234 lines
30 KiB
Plaintext
234 lines
30 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"#修改起始id,出处,文件名\n",
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"starting_id = 22048\n",
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"raworigin = \"\"\n",
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"filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目9.tex\"\n",
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"editor = \"20230319\\t王伟叶\"\n",
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"indexed = False\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"添加题号022048, 来源: 2024届空间向量校本作业\n",
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"添加题号022049, 来源: 2024届空间向量校本作业\n",
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"添加题号022050, 来源: 2024届空间向量校本作业\n",
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"添加题号022051, 来源: 2024届空间向量校本作业\n",
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"添加题号022052, 来源: 2024届空间向量校本作业\n",
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"添加题号022053, 来源: 2024届空间向量校本作业\n",
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"添加题号022054, 来源: 2024届空间向量校本作业\n",
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"添加题号022055, 来源: 2024届空间向量校本作业\n",
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"添加题号022056, 来源: 2024届空间向量校本作业\n",
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"添加题号022057, 来源: 2024届空间向量校本作业\n",
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"添加题号022058, 来源: 2024届空间向量校本作业\n",
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"添加题号022059, 来源: 2024届空间向量校本作业\n",
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"添加题号022060, 来源: 2024届空间向量校本作业\n",
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"添加题号022061, 来源: 2024届空间向量校本作业\n",
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"添加题号022062, 来源: 2024届空间向量校本作业\n",
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"添加题号022063, 来源: 2024届空间向量校本作业\n",
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"添加题号022064, 来源: 2024届空间向量校本作业\n",
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"添加题号022065, 来源: 2024届空间向量校本作业\n",
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"添加题号022066, 来源: 2024届空间向量校本作业\n",
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"添加题号022067, 来源: 2024届空间向量校本作业\n",
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"添加题号022068, 来源: 2024届空间向量校本作业\n",
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"添加题号022069, 来源: 2024届空间向量校本作业\n",
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"添加题号022070, 来源: 2024届空间向量校本作业\n",
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"添加题号022071, 来源: 2024届空间向量校本作业\n",
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"添加题号022072, 来源: 2024届空间向量校本作业\n",
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"添加题号022073, 来源: 2024届空间向量校本作业\n",
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"添加题号022074, 来源: 2024届空间向量校本作业\n",
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"添加题号022075, 来源: 2024届空间向量校本作业\n",
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"添加题号022076, 来源: 2024届空间向量校本作业\n",
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"添加题号022077, 来源: 2024届空间向量校本作业\n",
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"添加题号022078, 来源: 2024届空间向量校本作业\n",
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"添加题号022079, 来源: 2024届空间向量校本作业\n",
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"添加题号022080, 来源: 2024届空间向量校本作业\n",
