diff --git a/工具/修改题目数据库.py b/工具/修改题目数据库.py index 954edbf5..9bd32097 100644 --- a/工具/修改题目数据库.py +++ b/工具/修改题目数据库.py @@ -1,6 +1,6 @@ import os,re,json """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""" -problems = "514" +problems = "14841" def generate_number_set(string,dict): string = re.sub(r"[\n\s]","",string) diff --git a/工具/关键字筛选题号.py b/工具/关键字筛选题号.py index 279a7845..9a4921ef 100644 --- a/工具/关键字筛选题号.py +++ b/工具/关键字筛选题号.py @@ -2,7 +2,7 @@ import os,re,json """---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---""" keywords_dict_table = [ - {"edit":[r"\d{9,}"]} + {"origin":[r"2023"],"origin2":[r"二模"],"tags":[r"第九单元"]} ] """---关键字设置完毕---""" # 示例： keywords_dict_table = [ diff --git a/工具/单元标记转换.py b/工具/单元标记转换.py index cd0f4249..14f84037 100644 --- a/工具/单元标记转换.py +++ b/工具/单元标记转换.py @@ -1,4 +1,4 @@ -filename = r"D:\temp\tag.txt" +filename = r"D:\temp\units.txt" # 设置一个从excel文件复制出来的txt # 每一行的格式如下: <题号>\t<对应单元(若干位0-9的数字)> diff --git a/工具/批量生成题目pdf.py b/工具/批量生成题目pdf.py index 3e784d13..630f3b2a 100644 --- a/工具/批量生成题目pdf.py +++ b/工具/批量生成题目pdf.py @@ -11,50 +11,62 @@ answered = True #目录和文件的分隔务必用/ directory = "临时文件/" # filename = "高三二模前易错题" -filename = "2022学年度下学期高一高二新增题目及校本作业" +filename = "2023届二模分类题目汇编(缺奉贤徐汇)" """---设置文件名结束---""" """---设置题目列表---""" #字典字段为文件名, 之后为内容的题号 -problems_dict = { - -"2025届高一下学期校本作业":"21441:22047", -"2024届高二下学期周末卷01":"40001:40017", -"2025届高一下学期周末卷01":"40018:40036", -"2024届高二下学期周末卷02":"40037:40056", -"2025届高一下学期周末卷02":"40057:40082", -"2025届高一下学期周末卷03":"40083:40104", -"2025届高一下学期周末卷03小测":"40105:40112", -"2025届高一下学期周末卷04旧版":"40113:40130", -"2025届高一下学期周末卷04小测":"40131:40139", -"2024届高二下学期周末卷03":"40140:40160", -"2024届高二上学期期末考试":"31267:31287", -"2025届高一上学期期末考试":"31288:31308", -"2024届高二下学期周末卷04":"40161:40180", -"2025届高一下学期周末卷04":"40181:40201", -"2024届高二下学期周末卷05":"40202:40225", -"2025届高一下学期周末卷05":"40226:40245", -"2024届空间向量校本作业":"22048:22083", -"2024届二项式定理校本作业":"22084:22105", -"2025届高一下学期周末卷05小测":"40246:40255", -"2025届高一下学期周末卷06":"40256:40273", -"2025届高一下学期周末卷06小测":"40274:40282", -"2025届高一下学期期中复习一(集合逻辑不等式)":"40283:40298", -"2024届高二下学期周末卷06":"40299:40316", -"2024届高二下学期周末卷07":"40317:40335", -"2025届高一下学期测验01":"40336:40349", -"2025届高一下学期测验02":"40350:40367", -"2025届高一下学期期中复习二(幂指对函数)":"40368:40386", -"2025届高一下学期周末卷02小测":"40387:40395", -"2025届高一下学期周末卷07":"40396:40413", -"2025届高一下学期周末卷07小测":"40414:40421", -"2025届高一下学期周末卷08":"40527:40551", -"2024届高二下学期周末卷08":"40570:40587", -"2024届高二下学期周末卷09":"40588:40604" - +problems_dict = { +"第一单元":"014784,014796,014805,014807,014809,014828,014838,014996,015008,015017,015018,015029,015038,015039,015046,015050,015059,015084,015101,015122,015133,015143,015155,015164,015165,015185,015186,015206,015220", +"第二单元":"014788,014790,014798,014802,014814,014816,014817,014825,014831,014836,014837,014839,015004,015007,015011,015016,015019,015027,015030,015037,015044,015048,015058,015061,015065,015070,015079,015086,015087,015091,015100,015102,015108,015115,015121,015128,015135,015137,015142,015147,015153,015163,015171,015172,015173,015176,015178,015184,015188,015190,015192,015193,015194,015216,015221,015224,015226", +"第三单元":"014792,014794,014808,014822,014829,014846,015003,015005,015025,015033,015047,015054,015060,015067,015075,015096,015105,015107,015126,015127,015144,015148,015150,015158,015180,015205,015210,015215,015222", +"第四单元":"014786,014804,014820,014833,015012,015021,015024,015031,015053,015064,015074,015089,015095,015116,015117,015123,015138,015160,015179,015200,015214", +"第五单元":"014785,014795,014806,014815,014826,014830,014843,014997,015006,015020,015028,015041,015043,015049,015062,015069,015081,015082,015092,015103,015112,015113,015124,015129,015146,015154,015166,015175,015195,015198,015207,015208,015217", +"第六单元":"014789,014799,014801,014819,014821,014834,014840,014842,014998,015010,015013,015032,015034,015052,015055,015066,015073,015076,015080,015085,015088,015097,015110,015118,015131,015134,015139,015149,015156,015159,015167,015181,015189,015199,015202,015212,015223", +"第七单元":"014793,014803,014813,014824,014827,014845,015001,015015,015036,015040,015057,015063,015071,015078,015083,015090,015099,015104,015111,015120,015125,015136,015140,015152,015157,015162,015174,015183,015196,015204,015218,015225", +"第八单元":"014787,014800,014810,014818,014823,014832,014835,014841,014999,015000,015002,015014,015022,015023,015042,015045,015056,015068,015072,015093,015098,015109,015114,015132,015161,015168,015169,015170,015182,015187,015191,015201,015209,015211,015213", +"第九单元":"014791,014797,014811,014812,014823,014844,015009,015026,015035,015051,015077,015094,015098,015106,015119,015130,015141,015145,015151,015177,015197,015203,015219" } +# problems_dict = { + +# "2025届高一下学期校本作业":"21441:22047", +# "2024届高二下学期周末卷01":"40001:40017", +# "2025届高一下学期周末卷01":"40018:40036", +# "2024届高二下学期周末卷02":"40037:40056", +# "2025届高一下学期周末卷02":"40057:40082", +# "2025届高一下学期周末卷03":"40083:40104", +# "2025届高一下学期周末卷03小测":"40105:40112", +# "2025届高一下学期周末卷04旧版":"40113:40130", +# "2025届高一下学期周末卷04小测":"40131:40139", +# "2024届高二下学期周末卷03":"40140:40160", +# "2024届高二上学期期末考试":"31267:31287", +# "2025届高一上学期期末考试":"31288:31308", +# "2024届高二下学期周末卷04":"40161:40180", +# "2025届高一下学期周末卷04":"40181:40201", +# "2024届高二下学期周末卷05":"40202:40225", +# "2025届高一下学期周末卷05":"40226:40245", +# "2024届空间向量校本作业":"22048:22083", +# "2024届二项式定理校本作业":"22084:22105", +# "2025届高一下学期周末卷05小测":"40246:40255", +# "2025届高一下学期周末卷06":"40256:40273", +# "2025届高一下学期周末卷06小测":"40274:40282", +# "2025届高一下学期期中复习一(集合逻辑不等式)":"40283:40298", +# "2024届高二下学期周末卷06":"40299:40316", +# "2024届高二下学期周末卷07":"40317:40335", +# "2025届高一下学期测验01":"40336:40349", +# "2025届高一下学期测验02":"40350:40367", +# "2025届高一下学期期中复习二(幂指对函数)":"40368:40386", +# "2025届高一下学期周末卷02小测":"40387:40395", +# "2025届高一下学期周末卷07":"40396:40413", +# "2025届高一下学期周末卷07小测":"40414:40421", +# "2025届高一下学期周末卷08":"40527:40551", +# "2024届高二下学期周末卷08":"40570:40587", +# "2024届高二下学期周末卷09":"40588:40604" + +# } + """---设置题目列表结束---""" diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 50faba37..62dbcf00 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,542 +1,1180 @@ -usages +tags -000778 -20230412 2023届高三11班 0.792 +14784 +第一单元 -001253 -20230412 2023届高三11班 0.667 -001325 -20230412 2023届高三11班 0.667 +14785 +第五单元 -003747 -20230412 2023届高三11班 0.792 -013721 -20230412 2023届高三11班 0.542 +14786 +第四单元 -031392 -20230412 2023届高三11班 0.500 -010938 -20230412 2023届高三11班 0.583 +14787 +第八单元 -011148 -20230412 2023届高三11班 0.875 -004157 -20230412 2023届高三11班 0.833 +14788 +第二单元 -010197 -20230412 2023届高三11班 0.667 -030060 -20230412 2023届高三11班 0.417 +14789 +第六单元 -001339 -20230412 2023届高三11班 0.333 -002918 -20230412 2023届高三11班 0.458 +14790 +第二单元 -030327 -20230412 2023届高三11班 0.917 0.875 0.917 0.958 0.750 0.833 -004009 -20230412 2023届高三11班 0.500 +14791 +第九单元 -000778 -20230412 2023届高三10班 0.879 -001253 -20230412 2023届高三10班 0.818 +14792 +第三单元 -001325 -20230412 2023届高三10班 0.667 -003747 -20230412 2023届高三10班 0.818 +14793 +第七单元 -013721 -20230412 2023届高三10班 0.758 -031392 -20230412 2023届高三10班 0.455 +14794 +第三单元 -010938 -20230412 2023届高三10班 0.515 -011148 -20230412 2023届高三10班 0.849 +14795 +第五单元 -004157 -20230412 2023届高三10班 0.697 -010197 -20230412 2023届高三10班 0.667 +14796 +第一单元 -030060 -20230412 2023届高三10班 0.576 -001339 -20230412 2023届高三10班 0.333 +14797 +第九单元 -002918 -20230412 2023届高三10班 0.697 -030327 -20230412 2023届高三10班 0.879 0.970 0.939 0.939 0.849 0.879 +14798 +第二单元 -004009 -20230412 2023届高三10班 0.424 -000778 -20230412 2023届高三01班 0.871 +14799 +第六单元 -001253 -20230412 2023届高三01班 0.742 -001325 -20230412 2023届高三01班 0.935 +14800 +第八单元 -003747 -20230412 2023届高三01班 0.839 -013721 -20230412 2023届高三01班 0.742 +14801 +第六单元 -031392 -20230412 2023届高三01班 0.613 -010938 -20230412 2023届高三01班 0.839 +14802 +第二单元 -011148 -20230412 2023届高三01班 0.968 -004157 -20230412 2023届高三01班 0.710 +14803 +第七单元 -010197 -20230412 2023届高三01班 0.677 -030060 -20230412 2023届高三01班 0.710 +14804 +第四单元 -001339 -20230412 2023届高三01班 0.581 -002918 -20230412 2023届高三01班 0.774 +14805 +第一单元 -030327 -20230412 2023届高三01班 0.968 1.000 0.968 1.000 0.677 1.000 -004009 -20230412 2023届高三01班 0.645 +14806 +第五单元 -000778 -20230412 2023届高三12班 0.762 -001253 -20230412 2023届高三12班 0.762 +14807 +第一单元 -001325 -20230412 2023届高三12班 0.809 -003747 -20230412 2023届高三12班 0.905 +14808 +第三单元 -013721 -20230412 2023届高三12班 0.667 -031392 -20230412 2023届高三12班 0.714 +14809 +第一单元 -010938 -20230412 2023届高三12班 0.571 -011148 -20230412 2023届高三12班 0.905 +14810 +第八单元 -004157 -20230412 2023届高三12班 0.809 -010197 -20230412 2023届高三12班 0.476 +14811 +第九单元 -030060 -20230412 2023届高三12班 0.571 -001339 -20230412 2023届高三12班 0.286 +14812 +第九单元 -002918 -20230412 2023届高三12班 0.571 -030327 -20230412 2023届高三12班 0.857 1.000 0.857 1.000 0.905 0.952 +14813 +第七单元 -004009 -20230412 2023届高三12班 0.524 -000778 -20230412 2023届高三02班 0.806 +14814 +第二单元 -001253 -20230412 2023届高三02班 0.710 -001325 -20230412 2023届高三02班 0.871 +14815 +第五单元 -003747 -20230412 2023届高三02班 0.806 -013721 -20230412 2023届高三02班 0.484 +14816 +第二单元 -031392 -20230412 2023届高三02班 0.806 -010938 -20230412 2023届高三02班 0.871 +14817 +第二单元 -011148 -20230412 2023届高三02班 0.839 -004157 -20230412 2023届高三02班 0.806 +14818 +第八单元 -010197 -20230412 2023届高三02班 0.742 -030060 -20230412 2023届高三02班 0.742 +14819 +第六单元 -001339 -20230412 2023届高三02班 0.323 -002918 -20230412 2023届高三02班 0.806 +14820 +第四单元 -030327 -20230412 2023届高三02班 0.839 0.968 0.871 0.903 0.742 0.806 -004009 -20230412 2023届高三02班 0.710 +14821 +第六单元 -000778 -20230412 2023届高三03班 0.962 -001253 -20230412 2023届高三03班 0.500 +14822 +第三单元 -001325 -20230412 2023届高三03班 0.962 -003747 -20230412 2023届高三03班 0.731 +14823 +第八单元 +第九单元 -013721 -20230412 2023届高三03班 0.462 -031392 -20230412 2023届高三03班 0.692 +14824 +第七单元 -010938 -20230412 2023届高三03班 0.615 -011148 -20230412 2023届高三03班 0.962 +14825 +第二单元 -004157 -20230412 2023届高三03班 0.885 -010197 -20230412 2023届高三03班 0.692 +14826 +第五单元 -030060 -20230412 2023届高三03班 0.731 -001339 -20230412 2023届高三03班 0.615 +14827 +第七单元 -002918 -20230412 2023届高三03班 0.615 -030327 -20230412 2023届高三03班 0.885 1.000 0.654 1.000 0.885 0.923 +14828 +第一单元 -004009 -20230412 2023届高三03班 0.692 -000778 -20230412 2023届高三04班 0.809 +14829 +第三单元 -001253 -20230412 2023届高三04班 0.619 -001325 -20230412 2023届高三04班 0.809 +14830 +第五单元 -003747 -20230412 2023届高三04班 0.762 -013721 -20230412 2023届高三04班 0.667 +14831 +第二单元 -031392 -20230412 2023届高三04班 0.476 -010938 -20230412 2023届高三04班 0.571 +14832 +第八单元 -011148 -20230412 2023届高三04班 0.857 -004157 -20230412 2023届高三04班 0.857 +14833 +第四单元 -010197 -20230412 2023届高三04班 0.191 -030060 -20230412 2023届高三04班 0.524 +14834 +第六单元 -001339 -20230412 2023届高三04班 0.238 -002918 -20230412 2023届高三04班 0.714 +14835 +第八单元 -030327 -20230412 2023届高三04班 0.714 0.905 1.000 0.905 0.952 0.857 -004009 -20230412 2023届高三04班 0.381 +14836 +第二单元 -000778 -20230412 2023届高三05班 0.778 -001253 -20230412 2023届高三05班 0.722 +14837 +第二单元 -001325 -20230412 2023届高三05班 0.750 -003747 -20230412 2023届高三05班 0.806 +14838 +第一单元 -013721 -20230412 2023届高三05班 0.778 -031392 -20230412 2023届高三05班 0.472 +14839 +第二单元 -010938 -20230412 2023届高三05班 0.722 -011148 -20230412 2023届高三05班 0.944 +14840 +第六单元 -004157 -20230412 2023届高三05班 0.694 -010197 -20230412 2023届高三05班 0.556 +14841 +第八单元 -030060 -20230412 2023届高三05班 0.806 -001339 -20230412 2023届高三05班 0.361 +14842 +第六单元 -002918 -20230412 2023届高三05班 0.722 -030327 -20230412 2023届高三05班 0.944 0.833 0.861 0.944 0.611 0.889 +14843 +第五单元 -004009 -20230412 2023届高三05班 0.361 -000778 -20230412 2023届高三06班 0.974 +14844 +第九单元 -001253 -20230412 2023届高三06班 0.744 -001325 -20230412 2023届高三06班 0.872 +14845 +第七单元 -003747 -20230412 2023届高三06班 0.846 -013721 -20230412 2023届高三06班 0.590 +14846 +第三单元 -031392 -20230412 2023届高三06班 0.641 -010938 -20230412 2023届高三06班 0.744 +14996 +第一单元 -011148 -20230412 2023届高三06班 0.897 -004157 -20230412 2023届高三06班 0.769 +14997 +第五单元 -010197 -20230412 2023届高三06班 0.718 -030060 -20230412 2023届高三06班 0.667 +14998 +第六单元 -001339 -20230412 2023届高三06班 0.436 -002918 -20230412 2023届高三06班 0.795 +14999 +第八单元 -030327 -20230412 2023届高三06班 0.949 0.949 0.923 1.000 0.692 0.846 -004009 -20230412 2023届高三06班 0.692 +15000 +第八单元 -000778 -20230412 2023届高三07班 0.857 -001253 -20230412 2023届高三07班 0.571 +15001 +第七单元 -001325 -20230412 2023届高三07班 0.714 -003747 -20230412 2023届高三07班 0.893 +15002 +第八单元 -013721 -20230412 2023届高三07班 0.536 -031392 -20230412 2023届高三07班 0.643 +15003 +第三单元 -010938 -20230412 2023届高三07班 0.536 -011148 -20230412 2023届高三07班 0.821 +15004 +第二单元 -004157 -20230412 2023届高三07班 0.714 -010197 -20230412 2023届高三07班 0.107 +15005 +第三单元 -030060 -20230412 2023届高三07班 0.536 -001339 -20230412 2023届高三07班 0.250 +15006 +第五单元 -002918 -20230412 2023届高三07班 0.750 -030327 -20230412 2023届高三07班 0.821 0.964 0.964 1.000 0.893 0.964 +15007 +第二单元 -004009 -20230412 2023届高三07班 0.429 -000778 -20230412 2023届高三08班 0.933 +15008 +第一单元 -001253 -20230412 2023届高三08班 0.700 -001325 -20230412 2023届高三08班 0.833 +15009 +第九单元 -003747 -20230412 2023届高三08班 1.000 -013721 -20230412 2023届高三08班 0.500 +15010 +第六单元 -031392 -20230412 2023届高三08班 0.567 -010938 -20230412 2023届高三08班 0.767 +15011 +第二单元 -011148 -20230412 2023届高三08班 0.700 -004157 -20230412 2023届高三08班 0.767 +15012 +第四单元 -010197 -20230412 2023届高三08班 0.533 -030060 -20230412 2023届高三08班 0.400 +15013 +第六单元 -001339 -20230412 2023届高三08班 0.433 -002918 -20230412 2023届高三08班 0.500 +15014 +第八单元 -030327 -20230412 2023届高三08班 0.933 1.000 0.933 0.967 0.900 0.933 -004009 -20230412 2023届高三08班 0.533 +15015 +第七单元 -000778 -20230412 2023届高三09班 0.800 -001253 -20230412 2023届高三09班 0.867 +15016 +第二单元 -001325 -20230412 2023届高三09班 0.867 -003747 -20230412 2023届高三09班 0.967 +15017 +第一单元 -013721 -20230412 2023届高三09班 0.800 -031392 -20230412 2023届高三09班 0.667 +15018 +第一单元 -010938 -20230412 2023届高三09班 0.600 -011148 -20230412 2023届高三09班 0.900 +15019 +第二单元 -004157 -20230412 2023届高三09班 0.767 -010197 -20230412 2023届高三09班 0.467 +15020 +第五单元 -030060 -20230412 2023届高三09班 0.433 -001339 -20230412 2023届高三09班 0.433 +15021 +第四单元 -002918 -20230412 2023届高三09班 0.600 -030327 -20230412 2023届高三09班 0.700 0.967 1.000 0.833 0.867 0.933 +15022 +第八单元 + + +15023 +第八单元 + + +15024 +第四单元 + + +15025 +第三单元 + + +15026 +第九单元 + + +15027 +第二单元 + + +15028 +第五单元 + + +15029 +第一单元 + + +15030 +第二单元 + + +15031 +第四单元 + + +15032 +第六单元 + + +15033 +第三单元 + + +15034 +第六单元 + + +15035 +第九单元 + + +15036 +第七单元 + + +15037 +第二单元 + + +15038 +第一单元 + + +15039 +第一单元 + + +15040 +第七单元 + + +15041 +第五单元 + + +15042 +第八单元 + + +15043 +第五单元 + + +15044 +第二单元 + + +15045 +第八单元 + + +15046 +第一单元 + + +15047 +第三单元 + + +15048 +第二单元 + + +15049 +第五单元 + + +15050 +第一单元 + + +15051 +第九单元 + + +15052 +第六单元 + + +15053 +第四单元 + + +15054 +第三单元 + + +15055 +第六单元 + + +15056 +第八单元 + + +15057 +第七单元 + + +15058 +第二单元 + + +15059 +第一单元 + + +15060 +第三单元 + + +15061 +第二单元 + + +15062 +第五单元 + + +15063 +第七单元 + + +15064 +第四单元 + + +15065 +第二单元 + + +15066 +第六单元 + + +15067 +第三单元 + + +15068 +第八单元 + + +15069 +第五单元 + + +15070 +第二单元 + + +15071 +第七单元 + + +15072 +第八单元 + + +15073 +第六单元 + + +15074 +第四单元 + + +15075 +第三单元 + + +15076 +第六单元 + + +15077 +第九单元 + + +15078 +第七单元 + + +15079 +第二单元 + + +15080 +第六单元 + + +15081 +第五单元 + + +15082 +第五单元 + + +15083 +第七单元 + + +15084 +第一单元 + + +15085 +第六单元 + + +15086 +第二单元 + + +15087 +第二单元 + + +15088 +第六单元 + + +15089 +第四单元 + + +15090 +第七单元 + + +15091 +第二单元 + + +15092 +第五单元 + + +15093 +第八单元 + + +15094 +第九单元 + + +15095 +第四单元 + + +15096 +第三单元 + + +15097 +第六单元 + + +15098 +第八单元 +第九单元 + + +15099 +第七单元 + + +15100 +第二单元 + + +15101 +第一单元 + + +15102 +第二单元 + + +15103 +第五单元 + + +15104 +第七单元 + + +15105 +第三单元 + + +15106 +第九单元 + + +15107 +第三单元 + + +15108 +第二单元 + + +15109 +第八单元 + + +15110 +第六单元 + + +15111 +第七单元 + + +15112 +第五单元 + + +15113 +第五单元 + + +15114 +第八单元 + + +15115 +第二单元 + + +15116 +第四单元 + + +15117 +第四单元 + + +15118 +第六单元 + + +15119 +第九单元 + + +15120 +第七单元 + + +15121 +第二单元 + + +15122 +第一单元 + + +15123 +第四单元 + + +15124 +第五单元 + + +15125 +第七单元 + + +15126 +第三单元 + + +15127 +第三单元 + + +15128 +第二单元 + + +15129 +第五单元 + + +15130 +第九单元 + + +15131 +第六单元 + + +15132 +第八单元 + + +15133 +第一单元 + + +15134 +第六单元 + + +15135 +第二单元 + + +15136 +第七单元 + + +15137 +第二单元 + + +15138 +第四单元 + + +15139 +第六单元 + + +15140 +第七单元 + + +15141 +第九单元 + + +15142 +第二单元 + + +15143 +第一单元 + + +15144 +第三单元 + + +15145 +第九单元 + + +15146 +第五单元 + + +15147 +第二单元 + + +15148 +第三单元 + + +15149 +第六单元 + + +15150 +第三单元 + + +15151 +第九单元 + + +15152 +第七单元 + + +15153 +第二单元 + + +15154 +第五单元 + + +15155 +第一单元 + + +15156 +第六单元 + + +15157 +第七单元 + + +15158 +第三单元 + + +15159 +第六单元 + + +15160 +第四单元 + + +15161 +第八单元 + + +15162 +第七单元 + + +15163 +第二单元 + + +15164 +第一单元 + + +15165 +第一单元 + + +15166 +第五单元 + + +15167 +第六单元 + + +15168 +第八单元 + + +15169 +第八单元 + + +15170 +第八单元 + + +15171 +第二单元 + + +15172 +第二单元 + + +15173 +第二单元 + + +15174 +第七单元 + + +15175 +第五单元 + + +15176 +第二单元 + + +15177 +第九单元 + + +15178 +第二单元 + + +15179 +第四单元 + + +15180 +第三单元 + + +15181 +第六单元 + + +15182 +第八单元 + + +15183 +第七单元 + + +15184 +第二单元 + + +15185 +第一单元 + + +15186 +第一单元 + + +15187 +第八单元 + + +15188 +第二单元 + + +15189 +第六单元 + + +15190 +第二单元 + + +15191 +第八单元 + + +15192 +第二单元 + + +15193 +第二单元 + + +15194 +第二单元 + + +15195 +第五单元 + + +15196 +第七单元 + + +15197 +第九单元 + + +15198 +第五单元 + + +15199 +第六单元 + + +15200 +第四单元 + + +15201 +第八单元 + + +15202 +第六单元 + + +15203 +第九单元 + + +15204 +第七单元 + + +15205 +第三单元 + + +15206 +第一单元 + + +15207 +第五单元 + + +15208 +第五单元 + + +15209 +第八单元 + + +15210 +第三单元 + + +15211 +第八单元 + + +15212 +第六单元 + + +15213 +第八单元 + + +15214 +第四单元 + + +15215 +第三单元 + + +15216 +第二单元 + + +15217 +第五单元 + + +15218 +第七单元 + + +15219 +第九单元 + + +15220 +第一单元 + + +15221 +第二单元 + + +15222 +第三单元 + + +15223 +第六单元 + + +15224 +第二单元 + + +15225 +第七单元 + + +15226 +第二单元 -004009 -20230412 2023届高三09班 0.300 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 676d24f0..53f9bda4 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 @@ -014826,014827,014828,014829,014830,014831,014832,014833,014834,014835,014836,014837,014838,014839,014840,014841,014842,014843,014844,014845,014846,014847,014848,014849,014850,014851,014852,014853,014854,014855,014856,014857,014858,014859,014860,014861,014862,014863,014864,014865,014866,014867,014868,014869,014870,014871,014872,014873,014874,014875,014876,014877,014878,014879,014880,014881,014882,014883,014884,014885,014886,014887,014888,014889,014890,014891,014892,014893,014894,014895,014896,014897,014898,014899,014900,014901,014902,014903,014904,014905,014906,014907,014908,014909,014910,014911,014912,014913,014914,014915,014916,014917,014918,014919,014920,014921,014922,014923,014924,014925,014926,014927,014928,014929,014930,014931,014932,014933,014934,014935,014936,014937,014938,014939,014940,014941,014942,014943,014944,014945,014946,014947,014948,014949,014950,014951,014952,014953,014954,014955,014956,014957,014958,014959,014960,014961,014962,014963,014964,014965,014966,014967,014968,014969,014970,014971,014972,014973,014974,014975,014976,014977,014978,014979,014980,014981,014982,014983,014984,014985,014986,014987,014988,014989,014990,014991,014992,014993,014994,014995,014996,014997,014998,014999,015000,015001,015002,015003,015004,015005,015006,015007,015008,015009,015010,015011,015012,015013,015014,015015,015016,015017,015018,015019,015020,015021,015022,015023,015024,015025,015026,015027,015028,015029,015030,015031,015032,015033,015034,015035,015036,015037,015038,015039,015040,015041,015042,015043,015044,015045,015046,015047,015048,015049,015050,015051,015052,015053,015054,015055,015056,015057,015058,015059,015060,015061,015062,015063,015064,015065,015066,015067,015068,015069,015070,015071,015072,015073,015074,015075,015076,015077,015078,015079,015080,015081,015082,015083,015084,015085,015086,015087,015088,015089,015090,015091,015092,015093,015094,015095,015096,015097,015098,015099,015100,015101,015102,015103,015104,015105,015106,015107,015108,015109,015110,015111,015112,015113,015114,015115,015116,015117,015118,015119,015120,015121,015122,015123,015124,015125,015126,015127,015128,015129,015130,015131,015132,015133,015134,015135,015136,015137,015138,015139,015140,015141,015142,015143,015144,015145,015146,015147,015148,015149,015150,015151,015152,015153,015154,015155,015156,015157,015158,015159,015160,015161,015162,015163,040570,040571,040572,040573,040574,040575,040576,040577,040578,040579,040580,040581,040582,040583,040584,040585,040586,040587,040588,040589,040590,040591,040592,040593,040594,040595,040596,040597,040598,040599,040600,040601,040602,040603,040604 \ No newline at end of file +014791,014797,014811,014812,014823,014844,015009,015026,015035,015051,015077,015094,015098,015106,015119,015130,015141,015145,015151,015177,015197,015203,015219 \ No newline at end of file diff --git a/工具/题号选题pdf生成.py b/工具/题号选题pdf生成.py index becd9fa2..5178cd2e 100644 --- a/工具/题号选题pdf生成.py +++ b/工具/题号选题pdf生成.py @@ -7,13 +7,13 @@ import os,re,time,json,sys """---设置题目列表---""" #留空为编译全题库, a为读取文本文件中的题号筛选.txt文件生成题库 problems = r""" -4572:4618 +a """ """---设置题目列表结束---""" """---设置文件名---""" #目录和文件的分隔务必用/ -filename = "临时文件/临时" +filename = "临时文件/二模十四套" """---设置文件名结束---""" """---设置是否需要解答题的空格---""" diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 41874468..5a4ec8e5 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -365678,7 +365678,9 @@ "id": "014784", "content": "集合$A=\\{x | x^2-2 x-3=0\\}$, $B=\\{x | 2 \\leq x \\leq 4,\\ x \\in\\mathbf{R}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{3\\}$", "solution": "", @@ -365697,7 +365699,9 @@ "id": "014785", "content": "设$\\mathrm{i}$为虚数单位, 则复数$\\dfrac{3+4 \\mathrm{i}}{3-4 \\mathrm{i}}$的虚部是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac{24}{25}$", "solution": "", @@ -365716,7 +365720,9 @@ "id": "014786", "content": "已知等差数列$\\{a_n\\}$中, $a_3=7$, $a_7=3$, 则通项公式为$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$-n+10$", "solution": "", @@ -365735,7 +365741,9 @@ "id": "014787", "content": "设$(2 x+1)^5=a_5 x^5+a_4 x^4+a_3 x^3+\\cdots+a_1 x+a_0$, 则$a_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$80$", "solution": "", @@ -365754,7 +365762,9 @@ "id": "014788", "content": "函数$y=\\ln (2-3 x)$的导数是$y'=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{3}{3x-2}$($x<\\dfrac 23$)", "solution": "", @@ -365773,7 +365783,9 @@ "id": "014789", "content": "若圆锥的侧面积为$15 \\pi$, 高为 4 , 则圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$12\\pi$", "solution": "", @@ -365792,7 +365804,9 @@ "id": "014790", "content": "由函数的观点, 不等式$3^x+\\lg x \\leq 3$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,1]$", "solution": "", @@ -365811,7 +365825,9 @@ "id": "014791", "content": "某中学举办思维竞赛, 现随机抽取$50$名参学生的成绩制作成频率分布直方图 (如图), 估计: 学生的平均成绩为\\blank{50}分.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.08, yscale = 50]\n\\draw [->] (80,0) -- (155,0) node [below] {成绩/分};\n\\draw [->] (80,0) -- (80,0.055) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {90/0.03,100/0.04,110/0.015,120/0.01,130/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {90/0.03,100/0.04,110/0.015,120/0.01,130/0.005}\n{\\draw [dashed] (\\i,\\j) -- (80,\\j) node [left] {$\\k$};};\n\\draw (140,0) node [below] {$140$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$107$", "solution": "", @@ -365830,7 +365846,9 @@ "id": "014792", "content": "$\\triangle ABC$内角$A, B, C$的对边是$a, b, c$, 若$a=3$, $b=\\sqrt{6}$, $\\angle A=\\dfrac{\\pi}{3}$, 则$\\angle B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{\\pi}4$", "solution": "", @@ -365849,7 +365867,9 @@ "id": "014793", "content": "$F_1, F_2$分别是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的左右焦点, 过$F_1$的直线$l$与双曲线的左右两支分别交于$A, B$两点. 若$\\triangle ABF_2$为等边三角形, 则双曲线的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\sqrt{7}$", "solution": "", @@ -365868,7 +365888,9 @@ "id": "014794", "content": "若存在实数$\\varphi$, 使函数$f(x)=\\cos (\\omega x+\\varphi)-\\dfrac{1}{2}$($\\omega>0$)在$x \\in[\\pi, 3 \\pi]$上有且仅有$2$个零点, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac 13,\\dfrac 53)$", "solution": "", @@ -365887,7 +365909,9 @@ "id": "014795", "content": "已知非零平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 满足$|\\overrightarrow {a}|=5$, $2|\\overrightarrow {b}|=|\\overrightarrow {c}|$, 且$(\\overrightarrow {b}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {a})=0$, 则$|\\overrightarrow {b}|$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{5}$", "solution": "", @@ -365906,7 +365930,9 @@ "id": "014796", "content": "已知$a, b \\in \\mathbf{R}$, 则``$a>b$''是``$a^3>b^3$''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": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -365925,7 +365951,9 @@ "id": "014797", "content": "对成对数据$(x_1, y_1)$、$(x_2, y_2)$、$\\cdots \\cdots$、$(x_n, y_n)$用最小二乘法求回归方程是为了使\\bracket{20}.\n\\fourch{$\\displaystyle\\sum_{i=1}^n(y_i-\\overline {y})=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)=0$}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)$最小}{$\\displaystyle\\sum_{i=1}^n(y_i-\\hat{y}_i)^2$最小}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -365944,7 +365972,9 @@ "id": "014798", "content": "下列函数中, 既是偶函数, 又在区间$(-\\infty, 0)$上严格递减的是\\bracket{20}.\n\\fourch{$y=2^{|x|}$}{$y=\\ln (-x)$}{$y=x^{-\\frac{2}{3}}$}{$y=-\\sqrt{x^2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -365963,7 +365993,9 @@ "id": "014799", "content": "如图, 一个由四根细铁杆$PA$、$PB$、$PC$、$PD$组成的支架($PA$、$PB$、$PC$、$PD$按照逆时针排布), 若$\\angle APB=\\angle BPC=\\angle CPD=\\angle DPA=\\dfrac{\\pi}{3}$, 一个半径为$1$的球恰好放在支架上与四根细铁杆均有接触, 则球心$O$到点$P$的距离是\n\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\path [name path = circ, draw] (0,0) circle (1);\n\\filldraw (0,0) node [right] {$O$} circle (0.03);\n\\draw [dashed] (0,0) ellipse (1 and 0.25);\n\\draw [dashed] (0,{-sqrt(2)/2}) ellipse ({sqrt(2)/2} and {sqrt(2)/8});\n\\draw (0,{-sqrt(2)}) node [below] {$P$} coordinate (P);\n\\draw (P) --++ (-1.5,1.5) node [above] {$A$};\n\\draw (P) --++ (-0.6,2.5) node [above] {$B$};\n\\draw (P) --++ (1.5,1.5) node [above] {$C$};\n\\path [name path = PD] (P) --++ (0.6,2.5) node [above] {$D$} coordinate (D);\n\\path [name intersections = {of = PD and circ, by = {M,N}}];\n\\draw (M)--(D)(P)--(N);\n\\draw [dashed] (M)--(N);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\sqrt{3}$}{$\\sqrt{2}$}{2}{$\\dfrac{3}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -365982,7 +366014,9 @@ "id": "014800", "content": "已知一个随机变量$X$的分布为: $\\begin{pmatrix}6 & 7 & 8 & 9 & 10 \\\\ 0.1 & a & 0.2 & 0.3 & b\\end{pmatrix}$.\\\\\n(1) 已知$E[X]=\\dfrac{43}{5}$, 求$a, b$的值;\\\\\n(2) 记事件$A: X$为偶数; 事件$B: X \\leq 8$. 已知$P(A)=\\dfrac{1}{2}$, 求$P(B)$, $P(A \\cap B)$, 并判断$A, B$是否相互独立?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "(1) $a=0.1$, $b=0.3$; (2) $P(B)=0.5$, $P(A\\cap B)=0.3$, $P(A)P(B)=0.25\\ne 0.3$, 不相互独立", "solution": "", @@ -366001,7 +366035,9 @@ "id": "014801", "content": "四边形$ABCD$是边长为$1$的正方形, $AC$与$BD$交于$O$点, $PA \\perp$平面$ABCD$, 且二面角$P-BC-A$的大小为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (0.5,0,0.5) node [below] {$O$} coordinate (O);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (P)--(A)--(C)(B)--(D)(B)--(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A$到平面$PBD$的距离;\\\\\n(2) 求直线$AC$与平面$PCD$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}3$; (2) $\\dfrac\\pi 6$", "solution": "", @@ -366020,7 +366056,9 @@ "id": "014802", "content": "如图, 某国家森林公园的一区域$OAB$为人工湖, 其中射线$OA, OB$为公园边界. 已知$OA \\perp OB$, 以点$O$为坐标原点, 以$OB$为$x$轴正方向, 建立平面直角坐标系(单位: 千米), 曲线$AB$的轨迹方程为: $y=-x^2+4$($0 \\leq x \\leq 2$). 计划修一条与湖边$AB$相切于点$P$的直路\n$l$(宽度不计), 直路$l$与公园边界交于点$C, D$两点, 把人工湖围成一片景区$\\triangle OCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,5.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0.5,1,1.5,2,2.5,3,3.5}{\\draw (\\i,0.1)--(\\i,0) node [below] {\\tiny$\\i$};};\n\\foreach \\i in {0.5,1,1.5,2,2.5,3,3.5,4,4.5,5}{\\draw (0.1,\\i)--(0,\\i) node [left] {\\tiny$\\i$};};\n\\draw [domain = 0:2, samples = 100] plot (\\x,{4-\\x*\\x});\n\\filldraw (0,4) circle (0.05) node [below right] {$A$};\n\\filldraw (2,0) circle (0.05) node [above right] {$B$};\n\\filldraw (0.8,3.36) circle (0.05) node [above right] {$P$};\n\\filldraw (0,4.64) circle (0.05) node [above right] {$C$} coordinate (C);\n\\filldraw (2.9,0) circle (0.05) node [above right] {$D$} coordinate (D);\n\\draw ($(C)!-0.1!(D)$) node [left] {$l$}-- ($(C)!1.1!(D)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$P$点坐标为$(1,3)$, 计算直路$CD$的长度; (精确到$0.1$千米)\\\\\n(2) 若$P$为曲线$AB$(不含端点)上的任意一点, 求景区$\\triangle OCD$面积的最小值. (精确到$0.1$平方千米)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 约$5.6$千米($\\sqrt{31.25}$); (2) 约$6.2$平方千米($\\dfrac{32\\sqrt{3}}{9}$)", "solution": "", @@ -366039,7 +366077,9 @@ "id": "014803", "content": "已知椭圆$C: \\dfrac{x^2}{4 a^2}+\\dfrac{y^2}{3 a^2}=1$($a>0$)的右焦点为$F$, 直线$l: x+y-4=0$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [name path = elli, draw] (0,0) ellipse (2 and {sqrt(3)});\n\\filldraw (1,0) circle (0.03) node [below] {$F$} coordinate (F);\n\\path [name path = l,draw] (-1.5,2.1) -- (2.1,-1.5);\n\\path [name intersections = {of = l and elli, by = {A,B}}];\n\\draw (A) node [above] {$A$};\n\\draw (B) node [below] {$B$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$F$到直线$l$的距离为$2 \\sqrt{2}$, 求$a$;\\\\\n(2) 若直线$l$与椭圆$C$交于$A, B$两点, 且$\\triangle ABO$的面积为$\\dfrac{48}{7}$, 求$a$;\\\\\n(3) 若椭圆$C$上存在点$P$, 过$P$作直线$l$的垂线$l_1$, 垂足为$H$, 满足直线$l_1$和直线$FH$的夹角为$\\dfrac{\\pi}{4}$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $a=8$; (2) $a=2$; (3) $[\\dfrac{4\\sqrt{7}-8}3,4)\\cup (4,+\\infty)$", "solution": "", @@ -366058,7 +366098,9 @@ "id": "014804", "content": "已知数列$\\{a_n\\}$是由正实数组成的无穷数列, 满足$a_1=3$, $a_2=7$, $a_n=|a_{n+1}-a_{n+2}|$($n$为正整数).