From 212174f0bb06d281dd5b59483587156c44953073 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Thu, 3 Nov 2022 23:56:00 +0800 Subject: [PATCH] 20221103 night --- 工具/单元课时目标题目数据清点.ipynb | 6 +- 工具/寻找tex文件中未赋答案的题目.ipynb | 8 +- 工具/寻找阶段末尾空闲题号.ipynb | 2 +- 工具/批量添加题库字段数据.ipynb | 954 ++++- 工具/批量题号选题pdf生成.ipynb | 94 +- 工具/文本文件/metadata.txt | 3177 ++++++++++++++++- 工具/根据目标列表批量生成对应题目的字典.ipynb | 61 +- 工具/添加关联题目.ipynb | 12 +- 工具/添加题目到数据库.ipynb | 10 +- 工具/目标挂钩简要清点.ipynb | 66 +- 工具/讲义生成.ipynb | 31 +- 工具/识别题库中尚未标注的题目类型.ipynb | 161 +- 工具/课时目标及课时划分信息汇总.ipynb | 10 +- 工具/题号选题pdf生成.ipynb | 16 +- 文本处理工具/表格整理.ipynb | 6 +- 题库0.3/Problems.json | 1999 ++++++++--- 16 files changed, 5820 insertions(+), 793 deletions(-) diff --git a/工具/单元课时目标题目数据清点.ipynb b/工具/单元课时目标题目数据清点.ipynb index 9e97a3b7..214be97a 100644 --- a/工具/单元课时目标题目数据清点.ipynb +++ b/工具/单元课时目标题目数据清点.ipynb @@ -98,7 +98,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -112,12 +112,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/寻找tex文件中未赋答案的题目.ipynb b/工具/寻找tex文件中未赋答案的题目.ipynb index 252498ad..965f58ff 100644 --- a/工具/寻找tex文件中未赋答案的题目.ipynb +++ b/工具/寻找tex文件中未赋答案的题目.ipynb @@ -2,14 +2,14 @@ "cells": [ { "cell_type": "code", - "execution_count": 12, + "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "25_概率的概念及性质.tex\n" + "测验04.tex\n" ] } ], @@ -17,9 +17,9 @@ "import os,json,re\n", "\n", "#这里需要修改, 设定路径与选择文件\n", - "fileind = 25\n", + "fileind = 6\n", "# path = r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\第一轮复习讲义\"\n", - "path = r\"C:\\Users\\Weiye\\Documents\\wwy sync\\23届\\第一轮复习讲义\"\n", + "path = r\"C:\\Users\\Weiye\\Documents\\wwy sync\\23届\\上学期测验卷\"\n", "\n", "fileind = fileind - 1\n", "with open(\"../题库0.3/Problems.json\",\"r\",encoding = \"utf8\") as f:\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 dce691d8..b9183aaf 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 @@ -11,7 +11,7 @@ "text": [ "首个空闲id: 12030 , 直至 020000\n", "首个空闲id: 20227 , 直至 030000\n", - "首个空闲id: 30427 , 直至 999999\n" + "首个空闲id: 30451 , 直至 999999\n" ] } ], diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 6e5f35bb..97cd755c 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,37 +2,941 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 030276 , 字段: ans 中已修改数据: (1) 确定性现象; (2) 随机现象; (3) 确定性现象; (4) 随机现象\n", - "题号: 009736 , 字段: ans 中已修改数据: $\\Omega = \\{1,2,3,4,5,6\\}$, $A=\\{2,3,4,5,6\\}$, $B=\\{1,3,5\\}$, $C=\\{3,4,5,6\\}$\n", - "题号: 010539 , 字段: ans 中已修改数据: $\\Omega=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3,R_1R_2,R_1R_3,R_2R_3\\}$, $A=\\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3\\}$, $B=\\{W_1B_1,W_1B_2,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3\\}$. 或者$\\Omega' = \\{WB,WR,BB,BR,RR\\}$, $A'=\\{WB,WR\\}$, $B'=\\{WB,BB,BR\\}$\n", - "题号: 000218 , 字段: ans 中已修改数据: $\\dfrac 25$\n", - "题号: 000219 , 字段: ans 中已修改数据: $\\dfrac 3{10}$\n", - "题号: 000223 , 字段: ans 中已修改数据: (1) $\\Omega = \\{W_1B_1,W_1B_2,W_2B_1,W_2B_2,W_3B_1,W_3B_2,B_1B_2\\}$; (2) $\\dfrac 3{10}$; (3) $\\dfrac 35$\n", - "题号: 002662 , 字段: ans 中已修改数据: $\\dfrac 38$\n", - "题号: 002664 , 字段: ans 中已修改数据: $\\dfrac{16}{33}$\n", - "题号: 003660 , 字段: ans 中已修改数据: $\\dfrac 15$\n", - "题号: 010544 , 字段: ans 中已修改数据: B\n", - "题号: 000227 , 字段: ans 中已修改数据: $\\dfrac 23$\n", - "题号: 009745 , 字段: ans 中已修改数据: 证明略\n", - "题号: 010550 , 字段: ans 中已修改数据: $\\dfrac 35$\n", - "题号: 010540 , 字段: ans 中已修改数据: (1) $\\Omega = \\{BBGG,BGBG,BGGB,GGBB,GBGB,GBBG\\}$等; (2) $A=\\{BGBG,GBGB\\}$等; (3) $B=\\{BGBG,BGGB,GBGB,GBBG\\}$等\n", - "题号: 010535 , 字段: ans 中已修改数据: (1) 错误; (2) 错误; (3) 正确\n", - "题号: 000222 , 字段: ans 中已修改数据: $\\dfrac 3{10}$\n", - "题号: 003585 , 字段: ans 中已修改数据: $\\dfrac 3{10}$\n", - "题号: 003787 , 字段: ans 中已修改数据: B\n", - "题号: 004647 , 字段: ans 中已修改数据: $\\dfrac 34$\n", - "题号: 009349 , 字段: ans 中已修改数据: $\\dfrac 1{28}$, $\\dfrac 37$\n", - "题号: 000229 , 字段: ans 中已修改数据: $\\dfrac {13}{42}$\n", - "题号: 009741 , 字段: ans 中已修改数据: $P(A_0)=\\dfrac 13$, $P(A_1)=\\dfrac 12$, $P(A_2)=0$, $P(A_3)=\\dfrac 16$\n", - "题号: 010545 , 字段: ans 中已修改数据: $0.10$\n", - "题号: 009744 , 字段: ans 中已修改数据: 证明略\n" + "题号: 000333 , 字段: objs 中已有该数据: K0405002X\n", + "题号: 000333 , 字段: objs 中已有该数据: K0405003X\n", + "题号: 000336 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000353 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000370 , 字段: objs 中已有该数据: K0405002X\n", + "题号: 000370 , 字段: objs 中已有该数据: K0405003X\n", + "题号: 000376 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000403 , 字段: objs 中已有该数据: K0406004X\n", + "题号: 000408 , 字段: objs 中已有该数据: K0402005X\n", + "题号: 000430 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000443 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000455 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000457 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000475 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000488 , 字段: objs 中已有该数据: K0405002X\n", + "题号: 000488 , 字段: objs 中已有该数据: K0405003X\n", + "题号: 000499 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000514 , 字段: objs 中已有该数据: K0406003X\n", + "题号: 000516 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000527 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000534 , 字段: objs 中已有该数据: K0402001X\n", + "题号: 000534 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000543 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000546 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000548 , 字段: objs 中已有该数据: K0401003X\n", + "题号: 000548 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000562 , 字段: objs 中已有该数据: K0401003X\n", + "题号: 000562 , 字段: objs 中已有该数据: K0403001X\n", + "题号: 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K0405003X\n", + "题号: 000752 , 字段: objs 中已有该数据: K0401002X\n", + "题号: 000752 , 字段: objs 中已有该数据: K0403002X\n", + "题号: 000784 , 字段: objs 中已有该数据: K0403002X\n", + "题号: 000784 , 字段: objs 中已有该数据: K0404003X\n", + "题号: 000791 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000791 , 字段: objs 中已有该数据: K0405001X\n", + "题号: 000798 , 字段: objs 中已有该数据: K0403004X\n", + "题号: 000798 , 字段: objs 中已有该数据: K0404003X\n", + "题号: 000814 , 字段: objs 中已有该数据: K0403002X\n", + "题号: 000814 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000820 , 字段: objs 中已有该数据: K0403002X\n", + "题号: 000820 , 字段: objs 中已有该数据: K0404003X\n", + "题号: 000835 , 字段: objs 中已有该数据: K0403004X\n", + "题号: 000835 , 字段: objs 中已有该数据: K0119001B\n", + "题号: 000856 , 字段: objs 中已有该数据: KNONE\n", + "题号: 000867 , 字段: objs 中已有该数据: K0402005X\n", + "题号: 000867 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000875 , 字段: objs 中已有该数据: K0403006X\n", + "题号: 000875 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 000875 , 字段: objs 中已有该数据: K0403002X\n", + "题号: 000897 , 字段: objs 中已有该数据: K0401003X\n", + "题号: 000897 , 字段: objs 中已有该数据: K0826001X\n", + "题号: 000928 , 字段: objs 中已有该数据: K0405002X\n", + "题号: 000928 , 字段: objs 中已有该数据: K0405003X\n", + "题号: 000950 , 字段: objs 中已有该数据: K0407002X\n", + "题号: 000958 , 字段: objs 中已有该数据: K0405005X\n", + "题号: 000966 , 字段: objs 中已有该数据: K0405005X\n", + "题号: 001018 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001019 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001020 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001021 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001022 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001023 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001024 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001025 , 字段: objs 中已有该数据: K0408003X\n", + "题号: 001026 , 字段: objs 中已有该数据: K0409001X\n", + "题号: 001027 , 字段: objs 中已有该数据: K0409001X\n", + "题号: 001028 , 字段: objs 中已有该数据: K0409001X\n", + "题号: 001467 , 字段: objs 中已有该数据: K0409001X\n", + "题号: 001739 , 字段: objs 中已有该数据: K0406001X\n", + "题号: 001740 , 字段: objs 中已有该数据: K0406001X\n", + "题号: 001741 , 字段: objs 中已有该数据: K0406003X\n", + "题号: 001742 , 字段: objs 中已有该数据: K0406003X\n", + "题号: 001743 , 字段: objs 中已有该数据: K0406003X\n", + "题号: 001744 , 字段: objs 中已有该数据: K0406003X\n", + "题号: 001745 , 字段: objs 中已有该数据: K0402005X\n", + "题号: 001746 , 字段: objs 中已有该数据: K0402005X\n", + "题号: 001747 , 字段: objs 中已有该数据: K0401002X\n", + "题号: 001748 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001749 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001750 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001751 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001752 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001753 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001754 , 字段: objs 中已有该数据: K0401006X\n", + "题号: 001755 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001756 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001757 , 字段: objs 中已有该数据: K0401006X\n", + "题号: 001758 , 字段: objs 中已有该数据: K0401006X\n", + "题号: 001758 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001759 , 字段: objs 中已有该数据: K0401007X\n", + "题号: 001759 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 001760 , 字段: objs 中已有该数据: K0402004X\n", + "题号: 001761 , 字段: objs 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中已添加数据: K0404003X\n", + "题号: 010776 , 字段: objs 中已添加数据: K0407004X\n", + "题号: 010777 , 字段: objs 中已添加数据: K0406002X\n", + "题号: 010777 , 字段: objs 中已添加数据: K0406005X\n", + "题号: 010778 , 字段: objs 中已添加数据: K0406004X\n", + "题号: 010779 , 字段: objs 中已添加数据: K0407002X\n", + "题号: 010780 , 字段: objs 中已添加数据: K0407002X\n", + "题号: 010781 , 字段: objs 中已添加数据: K0403005X\n", + "题号: 010782 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010783 , 字段: objs 中已添加数据: K0408002X\n", + "题号: 010783 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010784 , 字段: objs 中已添加数据: K0408002X\n", + "题号: 010784 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010785 , 字段: objs 中已添加数据: K0408002X\n", + "题号: 010785 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010786 , 字段: objs 中已添加数据: K0408002X\n", + "题号: 010786 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010787 , 字段: objs 中已添加数据: K0408002X\n", + "题号: 010787 , 字段: objs 中已添加数据: K0408003X\n", + "题号: 010788 , 字段: objs 中已添加数据: K0409001X\n" ] } ], diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb index 