From 129205fe5f3b1976792a12bda0f79723f68e36d2 Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Thu, 9 Feb 2023 16:02:48 +0800 Subject: [PATCH] 20230209 afternoon --- 工具/寻找阶段末尾空闲题号.ipynb | 8 +- 工具/文本文件/题号筛选.txt | 2 +- 工具/新题比对.ipynb | 249 + 工具/添加题目到数据库.ipynb | 69 +- 工具/相似题目检测.ipynb | 6 +- 工具/识别题库中尚未标注的题目类型.ipynb | 327 +- 工具/题号选题pdf生成.ipynb | 6 +- 文本处理工具/剪贴板文本整理_mathpix.ipynb | 4 +- 文本处理工具/有用的文本/2025届高一下校本.txt | 1 + 题库0.3/Problems.json | 11533 +++++++++++++++++ 10 files changed, 11861 insertions(+), 344 deletions(-) create mode 100644 工具/新题比对.ipynb create mode 100644 文本处理工具/有用的文本/2025届高一下校本.txt diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index c7152324..c4ca9736 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 14379 , 直至 020000\n", - "首个空闲id: 21441 , 直至 030000\n", - "首个空闲id: 31222 , 直至 999999\n" + "首个空闲id: 14400 , 直至 020000\n", + "首个空闲id: 22022 , 直至 030000\n", + "首个空闲id: 31225 , 直至 999999\n" ] } ], diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 989839f6..e3dd97f2 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 @@ -000456,031204,000458,000459,000460,000461,000462,000463,000464,000465 \ No newline at end of file +21441:22021,21365,21366,21367,21368,21369,21370,21372,21371,21373,21374,21375,21376,21377,22022,21379,21382,22023,21383,21384,21385,21386,21387,21389,22024,21390,22026,21392,21393,21394,21395,21396,22027,21397,22028,21401,21403,22029,22030,22031,22032,22033,22034,21410,22035,22036,22037,21413,22038,22039,21415,22040,22041,21418,21420,21421,21422,21423,22042,22043,21427,21425,21428,22044,22045,21430,22046,22047,21432,21434,21433,21435,21436,21437,21438,21440, \ No newline at end of file diff --git a/工具/新题比对.ipynb b/工具/新题比对.ipynb new file mode 100644 index 00000000..9472afca --- /dev/null +++ b/工具/新题比对.ipynb @@ -0,0 +1,249 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.000\t1\t021365\n", + "1.000\t2\t021366\n", + "1.000\t3\t021367\n", + "1.000\t4\t021368\n", + "0.944\t5\t021369\n", + "0.947\t6\t021370\n", + "0.917\t7\t021372\n", + "1.000\t8\t021371\n", + "1.000\t9\t021373\n", + "1.000\t10\t021374\n", + "1.000\t11\t021375\n", + "1.000\t12\t021376\n", + "0.966\t13\t021377\n", + "1.000\t14\t022022\n", + "0.902\t15\t021379\n", + "0.865\t16\t021382\n", + "1.000\t17\t022023\n", + "0.987\t18\t021383\n", + "1.000\t19\t021384\n", + "1.000\t20\t021385\n", + "1.000\t21\t021386\n", + "1.000\t22\t021387\n", + "1.000\t23\t021389\n", + "1.000\t24\t022024\n", + "1.000\t25\t021390\n", + "1.000\t26\t022026\n", + "0.891\t27\t021392\n", + "0.965\t28\t021393\n", + "0.986\t29\t021394\n", + "0.940\t30\t021395\n", + "1.000\t31\t021396\n", + "1.000\t32\t022027\n", + "1.000\t33\t021397\n", + "1.000\t34\t022028\n", + "0.805\t35\t021401\n", + "1.000\t36\t021403\n", + "1.000\t37\t022029\n", + "1.000\t38\t022030\n", + "1.000\t39\t022031\n", + "1.000\t40\t022032\n", + "1.000\t41\t022033\n", + "1.000\t42\t022034\n", + "0.887\t43\t021410\n", + "1.000\t44\t022035\n", + "1.000\t45\t022036\n", + "1.000\t46\t022037\n", + "1.000\t47\t021413\n", + "0.959\t48\t022038\n", + "1.000\t49\t022039\n", + "1.000\t50\t021415\n", + "1.000\t51\t022040\n", + "1.000\t52\t022041\n", + "0.793\t53\t021418\n", + "0.807\t54\t021420\n", + "0.693\t55\t021421\n", + "1.000\t56\t021422\n", + "1.000\t57\t021423\n", + "1.000\t58\t022042\n", + "1.000\t59\t022043\n", + "0.805\t60\t021427\n", + "0.957\t61\t021425\n", + "0.770\t62\t021428\n", + "0.970\t63\t022044\n", + "1.000\t64\t022045\n", + "0.738\t65\t021430\n", + "1.000\t66\t022046\n", + "1.000\t67\t022047\n", + "0.792\t68\t021432\n", + "0.793\t69\t021434\n", + "0.721\t70\t021433\n", + "0.811\t71\t021435\n", + "0.728\t72\t021436\n", + "1.000\t73\t021437\n", + "0.989\t74\t021438\n", + "0.848\t75\t021440\n" + ] + } + ], + "source": [ + "import os,re,difflib,Levenshtein,time,json\n", + "\n", + "# 重要!!! 范围\n", + "old_problems_range = \"21365:21440,22022:22047\"\n", + "threshold = 0.85\n", + "\n", + "# 待比对的文件\n", + "filename = r\"D:\\temp\\derivatives.tex\"\n", + "\n", + "#生成数码列表, 逗号分隔每个区块, 区块内部用:表示整数闭区间\n", + "def generate_number_set(string):\n", + " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", + " string_list = string.split(\",\")\n", + " numbers_list = []\n", + " for s in string_list:\n", + " if not \":\" in s:\n", + " numbers_list.append(s.zfill(6))\n", + " else:\n", + " start,end = s.split(\":\")\n", + " for ind in range(int(start),int(end)+1):\n", + " numbers_list.append(str(ind).zfill(6))\n", + " return numbers_list\n", + "\n", + "#字符串预处理\n", + "def pre_treating(string):\n", + " string = re.sub(r\"\\\\begin\\{center\\}[\\s\\S]*?\\\\end\\{center\\}\",\"\",string)\n", + " string = re.sub(r\"(bracket\\{\\d+\\})|(blank\\{\\d+\\})|(fourch)|(twoch)|(onech)|(mathrm)|(text)\",\"\",string)\n", + " string = re.sub(r\"[\\s\\\\\\{\\}\\$\\(\\)\\[\\]]\",\"\",string)\n", + " string = re.sub(r\"[\\n\\t]\",\"\",string)\n", + " string = re.sub(r\"(displaystyle)|(overrightarrow)\",\"\",string)\n", + " string = re.sub(r\"[,\\.:;?]\",\"\",string)\n", + " return string\n", + "\n", + "#difflab字符串比较\n", + "def difflab_get_equal_rate(str1, str2):\n", + " return difflib.SequenceMatcher(None, str1, str2).ratio()\n", + "\n", + "#Levenshtein jaro字符串比较\n", + "def jaro_get_equal_rate(str1,str2):\n", + " return Levenshtein.jaro(str1,str2)\n", + "\n", + "#Levenshtein 字符串比较\n", + "def Lev_get_equal_rate(str1,str2):\n", + " return Levenshtein.ratio(str1,str2)\n", + "\n", + "def GenerateProblemListFromString(data):\n", + " try:\n", + " data = re.findall(r\"\\\\begin\\{document\\}([\\s\\S]*?)\\\\end\\{document\\}\",problems_string)[0]\n", + " except:\n", + " pass\n", + " data = re.sub(r\"\\n{2,}\",\"\\n\",data)\n", + " data = re.sub(r\"\\\\item\",r\"\\\\enditem\\\\item\",data)\n", + " data = re.sub(r\"\\\\end\\{enumerate\\}\",r\"\\\\enditem\",data)\n", + " ProblemList_raw = [p.strip() for p in re.findall(r\"\\\\item([\\s\\S]*?)\\\\enditem\",data)]\n", + " ProblemsList = []\n", + " for p in ProblemList_raw:\n", + " startpos = data.index(p)\n", + " tempdata = data[:startpos]\n", + " suflist = re.findall(r\"\\n\\%[\\dA-Za-z]+\",tempdata)\n", + " if len(suflist) > 0:\n", + " suffix = suflist[-1].replace(\"%\",\"\").strip()\n", + " else:\n", + " suffix = \"\"\n", + " ProblemsList.append((p,suffix))\n", + " return ProblemsList\n", + "\n", + "\n", + "#指定对比方法\n", + "sim_test = jaro_get_equal_rate\n", + "\n", + "#读入题库\n", + "with open(r\"../题库0.3/Problems.json\",\"r\",encoding = \"utf8\") as f:\n", + " database = f.read()\n", + "pro_dict = json.loads(database)\n", + "\n", + "with open(filename,\"r\",encoding=\"u8\") as f:\n", + " newdatabase = f.read()\n", + "new_pro_list = GenerateProblemListFromString(newdatabase)\n", + "\n", + "pro_dict_treated = {}\n", + "idrange = generate_number_set(old_problems_range)\n", + "for p in idrange:\n", + " pro_dict_treated[p] = pre_treating(pro_dict[p][\"content\"])\n", + "\n", + "new_dict_treated = {}\n", + "for i in range(len(new_pro_list)):\n", + " new_dict_treated[i+1] = pre_treating(new_pro_list[i][0])\n", + "\n", + "for i in new_dict_treated:\n", + " new_p = new_dict_treated[i]\n", + " maxsim = 0\n", + " for p in pro_dict_treated:\n", + " old_p = pro_dict_treated[p]\n", + " sim = sim_test(new_p,old_p)\n", + " if sim > maxsim:\n", + " maxsim = sim\n", + " argmax = p\n", + " print(\"%.3f\\t%d\\t%s\" %(maxsim,i,argmax))\n", + " # print(\"\\n新题: %s\" %new_pro_list[i-1][0])\n", + " # print(\"\\n原题: %s\\n\\n\\n\" %pro_dict[][\"content\"])\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "75" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "len(new_dict_treated)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "mathdept", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.15" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index f1a2e908..c729f9cc 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -2,48 +2,53 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 14379\n", - "raworigin = \"2023年空中课堂高三复习课\"\n", - "filename = r\"D:\\temp\\空中课堂第三批.tex\"\n", - "editor = \"20230203\\t王伟叶\"\n", + "starting_id = 22022\n", + "raworigin = \"2025届高一下校本作业\"\n", + "filename = r\"D:\\temp\\derivatives.tex\"\n", + "editor = \"20230209\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "添加题号014379, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014380, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014381, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014382, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014383, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014384, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014385, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014386, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014387, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014388, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014389, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014390, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014391, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014392, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014393, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014394, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014395, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014396, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014397, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014398, 来源: 2023年空中课堂高三复习题13\n", - "添加题号014399, 来源: 2023年空中课堂高三复习题13\n" + "添加题号022022, 来源: 2025届高一下校本作业\n", + "添加题号022023, 来源: 2025届高一下校本作业\n", + "添加题号022024, 来源: 2025届高一下校本作业\n", + "添加题号022025, 来源: 2025届高一下校本作业\n", + "添加题号022026, 来源: 2025届高一下校本作业\n", + "添加题号022027, 来源: 2025届高一下校本作业\n", + "添加题号022028, 来源: 2025届高一下校本作业\n", + "添加题号022029, 来源: 2025届高一下校本作业\n", + "添加题号022030, 来源: 2025届高一下校本作业\n", + "添加题号022031, 来源: 2025届高一下校本作业\n", + "添加题号022032, 来源: 2025届高一下校本作业\n", + "添加题号022033, 来源: 2025届高一下校本作业\n", + "添加题号022034, 来源: 2025届高一下校本作业\n", + "添加题号022035, 来源: 2025届高一下校本作业\n", + "添加题号022036, 来源: 2025届高一下校本作业\n", + "添加题号022037, 来源: 2025届高一下校本作业\n", + "添加题号022038, 来源: 2025届高一下校本作业\n", + "添加题号022039, 来源: 2025届高一下校本作业\n", + "添加题号022040, 来源: 2025届高一下校本作业\n", + "添加题号022041, 来源: 2025届高一下校本作业\n", + "添加题号022042, 来源: 2025届高一下校本作业\n", + "添加题号022043, 来源: 2025届高一下校本作业\n", + "添加题号022044, 来源: 2025届高一下校本作业\n", + "添加题号022045, 来源: 2025届高一下校本作业\n", + "添加题号022046, 来源: 2025届高一下校本作业\n", + "添加题号022047, 来源: 2025届高一下校本作业\n" ] } ], @@ -138,16 +143,16 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "'18'" + "''" ] }, - "execution_count": 5, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -166,7 +171,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -185,7 +190,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/工具/相似题目检测.ipynb b/工具/相似题目检测.ipynb index cb7b4779..21a48a43 100644 --- a/工具/相似题目检测.ipynb +++ b/工具/相似题目检测.ipynb @@ -24,7 +24,7 @@ } ], "source": [ - "from hashlib import new\n", + "# from hashlib import new\n", "import os,re,difflib,Levenshtein,time,json\n", "\n", "# 重要!!! 新题目的范围\n", @@ -48,10 +48,10 @@ "#字符串预处理\n", "def pre_treating(string):\n", " string = re.sub(r\"\\\\begin\\{center\\}[\\s\\S]*?\\\\end\\{center\\}\",\"\",string)\n", + " string = re.sub(r\"(bracket\\{\\d+\\})|(blank\\{\\d+\\})|(fourch)|(twoch)|(onech)\",\"\",string)\n", " string = re.sub(r\"[\\s\\\\\\{\\}\\$\\(\\)\\[\\]]\",\"\",string)\n", " string = re.sub(r\"[\\n\\t]\",\"\",string)\n", " string = re.sub(r\"(displaystyle)|(overrightarrow)\",\"\",string)\n", - " string = re.sub(r\"(bracket)|(blank)|(fourch)|(twoch)|(onech)\",\"\",string)\n", " string = re.sub(r\"[,\\.:;?]\",\"\",string)\n", " return string\n", "\n", @@ -170,7 +170,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.13" + "version": "3.9.13 (main, Aug 25 2022, 23:51:50) [MSC v.1916 64 bit (AMD64)]" }, "orig_nbformat": 4, "vscode": { diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index eb9ca978..3a7cda8d 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -9,303 +9,32 @@ "name": "stdout", "output_type": "stream", "text": [ - "014103 填空题\n", - "014104 填空题\n", - "014105 填空题\n", - "014106 选择题\n", - "014107 填空题\n", - "014108 解答题\n", - "014109 解答题\n", - "014110 解答题\n", - "014111 填空题\n", - "014112 填空题\n", - "014113 解答题\n", - "014114 解答题\n", - "014115 填空题\n", - "014116 填空题\n", - "014117 填空题\n", - "014118 解答题\n", - "014119 解答题\n", - "014120 解答题\n", - "014121 解答题\n", - "014122 选择题\n", - "014123 填空题\n", - "014124 填空题\n", - "014125 填空题\n", - "014126 填空题\n", - "014127 填空题\n", - "014128 解答题\n", - "014129 解答题\n", - "014130 解答题\n", - "014131 解答题\n", - "014132 解答题\n", - "014133 填空题\n", - "014134 填空题\n", - "014135 填空题\n", - "014136 填空题\n", - "014137 选择题\n", - "014138 填空题\n", - "014139 填空题\n", - "014140 填空题\n", - "014141 填空题\n", - "014142 填空题\n", - "014143 填空题\n", - "014144 解答题\n", - "014145 选择题\n", - "014146 填空题\n", - "014147 填空题\n", - "014148 填空题\n", - "014149 填空题\n", - "014150 填空题\n", - "014151 解答题\n", - "014152 解答题\n", - "014153 解答题\n", - "014154 选择题\n", - "014155 填空题\n", - "014156 填空题\n", - "014157 填空题\n", - "014158 选择题\n", - "014159 选择题\n", - "014160 填空题\n", - "014161 填空题\n", - "014162 解答题\n", - "014163 解答题\n", - "014164 选择题\n", - "014165 解答题\n", - "014166 填空题\n", - "014167 填空题\n", - "014168 填空题\n", - "014169 选择题\n", - "014170 解答题\n", - "014171 解答题\n", - "014172 解答题\n", - "014173 解答题\n", - "014174 填空题\n", - "014175 填空题\n", - "014176 填空题\n", - "014177 填空题\n", - "014178 填空题\n", - "014179 填空题\n", - "014180 填空题\n", - "014181 选择题\n", - "014182 解答题\n", - "014183 填空题\n", - "014184 解答题\n", - "014185 选择题\n", - "014186 选择题\n", - "014187 填空题\n", - "014188 填空题\n", - "014189 填空题\n", - "014190 解答题\n", - "014191 解答题\n", - "014192 解答题\n", - "014193 填空题\n", - "014194 填空题\n", - "014195 填空题\n", - "014196 选择题\n", - "014197 选择题\n", - "014198 填空题\n", - "014199 填空题\n", - "014200 解答题\n", - "014201 解答题\n", - "014202 选择题\n", - "014203 解答题\n", - "014204 填空题\n", - "014205 填空题\n", - "014206 填空题\n", - "014207 填空题\n", - "014208 填空题\n", - "014209 解答题\n", - "014210 解答题\n", - "014211 解答题\n", - "014212 解答题\n", - "014213 选择题\n", - "014214 填空题\n", - "014215 填空题\n", - "014216 填空题\n", - "014217 填空题\n", - "014218 填空题\n", - "014219 填空题\n", - "014220 解答题\n", - "014221 解答题\n", - "014222 解答题\n", - "014223 解答题\n", - "014224 填空题\n", - "014225 填空题\n", - "014226 填空题\n", - "014227 填空题\n", - "014228 选择题\n", - "014229 解答题\n", - "014230 填空题\n", - "014231 解答题\n", - "014232 填空题\n", - "014233 填空题\n", - "014234 选择题\n", - "014235 解答题\n", - "014236 填空题\n", - "014237 填空题\n", - "014238 解答题\n", - "014239 解答题\n", - "014240 解答题\n", - "014241 解答题\n", - "014242 选择题\n", - "014243 解答题\n", - "014244 填空题\n", - "014245 填空题\n", - "014246 填空题\n", - "014247 选择题\n", - "014248 填空题\n", - "014249 解答题\n", - "014250 解答题\n", - "014251 解答题\n", - "014252 解答题\n", - "014253 解答题\n", - "014254 选择题\n", - "014255 填空题\n", - "014256 填空题\n", - "014257 选择题\n", - "014258 解答题\n", - "014259 填空题\n", - "014260 填空题\n", - "014261 填空题\n", - "014262 选择题\n", - "014263 填空题\n", - "014264 解答题\n", - "014265 填空题\n", - "014266 解答题\n", - "014267 填空题\n", - "014268 填空题\n", - "014269 填空题\n", - "014270 选择题\n", - "014271 填空题\n", - "014272 解答题\n", - "014273 解答题\n", - "014274 解答题\n", - "014275 解答题\n", - "014276 填空题\n", - "014277 填空题\n", - "014278 填空题\n", - "014279 解答题\n", - "014280 填空题\n", - "014281 填空题\n", - "014282 填空题\n", - "014283 填空题\n", - "014284 解答题\n", - "014285 解答题\n", - "014286 填空题\n", - "014287 解答题\n", - "014288 填空题\n", - "014289 选择题\n", - "014290 填空题\n", - "014291 解答题\n", - "014292 填空题\n", - "014293 解答题\n", - "014294 解答题\n", - "014295 填空题\n", - "014296 填空题\n", - "014297 填空题\n", - "014298 选择题\n", - "014299 填空题\n", - "014300 填空题\n", - "014301 填空题\n", - "014302 填空题\n", - "014303 填空题\n", - "014304 解答题\n", - "014305 解答题\n", - "014306 解答题\n", - "014307 填空题\n", - "014308 填空题\n", - "014309 填空题\n", - "014310 填空题\n", - "014311 填空题\n", - "014312 填空题\n", - "014313 解答题\n", - "014314 填空题\n", - "014315 填空题\n", - "014316 填空题\n", - "014317 填空题\n", - "014318 填空题\n", - "014319 填空题\n", - "014320 解答题\n", - "014321 填空题\n", - "014322 填空题\n", - "014323 解答题\n", - "014324 选择题\n", - "014325 选择题\n", - "014326 填空题\n", - "014327 填空题\n", - "014328 选择题\n", - "014329 解答题\n", - "014330 解答题\n", - "014331 解答题\n", - "014332 选择题\n", - "014333 解答题\n", - "014334 填空题\n", - "014335 填空题\n", - "014336 填空题\n", - "014337 解答题\n", - "014338 解答题\n", - "014339 填空题\n", - "014340 解答题\n", - "014341 填空题\n", - "014342 解答题\n", - "014343 选择题\n", - "014344 填空题\n", - "014345 填空题\n", - "014346 填空题\n", - "014347 解答题\n", - "014348 解答题\n", - "014349 解答题\n", - "014350 填空题\n", - "014351 选择题\n", - "014352 选择题\n", - "014353 填空题\n", - "014354 填空题\n", - "014355 填空题\n", - "014356 解答题\n", - "014357 解答题\n", - "014358 解答题\n", - "014359 填空题\n", - "014360 解答题\n", - "014361 填空题\n", - "014362 填空题\n", - "014363 填空题\n", - "014364 解答题\n", - "014365 解答题\n", - "014366 解答题\n", - "014367 填空题\n", - "014368 填空题\n", - "014369 填空题\n", - "014370 选择题\n", - "014371 填空题\n", - "014372 填空题\n", - "014373 填空题\n", - "014374 填空题\n", - "014375 填空题\n", - "014376 解答题\n", - "014377 填空题\n", - "014378 解答题\n", - "014379 填空题\n", - "014380 填空题\n", - "014381 填空题\n", - "014382 填空题\n", - "014383 填空题\n", - "014384 解答题\n", - "014385 解答题\n", - "014386 解答题\n", - "014387 解答题\n", - "014388 填空题\n", - "014389 填空题\n", - "014390 解答题\n", - "014391 解答题\n", - "014392 填空题\n", - "014393 填空题\n", - "014394 填空题\n", - "014395 解答题\n", - "014396 解答题\n", - "014397 解答题\n", - "014398 解答题\n", - "014399 解答题\n" + "022022 解答题\n", + "022023 解答题\n", + "022024 解答题\n", + "022025 解答题\n", + "022026 选择题\n", + "022027 解答题\n", + "022028 解答题\n", + "022029 填空题\n", + "022030 解答题\n", + "022031 解答题\n", + "022032 解答题\n", + "022033 解答题\n", + "022034 解答题\n", + "022035 解答题\n", + "022036 解答题\n", + "022037 解答题\n", + "022038 解答题\n", + "022039 解答题\n", + "022040 解答题\n", + "022041 填空题\n", + "022042 解答题\n", + "022043 解答题\n", + "022044 填空题\n", + "022045 填空题\n", + "022046 解答题\n", + "022047 解答题\n" ] } ], @@ -347,7 +76,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -366,7 +95,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 3d030db5..2f2c0b17 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -9,9 +9,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/赋能13_教师用_20230208.tex\n", + "开始编译教师版本pdf文件: 临时文件/2025届高一下校本作业_教师用_20230209.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/赋能13_学生用_20230208.tex\n", + "开始编译学生版本pdf文件: 临时文件/2025届高一下校本作业_学生用_20230209.tex\n", "0\n" ] } @@ -33,7 +33,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/赋能13\"\n", + "filename = \"临时文件/2025届高一下校本作业\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/文本处理工具/剪贴板文本整理_mathpix.ipynb b/文本处理工具/剪贴板文本整理_mathpix.ipynb index c6ea49f4..ea51543a 100644 --- a/文本处理工具/剪贴板文本整理_mathpix.ipynb +++ b/文本处理工具/剪贴板文本整理_mathpix.ipynb @@ -384,7 +384,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -403,7 +403,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/文本处理工具/有用的文本/2025届高一下校本.txt b/文本处理工具/有用的文本/2025届高一下校本.txt new file mode 100644 index 00000000..6b1a85af --- /dev/null +++ b/文本处理工具/有用的文本/2025届高一下校本.txt @@ -0,0 +1 @@ +21441:22021,21365,21366,21367,21368,21369,21370,21372,21371,21373,21374,21375,21376,21377,22022,21379,21382,22023,21383,21384,21385,21386,21387,21389,22024,21390,22026,21392,21393,21394,21395,21396,22027,21397,22028,21401,21403,22029,22030,22031,22032,22033,22034,21410,22035,22036,22037,21413,22038,22039,21415,22040,22041,21418,21420,21421,21422,21423,22042,22043,21427,21425,21428,22044,22045,21430,22046,22047,21432,21434,21433,21435,21436,21437,21438,21440 \ No newline at end of file diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index a74e3c7d..c3231f69 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -383080,6 +383080,11539 @@ "remark": "", "space": "12ex" }, + "021441": { + "id": "021441", + "content": "判断下列命题是否正确:\\\\\n(1) 终边重合的两个角相等;\\blank{20}\\\\\n(2) 锐角是第一象限的角;\\blank{20}\\\\\n(3) 第二象限的角是钝角;\\blank{20}\\\\\n(4) 小于$90^{\\circ}$的角都是锐角.\\blank{20}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021442": { + "id": "021442", + "content": "下列各组的两个角中, 终边不重合的一组是\\bracket{20}.\n\\fourch{$-43^{\\circ}$与$677^{\\circ}$}{$900^{\\circ}$与$-1260^{\\circ}$}{$-120^{\\circ}$与$960^{\\circ}$}{$150^{\\circ}$与$630^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021443": { + "id": "021443", + "content": "如果$\\alpha$是锐角, 那么$2 \\alpha$是\\bracket{20}.