\\\\\n(1) 写出数列$\\{a_n\\}$前$4$项的所有可能取法;\\\\\n(2) 判断: 在满足条件的所有数列$\\{a_n\\}$中, 是否可能存在正整数$k$, 满足$a_k=1$, 并说明理由;\\\\\n(3) $c_n$为数列$\\{a_n\\}$的前$n$项中不同取值的个数, 求$c_{100}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $3,7,10,17$或$3,7,10,3$或$3,7,4,11$; (2) 不存在, 证明略; (3) $51$, 证明略", "solution": "", @@ -366077,7 +366119,9 @@ "id": "014805", "content": "若实数$x$满足$|x-2|<1$, 则$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(1,3)$", "solution": "", @@ -366096,7 +366140,9 @@ "id": "014806", "content": "设复数$z$满足$(1+\\mathrm{i}) z=2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$1+\\mathrm{i}$", "solution": "", @@ -366115,7 +366161,9 @@ "id": "014807", "content": "已知集合$A=\\{1,2\\}$, $B=\\{a, a^2+1\\}$, 若$A \\cap B=\\{1\\}$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -366134,7 +366182,9 @@ "id": "014808", "content": "已知函数$y=\\sin (2 \\omega x+\\varphi)$($\\omega>0$)的最小正周期为$1$, 则$\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2\\pi$", "solution": "", @@ -366153,7 +366203,9 @@ "id": "014809", "content": "已知正实数$a$、$b$满足$a b=1$, 则$a+4 b$的最小值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -366172,7 +366224,9 @@ "id": "014810", "content": "在$(x^4+\\dfrac{1}{x})^{10}$的展开式中常数项是\\blank{50}.(用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$45$", "solution": "", @@ -366191,7 +366245,9 @@ "id": "014811", "content": "以下数据为参加某次数学竞赛的$15$人的成绩 (单位: 分), 分数从低到高依次是: 56、70、72、78、79、80、81、83、84、86、88、90、91、94、98, 则这$15$人成绩的第$80$百分位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$90.5$", "solution": "", @@ -366210,7 +366266,9 @@ "id": "014812", "content": "某单位为了解用电量$y$度与气温$x^{\\circ} \\text{C}$之间的关系, 随机统计了某$4$天的用电量与当天气温.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 气温$({ }^{\\circ} \\text{C})$& 14 & 12 & 8 & 6 \\\\\n\\hline 用电量 (度) & 22 & 26 & 34 & 38 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n由表中数据所得回归直线方程为$y=-2 x+\\hat{b}$, 据此预测当气温为$5^{\\circ} \\text{C}$时, 用电量的度数约为\\blank{50}${ }^{\\circ} \\text{C}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$40$", "solution": "", @@ -366229,7 +366287,9 @@ "id": "014813", "content": "已知抛物线$x^2=2 y$上的两个不同的点$A$、$B$的横坐标恰好是方程$x^2+6 x+4=0$的根, 则直线$AB$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$y=-3x-2$", "solution": "", @@ -366248,7 +366308,9 @@ "id": "014814", "content": "在一个十字路口, 每次亮绿灯的时长为$30$秒, 那么, 每次绿灯亮时, 在一条直行道路上能有多少汽车通过? 这个问题涉及车长、车距、车速、堵塞的干扰等多种因素, 不同型号车的车长是不同的, 驾驶员的习惯不同也会使车距、车速不同, 行人和非机动车的干扰因素则复杂且不确定. 面对这些不同和不确定, 需要作出假设, 例如小明发现虽然通过路口的车辆各种各样, 但多数是小轿车, 因此小明给出如下假设: 通过路口的车辆长度都相等, 请写出一个你认为合理的假设\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "如: 等待时, 前后相邻两辆车的车距都相等; 绿灯亮后, 汽车都是在静止状态匀加速启动; 前一辆车启动后, 下一辆车启动的间隔时间相等; 车辆行驶秩序良好, 不会发生堵塞; 等等", "solution": "", @@ -366267,7 +366329,9 @@ "id": "014815", "content": "设平面向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足: $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=|\\overrightarrow {c}|$, $|\\overrightarrow {a}-\\overrightarrow {b}|=1$, $\\overrightarrow {b} \\perp \\overrightarrow {c}$, 则$|\\overrightarrow {b}-\\overrightarrow {c}|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[\\sqrt{2},3\\sqrt{2}]$", "solution": "", @@ -366286,7 +366350,9 @@ "id": "014816", "content": "若函数$y=\\begin{cases}\\dfrac{x^3}{\\mathrm{e}^x}, & x \\geq 0, \\\\ a x^2, & x<0\\end{cases}$的图像上点$A$与点$B$, 点$C$与点$D$分别关于原点对称且这四点两两均不重合, 除此之外, 不存在函数图像上的其它不重合的两点关于原点对称, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-\\dfrac{1}{\\mathrm{e}},0)$", "solution": "", @@ -366305,7 +366371,9 @@ "id": "014817", "content": "下列函数在其定义域上既是严格增函数, 又是奇函数的是\\bracket{20}.\n\\fourch{$f(x)=\\tan x$}{$f(x)=-\\dfrac{1}{x}$}{$f(x)=x-\\cos x$}{$f(x)=\\mathrm{e}^x-\\mathrm{e}^{-x}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -366324,7 +366392,9 @@ "id": "014818", "content": "设两个正态分布$N(\\mu_1, \\sigma_1^2)$($\\sigma_1>0$)和$N(\\mu_2, \\sigma_2^2)$($\\sigma_2>0$)的正态密度函数图像如图所示, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\def\\s{0.3}\n\\def\\m{0.5}\n\\draw [domain = -0.3:1.3, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\def\\s{0.2}\n\\def\\m{-0.1}\n\\draw [domain = -1:0.8, samples = 100] plot (\\x,{1/sqrt(2*pi)/\\s*exp(-(\\x-\\m)*(\\x-\\m)/\\s/\\s/2)});\n\\draw (-0.1,2) node [left] {$N(\\mu_1,\\sigma_1^2)$};\n\\draw (0.5,1.5) node [right] {$N(\\mu_2,\\sigma_2^2)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\mu_1<\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1<\\mu_2$, $\\sigma_1>\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1>\\sigma_2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -366343,7 +366413,9 @@ "id": "014819", "content": "《九章算术》中将底面为直角三角形且侧棱垂直于底面的三棱柱称为``堑堵''; 底面为矩形, 一条侧棱垂直于底面的四棱锥称之为``阳马''; 四个面均为直角三角形的四面体称为``鳖臑''. 如图, 在堑堵$ABC-A_1B_1C_1$中, $AC \\perp BC$, 且$AA_1=AB=2$. 下列说法错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!{2.5/7}!(A_1)$) node [right] {$F$} coordinate (F);\n\\draw (A_1)--(C)(A_1)--(A)--(C)--(B)--(B_1)--cycle(A_1)--(C_1)--(B_1)(C)--(C_1)(A)--(F);\n\\draw [dashed] (A_1)--(B)(A)--(B)(A)--(E)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\onech{四棱锥$B-A_1ACC_1$为``阳马''}{四面体$A_1C_1CB$为``鳖臑''}{四棱锥$B-A_1ACC_1$体积的最大值为$\\dfrac{2}{3}$}{过$A$点作$AE \\perp A_1B$于点$E$, 过$E$点作$EF \\perp A_1B$并交$A_1C$于点$F$, 则$A_1B \\perp$平面$AEF$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -366362,7 +366434,9 @@ "id": "014820", "content": "已知数列$\\{a_n\\}$是各项为正数的等比数列, 公比为$q$, 在$a_1$、$a_2$之间插入$1$个数, 使这$3$个数成等差数列, 记公差为$d_1$, 在$a_2$、$a_3$之间插入$2$个数, 使这$4$个数成等差数列, 公差为$d_2$, $\\cdots$, 在$a_n$、$a_{n+1}$之间插入$n$个数, 使这$n+2$个数成等差数列, 公差为$d_n$, 则\\bracket{20}.\n\\twoch{当$01$时, 数列$\\{d_n\\}$严格增}{当$d_1>d_2$时, 数列$\\{d_n\\}$严格减}{当$d_1=latex]\n\\filldraw (0,0) node [above left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2.3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O_1) ellipse (1.5 and 0.5);\n\\draw (O) ++ (1.5,0) node [right] {$B$} coordinate (B) --++ (0,2.3) node [right] {$B_1$} coordinate (B_1);\n\\draw (O) ++ (-1.5,0) node [left] {$A$} coordinate (A) --++ (0,2.3) node [left] {$A_1$} coordinate (A_1) -- (B_1);\n\\draw (A) arc (180:360:1.5 and 0.5);\n\\draw [dashed] (A) arc (180:0:1.5 and 0.5);\n\\draw (-60:1.5 and 0.5) node [below] {$P$} coordinate (P);\n\\draw [dashed] (A_1)--(P)(A_1)--(B)--(P)--(A)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1P$与平面$ABP$所成角的大小;\\\\\n(2) 求点$A$到平面$A_1BP$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\arctan\\dfrac{\\sqrt{3}}2$; (2) $\\dfrac{6\\sqrt{7}}7$", "solution": "", @@ -366400,7 +366476,9 @@ "id": "014822", "content": "在$\\triangle ABC$中, $a$、$b$、$c$分别是内角$A$、$B$、$C$的对边, $\\overrightarrow {m}=(2 a+c, b)$, $\\overrightarrow {n}=(\\cos B, \\cos C)$, $\\overrightarrow {m} \\cdot \\overrightarrow {n}=0$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 设$f(x)=2 \\cos x \\sin (x+\\dfrac{\\pi}{3})-2 \\sin ^2 x \\sin B+2 \\sin x \\cos x \\cos (A+C)$, 当$x \\in[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$时, 求$f(x)$的最小值及相应的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $B=\\dfrac{2\\pi}{3}$; (2) 最小值为$-2$, 取到最小值当且仅当$x=\\dfrac{7\\pi}{12}$", "solution": "", @@ -366419,7 +366497,10 @@ "id": "014823", "content": "某校工会开展健身健步走活动, 要求教职工上传$3$月$1$日至$3$月$7$日的微信运动步数信息, 下图是职工甲和职工乙微信运动步数情况:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {1,2,...,7} {\\draw ({\\i*0.7},0) node [below] {$\\i$};};\n\\draw (0,0) node [below] {$3$月};\n\\draw (0,0)--(5.5,0);\n\\foreach \\i/\\j in {1/8566,2/19891,3/16820,4/5207,5/13022,6/11860,7/15524}\n{\\draw ({\\i*0.7},2) node [above] {\\tiny$\\j$};};\n\\draw (0.7,0) -- (0.7,0.8566) -- (1.4,1.9891) -- (2.1,1.6820) -- (2.8,0.5207) -- (3.5,1.3022) -- (4.2,1.1860) -- (4.9,1.5524) -- (4.9,0);\n\\foreach \\i/\\j in {0.7/0.8566,1.4/1.9891,2.1/1.6820,2.8/0.5207,3.5/1.3022,4.2/1.1860,4.9/1.5524}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (2.75,-0.5) node [below] {职工甲};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {1,2,...,7} {\\draw ({\\i*0.7},0) node [below] {$\\i$};};\n\\draw (0,0) node [below] {$3$月};\n\\draw (0,0)--(5.5,0);\n\\foreach \\i/\\j in {1/11845,2/10577,3/9780,4/4872,5/17022,6/9655,7/12396}\n{\\draw ({\\i*0.7},2) node [above] {\\tiny$\\j$};};\n\\draw (0.7,0) -- (0.7,1.1845) -- (1.4,1.0577) -- (2.1,0.9780) -- (2.8,0.4872) -- (3.5,1.7022) -- (4.2,0.9655) -- (4.9,1.2396) -- (4.9,0);\n\\foreach \\i/\\j in {0.7/1.1845,1.4/1.0577,2.1/0.9780,2.8/0.4872,3.5/1.7022,4.2/0.9655,4.9/1.2396}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\draw (2.75,-0.5) node [below] {职工乙};\n\\end{tikzpicture}\n\\end{center}\n(1) 从$3$月$2$日至$3$月$7$日中任选一天, 求这一天职工甲和职工乙微信运动步数都不低于$10000$的概率;\\\\\n(2) 从$3$月$1$日至$3$月$7$日中任选两天, 记职工乙在这两天中微信运动步数不低于$10000$的天数为$X$, 求$X$的分布列及数学期望;\\\\\n(3) 下图是校工会根据$3$月$1$日至$3$月$7$日某一天的数据制作的全校$200$名教职工微信运动步数的频率分布直方图, 已知这一天甲和乙微信记步数在单位$200$名教职工中排名(按照从大到小排序)分别为第$68$和第$142$, 请指出这是根据哪一天的数据制作的频率分布直方图. (不用说明理由)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 30]\n\\draw [->] (0,0) -- (48,0) node [below] {微信计步数(单位:千步)};\n\\draw [->] (0,0) -- (0,0.08) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {0/0.02,5/0.04,10/0.06,15/0.05,20/0.03}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.02,5/0.04,10/0.06,15/0.05,20/0.03}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (25,0) node [below] {$25$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 分布列为$\\begin{pmatrix} 0 & 1 & 2 \\\\ \\dfrac 17 & \\dfrac 47 & \\dfrac 27\\end{pmatrix}$, $E[X]=\\dfrac 87$; (3) $3$月$3$日", "solution": "", @@ -366438,7 +366519,9 @@ "id": "014824", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{m^2}+\\dfrac{y^2}{2}=1$($m>0$, $m \\neq \\sqrt{2}$), 点$A$、$B$分别是椭圆$\\Gamma$与$y$轴的交点(点$A$在点$B$的上方), 过点$D(0,1)$且斜率为$k$的直线$l$交椭圆$\\Gamma$于$E$、$G$两点.\\\\\n(1) 若椭圆$\\Gamma$焦点在$x$轴上, 且其离心率是$\\dfrac{\\sqrt{2}}{2}$, 求实数$m$的值;\\\\\n(2) 若$m=k=1$, 求$\\triangle BEG$的面积;\\\\\n(3) 设直线$AE$与直线$y=2$交于点$H$, 证明: $B$、$G$、$H$三点共线.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $m=2$; (2) $\\dfrac{2\\sqrt{2}-2}3$; (3) 证明略", "solution": "", @@ -366457,7 +366540,9 @@ "id": "014825", "content": "已知定义域为$D$的函数$y=f(x)$, 其导函数为$y'=f'(x)$, 满足对任意的$x \\in D$都有$|f'(x)|<1$.\\\\\n(1) 若$f(x)=a x+\\ln x$, $x \\in[1,2]$, 求实数$a$的取值范围;\\\\\n(2) 证明: 方程$f(x)-x=0$至多只有一个实根;\\\\\n(3) 若$y=f(x)$, $x \\in \\mathbf{R}$是周期为$2$的周期函数, 证明: 对任意的实数$x_1$、$x_2$, 都有$|f(x_1)-f(x_2)|<1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $(-\\dfrac 32,0)$; (2) 证明略; (3) 证明略", "solution": "", @@ -366476,7 +366561,9 @@ "id": "014826", "content": "已知复数$z=3+4 \\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -366495,7 +366582,9 @@ "id": "014827", "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{7}=1$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$\\dfrac 43$", "solution": "", @@ -366514,7 +366603,9 @@ "id": "014828", "content": "已知$A=\\{x | \\dfrac{x-1}{x} \\leq 0\\}$, $B=\\{x | x \\geq 1\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{1\\}$", "solution": "", @@ -366533,7 +366624,9 @@ "id": "014829", "content": "函数$y=\\sin 2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -366552,7 +366645,9 @@ "id": "014830", "content": "$\\triangle ABC$是边长为$1$的等边三角形, 点$M$为边$AB$的中点, 则$\\overrightarrow{AC} \\cdot \\overrightarrow{AM}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac 14$", "solution": "", @@ -366571,7 +366666,9 @@ "id": "014831", "content": "已知函数$y=2 x+\\dfrac{1}{8 x}$, 定义域为$(0,+\\infty)$, 则该函数的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -366590,7 +366687,9 @@ "id": "014832", "content": "已知$n \\in \\mathbf{N}$, 若$\\mathrm{C}_6^n=\\mathrm{P}_5^2$, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -366609,7 +366708,9 @@ "id": "014833", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases}2 n, & n=1, \\\\ 2^{-n}, & n \\geq 2,\\end{cases}$ 前$n$项和为$S_n$, 则$\\displaystyle\\lim _{n \\to+\\infty} S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$\\dfrac 52$", "solution": "", @@ -366628,7 +366729,9 @@ "id": "014834", "content": "已知四棱锥$P-ABCD$的底面是边长为$\\sqrt{2}$的正方形, 侧棱长均为$\\sqrt{5}$. 若点$A$、$B$、$C$、$D$在圆柱的一个底面圆周上, 点$P$在圆柱的另一个底面内, 则该圆柱的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2\\pi$", "solution": "", @@ -366647,7 +366750,9 @@ "id": "014835", "content": "已知某产品的一类部件由供应商$A$和$B$提供, 占比分别为$\\dfrac{1}{3}$和$\\dfrac{2}{3}$, 供应商$A$提供的部件的良品率为$0.96$, 若该部件的总体良品率为$0.92$, 则供应商$B$提供的部件的良品率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$0.9$", "solution": "", @@ -366666,7 +366771,9 @@ "id": "014836", "content": "如图, 线段$AB$的长为$8$, 点$C$在线段$AB$上, $AC=2$. 点$P$为线段$CB$上任意一点, 点$A$绕着点$C$顺时针旋转, 点$B$绕着点$P$逆时针旋转. 若它们恰重合于点$D$, 则$\\triangle CDP$的面积的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (8,0) node [right] {$B$} coordinate (B);\n\\draw (5.5,0) node [below] {$P$} coordinate (P);\n\\draw ({24/7},{4*sqrt(6)/7}) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)(C)--(D)--(P);\n\\draw [dashed] (A) arc (180:{atan(2*sqrt(6)/5)}:2);\n\\draw [dashed] (B) arc (0:{180-atan(8*sqrt(6)/29)}:2.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$2\\sqrt{2}$", "solution": "", @@ -366685,7 +366792,9 @@ "id": "014837", "content": "若关于$x$的函数$y=\\dfrac{x^3+a}{\\mathrm{e}^x}$在$\\mathbf{R}$上存在极小值($\\mathrm{e}$为自然对数的底数), 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,4)$", "solution": "", @@ -366704,7 +366813,9 @@ "id": "014838", "content": "设$a \\in \\mathbf{R}$, 则``$a<1$''是``$a^2=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{{sqrt(5)}}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (A)--(E)--(D1)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 判断直线$AE$与$D_1F$的关系, 并说明理由;\\\\\n(2) 若直线$D_1E$与底面$ABCD$所成角为$\\dfrac{\\pi}{4}$, 求四棱柱$ABCD-A_1B_1C_1D_1$的全面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 相交; (2) $5\\sqrt{5}+8$", "solution": "", @@ -366799,7 +366919,9 @@ "id": "014843", "content": "已知向量$\\overrightarrow {a}=(\\sin x, 1+\\cos 2 x)$, $\\overrightarrow {b}=(\\cos x, \\dfrac{1}{2})$, $f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$.\\\\\n(1) 求函数$y=f(x)$的最大值及相应$x$的值;\\\\\n(2) 在$\\triangle ABC$中, 角$A$为锐角, 且$A+B=\\dfrac{7 \\pi}{12}$, $f(A)=1$, $BC=2$, 求边$AC$的长.