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 7e7d183a..89f8513a 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": 1, + "execution_count": 9, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 1, + "execution_count": 9, "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 = \"3138\"\n", + "problems = \"12759\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", @@ -51,7 +51,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 8, "metadata": {}, "outputs": [], "source": [ @@ -75,7 +75,7 @@ ], "metadata": { "kernelspec": { - "display_name": "base", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -89,12 +89,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.13" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ad2bdc8ecc057115af97d19610ffacc2b4e99fae6737bb82f5d7fb13d2f2c186" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/工具/分年级专用工具/模板文件/赋能template.txt b/工具/分年级专用工具/模板文件/赋能template.txt new file mode 100644 index 00000000..6c5f2195 --- /dev/null +++ b/工具/分年级专用工具/模板文件/赋能template.txt @@ -0,0 +1,69 @@ +\documentclass[10pt,a4paper]{article} +\usepackage[UTF8,fontset = windows]{ctex} +\setCJKmainfont[BoldFont=黑体,ItalicFont=楷体]{华文中宋} +\usepackage{amssymb,amsmath,amsfonts,amsthm,mathrsfs,dsfont,graphicx} +\usepackage{ifthen,indentfirst,enumerate,color,titletoc,xcolor} +\usepackage{tikz} +\usepackage{multicol} +\usepackage{makecell} +\usepackage{longtable} +\usepackage{diagbox} +\usetikzlibrary{arrows,calc,intersections,patterns,decorations.pathreplacing,3d,angles,quotes,positioning,shapes.geometric} +\usepackage[top=9.5cm, bottom=1in,left=0.8in,right=0.8in]{geometry} +\pagestyle{empty} +\usepackage{wallpaper} + +\renewcommand{\baselinestretch}{1.65} +\newtheorem{defi}{定义~} +\newtheorem{eg}{例~} +\newtheorem{ex}{~} +\newtheorem{rem}{注~} +\newtheorem{thm}{定理~} +\newtheorem{coro}{推论~} +\newtheorem{axiom}{公理~} +\newtheorem{prop}{性质~} +\newcommand{\blank}[1]{\underline{\hbox to #1pt{}}} +\newcommand{\bracket}[1]{(\hbox to #1pt{})} +\newcommand{\onech}[4]{\par\begin{tabular}{p{.9\textwidth}} +A.~#1\\ +B.~#2\\ +C.~#3\\ +D.~#4 +\end{tabular}} +\newcommand{\twoch}[4]{\par\begin{tabular}{p{.46\textwidth}p{.46\textwidth}} +A.~#1& B.~#2\\ +C.~#3& D.~#4 +\end{tabular}} +\newcommand{\vartwoch}[4]{\par\begin{tabular}{p{.46\textwidth}p{.46\textwidth}} +(1)~#1& (2)~#2\\ +(3)~#3& (4)~#4 +\end{tabular}} +\newcommand{\fourch}[4]{\par\begin{tabular}{p{.23\textwidth}p{.23\textwidth}p{.23\textwidth}p{.23\textwidth}} +A.~#1 &B.~#2& C.~#3& D.~#4 +\end{tabular}} +\newcommand{\varfourch}[4]{\par\begin{tabular}{p{.23\textwidth}p{.23\textwidth}p{.23\textwidth}p{.23\textwidth}} +(1)~#1 &(2)~#2& (3)~#3& (4)~#4 +\end{tabular}} +\begin{document} + + +\CenterWallPaper{0.95}{背景待替换} + +\begin{multicols}{2} + + + +\begin{enumerate}[1.] + +\vspace*{5cm} + + +题目待替换 + +\end{enumerate} + +\end{multicols} + + + +\end{document} \ No newline at end of file diff --git a/工具/分年级专用工具/赋能卷生成.ipynb b/工具/分年级专用工具/赋能卷生成.ipynb index 4a8eba6f..ddca3533 100644 --- a/工具/分年级专用工具/赋能卷生成.ipynb +++ b/工具/分年级专用工具/赋能卷生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 16, + "execution_count": 7, "metadata": {}, "outputs": [ { @@ -15,14 +15,14 @@ ], "source": [ "#在 临时文件/赋能答题纸 目录中保留一个pdf(赋能试卷的答题纸), 不留别的pdf文件. \n", - "#在 临时文件/赋能答题纸 目录中保留 赋能template.tex.\n", + "#在 模板文件 目录中保留 赋能template.tex.\n", "#在 临时文件/赋能答题纸 目录中保留output目录.\n", "\"\"\"---设置文件名---\"\"\"\n", - "filename = \"赋能12\"\n", + "filename = \"赋能17\"\n", "\n", "\"\"\"---设置题目列表---\"\"\"\n", "problems = r\"\"\"\n", - "446:448,30500,450,451,30498,435,454,30499\n", + "496:498,31205,500:505\n", "\"\"\"\n", "#完成后将含有 filename 的文件移至其它目录\n", "\n", @@ -66,7 +66,7 @@ "#替换tex模板中的内容\n", "problem_list = [id for id in generate_number_set(problems.strip(),pro_dict) if id in pro_dict]\n", "\n", - "with open(\"临时文件/赋能答题纸/赋能template.tex\",\"r\",encoding=\"utf8\") as f:\n", + "with open(\"模板文件/赋能template.txt\",\"r\",encoding=\"utf8\") as f:\n", " texdata = f.read()\n", "texdata = texdata.replace(\"背景待替换\",filename+\"答题纸.png\")\n", "data_problems = \"\\n\\n\"\n", @@ -85,6 +85,13 @@ "print(os.system(\"xelatex -interaction=batchmode \" + filename +\".tex\"))\n", "os.chdir(\"../../..\")" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/工具/寻找tex文件中未赋答案的题目.ipynb b/工具/寻找tex文件中未赋答案的题目.ipynb index cacf5bb6..58760ac1 100644 --- a/工具/寻找tex文件中未赋答案的题目.ipynb +++ b/工具/寻找tex文件中未赋答案的题目.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -24,6 +24,7 @@ "测验10.tex\n", "测验11.tex\n", "测验12.tex\n", + "测验13.tex\n", "线上测验01.tex\n", "周末卷01.tex\n", "周末卷02.tex\n", @@ -38,6 +39,8 @@ "周末卷10.tex\n", "周末卷11.tex\n", "周末卷12.tex\n", + "周末卷13.tex\n", + "周末卷14.tex\n", "国庆卷.tex\n", "01_集合.tex\n", "02_常用逻辑用语.tex\n", @@ -97,17 +100,22 @@ "2020届上海春季高考.tex\n", "2021届上海春季高考.tex\n", "2022届上海春季高考.tex\n", + "2023届上海春季高考.tex\n", "2023届嘉定区一模.tex\n", + "2023届奉贤区一模.tex\n", "2023届宝山区一模.tex\n", "2023届崇明区一模.tex\n", "2023届徐汇区一模.tex\n", "2023届普陀区一模.tex\n", "2023届杨浦区一模.tex\n", "2023届松江区一模.tex\n", + "2023届浦东新区一模.tex\n", "2023届虹口区一模.tex\n", + "2023届金山区一模.tex\n", "2023届长宁区一模.tex\n", "2023届闵行区一模.tex\n", - "2023届青浦区一模.tex\n" + "2023届青浦区一模.tex\n", + "2023届静安区一模.tex\n" ] } ], @@ -148,7 +156,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -162,12 +170,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, 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 09d59313..994257e4 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 13692 , 直至 020000\n", + "首个空闲id: 14103 , 直至 020000\n", "首个空闲id: 21441 , 直至 030000\n", - "首个空闲id: 31204 , 直至 999999\n" + "首个空闲id: 31206 , 直至 999999\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 377e2676..fafeaecc 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 @@ -9,9 +9,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/高三上末尾作业_教师用_20230108.tex\n", + "开始编译教师版本pdf文件: 临时文件/赋能_教师用_20230128.