diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index 896e510c..74b61ec9 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 5, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n", - "problems = \"4067\"\n", + "problems = \"014612\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index 5c5c95fd..79c9cf11 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 8, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 8, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -21,7 +21,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"content\":[r\"[\\s\\S]{300}\"],\"usages2\":[r\"0\\.[67]\\d{2}\"],\"genre\":[r\"解答题\"]},\n", + " {\"origin\":[\"空中课堂\"],\"origin2\":[\"2023\"]},\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\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 487c80ee..b094f73e 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: 14628 , 直至 020000\n", "首个空闲id: 22048 , 直至 030000\n", - "首个空闲id: 31309 , 直至 040000\n", + "首个空闲id: 31310 , 直至 040000\n", "首个空闲id: 40202 , 直至 999999\n" ] } diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 396287ab..76b02e4d 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ -000982,001124,001156,001173,001174,001198,001279,001384,001388,001602,001604,001621,001855,001885,001900,001991,002119,004075,004076,004077,004079,004099,004120,004162,004163,004205,004225,004226,004244,004246,004268,004288,004330,004331,004351,004369,004371,004373,004408,004445,004549,004595,004598,004599,004611,004616,004617,004618,004638,004656,004658,004677,004721,004722,004742,004743,010905,010906,010913,010916,010917,010918,012025,020144,020503,020520,020852,020866,020879,020883,020884,020885,020888,020896,020898,020899,020901,020902,020903,020907,020914,020915,020916,020925,020931,020947,020957,020962,020963,020964,020986,020994,021001,021126,021131,021170,021171,021176,021332,030521,030523,030525,030531,030582,030588,031284,031286,031306 \ No newline at end of file +014103,014104,014105,014106,014107,014108,014109,014110,014111,014112,014113,014114,014115,014116,014117,014118,014119,014120,014121,014122,014123,014124,014125,014126,014127,014128,014129,014130,014131,014132,014133,014134,014135,014136,014137,014138,014139,014140,014141,014142,014143,014144,014145,014146,014147,014148,014149,014150,014151,014152,014153,014154,014155,014156,014157,014158,014159,014160,014161,014162,014163,014164,014165,014166,014167,014168,014169,014170,014171,014172,014173,014174,014175,014176,014177,014178,014179,014180,014181,014182,014183,014184,014185,014186,014187,014188,014189,014190,014191,014192,014193,014194,014195,014196,014197,014198,014199,014200,014201,014202,014203,014204,014205,014206,014207,014208,014209,014210,014211,014212,014213,014214,014215,014216,014217,014218,014219,014220,014221,014222,014223,014224,014225,014226,014227,014228,014229,014230,014231,014232,014233,014234,014235,014236,014237,014238,014239,014240,014241,014242,014243,014244,014245,014246,014247,014248,014249,014250,014251,014252,014253,014254,014255,014256,014257,014258,014259,014260,014261,014262,014263,014264,014265,014266,014267,014268,014269,014270,014271,014272,014273,014274,014275,014276,014277,014278,014279,014280,014281,014282,014283,014284,014285,014286,014287,014288,014289,014290,014291,014292,014293,014294,014295,014296,014297,014298,014299,014300,014301,014302,014303,014304,014305,014306,014307,014308,014309,014310,014311,014312,014313,014314,014315,014316,014317,014318,014319,014320,014321,014322,014323,014324,014325,014326,014327,014328,014329,014330,014331,014332,014333,014334,014335,014336,014337,014338,014339,014340,014341,014342,014343,014344,014345,014346,014347,014348,014349,014350,014351,014352,014353,014354,014355,014356,014357,014358,014359,014360,014361,014362,014363,014364,014365,014366,014367,014368,014369,014370,014371,014372,014373,014374,014375,014376,014377,014378,014379,014380,014381,014382,014383,014384,014385,014386,014387,014388,014389,014390,014391,014392,014393,014394,014395,014396,014397,014398,014399,014400,014401,014402,014403,014404,014405,014406,014407,014408,014409,014410,014411,014412,014413,014414,014415,014416,014417,014418,014419,014420,014421,014422,014423,014424,014425,014426,014427,014428,014429,014430,014431,014432,014433,014434,014435,014436,014437,014438,014439,014440,014441,014442,014443,014444,014445,014446,014447,014448,014449,014450,014451,014452,014453,014454,014455,014456,014457,014458,014459,014460,014461,014462,014463,014464,014465,014466,014467,014468,014469,014470,014471,014472,014473,014474,014475,014476,014477,014478,014479,014480,014481,014482,014483,014484,014485,014486,014487,014488,014489,014490,014491,014492,014493,014494,014495,014496,014497,014498,014499,014500,014501,014502,014503,014504,014505,014506,014507,014508,014509,014510,014532,014533,014534,014535,014536,014537,014538,014539,014540,014541,014542,014543,014544,014545,014546,014547,014548,014549,014550,014551,014552,014553,014554,014555,014556,014557,014558,014559,014560,014561,014562,014563,014564,014565,014566,014567,014568,014569,014570,014571,014572,014573,014574,014575,014576,014577,014578,014579,014580,014581,014582,014583,014584,014585,014586,014587,014588,014589,014590,014591,014592,014593,014594,014595,014596,014597,014598,014599,014600,014601,014602,014603,014604,014605,014606,014607,014608,014609,014610,014611,014612,014613,014614,014615,014616,014617,014618,014619,014620,014621,014622,014623,014624,014625,014626,014627,031223,031224,031226,031227,031228,031229,031233 \ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index c767b3f7..dc09f6dd 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 @@ -7,10 +7,10 @@ "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 40181\n", - "raworigin = \"\"\n", - "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目4.tex\"\n", - "editor = \"20230308\\t王伟叶\"\n", + "starting_id = 14628\n", + "raworigin = \"2023年空中课堂高三复习课\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\空中课堂第六批.tex\"\n", + "editor = \"20230312\\t王伟叶\"\n", "indexed = False\n" ] }, @@ -23,27 +23,121 @@ "name": "stdout", "output_type": "stream", "text": [ - "添加题号040181, 来源: 2025届高一下学期周末卷04\n", - "添加题号040182, 来源: 2025届高一下学期周末卷04\n", - "添加题号040183, 来源: 2025届高一下学期周末卷04\n", - "添加题号040184, 来源: 2025届高一下学期周末卷04\n", - "添加题号040185, 来源: 2025届高一下学期周末卷04\n", - "添加题号040186, 来源: 2025届高一下学期周末卷04\n", - "添加题号040187, 来源: 2025届高一下学期周末卷04\n", - "添加题号040188, 来源: 2025届高一下学期周末卷04\n", - "添加题号040189, 来源: 2025届高一下学期周末卷04\n", - "添加题号040190, 来源: 2025届高一下学期周末卷04\n", - "添加题号040191, 来源: 2025届高一下学期周末卷04\n", - "添加题号040192, 来源: 2025届高一下学期周末卷04\n", - "添加题号040193, 来源: 2025届高一下学期周末卷04\n", - "添加题号040194, 来源: 2025届高一下学期周末卷04\n", - "添加题号040195, 来源: 2025届高一下学期周末卷04\n", - "添加题号040196, 来源: 2025届高一下学期周末卷04\n", - "添加题号040197, 来源: 2025届高一下学期周末卷04\n", - "添加题号040198, 来源: 2025届高一下学期周末卷04\n", - "添加题号040199, 来源: 2025届高一下学期周末卷04\n", - "添加题号040200, 来源: 2025届高一下学期周末卷04\n", - "添加题号040201, 来源: 2025届高一下学期周末卷04\n" + "添加题号014628, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014629, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014630, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014631, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014632, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014633, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014634, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014635, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014636, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014637, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014638, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014639, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014640, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014641, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014642, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014643, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014644, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014645, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014646, 来源: 2023年空中课堂高三复习课27\n", + "添加题号014647, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014648, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014649, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014650, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014651, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014652, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014653, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014654, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014655, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014656, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014657, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014658, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014659, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014660, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014661, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014662, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014663, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014664, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014665, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014666, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014667, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014668, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014669, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014670, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014671, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014672, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014673, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014674, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014675, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014676, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014677, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014678, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014679, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014680, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014681, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014682, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014683, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014684, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014685, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014686, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014687, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014688, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014689, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014690, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014691, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014692, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014693, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014694, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014695, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014696, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014697, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014698, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014699, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014700, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014701, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014702, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014703, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014704, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014705, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014706, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014707, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014708, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014709, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014710, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014711, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014712, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014713, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014714, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014715, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014716, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014717, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014718, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014719, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014720, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014721, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014722, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014723, 来源: 2023年空中课堂高三复习课28\n", + "添加题号014724, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014725, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014726, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014727, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014728, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014729, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014730, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014731, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014732, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014733, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014734, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014735, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014736, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014737, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014738, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014739, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014740, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014741, 来源: 2023年空中课堂高三复习课31\n", + "添加题号014742, 来源: 2023年空中课堂高三复习课31\n" ] } ], @@ -162,7 +256,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { 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 0dd85ec0..6bb5f7e8 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": 5, + "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/测验03选题_教师用_20230309.tex\n", + "开始编译教师版本pdf文件: 临时文件/8_概率统计备选_教师用_20230312.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/测验03选题_学生用_20230309.tex\n", + "开始编译学生版本pdf文件: 临时文件/8_概率统计备选_学生用_20230312.tex\n", "0\n" ] } @@ -26,13 +26,13 @@ "\"\"\"---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", - "a\n", + "13217:13239,14070:14080,14081:14094,14095:14102,14570:14590,14591:14608,14628:14646\n", "\"\"\"\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/测验03选题\"\n", + "filename = \"临时文件/8_概率统计备选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\"\"\"---设置是否需要解答题的空格---\"\"\"\n", @@ -146,7 +146,7 @@ " latex_raw = re.sub(r\"fontset[\\s]*=[\\s]*none\",\"fontset = fandol\",latex_raw)\n", " latex_raw = re.sub(r\"\\\\setCJKmainfont\",r\"% \\\\setCJKmainfont\",latex_raw)\n", "\n", - "latex_raw = re.sub(r\"\\tableofcontents\",\"\",latex_raw)\n", + "latex_raw = re.sub(r\"\\\\tableofcontents\",\"\",latex_raw)\n", "\n", "latex_teachers = latex_raw.replace(\"编译模板\",data_teachers)\n", "with open(teachers_latex_file,\"w\",encoding = \"utf8\") as f:\n", @@ -191,7 +191,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 3e851d20..532c0232 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -357238,6 +357238,2191 @@ "remark": "", "space": "12ex" }, + "014628": { + "id": "014628", + "content": "某校为了解高三年级学生体重情况, 从该年级$1000$名学生中抽取$125$名学生测量他们的体重进行分析. 在这项调查中, 抽取的$125$名学生的体重是\\bracket{20}.\n\\fourch{总体}{样本}{总体的容量}{样本容量}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014629": { + "id": "014629", + "content": "某校小卖部为研究学生对饮料的喜好情况, 从该校$450$名同学中用随机数法抽取$30$人参加这一项调查. 将这$450$名同学编号为$001,002, \\ldots, 449,450$, 在以下随机数表中从任意一个随机数开始读出三位数组, 假设从第$2$行第$7$列的数字开始, 则第$5$个被抽到的同学的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{llllll}\n16227794 & 39495443 & 54821737 & 93237887 & 35209643 & 84263491 \\\\ 64844217 & 55721754 & 55068331 & 04744767 & 21763350 & 25839212 \\\\ 06766301 & 63785916 & 95556719 & 98105071 & 75128673 & 58074439\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014630": { + "id": "014630", + "content": "某书店新进了一批书籍, 下表是某月中连续$6$天的销售情况记录:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 日期 & 6 日 & 7 日 & 8 日 & 9 日 & 10 日 & 11 日 \\\\\n\\hline 当日销售量/本 & 30 & 40 & 28 & 44 & 38 & 42 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表估计该书店该月(按$31$天计算)的销售总量约是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014631": { + "id": "014631", + "content": "某校课题小组为了研究高一年级学生甲、乙两门学科成绩的线性相关关系, 在高一第二学期期末考试后随机抽取了$5$名同学 (记为$A$、$B$、$C$、$D$、$E$) 的甲、乙两门学科成绩(满分均为$100$分), 如图. 