\n\\fourch{第一象限角}{第二象限角}{小于$180^{\\circ}$的正角}{大于直角的正角}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021444": { + "id": "021444", + "content": "如果$\\alpha$是钝角, 那么$\\dfrac{\\alpha}{2}$是\\bracket{20}.\n\\twoch{第一象限角}{第二象限角}{第二象限角或终边与坐标轴重合的角}{不小于直角的正角}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021445": { + "id": "021445", + "content": "在平面直角坐标系中, 下列结论正确的是\\bracket{20}.\n\\twoch{小于$90^{\\circ}$的角一定是锐角}{第一象限角必定大于$0^{\\circ}$且小于$90^{\\circ}$}{始边重合且相等的角, 终边一定重合}{始边重合且终边也重合的角一定相等}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021446": { + "id": "021446", + "content": "已知集合$A=\\{\\alpha|\\alpha\\text{是第一象限角}\\}$, $B=\\{\\alpha|\\alpha\\text{是锐角}\\}$, $C=\\{\\alpha|\\alpha\\text{是小于}90^\\circ\\text{的角}\\}$, 下列结论正确的是\\bracket{20}.\n\\fourch{$A=B=C$}{$A \\subset C$}{$A \\cap C=B$}{$B \\cup C=C$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021447": { + "id": "021447", + "content": "平面内一条射线绕着其端点先逆时针旋转$60^{\\circ}$, 再顺时针旋转$450^{\\circ}$, 已原射线为始边, 所得射线为终边所形成的角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021448": { + "id": "021448", + "content": "与角$1024^{\\circ}$终边重合的角中, 最小的正角为\\blank{50}, 最大的负角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021449": { + "id": "021449", + "content": "与角$576^{\\circ}$终边重合的角中, 绝对值最小的角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021450": { + "id": "021450", + "content": "若角$\\alpha$是第三象限的角, 则角$\\alpha+270^{\\circ}$是第\\blank{50}象限的角, 角$\\alpha-270^{\\circ}$是第\\blank{50}象限的角.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021451": { + "id": "021451", + "content": "写出与下列各角的终边重合的所有角组成的集合$S$, 并列举$S$中满足不等式$-360^{\\circ} \\leq \\alpha<720^{\\circ}$的所有元素$\\alpha$:\\\\\n(1) $60^{\\circ}$;\\\\\n(2) $-21^{\\circ}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021452": { + "id": "021452", + "content": "在平面直角坐标系中, 用阴影部分表示集合: $\\{\\alpha | 30^{\\circ}+k \\cdot 360^{\\circ} \\leq \\alpha \\leq 60^{\\circ}+k \\cdot 360^{\\circ},\\ k \\in \\mathbf{Z}\\}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021453": { + "id": "021453", + "content": "在同一天$10: 45$到$14: 20$期间, 时钟的分针转过的角度为\\blank{50}, 它是第\\blank{50}象限的角.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021454": { + "id": "021454", + "content": "(1) 终边在第一象限角平分线上的所有角组成的集合为\\blank{50};\\\\\n(2) 终边在第二、四象限角平分线上的所有角组成的集合为\\blank{50};\\\\\n(3) 终边在直线$y=x$或$y=-x$上的所有角组成的集合为\\blank{50};\\\\\n(4) 终边位于第三象限的所有角组成的集合为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021455": { + "id": "021455", + "content": "已知角$\\alpha$, 写出满足下列条件的角$\\beta$的集合(用$\\alpha$表示$\\beta$):\\\\\n(1) $\\beta$的终边是$\\alpha$终边的反向延长线: \\blank{50};\\\\\n(2) $\\beta$的终边与$\\alpha$终边垂直: \\blank{50};\\\\\n(3) $\\beta$的终边与$\\alpha$终边关于$x$轴对称: \\blank{50};\\\\\n(4) $\\beta$的终边与$\\alpha$终边关于直线$y=x$对称: \\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021456": { + "id": "021456", + "content": "如果存在角$\\beta \\in(-3 \\pi,-\\dfrac{5 \\pi}{2})$与角$\\alpha$的终边重合, 那么角$\\alpha$所在的象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021457": { + "id": "021457", + "content": "设$\\alpha=2019$, 则$\\alpha$的终边所在的象限为\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021458": { + "id": "021458", + "content": "完成下列角度与弧度的换算:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 角度数 &$15^{\\circ}$&$105^{\\circ}$&$225^{\\circ}$&\\blank{15} &\\blank{15} &\\blank{15} &\\blank{15} \\\\\n\\hline 弧度数 &\\blank{15} &\\blank{15} &\\blank{15} &$\\dfrac{5 \\pi}{3}$&$\\dfrac{9 \\pi}{5}$&$\\dfrac{7 \\pi}{4}$&$\\dfrac{3}{2}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021459": { + "id": "021459", + "content": "已知扇形的半径为$10$, 圆心角的大小为$\\dfrac{5 \\pi}{9}$, 则它的周长为\\blank{50}, 它的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021460": { + "id": "021460", + "content": "若一圆弧长等于其所在圆的内接正三角形的边长, 则其圆心角的弧度数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021461": { + "id": "021461", + "content": "在定圆中, 长度等于半径的弦所对圆心角的弧度数为\\blank{50}, 长度等于半径的$\\sqrt{3}$倍的弦所对的圆心角的弧度数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021462": { + "id": "021462", + "content": "把下列各角化成$2 k \\pi+\\alpha$($0 \\leq \\alpha<2 \\pi$, $k \\in \\mathbf{Z}$)的形式, 并指出其终边所在的象限:\\\\\n(1) $\\dfrac{50 \\pi}{3}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(2) $-\\dfrac{50 \\pi}{3}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(3) $-108^{\\circ}=$\\blank{50}, 第\\blank{50}象限;\\\\\n(4) $-225^{\\circ}=$\\blank{50}, 第\\blank{50}象限.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021463": { + "id": "021463", + "content": "已知扇形的周长为$10$, 面积为$4$, 求该扇形对应的圆心角的弧度数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021464": { + "id": "021464", + "content": "用弧度制表示下列角的集合:\\\\\n(1) 第四象限角的集合: \\blank{100};\\\\\n(2) 终边在坐标轴上的角的集合: \\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021465": { + "id": "021465", + "content": "(1) 角$\\alpha$的终边与角$\\beta$的终边重合, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(2) 角$\\alpha$的终边与角$\\beta$的终边关于$x$轴对称, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(3) 角$\\alpha$的终边与角$\\beta$的终边关于$y$轴对称, 则$\\alpha$与$\\beta$的关系是\\blank{50};\\\\\n(4) 角$\\alpha$的终边与角$\\beta$的终边关于原点对称, 则$\\alpha$与$\\beta$的关系是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021466": { + "id": "021466", + "content": "用弧度制写出下图中的阴影部分表示的角的集合(包括边界).\n(1) \\blank{100};\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!25] (0,0) --++ (-45:1.5) arc (-45:90:1.5);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$} coordinate (x);\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$} coordinate (y);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) --++ (-45:1.5) coordinate (T);\n\\draw pic [draw, \"$45^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = T--O--x};\n\\end{tikzpicture}\n\\end{center}\n(2)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!25] (0,0) --++ (30:1.5) arc (30:150:1.5);\n\\fill [gray!25] (0,0) --++ (-30:1.5) arc (-30:-150:1.5);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$} coordinate (x);\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$} coordinate (y);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) --++ (30:1.5) coordinate (T);\n\\draw (0,0) --++ (-30:1.5) (0,0) --++ (-150:1.5);\n\\draw (0,0) --++ (150:1.5) coordinate (S);\n\\draw pic [draw, \"$30^\\circ$\", scale = 0.5, angle eccentricity = 3] {angle = x--O--T};\n\\draw (-2,0) coordinate (x1);\n\\draw pic [draw, \"$30^\\circ$\", scale = 0.5, angle eccentricity = 3] {angle = S--O--x1};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021467": { + "id": "021467", + "content": "已知角$\\alpha$是第三象限的角.\\\\\n(1) 指出角$\\pi-\\alpha$、$\\dfrac{\\pi}{2}+\\alpha$的终边位置;\\\\\n(2) 讨论角$\\dfrac{\\alpha}{2}$、$2 \\alpha$的终边位置.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021468": { + "id": "021468", + "content": "已知集合$A=\\{\\alpha | k \\pi-\\dfrac{\\pi}{6}<\\alpha0$, 则$\\alpha$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021473": { + "id": "021473", + "content": "已知$-\\pi<\\alpha<-\\dfrac{\\pi}{2}$, $-\\dfrac{\\pi}{2}<\\beta<0$, 则$\\sin (\\alpha-\\beta)$\\blank{50}$0$.(选填``$>$''、``$<$''、``$\\geq$''、``$\\leq$'')", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021474": { + "id": "021474", + "content": "计算: $6 \\cos 270^{\\circ}+10 \\sin 0^{\\circ}-4 \\tan 180^{\\circ}+5 \\cos 360^{\\circ}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021475": { + "id": "021475", + "content": "计算: $\\sin ^2 \\dfrac{\\pi}{3}-\\cos ^2 \\dfrac{\\pi}{6}+2 \\tan ^3 \\dfrac{\\pi}{4}$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021476": { + "id": "021476", + "content": "已知常数$t \\in \\mathbf{R}$, 角$\\alpha$的终边经过点$P(-\\sqrt{3}, t)$, 且$\\sin \\alpha=\\dfrac{\\sqrt{2}}{4} t$, 求$\\cos \\alpha$和$\\tan \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021477": { + "id": "021477", + "content": "已知角$\\alpha$终边上一点$P$满足到$x$轴、$y$轴的距离之比为$4: 3$, 且$\\cos \\alpha<0$. 求$\\sin \\alpha$与$\\tan \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021478": { + "id": "021478", + "content": "若角$\\alpha$的终边在直线$y=-\\sqrt{3} x$上, 求$\\sin \\alpha \\cdot \\cos \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021479": { + "id": "021479", + "content": "根据下列条件, 分别确定角$\\alpha$所在的象限:\\\\\n(1) $\\sin \\alpha<0$且$\\cos \\alpha>0$;\\\\\n(2) $\\dfrac{\\sin \\alpha}{\\tan \\alpha}>0$;\\\\\n(3) $\\tan \\alpha+\\cot \\alpha>0$;\\\\\n(4) $\\sin 2 \\alpha>0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021480": { + "id": "021480", + "content": "已知集合$A=\\{y | y=\\dfrac{\\sin \\alpha}{|\\sin \\alpha|}+\\dfrac{|\\cos \\alpha|}{\\cos \\alpha}+\\dfrac{\\tan \\alpha}{|\\tan \\alpha|}+\\dfrac{|\\cot \\alpha|}{\\cot \\alpha}\\}$, 请用列举法表示集合$A$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021481": { + "id": "021481", + "content": "分别求下列式子有意义时, 角$x$的取值范围:\\\\\n(1) $\\sqrt{\\sin x}+\\sqrt{\\cos x}$;、、\n(2) $\\sqrt{\\sin x}+\\lg (9-x^2)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021482": { + "id": "021482", + "content": "在下表填入相应的正弦、余弦、正切和余切值.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline$\\alpha$&$\\dfrac{\\pi}{3}$&$\\dfrac{7 \\pi}{4}$&$\\dfrac{2021 \\pi}{2}$&$-\\dfrac{\\pi}{6}$&$-\\dfrac{22 \\pi}{3}$\\\\\n\\hline$\\sin \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\cos \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\tan \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline$\\cot \\alpha$&\\blank{30} &\\blank{30} & \\blank{30}&\\blank{30} &\\blank{30} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021483": { + "id": "021483", + "content": "已知$\\sin x=-\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 当$x \\in \\mathbf{R}$时, 则满足条件的$x$的集合是\\blank{50};\\\\\n(2) 当$x \\in[-2 \\pi, 4 \\pi]$时, 则满足条件的$x$的集合是\\blank{50};\\\\\n(3) 当$x$是第三象限角时, 则满足条件的$x$的集合是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021484": { + "id": "021484", + "content": "若$\\alpha$为第三象限的角, $\\cos \\alpha=-\\dfrac{\\sqrt{5}}{5}$, 则$\\sin \\alpha=$\\blank{50}, $\\tan \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021485": { + "id": "021485", + "content": "给出以下四个命题:\\\\\n\\textcircled{1} 如果$\\alpha \\neq \\beta$, 则$\\sin \\alpha \\neq \\sin \\beta$;\\\\\n\\textcircled{2} 如果$\\sin \\alpha \\neq \\sin \\beta$, 则$\\alpha \\neq \\beta$;\\\\\n\\textcircled{3} 如果$\\sin \\alpha>0$, 则$\\alpha$是第一或第二象限角;\\\\\n\\textcircled{4}如果$\\alpha$是第一或第二象限角, 则$\\sin \\alpha>0$.\\\\\n在以上四个命题中, 所有真命题的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021486": { + "id": "021486", + "content": "已知$\\cot \\alpha=\\dfrac{1}{3}$, 求$\\sin \\alpha$、$\\cos \\alpha$及$\\tan \\alpha$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021487": { + "id": "021487", + "content": "分别求$\\sin k \\pi$($k \\in \\mathbf{Z}$)和$\\cos k \\pi$($k \\in \\mathbf{Z}$)的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021488": { + "id": "021488", + "content": "利用单位圆, 求满足下列条件的角$\\theta$的取值范围.\\\\\n(1) $\\sin \\theta>\\dfrac{\\sqrt{3}}{2}$;\\\\\n(2) $\\tan \\theta \\leq-\\dfrac{\\sqrt{3}}{3}$;\\\\\n(3) $\\dfrac{1}{2} \\leq \\cos \\theta \\leq \\dfrac{1}{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021489": { + "id": "021489", + "content": "已知$\\theta$是第三象限角且$\\cos \\dfrac{\\theta}{2}<0$, 问$\\dfrac{\\theta}{2}$是第几象限角?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021490": { + "id": "021490", + "content": "已知$\\alpha$是第三象限角.\\\\\n(1) 设点$P(\\sin \\dfrac{\\alpha}{2}, \\cos \\dfrac{\\alpha}{2})$, 判断点$P$所在象限;\\\\\n(2) 判断$\\sin (\\cos \\alpha) \\cdot \\cos (\\sin \\alpha)$的符号.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021491": { + "id": "021491", + "content": "设$\\alpha$、$m$满足$\\sin (\\alpha+105^{\\circ})=\\dfrac{\\sqrt{2}}{4} m$, 且角$\\alpha+1905^{\\circ}$的终边上有一点$P(-\\sqrt{3}, m)$. 求$\\cos (\\alpha+1905^{\\circ})$与$\\tan (\\alpha-615^{\\circ})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021492": { + "id": "021492", + "content": "已知$\\alpha$满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{2}$, 则$\\sin \\alpha \\cos \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021493": { + "id": "021493", + "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, 且$\\dfrac{\\pi}{2}<\\alpha<\\dfrac{3 \\pi}{2}$, 则$\\cos \\alpha-\\tan \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021494": { + "id": "021494", + "content": "已知$\\cos \\alpha=-\\dfrac{2 \\sqrt{2}}{3}$, 且$\\sin \\alpha>0$, 则$\\tan \\alpha-\\cot \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021495": { + "id": "021495", + "content": "已知$\\tan \\alpha=2$, 且$\\sin \\alpha=a^2$, 则$\\sin \\alpha+\\cos \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021496": { + "id": "021496", + "content": "已知$\\tan \\alpha-\\cot \\alpha=3$, 则$\\tan ^2 \\alpha+\\cot ^2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021497": { + "id": "021497", + "content": "已知$\\tan \\alpha=-\\dfrac{1}{2}$, 求下列各式的值:\\\\\n(1) $\\dfrac{2 \\cos \\alpha-\\sin \\alpha}{\\sin \\alpha+\\cos \\alpha}=$\\blank{50};\\\\\n(2) $2 \\sin ^2 \\alpha+\\sin \\alpha \\cdot \\cos \\alpha-3 \\cos ^2 \\alpha=$\\blank{50};\\\\\n(3) $\\dfrac{2 \\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha}{1+\\cos ^2 \\alpha}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021498": { + "id": "021498", + "content": "化简: $\\cot ^2 \\alpha(\\tan ^2 \\alpha-\\sin ^2 \\alpha)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021499": { + "id": "021499", + "content": "化简: $\\sin ^4 x+\\cos ^4 x+\\dfrac{2 \\sin x \\cos x}{\\tan x+\\cot x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021500": { + "id": "021500", + "content": "证明: $\\tan ^2 \\alpha-\\sin ^2 \\alpha=\\tan ^2 \\alpha \\cdot \\sin ^2 \\alpha$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021501": { + "id": "021501", + "content": "证明: $\\dfrac{1-2 \\sin \\alpha \\cos \\alpha}{\\cos ^2 \\alpha-\\sin ^2 \\alpha}=\\dfrac{1-\\tan \\alpha}{1+\\tan \\alpha}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021502": { + "id": "021502", + "content": "已知$\\sin \\theta=\\dfrac{k-3}{k+5}$, $\\cos \\theta=\\dfrac{4-2 k}{k+5}$, 若$\\theta$是第二象限角, 求$\\cot \\theta$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021503": { + "id": "021503", + "content": "若$\\sin \\alpha \\cos \\alpha=\\dfrac{1}{8}$, $\\alpha \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\cos \\alpha-\\sin \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021504": { + "id": "021504", + "content": "已知$\\alpha \\in(0, \\pi)$, 满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{2}$.\\\\ \n(1) 求$\\sin \\alpha-\\cos \\alpha$的值;\\\\\n(2) 求$\\sin ^4 \\alpha-\\cos ^4 \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021505": { + "id": "021505", + "content": "已知$\\tan \\alpha$, $\\cot \\alpha$是关于$x$的一元二次方程$x^2+3 k x+k^2-8=0$的两个实数根, 且$3 \\pi<\\alpha<\\dfrac{7}{2} \\pi$, 求$\\sin \\alpha+\\cos \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021506": { + "id": "021506", + "content": "是否存在锐角$\\alpha$、$\\beta$满足$\\sin \\alpha=\\dfrac{7}{8} \\sin \\beta$, $\\cos \\alpha=\\dfrac{7}{2} \\cos \\beta$? 若存在, 求出$\\alpha$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021507": { + "id": "021507", + "content": "已知$\\alpha \\in[0,2 \\pi)$, 且满足$\\sqrt{\\dfrac{1-\\cos \\alpha}{1+\\cos \\alpha}}=\\dfrac{\\sin \\alpha}{1+\\cos \\alpha}$, 求$\\alpha$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021508": { + "id": "021508", + "content": "利用诱导公式求下列各值, 要求写出必要的步骤, 且不能使用计算器:\\\\\n(1) $\\sin (-\\dfrac{\\pi}{3})=$\\blank{200};\\\\\n(2) $\\cos (-\\dfrac{13 \\pi}{4})=$\\blank{200};\\\\\n(3) $\\tan \\dfrac{26 \\pi}{3}=$\\blank{200};\\\\\n(4) $\\cot (-\\dfrac{31 \\pi}{6})=$\\blank{200}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021509": { + "id": "021509", + "content": "分别根据下列条件, 找出满足条件的一个锐角$\\alpha$:\\\\\n(1) $\\sin \\alpha=\\sin 1551^{\\circ}, \\alpha=$\\blank{50};\\\\\n(2) $\\cos a=\\cos (-3312^{\\circ}), \\alpha=$\\blank{50};\\\\\n(3) $\\tan \\alpha=\\tan \\dfrac{190 \\pi}{9}, \\alpha=$\\blank{50};\\\\\n(4) $\\cot \\alpha=\\cot (-\\dfrac{158 \\pi}{15}), \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021510": { + "id": "021510", + "content": "化简: $\\dfrac{\\sin (\\alpha-\\pi) \\cot (\\alpha+\\pi)}{\\cos (\\alpha-\\pi) \\tan (\\alpha+\\pi)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021511": { + "id": "021511", + "content": "化简: $\\dfrac{\\cos (\\dfrac{7 \\pi}{10}+\\alpha)}{\\cos (\\alpha-\\dfrac{23 \\pi}{10})}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021512": { + "id": "021512", + "content": "化简: $\\dfrac{\\tan (177^{\\circ}-\\alpha)}{\\tan (543^{\\circ}+\\alpha)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021513": { + "id": "021513", + "content": "化简: $\\sqrt{1+2 \\sin (\\pi-2) \\cos (\\pi+2)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021514": { + "id": "021514", + "content": "化简: $\\sin (\\dfrac{5 \\pi}{6}+\\alpha)+\\cos (\\alpha-\\dfrac{2 \\pi}{3})+\\tan (\\dfrac{3 \\pi}{4}+\\alpha)+\\sin (\\alpha-\\dfrac{13 \\pi}{6})+\\cos (\\dfrac{\\pi}{3}+\\alpha)+\\tan (\\dfrac{5 \\pi}{4}-\\alpha)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021515": { + "id": "021515", + "content": "不使用计算器, 求$\\tan \\dfrac{\\pi}{11}+\\tan \\dfrac{2 \\pi}{11}+\\cdots+\\tan \\dfrac{10 \\pi}{11}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021516": { + "id": "021516", + "content": "设$\\cos 130^{\\circ}=a$, 用含$a$的代数式表示$\\tan 230^{\\circ}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021517": { + "id": "021517", + "content": "已知$\\cos (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{\\sqrt{3}}{3}$, 求$\\cos (\\alpha+\\dfrac{5 \\pi}{6})-\\sin ^2(\\alpha-\\dfrac{\\pi}{6})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021518": { + "id": "021518", + "content": "已知$\\cos (3 \\pi+\\alpha)=-\\dfrac{1}{2}$.\\\\\n(1) 若$\\alpha$为第四象限角, 求$\\sin (2 \\pi-\\alpha)$的值;\\\\\n(2) 设$n$是非零自然数, 求$\\dfrac{\\sin (n \\pi-\\alpha) \\cos [(n+1) \\pi+\\alpha]}{\\tan (\\alpha-n \\pi)}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021519": { + "id": "021519", + "content": "已知$\\sin \\alpha=\\dfrac{2}{3}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则:\\\\\n(1) $\\sin (3 \\pi+\\alpha)=$\\blank{50};\\\\\n(2) $\\cos (\\dfrac{5 \\pi}{2}-\\alpha)=$\\blank{50};\\\\\n(3) $\\sin (-\\dfrac{7}{2} \\pi+\\alpha)=$\\blank{50};\\\\\n(4) $\\tan (\\alpha-\\dfrac{3}{2} \\pi)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021520": { + "id": "021520", + "content": "用锐角的正弦、余弦、正切或余切表示下列各值:\\\\\n(1) $\\sin 1731^{\\circ}=$\\blank{50};\\\\\n(2) $\\cos (-3412^{\\circ})=$\\blank{50};\\\\\n(3) $\\tan \\dfrac{188 \\pi}{9}=$\\blank{50};\\\\\n(4) $\\cot (-\\dfrac{158 \\pi}{15})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021521": { + "id": "021521", + "content": "已知$\\tan (\\pi-\\alpha)=\\dfrac{1}{3}$, 则$1-2 \\sin \\alpha \\sin (\\alpha-\\dfrac{5 \\pi}{2})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021522": { + "id": "021522", + "content": "在平面直角坐标系中, 已知点$M(4,-3)$, 若将$OM$绕原点顺时针转$\\dfrac{3 \\pi}{2}$至$OM'$, 则点$M'$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021523": { + "id": "021523", + "content": "化简: $\\sin (21 \\pi-\\alpha)+\\cos (\\dfrac{9 \\pi}{2}-\\alpha)+\\tan (\\dfrac{9 \\pi}{4}-\\alpha)-\\sin (-\\alpha-19 \\pi)-\\cos (-\\alpha-\\dfrac{27 \\pi}{2})-\\tan (-\\alpha-\\dfrac{7 \\pi}{4})$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021524": { + "id": "021524", + "content": "化简: $\\dfrac{\\sin (\\pi-\\alpha) \\cos (\\dfrac{7 \\pi}{2}-\\alpha) \\cos (\\alpha-8 \\pi)}{\\sin (\\alpha-\\dfrac{5 \\pi}{2}) \\sin (-\\theta-5 \\pi)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021525": { + "id": "021525", + "content": "设$f(\\theta)=\\dfrac{\\cos ^2(2 \\pi-\\theta)+5 \\cos (-\\theta)-2 \\sin ^2 \\theta}{\\cos ^2(\\pi+\\theta)+\\sin (\\theta-\\dfrac{\\pi}{2})-6}$, 求$f(\\dfrac{\\pi}{3})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021526": { + "id": "021526", + "content": "不使用计算器, 求下列各式的值:\\\\\n(1) $\\tan \\dfrac{5 \\pi}{6} \\cdot \\cos \\dfrac{3 \\pi}{4}+\\tan \\dfrac{2 \\pi}{3} \\cdot \\cot \\dfrac{21 \\pi}{4}=$\\blank{50};\\\\\n(2) $\\dfrac{\\sin (-1234^{\\circ}) \\cos (-495^{\\circ}) \\tan 909^{\\circ}}{\\sin 694^{\\circ} \\cos 1230^{\\circ} \\tan 711^{\\circ}}=$\\blank{50};\\\\\n(3) $\\dfrac{\\sin (31^{\\circ}+\\alpha)}{\\tan (27^{\\circ}+\\alpha)} \\cdot \\dfrac{\\tan (747^{\\circ}+\\alpha)}{\\cos (36^{\\circ}+\\alpha)} \\cdot \\dfrac{\\cos (1116^{\\circ}+\\alpha)}{\\sin (751^{\\circ}+\\alpha)}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021527": { + "id": "021527", + "content": "已知角$\\alpha \\in[0,2 \\pi)$.