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/高三上末尾作业_学生用_20230108.tex\n", + "开始编译学生版本pdf文件: 临时文件/赋能_学生用_20230128.tex\n", "0\n" ] } @@ -26,13 +26,11 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#字典字段为文件名, 之后为内容的题号\n", "problems_dict = {\n", - "\"7.3.3-正态分布\":\"30515,30516,10903,30519,30534,30517,30518,30520,30535,30538,30540,30552\",\n", - "\"8.1.1-成对数据间的关系\":\"030554,030521,030522,030523,030524\",\n", - "\"8.1.2-相关系数\":\"10905,10906,10908,30525,30526,30591\",\n", - "\"8.2.1-一元线性回归分析的基本思想\":\"30527,30528,30567,10911,10912,30573\",\n", - "\"8.2.2-一元线性回归分析的应用举例\":\"10913,10915,10916,10917,10918\",\n", - "\"8.3.1-2乘2列联表独立性检验\":\"10920,10922,30530,30531,30532,30578,30579,30580,30593\",\n", - "\"8.3.2-独立性检验的具体应用\":\"30582,30588\"\n", + "\"fn13\":\"456:465\",\n", + "\"fn14\":\"466:475\",\n", + "\"fn15\":\"476:485\",\n", + "\"fn16\":\"486:495\",\n", + "\"fn17\":\"496:505\"\n", "\n", "}\n", "\n", @@ -41,7 +39,7 @@ "\"\"\"---设置文件保存路径---\"\"\"\n", "#目录和文件的分隔务必用/\n", "directory = \"临时文件/\"\n", - "filename = \"高三上末尾作业\"\n", + "filename = \"赋能\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "if directory[-1] != \"/\":\n", " directory += \"/\"\n", @@ -186,7 +184,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -200,12 +198,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, 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 525eecc1..4570bdb3 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 @@ -2,21 +2,21 @@ "cells": [ { "cell_type": "code", - "execution_count": 28, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 13692\n", - "origin = \"2021年空中课堂高三复习课\"\n", - "filename = r\"D:\\temp\\2022kzkt.tex\"\n", + "starting_id = 31205\n", + "origin = \"自拟题目\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目6.tex\"\n", "editor = \"20230128\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -105,6 +105,13 @@ "else:\n", " print(\"题号有重复, 请检查.\\n\"*5)" ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/工具/生成文件夹下的题号清单.ipynb b/工具/生成文件夹下的题号清单.ipynb index b0a66fa9..45a8b27d 100644 --- a/工具/生成文件夹下的题号清单.ipynb +++ b/工具/生成文件夹下的题号清单.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -94,6 +94,11 @@ "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能10.tex\n", "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能11.tex\n", "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能12.tex\n", + "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能13.tex\n", + "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能14.tex\n", + "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能15.tex\n", + "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能16.tex\n", + "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\赋能\\赋能17.tex\n", "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\一模后春考前试卷备选\\2017届上海春季高考.tex\n", "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\一模后春考前试卷备选\\2018届上海春季高考.tex\n", "C:\\Users\\weiye\\Documents\\wwy sync\\23届\\一模后春考前试卷备选\\2019届上海春季高考.tex\n", @@ -184,7 +189,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -198,12 +203,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, 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 331cab18..b9b93566 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 @@ -9,411 +9,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "013287 填空题\n", - "013288 填空题\n", - "013289 填空题\n", - "013290 填空题\n", - "013291 填空题\n", - "013292 填空题\n", - "013293 填空题\n", - "013294 填空题\n", - "013295 填空题\n", - "013296 填空题\n", - "013297 选择题\n", - "013298 选择题\n", - "013299 选择题\n", - "013300 解答题\n", - "013301 解答题\n", - "013302 填空题\n", - "013303 填空题\n", - "013304 填空题\n", - "013305 填空题\n", - "013306 填空题\n", - "013307 填空题\n", - "013308 填空题\n", - "013309 填空题\n", - "013310 填空题\n", - "013311 填空题\n", - "013312 选择题\n", - "013313 选择题\n", - "013314 选择题\n", - "013315 解答题\n", - "013316 解答题\n", - "013317 填空题\n", - "013318 填空题\n", - "013319 填空题\n", - "013320 填空题\n", - "013321 填空题\n", - "013322 填空题\n", - "013323 填空题\n", - "013324 填空题\n", - "013325 填空题\n", - "013326 填空题\n", - "013327 选择题\n", - "013328 选择题\n", - "013329 选择题\n", - "013330 解答题\n", - "013331 解答题\n", - "013332 填空题\n", - "013333 填空题\n", - "013334 填空题\n", - "013335 填空题\n", - "013336 填空题\n", - "013337 填空题\n", - "013338 填空题\n", - "013339 填空题\n", - "013340 填空题\n", - "013341 填空题\n", - "013342 选择题\n", - "013343 选择题\n", - "013344 选择题\n", - "013345 解答题\n", - "013346 解答题\n", - "013347 填空题\n", - "013348 填空题\n", - "013349 填空题\n", - "013350 填空题\n", - "013351 填空题\n", - "013352 填空题\n", - "013353 填空题\n", - "013354 填空题\n", - "013355 填空题\n", - "013356 填空题\n", - "013357 选择题\n", - "013358 选择题\n", - "013359 选择题\n", - "013360 解答题\n", - "013361 解答题\n", - "013362 填空题\n", - "013363 填空题\n", - "013364 填空题\n", - "013365 填空题\n", - "013366 填空题\n", - "013367 填空题\n", - "013368 填空题\n", - "013369 填空题\n", - "013370 填空题\n", - "013371 填空题\n", - "013372 选择题\n", - "013373 选择题\n", - "013374 选择题\n", - "013375 解答题\n", - "013376 解答题\n", - "013377 填空题\n", - "013378 填空题\n", - "013379 填空题\n", - "013380 填空题\n", - "013381 填空题\n", - "013382 填空题\n", - "013383 填空题\n", - "013384 填空题\n", - "013385 填空题\n", - "013386 填空题\n", - "013387 选择题\n", - "013388 选择题\n", - "013389 选择题\n", - "013390 解答题\n", - "013391 解答题\n", - "013392 填空题\n", - "013393 填空题\n", - "013394 填空题\n", - "013395 填空题\n", - "013396 填空题\n", - "013397 填空题\n", - "013398 填空题\n", - "013399 填空题\n", - "013400 填空题\n", - "013401 填空题\n", - "013402 选择题\n", - "013403 选择题\n", - "013404 选择题\n", - "013405 解答题\n", - "013406 解答题\n", - "013407 填空题\n", - "013408 填空题\n", - "013409 填空题\n", - "013410 填空题\n", - "013411 填空题\n", - "013412 填空题\n", - "013413 填空题\n", - "013414 填空题\n", - "013415 填空题\n", - "013416 填空题\n", - "013417 选择题\n", - "013418 选择题\n", - "013419 选择题\n", - "013420 解答题\n", - "013421 解答题\n", - "013422 填空题\n", - "013423 填空题\n", - "013424 填空题\n", - "013425 填空题\n", - "013426 填空题\n", - "013427 填空题\n", - "013428 填空题\n", - "013429 填空题\n", - "013430 填空题\n", - "013431 填空题\n", - "013432 选择题\n", - "013433 选择题\n", - "013434 选择题\n", - "013435 解答题\n", - "013436 解答题\n", - "013437 填空题\n", - "013438 填空题\n", - "013439 填空题\n", - "013440 填空题\n", - "013441 填空题\n", - "013442 填空题\n", - "013443 填空题\n", - "013444 填空题\n", - "013445 填空题\n", - "013446 填空题\n", - "013447 选择题\n", - "013448 选择题\n", - "013449 选择题\n", - "013450 解答题\n", - "013451 解答题\n", - "013452 填空题\n", - "013453 填空题\n", - "013454 填空题\n", - "013455 填空题\n", - "013456 填空题\n", - "013457 填空题\n", - "013458 填空题\n", - "013459 填空题\n", - "013460 填空题\n", - "013461 填空题\n", - "013462 选择题\n", - "013463 选择题\n", - "013464 选择题\n", - "013465 解答题\n", - "013466 解答题\n", - "013467 填空题\n", - "013468 