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"\"K0232002X\":\"004013,009918,009919,010819,010820,010823,010824,030343,030345,030347,030348,030349,030356,030360,030361,030375,030385,030387,030408,030409,030410,030416,030422\",\n", - "\"K0232003X\":\"030415\",\n", - "\"K0232004X\":\"030372,030379\",\n", - "\"K0233001X\":\"004007\",\n", - "\"K0233002X\":\"010825,010827,030377\",\n", - "\"K0233003X\":\"004005,004009,009920,009921,010803,010822,010828,030350,030351,030362,030411,030412,030417,030425\",\n", - "\"K0234001X\":\"009922,010821\",\n", - "\"K0234003X\":\"004006,004014,005114,005115,005116,005121,005135,009923,009924,010826,030051,030352,030354,030363,030378,030413,030418,030419,030420,030424,030426\",\n", - "\"K0234004X\":\"030346,030376\",\n", - "\"K0234005X\":\"030353,030355,030358,030369,030386,030414,030421,030423\",\n", - "\"K0235001X\":\"004010,004011,004012,009925,009926,010829,010830,030357,030359,030364,030365,030366,030367,030368,030380,030382,030383,030384\"\n", + "\"K0406001X\":\"001739,001740\",\n", + "\"K0406002X\":\"010770,010771\",\n", + "\"K0406003X\":\"000514,001741,001742,001743,001744,003202,008400,008401,008402,008406,008408,009890\",\n", + "\"K0406004X\":\"000403,003211,003219,003630,010778\",\n", + "\"K0406005X\":\"000575,003210,003215,003218,003226,009892,010772,010773,010777\",\n", + "\"K0407001X\":\"001821\",\n", + "\"K0407002X\":\"000307,000320,000574,000950,001795,001803,001813,001818,001823,001824,001825,003205,003206,003213,003214,003273,003309,003310,003319,004179,006968,006969,006974,006982,006983,008404,008405,008407,008409,009894,009895,010774,010775,010779,010780\",\n", + "\"K0407003X\":\"001804,001805,001806,001807,001810,001811,001814,001815,001816,001819,001820,001822,003312,003322,006973\",\n", + "\"K0407004X\":\"001809,003600,010776\",\n", + "\"K0408002X\":\"006910,006912\",\n", + "\"K0408003X\":\"000315,000322,001018,001019,001020,001021,001022,001023,001024,001025,003274,003275,003276,003277,003278,003282,003284,003285,003286,003287,003289,004981,006909,006911,006913,006915,006916,006917,006918,006919,006920,006921,006922,006923,006924,006925,006926,006927,006928,006929,006930,006931,006932,006933,006934,006935,006936,006937,006938,006939,006940,006941,006942,006943,006944,006945,006946,006947,006948,006949,006950,006951,006952,006953,006956,006988,006989,006990,008457,008458,008459,008461,008462,008463,008464,008465,008466,008467,008468,008469,008534,009896,009897,009898,010782,010783,010784,010785,010786,010787\",\n", + "\"K0409001X\":\"000316,000324,000595,001026,001027,001028,001467,001817,003279,003280,003283,003288,006914,006954,006955,006957,006958,006959,006960,008460,008470,008471,008472,008473,008522,008523,008524,008525,008535,009899,009900,010788\",\n", + "\"K0409002X\":\"000323,003281,008474,008475,009901\"\n", "}\n", "\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件保存路径---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "directory = \"临时文件/批量生成题目/29_\"\n", + "directory = \"临时文件/批量生成题目/递推数列/\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", - "\n", + "if directory[-1] != \"/\":\n", + " directory += \"/\"\n", "\n", "\n", "#生成数码列表, 逗号分隔每个区块, 区块内部用:表示整数闭区间\n", @@ -223,7 +229,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -237,12 +243,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index ea4d3449..ebb61e37 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,73 +1,3152 @@ -ans +objs +000333 +K0405002X +K0405003X -030276 -(1) 确定性现象; (2) 随机现象; (3) 确定性现象; (4) 随机现象 -009736 -$\Omega = \{1,2,3,4,5,6\}$, $A=\{2,3,4,5,6\}$, $B=\{1,3,5\}$, $C=\{3,4,5,6\}$ +000336 +KNONE -010539 -$\Omega=\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3,R_1R_2,R_1R_3,R_2R_3\}$, $A=\{W_1B_1,W_1B_2,W_1R_1,W_1R_2,W_1R_3\}$, $B=\{W_1B_1,W_1B_2,B_1B_2,B_1R_1,B_1R_2,B_1R_3,B_2R_1,B_2R_2,B_2R_3\}$. 或者$\Omega' = \{WB,WR,BB,BR,RR\}$, $A'=\{WB,WR\}$, $B'=\{WB,BB,BR\}$ -000218 -$\dfrac 25$ +000353 +KNONE -000219 -$\dfrac 3{10}$ -000223 -(1) $\Omega = \{W_1B_1,W_1B_2,W_2B_1,W_2B_2,W_3B_1,W_3B_2,B_1B_2\}$; (2) $\dfrac 3{10}$; (3) $\dfrac 35$ +000370 +K0405002X +K0405003X -002662 -$\dfrac 38$ -002664 -$\dfrac{16}{33}$ +000376 +KNONE -003660 -$\dfrac 15$ -010544 -B +000403 +K0406004X -000227 -$\dfrac 23$ -009745 -证明略 +000408 +K0402005X -010550 -$\dfrac 35$ -010540 -(1) $\Omega = \{BBGG,BGBG,BGGB,GGBB,GBGB,GBBG\}$等; (2) $A=\{BGBG,GBGB\}$等; (3) $B=\{BGBG,BGGB,GBGB,GBBG\}$等 +000430 +KNONE -010535 -(1) 错误; (2) 错误; (3) 正确 -000222 -$\dfrac 3{10}$ +000443 +KNONE -003585 -$\dfrac 3{10}$ -003787 -B +000455 +KNONE -004647 -$\dfrac 34$ -009349 -$\dfrac 1{28}$, $\dfrac 37$ +000457 +KNONE -000229 -$\dfrac {13}{42}$ -009741 -$P(A_0)=\dfrac 13$, $P(A_1)=\dfrac 12$, $P(A_2)=0$, $P(A_3)=\dfrac 16$ +000475 +KNONE + + +000488 +K0405002X +K0405003X + + +000499 +KNONE + + +000514 +K0406003X + + +000516 +KNONE + + +000527 +KNONE + + +000534 +K0402001X +K0402004X + + +000543 +KNONE + + +000546 +KNONE + + +000548 +K0401003X +K0402004X + + +000562 +K0401003X +K0403001X + + +000573 +K0401004X +K0401003X + + +000574 +K0407002X + + +000575 +K0406005X +K0406004X + + +000583 +K0403001X +K0316002B + + +000588 +KNONE + + +000591 +KNONE + + +000595 +K0409001X +K0407002X + + +000599 +KNONE + + +000600 +K0819005X +K0401002X + + +000606 +K0405001X + + +000633 +K0405003X + + +000638 +K0405001X + + +000662 +K0405002X + + +000674 +K0405001X + + +000681 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"cell_type": "code", - "execution_count": 5, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# 输入目标列表\n", - "t = \"\"\"K0232001X \n", - "K0232002X \n", - "K0232003X \n", - "K0232004X \n", - "K0233001X \n", - "K0233002X \n", - "K0233003X \n", - "K0234001X \n", - "K0234002X \n", - "K0234003X \n", - "K0234004X \n", - "K0234005X \n", - "K0235001X\"\"\"" + "t = \"\"\"K0624001X\n", + "K0624002X\n", + "K0624003X\n", + "K0625001X\n", + "K0625002X\n", + "K0625003X\n", + "K0625004X\n", + "K0626001X\n", + "K0626002X\n", + "K0626003X\n", + "K0626004X\n", + "K0627001X\n", + "K0627002X\n", + "K0627003X\n", + "K0627004X\n", + "K0627005X\n", + "K0627006X\n", + "K0627007X\n", + "K0628001X\n", + "K0628002X\n", + "K0628003X\n", + "K0628004X\n", + "K0628005X\"\"\"" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "\"K0232001X\":\"030344,030374,030407\",\n", - "\"K0232002X\":\"004013,009918,009919,010819,010820,010823,010824,030343,030345,030347,030348,030349,030356,030360,030361,030375,030385,030387,030408,030409,030410,030416,030422\",\n", - "\"K0232003X\":\"030415\",\n", - "\"K0232004X\":\"030372,030379\",\n", - "\"K0233001X\":\"004007\",\n", - "\"K0233002X\":\"010825,010827,030377\",\n", - "\"K0233003X\":\"004005,004009,009920,009921,010803,010822,010828,030350,030351,030362,030411,030412,030417,030425\",\n", - "\"K0234001X\":\"009922,010821\",\n", - "\"K0234003X\":\"004006,004014,005114,005115,005116,005121,005135,009923,009924,010826,030051,030352,030354,030363,030378,030413,030418,030419,030420,030424,030426\",\n", - "\"K0234004X\":\"030346,030376\",\n", - "\"K0234005X\":\"030353,030355,030358,030369,030386,030414,030421,030423\",\n", - "\"K0235001X\":\"004010,004011,004012,009925,009926,010829,010830,030357,030359,030364,030365,030366,030367,030368,030380,030382,030383,030384\",\n" + "\"K0624001X\":\"000291\",\n", + "\"K0625001X\":\"000292\",\n", + "\"K0625004X\":\"000301\",\n" ] } ], @@ -91,7 +92,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -105,12 +106,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index dd716a1f..2aa41758 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id---\"\"\"\n", - "old_id = \"4656\"\n", - "new_id = \"30282\"\n", + "old_id = \"30353\"\n", + "new_id = \"30451\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\n", "old_id = old_id.zfill(6)\n", @@ -50,7 +50,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -64,12 +64,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, 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 b29d9b99..e7986b13 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 @@ -2,20 +2,20 @@ "cells": [ { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 30388\n", - "origin = \"高中数学教与学例题与习题\"\n", + "starting_id = 30449\n", + "origin = \"自拟题目\"\n", "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目4.tex\"\n", - "editor = \"20221029\\t王伟叶\"" + "editor = \"20221103\\t王伟叶\"" ] }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": {}, "outputs": [], "source": [ diff --git a/工具/目标挂钩简要清点.ipynb b/工具/目标挂钩简要清点.ipynb index 76e9ae5e..763ffcc9 100644 --- a/工具/目标挂钩简要清点.ipynb +++ b/工具/目标挂钩简要清点.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ @@ -24,22 +24,22 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "第一单元 . 总题数: 1411 , 完成对应题数: 1397\n", - "第二单元 . 总题数: 1987 , 完成对应题数: 1985\n", - "第三单元 . 总题数: 1946 , 完成对应题数: 390\n", - "第四单元 . 总题数: 1023 , 完成对应题数: 594\n", - "第五单元 . 总题数: 1236 , 完成对应题数: 301\n", - "第六单元 . 总题数: 882 , 完成对应题数: 254\n", + "第一单元 . 总题数: 1417 , 完成对应题数: 1397\n", + "第二单元 . 总题数: 2144 , 完成对应题数: 2121\n", + "第三单元 . 总题数: 1958 , 完成对应题数: 390\n", + "第四单元 . 总题数: 1030 , 完成对应题数: 770\n", + "第五单元 . 总题数: 1242 , 完成对应题数: 303\n", + "第六单元 . 总题数: 957 , 完成对应题数: 311\n", "第七单元 . 总题数: 1219 , 完成对应题数: 59\n", - "第八单元 . 总题数: 1122 , 完成对应题数: 169\n", - "第九单元 . 总题数: 156 , 完成对应题数: 23\n" + "第八单元 . 总题数: 1154 , 完成对应题数: 190\n", + "第九单元 . 总题数: 236 , 完成对应题数: 100\n" ] } ], @@ -48,6 +48,46 @@ " print(units[u],\". 总题数:\",count1[u],\", 完成对应题数:\",count2[u])\n" ] }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "11357" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sum(count1)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "12682" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "len(pro_dict)" + ] + }, { "cell_type": "code", "execution_count": null, @@ -58,7 +98,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -72,12 +112,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index b70b1bd0..1ecb4fa4 100644 --- a/工具/讲义生成.ipynb +++ b/工具/讲义生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 8, "metadata": {}, "outputs": [ { @@ -11,9 +11,11 @@ "text": [ "正在处理题块 1 .\n", "题块 1 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/三角向量复数立几易错题_教师_20221101.tex\n", + "正在处理题块 2 .\n", + "题块 2 处理完毕.\n", + "开始编译教师版本pdf文件: 临时文件/线上测验02_教师_20221103.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/三角向量复数立几易错题_学生_20221101.tex\n", + "开始编译学生版本pdf文件: 临时文件/线上测验02_学生_20221103.tex\n", "0\n" ] } @@ -26,34 +28,35 @@ "\"\"\"---设置模式结束---\"\"\"\n", "\n", "\"\"\"---设置模板文件名---\"\"\"\n", - "# template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", + "template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", "# template_file = \"模板文件/测验周末卷模板.tex\"\n", - "template_file = \"模板文件/日常选题讲义模板.tex\"\n", + "# template_file = \"模板文件/日常选题讲义模板.tex\"\n", "\"\"\"---设置模板文件名结束---\"\"\"\n", "\n", "\"\"\"---设置其他预处理替换命令---\"\"\"\n", "#2023届第一轮讲义更换标题\n", - "# exec_list = [(\"标题数字待处理\",\"29\"),(\"标题文字待处理\",\"导数的应用\")] \n", - "# enumi_mode = 0\n", + "exec_list = [(\"标题数字待处理\",\"29\"),(\"标题文字待处理\",\"导数的应用\")] \n", + "enumi_mode = 0\n", "\n", "#2023届测验卷与周末卷\n", - "# exec_list = [(\"标题替换\",\"测验05\")]\n", + "# exec_list = [(\"标题替换\",\"线上测验02\")]\n", "# enumi_mode = 1\n", "\n", "# 日常选题讲义\n", - "exec_list = [(\"标题文字待处理\",\"三角向量复数立几易错题\")] \n", + "# exec_list = [(\"标题文字待处理\",\"三角向量复数立几易错题\")] \n", "# enumi_mode = 0\n", "\n", "\"\"\"---其他预处理替换命令结束---\"\"\"\n", "\n", "\"\"\"---设置目标文件名---\"\"\"\n", - "destination_file = \"临时文件/三角向量复数立几易错题\"\n", + "# destination_file = \"临时文件/线上测验02\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"000179,000749,001895,002004,002013,003113,003135,003356,003505,003522,003528,003985,004539,006147,008101,010222,010453,030058,030101,030106,030107\"\n", + "\"30374,30372,30343,30408,30410,30411,30377,9922,30352,30451,9926\",\n", + "\"30379,30407,30415,30356,4009,30418,30419,30421,30369,30382\"\n", "]\n", "\"\"\"---设置题号数据结束---\"\"\"\n", "\n", @@ -204,7 +207,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -218,12 +221,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index f36b517c..4abc7ea5 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -9,143 +9,29 @@ "name": "stdout", "output_type": "stream", "text": [ - "030290 解答题\n", - "030291 解答题\n", - "030292 解答题\n", - "030293 解答题\n", - "030294 解答题\n", - "030295 解答题\n", - "030296 解答题\n", - "030297 