrac{x^2}{m^2}+\\dfrac{y^2}{4}=1$经过点$M(-2,\\sqrt{3})$, 则它的焦距为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014471": { + "id": "014471", + "content": "以双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$的中心为顶点, 且以该双曲线的右焦点为焦点的抛物线的标准方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014472": { + "id": "014472", + "content": "抛物线$y^2=x$上到焦点的距离等于$2$的点的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014473": { + "id": "014473", + "content": "在平面直角坐标系$x O y$中, 点$F$是椭圆$x^2+\\dfrac{y^2}{b^2}=1$ ($00$, $b>0$)的左、右焦点分别为$F_1,F_2$, 过$F_2$且斜率为$-\\dfrac{\\sqrt{5}}{2}$的直线与双曲线$C$的左支交于点$A$. 若$(\\overrightarrow{F_1F_2}+\\overrightarrow{F_1A}) \\cdot \\overrightarrow{F_2A}=0$, 则双曲线$C$的渐近线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014476": { + "id": "014476", + "content": "在平面直角坐标系$x O y$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{2}+y^2=1$, 过右焦点$F$作两条互相垂直的弦$AB$、$CD$, 设$AB$、$CD$中点分别为$M$、$N$.\\\\\n(1) 证明: 直线$MN$必过定点, 并求出此定点坐标;\\\\\n(2) 若弦$AB$、$CD$的斜率均存在, 求$\\triangle FMN$面积的最大值.\n%20", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课19", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014477": { + "id": "014477", + "content": "已知直线$y=x-1$与椭圆$\\dfrac{x^2}{2}+y^2=1$交于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014478": { + "id": "014478", + "content": "若过点$P(0,-1)$的直线$l$与抛物线$y^2=2 x$恰有一个公共点, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014479": { + "id": "014479", + "content": "若$P(x, y)$是抛物线$y^2=x$上一动点, 则点$P$到直线$y=x+3$的距离的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014480": { + "id": "014480", + "content": "已知$t \\in \\mathbf{R}$, 直线$y=m x+1$($m>0$)与双曲线$x^2-y^2=1$相交于$A$、$B$两点, 若线段$AB$的垂直平分线与$x$轴交于点$(t, 0)$, 则$t$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014481": { + "id": "014481", + "content": "设$k \\in \\mathbf{R}$, 已知直线$l: y=k x+1$, 双曲线$C: 3x^2-y^2=1$, 若直线$l$与双曲线$C$有两个公共点$A$、$B$.\\\\\n(1) 若$A$、$B$均在双曲线$C$的右支上, 求$k$的取值范围;\\\\\n(2) 若以线段$AB$为直径的圆过坐标原点, 求$k$的值;\\\\\n(3) 若$\\angle AOB$为钝角 (其中$O$为坐标原点), 求$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014482": { + "id": "014482", + "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$($a>0$), 双曲线$\\Gamma$右支上的任意两点$P_1$、$P_2$的坐标分别为$(x_1, y_1) 、(x_2, y_2)$, 且满足$x_1 x_2-y_1 y_2>0$恒成立, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014483": { + "id": "014483", + "content": "设椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>0$), $F_1$、$F_2$分别是椭圆$\\Gamma$的左、右焦点, 椭圆$\\Gamma$的离心率为$\\dfrac{\\sqrt{2}}{2}$, 直线$l$与椭圆$\\Gamma$交于不同的两点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆$\\Gamma$的右焦点$F_2$, $P$、$Q$是椭圆$\\Gamma$上两点, 四边形$ABQP$是菱形, 求直线$l$的方程;\\\\\n(3) 已知直线$l$与$x$轴的正半轴和$y$轴分别交于点$M$、$N$, 且$\\overrightarrow{AN}=\\lambda \\overrightarrow{AM}$, $\\overrightarrow{BN}=\\mu \\overrightarrow{BM}$, 若$\\lambda+\\mu=3$, 证明: 直线$l$过定点, 并求此定点的坐标.