填空题\n", - "013469 填空题\n", - "013470 填空题\n", - "013471 填空题\n", - "013472 填空题\n", - "013473 填空题\n", - "013474 填空题\n", - "013475 填空题\n", - "013476 填空题\n", - "013477 选择题\n", - "013478 选择题\n", - "013479 选择题\n", - "013480 解答题\n", - "013481 解答题\n", - "013482 填空题\n", - "013483 填空题\n", - "013484 填空题\n", - "013485 填空题\n", - "013486 填空题\n", - "013487 填空题\n", - "013488 填空题\n", - "013489 填空题\n", - "013490 填空题\n", - "013491 填空题\n", - "013492 选择题\n", - "013493 选择题\n", - "013494 选择题\n", - "013495 解答题\n", - "013496 解答题\n", - "013497 填空题\n", - "013498 填空题\n", - "013499 填空题\n", - "013500 填空题\n", - "013501 填空题\n", - "013502 填空题\n", - "013503 填空题\n", - "013504 填空题\n", - "013505 填空题\n", - "013506 填空题\n", - "013507 选择题\n", - "013508 选择题\n", - "013509 选择题\n", - "013510 解答题\n", - "013511 解答题\n", - "013512 填空题\n", - "013513 填空题\n", - "013514 填空题\n", - "013515 填空题\n", - "013516 填空题\n", - "013517 填空题\n", - "013518 填空题\n", - "013519 填空题\n", - "013520 填空题\n", - "013521 填空题\n", - "013522 选择题\n", - "013523 选择题\n", - "013524 选择题\n", - "013525 解答题\n", - "013526 解答题\n", - "013527 填空题\n", - "013528 填空题\n", - "013529 填空题\n", - "013530 填空题\n", - "013531 填空题\n", - "013532 填空题\n", - "013533 填空题\n", - "013534 填空题\n", - "013535 填空题\n", - "013536 填空题\n", - "013537 选择题\n", - "013538 选择题\n", - "013539 选择题\n", - "013540 解答题\n", - "013541 解答题\n", - "013542 填空题\n", - "013543 填空题\n", - "013544 填空题\n", - "013545 填空题\n", - "013546 填空题\n", - "013547 填空题\n", - "013548 填空题\n", - "013549 填空题\n", - "013550 填空题\n", - "013551 填空题\n", - "013552 选择题\n", - "013553 选择题\n", - "013554 选择题\n", - "013555 解答题\n", - "013556 解答题\n", - "013557 填空题\n", - "013558 填空题\n", - "013559 填空题\n", - "013560 填空题\n", - "013561 填空题\n", - "013562 填空题\n", - "013563 填空题\n", - "013564 填空题\n", - "013565 填空题\n", - "013566 填空题\n", - "013567 选择题\n", - "013568 选择题\n", - "013569 选择题\n", - "013570 解答题\n", - "013571 解答题\n", - "013572 填空题\n", - "013573 填空题\n", - "013574 填空题\n", - "013575 填空题\n", - "013576 填空题\n", - "013577 填空题\n", - "013578 填空题\n", - "013579 填空题\n", - "013580 填空题\n", - "013581 填空题\n", - "013582 选择题\n", - "013583 选择题\n", - "013584 选择题\n", - "013585 解答题\n", - "013586 解答题\n", - "013587 填空题\n", - "013588 填空题\n", - "013589 填空题\n", - "013590 填空题\n", - "013591 填空题\n", - "013592 填空题\n", - "013593 填空题\n", - "013594 填空题\n", - "013595 填空题\n", - "013596 填空题\n", - "013597 填空题\n", - "013598 填空题\n", - "013599 选择题\n", - "013600 选择题\n", - "013601 选择题\n", - "013602 选择题\n", - "013603 解答题\n", - "013604 解答题\n", - "013605 解答题\n", - "013606 解答题\n", - "013607 解答题\n", - "013608 填空题\n", - "013609 填空题\n", - "013610 填空题\n", - "013611 填空题\n", - "013612 填空题\n", - "013613 填空题\n", - "013614 填空题\n", - "013615 填空题\n", - "013616 填空题\n", - "013617 填空题\n", - "013618 填空题\n", - "013619 填空题\n", - "013620 选择题\n", - "013621 选择题\n", - "013622 选择题\n", - "013623 选择题\n", - "013624 解答题\n", - "013625 解答题\n", - "013626 解答题\n", - "013627 解答题\n", - "013628 解答题\n", - "013629 填空题\n", - "013630 填空题\n", - "013631 填空题\n", - "013632 填空题\n", - "013633 填空题\n", - "013634 填空题\n", - "013635 填空题\n", - "013636 填空题\n", - "013637 填空题\n", - "013638 填空题\n", - "013639 填空题\n", - "013640 填空题\n", - "013641 选择题\n", - "013642 解答题\n", - "013643 选择题\n", - "013644 选择题\n", - "013645 解答题\n", - "013646 解答题\n", - "013647 解答题\n", - "013648 解答题\n", - "013649 解答题\n", - "013650 填空题\n", - "013651 填空题\n", - "013652 填空题\n", - "013653 填空题\n", - "013654 填空题\n", - "013655 填空题\n", - "013656 填空题\n", - "013657 填空题\n", - "013658 填空题\n", - "013659 填空题\n", - "013660 填空题\n", - "013661 填空题\n", - "013662 选择题\n", - "013663 选择题\n", - "013664 选择题\n", - "013665 选择题\n", - "013666 解答题\n", - "013667 解答题\n", - "013668 解答题\n", - "013669 解答题\n", - "013670 解答题\n", - "013671 填空题\n", - "013672 填空题\n", - "013673 填空题\n", - "013674 填空题\n", - "013675 填空题\n", - "013676 填空题\n", - "013677 填空题\n", - "013678 填空题\n", - "013679 填空题\n", - "013680 填空题\n", - "013681 填空题\n", - "013682 填空题\n", - "013683 选择题\n", - "013684 选择题\n", - "013685 选择题\n", - "013686 选择题\n", - "013687 解答题\n", - "013688 解答题\n", - "013689 解答题\n", - "013690 解答题\n", - "013691 解答题\n" + "031205 填空题\n" ] } ], @@ -455,7 +51,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -474,7 +70,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" } } }, diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 2e25e13b..11281d8c 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": 6, + "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/高一校本_教师用_20230125.tex\n", + "开始编译教师版本pdf文件: 临时文件/pudong_教师用_20230128.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/高一校本_学生用_20230125.tex\n", + "开始编译学生版本pdf文件: 临时文件/pudong_学生用_20230128.tex\n", "0\n" ] } @@ -26,14 +26,14 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n", "problems = r\"\"\"\n", - "a\n", + "12676:12696\n", "\n", "\"\"\"\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/高一校本\"\n", + "filename = \"临时文件/pudong\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", @@ -188,12 +188,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" + "version": "3.8.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" } } }, diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index d65a731b..13597d7a 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -13438,7 +13438,7 @@ }, "000461": { "id": "000461", - "content": "若从五个数$-1,0,1,2,3$中任选一个数$m$, 则使得函数$f(x)=(m^2-1)x+1$在$\\mathbf{R}$上单调递增的概率为\\blank{50}(结果用最简分数表示).", + "content": "若从五个数$-1,0,1,2,3$中任选一个数$m$, 则使得函数$f(x)=(m^2-1)x+1$在$\\mathbf{R}$上为严格增函数的概率为\\blank{50}(结果用最简分数表示).", "objs": [ "K0219001B", "K0803002B" @@ -13456,7 +13456,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [], "related": [], @@ -13465,7 +13466,7 @@ }, "000462": { "id": "000462", - "content": "在$(\\dfrac3{x^2}+\\sqrt{x})^n$的二项展开式中, 所有项的二项式系数之和为$1024$, 则常数项的值等于\\blank{50}.", + "content": "在$(\\dfrac3{x^2}+\\sqrt{x})^n$的二项展开式中, 所有项的系数之和为$1048576$, 则常数项的值等于\\blank{50}.", "objs": [ "KNONE" ], @@ -13482,7 +13483,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [], "related": [ @@ -13842,7 +13844,7 @@ }, "000476": { "id": "000476", - "content": "已知全集$U=\\mathbf{N}$, 集合$A=\\{1,2,3,4\\}$, 集合$B=\\{3,4,5\\}$, 则$(\\complement_U A)\\cap B=$\\blank{50}.", + "content": "已知全集$U=\\mathbf{N}$, 集合$A=\\{1,2,3,4\\}$, 