diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index be96c3e1..3c00d32e 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n", - "problems = \"21106\"\n", + "problems = \"14384\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", @@ -51,7 +51,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 25, "metadata": {}, "outputs": [], "source": [ 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 60354812..c7152324 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": 1, + "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 14103 , 直至 020000\n", + "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: 31206 , 直至 999999\n" + "首个空闲id: 31222 , 直至 999999\n" ] } ], @@ -45,7 +45,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "base", "language": "python", "name": "python3" }, @@ -59,12 +59,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.13" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "ad2bdc8ecc057115af97d19610ffacc2b4e99fae6737bb82f5d7fb13d2f2c186" } } }, 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 be8445a6..41298559 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 @@ -12781:12822,13692:13758,14103:14165 \ No newline at end of file +13008:13033,13034:13055,13147:13172,13196:13216,13845:13859,13860:13874,13875:13887,13888:13898,13899:13910,13911:13917,14204:14223,14224:14243,14244:14266,14267:14287,14325:14342,14379:14399 \ No newline at end of file diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index 69582b38..7137a771 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": null, + "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id---\"\"\"\n", - "old_id = \"12281\"\n", - "new_id = \"30505\"\n", + "old_id = \"3648\"\n", + "new_id = \"31221\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\n", "old_id = old_id.zfill(6)\n", @@ -50,7 +50,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.8.15 ('mathdept')", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -64,12 +64,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, 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 39126a87..bdd8bfd5 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 @@ -2,23 +2,51 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 14103\n", - "origin = \"2023年空中课堂高三复习题\"\n", - "filename = r\"D:\\temp\\空中课堂2023.tex\"\n", - "editor = \"20230131\\t王伟叶\"\n", + "starting_id = 14379\n", + "raworigin = \"2023年空中课堂高三复习题\"\n", + "filename = r\"D:\\temp\\空中课堂第三批.tex\"\n", + "editor = \"20230203\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 4, "metadata": {}, - "outputs": [], + "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" + ] + } + ], "source": [ "import os,re,json\n", "\n", @@ -84,9 +112,11 @@ " if pid in pro_dict:\n", " duplicate_flag = True\n", " if indexed == False:\n", - " NewProblem = CreateNewProblem(id = pid, content = p, origin = origin + suffix , dict = pro_dict,editor = editor)\n", + " origin = raworigin + suffix\n", " else:\n", - " NewProblem = CreateNewProblem(id = pid, content = p, origin = origin + suffix + \"试题\" + str(id- starting_id+1), dict = pro_dict,editor = editor)\n", + " origin = raworigin + suffix + \"试题\" + str(id- starting_id+1)\n", + " NewProblem = CreateNewProblem(id = pid, content = p, origin = origin, dict = pro_dict,editor = editor)\n", + " print(\"添加题号\"+pid+\", \"+\"来源: \" + origin)\n", " pro_dict[pid] = NewProblem\n", " id += 1\n", "\n", @@ -106,6 +136,26 @@ " print(\"题号有重复, 请检查.\\n\"*5)" ] }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'18'" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "suffix" + ] + }, { "cell_type": "code", "execution_count": null, @@ -130,12 +180,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/生成文件夹下的题号清单.ipynb b/工具/生成文件夹下的题号清单.ipynb index 45a8b27d..d610f816 100644 --- a/工具/生成文件夹下的题号清单.ipynb +++ b/工具/生成文件夹下的题号清单.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -189,7 +189,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -208,7 +208,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index 04146a33..50cfd609 100644 --- a/工具/讲义生成.ipynb +++ b/工具/讲义生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -15,9 +15,9 @@ "题块 2 处理完毕.\n", "正在处理题块 3 .\n", "题块 3 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/2023届静安区一模_教师_20230113.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/2023届静安区一模_学生_20230113.tex\n", + "开始编译教师版本pdf文件: 临时文件/寒假作业反馈练习_教师_20230202.tex\n", + "1\n", + "开始编译学生版本pdf文件: 临时文件/寒假作业反馈练习_学生_20230202.tex\n", "0\n" ] } @@ -41,7 +41,7 @@ "# enumi_mode = 0\n", "\n", "#2023届测验卷与周末卷\n", - "exec_list = [(\"标题替换\",\"2023届静安区一模\")]\n", + "exec_list = [(\"标题替换\",\"寒假作业反馈练习\")]\n", "enumi_mode = 1\n", "\n", "# 日常选题讲义\n", @@ -51,13 +51,13 @@ "\"\"\"---其他预处理替换命令结束---\"\"\"\n", "\n", "\"\"\"---设置目标文件名---\"\"\"\n", - "destination_file = \"临时文件/2023届静安区一模\"\n", + "destination_file = \"临时文件/寒假作业反馈练习\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"12760:12771\",\"12772:12775\",\"12776:12780\"\n", + "\"31206:31215\",\"31216:31218\",\"31219:31221\"\n", "\n", "\n", "]\n", @@ -210,7 +210,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.15 ('pythontest')", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -229,7 +229,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, 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 b9b93566..eb9ca978 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 @@ -9,7 +9,303 @@ "name": "stdout", "output_type": "stream", "text": [ - "031205 填空题\n" + "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 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"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" ] } ], @@ -51,7 +347,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -70,7 +366,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, 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 68bcf56d..7a4c2cfd 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/01_集合逻辑不等式备选_教师用_20230201.tex\n", + "开始编译教师版本pdf文件: 临时文件/03_三角平面向量复数备选_教师用_20230203.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/01_集合逻辑不等式备选_学生用_20230201.tex\n", + "开始编译学生版本pdf文件: 临时文件/03_三角平面向量复数备选_学生用_20230203.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/01_集合逻辑不等式备选\"\n", + "filename = \"临时文件/03_三角平面向量复数备选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", @@ -188,7 +188,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { diff --git a/文本处理工具/剪贴板圆圈数字生成.ipynb b/文本处理工具/剪贴板圆圈数字生成.ipynb index ee5595ce..79699604 100644 --- a/文本处理工具/剪贴板圆圈数字生成.ipynb +++ b/文本处理工具/剪贴板圆圈数字生成.ipynb @@ -45,7 +45,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -64,7 +64,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/文本处理工具/剪贴板文本整理_word文件.ipynb b/文本处理工具/剪贴板文本整理_word文件.ipynb index 2345d154..e0e56454 100644 --- a/文本处理工具/剪贴板文本整理_word文件.ipynb +++ b/文本处理工具/剪贴板文本整理_word文件.ipynb @@ -522,7 +522,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.15 ('pythontest')", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -536,12 +536,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15" + "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/文本处理工具/剪贴板文本整理_第三方输入.ipynb b/文本处理工具/剪贴板文本整理_第三方输入.ipynb new file mode 100644 index 00000000..1be9ccce --- /dev/null +++ b/文本处理工具/剪贴板文本整理_第三方输入.ipynb @@ -0,0 +1,428 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "import os,re\n", + "import win32clipboard as wc\n", + "import win32con\n", + "\n", + "# 获取剪切板内容\n", + "def getCopy():\n", + " wc.OpenClipboard()\n", + " t = wc.GetClipboardData(win32con.CF_UNICODETEXT)\n", + " wc.CloseClipboard()\n", + " return t\n", + "\n", + "# 写入剪切板内容\n", + "def setCopy(str):\n", + " wc.OpenClipboard()\n", + " wc.EmptyClipboard()\n", + " wc.SetClipboardData(win32con.CF_UNICODETEXT, str)\n", + " wc.CloseClipboard()\n", + "\n", + "def full_stop(matchobj):\n", + " if matchobj.group(1) == \"。\" or matchobj.group(1) == \"．\":\n", + " return \". \"\n", + " else:\n", + " return \".\\n\"\n", + "def refine_brackets(matchobj):\n", + " return matchobj.group(1)[1:-1]\n", + "def insert_a_blank(matchobj):\n", + " return matchobj.group(1)[:-1]+\" \"+matchobj.group(1)[-1]\n", + "def multiple_choice(matchobj):\n", + " string = \"\\\\fourch\" + \"{\" + matchobj.group(1) + \"}{\" + matchobj.group(2) + \"}{\" + matchobj.group(3) + \"}{\" + matchobj.group(4) + \"}\\n\"\n", + " return string\n", + "def boldsymbols(matchobj):\n", + " return \"\\\\i\"+matchobj.group(1)[:-1]+\"\\\\mathbf{\"+matchobj.group(1)[-1]+\"}\"\n", + "def boldsymbols_star(matchobj):\n", + " return \"\\\\in \\\\mathbf{\"+matchobj.group(1)+\"}^*\"\n", + "def singleboldsymbols(matchobj):\n", + " return \"$\\\\mathbf{\" + matchobj.group(1) + \"}$\"\n", + "def blackboardbold(matchobj):\n", + " string = \"\\\\mathbf\" + \"{\" + matchobj.group(1) + \"}\"\n", + " return string\n", + "def limit(matchobj):\n", + " return \"\\\\displaystyle\\\\lim_{\"+matchobj.group(1)+\"}\"\n", + "def replace_i(matchobj):\n", + " string = matchobj.group(1)\n", + " length = len(string)\n", + " for i in range(length-1,-1,-1):\n", + " if string[i] == \"i\" and not \"item\" in string[i:] and not \"overline\" in string[i:]:\n", + " string = string[:i] + \"\\\\mathrm{i}\" + string[i+1:]\n", + " return string\n", + "def refine_log(matchobj):\n", + " return r\"\\log_\"+matchobj.group(1)\n", + "def refine_powers(matchobj):\n", + " base = matchobj.group(1)\n", + " power = matchobj.group(2)\n", + " return base + \"^\" + power\n", + "def refine_sequences(matchobj):\n", + " return \"\\{\" + matchobj.group(1) + \"\\}\"\n", + "def refine_starting_brackets(matchobj):\n", + " return \"$\" + matchobj.group(1)\n", + "def refine_left_operating_brackets(matchobj):\n", + " obj = matchobj.group(2)\n", + " return matchobj.group(1)+obj\n", + "def refine_right_operating_brackets(matchobj):\n", + " obj = matchobj.group(1)\n", + " return obj + matchobj.group(2)\n", + "def refine_brackets_in_brackets(matchobj):\n", + " return matchobj.group(1) + matchobj.group(2) + matchobj.group(3)\n", + "def mathbf(matchobj):\n", + " return \"\\\\mathbf{\" + matchobj.group(1) + \"}^\" + matchobj.group(2)\n", + "#以上是202207之前的文本处理机制\n", + "global layer\n", + "def rename_bracket(matchobj):\n", + " return \"leftbracket\" + str(layer) + matchobj.group(1) + \"rightbracket\" + str(layer)\n", + "def frac_brackets(matchobj):\n", + " return \"frac{\"+matchobj.group(1)+\"}{\"+matchobj.group(2)+\"}\"\n", + "def frac_single_second_bracket(matchobj):\n", + " return \"frac \"+matchobj.group(1)+\"{\"+matchobj.group(2)+\"}\"\n", + "def recall_vital_bracket(matchobj):\n", + " return matchobj.group(1) + \"{\" + matchobj.group(2) + \"}\"\n", + "def sqrt_brackets(matchobj):\n", + " if matchobj.group(1) == None:\n", + " first_group = \"\"\n", + " else:\n", + " first_group = matchobj.group(1)\n", + " return \"sqrt \"+ first_group +\"{\" + matchobj.group(2) + \"}\"\n", + "#def refine_frac(string):\n", + "# for s in range(7):\n", + "# for t in range(7):\n", + "# string = re.sub(r\"frac[\\s]*leftbracket\"+str(s)+\"(.*?)\"+r\"rightbracket\"+str(s)+\"[\\s]*\"+r\"leftbracket\"+str(t)+\"(.*?)\"+r\"rightbracket\"+str(t),frac_brackets,string)\n", + "# return string\n", + "def refine_single_second_frac(string):\n", + " for s in range(7):\n", + " string = re.sub(r\"frac[\\s]*(\\w)[\\s]*leftbracket\"+str(s)+\"(.*?)\"+r\"rightbracket\"+str(s),frac_single_second_bracket,string)\n", + " return string\n", + "def refine_vital_bracket(string):\n", + " for s in range(7):\n", + " string = re.sub(r\"(frac)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(line)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(arrow)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(_)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(\\^)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(mathrm)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(mathbf)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(begin)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " string = re.sub(r\"(end)[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),recall_vital_bracket,string)\n", + " return string\n", + "def refine_sqrt(string):\n", + " for s in range(7):\n", + " string = re.sub(r\"sqrt[\\s]*(\\[\\w*\\])*[\\s]*leftbracket\"+str(s)+\"(.