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014484": { + "id": "014484", + "content": "已知$F$是椭圆$C_1: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$与抛物线$C_2: y^2=2 p x$($p>0$)的一个共同焦点, $C_1$与$C_2$相交于$A$、$B$两点, 则线段$AB$的长等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3} \\sqrt{6}$}{$\\dfrac{4}{3} \\sqrt{6}$}{$\\dfrac{5}{3}$}{$\\dfrac{10}{3}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014485": { + "id": "014485", + "content": "若经过点$F_2(2,0)$的直线$l$与双曲线$x^2-\\dfrac{y^2}{3}=1$相交于$A$、$B$两点, 且$|AB|=6$, 则满足条件的直线$l$共有\\blank{50}条.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014486": { + "id": "014486", + "content": "设$t \\in \\mathbf{R}$, 已知双曲线$C: x^2-y^2=1$, 过点$T(t, 0)$作直线$l$和双曲线$C$交于$A$、$B$两点.\\\\\n(1) 求双曲线$C$的焦点和它的渐近线方程;\\\\\n(2) 若$t=0$, 点$A$在第一象限, $AH \\perp x$轴, 垂足为$H$, 求直线$BH$斜率的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014487": { + "id": "014487", + "content": "椭圆$x^2+4 y^2=4$的长轴的一个端点为$A$, 以$A$为直角顶点作一个内接于此椭圆的等腰直角三角形, 则此三角形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014488": { + "id": "014488", + "content": "已知抛物线$y^2=8 x$的动弦$AB$的长为$12$, 则弦$AB$的中点$M$到$y$轴的最短距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014489": { + "id": "014489", + "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$的左、右焦点分别为$F_1$、$F_2$, 设点$P$是椭圆$\\Gamma$上一点, 且位于$x$轴的上方, 若$\\triangle PF_1F_2$是等腰三角形, 则点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014490": { + "id": "014490", + "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($02$. 在平面直角坐标系$x O y$中, 已知点$F(2,0)$, 直线$l: x=t$, 曲线$\\Gamma: y^2=8 x$($0 \\leq x \\leq t$, $y \\geq 0$). 直线$l$与$x$轴交于点$A$、与$\\Gamma$交于点$B$. $P$、$Q$分别是曲线$\\Gamma$与线段$AB$上的动点.\\\\\n(1) 用$t$表示点$B$到点$F$的距离;\\\\\n(2) 设$t=3$, $|FQ|=2$, 线段$OQ$的中点在直线$FP$上, 求$\\triangle AQP$的面积;\\\\\n(3) 设$t=8$, 是否存在以$FP$、$FQ$为邻边的矩形$FPEQ$, 使得点$E$在$\\Gamma$上? 若存在, 求点$P$的坐标; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014493": { + "id": "014493", + "content": "已知点$P(0,1)$, 椭圆$\\dfrac{x^2}{4}+y^2=m$($m>1$)上两点$A, B$满足$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 则当$m=$\\blank{50}时, 点$B$横坐标的绝对值最大.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014494": { + "id": "014494", + "content": "设直线$x-3 y+m=0$($m \\neq 0$)与双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{b}=1$($b>0$)的两条渐近线分别交于$A$、$B$两点. 若点$P(m, 0)$满足$|PA|=|PB|$, 则实数$b$的值是\\blank{50}.\n%21", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课20", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014495": { + "id": "014495", + "content": "某科研团队在基地$O$点西侧、东侧$20$千米处设有$A$、$B$两站点, 经测量发现点$P$满足$|PA|-|PB|=20$千米, 且点$P$在$O$点北偏东$60^{\\circ}$处, 则$O$、$P$之间的距离为\\blank{50}千米.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014496": { + "id": "014496", + "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{4}+y^2=1$, $Q(0,1)$为$\\Gamma$的上顶点, 过原点的直线$l$与$\\Gamma$交于不同的两点$A$、$B$, 则$\\triangle ABQ$面积的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014497": { + "id": "014497", + "content": "过抛物线$y^2=2 x$上一点$P(2,2)$作两条直线分别交抛物线于$A(x_1, y_1)$、$B(x_2, y_2)$两点.