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) $f(x)=\\dfrac{\\sqrt{2}}2\\sin (2x+\\dfrac\\pi 4)+\\dfrac 12$, 最大值为$\\dfrac{1+\\sqrt{2}}2$, 当且仅当$x=\\dfrac\\pi 8+k\\pi$, $k\\in \\mathbf{Z}$时取得; (2) $A=\\dfrac\\pi 4$, $B=\\dfrac\\pi 3$, $AC=\\sqrt{6}$", "solution": "", @@ -366818,7 +366940,9 @@ "id": "014844", "content": "李先生是一名上班族, 为了比较上下班的通勤时间, 记录了$20$天个工作日内, 家里到单位的上班时间以及同路线返程的下班时间(单位: 分钟), 如下茎叶图显示两类时间的共$40$个记录:\n\\begin{center}\n\\begin{tabular}{cccccccccccc|c|ccccccccccc}\n\\multicolumn{12}{r|}{上班时间} & & \\multicolumn{11}{l}{下班时间} \\\\\n& & & & & & & & 9 & 8 & 8 & 7 & 3 & 6 & 7 & 8 & 8 & 8 & 9 \\\\\n6 & 5 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & 0 & 4 & 0 & 0 & 1 & 3 & 3 & 3 & 3 & 4 & 4 & 5 & 5 \\\\\n& & & & & & & & 4 & 2 & 2 & 1 & 5 & 1 & 7\\\\\n& & & & & & & & & & & & 6 & 4\n\\end{tabular}\n\\end{center}\n(1) 求出这$40$个通勤记录的中位数$M$, 并完成下列$2 \\times 2$列联表:\n\\begin{center}\n\\begin{tabular}{|l|l|l|}\n\\hline & 超过$M$& 不超过$M$\\\\\n\\hline 上班时间 & & \\\\\n\\hline 下班时间 & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 根据列联表中的数据, 请问上下班的通勤时间是否有显著差异? 并说明理由.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, $n=a+b+c+d$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) 中位数$M=42.5$, 列联表如下: \\begin{tabular}{|c|c|c|}\n\\hline & 超过$M$& 不超过$M$\\\\\n\\hline 上班时间 & 10 & 10 \\\\\n\\hline 下班时间 & 11 & 9\\\\\n\\hline\n\\end{tabular}; (2) $\\chi^2=0.1$, 无显著差异", "solution": "", @@ -366837,7 +366961,9 @@ "id": "014845", "content": "若直线和抛物线的对称轴不平行且与抛物线只有一个公共点, 则称该直线是抛物线在该点处的切线, 该公共点为切点. 已知抛物线$C_1: y^2=4 a x$和$C_2: x^2=4 y$, 其中$a>0$. $C_1$与$C_2$在第一象限内的交点为$P$. $C_1$与$C_2$在点$P$处的切线分别为$l_1$和$l_2$, 定义$l_1$和$l_2$的夹角为曲线$C_1$、$C_2$的夹角.\\\\\n(1) 求点$P$的坐标;\\\\\n(2) 若$C_1$、$C_2$的夹角为$\\arctan \\dfrac{3}{4}$, 求$a$的值;\\\\\n(3) 若直线$l_3$既是$C_1$也是$C_2$的切线, 切点分别为$Q$、$R$, 当$\\triangle PQR$为直角三角形时, 求出相应的$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $P(4a^{\\frac 13},4a^{\\frac 23})$; (2) $1$; (3) $2\\sqrt{2}$或$\\dfrac{\\sqrt{2}}4$", "solution": "", @@ -366856,7 +366982,9 @@ "id": "014846", "content": "已知$f(x)=x+2 \\sin x$, 等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 记$T_n=\\displaystyle\\sum_{i=1}^n f(a_1)$.\\\\\n(1) 求证: 函数$y=f(x)$的图像关于点$(\\pi, \\pi)$中心对称;\\\\\n(2) 若$a_1$、$a_2$、$a_3$是某三角形的三个内角, 求$T_3$的取值范围;\\\\\n(3) 若$S_{100}=100 \\pi$, 求证: $T_{100}=100 \\pi$. 反之是否成立? 并请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 证明略 (2) $(\\pi,\\pi+3\\sqrt{3}]$; (3) 证明略, 反之不一定成立, 如取$a_n$是常数$a$, 满足$a+2\\sin a=\\pi$(这样的$a$有三个)", "solution": "", @@ -369706,7 +369834,9 @@ "id": "014996", "content": "已知集合$A=\\{x | x^2+x-6<0,\\ x \\in \\mathbf{R}\\}$, $B=\\{0,1,2\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369725,7 +369855,9 @@ "id": "014997", "content": "若复数$z$满足$z(1-\\mathrm{i})=1+2 \\mathrm{i}$($\\mathrm{i}$是虚数单位), 则复数$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369744,7 +369876,9 @@ "id": "014998", "content": "若圆柱的高为$10$, 底面积为$4 \\pi$, 则这个圆柱的侧面积为\\blank{50}.(结果保留$\\pi$)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369763,7 +369897,9 @@ "id": "014999", "content": "$(x+3)^5$的二项展开式中$x^2$项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369782,7 +369918,9 @@ "id": "015000", "content": "设随机变量$X$服从正态分布$N(0, \\sigma^2)$, 且$P(X>-2)=0.9$, 则$P(X>2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369801,7 +369939,9 @@ "id": "015001", "content": "双曲线$C: \\dfrac{x^2}{2}-\\dfrac{y^2}{4}=1$的右焦点$F$到其一条渐近线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369820,7 +369960,9 @@ "id": "015002", "content": "投掷一颗骰子, 记事件$A=\\{2,4,5\\}$, $B=\\{1,2,4,6\\}$, 则$P(A | B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369839,7 +369981,9 @@ "id": "015003", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别记为$a$、$b$、$c$, 若$5 a \\cos A=b \\cos C+c \\cos B$, 则$\\sin 2A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369858,7 +370002,9 @@ "id": "015004", "content": "函数$y=\\log _2 x+\\dfrac{1}{\\log _4(2 x)}$在区间$(\\dfrac{1}{2},+\\infty)$上的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369877,7 +370023,9 @@ "id": "015005", "content": "已知$\\omega \\in \\mathbf{R}$, $\\omega>0$, 函数$y=\\sqrt{3} \\sin \\omega x-\\cos \\omega x$在区间$[0,2]$上有唯一的最小值$-2$, 则$\\omega$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369896,7 +370044,9 @@ "id": "015006", "content": "已知边长为$2$的菱形$ABCD$中, $\\angle A=120^{\\circ}$, $P$、$Q$是菱形内切圆上的两个动点, 且$PQ \\perp BD$, 则$\\overrightarrow{AP} \\cdot \\overrightarrow{CQ}$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -369915,7 +370065,9 @@ "id": "015007", "content": "已知$01$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -369953,7 +370107,9 @@ "id": "015009", "content": "某种产品的广告支出$x$与销售额$y$(单位: 万元) 之间有下表关系, $y$与$x$的线性回归方程为$y=10.5 x+5.4$, 当广告支出$6$万元时, 随机误差的效应即离差(真实值减去预报值)为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$x$& 2 & 4 & 5 & 6 & 8 \\\\\n\\hline$y$& 30 & 40 & 60 & 70 & 80 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$1.6$}{$8.4$}{$11.6$}{$7.4$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -369972,7 +370128,9 @@ "id": "015010", "content": "在空间中, 下列命题为真命题的是\\bracket{20}.\n\\onech{若两条直线垂直于第三条直线, 则这两条直线互相平行}{若两个平面分别平行于两条互相垂直的直线, 则这两个平面互相垂直}{若两个平面垂直, 则过一个平面内一点垂直于交线的直线与另外一个平面垂直}{若一条直线平行于一个平面, 另一条直线与这个平面垂直, 则这两条直线互相垂直}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -369991,7 +370149,9 @@ "id": "015011", "content": "已知函数$y=f(x)$($x \\in \\mathbf{R}$), 其导函数为$y=f'(x)$, 有以下两个命题: \\textcircled{1} 若$y=f'(x)$为偶函数, 则$y=f(x)$为奇函数; \\textcircled{2} 若$y=f'(x)$为周期函数, 则$y=f(x)$也为周期函数. 那么\\bracket{20}.\n\\twoch{\\textcircled{1}是真命题, \\textcircled{2}是假命题}{\\textcircled{1}是假命题, \\textcircled{2}是真命题}{\\textcircled{1}、\\textcircled{2}都是真命题}{\\textcircled{1}、\\textcircled{2}都是假命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370010,7 +370170,9 @@ "id": "015012", "content": "已知数列$\\{a_n\\}$是首项为$9$, 公比为$\\dfrac{1}{3}$的等比数列.\\\\\n(1) 求$\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\dfrac{1}{a_3}+\\dfrac{1}{a_4}+\\dfrac{1}{a_5}$的值;\\\\\n(2) 设数列$\\{\\log _3 a_n\\}$的前$n$项和为$S_n$, 求$S_n$的最大值, 并指出$S_n$取最大值时$n$的取值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370029,7 +370191,9 @@ "id": "015013", "content": "如图, 三角形$EAD$与梯形$ABCD$所在的平面互相垂直, $AE \\perp AD$, $AB \\perp AD$, $BC\\parallel AD$, $AB=AE=BC=2$, $AD=4$, $F$、$H$分别为$ED$、$EA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$E$} coordinate (E);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw ($(E)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(E)$) node [right] {$H$} coordinate (H);\n\\draw (B)--(C)--(D)--(E)--cycle(C)--(F)(E)--(C);\n\\draw [dashed] (B)--(H)(E)--(A)--(B)(A)--(C)(A)--(D)(A)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BH\\parallel$平面$AFC$;\\\\\n(2) 求平面$ACF$与平面$EAB$所成锐二面角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370048,7 +370212,9 @@ "id": "015014", "content": "为了庆祝党的二十大顺利召开, 某学校特举办主题为``重温光辉历史展现坚定信心''的百科知识小测试比赛. 比赛分抢答和必答两个环节, 两个环节均设置$10$道题, 其中$5$道人文历史题和$5$道地理环境题.\\\\\n(1) 在抢答环节, 某代表队非常积极, 抢到$4$次答题机会, 求该代表队至少抢到$1$道地理环境题的概率;\\\\\n(2) 在必答环节, 每个班级从$5$道人文历史题和$5$道地理环境题各选$2$题, 各题答对与否相互独立, 每个代表队可以先选择人文历史题, 也可以先选择地理环境题开始答题. 若中间有一题答错就退出必答环节, 仅当第一类问题中$2$题均答对, 才有资格开始第二类问题答题. 已知答对$1$道人文历史题得$2$分, 答对$1$道地理环境题得$3$分. 假设某代表队答对人文历史题的概率都是$\\dfrac{3}{5}$, 答对地理环境题的概率都是$\\dfrac{1}{3}$. 请你为该代表队作出答题顺序的选择, 使其得分期望值更大, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370067,7 +370233,9 @@ "id": "015015", "content": "椭圆$C$的方程为$x^2+3 y^2=4$, $A$、$B$为椭圆的左右顶点, $F_1$、$F_2$为左右焦点, $P$为椭圆上的动点.\\\\\n(1) 求椭圆的离心率;\\\\\n(2) 若$\\triangle PF_1F_2$为直角三角形, 求$\\triangle PF_1F_2$的面积;\\\\\n(3) 若$Q$、$R$为椭圆上异于$P$的点, 直线$PQ$、$PR$均与圆$x^2+y^2=r^2$($0=latex, scale = 0.6]\n\\foreach \\i/\\j in {0/5,1/4,2/5,3/6,4/8,5/9}\n{\\draw (\\i,0) node {$\\j$};};\n\\foreach \\i/\\j in {0/9,1/2,2/4}\n{\\draw (\\i,-4) node {$\\j$};};\n\\draw (0,-1) node {$6$};\n\\draw (0,-2) node {$7$};\n\\draw (0,-3) node {$8$};\n\\draw (0.5,1) -- (0.5,-5);\n\\end{tikzpicture}\n\\phantom{宝山二模2023}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.02,60/0.024,70/0.036,80/0.012,90/0.008}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.02,60/0.024,70/0.036,80/0.012/x,90/0.008/y}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370295,7 +370485,9 @@ "id": "015027", "content": "已知函数$f(x)=\\dfrac{1}{a^x+1}-\\dfrac{1}{2}$($a>0$且$a \\neq 1)$, 若关于$x$的不等式$f(a x^2+b x+c)>0$的解集为$(1,2)$, 其中$b \\in(-6,1)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370314,7 +370506,9 @@ "id": "015028", "content": "已知非零平面向量$\\overrightarrow {a}, \\overrightarrow {b}$不平行, 且满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {a}^2=4$, 记$\\overrightarrow {c}=\\dfrac{3}{4} \\overrightarrow {a}+\\dfrac{1}{4} \\overrightarrow {b}$, 则当$\\overrightarrow {b}$与$\\overrightarrow {c}$的夹角最大时, $|\\overrightarrow {a}-\\overrightarrow {b}|$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370333,7 +370527,9 @@ "id": "015029", "content": "若$\\alpha: x^2=4$, $\\beta: x=2$, 则$\\alpha$是$\\beta$的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370352,7 +370548,9 @@ "id": "015030", "content": "已知定义在$\\mathbf{R}$上的偶函数$f(x)=|x-m+1|-2$, 若正实数$a$、$b$满足$f(a)+f(2 b)=m$, 则$\\dfrac{1}{a}+\\dfrac{2}{b}$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{9}{5}$}{$9$}{$\\dfrac{8}{5}$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370371,7 +370569,9 @@ "id": "015031", "content": "将正整数$n$分解为两个正整数$k_1$、$k_2$的积, 即$n=k_1 \\cdot k_2$, 当$k_1$、$k_2$两数差的绝对值最小时, 我们称其为最优分解. 如$20=1 \\times 20=2 \\times 10=4 \\times 5$, 其中$4 \\times 5$即为$20$的最优分解, 当$k_1$、$k_2$是$n$的最优分解时, 定义$f(n)=|k_1-k_2|$, 则数列$\\{f(5^n)\\}$的前$2023$项的和为\\bracket{20}.\n\\fourch{$5^{1012}$}{$5^{1012}-1$}{$5^{2023}$}{$5^{2023}-1$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370390,7 +370590,9 @@ "id": "015032", "content": "在空间直角坐标系$O-x y z$中, 已知定点$A(2,1,0)$、$B(0,2,0)$和动点$C(0, t, t+2)$($t \\geq 0$). 若$\\triangle OAC$的面积为$S$, 以$O$、$A$、$B$、$C$为顶点的锥体的体积为$V$, 则$\\dfrac{V}{S}$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{2}{15} \\sqrt{5}$}{$\\dfrac{1}{5} \\sqrt{5}$}{$\\dfrac{4}{15} \\sqrt{5}$}{$\\dfrac{4}{5} \\sqrt{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370409,7 +370611,9 @@ "id": "015033", "content": "已知函数$f(x)=\\sin x \\cos x-\\sqrt{3} \\cos ^2 x+\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 求函数$y=f(x)$的最小正周期和单调区间;\\\\\n(2) 若关于$x$的方程$f(x)-m=0$在$x \\in[0, \\dfrac{\\pi}{2}]$上有两个不同的实数解, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370428,7 +370632,9 @@ "id": "015034", "content": "四棱锥$P-ABCD$的底面是边长为$2$的菱形, $\\angle DAB=60^{\\circ}$, 对角线$AC$与$BD$相交于点$O$, $PO \\perp$底面$ABCD$, $PB$与底面$ABCD$所成的角为$60^{\\circ}$, $E$是$PB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw ({sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw ($(C)!2!(O)$) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw ($(B)!2!(O)$) node [above] {$D$} coordinate (D);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(B)--(D)--(E)(P)--(O)(P)--(D)(A)--(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$DE$与$PA$所成角的大小 (结果用反三角函数值表示);\\\\\n(2) 证明: $OE\\parallel$平面$PAD$, 并求点$E$到平面$PAD$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370447,7 +370653,9 @@ "id": "015035", "content": "下表是某工厂每月生产的一种核心产品的产量$x$($4 \\leq x \\leq 20$, $x \\in \\mathbf{Z}$)(件) 与相应的生产成本$y$(万元)的四组对照数据.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$& 4 & 6 & 8 & 10 \\\\\n\\hline$y$& 12 & 20 & 28 & 84 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 试建立$x$与$y$的线性回归方程;\\\\\n(2) 研究人员进一步统计历年的销售数据发现, 在供销平衡的条件下, 市场销售价格会波动变化. 经分析, 每件产品的销售价格$q$(万元) 是一个与产量$x$相关的随机变量, 分布为$\\begin{pmatrix} 100-x&90-x&80-x\\\\ \\dfrac{1}{4}&\\dfrac{1}{2}&\\dfrac{1}{4} \\end{pmatrix}$, 假设产品月利润$=$月销售量$\\times$销售价格$-$成本(其中月销售量$=$生产量). 根据(1)进行计算, 当产量$x$为何值时, 月利润的期望值最大? 最大值为多少?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370466,7 +370674,9 @@ "id": "015036", "content": "已知拋物线$\\Gamma: y^2=4 x$.\\\\\n(1) 求抛物线$\\Gamma$的焦点$F$的坐标和准线$l$的方程;\\\\\n(2) 过焦点$F$且斜率为$\\dfrac{1}{2}$的直线与抛物线$\\Gamma$交于两个不同的点$A$、$B$, 求线段$AB$的长;\\\\\n(3) 已知点$P(1,2)$, 是否存在定点$Q$, 使得过点$Q$的直线与抛物线$\\Gamma$交于两个不同的点$M$、$N$(均不与点$P$重合), 且以线段$MN$为直径的圆恒过点$P$? 若存在, 求出点$Q$的坐标; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370485,7 +370695,9 @@ "id": "015037", "content": "直线族是指具有某种共同性质的直线的全体. 如: 方程$y=k x+1$中, 当$k$取给定的实数时, 表示一条直线; 当$k$在实数范围内变化时, 表示过点$(0,1)$的直线族(不含$y$轴). 记直线族$2(a-2) x+4 y-4 a+a^2=0$(其中$a \\in \\mathbf{R}$)为$\\Psi$, 直线族$y=3 t^2 x-2 t^3$(其中$t>0)$为$\\Omega$.\\\\\n(1) 分别判断点$A(0,1), B(1,2)$是否在$\\Psi$的某条直线上, 并说明理由;\\\\\n(2) 对于给定的正实数$x_0$, 点$P(x_0, y_0)$不在$\\Omega$的任意一条直线上, 求$y_0$的取值范围(用$x_0$表示);\\\\\n(3) 直线族的包络被定义为这样一条曲线: 直线族中的每一条直线都是该曲线上某点处的切线, 且该曲线上每一点处的切线都是该直线族中的某条直线. 求$\\Omega$的包络和$\\Psi$的包络.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370504,7 +370716,9 @@ "id": "015038", "content": "已知集合$A=\\{-1,0\\}$, 集合$B=\\{2, a\\}$, 若$A \\cap B=\\{0\\}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370523,7 +370737,9 @@ "id": "015039", "content": "若实数$x$满足不等式$x^2-3 x+2<0$, 则$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370542,7 +370758,9 @@ "id": "015040", "content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的渐近线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370561,7 +370779,9 @@ "id": "015041", "content": "已知向量$\\overrightarrow {a}=(0,1,0)$, 向量$\\overrightarrow {b}=(1,1,0)$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370580,7 +370800,9 @@ "id": "015042", "content": "在$(2+x)^5$的二项展开式中, $x^4$项的系数为\\blank{50}(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370599,7 +370821,9 @@ "id": "015043", "content": "若复数$z=2+\\mathrm{i}$($\\mathrm{i}$是虚数单位), 则$z \\cdot \\overline {z}-z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370618,7 +370842,9 @@ "id": "015044", "content": "已知$y=f(x)$是定义域为$\\mathbf{R}$的奇函数, 当$x \\geq 0$时, $f(x)=2 x^3+2^x-1$, 则$f(-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370637,7 +370863,9 @@ "id": "015045", "content": "掷一颗骰子, 令事件$A=\\{1,2,3\\}$, $B=\\{1,2,5,6\\}$, 则$P(A | B)=$\\blank{50}(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370656,7 +370884,9 @@ "id": "015046", "content": "已知正实数$a$、$b$满足$\\dfrac{1}{a}+\\dfrac{2}{b}=1$, 则$2 a+b$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370675,7 +370905,9 @@ "id": "015047", "content": "若函数$y=\\sin (\\omega x-\\dfrac{\\pi}{3})$(常数$\\omega>0$)在区间$(0, \\pi)$没有最值, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -370694,7 +370926,9 @@ "id": "015048", "content": "已知函数$y=f(x)$和$y=g(x)$的表达式分别为$f(x)=\\sqrt{-x^2-4 x}$, $g(x)=x|x^2-a|$, 若对任意$x_1 \\in[1, \\sqrt{2}]$, 总存在$x_2 \\in[-3,0]$, 使得$g(x_1)b^2>0$, 则下列不等式中成立的是\\bracket{20}.\n\\fourch{$a>b$}{$2^a>2^b$}{$a>|b|$}{$\\log _2 a^2>\\log _2 b^2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370751,7 +370989,9 @@ "id": "015051", "content": "某社区通过公益讲座宣传交通法规. 为了解讲座效果, 随机抽取$10$位居民, 分别在讲座前、\n后各回答一份交通法规知识问卷, 满分为$100$分. 