\\\\\n(1) 若$\\alpha$的终边经过点$(\\cos (-\\dfrac{6 \\pi}{5}), \\sin \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50};\\\\\n(2) 若$\\alpha$的终边经过点$(\\cos \\dfrac{6 \\pi}{5}, \\sin (-\\dfrac{6 \\pi}{5}))$, 则$\\alpha=$\\blank{50};\\\\\n(3) 若$\\alpha$的终边经过点$(\\sin \\dfrac{6 \\pi}{5}, \\cos \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50};\\\\\n(4) 若$\\alpha$的终边经过点$(\\sin (-\\dfrac{6 \\pi}{5}), \\cos \\dfrac{6 \\pi}{5})$, 则$\\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021528": { + "id": "021528", + "content": "已知$\\cos (3 \\pi+\\alpha)=-\\dfrac{1}{2}$, $k \\in \\mathbf{Z}$.\\\\\n(1) 求$\\sin (2 \\pi-\\alpha)$;\\\\\n(2) 求$\\dfrac{1}{\\tan [\\dfrac{(2 k+1) \\pi}{2}+\\alpha]}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021529": { + "id": "021529", + "content": "根据下列条件, 分别写出角$x$的集合:\\\\\n(1) $\\sin x=\\dfrac{\\sqrt{2}}{2}$, $x$的集合为\\blank{100};\\\\\n(2) $\\cos x=-\\dfrac{1}{2}$, $x$的集合为\\blank{100};\\\\\n(3) $\\cot x=-\\sqrt{3}$, $x$的集合为\\blank{100};\\\\\n(4) $\\sin (x-\\dfrac{2 \\pi}{3})=\\dfrac{1}{2}$, $x$的集合为\\blank{100};\\\\\n(5) $\\tan (\\dfrac{\\pi}{4}-x)=\\dfrac{\\sqrt{3}}{2}$, $x$的集合为\\blank{100};\\\\\n(6) $\\cos (5 x+\\dfrac{\\pi}{4})=-\\dfrac{\\sqrt{3}}{2}$, $x$的集合为\\blank{100};\\\\\n(7) $\\sin (\\dfrac{4 x+7 \\pi}{2})=\\dfrac{\\sqrt{2}}{2}$, $x$的集合为\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021530": { + "id": "021530", + "content": "分别求满足下列条件的角$x$的集合:\\\\\n(1) $\\cos (x+\\dfrac{\\pi}{4})=\\dfrac{1}{2}$, $x \\in(0,2 \\pi)$;\\\\\n(2) $3 \\tan (x+\\dfrac{\\pi}{3})=\\sqrt{3}$, $x \\in(0, \\pi)$;\\\\\n(3) $2 \\sin 2 x-1=0$, $x \\in(0, \\dfrac{\\pi}{2})$;\\\\\n(4) $3 \\cot (\\dfrac{x}{2}-\\dfrac{\\pi}{12})=\\sqrt{3}$, $x \\in[0,2 \\pi)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021531": { + "id": "021531", + "content": "分别求满足下列条件的角$x$的集合:\\\\\n(1) $\\cos 2 x=\\cos 3 x$;\\\\\n(2) $\\sin x=\\cos 2 x$;\\\\\n(3) $2 \\sin ^2 x+\\sin x-1=0$;\\\\\n(4) $\\sin ^2 x-\\dfrac{2 \\sqrt{3}}{3} \\sin x \\cos x-\\cos ^2 x=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021532": { + "id": "021532", + "content": "设第四象限角$\\alpha$满足$\\cos \\alpha=\\dfrac{3}{5}$, 则$\\cos (\\alpha+\\dfrac{\\pi}{3})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021533": { + "id": "021533", + "content": "若$\\sin \\alpha \\cdot \\sin \\beta=1$, 则$\\cos (\\alpha+\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021534": { + "id": "021534", + "content": "已知$\\sin \\alpha+\\sin \\beta=\\dfrac{1}{5}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{4}{5}$, 则$\\cos (\\alpha-\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021535": { + "id": "021535", + "content": "不用计算器, 求下列各式的值:\\\\\n(1) $\\cos \\dfrac{5 \\pi}{12}$;\\\\\n(2) $\\cos 345^{\\circ}$;\\\\\n(3) $\\cos \\dfrac{7 \\pi}{10} \\cos (-\\dfrac{2 \\pi}{10})+\\sin \\dfrac{3 \\pi}{10} \\sin \\dfrac{2 \\pi}{10}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021536": { + "id": "021536", + "content": "化简下列各式:\\\\\n(1) $\\cos (\\dfrac{\\pi}{3}-\\alpha)-\\cos (\\dfrac{\\pi}{3}+\\alpha)$;\\\\\n(2) $\\sin (\\alpha-\\beta) \\sin \\beta+\\cos (\\alpha-\\beta) \\cos \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021537": { + "id": "021537", + "content": "设$\\alpha, \\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 满足$\\sin \\alpha=\\dfrac{8}{17}$, $\\tan \\beta=-\\dfrac{5}{12}$, 求$\\cos (\\alpha+\\beta)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021538": { + "id": "021538", + "content": "设$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且满足$\\sin (\\alpha-\\dfrac{\\pi}{6})=\\dfrac{1}{3}$, 求$\\cos \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021539": { + "id": "021539", + "content": "求证: $\\cos (\\alpha+\\beta) \\cos (\\beta-\\alpha)=\\cos ^2 \\alpha-\\sin ^2 \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021540": { + "id": "021540", + "content": "若$\\alpha$、$\\beta$是锐角, 则下列不等式一定成立的是\\bracket{20}.\n\\twoch{$\\cos (\\alpha+\\beta)>0$}{$\\cos (\\alpha+\\beta)<0$}{$\\cos (\\alpha-\\beta)>\\cos (\\alpha+\\beta)$}{$\\cos (\\alpha-\\beta)<\\cos (\\alpha+\\beta)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021541": { + "id": "021541", + "content": "若$\\cos (\\alpha+\\beta) \\cos (\\alpha-\\beta)=\\dfrac{1}{4}$, 则$\\cos ^2 \\alpha+\\cos ^2 \\beta=$\\bracket{20}.\n\\fourch{$\\dfrac{5}{4}$}{$1$}{$\\dfrac{1}{4}$}{$\\dfrac{3}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021542": { + "id": "021542", + "content": "已知$\\sin \\alpha=\\dfrac{2}{3}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\cos \\beta=-\\dfrac{3}{4}$, $\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\cos (\\alpha-\\beta+\\dfrac{\\pi}{4})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021543": { + "id": "021543", + "content": "已知$\\sin \\theta+\\cos \\theta=\\dfrac{1}{5}$, $\\theta \\in(0, \\pi)$, 求$\\cos (\\theta-\\dfrac{\\pi}{3})+\\cot \\theta$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021544": { + "id": "021544", + "content": "已知锐角$\\alpha$、$\\beta$满足$\\sin \\alpha-\\sin \\beta=\\dfrac{\\sqrt{2}}{2}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{\\sqrt{6}}{2}$, 求$\\alpha+\\beta$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021545": { + "id": "021545", + "content": "已知$\\sin \\alpha=\\dfrac{2}{3}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$\\sin (\\dfrac{\\pi}{3}-\\alpha)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021546": { + "id": "021546", + "content": "不使用计算器, 求值: $\\sin 28^{\\circ} \\cos 73^{\\circ}-\\sin 62^{\\circ} \\cos 17^{\\circ}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021547": { + "id": "021547", + "content": "化简: $\\sin (\\alpha+\\beta) \\cdot \\cos (\\alpha-\\beta)-\\cos (\\alpha+\\beta) \\cdot \\sin (\\alpha-\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021548": { + "id": "021548", + "content": "化简: $\\sin \\alpha+\\sin (\\alpha+\\dfrac{2 \\pi}{3})+\\sin (\\alpha+\\dfrac{4 \\pi}{3})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021549": { + "id": "021549", + "content": "不使用计算器, 求值: $\\dfrac{\\sin 9^{\\circ}+\\sin 6^{\\circ} \\cos 15^{\\circ}}{\\cos 9^{\\circ}-\\sin 6^{\\circ} \\sin 15^{\\circ}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021550": { + "id": "021550", + "content": "在$\\triangle ABC$中, 若$\\sin A=\\dfrac{4}{5}$, $\\cos B=-\\dfrac{5}{13}$, 则$\\sin C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021551": { + "id": "021551", + "content": "已知$\\tan \\alpha=-\\dfrac{4}{3}$, $\\sin \\beta=\\dfrac{3}{5}$, 且$\\alpha$、$\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\sin (\\alpha-\\beta)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021552": { + "id": "021552", + "content": "若$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, 0)$, $\\sin (\\alpha+\\beta)=-\\dfrac{\\sqrt{2}}{4}$, $\\cos \\alpha=\\dfrac{3}{5}$, 求$\\sin \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021553": { + "id": "021553", + "content": "已知点$P(4,3)$, 将$P$绕坐标原点$O$顺时针方向旋转$\\dfrac{\\pi}{6}$至点$P'$, 求$P'$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021554": { + "id": "021554", + "content": "在$\\triangle ABC$中, 若$\\cos A=\\dfrac{12}{13}$, $\\sin B=\\dfrac{3}{5}$, 则$\\cos C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021555": { + "id": "021555", + "content": "若$\\alpha$、$\\beta$为锐角, 则下列不等式中一定成立的是\\bracket{20}.\n\\twoch{$\\sin (\\alpha+\\beta)>\\sin \\alpha+\\sin \\beta$}{$\\sin (\\alpha+\\beta)<\\sin \\alpha+\\sin \\beta$}{$\\cos (\\alpha+\\beta)>\\cos \\alpha+\\cos \\beta$}{$\\cos (\\alpha+\\beta)>\\sin \\alpha+\\sin \\beta$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021556": { + "id": "021556", + "content": "$\\triangle ABC$中, 若两内角$A$、$B$满足$\\cot A \\cdot \\cot B>1$, 则$\\triangle ABC$的形状为\\bracket{20}三角形.\n\\fourch{锐角}{直角}{钝角}{无法确定的}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021557": { + "id": "021557", + "content": "已知$\\alpha, \\beta \\in(\\dfrac{3 \\pi}{4}, \\pi)$, $\\sin (\\alpha+\\beta)=-\\dfrac{3}{5}$, $\\sin (\\beta-\\dfrac{\\pi}{4})=\\dfrac{12}{13}$, 求$\\cos (\\alpha+\\dfrac{\\pi}{4})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021558": { + "id": "021558", + "content": "若$\\sin \\alpha=\\dfrac{4}{5}, \\cot \\beta=3$, 且$\\alpha$是第二象限的角, 则$\\tan (\\alpha-\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021559": { + "id": "021559", + "content": "若$\\tan \\alpha, \\tan \\beta$是方程$x^2-3 x-3=0$的两个实数解, 则$\\tan (\\alpha+\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021560": { + "id": "021560", + "content": "若$\\tan (\\theta+\\dfrac{\\pi}{6})=3$, 则$\\tan \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021561": { + "id": "021561", + "content": "化简: $\\dfrac{\\tan (\\alpha-\\beta)+\\tan \\beta}{1-\\tan (\\alpha-\\beta) \\tan \\beta}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021562": { + "id": "021562", + "content": "不用计算器, 求值: $\\tan 36^{\\circ}+\\sqrt{3} \\tan 24^{\\circ} \\tan 36^{\\circ}+\\tan 24^{\\circ}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021563": { + "id": "021563", + "content": "不用计算器, 求值: $\\dfrac{1-\\tan 75^{\\circ}}{1+\\tan 75^{\\circ}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021564": { + "id": "021564", + "content": "甲: $\\tan \\alpha+\\tan \\beta=0$, 乙: $\\tan (\\alpha+\\beta)=0$, 则甲是乙的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021565": { + "id": "021565", + "content": "已知$\\sin \\theta=-\\dfrac{7}{25}$, $\\theta \\in(\\pi, \\dfrac{3 \\pi}{2})$, 求$\\tan (\\theta-\\dfrac{\\pi}{4})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021566": { + "id": "021566", + "content": "已知$(1+\\tan \\alpha)(1+\\tan \\beta)=2$, 且$\\alpha$、$\\beta$都是锐角, 求$\\alpha+\\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021567": { + "id": "021567", + "content": "已知$\\tan (\\dfrac{\\pi}{4}+\\alpha)=2$, $\\tan \\beta=\\dfrac{1}{2}$. 求:\\\\\n(1) $\\tan \\alpha$;\\\\\n(2) $\\dfrac{\\sin (\\alpha+\\beta)-2 \\sin \\alpha \\cos \\beta}{2 \\sin \\alpha \\sin \\beta+\\cos (\\alpha+\\beta)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021568": { + "id": "021568", + "content": "已知$\\cos (\\alpha+\\beta)=\\dfrac{1}{2}$, $\\cos (\\alpha-\\beta)=\\dfrac{1}{3}$, 求$\\tan \\alpha \\tan \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021569": { + "id": "021569", + "content": "镜框$AB$的高为$0.96$米, 平挂在墙上, 与人眼的水平视线的高度差为$OB=1$米. 现人眼在点$C$处, 当人朝墙走去时, 在什么位置$\\tan \\angle ACB$最大?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [right] {$O$} coordinate (O);\n\\draw (0,1) node [right] {$B$} coordinate (B);\n\\draw (0,1.96) node [right] {$A$} coordinate (A);\n\\draw (-2,0) node [left] {$C$} coordinate (C);\n\\draw (C)--(O)--(A)--cycle(B)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021570": { + "id": "021570", + "content": "把下列各式化成$A \\sin (\\alpha+\\varphi)$($A>0$, $\\varphi \\in[0,2 \\pi)$)的形式:\\\\\n(1) $\\sqrt{3} \\sin \\alpha+\\cos \\alpha=$\\blank{50};\\\\\n(2) $\\sin x-\\cos x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021571": { + "id": "021571", + "content": "把$3 \\cos \\alpha-3 \\sqrt{3} \\sin \\alpha$化成$A \\cos (\\alpha+\\varphi)$($A>0$, $\\varphi \\in(-\\pi, \\pi)$)的形式, 则结果为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021572": { + "id": "021572", + "content": "若角$\\alpha$满足$\\dfrac{1}{2} \\cos \\alpha-\\dfrac{\\sqrt{3}}{2} \\sin \\alpha=1$, 则$\\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021573": { + "id": "021573", + "content": "下列关系中, 角$\\alpha$存在的是\\bracket{20}.\n\\twoch{$\\sin \\alpha+\\cos \\alpha=\\dfrac{3}{2}$}{$\\sin \\alpha+\\cos \\alpha=\\dfrac{4}{3}$}{$\\sin \\alpha=\\dfrac{1}{3}$且$\\cos \\alpha=\\dfrac{2}{3}$}{$\\cos \\alpha-\\sin \\alpha=-\\sqrt{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021574": { + "id": "021574", + "content": "若$\\tan (\\alpha+\\beta)=\\dfrac{3}{4}$, $\\tan (\\beta+\\dfrac{\\pi}{4})=\\dfrac{1}{3}$, 求$\\tan (\\alpha-\\dfrac{\\pi}{4})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021575": { + "id": "021575", + "content": "已知$\\alpha$是$\\triangle ABC$的一个内角, 且满足$\\sin \\alpha+\\cos \\alpha=\\dfrac{\\sqrt{6}}{2}$, 求$\\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021576": { + "id": "021576", + "content": "已知$\\sin (\\alpha+\\beta)=\\dfrac{1}{2}$, $\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, 求$\\tan \\alpha \\cot \\beta$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021577": { + "id": "021577", + "content": "已知$8 \\cos (2 \\alpha+\\beta)+5 \\cos \\beta=0$, 且$\\cos (\\alpha+\\beta) \\cos \\alpha \\neq 0$, 求$\\tan (\\alpha+\\beta) \\cdot \\tan \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021578": { + "id": "021578", + "content": "已知关于$x$的方程$x^2+p x+q=0$的两根是$\\tan \\alpha$和$\\tan \\beta$, 求$\\dfrac{\\sin (\\alpha+\\beta)}{\\cos (\\alpha-\\beta)}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021579": { + "id": "021579", + "content": "已知$\\tan (\\alpha+\\beta)=-2$, $\\tan (\\alpha-\\beta)=\\dfrac{1}{2}$, 求$\\dfrac{\\sin 2 \\alpha}{\\sin 2 \\beta}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021580": { + "id": "021580", + "content": "若$\\tan \\dfrac{3 \\alpha}{2}=\\dfrac{3}{4}$, 则$\\tan 3 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021581": { + "id": "021581", + "content": "已知$\\sin 2 \\alpha=\\dfrac{4}{5}$, $\\alpha \\in(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$, 则$\\sin 4 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021582": { + "id": "021582", + "content": "若$\\alpha \\in(\\pi, \\dfrac{3}{2} \\pi)$, $\\tan \\alpha=4$, 则$\\cos 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021583": { + "id": "021583", + "content": "已知$\\cos \\varphi=-\\dfrac{1}{3}$, 且$\\pi<\\varphi<\\dfrac{3 \\pi}{2}$, 求$\\sin 2 \\varphi$、$\\cos 2 \\varphi$和$\\tan 2 \\varphi$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021584": { + "id": "021584", + "content": "已知等腰三角形的底角的正弦值等于$\\dfrac{4}{5}$, 求这个三角形的顶角的正弦、余弦和正切的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021585": { + "id": "021585", + "content": "不用计算器, 求下列各式的值:\\\\\n(1) $\\dfrac{1}{1+\\tan 15^{\\circ}}-\\dfrac{1}{1-\\tan 15^{\\circ}}$;\\\\\n(2) $\\sin ^4 \\dfrac{\\pi}{8}+\\cos ^4 \\dfrac{\\pi}{8}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021586": { + "id": "021586", + "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, $\\tan (\\pi-\\beta)=\\dfrac{1}{2}$, 求$\\tan (\\alpha-2 \\beta)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021587": { + "id": "021587", + "content": "已知$\\sin 2 \\alpha=-\\dfrac{3}{5}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\cos \\alpha-\\sin \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021588": { + "id": "021588", + "content": "化简: $\\cos ^2(\\theta+15^{\\circ})+\\sin ^2(\\theta-15^{\\circ})+\\sin (\\theta+180^{\\circ}) \\cos (\\theta-180^{\\circ})$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021589": { + "id": "021589", + "content": "证明下列恒等式:\\\\\n(1) $1+\\sin \\alpha=(\\sin \\dfrac{\\alpha}{2}+\\cos \\dfrac{\\alpha}{2})^2$;\\\\\n(2) $\\dfrac{1+\\sin 2 \\alpha-\\cos 2 \\alpha}{1+\\sin 2 \\alpha+\\cos 2 \\alpha}=\\tan \\alpha$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021590": { + "id": "021590", + "content": "若$\\sin \\alpha=\\dfrac{8}{5} \\sin \\dfrac{\\alpha}{2}$, 求$\\cos \\alpha$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021591": { + "id": "021591", + "content": "若$\\theta \\in(-\\dfrac{3}{2} \\pi,-\\pi)$, 且$\\cos \\theta=a$, 则$\\sin \\dfrac{\\theta}{2}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021592": { + "id": "021592", + "content": "若$\\sin \\dfrac{\\alpha}{2}=-\\dfrac{5}{6}$, $\\cos \\dfrac{\\alpha}{2}=\\dfrac{\\sqrt{11}}{6}$, 则$\\alpha$的终边在第\\blank{50}象限.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021593": { + "id": "021593", + "content": "已知$\\cos \\theta=\\dfrac{1}{3}$, 且$\\theta$是第四象限的角, 求$\\sin \\dfrac{\\theta}{2}$、$\\cos \\dfrac{\\theta}{2}$和$\\tan \\dfrac{\\theta}{2}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021594": { + "id": "021594", + "content": "已知等腰三角形的顶角的余弦值等于$-\\dfrac{7}{25}$, 求这个三角形的底角的正弦、余弦的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021595": { + "id": "021595", + "content": "证明下列恒等式:\\\\\n(1) $8 \\sin ^4 \\alpha=\\cos 4 \\alpha-4 \\cos 2 \\alpha+3$;\\\\\n(2) $\\dfrac{1+\\sin \\alpha}{\\cos \\alpha}=\\dfrac{1+\\tan \\dfrac{\\alpha}{2}}{1-\\tan \\dfrac{\\alpha}{2}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021596": { + "id": "021596", + "content": "已知$\\dfrac{\\cos 2 x}{1+\\sin 2 x}=\\dfrac{1}{5}$, 求$\\tan x$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021597": { + "id": "021597", + "content": "已知$\\alpha \\in(\\dfrac{5}{4} \\pi, \\dfrac{3}{2} \\pi)$, 化简$\\sqrt{1-\\sin 2 \\alpha}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021598": { + "id": "021598", + "content": "化简: $\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos 2 \\alpha}},(\\pi<\\alpha<\\dfrac{3 \\pi}{2})$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021599": { + "id": "021599", + "content": "化简:\\\\\n(1) $\\dfrac{(\\sin \\theta+\\cos \\theta-1)(\\sin \\theta-\\cos \\theta+1)}{2 \\sin \\theta \\cos \\theta}$;\\\\\n(2) $\\dfrac{1+\\sin \\alpha}{2 \\cos ^2(\\dfrac{\\pi}{4}-\\dfrac{\\alpha}{2})}-2 \\sin ^2(\\dfrac{\\pi}{4}-\\dfrac{\\alpha}{2})$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021600": { + "id": "021600", + "content": "在$\\triangle ABC$中, 若$B=45^{\\circ}$, $C=60^{\\circ}$, $b=1$, 则边$c=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021601": { + "id": "021601", + "content": "在$\\triangle ABC$中, 若$a=1$, $b=\\sqrt{3}$, $A=30^{\\circ}$, 则角$C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021602": { + "id": "021602", + "content": "在$\\triangle ABC$中, 若$B=45^{\\circ}$, $C=15^{\\circ}$, $b=2$, 则该三角形的最长边长等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021603": { + "id": "021603", + "content": "在$\\triangle ABC$中, 若$A=60^{\\circ}$, $b=16$, $S_{\\triangle ABC}=220 \\sqrt{3}$, 则边$c=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021604": { + "id": "021604", + "content": "在$\\triangle ABC$中, 若$\\sqrt{3} a=2 b \\sin A$, 则$B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021605": { + "id": "021605", + "content": "若三角形的三内角之比为$1: 2: 3$, 则三边长之比为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021606": { + "id": "021606", + "content": "在$\\triangle ABC$中, 若$a^2+b^2=2 c^2$, 则$\\dfrac{\\sin ^2A+\\sin ^2B}{\\sin ^2C}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021607": { + "id": "021607", + "content": "在$\\triangle ABC$中, 已知$a=5$, $b=4$, $A=2B$, 则$\\cos B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021608": { + "id": "021608", + "content": "在$\\triangle ABC$中, 满足$\\dfrac{a}{b}=\\dfrac{\\cos A}{\\cos B}$的三角形是\\blank{50}三角形.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021609": { + "id": "021609", + "content": "在$\\triangle ABC$中, 三内角的度数比是$3: 4: 5$. 若最小边长为$3$, 则此三角形的外接圆的半径是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021610": { + "id": "021610", + "content": "在$\\triangle ABC$中, 已知$\\cos A=\\dfrac{\\sqrt{2}}{2}$, $B-C=\\dfrac{\\pi}{12}$, $a=\\sqrt{2}$, 求$c$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021611": { + "id": "021611", + "content": "在$\\triangle ABC$中, 若$a0$, 则$C$一定是锐角;\\\\\n\\textcircled{2} 在$\\triangle ABC$中, 若$a^2>b^2+c^2$, 则$A>B+C$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021618": { + "id": "021618", + "content": "若$a$、$a+1$、$a+2$是锐角三角形的三边长, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021619": { + "id": "021619", + "content": "在$\\triangle ABC$中, 已知$b=5$, $c=4$, $A=60^{\\circ}$, 求$a$和$\\sin B$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021620": { + "id": "021620", + "content": "在$\\triangle ABC$中, 已知$a, b, c$满足$(a+b+c)(a-b+c)=a c$, 求$B$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021621": { + "id": "021621", + "content": "在$\\triangle ABC$中, 已知$a=2 \\sqrt{3}$, $b=45^{\\circ}$, 面积$S=3+\\sqrt{3}$, 求$c$和$C$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021622": { + "id": "021622", + "content": "在$\\triangle ABC$中, $A=60^{\\circ}$, $b=3$, $c=2$, 求$BC$边上的中线长.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021623": { + "id": "021623", + "content": "已知三角形两边之和为$8$, 其夹角为$60^{\\circ}$, 分别求这个三角形周长的最小值和面积的最大值, 并指出面积最大时三角形的形状.