集合$B=\\{3,4,5\\}$, 则$\\overline{A}\\cap B=$\\blank{50}.", "objs": [ "K0104001B", "K0104006B" @@ -13859,7 +13861,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [], "related": [], @@ -14017,7 +14020,7 @@ }, "000482": { "id": "000482", - "content": "已知球主视图的面积等于$9\\pi$, 则该球的体积为\\blank{50}.", + "content": "已知球的大圆面积等于$9\\pi$, 则该球的体积为\\blank{50}.", "objs": [], "tags": [ "第六单元" @@ -14031,7 +14034,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [], "related": [ @@ -14385,7 +14389,7 @@ }, "000496": { "id": "000496", - "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x||x-1|>1\\}$, $B=\\{x|\\dfrac{x-3}{x+1}<0\\}$, 则$(\\complement_U A)\\cap B=$\\blank{50}.", + "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x||x-1|>1\\}$, $B=\\{x|\\dfrac{x-3}{x+1}<0\\}$, 则$\\overline{A}\\cap B=$\\blank{50}.", "objs": [ "K0116001B", "K0116002B", @@ -14406,7 +14410,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [ "010965" @@ -14519,7 +14524,7 @@ }, "000501": { "id": "000501", - "content": "过点$P(-2,1)$作圆$x^2+y^2=5$的切线, 则该切线的点法向式方程是\\blank{50}.", + "content": "过点$P(-2,1)$作圆$x^2+y^2=5$的切线, 则该切线的点法式方程是\\blank{50}.", "objs": [ "K0711003X" ], @@ -14549,7 +14554,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20230128\t王伟叶" ], "same": [ "010969" @@ -314892,7 +314898,7 @@ }, "012752": { "id": "012752", - "content": "根据下图判断, 下列选项错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {2017,2018,...,2021} {\\draw ({(\\i-2016)*2-1},0) node [below] {$\\i$年};};\n\\foreach \\i/\\j in {1/10000,2/20000,3/30000,4/40000,5/50000} {\\draw [gray] (10,\\i) -- (0,\\i) node [black,left] {$\\j$};}; \n\\foreach \\i/\\j/\\k in {1/1.312031/1.911751,2/1.366685/2.034308,3/1.372091/2.032591,4/1.372536/2.110311,5/1.571867/2.489168}\n{\\fill [white] ({2*\\i-1.5},0) --++ (0,\\j) --++ (1,0) --++ (0,{-\\j});\n\\fill [gray!50] ({2*\\i-1.5},\\j) --++ (0,\\k) --++ (1,0) --++ (0,{-\\k});\n\\draw ({2*\\i-1.5},0) --++ (0,\\j) --++ (1,0) --++ (0,{-\\j});\n\\draw ({2*\\i-1.5},\\j) --++ (0,\\k) --++ (1,0) --++ (0,{-\\k});\n};\n\\foreach \\i/\\j/\\k in {1/1.312031/13120.31,2/1.366685/13666.85,3/1.372091/13720.91,4/1.372536/13725.36,5/1.571867/15718.67}\n{\\draw ({2*\\i-1},{\\j/2}) node {\\tiny $\\k$};};\n\\foreach \\i/\\j/\\k/\\n in {1/1.312031/1.911751/19117.51,2/1.366685/2.034308/20343.08,3/1.372091/2.032591/20325.91,4/1.372536/2.110311/21103.11,5/1.571867/2.489168/24891.68}\n{\\draw ({2*\\i-1},{\\j+\\k/2}) node {\\tiny $\\n$};};\n\\filldraw [fill = gray!50] (10.5,3) rectangle (10.7,3.2) (10.9,3.1) node [right] {货物进口额};\n\\draw (10.5,2) rectangle (10.7,2.2) (10.9,2.1) node [right] {货物出口额};\n\\draw [->] (0,0) -- (10,0) node [below] {年份};\n\\draw [->] (0,0) -- (0,5.5) node [left] {亿元};\n\\draw (5,-1) node {2017-2021年上海市货物进出口总额};\n\\end{tikzpicture}\n\\end{center}\n\\onech{从 2018 年开始, 2021 的进出口总额增长率最大}{从 2018 年开始, 进出口总额逐年增大}{从 2018 年开始, 进口总额逐年增大}{从 2018 年开始, 2020 的进出口总额增长率最小}", + "content": "根据下图判断, 下列选项错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {2017,2018,...,2021} {\\draw ({(\\i-2016)*2-1},0) node [below] {$\\i$年};};\n\\foreach \\i/\\j in {1/10000,2/20000,3/30000,4/40000,5/50000} {\\draw [gray] (10,\\i) -- (0,\\i) node [black,left] {$\\j$};}; \n\\foreach \\i/\\j/\\k in {1/1.312031/1.911751,2/1.366685/2.034308,3/1.372091/2.032591,4/1.372536/2.110311,5/1.571867/2.489168}\n{\\fill [white] ({2*\\i-1.5},0) --++ (0,\\j) --++ (1,0) --++ (0,{-\\j});\n\\fill [gray!50] ({2*\\i-1.5},\\j) --++ (0,\\k) --++ (1,0) --++ (0,{-\\k});\n\\draw ({2*\\i-1.5},0) --++ (0,\\j) --++ (1,0) --++ (0,{-\\j});\n\\draw ({2*\\i-1.5},\\j) --++ (0,\\k) --++ (1,0) --++ (0,{-\\k});\n};\n\\foreach \\i/\\j/\\k in {1/1.312031/13120.31,2/1.366685/13666.85,3/1.372091/13720.91,4/1.372536/13725.36,5/1.571867/15718.67}\n{\\draw ({2*\\i-1},{\\j/2}) node {\\tiny $\\k$};};\n\\foreach \\i/\\j/\\k/\\n in {1/1.312031/1.911751/19117.51,2/1.366685/2.034308/20343.08,3/1.372091/2.032591/20325.91,4/1.372536/2.110311/21103.11,5/1.571867/2.489168/24891.68}\n{\\draw ({2*\\i-1},{\\j+\\k/2}) node {\\tiny $\\n$};};\n\\filldraw [fill = gray!50] (10.5,3) rectangle (10.7,3.2) (10.9,3.1) node [right] {货物进口额};\n\\draw (10.5,2) rectangle (10.7,2.2) (10.9,2.1) node [right] {货物出口额};\n\\draw [->] (0,0) -- (10,0) node [below] {年份};\n\\draw [->] (0,0) -- (0,5.5) node [left] {亿元};\n\\draw (5,-1) node {2017-2021年上海市货物进出口总额};\n\\end{tikzpicture}\n\\end{center}\n\\onech{从 2018 年开始, 2021 的进出口总额增长率最大}{从 2018 年开始, 进出口总额逐年增大}{从 2018 年开始, 进口总额逐年增大}{从 2018 年开始, 出口总额逐年增大}", "objs": [], "tags": [ "第九单元" @@ -314955,7 +314961,7 @@ }, "012755": { "id": "012755", - "content": "已知$PA \\perp$平面$ABC$, $AB \\perp AC$, $PA=AB=3$, $AC=4$, $M$为$BC$中点, 过点$M$分别作平行于平面$PAB$的直线交$AC$、$PC$于点$E$、$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(P)$) node [above] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (P)--(B)--(C)--cycle (E)--(M);\n\\draw [dashed] (P)--(A)--(B) (E)--(F)--(M) (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PM$与平面$ABC$所成的角;\\\\\n(2) 证明: 平面$MEF\\parallel$平面$PAB$, 并求直线$ME$到平面$PAB$的距离.", + "content": "已知$PA \\perp$平面$ABC$, $AB \\perp AC$, $PA=AB=3$, $AC=4$, $M$为$BC$中点, 过点$M$分别作平行于平面$PAB$的直线交$AC$、$PC$于点$E$、$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(P)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--cycle (F)--(M);\n\\draw [dashed] (P)--(A)--(B) (F)--(E)--(M) (A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PM$与平面$ABC$所成的角;\\\\\n(2) 证明: 平面$MEF\\parallel$平面$PAB$, 并求直线$ME$到平面$PAB$的距离.", "objs": [], "tags": [ "第六单元" @@ -314997,7 +315003,7 @@ }, "012757": { "id": "012757", - "content": "为了节能环保, 节约材料、定义建筑物的 ``体形系数'' 为$S=\\dfrac{F_0}{V_0}$, 其中$F_0$为建筑物暴露在空气中的面积 (单位: 平方米), $V_0$为建筑物的体积 (单位: 立方米).\\\\\n(1) 若有一圆柱形建筑物的底面半径为$R$, 高度为$H$, 求该建筑物的 ``体形系数'';(结果用含$R$、$H$的代数式表示)\\\\\n(2) 定义建筑物的 ``形状因子'' 为$f=\\dfrac{L^2}A$, 其中$A$为底面面积, $L$为建筑底面周长, 又定义$T$为总建筑面积, 即为每层建筑面积总和 (每层建筑面积为每一层的底面面积). 现有一垂直于地面的宿舍, 该宿舍的层高为$3$米, 有$n$层, 其 ``形状因子''$f=18$, 总建筑面积为$T=100000$平方米. 已知该建筑体推导得出$S=\\sqrt{\\dfrac{f\\cdot n}{T}}+\\dfrac 1{3n}$($n$为层数), 试求当该宿舍的层数$n$为多少时, ``体形系数''$S$最小.", + "content": "为了节能环保, 节约材料、定义建筑物的 ``体形系数'' 为$S=\\dfrac{F_0}{V_0}$, 其中$F_0$为建筑物暴露在空气中的面积 (单位: 平方米), $V_0$为建筑物的体积 (单位: 立方米).