他们得分的茎叶图如图所示 (``叶''是个位数字), 则下列选项叙述错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{ccc|c|cccc} \n\\multicolumn{3}{r|}{讲座前} & & \\multicolumn{4}{l}{讲座后} \\\\\n& 5 & 0 & 5 & & & & \\\\\n5 & 0 & 0 & 6 & & & & \\\\\n5 & 0 & 0 & 7 & & & & \\\\\n& & 0 & 8 & 0 & 5 & 5 & 5 \\\\\n& & 0 & 9 & 0 & 0 & 5 & 5 \\\\\n& & & 10 & 0 & 0 & &\n\\end{tabular}\n\\end{center}\n\\twoch{讲座后的答卷得分整体上高于讲座前的得分}{讲座前的答卷得分分布较讲座后分散}{讲座后答卷得分的第$80$百分位数为$95$}{讲座前答卷得分的极差大于讲座后得分的极差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370770,7 +371010,9 @@ "id": "015052", "content": "如图, 在矩形$ABCD$中, $E$、$F$分别为边$AD$、$BC$上的点, 且$AD=3AE$, $BC=3BF$, 设$P$、$Q$分别为线段$AF$、$CE$的中点, 将四边形$ABFE$沿着直线$EF$进行翻折, 使得点$A$不在平面$CDEF$上, 在这一过程中, 下列关系不能恒成立的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw (3,2) node [right] {$D$} coordinate (D);\n\\draw (1,2) node [above] {$E$} coordinate (E);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (B) rectangle (D) (E)--(F)(A)--(F)(E)--(C);\n\\filldraw ($(A)!0.5!(F)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(C)!0.5!(E)$) node [left] {$Q$} coordinate (Q) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$AB\\parallel$直线$CD$}{直线$PQ\\parallel$直线$ED$}{直线$AB \\perp$直线$PQ$}{直线$PQ\\parallel$平面$ADE$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -370789,7 +371031,9 @@ "id": "015053", "content": "设$\\{a_n\\}$是项数为$n_0$的有穷数列, 其中$n_0 \\geq 2$. 当$n \\leq \\dfrac{n_0}{2}$时, $a_n=\\dfrac{1}{2^n}$, 且对任意正整数$n \\leq n_0$都有$a_n+a_{n_0+1-n}=0$. 给出下列两个命题: \\textcircled{1} 若对任意正整数$n \\leq n_0$都有$\\displaystyle\\sum_{i=1}^n a_i \\leq \\dfrac{511}{512}$, 则$n_0$的最大值为$18$; \\textcircled{2} 对于任意满足$1 \\leq s=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (C) ++ (0,\\h) node [left] {$C_1$} coordinate (C_1);\n\\draw (A) ++ (0,\\h) node [below right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) -- (A) -- (B) (C) -- (C_1) (A) -- (A_1) (B) -- (B_1) (C_1) -- (A_1) -- (B_1) (C_1) -- (B_1);\n\\draw [dashed] (C) -- (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$D$} coordinate (D);\n\\draw (D)--(B_1);\n\\draw [dashed] (D)--(C)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$CC_1$与$DB_1$所成的角的大小;\\\\\n(2) 求证: 平面$CDB_1 \\perp$平面$ABB_1A_1$, 并求点$B$到平面$CDB_1$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370846,7 +371094,9 @@ "id": "015056", "content": "某网站计划$4$月份订购草莓在网络销售, 每天的进货量相同, 成本价为每盒$15$元. 决定每盒售价为$20$元, 未售出的草莓降价处理, 每盒$10$元. 假设当天进货能全部售完. 根据销售经验, 每天的购买量与网站每天的浏览量(单位: 万次) 有关. 为确定草莓的进货量, 相关人员统计了前两年$4$月份(共$60$天)网站每天的浏览量(单位: 万次)、购买草莓的数量(单位: 盒) 以及达到该流量的天数, 如下表所示:\n\\begin{center} \n\\begin{tabular}{|c|c|c|}\n\\hline 每天的浏览量 &$(0,1)$& {$[1,+\\infty)$} \\\\\n\\hline 每天的购买量 & 600 & 900 \\\\\n\\hline 天数 & 36 & 24 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n以每天的浏览量位于各区间的频率代替浏览量位于该区间的概率.\n(1) 求$4$月份草苺一天的购买量$X$(单位: 盒)的分布;\\\\\n(2) 设$4$月份销售草莓一天的利润为$Y$(单位: 元), 一天的进货量为$n$(单位: 盒), $n$为正整数且$n \\in[600,900]$, 当$n$为多少时, $Y$的期望达到最大值, 并求此最大值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -370865,7 +371115,9 @@ "id": "015057", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1$($0=latex]\n\\fill [pattern = north east lines] (-1,1) arc (180:360:1 and 0.25) -- (0,0) -- cycle;\n\\fill [pattern = north west lines] (0,1) ellipse (1 and 0.25);\n\\draw (0,1) ellipse (1 and 0.25);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25);\n\\draw (-1,0) --++ (0,1) (1,0) --++ (0,1);\n\\draw [dashed] (-1,1) -- (0,0) -- (1,1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371055,7 +371325,9 @@ "id": "015067", "content": "若函数$y=f(x)$的图像可由函数$y=3 \\sin 2 x-\\sqrt{3} \\cos 2 x$的图像向右平移$\\varphi$($0<\\varphi<\\pi$)个单位所得到, 且函数$y=f(x)$在区间$[0, \\dfrac{\\pi}{2}]$上是严格减函数, 则$\\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371074,7 +371346,9 @@ "id": "015068", "content": "若每经过一天某种物品的价格变为原来的$1.02$倍的概率为$0.5$, 变为原来的$0.98$倍的概率也为$0.5$, 则经过$6$天该物品的价格较原来价格增加的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371093,7 +371367,9 @@ "id": "015069", "content": "如图, 在直角梯形$ABCD$中, $AD\\parallel BC$, $\\angle ABC=90^{\\circ}$, $AD=2$, $BC=1$, 点$P$是腰$AB$上的动点, 则$|2 \\overrightarrow{PC}+\\overrightarrow{PD}|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2.5,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (2.5,1) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.6!(B)$) node [below] {$P$} coordinate (P);\n\\draw (D)--(A)--(B)--(C)--cycle;\n\\draw [->] (P) -- (C);\n\\draw [->] (P) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371112,7 +371388,9 @@ "id": "015070", "content": "已知实数$a, b, c$满足: $a+b+c=0$与$a^2-b c=3$, 则$a b c$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371131,7 +371409,9 @@ "id": "015071", "content": "若直线$(a-1) x+y-1=0$与直线$3 x-a y+2=0$垂直, 则实数$a$的值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{3}{2}$}{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371150,7 +371430,9 @@ "id": "015072", "content": "从装有两个红球和两个白球的口袋内任取两个球, 那么互斥而不对立的事件是\\bracket{20}.\n\\twoch{``恰好有一个白球''与``都是红球''}{``至多有一个白球''与``都是红球''}{``至多有一个白球''与``都是白球''}{``至多有一个白球''与``至多有一个红球''}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371169,7 +371451,9 @@ "id": "015073", "content": "如图, $\\triangle ABD$与$\\triangle BCD$都是等腰直角三角形, 其底边分别为$BD$与$BC$, 点$E$、$F$分别为线段$BD$、$AC$的中点, 设二面角$A-BD-C$的大小为$\\alpha$, 当$\\alpha$在区间$(0, \\pi)$内变化时, 下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(D)$) node [above] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [left] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C);\n\\draw [dashed] (B)--(D)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{存在某一$\\alpha$值, 使得$AC \\perp BD$}{存在某一$\\alpha$值, 使得$EF \\perp BD$}{存在某一$\\alpha$值, 使得$EF \\perp CD$}{存在某一$\\alpha$值, 使得$AB \\perp CD$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371188,7 +371472,9 @@ "id": "015074", "content": "设数列$\\{a_n\\}$的前$n$项的和为$S_n$, 若对任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$S_n=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$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$} coordinate (A1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1)--(C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (A1)--(B)--(C1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!{1/3}!(C1)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!{1/3}!(C1)$) node [left] {$F$} coordinate (F);\n\\draw (D1)--(E)--(F)--(C);\n\\draw [dashed] (C)--(D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF\\parallel A_1B$;\\\\\n(2) 若$C_1E=2EA_1$, 求点$E$到平面$A_1D_1CB$的距离以及$ED_1$与平面$A_1D_1CB$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371245,7 +371535,9 @@ "id": "015077", "content": "将某工厂的工人按年龄分成两组: ``$35$周岁及以上''、``$35$周岁以下'', 从每组中随机抽取$80$人, 将他们的绩效分数分成$5$组: $[50,60),[60,70),[70,80),[80,90),[90,100]$, 分别加以统计, 得到下列频率分布直方图. 该工厂规定绩效分数不少于$80$者为生产标兵.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {绩效分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.005,60/0.035,70/0.035,80/0.020,90/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {70/0.035,80/0.020,90/0.005}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\draw (75,-0.01) node {$35$周岁及以上组};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {绩效分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.005,60/0.025,70/0.0325,80/0.0325,90/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {60/0.025,80/0.0325,90/0.005}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\draw (75,-0.01) node {$35$周岁以下组};\n\\end{tikzpicture}\n\\end{center}\n(1) 请列出$2 \\times 2$列联表, 并判断能否有$95 \\%$的把握认为是否为生产标兵与工人所在的年龄组有关;\\\\\n(2) 若已知该工厂工人中生产标兵的占比为$30 \\%$, 试估计该厂$35$周岁以下的工人所占的百分比以及生产标兵中$35$周岁以下的工人所占的百分比.\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$P(x^2 \\geq k)$& 0.100 & 0.050 & 0.010 & 0.001 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371264,7 +371556,9 @@ "id": "015078", "content": "已知双曲线$C$的中心在坐标原点, 左焦点$F_1$与右焦点$F_2$都在$x$轴上, 离心率为$3$, 过点$F_2$的动直线$l$与双曲线$C$交于点$A$、$B$, 设$\\dfrac{|AF_2| \\cdot|BF_2|}{|AB|^2}=\\lambda$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw [->] (-6,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-12) -- (0,12) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({sqrt(1+\\x*\\x/8)},\\x);\n\\draw [domain = {-2*sqrt(30)}:{2*sqrt(30)}, samples = 100] plot ({-sqrt(1+\\x*\\x/8)},\\x);\n\\filldraw (3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\filldraw (-3,0) circle (0.15) node [below] {$F_2$} coordinate (F_2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求双曲线$C$的渐近线方程;\\\\\n(2) 若点$A$、$B$都在双曲线$C$的右支上, 求$\\lambda$的最大值以及$\\lambda$取最大值时$\\angle AF_1B$的正切值; (关于求$\\lambda$的最值, 某学习小组提出了如下的思路可供参考: \\textcircled{1} 利用基本不等式求最值; \\textcircled{2} 设$\\dfrac{|AF_2|}{|AB|}$为$\\mu$, 建立相应数量关系并利用它求最值; \\textcircled{3} 设直线$l$的斜率为$k$, 建立相应数量关系并利用它求最值)\\\\\n(3) 若点$A$在双曲线$C$的左支上(点$A$不是该双曲线的顶点), 且$\\lambda=1$, 求证: $\\triangle AF_1B$是等腰三角形, 且$AB$边的长等于双曲线$C$的实轴长的$2$倍.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371283,7 +371577,9 @@ "id": "015079", "content": "三个互不相同的函数$y=f(x), y=g(x)$与$y=h(x)$在区间$D$上恒有$f(x) \\geq h(x) \\geq g(x)$或恒有$f(x) \\leq h(x) \\leq g(x)$, 则称$y=h(x)$为$y=f(x)$与$y=g(x)$在区间$D$上的``分割函数''.\\\\\n(1) 设$h_1(x)=4 x$, $h_2(x)=x+1$, 试分别判断$y=h_1(x)$、$y=h_2(x)$是否是$y=2 x^2+2$与$y=-x^2+4 x$在区间$(-\\infty,+\\infty)$上的``分割函数'', 请说明理由;\\\\\n(2) 求所有的二次函数, 使得该函数是$y=2 x^2+2$与$y=4 x$在区间$(-\\infty,+\\infty)$上的``分割函数'';\\\\\n(3) 若$[m, n] \\subseteq[-2,2]$, 且存在实数$k, b$, 使得$y=k x+b$为$y=x^4-4 x^2$与$y=4 x^2-16$在区间$[m, n]$\n上的``分割函数'', 求$n-m$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371302,7 +371598,9 @@ "id": "015080", "content": "若空间中两条直线$a$、$b$确定一个平面, 则$a$、$b$的位置关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371321,7 +371619,9 @@ "id": "015081", "content": "已知复数$z$满足$\\overline {z} \\cdot \\mathrm{i}=4+3 \\mathrm{i}$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371340,7 +371640,9 @@ "id": "015082", "content": "已知向量$\\overrightarrow {a}=(1,0)$和$\\overrightarrow {b}=(\\sqrt{3}, 1)$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371359,7 +371661,9 @@ "id": "015083", "content": "过点$P(-1,3)$, 与直线$x+\\sqrt{3} y+1=0$垂直的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371378,7 +371682,9 @@ "id": "015084", "content": "已知集合$A=\\{x | y=\\ln (3-x)\\}$, $B=\\{x | x>a\\}$, 若$A \\cap B=\\varnothing$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371397,7 +371703,9 @@ "id": "015085", "content": "已知圆柱的底面直径和高都等于球的直径, 圆柱的体积为$16 \\pi$, 则球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371416,7 +371724,9 @@ "id": "015086", "content": "已知函数$y=a x^2+b x+c$的图像如图所示, 则不等式$(a x+b)(b x+c)(c x+a)<0$的解集是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.5:3.5] plot (\\x,{(\\x-1)*(\\x-2)});\n\\draw (1,0) node [below] {$1$} (2,0) node [below] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371435,7 +371745,9 @@ "id": "015087", "content": "已知函数$y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 且满足$f(2+x)=-f(2-x)$, $f(1)=1$, 则$f(1)+f(2)+\\cdots+f(2023)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371454,7 +371766,9 @@ "id": "015088", "content": "如图所示, 要在两山顶$M$、$N$间建一索道, 需测量两山顶$M$、$N$间的距离. 已知两山的海拔高度分别是$MC=100 \\sqrt{3}$米和$NB=50 \\sqrt{2}$米, 现选择海平面上一点$A$为观测点, 从$A$点测得$M$点的仰角$\\angle MAC=60^{\\circ}$, $N$点的仰角$\\angle NAB=30^{\\circ}$以及$\\angle MAN=45^{\\circ}$, 则$MN$等于\\blank{50}米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (1.7,-0.5) node [below] {$A$} coordinate (A);\n\\draw (C) ++ (0,3) node [above] {$M$} coordinate (M);\n\\draw (B) ++ (0,1.5) node [above] {$N$} coordinate (N);\n\\draw (C)--(A)--(B)--(N)--(M)--cycle(M)--(A)--(N);\n\\draw [dashed] (C)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371473,7 +371787,9 @@ "id": "015089", "content": "已知数列$\\{a_n\\}$满足$a_n=a n^2+n$, 若满足$a_1a_{n+1}$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371492,7 +371808,9 @@ "id": "015090", "content": "如图, 已知$F_1, F_2$分别是椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点, $M, N$为椭圆上两点, 满足$F_1M\\parallel F_2N$, 且$|F_2N|: |F_2M|: |F_1M|=1: 2: 3$, 则椭圆$C$的离心率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(5)} and {sqrt(3)});\n\\draw ({-sqrt(2)},0) node [below] {$F_1$} coordinate (F_1);\n\\draw ({sqrt(2)},0) node [below] {$F_2$} coordinate (F_2);\n\\draw ({sqrt(2)/2},{3*sqrt(3)/sqrt(10)}) node [above] {$M$} coordinate (M);\n\\draw ($(F_2)+1/3*(M)-1/3*(F_1)$) node [above] {$N$} coordinate (N);\n\\draw (F_1)--(M)--(F_2)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371511,7 +371829,9 @@ "id": "015091", "content": "已知函数$y=\\sqrt{1-x^2}$, $-\\dfrac{1}{2} \\leq x \\leq \\dfrac{1}{2}$的图像绕着原点按逆时针方向旋转$\\theta$($0 \\leq \\theta \\leq \\pi$)弧度, 若得到的图像仍是函数图像, 则$\\theta$可取值的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371530,7 +371850,9 @@ "id": "015092", "content": "设$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$是两个不平行的向量, 则下列四组向量中, 不能组成平面向量的一个基的是\\bracket{20}.\n\\fourch{$\\overrightarrow{e_1}+\\overrightarrow{e_2}$和$\\overrightarrow{e_1}-\\overrightarrow{e_2}$}{$\\overrightarrow{e_1}+2 \\overrightarrow{e_2}$和$\\overrightarrow{e_2}+2 \\overrightarrow{e_1}$}{$3 \\overrightarrow{e_1}-2 \\overrightarrow{e_2}$和$4 \\overrightarrow{e_2}-6 \\overrightarrow{e_1}$}{$\\overrightarrow{e_2}$和$\\overrightarrow{e_2}+\\overrightarrow{e_1}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371549,7 +371871,9 @@ "id": "015093", "content": "已知$n$为正整数, 则``$n$是$3$的倍数''是``$(x^4-\\dfrac{2}{x^2})^n$的二项展开式中存在常数项''的 \\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371568,7 +371892,9 @@ "id": "015094", "content": "某产品的广告费$x$(单位: 万元) 与销售额$y$(单位: 万元) 的统计数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 广告费$x$(万元) & 2 & 3 & 4 & 5 \\\\\n\\hline 销售额$y$(万元) & 26 & 39 & 49 & 54 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表可得回归方程$y=\\hat{a} x+\\hat{b}$中$\\hat{a}=9.4$, 据此模型可预测当广告费为$6$万元时, 销售额约为\\bracket{20}.\n\\fourch{$63.6$万元}{$65.5$万元}{$67.7$万元}{$72.0$万元}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371587,7 +371913,9 @@ "id": "015095", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}-a_n=(-\\dfrac{1}{2})^n$, 存在正偶数$n$使得$(a_n-\\lambda)(a_{n+1}+\\lambda)>0$, 且对任意正奇数$n$有$(a_n-\\lambda)(a_{n+1}+\\lambda)<0$, 则实数$\\lambda$的取值范围是\\bracket{20}.\n\\fourch{$(-\\dfrac{2}{3}, 1]$}{$(-\\infty,-\\dfrac{2}{3}] \\cup(1,+\\infty)$}{$(-\\dfrac{3}{4}, \\dfrac{2}{3})$}{$(-\\dfrac{3}{4},-\\dfrac{2}{3}]$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371606,7 +371934,9 @@ "id": "015096", "content": "已知函数$y=f(x)$的表达式为$f(x)=\\sqrt{3} \\sin (x+\\dfrac{\\pi}{6}) \\cos (x+\\dfrac{\\pi}{6})+\\cos ^2(x-\\dfrac{\\pi}{3})$.\\\\\n(1) 求函数$y=f(x)$的最小正周期及图像的对称轴的方程;\\\\\n(2) 求函数$y=f(x)$在$(0, \\dfrac{\\pi}{2})$上的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371625,7 +371955,9 @@ "id": "015097", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, 底面$\\triangle ABC$是等腰直角三角形, $AC=BC=AA_1=2$, $D$为侧棱$AA_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (2,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (0,2,2) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1)(D)--(B_1);\n\\draw [dashed] (A)--(C)--(B)(C)--(C_1)(C)--(B_1)(C)--(D)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$ACC_1A_1$;\\\\\n(2) 求二面角$B_1-CD-C_1$的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371644,7 +371976,10 @@ "id": "015098", "content": "在全民抗击新冠疫情期间, 某校开展了``停课不停学''活动, 一个星期后, 某校随机抽取了$100$名居家学习的高二学生进行问卷调查, 得到学生每天学习时间 (单位: $\\text{h}$) 的频率分布直方图如下, 若被抽取的这$100$名学生中, 每天学习时间不低于$8$小时有$30$人.