From 6b5a7aa72de93509f1e7fedda3883434023e178b Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Wed, 22 Feb 2023 12:13:49 +0800 Subject: [PATCH] 20230222 noon --- 工具/修改题目数据库.ipynb | 8 +- 工具/关键字筛选题号.ipynb | 10 +- 工具/寻找阶段末尾空闲题号.ipynb | 6 +- 工具/已用题号剔除.ipynb | 6 +- 工具/文本文件/题号筛选.txt | 2 +- 工具/添加关联题目.ipynb | 6 +- 工具/讲义生成.ipynb | 10 +- 工具/题号选题pdf生成.ipynb | 12 +- 题库0.3/Problems.json | 270 +++++++++++++++++++------------- 9 files changed, 192 insertions(+), 138 deletions(-) diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index a50abd8c..304c1113 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": 9, + "execution_count": 13, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 9, + "execution_count": 13, "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 = \"21481\"\n", + "problems = \"11388\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", @@ -51,7 +51,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 12, "metadata": {}, "outputs": [], "source": [ 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 451aba18..8c140445 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": 2, + "execution_count": 8, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 2, + "execution_count": 8, "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", - " {\"tags\":[\"第二单元\"],\"genre\":[\"选择题\"]}\n", + " {\"genre\":[\"解答题\"],\"tags\":[r\"第[二四]单元\"],\"content\":[r\"\\(3\\)\"],\"content_not\":[r\"\\(4\\)\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", @@ -96,7 +96,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -115,7 +115,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, 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 9e254d13..c0bbaccd 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 @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -11,8 +11,8 @@ "text": [ "首个空闲id: 14532 , 直至 020000\n", "首个空闲id: 22048 , 直至 030000\n", - "首个空闲id: 31236 , 直至 040000\n", - "首个空闲id: 40037 , 直至 999999\n" + "首个空闲id: 40037 , 直至 999999\n", + "首个空闲id: 31241 , 直至 040000\n" ] } ], diff --git a/工具/已用题号剔除.ipynb b/工具/已用题号剔除.ipynb index dae03779..e0af0205 100644 --- a/工具/已用题号剔除.ipynb +++ b/工具/已用题号剔除.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首行题目数量: 82\n", - "剩余题目数量: 78\n" + "首行题目数量: 301\n", + "剩余题目数量: 259\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 12021a25..24a39296 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 @@ -000049,000055,000062,000070,001152,001210,001287,002838,002853,002854,002863,002864,002865,002874,002875,002880,002902,002909,002913,002924,002925,002938,002954,002965,002987,003022,003601,003625,003667,003718,003730,003757,003770,003772,003801,003815,003862,003892,003907,003921,003936,003966,003980,003981,004074,004094,004116,004155,004157,004220,004284,004305,004347,004366,004368,004385,004401,004403,004422,004439,004440,004523,004525,004544,004546,004563,004631,004697,004739,004757,004760,004902,004907,005008,005016,005103,005104,005105,005193,005194,005195,005196,005197,005198,005283,005284,005285,005286,005287,005288,005296,005298,005299,005300,005301,005302,005303,005329,005330,005331,005332,005333,005362,005368,005369,005370,005371,005372,005373,005374,005375,005376,005377,005378,005379,005380,005381,005382,005383,005384,005385,005386,005424,005445,005446,005447,005448,005449,005450,005472,005473,005474,005475,005