若直线$PA$与$PB$的倾斜角互补, 则$\\dfrac{y_1+y_2}{2}$的值为\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2}$}{$-2$}{$2$}{无法确定}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014498": { + "id": "014498", + "content": "某科研团队在基地$O$点西侧、东侧$20$千米处设有$A$、$B$两站点, 南侧、北侧$15$千米处设有$C$、$D$两站点, 测量距离发现点$Q$满足$|QA|+|QB|=60$千米, $|QC|-|QD|=10$千米, 求$O$、$Q$之间的距离和$Q$点位置(结果精确到$1$千米, $1^{\\circ}$).", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014499": { + "id": "014499", + "content": "设$b>0$. 如图, 在平面直角坐标系$xOy$中, $A(x_A, y_A)$是双曲线$\\Gamma_1: \\dfrac{x^2}{4}-\\dfrac{y^2}{b^2}=1$和圆$\\Gamma_2: x^2+y^2=4+b^2$在第一象限内的交点, 曲线$\\Gamma$由$\\Gamma_1$中满足$|x|>x_A$的部分和$\\Gamma_2$中满足$|x|>x_A$的部分构成.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (0,0) circle ({2*sqrt(2)});\n\\draw [domain = -4:4,dashed] plot ({sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = -4:4,dashed] plot ({-sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({-sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({sqrt(\\x*\\x+4)},\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({-sqrt(\\x*\\x+4)},-\\x);\n\\draw [domain = {sqrt(2)}:4, thick] plot ({sqrt(\\x*\\x+4)},-\\x);\n\\draw [thick] ({sqrt(6)},{sqrt(2)}) coordinate (A1) arc (30:-30:{2*sqrt(2)}) coordinate (A2);\n\\draw [thick] ({-sqrt(6)},{-sqrt(2)}) coordinate (A3) arc (210:150:{2*sqrt(2)}) coordinate (A4);\n\\foreach \\i in {1,2,3,4}\n{\\filldraw [fill = white] (A\\i) circle (0.1);};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$x_A=\\sqrt{6}$, 求$b$的值;\\\\\n(2) 设$b=\\sqrt{5}$, $F_1$、$F_2$分别为$\\Gamma$与$x$轴的左、右两个交点. 第一象限内的点$P$也在$\\Gamma$上, 且$|PF_1|=8$, 求$\\angle F_1PF_2$的大小;\n(3) 若与$\\Gamma_1$的一条渐近线平行的直线$l$与$\\Gamma$在第一象限内恰有两个不同的公共点, 求$b^2$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014500": { + "id": "014500", + "content": "太平洋上有$A$、$B$两个岛屿, $B$岛在$A$岛正东$40$海里处. 经多年观察研究发现, 某种鱼群洄游的路线像一个椭圆, 其焦点恰好是$A$、$B$两岛. 曾有渔船在距$A$岛正西$20$海里处发现过鱼群. 某日, 研究人员在$A$、$B$两岛同时用声纳探测仪发出不同频率的探测信号(传播速度相同), $A$、$B$两岛收到鱼群反射信号的时间比为$5: 3$, 则鱼群此时与$A$岛的距离为\\blank{50}海里, 与$B$岛的距离为\\blank{50}海里.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014501": { + "id": "014501", + "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{9}+\\dfrac{y^2}{5}=1$的左、右焦点分别为$F_1$、$F_2$, 直线$y=k_1 x$($k_1 \\neq 0$)与$\\Gamma$相交于$A$、$B$两点.记$d$为$A$到直线$2 x+9=0$的距离, 当$k_1$变化时, 求证: $\\dfrac{|AF_1|}{d}$为定值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014502": { + "id": "014502", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $P(1,3)$、$Q(3,1)$、$M(-3,1)$、$N(0,2)$这四点中恰有三点在椭圆$C$上.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 点$E$是椭圆$C$上的一个动点, 求$\\triangle EMN$面积的最大值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014503": { + "id": "014503", + "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯口所在的圆面与反射镜的轴垂直, 灯泡位于抛物线的焦点处. 