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021624": { + "id": "021624", + "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\cos A=\\dfrac{\\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021625": { + "id": "021625", + "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\cos A=\\dfrac{2 \\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021626": { + "id": "021626", + "content": "在$\\triangle ABC$中, 若$\\sin B=\\dfrac{4}{5}$, $\\sin A=\\dfrac{2 \\sqrt{5}}{5}$, 则$\\cos C=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021627": { + "id": "021627", + "content": "在$\\triangle ABC$中, 已知$a=2$, $A=45^{\\circ}$. 若此三角形有两解, 则$b$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021628": { + "id": "021628", + "content": "根据下列条件, 判断$\\triangle ABC$的形状:\\\\\n(1) $\\cos ^2B-\\cos ^2C=\\sin ^2A$;\\\\\n(2) $a=2 b \\cos C$;\\\\\n(3) $\\tan B=\\dfrac{\\cos (B-C)}{\\sin A-\\sin (B-C)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021629": { + "id": "021629", + "content": "在$\\triangle ABC$中, $A=60^{\\circ}$, $b=1$, 且其面积为$\\sqrt{3}$, 求$a$和$\\triangle ABC$的外接圆半径$R$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021630": { + "id": "021630", + "content": "在$\\triangle ABC$中, 已知$\\sin A: \\sin C=5: 2$, $B=60^{\\circ}$, 且$S_{\\triangle ABC}=90 \\sqrt{3}$, 求$b$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021631": { + "id": "021631", + "content": "求分别满足下列条件的角:\\\\\n(1) $\\sin x=\\dfrac{2}{5}$, $x \\in[0, \\pi]$;\\\\\n(2) $\\cos x=-\\dfrac{2}{3}$, $x \\in[0,2 \\pi]$;\\\\\n(3) $\\tan x=-\\dfrac{1}{2}$, $x \\in \\mathbf{R}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021632": { + "id": "021632", + "content": "在平行四边形$ABCD$中, 已知$AB=10 \\sqrt{3}$, $B=60^{\\circ}$, $AC=30$, 求平行四边形的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021633": { + "id": "021633", + "content": "在$\\triangle ABC$中, 求证: $a \\cos ^2 \\dfrac{C}{2}+c \\cos ^2 \\dfrac{A}{2}=\\dfrac{1}{2}(a+b+c)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021634": { + "id": "021634", + "content": "在地面某处测得塔顶的仰角为$\\theta$, 由此向塔底沿直线走$3$千米, 测得塔顶的仰角为\n$2 \\theta$, 再向塔底沿同一直线走$\\sqrt{3}$千米, 测得塔顶仰角为$4 \\theta$(三个测量点都在塔的同一\n侧). 试求$\\theta$与塔高.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021635": { + "id": "021635", + "content": "如图, 为了测定对岸$A, B$两点之间的距离, 在河的一岸定一条基线$CD$, 测得$CD=100$米, $\\angle ACD=80^{\\circ}$, $\\angle BCD=45^{\\circ}$, $\\angle BDC=70^{\\circ}$, $\\angle ADC=33^{\\circ}$, 求$A, B$间的距离. (结果精确到$0.01$米)\n\\begin{center}\n\\begin{tikzpicture}\n\\clip (-1.5,-1) rectangle (6.5,4);\n\\fill [gray!60] (-1,1.1) rectangle (6,2.2);\n\\draw (0,0) node [below left] {$C$} coordinate (C) ++ (-8:5) node [below right] {$D$} coordinate (D);\n\\path [name path = lineBC] (C) --++ (37:10);\n\\path [name path = lineBD] (D) --++(102:10);\n\\path [name intersections={of = lineBC and lineBD, by=B}];\n\\draw (C) -- (B) node [above right] {$B$} --(D);\n\\path [name path = lineAC] (C) --++ (72:10);\n\\path [name path = lineAD] (D) --++(139:10);\n\\path [name intersections={of = lineAC and lineAD, by=A}];\n\\draw (B) -- (A) -- (C) -- (D) -- (A) node [above left] {$A$};\n\\draw (C) ++ (-8:0.3) arc (-8:72:0.3) node [above right] {$80^\\circ$};\n\\draw (C) ++ (-8:0.4) arc (-8:37:0.4) node [right] {$45^\\circ$};\n\\draw (D) ++ (172:0.4) arc (172:139:0.4) node [left] {$33^\\circ$};\n\\draw (D) ++ (172:0.3) arc (172:102:0.3) node [above left] {$70^\\circ$};\n\\draw (-1,1.1) -- (6,1.1) (-1,2.2) -- (6,2.2);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021636": { + "id": "021636", + "content": "如图, 某市郊外景区内一条笔直的公路$a$经过三个景点$A$、$B$、$C$. 景区管委会又开发了风景优美的景点$D$. 经测量景点$D$位于景点$A$的北偏东$30^{\\circ}$方向$8$千米处, 且位于景点$B$的正北方向, 还位于景点$C$的北偏西$75^{\\circ}$方向上. 已知$AB=5$千米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw [->] (6,8) -- (10,8) node [right] {东};\n\\draw [->] (6,8) -- (6,12) node [above] {北};\n\\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {$N$} coordinate (N);\n\\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n\\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n\\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n\\path [name path = DC] (D) --++ (-15:4);\n\\path [name intersections = {of = linea and DC, by = C}];\n\\draw (D) -- (C) node [below] {$C$};\n\\draw (60:2) arc (60:90:2);\n\\draw (75:4) node {$30^\\circ$};\n\\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点$D$向景点$B$修一条笔直的公路, 不考虑其他因素, 求出这条公路的长; (结果精确到$0.1$千米)\\\\\n(2) 求景点$C$与景点$D$之间的距离. (结果精确到$0.1$千米)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021637": { + "id": "021637", + "content": "如图所示, 某游客在$A$处望见一塔$B$在正北方向, 在北偏西$60^{\\circ}$方向的$C$处有一寺庙, 此游客乘车向西$1$千米后到达$D$处, 这时塔和寺庙分别在东北和西北方向.求塔与寺庙间的距离. (结果精确到$0.1$千米)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (-1,0) node [below] {$D$} coordinate (D);\n\\draw (0,1) node [right] {$B$} coordinate (B);\n\\path [name path = AC] (A)--++(150:3);\n\\path [name path = CD] (D)--++(135:2);\n\\path [name intersections = {of = AC and CD, by = C}];\n\\draw (C) node [above] {$C$};\n\\draw (C)--(D)--(A)--cycle (A)--(B)--(D);\n\\draw [->] (B)--($(B)!-1.5!(A)$) node [right] {北};\n\\draw pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 1.7] {angle = B--A--C};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021638": { + "id": "021638", + "content": "余弦函数$y=\\cos x$, $x \\in \\mathbf{R}$图像至少向右平移单位得到正弦函数$y=\\sin x$, $x \\in \\mathbf{R}$的图像.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021639": { + "id": "021639", + "content": "函数$y=\\cos (x+\\dfrac{\\pi}{2})$, $x \\in[-\\pi, \\pi]$的图像是\\bracket{20}.\n\\twoch{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below left] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{-sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [below left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{-cos(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,0.1) -- (-pi,0) node [below] {$-\\pi$};\n\\draw (pi,0.1) -- (pi,0) node [below] {$\\pi$};\n\\draw (0.1,1) -- (0,1) node [below left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = -pi:pi,samples = 100] plot (\\x,{cos(\\x/pi*180)});\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021640": { + "id": "021640", + "content": "作出下列函数的大致图像:\\\\\n(1) $y=1+\\sin x$, $x \\in[-\\pi, \\pi]$;\\\\\n(2) $y=-\\cos x$, $x \\in[0,2 \\pi]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021641": { + "id": "021641", + "content": "写出下列函数的定义域:\\\\\n(1) $y=\\dfrac{1}{1+\\sin x}$, 定义域为\\blank{100};\\\\\n(2) $y=\\sqrt{-2 \\cos x}$, 定义域为\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021642": { + "id": "021642", + "content": "已知$\\sin \\alpha \\geq \\dfrac{1}{2}$, 则在$[0,2 \\pi]$中的$\\alpha$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021643": { + "id": "021643", + "content": "已知函数$y=\\cos x$($0 \\leq x \\leq 2 \\pi$)的图像与直线$y=1$围成一个封闭的平面图形, 则该封闭图形的面积是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021644": { + "id": "021644", + "content": "函数$y=\\sin |x|$, $x \\in[-2 \\pi, 2 \\pi]$的图像是\\bracket{20}.\n\\twoch{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{sin(\\x/pi*180)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{abs(sin(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{sin(abs(\\x/pi*180))});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-7,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw ({-2*pi},0.1) -- ({-2*pi},0) node [below] {$-2\\pi$};\n\\draw ({2*pi},0.1) -- ({2*pi},0) node [below] {$2\\pi$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw (0.1,-1) -- (0,-1) node [left] {$-1$};\n\\draw [domain = {-2*pi}:{2*pi},samples = 100] plot (\\x,{-sin(abs(\\x/pi*180))});\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021645": { + "id": "021645", + "content": "下列各组函数中, 表示同一函数的是\\bracket{20} .\n\\twoch{$f(x)=\\sin x$, $g(x)=\\dfrac{x \\sin x}{x}$}{$f(x)=\\cos x$, $g(x)=\\sqrt{1-\\sin ^2 x}$}{$f(x)=1$, $g(x)=\\sin ^2 x+\\cos ^2 x$}{$f(x)=1$, $g(x)=\\tan x \\cdot \\cot x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021646": { + "id": "021646", + "content": "已知$a$为实常数, 作出相应的函数图像并讨论方程$f(x)=a$的实数解个数.\\\\\n(1) $f(x)=\\sin x$, $x \\in[0, \\dfrac{5}{4} \\pi]$;\\\\\n(2) $f(x)=\\begin{cases}-\\sin x,& x \\in(0, \\pi], \\\\ \\cos x,& x \\in[-\\pi, 0].\\end{cases}$", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021647": { + "id": "021647", + "content": "写出下列函数的最小正周期:\\\\\n(1) $y=\\sin \\dfrac{x}{4}$的最小正周期为\\blank{50};\\\\\n(2) $y=2 \\cos (2 x+\\dfrac{\\pi}{3})$的最小正周期为\\blank{50};\\\\\n(3) $y=\\cos ^2 x$的最小正周期为\\blank{50};\\\\\n(4) $y=\\sin x+\\sqrt{3} \\cos x$的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021648": { + "id": "021648", + "content": "若函数$y=f(x)$的周期为$3$, 则函数$y=f(x+1)$的一个周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021649": { + "id": "021649", + "content": "``$\\omega=1$''是``函数$y=\\cos \\omega x$的最小正周期为$2 \\pi$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021650": { + "id": "021650", + "content": "设常数$a \\neq 0$, 函数$y=\\sin (a \\pi x+1)$的周期为\\bracket{20}.\n\\fourch{$\\dfrac{2}{a}$}{$\\dfrac{a}{2}$}{$\\dfrac{2}{|a|}$}{$\\dfrac{2 \\pi}{|a|}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021651": { + "id": "021651", + "content": "判断以下命题的真假:\\\\\n(1) 对于函数$f(x)$, 如果存在一个常数$T$($T \\neq 0$), 使得当$x$取定义域$D$内的某一个值$x_0$时, 有$f(x_0+T)=f(x_0)$成立, 那么这个函数$f(x)$叫做周期函数;\\\\\n(2) 周期函数的定义域一定是$\\mathbf{R}$;\\\\\n(3) 每个周期函数都有无数个周期.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021652": { + "id": "021652", + "content": "$x$为实数, $[x]$表示不超过$x$的最大整数, 则函数$f(x)=x-[x]$在$\\mathbf{R}$上为\\bracket{20}.\n\\fourch{奇函数}{偶函数}{增函数}{周期函数}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021653": { + "id": "021653", + "content": "求下列函数的最小正周期:\\\\\n(1) $y=\\sin ^2 x$;\\\\\n(2) $y=\\sin ^2 x+2 \\sin x \\cos x$;\\\\\n(3) $y=\\sin ^4 x+\\cos ^4 x$;\\\\\n(4) $y=\\sin 2 a x$($a \\neq 0$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021654": { + "id": "021654", + "content": "已知函数$y=f(x)$的最小正周期$4$. 若此函数的最大值为$2$, 最小值为$-6$, 则这个函数的一个可能的解析式是$f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021655": { + "id": "021655", + "content": "函数$y=|\\sin x|$的最小正周期为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\pi$}{$2 \\pi$}{$4 \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021656": { + "id": "021656", + "content": "若$y=f(x) \\cdot \\cos x$是周期为$\\pi$, 且最大值为$1$, 则$y=f(x)$可能是\\bracket{20}.\n\\fourch{$y=\\cos x$}{$y=\\sin x$}{$y=-\\cos x$}{$y=-\\sin x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021657": { + "id": "021657", + "content": "已知函数$y=f(x)$满足$f(x+2)=-f(x)$.\\\\\n(1) 若当$x \\in[0,2)$时, $f(x)=x$, 求$f(3)$、$f(5)$、$f(7)$的值;\\\\\n(2) 求函数$y=f(x)$的一个周期, 并加以证明.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021658": { + "id": "021658", + "content": "函数$y=\\cos x+3$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021659": { + "id": "021659", + "content": "函数$y=4 \\sin ^2 x-2$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021660": { + "id": "021660", + "content": "函数$y=2 \\sin ^2 x+2 \\sin x-1$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021661": { + "id": "021661", + "content": "若$\\dfrac{\\pi}{6} \\leq \\theta<\\dfrac{4 \\pi}{3}$, 则$\\sin \\theta$的范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021662": { + "id": "021662", + "content": "函数$y=2-\\sin x$的最大值是\\blank{50}, 此时$x$的集合是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021663": { + "id": "021663", + "content": "函数$y=3 \\sin (2 x-\\dfrac{\\pi}{3})$的最小值是\\blank{50}, 此时$x$的集合是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021664": { + "id": "021664", + "content": "求函数$y=2 \\sin x+3 \\cos x$的最大值与最小值, 并指出何时取到.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021665": { + "id": "021665", + "content": "若直角三角形$ABC$的两锐角为$\\alpha$、$\\beta$, 则$\\sin \\alpha \\sin \\beta$的值域是\\bracket{20}.\n\\fourch{$[0,1]$}{$(0,1)$}{$(0, \\dfrac{1}{2})$}{$(0, \\dfrac{1}{2}]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021666": { + "id": "021666", + "content": "若存在$\\alpha$使得$\\sin \\alpha+\\sqrt{3} \\cos \\alpha=\\dfrac{2 m-4}{m+3}$, 则$m$的取值范围是\\bracket{20}.\n\\fourch{$(-3,-\\dfrac{1}{2}]$}{$(-3,+\\infty)$}{$[-\\dfrac{1}{2},+\\infty)$}{$(-\\infty,-3) \\cup(-3,-\\dfrac{1}{2}]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021667": { + "id": "021667", + "content": "如图, 太阳光斜照地面, 光线与水平面所成的角为$\\theta$, 长为$l$的竹竿与地面所成的角为$\\alpha$(其中$\\theta, l$为常数). 问当$\\alpha$为多少时, 竹竿的影子最长?\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\filldraw [gray!30] (-1.5,-0.3) rectangle (1.5,0);\n\\draw (-1.5,0) -- (1.5,0);\n\\draw [very thick] (-1,0) coordinate (A) -- (-0.3,1) coordinate (B);\n\\draw (-0.3,1) -- (1,0) coordinate (C);\n\\draw (A) pic [\"$\\alpha$\", angle radius = 0.3cm, draw, angle eccentricity = 1.5] {angle = C--A--B};\n\\draw (C) pic [\"$\\theta$\", angle radius = 0.3cm, draw, angle eccentricity = 1.5] {angle = B--C--A};\n\\draw ($(B)!0.25!(A)$) coordinate (B1) ++ (-0.65,0.5) coordinate (D1);\n\\draw ($(B)!0.5!(A)$) coordinate (B2) ++ (-0.65,0.5) coordinate (D2);\n\\draw ($(B)!0.75!(A)$) coordinate (B3) ++ (-0.65,0.5) coordinate (D3);\n\\draw (B) ++ (-0.65,0.5) coordinate (D);\n\\draw [->] (D) -- (B);\n\\draw [->] (D1) -- (B1);\n\\draw [->] (D2) -- (B2);\n\\draw [->] (D3) -- (B3);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021668": { + "id": "021668", + "content": "函数$y=\\sin (x^2+x+1)$的值域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021669": { + "id": "021669", + "content": "函数$y=\\dfrac{\\cos x+1}{\\cos x-1}$的定义域是值域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021670": { + "id": "021670", + "content": "函数$y=k \\sin x+b$的最大值为 2 , 最小值为$-4$, 求$k$、$b$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021671": { + "id": "021671", + "content": "设$-\\dfrac{\\pi}{6} \\leq x \\leq \\dfrac{\\pi}{4}$, 求函数$y=\\log _2(1+\\sin x)+\\log _2(1-\\sin x)$的最大值和最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021672": { + "id": "021672", + "content": "已知函数$y=2+a^2-2 a \\sin x-\\cos ^2 x$的最小值为$f(a)$, $a$为实数, 求$f(a)$的表达式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021673": { + "id": "021673", + "content": "判断下列函数的奇偶性, 并说明理由.\\\\\n(1) $y=|\\sin x|$;\\\\\n(2) $y=3 \\sin x+1$;\\\\\n(3) $y=\\sin x+\\sin 2 x$;\\\\\n(4) $y=\\sin ^2 x+\\cos 2 x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021674": { + "id": "021674", + "content": "判断以下命题的真假:\\\\\n(1) 函数$y=\\cos (2 \\sin x)$是偶函数, \\blank{20};\\\\\n(2) 函数$y=|\\sin x|+x^2$是偶函数, \\blank{20};\\\\\n(3) 若$\\alpha$、$\\beta$是第一象限的角, 且$\\alpha>\\beta$, 则$\\sin \\alpha>\\sin \\beta$, \\blank{20};\\\\\n(4) 函数$y=\\sin (\\dfrac{7 \\pi}{2}+3 x)$是偶函数, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021675": { + "id": "021675", + "content": "若函数$y=\\sqrt{3} \\cos (3 x-\\theta)-\\sin (3 x-\\theta)$是奇函数, 则$\\theta$的一个可能值是\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$-\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{3}$}{$-\\dfrac{\\pi}{6}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021676": { + "id": "021676", + "content": "函数$f(x)$具有下列性质: \\textcircled{1} $f(x)$是偶函数; \\textcircled{2} 对任意$x \\in \\mathbf{R}$, 都有$f(\\dfrac{\\pi}{4}-x)=f(\\dfrac{\\pi}{4}+x)$, 则函数$f(x)$的解析式可以是\\blank{50}. (写出一个答案即可)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021677": { + "id": "021677", + "content": "已知函数$f(x)=a x^3-b \\sin x-1$. 若$f(5)=5$, 则$f(-5)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021678": { + "id": "021678", + "content": "写出下列各函数的奇偶性:\\\\\n(1) 函数$y=\\cos x+\\sin x$的奇偶性为\\blank{50};\\\\\n(2) 函数$y=\\sin (x+\\dfrac{\\pi}{4})+\\cos (x+\\dfrac{\\pi}{4})$的奇偶性为\\blank{50};\\\\\n(3) 函数$y=\\sin x(|\\sin x-3|-3)$的奇偶性为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021679": { + "id": "021679", + "content": "判断下列命题的真假:\\\\\n(1) 函数$y=|\\sin x|+|\\cos x|$的最小正周期为$\\pi$, \\blank{20};\\\\\n(2) 存在实数使$\\sin \\alpha-\\cos \\alpha=\\dfrac{3}{2}$成立, \\blank{20};\\\\\n(3) 存在实数使$\\sin \\alpha \\cos \\alpha=\\dfrac{1}{3}$成立, \\blank{20};\\\\\n(4) 函数$y=\\cos ^22 x-\\sin ^22 x$是偶函数, 且最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021680": { + "id": "021680", + "content": "已知函数$f(x)=1+\\sin (x+\\theta)+\\sqrt{3} \\cos (x+\\theta)$, 是否存在实数$\\theta$, 使$f(x)$是奇函数? 若存在, 求出所有这样的$\\theta$, 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021681": { + "id": "021681", + "content": "若$\\alpha$、$\\beta$是锐角, $\\cos \\alpha<\\cos \\beta$, 则$\\alpha$、$\\beta$的大小关系为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021682": { + "id": "021682", + "content": "判断下列命题的真假:\\\\\n(1) $y=\\sin x$在第一象限内为增函数, \\blank{20};\\\\\n(2) 直线$x=\\dfrac{\\pi}{8}$是$y=\\sin (2 x+\\dfrac{\\pi}{4})$图像的一条对称轴, \\blank{20};\\\\\n(3) 函数$y=\\cos x+\\sin x$的值域为$[-2,2]$, \\blank{20};\\\\\n(4) 函数$y=\\sqrt{2} \\sin 2 x \\cos 2 x$的最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021683": { + "id": "021683", + "content": "求函数$y=\\sin x$, $x \\in[-2 \\pi, 0]$的单调递减区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021684": { + "id": "021684", + "content": "求函数$y=\\sin 2 x$的单调递增区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021685": { + "id": "021685", + "content": "已知$0 \\leq x \\leq 2 \\pi$, 写出适合下列条件的角$x$的区间:\\\\\n(1) 角$x$的正弦函数、余弦函数都是增函数, \\blank{100};\\\\\n(2) 角$x$的正弦函数是减函数, 余弦函数是增函数, \\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021686": { + "id": "021686", + "content": "求下列函数的单调递增区间:\\\\\n(1) $y=1-\\sin (x-\\dfrac{3 \\pi}{4})$;\\\\\n(2) $y=\\sqrt{3} \\cos x-\\sin x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021687": { + "id": "021687", + "content": "求函数$y=2 \\sin (\\dfrac{\\pi}{6}-4 x)+3$的递增区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021688": { + "id": "021688", + "content": "求函数$y=\\cos ^2 x+2$的递减区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021689": { + "id": "021689", + "content": "若$\\alpha$、$\\beta$是锐角, 且$\\sin \\alpha<\\cos \\beta$, 证明或否定: $\\alpha+\\beta$是锐角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021690": { + "id": "021690", + "content": "函数$y=3 \\sin (\\dfrac{x}{2}+\\dfrac{\\pi}{3})$的周期为\\blank{50}、振幅为\\blank{50}、初相为\\blank{50}、频率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021691": { + "id": "021691", + "content": "函数$y=\\dfrac{4}{3} \\sin 2 x$的周期为\\blank{50}, $y$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021692": { + "id": "021692", + "content": "函数$y=b+a \\sin \\dfrac{x}{2}(a \\neq 0)$的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021693": { + "id": "021693", + "content": "已知函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$)的振幅是$3$, 最小正周期是$\\dfrac{2 \\pi}{7}$, 初相是$\\dfrac{\\pi}{6}$, 写出这个函数的解析式: \\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021694": { + "id": "021694", + "content": "作出下列函数在长度为一个周期的闭区间上的大致图像:\\\\\n(1) $y=\\sin (2 x+\\dfrac{\\pi}{6})$;\\\\\n(2) $y=2 \\sin \\dfrac{x}{2}$;\\\\\n(3) $y=\\sin x \\cdot \\cos x$;\\\\\n(4) $y=5 \\sin (2 x-\\dfrac{\\pi}{3})$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021695": { + "id": "021695", + "content": "若函数$f(x)=2 a+b \\sin x$的最大值为$3$, 最小值为$1$, 则函数$g(x)=-4 a \\sin \\dfrac{b x}{2}$的周期是\\blank{50}, 最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021696": { + "id": "021696", + "content": "已知函数$y=A \\sin (x+\\varphi)$($A>0$, $\\omega>0$, $0<\\varphi<\\pi$)在同一周期内, 当$x=\\dfrac{\\pi}{3}$时, 取到最大值$4$, 当$x=\\dfrac{4}{3} \\pi$时, 取到最小值$-4$. 求函数的解析式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021697": { + "id": "021697", + "content": "如图是某个定义在$\\mathbf{R}$上的函数$f(x)=A \\sin (\\omega x+\\varphi)+B$, ($A>0$, $\\omega>0$, $0<\\varphi<2 \\pi$)的一部分图像.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {pi/3}:pi] plot (\\x,{sqrt(3)/2+sqrt(3)/2*sin(3*\\x/pi*180-180)});\n\\draw [dashed] ({pi/3},0) -- ({pi/3},{sqrt(3)/2});\n\\draw [dashed] (pi,0) -- (pi,{sqrt(3)/2}) -- (0,{sqrt(3)/2});\n\\draw [dashed] ({pi/2},{sqrt(3)}) -- (0,{sqrt(3)});\n\\draw (pi,0) node [below] {$\\pi$};\n\\draw ({pi/3},0) node [below] {$\\dfrac \\pi 3$};\n\\draw (0,{sqrt(3)/2}) node [left] {$\\dfrac{\\sqrt{3}}2$};\n\\draw (0,{sqrt(3)}) node [left] {$\\sqrt{3}$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$f(x)$的解析式;\\\\\n(2) 求该函数的单调递增区间;\\\\\n(3) 求该函数的最大值及取最大值时自变量的取值集合.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021698": { + "id": "021698", + "content": "函数$y=4 \\sin \\dfrac{x}{2}$的对称轴为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021699": { + "id": "021699", + "content": "要得到函数$y=3 \\sin x$的图像, 只需要把函数$y=\\sin x$的图像上的对应点的坐标\\blank{30}(``伸长''或``缩短'')到原来的\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021700": { + "id": "021700", + "content": "函数$f(x)=\\dfrac{1}{3} \\sin 2 x$的图像可由$y=\\sin x$的图像上所有点的横坐标\\blank{30}(``伸长''或\n``缩短'')到原来的\\blank{50}, 纵坐标\\blank{30}(``伸长''或``缩短'')到原来的\\blank{50}得到.