476,005477,005491,005492,005493,005494,005495,005496,005501,005502,005503,005504,005505,005506,005507,005508,005523,005524,005525,005526,005527,005528,005529,005530,005541,005542,005543,005563,005564,005565,005566,005567,005568,005569,005570,005571,005572,005573,005574,005612,005613,005614,005615,005616,005617,005618,005619,005620,005621,005653,005654,005655,005656,005657,005658,005659,005660,005687,005688,005689,005690,005691,005692,005693,005694,005695,005696,005713,005714,005718,005719,005720,005721,005722,005723,005724,005758,005759,005760,005761,005762,005763,005764,005782,005783,005784,005785,005786,007861,007892,007958,007959,007966,007984,008036,008037,008081,008082,008094,008095,008371,008372,008387,008388,008389,009476,009508,009537,010137,010149,010150,010173,010938,010977,010999,011041,011042,011043,011126,011145,011147,011148,011196,011212,011215,011233,011236,011256,011320,011404,011446,011529,011548,011594,011616,011687,011733,011911,011915,011927,011929,011947,011985,012001,012003,012022,012023,012090,012130,012132,012153,012173,012194,012215,012257,012259,012278,012300,012301,012302,012320,012344,012365,012367,012371,012374,012399,012401,012428,012430,012466,012471,012502,012543,012564,012605,012626,012628,012709,012733,012751,012783,012792,012836,012843,012849,012850,012860,012861,012863,012884,012885,012890,012895,012904,012905,012906,012907,012913,013297,013314,013342,013388,013418,013433,013447,013449,013477,013479,013508,013523,013539,013553,013554,013583,013600,013621,013768,013798,013799,013801,013829,013841,020358,020359,020365,020366,020377,020378,020379,020386,020387,020406,020407,020408,020417,020418,020419,020439,020440,020441,020442,020451,020465,020476,020477,020478,020487,020498,020499,020504,020524,020525,020526,020529,020554,020555,020561,020562,020566,020678,020679,020680,020681,021369,021388,030300,030307,030372,030379,030393,030400,030442,030667,030668,030671,030672,030673,030674,030675,030689,030690,030691,030692,030701,030722,030723,030728,030729,030755,030756 +000063,000064,000077,000325,001157,001185,001186,001212,001322,001820,002841,002906,002948,003033,003248,003308,003325,003609,003630,003651,003672,003693,003696,003924,003954,004005,004006,004079,004100,004121,004142,004163,004183,004205,004226,004268,004289,004310,004331,004352,004373,004407,004408,004445,004488,004530,004551,004571,004638,004660,004723,004744,004765,004980,005136,005549,005596,005597,005607,005716,005814,005830,005832,005834,005850,006728,006734,006799,006823,006824,006955,006959,006963,006974,006977,007867,007879,007920,007947,007970,008017,008018,008020,008050,008089,008377,008378,008394,008404,008420,009480,009486,009494,009889,009905,009918,010004,010117,010118,010120,010124,010130,010142,010157,010164,010170,010193,010199,010200,010757,010769,010790,010791,010795,010799,010800,010808,011005,011006,011047,011069,011111,011219,011220,011241,011262,011304,011325,011346,011367,011388,011408,011409,011430,011451,011472,011492,011493,011514,011535,011556,011577,011622,011646,011669,011692,011714,011715,011737,011738,011861,011865,011868,011874,011897,011907,011913,011925,011979,012007,012008,012028,012137,012159,012160,012201,012223,012264,012265,012286,012307,012326,012328,012350,012382,012505,012507,012528,012570,012591,012612,012654,012672,012675,