\\\\\n(1) 若有一圆柱形建筑物的底面半径为$R$, 高度为$H$, 求该建筑物的 ``体形系数'';(结果用含$R$、$H$的代数式表示)\\\\\n(2) 定义建筑物的 ``形状因子'' 为$f=\\dfrac{L^2}A$, 其中$A$为底面面积, $L$为建筑底面周长, 又定义$T$为总建筑面积, 即为每层建筑面积总和 (每层建筑面积为每一层的底面面积). 现有一垂直于地面的宿舍, 该宿舍的层高为$3$米, 有$n$层, 其 ``形状因子''$f=18$, 总建筑面积为$T=10000$平方米. 试求当该宿舍的层数$n$为多少时, ``体形系数''$S$最小.", "objs": [], "tags": [ "第六单元", @@ -315040,7 +315046,7 @@ }, "012759": { "id": "012759", - "content": "设函数$f(x)=a x^3-(a+1) x^2+x$, $g(x)=k x+m$, 其中$a \\geq 0$, $k$、$m \\in \\mathbf{R}$, 若对任意$x \\in[0,1]$均有$f(x) \\leq g(x)$, 则称函数$y=g(x)$是函数$y=f(x)$的 ``控制函数''. 对于$x_0\\in [0,1]$以及定义域包含$[0,1]$上的函数$y=f(x)$, 若对所有$y=f(x)$的控制函数$y=g(x)$, $g(x_0)$有最小值, 就将该最小值定义为$\\overline f(x_0)$.\\\\\n(1) 若$a=2$, $g(x)=x$, 试问$y=g(x)$是否为函数$y=f(x)$的 ``控制函数'';\\\\\n(2) 若$a=0$, 使得直线$y=h(x)$是曲线$y=f(x)$在$x=\\dfrac 14$处的切线. 证明: 函数$y=h(x)$为函数$y=f(x)$的 ``控制函数'', 并求$\\overline f(\\dfrac 14)$的值;\\\\\n(3) 若曲线$y=f(x)$在$x=x_0$, $x_0 \\in(0,1)$处的切线过点$(1,0)$, 且$c \\in[x_0,1]$. 证明: 当且仅当$c=x_0$或$c=1$时, $\\overline f(c)=f(c)$.", + "content": "设函数$f(x)=a x^3-(a+1) x^2+x$, $g(x)=k x+m$, 其中$a \\geq 0$, $k$、$m \\in \\mathbf{R}$, 若对任意$x \\in[0,1]$均有$f(x) \\leq g(x)$, 则称函数$y=g(x)$是函数$y=f(x)$的 ``控制函数''. 对于$x_0\\in [0,1]$以及定义域包含$[0,1]$的函数$y=f(x)$, 若对所有$y=f(x)$的控制函数$y=g(x)$, $g(x_0)$有最小值, 就将该最小值定义为$\\overline f(x_0)$.\\\\\n(1) 若$a=2$, $g(x)=x$, 试问$y=g(x)$是否为函数$y=f(x)$的 ``控制函数'';\\\\\n(2) 若$a=0$, 使得直线$y=h(x)$是曲线$y=f(x)$在$x=\\dfrac 14$处的切线. 证明: 函数$y=h(x)$为函数$y=f(x)$的 ``控制函数'', 并求$\\overline f(\\dfrac 14)$的值;\\\\\n(3) 若曲线$y=f(x)$在$x=x_0$, $x_0 \\in(0,1)$处的切线过点$(1,0)$, 且$c \\in[x_0,1]$. 证明: 当且仅当$c=x_0$或$c=1$时, $\\overline f(c)=f(c)$.", "objs": [], "tags": [ "第二单元" @@ -333843,7 +333849,7 @@ "content": "已知集合$A=\\{x | x<2\\}$, 集合$B=\\{x | 3-2 x \\geq 0\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -333862,7 +333868,7 @@ "content": "已知集合$A=\\{1,2\\}, B=\\{a, a^2, 3\\}, A \\cap B=\\{1\\}$, 则$a$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -333881,7 +333887,7 @@ "content": "设集合$A=\\{x | x=\\sqrt{5 k+1},\\ k \\in \\mathrm{N}, \\ k\\ge 1\\}, B=\\{x | x \\leq 6,\\ x \\in \\mathbf{Q}\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -333900,7 +333906,7 @@ "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x |-2 \\leq x \\leq 2\\}, B=\\{y | y=\\sqrt{x}, 0 \\leq x \\leq 4\\}$, 则下列关系正确的是\\bracket{20}.\n\\fourch{$A \\subseteq \\overline B$}{$B \\subseteq \\overline A$}\n{$\\overline A \\subseteq \\overline B$}{$A \\cup B=\\mathbf{R}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -333919,7 +333925,7 @@ "content": "已知$p: (x-3)(x+1)>0$, $q: x^2-2 x+1-m^2>0$($m>0$), 若$p$是$q$的充分不必要条件, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -333938,7 +333944,7 @@ "content": "集合$A=\\{x | x^2-5 x-6=0\\}$, $B=\\{x | a x^2-x+6=0, x \\in \\mathbf{R}\\}$, 且$A \\cap B=B$, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -333950,14 +333956,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013698": { "id": "013698", "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$, 求证: ``$S_n=a n^2+b n$($a, b \\in \\mathbf{R}$)''是``$\\{a_n\\}$为等差数列''的充要条件.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -333969,14 +333975,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013699": { "id": "013699", "content": "已知$m \\in \\mathbf{R}$, 设$p: x_1$、$x_2$是方程$x^2-a x-2=0$的两个实根, 不等式$|m^2-5 m-3| \\geq|x_1-x_2|$对任意实数$a \\in[-1,1]$恒成立. $q$: 函数$y=3 x^2+2 m x+m+\\dfrac{4}{3}$图像与$x$轴有两个不同交点, 求使$p$、$q$都是真命题的$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -333988,14 +333994,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013700": { "id": "013700", "content": "已知$f(x)=(a^2-1) x^2+(a-1) x+2$, 写出``$f(x)>0$($x \\in \\mathbf{R}$)恒成立''的充要条件是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334014,7 +334020,7 @@ "content": "$A=\\{-3,-2,-1,0,1,2,3\\}$, $a$、$b \\in A$, 则$|a| < |b|$的情况有\\blank{50}种.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334033,7 +334039,7 @@ "content": "设$A, B$两点坐标分别为$(-1,0),(1,0)$. 条件甲: $A$、$B$、$C$三点构成以$\\angle C$为钝角的三角形; 条件乙: 点$C$的坐标是方程$x^2+2 y^2=1$($y \\neq 0$)的解, 则甲是乙的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -334052,7 +334058,7 @@ "content": "在直角坐标系$xOy$中, 直线$l$与拋物线$y^2=2x$相交于$A$、$B$两点.\\\\\n(1) 求证: ``如果直线$l$过点$T(3,0)$, 那么$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$''是真命题;\\\\\n(2) ``若$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=3$, 则直线$l$过点$T(3,0)$''是真命题还是假命题, 并说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334064,14 +334070,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013704": { "id": "013704", "content": "已知全集为$\\mathbf{R}$, 集合$A=\\{x | y=\\sqrt{1-x}\\}$, 集合$B=\\{x | 02 x+3 y$, $N: \\dfrac{x^2}{9}+\\dfrac{y^2}{4}<1$, 则$M$是$N$的\\bracket{20}.\n\\twoch{充要非必要条件}{必要非充分条件}{必要条件}{既不充分也不必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -334166,7 +334172,7 @@ "content": "已知集合$P=\\{x \\| x-a |<4\\}$, $Q=\\{x | x^2-4 x+3<0\\}$, 且``$x \\in P$''是``$x \\in Q$''的必要非充分条件, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334185,7 +334191,7 @@ "content": "设边长为$1$的正六边形的顶点分别为$A_i$($i=1,2, \\cdots, 6$), 集合$M=\\{\\overrightarrow {a} | \\overrightarrow {a}=\\overline{A_i A_j}(i, j=1,2, \\cdots, 6, i \\neq j)\\}$, 在集合$M$中任取两个不同的元素$\\overline {m}, \\overline {n}$, 满足$\\overrightarrow {m} \\cdot \\overrightarrow {n}=0$的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334204,7 +334210,7 @@ "content": "已知集合$M=\\{1,2,3, \\cdots, 10\\}$, 集合$A \\subseteq M$, 定义$M(A)$为$A$中元素的最小值, 当$A$取谝$M$的所有非空子集时, 对应的$M(A)$的和记为$S_{10}$, 则$S_{10}=$\\blank{50}.\n%02", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334223,7 +334229,7 @@ "content": "判断下列命题的真假:\\\\\n(1) 若$a>b$, 则$a m^2>b m^2$;\\\\\n(2) 若$a m^2>b m^2$, 则$a>b$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334235,14 +334241,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013713": { "id": "013713", "content": "比较大小: $(a+1)(a-3)$\\blank{50}$a-9$.(用``$<$''、``$>$''连接)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334261,7 +334267,7 @@ "content": "函数$f(x)=x+\\dfrac{1}{x}$的值域为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334280,7 +334286,7 @@ "content": "若$00$, $b>0$, $a+b=2$, 则下列不等式对满足条件的$a, b$恒成立的序号是\\blank{50}.