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021701": { + "id": "021701", + "content": "将函数$y=\\sin x$的图像向右平移$\\dfrac{\\pi}{3}$个单位, 再将所得图像上所有点的横坐标伸长到原来的$2$倍, 所得的曲线是$y=f(x)$的图像, 则$f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021702": { + "id": "021702", + "content": "将函数$y=\\sin x$的图像上所有点的横坐标伸长到原来的$2$倍, 再向右平移$\\dfrac{\\pi}{3}$个单位, 所得的曲线是$y=f(x)$的图像, 则$f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021703": { + "id": "021703", + "content": "将函数$y=f(x)$的图像上所有点的横坐标缩短到原来的$\\dfrac{1}{3}$倍, 再向右平移$\\dfrac{\\pi}{6}$个单位, 所得的曲线是$y=2 \\sin x$的图像, 则函数$f(x)$的解析式为$f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021704": { + "id": "021704", + "content": "函数$y=3 \\sin (\\dfrac{x}{2}+\\dfrac{\\pi}{3})$的所有垂直于$x$轴的对称轴是直线\\blank{50}, 对称中心是点\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021705": { + "id": "021705", + "content": "函数$y=3 \\sin (2 x+5 \\theta)$的图像关于$y$轴对称的充要条件是\\bracket{20}.\n\\fourch{$\\theta=\\dfrac{2 k \\pi}{5}+\\dfrac{\\pi}{10}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{2 k \\pi}{5}+\\dfrac{\\pi}{5}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{k \\pi}{5}+\\dfrac{\\pi}{10}$($k \\in\\mathbf{Z}$)}{$\\theta=\\dfrac{k \\pi}{5}+\\dfrac{\\pi}{5}$($k \\in\\mathbf{Z}$)}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021706": { + "id": "021706", + "content": "若将函数$y=\\dfrac{1}{2} \\sin (2 x-\\dfrac{\\pi}{4})$的图像向\\blank{20}(左、右)平移\\blank{50}个单位, 则可得到函数$y=\\dfrac{1}{2} \\sin 2 x$的图像.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021707": { + "id": "021707", + "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)的周期为$\\pi$, 图像的一个对称中心为$(\\dfrac{\\pi}{4}, 0)$. 将函数$f(x)$图像上所有点的横坐标伸长到原来的$2$倍(纵坐标不变), 再将所得到的图像向右平移$\\dfrac{\\pi}{2}$个单位长度后得到函数$g(x)$的图像, 求函数$f(x)$与$g(x)$的解析式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021708": { + "id": "021708", + "content": "已知函数$f(x)=\\sqrt{3} \\sin (\\omega x+\\varphi)-\\cos (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)为偶函数, 且函数$y=f(x)$图像的两相邻对称轴间的距离为$\\dfrac{\\pi}{2}$.\\\\\n(1) 求$f(\\dfrac{\\pi}{8})$的值;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{6}$个单位后, 再将得到的图像上各点的横坐标伸长到原来的$4$倍, 纵坐标不变, 得到函数$y=g(x)$的图像, 求$g(x)$的单调递减区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021709": { + "id": "021709", + "content": "写出下列各函数的最小正周期:\\\\\n(1) $y=\\tan \\dfrac{x}{2}$的最小正周期为\\blank{50};\\\\\n(2) $y=\\tan \\pi x$的最小正周期为\\blank{50};\\\\\n(3) $y=\\tan (2 x-\\dfrac{\\pi}{4})$的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021710": { + "id": "021710", + "content": "已知$0 \\leq x \\leq 2 \\pi$, 写出适合下列条件的角$x$的区间:\\\\\n(1) 角$x$的正弦函数, 正切函数都是增函数: \\blank{100};\\\\\n(2) 角$x$的余弦函数是减函数, 正切函数是增函数: \\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021711": { + "id": "021711", + "content": "判断下列函数的奇偶性, 并说明理由.\\\\\n(1) $f(x)=-2 \\tan 3 x$;\\\\\n(2) $f(x)=x \\tan x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021712": { + "id": "021712", + "content": "函数$y=\\tan ^2 x+4 \\tan x-1$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021713": { + "id": "021713", + "content": "比较下列各组数的大小 (填: ``$<$''``$>$''``$=$''), 不得使用计算器:\\\\\n(1) $\\tan \\dfrac{2 \\pi}{7}$\\blank{20}$\\tan \\dfrac{2 \\pi}{5}$;\\\\\n(2) $\\tan (-\\dfrac{2 \\pi}{7})$\\blank{20}$\\tan (-\\dfrac{2 \\pi}{5})$;\\\\\n(3) $\\cot 231^{\\circ}$\\blank{20}$\\cot 237^{\\circ}$;\\\\\n(4) $\\tan (k \\pi-\\dfrac{\\pi}{3})$\\blank{20}$\\tan (k \\pi+\\dfrac{\\pi}{3})$, $k \\in\\mathbf{Z}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021714": { + "id": "021714", + "content": "下列四组函数$f(x)$与$g(x)$表示同一函数的是\\blank{50}.\\\\\n\\textcircled{1}$f(x)=\\sin x$与$g(x)=\\tan x \\cos x$; \\textcircled{2}$f(x)=\\tan \\dfrac{x}{2}$与$g(x)=\\dfrac{1-\\cos x}{\\sin x}$; \\textcircled{3}$f(x)=\\tan \\dfrac{x}{2}$与$g(x)=\\dfrac{\\sin x}{1+\\cos x}$; \\textcircled{4}$f(x)=\\tan (x+\\dfrac{\\pi}{4})$与$g(x)=\\dfrac{1+\\tan x}{1-\\tan x}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021715": { + "id": "021715", + "content": "求函数$f(x)=\\tan (x+\\dfrac{\\pi}{6})$, $x \\in[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{3})$的最小值, 并指出使其取得最小值时$x$的所\n有值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021716": { + "id": "021716", + "content": "已知函数$f(x)=\\tan x-\\cot x$.\\\\\n(1) 求函数$y=f(x)$的定义域;\\\\\n(2) 求函数$y=f(x)$的单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021717": { + "id": "021717", + "content": "函数$y=\\tan (2 x+1)$的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021718": { + "id": "021718", + "content": "函数$y=\\tan (3 x+\\dfrac{\\pi}{4})$的单调递增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021719": { + "id": "021719", + "content": "已知$\\alpha$、$\\beta \\in(\\dfrac{\\pi}{2}, \\pi)$. 若$\\tan \\alpha<\\cot \\beta$, 则\\bracket{20}.\n\\fourch{$\\alpha<\\beta$}{$\\alpha+\\beta<\\dfrac{3 \\pi}{2}$}{$\\alpha>\\beta$}{$\\alpha+\\beta>\\dfrac{3 \\pi}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021720": { + "id": "021720", + "content": "求函数$y=4 \\tan (\\dfrac{x}{2}-\\dfrac{\\pi}{5})$的定义域和单调区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021721": { + "id": "021721", + "content": "求函数$y=4 \\tan (\\dfrac{x}{2}-\\dfrac{\\pi}{5})$的零点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021722": { + "id": "021722", + "content": "判断下列命题的真假:\\\\\n(1) 函数$y=\\tan |x|$既是偶函数又是周期函数, \\blank{20};\\\\\n(2)函数$y=\\tan x$既是奇函数, 又是增函数, \\blank{20};\\\\\n(3)函数$y=\\dfrac{4 \\tan x}{2-\\sec x}$的最小正周期为$\\dfrac{\\pi}{2}$, \\blank{20};\\\\\n(4) 函数$y=\\lg [\\cos x(1+\\sqrt{3} \\tan x)]$的最大值为$\\lg 2$, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021723": { + "id": "021723", + "content": "求函数$y=\\tan ^2 x+4 \\tan x-1$, $x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$的值域.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021724": { + "id": "021724", + "content": "在墙壁上挂有一个表面直径为$0.3$米的时钟. 已知时钟距离地面的最近点到地面距离为$2$米, 某位学生的眼睛与地面距离为$1.7$米. 求该学生观察时钟表面的最大张角的正切值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-3,0) --++ (0,1.7) coordinate (A) --++ (3,0.3) coordinate (B) (-3,1.7)--++ (3,0.6) coordinate (C);\n\\filldraw [pattern = north east lines] (-3.5,0) -- (0,0) -- (0,2.5) -- (0.5,2.5) -- (0.5,-0.5) -- (-3.5,-0.5) -- cycle;\n\\draw pic [draw] {angle = B--A--C};\n\\draw (-2.5,2) node {$\\theta$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021725": { + "id": "021725", + "content": "按要求, 分别以$A$、$B$、$C$为向量的起点, 在右图中画出以下向量. (图中每个小正方形的边长为$1$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale= 0.5]\n\\foreach \\i in {0,1,...,8}\n{\\draw [dashed] (0,\\i) -- (8,\\i) (\\i,0) -- (\\i,8);};\n\\filldraw (1,2) node [below left] {$A$} coordinate (A) circle (0.06);\n\\filldraw (7,3) node [below left] {$B$} coordinate (B) circle (0.06);\n\\filldraw (3,5) node [below left] {$C$} coordinate (C) circle (0.06);\n\\draw [->] (8.5,5) -- (8.5,7) node [right] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 正北方向, 且模为$2$的向量$\\overrightarrow{AE}$;\\\\\n(2) 长度为$2 \\sqrt{2}$, 方向为北偏西$45^{\\circ}$的向量$\\overrightarrow{BF}$;\\\\\n(3) (2)中$\\overrightarrow{BF}$向量的负向量$\\overrightarrow{CG}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021726": { + "id": "021726", + "content": "``$\\overrightarrow {a}=\\overrightarrow {b}$''是``$\\overrightarrow {a}\\parallel \\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021727": { + "id": "021727", + "content": "``$|\\overrightarrow {a}|=0$''是``$\\overrightarrow {a}=\\overrightarrow{0}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021728": { + "id": "021728", + "content": "已知非零向量$\\overrightarrow {a}$和$\\overrightarrow {b}$, ``$|\\overrightarrow {a}|=|\\overrightarrow {b}|$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021729": { + "id": "021729", + "content": "把平面上一切单位向量放置到共同的始点, 那么这些向量的终点所构成的图形是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021730": { + "id": "021730", + "content": "现有以下命题:\n\\textcircled{1}向量的模是一个正实数;\n\\textcircled{2}所有的单位向量都相等;\n\\textcircled{3}零向量与任意非零向量平行. 其中真命题的个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021731": { + "id": "021731", + "content": "如图, 在四边形$ABCD$中, $\\overrightarrow{CB}+\\overrightarrow{AD}+\\overrightarrow{BA}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0.2) node [right] {$D$} coordinate (D);\n\\draw (1,0.8) node [above] {$A$} coordinate (A);\n\\draw (1.3,-0.5) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021732": { + "id": "021732", + "content": "向量$(\\overrightarrow{AB}+\\overrightarrow{MB})+(\\overrightarrow{BO}+\\overrightarrow{BC})+\\overrightarrow{OM}$化简后等于\\bracket{20}.\n\\fourch{$\\overrightarrow {BC}$}{$\\overrightarrow {AB}$}{$\\overrightarrow{AC}$}{$\\overrightarrow{AM}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021733": { + "id": "021733", + "content": "判断下列命题的真假:\\\\\n(1) 长度相等的向量都相等, \\blank{20};\\\\\n(2) 若$\\overrightarrow {a}=\\overrightarrow {b}, \\overrightarrow {c}=\\overrightarrow {b}$, 则$\\overrightarrow {a}=\\overrightarrow {c}$, \\blank{20};\\\\\n(3) 若四边形$ABCD$是平行四边形, 则$\\overrightarrow{AB}=\\overrightarrow{CD}$, \\blank{20};\\\\\n(4) 已知$A,B,C,D$四点两两不重合, 若$\\overrightarrow{AB}=\\overrightarrow{DC}$, 则$|\\overrightarrow{AB}|=|\\overrightarrow{CD}|$且直线$AB\\parallel CD$, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021734": { + "id": "021734", + "content": "如图, 已知$D$、$E$、$F$分别是$\\triangle ABC$的$AB$、$BC$、$CA$边的中点. 以图中的点为始点和终点, 写出所有\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,1) node [right] {$C$} coordinate (C);\n\\draw (1,2) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [above left] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--cycle (D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 与$\\overrightarrow{AD}$相等的向量;\\\\\n(2) $\\overrightarrow{DE}$的负向量;\\\\\\\n(3) 与$\\overrightarrow{FE}$平行的非零向量.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021735": { + "id": "021735", + "content": "如图, 在$2 \\times 4$的矩形中, 起、终点都在小方格顶点、模与$|\\overrightarrow{AB}|$相等的向量共有几个?\n($\\overrightarrow{AB}$也算一个)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,...,4}\n{\\draw (\\i,0) -- (\\i,2);};\n\\foreach \\i in {0,1,2}\n{\\draw (0,\\i) -- (4,\\i);};\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,2) node [above] {$B$} coordinate (B);\n\\draw [->] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021736": { + "id": "021736", + "content": "如图, 在$3 \\times 4$的矩形中, 起、终点都在小方格顶点、模与$|\\overrightarrow{AB}|$相等的向量共有几个?\n($\\overrightarrow{AB}$也算一个)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\foreach \\i in {0,1,...,4}\n{\\draw (\\i,0) -- (\\i,3);};\n\\foreach \\i in {0,1,2,3}\n{\\draw (0,\\i) -- (4,\\i);};\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,3) node [above] {$B$} coordinate (B);\n\\draw [->] (A)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021737": { + "id": "021737", + "content": "已知菱形$ABCD$的边长为$2$, 求向量$\\overrightarrow{AB}-\\overrightarrow{CB}+\\overrightarrow{CD}$的模.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021738": { + "id": "021738", + "content": "如图, 已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$, 分别以$O_1,O_2$为始点, 作出向量$\\overrightarrow {a}+\\overrightarrow {c}-\\overrightarrow {b}$和$\\overrightarrow {a}+(\\overrightarrow {c}-\\overrightarrow {b})$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (1,0.7) node [midway, above] {$\\overline{a}$};\n\\draw [->] (1.2,0) -- (2,0) node [midway, above] {$\\overline{b}$};\n\\draw [->] (3,0) -- (2.4,0.6) node [midway, above] {$\\overline{c}$};\n\\filldraw (6,0) node [below] {$O_1$} coordinate (O_1) circle (0.03);\n\\filldraw (9,0) node [below] {$O_2$} coordinate (O_2) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021739": { + "id": "021739", + "content": "化简: $\\dfrac{1}{2}(2 \\overrightarrow {a}+8 \\overrightarrow {b})-(4 \\overrightarrow {a}-2 \\overrightarrow {b})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021740": { + "id": "021740", + "content": "化简: $3(\\overrightarrow {a}-2 \\overrightarrow {b}+\\overrightarrow {c})-4(-\\overrightarrow {a}-\\overrightarrow {b}+\\overrightarrow {c})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021741": { + "id": "021741", + "content": "判断题:(填``真''、``假'')\\\\\n(1) $0 \\overrightarrow {a}=0$, \\blank{20};\\\\\n(2) 对于实数$m$和向量$\\overrightarrow {a}$、$\\overrightarrow {b}$, 恒有$m(\\overrightarrow {a}-\\overrightarrow {b})=m \\overrightarrow {a}-m \\overrightarrow {b}$, \\blank{20};\\\\\n(3) 若$m \\overrightarrow {a}=m \\overrightarrow {b}(m \\in \\mathbf{R})$, 则$\\overrightarrow {a}=\\overrightarrow {b}$, \\blank{20};\\\\\n(4) 若$m \\overrightarrow {a}=n \\overrightarrow {a}$($m$、$n \\in \\mathbf{R}$且$\\overrightarrow {a} \\neq \\overrightarrow{0}$), 则$m=n$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021742": { + "id": "021742", + "content": "已知四边形$ABCD$是平行四边形, $\\overrightarrow{AC}=\\overrightarrow {a}$, $\\overrightarrow{BD}=\\overrightarrow {b}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow {AB}$、$\\overrightarrow {BC}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021743": { + "id": "021743", + "content": "已知$\\overrightarrow{AB}=-\\dfrac{3}{4} \\overrightarrow{BC}$, $\\overrightarrow{BC} \\neq \\overrightarrow{0}$, 记$\\overrightarrow{AC}=\\lambda \\overrightarrow{BA}$, 求实数$\\lambda$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021744": { + "id": "021744", + "content": "设$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个不平行的非零向量, $x(2 \\overrightarrow {a}+\\overrightarrow {b})+y(3 \\overrightarrow {a}-2 \\overrightarrow {b})=7 \\overrightarrow {a}$, 其中$x$、$y \\in \\mathbf{R}$, 求$x$、$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021745": { + "id": "021745", + "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个不平行的非零向量.\\\\\n(1) 若向量$\\overrightarrow{AB}=\\overrightarrow {a}-\\overrightarrow {b}, \\overrightarrow{BC}=2 \\overrightarrow {a}-8 \\overrightarrow {b}, \\overrightarrow {C} \\overrightarrow {D}=3 \\overrightarrow {a}+3 \\overrightarrow {b}$, 证明: $A$、$B$、$D$三点共线;\\\\\n(2) 若向量$m \\overrightarrow {a}-\\overrightarrow {b}$与$\\overrightarrow {a}-m \\overrightarrow {b}$平行, 求实数$m$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021746": { + "id": "021746", + "content": "梯形$ABCD$中, $AB\\parallel CD$, 且$AB=2CD, M$、$N$分别是$DC$、$AB$的中点, 已知$\\overrightarrow{AB}=\\overrightarrow {a}, \\overrightarrow{AD}=\\overrightarrow {b}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$分别表示$\\overrightarrow{DC}$、$\\overrightarrow{BC}$、$\\overrightarrow{MN}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021747": { + "id": "021747", + "content": "设$G$是$\\triangle ABC$的重心, $AB$、$BC$、$CA$的中点分别为$D$、$E$、$F$, 则$\\overrightarrow{GD}+\\overrightarrow{GE}+\\overrightarrow{GF}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021748": { + "id": "021748", + "content": "在平行四边形$ABCD$中, $AC$与$BD$交于点$O$, $E$是线段$OD$的中点, $AE$的延长线与$CD$交于点$F$, 若$\\overrightarrow{AC}=\\overrightarrow {a}$, $\\overrightarrow{BD}=\\overrightarrow {b}$, 则$\\overrightarrow{AF}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021749": { + "id": "021749", + "content": "如果$|\\overrightarrow {a}|=2,|\\overrightarrow {b}|=\\dfrac{1}{2}$, $\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$60^{\\circ}$, 那么$\\overrightarrow {a}$与$\\overrightarrow {b}$的数量积等于\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{1}{4}$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021750": { + "id": "021750", + "content": "已知$|\\overrightarrow {b}|=3$. 如果$\\overrightarrow {a}$在$\\overrightarrow {b}$方向上的投影是$\\dfrac{1}{2}\\overrightarrow{b}$, 那么$\\overrightarrow {a} \\cdot \\overrightarrow {b}$为\\bracket{20}.\n\\fourch{$3$}{$\\dfrac{9}{2}$}{$2$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021751": { + "id": "021751", + "content": "如果$\\overrightarrow {a}$、$\\overrightarrow {b}$是两个非零向量, 那么``$(\\overrightarrow {a}+\\overrightarrow {b})^2=\\overrightarrow {a}^2+\\overrightarrow {b}^2$''是``$\\overrightarrow {a} \\perp \\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021752": { + "id": "021752", + "content": "已知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$. 若$\\overrightarrow {a}$与$\\overrightarrow {b}$之间的夹角为$\\dfrac{\\pi}{3}$, 则向量$\\overrightarrow {m}=\\overrightarrow {a}-\\overrightarrow {b}$的模为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021753": { + "id": "021753", + "content": "在$\\triangle ABC$中, 已知$\\angle A$、$\\angle B$、$\\angle C$的对边长分别为$a$、$b$、$c$. 若$a=3$, $b=1$, $\\angle C=30^{\\circ}$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021754": { + "id": "021754", + "content": "若$|\\overrightarrow{AB}|=|\\overrightarrow{AC}|=6$, $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=18$, 则$\\triangle ABC$的形状是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021755": { + "id": "021755", + "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=\\sqrt{2}$. 若$(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {a}$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021756": { + "id": "021756", + "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=3$, $|2 \\overrightarrow {a}+\\overrightarrow {b}|=\\sqrt{7}$, 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021757": { + "id": "021757", + "content": "在$\\triangle ABC$中, 若$BC=5$, $AC=4$, $\\angle C=45^{\\circ}$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021758": { + "id": "021758", + "content": "已知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$. 若$(\\overrightarrow {a}+k \\overrightarrow {b}) \\perp(\\overrightarrow {a}-3 \\overrightarrow {b})$, $\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则实数$k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021759": { + "id": "021759", + "content": "已知$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$. 若$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$120^{\\circ}$, 则$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的投影为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021760": { + "id": "021760", + "content": "下列等式中一定成立的是\\bracket{20}.\n\\twoch{$|\\overrightarrow {a}| \\cdot \\overrightarrow {a}=(\\overrightarrow {a})^2$}{$\\overrightarrow {a}(\\overrightarrow {b} \\cdot \\overrightarrow {b})=\\overrightarrow {a}(\\overrightarrow {b})^2$}{$\\overrightarrow {a}(\\overrightarrow {a} \\cdot \\overrightarrow {b})=(\\overrightarrow {a})^2 \\cdot \\overrightarrow {b}$}{$(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2=(\\overrightarrow {a})^2(\\overrightarrow {b})^2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021761": { + "id": "021761", + "content": "在边长为$1$的正三角形$ABC$中, 若$\\overrightarrow{BC}=\\overrightarrow {a}$, $\\overrightarrow{CA}=\\overrightarrow {b}$, $\\overrightarrow{AB}=\\overrightarrow {c}$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {c} \\cdot \\overrightarrow {a}=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{2}$}{$-\\dfrac{3}{2}$}{$0$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021762": { + "id": "021762", + "content": "若$\\overrightarrow{AB} \\cdot \\overrightarrow{BC}+\\overrightarrow{AB}^2=0$, 则$\\triangle ABC$为\\bracket{20}.\n\\fourch{直角三角形}{钝角三角形}{锐角三角形}{等腰直角三角形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021763": { + "id": "021763", + "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}+\\overrightarrow {b}|=8$, $|\\overrightarrow {a}-\\overrightarrow {b}|=6$, 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021764": { + "id": "021764", + "content": "已知$|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=5$, $|2 a-3 \\overrightarrow {b}|=7$, 求$|2 \\overrightarrow {b}-\\overrightarrow{3 a}|$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021765": { + "id": "021765", + "content": "已知平面上的三点$A$、$B, C$与$BD$满足$(\\overrightarrow{BC} \\cdot \\overrightarrow{CA}): (\\overrightarrow{AB} \\cdot \\overrightarrow{CA}): (\\overrightarrow{BC} \\cdot \\overrightarrow{AB})=3: 4: 5$, 则\n$A$、$B$、$C$这三点的关系是\\bracket{20}.\n\\twoch{是一个直角三角形的三个顶点}{是一个钝角三角形的三个顶点}{是一个锐角三角形的三个顶点}{三点共线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021766": { + "id": "021766", + "content": "已知$O$、$N$、$P$在$\\triangle ABC$所在的平面内, 且$|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=|\\overrightarrow{OC}|$, $\\overrightarrow{NA}+\\overrightarrow{NB}+\\overrightarrow{NC}=\\overrightarrow{0}$, $\\overrightarrow{PA} \\cdot \\overrightarrow{PB}=\\overrightarrow{PB} \\cdot \\overrightarrow{PC}=\\overrightarrow{PA} \\cdot \\overrightarrow{PC}$, 则点$O$、$N$、$P$依次是$\\triangle ABC$的\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021767": { + "id": "021767", + "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$为非零向量, 且$(\\overrightarrow {a}-4 \\overrightarrow {b}) \\perp(7 \\overrightarrow {a}-2 \\overrightarrow {b})$, $(\\overrightarrow {a}+3 \\overrightarrow {b}) \\perp(7 \\overrightarrow {a}-5 \\overrightarrow {b})$. 