012696,012759,012780,012845,012880,013007,013239,013526,013607,013649,013670,013691,013922,013943,013945,013946,013952,013953,013954,013959,013960,020409,020421,020443,020503,020514,020545,020559,020636,020649,020786,021372,021378,021380,021398,021413,030310,030311,030313,030325,030339,030341,030447,030727,030730,030731,030732,030733,030734,030735,030736,030737,030738,030739,030740,030882,030908,030909,030910,030911,030913,030914,030915,030916,030917,030918,030919,030920,030921,030922,030923,030924,030925,030926,030927 diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index 1eff4444..36ad0247 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -9,8 +9,8 @@ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id列表, 新id的数目不能小于旧id的数目---\"\"\"\n", - "old_ids = \"221,7528,8380,9626\"\n", - "new_ids = \"31237:31240\"\n", + "old_ids = \"12847,12851\"\n", + "new_ids = \"31241:31242\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\"\"\"---完成编辑后记得运行第二个单元格---\"\"\"\n", "\n", @@ -82,7 +82,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index cc409d8f..0a424239 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": 5, "metadata": {}, "outputs": [ { @@ -15,9 +15,9 @@ "题块 2 处理完毕.\n", "正在处理题块 3 .\n", "题块 3 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/高三下月考01预选_教师_20230220.tex\n", + "开始编译教师版本pdf文件: 临时文件/高三下学期测验02_教师_20230222.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/高三下月考01预选_学生_20230220.tex\n", + "开始编译学生版本pdf文件: 临时文件/高三下学期测验02_学生_20230222.tex\n", "0\n" ] } @@ -35,7 +35,7 @@ "\"\"\"---设置题块编号---\"\"\"\n", "\n", "problems = [\n", - "\"3675,12165,31237,31238,31239,4252,4256,31240,4499,4132,4217,4235\",\"8972,9082,12445,30691\",\"11301,10000,3691,11450,31236\"\n", + "\"4292,4143,4190,4169,31241,4128,4189,4129,31242,4130,4233,4174\",\"12021,12437,14087,12346\",\"20958,4657,11218,4508,11388\"\n", "]\n", "\n", "\"\"\"---设置结束---\"\"\"\n", @@ -49,7 +49,7 @@ "elif paper_type == 2:\n", " enumi_mode = 1 #设置模式(1为整卷统一编号, 0为每一部分从1开始编号)\n", " template_file = \"模板文件/测验周末卷模板.txt\" #设置模板文件名\n", - " exec_list = [(\"标题替换\",\"高三下月考01预选\")] #设置讲义标题\n", + " exec_list = [(\"标题替换\",\"高三下学期测验02\")] #设置讲义标题\n", " destination_file = \"临时文件/\"+exec_list[0][1] # 设置输出文件名\n", "elif paper_type == 3:\n", " enumi_mode = 0 #设置模式(1为整卷统一编号, 0为每一部分从1开始编号)\n", 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 8cdfbcc1..c7742398 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": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/高一前三次作业_教师用_20230221.tex\n", + "开始编译教师版本pdf文件: 临时文件/预选_教师用_20230222.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/高一前三次作业_学生用_20230221.tex\n", + "开始编译学生版本pdf文件: 临时文件/预选_学生用_20230222.tex\n", "0\n" ] } @@ -26,13 +26,15 @@ "\"\"\"---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", - "021441,021442,021443,021444,021445,021446,021447,021448,021449,021450,021451,021452,021453,021454,021455,021469,021470,021471,021472,021473,021474,021475,021476,021477,021478,021479,021480,021481\n", + "a\n", + "\n", + "\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/高一前三次作业\"\n", + "filename = \"临时文件/预选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 483362ff..08b4e54e 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -112124,7 +112124,7 @@ }, "004189": { "id": "004189", - "content": "已知无穷数列$a_n=\\dfrac 2{(-3)^n}$, $n\\in \\mathbf{N}^*$, 