*?)rightbracket\"+str(s),sqrt_brackets,string)\n", + " return string\n", + "def give_blanks(string):\n", + " string = re.sub(r\"(sqrt[\\w])\",insert_a_blank,string)\n", + " string = re.sub(r\"(frac[\\w])\",insert_a_blank,string)\n", + " return string\n", + "def give_brackets(string):\n", + " string = re.sub(r\"leftbracket\\d\",\"\",string)\n", + " string = re.sub(r\"rightbracket\\d\",\"\",string)\n", + " string = re.sub(r\"leftset\",r\"\\{\",string)\n", + " string = re.sub(r\"rightset\",r\"\\}\",string)\n", + " return string\n", + "#以上是20220715新加的文本处理机制\n", + "def initial_bracket_search(string):\n", + " t = re.search(r\"^[\\s]*?leftbracket(\\d)\",string)\n", + " if t == None:\n", + " return -1\n", + " else:\n", + " return t.span()[1]\n", + "def initial_brackets_pair_search(string,d):\n", + " t = re.search(\"rightbracket\"+d,string)\n", + " if t == None:\n", + " return -1\n", + " else:\n", + " return t.span()[1]\n", + "def refine_frac(string):\n", + " eq_left = \"\"\n", + " eq_right = string\n", + " while re.search(\"frac\",eq_right) != None:\n", + " pos = re.search(\"frac\",eq_right)\n", + " eq_left += eq_right[:pos.span()[1]]\n", + " eq_right = eq_right[pos.span()[1]:]\n", + " if initial_bracket_search(eq_right)>0:\n", + " pos = initial_brackets_pair_search(eq_right,eq_right[initial_bracket_search(eq_right)-1])\n", + " first_bracket = eq_right[:pos]\n", + " first_layer = first_bracket[-1]\n", + " eq_remain = eq_right[pos:]\n", + " if initial_bracket_search(eq_remain)>0:\n", + " pos = initial_brackets_pair_search(eq_remain,eq_remain[initial_bracket_search(eq_remain)-1])\n", + " second_bracket = eq_remain[:pos]\n", + " second_layer = second_bracket[-1]\n", + " first_bracket = re.sub(r\"leftbracket\"+first_layer,\"{\",first_bracket)\n", + " second_bracket = re.sub(r\"leftbracket\"+second_layer,\"{\",second_bracket)\n", + " first_bracket = re.sub(r\"rightbracket\"+first_layer,\"}\",first_bracket)\n", + " second_bracket = re.sub(r\"rightbracket\"+second_layer,\"}\",second_bracket)\n", + " eq_right = first_bracket+second_bracket+eq_remain[pos:]\n", + " return eq_left+eq_right\n", + "#以上是20220718修改的大括号处理机制, 修复了一个bug\n", + "def reduce_blank(matchobj):\n", + " return matchobj.group(1).replace(\" \",\"\")\n", + "def add_dollars(matchobj):\n", + " return matchobj.group(1)[0] + r\"$\" + matchobj.group(1)[1:-1] + r\"$\" + matchobj.group(1)[-1]\n", + "def del_first_char(matchobj):\n", + " return matchobj.group(1)[1:]\n", + "def add_underline(matchobj):\n", + " return matchobj.group(1)[0] + \"_\" + matchobj.group(1)[-1]\n", + "def brackets_to_cwords(matchobj):\n", + " return \"左括号\"+matchobj.group(1)+\"右括号\"\n", + "def cwords_to_brackets(matchobj):\n", + " return \"(\"+matchobj.group(1)+\")\"\n", + "def circled_brackets(matchobj):\n", + " return matchobj.group(1)[:-1]+\"{\"+matchobj.group(1)[-1] + \"}\"\n", + "\n", + "# try:\n", + "# os.chdir(r\"D:\\mathdept\\mathdept\\文本处理程序等\")\n", + "# except:\n", + "# os.chdir(r\"D:\\mathdept\\文本处理程序等\")\n", + "# with open(\"textfile.txt\", \"r\", encoding = \"utf8\") as textfile:\n", + "# data = textfile.read()\n", + "\n", + "filename = getCopy().strip()\n", + "if not filename[-4:] == \".tex\":\n", + " filename += \".tex\"\n", + "with open(r\"C:\\Users\\weiye\\Documents\\wwy sync\\待整理word题目\\高考数学风暴tex/\"+filename,\"r\",encoding = \"u8\") as f:\n", + " raw_data = f.read()\n", + "data = re.findall(r\"\\\\begin\\{enumerate\\}([\\s\\S]*?)\\\\end\\{enumerate\\}\",raw_data)[0]\n", + "\n", + "\n", + "#去除左右括号的前缀\n", + "data = data.replace(r\"\\rightarrow\",r\"\\to\")\n", + "data = data.replace(r\"\\left.\",\"\").replace(r\"\\left\",\"\").replace(r\"\\right.\",\"\").replace(r\"\\right\",\"\")\n", + "\n", + "#全角半角符号替换\n", + "data = re.sub(\"　\",\" \",data)\n", + "data = re.sub(\"(。[\\n]*)\",full_stop,data)\n", + "data = re.sub(\"(．[\\n]*)\",full_stop,data)\n", + "data = re.sub(\"，\",\", \",data)\n", + "data = re.sub(\"：\",\": \",data)\n", + "data = re.sub(\"；\",\"; \",data)\n", + "data = re.sub(\"（\",\"(\",data)\n", + "data = re.sub(\"）\",\")\",data)\n", + "data = re.sub(\"？\",\"? \",data)\n", + "data = re.sub(\"“\",\"``\",data)\n", + "data = re.sub(\"”\",\"''\",data)\n", + "data = re.sub(\" ``\",\"``\",data)\n", + "data = re.sub(\"'' \",\"''\",data)\n", + "\n", + "#替换题号\n", + "data = re.sub(\"(^[例]*[0-9]+[\\s]*\\.[\\s]+)\",\"\\\\n\\\\\\\\item \",data)\n", + "data = re.sub(\"(\\\\n[例]*[0-9]+[\\s]*\\.[\\s]+)\",\"\\\\n\\\\\\\\item \",data)\n", + "\n", + "#公式标志换成$符号\n", + "data = re.sub(\"\\\\\\\\\\[\",r\"$\",data)\n", + "data = re.sub(\"\\\\\\\\\\]\",r\"$\",data)\n", + "data = re.sub(\"\\$\\$\",\"\",data)\n", + "\n", + "#标点和$符号分开\n", + "data = re.sub(r\"([,.:;])\\$\",lambda x:x.group(1)+\" $\",data)\n", + "\n", + "#选择题替换成标准格式\n", + "data = re.sub(\"A\\.([\\s\\S]*?)B\\.([\\s\\S]*?)C\\.([\\s\\S]*?)D\\.([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", + "data = re.sub(\"\\(A\\)([\\s\\S]*?)\\(B\\)([\\s\\S]*?)\\(C\\)([\\s\\S]*?)\\(D\\)([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", + "data = re.sub(\"Ａ\\.([\\s\\S]*?)Ｂ\\.([\\s\\S]*?)Ｃ\\.([\\s\\S]*?)Ｄ\\.([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", + "data = re.sub(\"\\(Ａ\\)([\\s\\S]*?)\\(Ｂ\\)([\\s\\S]*?)\\(Ｃ\\)([\\s\\S]*?)\\(Ｄ\\)([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", + "data = re.sub(\"\\$[ ]+\\}\",\"$}\",data)\n", + "data = re.sub(\"\\{[ ]+\\$\",\"{$\",data)\n", + "\n", + "\n", + "data = re.sub(r\"\\\\_\\\\_\\\\_\\\\_\", r\"\\\\blank{50}\",data)\n", + "data = re.sub(r\"\\\\[\\(\\)]\",\"$\",data)\n", + "datalines = [l.strip() for l in data.split(\"\\n\") if len(l.strip())>0]\n", + "data = \"\\n\".join(datalines)\n", + "\n", + "#替换frac为dfrac\n", + "data = data.replace(\"\\\\frac\",\"\\\\dfrac\")\n", + "\n", + "#替换多余的空行\n", + "for i in range(20):\n", + " data = re.sub(\"\\n[\\t ]*\\n\",\"\\n\",data)\n", + "\n", + "#删除\\quad\n", + "data = re.sub(r\"\\\\q+uad\",\"\",data)\n", + "\n", + "#删除~\n", + "data = re.sub(r\"~\",\"\",data)\n", + "\n", + "\n", + "\n", + "data1 = data #替换后暂存data1\n", + "\n", + "#分离文字和公式\n", + "raw_texts = [] #文字数组\n", + "raw_equations = [] #公式数组\n", + "d = data\n", + "while len(d) > 0:\n", + " interval = re.search(r\"\\$[\\s\\S]*?\\$\",d)\n", + " if not interval == None:\n", + " (start, end) = interval.span()\n", + " raw_texts.append(d[:start])\n", + " raw_equations.append(d[start:end])\n", + " d = d[end:]\n", + " else:\n", + " raw_texts.append(d)\n", + " d = \"\"\n", + "#至此已经分离了文字和公式，公式在两个$之内，包含两个$\n", + "\n", + "modified_texts = []\n", + "modified_equations = []\n", + "\n", + "for text in raw_texts:\n", + " text1 = text\n", + " #删除选项中无用的空格\n", + " text1 = re.sub(\"\\{[\\s]+?\",\"{\",text1)\n", + " text1 = re.sub(\"[\\s]+?\\}\",\"}\",text1)\n", + " text1 = re.sub(r\"\\\\ \",\" \",text1)\n", + " #填空题的处理\n", + " # text1 = re.sub(\"[ _]{2,}\",r\"\\\\blank{50}\",text1)\n", + " #选择题的处理\n", + " text1 = re.sub(r\"\\(\\\\blank\\{50\\}\\)\",\"\\\\\\\\bracket{20}\",text1)\n", + " text1 = re.sub(r\"\\([\\s]{1,10}\\)\",\"\\\\\\\\bracket{20}\",text1)\n", + " #逗号后面加空格\n", + " text1 = re.sub(\",[ ]*\",\", \",text1)\n", + " text1 = re.sub(r\"\\.\\}\",\"}\",text1)\n", + " text1 = re.sub(r\"\\n\\d{1,3}\\.\",r\"\\n\\\\item \",text1)\n", + " # text1 = re.sub(r\"\\s{2,}\\.\",r\"\\\\blank{50}.\",text1)\n", + " # text1 = re.sub(r\"\\s{2,}\\,\",r\"\\\\blank{50},\",text1)\n", + " text1 = re.sub(r\"\\s*\\\\bracket\\{20\\}\\s*\\n\",r\"\\\\bracket{20}.\\n\",text1)\n", + " #改非规范选择题\n", + " text1 = re.sub(r\"[\\.;]\\}\",\"}\",text1)\n", + " text1 = re.sub(r\"([\\u4e00-\\u9fa5])[\\s]+([\\d]{1,6})[\\s]+([\\u4e00-\\u9fa5])\",lambda x:x.group(1)+\"$\"+x.group(2)+\"$\"+x.group(3),text1)\n", + " modified_texts.append(text1)\n", + " \n", + "\n", + "for equation in raw_equations:\n", + " equation1 = equation\n", + " if equation1[1] == \"{\" and equation1[-2] == \"}\":\n", + " equation1 = \"$\" + equation1[2:-2] + \"$\"\n", + " #删除一些无效大括号\n", + " for i in range(3):\n", + " equation1 = re.sub(r\"_\\{([0-9a-zA-Z])\\}\",lambda x:\"_\"+x.group(1),equation1)\n", + " equation1 = re.sub(r\"\\^\\{([0-9a-zA-Z])\\}\",lambda x:\"^\"+x.group(1),equation1)\n", + " #合并一些公式中的无效空格\n", + " for i in range(2):\n", + " equation1 = re.sub(r\"([0-9A-Z])\\s+([0-9A-Z])\",lambda x:x.group(1)+x.group(2),equation1)\n", + " #改变组合数和排列数\n", + " equation1 = re.sub(r\"([CP])(_[^_\\^]{,5}\\^)\",lambda x:r\"\\mathrm{\"+x.group(1)+\"}\"+x.group(2),equation1)\n", + " #改单位\n", + " equation1 = re.sub(r\"mathrm\\{cm\\}\",\"text{cm}\",equation1)\n", + " equation1 = re.sub(r\"mathrm\\{km\\}\",\"text{km}\",equation1)\n", + " modified_equations.append(equation1)\n", + "\n", + "\n", + "#整合修改过的文本和公式 \n", + "modified_data = \"\"\n", + "for i in range(len(modified_texts)):\n", + " try:\n", + " modified_data += modified_texts[i]\n", + " except:\n", + " a = 1\n", + " try:\n", + " modified_data += modified_equations[i]\n", + " except:\n", + " a = 1\n", + "modified_data = re.sub(r\"[ ]+\\n\",\"\\n\",modified_data)\n", + "modified_data = re.sub(r\"\\$[\\s]*?\\\\parallel[\\s]*?\\$\",r\"\\\\parallel\",modified_data)\n", + "modified_data = re.sub(r\"\\n例\\s*?\\d{1,3}\\s*\",r\"\\n\\\\item \",modified_data)\n", + "modified_data = re.sub(r\"(\\$[\\,\\.:;]\\$)\",refine_brackets,modified_data)\n", + "\n", + "\n", + "#以下是mathpix之后的空格去除\n", + "for i in range(3):\n", + " modified_data = re.sub(r\"([\\u4e00-\\u9fa5])( )([\\u4e00-\\u9fa5])\",lambda x:x.group(1)+x.group(3),modified_data)\n", + " modified_data = re.sub(r\"\\$ \",\"$\",modified_data)\n", + " modified_data = re.sub(r\" \\$\",\"$\",modified_data)\n", + "#mathpix的错别字修改\n", + "modified_data = modified_data.replace(\"雉\",\"锥\")\n", + "modified_data = re.sub(\"[粗秿]圆\",\"椭圆\",modified_data)\n", + "modified_data = modified_data.replace(\"针角\",\"钝角\")\n", + "modified_data = re.sub(\"投郑\",\"投掷\",modified_data)\n", + "modified_data = re.sub(\"抛郑\",\"抛掷\",modified_data)\n", + "modified_data = re.sub(\"范目\",\"范围\",modified_data)\n", + "modified_data = re.sub(\"揷\",\"插\",modified_data)\n", + "#mathpix的自由向量修改\n", + "modified_data = modified_data.replace(r\"\\vec\",r\"\\overrightarrow \")\n", + "modified_data = modified_data.replace(r\"\\bar\",r\"\\overline \")\n", + "#mathpix的极限修改\n", + "modified_data = re.sub(r\"\\\\lim[\\s]*_\\{n \\\\to \\\\infty\\}\",r\"\\\\displaystyle\\\\lim_{n\\\\to\\\\infty}\",modified_data)\n", + "#mathpix的顿号修改\n", + "modified_data = modified_data.replace(r\" 、 \",r\"$、$\")\n", + "#改slant等\n", + "modified_data = modified_data.replace(r\"slant\",\"\")\n", + "modified_data = modified_data.replace(r\"\\mid\",\"|\")\n", + "modified_data = re.sub(r\"\\\\mathrm\\{\\\\mathrm\\{i\\}\\}\",r\"\\\\mathrm{i}\",modified_data)\n", + "modified_data = modified_data.replace(\",$\",\", $\")\n", + "modified_data = modified_data.replace(\" / /\",r\"\\parallel\")\n", + "modified_data = modified_data.replace(\"mathrmR\",r\"mathbf{R}\")\n", + "modified_data = modified_data.replace(r\"^{\\prime}\",\"'\")\n", + "modified_data = re.sub(r\"\\^\\{\\\\dfrac\",r\"^{\\\\frac\",modified_data)\n", + "modified_data = re.sub(r\"\\^\\{-\\\\dfrac\",r\"^{-\\\\frac\",modified_data)\n", + "modified_data = re.sub(r\"_\\{\\\\dfrac\",r\"_{\\\\frac\",modified_data)\n", + "modified_data = re.sub(r\"_\\{-\\\\dfrac\",r\"_{-\\\\frac\",modified_data)\n", + "\n", + "#改分段函数等\n", + "modified_data = re.sub(r\"\\\\{\\\\begin\\{array\\}\\{[rcl]*\\}\",r\"\\\\begin{cases}\",modified_data)\n", + "modified_data = re.sub(r\"\\\\end{array}\",r\"\\\\end{cases}\",modified_data)\n", + "\n", + "#冒号后加空格\n", + "modified_data = re.sub(r\":([\\S])\", lambda x:\": \"+x.group(1),modified_data)\n", + "\n", + "#识别填空题加空格\n", + "modified_data = re.sub(r\"([\\u4e00-\\u9fa5\\$])[\\s]*\\n\\\\item\",lambda x: x.group(1)+\"\\\\blank{50}.\\n\\\\item\",modified_data)\n", + "\n", + "#识别选择题加括号\n", + "modified_data = re.sub(r\"\\$\\(\\s*\\)\\$\",r\"\\\\bracket{20}\",modified_data)\n", + "modified_data = re.sub(r\"([\\u4e00-\\u9fa5\\$])[\\s]*\\n\\\\fourch\",lambda x: x.group(1)+\"\\\\bracket{20}.\\n\\\\fourch\",modified_data)\n", + "\n", + "#改圆弧\n", + "modified_data = re.sub(r\"overparen\",r\"overset\\\\frown\",modified_data)\n", + "\n", + "\n", + "\n", + "setCopy(modified_data)\n", + "\n", + "with open(\"临时文件/outputfile.txt\",\"w\",encoding = \"utf8\") as f:\n", + " f.write(modified_data)" + ] + }, + { + "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": "3.9.15" + }, + "orig_nbformat": 4, + "vscode": { + "interpreter": { + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + } + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 6b765ed5..7aeeb2fa 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -10133,7 +10133,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "031207" + ], "remark": "", "space": "" }, @@ -98086,7 +98088,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031206" + ], "remark": "", "space": "" }, @@ -98507,7 +98511,8 @@ ], "same": [], "related": [ - "030050" + "030050", + "031221" ], "remark": "", "space": "12ex" @@ -98761,7 +98766,9 @@ "same": [ "002911" ], - "related": [], + "related": [ + "031209" + ], "remark": "", "space": "" }, @@ -98866,7 +98873,9 @@ "20220701\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031212" + ], "remark": "", "space": "" }, @@ -124163,7 +124172,9 @@ "20220710\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031220" + ], "remark": "", "space": "12ex" }, @@ -249800,7 +249811,9 @@ "20220730\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031210" + ], "remark": "", "space": "" }, @@ -290235,7 +290248,9 @@ "20220819\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031214" + ], "remark": "", "space": "" }, @@ -296906,7 +296921,9 @@ "20221027\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031208" + ], "remark": "", "space": "" }, @@ -299588,7 +299605,9 @@ "20221121\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031215" + ], "remark": "", "space": "" }, @@ -302870,7 +302889,9 @@ "20221209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031211" + ], "remark": "", "space": "" }, @@ -303109,7 +303130,9 @@ "20221209\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031216" + ], "remark": "", "space": "" }, @@ -343332,7 +343355,7 @@ "content": "已知集合$A=\\{1,2\\}$, $B=\\{a, a^2, 3\\}$, 若$A \\cap B=\\{1\\}$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343351,7 +343374,7 @@ "content": "若用列举法表示集合$A=\\{x \\in \\mathbf{N} | \\dfrac{8}{6-x} \\in \\mathbf{N}\\}$, 则$A=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343370,7 +343393,7 @@ "content": "设全集为$\\mathbf{R}$, 集合$A=\\{x | 03$或$x<1\\}$, 则$A \\cup \\overline {B}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343389,7 +343412,7 @@ "content": "已知点集$M=\\{(x, y) | x^2+y^2<2\\}$, $N=\\{(x, y)|| x|<\\sqrt{2}|,\\ |y |<\\sqrt{2}\\}$. 若$\\alpha$: 点$P \\in M$, $\\beta$: 点$P \\in N$. 则$\\alpha$是$\\beta$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": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -343408,7 +343431,7 @@ "content": "用反证法证明命题``设$x_1, x_2, \\cdots, x_n \\in \\mathbf{R}$($n$是正整数). 求证: 若$x_1+x_2+\\cdots+x_n>n$, 则$x_1, x_2, \\cdots, x_n$中至少有一个大于$1$.''时, 第一步需假设\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343427,7 +343450,7 @@ "content": "集合$A=\\{x | x^2-5 x-6=0\\}$, $B=\\{x | a x^2-x+6=0,\\ x \\in \\mathbf{R}\\}$, 且$A \\cup B=A$. 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343439,14 +343462,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014109": { "id": "014109", "content": "已知$m, a \\in \\mathbf{R}$, $f(x)=x^2+(a-1) x+1$, $g(x)=m x^2+2 a x+\\dfrac{m}{4}$, 若``对于一切实数$x, f(x)>0$''是``对一切实数$x, g(x)>0$''的充分条件, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343458,14 +343481,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014110": { "id": "014110", "content": "已知函数$y=f(x)$是$\\mathbf{R}$上的严格增函数, 若$a, b \\in \\mathbf{R}$, 求证: ``$a+b \\geq 0$''是``$f(a)+f(b) \\geq f(-a)+f(-b)$''的充要条件.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343477,14 +343500,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014111": { "id": "014111", "content": "集合$A=\\{x | a x^2+2 x+1=0, x \\in \\mathbf{R}\\}$的元素个数组成的集合为\\blank{50}.(用列举法表示)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343503,7 +343526,7 @@ "content": "设$a, b$是实数, \\textcircled{1} $a+b>1$; \\textcircled{2} $a+b>2$; \\textcircled{3} $a^2+b^2>2$; \\textcircled{4} $a b>1$, 其中能作为``$a, b$中至少有一个大于$1$''的充分条件的是\\blank{50}.(写出所有正确结论的序号)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343522,7 +343545,7 @@ "content": "已知集合$A=\\{x | x^2+3 x+2<0\\}, B=\\{x | x^2-4 a x+3 a^2<0\\}$, 且$A \\subset B$, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343534,14 +343557,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014114": { "id": "014114", "content": "若正数$x, y$满足$x y-1=x+y$, 求证: $x, y$均大于$1$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343553,14 +343576,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014115": { "id": "014115", "content": "设集合$A=\\{a+1, a^2-1, a^2-a-1\\}$, 若$1 \\in A$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343579,7 +343602,7 @@ "content": "已知集合$A=\\{x | \\dfrac{6}{5-x} \\in \\mathbf{Z}, \\ x \\in \\mathbf{N}\\}$, 试用列举法表示集合$A$, 则$A=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343598,7 +343621,7 @@ "content": "若集合$M=\\{y | y=x^2,\\ x \\in \\mathbf{R}\\}$, $N=\\{y | y=-3 x^2+4, \\ x \\in \\mathbf{R}\\}$, 则$M \\cap N=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343617,7 +343640,7 @@ "content": "判断下列各题中甲是乙的什么条件(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), 并说明理由.\\\\\n(1) 已知$\\triangle ABC$. 甲: $A>B$; 乙: $\\sin A>\\sin B$;\\\\\n(2) 已知$\\overrightarrow {a}$, $\\overrightarrow {b}$, $\\overrightarrow {c}$是非零的共面向量. 甲: $\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$; 乙: $\\overrightarrow {a}=\\overrightarrow {c}$;\\\\\n(3) 已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 且存在导函数$f'(x)$. 甲: $f'(x)>0$恒成立; 乙: $y=f(x)$是$\\mathbf{R}$上的严格增函数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343629,14 +343652,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014119": { "id": "014119", "content": "若不等式$\\dfrac{x-m+1}{x-2 m}<0$成立的一个充分非必要条件是$\\dfrac{1}{3}0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a, b \\in \\mathbf{R}$. 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}\n%02", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -343712,7 +343735,7 @@ "content": "设$a$、$b \\in \\mathbf{R}$, 若关于$x$的方程$x^2+a x+b=0$的解集为$\\{1,3\\}$, 则$a-b=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343731,7 +343754,7 @@ "content": "若关于$x$与$y$的二元一次方程组$\\begin{cases}x+m y=2, \\\\ 2 x+4 y=3\\end{cases}$的解集为$\\varnothing$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343750,7 +343773,7 @@ "content": "不等式$\\dfrac{2 x+1}{x-1} \\geq 1$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343769,7 +343792,7 @@ "content": "不等式$1<|x-1| \\leq 2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343788,7 +343811,7 @@ "content": "设$a$、$b \\in \\mathbf{R}$, 若关于$x$的不等式$a x-b<0$的解集为$(1,+\\infty)$, 则不等式$(a x+b)(x-2)>0$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343807,7 +343830,7 @@ "content": "设$m \\in \\mathbf{R}$, 若函数$y=\\sqrt{3 x^2-m x+m}$的定义域为$\\mathbf{R}$, 求$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343819,14 +343842,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014129": { "id": "014129", "content": "若函数$y=(m+1) x^2+m x+(m-1)$的图像都在$x$轴下方 (不含$x$轴), 求$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343838,14 +343861,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014130": { "id": "014130", "content": "设$a \\in \\mathbf{R}$, 解关于$x$的不等式: $a x^2-(a+1) x+1<0$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343857,14 +343880,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014131": { "id": "014131", "content": "设$a \\in \\mathbf{R}$, 关于$x$的不等式$|x-3|<\\dfrac{x+a}{2}$的解集为$A$.\\\\\n(1) 若$a=2$, 求$A$;\\\\\n(2) 若$A=\\varnothing$, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343876,14 +343899,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014132": { "id": "014132", "content": "设$a \\in \\mathbf{R}$, 关于$x$的不等式$|x-3|<\\dfrac{x+a}{2}$的解集为$A$. 若$A \\cap \\mathbf{Z}=\\{3,4\\}$, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -343895,14 +343918,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014133": { "id": "014133", "content": "不等式$\\dfrac{x+5}{x^2-2 x+1} \\geq 1$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343921,7 +343944,7 @@ "content": "不等式$|x-2|+2|x+1|<5$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343940,7 +343963,7 @@ "content": "设$a \\in \\mathbf{R}$, 若关于$x$的不等式$x^2-a x<0$的解集是区间$(0,1)$的真子集, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343959,7 +343982,7 @@ "content": "若关于$x$的不等式$0 \\leq x^2-m x+2 \\leq 1$有且仅有一个实数解, 则实数$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -343978,7 +344001,7 @@ "content": "下列不等式中, 解集为$\\{x |-1b>c>d$, 则下列不等式恒成立的是\\bracket{20}.\n\\fourch{$a+d>b+c$}{$a+c>b+d$}{$a c>b d$}{$a d>b c$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344149,7 +344172,7 @@ "content": "若$x \\in \\mathbf{R}$, 比较大小: $2 x^2+5 x+3$\\blank{50}$x^2+4 x+2$. (用``$<$''、``$>$''、``$=$''连接)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344168,7 +344191,7 @@ "content": "若正数$x$、$y$满足$\\dfrac{1}{x}+y=4$, 则$\\dfrac{y}{x}$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344187,7 +344210,7 @@ "content": "若$x<\\dfrac{5}{4}$, 则函数$y=4 x-2+\\dfrac{1}{4 x-5}$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344206,7 +344229,7 @@ "content": "若正数$x$、$y$满足$\\dfrac{1}{x}+\\dfrac{9}{y}=1$, 则$x+y$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344225,7 +344248,7 @@ "content": "已知$\\dfrac{1}{a}<\\dfrac{1}{b}<0$, 有下列不等式: \\textcircled{1} $\\dfrac{1}{a+b}<\\dfrac{1}{a b}$; \\textcircled{2} $|a|+b>0$;\n\\textcircled{3} $a-\\dfrac{1}{a}\\ln b^2$. 其中所有恒成立的不等式的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344244,7 +344267,7 @@ "content": "已知$x>0, y>0$, 且$x+y+x y=8$, 求$x+y$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344256,14 +344279,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014152": { "id": "014152", "content": "已知$x>0, y>0$, 且$x+y+x y=8$, 求$4 x+y$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344275,14 +344298,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014153": { "id": "014153", "content": "已知$f(x)=2 \\lg x-1, g(x)=2 \\lg x-3$.\\\\\n(1) 若$|f(x)+g(x)|=|f(x)|+|g(x)|$, 求满足条件的$x$的取值范围;\\\\\n(2) 若$|f(x)|+|g(x)|$的最小值为$M, 0b$成立的充要条件为\\bracket{20}.\n\\fourch{$a^2>b^2$}{$a^3>b^3$}{$a>b-1$}{$a>b+1$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344320,7 +344343,7 @@ "content": "已知$a>0, b>0$, 且$a+b=1$, 有下列不等式: \\textcircled{1} $a^2+b^2 \\geq \\dfrac{1}{2}$, \\textcircled{2} $2^{a-b} \\geq \\dfrac{1}{2}$, \\textcircled{3} $\\log _2 a+\\log _2 b \\geq-2$, \\textcircled{4}$\\sqrt{a}+\\sqrt{b} \\leq \\sqrt{2}$. 其中所有正确的不等式的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344339,7 +344362,7 @@ "content": "对任意$x \\in \\mathbf{R}$, 不等式$|x-2|+|x-3| \\geq 2 a^2+a$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344358,7 +344381,7 @@ "content": "若正数$a$、$b$满足$a b=1$, 则$\\dfrac{1}{2 a}+\\dfrac{1}{2 b}+\\dfrac{8}{a+b}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344377,7 +344400,7 @@ "content": "已知$a, b$为非零实数, 则``$a>b$''是``$\\dfrac{1}{a}<\\dfrac{1}{b}$''的\\bracket{20}.\n\\fourch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分也非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344396,7 +344419,7 @@ "content": "已知$a>b>0$, 下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$a+b>2 \\sqrt{a b}$}{$a+b<2 \\sqrt{a b}$}{$\\dfrac{a}{2}+2 b>2 \\sqrt{a b}$}{$\\dfrac{a}{2}+2 b<2 \\sqrt{a b}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344415,7 +344438,7 @@ "content": "已知实数$a, b$满足$|\\lg a|=|\\lg b|$, 且$a \\neq b$, 那么$\\dfrac{1}{a}+\\dfrac{4}{b}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344434,7 +344457,7 @@ "content": "若函数$y=|2 x-a|+2|x+3|$($a>0$)的最小值为$9$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344453,7 +344476,7 @@ "content": "已知$x>0$, $y>0$, 且$x+3 y=x y$, 求$x+y$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344465,14 +344488,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014163": { "id": "014163", "content": "设$a, b, c \\in \\mathbf{R}$, 比较$a^2+b^2+c^2$与$a b+b c+c a$的值的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344484,14 +344507,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014164": { "id": "014164", "content": "若$x$、$y$、$z$是互不相等的正数, 则在$x(1-y)$、$y(1-z)$、$z(1-x)$三个值中, 大于$\\dfrac{1}{4}$的个数的最大值是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344510,7 +344533,7 @@ "content": "冬奥会期间, 冰墩墩成热销商品, 一家冰墩墩生产公司为加大生产, 计划租地建造临时仓库储存货物, 若记仓库到车站的距离为$x$(单位: $\\text{km}$), 经过市场调查了解到: 每月土地占地费$y_1$(单位: 万元) 与$x+1$成反比, 每月库存货物费$y_2$(单位: 万元) 与$4 x+1$成正比; 若在距离车站$5 \\text{km}$处建仓库, 则$y_1$与$y_2$分别为$12.5$万元和 7 万元. 记两项费用之和为$w$. 问这家公司应该把仓库建在距离车站多少千米处, 才能使两项费用之和最小? 并求出最小值.\n%04", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344522,14 +344545,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014166": { "id": "014166", "content": "已知$\\log _25=a$, 则$\\log _510=$\\blank{50}(用含$a$的代数式表示).", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344548,7 +344571,7 @@ "content": "函数$y=\\log _2 \\dfrac{x-1}{x+1}$的定义域是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344567,7 +344590,7 @@ "content": "若指数函数$y=a^x$($a>0$, $a \\neq 1$)在区间$[1,2]$上的最大值与最小值之和等于$12$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344586,7 +344609,7 @@ "content": "设幂函数$y=x^a$, $a \\in\\{-2,-1, \\dfrac{1}{2}, 1,2,3\\}$, 则``函数$y=x^a$的图像经过点$(-1,-1)$''是``函数$y=x^a$为奇函数''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": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344605,7 +344628,7 @@ "content": "证明: 函数$y=\\log _2 \\dfrac{x-1}{x+1}$在区间$(1,+\\infty)$上是严格增函数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344617,14 +344640,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014171": { "id": "014171", "content": "设非零实数$x, y, z$满足$3^x=4^y=6^z$, 求$\\dfrac{2 z}{x}+\\dfrac{z}{y}$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344636,14 +344659,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014172": { "id": "014172", "content": "设正数$p$、$q$满足$\\log _{16} p=\\log _{20} q=\\log _{25}(p+q)$, 求$\\dfrac{p}{q}$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344655,14 +344678,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014173": { "id": "014173", "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述. 两颗星的星等与亮度满足$m_2-m_1=\\dfrac{5}{2} \\lg \\dfrac{E_1}{E_2}$, 其中星等为$m_k$的星的亮度为$E_k$($k=1,2$). 已知太阳的星等是$-26.7$, 天狼星的星等是$-1.45$, 求太阳与天狼星的亮度的比值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344674,14 +344697,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014174": { "id": "014174", "content": "若正实数$x$、$y$满足$\\lg x=m$, $y=10^{m-1}$, 则$\\dfrac{x}{y}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344700,7 +344723,7 @@ "content": "若幂函数$y=x^a$的图像经过点$(\\sqrt[4]{3}, 3)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344719,7 +344742,7 @@ "content": "研究发现, 某昆虫释放信息素$t$秒后, 在距释放处$x$米的地方测得的信息素浓度$y$满足$\\ln y=-\\dfrac{1}{2} \\ln t-\\dfrac{k}{t} x^2+a$, 其中$k, a$为非零常数. 已知该昆虫释放信息素$1$秒后, 在距释放处$2$米的地方测得信息素浓度为$m$, 则该昆虫释放信息素$4$秒后, 距释放处的\\blank{50}米的位置, 信息素浓度为$\\dfrac{m}{2}$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344738,7 +344761,7 @@ "content": "若对任意负数$x$, 代数式$|x|+2 \\cdot \\sqrt[2022]{x^{2002}}+a \\cdot \\sqrt[2003]{x^{2023}}$恒为定值, 则实数$a$的取值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344757,7 +344780,7 @@ "content": "若$g(x)=\\begin{cases}\\log _2(x+1), & x \\geq 0, \\\\ 2^x+1, & x<0,\\end{cases}$则满足方程$g(x)=2$的$x$的值为\\blank{50}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344776,7 +344799,7 @@ "content": "设$y=x^{\\frac{1}{2}}-x^3$, 则满足不等式$y<0$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344795,7 +344818,7 @@ "content": "服用某种感冒药, 每次服用的药物含量为$a$, 随着时间$t$的变化, 体内的药物含量为$y=0.57^a$(其中$t$以小时为单位), 则服药$2$小时后体内药物的含量为服药 $4$小时后体内药物含量的\\blank{50}倍.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344814,7 +344837,7 @@ "content": "下列命题中正确的是\\bracket{20}.\n\\twoch{若$a>0$, $x_2>x_1>0$, 则$(\\dfrac{x_2}{x_1})^a<1$}{若$a>0$, $x_1>x_2>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}{若$a<0$, $x_2>x_1>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}{若$a<0$, $x_1>x_2>0$, 则$(\\dfrac{x_2}{x_1})^a>1$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344833,7 +344856,7 @@ "content": "已知常数$a>0$且$a \\neq 1$, $b \\in \\mathbf{R}$, 函数$y=a^x+b$的定义域和值域都是$[-1,0]$, 求$a+b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344845,14 +344868,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014183": { "id": "014183", "content": "定义: 若一个对数可以用$\\lg 2$、$\\lg 3$和整数的线性组合表示, 则称$\\lg 2$和$\\lg 3$为该对数的``基本对数''. 如: $\\lg 12=2 \\lg 2+\\lg 3$, 故$\\lg 2$和$\\lg 3$为$\\lg 12$的``基本对数''. 下列对数中, 所有以$\\lg 2$和$\\lg 3$为``基本对数''的对数的序号为\\blank{50}.\n\\textcircled{1} $\\log _34$; \\textcircled{2} $\\log _{42} 56$; \\textcircled{3} $\\lg 15$; \\textcircled{4} $\\log _2225$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -344871,7 +344894,7 @@ "content": "我们知道当$a>0$时, $a^{m+n}=a^m \\cdot a^n$对一切$m, n \\in \\mathbf{R}$都成立. 学生小贤在进一步研究指数幂的性质时, 发现有这么一个等式$2^{1+1}=2^1+2^1$, 带着好奇, 他进一步对$2^{m+n}=2^m+2^n$进行深入研究.\\\\\n(1) 当$m=2$时, 求$n$的值;\\\\\n(2) 当$m \\leq 0$时, 求证: $n$是不存在的.\n%05", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -344883,14 +344906,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014185": { "id": "014185", "content": "下列四组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\twoch{$y=x$与$y=(\\dfrac{1}{x})^{-1}$}{$y=|x|$与$y=\\begin{cases}x, & x>0, \\\\ -x, & x \\leq 0\\end{cases}$}{$y=2 \\ln x$与$y=\\ln x^2$}{$y=x$与$y=\\sqrt[6]{x^6}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344909,7 +344932,7 @@ "content": "下列函数是偶函数的为\\bracket{20}.\n\\fourch{$y=\\sin x$}{$y=\\cos x$}{$y=x^3$}{$y=2^x$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -344928,7 +344951,7 @@ "content": "某企业经营一款节能环保产品, 其成本由研发成本与生产成本两部分构成. 研发成本固定为 60 万元, 生产成本为每台 130 元. 根据市场调研, 若该产品产量为$x$万台时, 每万台产品的销售收入为$220-x$($00$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345035,14 +345058,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014193": { "id": "014193", "content": "若函数$y=a-\\dfrac{2}{2^x+1}$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345061,7 +345084,7 @@ "content": "满足不等式$\\ln x+x<1$的实数$x$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345080,7 +345103,7 @@ "content": "将函数$y=|x|$的图像绕坐标原点逆时针方向旋转角$\\theta$($0<\\theta<\\alpha$), 得到曲线$C$. 若对于每一个旋转角$\\theta$, 曲线$C$都可以看成是某一个函数的图像, 则$\\alpha$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345099,7 +345122,7 @@ "content": "下列各组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\twoch{$y=\\dfrac{x^2-1}{x-1}$与$y=x+1$}{$y=x-1$与$y=\\sqrt{x^2-2 x+1}$}{$y=\\sqrt{x^2}$与$y=(\\sqrt{x})^2$}{$y=|x+1|$与$y=\\begin{cases}x+1, & x \\geq-1, \\\\ -x-1, & x<-1\\end{cases}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345118,7 +345141,7 @@ "content": "下列函数中, 既是奇函数又在区间$(0,1)$上是严格增函数的是\\bracket{20}.\n\\fourch{$y=\\sqrt{x}$}{$y=-x^3$}{$y=\\lg x$}{$y=\\sin x$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345137,7 +345160,7 @@ "content": "若函数$y=\\lg \\dfrac{x+1}{x+a}$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345156,7 +345179,7 @@ "content": "已知集合$A=\\{x | \\dfrac{2}{x-2} \\geq 1, \\ x \\in \\mathbf{R}\\}$, 设函数$y=\\log _{\\frac{1}{2}} x+a,\\ x \\in A$的值域为$B$, 若$B \\subseteq A$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345175,7 +345198,7 @@ "content": "若函数$y=\\dfrac{1}{2} x^2-x+\\dfrac{3}{2}$的定义域和值域都是$[1, b]$, 求实数$b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345187,14 +345210,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014201": { "id": "014201", "content": "已知函数$y=x^3+a \\sin x-2$, 当$x=2023$时, 其函数值为$3$, 求当$x=-2023$时该函数的函数值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345206,14 +345229,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014202": { "id": "014202", "content": "命题$\\alpha$: 定义在$\\mathbf{R}$上的函数$y=f(x)$一定能表示成一个定义在$\\mathbf{R}$上的偶函数$y=g(x)$与定义在$\\mathbf{R}$上的奇函数$y=h(x)$的和(即$f(x)=g(x)+h(x)$, 下同); 命题$\\beta$: 定义在$\\mathbf{R}$上的严格增函数$y=f(x)$一定能表示成一个定义在$\\mathbf{R}$上的严格增函数$y=p(x)$与定义在$\\mathbf{R}$上的严格减函数$y=q(x)$的和. 下列判断正确的是\\bracket{20}.\n\\twoch{$\\alpha$、$\\beta$均为真命题}{$\\alpha$、$\\beta$均为假命题}{$\\alpha$为真命题, $\\beta$为假命题}{$\\alpha$为假命题, $\\beta$为真命题}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345232,7 +345255,7 @@ "content": "求满足方程$12^x+5^x=13^x$的实数$x$的值.\n%08", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345244,14 +345267,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014204": { "id": "014204", "content": "若扇形$AOB$的周长是$32$, 则该扇形面积的最大值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345270,7 +345293,7 @@ "content": "在直角坐标系$xOy$中, 角$\\alpha$的顶点与坐标原点重合, 始边与$x$轴的正半轴重合. 若角$\\alpha$的终边经过点$(-3,4)$, 则$\\sin (\\alpha+\\pi)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345289,7 +345312,7 @@ "content": "若$\\sin (\\alpha+\\beta)=\\dfrac{1}{3}$, $\\sin (\\alpha-\\beta)=\\dfrac{1}{2}$, 则$\\sin \\alpha \\cos \\beta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345308,7 +345331,7 @@ "content": "若$\\cos \\alpha=-\\dfrac{\\sqrt{5}}{5}$,$\\alpha \\in(0, \\pi)$, 则$\\tan 2 \\alpha=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345327,7 +345350,7 @@ "content": "若角$x$满足$3 \\sin 2 x=2 \\sin x$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345346,7 +345369,7 @@ "content": "证明下列恒等式:\\\\\n(1) $\\dfrac{1+\\sin 2 \\alpha-\\cos 2 \\alpha}{1+\\sin 2 \\alpha+\\cos 2 \\alpha}=\\tan \\alpha$;\\\\\n(2) $\\dfrac{\\cos ^2 \\alpha-\\cos ^2 \\beta}{\\cot ^2 \\alpha-\\cot ^2 \\beta}=\\sin ^2 \\alpha \\sin ^2 \\beta$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345358,14 +345381,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014210": { "id": "014210", "content": "已知$\\dfrac{1-\\cos 2 \\alpha}{\\sin \\alpha \\cos \\alpha}=1$, $\\tan (\\beta-\\alpha)=-\\dfrac{1}{3}$, 求$\\tan (\\beta-2 \\alpha)$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345377,14 +345400,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014211": { "id": "014211", "content": "已知$\\dfrac{1-\\cos 2 \\alpha}{\\sin \\alpha \\cos \\alpha}=1$, $\\tan (\\beta-\\alpha)=-\\dfrac{1}{3}$, $\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 求$\\beta-2 \\alpha$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345396,14 +345419,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014212": { "id": "014212", "content": "如图, 在平面直角坐标系$xOy$中, 点$A(\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2})$在以原点$O$为圆心的单位圆上, 将射线$OA$绕原点$O$逆时针方向旋转$\\alpha$后交该圆于点$B$, 设点$B$的横坐标为$f(\\alpha)$, 纵坐标为$g(\\alpha)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (45:1) node [above] {$A$} coordinate (A);\n\\draw (125:1) node [above] {$B$} coordinate (B);\n\\draw (0,0) circle (1);\n\\draw (O) pic [draw, \"$\\alpha$\", scale = 0.5, angle eccentricity = 1.5] {angle = A--O--B};\n\\draw (A)--(O)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 化简$f(\\alpha)+g(\\alpha)$;\\\\\n(2) 如果$\\dfrac{f(\\alpha)}{g(\\alpha)}=2$, 求$f(\\alpha) \\cdot g(\\alpha)$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345415,14 +345438,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014213": { "id": "014213", "content": "``$\\alpha=\\beta$''是``$\\sin ^2 \\alpha+\\cos ^2 \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345441,7 +345464,7 @@ "content": "已知点$(-2, y)$在角$\\alpha$的终边上, 若$\\tan (\\pi-\\alpha)=2 \\sqrt{2}$, 则$\\sin \\alpha=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345460,7 +345483,7 @@ "content": "若角$x$满足$2 \\cos (x-\\dfrac{\\pi}{4})=1$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345479,7 +345502,7 @@ "content": "若角$\\alpha$的终边与单位圆$x^2+y^2=1$交于点$P(\\dfrac{1}{2}, y)$, 则$\\sin (\\dfrac{\\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345498,7 +345521,7 @@ "content": "若角$\\alpha$的终边经过点$P(3,4)$, 将角$\\alpha$的终边绕原点$O$逆时针旋转$\\dfrac{\\pi}{2}$得到角$\\beta$的终边, 则$\\tan \\beta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345517,7 +345540,7 @@ "content": "已知$\\alpha \\in(0, \\pi)$, 若$1-2 \\sin 2 \\alpha=\\cos 2 \\alpha$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345536,7 +345559,7 @@ "content": "已知关于$x$的方程$x^2+3 a x+3 a+1=0$($a>2$)的两实数根分别是$\\tan \\alpha, \\tan \\beta$, 且$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 则$\\alpha+\\beta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345555,7 +345578,7 @@ "content": "已知$\\cos \\theta=-\\dfrac{\\sqrt{2}}{3}$, $\\theta \\in(\\dfrac{\\pi}{2}, \\pi)$, 求$\\dfrac{2}{\\sin 2 \\theta}-\\dfrac{\\cos \\theta}{\\sin \\theta}$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345567,14 +345590,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014221": { "id": "014221", "content": "已知$\\cos (2 \\alpha-\\beta)=-\\dfrac{11}{14}$, $\\sin (\\alpha-2 \\beta)=\\dfrac{4 \\sqrt{3}}{7}$, $0<\\beta<\\dfrac{\\pi}{4}<\\alpha<\\dfrac{\\pi}{2}$, 求$\\alpha+\\beta$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345586,14 +345609,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014222": { "id": "014222", "content": "已知$\\tan \\alpha$是关于$x$的方程$x^2+\\dfrac{2 x}{\\cos \\alpha}+1=0$两个实数根中较小的根, 求$\\alpha$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345605,14 +345628,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014223": { "id": "014223", "content": "设$\\alpha, \\beta$均为锐角, 且满足$3 \\sin ^2 \\alpha+2 \\sin ^2 \\beta=1$, $2 \\sin 2 \\beta-3 \\sin 2 \\alpha=0$. 求证: $\\alpha+2 \\beta=\\dfrac{\\pi}{2}$.\n%09", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345624,14 +345647,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014224": { "id": "014224", "content": "在$\\triangle ABC$中, 若$AB=\\sqrt{2}$, $AC=2$, $A=45^{\\circ}$, 则$\\triangle ABC$的面积$S=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345650,7 +345673,7 @@ "content": "在$\\triangle ABC$中, 若$AB=4 \\sqrt{3}$, $A=45^{\\circ}$, $C=60^{\\circ}$, 则边$BC=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345669,7 +345692,7 @@ "content": "在$\\triangle ABC$中, 若$AB=2$, $AC=2$, $A=60^{\\circ}$, 则$\\triangle ABC$外接圆的半径为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345688,7 +345711,7 @@ "content": "已知角$A$、$B$、$C$是$\\triangle ABC$的三个内角, 若$\\sin A: \\sin B: \\sin C=4: 5: 7$, 则该三角形的最大内角等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345707,7 +345730,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$b \\cos C+c \\cos B=a \\sin A$, 则$\\triangle ABC$的形状为\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{不确定}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345726,7 +345749,7 @@ "content": "如图, 某观测站$C$在$A$城的南偏西$20^{\\circ}$方向上, 由$A$城出发有一条公路走向是南偏东$40^{\\circ}$, 测得距$C$点$31$千米的$B$处有一人开车正沿公路向$A$城行驶, 行驶了$20$千米后到达$D$处, 此时$C$、$D$间的距离为$21$千米. 问: 此人还需行驶多少千米才到$A$城?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-110:2.4) node [left] {$C$} coordinate (C);\n\\draw (-50:3.5) node [right] {$B$} coordinate (B);\n\\draw (-50:1.5) node [above right] {$D$} coordinate (D);\n\\draw (A)--(C)--(B)--cycle (C)--(D);\n\\draw [->] (1.7,-1) -- (2.6,-1) node [right] {东};\n\\draw [->] (2,-1.3) -- (2,-0.4) node [above] {北}; \n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345738,14 +345761,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014230": { "id": "014230", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若满足$A=\\dfrac{\\pi}{3}$, $a=4$的$\\triangle ABC$恰有一个, 则$c$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345764,7 +345787,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$.\\\\\n(1) 若$a=3$, $b=2c$, $2 \\sin B-\\sin C=1$, 求$\\triangle ABC$的周长;\\\\\n(2) 若$B=\\dfrac{2 \\pi}{3}$, $b=2 \\sqrt{3}$, 求$\\triangle ABC$面积的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345776,14 +345799,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014232": { "id": "014232", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$a=8$, $b=5$, $c=\\sqrt{153}$则$\\triangle ABC$的面积$S=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345802,7 +345825,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若满足$b \\sin A=a \\cos (B-\\dfrac{\\pi}{6})$. 则角$B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345821,7 +345844,7 @@ "content": "已知$\\triangle ABC$, 那么``$|\\overrightarrow{AC}|^2+|\\overrightarrow{AB}|^2-|\\overrightarrow{BC}|^2<0$''是``$\\triangle ABC$为钝角三角形''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": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345840,7 +345863,7 @@ "content": "某公司要在$A$、$B$两地连线上的定点$C$处建造广告牌$CD$, 其中$D$为顶端, $AC$长$35$米, $CB$长$80$米, 设$A$、$B$在同一水平面上, 从$A$和$B$看$D$的仰角分别为$\\alpha$和$\\beta$, 现测得$\\alpha=28.12^{\\circ}$, $\\beta=18.45^{\\circ}$, 求$AD$与$CD$的长. (结果精确到$0.01$米)", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345852,14 +345875,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014236": { "id": "014236", "content": "在$\\triangle ABC$中, 若$AC=3$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac{1}{4}$, 则$AB=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345878,7 +345901,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$. 若$b \\cos A+(a+4 c) \\cos B=0$, 则$B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -345897,7 +345920,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 其中$a=4, c=6, \\cos C=\\dfrac{1}{8}$.\\\\\n(1) 求$\\sin A$的值;\\\\\n(2) 求$b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345909,14 +345932,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014239": { "id": "014239", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$.\\\\\n(1) 若$\\triangle ABC$的面积$S=\\dfrac{a^2+c^2-b^2}{4}$, 求$B$;\\\\\n(2) 若$a c=\\sqrt{3}$, $\\sin A=\\sqrt{3} \\sin B$, $C=\\dfrac{\\pi}{6}$, 求$c$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345928,14 +345951,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014240": { "id": "014240", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$c \\sin C-b \\sin B=a(\\sin A-\\sin B)$.\\\\\n(1) 求角$C$的值;\\\\\n(2) 若$c=3$, 求$\\triangle ABC$周长的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345947,14 +345970,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014241": { "id": "014241", "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$b=2$.\\\\\n(1) 若$A+C=\\dfrac{2 \\pi}{3}$, 且$a=2c$, 求$c$的值;\\\\\n(2) 若$A-C=\\dfrac{\\pi}{12}$, $a=\\sqrt{2} c \\sin A$, 求$\\triangle ABC$面积$S$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -345966,14 +345989,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014242": { "id": "014242", "content": "在$\\triangle ABC$中, $A=\\dfrac{\\pi}{3}$, 则``$\\sin B<\\dfrac{1}{2}$''是``$\\triangle ABC$是钝角三角形''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": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -345992,7 +346015,7 @@ "content": "在临港滴水湖畔拟建造一个四边形的露营基地, 如图$ABCD$所示. 为考虑露营客人娱乐休闲的需求, 在四边形$ABCD$区域中, 将$\\triangle ABD$区域设立成花卉观赏区, $\\triangle BCD$区域设立成烧烤区, 边$AB$、$BC$、$CD$、$DA$修建成观赏步道, 边$BD$修建隔离防护栏. 其中$CD=100$米, $BC=200$米, $\\angle A=\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw (0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0) node [left] {$D$} coordinate (D);\n\\draw (-80:2) node [right] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(3)}!30:(D)$) coordinate (O);\n\\draw ($(O)!1!100:(D)$) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (D) -- (B);\n\\draw (barycentric cs:A=1,B=1,D=1) node {花卉观赏区};\n\\draw (barycentric cs:B=1,C=1,D=1) node {烧烤区};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果烧烤区是一个占地面积为$9600$平方米的钝角三角形, 那么需要修建多长的隔离防护栏(精确到$0.1$米)?\\\\\n(2) 考虑到烧烤区的安全性, 在规划四边形$ABCD$区域时, 首先保证烧烤区的占地面积最大时, 再使得花卉观赏区的面积尽可能大, 则应如何设计观赏步道?\n%10", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346004,14 +346027,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014244": { "id": "014244", "content": "函数$y=\\tan (\\dfrac{\\pi}{3} x-\\dfrac{\\pi}{5})$的单调区间是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346030,7 +346053,7 @@ "content": "若函数$y=\\sin (\\omega x)$(其中常数$\\omega \\neq 0$) 的最小正周期为$\\dfrac{\\pi}{3}$, 则$\\omega=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346049,7 +346072,7 @@ "content": "函数$y=2 \\sin (x-\\dfrac{\\pi}{3})$, $x \\in[0, \\dfrac{\\pi}{2}]$的值域是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346068,7 +346091,7 @@ "content": "在下列函数中, 既在区间$(0, \\dfrac{\\pi}{2})$上是严格增函数, 又是以$\\pi$为最小正周期的偶函数的是\\bracket{20}\n\\fourch{$y=\\sin x$}{$y=\\cos 2 x$}{$y=|\\sin x|$}{$y=\\sin |2 x|$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -346087,7 +346110,7 @@ "content": "已知$f(x)=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$, 若函数$y=f(x)-k$在区间$[0, \\dfrac{\\pi}{4}]$上有两个不同的零点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346106,7 +346129,7 @@ "content": "已知$f(x)=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$.\\\\\n(1) 求函数$y=f(x)$的单调减区间;\\\\\n(2) 求函数$y=f(x)$在区间$[0, \\dfrac{\\pi}{4}]$上的最大值和最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346118,14 +346141,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014250": { "id": "014250", "content": "求函数$y=2 \\sin x \\cos x+\\sqrt{3} \\cos 2 x$在区间$(0, \\pi)$上的单调减区间和单调增区间;", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346137,14 +346160,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014251": { "id": "014251", "content": "已知$f(x)=\\sin (\\pi x+\\varphi)-\\sqrt{3} \\cos (\\pi x+\\varphi)$($0<\\varphi<\\pi$).\\\\\n(1) 若函数$y=f(x)$是偶函数, 求$\\varphi$的值;\\\\\n(2) 若函数$y=f(x)$的图像关于直线$x=1$对称, 求$\\sin 2 \\varphi$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346156,14 +346179,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014252": { "id": "014252", "content": "已知$f(x)=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$), 函数$y=f(x)$的图像与$y$轴的交点为$(0,-\\sqrt{3})$, 它在$y$轴右侧的第一个最高点和第一个最低点的坐标分别为$(x_0, 2)$和$(x_0+\\dfrac{\\pi}{2},-2)$.\\\\\n(1) 求函数$y=f(x)$的表达式及$x_0$的值;\\\\\n(2) 设函数$y=a f(x)+b$, $x\\in [-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$的值域为$[0,3]$, 求实数$a, b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346175,14 +346198,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014253": { "id": "014253", "content": "设函数$y=-2\\sin(2x-\\dfrac \\pi 3)+5+(-1)^n \\cdot m$($m \\in \\mathbf{R}$, $n$是正整数), 若该函数对任意的$x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$均有$y>0$恒成立, 求实数$m$取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346194,14 +346217,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014254": { "id": "014254", "content": "函数$y=\\sin 3 x+|\\sin 3 x|$\\bracket{20}.\n\\twoch{为周期函数, 且最小正周期为$\\dfrac{\\pi}{3}$}{为周期函数, 且最小正周期为$\\dfrac{2 \\pi}{3}$}{为周期函数, 且最小正周期为$2 \\pi$}{不是周期函数}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -346220,7 +346243,7 @@ "content": "若函数$y=3 \\cos (2 x+\\varphi)$的图像关于点$(\\dfrac{\\pi}{3}, 0)$对称, 则$|\\varphi|$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346239,7 +346262,7 @@ "content": "已知$f(x)=\\sin 2 x+\\sin (2 x+\\dfrac{\\pi}{3})$, 若将函数$y=f(x)$的图像向左平移$\\varphi$($\\varphi>0$)个单位长度后得到的图像关于$y$轴对称, 则$\\varphi$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346258,7 +346281,7 @@ "content": "若函数$y=\\cos x-\\sin x$在区间$[-a, a]$上是严格减函数, 则$a$的最大值是\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{3 \\pi}{4}$}{$\\pi$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -346277,7 +346300,7 @@ "content": "已知$f(x)=\\sin x+a \\cos x$($a$为常数) 满足$f(x) \\leq f(\\dfrac{\\pi}{6})$. 若函数$y=f(x)$在区间$[x_1, x_2]$上具有单调性, 且满足$f(x_1)+f(x_2)=0$, 求$|x_1+x_2|$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346289,14 +346312,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014259": { "id": "014259", "content": "函数$y=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346315,7 +346338,7 @@ "content": "函数$y=\\tan 2 x$在区间$(-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4})$上的零点为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346334,7 +346357,7 @@ "content": "已知函数$y=\\cos (\\omega x+\\varphi)$($\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$), 若该函数图像的对称中心到其对称轴$x=\\dfrac{\\pi}{6}$的距离的最小值为$\\dfrac{\\pi}{8}$, 则该函数的表达式为$y=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346353,7 +346376,7 @@ "content": "已知$f(x)=(x-6)^2 \\cdot \\sin (\\omega x)$($\\omega \\in \\mathbf{R}$), 若存在常数$a \\in \\mathbf{R}$, 使得$y=f(x+a)$为偶函数, 则$\\omega$的值可以为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{5}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -346372,7 +346395,7 @@ "content": "已知$f(x)=\\sin (\\omega x-\\dfrac{\\pi}{6})+k$($\\omega>0$), 若$f(x) \\leq f(\\dfrac{\\pi}{3})$对任意的实数$x$成立, 则$\\omega$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346391,7 +346414,7 @@ "content": "已知$f(x)=\\sin x \\cos x-\\sin ^2 x$.\\\\\n(1) 求函数$y=f(x)$的最小值, 并求出此时相应的$x$的值;\\\\\n(2) 写出函数$y=f(x)$, $x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$的单调区间, 并求该函数的值域.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346403,14 +346426,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014265": { "id": "014265", "content": "已知$f(x)=\\sin x$, 若存在$x_1, x_2, \\cdots, x_m$满足$0 \\leq x_10$, $0<\\varphi<\\pi$), 函数$y=f(x)$的最小正周期为$\\pi$, 且直线$x=-\\dfrac{\\pi}{2}$是其图像的一条对称轴.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{4}$个单位, 再将所得的图像上每一点的纵坐标不变, 横坐标伸长为原来的$2$倍后得到函数$y=g(x)$的图像, 设常数$\\lambda \\in \\mathbf{R}$, $n$为正整数, 且函数$y=f(x)+\\lambda g(x)$在区间$(0, n \\pi)$内恰有$2023$个零点, 求常数$\\lambda$与$n$的值.\n%11", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346441,14 +346464,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014267": { "id": "014267", "content": "若角$\\alpha$的终边经过点$P(6,-8)$, 则$\\sin (\\dfrac{3 \\pi}{2}+\\alpha)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346467,7 +346490,7 @@ "content": "已知$\\triangle ABC$的面积是$\\dfrac{1}{2}$, $AB=1$, $BC=\\sqrt{2}$, 则$AC=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346486,7 +346509,7 @@ "content": "若函数$y=\\sin (x+\\theta)$(其中常数$\\theta \\in[0, \\pi)$) 是$\\mathbf{R}$上的偶函数, 则满足$\\sin (x+\\theta)=\\dfrac{1}{2}$的角$x$的集合为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346505,7 +346528,7 @@ "content": "已知函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $|\\varphi| \\leq \\dfrac{\\pi}{2}$)图像的一部分如图所示, 则$y=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:3, samples = 100] plot (\\x,{2*sin(2*\\x/pi*180-30)});\n\\draw ({-5*pi/12},0) node [below left] {$-\\dfrac{5\\pi}{12}$};\n\\draw ({pi/3},0) node [below] {$\\dfrac\\pi 3$};\n\\draw [dashed] ({pi/3},0) --++ (0,2) -- (0,2) node [left] {$2$};\n\\draw [dashed] ({-pi/6},-2) -- (0,-2) node [right] {$-2$};\n\\draw (0,-1) node [left] {$-1$};\n\\filldraw (0,-1) circle (0.03);\n\\filldraw ({-5*pi/12},0) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2 \\sin (2 x-\\dfrac{\\pi}{6})$}{$2 \\sin (2 x+\\dfrac{\\pi}{6})$}{$2 \\sin (2 x-\\dfrac{\\pi}{3})$}{$2 \\sin (2 x+\\dfrac{\\pi}{3})$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -346524,7 +346547,7 @@ "content": "函数$y=-\\sin ^2 x-4 \\cos x+6$的值域是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346543,7 +346566,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 且$\\triangle ABC$的面积$S$满足$S=\\dfrac{\\sqrt{3}}{2} \\overrightarrow{BA} \\cdot \\overrightarrow{BC}$, $b=2$, 求$a+c$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346555,14 +346578,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014273": { "id": "014273", "content": "已知$f(x)=2 \\sin (x+\\dfrac{\\pi}{3})$.\\\\\n(1) 若对任意的$x \\in[-\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3}]$, 不等式$|f(x)-m| \\leq 3$恒成立, 求实数$m$的取值范围;\\\\\n(2) 画出函数$y=f(x-\\dfrac{\\pi}{3})+f(x)$, $x \\in[0, \\dfrac{\\pi}{2}]$的大致图像, \n并写出满足$f(x-\\dfrac{\\pi}{3})+f(x)=\\sqrt{10}$的锐角$x$的集合.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346574,14 +346597,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014274": { "id": "014274", "content": "已知$f(x)=2 \\sin (x+\\dfrac{\\pi}{3})$.\\\\\n(1) 若满足$f(x-\\dfrac{\\pi}{3})+f(x)=a$的锐角$x$有两个, 求实数$a$的取值范围;\\\\\n(2) 若满足$f(x-\\dfrac{\\pi}{3})+f(x)=a$的锐角$x$只有一个, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346593,14 +346616,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014275": { "id": "014275", "content": "如图, 某污水处理厂要在一个矩形污水处理池$ABCD$的池底水平铺设污水净化管道$EH$、$FH$、$EF$来处理污水, 管道越长, 污水净化效果越好. 要求管道的接口$H$是$AB$的中点, $F$分别落在线段$BC$、$AD$上(含线段两端点), $FH \\perp HE$. 已知$AB=40$米, $AD=20 \\sqrt{3}$米, 记$\\angle BHE=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\draw (4,{2*sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (0,{2*sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) rectangle (C);\n\\draw ($(A)!0.5!(B)$) node [below] {$H$} coordinate (H);\n\\draw [dashed] (H) --++ (-2,{4/3}) node [left] {$F$} coordinate (F);\n\\draw [dashed] (H) --++ (2,3) node [right] {$E$} coordinate (E);\n\\draw [dashed] (E)--(F);\n\\draw (H) pic [draw, \"$\\theta$\", angle eccentricity = 1.5] {angle = B--H--E};\n\\draw (H) pic [draw, scale = 0.5] {right angle = E--H--F};\n\\end{tikzpicture}\n\\end{center}\n(1) 试将污水净化管道的总长度$L$(即$\\triangle FHE$的周长)表示为$\\theta$的函数, 并求出其定义域;\\\\\n(2) 问$\\theta$取何值时, 污水净化效果最好? 并求出此时管逭的总长度$L$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346612,14 +346635,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014276": { "id": "014276", "content": "若角$\\alpha$的终边经过点$P(4,-3)$, 则$\\sin (\\dfrac{\\pi}{2}+2 \\alpha)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346638,7 +346661,7 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$A=60^{\\circ}, a=\\sqrt{6}, b=2$, 则$\\cos C=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346657,7 +346680,7 @@ "content": "若对任意实数$x$, 不等式$m+\\cos ^2 x<3+2 \\sin x$恒成立, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346676,7 +346699,7 @@ "content": "已知向量$\\overrightarrow {m}=(\\dfrac{1}{2}, \\dfrac{1}{2} \\sin 2 x+\\dfrac{\\sqrt{3}}{2} \\cos 2 x)$, $\\overrightarrow {n}=(f(x),-1)$, 且$\\overrightarrow {m} \\perp \\overrightarrow {n}$.\\\\\n(1) 求函数$y=f(x), x \\in[0, \\pi]$的单调增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$f(A-\\dfrac{\\pi}{12})=1$, $BC=\\sqrt{3}$, 求$\\triangle ABC$面积的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346688,14 +346711,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014280": { "id": "014280", "content": "若角$\\alpha$的终边经过点$P(-x,-6)$, 且$\\cos \\alpha=-\\dfrac{5}{13}$, 则$\\tan \\alpha=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346714,7 +346737,7 @@ "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{12}=7 \\pi$, 则$\\cos (a_6+a_7)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346733,7 +346756,7 @@ "content": "已知函数$y=6 \\cos ^2 \\omega x+\\sqrt{3} \\sin 2 \\omega x-3$(其中常数$\\omega>0$) 的最小正周期为$8$, 则函数的单调减区间是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346752,7 +346775,7 @@ "content": "设$\\omega>0$, 若函数$y=\\sin \\omega x$在区间$[0, \\pi]$上恰有两个零点, 则实数$\\omega$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346771,7 +346794,7 @@ "content": "已知$f(x)=2 \\sin \\omega x \\cos \\omega x-2 \\sqrt{3} \\cos ^2 \\omega x+\\sqrt{3}$(其中常数$\\omega>0$), 函数$y=f(x)$图像的两条相邻的对称轴之间的距离为$\\dfrac{\\pi}{2}$.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 角$C$为锐角, 且$f(C)=\\sqrt{3}$, $c=3$, $\\sin B=2 \\sin A$, 求$\\triangle ABC$的周长.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346783,14 +346806,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014285": { "id": "014285", "content": "如图, 某地有三家工厂分别位于矩形$ABCD$的两个顶点$A$、$B$及$CD$的中点$P$处. $AB=20 \\text{km}, BC=10 \\text{km}$. 为了处理这三家工厂的污水, 现要在该矩形区域内 (含边界) 且与$A$、$B$等距离的一点$O$处, 建造一个污水处理厂, 并铺设三条排污管道$OA$、$OB$、$OP$. 记排污管道的总长度为$y \\text{km}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (3,1.5) node [right] {$C$} coordinate (C);\n\\draw (0,1.5) node [left] {$D$} coordinate (D);\n\\draw (1.5,0.75) node [below] {$O$} coordinate (O);\n\\draw ($(C)!0.5!(D)$) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (P) -- (O) (A) -- (O) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$\\angle BAO=\\theta$, 将$y$表示成$\\theta$的函数并求其定义域;\\\\\n(2) 确定污水处理厂的位置, 使排污管道的总长度$y$最短, 并求出其值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346802,14 +346825,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014286": { "id": "014286", "content": "已知函数$y=\\sin (\\omega x-\\dfrac{\\pi}{3})$($\\omega>0$), 记$y=f(x)$. 对任意$x_1, x_2 \\in \\mathbf{R}$, 当$|f(x_1)-f(x_2)|=2$时, $|x_1-x_2|$的最小值是$\\dfrac{\\pi}{3}$, 则函数$y=f(x)$, $x \\in[0, \\dfrac{\\pi}{2}]$的单调减区间是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -346828,7 +346851,7 @@ "content": "在锐角$\\triangle ABC$中, 角$A$、$B$及$C$所对边的边长分别为$a$、$b$及$c$, 若$A=2B$.\\\\\n(1) 求$B$的取值范围;\\\\\n(2) 求$\\dfrac{a}{b}+\\dfrac{b}{a}$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -346840,8 +346863,2136 @@ "same": [], "related": [], "remark": "", + "space": "12ex" + }, + "014288": { + "id": "014288", + "content": "函数$y=\\begin{cases}x^2-2, & x \\leq 0, \\\\ 2 x-6+\\ln x, & x>0\\end{cases}$的零点的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", "space": "" }, + "014289": { + "id": "014289", + "content": "已知$a$、$b$、$\\alpha$、$\\beta$为实数, $a\\log _{\\frac{1}{2}}(3 x)$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014291": { + "id": "014291", + "content": "已知$f(x)=2 x^2-x$, 证明: 函数$y=f(x)$图像上的点都在直线$y=3 x-3$的上方.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014292": { + "id": "014292", + "content": "已知$a \\in \\mathbf{R}$, $3$是函数$y=f(x)$, $x \\in \\mathbf{R}$的一个周期, 当$x \\in[0,3)$时, $f(x)=|x^2-2 x+\\dfrac{1}{2}|$. 若函数$y=f(x)-a$在区间$[-3,4]$上有$10$个零点(互不相同), 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014293": { + "id": "014293", + "content": "已知$a \\in \\mathbf{R}$, $f(x)=x^2+a(\\mathrm{e}^x+\\mathrm{e}^{-x})-1$, 试判断是否存在实数$a$, 使方程$f(x)=0$有且只有一个实数解? 若存在, 求$a$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014294": { + "id": "014294", + "content": "已知$a \\in \\mathbf{R}$, 定义域为$\\mathbf{R}$的函数$y=f(x)$满足: 对任意$x_1a$成立, 且$f(0)=-a^2$. 证明: 对任意常数$a$, 关于$x$的方程$f(x+a)=a x$有且仅有$1$个解.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014295": { + "id": "014295", + "content": "函数$y=x^2-3 x-\\dfrac{2}{x}$的零点的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014296": { + "id": "014296", + "content": "已知$a \\in \\mathbf{R}$, 若关于$x$的方程$|3^x-1|-2 a=0$有唯一解, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014297": { + "id": "014297", + "content": "已知$t \\in \\mathbf{R}$, 若关于$x$的方程$(x-1)^2+2 t^2+6 t=2 t \\cos (x-1)+16$有且仅有一个实数解, 则$t$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014298": { + "id": "014298", + "content": "设$\\theta \\in \\mathbf{R}$. 若$f(x)=3 \\sin x+2$满足: 对任意$x_1 \\in[0, \\dfrac{\\pi}{2}]$, 都存在$x_2 \\in[0, \\dfrac{\\pi}{2}]$, 使得$f(x_1)+2 f(x_2+\\theta)=3$, 则$\\theta$可以是\\bracket{20}.\n\\fourch{$\\dfrac{3 \\pi}{5}$}{$\\dfrac{4 \\pi}{5}$}{$\\dfrac{6 \\pi}{5}$}{$\\dfrac{7 \\pi}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014299": { + "id": "014299", + "content": "已知$f(x)=(\\dfrac{1}{2})^x-\\log _3 x$, $x_0$是函数$y=f(x)$的零点, 实数$a$、$b$、$c$满足$0c$. 其中一定不成立的不等式的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014300": { + "id": "014300", + "content": "已知$k \\in \\mathbf{R}$, 当$0 \\leq x \\leq 1$时, 不等式$\\sin \\dfrac{\\pi x}{2} \\geq k x$成立, 则$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014301": { + "id": "014301", + "content": "已知$f(x)=x-[x]$, 其中$[x]$表示不大于$x$的最大整数, 则方程$5 f(x)-x-2=0$的解的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014302": { + "id": "014302", + "content": "设$a, b \\in \\mathbf{Z}$, 若对任意$x \\leq 0$, 都有$(a x+2)(x^2+2 b) \\leq 0$, 则$a+b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014303": { + "id": "014303", + "content": "已知$a \\in \\mathbf{R}, y=f(x)$是定义在区间$[-2,2]$上的奇函数, 在区间$[0,2]$上是严格增函数, 若$f(a^2+2 a-3)+f(2-2 a^2)<0$, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014304": { + "id": "014304", + "content": "已知$f(x)=\\dfrac{2-x}{x+1}$.\\\\\n(1) 证明: 函数$y=f(x)$在区间$(-1,+\\infty)$上为严格减函数;\\\\\n(2) 是否存在负数$x_0$, 使得$f(x_0)=2^{\\circ}$成立, 若存在求出$x_0$: 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014305": { + "id": "014305", + "content": "方程$\\ln x-1=\\dfrac{1-x}{\\mathrm{e}^x}$是否有整数解? 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题06", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014306": { + "id": "014306", + "content": "已知$a \\in \\mathbf{R}$, 若对任意$x_1, x_2 \\in[1,+\\infty)$, 当$x_1a$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014308": { + "id": "014308", + "content": "若关于$x$的不等式$x^2+x>a$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014309": { + "id": "014309", + "content": "若关于$x$的不等式$x^2+x>a$对任意的$x \\in(0, \\dfrac{1}{2})$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014310": { + "id": "014310", + "content": "若关于$x$的方程$x^2+x=a$在区间$(0, \\dfrac{1}{2})$上有解, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014311": { + "id": "014311", + "content": "若关于$x$的不等式$x^2+a x+1>0$对任意的$x \\in[0, \\dfrac{1}{2}]$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014312": { + "id": "014312", + "content": "若关于$x$的方程$x^2+a x+1=0$在区间$(0, \\dfrac{1}{2})$上有解, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014313": { + "id": "014313", + "content": "已知定义在区间$[0,2]$上的两个函数$y=f(x)$和$y=g(x)$, 其中$f(x)=x^2-a x+4$($a \\geq 2$), $g(x)=\\dfrac{x^2}{x+1}$, 若对于任意的$x_1, x_2 \\in[0,2]$, $f(x_1) \\geq g(x_2)$恒成立, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014314": { + "id": "014314", + "content": "已知$f(x)=a x+\\dfrac{1}{x+1}$, $a \\in \\mathbf{R}$, 若函数$y=f(x)$区间$[1,2]$上有零点, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014315": { + "id": "014315", + "content": "已知定义在区间$[0,2]$上的两个函数$y=f(x)$和$y=g(x)$, 其中$f(x)=x^2-a x+4$($a \\geq 2$), $g(x)=\\dfrac{x^2}{x+1}$, 若对于任意的$x_1 \\in[0,2]$, 总存在$x_2 \\in[0,2]$, 使得$f(x_1) \\geq g(x_2)$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014316": { + "id": "014316", + "content": "设$a \\in \\mathbf{R}$, $f(x)=\\dfrac{2^x+a}{2^x+1}$, 若$f(x)<\\dfrac{a+2}{2}$对任意的$x \\in \\mathbf{R}$成立, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014317": { + "id": "014317", + "content": "若关于$x$的不等式$x^2-3>a x-a$对任意的$x \\in[3,4]$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014318": { + "id": "014318", + "content": "若关于$x$的不等式$2^x-\\dfrac{2}{x}-a>0$在区间$[1,2]$上有实数解, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014319": { + "id": "014319", + "content": "设$f(x)=\\begin{cases}|x^2+2 x-1|,& x \\leq 0, \\\\ 2^{x-1}+a & , x>0.\\end{cases}$ 若函数$y=f(x)$有两个不同的零点, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014320": { + "id": "014320", + "content": "设$f(x)=\\dfrac{-4 x+5}{x+1}$, $g(x)=a \\sin (\\dfrac{\\pi}{3} x)+2 a$, $a>0$, 若对任意$x_1 \\in[0,2]$, 总存在$x_2 \\in[0,2]$, 使得$g(x_1)=f(x_2)$成立, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014321": { + "id": "014321", + "content": "设$f(x)=|2 x+1|$, 若不等式$| f(x)-2 f(\\dfrac{x}{2})| \\leq k$对任意的$x \\in \\mathbf{R}$成立, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014322": { + "id": "014322", + "content": "设$f(x)=\\begin{cases}x^2-x+3, & x \\leq 1, \\\\ x+\\dfrac{2}{x}, & x>1.\\end{cases}$若关于$x$的不等式$f(x)>|\\dfrac{x}{2}+a|$对任意的$x \\in \\mathbf{R}$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014323": { + "id": "014323", + "content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(\\dfrac{1}{3},+\\infty)$, 使得$\\{y | y=f(x),\\ x \\in[a, b]\\}=[m a, m b]$, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014324": { + "id": "014324", + "content": "设$f(x)=\\begin{cases}|x+2|, & x<0, \\\\ x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$. 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围是\\bracket{20}.\n\\fourch{$(-2, \\dfrac{1}{2})$}{$(-6, \\dfrac{1}{2})$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac{1}{2},+\\infty)$}\n \n%12", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题07", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014325": { + "id": "014325", + "content": "已知$\\overrightarrow {a}, \\overrightarrow {b}$为两个单位向量, 下列结论中正确的是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}=\\overrightarrow {b}$}{如果$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 那么$\\overrightarrow {a}=\\overrightarrow {b}$}{$\\overrightarrow {a}\\parallel \\overrightarrow {b}$}{如果$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 那么$\\overrightarrow {a}=\\overrightarrow {b}$或$\\overrightarrow {a}=-\\overrightarrow {b}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014326": { + "id": "014326", + "content": "已知在平行四边形$ABCD$中, $|\\overrightarrow{AB}|=2$, $|\\overrightarrow{AD}|=3$, $M$为边$CD$的中点, \n若$\\overrightarrow{AM} \\cdot \\overrightarrow{AD}=\\dfrac{21}{2}$, 则向量$\\overrightarrow{AB}$与$\\overrightarrow{AD}$的夹角大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014327": { + "id": "014327", + "content": "已知$\\overrightarrow {b}=(1,2)$, 若向量$\\overrightarrow {a}$满足$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 且$|\\overrightarrow {a}|=5$, 则$\\overrightarrow {a}$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014328": { + "id": "014328", + "content": "在$\\triangle ABC$中, 若$|\\overrightarrow{AB}+\\overrightarrow{AC}|=|\\overrightarrow{AB}-\\overrightarrow{AC}|$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{等腰三角形}{直角三角形}{等边三角形}{等腰直角三角形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014329": { + "id": "014329", + "content": "已知向量$\\overrightarrow {a}=(-1,2)$, $\\overrightarrow {b}=(1,1)$.\\\\\n(1) 已知$k \\in \\mathbf{R}$, 若$\\overrightarrow {c}=k \\overrightarrow {a}+(1-k) \\overrightarrow {b}$, 且$\\overrightarrow {b} \\perp \\overrightarrow {c}$, 求$k$的值;\\\\\n(2) 若$\\overrightarrow{AB}=\\overrightarrow {a}+\\overrightarrow {b}$, $\\overrightarrow{BC}=\\overrightarrow {a}-2 \\overrightarrow {b}$, $\\overrightarrow{CD}=4 \\overrightarrow {a}-2 \\overrightarrow {b}$, 求证: $A$、$C$、$D$三点共线.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014330": { + "id": "014330", + "content": "已知平面向量$\\overrightarrow {a}$、$\\overrightarrow {b}$满足$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角是$120^{\\circ}$, 问$t$为何实数值时, $|\\overrightarrow {a}-t \\overrightarrow {b}|$的值最小? 并求此时$\\overrightarrow {b}$与$\\overrightarrow {a}-t \\overrightarrow {b}$夹角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014331": { + "id": "014331", + "content": "如图, 在直角三角形$ABC$中, $|CA|=|CB|=2, M$、$N$是斜边$AB$上的两个动点(点$M$靠在线段$BN$上), 且$|MN|=\\sqrt{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0) node [right] {$A$} coordinate (A);\n\\draw (0,2) node [above] {$B$} coordinate (B);\n\\draw ($(A)!0.2!(B)$) node [above right] {$M$} coordinate (M);\n\\draw ($(A)!0.7!(B)$) node [above right] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (C)--(M)(C)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求向量$\\overrightarrow{MN}$在$\\overrightarrow{CB}$方向上的投影与数量投影;\\\\\n(2) 求$\\overrightarrow{CM} \\cdot \\overrightarrow{CN}$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014332": { + "id": "014332", + "content": "如图, 在正方形$ABCD$中, $E$为$AB$的中点, $F$为$CE$的中点, 则$\\overrightarrow{BF}$等于\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(E)$) node [left] {$F$} coordinate (F);\n\\draw (A) rectangle (C) --(E) (B)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{3}{4} \\overrightarrow{AB}+\\dfrac{1}{4} \\overrightarrow{AD}$}{$-\\dfrac{1}{4} \\overrightarrow{AB}+\\dfrac{1}{2} \\overrightarrow{AD}$}{$\\dfrac{1}{2} \\overrightarrow{AB}+\\overrightarrow{AD}$}{$\\dfrac{1}{4} \\overrightarrow{AB}+\\dfrac{3}{4} \\overrightarrow{AD}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014333": { + "id": "014333", + "content": "已知向量$\\overrightarrow {a}=(1,0), \\overrightarrow {b}=(1,1)$, 当$k$为何实数时:\\\\\n(1) $\\overrightarrow {a}+k \\overrightarrow {b}$与$\\overrightarrow {a}+\\overrightarrow {b}$垂直;\\\\\n(2) $\\overrightarrow {a}+k \\overrightarrow {b}$与$\\overrightarrow {a}+\\overrightarrow {b}$平行.