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1, yscale = 5]\n\\draw [->] (5,0) -- (5.2,0) -- (5.3,0.04) -- (5.5,-0.04) -- (5.6,0) -- (10,0) node [below right] {每天学习时间(h)};\n\\draw [->] (5,0) -- (5,0.75) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (5,0) node [below left] {$O$};\n\\foreach \\i/\\j in {6/0.14,6.5/0.26,7/0.42,7.5/0.58,8/0.38,8.5/0.22}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {6/0.14,6.5/0.26/a,7/0.42,7.5/0.58,8/0.38/b,8.5/0.22}\n{\\draw [dashed] (\\i,\\j) -- (5,\\j) node [left] {$\\k$};};\n\\draw (9,0) node [below] {$9$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求频率分布直方图中实数$a, b$的值;\\\\\n(2) 每天学习时间在$[6.0,6.5)$的$7$名学生中, 有$4$名男生, $3$名女生, 现从中抽$2$人进行电话访谈, 已知抽取的学生有男生, 求抽取的$2$人恰好为一男一女的概率;\\\\\n(3) 依据所抽取的样本, 从每天学习时间在$[6.0,6.5)$和$[7.0,7.5)$的学生中按比例分层抽样抽取$8$人, 再从这$8$人中选$3$人进行电话访谈, 求抽取的$3$人中每天学习时间在$[6.0,6.5)$的人数$X$的分布和数学期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371663,7 +371998,9 @@ "id": "015099", "content": "如图, 已知$A$、$B$、$C$是抛物线$\\Gamma_1: x^2=y$上的三个点, 且直线$CB$、$CA$分别与抛物线$\\Gamma_2: y^2=4 x$相切, $F$为抛物线$\\Gamma_1$的焦点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -2:2, samples = 100] plot (\\x,{\\x*\\x});\n\\draw [domain = {-sqrt(12)}:{sqrt(12)}, samples = 100] plot ({\\x*\\x/4},\\x);\n\\filldraw (0,0.25) node [right] {$F$} coordinate (F) circle (0.03);\n\\draw (-1,1) node [below] {$C$} coordinate (C);\n\\draw ({(1+sqrt(5))/2},{(3+sqrt(5))/2}) node [above] {$B$} coordinate (B);\n\\draw ({(1-sqrt(5))/2},{(3-sqrt(5))/2}) node [below] {$A$} coordinate (A);\n\\draw [thick] ($(B)!-0.5!(C)$) -- ($(B)!1.2!(C)$);\n\\draw [thick] ($(A)!-5!(C)$) -- ($(A)!1.5!(C)$);\n\\draw [thick] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$C$的横坐标为$x_3$, 用$x_3$表示线段$CF$的长;\\\\\n(2) 若$CA \\perp CB$, 求点$C$的坐标;\\\\\n(3) 证明: 直线$AB$与抛物线$\\Gamma_2$相切.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371682,7 +372019,9 @@ "id": "015100", "content": "设$y=f(x)$、$y=g(x)$是定义域为$\\mathbf{R}$的函数, 当$g(x_1) \\neq g(x_2)$时, 记$\\delta(x_1, x_2)=\\dfrac{f(x_1)-f(x_2)}{g(x_1)-g(x_2)}$.\\\\\n(1) 已知$y=g(x)$在区间$I$上严格增, 且对任意$x_1, x_2 \\in I$, $x_1 \\neq x_2$, 有$\\delta(x_1, x_2)>0$, 证明: 函数$y=f(x)$在区间$I$上严格增;\\\\\n(2) 已知$g(x)=\\dfrac{1}{3} x^3+a x^2-3 x$, 且对任意$x_1, x_2 \\in \\mathbf{R}$, 当$g(x_1) \\neq g(x_2)$时, 有$\\delta(x_1, x_2)>0$, 若当$x=1$时, 函数$y=f(x)$取得极值, 求实数$a$的值;\\\\\n(3) 已知$g(x)=\\sin x$, $f(\\dfrac{\\pi}{2})=1$, $f(-\\dfrac{\\pi}{2})=-1$, 且对任意$x_1, x_2 \\in \\mathbf{R}$, 当$g(x_1) \\neq g(x_2)$时, 有$|\\delta(x_1, x_2)| \\leq 1$, 证明: $f(x)=\\sin x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -371701,7 +372040,9 @@ "id": "015101", "content": "已知集合$A=\\{x |-20$时, $f(x)=2^x+\\dfrac{9}{2^x+1}$, 则该函数的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371853,7 +372208,9 @@ "id": "015109", "content": "端午节吃粽子是我国的传统习俗. 一盘中放有$10$个外观完全相同的粽子, 其中豆沙粽$3$个, 肉粽$3$个, 白米粽$4$个, 现从盘子任意取出$3$个, 则取到白米粽的个数的数学期望为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371872,7 +372229,9 @@ "id": "015110", "content": "已知$A, B$是球$O$的球面上两点, $\\angle AOB=60^{\\circ}$, $P$为该球面上的动点, 若三棱锥$P-OAB$体积的最大值为$6$, 则球$O$的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371891,7 +372250,9 @@ "id": "015111", "content": "过原点的直线$l$与双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左、右两支分别交于$M, N$两点, $F(2,0)$为$C$的右焦点, 若$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=0$, 且$|\\overrightarrow{FM}|+|\\overrightarrow{FN}|=2 \\sqrt{5}$, 则双曲线$C$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371910,7 +372271,9 @@ "id": "015112", "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}, \\overrightarrow {e}$满足$|\\overrightarrow {a}|=3$, $|\\overrightarrow {e}|=1$, $|\\overrightarrow {b}-\\overrightarrow {a}|=1$, $\\langle\\overrightarrow {a}, \\overrightarrow {e}\\rangle=\\dfrac{2 \\pi}{3}$, 且对任意的实数$t$, 均有$|\\overrightarrow {c}-t \\overrightarrow {e}| \\geq|\\overrightarrow {c}-2 \\overrightarrow {e}|$, 则$|\\overrightarrow {c}-\\overrightarrow {b}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -371929,7 +372292,9 @@ "id": "015113", "content": "已知复数$z=\\dfrac{1}{1-\\mathrm{i}}-\\mathrm{i}$($\\mathrm{i}$为虚数单位$)$, 则$z \\cdot \\overline {z}=$\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}{2}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371948,7 +372313,9 @@ "id": "015114", "content": "某同学上学路上有$4$个红绿灯的路口, 假设他走到每个路口遇到绿灯的概率为$\\dfrac{2}{3}$, 且在各个路口遇到红灯或绿灯互不影响, 则该同学上学路上至少遇到$2$次绿灯的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{3}{8}$}{$\\dfrac{7}{8}$}{$\\dfrac{8}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371967,7 +372334,9 @@ "id": "015115", "content": "对于函数$f(x)=\\sqrt{3} \\sin x \\cos x+\\sin ^2 x-\\dfrac{1}{2}$, 给出下列结论:\\\\\n\\textcircled{1} 函数$y=f(x)$的图像关于点$(\\dfrac{5 \\pi}{12}, 0)$对称;\\\\\n\\textcircled{2} 函数$y=f(x)$在区间$[\\dfrac{\\pi}{6}, \\dfrac{2 \\pi}{3}]$上的值域为$[-\\dfrac{1}{2}, 1]$;\\\\\n\\textcircled{3} 将函数$y=f(x)$的图像向左平移$\\dfrac{\\pi}{3}$个单位长度得到函数$y=-\\cos 2 x$的图像;\\\\\n\\textcircled{4} 曲线$y=f(x)$在$x=\\dfrac{\\pi}{4}$处的切线的斜率为$1$.\n则所有正确的结论是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -371986,7 +372355,9 @@ "id": "015116", "content": "在数列$\\{b_n\\}$中, 若有$b_m=b_n$($m, n$均为正整数, 且$m \\neq n$), 就有$b_{m+1}=b_{n+1}$, 则称数列$\\{b_n\\}$为``递等数列''. 已知数列$\\{a_n\\}$满足$a_5=5$, 且$a_n=n(a_{n+1}-a_n)$, 将``递等数列''$\\{b_n\\}$的前$n$项和记为$S_n$, 若$b_1=a_1=b_4$, $b_2=a_2$, $S_5=a_{10}$, 则$S_{2023}=$\\bracket{20}.\n\\fourch{$4720$}{$4719$}{$4718$}{$4716$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372005,7 +372376,9 @@ "id": "015117", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知$a_1=2$, $a_{n+1}=S_n$($n$为正整数).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\log _2 a_n$, 若$b_m+b_{m+1}+b_{m+2}+\\cdots+b_{m+9}=145$, 求正整数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372024,7 +372397,9 @@ "id": "015118", "content": "如图, 在圆锥$PO$中, $AB$是底面的直径, $C$是底面圆周上的一点, 且$PO=3$, $AB=4$, $\\angle BAC=30^{\\circ}$, $M$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) arc (180:360:2 and 0.5) -- (P)--cycle;\n\\draw [dashed] (A) arc (180:0:2 and 0.5) -- cycle(O)--(P);\n\\draw (-60:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw (P)--(C);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (B)--(C)(O)--(C)(P)--(M)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$PBC \\perp$平面$POM$;\\\\\n(2) 求二面角$O-PB-C$的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372043,7 +372418,9 @@ "id": "015119", "content": "电解电容是常见的电子元件之一. 检测组在$85^{\\circ} \\text{C}$的温度条件下对电解电容进行质量检测, 按检测结果将其分为次品、正品, 其中正品分合格品、优等品两类.\\\\\n(1) 铝箔是组成电解电容必不可少的材料. 现检测组在$85^{\\circ} \\text{C}$的温度条件下, 对铝箔质量与电解电容质量进行测试, 得到如下$2 \\times 2$列联表, 那么他们是否有$99.9 \\%$的把握认为电解电容质量与铝箔质量有关? 请说明理由;\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 电解电容为次品 & 电解电容为正品 \\\\\n\\hline 铝箔为次品 & 174 & 76 \\\\\n\\hline 铝箔为正品 & 108 & 142 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 电解电容经检验为正品后才能装箱, 已知两箱电解电容 (每箱$50$个), 第一箱和第二箱中分别有优等品$8$件与$9$件. 现用户从两箱中随机挑选出一箱, 并从该箱中先后随机抽取两个元件, 求在第一次取出的是优等品的情况下, 第二次取出的是合格品的概率.\\\\\n附录: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中 $n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $P\\left(\\chi^2 \\geq k\\right)$ & 0.100 & 0.050 & 0.025 & 0.010 & 0.001 \\\\\n\\hline $k$ & 2.706 & 3.841 & 5.024 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372062,7 +372439,9 @@ "id": "015120", "content": "已知动点$R(x, y)$到点$F(1,0)$的距离和它到直线$x=2$的距离之比等于$\\dfrac{\\sqrt{2}}{2}$, 动点$M$的轨迹记为曲线$C$, 过点$F$的直线$l$与曲线$C$相交于$P, Q$两点.\\\\\n(1) 求曲线$C$的方程;\\\\\n(2) 若$\\overrightarrow{FP}=-2 \\overrightarrow{FQ}$, 求直线$l$的方程;\\\\\n(3) 已知$A(-\\sqrt{2}, 0)$, 直线$AP, AQ$分别与直线$x=2$相交于$M, N$两点, 求证: 以$MN$为直径的圆经过点$F$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372081,7 +372460,9 @@ "id": "015121", "content": "设$f(x)=\\mathrm{e}^x$, $g(x)=\\ln x$, $h(x)=\\sin x+\\cos x$.\\\\\n(1) 求函数$y=\\dfrac{h(x)}{f(x)}$, $x \\in(-\\pi, 3 \\pi)$的单调区间和极值;\\\\\n(2) 若关于$x$不等式$f(x)+h(x) \\geq a x+2$在区间$[0,+\\infty)$上恒成立, 求实数$a$的取值范围;\\\\\n(3) 若存在直线$y=t$, 其与曲线$y=\\dfrac{x}{f(x)}$和$y=\\dfrac{g(x)}{x}$共有$3$个不同交点$A(x_1, t)$, $B(x_2, t)$, $C(x_3, t)$($x_10$)为偶函数, 则函数$f(x)$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372233,7 +372628,9 @@ "id": "015129", "content": "已知向量$\\overrightarrow {a}=(1, \\sqrt{3})$, 且$\\overrightarrow {a}, \\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, $(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot(2 \\overrightarrow {a}-3 \\overrightarrow {b})=4$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影\n向量等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372252,7 +372649,9 @@ "id": "015130", "content": "某运动生理学家在一项健身活动中选择了$10$名男性参与者, 以他们的皮下脂肪厚度来估计身体的脂肪含量, 其中脂肪含量以占体重 (单位: $\\text{kg}$) 的百分比表示. 得到脂肪含量和体重的数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n个体编号 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\\hline\n体重$x$($\\text{kg}$) & 89 & 88 & 66 & 59 & 93 & 73 & 82 & 77 & 100 & 67\\\\\\hline\n脂肪含量$y$($\\%$) & 28 & 27 & 24 & 23 & 29 & 25 & 29 & 25 & 30 & 23\\\\\\hline\n\\end{tabular}\n\\end{center}\n建立男性体重与脂肪含量的回归方程为: \\blank{100}(结果中回归系数保留三位小数).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372271,7 +372670,9 @@ "id": "015131", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, $E$为$AB$的中点, $F$为正方形$BCC_1B_1$的中心, 则直线$EF$与侧面$BB_1C_1C$所成角的正切值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(B_1)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\filldraw (E) circle (0.03) (F) circle (0.03);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372290,7 +372691,9 @@ "id": "015132", "content": "今年是农历癸卯兔年, 一种以兔子形象命名的牛奶糖深受顾客欢迎. 标识质量为$500 \\text{g}$的这种袋装奶糖的质量指标$X$是服从正态分布$N(500,2.5^2)$的随机变量. 若质量指标介于$495 \\text{g}$(含) 至$505 \\text{g}$(含) 之间的产品包装为合格包装, 则随意买一包这种袋装奶糖, 是合格包装的可能性大小为\\blank{50}$\\%$. (结果保留一位小数)\\\\\n(已知$\\Phi(1) \\approx 0.8413$, $\\Phi(2) \\approx 0.9772$, $\\Phi(3) \\approx 0.9987$. $\\Phi(x)$表示标准正态分布的密度函数从$-\\infty$到$x$的累计面积)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372309,7 +372712,9 @@ "id": "015133", "content": "若$10^x-10^y=10$, 其中$x, y \\in \\mathbf{R}$, 则$2 x-y$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372328,7 +372733,9 @@ "id": "015134", "content": "若直线$l$的方向向量为$\\overrightarrow {a}$, 平面$\\alpha$的法向量为$\\overrightarrow {n}$, 则能使$l\\parallel \\alpha$的是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}=(1,0,0)$, $\\overrightarrow {n}=(-2,0,0)$}{$\\overrightarrow {a}=(1,3,5)$, $\\overrightarrow {n}=(1,0,1)$}{$\\overrightarrow {a}=(1,-1,3)$, $\\overrightarrow {n}=(0,3,1)$}{$\\overrightarrow {a}=(0,2,1)$, $\\overrightarrow {n}=(-1,0,-1)$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372347,7 +372754,9 @@ "id": "015135", "content": "摩天轮常被当作一个城市的地标性建筑, 如静安大悦城的``Sky Ring''摩天轮是上海首个悬臂式屋顶摩天轮. 摩天轮最高点离地面高度$106$米, 转盘直径$56$米, 轮上设置$30$个极具时尚感的$4$人轿舱, 拥有$360$度的绝佳视野. 游客从离楼顶屋面最近的平台位置进入轿舱, 开启后按逆时针匀速旋转$t$分钟后, 游客距离地面的高度为$h$米, $h=$$-28 \\cos (\\dfrac{\\pi t}{6})+78$. 若在$t_1, t_2$时刻, 游客距离地面的高度相等, 则$t_1+t_2$的最小值为\\bracket{20}.\n\\fourch{$6$}{$12$}{$18$}{$24$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372366,7 +372775,9 @@ "id": "015136", "content": "设直线$l_1: x-2 y-2=0$与$l_2$关于直线$l: 2 x-y-4=0$对称, 则直线$l_2$的方程是\\bracket{20}.\n\\fourch{$11 x+2 y-22=0$}{$11 x+y+22=0$}{$5 x+y-11=0$}{$10 x+y-22=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372385,7 +372796,9 @@ "id": "015137", "content": "函数$y=x \\ln x$\\bracket{20}.\n\\onech{是严格增函数}{在$(0, \\dfrac{1}{\\mathrm{e}})$上是严格增函数, 在$(\\dfrac{1}{\\mathrm{e}},+\\infty)$上是严格减函数}{是严格减函数}{在$(0, \\dfrac{1}{\\mathrm{e}})$上是严格减函数, 在$(\\dfrac{1}{\\mathrm{e}},+\\infty)$上是严格增函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372404,7 +372817,9 @@ "id": "015138", "content": "已知各项均为正数的数列$\\{a_n\\}$满足$a_1=1$, $a_n=2 a_{n-1}+3$(正整数$n \\geq 2$).\\\\\n(1) 求证: 数列$\\{a_n+3\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的前$n$项和$S_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372423,7 +372838,9 @@ "id": "015139", "content": "如图, 在五面体$ABCDEF$中, $FA \\perp$平面$ABCD$, $AD\\parallel BC\\parallel FE$, $AB \\perp AD$, 若$AD=2$, $AF=AB=BC=FE=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$F$} coordinate (F);\n\\draw (1,1,0) node [above] {$E$} coordinate (E);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(C)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw (B)--(C)--(D)--(E)--(F)--cycle(E)--(C)(M)--(D);\n\\draw [dashed] (F)--(A)--(B)(A)--(D)(A)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求五面体$ABCDEF$的体积;\\\\\n(2) 若$M$为$EC$的中点, 求证: 平面$CDE \\perp$平面$AMD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372442,7 +372859,9 @@ "id": "015140", "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$(其中$a>0, b>0$)的左、右焦点分别为$F_1(-c, 0)$、$F_2(c, 0)$(其中$c>0$).\\\\\n(1) 若双曲线$\\Gamma$过点$(2,1)$且一条渐近线方程为$y=\\dfrac{\\sqrt{2}}{2} x$; 直线$l$的倾斜角为$\\dfrac{\\pi}{4}$, 在$y$轴上的截距为$-2$. 直线$l$与该双曲线$\\Gamma$交于两点$A$、$B, M$为线段$AB$的中点, 求$\\triangle MF_1F_2$的面积;\\\\\n(2) 以坐标原点$O$为圆心, $c$为半径作圆, 该圆与双曲线$\\Gamma$在第一象限的交点为$P$. 过$P$作圆的切线, 若切线的斜率为$-\\sqrt{3}$, 求双曲线$\\Gamma$的离心率.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372461,7 +372880,9 @@ "id": "015141", "content": "概率统计在生产实践和科学实验中应用广泛. 请解决下列两个问题.\\\\\n(1) 随着中小学``双减''政策的深入人心, 体育教学和各项体育锻炼迎来时间充沛的春天. 某初中学校学生篮球队从开学第二周开始每周进行训练, 第一次训练前共有$6$个篮球, 其中$3$个是新球(即没有用过的球), $3$个是旧球(即至少用过一次的球). 每次训练, 都是从中不放回任意取出$2$个篮球, 训练结束后放回原处. 设第一次训练时取到的新球个数为$X$, 求随机变量$X$的分布和期望;\\\\\n(2) 由于手机用微波频率信号传递信息, 那么长时间使用手机是否会增加得脑瘤的概率? 研究者针对这个问题, 对脑瘤病人进行问卷调查, 询问他们是否总是习惯在固定的一侧接听电话? 如果是, 是哪边? 结果有$88$人喜欢用固定的一侧接电话. 其中脑瘤部位在左侧的病人习惯固定在左侧接听电话的有$14$人, 习惯固定在右侧接听电话的有$28$人; 脑瘤部位在右侧的病人习惯固定在左侧接听电话的有$19$人, 习惯固定在右侧接听电话的有$27$人.