求向量$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021768": { + "id": "021768", + "content": "已知$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 且$|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=3$, $|\\overrightarrow {c}|=5$. 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {c} \\cdot \\overrightarrow {a}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021769": { + "id": "021769", + "content": "已知向量$\\overrightarrow{AB}$、$\\overrightarrow{AC}$的夹角为$120^{\\circ}$, 且$|\\overrightarrow{AB}|=3$, $|\\overrightarrow{AC}|=2$, 若$\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\overrightarrow{AC}$, 且\n$\\overrightarrow{AP} \\perp \\overrightarrow{BC}$, 求实数$\\lambda$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021770": { + "id": "021770", + "content": "在平行四边形$ABCD$中, $AD=1$, $\\angle BAD=60^{\\circ}$, $E$为$CD$的中点, 若$\\overrightarrow{AD} \\cdot \\overrightarrow{BE}=1$, 求$AB$的长.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021771": { + "id": "021771", + "content": "已知$\\overrightarrow {a} \\overrightarrow {b}$是两个不共线的向量, 若它们起点相同, $\\overrightarrow {a}$、$\\dfrac{\\overrightarrow {b}}{2}$、$t(\\overrightarrow {a}+\\overrightarrow {b})$三向量的终点在一条直线上, 求实数$t$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021772": { + "id": "021772", + "content": "已知平面上三个向量$\\overrightarrow {a},\\overrightarrow {b},\\overrightarrow {c}$的模均为 1 , 它们相互之间的夹角均为$\\dfrac{2 \\pi}{3}$.\\\\\n(1) 求证: $(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {c}$;\\\\\n(2) 若$|k \\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}|>1$, $k \\in \\mathbf{R}$, 求$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021773": { + "id": "021773", + "content": "在平行四边形$ABCD$中, $A=\\dfrac{\\pi}{3}$, 边$AB$、$AD$的长分别为$2$和$1$, 若$M$、$N$分别是边$BC$、$CD$上的点, 且$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 求$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021774": { + "id": "021774", + "content": "在等腰$\\triangle ABC$中, 两条腰上的中线$BD$, $CE$互相垂直, 求$\\angle A$的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021775": { + "id": "021775", + "content": "如图, 设$\\overrightarrow {a}=\\overrightarrow{OA}=3 \\overrightarrow{OC}$, $\\overrightarrow {b}=\\overrightarrow{OB}=4 \\overrightarrow{OD}$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$不共线, $AD$、$BC$交于点$P$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow{OP}$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (4,1) node [below] {$A$} coordinate (A);\n\\draw (2,3) node [right] {$B$} coordinate (B);\n\\draw ($(O)!{1/3}!(A)$) node [below] {$C$} coordinate (C);\n\\draw ($(O)!{1/4}!(B)$) node [left] {$D$} coordinate (D);\n\\draw (B)--(O)--(A)(B)--(C)(A)--(D);\n\\draw ($(C)!{2/11}!(B)$) node [above right] {$P$} coordinate (P);\n\\draw (O)--(P);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021776": { + "id": "021776", + "content": "用坐标表示下列向量:\\\\\n(1) $-\\overrightarrow {i}=$\\blank{50};\\\\\n(2) $2 \\overrightarrow {i}+\\dfrac{1}{2} \\overrightarrow {j}=$\\blank{50};\\\\\n(3) 与$x$轴平行、模为$2$的向量的坐标为\\blank{50};\\\\\n(4) 向东南方向前进$3$个单位长度, 对应向量的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021777": { + "id": "021777", + "content": "已知向量$\\overrightarrow {a}=(-3,1)$、$\\overrightarrow {b}=(-1,-3)$.\\\\\n(1) 求$|3 \\overrightarrow {a}-\\overrightarrow {b}|$;\\\\\n(2) 求$3 \\overrightarrow {a}-\\overrightarrow {b}$的单位向量的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021778": { + "id": "021778", + "content": "已知$\\overrightarrow {a}=(x, 3)$, $\\overrightarrow {b}=(1, y)$, $\\overrightarrow {a}-2 \\overrightarrow {b}=(2,5)$, 求实数$x$、$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021779": { + "id": "021779", + "content": "若$\\overrightarrow {a}=3 \\overrightarrow {i}-4 \\overrightarrow {j}$, 则$\\overrightarrow {a}$的单位向量是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021780": { + "id": "021780", + "content": "若$\\overrightarrow {a}=\\overrightarrow {i}-3 \\overrightarrow {j}, \\overrightarrow {b}=-2 \\overrightarrow {i}+2 \\overrightarrow {j}$, 则$2 \\overrightarrow {a}-\\overrightarrow {b}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021781": { + "id": "021781", + "content": "若向量$\\overrightarrow{AB}=(2-x) \\overrightarrow {i}+(1-x) \\overrightarrow {j}$的坐标所表示的点在第四象限内, 则$x$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021782": { + "id": "021782", + "content": "化简$\\overrightarrow{AB}-\\overrightarrow{AC}-\\overrightarrow{BC}$等于\\bracket{20}.\n\\fourch{$2 \\overrightarrow{BC}$}{$2 \\overrightarrow{AC}$}{$-2 \\overrightarrow{BC}$}{$\\overrightarrow{0}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021783": { + "id": "021783", + "content": "若$A(0,-3)$, $B(3,3)$, $C(x, 1)$, 且$\\overrightarrow{AB}\\parallel 2 \\overrightarrow{BC}$, 则$x=$\\bracket{20}.\n\\fourch{$2$}{$1$}{$-1$}{$-2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021784": { + "id": "021784", + "content": "已知$\\overrightarrow {m}=(-1, a)$, $\\overrightarrow {n}=(2 a, 4)$. 若$\\overrightarrow {p}=\\overrightarrow {m}+\\dfrac{1}{2} \\overrightarrow {n}$, 且$|\\overrightarrow {p}|=3$, 则实数$a$的值等于\\bracket{20}.\n\\fourch{$1$或$2$}{$-2$或$1$}{$\\pm 1$}{$\\pm 2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021785": { + "id": "021785", + "content": "已知$O(0,0)$、$A(1,2)$、$B(4,6)$、$C(3,4)$, 求证: 四边形$OABC$为平行四边形.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021786": { + "id": "021786", + "content": "已知$\\overrightarrow {a}=(\\sin 30^{\\circ}, \\cos 30^{\\circ})$, $\\overrightarrow {b}=(\\cos 30^{\\circ}, \\sin 30^{\\circ}), \\overrightarrow {m}=(1,1)$. 是否存在非零实数$\\lambda$、$\\mu$, 使得$\\overrightarrow {m}\\parallel(\\lambda \\overrightarrow {a}+\\mu \\overrightarrow {b})$? 若存在, 分别求出$\\lambda$、$\\mu$的值; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021787": { + "id": "021787", + "content": "已知点$O$、$A$、$B$的坐标分别为$(0,0)$、$(1,2)$、$(4,5)$, 且$\\overrightarrow{OP}=\\overrightarrow{OA}+t \\overrightarrow{AB}$.\\\\\n(1) 当$t$分别为何值时, 点$P$在$x$轴上? 点$P$在$y$轴上? 点$P$在第二象限?\\\\\n(2) 四边形$OABP$能否为平行四边形? 若能, 求出相应的$t$值; 若不能, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021788": { + "id": "021788", + "content": "在四边形$ABCD$中, $\\overrightarrow{AB}=(6,1)$, $\\overrightarrow{BC}=(x,y)$, $\\overrightarrow{CD}=(-2,-3)$.\n(1) 若$\\overrightarrow{BC}\\parallel \\overrightarrow{DA}$, 试求$x$、$y$满足的关系式;\\\\\n(2) 满足(1)的同时, 又有$\\overrightarrow{AC} \\perp \\overrightarrow{BD}$, 求$x$、$y$的值及四边形$ABCD$的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021789": { + "id": "021789", + "content": "已知$M$、$N$两点的坐标分别是$(3,-2)$、$(-5,-1)$, 且$\\overrightarrow{MP}=\\overrightarrow{PN}$, 求点$P$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021790": { + "id": "021790", + "content": "已知$A$、$B$、$C$三点的坐标分别是$(0,1)$、$(1,2)$、$(3,4)$, 求证: $A$、$B$、$C$三点共线.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021791": { + "id": "021791", + "content": "已知$A$、$B$两点的坐标分别是$(3,-1)$、$(-4,-2)$, $P$是直线$AB$上的点, 根据下列条件求点$P$的坐标:\\\\\n(1) $\\overrightarrow{AP}=2 \\overrightarrow{PB}$;\\\\\n(2) $2 \\overrightarrow{AP}=3 \\overrightarrow{BP}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021792": { + "id": "021792", + "content": "在向量$\\overrightarrow {a}=(2,-6)$, $\\overrightarrow {b}=(-5,-3)$, $\\overrightarrow {c}=(-1,3)$, $\\overrightarrow {d}=(\\dfrac{5}{3}, 1)$中, 相互平行的向量是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021793": { + "id": "021793", + "content": "$\\overrightarrow {b}=(3,4)$的单位向量$\\overrightarrow {b_0}$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021794": { + "id": "021794", + "content": "已知$P_1(-3,4)$, $P_2(x, y)$, 若线段$P_1P_2$的中点坐标为$(1,-1)$, 则点$P_2$的坐标为\\bracket{20}.\n\\fourch{$(-5,6)$}{$(5,-6)$}{$(-5,-6)$}{$(5,6)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021795": { + "id": "021795", + "content": "已知在$\\triangle ABC$中, $M$、$N$是$AB$上的三等分点 ($M$靠近$A$点、$N$靠近$B$点), $P$、$Q$是$AC$\n上的三等分点($P$靠近$A$点、$Q$靠近$C$点), 且$\\overrightarrow{AB}=\\overrightarrow {c}$, $\\overrightarrow{AC}=\\overrightarrow {b}$. 试用$\\overrightarrow {b}$、$\\overrightarrow {c}$表示$\\overrightarrow{PM}$与$\\overrightarrow{QB}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021796": { + "id": "021796", + "content": "已知平面上$A$、$B$、$C$三点的坐标分别为$(-2,1)$、$(-1,3)$、$(3,4)$, 且$A$、$B$、$C$、$D$这四点可以构成平行四边形的四个顶点, 求点$D$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021797": { + "id": "021797", + "content": "已知$A$、$B$两点的坐标分别是$(1,2)$、$(-1,4)$, $\\overrightarrow{AC}=\\dfrac{1}{2} \\overrightarrow{AB}$, $\\overrightarrow{AD}=2 \\overrightarrow{BC}$.\\\\\n(1) 求$C$、$D$的坐标;\\\\\n(2) 求向量$\\overrightarrow{BD}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021798": { + "id": "021798", + "content": "已知向量$\\overrightarrow {a}=(-2,3)$, 点$A(2,-1)$, 向量$\\overrightarrow{AB}$与$\\overrightarrow {a}$平行, 且$|\\overrightarrow{AB}|=2 \\sqrt{13}$, 求向量$\\overrightarrow{OB}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021799": { + "id": "021799", + "content": "正方形$ABCD$边长为$1$, 点$P$在线段$AC$上运动, 求$\\overrightarrow{AP} \\cdot(\\overrightarrow{PB}+\\overrightarrow{PD})$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021800": { + "id": "021800", + "content": "已知向量$\\overrightarrow {a}=(3,4)$与$\\overrightarrow {b}=(5,-12)$, 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$与$|\\overrightarrow {a}-\\overrightarrow {b}|$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021801": { + "id": "021801", + "content": "已知向量$\\overrightarrow {a}=(5,12)$与$\\overrightarrow {b}=(4,6)$, 求$\\overrightarrow {a}+\\overrightarrow {b}$与$2 \\overrightarrow {a}-3 \\overrightarrow {b}$的夹角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021802": { + "id": "021802", + "content": "已知$A$、$B$两点的坐标分别是$(1,2)$、$(4,-1)$. 能否在$y$轴上找到一点$C$, 使$\\angle ACB=90^{\\circ}$? 若不能, 说明理由; 若能, 求点$C$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021803": { + "id": "021803", + "content": "若$|\\overrightarrow {a}|=2 \\sqrt{13}$, $\\overrightarrow {b}=(-2,3)$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$\\overrightarrow {a}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021804": { + "id": "021804", + "content": "若$\\overrightarrow {a}=(-1,-3)$, $\\overrightarrow {b}=(2,-5)$, 且$\\overrightarrow {a} \\cdot \\overrightarrow {c}=5$, $\\overrightarrow {b} \\cdot \\overrightarrow {c}=1$, 则$\\overrightarrow {c}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021805": { + "id": "021805", + "content": "已知$A(2,-1)$, $B(-1,3)$. 若点$C$在$y$轴上, 且满足$\\overrightarrow{AC} \\cdot \\overrightarrow{BC}=3$, 则点$C$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021806": { + "id": "021806", + "content": "若$\\overrightarrow {a}=(x, \\log _2 x)$, $\\overrightarrow {b}=(-1,-\\dfrac{1}{2})$, 且$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021807": { + "id": "021807", + "content": "若$|\\overrightarrow {a}-\\overrightarrow {b}|^2=41-20 \\sqrt{3}$, $|\\overrightarrow {a}|=4$, $|\\overrightarrow {b}|=5$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}=$\\bracket{20}.\n\\fourch{$10 \\sqrt{3}$}{$-10 \\sqrt{3}$}{$10 \\sqrt{2}$}{$10$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021808": { + "id": "021808", + "content": "已知$|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=4$, 则$\\overrightarrow {a}+\\dfrac{3}{4} \\overrightarrow {b}$与$\\overrightarrow {a}-\\dfrac{3}{4} \\overrightarrow {b}$的位置关系为\\bracket{20}.\n\\fourch{平行}{垂直}{夹角为$60^{\\circ}$}{既不平行也不垂直}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021809": { + "id": "021809", + "content": "已知$\\triangle ABC$的三个顶点$A(2-1)$、$B(3,2)$、$C(-3,-1)$, 边$BC$上的高为$AD$, 求点$D$和向量$\\overrightarrow{AD}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021810": { + "id": "021810", + "content": "已知$O$为原点, $\\overrightarrow{OA}=(3,1)$, $\\overrightarrow{OB}=(-1,2)$, 且$\\overrightarrow{OC} \\perp \\overrightarrow{OB}$, $\\overrightarrow{BC}\\parallel \\overrightarrow{OA}$, $\\overrightarrow{OD}+\\overrightarrow{OA}=\\overrightarrow{OC}$, \n求向量$\\overrightarrow{OD}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021811": { + "id": "021811", + "content": "已知位置向量$\\overrightarrow {a}=(2,2)$、$\\overrightarrow {b}=(-3,3)$、$\\overrightarrow {c}=(-1,0)$的终点分别为$A$、$B$、$C$, 试判断$\\triangle ABC$的形状.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021812": { + "id": "021812", + "content": "已知向量$\\overrightarrow {a}=(x, 2)$与$\\overrightarrow {b}=(-3,-5)$, $\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为钝角, 求$x$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021813": { + "id": "021813", + "content": "在$\\triangle ABC$中, $M$是$BC$的中点, $AM=1$, 点$P$在$AM$上且满足$\\overrightarrow{AP}=2 \\overrightarrow{PM}$, 求$\\overrightarrow{PA} \\cdot(\\overrightarrow{PB}+\\overrightarrow{PC})$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021814": { + "id": "021814", + "content": "已知向量$\\overrightarrow {a}=(2,3)$与$\\overrightarrow {b}=(4,-1+y)$, 且$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 求实数$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021815": { + "id": "021815", + "content": "已知$P_1(3,2)$, $P_2(-8,3)$, $P(\\dfrac{1}{2}, y)$.若$\\overrightarrow{P_1P}=\\lambda \\overrightarrow{PP_2}$, 求实数$\\lambda$和$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021816": { + "id": "021816", + "content": "已知平面上点$A, B$的坐标分别是$(0,-1)$、$(-5,1)$, 点$C$在直线$AB$上, 且$|\\overrightarrow{AC}|=\\dfrac{2}{3}|\\overrightarrow{CB}|$, 求点$C$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021817": { + "id": "021817", + "content": "已知平面内三个点$A$、$B$、$C$坐标分别是$(-1,2)$、$(10,-1)$、$(-4,3)$, $G$是已知平面内的一点, 且$\\overrightarrow{AG}+\\overrightarrow{BG}+\\overrightarrow{CG}=\\overrightarrow{0}$, 求点$G$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021818": { + "id": "021818", + "content": "在$\\triangle ABC$中, 已知$A$、$B$两点的坐标分别为$(2,1)$、$(-3,4)$, $\\triangle ABC$的重心坐标为$(-\\dfrac{2}{3}, \\dfrac{4}{3})$, 求点$C$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021819": { + "id": "021819", + "content": "以原点$O$和点$A(5,2)$为顶点作等腰直角$\\triangle ABO$, 使$\\angle B=90^{\\circ}$, 求向量$\\overrightarrow{OB}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021820": { + "id": "021820", + "content": "如图, $A$、$M$、$B$在同一直线上, 且$\\overrightarrow{AM}=\\dfrac{1}{3} \\overrightarrow{AB}$, 设$\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OM}=\\overrightarrow {c}$, 用\n$\\overrightarrow {a}$、$\\overrightarrow {b}$表示$\\overrightarrow {c}$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (3,0) node [above] {$B$} coordinate (B);\n\\draw (2,-2) node [below] {$O$} coordinate (O);\n\\draw ($(A)!{1/3}!(B)$) node [above] {$M$} coordinate (M);\n\\draw [->] (A)--(M);\n\\draw [->] (M)--(B);\n\\draw [->] (O)--(M);\n\\draw (O)--(A)(O)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021821": { + "id": "021821", + "content": "已知$\\overrightarrow{e_1}$与$\\overrightarrow{e_2}$不平行, 实数$x$、$y$满足: $3 x \\overrightarrow{e_1}+(10-y) \\overrightarrow{e_2}=(4 y+7) \\overrightarrow{e_1}+2 x \\overrightarrow{e_2}$, 求$5 x-3 y$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021822": { + "id": "021822", + "content": "如图, 已知$\\triangle AOB$的重心为$G$, 过点$G$的直线与边$OA$、$OB$分别交于点$P$、$Q$, 设$\\overrightarrow{OP}=h \\overrightarrow{OA}$, $\\overrightarrow{OQ}=k \\overrightarrow{OB}$, $\\triangle AOB$、$\\triangle POQ$的面积分别为$S$、$T$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (2,2.5) node [above] {$O$} coordinate (O);\n\\filldraw ($1/3*(A)+1/3*(B)+1/3*(O)$) node [below] {$G$} coordinate (G) circle (0.03);\n\\draw (0,0.2) coordinate (S);\n\\draw ($(S)!2!(G)$) coordinate (T);\n\\path [name path = PQ, draw] (S)--(T);\n\\path [name path = AOB, draw] (A)--(O)--(B)--cycle;\n\\path [name intersections = {of = PQ and AOB, by = {P,Q}}];\n\\draw (P) node [above] {$P$};\n\\draw (Q) node [above] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $\\dfrac{1}{h}+\\dfrac{1}{k}=3$;\\\\\n(2) 求证: $\\dfrac{4}{9} S \\leq T \\leq \\dfrac{1}{2} S$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021823": { + "id": "021823", + "content": "两块斜边长相等的直角三角板如图拼在一起, 若$\\overrightarrow{AD}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 求$x, y$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$C$} coordinate (C);\n\\draw (B) ++ (45:{sqrt(6)}) node [right] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw (C)--(A)--(B)--(D)--(E)(C)--(B);\n\\draw pic [draw,scale = 0.5] {right angle = B--A--C};\n\\draw pic [draw,scale = 0.5,\"$45^\\circ$\", angle eccentricity = 2.5] {angle = A--C--B};\n\\draw pic [draw,scale = 0.5,\"$60^\\circ$\", angle eccentricity = 2.5] {angle = B--E--D};\n\\draw (A)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021824": { + "id": "021824", + "content": "用向量法证明: 平行四边形的对角线的平方和等于四边的平方和.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021825": { + "id": "021825", + "content": "用向量法证明: 在正方形$ABCD$中, $E$、$F$分别为边$AB$、$BC$的中点, 有$AF \\perp DE$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021826": { + "id": "021826", + "content": "已知三角形$\\triangle ABC$的三个顶点坐标分别为$A(2,5)$、$B(3,1)$、$C(-1,4)$, 求该三角形面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021827": { + "id": "021827", + "content": "已知$\\triangle ABC$的重心为$G$, $O$为三角形外的任一点, $\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$表示$\\overrightarrow{OG}$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021828": { + "id": "021828", + "content": "试用向量法解决下列问题:\\\\\n(1) 求函数$y=2 \\sqrt{3-x}+5 \\sqrt{x+8}$的最大值;\\\\\n(2) 求函数$y=\\sqrt{x^2-2 x+2}+\\sqrt{x^2-10 x+34}$的最小值;\\\\\n(3) 求函数$y=\\sqrt{x^2-12 x+52}-\\sqrt{x^2-4 x+5}$的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021829": { + "id": "021829", + "content": "$O$是平面上一点, $A$、$B$、$C$是平面上不共线的三个点, 动点$P$满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\lambda(\\dfrac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|}+\\dfrac{\\overrightarrow{AC}}{|\\overrightarrow{AC}|})$, $\\lambda \\in[0,+\\infty)$, 则$P$点轨迹一定通过$\\triangle ABC$的\\bracket{20}.\n\\fourch{外心}{内心}{重心}{垂心}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021830": { + "id": "021830", + "content": "计算:\n(1) $(\\dfrac{1}{4}-\\dfrac{13}{5} \\mathrm{i})+(\\dfrac{2}{3}+\\dfrac{5}{2} \\mathrm{i})=$\\blank{50};\\\\\n(2) $-1-\\mathrm{i}-(2+3 \\mathrm{i})+4 \\mathrm{i}=$\\blank{50};\\\\\n(3) $[(a+b)+(a-b) \\mathrm{i}]-[(a-b)-(a+b) \\mathrm{i}]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021831": { + "id": "021831", + "content": "计算:\n(1) $(\\dfrac{\\sqrt{2}}{2}-\\dfrac{\\sqrt{2}}{2} \\mathrm{i})^2=$\\blank{50};\\\\\n(2) $(2-5 \\mathrm{i})(1+2 \\mathrm{i})(12+\\mathrm{i})=$\\blank{50};\\\\\n(3) $(\\dfrac{1-\\mathrm{i}}{1+\\mathrm{i}})^3=$\\blank{50};\\\\\n(4) $\\dfrac{-2+2 \\sqrt{3} \\mathrm{i}}{(\\sqrt{3}+\\mathrm{i})^2}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021832": { + "id": "021832", + "content": "计算:\\\\\n(1) $\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\cdots+\\mathrm{i}^{200}$;\\\\\n(2) $(1+\\mathrm{i})^{10}-(1-\\mathrm{i})^{10}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021833": { + "id": "021833", + "content": "已知$z=1-\\mathrm{i}$, 求$\\dfrac{z^2-z+1}{z^2+z+1}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021834": { + "id": "021834", + "content": "已知$z_1=5+10 \\mathrm{i}$, $z_2=3-4 \\mathrm{i}$, 复数$z$满足$\\dfrac{1}{z}=\\dfrac{1}{z_1}+\\dfrac{1}{z_2}$, 求$z$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021835": { + "id": "021835", + "content": "若复数$z=x^2-y^2-7+(x-y-3) \\mathrm{i}$等于$-2 \\mathrm{i}$, 求实数$x$、$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021836": { + "id": "021836", + "content": "已知实数$x$、$y$满足$\\dfrac{x}{1-\\mathrm{i}}+\\dfrac{y}{1-2 \\mathrm{i}}=\\dfrac{5}{1-3 \\mathrm{i}}$, 求$x$、$y$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021837": { + "id": "021837", + "content": "从很久以前开始, 我们先后学习了整数、有理数、实数, 到了高中, 我们分别用记号$\\mathbf{Z}, \\mathbf{Q}, \\mathbf{R}$表示相应的数的集合, 现在, 我们学习了复数集合, 并用记号$\\mathbf{C}$表示, 用集合中的真子集的关系表示这四个集合的关系为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021838": { + "id": "021838", + "content": "若复数$z=a+b \\mathrm{i}$($a, b \\in \\mathbf{R}$)是虚数, 则$a, b$应满足的条件是\\bracket{20}.