解答题\n", - "030298 解答题\n", - "030299 解答题\n", - "030300 选择题\n", - "030301 解答题\n", - "030302 解答题\n", - "030303 解答题\n", - "030304 解答题\n", - "030305 解答题\n", - "030306 解答题\n", - "030307 选择题\n", - "030308 解答题\n", - "030309 解答题\n", - "030310 解答题\n", - "030311 解答题\n", - "030312 解答题\n", - "030313 解答题\n", - "030314 解答题\n", - "030315 解答题\n", - "030316 解答题\n", - "030317 解答题\n", - "030318 解答题\n", - "030319 解答题\n", - "030320 解答题\n", - "030321 解答题\n", - "030322 解答题\n", - "030323 解答题\n", - "030324 解答题\n", - "030325 解答题\n", - "030326 解答题\n", - "030327 解答题\n", - "030328 解答题\n", - "030329 解答题\n", - "030330 解答题\n", - "030331 解答题\n", - "030332 解答题\n", - "030333 解答题\n", - "030334 解答题\n", - "030335 解答题\n", - "030336 解答题\n", - "030337 解答题\n", - "030338 解答题\n", - "030339 解答题\n", - "030340 解答题\n", - "030341 解答题\n", - "030342 解答题\n", - "030343 解答题\n", - "030344 解答题\n", - "030345 解答题\n", - "030346 解答题\n", - "030347 解答题\n", - "030348 解答题\n", - "030349 解答题\n", - "030350 解答题\n", - "030351 解答题\n", - "030352 解答题\n", - "030353 解答题\n", - "030354 解答题\n", - "030355 解答题\n", - "030356 解答题\n", - "030357 解答题\n", - "030358 解答题\n", - "030359 解答题\n", - "030360 解答题\n", - "030361 解答题\n", - "030362 解答题\n", - "030363 解答题\n", - "030364 解答题\n", - "030365 解答题\n", - "030366 解答题\n", - "030367 解答题\n", - "030368 解答题\n", - "030369 解答题\n", - "030370 解答题\n", - "030371 解答题\n", - "030372 选择题\n", - "030373 解答题\n", - "030374 解答题\n", - "030375 解答题\n", - "030376 解答题\n", - "030377 解答题\n", - "030378 解答题\n", - "030379 选择题\n", - "030380 解答题\n", - "030381 解答题\n", - "030382 解答题\n", - "030383 解答题\n", - "030384 解答题\n", - "030385 解答题\n", - "030386 解答题\n", - "030387 解答题\n", - "030388 填空题\n", - "030389 填空题\n", - "030390 解答题\n", - "030391 解答题\n", - "030392 解答题\n", - "030393 选择题\n", - "030394 解答题\n", - "030395 解答题\n", - "030396 解答题\n", - "030397 填空题\n", - "030398 填空题\n", - "030399 填空题\n", - "030400 选择题\n", - "030401 填空题\n", - "030402 填空题\n", - "030403 解答题\n", - "030404 解答题\n", - "030405 解答题\n", - "030406 解答题\n", - "030407 填空题\n", - "030408 填空题\n", - "030409 解答题\n", - "030410 解答题\n", - "030411 解答题\n", - "030412 解答题\n", - "030413 解答题\n", - "030414 解答题\n", - "030415 填空题\n", - "030416 填空题\n", - "030417 填空题\n", - "030418 填空题\n", - "030419 填空题\n", - "030420 填空题\n", - "030421 填空题\n", - "030422 解答题\n", - "030423 解答题\n", - "030424 解答题\n", - "030425 解答题\n", - "030426 解答题\n" + "030428 填空题\n", + "030429 填空题\n", + "030430 填空题\n", + "030431 填空题\n", + "030432 填空题\n", + "030433 填空题\n", + "030434 填空题\n", + "030435 填空题\n", + "030436 填空题\n", + "030437 填空题\n", + "030438 填空题\n", + "030439 填空题\n", + "030440 选择题\n", + "030441 选择题\n", + "030442 选择题\n", + "030443 选择题\n", + "030444 解答题\n", + "030445 解答题\n", + "030446 解答题\n", + "030447 解答题\n", + "030448 解答题\n", + "030449 填空题\n", + "030450 填空题\n" ] } ], @@ -168,6 +54,7 @@ " print(p,\"填空题\")\n", " else:\n", " pro_dict[p][\"genre\"] = \"解答题\"\n", + " pro_dict[p][\"space\"] = \"12ex\"\n", " print(p,\"解答题\")\n", "\n", "#将修改结果写入json数据库\n", diff --git a/工具/课时目标及课时划分信息汇总.ipynb b/工具/课时目标及课时划分信息汇总.ipynb index 995d8f2d..fc0d491c 100644 --- a/工具/课时目标及课时划分信息汇总.ipynb +++ b/工具/课时目标及课时划分信息汇总.ipynb @@ -9,8 +9,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标表_20221014.tex\n", - "开始编译课时划分信息pdf文件: 临时文件/按课时分类目标及题目清单_20221014.tex\n" + "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标表_20221102.tex\n", + "开始编译课时划分信息pdf文件: 临时文件/按课时分类目标及题目清单_20221102.tex\n" ] }, { @@ -172,7 +172,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -186,12 +186,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, 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 b426fa18..ecf90d20 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": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/29预选_教师用_20221031.tex\n", + "开始编译教师版本pdf文件: 临时文件/赋能05_教师用_20221103.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/29预选_学生用_20221031.tex\n", + "开始编译学生版本pdf文件: 临时文件/赋能05_学生用_20221103.tex\n", "0\n" ] } @@ -26,7 +26,7 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n", "problems = r\"\"\"\n", - "30374,30407,9919,10824,30343,30347,30356,30385,30408,30410,30415,30372,30379,30377,4009,10803,10828,30411,9922,5116,30352,30418,30419,30353,30355,30369,30421,9926,30359,30382\n", + "000366,000367,000368,000369,030281,000371,000372,000373,000374,000375\n", "\n", "\n", "\n", @@ -35,7 +35,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/29预选\"\n", + "filename = \"临时文件/赋能05\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", @@ -176,7 +176,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -190,12 +190,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/文本处理工具/表格整理.ipynb b/文本处理工具/表格整理.ipynb index 61db38ca..c13373f1 100644 --- a/文本处理工具/表格整理.ipynb +++ b/文本处理工具/表格整理.ipynb @@ -68,7 +68,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -82,12 +82,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 11839a40..7e6a3134 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -5830,7 +5830,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "$0.95$", "solution": "", "duration": -1, "usages": [ @@ -5886,7 +5886,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "$\\dfrac 9{16}$", "solution": "", "duration": -1, "usages": [ @@ -5993,7 +5993,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "$6\\%$", "solution": "", "duration": -1, "usages": [ @@ -6075,7 +6075,7 @@ "第九单元" ], "genre": "选择题", - "ans": "", + "ans": "A", "solution": "", "duration": -1, "usages": [], @@ -6122,7 +6122,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) \\begin{tabular}{|c|c|c|}\n\\hline\n分组区间 & 频数 & 频率 \\\\ \\hline\n$[11.5,13.5)$ & $2$ & $\\dfrac 1{15}$ \\\\ \\hline\n$[13.5,15.5)$ & $8$ & $\\dfrac 4{15}$ \\\\ \\hline\n$[15.5,17.5)$ & $11$ & $\\dfrac {11}{30}$ \\\\ \\hline\n$[17.5,19.5)$ & $5$ & $\\dfrac 1{6}$ \\\\ \\hline\n$[11.5,13.5)$ & $4$ & $\\dfrac 2{15}$ \\\\ \\hline\n\\end{tabular}; (2) \\begin{tikzpicture}[>=latex,xscale = {8/15}, yscale = {5/0.25}]\n\\draw [->] (0,0) -- (0.25,0) -- (0.5,0.01) -- (0.75,-0.01) -- (1,0) -- (15,0) node [below] {距离};\n\\draw [->] (0,0) -- (0,0.25) node [left] {频率/组距};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\n in {1.5/{1/30}/11.5,3.5/{4/30}/13.5,5.5/{11/60}/15.5,7.5/{5/60}/17.5,9.5/{4/60}/19.5} {\\draw (\\i,0) --++ (0,\\j) --++ (2,0) --++ (0,{-\\j}); \\draw (\\i,0) node [below] {$\\n$};};\n\\draw (11.5,0) node [below] {$21.5$};\n\\draw [dashed] (0,{2/60}) node [left] {$1/30$} --++ (1.5,0);\n\\draw [dashed] (0,{2/15}) node [left] {$2/15$} --++ (3.5,0);\n\\draw [dashed] (0,{11/60}) node [left] {$11/60$} --++ (5.5,0);\n\\draw [dashed] (0,{1/12}) node [left] {$1/12$} --++ (7.5,0);\n\\draw [dashed] (0,{1/15}) node [left] {$1/15$} --++ (9.5,0);\n\\end{tikzpicture}", "solution": "", "duration": -1, "usages": [], @@ -6145,7 +6145,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "\\begin{tabular}{c|cccccccccccc}\n$6$ & $4$ & $7$\\\\\n$7$ & $0$ & $2$ & $4$ & $6$ & $6$ & $9$ \\\\\n$8$ & $0$ & $1$ & $2$ & $2$ & $3$ & $5$ & $6$ & $8$ \\\\\n$9$ & $1$ & $1$ & $2$ & $3$ & $3$ & $3$ & $5$ & $6$ & $6$ & $7$ & $7$ & $9$ \\\\\n$10$ & $0$ & $0$ & $2$ & $4$ & $6$ & $6$ & $7$ & $8$ & $8$ \\\\\n$11$ & $2$ & $2$ & $4$ & $6$ & $8$ & $9$ & $9$ \\\\\n$12$ & $2$ & $3$ & $5$ & $6$ & $8$\\\\\n$13$ & $3$\n\\end{tabular}", "solution": "", "duration": -1, "usages": [], @@ -43181,7 +43181,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\dfrac{\\sqrt{7}}5$", "solution": "", "duration": -1, "usages": [ @@ -44129,7 +44129,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\dfrac{5\\pi}{12}$", "solution": "", "duration": -1, "usages": [ @@ -44329,7 +44329,7 @@ "第六单元" ], "genre": "选择题", - "ans": "", + "ans": "A", "solution": "", "duration": -1, "usages": [ @@ -45553,7 +45553,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$8\\pi$", "solution": "", "duration": -1, "usages": [ @@ -45604,7 +45604,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\arctan \\dfrac 32$", "solution": "", "duration": -1, "usages": [ @@ -71575,7 +71575,7 @@ "030032" ], "remark": "", - "space": "" + "space": "12ex" }, "002748": { "id": "002748", @@ -93437,10 +93437,19 @@ "第五单元" ], "genre": "填空题", - "ans": "", + "ans": "$5-\\mathrm{i}$", "solution": "", "duration": -1, - "usages": [], + "usages": [ + "20221102\t2023届高三12班\t1.000", + "20221102\t2023届高三01班\t1.000", + "20221102\t2023届高三04班\t0.968", + "20221102\t2023届高三02班\t1.000", + "20221102\t2023届高三05班\t1.000", + "20221102\t2023届高三06班\t0.975", + "20221102\t2023届高三07班\t1.000", + "20221102\t2023届高三09班\t0.969" + ], "origin": "上海2019年秋季高考试题2", "edit": [ "20220701\t王伟叶" @@ -93531,10 +93540,19 @@ "第二单元" ], "genre": "填空题", - "ans": "", + "ans": "$-1$", "solution": "", "duration": -1, - "usages": [], + "usages": [ + "20221102\t2023届高三12班\t0.955", + "20221102\t2023届高三01班\t1.000", + "20221102\t2023届高三04班\t1.000", + "20221102\t2023届高三02班\t1.000", + "20221102\t2023届高三05班\t1.000", + "20221102\t2023届高三06班\t1.000", + "20221102\t2023届高三07班\t0.968", + "20221102\t2023届高三09班\t1.000" + ], "origin": "上海2019年秋季高考试题6", "edit": [ "20220701\t王伟叶" @@ -94049,10 +94067,19 @@ "第二单元" ], "genre": "填空题", - "ans": "", + "ans": "$-1$", "solution": "", "duration": -1, - "usages": [], + "usages": [ + "20221102\t2023届高三12班\t1.000", + "20221102\t2023届高三01班\t1.000", + "20221102\t2023届高三04班\t0.968", + "20221102\t2023届高三02班\t0.906", + "20221102\t2023届高三05班\t0.974", + "20221102\t2023届高三06班\t0.975", + "20221102\t2023届高三07班\t0.968", + "20221102\t2023届高三09班\t0.906" + ], "origin": "上海2018年秋季高考试题7", "edit": [ "20220701\t王伟叶" @@ -97032,7 +97059,7 @@ }, "003786": { "id": "003786", - "content": "若数列$\\{b_n\\}$为等比数列, 其前$n$项的和为$S_n$, 若对任意$n\\in \\mathbf{N}^*$, 点$(n,S_n)$均在函数$y=bx+r$($b>0, \\ b\\ne 1, \\ b,r$为常数)的图像上, 则$r=$\\bracket{20}.\n\\fourch{$0$}{$-1$}{$1$}{$2$}", + "content": "若数列$\\{b_n\\}$为等比数列, 其前$n$项的和为$S_n$, 若对任意$n\\in \\mathbf{N}^*$, 点$(n,S_n)$均在函数$y=b^x+r$($b>0, \\ b\\ne 1, \\ b,r$为常数)的图像上, 则$r=$\\bracket{20}.\n\\fourch{$0$}{$-1$}{$1$}{$2$}", "objs": [], "tags": [ "第四单元" @@ -97602,7 +97629,7 @@ "content": "若$(1-2x)^{2014}=a_0+a_1x+a_2x^2+\\cdots+a_{2014}x^{2014} \\ (x\\in \\mathbf{R})$, 则$\\dfrac{a_1}{2}+\\dfrac{a_2}{2^2}+\\cdots+\\dfrac{a_{2014}}{2^{2014}}$的值为\\blank{50}.", "objs": [], "tags": [ - "第四单元" + "第八单元" ], "genre": "填空题", "ans": "", @@ -98104,7 +98131,7 @@ }, "003834": { "id": "003834", - "content": "已知$\\{a_n\\}$为无穷等比数列, 数列$\\{b_n\\}$满足$b_1+b_2+\\cdots+b_n=\\dfrac{n}{n+1} \\ (n\\in \\mathbf{N}^*)$, 且$a_1+3b_2=2$, $\\displaystyle_{n\\to \\infty}(a_1+a_2+a_3+\\cdots+a_n)=\\dfrac 56$. \\\\\n(1) 求数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式;\\\\\n(2) 是否存在$m\\in \\mathbf{N}^*$, 使得当正整数$n\\ge m$时, 总有$a_n1$时, 和为$171\\pi$, 当$a=1$时, 和为$266\\pi$, 当$0=latex]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) -- (2,0) --++ (0,1) --++ (0.5,0) --++ (0,0.25) --++ (0.125,0) --++ (0,0.0625) --++ (0.03125,0);\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0405005X" + ], "tags": [ "第四单元" ], @@ -170928,7 +171094,9 @@ "006906": { "id": "006906", "content": "设扇形$AOB$的半径为$R$, 中心角为$\\theta$($0<\\theta <\\dfrac{\\pi }2$), 由$A$向半径$OB$作垂线$AB_1$, 由垂足$B_1$引弦$AB$的平行线交$OA$于点$A_1$, 再由$A_1$向$OB$作垂线$A_1B_2$, 由垂足$B_2$引弦$AB$的平行线交$OA$于点$A_2$(如图), 这样无限地继续下去, 在$OA$, $OB$上得到的点列$\\{A_n\\}$、$\\{B_n\\}$, 设$\\triangle ABB_1,\\triangle A_1B_1B_2,\\cdots,\\triangle A_nB_nB_{n+1},\\cdots$的面积为$S_1,S_2,\\cdots,S_{n+1},\\cdots$, 求$S=\\displaystyle \\lim_{n\\to \\infty} \\sum\\limits_{k=1}^nS_k$.