(写出所以正确命题的序号)\n\\textcircled{1} $a b \\leq 1$; \\textcircled{2} $\\sqrt{a}+\\sqrt{b} \\leq \\sqrt{2}$;\n\\textcircled{3} $a^2+b^2 \\geq 2$; \\textcircled{4} $a^3+b^3 \\geq 3$; \\textcircled{5} $\\dfrac{1}{a}+\\dfrac{1}{b} \\geq 2$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334318,7 +334324,7 @@ "content": "若$\\dfrac{1}{a}<\\dfrac{1}{b}<0$, 则下列不等式: \\textcircled{1} $\\dfrac{1}{a+b}<\\dfrac{1}{a b}$、\n\\textcircled{2} $|a|+b>0$、\n\\textcircled{3} $a-\\dfrac{1}{a}>b-\\dfrac{1}{b}$\n\\textcircled{4} $\\ln a^2>\\ln b^2$中正确的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334337,7 +334343,7 @@ "content": "求使$\\sqrt{x}+\\sqrt{y} \\leq a \\sqrt{x+y}$($x>0$, $y>0$)恒成立的$a$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334349,14 +334355,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013719": { "id": "013719", "content": "已知圆$O$的半径为$1$, $PA, PB$为该圆的两条切线, $A$、$B$为两个切点, 求$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334368,14 +334374,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013720": { "id": "013720", "content": "设$a$、$b$、$c\\in (0,+\\infty)$且$\\dfrac{1}{a}+\\dfrac{9}{b}=1$, 则不等式$a+b-c \\geq 0$对任意满足条件的$a$、$b$恒成立的$c$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334394,7 +334400,7 @@ "content": "已知函数$f(x)=|\\lg x|$, 若$00$, $y>0$, 且$x+2 y+2 x y=8$, 则$x+2 y$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334432,7 +334438,7 @@ "content": "若$\\theta \\in(0, \\dfrac{\\pi}{2})$, $a>b>0$, 则$\\dfrac{a^2}{\\cos^2\\theta}+\\dfrac{b^2}{\\sin^2\\theta}$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334451,7 +334457,7 @@ "content": "若$a$、$b$是任意非零实数且$a>b$, 则下列不等式 \\textcircled{1} $\\dfrac{1}{a}<\\dfrac{1}{b}$; \\textcircled{2} $\\dfrac{b}{a}<1$; \\textcircled{3} $\\lg (a-b)>0$; \\textcircled{4} $3^{-a}<3^{-b}$中正确的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334470,7 +334476,7 @@ "content": "已知不等式$a \\leq \\dfrac{x^2+2}{|x|}$对一切负数$x$恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334489,7 +334495,7 @@ "content": "已知二次函数$f(x)=a a^2+b x$($a \\neq 0$)满足$1 \\leq f(-1) \\leq 2$,$3 \\leq f(1) \\leq 4$, 则$f(-2)$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334508,7 +334514,7 @@ "content": "正方形$ABCD$的边长为$1, K$为对角线$BD$上一动点, 联结$CK$并延长, 交$BA$于点$M$, 则$\\triangle CKD$与$\\triangle BKM$的面积之和最小时, $DK=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334527,7 +334533,7 @@ "content": "$a>1$, $b>1$且$a b-(a+b)=1$, 那么\\bracket{20}.\n\\twoch{$a+b$有最小值$2+2 \\sqrt{2}$}{$a+b$有最大值$2+2 \\sqrt{2}$}{$a b$有最大值$\\sqrt{2}+1$}{$ab$有最小值$2+2 \\sqrt{2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -334546,7 +334552,7 @@ "content": "若不等式$a^2+3 b^2 \\geq \\lambda b(a+b)$对任意实数$a$、$b$恒成立, 则$\\lambda$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334565,7 +334571,7 @@ "content": "$a$、$b \\in \\mathbf{R}$且$ab>0$, 则$\\dfrac{a^4+4 b^4+1}{ab}$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334584,7 +334590,7 @@ "content": "$a>b>0$, 那么当代数式$a^2+\\dfrac{16}{b(a-b)}$取得最小值时点$P(a, b)$的坐标为\\blank{50}.\n%03", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334603,7 +334609,7 @@ "content": "不等式$\\begin{vmatrix}x & 1 \\\\ x & x\\end{vmatrix}<2$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334622,7 +334628,7 @@ "content": "不等式$x^2-2|x|-3>0$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334641,7 +334647,7 @@ "content": "不等式$\\dfrac{(x+a)(x+b)}{x-c} \\geq 0$的解集是$[-1,2) \\cup[3,+\\infty)$, 则$a+b=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334660,7 +334666,7 @@ "content": "不等式$2 \\leq x^2+m x+10 \\leq 6$有且仅有一解, 则实数$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334679,7 +334685,7 @@ "content": "已知关于$x$的方程$a x+2 a+1=0$. 若该方程在区间$[-1,1]$上有解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334698,7 +334704,7 @@ "content": "求函数$f(x)=\\sqrt{3 x^2-a x-a}$的定义域.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334710,14 +334716,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013738": { "id": "013738", "content": "若$1$是关于$x$的不等式$\\log _a(\\dfrac{x^2}{2}-\\dfrac{1}{4})>\\log _a(x-a)$的唯一整数解, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334729,14 +334735,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013739": { "id": "013739", "content": "已知$a_n=(n-1) n$, $b_n=n^2$, $n \\in \\mathbf{N}$, $n \\ge 1$.\\\\\n(1) 若$ma_n^2+b_m^2-2 a_m b_n$;\\\\\n(2) 求最小的自然数$k$, 使得当$n \\geq k$时, 对任意实数$\\lambda \\in[0,1]$, 不等式$(2 \\lambda-3) b_n \\geq(2 \\lambda-4) a_n+(\\lambda-3)$恒成立.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334748,14 +334754,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013740": { "id": "013740", "content": "不等式$\\dfrac{x+5}{(x-1)^2} \\geq 2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334774,7 +334780,7 @@ "content": "$f(x)=(x-2)(a x+b)$为偶函数, 且在$(0,+\\infty)$上是严格增函数, 则不等式$f(2-x)>0$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334793,7 +334799,7 @@ "content": "函数$f(x)=\\begin{cases}2 e^{x-1}, & x<2, \\\\ \\log _3(x^2-1), & x \\geq 2.\\end{cases}$ 则不等式$f(x)>2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334812,7 +334818,7 @@ "content": "设函数$f(x)=m x^2-m x-1$.\\\\\n(1) 若对于一切实数$x$, $f(x)<0$恒成立, 求$m$的取值范围;\\\\\n(2) 若对于$x \\in[1,3]$, $f(x)<-m+5$恒成立, 求$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -334824,14 +334830,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013744": { "id": "013744", "content": "不等式$\\dfrac{2}{3-5 x} \\geq 3$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334850,7 +334856,7 @@ "content": "不等式$|x+3|<|2 x-1 |+\\dfrac{x}{2}+1$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334869,7 +334875,7 @@ "content": "关于$x$的不等式$a x-b<0$的解集是$(1,+\\infty)$, 则关于$x$的不等式$(a x+b)(x-3)>0$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334888,7 +334894,7 @@ "content": "不等式$\\log _2^2 x+p \\log _2 x+1>2 \\log _2 x+p$对任意的$p \\in(-2,2)$恒成立, 则实数$x$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334907,7 +334913,7 @@ "content": "若不等式$\\sqrt{9-x^2} \\leq k(x+2)-\\sqrt{2}$的解集为$[a, b]$, 且$b-a=2$, 则$k=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334926,7 +334932,7 @@ "content": "已知关于$x$的不等式$\\dfrac{a x-5}{x^2-a}<0$的解集为$M$, 若$3 \\in M$且$5 \\notin M$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334945,7 +334951,7 @@ "content": "设函数$f(x)=\\begin{cases}x+1, & x \\leq 0, \\\\ 2^x, & x>0.