后来发现$D$同学数据记录有误, 那么去掉数据$D(82,88)$后, 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (80,75) -- (95,75) node [below] {学科甲成绩$x$};\n\\draw [->] (80,75) -- (80,90) node [left] {学科乙成绩$y$};\n\\filldraw (81,78) circle (0.1) node [right] {$A(81,78)$};\n\\filldraw (83,81) circle (0.1) node [right] {$E(83,81)$};\n\\filldraw (87,84) circle (0.1) node [right] {$B(87,84)$};\n\\filldraw (89,87) circle (0.1) node [right] {$C(89,87)$};\n\\filldraw (82,88) circle (0.1) node [right] {$D(82,88)$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{样本线性相关系数$r$变大}{最小二乘拟合误差变大}{变量$x$、$y$的相关程度变高}{线性相关系数$r$越接近于$1$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014632": { + "id": "014632", + "content": "甲、乙两城之间的长途客车均由$A$和$B$两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的$500$个班次, 得到下面的$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 准点班次数 & 未准点班次数 \\\\\n\\hline$A$& 240 & 20 \\\\\n\\hline$B$& 210 & 30 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表数据, 判断甲、乙两城之间的长途客车准点情况与客车所属公司\\blank{50}. (填写``有关''或``无关'')\\\\\n($\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 显著水平取$0.05$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$).", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014633": { + "id": "014633", + "content": "深入实施科教兴国战略是中华人民伟大复兴的必由之路. $2020$年第七次全国人口普查中(未包括中国香港、澳门特别行政区和台湾省的人口数据), 我国$31$个省级行政区$15$岁及以上男性和女性的文盲人口比重($\\%$)情况, 经统计得到如下的茎叶图(其中$a$是$0$至$9$中的一个数字).\n\\begin{center}\n\\begin{tabular}{cccccccc|c|cccccccccccc}\n&&&&&&&女&&男\\\\\n&&&&&&&&0&4&6&7&7&7&8&9&9&9\\\\\n&&&&&&5&4&1&0&0&0&1&2&3&3&5&5&8&8&9\\\\\n&&9&9&2&1&1&1&2&1&6&7&7&7\\\\\n&&&&&2&0&0&3&3\\\\\n9&9&9&8&4&3&3&0&4&1&7\\\\\n&&&&&&7&0&5\\\\\n&&&&&8&5&2&6&2\\\\\n&&&&&&&6&7\\\\\n&&&&&&5&5&8\\\\\n&&&&&&&1&12\\\\\n&&&&&&&5&13\\\\\n&&&&&&&1&14\\\\\n&&&&&&&&20&4\\\\\n&&&&&&&$a$&36\n\\end{tabular}\n\\end{center}\n(1) 根据茎叶图判断男性样本数据和女性样本数据的离散程度, 并求离散程度较小的样本数据的第$80$百分位数;\\\\\n(2) 若女性样本数据的极差为$35.3(\\%)$, 求该样本数据的的平均数与方差(结果精确到$0.1$);\\\\\n(3) 为了调查今年某地区$15$岁及以上男性和女性文盲人口情况, 研究小组准备采用分层随机抽样方法抽取$5000$人进行调查. 已知该地区$15$岁及以上的男性约有$4.2$百万人, 女性约有$3.8$百万人. 分别求出抽取的男性人数和女性人数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014634": { + "id": "014634", + "content": "某研究小组通过部分直辖市的统计年鉴, 整理了各区$15$岁及以上人口中大学专科、 本科和研究生学历的人口比重($\\%$)如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} & 地区 & \\makecell{专科\\\\学历\\\\比重} & \\makecell{本科\\\\学历\\\\比重} & \\makecell{研究生\\\\学历\\\\比重} \\\\\n\\hline 1 区 & 12.37 & 18.14 & 4.84 & 12 区 & 10.74 & 10.22 & 1.01 & 23 区 & 16.37 & 17.21 & 3.15 \\\\\n\\hline 2 区 & 14.69 & 26.5 & 9.74 & 13 区 & 13.27 & 17.92 & 3.12 & 24 区 & 15.07 & 17.1 & 3.21 \\\\\n\\hline 3 区 & 14.63 & 26.74 & 8.28 & 14 区 & 11.35 & 12.1 & 1.56 & 25 区 & 16.65 & 20.04 & 3.29 \\\\\n\\hline 4 区 & 15.67 & 23.65 & 5.38 & 15 区 & 10.46 & 11.86 & 1.13 & 26 区 & 13.62 & 17.3 & 2.72 \\\\\n\\hline 5 区 & 15.89 & 23.47 & 5.77 & 16 区 & 7.03 & 6.41 & 0.42 & 27 区 & 16.7 & 26.34 & 6.46 \\\\\n\\hline 6 区 & 15.33 & 23.92 & 5.43 & 17 区 & 15.82 & 28.71 & 8.44 & 28 区 & 16.05 & 19.51 & 4.23 \\\\\n\\hline 7 区 & 14.52 & 23.55 & 8.01 & 18 区 & 15.01 & 29.25 & 12.28 & 29 区 & 12.74 & 12.62 & 3.84 \\\\\n\\hline 8 区 & 14.68 & 22.21 & 5.89 & 19 区 & 15.28 & 30.76 & 9.56 & 30 区 & 13.7 & 11.41 & 0.98 \\\\\n\\hline 9 区 & 15.45 & 17.82 & 2.83 & 20 区 & 16.29 & 26.37 & 7.27 & 31 区 & 13 & 11.73 & 1.14 \\\\\n\\hline 10 区 & 13.01 & 14.93 & 3.03 & 21 区 & 16.97 & 28.17 & 8.21 & 32 区 & 12.79 & 12.51 & 1.26 \\\\\n\\hline 11 区 & 13.91 & 19.98 & 6.14 & 22 区 & 13.08 & 33.14 & 17.81 &&&&\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请从以下几个方面对上表中的数据进行统计分析:\\\\\n(1) 各类学历的频率分布直方图;\\\\\n(2) 各类学历的集中趋势和离散程度;\\\\\n(3) 相关指标的相关性分析.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014635": { + "id": "014635", + "content": "某同学要调查一个地区博物馆的展品数量, 有以下说法:\n\\textcircled{1} 可以通过调查获取相关数据;\n\\textcircled{2} 可以通过统计报表获取相关数据;\n\\textcircled{3} 可以通过实验获取相关数据;\n\\textcircled{4} 可以通过互联网获取相关数据. 其中所有正确说法的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014636": { + "id": "014636", + "content": "分别统计了甲、乙两位同学$16$周的各周课外体育运动时长(单位: $\\text{h}$), 得如下茎叶图: 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccc|c|cccccccc}\n&&&甲&&乙\\\\\n&&6&1&5&\\\\\n8&5&3&0&6&3\\\\\n7&5&3&2&7&4&6\\\\\n6&4&2&1&8&1&2&2&5&6&6&6&6\\\\\n&&4&2&9&0&2&3&8\\\\\n&&&&10&1\n\\end{tabular}\n\\end{center}\n\\onech{甲同学周课外体育运动时长的样本中位数为$7.4$}{乙同学周课外体育运动时长的样本平均数大于$8$}{甲同学周课外体育运动时长大于$8$的概率的估计值大于$0.4$}{乙同学周课外体育运动时长大于$8$的概率的估计值大于$0.6$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014637": { + "id": "014637", + "content": "为了解甲、乙两种离子在小鼠体内的残留程度, 进行如下试验: 将$200$只小鼠随机分成$A$、$B$两组, 每组$100$只, 其中给$A$组小鼠服甲离子溶液, 给$B$组小鼠服乙离子溶液, 给每只小鼠服的溶液体积相同、摩尔浓度相同. 经过一段时间后, 用某种科学方法测算出残留在小鼠体内离子的百分比. 根据试验数据分别得到如下直方图:\n记$C$为事件: ``乙离子残留在体内的百分比不低于$5.5$'', 根据直方图得到概率$P(C)$的估计值为$0.70$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {6.5/0.05/0.05,5.5/0.1/0.1,1.5/0.15/0.15,4.5/0.2/0.2,3.5/0.3/0.3}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {1.5/0.15/0.15,2.5/0.2/0.2,3.5/0.3/0.3,4.5/0.2/0.2,5.5/0.1/0.1,6.5/0.05/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$7.5$};\n\\draw (4.5,-0.1) node {甲离子残留百分比直方图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/b,7.5/0.15/0.15,6.5/0.2/0.2,5.5/0.35/a}\n{\\draw [dashed] ({\\i-1},\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/0.1,4.5/0.15/0.15,5.5/0.35/0.35,6.5/0.2/0.2,7.5/0.15/0.15}\n{\\draw ({\\i-1},0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$8.5$};\n\\draw (4.5,-0.1) node {乙离子残留百分比直方图};\n\\end{tikzpicture}\n\\end{center}\n(1) 求乙离子残留百分比直方图中$a$、$b$的值;\\\\\n(2) 分别估计甲、乙离子残留百分比的平均值(同一组中的数据用该组区间的中点值为代表).", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014638": { + "id": "014638", + "content": "通过随机抽样, 我们获得某种产品加工前含水率与加工后含水率的一组测试数据, 如下表所示.\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline 加工前含水率 ($\\%$) & 16.7 & 18.2 & 18.2 & 17.9 & 17.4 & 16.6 & 17.2 & 17.7 & 15.7 & 17.1 \\\\\n\\hline 加工后含水率($\\%$)& 17.1 & 18.4 & 18.6 & 18.5 & 18.2 & 17.1 & 18.0 & 18.2 & 16.0 & 17.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请绘制上述数据的散点图, 并依据散点图观察两组数据的相关性;\\\\\n(2) 计算产品加工前含水率与加工后含水率之间的相关系数, 并判断两个变量的相关程度;\\\\\n(3) 依据表中给出的某种产品加工前含水率与加工后含水率的一组测试数据, 试预测当产品加工前含水率为$19 \\%$时, 加工后含水率的数值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014639": { + "id": "014639", + "content": "一支田径队有男运动员$48$人, 女运动员$36$人, 若用分层抽样的方法从全体运动员中抽取一个容量为$21$的样本, 则抽取男运动员的人数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014640": { + "id": "014640", + "content": "甲、乙两城市某月初连续$7$天的日均气温数据如图所示, 则在这$7$天中, 有以下说法:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\foreach \\i in {1,2,...,7} {\n\\draw [gray] (\\i,0) -- (\\i,7) (0,\\i) -- (7,\\i); \n\\draw (\\i,0.2) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.2,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\draw [->] (0,0) -- (8,0) node [below] {日期};\n\\draw [->] (0,0) -- (0,8) node [left] {气温$^\\circ\\text{C}$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,5) -- (2,3) -- (3,6) -- (4,3) -- (5,7) -- (6,5) -- (7,6);\n\\draw [dashed] (1,5) -- (2,4) -- (3,6) -- (4,5) -- (5,5) -- (6,4) -- (7,6);\n\\draw (7.5,5.5) -- (9.5,5.5) node [right] {甲};\n\\draw [dashed] (7.5,4) -- (9.5,4) node [right] {乙};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲城市日均气温的平均数与中位数相等;\\\\\n\\textcircled{2} 甲城市的日均气温比乙城市的日均气温稳定;\\\\\n\\textcircled{3} 乙城市日均气温的极差为$3^{\\circ} \\text{C}$;\\\\\n\\textcircled{4} 乙城市日均气温的众数为$5^{\\circ} \\text{C}$.\n其中所有正确说法的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014641": { + "id": "014641", + "content": "某校抽取$100$名学生测身高, 其中身高最大值为$186 \\text{cm}$, 最小值为$154 \\text{cm}$, 根据身高数据绘制频率组距分布直方图, 组距为$5$, 且第一组下限为$153.5$, 则组数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014642": { + "id": "014642", + "content": "翠冠梨是盛产于江南夏季的一种丰水蜜梨, 是盛夏解渴消暑的时令佳品. 某产区标准化的果园每亩梨树$42$株. 为调查该产区$8000$亩翠冠梨的产量, 在收获期从绿港村$600$亩梨园中随机抽取了$10$棵梨树, 测得其产量 (单位: $\\text{kg}$) 分别为$72$、$75$、$84$、$94$、$60$、$78$、$99$、$78$、$91$、$90$. 预估该产区翠冠梨的总产量为\\blank{50}吨.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014643": { + "id": "014643", + "content": "某国家足球队$26$名球员的年龄分布茎叶图如图所示:\n\\begin{center}\n\\begin{tabular}{c|ccccccccccccccccc}\n1 & 8 & 9 \\\\\n2 & 1 & 2 & 3 & 3 & 4 & 5 & 5 & 5 & 6 & 6 & 7 & 8 & 8 & 8 & 9 & 9 & 9 \\\\ \n3 & 0 & 1 & 2 & 2 & 2 & 3 & 4 \n\\end{tabular}\n\\end{center}\n该国家足球队$25$岁的球员共有\\blank{50}位, 该国家足球队球员年龄的第$75$百分位数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014644": { + "id": "014644", + "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山. 为估计一林区某种树木的总材积量, 随机选取了$10$棵这种树木, 测量每棵树的根部横截面积 (单位: $\\text{m}^2$) 和材积量 (单位: $\\text{m}^3$), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 样本号$i$& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\\hline\n\\makecell{根部横截\\\\面积$x_i$} & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\\hline\n材积量$y_i$ & 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数(精确到$0.01$);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积, 并得到所有这种树木的根部横截面积总和为$186 \\text{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014645": { + "id": "014645", + "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯(卫生习惯分为良好和不够良好两类) 的关系, 在已患该疾病的病例中随机调查了$100$例 (称为病例组), 同时在未患该疾病的人群中随机调查了$100$人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 不够良好 & 良好 \\\\\n\\hline 病例组 & 40 & 60 \\\\\n\\hline 对照组 & 10 & 90 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 判断患该疾病群体与未患该疾病群体的卫生习惯是否有关?\\\\\n(2) 从该地的人群中任选一人, $A$表示事件``选到的人卫生习惯不够良好'', $B$表示事件``选到的人患有该疾病''. $\\dfrac{P(B | A)}{P(\\overline {B} | A)}$与$\\dfrac{P(B | \\overline {A})}{P(\\overline {B} | \\overline {A})}$的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标, 记该指标为$R$.\\\\\n(i) 证明: $R=\\dfrac{P(A | B)}{P(\\overline {A} | B)} \\cdot \\dfrac{P(\\overline {A} | \\overline {B})}{P(A | \\overline {B})}$;\\\\\n(ii) 利用该调查数据, 给出$P(A | B), P(A | \\overline {B})$的估计值, 并利用 (i) 的结果给出$R$的估计值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014646": { + "id": "014646", + "content": "本市某区对全区高中生的身高(单位: 厘米) 进行统计, 得到如下的频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 120]\n\\draw [->] (0,0) -- (0,0.035) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.002) -- (0.4,0.002) -- (0.6,-0.002) -- (0.8,0) -- (10,0) node [below] {身高/厘米};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2/0.022/0.022,3/0.027/0.027,4/0.025/0.025,5/0.015/x,6/0.01/0.01,7/0.001/0.001}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};}; \n\\foreach \\i/\\j/\\k in {2/0.022/150,3/0.027/160,4/0.025/170,5/0.015/180,6/0.01/190,7/0.001/200}\n{\\draw (\\i,0) node [below] {$\\k$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);};\n\\draw (8,0) node [below] {$210$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若数据分布均匀, 记随机变量$X$为各区间中点所代表的身高, 写出$X$的分布及期望;\\\\\n(2) 已知本市身高在区间$[180,210]$的市民人数约占全市总人数的$10 \\%$, 且全市高中生约占全市总人数的$1.2 \\%$. 现在要以该区本次统计数据估算全市高中生身高情况, 从本市市民中任取$1$人, 若此人的身高位于区间$[180,210]$, 试估计此人是高中生的概率;\\\\\n(3) 现从身高在区间$[170,190)$的高中生中分层抽样抽取一个$80$人的样本. 若身高在$[170,180)$中样本的均值为$176$厘米, 方差为$10$; 身高在区间$[180,190)$中样本的均值为$184$厘米, 方差为$16$, 试求这$80$人身高的方差.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课27", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014647": { + "id": "014647", + "content": "已知集合$A=\\{-1,3,2 m-1\\}$, $B=\\{3, m^2\\}$. 若$B \\subseteq A$, 则实数$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014648": { + "id": "014648", + "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{3,4,5\\}$, 则$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014649": { + "id": "014649", + "content": "若全集$U=\\mathbf{R}$, 集合$A=\\{x | x \\geq 1\\} \\cup\\{x | x \\leq 0\\}$, 则$\\overline {A}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014650": { + "id": "014650", + "content": "已知$a\\in \\mathbf{R}$, 那么``$a>1$''是``$\\dfrac 1a<1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014651": { + "id": "014651", + "content": "设$a$、$b$是非零实数, 若$a0$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014653": { + "id": "014653", + "content": "设$x \\in \\mathbf{R}$, 不等式$|x-3|<1$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014654": { + "id": "014654", + "content": "若实数$x$、$y$满足$x y=1$, 则$x^2+2 y^2$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014655": { + "id": "014655", + "content": "若正实数$x$、$y$满足$x+4 y=1$, 则$x y$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014656": { + "id": "014656", + "content": "已知$\\alpha \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2} 1,2,3\\}$. 若幂函数$y=x^\\alpha$为奇函数, 且在区间$(0,+\\infty)$上为严格减函数, 则$\\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014657": { + "id": "014657", + "content": "已知$f(x)=\\mathrm{e}^{|x-a|}$($a$为常数). 若$y=f(x)$在区间$[1,+\\infty)$上是严格增函数, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014658": { + "id": "014658", + "content": "设函数$y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 当$x \\in(0,+\\infty)$时, $f(x)=\\lg x$, 则满足$f(x)>0$的$x$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014659": { + "id": "014659", + "content": "已知函数$y=f(x)$的图像是折线段$ABC$, 其中$A(0,0)$、$B(\\dfrac{1}{2}, 5)$、$C(1,0)$. 函数$y=x f(x)$($0 \\leq x \\leq 1$)的图像与$x$轴围成的图形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014660": { + "id": "014660", + "content": "设常数$a \\geq 0$, 根据$a$的不同取值, 讨论函数$y=\\dfrac{2^x+a}{2^x-a}$的奇偶性, 并说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014661": { + "id": "014661", + "content": "设常数$a \\in \\mathbf{R}$, 根据$a$的不同取值, 讨论函数$y=x^2+\\dfrac{a}{x}$($x \\neq 0$)的奇偶性, 并说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014662": { + "id": "014662", + "content": "对于定义在$\\mathbf{R}$上的函数$y=f(x)$, 考察以下性质, $p$: 存在非零实数$a$, 使得$f(x+a)0.\\end{cases}$若$f(0)$是函数$y=f(x)$的最小值, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$[-1,2]$}{$[-1,0]$}{$[1,2]$}{$[0,2]$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014664": { + "id": "014664", + "content": "如图, $A, B, C$三地有直道相通, $AB=5$千米, $AC=3$千米, $BC=4$千米. 现甲、乙两警员同时从$A$地出发匀速前往$B$地, 经过$t$小时, 他们之间的距离为$f(t)$(单位: 千米). 甲的路线是$AB$, 速度为$5$千米/小时, 乙的路线是$ACB$, 速度为$8$千米/小时. 乙到达$B$地后在原地等待. 设$t=t_1$时, 乙到达$C$地. 已知警员的对讲机的有效通话距离是$3$千米. 当$t_1 \\leq t \\leq 1$时, 求$f(t)$的表达式, 并判断$f(t)$在区间$[t_1, 1]$上的最大值是否超过$3$? 说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (5,0) node [right] {$B$} coordinate (B);\n\\draw (1.8,2.4) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014665": { + "id": "014665", + "content": "已知$f(x)=\\begin{cases}|\\lg | x-1 \\|, & x \\neq 1, \\\\ 0, & x=1.\\end{cases}$关于$x$的方程$f^2(x)+b f(x)+c=0$有$7$个不同实数解的一个充要条件是\\bracket{20}.\n\\fourch{$b<0$且$c>0$}{$b>0$且$c<0$}{$b<0$且$c=0$}{$b \\geq 0$且$b<0$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014666": { + "id": "014666", + "content": "设$a \\in \\mathbf{R}$. 若存在唯一的$m\\in \\mathbf{Z}$使得关于$x$的不等式组$\\dfrac{1}{2} x^2-\\dfrac{1}{2}0, f(x)=2 \\sin (\\omega x)$, 若$y=f(x)$在区间$[-\\dfrac{\\pi}{4}, \\dfrac{2 \\pi}{3}]$上是严格增函数, 则$\\omega$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014677": { + "id": "014677", + "content": "已知等差数列$\\{a_n\\}$的公差不为零, $S_n$为其前$n$项和. 若$S_5=0$, 则$S_1, S_2, \\cdots, S_{100}$这$100$个数中所有不同数值的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014678": { + "id": "014678", + "content": "已知数列$\\{a_n\\}$前$n$项和为$S_n$, 若$S_n+a_n=2$, 则$S_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014679": { + "id": "014679", + "content": "某公司积极投入本市``五个新城''建设, 将总部迁入其中一个新城. 该公司$2021$年第一季度的营业收入为$1.1$亿元, 利润为$0.16$亿元. 预测显示: 在以$2021$年为第一年的未来十年(每年$4$个季度, 共$40$个季度)内, 该公司每一季度的营业收入比上一季度增加$0.05$亿元, 利润比上一季度增长$4 \\%$. 据此预测, 解答以下问题:\\\\\n(1) 求$2021$年至$2025$年, 该公司五年共$20$个季度营业收入共计多少亿元?\\\\\n(2) 该公司在哪年哪季度的利润将首次超过该季度的营业收入的$18 \\%$?", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014680": { + "id": "014680", + "content": "已知$f(x)=x^3-x$, $g(x)=x^2+a$, 若曲线$y=f(x)$在点$(-1, f(-1))$处的切线也是曲线$y=g(x)$的切线, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014681": { + "id": "014681", + "content": "若$x=-2$是函数$y=(x^2+a x-1) \\mathrm{e}^{x-1}$的极值点, 则该函数的极小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014682": { + "id": "014682", + "content": "已知$k$为常数, 若$(x-k)^2 \\mathrm{e}^{\\frac{x}{k}} \\leq \\dfrac{1}{\\mathrm{e}}$对任意的$x \\in(0,+\\infty)$成立, 求$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014683": { + "id": "014683", + "content": "在正三角形$ABC$中, $D$是$BC$上的点, 若$|AB|=3$, $|BD|=1$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AD}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014684": { + "id": "014684", + "content": "直角坐标系$xOy$中, $\\overrightarrow {i}$、$\\overrightarrow {j}$分别是与$x$、$y$轴正方向同向的单位向量. 在直角$\\triangle ABC$中, 若$\\overrightarrow{AB}=2 \\overrightarrow {i}+\\overrightarrow {j}$, $\\overrightarrow{AC}=3 \\overrightarrow {i}+k \\overrightarrow {j}$, 则$k$的可能值的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014685": { + "id": "014685", + "content": "已知圆心为$O$、半径为$1$的圆上有三点$A$、$B$、$C$. 若$7 \\overrightarrow{OA}+5 \\overrightarrow{OB}+8 \\overrightarrow{OC}=\\overrightarrow{0}$, 则$|\\overrightarrow{BC}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014686": { + "id": "014686", + "content": "设$m \\in \\mathbf{R}$, $m^2+m-2+(m^2-1) \\mathrm{i}$是纯虚数, 其中$\\mathrm{i}$是虚数单位, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014687": { + "id": "014687", + "content": "若复数$z=1+2 \\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$(z+\\dfrac{1}{\\overline {z}}) \\cdot \\overline {z}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014688": { + "id": "014688", + "content": "已知$a$、$b \\in \\mathbf{R}$, 若$2+a \\mathrm{i}, b+\\mathrm{i}$($\\mathrm{i}$是虚数单位)是实系数一元二次方程$x^2+p x+q=0$的两个根, 那么$p$、$q$的值分别是 \\bracket{20}.\n\\fourch{$p=-4$, $q=5$}{$p=-4$, $q=3$}{$p=4$, $q=5$}{$p=4$, $q=3$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014689": { + "id": "014689", + "content": "在如图所示的棱长为$10$的正方体$ABCD-A_1B_1C_1D_1$中, 一条平行于$A_1C$的直线与正方体的表面交于$P$、$Q$两点, 其中点$P$在侧面$ADD_1A_1$上, 且到$A_1D_1$的距离为$3$, 到$AA_1$的距离为$2$, 则点$Q$所在的面是 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$ABCD$}{$ABB_1A_1$}{$BCC_1B_1$}{$CDD_1C_1$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014690": { + "id": "014690", + "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马. 设$AA_1$是正六棱柱的一条侧棱, 如图, 若阳马以该六棱柱的顶点为顶点, 以$AA_1$为底面矩形的一边, 则这样的阳马的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}\n\\coordinate (A) at (0,0,0) node [below] {$A$};\n\\path (A) --++ (45:{sqrt(3)/2}) --++ (1,0) coordinate (B);\n\\path (A) --++ (45:{sqrt(3)}) coordinate (C);\n\\path (C) --++ (-2,0) coordinate (D);\n\\path (B) --++ (-4,0) coordinate (E);\n\\coordinate (F) at (-2,0);\n\\draw (E) -- (F) -- (A) -- (B);\n\\draw [dashed] (B) -- (C) -- (D) -- (E) -- (F);\n\\foreach \\i in {(A),(B),(E),(F)}{\\draw \\i --++ (0,2);};\n\\foreach \\i in {(D),(C)}{\\draw [dashed] \\i --++ (0,2);};\n\\path (A) --++ (0,2) coordinate (A1) node [above] {$A_1$};\n\\path (B) --++ (0,2) coordinate (B1);\n\\path (C) --++ (0,2) coordinate (C1);\n\\path (D) --++ (0,2) coordinate (D1);\n\\path (E) --++ (0,2) coordinate (E1);\n\\path (F) --++ (0,2) coordinate (F1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- (E1) -- (F1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014691": { + "id": "014691", + "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$的棱所在的直线中, 与直线$BC_1$异面的直线的条数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{1.5}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw (B) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014692": { + "id": "014692", + "content": "若圆柱的高为$4$, 底面积为$9 \\pi$, 则该圆柱的侧面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014693": { + "id": "014693", + "content": "若正三棱柱的所有棱长均为$a$, 且其体积为$16 \\sqrt{3}$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014694": { + "id": "014694", + "content": "若一个圆锥的侧面展开图是面积为$2 \\pi$的半圆面, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014695": { + "id": "014695", + "content": "若三个球的半径$R_1, R_2, R_3$满足$R_1+2R_2=3R_3$, 则它们的表面积$S_1, S_2, S_3$, 满足的等量关系是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014696": { + "id": "014696", + "content": "在四面体$ABCD$中, 已知$AB \\perp CD, BC \\perp DA$. 求证: $BD \\perp AC$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014697": { + "id": "014697", + "content": "如图, 在正四棱柱$ABCD-A_1B_1C_1D_1$中, 底面$ABCD$的边长为$3$, $BD_1$与底面所成角的大小为$\\arctan \\dfrac{2}{3}$, 则该正四棱柱的高等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (D)--(B1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014698": { + "id": "014698", + "content": "若$\\overrightarrow {n}=(-2,1)$是直线$l$的一个法向量, 则$l$的倾斜角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014699": { + "id": "014699", + "content": "若直线$2 x+m y+1=0$与直线$y=3 x-1$平行, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014700": { + "id": "014700", + "content": "已知$P_1(a_1, b_1)$与$P_2(a_2, b_2)$是直线$y=k x+1$($k$为常数) 上两个不同的点, 则关于$x$和$y$的方程组$\\begin{cases}a_1 x+b_1 y=1, \\\\ a_2 x+b_2 y=1\\end{cases}$的解的情况是\\bracket{20} .\n\\twoch{无论$k$、$P_1$、$P_2$如何, 总是无解}{无论$k$、$P_1$、$P_2$如何, 总有唯一解}{存在$k$、$P_1$、$P_2$, 使之恰有两解}{存在$k$、$P_1$、$P_2$, 使之有无穷多解}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014701": { + "id": "014701", + "content": "已知平行直线$l_1: 2 x+y-1=0$, $l_2: 2 x+y+1=0$, 则$l_1$与$l_2$的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014702": { + "id": "014702", + "content": "已知椭圆的中心在原点, 一个焦点为$F(-2 \\sqrt{3}, 0)$, 且长轴长是短轴长的$2$倍, 则该椭圆的标准方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014703": { + "id": "014703", + "content": "设$m$为常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014704": { + "id": "014704", + "content": "抛物线$y^2=2 p x$($p>0$)上的动点$Q$到焦点的距离的最小值为$1$, 则$p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014705": { + "id": "014705", + "content": "双曲线$x^2-\\dfrac{y^2}{b^2}=1(b>0)$的左、右焦点分别为$F_1$、$F_2$, 直线$l$过$F_2$且与双曲线交于$A$、$B$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi}{2}$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt{3}$. 