\n\\begin{center}\n \\begin{tikzpicture}[scale = 5]\n \\draw (0,0) node [left] {$O$} coordinate (O);\n \\draw (1,0) node [right] {$A$} coordinate (A);\n \\draw (35:1) node [above] {$B$} coordinate (B);\n \\draw (O) -- (A) -- (B) -- cycle (A) arc (0:35:1);\n \\draw ($(O)!{cos(35)}!(B)$) node [above left] {$B_1$} coordinate (B1) ($(O)!{cos(35)}!(A)$) node [below] {$A_1$} coordinate (A1);\n \\draw ($(O)!{cos(35)}!(B1)$) node [above left] {$B_2$} coordinate (B2) ($(O)!{cos(35)}!(A1)$) node [below] {$A_2$} coordinate (A2);\n \\draw ($(O)!{cos(35)}!(B2)$) node [above left] {$B_3$} coordinate (B3) ($(O)!{cos(35)}!(A2)$) node [below] {$A_3$} coordinate (A3);\n \\draw (A) -- (B1) -- (A1) -- (B2) -- (A2) -- (B3) -- (A3);\n \\pic [draw] {angle = A--O--B}; \n \\draw (0.15,0) node [above] {$\\theta$};\n \\pic [draw, angle radius = 0.2cm] {right angle = A--B1--B};\n \\pic [draw, angle radius = 0.2cm] {right angle = A1--B2--B1};\n \\pic [draw, angle radius = 0.2cm] {right angle = A2--B3--B2}; \n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0405005X" + ], "tags": [ "第四单元" ], @@ -170949,7 +171117,9 @@ "006907": { "id": "006907", "content": "如图, 在Rt$\\triangle ABC$中排列着无限个正方形$S_1,S_2,S_3,S_4,\\cdots$, 且已知直角边$BC=a$, 这无限个正方形的面积之和正好是这个直角三角形面积的一半, 求另一直角边$AC$的长.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [left] {$C$} coordinate (C);\n \\draw (4,0) node [right] {$A$} coordinate (A);\n \\draw (0,3) node [left] {$B$} coordinate (B);\n \\draw (A) -- (C) -- (B) -- cycle;\n \\draw ({12/7},0) coordinate (X1) -- ({12/7},{12/7}) -- (0,{12/7});\n \\draw ({6/7},{6/7}) node {$S_1$};\n \\draw (X1) ++ ({12/7*4/7},0) coordinate (X2) --++ (0,{12/7*4/7}) --++ ({-12/7*4/7},0);\n \\draw (X1) ++ ({12/7*4/7/2},{12/7*4/7/2}) node {$S_2$};\n \\draw (X2) ++ ({12/7*4/7*4/7},0) coordinate (X3) --++ (0,{12/7*4/7*4/7}) --++ ({-12/7*4/7*4/7},0);\n \\draw (X2) ++ ({12/7*4/7*4/7/2},{12/7*4/7*4/7/2}) node {$S_3$};\n \\draw (X3) ++ ({12/7*4/7*4/7*4/7},0) coordinate (X3) --++ (0,{12/7*4/7*4/7*4/7}) --++ ({-12/7*4/7*4/7*4/7},0);\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0405005X" + ], "tags": [ "第四单元" ], @@ -170970,7 +171140,9 @@ "006908": { "id": "006908", "content": "在半径为$r$的球内作正方体, 然后在正方体内再作内切球, 在内切球内再作内接正方体, 然后再作它的内切球, 如此无限地作下去, 求所有这些球的表面积之和(包括半径为$r$的球).", - "objs": [], + "objs": [ + "K0405005X" + ], "tags": [ "第四单元" ], @@ -170991,7 +171163,9 @@ "006909": { "id": "006909", "content": "用数学归纳法证明: $1+2+\\cdots +2n=n(2n+1)$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171014,7 +171188,9 @@ "006910": { "id": "006910", "content": "用数学归纳法证明: $\\sqrt {1\\times 2}+\\sqrt {2\\times 3}+\\cdots +\\sqrt {n(n+1)}>\\dfrac{n(n+1)}2$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X" + ], "tags": [ "第四单元" ], @@ -171035,7 +171211,9 @@ "006911": { "id": "006911", "content": "用数学归纳法证明: $1\\times n+2(n-1)+\\cdots +n\\times 1=\\dfrac{n(n+1)(n+2)}6$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171056,7 +171234,9 @@ "006912": { "id": "006912", "content": "记$S_n=1+\\dfrac 12+\\dfrac 13+\\cdots +\\dfrac 1n$($n>1$, $n\\in \\mathbf{N}^*$), 求证: $S_{2^n}>1+\\dfrac n2$($n\\ge 2$, $n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X" + ], "tags": [ "第四单元" ], @@ -171077,7 +171257,9 @@ "006913": { "id": "006913", "content": "求证: $a^{n+1}+(a+1)^{2n-1}$($n\\in \\mathbf{N}^*$)能被$a^2+a+1$整除.", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171098,7 +171280,9 @@ "006914": { "id": "006914", "content": "已知数列$\\{a_n\\}$满足$a_1=a$, $a_{n+1}=\\dfrac 1{2-a_n}$.\\\\\n(1) 求$a_2,a_3,a_4$;\\\\\n(2) 推测通项$a_n$的表达式, 并用数学归纳法加以证明.", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -171121,7 +171305,10 @@ "006915": { "id": "006915", "content": "平面内有$n$个圆, 其中每两个圆都交于两点, 且无任何三个圆交于一点, 求证: 这$n$个圆将平面分成$n^2-n+2$个部分.", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171142,7 +171329,10 @@ "006916": { "id": "006916", "content": "利用数学归纳法证明``$1+a+a^2+\\cdots +a^{n+1}=\\dfrac{1-{a^{n+2}}}{1-a}$($a\\ne 1$, $n\\in \\mathbf{N}^*$)''时, 在验证$n=1$成立时, 左边应该是\\bracket{20}.\n\\fourch{$1$}{$1+a$}{$1+a+a^2$}{$1+a+a^2+a^3$}", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171165,7 +171355,10 @@ "006917": { "id": "006917", "content": "欲用数学归纳法证明``对于足够大的自然数$n$, 总有$2^n>n^3$'', 则验证不等式成立所取的第一个$n$值, 最小应当是\\bracket{20}.\n\\twoch{$1$}{大于$1$且小于$6$的某个自然数}{$10$}{大于$5$且小于$10$的某个自然数}", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171186,7 +171379,10 @@ "006918": { "id": "006918", "content": "利用数学归纳法证明``对任意偶数$n$, $a^n-b^n$能被$a+b$整除''时, 其第二步论证, 应该是\\bracket{20}.\n\\onech{假设$n=k$时命题成立, 再证$n=k+1$时命题也成立}{假设$n=2k$时命题成立, 再证$n=2k+1$时命题也成立}{假设$n=k$时命题成立, 再证$n=k+2$时命题也成立}{假设$n=2k$时命题成立, 再证$n=2(k+1)$时命题也成立}", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171207,7 +171403,10 @@ "006919": { "id": "006919", "content": "利用数学归纳法证明``$(n+1)(n+2)(n+3)\\cdots (n+n)=2^n\\times 1\\times 3\\times \\cdots \\times (2n-1)$($n\\in \\mathbf{N}^*$)''时, 从``$n=k$''变到``$n=k+1$''时, 左边应增添的因式是\\bracket{20}.\n\\fourch{$2k+1$}{$\\dfrac{2k+1}{k+1}$}{$\\dfrac{(2k+1)(2k+2)}{k+1}$}{$\\dfrac{2k+3}{k+1}$}", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171228,7 +171427,10 @@ "006920": { "id": "006920", "content": "利用数学归纳法证明``$\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots +\\dfrac 1{2n}>\\dfrac{13}{24}$($n\\ge 2$, $n\\in \\mathbf{N}^*$)''的过程中, 由``$n=k$''变到``$n=k+1$''时, 不等式左边的变化是\\bracket{20}.\n\\twoch{增加$\\dfrac 1{2(k+1)}$}{增加$\\dfrac 1{2k+1}$和$\\dfrac 1{2k+2}$}{增加$\\dfrac 1{2k+2}$并减少$\\dfrac 1{k+1}$}{增加$\\dfrac 1{2k+1}$和$\\dfrac 1{2k+2}$, 并减少$\\dfrac 1{k+1}$.}", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171249,7 +171451,10 @@ "006921": { "id": "006921", "content": "利用数学归纳法证明不等式``$\\sqrt {n^2+n}\\dfrac{13}{24}$($n\\ge 2$, $n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171554,7 +171798,10 @@ "006935": { "id": "006935", "content": "利用数学归纳法证明: $\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots +\\dfrac 1{3n+2}>1$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171575,7 +171822,10 @@ "006936": { "id": "006936", "content": "利用数学归纳法证明: $\\dfrac 1n+\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots +\\dfrac 1{n^2}>1$($n\\ge 2$, $n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171596,7 +171846,10 @@ "006937": { "id": "006937", "content": "利用数学归纳法证明: $1+\\dfrac 12+\\dfrac 13+\\cdots +\\dfrac 1{2^n-1}\\dfrac n{n+1}$($n\\ge 3$, $n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171701,7 +171966,10 @@ "006942": { "id": "006942", "content": "利用数学归纳法证明:$\\dfrac{2^n+4^n}2\\ge 3^n$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171722,7 +171990,10 @@ "006943": { "id": "006943", "content": "利用数学归纳法证明:$\\dfrac{a^n+b^n}2\\ge (\\dfrac{a+b}2)^n$($a,b\\in \\mathbf{R}^+$, $n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -171743,7 +172014,10 @@ "006944": { "id": "006944", "content": "利用数学归纳法证明:$(2n+1)(1-x)x^n<1-x^{2n+1}$($00$, $b>0$, 数列$\\{a_n\\}$满足$a_1=\\dfrac 12(a+\\dfrac ba),a_2=\\dfrac 12(a_1+\\dfrac b{a_1}),a_3=\\dfrac 12(a_2+\\dfrac b{a_2}),\\cdots,a_n=\\dfrac 12(a_{n-1}+\\dfrac b{a_{n-1}})$.\\\\\n(1) 求证: $\\dfrac{{a_n}-\\sqrt b}{{a_n}+\\sqrt b}=(\\dfrac{a-\\sqrt b}{a+\\sqrt b})^{2n}$;\\\\\n(2) 求$\\displaystyle \\lim_{n\\to \\infty} a_n$.", - "objs": [], + "objs": [ + "K0408003X", + "K0405001X" + ], "tags": [ "第四单元" ], @@ -172018,7 +172326,9 @@ "006957": { "id": "006957", "content": "已知正数数列$\\{a_n\\}$满足$2\\sqrt {S_n}=a_n+1$($n\\in \\mathbf{N}^*$).\\\\\n(1) 求$a_1,a_2,a_3$;\\\\\n(2) 猜测$a_n$的表达式, 并证明你的结论.", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -172039,7 +172349,9 @@ "006958": { "id": "006958", "content": "已知正数数列$\\{a_n\\}$的前$n$项和$S_n$满足$S_n=\\dfrac 12(a_n+\\dfrac 1{a_n})$, 求$a_n$.", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -172060,7 +172372,10 @@ "006959": { "id": "006959", "content": "已知正数数列$\\{a_n\\}$的前$n$项和$S_n=\\dfrac 12(a_n+\\dfrac 1{a_n})$.\\\\\n(1) 求$S_1,S_2,S_3$;\\\\\n(2) 写出$S_n$的表达式, 并证明你的结论;\\\\\n(3) 求$\\displaystyle \\lim_{n\\to \\infty} a_n$.", - "objs": [], + "objs": [ + "K0409001X", + "K0405001X" + ], "tags": [ "第四单元" ], @@ -172081,7 +172396,9 @@ "006960": { "id": "006960", "content": "已知正数数列$\\{a_n\\}$的前$n$项和为$S_n$, 且对任何自然数$n$, $a_n$与$2$的等差中项等于$S_n$与$2$的正的等比中项.\\\\\n(1) 写出数列$\\{a_n\\}$的前三项;\\\\\n(2)求数列$\\{a_n\\}$的通项公式(写出证明过程).", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -172249,7 +172566,9 @@ "006968": { "id": "006968", "content": "在数列$\\{a_n\\}$中, 已知$a_1=2$, $a_{n+1}=a_n+2n$, 则$a_{100}$等于\\bracket{20}.\n\\fourch{$9900$}{$9902$}{$9904$}{$10100$}", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -172270,7 +172589,9 @@ "006969": { "id": "006969", "content": "已知数列$\\{a_n\\}$满足$a_1=4$, $a_2=2$, $a_3=1$, 又数列$\\{a_{n+1}-a_n\\}$成等差数列, 则$a_n$等于\\bracket{20}.\n\\twoch{$n-3$}{$\\dfrac 12(n^3-8n^2+13n+2)$}{$\\dfrac 12(2n^3-17n^2+33n-10)$}{$\\dfrac 12(n^2-7n+14)$}", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -172291,7 +172612,10 @@ "006970": { "id": "006970", "content": "求数列$23, 2323, 232323, \\cdots$的通项公式$a_n$.", - "objs": [], + "objs": [ + "K0403005X", + "K0404003X" + ], "tags": [ "第四单元" ], @@ -172354,7 +172678,10 @@ "006973": { "id": "006973", "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2a_n+3$, 且$a_1\\ne -3$.\\\\\n(1) 求证: 数列$\\{a_n+3\\}$成等比数列;\\\\\n(2) 若$a_1=5$, 求$a_n$.", - "objs": [], + "objs": [ + "K0403003X", + "K0407003X" + ], "tags": [ "第四单元" ], @@ -172375,7 +172702,11 @@ "006974": { "id": "006974", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=S_n+(n+1)$.\\\\\n(1) 用$a_n$表示$a_{n+1}$;\\\\\n(2) 求证: 数列$\\{a_n+1\\}$成等比数列;\\\\\n(3) 求$a_n$和$S_n$.", - "objs": [], + "objs": [ + "K0407002X", + "K0403003X", + "K0404003X" + ], "tags": [ "第四单元" ], @@ -172398,7 +172729,9 @@ "006975": { "id": "006975", "content": "已知数列$\\{a_n\\}$满足$a_1=\\dfrac 56$, 且关于$x$的二次方程$a_{k-1}x^2-a_kx+1=0$的两根$\\alpha ,\\beta$满足$3\\alpha -\\alpha \\beta +3\\beta =1$($k\\ge 2$, $k\\in \\mathbf{N}^*$), 求证: 数列$\\{a_n-\\dfrac 12\\}$是等比数列, 并求出通项$a_n$.", - "objs": [], + "objs": [ + "K0403003X" + ], "tags": [ "第四单元" ], @@ -172419,7 +172752,10 @@ "006976": { "id": "006976", "content": "求和: $\\dfrac 12+(\\dfrac 13+\\dfrac 23)+(\\dfrac 14+\\dfrac 24+\\dfrac 34)+\\cdots +(\\dfrac 1{100}+\\dfrac 2{100}+\\dfrac 3{100}+\\cdots +\\dfrac{99}{100})$.", - "objs": [], + "objs": [ + "K0402004X", + "K0401007X" + ], "tags": [ "第四单元" ], @@ -172482,7 +172818,11 @@ "006979": { "id": "006979", "content": "已知数列$\\{a_n\\}$的前$n$项之和$S_n$与$a_n$之间满足$2S_n^2=2a_nS_n-a_n$($n\\ge 2$), 且$a_1=2$.\\\\\n(1) 求证: 数列$\\{\\dfrac 1{S_n}\\}$是以$2$为公差的等差数列;\\\\\n(2) 求$S_n$和$a_n$.", - "objs": [], + "objs": [ + "K0402002X", + "K0402004X", + "K0402005X" + ], "tags": [ "第四单元" ], @@ -172503,7 +172843,11 @@ "006980": { "id": "006980", "content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_n=\\dfrac{2S_n^2}{2{S_n}-1}$($n\\ge 2$).\\\\\n(1) 求证: $\\{\\dfrac 1{S_n}\\}$成等差数列;\\\\\n(2) 求通项$a_n$的表达式.", - "objs": [], + "objs": [ + "K0402002X", + "K0402004X", + "K0402005X" + ], "tags": [ "第四单元" ], @@ -172545,7 +172889,9 @@ "006982": { "id": "006982", "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=3^na_n$, 且$a_1=1$, 求$a_n$.", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -172566,7 +172912,9 @@ "006983": { "id": "006983", "content": "已知数列$\\{a_n\\}$满足$a_1=\\dfrac 12$, $S_n=n^2a_n$($S_n$是前$n$项之和), 求$a_n$.", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -172671,7 +173019,9 @@ "006988": { "id": "006988", "content": "已知各项为正数的数列$\\{a_n\\}$满足$a_n^2\\le a_n-a_{n+1}$, 求证$a_n<\\dfrac 1n$.", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -172692,7 +173042,9 @@ "006989": { "id": "006989", "content": "已知各项为正数的数列$\\{a_n\\}$满足$a_1+a_2+\\cdots +a_n=1$, 求证: $a_1^2+a_2^2+\\cdots +a_n^2\\ge \\dfrac 1n$($n\\ge 2$).", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -172713,7 +173065,9 @@ "006990": { "id": "006990", "content": "已知$\\dfrac 12\\le a_k\\le 1$($k\\in \\mathbf{N}^*$), 求证: $a_1a_2\\cdots a_n+(1-a_1)(1-a_2)\\cdots (1-a_n)\\ge \\dfrac 1{2^{n-1}}$.", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -206497,7 +206851,9 @@ "008476": { "id": "008476", "content": "下列数列中, 存在极限的数列是\\bracket{20}.\n\\twoch{$-2,0,-2,0,\\cdots ,(-1)^n-1,\\cdots$}{$\\dfrac 32,(\\dfrac 32)^2,(\\dfrac 32)^3,\\cdots ,(\\dfrac 32)^n,\\cdots$}{$\\dfrac 53,\\dfrac 96,\\dfrac{13}9,\\cdots ,\\dfrac{2n+3}{3n},\\cdots$}{$\\dfrac 12,\\dfrac 43,\\dfrac 94,\\cdots ,\\dfrac{n^2}{n+1},\\cdots$}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206518,7 +206874,9 @@ "008477": { "id": "008477", "content": "分别判断数列$\\{a_n\\}$是否有极限, 并说明理由.\\\\\n(1) $a_n=\\dfrac{n+1}n$;\\\\\n(2) $a_n=1+(-\\dfrac 12)^n$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206539,7 +206897,9 @@ "008478": { "id": "008478", "content": "判断下列关于数列极限的叙述是否正确, 并说明理由.\\\\\n(1) 因为$\\dfrac 1n>\\dfrac 1{2n}$, 所以$\\displaystyle\\lim_{n\\to\\infty}\\dfrac 1n>\\displaystyle\\lim_{n\\to\\infty}\\dfrac 1{2n}$;\\\\\n(2) 如果$\\displaystyle\\lim_{n\\to\\infty}a_n=A$, 那么对一切正整数$n$, 都有$a_n\\le A$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206560,7 +206920,9 @@ "008479": { "id": "008479", "content": "举出两个极限为$2$的数列.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206581,7 +206943,9 @@ "008480": { "id": "008480", "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=\\dfrac{3n+2}{2n-1}$, 填写下表, 并判断这个数列是否有极限.\n\\begin{center}\n\\begin{tabular}{|p{.07\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|}\n \\hline\n $n$ & $1$ & $2$ & $3$ & $4$ & $\\cdots$ & $10$ & $\\cdots$ & $50$ & $\\cdots$ & $100$ & $\\cdots$ & $1000$ & $\\cdots$ \\\\\n \\hline $a_n$ &&&&&&&&&&&&&\\\\ \\hline\n $|a_n-\\dfrac 32|$ &&&&&&&&&&&&& \\\\ \\hline \n\\end{tabular}\n\\end{center}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206606,7 +206970,9 @@ "008481": { "id": "008481", "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=\\dfrac{2n^2+1}{n^2}$, 填写下表, 并判断这个数列是否有极限.\n\\begin{center}\n \\begin{tabular}{|p{.07\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|}\n \\hline\n $n$ & $1$ & $2$ & $3$ & $4$ & $\\cdots$ & $10$ & $\\cdots$ & $50$ & $\\cdots$ & $100$ & $\\cdots$ & $1000$ & $\\cdots$ \\\\\n \\hline $a_n$ &&&&&&&&&&&&&\\\\ \\hline\n $|a_n-2|$ &&&&&&&&&&&&& \\\\ \\hline \n \\end{tabular}\n\\end{center}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206631,7 +206997,9 @@ "008482": { "id": "008482", "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=(-\\dfrac 34)^n$, 填写下表, 并判断这个数列是否有极限.\n\\begin{center}\n \\begin{tabular}{|p{.07\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|}\n \\hline\n $n$ & $1$ & $2$ & $3$ & $4$ & $\\cdots$ & $9$ & $10$ & $\\cdots$ & $19$ & $\\cdots$ \\\\\n \\hline $a_n$ &&&&&&&&&& \\\\ \\hline\n $|a_n-0|$ &&&&&&&&&& \\\\ \\hline \n \\end{tabular}\n\\end{center}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206656,7 +207024,9 @@ "008483": { "id": "008483", "content": "选择题:\n下列四个命题中, 正确的是\\bracket{20}.\n\\twoch{若$\\displaystyle\\lim_{n\\to\\infty}a_n^2=A^2$, 则$\\displaystyle\\lim_{n\\to\\infty}a_n=A$}{若$a_n>0$, $\\displaystyle\\lim_{n\\to\\infty}a_n=A$, 则$A>0$}{若$\\displaystyle\\lim_{n\\to\\infty}a_n=A$, 则$\\displaystyle\\lim_{n\\to\\infty}a_n^2=A^2$}{若$\\displaystyle\\lim_{n\\to\\infty}a_n=A$, 则$\\displaystyle\\lim_{n\\to\\infty}na_n=nA$}", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第四单元" ], @@ -206677,7 +207047,9 @@ "008484": { "id": "008484", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}(\\dfrac 1{n^2}+\\dfrac 2n-3)=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206713,7 +207085,9 @@ "008485": { "id": "008485", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{7n+4}{5-3n}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206750,7 +207124,9 @@ "008486": { "id": "008486", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}[(3-\\dfrac 2n)(5+\\dfrac 3n)]=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206776,7 +207152,9 @@ "008487": { "id": "008487", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2n^2+n-3}{3n^2+n-2}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206809,7 +207187,9 @@ "008488": { "id": "008488", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{(n+3)(n-4)}{(n-1)(3-2n)}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206846,7 +207226,9 @@ "008489": { "id": "008489", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{1+4+7+\\cdots +(3n-2)}{1+5+9+\\cdots +(4n-3)}$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206867,7 +207249,9 @@ "008490": { "id": "008490", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}(\\dfrac 1{n^2+1}+\\dfrac 2{n^2+1}+\\dfrac 3{n^2+1}+\\cdots +\\dfrac{2n}{n^2+1})$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206891,7 +207275,9 @@ "008491": { "id": "008491", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{1-2^n}{3^n+1}$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206916,7 +207302,9 @@ "008492": { "id": "008492", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{4-2^{n+1}}{2^n+2^{n+2}}$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206940,7 +207328,9 @@ "008493": { "id": "008493", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2^n-3^n}{2^{n+1}+3^{n+1}}$.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206966,7 +207356,9 @@ "008494": { "id": "008494", "content": "已知$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{an^2+bn-100}{3n-1}=2$, 求$a,b$的值.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -206987,7 +207379,9 @@ "008495": { "id": "008495", "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=\\dfrac{2n}{3n^2+1}$, 填写下表, 并判断这个数列是否有极限.\n\\begin{center}\n \\begin{tabular}{|p{.07\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|p{.04\\textwidth}<{\\centering}|}\n \\hline\n $n$ & $1$ & $2$ & $3$ & $4$ & $\\cdots$ & $10$ & $\\cdots$ & $50$ & $\\cdots$ & $100$ & $\\cdots$ & $1000$ & $\\cdots$ \\\\\n \\hline $a_n$ &&&&&&&&&&&&&\\\\ \\hline\n $|a_n-0|$ &&&&&&&&&&&&& \\\\ \\hline \n \\end{tabular}\n\\end{center}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207012,7 +207406,9 @@ "008496": { "id": "008496", "content": "已知数列$\\{a_n\\}$的通项公式是$a_n=\\dfrac{(-1)^n\\cdot n}{2n-1}$, 判断这个数列是否有极限.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207033,7 +207429,9 @@ "008497": { "id": "008497", "content": "已知数列$\\{a_n\\}$的极限为$A$.如果数列$\\{b_n\\}$满足$b_n=\\begin{cases}\n \\dfrac 23a_n, & n\\le 10^6, \\\\ 3a_n, & n>10^6, \\end{cases}$ 那么数列$\\{b_n\\}$的极限是\\bracket{20}.\n\\fourch{$A$}{$\\dfrac 23A$}{$3A$}{不存在}", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207054,7 +207452,9 @@ "008498": { "id": "008498", "content": "下列命题中, 正确的是\\bracket{20}.\n\\onech{若$\\displaystyle\\lim_{n\\to\\infty}(a_n\\cdot b_n)=a\\ne 0$, 则$\\displaystyle\\lim_{n\\to\\infty}a_n\\ne 0$且$\\displaystyle\\lim_{n\\to\\infty}b_n\\ne 0$}{若$\\displaystyle\\lim_{n\\to\\infty}(a_n\\cdot b_n)=0$, 则$\\displaystyle\\lim_{n\\to\\infty}a_n=0$或$\\displaystyle\\lim_{n\\to\\infty}b_n=0$}{若无穷数列$\\{a_n\\}$有极限, 且它的前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty}S_n=\\displaystyle\\lim_{n\\to\\infty}a_1+\\displaystyle\\lim_{n\\to\\infty}a_2+\\cdots$ $+\\displaystyle\\lim_{n\\to\\infty}a_n$}{若无穷数列$\\{a_n\\}$有极限$A$, 则$\\displaystyle\\lim_{n\\to\\infty}a_n=\\displaystyle\\lim_{n\\to\\infty}a_{n+1}$}", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第四单元" ], @@ -207075,7 +207475,9 @@ "008499": { "id": "008499", "content": "若数列$\\{a_n\\}$的通项是$a_n=\\dfrac 1{n(n+1)}$, 则$\\displaystyle\\lim_{n\\to\\infty}(a_1+n^2a_n)=$\\blank{50}.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第四单元" ], @@ -207096,7 +207498,9 @@ "008500": { "id": "008500", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{3^n-4^n}{3^{n+1}+4^{n+1}}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207126,7 +207530,9 @@ "008501": { "id": "008501", "content": "若$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2^n}{2^{n+1}+a^n}=\\dfrac 12$, 则实数$a$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207150,7 +207556,9 @@ "008502": { "id": "008502", "content": "是否存在无穷数列$\\{a_n\\}$、$\\{b_n\\}$, 满足条件: $\\displaystyle\\lim_{n\\to\\infty}a_n$存在, $\\displaystyle\\lim_{n\\to\\infty}b_n$不存在, 而$\\displaystyle\\lim_{n\\to\\infty}(a_n\\cdot b_n)$存在? 若存在, 请举例.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207171,7 +207579,11 @@ "008503": { "id": "008503", "content": "化下列小数为分数.\\\\\n(1) $0.\\dot4$;\\\\\n(2) $0.\\dot3\\dot8$.", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X", + "K0405004X" + ], "tags": [ "第四单元" ], @@ -207192,7 +207604,9 @@ "008504": { "id": "008504", "content": "求无穷等比数列$\\{(-1)^{n-1}\\cdot \\dfrac 1{3^{n-1}}\\}$各项的和.", - "objs": [], + "objs": [ + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207213,7 +207627,10 @@ "008505": { "id": "008505", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{1+\\dfrac 13+\\dfrac 19+\\cdots +\\dfrac 1{3^{n-1}}}{1+\\dfrac 12+\\dfrac 14+\\cdots +\\dfrac 1{2^{n-1}}}$.", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207234,7 +207651,9 @@ "008506": { "id": "008506", "content": "已知数列$\\{a_n\\}$是无穷等比数列, 且$a_1+a_2+a_3+\\cdots +a_n+\\cdots =\\dfrac 1{a_1}$, 求实数$a_1$的取值范围.", - "objs": [], + "objs": [ + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207255,7 +207674,10 @@ "008507": { "id": "008507", "content": "如图, 在边长为$a$的正方形$ABCD$的四条边上分别取点$A_1B_1C_1D_1$, 使四边形$A_1B_1C_1D_1$仍为正方形, 且$AA_1=\\dfrac 13a$; 再作正方形$A_2B_2C_2D_2$, 使顶点$A_2B_2C_2D_2$分别在正方形$A_1B_1C_1D_1$的四条边上, 且$A_1A_2=\\dfrac 13A_1B_1$; 然后用同样的方法如此无限继续下去.求所有这些正方形的面积之和.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,3) node [above left] {$A$} coordinate (A) -- (0,0) node [below left] {$B$} coordinate (B) -- (3,0) node [below right] {$C$} coordinate (C) -- (3,3) node [above right] {$D$} coordinate (D) -- cycle;\n \\draw ($(A)!{1/3}!(B)$) node [left] {$A_1$} coordinate (A1) -- ($(B)!{1/3}!(C)$) node [below] {$B_1$} coordinate (B1) -- ($(C)!{1/3}!(D)$) node [right] {$C_1$} coordinate (C1) -- ($(D)!{1/3}!(A)$) node [above] {$D_1$} coordinate (D1) -- cycle;\n \\draw ($(A1)!{1/3}!(B1)$) node [right] {$A_2$} coordinate (A2) -- ($(B1)!{1/3}!(C1)$) node [above] {$B_2$} coordinate (B2) -- ($(C1)!{1/3}!(D1)$) node [left] {$C_2$} coordinate (C2) -- ($(D1)!{1/3}!(A1)$) node [below] {$D_2$} coordinate (D2) -- cycle;\n \\draw (1.5,1.5) node {$\\cdots$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207276,7 +207698,10 @@ "008508": { "id": "008508", "content": "如图, 直线$m$: $3x+4y-4=0$与以$O_1,O_2,\\cdots, O_n,\\cdots$为圆心, 且依次外切的半圆都相切, 其中半圆$O_1$与$y$轴相切, 半圆圆心都在$x$轴的正半轴上, 半径分别为$r_1,r_2,\\cdots, r_n,\\cdots$, 求所有半圆弧长的总和$L$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 3]\n \\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n \\draw [->] (0,-0.2) -- (0,1.2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = -0.2:1.45] plot (\\x,{1-3*\\x/4});\n \\draw (0,0) arc (180:0:0.5) arc (180:0:{0.5/4}) arc (180:0:{0.5/16});\n \\filldraw (0.5,0) circle (0.01) node [below] {$O_1$} (1.125,0) circle (0.01) node [below] {$O_2$} (1.25+0.5/16,0) circle (0.01) node [below] {$O_3$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207297,7 +207722,10 @@ "008509": { "id": "008509", "content": "从一点$O$顺次引出八条射线$OA,OB,OC,OD,OE,OF,OG,OH$, 其中每相邻两条射线的夹角都是$45^{\\circ}$.在$OA$上取$OA=a$, 由$A$作$OB$的垂线$AA_1$, $A_1$是垂足; 由点$A_1$作$OC$的垂线$A_1A_2$, $A_2$是垂足, 由点$A_2$作$OD$的垂线$A_2A_3$, $A_3$是垂足; 然后用同样的方法如此无限继续下去.求所得折线$A_1A_2A_3A_4\\cdots$的长度.", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207318,7 +207746,10 @@ "008510": { "id": "008510", "content": "已知$S_n=\\dfrac 15+\\dfrac 2{5^2}+\\dfrac 1{5^3}+\\dfrac 2{5^4}+\\cdots +\\dfrac 1{5^{2n-1}}+\\dfrac 2{5^{2n}}$($n\\in \\mathbf{N}^*$), 求$\\displaystyle\\lim_{n\\to\\infty}S_n$.", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207342,7 +207773,9 @@ "008511": { "id": "008511", "content": "已知无穷等比数列$\\{\\dfrac 1{2^{n-1}}\\cos ^{n-1}\\theta\\}$的各项和等于$\\dfrac 43$, 其中$-\\dfrac{\\pi }2<\\theta <\\dfrac{\\pi }2$, 求$\\theta$的值.", - "objs": [], + "objs": [ + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207363,7 +207796,10 @@ "008512": { "id": "008512", "content": "已知正方形的边长为$a$, 作正方形的内切圆, 在此内切圆内作新的内接正方形, 这样一直无限地继续下去.\\\\\n(1) 求所有这些内切圆周长的和;\\\\\n(2) 求所有这些内切圆面积的和.", - "objs": [], + "objs": [ + "K0405005X", + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207384,7 +207820,9 @@ "008513": { "id": "008513", "content": "对于数列$\\dfrac 12,\\dfrac 14,\\cdots ,\\dfrac 1{2^n},\\cdots$, 试从其中找出无限项构成一个新的等比数列, 使新数列的各项和为$\\dfrac 17$, 并求新数列的首项与公比.", - "objs": [], + "objs": [ + "K0405003X" + ], "tags": [ "第四单元" ], @@ -207407,7 +207845,9 @@ "008514": { "id": "008514", "content": "已知数列$\\{a_n\\}$是等差数列, 数列$\\{b_n\\}$分别满足下列各式, 其中数列$\\{b_n\\}$必为等差数列的是\\bracket{20}.\n\\fourch{$b_n=|a_n|$}{$b_n=a_n^2$}{$b_n=\\dfrac 1{a_n}$}{$b_n=-\\dfrac{a_n}2$}", - "objs": [], + "objs": [ + "K0402002X" + ], "tags": [ "第四单元" ], @@ -207428,7 +207868,9 @@ "008515": { "id": "008515", "content": "如果数列$\\{a_n\\}$是一个以$q$为公比的等比数列, $b_n=-2a_n$, 那么数列$\\{b_n\\}$是 \\bracket{20}.\n\\twoch{以$q$为公比的等比数列}{以$-q$为公比的等比数列}{以$2q$为公比的等比数列}{以$-2q$为公比的等比数列}", - "objs": [], + "objs": [ + "K0403003X" + ], "tags": [ "第四单元" ], @@ -207449,7 +207891,9 @@ "008516": { "id": "008516", "content": "在等差数列$\\{a_n\\}$中, 已知公差$d=\\dfrac 12$, 且$a_1+a_3+a_5+\\cdots +a_{99}=60$, 求$a_1+a_2+a_3+\\cdots +a_{99}+a_{100}$的值.", - "objs": [], + "objs": [ + "K0401007X" + ], "tags": [ "第四单元" ], @@ -207472,7 +207916,9 @@ "008517": { "id": "008517", "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n=t\\cdot n^2+(t-9)n+t-\\dfrac 32$($t$为常数), 求数列$\\{a_n\\}$的通项公式.", - "objs": [], + "objs": [ + "K0402006X" + ], "tags": [ "第四单元" ], @@ -207493,7 +207939,10 @@ "008518": { "id": "008518", "content": "图中的离散点是$(n,a_n)$的图像, 其中$n\\in \\mathbf{N}^*$, 如$(1,4)$是第$1$点, 表示$a_1=4$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,yscale = 0.2]\n \\foreach \\i in {1,2,3,4}\n {\n \\draw [gray!50] (\\i,0) -- (\\i,20);\n \\draw [gray!50] (0,{\\i*5}) -- (4,{\\i*5});\n \\filldraw (\\i,{5*\\i-1}) ellipse (0.05 and 0.25);\n \\draw (\\i,0) node [below] {$\\i$};\n };\n \\draw [->] (0,0) -- (5,0) node [below] {$n$};\n \\draw [->] (0,0) -- (0,22) node [left] {$a_n$};\n \\draw (0,0) node [below left] {$O$};\n \\foreach \\i in {5,10,15,20}\n {\n \\draw (0,\\i) node [left] {$\\i$};\n };\n \\draw (1,4) node [right] {$(1,4)$};\n \\draw (2,9) node [right] {$(2,9)$};\n \\draw (3,14) node [right] {$(3,14)$};\n \\draw (4,19) node [right] {$(4,19)$};\n \\end{tikzpicture}\n\\end{center}\n(1) 写出数列$\\{a_n\\}$的一个通项公式;\\\\\n(2) 计算从第$1$点起的前$46$个点的纵坐标之和.", - "objs": [], + "objs": [ + "K0401007X", + "K0402004X" + ], "tags": [ "第四单元" ], @@ -207684,7 +208133,9 @@ "008526": { "id": "008526", "content": "若$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2^n}{2^{n+1}+a^n}=0$, 则实数$a$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207707,7 +208158,9 @@ "008527": { "id": "008527", "content": "若$a>0$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{3^n-a^n}{3^{n+1}+a^{n+1}}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207728,7 +208181,10 @@ "008528": { "id": "008528", "content": "已知数列$\\{a_n\\}$是等差数列, 且$a_1\\ne 0$, $S_n$为这个数列的前$n$项和.\\\\\n(1) 求$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{na_n}{S_n}$;\\\\\n(2) 求$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{S_n+S_{n+1}}{S_n+S_{n-1}}$.", - "objs": [], + "objs": [ + "K0405001X", + "K0402004X" + ], "tags": [ "第四单元" ], @@ -207749,7 +208205,10 @@ "008529": { "id": "008529", "content": "已知等比数列$\\{a_n\\}$的首项为$1$, 公比为$q(q>0)$, 它的前$n$项和为$S_n$, 且$T_n=\\dfrac{S_n}{S_{n+1}}$, 求$\\displaystyle\\lim_{n\\to\\infty}T_n$的值.", - "objs": [], + "objs": [ + "K0404003X", + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207770,7 +208229,11 @@ "008530": { "id": "008530", "content": "已知数列$\\{\\log _3a_n\\}$是等差数列, 且$\\log _3a_1+\\log _3a_2+\\cdots +\\log _3a_{10}=10$, 求$a_5\\cdot a_6$.", - "objs": [], + "objs": [ + "K0402001X", + "K0204004B", + "K0205001B" + ], "tags": [ "第四单元" ], @@ -207856,7 +208319,10 @@ "008534": { "id": "008534", "content": "用数学归纳法证明:\n$1-\\dfrac 12+\\dfrac 13-\\dfrac 14+\\cdots +\\dfrac 1{2n-1}-\\dfrac 1{2n}=\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots +\\dfrac 1{2n}$($n\\in \\mathbf{N}^*$).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -207883,7 +208349,9 @@ "008535": { "id": "008535", "content": "依次计算数列: $(1-\\dfrac 14),(1-\\dfrac 14)(1-\\dfrac 19),(1-\\dfrac 14)(1-\\dfrac 19)(1-\\dfrac 1{16}),(1-\\dfrac 14)(1-\\dfrac 19)$ $(1-\\dfrac 1{16})(1-\\dfrac 1{25}),\\cdots$的前$4$项的值, 由此猜想$(1-\\dfrac 14)(1-\\dfrac 19)(1-\\dfrac 1{16})(1-\\dfrac 1{25})\\cdots$ $[1-\\dfrac 1{(n+1)^2}]$($n\\in \\mathbf{N}^*$)的结果, 并用数学归纳法加以证明.", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -207904,7 +208372,9 @@ "008536": { "id": "008536", "content": "计算: $\\displaystyle\\lim_{n\\to\\infty}(\\dfrac 1{n^2+1}+\\dfrac 2{n^2+1}+\\dfrac 3{n^2+1}+\\cdots +\\dfrac{2k}{n^2+1})$(其中$k$为与$n$无关的正整数).", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207927,7 +208397,9 @@ "008537": { "id": "008537", "content": "$\\displaystyle\\lim_{n\\to\\infty}n^2(\\dfrac kn-\\dfrac 1{n+1}-\\dfrac 1{n+2}-\\cdots -\\dfrac 1{n+k})$(其中$k$为与$n$无关的正整数).", - "objs": [], + "objs": [ + "K0405001X" + ], "tags": [ "第四单元" ], @@ -207948,7 +208420,9 @@ "008538": { "id": "008538", "content": "已知$a=0.4\\dot1\\dot8$, 若将$a$写成最简分数$\\dfrac mn$, 求$n-m$的值.", - "objs": [], + "objs": [ + "K0405004X" + ], "tags": [ "第四单元" ], @@ -225692,7 +226166,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "$0.97\\dot{3}$", "solution": "", "duration": -1, "usages": [ @@ -234609,7 +235083,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "约$6.7\\times 10^4$条", "solution": "", "duration": -1, "usages": [], @@ -234657,7 +235131,7 @@ "第八单元" ], "genre": "填空题", - "ans": "", + "ans": "$0.12$", "solution": "", "duration": -1, "usages": [ @@ -234686,7 +235160,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 证明略; (2) 证明略", "solution": "", "duration": -1, "usages": [], @@ -234711,7 +235185,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $pq$; (2) $1-p-q+pq$; (3) $p+q-pq$; (4) $1-pq$", "solution": "", "duration": -1, "usages": [], @@ -237939,7 +238413,9 @@ "009902": { "id": "009902", "content": "在计算$\\sqrt 2$的巴比伦算法中, 若选取初值$x_1=-2$, 通过计算器操作, 写出迭代序列的前$5$项.", - "objs": [], + "objs": [ + "K0410001X" + ], "tags": [ "第四单元" ], @@ -237960,7 +238436,9 @@ "009903": { "id": "009903", "content": "选取初值$x_1=-2$, 利用递推公式$x_{n+1}=1+ \\dfrac1{\nx_n+1}$, 通过计算器操作, 写出迭代序列的前$8$项.", - "objs": [], + "objs": [ + "K0410002X" + ], "tags": [ "第四单元" ], @@ -237981,7 +238459,9 @@ "009904": { "id": "009904", "content": "仿照计算$\\sqrt 2$的巴比伦算法, 构造计算$\\sqrt 3$的迭代算法的递推公式, 并选取初值$x_1=1$, 通过计算器操作, 列出该迭代序列的前$5$项.", - "objs": [], + "objs": [ + "K0410002X" + ], "tags": [ "第四单元" ], @@ -240057,7 +240537,7 @@ "第六单元" ], "genre": "选择题", - "ans": "", + "ans": "D", "solution": "", "duration": -1, "usages": [ @@ -240103,7 +240583,7 @@ "第六单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $1$; (2) $\\arctan\\dfrac{\\sqrt{39}}{13}$", "solution": "", "duration": -1, "usages": [ @@ -245283,7 +245763,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "010211": { "id": "010211", @@ -250982,7 +251462,7 @@ "第六单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [ @@ -251343,7 +251823,7 @@ "第六单元" ], "genre": "选择题", - "ans": "", + "ans": "B", "solution": "", "duration": -1, "usages": [ @@ -251431,7 +251911,7 @@ "第六单元" ], "genre": "选择题", - "ans": "", + "ans": "D", "solution": "", "duration": -1, "usages": [ @@ -251566,7 +252046,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$4\\pi$", "solution": "", "duration": -1, "usages": [ @@ -251676,7 +252156,7 @@ "第六单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\arctan \\sqrt{5}$", "solution": "", "duration": -1, "usages": [ @@ -251827,13 +252307,13 @@ }, "010508": { "id": "010508", - "content": "如图, 已知三棱锥$PABC$中, $PA$垂直于平面$ABC$,$ AB\\perp BC$, $PA=4$, $AB=3$, $AC=5$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,4,0) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,-4) node [right] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (C) (P) -- (B);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A$到平面$PBC$的距离;\\\\\n(2) 求三棱锥$PABC$的表面积.", + "content": "如图, 已知三棱锥$P-ABC$中, $PA$垂直于平面$ABC$,$ AB\\perp BC$, $PA=4$, $AB=3$, $AC=5$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,4,0) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,-4) node [right] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (C) (P) -- (B);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A$到平面$PBC$的距离;\\\\\n(2) 求三棱锥$P-ABC$的表面积.", "objs": [], "tags": [ "第六单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\dfrac {12}5$; (2) $32$", "solution": "", "duration": -1, "usages": [ @@ -252464,7 +252944,7 @@ "第六单元" ], "genre": "解答题", - "ans": "", + "ans": "$2\\pi R^2$", "solution": "", "duration": -1, "usages": [ @@ -252954,7 +253434,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $0.75$; (2) $15$", "solution": "", "duration": -1, "usages": [], @@ -252978,7 +253458,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\dfrac 79$; (2) 作``取两个球''的操作$n$次($n$充分大), 记录颜色不同的次数$S_n$, 计算$\\dfrac{S_n}n$", "solution": "", "duration": -1, "usages": [ @@ -253007,7 +253487,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "$0.75$", "solution": "", "duration": -1, "usages": [ @@ -253061,7 +253541,7 @@ "第八单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [ @@ -253092,7 +253572,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\dfrac 56$; (2) $\\dfrac 16$; (3) $\\dfrac 23$; (4) $\\dfrac 12$", "solution": "", "duration": -1, "usages": [], @@ -256999,7 +257479,9 @@ "010741": { "id": "010741", "content": "分别求下列两数的等差中项:\\\\\n(1) $\\dfrac{8-\\sqrt 2}2$与$\\dfrac{8+\\sqrt 2}2$;\\\\\n(2) $(a+b)^2$与 $(a-b)^2$.", - "objs": [], + "objs": [ + "K0401002X" + ], "tags": [ "第四单元" ], @@ -257022,7 +257504,9 @@ "010742": { "id": "010742", "content": "设数列$\\{a_n\\}$为等差数列, 其公差为$d$.\\\\\n(1) 已知$a_1=2$, $d=3$, 求$a_{10}$;\\\\\n(2) 已知$a_1=3$, $a_n=21, d=2$, 求$n$;\\\\\n(3) 已知$a_1=12$, $a_6=27$, 求$d$;\\\\\n(4) 已知$a_6=9$, $d=-\\dfrac 12$, 求$a_1$.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第四单元" ], @@ -257043,7 +257527,9 @@ "010743": { "id": "010743", "content": "已知数列$\\{a_n\\}$为等差数列, 其公差为$d$. 求证:对任意给定的正整数$m$、$n$, 都有$a_n=a_m+(n-m)d$.", - "objs": [], + "objs": [ + "K0401003X" + ], "tags": [ "第四单元" ], @@ -257064,7 +257550,9 @@ "010744": { "id": "010744", "content": "设数列$\\{a_n\\}$为等差数列, 其公差为$d$.\\\\\n(1) 已知$a_2=31$, $a_7=76$, 求$a_1$及$d$;\\\\\n(2) 已知$a_1+a_6=12$, $a4=7$, 求$a_9$.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第四单元" ], @@ -257085,7 +257573,9 @@ "010745": { "id": "010745", "content": "设数列$\\{a_n\\}$为等差数列, 其公差为$d$, 前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=20$, $a_n=54$, $S_n=999$, 求$d$及$n$;\\\\\n(2) 已知$d=\\dfrac 13$, $S_{37}=629$, 求$a_1$及$a_{37}$;\\\\\n(3) 已知$a_1=\\dfrac 56$, $d=-\\dfrac 16$, $S_n=-5$, 求$n$及$a_n$;\\\\\n(4) 已知$d=2$, $a_{15}=-10$, 求$a_1$及$S_15$.", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257106,7 +257596,9 @@ "010746": { "id": "010746", "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_6=10$, $S_5=5$, 求$S_8$;\\\\\n(2) 已知$S_4=2$, $S_9=-6$, 求$S_{12}$.", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257127,7 +257619,10 @@ "010747": { "id": "010747", "content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_4+a_{14}=1$, 求$S_{17}$;\\\\\n(2) 已知$S_{21}=420$, 求$a_{11}$;\n(3) 已知$a_1+a_2+a_3=-3$, $a_{18}+a_{19}+a_{20}=6$, 求$S_{20}$;\\\\\n(4) 已知$S_4=2$, $S_8=6$, 求$S_{16}$.", - "objs": [], + "objs": [ + "K0401007X", + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257148,7 +257643,9 @@ "010748": { "id": "010748", "content": "求证: ``$\\triangle ABC$三个内角的度数可以构成等差数列''是``$\\triangle ABC$中有一个内角为$60^\\circ$''的充要条件.", - "objs": [], + "objs": [ + "K0401002X" + ], "tags": [ "第四单元" ], @@ -257169,7 +257666,9 @@ "010749": { "id": "010749", "content": "《九章算术》中的``竹九节''问题: 现有一根$9$节的竹子, 自上而下各节的容积成等差数列. 若最上面$4$节的容积共$3$升, 最下面$3$节的容积共$4$升, 则第$5$节的容积为多少升?", - "objs": [], + "objs": [ + "K0401007X" + ], "tags": [ "第四单元" ], @@ -257190,7 +257689,9 @@ "010750": { "id": "010750", "content": "(1)在等差数列$\\{a_n\\}$中, 等式$a_n=\\dfrac{a_{n-1}+a_{n+1}}2$\n($n\\ge 2$)是否都成立?\\\\\n(2) 在数列$\\{a_n\\}$中, 如果对于任意的正整数$n$($n\\ge 2$), 都有$a_n=\\dfrac{a_{n-1}+a_{n+1}}2$, 那么数列$\\{a_n\\}$一定是等差数列吗?", - "objs": [], + "objs": [ + "K0401002X" + ], "tags": [ "第四单元" ], @@ -257211,7 +257712,11 @@ "010751": { "id": "010751", "content": "在等差数列$\\{a_n\\}$中, 其前$n$项和为$S_n$. 已知公差$d=24$, $S_{20}=400$.\\\\\n(1) 写出$\\displaystyle \\sum_{i=1}^{10}a_{2i-1}$的具体展开式, 并求其值;\\\\\n(2) 用求和符号表示$a_2+a_4+a_6+\\cdots+a_{20}$, 并求其值.", - "objs": [], + "objs": [ + "K0402003X", + "K0401007X", + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257232,7 +257737,9 @@ "010752": { "id": "010752", "content": "在等差数列$\\{a_n\\}$中, 已知$a_1=-3$, $11a_5=5a_8$. 求数列$\\{a_n\\}$的前$n$项和$S_n$的最小值.", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257253,7 +257760,9 @@ "010753": { "id": "010753", "content": "已知等差数列$\\{a_n\\}$, 其前$n$项和为$S_n$. 若存在两个不相等的正整数$p$和$q$, 满足$S_p=q$, $S_q=p$, 求$S_{p+q}$.", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257274,7 +257783,9 @@ "010754": { "id": "010754", "content": "已知一个凸多边形各个内角的度数可以排列成一个公差为$5$的等差数列, 且最小角为$120^\\circ$, 该多边形是几边形?", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257295,7 +257806,9 @@ "010755": { "id": "010755", "content": "某产品按质量分成$10$个档次, 生产最低档次产品的利润是$8$元/件. 每提高一个档次, 每件产品的利润增加$2$元, 但产量每天减少$3$件. 如果在某段时间内, 最低档次(记作第$1$档次)的产品每天可生产$60$件, 那么在该段时间内, 生产第几档次的产品可获得最大利润?", - "objs": [], + "objs": [ + "K0402004X" + ], "tags": [ "第四单元" ], @@ -257316,7 +257829,9 @@ "010756": { "id": "010756", "content": "求下列各组数的等比中项:\\\\\n(1) $\\sqrt 3+1$与$\\sqrt 3-1$;\\\\\n(2) $a^4+a^2b^2$与$b^4+a^2b^2$($a\\ne 0$, $b\\ne 0$).", - "objs": [], + "objs": [ + "K0403001X" + ], "tags": [ "第四单元" ], @@ -257337,7 +257852,9 @@ "010757": { "id": "010757", "content": "设数列$\\{a_n\\}$为等比数列, 其公比为$q$.\\\\\n(1) 已知$a_5=8$, $a_8=1$, 求$a_1$、$q$;\\\\\n(2) 已知$a_3=2$, $q=-1$, 求$a_{15}$;\\\\\n(3) 已知$a_4=12$, $a_8=6$, 求$a_{12}$.", - "objs": [], + "objs": [ + "K0403001X" + ], "tags": [ "第四单元" ], @@ -257358,7 +257875,9 @@ "010758": { "id": "010758", "content": "已知数列$\\{a_n\\}$为等比数列, 其公比为$q$. 判断下列数列是否为等比数列. 如果是, 求其公比; 如果不是, 请说明理由.\\\\\n(1) 数列$\\{2a_n\\}$;\\\\\n(2) 数列$\\{a_n+a_{n+1}\\}$.", - "objs": [], + "objs": [ + "K0403003X" + ], "tags": [ "第四单元" ], @@ -257379,7 +257898,9 @@ "010759": { "id": "010759", "content": "已知数列$\\{a_n\\}$和数列$\\{b_n\\}$为项数相同的等比数列, 公比分别为$q_1$和$q_2$. 求证:数列$\\{a_nb_n\\}$为等比数列, 其公比为$q_1q_2$.", - "objs": [], + "objs": [ + "K0403003X" + ], "tags": [ "第四单元" ], @@ -257400,7 +257921,9 @@ "010760": { "id": "010760", "content": "已知直角三角形的斜边长为$c$, 两条直角边长分别为$a$和$b$($a=latex,scale = 0.5]\n\\draw (0,5) node [left] {$A$} -- (0,0) node [left] {$O$} -- (12,0) node [right] {$B$}-- cycle;\n\\draw (0,0) arc (180:0:{10/3}) coordinate (A);\n\\draw (A) arc (180:0:{10/3*4/9}) coordinate (B);\n\\draw (B) arc (180:0:{10/3*4/9*4/9}) coordinate (C);\n\\draw (C) arc (180:0:{10/3*4/9*4/9*4/9}) coordinate (D);\n\\filldraw ($(0,0)!0.5!(A)$) circle (0.06) node [below] {$O_1$};\n\\filldraw ($(B)!0.5!(A)$) circle (0.06) node [below] {$O_2$};\n\\filldraw ($(B)!0.5!(C)$) circle (0.06) node [below] {$O_3$};\n\\filldraw ($(D)!0.5!(C)$) circle (0.06) node [below] {$O_4$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0404003X", + "K0403003X", + "K0405002X", + "K0405005X" + ], "tags": [ "第四单元" ], @@ -257612,7 +258160,9 @@ "010770": { "id": "010770", "content": "已知下列数列$\\{a_n\\}$的通项公式, 写出它的前$4$项.\\\\\n(1) $a_n=n^2-5n$;\\\\\n(2) $a_n=\\dfrac{\\cos n\\pi} 2$.", - "objs": [], + "objs": [ + "K0406002X" + ], "tags": [ "第四单元" ], @@ -257633,7 +258183,9 @@ "010771": { "id": "010771", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\dfrac{n^2+n-1}3$, $79\\dfrac 23$是否是该数列中的项? 若是, 是第几项?", - "objs": [], + "objs": [ + "K0406002X" + ], "tags": [ "第四单元" ], @@ -257654,7 +258206,10 @@ "010772": { "id": "010772", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2-8n+5$.\\\\\n(1) 写出这个数列的前$5$项;\\\\\n(2) 这个数列有没有最小项? 如果有, 是第几项?", - "objs": [], + "objs": [ + "K0406005X", + "K0406002X" + ], "tags": [ "第四单元" ], @@ -257675,7 +258230,11 @@ "010773": { "id": "010773", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(3n-2)(\\dfrac 35)^n$, 试问:该数列是否有最大项、最小项? 若有, 分别指出第几项最大、最小; 若没有, 试说明理由.", - "objs": [], + "objs": [ + "K0406002X", + "K0406004X", + "K0406005X" + ], "tags": [ "第四单元" ], @@ -257696,7 +258255,10 @@ "010774": { "id": "010774", "content": "已知数列$\\{a_n\\}$的首项$a_1=1$, 且$a_n=2^{n-1}\\cdot a_{n-1}$($n\\ge 2$). 求数列$\\{a_n\\}$的通项公式.", - "objs": [], + "objs": [ + "K0403005X", + "K0407002X" + ], "tags": [ "第四单元" ], @@ -257717,7 +258279,10 @@ "010775": { "id": "010775", "content": "已知数列$\\{a_n\\}$满足$a_1=33$, 且$a_n-a_{n-1}=2(n-1)$($n\\ge 2$). 求数列$\\{\\dfrac{a_n}n\\}$的最小项.", - "objs": [], + "objs": [ + "K0407002X", + "K0406005X" + ], "tags": [ "第四单元" ], @@ -257738,7 +258303,12 @@ "010776": { "id": "010776", "content": "一个正方形被等分成九个相等的小正方形, 将最中间的一个正方形挖掉, 得图\\textcircled{1}; 再将剩下的每个正方形都分成九个相等的小正方形, 并将其最中间的一个正方形挖掉, 得图\\textcircled{2}; 如此继续下去$\\cdots\\cdots$\\\\\n(1) 图\\textcircled{3}中共挖掉了多少个正方形?\\\\\n(2) 求每次挖掉的正方形个数所构成的数列的一个递推公式.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) rectangle (3,3);\n\\filldraw [white] (1,1) rectangle++ (1,1);\n\\filldraw (4,0) rectangle (7,3);\n\\filldraw [white] (5,1) rectangle++ (1,1);\n\\foreach \\i in {{4+1/3},{5+1/3},{6+1/3}} {\\foreach \\j in {{1/3},{1+1/3},{2+1/3}} {\\filldraw [white] (\\i,\\j) rectangle++ ({1/3},{1/3});};};\n\\filldraw (8,0) rectangle (11,3);\n\\filldraw [white] (9,1) rectangle++ (1,1);\n\\foreach \\i in {{8+1/3},{9+1/3},{10+1/3}} {\\foreach \\j in {{1/3},{1+1/3},{2+1/3}} {\\filldraw [white] (\\i,\\j) rectangle++ ({1/3},{1/3});};};\n\\foreach \\i in {{8+1/9},{8+1/3+1/9},{8+2/3+1/9},{9+1/9},{9+1/3+1/9},{9+2/3+1/9},{10+1/9},{10+1/3+1/9},{10+2/3+1/9}} {\\foreach \\j in {{1/9},{1/3+1/9},{2/3+1/9},{1+1/9},{1+1/3+1/9},{1+2/3+1/9},{2+1/9},{2+1/3+1/9},{2+2/3+1/9}} {\\filldraw [white] (\\i,\\j) rectangle++ ({1/9},{1/9});};};\n\\foreach \\i in {1,2} {\\draw [white] (\\i,0) -- (\\i,3); \\draw [white] (0,\\i) -- (3,\\i);};\n\\foreach \\i in {1,2,...,8} {\\draw [white] ({4+\\i/3},0) -- ({4+\\i/3},3); \\draw [white] (4,{\\i/3}) -- (7,{\\i/3});};\n\\foreach \\i in {1,2,...,26} {\\draw [white] ({8+\\i/9},0) -- ({8+\\i/9},3); \\draw [white] (8,{\\i/9}) -- (11,{\\i/9});};\n\\draw (1.5,-0.5) node {\\textcircled{1}};\n\\draw (5.5,-0.5) node {\\textcircled{2}};\n\\draw (9.5,-0.5) node {\\textcircled{3}};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0407002X", + "K0403005X", + "K0404003X", + "K0407004X" + ], "tags": [ "第四单元" ], @@ -257759,7 +258329,10 @@ "010777": { "id": "010777", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\dfrac{n-\\sqrt {97}}{n-\\sqrt {98}}$, 试问: 该数列是否有最大项、最小项? 若有, 分别指出第几项最大、最小; 若没有, 试说明理由.", - "objs": [], + "objs": [ + "K0406002X", + "K0406005X" + ], "tags": [ "第四单元" ], @@ -257780,7 +258353,9 @@ "010778": { "id": "010778", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+\\lambda n$, 其中$\\lambda$是常数. 若数列$\\{a_n\\}$为严格增数列, 求$\\lambda$的取值范围.", - "objs": [], + "objs": [ + "K0406004X" + ], "tags": [ "第四单元" ], @@ -257801,7 +258376,9 @@ "010779": { "id": "010779", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $a_{n+1}=2S_n$($n$为正整数). 求数列$\\{a_n\\}$的通项公式.", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -257824,7 +258401,9 @@ "010780": { "id": "010780", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=12-12\\cdot(\\dfrac 23)^n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$\\{b_n\\}$满足$b_n=(2n-1)a_n$, 问是否存在正整数$m$, 使得$b_m\\ge 9$成立, 并说明理由.", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ "第四单元" ], @@ -257845,7 +258424,9 @@ "010781": { "id": "010781", "content": "某皮革厂第$1$年初有资金$1000$万元, 由于引进了先进的生产设备, 资金年平均增长率可达到$50\\%$. 每年年底定额扣除下一年的消费基金后, 将剩余资金投入再生产. 这家皮革厂每年应扣除多少消费基金, 才能实现资金在第$5$年年底扣除消费基金后达到$2000$万元的目标? (结果精确到$1$万元)", - "objs": [], + "objs": [ + "K0403005X" + ], "tags": [ "第四单元" ], @@ -257866,7 +258447,9 @@ "010782": { "id": "010782", "content": "用数学归纳法证明$1+a+a^2+\\cdots+a^{n+1}=\\dfrac{1-a^{n+2}}{1-a}$ ($a\\ne 1$, $n$为正整数). 在验证$n=1$等式成立时, 等式左边为\\bracket{20}.\n\\fourch{$1$}{$1+a$}{$1+a+a^2$}{$1+a+a^2+a^3$}", - "objs": [], + "objs": [ + "K0408003X" + ], "tags": [ "第四单元" ], @@ -257889,7 +258472,10 @@ "010783": { "id": "010783", "content": "用数学归纳法证明: $1\\times 2+2\\times 5+\\cdots+n(3n-1)=n^2(n+1)$($n$为正整数).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -257912,7 +258498,10 @@ "010784": { "id": "010784", "content": "用数学归纳法证明: $\\dfrac 1{1\\times 3}+ \\dfrac 1{3\\times 5}+ \\dfrac 1{5\\times 7}+\\cdots+\\dfrac 1{(2n-1)(2n+1)}= \\dfrac n{2n+1}$($n$为正整数).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -257933,7 +258522,10 @@ "010785": { "id": "010785", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, 设该数列的前$n$项和为$S_n$, 且$S_n, S_{n+1}, 2a_1$成等差数列. 用数学归纳法证明: $S_n=\\dfrac{2^n-1}{2^{n-1}}$($n$为正整数).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -257956,7 +258548,10 @@ "010786": { "id": "010786", "content": "用数学归纳法证明: $1\\cdot n+2\\cdot (n-1) +3\\cdot (n-2)+\\cdots+n\\cdot 1=\\dfrac 16n(n+1)(n+2)$($n$为正整数).", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -257979,7 +258574,10 @@ "010787": { "id": "010787", "content": "已知数列$\\{a_n\\}$满足$a_1=\\dfrac 12$, 且对任意正整数$n$, $a_1+a_2+\\cdots+a_n=n^2a_n$成立. 试用数学归纳法证明: $a_n=\\dfrac1{n(n+1)}$.", - "objs": [], + "objs": [ + "K0408002X", + "K0408003X" + ], "tags": [ "第四单元" ], @@ -258000,7 +258598,9 @@ "010788": { "id": "010788", "content": "设$a_n=1+\\dfrac 12+\\dfrac 13+\\cdots+\\dfrac 1n$($n$为正整数), 是否存在一次函数$g(x)=kx+b$, 使得等式$a_1+a_2+a_3+\\cdots+a_{n-1}=g(n)(a_n-1)$对大于$1$的正整数$n$都成立? 证明你的结论.", - "objs": [], + "objs": [ + "K0409001X" + ], "tags": [ "第四单元" ], @@ -258698,7 +259298,9 @@ "20220806\t王伟叶" ], "same": [], - "related": [], + "related": [ + "030427" + ], "remark": "", "space": "12ex" }, @@ -285091,7 +285693,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012005": { "id": "012005", @@ -285112,7 +285714,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012006": { "id": "012006", @@ -285133,7 +285735,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012007": { "id": "012007", @@ -285154,7 +285756,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012008": { "id": "012008", @@ -285176,7 +285778,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012009": { "id": "012009", @@ -285534,7 +286136,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012026": { "id": "012026", @@ -285555,7 +286157,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012027": { "id": "012027", @@ -285576,7 +286178,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012028": { "id": "012028", @@ -285597,7 +286199,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012029": { "id": "012029", @@ -285618,7 +286220,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "020001": { "id": "020001", @@ -293519,7 +294121,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030079": { "id": "030079", @@ -293540,7 +294142,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030080": { "id": "030080", @@ -293597,7 +294199,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030082": { "id": "030082", @@ -293624,7 +294226,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030083": { "id": "030083", @@ -293649,7 +294251,7 @@ "same": [], "related": [], "remark": "", - "space": "" 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"related": [], "remark": "", - "space": "" + "space": "12ex" }, "030096": { "id": "030096", @@ -294049,7 +294651,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030098": { "id": "030098", @@ -294566,7 +295168,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030114": { "id": "030114", @@ -294694,7 +295296,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030119": { "id": "030119", @@ -294999,7 +295601,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030130": { "id": "030130", @@ -295093,7 +295695,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030134": { "id": "030134", @@ -296551,7 +297153,7 @@ "第八单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\dfrac 13$", "solution": "", "duration": -1, "usages": [], @@ -296574,7 +297176,7 @@ "第八单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\dfrac 8{81}$", "solution": "", "duration": -1, "usages": [], @@ -296597,7 +297199,7 @@ "第八单元" ], "genre": "填空题", - "ans": "", + "ans": "\\textcircled{1}\\textcircled{2}\\textcircled{4}", "solution": "", "duration": -1, "usages": [ @@ -296626,7 +297228,7 @@ "第八单元" ], "genre": "选择题", - "ans": "", + "ans": "B", "solution": "", "duration": -1, "usages": [ @@ -296656,7 +297258,7 @@ "第八单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], @@ -296677,7 +297279,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\dfrac 1{16}$; (2) $\\dfrac 34$; (3) $\\dfrac 7{16}$", "solution": "", "duration": -1, "usages": [ @@ -296727,7 +297329,7 @@ "第八单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\dfrac 12$", "solution": "", "duration": -1, "usages": [ @@ -296757,7 +297359,7 @@ "第八单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\dfrac 16$; (2) $\\dfrac 5{72}$", "solution": "", "duration": -1, "usages": [ @@ -296812,7 +297414,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 分层抽样, 因不同区域的体验不同; (2) 简单随机抽样, 因高峰时无论在地铁何区域体验都基本相同", "solution": "", "duration": -1, "usages": [], @@ -296904,7 +297506,7 @@ "第九单元" ], "genre": "填空题", - "ans": "", + "ans": "$24$与$30$", "solution": "", "duration": -1, "usages": [], @@ -296950,7 +297552,7 @@ "第九单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], @@ -297064,7 +297666,7 @@ "第九单元" ], "genre": "填空题", - "ans": "", + "ans": "$32$", "solution": "", "duration": -1, "usages": [], @@ -297088,7 +297690,7 @@ "第九单元" ], "genre": "填空题", - "ans": "", + "ans": "$0.32$, $91$, $60$, $75$, $74.84$, $8.54$", "solution": "", "duration": -1, "usages": [], @@ -297111,7 +297713,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\overline{x}=10.0$, $\\overline{y}=10.3$, $S_1^2=0.036$, $S_2^2=0.04$; (2) 不认为新设备的各项指标均值有显著提高($0.3<0.39$)", "solution": "", "duration": -1, "usages": [], @@ -297158,7 +297760,7 @@ "第九单元" ], "genre": "填空题", - "ans": "", + "ans": "$3$与$5$", "solution": "", "duration": -1, "usages": [], @@ -297181,7 +297783,7 @@ "第九单元" ], "genre": "填空题", - "ans": "", + "ans": "存在", "solution": "", "duration": -1, "usages": [], @@ -297227,7 +297829,7 @@ "第九单元" ], "genre": "选择题", - "ans": "", + "ans": "A", "solution": "", "duration": -1, "usages": [], @@ -297250,7 +297852,7 @@ "第九单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], @@ -297411,7 +298013,7 @@ "第九单元" ], "genre": "选择题", - "ans": "", + "ans": "D", "solution": "", "duration": -1, "usages": [], @@ -297526,7 +298128,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $165.406\\text{cm}$; (2) $165.406\\text{cm}$", "solution": "", "duration": -1, "usages": [], @@ -297802,7 +298404,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $9$, $12$; (2) 说明时长在$[20,25)$与$[25,30]$分的通话次数(频数, 频率)都小于$[15,20)$中的.", "solution": "", "duration": -1, "usages": [], @@ -297899,7 +298501,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "$\\text{P}25=155.5$, $\\text{P}50=161.0$, $\\text{P}75=164.0$", "solution": "", "duration": -1, "usages": [], @@ -297969,7 +298571,7 @@ "第九单元" ], "genre": "解答题", - "ans": "", + "ans": "$S_1^20$''是``$\\dfrac ab+\\dfrac ba>2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题14", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030442": { + "id": "030442", + "content": "符号$[x]$表示不超过$x$的最大整数, 如$[\\pi]=3$, $[-1.08]=2$, 定义函数$\\{x\\}=x-[x]$, 那么下列命题中正确的序号是\\bracket{20}.\\\\\n\\textcircled{1} 函数$\\{x\\}$的定义域为$\\mathbf{R}$, 值域为$[0,1]$; \\textcircled{2} 方程$\\{x\\}=\\dfrac 12$有无数解; \\textcircled{3} 函数$\\{x\\}$是周期函数; \\textcircled{4} 函数$\\{x\\}$是增函数.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{4}\\textcircled{1}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题15", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030443": { + "id": "030443", + "content": "如图所示, 已知$PA\\perp$平面$ABC$, $AD\\perp BC$于$D$, $BC=CD=AD=1$. 令$PD=x$, $\\angle BPC=\\theta$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (-1,0,2) node [below] {$D$} coordinate (D);\n\\draw (-5,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(D)!0.5!(B)$) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [right] {$P$} coordinate (P);\n\\draw (B) -- (D) -- (A) -- (P) (B) -- (P) (C) -- (P) (D) -- (P);\n\\draw [dashed] (B) -- (A) (C) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan\\theta = \\dfrac{x}{x^2+2}$}{$\\tan\\theta = \\dfrac{x}{x^2+1}$}{$\\tan\\theta = \\dfrac{1}{x^2+2}$}{$\\tan\\theta = \\dfrac{1}{x^2+1}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题16", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030444": { + "id": "030444", + "content": "在$\\triangle ABC$中, 角$A,B,C$所对的边分别为$a,b,c$. $b=\\sqrt{5}$, $B=\\dfrac\\pi 4$.\\\\\n(1) 若$a=3$, 求$\\sin A$的值;\\\\\n(2) 若$\\triangle ABC$的面积等于$1$, 求$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题17", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "030445": { + "id": "030445", + "content": "如图, 圆锥的顶点是$P$, 底面中心是$O$, 已知$OP=\\sqrt{2}$, 圆$O$的直径是$AB=2$, 点$C$在弧$AB$上, 且$\\angle CAB=30^\\circ$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) node [left] {$A$} coordinate (A) (1,0) node [right] {$B$} coordinate (B) (0,0) node [above right] {$O$} coordinate (O) (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw ({cos(-60)},{0.3*sin(-60)}) node [below] {$C$} coordinate (C);\n\\draw (A) arc (180:360:1 and 0.3);\n\\draw [dashed] (A) arc (180:0:1 and 0.3) (A) -- (B) (O) -- (P) (A) -- (C);\n\\draw (A) -- (P) (B) -- (P) (C) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积;\\\\\n(2) 求$O$到平面$APC$的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题18", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "030446": { + "id": "030446", + "content": "科学家发现某种特别物质的温度$y$(单位: 摄氏度)随时间$x$(单位: 分钟)的变化规律满足关系式: $y=m\\cdot 2^x+2^{1-x}$($0\\le x\\le 4$, $m>0$).\\\\\n(1) 若$m=2$, 求经过多少分钟, 该物质的温度为$5$摄氏度;\\\\\n(2) 如果该物质温度总不低于$2$摄氏度, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题19", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "030447": { + "id": "030447", + "content": "已知函数$f(x)=\\log_2(ax^2+2x-a)$.\\\\\n(1) 当$a=-1$时, 求该函数的定义域;\\\\\n(2) 当$a\\le 0$时, 如果$f(x)\\ge 1$对任何$x\\in [2,3]$都成立, 求实数$a$的取值范围;\\\\\n(3) 若$a<0$, 将函数$f(x)$的图像沿$x$轴或其相反方向平移, 得到一个偶函数$g(x)$的图像, 设函数$g(x)$的最大值为$h(a)$, 求$h(a)$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题20", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "030448": { + "id": "030448", + "content": "记$f_k(x)=x^k$($x>0$, $k\\in \\mathbf{Z}$).\\\\\n(1) 求函数$F(x)=f_2(x-1)-1$的零点;\\\\\n(2) 设$\\xi,\\eta,\\mu$均为正整数, 且$\\sqrt{\\mu}$为最简根式, 若存在$n_0\\in \\mathbf{N}^*$, 使得$f_{n_0}(\\xi+\\eta\\sqrt{\\mu})$可唯一表示为$\\sqrt{\\tau}+\\sqrt{\\tau-1}$的形式($\\tau\\in \\mathbf{N}^*$). 求证: $|\\xi^2-\\eta^2\\mu|=1$;\\\\\n(3) 已知$f_{-1}(t)+f_{-1}(s)=1$, 是否存在$n_1\\in \\mathbf{N}^*$, 使得$\\dfrac{f_{n_1}(t+s)-f_{n_1}(t)-f_{n_1}(s)+f_{n_1}(2)}{f_{n_1}(4)-f_{n_1}(2)}\\ge 1$成立. 若存在, 试求出$n_1$的值; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题21", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "030449": { + "id": "030449", + "content": "从小到大排列的$9$个数据$1.23,1.35,2.14,2.55,3.67,3.89,4.21,4.43,5.51$的第$60$百分位数$\\mathrm{P}60$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030450": { + "id": "030450", + "content": "函数$y=\\sqrt{\\dfrac 1{x^2-1}}$的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20221103\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030451": { + "id": "030451", + "content": "证明: 当$x>0$时, $1-\\dfrac 1x\\le \\ln x\\le x-1$.", + "objs": [ + "K0234005X" + ], + "tags": [ + "第二单元", + "导数" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题-20221103修改", + "edit": [ + "20221028\t王伟叶", + "20221103\t余利成" + ], + "same": [], + "related": [ + "030353", + "030355" + ], + "remark": "", + "space": "12ex" } } \ No newline at end of file