\n\\fourch{$a=0$且$b \\neq 0$}{$a \\neq 0$}{$a \\neq 0$且$b \\neq 0$}{$b \\neq 0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021839": { + "id": "021839", + "content": "设$z_1=a+b \\mathrm{i}$($a, b \\in \\mathbf{R}$), $z_2=c+d\\mathrm{i}$($c, d \\in \\mathbf{R}$), 则``$a=c$''是``$z_1=z_2$''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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021840": { + "id": "021840", + "content": "下列有关复数的描述中, 正确的是\\bracket{20}.\n\\twoch{$\\mathrm{i}$是$-1$的一个平方根}{$-2 \\mathrm{i}<-\\mathrm{i}$}{$b \\mathrm{i}(b \\in \\mathbf{R})$表示纯虚数}{若$z=3-4 \\mathrm{i}$, 则$\\mathrm{Im} z=4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021841": { + "id": "021841", + "content": "复数$z$与$\\overline {z}$在复平面$xOy$上所对应的点关于\\blank{50}对称.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021842": { + "id": "021842", + "content": "当且仅当$m \\in$\\blank{50}时, $(m^2+3 m-4)+(m^2+5 m-6) i$($m \\in \\mathbf{R}$)是纯虚数.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021843": { + "id": "021843", + "content": "已知复数$(3 m+2 n-5)+(-m+4 n+7) \\mathrm{i}$是纯虚数, 复数$(2 m-n-1)+(m+n+1) \\mathrm{i}$是实数, 求实数$m, n$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021844": { + "id": "021844", + "content": "判断是否存在实数$m$, 使复数$z=m^2-2 m-15+\\dfrac{m^2+5 m+6}{m^2-25} \\mathrm{i}$分别满足下列条件. 若存在, 求出$m$的值; 若不存在, 请说明理由.\\\\\n(1) $z$是实数;\\\\\n(2) $z$是虚数;\\\\\n(3) $z$是纯虚数;\\\\\n(4) $z$是零.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021845": { + "id": "021845", + "content": "设$z_1=2 a(a-1)+a^2 \\mathrm{i}$, $z_2=(a-1)+a(a-1) \\mathrm{i}$, 其中$a \\in \\mathbf{R}$. 若$\\overline{z_1}=z_2$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021846": { + "id": "021846", + "content": "求分别满足下列各等式的实数$x$与$y$的值.\\\\\n(1) $(x+y)-x y \\mathrm{i}=-5+24 \\mathrm{i}$;\\\\\n(2) $2 x^2-5 x+2+(y^2+y-2) \\mathrm{i}=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021847": { + "id": "021847", + "content": "设$z$是复数, 求证: ``$z$是纯虚数''的一个充要条件是``$z+\\overline {z}=0$且$z \\neq 0$''.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021848": { + "id": "021848", + "content": "下列命题中, 是真命题的命題的序号是\\blank{50}.\\\\\n\\textcircled{1} 在复平面上, 表示实数的点都在实轴上, 表示纯虚数的点都在虚轴上;\\\\\n\\textcircled{2} 在复平面上, 表示虚数的点都落在四个象限内;\\\\\n\\textcircled{3} 复数的模表示该复数在复平面上所对应的点到原点的距离.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021849": { + "id": "021849", + "content": "复数$-4+3 \\mathrm{i}$、$4-\\mathrm{i}$在复平面上对应点的象限分别为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021850": { + "id": "021850", + "content": "复数$-3 \\mathrm{i}$、$-4-\\mathrm{i}$在复平面上对应点的坐标分别为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021851": { + "id": "021851", + "content": "复数$-4+3 \\mathrm{i}$、$-6 \\mathrm{i}$、$2-\\mathrm{i}$的模分别为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021852": { + "id": "021852", + "content": "若$\\overrightarrow{OA}=(5,-1)$, $\\overrightarrow{OB}=(3,2)$, 则$\\overrightarrow{AB}$在复平面上所对应的复数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021853": { + "id": "021853", + "content": "复平面上向量$\\overrightarrow{OC}$, $\\overrightarrow{CD}$对应的复数分别为$-1-2 \\mathrm{i}$, $2-\\mathrm{i}$, 则$D$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021854": { + "id": "021854", + "content": "在复平面上, 平行于$y$轴的非零向量所对应的复数的集合是\\bracket{20}.\n\\twoch{实数集}{虚数集}{纯虚数集}{实数集与纯虚数集的并集}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021855": { + "id": "021855", + "content": "在复平面上, 复数$z$对应点$Z$落在以原点为圆心的单位圆上, 下列复数中, 其对应点总落在以原点为圆心, 半径为$2$的圆上的是\\blank{50}(填上正确的序号).\\\\\n\\textcircled{1} $1+z$; \\textcircled{2} $2 z$; \\textcircled{3} $\\dfrac{2}{z}$; \\textcircled{4} $z+\\overline {z}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021856": { + "id": "021856", + "content": "求实数$m$取何值时, 复数$z=(m^2-8 m+15)+(m^2-m-6) \\mathrm{i}$在复平面上所对应的点$Z$分别满足下列条件.\\\\\n(1) 点$Z$在实轴上;\\\\\n(2) 点$Z$在虚轴上;\\\\\n(3) 点$Z$在第四象限.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021857": { + "id": "021857", + "content": "设复数$z_1=m+8 \\mathrm{i}$, $m \\in \\mathbf{R}$, $z_2+\\mathrm{i} z_1=0$, 在复平面$xOy$上, $z_1, z_2$所对应的点分别为$Z_1, Z_2$.\\\\\n(1) 用$m$表示$z_2$;\\\\\n(2) 求$\\angle Z_1OZ_2$;\\\\\n(3) 若三角形$Z_1OZ_2$的面积为$50$, 求$m$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021858": { + "id": "021858", + "content": "已知复数$z_1=-4+3 \\mathrm{i}$, $z_2=4 \\mathrm{i}$. 设复数$z_k$($k=1,2,3$)在复平面$xOy$上所对应的点$Z_k$.\\\\\n(1) 若四边形$OZ_1Z_2Z_3$是一个平行四边形, 求$z_3$;\\\\\n(2) 若$O, Z_1, Z_2, Z_3$是一个平行四边形的四个顶点, 求$z_3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021859": { + "id": "021859", + "content": "已知复数$z=m+(2 m-1) \\mathrm{i}$, $m \\in \\mathbf{R}$. 设$z$在复平面上对应的点为$Z$.\\\\\n(1) 若点$Z$到原点的距离为$2$, 求$m$的值;\\\\\n(2) 问: 无论$m$取何值, 点$Z$总不落在第几象限? 为什么?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021860": { + "id": "021860", + "content": "若复数$z$满足$z-3+\\mathrm{i}=5-\\mathrm{i}$, 则$|\\mathrm{i} z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021861": { + "id": "021861", + "content": "若复数$z$满足$z+\\dfrac{2}{z}=0$, 则$|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021862": { + "id": "021862", + "content": "若复数$z$满足$|z|=3$, $\\mathrm{Re} z=2$, 则$\\mathrm{Im} z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021863": { + "id": "021863", + "content": "计算: $|\\mathrm{i} \\cdot \\mathrm{i}^3 \\cdot \\mathrm{i}^5 \\cdot \\cdots \\cdot \\mathrm{i}^{2021} \\cdot \\mathrm{i}^{2022}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021864": { + "id": "021864", + "content": "计算: $|(\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}})^{2022}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021865": { + "id": "021865", + "content": "计算: $|(1+\\mathrm{i})^2(\\mathrm{i}-2 \\sqrt{2})^3|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021866": { + "id": "021866", + "content": "计算: $|\\dfrac{(\\sqrt{5}-2 \\mathrm{i})(1+\\sqrt{3} \\mathrm{i})^2}{\\sqrt{13}+\\sqrt{23} \\mathrm{i}}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021867": { + "id": "021867", + "content": "复数$z$分别满足下列条件, 复数$z$在复平面上对应点$Z$, 画出点$Z$的集合对应的图形.\\\\\n(1) $|z|=3$;\\\\\n(2) $1<\\mathrm{Re} z<2$且$1<\\mathrm{Im} z<2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021868": { + "id": "021868", + "content": "已知复数$z_1=2+\\mathrm{i}$, $|z_2|=5$, $z_2 z_1$是负实数, 求复数$z_2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021869": { + "id": "021869", + "content": "已知复数$z=m+(3 m-5) \\mathrm{i}$, $m \\in \\mathbf{R}$.\\\\\n(1) 若$|z| \\leq 5$, 求$m$的取值范围;\\\\\n(2) 求$z$的模的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021870": { + "id": "021870", + "content": "``复数$z_1, z_2$互为共轭''是``$|z_1|=|z_2|$''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": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021871": { + "id": "021871", + "content": "设$z_1, z_2, z$是复数, 有下列五个命题: \\textcircled{1} 若$z_1^2+z_2^2=0$, 则$z_1=z_2=0$.\n\\textcircled{2} 若$|z_1|=|z_2|$, 则$z_1^2=z_2^2$. \\textcircled{3} 若$z_1^2=z_2^2$, 则$|z_1|=|z_2|$.\n\\textcircled{4} 若$|z|=1$, 则$z=1$或$z=-1$或$z=\\mathrm{i}$或$z=-\\mathrm{i}$. \\textcircled{5} 若$|z+\\mathrm{i}|=|z-\\mathrm{i}|$, 则$z$为实数. 其中, 正确的命题的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021872": { + "id": "021872", + "content": "设$z_1, z_2$是复数, 有下列五个条件:\\\\\n\\textcircled{1} $|z_1-\\overline{z_2}|=0$; \\textcircled{2} $|z_1|=|z_2|$; \\textcircled{3} $z_1=z_2$; \\textcircled{4} $\\overline {z}_1=z_2$; \\textcircled{5} $z_1^2=\\overline{z_2^2}$.\\\\\n则可以成为$z_1=\\overline{z_2}$的必要非充分条件的是\\blank{50}(填正确的条件的序号).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021873": { + "id": "021873", + "content": "已知$z \\in \\mathbf{C}$, $z+\\dfrac{4}{z} \\in \\mathbf{R}$, $|z-2|=2$, 求$z$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021874": { + "id": "021874", + "content": "复数$-1$的平方根是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021875": { + "id": "021875", + "content": "设$a<0$. 则$a$的平方根是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021876": { + "id": "021876", + "content": "若复数$z$满足$z^2 \\in[0,+\\infty)$, 则$z$的集合为\\bracket{20}.\n\\twoch{实数集}{虚数集}{纯虚数集}{实数集与纯虚数集的并集}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021877": { + "id": "021877", + "content": "一个非零复数的平方根恰有两个, 求下列复数的平方根.\\\\\n(1) $5\\mathrm{i}$;\\\\\n(2) $5-12\\mathrm{i}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021878": { + "id": "021878", + "content": "已知复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$)满足$|z-1|=1$.\\\\\n(1) 求$x, y$满足的关系式;\\\\\n(2) 求复数$z$的模的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021879": { + "id": "021879", + "content": "设$z$为复数. 若$\\dfrac{z-3}{z+3}$为纯虚数, 求$|z|$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021880": { + "id": "021880", + "content": "已知复数$z_1, z_2$满足$|z_1|=|\\overline{z_2}|=1$, 且$z_1+z_2=-\\mathrm{i}$, 求$z_1, z_2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021881": { + "id": "021881", + "content": "已知复数$z$的平方等于$8+6 \\mathrm{i}$, 求$z^3-16 z-\\dfrac{100}{z}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021882": { + "id": "021882", + "content": "设$z_1, z_2 \\in \\mathbf{C}$, 且$|z_1|=|z_2|=1$, $|z_1+z_2|=\\sqrt{2}$, 求$|z_1-z_2|$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021883": { + "id": "021883", + "content": "下列命题中, 是真命题的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 在复数范围内, 方程$a x^2+b x+c=0$($a, b, c \\in \\mathbf{R}$, $a \\neq 0$)总有两个根;\\\\\n\\textcircled{2} 设$p, q \\in \\mathbf{C}$. 若$3+2 \\mathrm{i}$是方程$x^2+p x+q=0$的一个根, 则该方程的另一个根是$3-2 \\mathrm{i}$;\\\\\n\\textcircled{3} 设$p, q \\in \\mathbf{C}$. 若方程$x^2+p x+q=0$有两个共轭虚数根, 则$p, q$均为实数.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021884": { + "id": "021884", + "content": "若实系数方程$x^2+b x+c=0$的一个根是$\\dfrac{1}{3}+\\dfrac{\\sqrt{7}}{3} \\mathrm{i}$, 则实数数对$(b, c)$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021885": { + "id": "021885", + "content": "若复系数方程$x^2+m x+n=0$的根为$-\\mathrm{i}$与$1+\\mathrm{i}$, 则复数数对$(m, n)$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021886": { + "id": "021886", + "content": "设$a \\in \\mathbf{R}$. 若方程$x^2-a x+2=0$有虚数根$z$, 则$|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021887": { + "id": "021887", + "content": "设$m \\in \\mathbf{R}$. 若关于$x$的方程$x^2+m x+m=0$有虚数根, 则$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021888": { + "id": "021888", + "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+(k+2 \\mathrm{i}) x+2+k \\mathrm{i}=0$有实数根, 则$k$值的集合为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021889": { + "id": "021889", + "content": "在复数集中解下列一元二次方程:\\\\\n(1) $4 x^2+9=0$;\\\\\n(2) $x^2+4 x+12=0$;\\\\\n(3) $x^2+x+1=0$;\\\\\n(4) $x^2-(1+\\mathrm{i}) x+\\mathrm{i}=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021890": { + "id": "021890", + "content": "在复数集中分解因式:\\\\\n(1) $x^2+2 x y+3 y^2$;\\\\\n(2) $x^3+y^3$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021891": { + "id": "021891", + "content": "已知两个数的和等于$\\sqrt{3}$, 它们的积等于$3$, 求这两个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021892": { + "id": "021892", + "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+3=0$有两个虚根$\\alpha$和$\\beta$, 且$|\\alpha-\\beta|=2 \\sqrt{2}$, 求$k$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021893": { + "id": "021893", + "content": "设$p \\in \\mathbf{R}$, 关于$x$的方程$2 x^2-p x+p=0$的两个根为$x_1$和$x_2$.\\\\\n(1) 若$p=3$, 求$x_1^2+x_2^2$的值;\\\\\n(2) 若$x_1^2+x_2^2=3$, 求$p$的值;\\\\\n(3) 若$|x_1|+|x_2|=3$, 求$p$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021894": { + "id": "021894", + "content": "在复数集中解下列方程:\\\\\n(1) $(x-3)(x-5)+2=0$;\\\\\n(2) $x^4-16=0$;\\\\\n(3) $\\dfrac{1}{x+3}-\\dfrac{1}{x}=1$;\\\\\n(4) $x^2-(2+2 \\mathrm{i}) x+4 \\mathrm{i}=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021895": { + "id": "021895", + "content": "在复数集中分解因式:\\\\\n(1) $a^2+2 a b+b^2+c^2$;\\\\\n(2) $x^4+3 x^2-10$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021896": { + "id": "021896", + "content": "设$p \\in \\mathbf{R}$. 若关于$x$的方程$x^2+(4+\\mathrm{i}) x+3+p \\mathrm{i}=0$有实数根, 求$p$的值, 并解这个方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021897": { + "id": "021897", + "content": "复数$w$满足$w-4=(3-2 w) \\mathrm{i}$, $z=\\dfrac{5}{w}+|w-2|$. 写出一个以$z$为根的实系数一元二次方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021898": { + "id": "021898", + "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+5=0$有两根$\\alpha$和$\\beta$, 且$|\\alpha-\\beta|=2$, 求$k$的值的集合.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021899": { + "id": "021899", + "content": "已知虚数$z_1, z_2$满足$z_1^2=z_2$.\\\\\n(1) 设$z_1, z_2$是一个实系数一元二次方程的两个根, 求$z_1, z_2$;\\\\\n(2) 设$z_1=1+m \\mathrm{i}$, $m>0$, $|z_1| \\leq \\sqrt{2}$, 复数$\\omega=z_2+3$, 求$|\\omega|$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021900": { + "id": "021900", + "content": "设$k \\in \\mathbf{R}$. 若关于$x$的方程$x^2+k x+k^2-2 k=0$有一个模为$1$的虚根, 求$k$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021901": { + "id": "021901", + "content": "把下列复数化为三角形式:(辐角用主值)\\\\\n(1) $-2 \\mathrm{i}=$\\blank{100};\\\\\n(2) $-1=$\\blank{100};\\\\\n(3) $-1+\\mathrm{i}=$\\blank{100};\\\\\n(4) $-1-\\sqrt{3} \\mathrm{i}=$\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021902": { + "id": "021902", + "content": "把下列复数化为三角形式:(辐角用主值)\\\\\n(1) $3-4 \\mathrm{i}$;\\\\\n(2) $\\cos \\dfrac{\\pi}{5}-\\mathrm{i} \\sin \\dfrac{\\pi}{5}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021903": { + "id": "021903", + "content": "设复数$\\sqrt{3}-\\mathrm{i}$在复平面上对应的向量为$\\overrightarrow{OA}$, 将$\\overrightarrow{OA}$绕原点$O$逆时针旋转$120^{\\circ}$, 且模缩小到原来$\\dfrac{1}{2}$得到向量$\\overrightarrow{OB}$, 求点$B$对应的复数的代数形式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021904": { + "id": "021904", + "content": "设复数$z_1=2+\\mathrm{i}$与复数$z_2=1+3 \\mathrm{i}$.\\\\\n(1) 将$\\dfrac{z_2}{z_1}$表示为复数三角形式;\\\\\n(2) 在复平面$xOy$上, $z_1$、$z_2$所对应的点为$Z_1$、$Z_2$, 求$\\angle Z_1OZ_2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021905": { + "id": "021905", + "content": "计算(结果用代数形式).\\\\\n(1) $\\sqrt{6}(\\cos \\dfrac{\\pi}{8}+\\mathrm{i} \\sin \\dfrac{\\pi}{8}) \\cdot \\sqrt{10}(\\cos \\dfrac{\\pi}{12}+\\mathrm{i} \\sin \\dfrac{\\pi}{12}) \\cdot \\sqrt{15}(\\cos \\dfrac{\\pi}{24}+\\mathrm{i} \\sin \\dfrac{\\pi}{24})$;\\\\\n(2) $\\dfrac{\\cos \\dfrac{8\\pi}{15}+\\mathrm{i}\\sin\\dfrac{8\\pi}{15}}{\\sqrt{3}(\\cos \\dfrac\\pi 5+\\mathrm{i}\\sin\\dfrac\\pi 5)}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021906": { + "id": "021906", + "content": "计算(结果用代数形式).\\\\\n(1) $(1+\\mathrm{i})^{20}$;\\\\\n(2) $(-\\sqrt{3}+\\mathrm{i})^{11}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021907": { + "id": "021907", + "content": "将向量$\\overrightarrow{OP}$按逆时针方向旋转$\\dfrac{2 \\pi}{3}$后再将其模伸长至原来的$3$倍, 所得向量$\\overrightarrow{OQ}$对应的复数$-6 \\mathrm{i}$, 则向量$\\overrightarrow{OP}$对应的复数为\\blank{50}(结果用代数形式表示).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021908": { + "id": "021908", + "content": "设$m, n$是正整数, 求满足$(\\sqrt{3}+\\mathrm{i})^m=(1+\\mathrm{i})^n$的$m+n$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021909": { + "id": "021909", + "content": "判断下列命题的真假:\\\\\n(1) 若一条直线的倾斜角为$90^{\\circ}$, 则这条直线与$y$轴平行, \\blank{20};\\\\\n(2) 若直线经过$(x_1, y_1),(x_2, y_2)$, 则该直线的斜率$k=\\dfrac{y_2-y_1}{x_2-x_1}$, \\blank{20};\\\\\n(3) 直线的倾斜角的变化范围是$[0, \\dfrac{\\pi}{2}) \\cup(\\dfrac{\\pi}{2}, \\pi)$, \\blank{20};\\\\\n(4) 倾斜角为$0$的直线是$x$轴, \\blank{20}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021910": { + "id": "021910", + "content": "经过两个点$P(2,1)$, $Q(0,2)$的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021911": { + "id": "021911", + "content": "经过两个点$P(2,1)$, $Q(a,-2)$(其中实常数$a>2$)的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021912": { + "id": "021912", + "content": "经过两个点$P(2,1)$, $Q(a,-2)$(其中实常数$a<2$)的直线的斜率是\\blank{50}, 倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021913": { + "id": "021913", + "content": "若直线$l$的倾斜角的取值范围是$[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$, 则$l$的斜率的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021914": { + "id": "021914", + "content": "已知直线$l$的倾斜角的取值范围是$[0, \\arccos \\dfrac{3}{5}] \\cup[\\pi-\\arccos \\dfrac{3}{5}, \\pi)$, 则$l$的斜率的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021915": { + "id": "021915", + "content": "若直线$l$经过$P(0,0)$, $Q(\\cos \\dfrac{7 \\pi}{5}, \\sin \\dfrac{7 \\pi}{5})$, 则直线$l$的倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021916": { + "id": "021916", + "content": "若直线$l$经过$P(0,0)$, $Q(\\sin \\dfrac{7 \\pi}{5}, \\cos \\dfrac{7 \\pi}{5})$, 则直线$l$的倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021917": { + "id": "021917", + "content": "已知斜率为$2$的直线过点$(2,2)$和$(x, 3)$, 则实数$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021918": { + "id": "021918", + "content": "经过点$P(2,0)$的直线$l$的倾斜角为$60^{\\circ}$. 若$l$绕$P$沿顺时针方向转过$90^{\\circ}$后所得直线$l'$的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021919": { + "id": "021919", + "content": "经过点$P(2,0)$的直线$l$的倾斜角为$30^{\\circ}$. 若$l$绕$P$沿顺时针方向转过$60^{\\circ}$后所得直线$l'$的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021920": { + "id": "021920", + "content": "已知$3$个不同点$A(2, a+1)$、$B(a+2,2 a+3)$、$C(-4,-a)$在同一条直线上, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021921": { + "id": "021921", + "content": "平面直角坐标系中有一个边长为$1$的正方形$OABC$, 其中点$O$为坐标原点, 点$A$、$C$分别在$x$轴和$y$轴上.\\\\\n(1) 若点$B$在第一象限, 求直线$OB$和$AC$的斜率;\\\\\n(2) 若点$B$不在第一象限, 求直线$AC$的斜率的所有可能值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021922": { + "id": "021922", + "content": "设一次函数$y=k x+b$($k \\neq 0$)的图像表示直线$l$.\\\\\n(1) 利用符号$\\arctan$, 试着用$k$表示$l$的倾斜角$\\alpha$;\\\\\n(2) 设$l$经过点$A(x_1, y_1)$、$B(x_2, y_2)$, 向量$\\overrightarrow {d}=(1, k)$, 求证: $\\overrightarrow {d}\\parallel \\overrightarrow{AB}$;\\\\\n(3) 设点$D$为$(0, b)$, 点$P$为$(x_0, y_0)$($x_0 \\neq 0$). 若直线$DP$的斜率为$k$, 求证: $y_0=k x_0+b$;\\\\\n(4) 设点$D$为$(0, b)$, 点$P$为$(x_0, y_0)$($x_0 \\neq 0$). 若$y_0=k x_0+b$, 求证: 直线$DP$的斜率为$k$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021923": { + "id": "021923", + "content": "在平面直角坐标系内, 非零向量$\\overrightarrow {d}=(a, b)$在直线$l$上, $a \\neq 0$.\\\\\n(1) 求证: $\\overrightarrow {d}$在$x$轴上投影为$(a, 0)$;\\\\\n(2) 求证: $l$的斜率为$\\dfrac{b}{a}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021924": { + "id": "021924", + "content": "已知直线$l$倾斜角$\\theta$的取值范围是$(\\dfrac{\\pi}{2}, \\dfrac{2 \\pi}{3})$, 则$l$的斜率的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021925": { + "id": "021925", + "content": "已知三个点$A(-1,-1)$、$B(1,1), C(0,3)$. 若点$M$是线段$BC$上任意一点, 则直线$AM$的斜率$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021926": { + "id": "021926", + "content": "设$\\theta \\in[\\pi, 2 \\pi)$. 若直线$l$经过$P(0,0)$, $Q(\\cos \\theta, \\sin \\theta)$, 则用$\\theta$表示$l$的倾斜角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021927": { + "id": "021927", + "content": "已知直线$l$经过$P(0,0)$, $Q(\\cos \\theta, \\sin \\theta)$. 若$\\theta$的取值范围为$[-\\dfrac{\\pi}{3}, 0)$, 则$l$的倾斜角取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021928": { + "id": "021928", + "content": "设$\\theta \\in(\\pi, \\dfrac{3 \\pi}{2})$. 若直线$l$经过$P(0,0)$, $Q(\\sin \\theta, \\cos \\theta)$, 则用$\\theta$表示$l$的倾斜角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021929": { + "id": "021929", + "content": "已知$A(0,1)$、$B(a, 2)$、$C(2 a, 4)$是某三角形的三个顶点, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021930": { + "id": "021930", + "content": "若直线$l$的倾斜角$\\alpha$满足$-2 \\leq \\tan \\alpha \\leq 1$, 则$\\alpha$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021931": { + "id": "021931", + "content": "已知两点$A(\\tan ^2 \\alpha, 0)$、$B(1,2 \\tan \\alpha)$.\\\\\n(1) 求证: $A$、$B$是两个不同的点, 并用$\\alpha$表示$|AB|$;\\\\\n(2) 若$0<\\alpha<\\dfrac{\\pi}{2}$, 求用$\\alpha$表示直线$AB$的倾斜角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021932": { + "id": "021932", + "content": "已知$A(a+2, a)$、$B(1,-a)$、$C(a-4, a-1)$是某三角形的三个顶点, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021933": { + "id": "021933", + "content": "已知直线$l$经过点$P(1, a)$, 其中常数$a>0$, 且直线$l$与$x$轴、$y$轴分别交于$A$、$B$两个不同的点. 若$P$恰为$AB$中点, 求直线$l$的斜率和倾斜角\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021934": { + "id": "021934", + "content": "已知直线$l_1$经过点$P$, 斜率为$k_1$($k_1\\ne 1$). 若$l_1$绕着点$P$沿逆时针方向转过$45^\\circ$后与直线$l$重合, 则$l$的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021935": { + "id": "021935", + "content": "已知直线$l_2$经过点$P$, 斜率为$k_2$($k_2\\ne -\\sqrt{3}$). 若直线$l$绕着点$P$沿逆时针方向转过$30^\\circ$后与直线$l_2$重合, 则$l$的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021936": { + "id": "021936", + "content": "已知三个点$A(-1,-1)$、$B(1,1), C(0,3)$, 点$M$在直线$BC$上, 设直线$AM$的斜率为$k$. 求证: 当且仅当$k \\in[1,4]$时, 存在$\\lambda \\in[0,1]$, 使得$\\overrightarrow{BM}=\\lambda \\overrightarrow{BC}$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021937": { + "id": "021937", + "content": "设直线$l$在平面直角坐标系内的斜率是$k$, 非零向量$\\overrightarrow {d}$在$l$上, 试用$k$、$|\\overrightarrow {d}|$表示$\\overrightarrow {d}$在$x$轴上投影与数量投影.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021938": { + "id": "021938", + "content": "若直线$l$经过两点$A(1,-2)$、倾斜角为$\\dfrac{\\pi}{3}$, 则直线$l$的点斜式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021939": { + "id": "021939", + "content": "若直线$l$经过两点$A(1,-2)$、$B(3,1)$, 则直线$l$的两点式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021940": { + "id": "021940", + "content": "若直线$l$在$x$轴上的截距是$-2$, $y$轴上的截距是$5$, 则直线$l$的斜截式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021941": { + "id": "021941", + "content": "已知直线$l: y+1=\\dfrac{2}{3}(x-2)$, 则它在$x$轴上的截距是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021942": { + "id": "021942", + "content": "已知直线$l$的倾斜角为$\\alpha$, $\\sin \\alpha=\\dfrac{3}{5}$, 且经过点$P(3,5)$, 则直线$l$的点斜式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021943": { + "id": "021943", + "content": "过点$M(-2,3)$, 且垂直于$x$轴的直线的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021944": { + "id": "021944", + "content": "已知直线$l: y=k x+2$经过点$(1,-3)$, 则$l$的倾斜角的大小是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021945": { + "id": "021945", + "content": "若直线$x-2 y+c=0$与两坐标轴围成的三角形的面积不大于$3$, 则实数$c$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021946": { + "id": "021946", + "content": "已知$\\triangle ABC$的两个顶点的坐标分别是$A(2,2)$、$B(3,0)$, 此三角形的重心坐标为$(3,1)$.\\\\\n(1) 求此三角形的三边所在直线的方程;\\\\\n(2) 求此三角形的三条中线所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021947": { + "id": "021947", + "content": "已知平行四边形$ABCD$中, 三个顶点的坐标分别为$A(1,2)$、$B(3,4)$、$C(2,6)$, 求$AD$与$CD$边所在的直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021948": { + "id": "021948", + "content": "已知梯形$ABCD$的三个顶点的坐标分别为$A(2,3)$、$B(-2,1)$、$C(4,5)$, 求此梯形中位线所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021949": { + "id": "021949", + "content": "已知直线$l$过点$(1,2)$, 且$M(2,3)$、$N(4,-5)$两点到直线$l$的距离相等, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021950": { + "id": "021950", + "content": "直线$l: y=k x+b$($k, b \\in \\mathbf{R}$)与线段$AB$相交, 其中$A(4,2)$, $B(1,5)$.\\\\\n(1) 当$k=1$时, 求$l$的取值范围;\\\\\n(2) 当$b=-1$时, 求$l$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021951": { + "id": "021951", + "content": "若直线$l$过点$(2,-3)$, 它的一个法向量为$\\overrightarrow {n}=(3,4)$, 则直线$l$的点法式方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021952": { + "id": "021952", + "content": "已知点$A(2,1)$、$B(5,3)$, 则$AB$的垂直平分线的点法式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021953": { + "id": "021953", + "content": "若直线$l: (2-m) x+m y+3=0$的纵截距为$3$, 则$m=$\\blank{50}; 若直线$l$的倾斜角为$\\dfrac{\\pi}{3}$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021954": { + "id": "021954", + "content": "已知直线$l_1$与直线$l_2: 2 x-3 y+4=0$有相同法向量, 且直线$l_1$在$y$轴上的截距为$-2$, 则直线$l_1$的一般式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021955": { + "id": "021955", + "content": "求过点$M(1,-2)$, 且与两坐标轴围成等腰直角三角形的直线$l$的一般式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021956": { + "id": "021956", + "content": "直线$l: 2 x-y-4=0$围绕它与$x$轴的交点$M$逆时针方向旋转$45^{\\circ}$, 则得到的直线的一般式方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021957": { + "id": "021957", + "content": "直线$l$的方程是$3 x-4 y+5=0$, 则$l$的一个法向量是\\bracket{20} .\n\\fourch{$(3,4)$}{$(-4,3)$}{$(3,-4)$}{$(4,3)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021958": { + "id": "021958", + "content": "直线$x=-2$的一个法向量$\\overrightarrow {n}$的坐标是\\bracket{20}.\n\\fourch{$(4,0)$}{$(0,3)$}{$(-4,-2)$}{$(0,-2)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021959": { + "id": "021959", + "content": "直线$x-a y+2=0(a<0)$的倾斜角是\\bracket{20}.\n\\fourch{$\\arctan \\dfrac{1}{a}$}{$-\\arctan \\dfrac{1}{a}$}{$\\pi-\\arctan \\dfrac{1}{a}$}{$\\pi+\\arctan \\dfrac{1}{a}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021960": { + "id": "021960", + "content": "已知四边形$ABCD$是平行四边形, $AB$边所在直线的方程是$x+y-1=0$, $AD$边所在直线的方程是$3 x-y+4=0$, 顶点$C$的坐标是$(3,3)$, 求这个平行四边形其他两条边所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021961": { + "id": "021961", + "content": "已知$\\triangle ABC$的三个顶点的坐标分别$A(4,0)$、$B(6,7)$、$C(0,3)$, 求此三角形的三条高所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021962": { + "id": "021962", + "content": "求证: 直线$2 x+(1-\\cos 2 \\theta) y-\\sin \\theta=0$($\\theta \\in \\mathbf{R}$, 且$\\theta$不是$\\pi$的整数倍)与两坐标轴围成的图形面积是定值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021963": { + "id": "021963", + "content": "已知直线$l_1: 3 k x-(k+2) y+6=0$, 直线$l_2: k x+(2 k-3) y+2=0$, 若这两条直线的倾斜角互补, 求$k$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021964": { + "id": "021964", + "content": "已知直线$l: (a-1) x+(3-2 a) y+a+1=0$.\\\\\n(1) 若直线$l$的斜率$k \\in[-1,2]$, 求实数$a$的取值范围;\\\\\n(2) 证明: 对任意实数$a$, 直线$l$都经过一个确定的点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021965": { + "id": "021965", + "content": "已知向量$\\overrightarrow {n}=(5,-1)$是直线$l$的一个法向量, 在下列条件下分别求直线$l$的方程:\n(1) 在$x$轴、$y$轴上的截距之和为$4$;\\\\\n(2) 与$x$轴、$y$轴围成的三角形面积为$20$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021966": { + "id": "021966", + "content": "已知直线$l_1: a x-2 y-1=0$和直线$l_2: 6 x-4 y-b=0$. 若直线$l_1$与直线$l_2$平行, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021967": { + "id": "021967", + "content": "已知直线$l_1: m x+y-m-1=0$和直线$l_2: x+m y-2 m=0$. 当且仅当$m=$\\blank{50}时, $l_1$平行于$l_2$; 当且仅当$m=$\\blank{50}时, $l_1$与$l_2$重合, 当且仅当$m \\in$\\blank{50}时.$l_1$与$l_2$相交.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021968": { + "id": "021968", + "content": "已知过点$A(-2, m)$, $B(m, 4)$的直线$l_1$与直线$l_2: 2 x+y-1=0$平行, 则实数$m$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021969": { + "id": "021969", + "content": "``直线$l_1$与直线$l_2$平行''是``直线$l_1$与直线$l_2$的斜率相等''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021970": { + "id": "021970", + "content": "直线$3 x-2 y+m=0$和直线$6 x-4 y+5=0$的位置关系为\\bracket{20}.\n\\fourch{平行}{平行或重合}{相交}{相交、平行和重合都有可能}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021971": { + "id": "021971", + "content": "若直线$l_1: a x+2 y+6=0$与直线$l_2: x+(a-1) y+a^2-1=0$平行, 则$a$的值为\\bracket{20}.\n\\fourch{$-1$或$2$}{$-1$}{$2$}{$\\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021972": { + "id": "021972", + "content": "已知集合$A=\\{(x, y) | 2 x-a(a+1) y-1=0\\}$, $B=\\{(x, y) | a x-y+1=0\\}$, 且$A \\cap B=\\varnothing$, 求实数$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021973": { + "id": "021973", + "content": "已知四边形$ABCD$的四个顶点分别为$A(-1,2)$、$B(3,4)$、$C(3,2)$、$D(1,1)$, 求证: 四边形$ABCD$是梯形.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021974": { + "id": "021974", + "content": "已知直线$l$经过直线$l_1: 2 x-5 y-1=0$和直线$l_2: x+4 y-7=0$的交点, 且直线$l$与线段$AB$的交点$P$, 满足$|AP|: |BP|=2: 3$, 其中点$A(4,-3)$、$B(-1,2)$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021975": { + "id": "021975", + "content": "已知直线$l_1: y=a x+b$, 直线$l_2: y=b x-a$, 若$l_1$的倾斜角为$\\dfrac{3 \\pi}{4}$, 且与$l_2$的交点落在第二象限, 求实数$b$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021976": { + "id": "021976", + "content": "如图, 在一本打开的书的封面上有一只蚂蚁, 在封底有一小块饼干, 蚂蚁想爬过书脊到达饼干处. 若蚂蚁和饼干离书脊的距离分别是$4 \\text{cm}$和$3 \\text{cm}$, 书脊的长度是$20 \\text{cm}$, 求蚂蚁爬行的最短路线和最短距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,3) ++ (2,0.5) coordinate (T) (0,3) ++ (-2,0.5) coordinate (S);\n\\filldraw [gray!30] (S) --++ (0.2,0.2) -- (0,3.3) -- ($(T)+(-0.2,0.2)$) -- (T) -- (0,3);\n\\draw (0,0) coordinate (O) -- (0,3) coordinate (O1);\n\\draw (0,3) --++ (2,0.5) --++ (0,-2.7) coordinate (A) -- (0,0);\n\\draw (0,3) --++ (-2,0.5) coordinate (B) --++ (0,-2.7) -- (0,0);\n\\draw [dashed] ($(O)!0.3!(A)$) node [below] {蚂蚁} -- (0,0.7) -- ($(O1)!0.4!(B)$) node [above] {饼干};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021977": { + "id": "021977", + "content": "判断下列各组直线的位置关系, 若相交, 求出它们的夹角.\\\\\n(1) $l_1: 2 x-3 y-1=0$, $l_2: 4 x-6 y-2=0$;\\\\\n(2) $l_1: y=\\dfrac{1}{3} x+1$, $l_2: x-6 y-2=0$;\\\\\n(3) $l_1: x+2=0$, $l_2: 2 x-3 y+1=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021978": { + "id": "021978", + "content": "已知直线$l_1: (a-2) x+a y-2=0$与$l_2: (1-a) x+(a+1) y+1=0$垂直, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021979": { + "id": "021979", + "content": "``两条直线的斜率的乘积等于$-1$''是``这两条直线互相垂直''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021980": { + "id": "021980", + "content": "若直线$l$过点$(3,4)$, 且与直线$x+2 y-1=0$的夹角为$\\arctan \\dfrac{1}{2}$, 则直线$l$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021981": { + "id": "021981", + "content": "已知直线$l_1: \\sqrt{3} x+y=0$与直线$l_2: k x-y+1=0$. 若直线$l_1$和直线$l_2$的夹角为$60^{\\circ}$, 则$k$的值为\\bracket{20}.\n\\fourch{$\\sqrt{3}$或$0$}{$-\\sqrt{3}$或$0$}{$\\sqrt{3}$}{$-\\sqrt{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021982": { + "id": "021982", + "content": "若直线$l_1$和$l_2$的斜率是方程$6 x^2+x-1=0$的两根, 则$l_1$与$l_2$的夹角等于\\bracket{20}.\n\\fourch{$15^{\\circ}$}{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021983": { + "id": "021983", + "content": "已知等腰直角三角形$ABC$的斜边$AB$所在直线的方程为$3 x-y-5=0$, 直角顶点为$C(4,-1)$, 求两条直角边所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021984": { + "id": "021984", + "content": "一直线$l$被两直线$4 x+y+6=0$和$3 x-5 y-6=0$截得的线段中点恰好是坐标原点, 则直线$l$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021985": { + "id": "021985", + "content": "在$\\triangle ABC$中, 已知点$A(3,-1)$和点$B(10,5)$, $\\angle B$的平分线所在直线$BD$的方程为$x-4 y+10=0$, 求边$BC$所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021986": { + "id": "021986", + "content": "一条光线从点$M(5,3)$射出, 被直线$l: x+y+1=0$反射, 入射光线与$l$的夹角为$\\arctan 2$. 求入射光线和反射光线分别所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021987": { + "id": "021987", + "content": "已知直线$l_1: 6 x+(t-1) y-8=0$, 直线$l_2: (t+4) x+(t+6) y-16=0$, 当且仅当$t=$时, $l_1$与$l_2$平行, 当且仅当$t=$时, $l_1$与$l_2$重合, 当且仅当$t \\in$\\blank{50}时, $l_1$与$l_2$相交.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021988": { + "id": "021988", + "content": "已知两条直线$l_1: m x+8 y+n=0$和$l_2: 2 x+m y-1=0$, 且它们的交点为$(m,-1)$, 则$(m, n)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021989": { + "id": "021989", + "content": "若$a$、$b$、$c$分别为$\\triangle ABC$中$\\angle A$、$\\angle B$、$\\angle C$所对边的长, 则直线$l: x \\sin A-a y+2 c=0$与直线$m: b x+y \\sin B+2 \\sin C=0$的位置关系是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021990": { + "id": "021990", + "content": "已知三条直线$l_1: 2 x+1=0$, 直线$l_2: m x+y=0$, 直线$l_3: x+m y-1=0$, 若这三条直线中有且仅有两条直线平行, 则实数$m$所有可能的值的个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021991": { + "id": "021991", + "content": "设$f(x, y)=a x+b y+c$, 方程$f(x, y)=0$表示定直线$l, M(x_0, y_0)$为不在直线$l$上的定点, 则方程$f(x, y)-f(x_0, y_0)=0$一定是\\bracket{20}.\n\\twoch{经过点$M$且与直线$l$斜交的直线}{经过点$M$且与直线$l$平行的直线}{经过点$M$且与直线$l$垂直的直线}{不经过点$M$的直线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021992": { + "id": "021992", + "content": "分别求经过直线$l_1: 5 x+2 y-3=0$和$l_2: 3 x-5 y-8=0$的交点, 且与直线$x+4 y-7=0$垂直或平行时的直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021993": { + "id": "021993", + "content": "与一条直线平行的非零向量称为它的方向向量.\\\\\n(1) 写出直线$a x+b y+c=0$($a^2+b^2 \\neq 0$)的一个方向向量;\\\\\n(2) 用直线的方向向量推导两直线夹角的余弦公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021994": { + "id": "021994", + "content": "已知$a, b \\in \\mathbf{R}$, 且$2 a+4 b=5$. 当$a, b$变化时, 直线$a x+b y-1=0$是否一定过平面上的一个定点? 若是, 求出这个定点的坐标; 若否, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021995": { + "id": "021995", + "content": "给定直线$l_1: 2 x+y-4=0$和$l_2: x-y+2=0$, $\\lambda$是任意实数, 求证: 无论$\\lambda$取何值, 直线$l: 2 x+y-4+\\lambda(x-y+2)=0$一定经过平面上的定点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021996": { + "id": "021996", + "content": "已知定点$A(0,3)$, 动点$B$在直线$l_1: y=1$上移动, 动点$C$在直线$l_2: y=-1$上移动, 且$\\angle BAC=90^{\\circ}$, 求$\\triangle ABC$的面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021997": { + "id": "021997", + "content": "一束光线经过点$(-2,1)$, 由直线$l: y=x$反射后经过点$(3,5)$射出, 求反射光线所在的直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "021998": { + "id": "021998", + "content": "点$P(3,2)$到直线$l: 3 x-2 y=13$的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "021999": { + "id": "021999", + "content": "直线$3 x-y+4=0$与直线$6 x-2 y-1=0$的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022000": { + "id": "022000", + "content": "与直线$6 x-8 y+3=0$垂直、且与原点距离等于$1$的直线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022001": { + "id": "022001", + "content": "平行于直线$x-y-2=0$、且与它的距离等于$\\sqrt{2}$的直线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022002": { + "id": "022002", + "content": "若点$(1, \\cos \\theta)$到直线$x \\sin \\theta+y \\cos \\theta=1$($0 \\leq \\theta \\leq \\dfrac{\\pi}{2}$)的距离等于$\\dfrac{1}{4}$, 则$\\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022003": { + "id": "022003", + "content": "若实数$x$、$y$满足关系式$4 x+3 y-12=0$, 则$\\sqrt{(x-2)^2+(y+3)^2}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022004": { + "id": "022004", + "content": "若点$A(-1,6)$与点$B(13,6)$到直线$l$的距离都等于$7$, 则直线$l$的不同位置有\\bracket{20}.\n\\fourch{$1$种}{$2$种}{$3$种}{$4$种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022005": { + "id": "022005", + "content": "``$a=b$''是``点$(a, b)$到直线$y=x+2$的距离是$\\sqrt{2}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022006": { + "id": "022006", + "content": "已知正方形$ABCD$的中心的坐标为点$P(1,1), AB$边所在直线的方程为$x+2 y+3=0$. 求这个正方形的其他三边所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022007": { + "id": "022007", + "content": "已知直线$l_1: 2 x-y+a=0$与直线$l_2: -4 x+2 y+1=0$, 且直线$l_1$与直线$l_2$的距离为$\\dfrac{7 \\sqrt{5}}{10}$, 求实数$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022008": { + "id": "022008", + "content": "证明: 点$A(-2,2)$到直线$(m+2) x-(m+1) y-2(2 m+3)=0$($m \\in \\mathbf{R}$)的距离$d$恒小于$4 \\sqrt{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022009": { + "id": "022009", + "content": "已知点$A(-4,5)$、$B(2,1)$试在$x$轴上求一点$M$, 使得$|MA|+|MB|$最小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022010": { + "id": "022010", + "content": "已知直线$l$经过点$P(1,2)$, 且被两平行直线$l_1: 4 x+3 y+1=0$与$l_2: 4 x+3 y+6=0$截得的线段长$|AB|=\\sqrt{2}$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022011": { + "id": "022011", + "content": "已知两相互平行的直线分别过点$A(6,2)$与$B(3,-1)$. 它们以相同的角速度旋转, 在旋转过程中, 则这两条平行直线间的距离$d$的取值范围是 , 当$d$取到最大值时, 过点$A$直线的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022012": { + "id": "022012", + "content": "已知$A(2,3)$、$B(-4,8)$两点, 直线$l$经过原点, 且$A$、$B$两点到直线$l$的距离相等, 则直线$l$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022013": { + "id": "022013", + "content": "已知平行直线$l_1$与$l_2$的距离为$\\sqrt{5}$, 且直线$l_1$经过原点, 直线$l_2$经过点$(1,3)$, 则直线$l_2$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022014": { + "id": "022014", + "content": "已知直线$l$经过点$P(0,-1)$, 且它与以$A(3,2)$、$B(2,-3)$为端点的线段$AB$有交点, 求直线$l$斜率的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022015": { + "id": "022015", + "content": "点$P(-2,-1)$关于点$Q(3,5)$的对称点是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022016": { + "id": "022016", + "content": "点$P(-2,-1)$关于直线$x+2 y-2=0$的对称点的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022017": { + "id": "022017", + "content": "直线$l: x+2 y-11=0$关于点$(-1,1)$对称的直线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022018": { + "id": "022018", + "content": "直线$y=2 x-3$关于$x$轴对称的直线方程为\\blank{50}, 直线$y=2 x-3$关于直线$y=x$对称的直线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022019": { + "id": "022019", + "content": "点到直线的距离是该点到直线上任意一点距离的最小值. 如果把一个给定点到线段上任意一点的距离的最小值定义为该点到该线段的距离. 试求点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022020": { + "id": "022020", + "content": "已知直线$l: 2 x-3 y+1=0$.\\\\\n(1) 求与直线$l$关于$x$轴对称的直线的方程;\\\\\n(2) 求与直线$l$关于$y$轴对称的直线的方程;\\\\\n(3) 求与直线$l$关于原点对称的直线的方程;\\\\\n(4)求与直线$l$关于$y=x$对称的直线的方程;\\\\\n(5) 求与直线$l$关于$y=-x$对称的直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022021": { + "id": "022021", + "content": "如图, $\\angle BAC$为伸入江中的半岛, $AB$和$AC$为两江岸, $M$处为水文站, $N$处为电讯局, 现欲在两江岸$AB$和$AC$上各建一个水文观测点$P$、$Q$. 现测得$\\angle BAC=45^{\\circ}$, 当直角坐标系以点$A$为坐标原点且以直线$BA$为$x$轴时, 测得$M(-4,1)$、$N(-3,2)$. $P$、$Q$两点应建在何处才能使路程$MPQN$最短?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-6,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$A$} node [below right] {($O$)};\n\\draw (-6,0) node [below] {$B$} coordinate (B);\n\\draw (0,0) -- (-6,6) node [left] {$C$} coordinate (C);\n\\draw (-4,1) node [left] {$M$} coordinate (M);\n\\draw (-3,2) node [left] {$N$} coordinate (N);\n\\draw (-3,0) node [below] {$P$} coordinate (P);\n\\draw (-2,2) node [above right] {$Q$} coordinate (Q);\n\\draw (M)--(P)--(Q)--(N);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022022": { + "id": "022022", + "content": "借助函数图像, 判断下列导数的正负:(用铅笔作图)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}\n\\end{center}\n(1) $f'(-\\dfrac{\\pi}{4})$, 其中$f(x)=\\cos x$;\\\\\n(2) $f'(3)$, 其中$f(x)=\\ln x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022023": { + "id": "022023", + "content": "过曲线$y=x^2$上某点$P$的切线满足下列条件, 分别求出$P$点.\\\\\n(1) 平行于直线$y=4 x-5$;\\\\\n(2) 垂直于直线$2 x-6 y+5=0$;\\\\ \n(3) 切线的倾斜角为$135^{\\circ}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022024": { + "id": "022024", + "content": "证明: 函数$f(x)=\\ln x$与函数$g(x)=e^x$没有驻点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022025": { + "id": "022025", + "content": "求余弦函数$y=\\cos x$的所有驻点.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022026": { + "id": "022026", + "content": "设$f_0(x)=\\sin x , f_1(x)=f_0'(x) , f_2(x)=f_1'(x), \\ldots, f_{n+1}(x)=f_n'(x), n \\in \\mathbf{N}$, 则$f_{2022}(x)=()$\\bracket{20}.\n\\fourch{$\\sin x$}{$-\\sin x$}{$\\cos x$}{$-\\cos x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022027": { + "id": "022027", + "content": "求下列函数的导数:\\\\\n(1) $y=\\sqrt{x}-\\ln x$;\\\\\n(2) $y=(x^2+1)(x-1)$;\\\\\n(3) $y=x^2 \\mathrm{e}^x$;\\\\\n(4) $y=\\sqrt{x} \\sin x$;\\\\\n(5) $y=x \\ln x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022028": { + "id": "022028", + "content": "求下列函数的导数:\\\\\n(1) $y=\\dfrac{\\sin x}{x}$;\\\\\n(2) $y=\\dfrac{x^2}{\\ln x}$;\\\\\n(3) $y=\\lg x$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022029": { + "id": "022029", + "content": "曲线$y=\\mathrm{e}^{-2 x}+1$在点$(0,2)$处的切线与直线$y=0$和$y=x$围成的三角形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022030": { + "id": "022030", + "content": "求下列函数的导数.\\\\\n(1) $y=(5 x-3)^4$;\\\\\n(2) $y=(3 x+2)^5$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022031": { + "id": "022031", + "content": "求下列函数的导数.\\\\\n(1) $y=\\dfrac{1}{(1-3 x)^4}$;\\\\\n(2) $y=\\sqrt[4]{\\dfrac{1}{3 x+1}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022032": { + "id": "022032", + "content": "求下列函数的导数.\\\\\n(1) $y=\\sin (3 x-\\dfrac{\\pi}{6})$;\\\\\n(2) $y=\\cos ^2 x-\\sin ^2 x$;\\\\\n(3) $y=\\ln \\sqrt{1+2 x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022033": { + "id": "022033", + "content": "求下列曲线在点$P$处的切线方程.\\\\\n(1) $y=\\sin 2 x$, $P(\\dfrac{\\pi}{3}, \\dfrac{\\sqrt{3}}{2})$;\\\\\n(2) $y=2^{1-3 x}$, $P(0,2)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022034": { + "id": "022034", + "content": "求下列函数的导数.\\\\\n(1) $y=\\mathrm{e}^{2 x} \\sin 3 x$;\\\\\n(2) $y=\\ln \\sqrt{\\dfrac{1+x}{1-x}}$;\\\\\n(3) $y=\\dfrac{x^2}{(2 x+1)^3}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022035": { + "id": "022035", + "content": "利用导数研究下列函数的单调性, 并说明所得结果与你之前的认识是否一致.\\\\\n(1) $y=2^x$;\\\\\n(2) $y=x-\\dfrac{1}{x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022036": { + "id": "022036", + "content": "利用导数研究下列函数的单调性.\\\\\n(1) $y=x+\\dfrac{1}{x}$;\\\\\n(2) $y=x+\\dfrac{1}{x^2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022037": { + "id": "022037", + "content": "研究函数$y=x^3-9 x^2+24 x+1$的单调性.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022038": { + "id": "022038", + "content": "已知$f(x)=x^3-a x-1$.\\\\\n(1) 若函数$y=f(x)$在$\\mathbf{R}$上为严格增函数, 求实数$a$的取值范围;\\\\\n(2) 若函数$y=f(x)$在$(-1,1)$上为严格减函数, 求实数$a$的取值范围;\\\\\n(3) 若函数$y=f(x)$的单调递减区间为$[-1,1]$, 求实数$a$的值;\\\\\n(4) 若函数$y=f(x)$在区间$(-1,1)$上不单调, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下校本作业", + "edit": [ + "20230209\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "022039": { + "id": "022039", + "content": "研究下列函数的单调性.\\\\\n(1) $y=\\dfrac{\\ln (1+x)}{x}$($x>0$);\\\\\n(2) $y=\\dfrac{\\sin x}{x}$($0| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0