\\end{cases}$ 则满足$f(x)+f(x-\\dfrac{1}{2})>1$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -334964,7 +334970,7 @@ "content": "关于$x$的不等式$(2 x-1)^2-1$, 且当$x \\in[-\\dfrac{a}{2}, \\dfrac{1}{2})$时, $f(x) \\leq g(x)$恒成立, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -335071,14 +335077,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013757": { "id": "013757", "content": "设$O$为坐标原点, $P$是以$F$为焦点的抛物线$y^2=2 p x$($p>0$)上任意一点, $M$是线段$PF$上的点, 且$|PM|=2|MF|$, 则直线$OM$斜率的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335097,7 +335103,7 @@ "content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{x \\cdot y | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.\n%05", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -335109,14 +335115,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013759": { "id": "013759", "content": "已知函数$y=f(x)+x^3$是偶函数, 且$f(2)=1$, 则$f(-2)$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335135,7 +335141,7 @@ "content": "已知函数$y=f(x)$存在反函数$y=f^{-1}(x)$, 且$f(x)+f(-x)=2020$, 则$f^{-1}(x)+f^{-1}(2020-x)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335154,7 +335160,7 @@ "content": "已知函数$f(x)=\\begin{cases}\\sin \\pi x, & x \\leq 0, \\\\ f(x-1)-1, & x>0,\\end{cases}$ 则$f(-\\dfrac{11 \\pi}{6})+f(\\dfrac{11 \\pi}{6})=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335173,7 +335179,7 @@ "content": "记函数$y=f(x)$的反函数为$y=f^{-1}(x)$, 如果函数$y=f(x)$的图像过点$(1,2)$, 那么函数$y=f^{-1}(x)+1$的图像过点\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335192,7 +335198,7 @@ "content": "已知$a \\in \\mathbf{R}$, 函数$f(x)=\\begin{cases}x^2+2 x+a-2, & x \\leq 0, \\\\ -x^2+2 x-2 a, & x>0,\\end{cases}$ 若对任意$x \\in[-3,+\\infty)$, $f(x) \\leq|x|$恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335211,7 +335217,7 @@ "content": "若函数$f(x)=1+\\dfrac{1}{x}$($x>0$)的反函数为$f^{-1}(x)$, 则不等式$f^{-1}(x)>2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335230,7 +335236,7 @@ "content": "若函数$f(x)=1+\\dfrac{1}{x}$($x>0$)的反函数为$f^{-1}(x)$, 则不等式$f^{-1}(x)>x$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335249,7 +335255,7 @@ "content": "若函数为$f(x)=1+\\dfrac{1}{x}+\\dfrac{1}{x^3}$($x>0$), 则不等式$f^{-1}(x)>2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335268,7 +335274,7 @@ "content": "若函数$y=f(x+2)$为奇函数, 函数$y=g(x)$的图像与$y=f(x)$的图像关于直线$y=x$成轴对称, 若$x_1+x_2=0$, 则$g(x_1)+g(x_2)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335287,7 +335293,7 @@ "content": "若定义在$\\mathbf{R}$上的函数$f(x)$满足: 对任意$x_1, x_2 \\in \\mathbf{R}$有$f(x_1+x_2)=f(x_1)+f(x_2)+1$, 则下列说法一定正确的是\\bracket{20}.\n\\fourch{$f(x)$为奇函数}{$f(x)$为偶函数}{$f(x)+1$为奇函数}{$f(x)+1$为偶函数}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -335306,7 +335312,7 @@ "content": "对于函数$y=f(x)$($x \\in D$), 若同时满足以下条件: \\textcircled{1} $f(x)$在$D$上是严格增函数; \\textcircled{2} 存在区间$[a, b] \\subseteq D$, 使$f(x)$在$[a, b]$上的值域是$[a, b]$, 那么, 我们把函数$y=f(x)$叫做闭函数. 若$y=k+\\sqrt{x+2}$是闭函数, 求实数$k$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -335318,14 +335324,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013770": { "id": "013770", "content": "函数$y=\\sqrt{3-2 x-x^2}$的定义域是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335344,7 +335350,7 @@ "content": "如果函数$y=f(x-1)$的图像与函数$y=\\ln \\sqrt{x}+1$的图像关于直线$y=x$对称, 那么函数$f(x)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335363,7 +335369,7 @@ "content": "设定义在$\\mathbf{R}$上的函数$f(x)$满足$f(x) \\cdot f(x+2)=13$, 若$f(1)=2$, 则$f(99)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335382,7 +335388,7 @@ "content": "设函数$f(x)$的图像关于点$(1,2)$对称, 且存在反函数$f^{-1}(x)$, $f(4)=0$, 求$f^{-1}(4)$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -335394,14 +335400,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013774": { "id": "013774", "content": "定义一种运算``$\\ast$'', 对于正整数满足以下运算性质: \\textcircled{1} $2\\ast 2018=1$; \\textcircled{2} $(2 n+2) \\ast 2018=3 \\cdot[(2 n) \\ast 2018]$, 则$2020 \\ast 2018$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335420,7 +335426,7 @@ "content": "已知函数$f(x)$的周期为$1$, 且当$00\\end{cases}$为奇函数, 则$f^{-1}(x)=2$付解为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335458,7 +335464,7 @@ "content": "若存在$a, b(b>a \\geq 1)$, 使得函数$f(x)=\\sqrt{x-1}+m$在区间$[a, b]$上的值域为$[\\dfrac{a}{2}, \\dfrac{b}{2}]$, 则实数$m$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335477,7 +335483,7 @@ "content": "已知$f(x)=\\begin{cases}-2(x-0.5)^2+1, & 0 \\leq x \\leq 0.5, \\\\ -2 x+2, & 0.50$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335572,7 +335578,7 @@ "content": "已知$a, b$为正实数, 函数$f(x)=a x^3+b x+2^x$在$[0,1]$上的最大值为 $4$, 则$f(x)$在$[-1,0]$上的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335591,7 +335597,7 @@ "content": "设$f(x)$是定义在$\\mathbf{R}$上的以$2$为最小正周期的偶函数, 在区间$[0,1]$上严格减, 且满足$f(\\pi)=1, f(2 \\pi)=2$, 则不等式组$\\begin{cases}1 \\leq x \\leq 2, \\\\ 1 \\leq f(x) \\leq 2\\end{cases}$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335610,7 +335616,7 @@ "content": "已知函数$f(x)=\\begin{cases}(2 a-1) x+7 a-2, & x<1, \\\\ a^x, & x \\geq 1\\end{cases}$在$(-\\infty,+\\infty)$上是严格减函数, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335629,7 +335635,7 @@ "content": "设$f(x)$是定义在$\\mathbf{R}$上且周期为$2$的函数, 在区间$[-1,1)$上, $f(x)=\\begin{cases}x+a, & -1 \\leq x<0, \\\\ |\\dfrac{2}{5}-x|, & 0 \\leq x<1,\\end{cases}$ 其中$a \\in \\mathbf{R}$. 若$f(-\\dfrac{5}{2})=f(\\dfrac{9}{2})$, 则$f(5 a)$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335648,7 +335654,7 @@ "content": "如果函数$f(x)=a^x(a^x-3 a^2-1)$($a>0$且$a \\neq 1$)在区间$[0,+\\infty)$上是严格增函数, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335667,7 +335673,7 @@ "content": "已知函数$f(x)=\\begin{cases}|x+a|+|x-1|, & x>0, \\\\ x^2-a x+2, & x \\leq 0\\end{cases}$的最小值为$a$, 则实数$a$的取值集合为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335686,7 +335692,7 @@ "content": "已知$a>\\dfrac{1}{3}$, 函数$f(x)=\\lg (|x-a|+1)$在区间$[0,3 a-1]$上的最小值为$0$且最大值为$\\lg (a+1)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335705,7 +335711,7 @@ "content": "设$f(x)$是$\\mathbf{R}$上的奇函数, 且$f(x+3)=-f(x)$, 当$0 \\leq x \\leq \\dfrac{3}{2}$时, $f(x)=x$, 则$f(2020)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335724,7 +335730,7 @@ "content": "设$f(x)$是定义在$\\mathbf{R}$上且周期为$4$的函数, 在区间$(-2,2]$上, 其函数解析式是$f(x)=\\begin{cases}x+a, & -20, \\\\ 4^x, & x \\leq 0.