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"添加题号022081, 来源: 2024届空间向量校本作业\n",
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"添加题号022082, 来源: 2024届空间向量校本作业\n",
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"添加题号022083, 来源: 2024届空间向量校本作业\n",
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"添加题号022084, 来源: 2024届二项式定理校本作业\n",
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"添加题号022085, 来源: 2024届二项式定理校本作业\n",
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"添加题号022086, 来源: 2024届二项式定理校本作业\n",
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"添加题号022087, 来源: 2024届二项式定理校本作业\n",
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"添加题号022088, 来源: 2024届二项式定理校本作业\n",
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"添加题号022089, 来源: 2024届二项式定理校本作业\n",
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"添加题号022090, 来源: 2024届二项式定理校本作业\n",
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"添加题号022091, 来源: 2024届二项式定理校本作业\n",
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"添加题号022092, 来源: 2024届二项式定理校本作业\n",
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"添加题号022093, 来源: 2024届二项式定理校本作业\n",
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"添加题号022094, 来源: 2024届二项式定理校本作业\n",
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"添加题号022095, 来源: 2024届二项式定理校本作业\n",
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"添加题号022096, 来源: 2024届二项式定理校本作业\n",
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"添加题号022097, 来源: 2024届二项式定理校本作业\n",
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"添加题号022098, 来源: 2024届二项式定理校本作业\n",
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"添加题号022099, 来源: 2024届二项式定理校本作业\n",
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"添加题号022100, 来源: 2024届二项式定理校本作业\n",
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"添加题号022101, 来源: 2024届二项式定理校本作业\n",
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"添加题号022102, 来源: 2024届二项式定理校本作业\n",
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"添加题号022103, 来源: 2024届二项式定理校本作业\n",
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"添加题号022104, 来源: 2024届二项式定理校本作业\n",
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"添加题号022105, 来源: 2024届二项式定理校本作业\n"
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]
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}
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],
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"source": [
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"import os,re,json\n",
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"\n",
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"\n",
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"#从enumerate环境的字符串生成题目列表\n",
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"def GenerateProblemListFromString(data):\n",
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" try:\n",
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" data = re.findall(r\"\\\\begin\\{document\\}([\\s\\S]*?)\\\\end\\{document\\}\",problems_string)[0]\n",
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" except:\n",
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" pass\n",
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" data = re.sub(r\"\\n{2,}\",\"\\n\",data)\n",
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" data = re.sub(r\"\\\\item\",r\"\\\\enditem\\\\item\",data)\n",
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" data = re.sub(r\"\\\\end\\{enumerate\\}\",r\"\\\\enditem\",data)\n",