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014334": { + "id": "014334", + "content": "在矩形$ABCD$中, $|AB|=\\sqrt{2}$, $|BC|=2$, 点$E$为边$BC$的中点, 点$F$在边$CD$上, $\\overrightarrow{AB} \\cdot \\overrightarrow{AF}=\\sqrt{2}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014335": { + "id": "014335", + "content": "已知四边形$ABCD$是边长为$1$的正方形, 则$|\\overrightarrow{AB}+\\overrightarrow{CA}-\\overrightarrow{DC}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014336": { + "id": "014336", + "content": "已知点$A(2,3)$, $B(6,-3)$, 且$\\overrightarrow{AB}=3 \\overrightarrow{AP}$, 则点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014337": { + "id": "014337", + "content": "设向量$\\overrightarrow {a}=(1,-1)$, $\\overrightarrow {b}=(3,5)$, 求$\\overrightarrow {b}$在$\\overrightarrow {a}$方向上的数量投影与投影的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014338": { + "id": "014338", + "content": "已知向量$\\overrightarrow{OA}=(4,1)$, $\\overrightarrow{OB}=(-1,3)$且$\\overrightarrow{OC} \\perp \\overrightarrow{OB}$, $\\overrightarrow{BC}\\parallel \\overrightarrow{OA}$, 求向量$\\overrightarrow{OC}$的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014339": { + "id": "014339", + "content": "已知向量$\\overrightarrow {a}=(1,1)$, $\\overrightarrow {b}=(-2,1)$, $\\lambda \\in \\mathbf{R}$. 若$\\overrightarrow {a}+\\overrightarrow {b}$与$2 \\overrightarrow {a}+\\lambda \\overrightarrow {b}$的夹角为锐角, 则$\\lambda$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014340": { + "id": "014340", + "content": "如图, 两块斜边长为$\\sqrt{2}$的直角三角板拼在一起, 求$\\overrightarrow{OD} \\cdot \\overrightarrow{AB}$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (0,2) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (45:{sqrt(6)}) node [right] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(B)$) node [below left] {$C$} coordinate (C);\n\\draw (B)--(O)--(A)--(D)--(C)(A)--(B);\n\\draw pic [draw,scale = 0.5] {right angle = A--O--B};\n\\draw pic [draw,scale = 0.5,\"$45^\\circ$\", angle eccentricity = 2.5] {angle = B--A--O};\n\\draw pic [draw,scale = 0.5,\"$60^\\circ$\", angle eccentricity = 2.5] {angle = A--C--D};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014341": { + "id": "014341", + "content": "在$\\triangle ABC$中, $|AB|=5$, $|AC|=6$, $\\cos A=\\dfrac{1}{5}$, $O$是$\\triangle ABC$的外心, 若$\\overrightarrow{OP}=x \\overrightarrow{OB}+y \\overrightarrow{OC}$, 其中$x, y \\in[0,1]$, 则动点$P$的轨迹所覆盖图形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014342": { + "id": "014342", + "content": "如图, 已知$\\triangle ABC$的三边长$|AB|=8$, $|BC|=7$, $|AC|=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:3) node [below] {$C$} coordinate (C);\n\\draw (-120:8) node [left] {$B$} coordinate (B);\n\\draw (10:3) node [right] {$Q$} coordinate (Q);\n\\draw (190:3) node [left] {$P$} coordinate (P);\n\\draw (A) circle (3);\n\\draw (P)--(Q)--(C)--(B)--cycle (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 圆$A$的半径为$3$, 设$PQ$是圆$A$的一条直径, 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值和最小值.\n%17", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题12", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014343": { + "id": "014343", + "content": "如图, 点$N$为正方形$ABCD$的中心, $\\triangle ECD$为正三角形, $M$是线段$ED$上任一点, 则直线$BM$、$EN$的位置关系是\\bracket{20}. (填``相交''、``平行''、或``异面'')\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$A$} coordinate (A);\n\\draw ($(C)!0.5!(D)$) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.5!(C)$) node [right] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--(D)--cycle (B)--(M)(E)--(N)(D)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014344": { + "id": "014344", + "content": "已知等边$\\triangle ABC$的面积为$\\dfrac{9 \\sqrt{3}}{4}$, 其顶点都在球$O$的表面上, 若球心$O$到平面$ABC$的距离为$1$, 则球$O$的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014345": { + "id": "014345", + "content": "如图所示, 在正三棱柱$ABC-A_1B_1C_1$中, $AB=3$, $AA_1=4$, $M$为$AA_1$的中点, $P$是$BC$上一点, 且由$P$沿棱柱侧面经过棱$CC_1$到$M$的最短距离为$\\sqrt{29}$, 则$PC$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (1.5,0,{1.5*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,4,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,4,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,4,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$M$} coordinate (M);\n\\draw ($(C)!0.2!(C_1)$) coordinate (N);\n\\draw ($(B)!{1/3}!(C)$) node [below right] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle (A_1)--(B_1)--(C_1) (B)--(B_1) (P)--(N);\n\\draw [dashed] (M)--(N)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014346": { + "id": "014346", + "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 正方体上底面$A_1B_1C_1D_1$内(含边界)一动点$P$到点$A$的距离为$2 \\sqrt{2}$, 点$P$的轨迹形成一条曲线, 这条曲线的长度为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014347": { + "id": "014347", + "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正$\\triangle ABC$内接于圆柱的下底面圆$O$, 点$O_1$是圆柱的上底面的圆心, 线段$AA_1$是圆柱的母线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [below] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O_1) ellipse (2 and 0.5);\n\\draw (O)++(-2,0) arc (180:360:2 and 0.5);\n\\draw [dashed] (O)++(-2,0) arc (180:0:2 and 0.5);\n\\draw (2,0) -- (2,2) (-2,0) -- (-2,2);\n\\draw (20:2 and 0.5) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw [dashed] (A)--(A_1);\n\\draw (140:2 and 0.5) node [above] {$C$} coordinate (C);\n\\draw (265:2 and 0.5) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below right] {$M$} coordinate (M);\n\\draw [dashed] (A)--(B)--(C)--cycle(A)--(B)(A_1)--(B);\n\\draw [dashed] (C)--(M)(B)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上存在一点$D$, 满足$\\angle BOD=\\dfrac{\\pi}{6}$, 证明$O_1D\\parallel$平面$A_1AB$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014348": { + "id": "014348", + "content": "如图, 在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$、$P$分别是$C_1D_1$、$C_1C$、$A_1A$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(A1)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\draw ($(C)!0.5!(C1)$) node [right] {$N$} coordinate (N) circle (0.03);\n\\draw ($(C1)!0.5!(D1)$) node [above] {$M$} coordinate (M) circle (0.03);\n\\draw (M)--(A1)--(B)--(N) (P)--(B);\n\\draw [dashed] (M)--(N) (C)--(D1) (P)--(D1) (P)--(M)(P)--(N)(M)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $M$、$N$、$A_1$、$B$四点共面;\\\\\n(2) 求异面直线$PD_1$与$MN$所成角的余弦值;\\\\\n(3) 求三棱锥$P-MNB$的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014349": { + "id": "014349", + "content": "如图甲所示, 在平面五边形$PABCD$中, $PD=PA$, $AC=CD=BD=\\sqrt{5}$, $AB=1$, $AD=2$, $PD \\perp PA$, 现将图甲所示中的$\\triangle PAD$沿$AD$边折起, 使平面$PAD \\perp$平面$ABCD$得到四棱锥$P-ABCD$, 如图乙所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0) node [right] {$A$} coordinate (A);\n\\draw (1,1) node [above] {$P$} coordinate (P);\n\\draw (2,-1) node [right] {$B$} coordinate (B);\n\\draw (1,{-sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--(P)--cycle(B)--(D)--(A)--(C);\n\\draw (1,-2.5) node [below] {图甲};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (1,1,0) node [above] {$P$} coordinate (P);\n\\draw (2,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--(P)--cycle (B)--(P)--(C);\n\\draw [dashed] (B)--(D)--(A)--(C);\n\\draw (1,-2.5) node [below] {图乙};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PD \\perp$平面$PAB$;\\\\\n(2) 求二面角$A-PB-C$的大小;\\\\\n(3) 在棱$PA$上是否存在点$M$使得$BM$与平面$PCB$所成的角的正弦值为$\\dfrac{1}{3}$? 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014350": { + "id": "014350", + "content": "若一个圆锥的侧面展开图为半径为$2$的半圆形, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014351": { + "id": "014351", + "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$, $M$, $N$分别是$A_1D$, $D_1B$的中点, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(D)$) node [below left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(D1)$) node [right] {$N$} coordinate (N);\n\\draw (B1)--(D1);\n\\draw [dashed] (A1)--(D1)(B)--(D)(A1)--(D)(D1)--(B)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n\\onech{直线$A_1D$与直线$D_1B$垂直, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$平行, 直线$MN\\parallel$平面$BDD_1B_1$}{直线$A_1D$与直线$D_1B$相交, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$异面, 直线$MN\\parallel$平面$BDD_1B_1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014352": { + "id": "014352", + "content": "在正四棱锥$S-ABCD$中, $E$是线段$AB$上的点(不含端点). 设$SE$与$BC$所成的角为$\\alpha$, $SE$与平面$ABCD$所成的角为$\\beta$, 二面角$S-AB-C$的平面角为$\\gamma$, 则\\bracket{20}.\n\\fourch{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}{$\\gamma \\leq \\beta \\leq \\alpha$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014353": { + "id": "014353", + "content": "在长方体$ABCD-A_1B_1C_1D_1$的各条棱所在直线中, 与直线$AB$异面且垂直的直线有\\blank{50}条.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014354": { + "id": "014354", + "content": "正三棱锥$S-ABC$中, $\\angle BSC=40^{\\circ}$, $SB=2$, \n一质点自点$B$出发, 沿着三棱锥的侧面绕行一周回到点$B$的最短路线的长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014355": { + "id": "014355", + "content": "若三棱锥$S-ABC$的所有的顶点都在球$O$的球面上, 且$SA \\perp$平面$ABC, AB=2$, $SA=AC=4$, $\\angle BAC=\\dfrac{\\pi}{3}$, 则球$O$的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014356": { + "id": "014356", + "content": "如图, 在矩形$ABCD$中, $AB=4$, $AD=2$, $E$为$AB$边的中点, 将$\\triangle ADE$沿$DE$翻折, 得到四棱锥$A_1-DEBC$. 设线段$A_1C$的中点为$M$, 在翻折过程中, 是否总有$BM\\parallel$平面$A_1DE$? 如果有, 证明你的结论.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (4,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) ++ (0,{sqrt(2)},0) coordinate (T);\n\\draw ($0.3*(A)+sqrt(0.91)*(T)$) node [above] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)!0.5!(C)$) node [above] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(A_1)--(D)--(A)(D)--(E)--(A_1)--(B)--(M);\n\\draw [dashed] (D)--(C)--(E);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014357": { + "id": "014357", + "content": "如图, $AB$是圆$O$的直径, 点$C$是圆$O$上异于$A, B$的点, $PO$垂直于圆$O$所在的平面, 且$PO=OB=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (-50:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(P)--(B)(P)--(C);\n\\draw [dashed] (A)--(B)(P)--(O)(A)--(C)--(B)(P)--(D)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$D$为线段$AC$的中点, 求证: $AC \\perp$平面$PDO$;\\\\\n(2) 求三棱锥$P-ABC$体积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014358": { + "id": "014358", + "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面$ABC$是以$AC$为斜边的等腰直角三角形, 侧面$AA_1C_1C$为菱形, 点$A_1$在底面上的投影为$AC$的中点$D$, 且$AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},0) node [above] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (2,0,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw ($(A_1)+(B)-(A)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)!0.4!(B_1)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A_1)--(D)--(E)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD \\perp CC_1$;\\\\\n(2) 求点$C$到侧面$AA_1B_1B$的距离;\\\\\n(3) 在线段$A_1B_1$上是否存在点$E$, 使得直线$DE$与侧面$AA_1B_1B$所成角的正弦值为$\\dfrac{\\sqrt{6}}{7}$? 若存在, 请求出$A_1E$的长; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014359": { + "id": "014359", + "content": "在直三棱柱$ABC-A_1B_1C_1$中, 底面为直角三角形, $\\angle ACB=90^{\\circ}$, $AC=6$, $BC=CC_1=\\sqrt{2}$, $P$是$BC_1$上一动点, 则$CP+PA_1$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014360": { + "id": "014360", + "content": "如图$1$是由矩形$ADEB$, Rt$\\triangle ABC$和菱形$BFGC$组成的一个平面图形, 其中$AB=1$, $BE=BF=2$, $\\angle FBC=60^{\\circ}$, 将其沿$AB, BC$折起使得$BE$与$BF$重合, 连结$DG$, 如图$2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (-2,0) node [below] {$E$} coordinate (E);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$A$} coordinate (A);\n\\draw (-2,1) node [above] {$D$} coordinate (D);\n\\draw (-60:2) node [below] {$F$} coordinate (F);\n\\draw (F) ++ (2,0) node [below] {$G$} coordinate (G);\n\\draw (E)--(C)(D)--(A)--(C)(D)--(E)(A)--(B)--(F)--(G)(C)--(G);\n\\draw (0.5,-2.5) node {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(120:0.5cm)}]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (B) ++ (1,{sqrt(3)},0) node [below right] {$E$($F$)} coordinate (E);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(G)--(D)--cycle (D)--(E)--(G) (E)--(B);\n\\draw [dashed] (A)--(C);\n\\draw (0.875,-0.75) node {图$2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 图$2$中的$A, C, G, D$四点共面, 且平面$ABC \\perp$平面$BCGE$;\\\\\n(2) 求图$2$中的二面角$B-CG-A$的大小.\n%18", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题17", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014361": { + "id": "014361", + "content": "已知直线$l$经过点$A(-3, \\sqrt{3})$、$B(\\sqrt{3},-1)$.\\\\\n(1) 直线$l$的斜率是\\blank{50};\\\\\n(2) 直线$l$的倾斜角是\\blank{50};\\\\\n(3) 直线$l$的点斜式方程是\\blank{50};\\\\\n(4) 直线$l$的一个法向量是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014362": { + "id": "014362", + "content": "圆$C: x^2+y^2-4 x-6 y+9=0$的圆心到直线$l: 3 x-4 y+2=0$的距离$d=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014363": { + "id": "014363", + "content": "若直线$x=-1$与曲线$(x-\\dfrac{m^2}{4})^2+(y-m)^2=1$仅有一个公共点, 则实数$m$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014364": { + "id": "014364", + "content": "已知点$A(1,0)$, $B(-1,2)$.\\\\\n(1) 若直线$AB$与直线$x-m y+1=0$垂直, 求实数$m$的值;\\\\\n(2) 若直线$AB$与直线$x-m y+1=0$平行, 求实数$m$的值, 以及这两条平行直线之间的距离;\\\\\n(3) 求过点$B$与直线$2 x-y+1=0$夹角的余弦值为$\\dfrac{2 \\sqrt{5}}{5}$的直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014365": { + "id": "014365", + "content": "分别求满足下列条件的圆的方程:\\\\\n(1) 经过点$P(2,2)$, $Q(5,3)$, $R(3,-1)$;\\\\\n(2) 经过点$A(-3,0)$与$B(0,-\\sqrt{3})$, 且圆心在直线$x+y+1=0$上.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014366": { + "id": "014366", + "content": "已知圆$N: (x-2)^2+(y+1)^2=4$.\\\\\n(1) 直线$l$经过点$A(3,2)$, 且被圆$N$截得长为$2 \\sqrt{2}$的弦, 求直线$l$的方程;\\\\\n(2) 过点$B(3,0)$的直线$l$与圆$N$交于$P$、$Q$两点, 分别求弦$PQ$最短和最长时其所在直线的方程;\\\\\n(3) 讨论圆$C: (x-a)^2+y^2=1$($a \\in \\mathbf{R}$)与圆$N$的位置关系.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014367": { + "id": "014367", + "content": "若方程$x^2+y^2-x+y+k=0$表示圆, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014368": { + "id": "014368", + "content": "若直线$l$过点$P(1,2)$且垂直于直线$5 x-2 y+3=0$, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014369": { + "id": "014369", + "content": "设$\\theta \\in \\mathbf{R}$, 直线$x \\cos \\theta+y-1=0$的倾斜角的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014370": { + "id": "014370", + "content": "已知圆$C_1$的半径为$3$, 圆$C_2$的半径为$7$, 若两圆相交, 则两圆的圆心距可能是\\bracket{20}.\n\\fourch{$0$}{$4$}{$8$}{$12$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014371": { + "id": "014371", + "content": "已知圆$C$的一般方程为$x^2+2 x+y^2=0$, 则圆$C$的半径为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014372": { + "id": "014372", + "content": "已知直线$l_1$与$l_2: 4 x+3 y+5=0$有相同的法向量, 且直线$l_1$在$x$轴上的截距为$-2$, 则直线$l_1$的点法式方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014373": { + "id": "014373", + "content": "若直线$l_1: 2 x+y-3=0$与直线$l_2: 4 x+2 y+a=0$的距离为$\\dfrac{\\sqrt{5}}{2}$, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014374": { + "id": "014374", + "content": "若圆$\\mathrm{C}_1: x^2+y^2=4$与圆$C_2: (x-3)^2+(y+m)^2=25$外切, 则实数$m$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014375": { + "id": "014375", + "content": "已知圆$C: x^2+y^2-2 y-4=0$, 直线$l: 3 x+y-6=0$, 则直线$l$被圆$C$所截得的弦长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014376": { + "id": "014376", + "content": "已知$\\triangle ABC$的三个顶点的坐标分别是$A(0,1)$, $B(-2,-1)$, $C(5,3)$.\\\\\n(1) 求边$AB$上的中线所在直线的方程;\\\\\n(2) 求$\\triangle ABC$的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014377": { + "id": "014377", + "content": "如图, 动点$C$在以$AB$为直径的半圆$O$上(异于$A, B$), $\\angle DCB=\\dfrac{\\pi}{2}$, 且$DC=CB$, 若$|AB|=2$, 则$\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$的取值范围为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw ($(O)!1!110:(B)$) node [above left] {$C$} coordinate (C);\n\\draw ($(C)!1!90:(B)$) node [above] {$D$} coordinate (D);\n\\draw (O)--(C)--(D)--cycle(A)--(B)--(C) (A)arc(180:0:1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014378": { + "id": "014378", + "content": "已知圆$M$与直线$x=2$相切, 圆心$M$在直线$x+y=0$上, 且直线$x-y-2=0$被圆$M$截得的弦长为$2 \\sqrt{2}$.\\\\\n(1) 求圆$M$的标准方程, 并判断圆$M$与圆$N: x^2+y^2-6 x+8 y+15=0$的位置关系;\\\\\n(2) 若不与坐标轴垂直的直线$l$在$x$轴上的截距为$1$, 且直线$l$与圆$M$交于$A$、$B$两点, 在$x$轴上是否存在定点$Q$, 使得直线$AQ$的倾斜角与直线$BQ$的倾斜角互补, 若存在, 求出点$Q$的坐标; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题18", + "edit": [ + "20230202\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014379": { + "id": "014379", + "content": "计算$\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\mathrm{i}^4=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014380": { + "id": "014380", + "content": "已知$m$为实数, 复数$z=m^2+m-2+(m^2-4) \\mathrm{i}$.\\\\\n(1) 当$z$为实数时, $m=$\\blank{50};\\\\\n(2) 当$z$为纯虚数时, $m=$\\blank{50};\\\\\n(3) 当$z=0$时, $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014381": { + "id": "014381", + "content": "设复数$3-4 \\mathrm{i}$与$5-6 \\mathrm{i}$在复平面上所对应的向量分别为$\\overrightarrow{OA}$与$\\overrightarrow{OB}$, 则向量$\\overrightarrow{AB}$所对应的复数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014382": { + "id": "014382", + "content": "如果复数$z$满足$(1+2 \\mathrm{i}) \\overline {z}=4+3 \\mathrm{i}$, 则$|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014383": { + "id": "014383", + "content": "若关于$x$的实系数一元二次方程$x^2-b x+c=0$的一个根为$1-3 \\mathrm{i}$, 则$3 b+c=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014384": { + "id": "014384", + "content": "若复数$z$满足: $z \\overline{z}+(z+\\overline {z}) \\mathrm{i}=\\dfrac{3-\\mathrm{i}}{2+\\mathrm{i}}$, 求复数$z$及$z+z+z^2$的值, 并证明$\\dfrac{1+z}{1-z}$是纯虚数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014385": { + "id": "014385", + "content": "已知$\\alpha$、$\\beta$是实系数一元二次方程$x^2-2 x+m=0$, $m \\in \\mathbf{R}$的两根, 求: $|\\alpha|+|\\beta|$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014386": { + "id": "014386", + "content": "已知复数$z$满足下列条件, 根据条件分别求复数$z$在复平面上对应点的轨迹方程, 并分别求出$|z|$的最大值.\\\\\n(1) $|z-1-\\mathrm{i}|=1$;\\\\\n(2) $|z-3 \\mathrm{i}|+|z+3 \\mathrm{i}|=10$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014387": { + "id": "014387", + "content": "已知复数$z$满足$|z-3 \\mathrm{i}|+|z+3 \\mathrm{i}|=10$, 求$|z-\\mathrm{i}|$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014388": { + "id": "014388", + "content": "已知复数$z=\\dfrac{(1+3 \\mathrm{i})^2(3-\\mathrm{i})}{(1-2 \\mathrm{i})^2}$, 则$|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014389": { + "id": "014389", + "content": "已知$z_1$、$z_2$是方程$x^2+2 x+3=0$在复数范围内的两个根, 则$|z_1-z_2|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014390": { + "id": "014390", + "content": "已知复平面上平行四边形$ABCD$的顶点$A$、$B$、$C$的坐标分别为$(-2,-1)$、$(7,3)$、$(12,9)$, 求向量$\\overrightarrow{AD}$所对应的复数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014391": { + "id": "014391", + "content": "已知$z_1$、$z_2 \\in \\mathbf{C}$, 且$z_1 z_2=0$, 求证: $z_1=0$或$z_2=0$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014392": { + "id": "014392", + "content": "若复数$z=(m^2-5 m+6)+(2 m^2-5 m+2) \\mathrm{i}$为纯虚数, 则实数$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014393": { + "id": "014393", + "content": "已知$|z_1|=3$, $|z_2|=4$, $|z_1+z_2|=5$, , 则$|z_1-z_2|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014394": { + "id": "014394", + "content": "已知复数$z=\\dfrac{\\mathrm{i}+\\mathrm{i}^2++\\mathrm{i}^3+\\cdots+\\mathrm{i}^{2003}}{1+\\mathrm{i}}$, 则复数$z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014395": { + "id": "014395", + "content": "设复数$z_1=1-\\mathrm{i}$, $z_2=\\cos \\theta+\\mathrm{i} \\sin \\theta$, 其中$\\theta \\in[0, \\pi]$.\\\\\n(1) 若复数$z=\\overline{z_1} \\cdot z_2$为实数, 求$\\theta$的值;\\\\\n(2) 求$|3 z_1+z_2|$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014396": { + "id": "014396", + "content": "已知复数$z_1=2-5 \\mathrm{i}$, $z_2=1+(2 \\cos \\theta) \\mathrm{i}$.\\\\\n(1) 求$z_1 \\cdot \\overline{z_1}$;\\\\\n(2) 复数$z_1$、$z_2$对应的向量分别是$\\overrightarrow{OZ_1}$, $\\overrightarrow{OZ_2}$, 其中$O$为坐标原点, 当$\\theta=\\dfrac{\\pi}{3}$时, 求$\\overrightarrow{OZ_1} \\cdot \\overrightarrow{OZ_2}$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014397": { + "id": "014397", + "content": "已知$z$是虚数, $z+\\dfrac{1}{z}$是实数.\\\\\n(1) 求$z$为何值时, $|z+2-\\mathrm{i}|$有最小值, 并求出$|z+2-\\mathrm{i}|$的最小值;\\\\\n(2) 设$\\mu=\\dfrac{1-z}{1+z}$, 求证: $\\mu$为纯虚数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014398": { + "id": "014398", + "content": "已知$z_1$、$z_2$是实系数一元二次方程的两个虚根, 且$z_1^2=z_2$, 求$z_1$、$z_2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014399": { + "id": "014399", + "content": "关于复数$z$的方程$z^2-(a+\\mathrm{i}) z-\\mathrm{i}-2=0$($a \\in \\mathbf{R}$).\\\\\n(1) 若此方程有实数解, 求$a$的值;\\\\\n(2) 用反证法证明: 对任意的实数$a$, 原方程不可能有纯虚数根.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习题13", + "edit": [ + "20230203\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.", @@ -390608,7 +392759,9 @@ "20221028\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031219" + ], "remark": "", "space": "12ex" }, @@ -393429,7 +395582,9 @@ "20221103\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031213" + ], "remark": "", "space": "" }, @@ -393497,7 +395652,9 @@ "20221103\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031218" + ], "remark": "", "space": "" }, @@ -393531,7 +395688,9 @@ "20221103\t王伟叶" ], "same": [], - "related": [], + "related": [ + "031217" + ], "remark": "", "space": "" }, @@ -410533,5 +412692,416 @@ "related": [], "remark": "", "space": "" + }, + "031206": { + "id": "031206", + "content": "已知$z\\in \\mathbf{C}$. 若$\\dfrac{1}{2z-3}=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "上海2019年秋季高考试题2-20230202修改", + "edit": [ + "20220701\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "003632" + ], + "remark": "", + "space": "" + }, + "031207": { + "id": "031207", + "content": "复数$(1+2\\mathrm{i})(3+\\mathrm{i})$的虚部为\\blank{50}.", + "objs": [ + "K0511005B", + "K0512002B" + ], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20230202修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "000366" + ], + "remark": "", + "space": "" + }, + "031208": { + "id": "031208", + "content": "已知集合$A=(-\\infty ,a]$, $B=[2,5]$且$A\\cap B$非空, 则实数$a$的取值范围\\blank{50}.", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2021届杨浦高三基础考试题2-20230202修改", + "edit": [ + "20221027\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "011989" + ], + "remark": "", + "space": "" + }, + "031209": { + "id": "031209", + "content": "已知$\\alpha\\in \\left\\{-2,-1,-\\dfrac{1}{2},\\dfrac{1}{2},1,2,3\\right\\}$. 若幂函数$f(x)=x^{\\alpha}$为偶函数, 且在$(0,+\\infty)$上递减, 则$\\alpha=$\\blank{50}.", + "objs": [ + "K0208002B", + "K0217004B" + ], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "上海2018年秋季高考试题7-20230202修改", + "edit": [ + "20220701\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "003658" + ], + "remark": "", + "space": "" + }, + "031210": { + "id": "031210", + "content": "双曲线$\\dfrac{x^2}{9}-y^2=1$的焦距为\\blank{50}.", + "objs": [ + "K0717002X" + ], + "tags": [ + "第七单元", + "双曲线" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "上海2022年秋季高考试题2-20230202修改", + "edit": [ + "20220730\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "009985" + ], + "remark": "", + "space": "" + }, + "031211": { + "id": "031211", + "content": "在$\\triangle ABC$中, $AC=6$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac 14$, 则$AB=$\\blank{50}.", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届春季高考试题8-20230202修改", + "edit": [ + "20221209\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "012210" + ], + "remark": "", + "space": "" + }, + "031212": { + "id": "031212", + "content": "已知常数$a>0$, 函数$f(x)=\\dfrac{2^x}{2^x+ax}$的图像经过点$P\\left(p,\\dfrac{6}{5}\\right)$, $Q\\left(q,-\\dfrac{1}{5}\\right)$. 若$2^{p+q}=5pq$, 则$a=$\\blank{50}.", + "objs": [ + "K0203005B" + ], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "$$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "上海2018年秋季高考试题11-20230202修改", + "edit": [ + "20220701\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "003662" + ], + "remark": "", + "space": "" + }, + "031213": { + "id": "031213", + "content": "已知$f(x)$是定义域为$\\mathbf{R}$的奇函数, 满足$f(1+x)=f(1-x)$. 若$f(1)=2$, 则$f(1)+f(2)+f(3)+\\cdots+f(2023)=$\\blank{50}.", + "objs": [], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题11-20230202修改", + "edit": [ + "20221103\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "030438" + ], + "remark": "", + "space": "" + }, + "031214": { + "id": "031214", + "content": "为强化安全意识, 某商场拟在未来的连续$10$天中随机选择$3$天进行紧急疏散演练, 则选择的$3$天不为连续$3$天的概率是\\blank{50}(结果用最简分数表示).", + "objs": [], + "tags": [ + "第八单元", + "概率" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "上海2014年秋季高考试题10-20230202修改", + "edit": [ + "20220819\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "011679" + ], + "remark": "", + "space": "" + }, + "031215": { + "id": "031215", + "content": "设区间$(m,n)$($m0$''是``$\\dfrac ab+\\dfrac ba>1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [ + "第一单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题14-20230202修改", + "edit": [ + "20221103\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "030441" + ], + "remark": "", + "space": "" + }, + "031218": { + "id": "031218", + "content": "为了得到函数$y=\\sin(2x+\\dfrac{5\\pi}6)$的图像, 可将函数$y=\\sin x$的图像\\bracket{20}.\n\\twoch{左移$\\dfrac{5\\pi}6$个长度}{右移$\\dfrac{5\\pi}6$个长度}{左移$\\dfrac{5\\pi}{12}$个长度}{右移$\\dfrac{5\\pi}{12}$个长度}", + "objs": [], + "tags": [ + "第三单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2019届高三基础考试题13-20230202修改", + "edit": [ + "20221103\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "030440" + ], + "remark": "", + "space": "" + }, + "031219": { + "id": "031219", + "content": "已知函数$f(x)=\\dfrac{x^2}2+3x-4\\ln x$, 求$f(x)$的导数, 并求出$f'(x)>0$的解集.", + "objs": [ + "K0229004X", + "K0230001X" + ], + "tags": [ + "第二单元", + "导数" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "人教版A版教材例题与习题-20230202修改", + "edit": [ + "20221028\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "030337" + ], + "remark": "", + "space": "12ex" + }, + "031220": { + "id": "031220", + "content": "已知$a$是常数, 设函数$f(x)=(a+2)x^2+2(a+2)x-4$.\\\\\n(1) 解不等式: $f(x)>-4$;\\\\\n(2) 求实数$a$的取值范围, 使得$f(x)<0$对任意$x\\in [1,3]$恒成立.", + "objs": [ + "K0115001B", + "K0114001B", + "KNONE" + ], + "tags": [ + "第一单元", + "第二单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三上月考卷01第18题-20230202修改", + "edit": [ + "20220710\t王伟叶", + "20230202\t王伟叶" + ], + "same": [], + "related": [ + "004636" + ], + "remark": "", + "space": "12ex" + }, + "031221": { + "id": "031221", + "content": "已知$f(x)=ax+\\dfrac{1}{x+1}, \\ a\\in \\mathbf{R}$.\\\\\n(1) 已知$a=1$, 求不等式$f(x)+2