则数列$\\{a_n\\}$的各项和为\\blank{50}.", + "content": "已知无穷数列$a_n=\\dfrac 2{(-3)^n}$, $n\\in \\mathbf{N}$, $n\\ge 1$, 则数列$\\{a_n\\}$的各项和为\\blank{50}.", "objs": [ "K0405003X" ], @@ -114833,7 +114833,7 @@ }, "004292": { "id": "004292", - "content": "已知集合$P=\\{x|(x+1)(x–3)<0\\}$, $Q=\\{x||x|>2\\}$, 则$P\\cap Q=$\\blank{50}.", + "content": "已知集合$P=\\{x|(x+1)(x-3)<0\\}$, $Q=\\{x||x|>2\\}$, 则$P\\cap Q=$\\blank{50}.", "objs": [ "K0114001B", "K0117001B", @@ -283847,7 +283847,7 @@ }, "011388": { "id": "011388", - "content": "给定数列$\\{a_n\\}$, 若满足$a_1=a$($a>0$且$a\\ne 1$), 对于任意的$n, m\\in \\mathbf{N}^*$, 都有$a_{n+m}=a_n\\cdot a_m$, 则称数列$\\{a_n\\}$为指数数列.\\\\\n(1) 已知数列$\\{a_n\\}$, $\\{b_n\\}$的通项公式分别为$a_n=3\\cdot 2^{n-1}$, $b_n=3^n$, 试判断$\\{a_n\\}$, $\\{b_n\\}$是不是指数数列(需说明理由);\\\\\n(2) 若数列$\\{a_n\\}$满足: $a_1=2$, $a_2=4$, $a_{n+2}=3a_{n+1}-2a_n$, 证明: $\\{a_n\\}$是指数数列;\\\\\n(3) 若数列$\\{a_n\\}$是指数数列, $a_1=\\dfrac{t+3}{t+4}$($t\\in \\mathbf{N}^*$), 证明: 数列$\\{a_n\\}$中任意三项都不能构成等差数列.", + "content": "给定数列$\\{a_n\\}$, 若满足$a_1=a$($a>0$且$a\\ne 1$), 对于任意正整数$n, m$, 都有$a_{n+m}=a_n\\cdot a_m$, 则称数列$\\{a_n\\}$为指数数列.\\\\\n(1) 已知数列$\\{a_n\\}$, $\\{b_n\\}$的通项公式分别为$a_n=3\\cdot 2^{n-1}$, $b_n=3^n$, 试判断$\\{a_n\\}$, $\\{b_n\\}$是不是指数数列(需说明理由);\\\\\n(2) 若数列$\\{a_n\\}$满足: $a_1=2$, $a_2=4$, $a_{n+2}=3a_{n+1}-2a_n$, 证明: $\\{a_n\\}$是指数数列;\\\\\n(3) 若数列$\\{a_n\\}$是指数数列, $a_1=\\dfrac{t+3}{t+4}$($t\\in \\mathbf{N}$, $t\\ge 1$), 证明: 数列$\\{a_n\\}$中任意三项都不能构成等差数列.", "objs": [], "tags": [ "第四单元" @@ -317337,7 +317337,9 @@ "20230118\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031241" + ], "remark": "", "space": "" }, @@ -317473,7 +317475,9 @@ "20230118\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031242" + ], "remark": "", "space": "" }, @@ -429276,6 +429280,158 @@ "remark": "", "space": "" }, + "031237": { + "id": "031237", + "content": "从编号分别为$1$、$2$、$3$、$4$、$5$、$6$的$6$个大小与质地相同的小球中随机取出$3$个, 则恰有$2$个小球编号相邻的概率为\\blank{50}.", + "objs": [ + "K0803002B", + "K0818001X" + ], + "tags": [ + "第八单元", + "概率" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "教材复习题-20230220修改", + "edit": [ + "20220624\t王伟叶, 余利成", + "20230220\t王伟叶" + ], + "same": [], + "related": [ + "000221" + ], + "remark": "", + "space": "" + }, + "031238": { + "id": "031238", + "content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{100}$的展开式中, 有理项有\\blank{50}项.", + "objs": [ + "K0819002X" + ], + "tags": [ + "第八单元", + "二项式定理" + ], + "genre": "填空题", + "ans": "有$17$项是有理项", + "solution": "考虑$(x^{\\frac 12}+x^{-\\frac 13})^{100}$展开式的通项$T_{r+1}=\\mathrm{C}_{100}^rx^{\\frac{100-r}2}\\cdot (x^{-\\frac 13})^r=\\mathrm{C}_{100}^rx^{50-\\dfrac{3r}6}$.\n令$r=6k$($k\\in \\mathbf{Z}$), 则$0\\le 6k\\le 100$, 即$r=0,6,12,\\cdots ,96$.\n因此共有$17$个有理项.", + "duration": -1, + "usages": [], + "origin": "代数精编第九章排列组合-20230220修改", + "edit": [ + "20220720\t王伟叶", + "20230220\t王伟叶" + ], + "same": [], + "related": [ + "007528" + ], + "remark": "", + "space": "" + }, + "031239": { + "id": "031239", + "content": "已知$\\tan \\alpha =3$, 则$\\dfrac 1{\\sin ^2\\alpha +2\\sin \\alpha \\cos \\alpha}$的值为\\blank{50}.