\n根据上述信息写出下面这张$2 \\times 2$列联表中字母所表示的数据, 并对患脑瘤在左右侧的部位是否与习惯在该侧接听手机电话相关进行独立性检验. (显著性水平$\\alpha=0.05$)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 习惯固定在左侧接听电话 & 习惯固定在右侧接听电话 & 总计 \\\\\n\\hline\n脑瘤部位在左侧的病人 & $a$ & $b$ & $42$ \\\\\n\\hline\n脑瘤部位在右侧的病人 & $c$ & $d$ & $46$ \\\\\n\\hline\n总计 & $a+c$ & $b=d$ & $88$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n参考公式及数据: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$, $P(\\chi^2 \\geq 3.841) \\approx 0.05$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372480,7 +372901,9 @@ "id": "015142", "content": "已知函数$f(x)=\\dfrac{1}{2} x^2-(a+1) x+a \\ln x$. (其中$a$为常数)\\\\\n(1) 若$a=-2$, 求曲线$y=f(x)$在点$(2, f(2))$处的切线方程;\\\\\n(2) 当$a<0$时, 求函数$y=f(x)$的最小值;\\\\\n(3) 当$0 \\leq a<1$时, 试讨论函数$y=f(x)$的零点个数, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372499,7 +372922,9 @@ "id": "015143", "content": "设全集$U=\\mathbf{R}$, 若集合$A=\\{x \\| x | \\geq 1, x \\in \\mathbf{R}\\}$, 则$\\overline {A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372518,7 +372943,9 @@ "id": "015144", "content": "函数$y=\\cos ^2 x-\\sin ^2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372537,7 +372964,9 @@ "id": "015145", "content": "现有一组数$1,1,2,2,3,5,6,7,9,9$, 则该组数的第$25$百分位数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372556,7 +372985,9 @@ "id": "015146", "content": "设$3 \\mathrm{i}$($\\mathrm{i}$为虚数单位)是关于$x$的方程$x^2+m=0$($m \\in \\mathbf{R})$的根, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372575,7 +373006,9 @@ "id": "015147", "content": "函数$y=\\sqrt{3-\\dfrac{1}{x}}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372594,7 +373027,9 @@ "id": "015148", "content": "若$\\pi<\\theta<\\dfrac{3 \\pi}{2}$且$\\sin \\theta=-\\dfrac{3}{5}$, 则$\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372613,7 +373048,9 @@ "id": "015149", "content": "现有一个底面半径为$2 \\text{cm}$、高为$9 \\text{cm}$的圆柱形铁料, 若将其熔铸成一个球形实心工件, 则该工件的表面积为\\blank{50}$\\text{cm}^2$(损耗忽略不计).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372632,7 +373069,9 @@ "id": "015150", "content": "设$\\triangle ABC$的三边$a, b, c$满足$a: b: c=7: 5: 3$, 且$S_{\\triangle ABC}=15 \\sqrt{3}$, 则此三角形最长的边长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372651,7 +373090,9 @@ "id": "015151", "content": "``民生''供电公司为了分析``康居''小区的用电量$y$(单位: $\\text{kW}\\cdot\\text{h}$)与气温$x$(单位: ${}^\\circ\\text{C})$之间的关系, 随机统计了$4$天的用电量与当天的气温, 这两者之间的对应关系见下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 气温$x$& 18 & 13 & 10 & -1 \\\\\n\\hline 用电量$y$& 24 & 34 & 38 & 64 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n若上表中的数据可用回归方程$y=-2 x+b$($b \\in \\mathbf{R}$)来预测, 则当气温为$-4^{\\circ} \\text{C}$时该小区相应的用电量约为$\\text{kW} \\cdot \\text{h}$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372670,7 +373111,9 @@ "id": "015152", "content": "设$F_1$、$F_2$为双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1(a>0)$左、右焦点, 且$\\Gamma$的离心率为$\\sqrt{5}$, 若点$M$在$\\Gamma$的右支上, 直线$F_1M$与$\\Gamma$的左支相交于点$N$, 且$|MF_2|=|MN|$, 则$|F_1N|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372689,7 +373132,9 @@ "id": "015153", "content": "设$a>0$且$a \\neq 1$, 若在平面直角坐标系$xOy$中, 函数$y=\\log _a(a x+2)$与$y=\\log _a \\dfrac{1}{2 x+a}$的图像关于直线$l$对称, 则$l$与这两个函数图像的公共点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372708,7 +373153,9 @@ "id": "015154", "content": "设$x$、$y \\in \\mathbf{R}$, 若向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$\\overrightarrow {a}=(x, 1)$, $\\overrightarrow {b}=(2, y)$, $\\overrightarrow {c}=(1,1)$, 且向量$\\overrightarrow {a}-\\overrightarrow {b}$与$\\overrightarrow {c}$互相平行, 则$|\\overrightarrow {a}|+2|\\overrightarrow {b}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372727,7 +373174,9 @@ "id": "015155", "content": "设$a$、$b$为实数, 则``$a>b>0$''的一个充分非必要条件是\\bracket{20}.\n\\fourch{$\\sqrt{a-1}>\\sqrt{b-1}$}{$a^2>b^2$}{$\\dfrac{1}{b}>\\dfrac{1}{a}$}{$a-b>b-a$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372746,7 +373195,9 @@ "id": "015156", "content": "设$a$、$b$表示空间的两条直线, $\\alpha$表示平面, 给出下列结论:\\\\\n\\textcircled{1} 若$a\\parallel b$且$b \\subset \\alpha$, 则$a\\parallel \\alpha$;\\\\\n\\textcircled{2} 若$a\\parallel \\alpha$且$b \\subset \\alpha$, 则$a\\parallel b$;\\\\\n\\textcircled{3} 若$a\\parallel b$且$a\\parallel \\alpha$, 则$b\\parallel \\alpha$;\\\\\n\\textcircled{4} 若$a\\parallel \\alpha$且$b\\parallel \\alpha$, 则$a\\parallel b$.\\\\\n其中不正确的个数是\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372765,7 +373216,9 @@ "id": "015157", "content": "设$P$为曲线$C: y^2=4 x$上的任意一点, 记$P$到$C$的准线的距离为$d$. 若关于点集$A=\\{M| |MP |=d\\}$和$B=\\{(x, y) |(x-1)^2+(y-1)^2=r^2\\}$, 给出如下结论:\\\\\n\\textcircled{1} 任意$r \\in(0,+\\infty)$, $A \\cap B$中总有$2$个元素;\\\\ \\textcircled{2} 存在$r \\in(0,+\\infty)$, 使得$A \\cap B=\\varnothing$.\\\\\n其中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -372784,7 +373237,9 @@ "id": "015158", "content": "设$\\omega>0$, 若在区间$[\\pi, 2 \\pi)$上存在$a, b$且$a=latex, scale = 0.6]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,4) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,3) node [above] {$C_1$} coordinate (C_1);\n\\draw (A)--(B)--(B_1)--(C_1)--(A_1)--cycle(A_1)--(B_1);\n\\draw [dashed] (A)--(C)--(B)--(C_1)--(C)(A)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AC \\perp BC_1$;\\\\\n(2) 设$AC_1$与底面$ABC$所成角的大小为$60^{\\circ}$, 求三梭锥$C-ABC_1$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372822,7 +373279,9 @@ "id": "015160", "content": "已知$a>b$均为不是$1$的正实数, 设函数$y=f(x)$的表达式为$f(x)=a \\cdot b^x$($x \\in \\mathbf{R}$).\\\\\n(1) 设$a>b$且$f(x) \\leq b \\cdot a^x$, 求$x$的取值范围;\\\\\n(2) 设$a=\\dfrac{1}{16}$, $b=4$, 记$a_n=\\log _2 f(n)$, $b_n=f(n)$, 现将数列$\\{a_n\\}$中剔除$\\{b_n\\}$的项后、不改变其原来顺序所组成的数列记为$\\{c_n\\}$, 求$\\displaystyle\\sum_{i=1}^{100} c_i$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372841,7 +373300,9 @@ "id": "015161", "content": "现有$3$个盒子, 其中第一个盒子中装有$1$个白球、$4$个黑球; 第二个盒子装有$2$个白球、$3$个黑球; 第三个盒子装有$3$个白球、$2$个黑球. 现任取一个盒子, 从中任取$3$个球.\\\\\n(1) 求取到的白球数不少于$2$个的概率;\\\\\n(2) 设$X$为所取到的白球数, 求取到的白球数的期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372860,7 +373321,9 @@ "id": "015162", "content": "在$xOy$平面上, 设椭圆$\\Gamma: \\dfrac{x^2}{m^2}+y^2=1$($m>1$), 梯形$ABCD$的四个顶点均在$\\Gamma$上, 且$AB\\parallel CD$. 设直线$AB$的方程为$y=k x$($k \\in \\mathbf{R}$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (2 and 1);\n\\draw (-2,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0.5) node [above] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0.5) node [above] {$D$} coordinate (D);\n\\filldraw ({sqrt(3)},0) node [below] {$F$} coordinate (F) circle (0.03);\n\\draw (B)--(C)--(D)--(A);\n\\draw (0,-1.5) node [below] {第(1)小题}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\filldraw (0,2) node [right] {$P$} coordinate (P) circle (0.03);\n\\path [name path = DP] (P) --++ (-2,-4);\n\\path [name intersections = {of = DP and elli, by = {C,D}}];\n\\path [name path = BA] (1,2) -- (-0.6,-1.2);\n\\path [name intersections = {of = BA and elli, by = {B,A}}];\n\\draw (C) node [above] {$C$};\n\\draw (D) node [below] {$D$};\n\\draw (A) node [below] {$A$};\n\\draw (B) node [above] {$B$};\n\\draw (P)--(D);\n\\draw (B)--(C)--(D)--(A)--cycle;\n\\draw [->] (O)--(C);\n\\draw [->] (O)--(D);\n\\draw (0,-1.5) node [below] {第(2)小题}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\draw ({2/sqrt(3)},{1/sqrt(3)}) node [above] {$B$} coordinate (B);\n\\draw ($(B)!2!(O)$) node [below] {$A$} coordinate (A);\n\\draw ({-sqrt(2)},{sqrt(2)}) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(M)$) node [left] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(M)$) node [above] {$C$} coordinate (C);\n\\draw (M)--(A)--(B)--cycle(C)--(D);\n\\draw (0,-1.5) node [below] {第(3)小题}; \n\\end{tikzpicture}\n\\end{center}\n(1) 若$AB$为$\\Gamma$的长轴, 梯形$ABCD$的高为$\\dfrac{1}{2}$, 且$C$在$AB$上的射影为$\\Gamma$的焦点, 求$m$的值;\\\\\n(2) 设$m=\\sqrt{2}$, 直线$CD$经过点$P(0,2)$, 求$\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$的取值范围;\\\\\n(3) 设$m=\\sqrt{2}$, $|AB|=2|CD|$, $AD$与$BC$的延长线相交于点$M$, 当$k$变化时, $\\triangle MAB$的面积是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372879,7 +373342,9 @@ "id": "015163", "content": "已知$a$、$b \\in \\mathbf{R}$, 设函数$y=f(x)$的表达式为$f(x)=a \\cdot x^2-b \\cdot \\ln x$(其中$x>0$).\\\\\n(1) 设$a=1$, $b=0$, 当$f(x)>x^{-1}$时, 求$x$的取值范围;\\\\\n(2) 设$a=2$, $b>4$, 集合$D=(0,1]$, 记$g(x)=2 c x-\\dfrac{1}{x^2}$($c \\in \\mathbf{R}$), 若$y=g(x)$在$D$上为严格增函数且对$D$上的任意两个变量$s, t$, 均有$f(s) \\geq g(t)$成立, 求$c$的取值范围;\\\\\n(3) 当$a=0$, $b<0$, $x>1$时, 记$h_n(x)=[f(x)]^n+\\dfrac{1}{[f(x)]^n}$, 其中$n$为正整数. 求证: $[h_1(x)]^n+2 \\geq h_n(x)+2^n$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -372898,7 +373363,9 @@ "id": "015164", "content": "设全集$U=\\{-2,-1,0,1,2\\}$, 集合$A=\\{-2,0,2\\}$, 则$\\overline {A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372917,7 +373384,9 @@ "id": "015165", "content": "若实数$x$、$y$满足$\\lg x=m$、$y=10^{1-m}$, 则$x y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372936,7 +373405,9 @@ "id": "015166", "content": "已知复数$z$满足$z(1-\\mathrm{i})=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z$的虚部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372955,7 +373426,9 @@ "id": "015167", "content": "已知圆柱的底面积为$9 \\pi$, 侧面积为$12 \\pi$, 则该圆柱的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372974,7 +373447,9 @@ "id": "015168", "content": "已知常数$m>0$, $(x+\\dfrac{m}{x})^6$的二项展开式中$x^2$项的系数是$60$, 则$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -372993,7 +373468,9 @@ "id": "015169", "content": "已知事件$A$与事件$B$互斥, 如果$P(A)=0.3$, $P(B)=0.5$, 那么$P(\\overline{A \\cup B})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373012,7 +373489,9 @@ "id": "015170", "content": "今年春季流感爆发期间, 某医院准备将$2$名医生和$4$名护士分配到两所学校, 给学校老师和学生接种流感疫苗. 若每所学校分配$1$名医生和$2$名护士, 则不同的分配方法数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373031,7 +373510,9 @@ "id": "015171", "content": "$\\displaystyle\\lim _{h \\to 0} \\dfrac{\\ln (h+4)-2 \\ln 2}{h}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373050,7 +373531,9 @@ "id": "015172", "content": "若关于$x$的方程$(\\dfrac{1}{2})^x+m=\\sqrt{x+1}$在实数范围内有解, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373069,7 +373552,9 @@ "id": "015173", "content": "已知在等比数列$\\{a_n\\}$中, $a_3$、$a_7$分别是函数$y=x^3-6 x^2+6 x-1$的两个驻点, 则$a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373088,7 +373573,9 @@ "id": "015174", "content": "已知抛物线$C_1: y^2=8 x$, 圆$C_2: (x-2)^2+y^2=1$, 点$M$的坐标为$(4,0)$, $P$、$Q$分别为$C_1$、$C_2$上的动点, 且满足$|PM|=|PQ|$, 则点$P$的横坐标的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373107,7 +373594,9 @@ "id": "015175", "content": "平面上有一组互不相等的单位向量$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$, 若存在单位向量$\\overrightarrow{OP}$满足$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_1}+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_2}+\\cdots+\\overrightarrow{OP} \\cdot \\overrightarrow{OA_n}=0$, 则称$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\cdots, \\overrightarrow{OA_n}$的平衡向量. 已知$\\langle\\overrightarrow{OA_1}, \\overrightarrow{OA_2}\\rangle=\\dfrac{\\pi}{3}$, 向量$\\overrightarrow{OP}$是向量组$\\overrightarrow{OA_1}, \\overrightarrow{OA_2}, \\overrightarrow{OA_3}$的平衡向量, 当$\\overrightarrow{OP} \\cdot \\overrightarrow{OA_3}$取得最大值时, $\\overrightarrow{OA_1} \\cdot \\overrightarrow{OA_3}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373126,7 +373615,9 @@ "id": "015176", "content": "下列函数中, 既不是奇函数, 也不是偶函数的为\\bracket{20}.\n\\fourch{$y=0$}{$y=\\dfrac{1}{x}$}{$y=x^2$}{$y=2^x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373145,7 +373636,9 @@ "id": "015177", "content": "在某区高三年级举行的一次质量检测中, 某学科共有$3000$人参加考试. 为了解本次考试学生的成绩情况, 从中抽取了部分学生的成绩 (成绩均为正整数, 满分为$100$分) 作为样本进行统计, 样本容量为$n$. 按照$[50,60),[60,70),[70,80),[80,90),[90,100]$的分组作出频率分布直方图 (如图所示). 已知成绩落在$[50,60)$内的人数为$16$, 则下列结论正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (40,0) -- (42,0) -- (44,-0.003) -- (46,0.003) -- (48,0)-- (120,0) node [below] {成绩(分)};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.016,60/0.03,70/0.04,80/0.01,90/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.016,60/0.03/x,70/0.04,80/0.01,90/0.004}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{样本容量$n=1000$}{图中$x=0.025$}{估计全体学生该学科成绩的平均分为$70.6$分}{若将该学科成绩由高到低排序, 前$15 \\%$的学生该学科成绩为A等, 则成绩为$78$分的学生该学科成绩肯定不是A等}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373164,7 +373657,9 @@ "id": "015178", "content": "已知$f(x)=\\cos 2 x-a \\sin x$, 若存在正整数$n$, 使函数$y=f(x)$在区间$(0, n \\pi)$内有$2023$个零点, 则实数$a$所有可能的值为\\bracket{20}.\n\\fourch{$1$}{$-1$}{$0$}{$1$或$-1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373183,7 +373678,9 @@ "id": "015179", "content": "若数列$\\{b_n\\}$、$\\{c_n\\}$均为严格增数列, 且对任意正整数$n$, 都存在正整数$m$, 使得$b_m \\in[c_n, c_{n+1}]$, 则称数列$\\{b_n\\}$为数列$\\{c_n\\}$的``M数列''. 已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 则下列选项中为假命题的是\\bracket{20}.\n\\onech{存在等差数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{a_n\\}$是$\\{S_n\\}$的``M数列''}{存在等差数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}{存在等比数列$\\{a_n\\}$, 使得$\\{S_n\\}$是$\\{a_n\\}$的``M数列''}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373202,7 +373699,9 @@ "id": "015180", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 已知$\\sin A=\\sin 2B, a=4$, $b=6$.\\\\\n(1) 求$\\cos B$的值;\\\\\n(2) 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373221,7 +373720,9 @@ "id": "015181", "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PD \\perp$平面$ABCD$, $PD=AD=2$, $AB=4$, 点$E$在线段$AB$上, 且$BE=\\dfrac{1}{4} AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (4,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.75!(B)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(E)(P)--(B);\n\\draw [dashed] (P)--(D)--(A)(D)--(C)(D)--(B)(A)--(C)(C)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CE \\perp$平面$PBD$;\\\\\n(2) 求二面角$P-CE-A$的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373240,7 +373741,9 @@ "id": "015182", "content": "在临床检测试验中, 某地用某种抗原来诊断试验者是否患有某种疾病. 设事件$A$表示试验者的检测结果为阳性, 事件$B$表示试验者患有此疾病. 据临床统计显示, $P(A | B)$$=0.99, P(\\overline {A} | \\overline {B})=0.98$. 已知该地人群中患有此种疾病的概率为$0.001$. (下列两小题计算结果中的概率值精确到$0.00001$)\\\\\n(1) 对该地某人进行抗原检测, 求事件$A$与$\\overline {B}$同时发生的概率;\\\\\n(2) 对该地$3$个患有此疾病的患者进行抗原检测, 用随机变量$X$表示检测结果为阳性的人数, 求$X$的分布和期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373259,7 +373762,9 @@ "id": "015183", "content": "已知$O$为坐标原点, 曲线$C_1: \\dfrac{x^2}{a^2}-y^2=1$($a>0$)和曲线$C_2: \\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$有公共点, 直线$l_1: y=k_1 x+b_1$与曲线$C_1$的左支相交于$A$、$B$两点, 线段$AB$的中点为$M$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\path [draw, name path = hypbo, samples = 100, domain = -2:3] plot ({-sqrt(\\x*\\x+1)},\\x); \n\\path [draw, name path = hypbo2, samples = 100, domain = -2:2] plot ({sqrt(\\x*\\x+1)},\\x); \n\\path [name path = AB] (-3,3) -- (-1.75,-2);\n\\path [name path = CD] (0,-2) -- (1,2);\n\\path [draw, name path = MN] (-3,0.75) -- (3,-0.75);\n\\path [name intersections = {of = AB and hypbo, by = {B,A}}];\n\\path [name intersections = {of = CD and elli, by = {C,D}}];\n\\path [name intersections = {of = AB and MN, by = M}];\n\\path [name intersections = {of = CD and MN, by = N}];\n\\draw (A) node [above] {$A$} -- (B) node [below] {$B$};\n\\draw (C) node [above] {$C$} -- (D) node [below] {$D$};\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若曲线$C_1$和$C_2$有且仅有两个公共点, 求曲线$C_1$的离心率和渐近线方程;\\\\\n(2) 若直线$OM$经过曲线$C_2$上的点$T(\\sqrt{2},-1)$, 且$a^2$为正整数, 求$a$的值;\\\\\n(3) 若直线$l_2: y=k_2 x+b_2$与曲线$C_2$相交于$C$、$D$两点, 且直线$OM$经过线段$CD$中点$N$, 求证: $k_1^2+k_2^2>1$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373278,7 +373783,9 @@ "id": "015184", "content": "如果曲线$y=f(x)$存在相互垂直的两条切线, 称函数$y=f(x)$是``正交函数''. 