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" ProblemList_raw = [p.strip() for p in re.findall(r\"\\\\item([\\s\\S]*?)\\\\enditem\",data)]\n",
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" ProblemsList = []\n",
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" for p in ProblemList_raw:\n",
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" startpos = data.index(p)\n",
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" tempdata = data[:startpos]\n",
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" suflist = re.findall(r\"\\n(\\%\\s{0,1}[\\S]+)\\n\",tempdata)\n",
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" if len(suflist) > 0:\n",
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" suffix = suflist[-1].replace(\"%\",\"\").strip()\n",
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" else:\n",
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" suffix = \"\"\n",
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" p_strip = re.sub(r\"\\n(\\%[\\S]+)$\",\"\",p).strip()\n",
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" ProblemsList.append((p_strip,suffix))\n",
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" return ProblemsList\n",
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"\n",
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"# 创建新的空题目\n",
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"def CreateEmptyProblem(problem):\n",
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" NewProblem = problem.copy()\n",
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" for field in NewProblem:\n",
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" if type(NewProblem[field]) == str:\n",
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" NewProblem[field] = \"\"\n",
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" elif type(NewProblem[field]) == list:\n",
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" NewProblem[field] = []\n",
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" elif type(NewProblem[field]) == int or type(NewProblem[field]) == float:\n",
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" NewProblem[field] = -1\n",
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" return NewProblem\n",
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"\n",
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"# 创建新题目\n",
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"def CreateNewProblem(id,content,origin,dict,editor):\n",
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" NewProblem = CreateEmptyProblem(dict[\"000001\"])\n",
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" NewProblem[\"id\"] = str(id).zfill(6)\n",
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" NewProblem[\"content\"] = content\n",
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" NewProblem[\"origin\"] = origin\n",
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" NewProblem[\"edit\"] = [editor]\n",
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" return NewProblem\n",
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"\n",
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"duplicate_flag = False\n",
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"\n",
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"with open(r\"../题库0.3/Problems.json\",\"r\",encoding = \"utf8\") as f:\n",
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" database = f.read()\n",
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"pro_dict = json.loads(database)\n",
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"\n",
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"with open(filename,\"r\",encoding = \"utf8\") as f:\n",
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" problems_string = f.read()\n",
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"problems = GenerateProblemListFromString(problems_string)\n",
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"\n",
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"\n",
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"id = starting_id\n",
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"for p_and_suffix in problems:\n",
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" p = p_and_suffix[0]\n",
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" suffix = p_and_suffix[1]\n",
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" pid = str(id).zfill(6)\n",
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" if pid in pro_dict:\n",
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" duplicate_flag = True\n",
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" if indexed == False:\n",
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" origin = raworigin + suffix\n",
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" else:\n",
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" origin = raworigin + suffix + \"试题\" + str(id- starting_id+1)\n",
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" NewProblem = CreateNewProblem(id = pid, content = p, origin = origin, dict = pro_dict,editor = editor)\n",
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" print(\"添加题号\"+pid+\", \"+\"来源: \" + origin)\n",
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" pro_dict[pid] = NewProblem\n",
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" id += 1\n",
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"\n",
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"#按id排序生成字典\n",
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"sorted_dict_id = sorted(pro_dict)\n",
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"sorted_dict = {}\n",
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"for id in sorted_dict_id:\n",
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" sorted_dict[id] = pro_dict[id]\n",
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"#将排序后的字典转为json\n",
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"\n",
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"if not duplicate_flag:\n",
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" new_database = json.dumps(sorted_dict,indent = 4,ensure_ascii=False)\n",
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" #写入json数据库文件\n",
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" with open(r\"../题库0.3/Problems.json\",\"w\",encoding = \"utf8\") as f:\n",
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" f.write(new_database)\n",
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"else:\n",
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" print(\"题号有重复, 请检查.\\n\"*5)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\"\\\\documentclass[10pt,a4paper]{article}\\n\\\\usepackage[UTF8,fontset = windows]{ctex}\\n\\\\setCJKmainfont[BoldFont=黑体,ItalicFont=楷体]{华文中宋}\\n\\\\usepackage{amssymb,amsmath,amsfonts,amsthm,mathrsfs,dsfont,graphicx}\\n\\\\usepackage{ifthen,indentfirst,enumerate,color,titletoc}\\n\\\\usepackage{tikz}\\n\\\\usepackage{multicol}\\n\\\\usepackage{multirow}\\n\\\\usepackage{makecell}\\n\\\\usepackage{longtable}\\n\\\\usepackage{diagbox}\\n\\\\usepackage{picinpar}\\n\\\\usetikzlibrary{arrows,calc,intersections,patterns,decorations.pathreplacing,3d,angles,quotes,positioning,shapes.geometric}\\n\\\\usepackage[bf,small,indentafter,pagestyles]{titlesec}\\n\\\\usepackage[top=1in, bottom=1in,left=0.8in,right=0.8in]{geometry}\\n\\\\renewcommand{\\\\baselinestretch}{1.65}\\n\\\\newtheorem{defi}{定义~}\\n\\\\newtheorem{eg}{例~}\\n\\\\newtheorem{ex}{~}\\n\\\\newtheorem{rem}{注~}\\n\\\\newtheorem{thm}{定理~}\\n\\\\newtheorem{coro}{推论~}\\n\\\\newtheorem{axiom}{公理~}\\n\\\\newtheorem{prop}{性质~}\\n\\\\newcommand{\\\\blank}[1]{\\\\underline{\\\\hbox to #1pt{}}}\\n\\\\newcommand{\\\\bracket}[1]{(\\\\hbox to #1pt{})}\\n\\\\newcommand{\\\\onech}[4]{\\\\par\\\\begin{tabular}{p{.9\\\\linewidth}}\\nA.~#1\\\\\\\\\\nB.~#2\\\\\\\\\\nC.~#3\\\\\\\\\\nD.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\twoch}[4]{\\\\par\\\\begin{tabular}{p{.46\\\\linewidth}p{.46\\\\linewidth}}\\nA.~#1& B.~#2\\\\\\\\\\nC.~#3& D.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\vartwoch}[4]{\\\\par\\\\begin{tabular}{p{.46\\\\linewidth}p{.46\\\\linewidth}}\\n(1)~#1& (2)~#2\\\\\\\\\\n(3)~#3& (4)~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\fourch}[4]{\\\\par\\\\begin{tabular}{p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}}\\nA.~#1 &B.~#2& C.~#3& D.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\varfourch}[4]{\\\\par\\\\begin{tabular}{p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}}\\n(1)~#1 &(2)~#2& (3)~#3& (4)~#4\\n\\\\end{tabular}}\\n\\\\begin{document}\\n\\n\\n\\\\begin{enumerate}\\n\\n% 2024届空间向量校本作业\\n\\n\\\\item 如图, 已知平行六面体$ABCD-A_1B_1C_1D_1$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [below] {$D_1$} coordinate (D_1);\\n\\\\draw (2,0,0) node [below] {$A_1$} coordinate (A_1);\\n\\\\draw (0.5,0,-2) node [below] {$C_1$} coordinate (C_1);\\n\\\\draw ($(A_1)+(C_1)-(D_1)$) node [right] {$B_1$} coordinate (B_1);\\n\\\\draw (1,2,0) node [left] {$D$} coordinate (D);\\n\\\\draw ($(D)+(A_1)-(D_1)$) node [above] {$A$} coordinate (A);\\n\\\\draw ($(A)+(C_1)-(D_1)$) node [above] {$B$} coordinate (B);\\n\\\\draw ($(D)+(B)-(A)$) node [above] {$C$} coordinate (C);\\n\\\\draw ($(D_1)!0.5!(A_1)$) node [below] {$M$} coordinate (M);\\n\\\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\\n\\\\draw (D_1)--(A_1)--(B_1)--(B)--(C)--(D)--cycle(A)--(B)(A)--(A_1)(A)--(D);\\n\\\\draw [dashed] (C_1)--(C)(C_1)--(B_1)(C_1)--(D_1)(M)--(N);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 写出以该平行六面体的顶点为起点与终点, 且与$\\\\overrightarrow{AB}$相等的向量;\\\\\\\\\\n(2) 写出以该平行六面体的顶点为起点与终点的$\\\\overrightarrow{AA_1}$的负向量;\\\\\\\\\\n(3) 写出以该平行六面体的顶点为起点与终点, 且与$\\\\overrightarrow{AD}$平行的向量;\\\\\\\\\\n(4) 设$M$、$N$分别是$A_1D_1$和$DC$的中点, 用$\\\\overrightarrow{AB}$、$\\\\overrightarrow{AA_1}$、$\\\\overrightarrow{AD}$表示向量$\\\\overrightarrow{MN}$.\\n\\\\item 对于平行六面体$ABCD-A_1B_1C_1D_1$, 求证: $\\\\overrightarrow{AB_1}+\\\\overrightarrow{AC}+\\\\overrightarrow{AD_1}=2 \\\\overrightarrow{AC_1}$.\\n\\\\item 在三棱锥$O-ABC$中, $G$是三角形$ABC$的重心, 用向量$\\\\overrightarrow{OA}$、$\\\\overrightarrow{OB}$、$\\\\overrightarrow{OC}$表示向量$\\\\overrightarrow{OG}$.\\n\\\\item 已知向量$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$、$\\\\overrightarrow {c}$两两垂直, 且$|\\\\overrightarrow {a}|=1$, $|\\\\overrightarrow {b}|=2$, $|\\\\overrightarrow {c}|=3$, $\\\\overrightarrow {m}=\\\\overrightarrow {a}+\\\\overrightarrow {b}+\\\\overrightarrow {c}$.\\\\\\\\\\n(1) 求$|\\\\overrightarrow {m}|$;\\\\\\\\\\n(2) 分别求$\\\\overrightarrow {m}$与$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$、$\\\\overrightarrow {c}$的夹角.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$分别是$A_1B_1$、$CD$的中点, $R$、$S$分别是棱$AA_1$、棱$CC_1$上的点, 且$AR=2RA_1$, $C_1S=2SC$, 求证: $PS\\\\parallel RQ$.\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$的边长为$1$. 求:\\\\\\\\\\n(1) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{AA_1}$;\\\\\\\\\\n(2) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{A_1C_1}$;\\\\\\\\\\n(3) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{AC_1}$.\\n\\\\item 在长方体$ABCD-A' B' C' D'$中, $A' C'$和$B' D'$相交于$O'$, 求证$DO'\\\\parallel$平面$ACB'$.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, $G$是三角形$ACD_1$的重心. 求证: $3 \\\\overrightarrow{DG}=\\\\overrightarrow{DB_1}$.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, 已知$AB=6$, $AD=2$, $AA_1=1$, $P$是棱$AB$上的点且$PB=2AP$, $M$是棱$DC$上的点, 且$DM=2MC$, $N$是$B_1C_1$的中点, 求直线$PD_1$与$MN$所成的角$\\\\theta$的大小.\\n\\\\item 已知棱长为$1$的正四面体$A-BCD$中, $E$、$F$分别在$AB$、$CD$上, 且$\\\\overrightarrow{AE}=\\\\dfrac{1}{4} \\\\overrightarrow{AB}$, \\n$\\\\overrightarrow{CF}=\\\\dfrac{1}{3} \\\\overrightarrow{CD}$.\\\\\\\\\\n(1) 求直线$DE$和$BF$所成的角的大小;\\\\\\\\\\n(2) 求$|\\\\overrightarrow{EF}|$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的高为$h$, 上、下底面是边长为$a$的正方形, 坐标原点$O$设在下底面的中心, $x$轴、$y$轴分别与下底面的对角线重合, $z$轴垂直于底面(如图). 写出下列点的坐标以及向量的坐标:\\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 ] {$B$} 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 [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $A$的坐标: \\\\blank{50}; (2) $D$的坐标: \\\\blank{50}; (3) $B_1$的坐标: \\\\blank{50};\\\\\\\\\\n(4) $\\\\overrightarrow{OA}$的坐标: \\\\blank{50}; (5) $\\\\overrightarrow{D_1A_1}$的坐标: \\\\blank{50}; (6) $\\\\overrightarrow{B_1D}$的坐标: \\\\blank{50}.\\n\\\\item 已知$\\\\overrightarrow {a}=(1,-5,4)$, $\\\\overrightarrow {b}=(2,1,7)$.\\\\\\\\\\n(1) 求$3 \\\\overrightarrow {a}+2 \\\\overrightarrow {b}$的坐标;\\\\\\\\\\n(2) 求$|\\\\overrightarrow {a}+\\\\overrightarrow {b}|$.\\n\\\\item 已知$\\\\overrightarrow {a}=(2,1,-2)$, $\\\\overrightarrow {b}=(5,-4,3)$, $\\\\overrightarrow {c}=(-8,4,1)$.\\\\\\\\\\n(1) 求证: $\\\\overrightarrow {a} \\\\perp \\\\overrightarrow {b}$;;\\n(2) 设$\\\\overrightarrow {a}$与$\\\\overrightarrow {c}$的夹角为$\\\\theta$, 求$\\\\cos \\\\theta$.\\n\\\\item 已知$P_1(2,5,4)$, $P_2(6,4,7)$, 设$\\\\overrightarrow {a}=\\\\overrightarrow{P_1P_2}$, 求$\\\\overrightarrow {a}$、$-\\\\overrightarrow {a}$和单位向量$\\\\overrightarrow{a_0}$的坐标.\\n\\\\item 已知$P_1(2,5,-6)$, 在$y$轴上求一点$P_2$, 使$|P_1P_2|=7$.\\n\\\\item 已知$P_1(1,2,3), P_2(5,4,7)$, 在$y$轴上求一点$Q$, 使$|P_1Q|=|P_2Q|$.\\n\\\\item 已知向量$\\\\overrightarrow {a}=(1,-3,2)$, $\\\\overrightarrow {b}=(2,0,-8)$, 求单位向量$\\\\overrightarrow {c}$, 使$\\\\overrightarrow {c}$与向量$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$都垂直. \\n\\\\item 已知平面$\\\\alpha$经过点$A(3,1,-1)$、$B(1,-1,0)$, 且平行于向量$\\\\overrightarrow {a}=(-1,0,2)$, 求平面$\\\\alpha$的一个法向量.\\n\\\\item 已知点$A$、$B$、$C$的坐标分别为$(x_1, y_1, z_1)$、$(x_2, y_2, z_2)$、$(x_3, y_3, z_3)$, $G$是$\\\\triangle ABC$的重心, 求点$G$的坐标.\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$, 求证: $BD_1 \\\\perp C_1D$.\\n\\\\item 正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\\\dfrac{\\\\sqrt{6}}{2}$.\\\\\\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\\\circ}$角?\\n\\\\item 如图, 平面$ABEF \\\\perp$平面$ABCD$, 四边形$ABEF$与$ABCD$都是直角梯形, \\n$\\\\angle BAD=\\\\angle FAB=90^{\\\\circ}$, $BC =\\\\dfrac 12 AD$且$BC\\\\parallel AD$, $BE = \\\\dfrac{1}{2} AF$且$BE\\\\parallel AF$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [above right] {$A$} coordinate (A);\\n\\\\draw (3,0,0) node [right] {$D$} coordinate (D);\\n\\\\draw (0,3,0) node [above] {$F$} coordinate (F);\\n\\\\draw (0,0,1.5) node [below] {$B$} coordinate (B);\\n\\\\draw (B) ++ (1.5,0,0) node [below] {$C$} coordinate (C);\\n\\\\draw (B) ++ (0,1.5,0) node [left] {$E$} coordinate (E);\\n\\\\draw (B)--(C)--(D)--(F)--(E)--cycle(E)--(C);\\n\\\\draw [dashed] (A)--(D)(A)--(F)(A)--(B);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 证明: $C, D, F, E$四点共面;\\\\\\\\\\n(2) 设$AB=BC=BE$, 求二面角$A-ED-B$的大小; \\n\\\\item 如图, 已知四棱锥$P-ABCD$的底面$ABCD$为等腰梯形, $AB\\\\parallel DC$, $AC \\\\perp BD$, $AC$与$BD$相交于点$O$, 且顶点$P$在底面上的射影恰为$O$点, 又$BO=2$, $PO=\\\\sqrt{2}$, $PB \\\\perp PD$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [left] {$O$} coordinate (O);\\n\\\\draw (-2,0,2) node [left] {$A$} coordinate (A);\\n\\\\draw (2,0,2) node [right] {$B$} coordinate (B);\\n\\\\draw ($(O)!-0.5!(B)$) node [left] {$D$} coordinate (D);\\n\\\\draw ($(O)!-0.5!(A)$) node [right] {$C$} coordinate (C);\\n\\\\draw (O) ++ (0,2,0) node [above] {$P$} coordinate (P);\\n\\\\draw (A)--(B)--(C)--(P)--cycle(B)--(P);\\n\\\\draw [dashed] (P)--(O)(P)--(D)--(B)(A)--(C)(A)--(D)--(C);\\n\\\\draw (O) pic [draw, scale = 0.3] {right angle = B--O--A};\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 求异面直线$PD$与$BC$所成角的余弦值;\\\\\\\\\\n(2) 求二面角$P-AB-C$的大小;\\\\\\\\\\n(3) 设点$M$在棱$PC$上, 且$\\\\dfrac{PM}{MC}=\\\\lambda$, 问$\\\\lambda$为何值时, $PC \\\\perp$平面$BMD$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列直线的一个方向向量.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} 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 [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $AD_1$;\\\\\\\\\\n(2) $AA_1$;\\\\\\\\\\n(3) $AC_1$;\\\\\\\\\\n(4) $AB_1$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列向量是图中哪些经过两个顶点的直线的一个方向向量?\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} 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 [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $\\\\overrightarrow {a}=(1,0,0)$;\\\\\\\\\\n(2) $\\\\overrightarrow {b}=(0,1,0)$;\\\\\\\\\\n(3) $\\\\overrightarrow {c}=(3 \\\\sqrt{2}, 0,4)$;\\\\\\\\\\n(4) $\\\\overrightarrow {d}=(0,3 \\\\sqrt{2}, 8)$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列平面的一个法向量.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} 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 [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 平面$AA_1D_1D$;\\\\\\\\\\n(2) 平面$BB_1D_1D$.\\n\\\\item 已知点$A(0,-7,0)$、$B(2,-1,1)$、$C(2,2,2)$, 求平面$ABC$的一个法向量.\\n\\\\item 已知点$S(0,6,4)$、$A(3,5,3)$、$B(-2,11,-5)$、$C(1,-1,4)$, 求点$S$到平面$ABC$的距离.\\n\\\\item 已知平面$\\\\alpha$的一个法向量$\\\\overrightarrow {n}=(3,-2,6)$, 且经过点$A(0,7,0)$, 求原点到平面$\\\\alpha$的距离.\\n\\\\item 已知三棱锥$A-BCD$的三条侧棱$AB$、$AC$、$AD$两两垂直, 且$AB=1$, $AC=2$, $AD=3$, 求顶点$A$到平面$BCD$的距离.\\n\\\\item 正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\\\dfrac{\\\\sqrt{6}}{2}$.\\\\\\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\\\circ}$角?\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$, 求二面角$B-AC-D_1$的大小.\\n\\\\item 已知$ABCD-A_1B_1C_1D_1$为正方体.\\\\\\\\\\n(1) 求直线$AC$与$B_1D$所成的角的大小;\\\\\\\\\\n(2) 求直线$B_1D$与平面$ACD_1$所成的角的大小;\\\\\\\\\\n(3) 求平面$ACD_1$与平面$B_1CD_1$所成的二面角的大小.\\n\\\\item 已知正三棱锥的底面边长和高都为$a$. 求侧面与底面所成的二面角的大小.\\n\\\\item 在三棱锥$P-ABC$中, 已知底面$ABC$是以$C$为直角的直角三角形, $PC \\\\perp$平面$ABC$, $AC=18$, $PC=6$, $BC=9$, $G$是$\\\\triangle PAB$的重心, $M$是棱$AC$的中点, 求直线$CG$与直线$BM$所成的角$\\\\theta$的大小.\\n\\\\item 已知矩形$ABCD$, 且$PD \\\\perp$平面$ABCD$, 若$PB=2$, $PB$与平面$PCD$所成的角为$45^{\\\\circ}$. $PB$与平面$ABD$所成的角为$30^{\\\\circ}$, 求:\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 1.3]\\n\\\\draw (0,0,0) node [below] {$D$} coordinate (D);\\n\\\\draw (2,0,0) node [right] {$C$} coordinate (C);\\n\\\\draw (2,0,2) node [below] {$B$} coordinate (B);\\n\\\\draw (0,0,2) node [left] {$A$} coordinate (A);\\n\\\\draw (0,2,0) node [above] {$P$} coordinate (P);\\n\\\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\\n\\\\draw [dashed] (A)--(D)--(C)(B)--(D)--(P);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $CD$的长;\\\\\\\\\\n(2) 求$PB$与$CD$所成的角;\\\\\\\\\\n(3) 求二面角$C-PB-D$的余弦值.\\n\\n% 2024届二项式定理校本作业\\n\\n\\\\item 用二项式定理展开下列两式:\\\\\\\\\\n(1) $(a+2 b)^6$;\\\\\\\\\\n(2) $(1-\\\\dfrac{1}{x})^5$.\\n\\\\item 化简:\\\\\\\\ \\n(1) $(1+\\\\sqrt{x})^5+(1-\\\\sqrt{x})^5$;\\\\\\\\\\n(2) $(2 x+y)^4-(2 x-y)^4$.\\n\\\\item 分别写出$(x-1)^{15}$的二项展开式中的前$4$项.\\n\\\\item 求$(2 a^3-3 b^2)^{10}$的二项展开式中的第$8$项.\\n\\\\item $(x-1)^n$的二项展开式中第$m$项($1 \\\\leq m \\\\leq n$且$m$、$n \\\\in \\\\mathbf{N}$)的二项式的系数是\\\\bracket{20}.\\n\\\\fourch{$\\\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1} \\\\mathrm{C}_n^m$}{$\\\\mathrm{C}_n^m$}{$(-1)^m \\\\mathrm{C}_n^m$}\\n\\\\item 求$(3 x^3-\\\\dfrac{1}{3 x^3})^{10}$的二项展开式中的常数项.\\n\\\\item 已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的系数相等, 求这两项的系数.\\n\\\\item 在$(\\\\sqrt[3]{x}-\\\\dfrac{2}{\\\\sqrt{x}})^{11}$的二项展开式中,\\\\\\\\\\n(1) 求含$x^2$项的二项式系数;\\\\\\\\\\n(2) 含$x^{\\\\frac{1}{3}}$的项是第几项? 并写出这项的系数.\\n\\\\item 已知$(x \\\\sin \\\\theta+1)^6$的二项展开式$x^2$项的系数与$(x-\\\\dfrac{15}{2} \\\\cos \\\\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\\\cos \\\\theta$的值.\\n\\\\item 求证: 当$n$为正整数时, $2^n-\\\\mathrm{C}_n^1 \\\\cdot 2^{n-1}+\\\\mathrm{C}_n^2 \\\\cdot 2^{n-2}+\\\\cdots+\\\\mathrm{C}_n^{n-1} \\\\cdot 2+(-1)^n=1$.\\n\\\\item 求$(1+2 x)^3(1-x)^4$展开式中$x^6$的系数.\\n\\\\item 在$(3 x-2 y)^9$的展开式中, 二项式系数的和是\\\\blank{50}, 各项系数的和是各项系数的绝对值之和是\\\\blank{50}.\\n\\\\item $\\\\mathrm{C}_n^1+3\\\\mathrm{C}_n^2+9\\\\mathrm{C}_n^3+\\\\cdots+3^{n-1} \\\\mathrm{C}_n^n$等于\\\\bracket{20}.\\n\\\\fourch{$4^n$}{$\\\\dfrac{4^n}{3}$}{$\\\\dfrac{4^n}{3}-1$}{$\\\\dfrac{4^n-1}{3}$}\\n\\\\item 求$(\\\\dfrac{\\\\sqrt{x}}{2}-\\\\dfrac{2}{\\\\sqrt{x}})^{10}$的二项展开式的正中间一项.\\n\\\\item 求$(x \\\\sqrt{y}-y \\\\sqrt{x})^{11}$的二项展开式的正中间两项.\\n\\\\item 用二项式定理证明: $99^{10}-1$能被$1000$整除.\\n\\\\item 求$77^{77}-15$除以$19$的余数.\\n\\\\item 求$(1+2 x+x^2)^{10}(1-x)^6$的展开式中各项系数之和.\\n\\\\item 在$(x^2-\\\\dfrac{3}{x})^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求:\\n$\\\\mathrm{C}_n^0-\\\\dfrac{1}{2} \\\\mathrm{C}_n^1+\\\\dfrac{1}{4} \\\\mathrm{C}_n^2-\\\\cdots+(-1)^n \\\\cdot \\\\dfrac{1}{2^n} \\\\mathrm{C}_n^n$.\\n\\\\item 在$(1+3 x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\\\\\\n(1) 求二项展开式中二项式系数最大的项;\\\\\\\\\\n(2) 求二项展开式中系数最大的项.\\n\\\\item 已知$(x+1)^n=x^n+\\\\cdots+a x^3+b x^2+c x+1$($n \\\\geq 1$, $n \\\\in \\\\mathrm{N}$), 且$a: b=3: 1$, 求$c$的值.\\n\\\\item 已知$n$为大于$1$的自然数, 用二项式定理证明: $(1+\\\\dfrac{1}{n})^n>2$.\\n\\n\\n\\\\end{enumerate}\\n\\n\\n\\n\\n\\n\\\\end{document}\""
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