若$l$的斜率存在, 且$(\\overrightarrow{F_1A}+\\overrightarrow{F_1B}) \\cdot \\overrightarrow{AB}=0$, 求$l$的斜率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014706": { + "id": "014706", + "content": "在报名的$3$名男教师和$6$名女教师中, 选取$5$人参加义务献血, 若要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}.(结果用数值表示)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014707": { + "id": "014707", + "content": "若排列数$\\mathrm{P}_6^m=6 \\times 5 \\times 4$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014708": { + "id": "014708", + "content": "组合数$\\mathrm{C}_n^r$($n>r \\geq 1$, $n$、$r \\in \\mathbf{Z})$恒等于\\bracket{20}.\n\\fourch{$\\dfrac{r+1}{n+1} \\mathrm{C}_{n-1}^{r-1}$}{$(n+1)(r+1) \\mathrm{C}_{n-1}^{r-1}$}{$n r \\mathrm{C}_{n-1}^{r-1}$}{$\\dfrac{n}{r} \\mathrm{C}_{n-1}^{r-1}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014709": { + "id": "014709", + "content": "若$(1-2 x)^4=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4$, 则$a_0+a_4=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014710": { + "id": "014710", + "content": "从一副混合后的扑克牌($52$张)中随机抽取$1$张, 设$A$表示事件``抽得红桃K'', $B$表示事件``抽得为黑桃'', 则$P(A \\cup B)=$\\blank{50}. (结果用最简分数表示)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014711": { + "id": "014711", + "content": "已知事件$A$, 其对立事件记为$\\overline {A}$, 若$P(A)=0.5$, 则$P(\\overline {A})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014712": { + "id": "014712", + "content": "若事件$E$与$F$相互独立, 且$P(E)=P(F)=\\dfrac{1}{4}$, 则$P(E \\cap F)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014713": { + "id": "014713", + "content": "从$1,2,3,4,5$中任取$2$个不同的数, 事件$A$表示``取到的$2$个数之和为偶数'', 事件$B$表示``取到的$2$个数均为偶数'', 则$P(B | A)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014714": { + "id": "014714", + "content": "设$A$、$B$为两个事件, 若$P(B)=0.4$, $P(A)=0.5$, $P(B | A)=0.3$, 则$P(B | \\overline {A})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014715": { + "id": "014715", + "content": "一批产品的二等品率为$0.02$, 从这批产品中每次随机取一件, 有放回地抽取$100$次, 若用$X$表示抽到的二等品件数, 则$D[X]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014716": { + "id": "014716", + "content": "袋中有大小与质地均相同的$4$个红球、$m$个黄球、$n$个绿球, 从该袋内随机取出两个球, 记取出的红球数为$X$. 若取出的两个球都是红球的概率为$\\dfrac{1}{6}$, 一个红球和一个黄球的概率为$\\dfrac{1}{3}$, 则$E[X]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课28", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014717": { + "id": "014717", + "content": "若随机变量$X$服从正态分布$N(2, \\sigma^2)$, 且$P(X<4)=0.8$, 则$P(0|PF_2|$, 求$\\dfrac{|PF_1|}{|PF_2|}$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014729": { + "id": "014729", + "content": "从集合$\\{a, b, c\\}$的所有子集中依次随机取出$2$个不同的子集, 记为$M$和$N$, 求$M \\subseteq N$的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014730": { + "id": "014730", + "content": "从集合$\\{a, b, c\\}$的所有子集中随机取出$3$个不同的集合, 求满足``任取其中两个集合$M$和$N$, 总成立$M \\subseteq N$或$N \\subseteq M$''的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014731": { + "id": "014731", + "content": "设$a \\in \\mathbf{R}$, 求函数$y=x^2+|x-a|+1$, $x \\in \\mathbf{R}$的最小值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014732": { + "id": "014732", + "content": "已知圆$O: x^2+y^2=1$, 点$A(1,0)$绕圆心$O$旋转$\\dfrac{\\pi}{6}$至点$B$, 点$C(0,1)$, 则$|BC|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014733": { + "id": "014733", + "content": "设$a \\in \\mathbf{R}$, 求函数$y=x^3-a x$的单调增区间.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014734": { + "id": "014734", + "content": "已知数列$\\{a_n\\}$, $a_n=2^n+1$, 试判断$a_n$是否可能为$7$的倍数, 并说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014735": { + "id": "014735", + "content": "已知三个两两不重合的平面$\\alpha$、$\\beta$和$\\gamma$, 三者之间的交线共有\\blank{50}条.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014736": { + "id": "014736", + "content": "已知$a \\in \\mathbf{R}$, 记函数$y=\\sqrt{2-\\dfrac{x+3}{x+1}}$的定义域为$A$, 函数$y=\\lg [(x-a-1)(2 a-x)]$的定义域为$B$. 若$B \\subseteq A$, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014737": { + "id": "014737", + "content": "已知$a$、$b \\in \\mathbf{R}$, 讨论函数$y=a \\sin x+b$的奇偶性.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014738": { + "id": "014738", + "content": "设$a \\in \\mathbf{R}$, 方程$\\dfrac{1}{\\sqrt{\\dfrac{1}{x}+a}}=\\dfrac{1}{\\sqrt{(a-3) x+2 a-4}}$的解集中恰有一个元素, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014739": { + "id": "014739", + "content": "已知数列$\\{a_n\\}$是等比数列, $a_1=1, a_2=a$, 数列$\\{b_n\\}$满足$b_n=a_n \\cdot a_{n+1}$, 求数列$\\{b_n\\}$的前$n$项和$S_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014740": { + "id": "014740", + "content": "已知$a \\in \\mathbf{R}$, 设$x_1$、$x_2$是方程$2 x^2+3 a x+a^2-a=0$的两根, 试将$|x_1|+|x_2|$表示为$a$的函数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014741": { + "id": "014741", + "content": "若四面体$ABCD$各棱的长为$1$或$2$, 且该四面体不是正四面体, 求四面体体积$V$的所有可能的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课31", + "edit": [ + "20230312\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014742": { + "id": "014742", + "content": "已知命题$A$: 若$3 \\leq ab^a$, 请据此证明: 若$a$、$b$是正整数, $a