已知$f(x)=x^2+a x+2 \\ln x$, 设曲线$y=f(x)$在点$M(x_0, f(x_0))$处的切线为$l_1$.\\\\\n(1) 当$f'(1)=0$时, 求实数$a$的值;\\\\\n(2) 当$a=-8$, $x_0=8$时, 是否存在直线$l_2$满足$l_1 \\perp l_2$, 且$l_2$与曲线$y=f(x)$相切? 请说明理由;\\\\\n(3) 当$a \\geq-5$时, 如果函数$y=f(x)$是``正交函数'', 求满足要求的实数$a$的集合$D$; 若对任意$a \\in D$, 曲线$y=f(x)$都不存在与$l_1$垂直的切线$l_2$, 求$x_0$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373297,7 +373804,9 @@ "id": "015185", "content": "已知集合$A=\\{1,2,3,4,5\\}$, $B=\\{2,4,6,8\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373316,7 +373825,9 @@ "id": "015186", "content": "若``$x=1$''是``$x>a$''的充分条件, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373335,7 +373846,9 @@ "id": "015187", "content": "已知事件$A$与事件$B$相互独立, 如果$P(A)=0.5$, $P(A \\cap \\overline {B})=0.4$, 那么$P(B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373354,7 +373867,9 @@ "id": "015188", "content": "当$x \\in[a,+\\infty)$时, 幂函数$y=x^2$的图像总在$y=x^{\\frac{1}{2}}$的图像上方, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373373,7 +373888,9 @@ "id": "015189", "content": "已知圆锥侧面展开图的圆心角为$\\dfrac{2 \\pi}{3}$, 底面周长为$2 \\pi$. 则这个圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373392,7 +373909,9 @@ "id": "015190", "content": "若函数$y=\\ln (1+x)-a \\ln (1-x)$为奇函数, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373411,7 +373930,9 @@ "id": "015191", "content": "设随机变量$X$服从正态分布$N(2, \\sigma^2)$, 若$P(X \\leq 1)=0.2$, 则$P(X<3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373430,7 +373951,9 @@ "id": "015192", "content": "某小学开展劳动教育, 欲在围墙边用栅栏围成一个$2$平方米的矩形植物种植园, 矩形的一条边为围墙, 如图. 则至少需要\\blank{50}米栅栏.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (-0.2,0.2) rectangle (3.2,0);\n\\draw (-0.2,0) -- (3.2,0);\n\\draw (0,0) rectangle (3,-1.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373449,7 +373972,9 @@ "id": "015193", "content": "已知函数$y=f(x)$, $y=g(x)$满足$f(x)+x g(x)=x^2-1$, 若$f(1)=1$, 则$f'(1)+g'(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373468,7 +373993,9 @@ "id": "015194", "content": "若对任意$x \\in[1,2]$, 均有$|x^2-a|+|x+a|=|x^2+x|$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373487,7 +374014,9 @@ "id": "015195", "content": "已知空间向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$、$\\overrightarrow {d}$满足: $|\\overrightarrow {a}-\\overrightarrow {b}|=1$, $|\\overrightarrow {b}-\\overrightarrow {c}|=2$, $(\\overrightarrow {a}-\\overrightarrow {b})\\parallel(\\overrightarrow {b}-\\overrightarrow {c})$, $(\\overrightarrow {a}-\\overrightarrow {d}) \\cdot(\\overrightarrow {b}-\\overrightarrow {d})=0$, 则$|\\overrightarrow {c}-\\overrightarrow {d}|$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373506,7 +374035,9 @@ "id": "015196", "content": "已知$F_1$、$F_2$是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, $l$是$\\Gamma$的一条渐近线, 以$F_2$为圆心的圆与$l$相切于点$P$. 若双曲线$\\Gamma$的离心率为$2$, 则$\\sin \\angle PF_1F_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373525,7 +374056,9 @@ "id": "015197", "content": "在下列统计量中, 用来描述一组数据离散程度的量是\\bracket{20}.\n\\fourch{平均数}{众数}{百分位数}{标准差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373544,7 +374077,9 @@ "id": "015198", "content": "设复平面上表示$2-\\mathrm{i}$和$3+4 \\mathrm{i}$的点分别为点$A$和点$B$, 则表示向量$\\overrightarrow{AB}$的复数在复平面上所对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373563,7 +374098,9 @@ "id": "015199", "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 点$P$在直线$AD_1$上, $Q$为线段$BD$的中点. 则下列说法不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.6!(D1)$) node [left] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(D)$) node [left] {$Q$} coordinate (Q);\n\\draw [dashed] (B)--(D)(A)--(D1)(P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{存在点$P$, 使得$PQ \\perp A_1C_1$}{存在点$P$, 使得$PQ\\parallel A_1B$}{直线$PQ$始终与直线$CC_1$异面}{直线$PQ$始终与直线$BC_1$异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373582,7 +374119,9 @@ "id": "015200", "content": "设各项均为实数的等差数列$\\{a_n\\}$和$\\{b_n\\}$的前$n$项和分别为$S_n$和$T_n$, 对于方程\\textcircled{1} $2023 x^2-S_{2023} x+T_{2023}=0$, \\textcircled{2} $x^2-a_1 x+b_1=0$, \\textcircled{3} $x^2+a_{2023} x+b_{2023}=0$.\n下列判断正确的是\\bracket{20}.\n\\twoch{若\\textcircled{1}有实根, \\textcircled{2}有实根, 则\\textcircled{3}有实根}{若\\textcircled{1}有实根, \\textcircled{2}无实根, 则\\textcircled{3}有实根}{若\\textcircled{1}无实根, \\textcircled{2}有实根, 则\\textcircled{3}无实根}{若\\textcircled{1}无实根, \\textcircled{2}无实根, 则\\textcircled{3}无实根}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373601,7 +374140,9 @@ "id": "015201", "content": "盒子中有$5$个乒乓球, 其中$2$个次品, $3$个正品. 现从中不放回地随机摸取$2$次小球, 每次一个.\\\\\n(1) 记``第二次摸出的小球是正品''为事件$B$, 求证: $P(B)=\\dfrac{3}{5}$;\\\\\n(2) 用$X$表示摸出的$2$个小球中次品的个数, 求$X$的分布和期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373620,7 +374161,9 @@ "id": "015202", "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为直角梯形, $AD\\parallel BC$, $AB \\perp BC$, $AB=AD$, $BC=2AB, E$、$F$分别为棱$BC$、$BP$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (-2,0,2) node [below] {$D$} coordinate (D);\n\\draw (-4,0,0) node [left] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(P)$) node [right] {$F$} coordinate (F);\n\\draw (C)--(D)--(A)--(B)--(P)--cycle(P)--(D)(P)--(A)--(F);\n\\draw [dashed] (C)--(B)(E)--(A)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AEF\\parallel$平面$DCP$;\\\\\n(2) 若平面$PBC \\perp$乎面$ABCD$, 直线$AP$与平面$PBC$所成的角为$45^{\\circ}$, 且$CP \\perp PB$, 求二面角$P-AB-C$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373639,7 +374182,9 @@ "id": "015203", "content": "某地新能源汽车保有量符合阻滞型增长模型$x(t)=\\dfrac{M}{1+\\lambda e^{-r t}}$, 其中$x(t)$为自统计之日起, 经过$t$年后该地新能源汽车保有量, $\\lambda$和$r$为增长系数, $M$为饱和量. 下表是该地近$6$年年底的新能源汽车的保有量 (万辆) 的统计数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 年份 & 2018 & 2019 & 2020 & 2021 & 2022 \\\\\n\\hline $t$& 0 & 1 & 2 & 3 & 4 \\\\\n\\hline 保有量$x(t)$& 9.6 & 12.9 & 17.1 & 23.2 & 31.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n假设该地新能源汽车饱和量$M=290$万辆.\\\\\n(1) 若$r=0.31$, 假定$2018$年数据满足公式$x(t)=\\dfrac{M}{1+\\lambda e^{-r t}}$, 计算$\\lambda$的值(精确到$0.01$), 并估算$2023$年年底该地新能源汽车保有量(精确到$0.1$万辆);\\\\\n(2) 设$y=\\dfrac{M}{x(t)}-1$, 则$\\ln y$与$t$线性相关, 请依据以上表格中相关数据, 利用线性回归分析确定$\\lambda$和$r$的值(精确到$0.01$).\\\\\n附: 线性回归方程$y=\\hat{a} x+\\hat{b}$中回归系数计算公式如下:\n$\\hat{a}=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})(y_i-\\overline {y})}{\\displaystyle\\sum_{i=1}^n(x_i-\\overline {x})^2}$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373658,7 +374203,9 @@ "id": "015204", "content": "已知拋物线$\\Gamma: y^2=4 x$的焦点为$F$, 准线为$l$, 直线$l'$经过点$F$且与$\\Gamma$交于点$A$、$B$.\\\\\n(1) 求以$F$为焦点, 坐标轴为对称轴, 离心率为$\\dfrac{1}{2}$的椭圆的标准方程;\\\\\n(2) 若$|AB|=5$, 求线段$AB$的中点到$x$轴的距离;\\\\\n(3) 设$O$为坐标原点, $M$为$\\Gamma$上的动点, 直线$AM$、$BM$分别与准线$l$交于点$C$、$D$. 求证: $\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$为常数.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373677,7 +374224,9 @@ "id": "015205", "content": "(1) 求简谐振动$y=\\sin x+\\cos x$的振幅、周期和初相位$\\varphi$($\\varphi \\in[0,2 \\pi)$);\\\\\n(2) 若函数$y=\\sin \\dfrac{1}{2} x+\\dfrac{1}{2} \\cos x$在区间$(0, m)$上有唯一的极大值点, 求实数$m$的取值范围;\\\\\n(3) 设$a>0$, $f(x)=\\sin a x-a \\sin x$, 若函数$y=f(x)$在区间$(0, \\pi)$上是严格增函数, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -373696,7 +374245,9 @@ "id": "015206", "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{x | \\dfrac{2}{x}>1\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373715,7 +374266,9 @@ "id": "015207", "content": "若复数$z$满足$\\mathrm{i} \\cdot z=3-4 \\mathrm{i}$, 则$|\\overline {z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373734,7 +374287,9 @@ "id": "015208", "content": "已知空间向量$\\overrightarrow {a}=(1,2,3)$, $\\overrightarrow {b}=(2,-2,0)$, $\\overrightarrow {c}=(1,1, \\lambda)$, 若$\\overrightarrow {c} \\perp(2 \\overrightarrow {a}+\\overrightarrow {b})$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373753,7 +374308,9 @@ "id": "015209", "content": "已知随机变量$X$服从正态分布$N(0,1)$, 若$P(X<-1.96)=0.03$, 则$P(|X|<1.96)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373772,7 +374329,9 @@ "id": "015210", "content": "已知$\\dfrac{\\pi}{2}<\\theta<\\pi$, 且$\\cos \\theta=-\\dfrac{4}{5}$, 则$\\tan 2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373791,7 +374350,9 @@ "id": "015211", "content": "在二项式$(x-\\dfrac{1}{x})^8$的展开式中, 含$x^4$的项的系数是\\blank{50}.(用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373810,7 +374371,9 @@ "id": "015212", "content": "将右图所示的圆锥形容器内的液体全部倒入底面半径为$50 \\text{mm}$的直立的圆柱形容器内, 则液面高度为\\blank{50}$\\text{mm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [pattern = north east lines] (-0.5,1.5) arc (180:0:0.5 and 0.125) -- (0,0)--cycle;\n\\draw (0,0) -- (1,3) (0,0) -- (-1,3) -- (1,3);\n\\draw (0,3) ellipse (1 and 0.25);\n\\draw (-0.5,1.5) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (-0.5,1.5) arc (180:0:0.5 and 0.125);\n\\draw (-1.1,0) -- (-1.5,0) (-1.1,3) -- (-1.5,3);\n\\draw [<->] (-1.3,0) -- (-1.3,3) node [midway, fill = white] {\\tiny$300\\text{mm}$};\n\\draw (-1,3.1) -- (-1,3.7) (1,3.1) -- (1,3.7);\n\\draw [<->] (-1,3.5) -- (1,3.5) node [midway, fill=white] {\\tiny$200\\text{mm}$};\n\\draw (0.8,1.5) -- (1.2,1.5) (0.8,0) -- (1.2,0);\n\\draw [<->] (1,1.5) -- (1,0) node [midway, fill=white] {\\tiny$150\\text{mm}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373829,7 +374392,9 @@ "id": "015213", "content": "从$4$名男生和$3$名女生中抽取两人加入志愿者服务队. 用$A$表示事件``抽到的两名学生性别相同'', 用$B$表示事件``抽到的两名学生都是女生'', 则$P(B | A)=$\\blank{50}.(结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373848,7 +374413,9 @@ "id": "015214", "content": "参考《九章算术》中``竹九节''问题, 提出: 一根$9$节的竹子, 自上而下各节的容积成等差数列, 上面$4$节的容积共$2$升, 下面$3$节的容积共$3$升, 则第$5$节的容积为\\blank{50}升.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373867,7 +374434,9 @@ "id": "015215", "content": "已知$x \\in(0, \\dfrac{\\pi}{2})$, 则$\\dfrac{1}{\\sin ^2 x}+\\dfrac{4}{\\cos ^2 x}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373886,7 +374455,9 @@ "id": "015216", "content": "已知函数$y=f(x)$为$\\mathbf{R}$上的奇函数, 且$f(x)+f(2-x)=0$, 当$-10$). 若存在$m, n \\in \\mathbf{R}$, 使得$m \\overrightarrow{AB}+\\overrightarrow{OA}$与$n \\overrightarrow{AB}+\\overrightarrow{OB}$垂直, 且$|(m \\overrightarrow{AB}+\\overrightarrow{OA})-(n \\overrightarrow{AB}+\\overrightarrow{OB})|=a$, 则$|\\overrightarrow{AB}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -373924,7 +374497,9 @@ "id": "015218", "content": "已知直线$l_1: a x+y+1=0$与直线$l_2: x+a y-2=0$, 则``$l_1\\parallel l_2$''是``$a=1$''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": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373943,7 +374518,9 @@ "id": "015219", "content": "为了解某社区居民的家庭年收入与年支出的关系, 随机调查了该社区$5$户家庭, 得到如下统计数据表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 收入$x$(万元) & 8.2 & 8.6 & 10.0 & 11.3 & 11.9 \\\\\n\\hline 支出$y$(万元) & 6.2 & 7.5 & 8.0 & 8.5 & 9.8 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n根据上表可得回归直线方程$y=\\hat{a} x+\\hat{b}$, 其中$\\hat{a}=0.76$, $\\hat{b}=\\overline {y}-\\hat{a} \\overline {x}$, 据此估计, 该社区一户收入为$15$万元家庭年支出为\\bracket{20}.\n\\fourch{$11.4$万元}{$11.8$万元}{$12.0$万元}{$12.2$万元}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373962,7 +374539,9 @@ "id": "015220", "content": "若方程$f(x) \\cdot g(x)=0$的解集为$M$, 则以下结论一定正确的是\\bracket{20}.\\\\\n\\textcircled{1} $M=\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{2} $M=\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$;\\\\\n\\textcircled{3} $M \\subseteq\\{x | f(x)=0\\} \\cup\\{x | g(x)=0\\}$;\\\\\n\\textcircled{4} $M \\supseteq\\{x | f(x)=0\\} \\cap\\{x | g(x)=0\\}$.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -373981,7 +374560,9 @@ "id": "015221", "content": "已知函数$y=\\dfrac{1}{3} x^3-x^2-3 x+a$, $a \\in \\mathbf{R}$, 在区间$(t-3, t+5)$上有最大值, 则实数$t$的取值范围是\\bracket{20}.\n\\fourch{$-6=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,0,{2*sin(22.5)}) node [below] {$A$} coordinate (A);\n\\draw (0,0,{-2*sin(22.5)}) node [below] {$C$} coordinate (C);\n\\draw ({-2*cos(22.5)},0,0) node [left] {$D$} coordinate (D);\n\\draw ({2*cos(22.5)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(D)$) node [left] {$M$} coordinate (M);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A)(M)--(A);\n\\draw [dashed] (D)--(C)--(B)(D)--(B)(A)--(C)(P)--(O)(P)--(C)(C)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PB\\parallel$平面$ACM$;\\\\\n(2) 求直线$AM$与平面$ABCD$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374038,7 +374623,9 @@ "id": "015224", "content": "某城市响应国家号召, 积极调整能源结构, 推出多种价位的新能源电动汽车. 根据前期市场调研, 有购买新能源车需求的约有$2$万人, 他们的选择意向统计如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 车型 &A&B&C&D&E&F\\\\\n\\hline 价格 & 9 万元 & 12 万元 & 18 万元 & 24 万元 & 30 万元 & 40 万元 \\\\\n\\hline 占比 &$5 \\%$&$15 \\%$&$25 \\%$&$35 \\%$&$15 \\%$&$5 \\%$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(1) 如果有购车需求的这些人今年都购买了新能源车, 今年新能源车的销售额预计约为多少亿元?\\\\\n(2) 车企推出两种付款方式: 全款购车: 购车时一次性付款可优惠车价的$3 \\%$; 分期付款: 无价格优惠, 购车时先付车价的一半, 余下的每半年付一次, 分$4$次付完, 每次付车价的$\\dfrac{1}{8}$.\\\\\n(i) 某位顾客现有$a$万元现金, 欲购买价值$a$万元的某款车, 付款后剩余的资金全部用于购买半年期的理财产品(该理财产品半年期到期收益率为$1.8 \\%$), 到期后, 可用资金(含理财收益)继续购买半年期的理财产品. 问: 顾客选择哪一种付款方式收益更多? (计算结果精确到$0.0001$)\\\\\n(ii) 为了激励购买理财产品, 银行对采用分期付款方式的顾客, 赠送价值$1888$元的大礼包, 试问: 这一措施对哪些车型有效? (计算结果精确到$0.0001$)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374057,7 +374644,9 @@ "id": "015225", "content": "已知椭圆$C_1: \\dfrac{x^2}{2}+\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_1$、$F_2$, 离心率为$e_1$; 双曲线\n$C_2: \\dfrac{x^2}{2}-\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_3$、$F_4$, 离心率为$e_2$, $e_1 \\cdot e_2=\\dfrac{\\sqrt{3}}{2}$. 过点$F_1$作不垂直于$y$轴的直线$l$交曲线$C_1$于点$A$、$B$, 点$M$为线段$AB$的中点, 直线$OM$交曲线$C_2$于$P$、$Q$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({sqrt(2*\\x*\\x+2)},\\x);\n\\draw [domain = -1.6:1.6, samples = 100] plot ({-sqrt(2*\\x*\\x+2)},\\x);\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\filldraw ({-sqrt(3)},0) circle (0.03) node [below] {$F_3$} coordinate (F_3);\n\\filldraw ({sqrt(3)},0) circle (0.03) node [below] {$F_4$} coordinate (F_4);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$C_1$、$C_2$的方程;\\\\\n(2) 若$\\overrightarrow{AF_1}=3 \\overrightarrow{F_1B}$, 求直线$PQ$的方程;\\\\\n(3) 求四边形$APBQ$面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -374076,7 +374665,9 @@ "id": "015226", "content": "已知$x>0$, 记$f(x)=e^x$, $g(x)=x^x$, $h(x)=\\ln g(x)$.\\\\\n(1) 试将$y=f(x)$、$y=g(x)$、$y=h(x)$中的一个函数表示为另外两个函数复合而成的复合函数;\\\\\n(2) 借助 (1) 的结果, 求函数$y=g(2 x)$的导函数和最小值;\\\\\n(3) 记$H(x)=\\dfrac{f(x)-h(x)}{x}+x+a$, $a$是实常数, 函数$y=H(x)$的导函数是$y'=H'(x)$. 已知函数$y=H(x) \\cdot H'(x)$有三个不相同的零点$x_1$、$x_2$、$x_3$. 求证: $x_1 \\cdot x_2 \\cdot x_3<1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "",