diff --git a/工具/tools panel.py b/工具/tools panel.py new file mode 100644 index 00000000..668c4389 --- /dev/null +++ b/工具/tools panel.py @@ -0,0 +1,50 @@ +from tkinter import * +import os +from subprocess import call + +#设置根目录名 +rootname = "mathdeptv2" +rawoutputdirectory = "d:/temp/tkoutput" + + +try: + os.mkdir(rawoutputdirectory) +except: + pass + +def testcall(): + LabelTool.config(text = "测试") + LabelOutputDir.insert(0,"1") + print("按钮") + +def SetOutputDir(): + try: + os.mkdir(LabelOutputDir.get()) + except: + pass + print("输出目录设为",LabelOutputDir.get()) + + +root = Tk() +outputdirectory = StringVar() +outputdirectory.set(rawoutputdirectory) +root.geometry("800x600") +LabelTool = Label(root, text = "工具选择待定", height = 1, width = 10) +LabelTool.place(x=420,y=50) + +# 设置输出目录名 +LabelOutputDir = Entry(root,textvariable=outputdirectory) +LabelOutputDir.place(x = 420, y = 120, width = 250) + +# 修改输出目录按钮 +ButtonDir = Button(root,text = "输出目录确定", command = SetOutputDir) +ButtonDir.place(x=700, y=120) + +# 运行按钮 +ButtonGo = Button(root, text ="运行", height = 1, width = 5, command = testcall) +ButtonGo.place(x=600, y= 550) + +# 编译 + +root.title(rootname + " 题库工具一览") +root.mainloop() \ No newline at end of file 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 5df8660f..8feb0edc 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 2, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n", - "problems = \"77\"\n", + "problems = \"31321\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", diff --git a/工具/单元标记.ipynb b/工具/单元标记.ipynb index 517fcbca..3ee76940 100644 --- a/工具/单元标记.ipynb +++ b/工具/单元标记.ipynb @@ -12,7 +12,7 @@ "pros = data.strip().split(\"\\n\")\n", "dic= {}\n", "for p in pros:\n", - " a,b=p.split(\",\")\n", + " a,b=p.split(\"\\t\")\n", " dic[a] = \"\"\n", " for t in b:\n", " dic[a] += taglist[int(t)]+\"\\n\"\n", diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index cf7b2dd4..8bdda52e 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 @@ -12,7 +12,7 @@ "首个空闲id: 14784 , 直至 020000\n", "首个空闲id: 22106 , 直至 030000\n", "首个空闲id: 31382 , 直至 040000\n", - "首个空闲id: 40336 , 直至 999999\n" + "首个空闲id: 40387 , 直至 999999\n" ] } ], diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 236b7318..c9c34ad3 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,34 +2,167 @@ "cells": [ { "cell_type": "code", - "execution_count": 12, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 000080 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.892\t0.919\n", - "题号: 000081 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t1.000\n", - "题号: 000084 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.919\n", - "题号: 000085 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.960\n", - "题号: 000086 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.960\n", - "题号: 000087 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.960\n", - "题号: 000088 , 字段: usages 中已添加数据: 20230300\t2025届高一11班\t0.878\n", - "题号: 000080 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.947\t0.974\n", - "题号: 000081 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.895\n", - "题号: 000084 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.829\n", - "题号: 000085 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.934\n", - "题号: 000086 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.947\n", - "题号: 000087 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.908\n", - "题号: 000088 , 字段: usages 中已添加数据: 20230300\t2025届高一12班\t0.987\n", - "题号: 000080 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.969\t0.938\n", - "题号: 000081 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.656\n", - "题号: 000084 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.656\n", - "题号: 000085 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.828\n", - "题号: 000086 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.875\n", - "题号: 000087 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.719\n", - "题号: 000088 , 字段: usages 中已添加数据: 20230300\t2025届高一02班\t0.922\n" + "题号: 014105 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014107 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014109 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014111 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014112 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014117 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014118 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014120 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014122 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014133 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014134 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014136 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014153 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014157 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014164 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014165 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 014166 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014167 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014168 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014169 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014170 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014171 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014172 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014174 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014175 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014176 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014177 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014178 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014179 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014180 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014181 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014182 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014183 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014190 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014191 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014192 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014193 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014194 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014199 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014203 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014204 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014205 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014206 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014207 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014208 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014217 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014221 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014226 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014227 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014231 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014245 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014246 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014249 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014255 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014259 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014260 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014262 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014262 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014263 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014265 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014266 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014269 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014270 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014273 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014283 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014285 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014286 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 014288 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014290 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 014330 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014331 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014339 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014346 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014348 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014349 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014352 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014354 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014359 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014360 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014364 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014366 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014367 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014369 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014373 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014377 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014380 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014381 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014382 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014383 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014386 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014388 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014392 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014393 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014394 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 014410 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014412 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014415 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014419 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014423 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014426 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014427 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014432 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014435 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014436 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014437 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014438 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014439 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014440 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014441 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014443 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014448 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014451 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014452 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014454 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014455 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 014464 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014478 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014483 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014503 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014505 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 014532 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 014533 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 014534 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 014538 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 014540 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 014546 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 031223 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 031227 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031236 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 031266 , 字段: tags 中已添加数据: 第九单元\n", + "题号: 031282 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031311 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 031312 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 031313 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 031314 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 031315 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031316 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 031317 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031317 , 字段: tags 中已添加数据: 第五单元\n", + "题号: 031318 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 031319 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 031320 , 字段: tags 中已添加数据: 第八单元\n", + "题号: 031321 , 字段: tags 中已添加数据: 第一单元\n", + "题号: 031321 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 031322 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031323 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031324 , 字段: tags 中已添加数据: 第二单元\n", + "题号: 031325 , 字段: tags 中已添加数据: 第六单元\n", + "题号: 031326 , 字段: tags 中已添加数据: 第四单元\n", + "题号: 031327 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031328 , 字段: tags 中已添加数据: 第九单元\n", + "题号: 031329 , 字段: tags 中已添加数据: 第三单元\n", + "题号: 031330 , 字段: tags 中已添加数据: 第七单元\n", + "题号: 031331 , 字段: tags 中已添加数据: 第二单元\n" ] } ], diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb index 348d9c40..0983b36f 100644 --- a/工具/批量题号选题pdf生成.ipynb +++ b/工具/批量题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/2022学年下学期高一高二材料_教师用_20230326.tex\n", + "开始编译教师版本pdf文件: 临时文件/2022学年下学期高一高二材料_教师用_20230328.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/2022学年下学期高一高二材料_学生用_20230326.tex\n", + "开始编译学生版本pdf文件: 临时文件/2022学年下学期高一高二材料_学生用_20230328.tex\n", "0\n" ] } @@ -54,7 +54,8 @@ "\"2024届高二下学期周末卷07\":\"40317:40335\",\n", "\"2025届高一下学期测验01\":\"40336:40349\",\n", "\"2025届高一下学期测验02\":\"40350:40367\",\n", - "\"2025届高一下学期期中复习二(幂指对函数)\":\"40368:40386\"\n", + "\"2025届高一下学期期中复习二(幂指对函数)\":\"40368:40386\",\n", + "\"2025届高一下学期周末卷02小测\":\"40387:40395\"\n", "\n", "}\n", "\n", diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 9009536c..19c66b29 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,65 +1,609 @@ -usages +tags -000080 -20230300 2025届高一11班 0.892 0.919 +14105 +第一单元 -000081 -20230300 2025届高一11班 1.000 -000084 -20230300 2025届高一11班 0.919 +14107 +第一单元 -000085 -20230300 2025届高一11班 0.960 -000086 -20230300 2025届高一11班 0.960 +14109 +第一单元 -000087 -20230300 2025届高一11班 0.960 -000088 -20230300 2025届高一11班 0.878 +14111 +第一单元 -000080 -20230300 2025届高一12班 0.947 0.974 -000081 -20230300 2025届高一12班 0.895 +14112 +第一单元 -000084 -20230300 2025届高一12班 0.829 -000085 -20230300 2025届高一12班 0.934 +14117 +第一单元 -000086 -20230300 2025届高一12班 0.947 -000087 -20230300 2025届高一12班 0.908 +14118 +第一单元 -000088 -20230300 2025届高一12班 0.987 -000080 -20230300 2025届高一02班 0.969 0.938 +14120 +第二单元 -000081 -20230300 2025届高一02班 0.656 -000084 -20230300 2025届高一02班 0.656 +14122 +第一单元 -000085 -20230300 2025届高一02班 0.828 -000086 -20230300 2025届高一02班 0.875 +14133 +第一单元 -000087 -20230300 2025届高一02班 0.719 -000088 -20230300 2025届高一02班 0.922 +14134 +第一单元 + + +14136 +第一单元 + + +14153 +第一单元 + + +14157 +第一单元 + + +14164 +第一单元 + + +14165 +第一单元 + + +14166 +第二单元 + + +14167 +第二单元 + + +14168 +第二单元 + + +14169 +第二单元 + + +14170 +第二单元 + + +14171 +第二单元 + + +14172 +第二单元 + + +14174 +第二单元 + + +14175 +第二单元 + + +14176 +第二单元 + + +14177 +第二单元 + + +14178 +第二单元 + + +14179 +第二单元 + + +14180 +第二单元 + + +14181 +第二单元 + + +14182 +第二单元 + + +14183 +第二单元 + + +14190 +第二单元 + + +14191 +第二单元 + + +14192 +第二单元 + + +14193 +第二单元 + + +14194 +第二单元 + + +14199 +第二单元 + + +14203 +第二单元 + + +14204 +第三单元 + + +14205 +第三单元 + + +14206 +第三单元 + + +14207 +第三单元 + + +14208 +第三单元 + + +14217 +第三单元 + + +14221 +第三单元 + + +14226 +第三单元 + + +14227 +第三单元 + + +14231 +第三单元 + + +14245 +第三单元 + + +14246 +第三单元 + + +14249 +第三单元 + + +14255 +第三单元 + + +14259 +第三单元 + + +14260 +第三单元 + + +14262 +第二单元 +第三单元 + + +14263 +第三单元 + + +14265 +第三单元 + + +14266 +第三单元 + + +14269 +第三单元 + + +14270 +第三单元 + + +14273 +第三单元 + + +14283 +第三单元 + + +14285 +第三单元 + + +14286 +第三单元 + + +14288 +第二单元 + + +14290 +第二单元 + + +14330 +第五单元 + + +14331 +第五单元 + + +14339 +第五单元 + + +14346 +第六单元 + + +14348 +第六单元 + + +14349 +第六单元 + + +14352 +第六单元 + + +14354 +第六单元 + + +14359 +第六单元 + + +14360 +第六单元 + + +14364 +第七单元 + + +14366 +第七单元 + + +14367 +第七单元 + + +14369 +第七单元 + + +14373 +第七单元 + + +14377 +第五单元 + + +14380 +第五单元 + + +14381 +第五单元 + + +14382 +第五单元 + + +14383 +第五单元 + + +14386 +第五单元 + + +14388 +第五单元 + + +14392 +第五单元 + + +14393 +第五单元 + + +14394 +第五单元 + + +14410 +第六单元 + + +14412 +第六单元 + + +14415 +第六单元 + + +14419 +第六单元 + + +14423 +第六单元 + + +14426 +第六单元 + + +14427 +第六单元 + + +14432 +第六单元 + + +14435 +第六单元 + + +14436 +第六单元 + + +14437 +第六单元 + + +14438 +第六单元 + + +14439 +第六单元 + + +14440 +第六单元 + + +14441 +第六单元 + + +14443 +第六单元 + + +14448 +第六单元 + + +14451 +第六单元 + + +14452 +第六单元 + + +14454 +第六单元 + + +14455 +第六单元 + + +14464 +第七单元 + + +14478 +第七单元 + + +14483 +第七单元 + + +14503 +第七单元 + + +14505 +第七单元 + + +14532 +第四单元 + + +14533 +第四单元 + + +14534 +第四单元 + + +14538 +第四单元 + + +14540 +第四单元 + + +14546 +第四单元 + + +31223 +第一单元 + + +31227 +第七单元 + + +31236 +第二单元 + + +31266 +第九单元 + + +31282 +第七单元 + + +31311 +第一单元 + + +31312 +第二单元 + + +31313 +第五单元 + + +31314 +第三单元 + + +31315 +第七单元 + + +31316 +第二单元 + + +31317 +第七单元 +第五单元 + + +31318 +第六单元 + + +31319 +第三单元 + + +31320 +第八单元 + + +31321 +第一单元 +第二单元 + + +31322 +第七单元 + + +31323 +第七单元 + + +31324 +第二单元 + + +31325 +第六单元 + + +31326 +第四单元 + + +31327 +第七单元 + + +31328 +第九单元 + + +31329 +第三单元 + + +31330 +第七单元 + + +31331 +第二单元 + 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 e61aee40..4f5a8da7 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 @@ -010171,010192,010197,020137,020143,020144,020147,020277,020290,020293,020297,020302,020307,020308,020309,020459,020471,020475,020482,020483,020487,020490,020492,020494,020495,020503,020506,020507,020519,020527,020528,020531,020532,020538,020541,020561,020563,020567,020620,020624,020626,021453,021461,021467,021476,021479,021488,021495,021505,021507,021521,021527,021553,021554,021564,021566,021570,021571,021579,021590,021593,021615,021631,031290,031297,031298,031299,031303,031307,031308 \ No newline at end of file +000321,000322,000403,001769,001781,001788,001809,001810,003210,003219,003239,003253,003281,003298,003309,003310,003312,004128,004472,004476,006717,006760,006793,006911,008456,010773,010777,010787,011059,011109,012067,012239,012729,012920,012933,013007,013921,014538,030072,030281,030473,030478,030496 \ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index 608b3ec1..f1722cf4 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 @@ -2,78 +2,36 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 40336\n", + "starting_id = 40387\n", "raworigin = \"\"\n", "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目9.tex\"\n", - "editor = \"20230326\\t王伟叶\"\n", + "editor = \"20230328\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "添加题号040336, 来源: 2025届高一下学期测验01\n", - "添加题号040337, 来源: 2025届高一下学期测验01\n", - "添加题号040338, 来源: 2025届高一下学期测验01\n", - "添加题号040339, 来源: 2025届高一下学期测验01\n", - "添加题号040340, 来源: 2025届高一下学期测验01\n", - "添加题号040341, 来源: 2025届高一下学期测验01\n", - "添加题号040342, 来源: 2025届高一下学期测验01\n", - "添加题号040343, 来源: 2025届高一下学期测验01\n", - "添加题号040344, 来源: 2025届高一下学期测验01\n", - "添加题号040345, 来源: 2025届高一下学期测验01\n", - "添加题号040346, 来源: 2025届高一下学期测验01\n", - "添加题号040347, 来源: 2025届高一下学期测验01\n", - "添加题号040348, 来源: 2025届高一下学期测验01\n", - "添加题号040349, 来源: 2025届高一下学期测验01\n", - "添加题号040350, 来源: 2025届高一下学期测验02\n", - "添加题号040351, 来源: 2025届高一下学期测验02\n", - "添加题号040352, 来源: 2025届高一下学期测验02\n", - "添加题号040353, 来源: 2025届高一下学期测验02\n", - "添加题号040354, 来源: 2025届高一下学期测验02\n", - "添加题号040355, 来源: 2025届高一下学期测验02\n", - "添加题号040356, 来源: 2025届高一下学期测验02\n", - "添加题号040357, 来源: 2025届高一下学期测验02\n", - "添加题号040358, 来源: 2025届高一下学期测验02\n", - "添加题号040359, 来源: 2025届高一下学期测验02\n", - "添加题号040360, 来源: 2025届高一下学期测验02\n", - "添加题号040361, 来源: 2025届高一下学期测验02\n", - "添加题号040362, 来源: 2025届高一下学期测验02\n", - "添加题号040363, 来源: 2025届高一下学期测验02\n", - "添加题号040364, 来源: 2025届高一下学期测验02\n", - "添加题号040365, 来源: 2025届高一下学期测验02\n", - "添加题号040366, 来源: 2025届高一下学期测验02\n", - "添加题号040367, 来源: 2025届高一下学期测验02\n", - "添加题号040368, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040369, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040370, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040371, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040372, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040373, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040374, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040375, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040376, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040377, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040378, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040379, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040380, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040381, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040382, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040383, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040384, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040385, 来源: 2025届高一下学期期中复习二(幂指对函数)\n", - "添加题号040386, 来源: 2025届高一下学期期中复习二(幂指对函数)\n" + "添加题号040387, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040388, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040389, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040390, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040391, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040392, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040393, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040394, 来源: 2025届高一下学期周末卷02小测\n", + "添加题号040395, 来源: 2025届高一下学期周末卷02小测\n" ] } ], 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 16159546..72671524 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 @@ -9,57 +9,15 @@ "name": "stdout", "output_type": "stream", "text": [ - "040336 填空题\n", - "040337 填空题\n", - "040338 填空题\n", - "040339 填空题\n", - "040340 填空题\n", - "040341 填空题\n", - "040342 填空题\n", - "040343 填空题\n", - "040344 填空题\n", - "040345 解答题\n", - "040346 选择题\n", - "040347 选择题\n", - "040348 解答题\n", - "040349 解答题\n", - "040350 填空题\n", - "040351 填空题\n", - "040352 填空题\n", - "040353 填空题\n", - "040354 填空题\n", - "040355 填空题\n", - "040356 填空题\n", - "040357 填空题\n", - "040358 填空题\n", - "040359 填空题\n", - "040360 填空题\n", - "040361 填空题\n", - "040362 选择题\n", - "040363 选择题\n", - "040364 解答题\n", - "040365 解答题\n", - "040366 解答题\n", - "040367 解答题\n", - "040368 填空题\n", - "040369 填空题\n", - "040370 解答题\n", - "040371 解答题\n", - "040372 解答题\n", - "040373 解答题\n", - "040374 解答题\n", - "040375 解答题\n", - "040376 解答题\n", - "040377 填空题\n", - "040378 填空题\n", - "040379 填空题\n", - "040380 填空题\n", - "040381 选择题\n", - "040382 选择题\n", - "040383 解答题\n", - "040384 解答题\n", - "040385 解答题\n", - "040386 解答题\n" + "040387 填空题\n", + "040388 填空题\n", + "040389 填空题\n", + "040390 填空题\n", + "040391 填空题\n", + "040392 填空题\n", + "040393 填空题\n", + "040394 填空题\n", + "040395 解答题\n" ] } ], diff --git a/工具/试卷答案生成.ipynb b/工具/试卷答案生成.ipynb index c61b5c01..54dc42a0 100644 --- a/工具/试卷答案生成.ipynb +++ b/工具/试卷答案生成.ipynb @@ -2,40 +2,14 @@ "cells": [ { "cell_type": "code", - "execution_count": 11, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "正在生成: 赋能17答案\n", - "0\n", - "正在生成: 赋能18答案\n", - "0\n", - "正在生成: 赋能19答案\n", - "0\n", - "正在生成: 赋能20答案\n", - "0\n", - "正在生成: 赋能21答案\n", - "0\n", - "正在生成: 赋能22答案\n", - "0\n", - "正在生成: 赋能23答案\n", - "0\n", - "正在生成: 赋能24答案\n", - "0\n", - "正在生成: 赋能25答案\n", - "0\n", - "正在生成: 赋能26答案\n", - "0\n", - "正在生成: 赋能27答案\n", - "0\n", - "正在生成: 赋能28答案\n", - "0\n", - "正在生成: 赋能29答案\n", - "0\n", - "正在生成: 赋能30答案\n", + "正在生成: 高三下学期周末卷04答案\n", "0\n" ] } @@ -44,20 +18,8 @@ "import os,re,json\n", "\n", "# filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\第二轮复习讲义\\09_立体几何综合.tex\"\n", - "filelist = [r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能17.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能18.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能19.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能20.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能21.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能22.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能23.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能24.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能25.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能26.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能27.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能28.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能29.tex\",\n", - "r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能30.tex\"\n", + "filelist = [r\"C:\\Users\\weiye\\Documents\\wwy sync\\23届\\下学期周末卷\\高三下学期周末卷04.tex\",\n", + "\n", "]\n", "pdf_dir = \"临时文件\"\n", "\n", diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 622bea8c..532bd2d5 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/2025届1112班较难题_教师用_20230327.tex\n", + "开始编译教师版本pdf文件: 临时文件/第四单元易错题_教师用_20230328.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/2025届1112班较难题_学生用_20230327.tex\n", + "开始编译学生版本pdf文件: 临时文件/第四单元易错题_学生用_20230328.tex\n", "0\n" ] } @@ -32,7 +32,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/2025届1112班较难题\"\n", + "filename = \"临时文件/第四单元易错题\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\"\"\"---设置是否需要解答题的空格---\"\"\"\n", @@ -177,7 +177,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -191,12 +191,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/工具/题号选题pdf生成.py b/工具/题号选题pdf生成.py index 8aef9f4b..218fb9c1 100644 --- a/工具/题号选题pdf生成.py +++ b/工具/题号选题pdf生成.py @@ -13,7 +13,7 @@ a """---设置文件名---""" #目录和文件的分隔务必用/ -filename = "临时文件/2025届1112班较难题" +filename = "临时文件/易错题" """---设置文件名结束---""" """---设置是否需要解答题的空格---""" diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index b6d3c73a..b5a7ad82 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -348248,7 +348248,9 @@ "id": "014105", "content": "设全集为$\\mathbf{R}$, 集合$A=\\{x | 03$或$x<1\\}$, 则$A \\cup \\overline {B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(0,3]$", "solution": "", @@ -348297,7 +348299,9 @@ "id": "014107", "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": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$x_1,x_2,\\cdots,x_n$均小于等于$1$", "solution": "", @@ -348343,7 +348347,9 @@ "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": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "$[6,+\\infty)$", "solution": "", @@ -348387,7 +348393,9 @@ "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": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{0,1,2\\}$", "solution": "", @@ -348412,7 +348420,9 @@ "id": "014112", "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": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "\\textcircled{2}", "solution": "", @@ -348518,7 +348528,9 @@ "id": "014117", "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": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$[0,4]$", "solution": "", @@ -348543,7 +348555,9 @@ "id": "014118", "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": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "(1) 充要条件, 理由略; (2) 必要非充分条件, 理由略; (3) 充分非必要条件, 理由略", "solution": "", @@ -348592,7 +348606,9 @@ "id": "014120", "content": "给定实数$a$, $a \\neq 0$且$a \\neq 1$, 设函数$y=\\dfrac{x-1}{a x-1}$(其中$x \\in \\mathbf{R}$且$x \\neq \\dfrac{1}{a}$). 求证: 经过这个函数图像上任意两个不同点的直线不平行于$x$轴.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -348641,7 +348657,9 @@ "id": "014122", "content": "设集合$P_1=\\{x | x^2+a x+1>0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 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}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -348861,7 +348879,9 @@ "id": "014133", "content": "不等式$\\dfrac{x+5}{x^2-2 x+1} \\geq 1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$[-1,1)\\cup (1,4]$", "solution": "", @@ -348891,7 +348911,9 @@ "id": "014134", "content": "不等式$|x-2|+2|x+1|<5$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$(-\\dfrac 53,1)$", "solution": "", @@ -348932,7 +348954,9 @@ "id": "014136", "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": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$2$或$-2$", "solution": "", @@ -349258,7 +349282,9 @@ "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$M, 00$, $a \\neq 1$)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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -349629,7 +349667,9 @@ "id": "014169", "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": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -349661,7 +349701,9 @@ "id": "014170", "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": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "证明略", "solution": "", @@ -349693,7 +349735,9 @@ "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": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$2$", "solution": "", @@ -349725,7 +349769,9 @@ "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": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$\\dfrac{\\sqrt{5}-1}2$", "solution": "", @@ -349776,7 +349822,9 @@ "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$10$", "solution": "", @@ -349808,7 +349856,9 @@ "id": "014175", "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -349840,7 +349890,9 @@ "id": "014176", "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$4$", "solution": "", @@ -349872,7 +349924,9 @@ "id": "014177", "content": "若对任意负数$x$, 代数式$|x|+2 \\cdot \\sqrt[2022]{x^{2022}}+a \\cdot \\sqrt[2023]{x^{2023}}$恒为定值, 则实数$a$的取值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -349904,7 +349958,9 @@ "id": "014178", "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -349936,7 +349992,9 @@ "id": "014179", "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(1,+\\infty)$", "solution": "", @@ -349968,7 +350026,9 @@ "id": "014180", "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$y=0.57^t\\cdot a$(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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$\\dfrac{1}{0.57^2}$", "solution": "", @@ -350000,7 +350060,9 @@ "id": "014181", "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": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -350032,7 +350094,9 @@ "id": "014182", "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": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$-\\dfrac 32$", "solution": "", @@ -350064,7 +350128,9 @@ "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(系数也为整数), 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} $\\lg 4$; \\textcircled{2} $\\lg 56$; \\textcircled{3} $\\lg 15$; \\textcircled{4} $\\lg 225$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "\\textcircled{1}\\textcircled{3}\\textcircled{4}", "solution": "", @@ -350210,7 +350276,9 @@ "id": "014190", "content": "(1) 是否存在实数$a$, 使函数$y=\\lg |2 x-a|$为偶函数?\\\\\n(2) 是否存在实数$a$、$b$, 使函数$y=\\lg |a+\\dfrac{4}{2-x}|+b$为奇函数?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 存在, $a=0$; (2) 存在, $a=-1$, $b=0$", "solution": "", @@ -350242,7 +350310,9 @@ "id": "014191", "content": "设$a$、$b$为常数, 函数$y=x^2-2 b x+1, x \\in[a, a+2]$.\\\\\n(1) 当$b=1$时, 若该函数在定义域上为单调函数, 求实数$a$的取值范围;\\\\\n(2) 当$a=0$时, 将该函数的最小值记为$f(b)$, 求$f(b)$的表达式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $(-\\infty,-1]\\cup [1,+\\infty)$; (2) $f(b)=\\begin{cases}1, & b\\le 0,\\\\1-b^2, & 0\\le b\\le 2,\\\\ 5-4b, & b\\ge 2.\\end{cases}$", "solution": "", @@ -350274,7 +350344,9 @@ "id": "014192", "content": "请利用函数$y=x^3+x$解决下列问题:\\\\\n(1) 方程$x^3+x=2023$是否有整数解? 请说明理由;\\\\\n(2) 解不等式:$(x+2023)^3+x^3+2 x+2023>0$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) 无整数解, 理由略; (2) $(-\\dfrac{2023}2,+\\infty)$", "solution": "", @@ -350306,7 +350378,9 @@ "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -350338,7 +350412,9 @@ "id": "014194", "content": "满足不等式$\\ln x+x<1$的实数$x$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(0,1)$", "solution": "", @@ -350446,7 +350522,9 @@ "id": "014199", "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(4,5]$", "solution": "", @@ -350535,7 +350613,9 @@ "id": "014203", "content": "求满足方程$12^x+5^x=13^x$的实数$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "$2$", "solution": "", @@ -350567,7 +350647,9 @@ "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$64$", "solution": "", @@ -350599,7 +350681,9 @@ "id": "014205", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 45$", "solution": "", @@ -350631,7 +350715,9 @@ "id": "014206", "content": "若$\\sin (\\alpha+\\beta)=\\dfrac{1}{3}$, $\\sin (\\alpha-\\beta)=\\dfrac{1}{2}$, 则$\\sin \\alpha \\cos \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{5}{12}$", "solution": "", @@ -350663,7 +350749,9 @@ "id": "014207", "content": "若$\\cos \\alpha=-\\dfrac{\\sqrt{5}}{5}$,$\\alpha \\in(0, \\pi)$, 则$\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 43$", "solution": "", @@ -350695,7 +350783,9 @@ "id": "014208", "content": "若角$x$满足$3 \\sin 2 x=2 \\sin x$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\arccos \\dfrac 13$", "solution": "", @@ -350879,7 +350969,9 @@ "id": "014217", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$-\\dfrac 34$", "solution": "", @@ -350968,7 +351060,9 @@ "id": "014221", "content": "已知$\\cos (2 \\alpha-\\beta)=-\\dfrac{2\\sqrt{7}}{7}$, $\\sin (\\alpha-2 \\beta)=\\dfrac{\\sqrt{21}}{14}$, $0<\\beta<\\dfrac{\\pi}{4}<\\alpha<\\dfrac{\\pi}{2}$, 求$\\alpha+\\beta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "$\\dfrac{2\\pi}3$", "solution": "", @@ -351077,7 +351171,9 @@ "id": "014226", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac{2\\sqrt{3}}3$", "solution": "", @@ -351109,7 +351205,9 @@ "id": "014227", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi-\\arccos\\dfrac 15$", "solution": "", @@ -351198,7 +351296,9 @@ "id": "014231", "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": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $3+4\\sqrt{2}\\pm \\sqrt{5}$; (2) $\\sqrt{3}$", "solution": "", @@ -351477,7 +351577,9 @@ "id": "014245", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$6$或$-6$", "solution": "", @@ -351508,7 +351610,9 @@ "id": "014246", "content": "函数$y=2 \\sin (x-\\dfrac{\\pi}{3})$, $x \\in[0, \\dfrac{\\pi}{2}]$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[-\\sqrt{3},1]$", "solution": "", @@ -351577,7 +351681,9 @@ "id": "014249", "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": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[\\dfrac\\pi{12}+k\\pi,\\dfrac{7\\pi}{12}+k\\pi]$, $k\\in \\mathbf{Z}$; (2) 最大值为$2$, 最小值为$1$", "solution": "", @@ -351716,7 +351822,9 @@ "id": "014255", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$", "solution": "", @@ -351817,7 +351925,9 @@ "id": "014259", "content": "函数$y=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac \\pi 2$", "solution": "", @@ -351861,7 +351971,9 @@ "id": "014260", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$0$", "solution": "", @@ -351911,7 +352023,10 @@ "id": "014262", "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": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -351955,7 +352070,9 @@ "id": "014263", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -352005,7 +352122,9 @@ "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.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $f(x)=\\cos 2x$; (2) $\\lambda=1$, $n=1349$", "solution": "", @@ -352105,7 +352226,9 @@ "id": "014269", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\{x|x=\\pm \\dfrac\\pi 3+2k\\pi, \\ k\\in \\mathbf{Z}\\}$", "solution": "", @@ -352149,7 +352272,9 @@ "id": "014270", "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": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -352231,7 +352356,9 @@ "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": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $[-1,4]$; (2) 图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-0.5,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:{pi/2}] plot (\\x,{2*(sin(\\x/pi*180)+sin(\\x/pi*180+60))});\n\\draw [dashed] (0,{2*sqrt(3)}) node [left] {$2\\sqrt{3}$} -- ({pi/3},{2*sqrt(3)}) --({pi/3},0) node [below] {$\\dfrac\\pi 3$};\n\\draw [dashed] (0,3) node [left] {$3$} -- ({pi/2},3) --({pi/2},0) node [below] {$\\dfrac\\pi 2$};\n\\draw (0,{sqrt(3)}) node [left] {$\\sqrt{3}$};\n\\end{tikzpicture}\n\\end{center}\n满足条件的锐角的集合为$\\{\\arcsin\\dfrac{\\sqrt{30}}6-\\dfrac\\pi 6,\\dfrac{5\\pi}6-\\arcsin\\dfrac{\\sqrt{30}}6\\}$", "solution": "", @@ -352446,7 +352573,9 @@ "id": "014283", "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[1,2)$", "solution": "", @@ -352509,7 +352638,9 @@ "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": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $y=10+\\dfrac{20}{\\cos\\theta}-10\\tan\\theta$, $\\theta\\in [0,\\dfrac\\pi 4)$; (2) 应使$|OP|=10-\\dfrac{10\\sqrt{3}}3\\text{km}$, 此时总长度的最小值为$10\\sqrt{3}+10\\text{km}$.", "solution": "", @@ -352553,7 +352684,9 @@ "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": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$[\\dfrac{5\\pi}{18},\\dfrac{\\pi}2]$", "solution": "", @@ -352603,7 +352736,9 @@ "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": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -352654,7 +352789,9 @@ "id": "014290", "content": "不等式$\\log _{\\frac{1}{2}}(1-2 x)>\\log _{\\frac{1}{2}}(3 x)$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(\\dfrac 15,\\dfrac 12)$", "solution": "", @@ -353427,7 +353564,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "当$t=-\\dfrac 12$时$|\\overrightarrow{a}-t\\overrightarrow{b}|$的值最小, 此时夹角的大小为$\\dfrac \\pi 2$", "solution": "", @@ -353459,7 +353598,9 @@ "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$N$在线段$BM$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": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) 投影为$\\dfrac 12\\overrightarrow{CB}$, 数量投影为$1$; (2) $[\\dfrac 32,2]$", "solution": "", @@ -353624,7 +353765,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(-\\dfrac 12,2)\\cup (2,+\\infty)$", "solution": "", @@ -353769,7 +353912,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -353820,7 +353965,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{10}}{10}$; (3) $\\dfrac 13$", "solution": "", @@ -353852,7 +353999,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{5\\pi}6$; (3) 存在($M$满足$\\overrightarrow{OM}=(4-\\sqrt{10})\\overrightarrow{OA}+(\\sqrt{10}-3)\\overrightarrow{OP}$)", "solution": "", @@ -353921,7 +354070,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -353972,7 +354123,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2\\sqrt{3}$", "solution": "", @@ -354080,7 +354233,9 @@ "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": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$5\\sqrt{2}$", "solution": "", @@ -354112,7 +354267,9 @@ "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.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac\\pi 6$", "solution": "", @@ -354201,7 +354358,9 @@ "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": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $1$; (2) $m=-1$, 距离为$\\sqrt{2}$; (3) $x+1=0$或$3x-4y+11=0$", "solution": "", @@ -354254,7 +354413,9 @@ "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": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $x-y-1=0$或$7x+2y-23=0$; (2) 弦$PQ$最短时, 所在直线的方程为$x+y-3=0$; 弦$PQ$最长时, 所在直线的方程为$x-y+3=0$; (3) 当$a=2$时, 两圆内切; 当$a\\in (2-2\\sqrt{2},2)\\cup (2,2+2\\sqrt{2})$时, 两圆相交; 当$a=2\\pm 2\\sqrt{2}$时, 两圆外切; 当$a\\in (-\\infty,2-2\\sqrt{2})\\cup (2+2\\sqrt{2},+\\infty)$时, 两圆外离", "solution": "", @@ -354286,7 +354447,9 @@ "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": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$(-\\infty,\\dfrac 12)$", "solution": "", @@ -354337,7 +354500,9 @@ "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": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$[0,\\dfrac\\pi 4]\\cup (\\dfrac {3\\pi}4,\\pi)$", "solution": "", @@ -354426,7 +354591,9 @@ "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": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$-1$或$-11$", "solution": "", @@ -354515,7 +354682,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(1,2]$", "solution": "", @@ -354585,7 +354754,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "(1) $2$或$-2$; (2) $1$; (3) $-2$", "solution": "", @@ -354616,7 +354787,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$2-2\\mathrm{i}$", "solution": "", @@ -354647,7 +354820,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\sqrt{5}$", "solution": "", @@ -354678,7 +354853,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$16$", "solution": "", @@ -354747,7 +354924,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "(1) 轨迹方程为$(x-1)^2+(y-1)^2=1$, $|z|$的最大值为$1+\\sqrt{2}$; (2) 轨迹方程为$\\dfrac{y^2}{25}+\\dfrac{x^2}{16}=1$, $|z|$的最大值为$5$", "solution": "", @@ -354797,7 +354976,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$2\\sqrt{10}$", "solution": "", @@ -354886,7 +355067,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -354918,7 +355101,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -354950,7 +355135,9 @@ "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": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac 12+\\dfrac 12\\mathrm{i}$", "solution": "", @@ -355267,7 +355454,9 @@ "id": "014410", "content": "如图, 在一个$60^{\\circ}$的二面角$\\alpha-l-\\beta$的棱上有两个点$A$、$B$, 其中$AC$、$BD$分别是在这个二面角的两个半平面内垂直于$AB$的线段, 且$AB=4$, $AC=6$, $BD=8$, 则$CD$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (0,0,0) -- (3,0,0) --++ (0,0,3) --++ (-3,0,0) coordinate (S) -- cycle;\n\\draw (0,0,0) --++ (0,{sqrt(3)},1) coordinate (T) --++ (3,0,0) --++ (0,{-sqrt(3)},-1);\n\\draw (1,0,0) node [below] {$A$} coordinate (A);\n\\draw (A) --++ (0,{0.6*sqrt(3)},0.6) node [above] {$C$} coordinate (C);\n\\draw (1.8,0,0) node [above] {$B$} coordinate (B);\n\\draw (B) --++ (0,0,1.6) node [below] {$D$} coordinate (D);\n\\draw (C)--(D);\n\\draw (0.5,0,0) node [above] {$l$} coordinate (l);\n\\draw (S) ++ (0.3,0,-0.3) node {$\\beta$};\n\\draw (T) ++ (0.3,{-0.15*sqrt(3)},-0.15) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2\\sqrt{17}$", "solution": "", @@ -355318,7 +355507,9 @@ "id": "014412", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 给出下列四个命题:\\\\\n\\textcircled{1} 点$P$在直线$BC_1$上运动时, 三棱锥$A-D_1PC$的体积不变;\\\\\n\\textcircled{2} 点$P$在直线$BC_1$上运动时, 直线$AP$与平面$ACD_1$所成的角的大小不变;\\\\\n\\textcircled{3} 点$P$在直线$BC_1$上运动时, 二面角$P-AD_1-C$的大小不变;\\\\\n\\textcircled{4} 点$P$是平面$ABCD$上到点$D$和$C_1$距离相等的动点, 则$P$的轨迹是过点$B$的直线.\n其中的真命题是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -355388,7 +355579,9 @@ "id": "014415", "content": "如图, 已知圆柱$OO_1$的底面半径为$1$, 正三角形$ABC$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 [left] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [left] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (1,0) node [right] {$A$} coordinate (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,0) -- (-1,2);\n\\draw (O_1) ellipse (1 and 0.3);\n\\draw (A) arc (0:-180:1 and 0.3);\n\\draw [dashed] (A) arc (0:180:1 and 0.3);\n\\draw (135:1 and 0.3) node [above] {$C$} coordinate (C);\n\\draw (-105:1 and 0.3) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$C$到平面$A_1AB$的距离;\\\\\n(2) 在劣弧$\\overset\\frown{BC}$上是否存在一点$D$, 满足$O_1D\\parallel$平面$A_1AB$? 若存在, 求出$\\angle BOD$的大小; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 32$; (2) 存在, $\\angle BOD=\\dfrac\\pi 6$", "solution": "", @@ -355477,7 +355670,9 @@ "id": "014419", "content": "如图, 已知正四面体$A-BCD$的棱长为$2$, 用平行于底面$BCD$的平面截这个棱锥, 得到一个小棱锥和一个棱台. 若截面与底面之间的距离为$\\dfrac{\\sqrt{6}}{2}$, 则棱台的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-2/sqrt(3)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({1/sqrt(3)},0,1) node [below] {$C$} coordinate (C);\n\\draw (C)++(0,0,-2) node [right] {$D$} coordinate (D);\n\\draw (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)(A)--(C)(A)--(D)(B)--(C)--(D);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(C)$) -- ($(A)!{1/4}!(D)$);\n\\draw [dashed] ($(A)!{1/4}!(B)$) -- ($(A)!{1/4}!(D)$);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{21\\sqrt{2}}{32}$", "solution": "", @@ -355566,7 +355761,9 @@ "id": "014423", "content": "已知球$O$的表面积为$900 \\pi$, $ABCD-A_1B_1C_1D_1$是该球的内接长方体(即该长方体的八个顶点均在球面上).\\\\\n(1) 若$AB=12$, $BC=9$, 求球心$O$到平面$ABCD$的距离;\\\\\n(2) 若$ABCD-A_1B_1C_1D_1$是正四棱柱, 当该正四棱柱的侧面积最大时, 求其体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{15}2\\sqrt{3}$; (2) $3375\\sqrt{2}$", "solution": "", @@ -355636,7 +355833,9 @@ "id": "014426", "content": "如图, 在直径$AB=4$的半圆$O$内作一个内接直角三角形$ABC$, 使$\\angle BAC=30^{\\circ}$, 将图中阴影部分以直线$AB$为旋转轴旋转$180^{\\circ}$形成一个几何体, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw (0,1.5) node [above] {$A$} coordinate (A);\n\\draw (0,-1.5) node [below] {$B$} coordinate (B);\n\\draw (-30:1.5) node [right] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A)--(C) arc (-30:90:1.5);\n\\fill [pattern = north east lines] (B)--(C) arc (-30:-90:1.5);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A) arc (90:-90:1.5);\n\\draw pic [draw, \"$30^\\circ$\", angle eccentricity = 2] {angle = B--A--C};\n\\draw pic [draw, \"$60^\\circ$\", angle eccentricity = 1.5] {angle = C--B--A};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{10}3\\pi$", "solution": "", @@ -355668,7 +355867,9 @@ "id": "014427", "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 体积为$27$, $E$、$F$都是棱$BC$上的任意点且$EF=1$, $P$、$Q$分别在棱$A_1D_1$、$D_1C_1$上运动, 则四面体$P-EFQ$的体积\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.4!(D1)$) node [left] {$P$} coordinate (P);\n\\draw ($(C1)!0.3!(D1)$) node [above] {$Q$} coordinate (Q);\n\\draw ($(B)!0.2!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(B)!0.7!(C)$) node [right] {$F$} coordinate (F);\n\\draw (P)--(Q);\n\\draw [dashed] (P)--(E)(P)--(F)(Q)--(E)(Q)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与点$E$、$F$、$P$、$Q$位置都有关}{与点$P$位置有关, 与点$E$、$F$、$Q$位置无关}{与点$Q$位置有关, 与点$E$、$F$、$P$位置无关}{与点$E$、$F$、$P$、$Q$位置都无关}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -355776,7 +355977,9 @@ "id": "014432", "content": "《九章算术》是我国古代的数学巨著, 其卷第五``商功''有如下的问题: ``今有刍甍, 下广三丈, 袤四丈, 上袤二丈, 无广, 高一丈. 问积几何? ''意思为: 今有底面为矩形的屋脊形状的多面体(如图), 下底面宽$AD=3$丈, 长$AB=4$丈, 上棱$EF=2$丈, $EF\\parallel AB$, $EF$与平面$ABCD$的距离为$1$丈, 则它的体积是\\blank{50}(立方丈).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\draw (-2,0,1.5) node [below] {$A$} coordinate (A);\n\\draw (A)++(4,0,0) node [below] {$B$} coordinate (B);\n\\draw (A)++(0,0,-3) node [left] {$D$} coordinate (D);\n\\draw (D)++(4,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,1,0) node [above] {$E$} coordinate (E);\n\\draw (1,1,0) node [above] {$F$} coordinate (F);\n\\draw (E)--(F)(E)--(A)(F)--(B)(F)--(C)(A)--(B)--(C)(A)--(D)--(E);\n\\draw [dashed] (D)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -355846,7 +356049,9 @@ "id": "014435", "content": "已知正四棱锥的侧棱长为$l$, 其各顶点都在同一球面上. 若该球的体积为$36 \\pi$, 且$3 \\leq l \\leq 3 \\sqrt{3}$, 求该正四棱锥体积的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "$[\\dfrac{27}4,\\dfrac{64}3]$", "solution": "", @@ -355878,7 +356083,9 @@ "id": "014436", "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, $AC$与$BD$的交点为$M$, 设$\\overrightarrow{A_1B_1}=\\overrightarrow {a}$, $\\overrightarrow{A_1D_1}=\\overrightarrow {b}$, $\\overrightarrow{A_1A}=\\overrightarrow {c}$, 则下列向量中与$\\overrightarrow{B_1M}$相等的向量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw ($(A)+(C)-(B)$) node [above left] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(3)},-1) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(B)-(A)+(A_1)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)-(A)+(A_1)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(D)-(A)+(A_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(A_1)--(B_1)--(B)--cycle (A_1)--(D_1)--(C_1)--(C)--(B)(B_1)--(C_1);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (A)--(C)(B)--(D)(M)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$-\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$-\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -355909,7 +356116,9 @@ "id": "014437", "content": "在如图所示的空间直角坐标系中, $ABCD-A_1B_1C_1D_1$为长方体, $AB=BC=1$, $AA_1=2$, 点$A$关于$x$轴对称的点$A'$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (B1) -- ($(C1)!2!(B1)$) node [below] {$x$};\n\\draw [->] (C1) -- ($(D1)!1.5!(C1)$) node [below] {$y$};\n\\draw [->] (C1) -- ($(C)!1.25!(C1)$) node [left] {$z$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(1,1,2)$", "solution": "", @@ -355940,7 +356149,9 @@ "id": "014438", "content": "已知向量$\\overrightarrow {a}=(1,-1,3)$, $\\overrightarrow {b}=(-1,4,-2)$, $\\overrightarrow {c}=(1,5,x)$, 若$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$共面, 则实数$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$5$", "solution": "", @@ -355972,7 +356183,9 @@ "id": "014439", "content": "已知空间三点$A(-2,0,2)$, $B(-1,1,2)$, $C(-3,0,4)$. 设$\\overrightarrow {a}=\\overrightarrow{AB}$, $\\overrightarrow {b}=\\overrightarrow{AC}$. 若向量$k \\overrightarrow {a}+\\overrightarrow {b}$与$k \\overrightarrow {a}-2 \\overrightarrow {b}$互相垂直, 则实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2$或$-\\dfrac 52$", "solution": "", @@ -356004,7 +356217,9 @@ "id": "014440", "content": "已知平面$ABC$中, $\\overrightarrow{AC}=(-1,-1,0)$, $\\overrightarrow{AB}=(0,1,2)$, 则平面$ABC$的一个法向量为$\\overrightarrow {n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(2,-2,1)$", "solution": "", @@ -356036,7 +356251,9 @@ "id": "014441", "content": "如图, 在正四面体$ABCD$中, $E$、$F$分别为棱$DA$、$BC$的中点, 又设$\\overrightarrow{DA}=\\overrightarrow {a}$, $\\overrightarrow{DB}=\\overrightarrow {b}$, $\\overrightarrow{DC}=\\overrightarrow {c}$.\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,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (${1/3}*(A)+{1/3}*(B)+{1/3}*(C)+(0,{2*sqrt(6)/3},0)$) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle(B)--(D);\n\\draw [dashed] (A)--(C);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below right] {$F$} coordinate (F);\n\\draw (E)--(B)(F)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow {a},\\overrightarrow {b},\\overrightarrow {c}$的线性组合表示向量$\\overrightarrow{BE}$, $\\overrightarrow{DF}$;\\\\\n(2) 求$\\langle\\overrightarrow{BE},\\overrightarrow{DF}\\rangle$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\overrightarrow{BE}=\\dfrac 12\\overrightarrow{a}-\\overrightarrow{b}$, $\\overrightarrow{DF}=\\dfrac 12\\overrightarrow{b}+\\dfrac 12\\overrightarrow{c}$; (2) $\\pi-\\arccos \\dfrac 23$", "solution": "", @@ -356087,7 +356304,9 @@ "id": "014443", "content": "如图, 在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$M$、$N$分别是棱$AB$、$AD$、$A_1B_1$、$A_1D_1$的中点, 点$P$、$Q$分别在棱$DD_1$、$BB_1$上移动, 且$DP=BQ=\\lambda$($0<\\lambda<2$).\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 [below] {$D$} 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 ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$M$} coordinate (M);\n\\draw ($(A1)!0.5!(D1)$) node [left] {$N$} coordinate (N);\n\\draw ($(B)!0.6!(B1)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(D)!0.5!(D1)$) node [left] {$P$} coordinate (P);\n\\draw (B)--(C1)(E)--(Q)--(M)--(N);\n\\draw [dashed] (E)--(F)--(P)--(N)(P)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda=1$时, 证明: 直线$BC_1\\parallel$平面$EFPQ$;\\\\\n(2) 是否存在$\\lambda$, 使平面$EFPQ$与平面$PQMN$所成的二面角为直二面角? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) 存在, $\\lambda=1\\pm \\dfrac{\\sqrt{2}}2$", "solution": "", @@ -356195,7 +356414,9 @@ "id": "014448", "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{AC}_1$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below right] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A)-- ($(D)!1.5!(A)$) node [below left] {$x$};\n\\draw [->] (C)-- ($(D)!1.4!(C)$) node [below] {$y$};\n\\draw [->] (D1)-- ($(D)!1.5!(D1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(-4,3,2)$", "solution": "", @@ -356264,7 +356485,9 @@ "id": "014451", "content": "在空间直角坐标系中, 点$A(-1,3,1)$、点$B(2,4,0)$、点$C(0,2,4)$, 则以$\\overrightarrow{AB}$、$\\overrightarrow{AC}$为一组邻边的平行四边形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$2\\sqrt{30}$", "solution": "", @@ -356295,7 +356518,9 @@ "id": "014452", "content": "在空间直角坐标系中, 点$A(1,0,0)$, 点$B(5,-4,3)$, 点$C(2,0,1)$, 则$\\overrightarrow{AB}$在$\\overrightarrow{CA}$方向上的投影向量的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(\\dfrac 72,0,\\dfrac 72)$", "solution": "", @@ -356345,7 +356570,9 @@ "id": "014454", "content": "已知半径为$1$的球$O$内切于正四面体$ABCD$, 线段$MN$是球$O$的一条动直径 ($M$、$N$是直径的两端点), 点$P$是正四面体$ABCD$的表面上的一个动点, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$[0,8]$", "solution": "", @@ -356377,7 +356604,9 @@ "id": "014455", "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": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac{2\\sqrt{42}}7$; (3) 存在, $A_1E=1$", "solution": "", @@ -356560,7 +356789,9 @@ "id": "014464", "content": "已知抛物线$C: y^2=4 x$的焦点为$F$.\\\\\n(1) 若抛物线$C$的焦点$F$为双曲线$\\Gamma: \\dfrac{x^2}{a^2}-2 y^2=1$($a>0$)的一个焦点, 求双曲线$\\Gamma$的离心率$e$;\\\\\n(2) 设抛物线$C$的准线$l$与$x$轴的交点为$E$, 点$P$在抛物线$C$上, 且在第一象限, 若$\\dfrac{|PF|}{|PE|}=\\dfrac{\\sqrt{2}}{2}$, 求直线$PE$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\sqrt{2}$; (2) $y=x+1$", "solution": "", @@ -356838,7 +357069,9 @@ "id": "014478", "content": "若过点$P(0,-1)$的直线$l$与抛物线$y^2=2 x$恰有一个公共点, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -356945,7 +357178,9 @@ "id": "014483", "content": "设椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>0$), $F_1$、$F_2$分别是椭圆$\\Gamma$的左、右焦点, 椭圆$\\Gamma$的离心率为$\\dfrac{\\sqrt{2}}{2}$, 直线$l$与椭圆$\\Gamma$交于不同的两点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆$\\Gamma$的右焦点$F_2$, $P$、$Q$是椭圆$\\Gamma$上两点, 四边形$ABQP$是菱形, 求直线$l$的方程;\\\\\n(3) 已知直线$l$与$x$轴的正半轴和$y$轴分别交于点$M$、$N$, 且$\\overrightarrow{AN}=\\lambda \\overrightarrow{AM}$, $\\overrightarrow{BN}=\\mu \\overrightarrow{BM}$, 若$\\lambda+\\mu=3$, 证明: 直线$l$过定点, 并求此定点的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -357337,7 +357572,9 @@ "id": "014503", "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯口所在的圆面与反射镜的轴垂直, 灯泡位于抛物线的焦点处. 经灯口直径是$24$厘米, 灯深$10$厘米, 则灯泡与反射镜顶点的距离是\\blank{50}厘米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\filldraw (3.6, 0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10, 0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x, 2)/14.4}, \\x);\n\\draw (10, 0) ellipse (3 and 12);\n\\draw (0, 0) node [left] {$O$} coordinate (O) --++ (0, -14);\n\\draw (10, -12) --++ (0, -2);\n\\draw [<->] (0, -13) -- (10, -13) node [midway, below] {$10\\text{cm}$};\n\\draw (10, -12) --++ (5, 0) (10, 12) --++ (5, 0);\n\\draw [<->] (14, -12) -- (14, 12) node [midway, right] {\\rotatebox{90}{$24\\text{cm}$}};\n\\draw [dashed] (10, 0) --++ (0, -12);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -357387,7 +357624,9 @@ "id": "014505", "content": "直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点, 与$x$轴正半轴不相交. 若$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 其中$O$为坐标原点, 则直线$l$过定点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -357912,7 +358151,9 @@ "id": "014532", "content": "已知等差数列$\\{a_n\\}$, 前$n$项和为$S_n$, 若$a_7+a_9=18, a_4=1$, 则$a_{12}=$\\blank{50}, $S_{15}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -357944,7 +358185,9 @@ "id": "014533", "content": "已知无穷等比数列$\\{a_n\\}$, 前$n$项和为$S_n$, 公比为$q$, $a_8=\\dfrac{1}{16}$, $q=\\dfrac{1}{2}$, 则$a_4=$\\blank{50}, $S_8=$\\blank{50}, $\\displaystyle \\lim _{n \\to+\\infty} S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -357976,7 +358219,9 @@ "id": "014534", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=2 \\times 3^n+a$. 当常数$a=$\\blank{50}时, 数列$\\{a_n\\}$为等比数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -358065,7 +358310,9 @@ "id": "014538", "content": "某地区$2020$年产生的生活垃圾为$20$万吨, 其中$6$万吨垃圾以环保方式处理, 剩余$14$万吨垃圾以填埋方式处理. 预测显示: 在以$2020$年为第一年的未来十年内, 该地区每年产生的生活垃圾量比上一年增长$5 \\%$, 同时, 通过环保方式处理的垃圾量比上一年增加$1.5$万吨, 剩余的垃圾以填埋方式处理. 根据预测, 解答下列问题:\\\\\n(1) 求$2021$年至$2023$年, 该地区三年通过填埋方式处理的垃圾共计多少万吨? (结果精确到$0.1$万吨)\\\\\n(2) 该地区在哪一年通过环保方式处理的垃圾量首次超过这一年产生的生活垃圾量的$50 \\%$?", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -358116,7 +358363,9 @@ "id": "014540", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_4=10$.\\\\\n(1) 若$S_{20}=590$, 求等差数列$\\{a_n\\}$的公差;\\\\\n(2) 若$a_1 \\in \\mathbf{Z}$, 且$S_7$是数列$\\{S_n\\}$中最大的项, 求$a_1$所有可能的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -358243,7 +358492,9 @@ "id": "014546", "content": "已知无穷等比数列$\\{a_n\\}$的公比为$q$, 前$n$项和为$S_n$, 且$\\displaystyle\\lim _{n \\to+\\infty} S_n=S$. 下列条件中, 使得$2S_n0$, $0.60$, $0.7f(s)+f(t)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=x$; (2) $g'(x)=\\mathrm{e}^x(\\ln (x+1)+\\dfrac 1{x+1}+\\dfrac{x}{(x+1)^2})$, 故$g(x)$在$[1,+\\infty)$上是严格增函数; (3) 证明略.", "solution": "(1) $f'(0)=\\mathrm{e}^0(\\dfrac{1}{1+0}+\\ln (1+0))=1$, 又$f(0)=0$, 故所求切线的方程为$y=x$.\\\\\n(2) $g(x)=f'(x)=\\mathrm{e}^x(\\dfrac{1}{1+x}+\\ln (1+x))$. $g'(x)=\\mathrm{e}^x(\\ln (1+x)+\\dfrac{2}{1+x}-\\dfrac{1}{(1+x)^2})$, 当$x>0$时, $\\mathrm{e}^x>1>0$, $\\ln(1+x)>0$, 且$\\dfrac{2}{1+x}-\\dfrac{1}{(1+x)^2}=\\dfrac{2x+1}{(1+x)^2}>0$, 所以$g'(x)$在$(0,+\\infty)$上恒正. 因此$g(x)$在$[0,+\\infty)$上是严格增函数.\\\\\n(3) 对于固定的常数$t$, 令$h(x)=f(x+t)-f(x)-f(t)$, 则$h(t)=0$, 而$h'(x)=f'(x+t)-f'(x)=g(x+t)-g(x)$. 由(2)可知$g(x+t)>g(x)$, 故$h'(x)$在$(0,+\\infty)$上恒正, 于是有$h(s)>h(0)=0$, 即$f(s+t)>f(s)+f(t)$.", @@ -442981,7 +443238,9 @@ "id": "031266", "content": "已知$18$个整数的中位数为$5$, 第$75$百分位数也为$5$, 那么这$18$个数中, $5$的个数的最小可能值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$6$", "solution": "从小到大排列后, 第$9$和第$10$个数的算术平均数为$5$. $18\\cdot 0.75=13.5$, 故第$14$个数也为$5$, 从而第$10$个数小于等于$5$, 因此它等于$5$, 第九个数也必须等于$5$, 这样, 第$9,10,11,12,13,14$这六个数都等于$5$. 另一方面, $8$个$4$, $6$个$5$, $4$个$6$的数据满足要求.", @@ -443493,7 +443752,9 @@ "id": "031282", "content": "设$a, b$均为非零实数, 则直线$y=a x+b$和曲线$a y^2-b x^2=a b$在同一坐标系下的图形可能是\\bracket{20}.\n\\fourch{\\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] {$O$};\n\\draw (0,0) ellipse ({sqrt(1.3)} and {sqrt(0.6)});\n\\draw (-0.6,{-1.3*(-0.6)+0.6}) -- (1.2,{-1.3*1.2+0.6});\n\\end{tikzpicture}}{\\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] {$O$};\n\\draw [domain = -0.9:0.9] plot ({sqrt(0.5*(1+\\x*\\x/0.3))},\\x) plot ({-sqrt(0.5*(1+\\x*\\x/0.3))},\\x);\n\\draw (-1.5,{0.3*(-1.5)-0.3}) -- (1.5,{0.3*1.5-0.3});\n\\end{tikzpicture}}{\\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] {$O$};\n\\draw (0,0) ellipse ({sqrt(0.6)} and {sqrt(1.3)});\n\\draw (-1,{-1.5*(-1)-0.3}) -- (0.7,{-1.5*0.7-0.3});\n\\end{tikzpicture}}{\\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] {$O$};\n\\draw [domain = -1.5:1.5] plot (\\x,{sqrt(0.4*(1+\\x*\\x/0.8))}) plot (\\x,{-sqrt(0.4*(1+\\x*\\x/0.8))});\n\\draw (-0.2,{2*(-0.2)-1.1}) -- (1.3,{2*1.3-1.1});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -444434,7 +444695,9 @@ "id": "031311", "content": "已知集合$A=\\{x | x^2-4 x<0\\}$, $B=\\{1,2,3,4,5\\}$, 则$\\overline {A} \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{4,5\\}$", "solution": "", @@ -444466,7 +444729,9 @@ "id": "031312", "content": "不等式$\\lg (x-1)<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(1,11)$", "solution": "", @@ -444498,7 +444763,9 @@ "id": "031313", "content": "已知复数$z=1+2 \\mathrm{i}$, 则$z\\cdot (\\overline{z})^{-1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$-\\dfrac 35+\\dfrac 45\\mathrm{i}$", "solution": "", @@ -444530,7 +444797,9 @@ "id": "031314", "content": "函数$y=\\sin ^2(\\pi x)$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -444562,7 +444831,9 @@ "id": "031315", "content": "平行直线$x+\\sqrt{3} y+\\sqrt{3}=0$与$\\sqrt{3} x+3 y-9=0$之间的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$2\\sqrt{3}$", "solution": "", @@ -444594,7 +444865,9 @@ "id": "031316", "content": "若$12^a=3^b=m$, 且$\\dfrac{1}{a}-\\dfrac{1}{b}=2$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -444626,7 +444899,10 @@ "id": "031317", "content": "向量$\\overrightarrow {n}=(a, 2)$为直线$x-2 y+2=0$的法向量, 则向量$(1,1)$在$\\overrightarrow {n}$方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第五单元" + ], "genre": "填空题", "ans": "$(-\\dfrac 15,\\dfrac 25)$", "solution": "", @@ -444658,7 +444934,9 @@ "id": "031318", "content": "长方体$ABCD-A_1B_1C_1D_1$为不计容器壁厚度的密封容器, 里面盛有体积为$V$的水, 已知$AB=3$, $AA_1=2$, $AD=1$, 如果将该密封容器任意摆放均不能使水面呈三角形, 则$V$的取值范围为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale= 0.6]\n\\def\\l{3}\n\\def\\m{2}\n\\def\\n{2}\n\\begin{scope}[x = {(10:1cm)}, y = {(100:1cm)}]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\end{scope}\n\\draw ($(A)!0.6!(A1)$) coordinate (P1);\n\\draw (P1) ++ (0,0,-\\m) coordinate (P4);\n\\draw (P1) ++ ({\\l/cos(10)},0) coordinate (P2);\n\\draw (P2) ++ (0,0,-\\m) coordinate (P3);\n\\fill [gray!30] (P1)--(P4)--(P3)--(C)--(B)--(A)--cycle;\n\\draw [thick] (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw [thick] (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [thick, dashed] (D) -- (D1);\n\\draw [thick] (A) -- (B) -- (C);\n\\draw [thick, dashed] (A) -- (D) -- (C);\n\\draw (P1)--(P2)--(P3);\n\\draw [dashed] (P1)--(P4)--(P3);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(1,5)$", "solution": "", @@ -444690,7 +444968,9 @@ "id": "031319", "content": "在$\\triangle ABC$中, $\\angle A=150^{\\circ}, D_1, D_2, \\cdots, D_{2022}$依次为边$BC$上的点, \n且$BD_1=D_1D_2=D_2D_3=\\cdots=D_{2021} D_{2022}=D_{2022} C$, 设$\\angle BAD_1=\\alpha_1$, $\\angle D_1AD_2=\\alpha_2$, $\\cdots$, \n$\\angle D_{2021} AD_{2022}=\\alpha_{2022}$, $\\angle D_{2022} AC=\\alpha_{2023}$, 则$\\dfrac{\\sin \\alpha_1 \\sin \\alpha_3 \\cdots \\sin \\alpha_{2023}}{\\sin \\alpha_2 \\sin \\alpha_4 \\cdots \\sin \\alpha_{2022}}$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 1{4046}$", "solution": "", @@ -444722,7 +445002,9 @@ "id": "031320", "content": "上海电视台五星体育频道有一档四人扑克牌竞技节目``上海三打一'', 在打法中有一种``三带二''的牌型, 即点数相同的三张牌外加一对牌, (三张牌的点数必须和对牌的点数不同). 在一副不含大小王的$52$张扑克牌中不放回的抽取五次, 已知前三次抽到两张A, 一张K, 则接下来两次抽取能抽到``三带二''的牌型(AAAKK或KKKAA)的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 3{392}$", "solution": "", @@ -444754,7 +445036,10 @@ "id": "031321", "content": "函数$y=f(x)$的表达式为$f(x)=2 x^3-5 x^2-4 x$, 如果$f(a)=f(b)=f(c)$且$a0$), $P(2,1)$为抛物线内一点, 不经过点$P$的直线$l: y=2 x+m$与抛物线相交于$A, B$两点, 直线$AP, BP$分别交抛物线于$C, D$两点, 若对任意直线$l$, 总存在$\\lambda$, 使得$\\overrightarrow{AP}=\\lambda \\overrightarrow{PC}$, $\\overrightarrow{BP}=\\lambda \\overrightarrow{PD}$($\\lambda>0$, $\\lambda \\neq 1$)成立, 则$p=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-2,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-6.5) -- (0,6.5) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -6.3:6.3] plot ({\\x*\\x/4},\\x);\n\\draw (2,1) node [right] {$P$} coordinate (P);\n\\draw (4,-4) node [below] {$C$} coordinate (C);\n\\draw (9,6) node [above] {$D$} coordinate (D);\n\\draw (1.44,2.4) node [above left] {$A$} coordinate (A);\n\\draw (0.04,-0.4) node [below left] {$B$} coordinate (B);\n\\draw ($(A)!-0.5!(B)$)--($(A)!1.5!(B)$) (C)--(D) (B)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -444818,7 +445105,9 @@ "id": "031323", "content": "已知直线$l_1: x+a y-2=0$, $l_2: (a+1) x-a y+1=0$, 则$a=-2$是$l_1\\parallel l_2$的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -444850,7 +445139,9 @@ "id": "031324", "content": "已知函数$y=\\dfrac{\\mathrm{e}^x}{\\mathrm{e}^x-1}$, 则其图像大致是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3.5:-0.1, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3:-0.1, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3.5:-0.1, samples = 100] plot (\\x+0.6,{exp(\\x)/(exp(\\x)-1)+1.3});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x+0.5,{exp(\\x)/(exp(\\x)-1)+0.3});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\begin{scope}\n\\clip (-3,-3) rectangle (3,3);\n\\draw [domain = -3:-0.1, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)-1});\n\\draw [domain = 0.1:3, samples = 100] plot (\\x,{exp(\\x)/(exp(\\x)-1)-1});\n\\end{scope}\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -444882,7 +445173,9 @@ "id": "031325", "content": "如图所示, 正三棱柱$ABC-A_1B_1C_1$的所有棱长均为$1$, 点$P$、$M$、$N$分别为棱$AA_1$、$AB$、$A_1B_1$的中点, 点$Q$为线段$MN$上的动点. 当点$Q$由点$N$出发向点$M$运动的过程中, 以下结论中正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(N)!0.3!(M)$) node [left] {$Q$} coordinate (Q);\n\\draw (A)--(B)--(B_1)--(A_1)--cycle(A_1)--(C_1)--(B_1)(M)--(N)(P)--(B);\n\\draw [dashed] (A)--(C)--(B)(C)--(P)(C)--(C_1)--(Q);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$C_1Q$与直线$CP$可能相交}{直线$C_1Q$与直线$CP$始终异面}{直线$C_1Q$与直线$CP$可能垂直}{直线$C_1Q$与直线$BP$不可能垂直}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -444914,7 +445207,9 @@ "id": "031326", "content": "下列用递推公式表示的数列中, 使得$\\displaystyle\\lim _{n \\to+\\infty} a_n=\\sqrt{2}$成立的是\\bracket{20}.\n\\twoch{$\\begin{cases}a_n=\\dfrac{1}{2}(a_{n-1}+\\dfrac{2}{a_{n-1}})(n \\geq 2), \\\\ a_1=-1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{2-3 a_{n-1}}{a_{n-1}-3}(n \\geq 2), \\\\ a_1=1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{a_{n-1}+99}{49 a_{n-1}+1}(n \\geq 2), \\\\ a_1=1\\end{cases}$}{$\\begin{cases}a_n=\\dfrac{2+a_{n-1} \\ln a_{n-1}}{a_{n-1}+\\ln a_{n-1}}(n \\geq 2), \\\\ a_1=1\\end{cases}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -444946,7 +445241,9 @@ "id": "031327", "content": "如图, 四棱锥$P-ABCD$中, 等腰$\\triangle PAB$的边长分别为$PA=PB=5$, $AB=6$, 矩形$ABCD$所在的平面与平面$PAB$垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (6,0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3,0) node [left] {$D$} coordinate (D);\n\\draw (B) ++ (0,3,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,5) node [below] {$P$} coordinate (P);\n\\draw (A)--(P)--(B)--(C)--(D)--cycle(D)--(P)--(C);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 如果$BC=3$, 求直线$PC$与平面$PAB$所成的角的大小;\\\\\n(2) 如果$PC \\perp BD$, 求$BC$的长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\arctan 35$; (2) $3\\sqrt{2}$", "solution": "", @@ -444978,7 +445275,9 @@ "id": "031328", "content": "自$2015$年上海启动 《上海绿道专项规划(2035)》至今上海已建成绿道总长度近$1600$公里. 根据 《上海市生态空间专项规划(2021-2035)》, 到$2035$年, 上海绿道总长度将超过$2000$公里. 届时, 绿道会像城市的毛细血管一样, 延伸到市民生活的各个角落. 绿荫下的绿道 (步道、骑行道) 给市民提供了散步休憩、跑步骑行运动的生态空间. 某一线品牌自行车制造商在布局线下自行车体验与销售店时随机调研了$1000$位市民, 调研数据如左下表所示. $166$位有意愿购买万元级运动自行车的受访者的年龄(单位: 岁), 在各区间内的频数记录如右下表所示.\\\\\n(1) 试估计有意愿购买万元级运动自行车人群的平均年龄 (结果精确到$0.1$岁).\\\\\n(2) 将表$1$的$2 \\times 2$列联表中的数据补充完整, 并判断是否有$95\\%$的把握认为``离家附近($2$千米内)有骑行绿道与万元级运动自行车消费有关''? \\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$, 其中$n=a+b+c+d$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$& 0.10 & 0.05 & 0.01 & 0.005 \\\\\n\\hline$k$& 2.706 & 3.841 & 6.635 & 7.879 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline &\\makecell{有意愿购买 \\\\ 万元级 \\\\ 运动自行车 }& \\makecell{没有意愿购买 \\\\ 万元级 \\\\ 运动自行车 }& 总计 \\\\ \\hline\n\\makecell{距家$2$千米内\\\\有骑行绿道}& 118 & 270 & \\\\ \\hline\n\\makecell{距家$2$千米内\\\\无骑行绿道}& & & \\\\\\hline \n总计 & 166 & & 1000 \\\\\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|}\n\\hline\n\\makecell{ 年龄分 \\\\ 组区间 } & 频数 \\\\\n\\hline$[12,18)$& 16 \\\\\n\\hline$[18,24)$& 24 \\\\\n\\hline$[24,30)$& 35 \\\\\n\\hline$[30,36)$& 30 \\\\\n\\hline$[36,42)$& 21 \\\\\n\\hline$[42,48)$& 15 \\\\\n\\hline$[48,54)$& 11 \\\\\n\\hline$[54,60)$& 6 \\\\\n\\hline$[60,66)$& 5 \\\\\n\\hline$[66,72)$& 3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "(1) $33.7$岁; (2) $\\chi^2\\approx 87.366>3.841$, 所以有骑行绿道与万元级运动自行车购买意愿有关", "solution": "", @@ -445010,7 +445309,9 @@ "id": "031329", "content": "雨天外出虽然有撑雨伞, 时常却总免不了淋湿衣袖、裤脚、背包等, 小明想通过数学建模的方法研究如何撑伞可以让淋湿的面积尽量小. 为了简化问题小明做出下列假设:\\\\\n假设 1: 在网上查阅了人均身高和肩宽的数据后, 小明把人假设为身高、肩宽分别为$170 \\text{cm}$、$40 \\text{cm}$的矩形``纸片人'';\\\\\n假设 2: 受风的影响, 雨滴下落轨迹视为与水平地面所成角为$60^{\\circ}$的直线;\\\\\n假设 3: 伞柄$OT$长为$60 \\text{cm}$, 可绕矩形``纸片人''上点$O$旋转;\\\\\n假设 4: 伞面为被伞柄$OT$垂直平分的线段$AB$, $AB=120 \\text{cm}$.\\\\\n以如图$1$方式撑伞矩形``纸片人''将淋湿``裤脚''; 以如图$2$方式撑伞矩形``纸片人''将淋湿``头和肩膀''(``裤脚''也会有一小部分被淋湿).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3,6);\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (60:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (150:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ (-30:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3,6);\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (20:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (110:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ (-70:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$2$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\begin{scope}\n\\clip (-3,0) rectangle (3.5,6);\n\\filldraw [gray!50] (0.62586,0) -- (0.8,0) -- (0.8,0.3016) -- cycle;\n\\draw (0,0) -- (0,3.4) -- (0.8,3.4) node [below right] {$O$} coordinate (O) -- (0.8,0);\n\\draw (O) --++ (39.09484:1.2) node [above right] {$T$} coordinate (T);\n\\draw (T) --++ (129.09484:1.2) node [left] {$A$} coordinate (A);\n\\draw (T) --++ ({39.09484-90}:1.2) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) --++ (60:-15);\n\\draw (B) ++ (60:1) --++ (60:-15);\n\\end{scope}\n\\draw (-3,0) -- (3,0);\n\\draw (0,0) node [below] {图$3$};\n\\end{tikzpicture}\n\\end{center}\n(1) 如图$3$在矩形``纸片人''上身恰好不被淋湿时, 求其``裤脚''被淋湿 (阴影) 部分的面积(结果精确到$0.1 \\text{cm}^2)$;\\\\\n(2) 请根据你的生活经验对小明建立的数学模型提两条改进建议. (无需求解改进后的模型, 如果建议超过两条仅对前两条评分)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 约为$65.7\\text{cm}^2$; (2) 参考改进建议: \\textcircled{1} 雨伞不遮挡视线; \\textcircled{2} 伞面为弧形，改进模型将伞设为一段圆弧; \\textcircled{3} 考虑伞柄可以伸缩; \\textcircled{4} 人体改进为立体模型; \\textcircled{5} 考虑风速、风向; \\textcircled{6} 考虑撑伞的省力、稳定等.", "solution": "", @@ -445042,7 +445343,9 @@ "id": "031330", "content": "已知椭圆$c: \\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1$($00$, 记直线$QF_1$与椭圆$C$在$x$轴上方的交点为$A(x_1, y_1)$, 直线$QF_2$与椭圆$c$在$x$轴上方的交点为$B(x_2, y_2)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (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 [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\filldraw (-1,0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (1,0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = line1] (F_1) --++ (50:2.5);\n\\path [name path = line2] (F_2) --++ (50:1.5);\n\\path [name intersections = {of = line1 and elli, by = B}];\n\\path [name intersections = {of = line2 and elli, by = A}];\n\\path [draw, name path = line3] (F_1) -- (A) node [above] {$A$};\n\\path [draw, name path = line4] (F_2) -- (B) node [above] {$B$};\n\\path [name intersections = {of = line3 and line4, by = Q}];\n\\draw (Q) node [below] {$Q$};\n\\draw (F_1)--(B) (F_2)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$c$的的离心率;\\\\\n(2) 若$AF_2\\parallel BF_1$, 证明: $\\dfrac{1}{y_1}+\\dfrac{1}{y_2}=\\dfrac{1}{y_0}$;\\\\\n(3) 若$\\dfrac{1}{y_1}+\\dfrac{1}{y_2}=\\dfrac{4}{3 y_0}$, 求点$Q$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 12$; (2) 证明略; (3) $\\dfrac 49x^2+\\dfrac 45 y^2=1$($y>0$)", "solution": "", @@ -445074,7 +445377,9 @@ "id": "031331", "content": "已知函数$y=f(x)$的表达式为$f(x)=\\dfrac{1}{2} a x^2+(a+1) x+\\ln x$($a \\in \\mathbf{R}$).\\\\\n(1) 若$1$是$f(x)$的极值点, 求$a$的值;\\\\\n(2) 求$f(x)$的单调区间;\\\\\n(3) 若$f(x)=\\dfrac{1}{2} a x^2+x$有两个实数解$x_1, x_2$($x_1