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "二期课改练习册高一第二学期-20230220修改", + "edit": [ + "20220726\t王伟叶", + "20230220\t王伟叶" + ], + "same": [], + "related": [ + "009553", + "008380" + ], + "remark": "", + "space": "" + }, + "031240": { + "id": "031240", + "content": "设向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=5$, $|\\overrightarrow b|=6$, $(\\overrightarrow a+\\overrightarrow b)\\cdot \\overrightarrow b=21$, 则$\\langle \\overrightarrow a, \\overrightarrow b\\rangle =$\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "新教材必修第二册课堂练习-20230220修改", + "edit": [ + "20220730\t王伟叶", + "20230220\t王伟叶" + ], + "same": [], + "related": [ + "009626" + ], + "remark": "", + "space": "" + }, + "031241": { + "id": "031241", + "content": "函数$y=\\dfrac{x^2+7}{\\sqrt{x^2+4}}$的值域是\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义-20230222修改", + "edit": [ + "20230118\t王伟叶", + "20230222\t王伟叶" + ], + "same": [], + "related": [ + "012847" + ], + "remark": "", + "space": "" + }, + "031242": { + "id": "031242", + "content": "已知$f(x)=\\begin{cases}(2-a) x+1, & x<1, \\\\ 2ax, & x \\geq 1\\end{cases}$满足: 对任意$x_1 \\neq x_2$, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义-20230222修改", + "edit": [ + "20230118\t王伟叶", + "20230222\t王伟叶" + ], + "same": [], + "related": [ + "012851" + ], + "remark": "", + "space": "" + }, "040001": { "id": "040001", "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}", @@ -429959,109 +430115,5 @@ "related": [], "remark": "", "space": "" - }, - "031237": { - "id": "031237", - "content": "从编号分别为$1$、$2$、$3$、$4$、$5$、$6$的$6$个大小与质地相同的小球中随机取出$3$个, 则恰有$2$个小球编号相邻的概率为\\blank{50}.", - "objs": [ - "K0803002B", - "K0818001X" - ], - "tags": [ - "第八单元", - "概率" - ], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "教材复习题-20230220修改", - "edit": [ - "20220624\t王伟叶, 余利成", - "20230220\t王伟叶" - ], - "same": [], - "related": [ - "000221" - ], - "remark": "", - "space": "" - }, - "031238": { - "id": "031238", - "content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{100}$的展开式中, 有理项有\\blank{50}项.", - "objs": [ - "K0819002X" - ], - "tags": [ - "第八单元", - "二项式定理" - ], - "genre": "填空题", - "ans": "有$17$项是有理项", - "solution": "考虑$(x^{\\frac 12}+x^{-\\frac 13})^{100}$展开式的通项$T_{r+1}=\\mathrm{C}_{100}^rx^{\\frac{100-r}2}\\cdot (x^{-\\frac 13})^r=\\mathrm{C}_{100}^rx^{50-\\dfrac{3r}6}$.\n令$r=6k$($k\\in \\mathbf{Z}$), 则$0\\le 6k\\le 100$, 即$r=0,6,12,\\cdots ,96$.\n因此共有$17$个有理项.", - "duration": -1, - "usages": [], - "origin": "代数精编第九章排列组合-20230220修改", - "edit": [ - "20220720\t王伟叶", - "20230220\t王伟叶" - ], - "same": [], - "related": [ - "007528" - ], - "remark": "", - "space": "" - }, - "031239": { - "id": "031239", - "content": "已知$\\tan \\alpha =3$, 则$\\dfrac 1{\\sin ^2\\alpha +2\\sin \\alpha \\cos \\alpha}$的值为\\blank{50}.", - "objs": [], - "tags": [ - "第三单元" - ], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "二期课改练习册高一第二学期-20230220修改", - "edit": [ - "20220726\t王伟叶", - "20230220\t王伟叶" - ], - "same": [], - "related": [ - "009553", - "008380" - ], - "remark": "", - "space": "" - }, - "031240": { - "id": "031240", - "content": "设向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=5$, $|\\overrightarrow b|=6$, $(\\overrightarrow a+\\overrightarrow b)\\cdot \\overrightarrow b=21$, 则$\\langle \\overrightarrow a, \\overrightarrow b\\rangle =$\\blank{50}.", - "objs": [], - "tags": [ - "第五单元" - ], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "新教材必修第二册课堂练习-20230220修改", - "edit": [ - "20220730\t王伟叶", - "20230220\t王伟叶" - ], - "same": [], - "related": [ - "009626" - ], - "remark": "", - "space": "" } } \ No newline at end of file