经灯口直径是$24$厘米, 灯深$10$厘米, 则灯泡与反射镜顶点的距离是\\blank{50}厘米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\filldraw (3.6, 0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10, 0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x, 2)/14.4}, \\x);\n\\draw (10, 0) ellipse (3 and 12);\n\\draw (0, 0) node [left] {$O$} coordinate (O) --++ (0, -14);\n\\draw (10, -12) --++ (0, -2);\n\\draw [<->] (0, -13) -- (10, -13) node [midway, below] {$10\\text{cm}$};\n\\draw (10, -12) --++ (5, 0) (10, 12) --++ (5, 0);\n\\draw [<->] (14, -12) -- (14, 12) node [midway, right] {\\rotatebox{90}{$24\\text{cm}$}};\n\\draw [dashed] (10, 0) --++ (0, -12);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014504": { + "id": "014504", + "content": "在椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$上任意一点$P, Q$与$P$关于$x$轴对称, 若$\\overrightarrow{F_1P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则$\\langle\\overrightarrow{F_1P}, \\overrightarrow{F_2Q}\\rangle$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014505": { + "id": "014505", + "content": "直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点, 与$x$轴正半轴不相交. 若$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 其中$O$为坐标原点, 则直线$l$过定点\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014506": { + "id": "014506", + "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}的一部分.\n\\fourch{椭圆}{双曲线}{圆}{抛物线}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014507": { + "id": "014507", + "content": "在相距$1800 \\text{m}$的两个观察站$A$、$B$先后听到远处传来的爆炸声, 已知$A$站听到的时间比$B$站早$5$秒, 声速是$340 \\text{m} / \\text{s}$. 建立适当的平面直角坐标系, 判断爆炸点$P$可能分布在什么样的轨迹上, 并求该轨迹的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014508": { + "id": "014508", + "content": "已知抛物线$y^2=x$上的动点$M(x_0, y_0)$, 过$M$分别作两条直线交抛物线于$P$、$Q$两点, 交直线$x=-1$于$A$、$B$两点.\\\\\n(1) 若点$M$纵坐标为$\\sqrt{2}$, 求$M$点与焦点的距离;\\\\\n(2) 若$P(1,1)$, $Q(1,-1)$, 求证: $y_A \\cdot y_B$为常数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014509": { + "id": "014509", + "content": "已知$F_1, F_2$是双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右焦点, 过$F_2$的直线交双曲线的右支于$A, B$两点, 且$|AF_1|=2|AF_2|$, $\\angle AF_1F_2=\\angle F_1BF_2$, 则在下列结论中, 正确结论的序号为\\blank{50}.\\\\\n\\textcircled{1} 双曲线$C$的离心率为$2$;\\\\\n\\textcircled{2} 双曲线$C$的一条渐近线的斜率为$\\sqrt{2}$;\\\\\n\\textcircled{3} 线段$AB$的长为$6 a$;\\\\\n\\textcircled{4} $\\triangle AF_1F_2$的面积为$\\sqrt{15} a^2$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014510": { + "id": "014510", + "content": "已知抛物线$y^2=x$.\\\\\n(1) 过抛物线焦点$F$的直线交抛物线于$A$、$B$两点, 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值(其中$O$为坐标原点);\\\\\n(2) 过抛物线上的一点$C(x_0, y_0)$, 分别作两条直线交抛物线于另外两点$P(x_P, y_P)$、$Q(x_Q, y_Q)$, 交直线$x=-1$于$A_1(-1,1)$、$B_1(-1,-1)$两点, 求证: $y_P \\cdot y_Q$为常数;\\\\\n(3) 已知点$D(1,1)$, 在抛物线上是否存在异于点$D$的两个不同点$M$、$N$, 使得$DM \\perp MN$? 若存在, 求$N$点纵坐标的取值范围; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课21", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",