\\end{cases}$ 若函数$g(x)=f(x)-k$存在两个零点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335952,7 +335958,7 @@ "content": "若不等式$x^2+a x+1 \\geq 0$对于一切$x \\in(0, \\dfrac{1}{2})$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335971,7 +335977,7 @@ "content": "若不等式$x^2+a x+1>0$对于一切$x \\in(0, \\dfrac{1}{2})$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -335990,7 +335996,7 @@ "content": "若函数$y=x^2+a x+1$在$x \\in(0, \\dfrac{1}{2})$的图像恒在$x$轴的上方, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336009,7 +336015,7 @@ "content": "若方程$x^2+a x+1=0$在$x \\in(0, \\dfrac{1}{2})$时无实数根, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336028,7 +336034,7 @@ "content": "若$f(x)=a x^2+x+1$在$x \\in(0,1)$时恒正, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336047,7 +336053,7 @@ "content": "已知$f(x)$是定义在$\\mathbf{R}$上且周期为$3$的函数, 当$x \\in[0,3)$时, $f(x)=|x^2-2 x+\\dfrac{1}{2}|$. 若函数$y=f(x)-a$在区间$[-3,4]$上有$10$个零点(互不相同), 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336066,7 +336072,7 @@ "content": "函数$f(x)=2 \\sin \\pi x$与函数$g(x)=\\sqrt[3]{x-1}$的图像所有交点的横坐标之和为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336085,7 +336091,7 @@ "content": "函数$y=x^2-3 x-\\dfrac{2}{x}$零点的个数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336104,7 +336110,7 @@ "content": "已知函数$f(x)=\\log _a x+x-b$($a>0$, 且$a \\neq 1$). 当$2b.\\end{cases}$ 设$f(x)=(2 x-1) \\ast (x-1)$, 且关于$x$的方程为$f(x)=m$($m \\in \\mathbf{R}$)恰有三个互不相等的实数根$x_1, x_2, x_3$, 则$x_1 x_2 x_3$的取值范围是\\blank{50}.\n%08", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336275,7 +336281,7 @@ "content": "解方程: $\\log _2(x-3)=2-\\log _2 x$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336287,14 +336293,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013821": { "id": "013821", "content": "若关于$x$的方程$\\log _2(a x+3)=2-\\log _2 x$有实数解, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336306,14 +336312,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013822": { "id": "013822", "content": "已知函数$f(x)=a x^2-\\dfrac{a}{3}+1$, $g(x)=x+\\dfrac{a}{x}$($a \\in \\mathrm{R}$).\\\\\n(1) 若$f(x)>0$在$x \\in[1,2)$上恒成立, 求$a$的取值范围;\\\\\n(2) 当$a>0$时, 对任意的$x_1 \\in[1,2]$, 存在$x_2 \\in[1,2]$, 使得$f(x_1) \\geq g(x_2)$恒成立, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336325,14 +336331,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013823": { "id": "013823", "content": "已知函数$f(x)=a x^2-\\dfrac{a}{3}+1$, $g(x)=x+\\dfrac{a}{x}$($a \\in \\mathrm{R}$). 若$a>0$, 且对任意的$x_1 \\in[1,2]$, 存在$x_2 \\in[1,2]$, 使得$f(x_1) \\leq g(x_2)$恒成立, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336344,14 +336350,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013824": { "id": "013824", "content": "已知函数$f(x)=a x^2-\\dfrac{a}{3}+1$, $g(x)=x+\\dfrac{a}{x}$($a \\in \\mathrm{R}$). 是否存在$a>0$, 使对任意的$x_1, x_2 \\in[1,2]$, 都有$f(x_1) \\geq g(x_2)$恒成立? 若存在, 求出$a$的取值范围; 若不存在, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336363,14 +336369,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013825": { "id": "013825", "content": "已知函数$f(x)=a x^2-\\dfrac{a}{3}+1$, $g(x)=x+\\dfrac{a}{x}$($a \\in \\mathrm{R}$). 若$a>0$, 且对任意的$x_1 \\in[1,2]$, 都存在$x_2 \\in[1,2]$, 使得$f(x_2)=g(x_1)$恒成立, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336382,14 +336388,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013826": { "id": "013826", "content": "对于定义在$D$上的函数$f(x)$和实数$q$, 若$f(x) \\leq q$恒成立, 就称$q$是函数$f(x)$是$D$上的一个上界. 已知函数$f(x)=1+a \\cdot(\\dfrac{1}{2})^x+(\\dfrac{1}{4})^x$, 若$3$是函数$y=|f(x)|$在$[0,+\\infty)$上的一个上界, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336401,14 +336407,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013827": { "id": "013827", "content": "已知集合$A=\\{x | \\dfrac{2 x+1}{x+2}<1,\\ x \\in \\mathbf{R}\\}$, 函数$f(x)=|m x+1|$($m \\in \\mathbf{R}$), 若不等式$f(x)<3$的解集为$A$, 则$m$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336427,7 +336433,7 @@ "content": "对任意两个实数$x_1, x_2$, 定义义$\\max (x_1, x_2)=\\begin{cases}\nx_1, & x_1\\ge x_2, \\\\ x_2, & x_10$对$x \\in(1,2)$恒成立, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336522,7 +336528,7 @@ "content": "设函数$f(x)=|2 x+1|$, 若$|f(x)-2 f(\\dfrac{x}{2})| \\leq k$恒成立, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336541,7 +336547,7 @@ "content": "设不等式$x^2-2 a x+a+2 \\leq 0$的解集为$M$, 若$M \\subseteq[1,4]$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336560,7 +336566,7 @@ "content": "已知$y=f(x)$是偶函数, 当$x>0$时, $f(x)=x+\\dfrac{4}{x}$, 且当$x \\in[-3,-1]$时, $n \\leq f(x) \\leq m$恒成立, 则$m-n$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336579,7 +336585,7 @@ "content": "若集合$M=\\{x | x^2+x-2^x \\lambda \\geq 0,\\ x \\in \\mathbf{N}, \\ x \\ge 1\\}$中的元素个数为$4$, 则实数$\\lambda$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336598,7 +336604,7 @@ "content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(\\dfrac{1}{3},+\\infty)$, 使得$\\{y | y=f(x), x \\in[a, b]\\}=[m a, m b]$, 则实数$m$的取值范围是\\blank{50}.\n%09", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336617,7 +336623,7 @@ "content": "给出四个函数: \\textcircled{1} $f(x)=x+\\dfrac{1}{x}$; \\textcircled{2} $f(x)=3^x+3^{-x}$; \\textcircled{3} $f(x)=x^3$; \\textcircled{4} $f(x)=\\sin x$; 其中满足条件: 对任意实数$x$及任意正数$m$, 都有$f(-x)+f(x)=0$及$f(x+m)>f(x)$的函数为\\blank{50}.(写出所有满足条件的函数的序号)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336636,7 +336642,7 @@ "content": "设定义域为$\\mathbf{R}$的函数$f(x)=\\begin{cases}|\\lg x |, & x>0, \\\\ -x^2-2 x, & x \\leq 0,\\end{cases}$ 若关于$x$的函数$y=2 f^2(x)+2 b f(x)+1$有$8$个不同的零点, 则实数$b$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336655,7 +336661,7 @@ "content": "已知函数$f(x)=\\begin{cases}|x|+2, & x<1, \\\\ x+\\dfrac{2}{x}, & x \\geq 1,\\end{cases}$ 设$a \\in \\mathbf{R}$, 若关于$x$的不等式$f(x) \\geq|\\dfrac{x}{2}+a|$在$\\mathbf{R}$上恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336674,7 +336680,7 @@ "content": "已知$A(1,0)$, 点$B$在曲线$G: y=\\ln (x+1)$上, 若线段$AB$与曲线$M: y=\\dfrac{1}{x}$相交且交点恰为线段$AB$的中点, 则称点$B$为曲线$G$关于曲线$M$的一个关联点. 记曲线$G$关于曲线$M$的关联点的个数为$a$, 则\\bracket{20}.\n\\fourch{$a=0$}{$a=1$}{$a=2$}{$a>2$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -336693,7 +336699,7 @@ "content": "已知函数$f(x)=\\log _4(4^x+1)+k x$是偶函数.\\\\\n(1) 求$k$的值;\\\\\n(2) 若方程$f(x)-m=0$有解, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336705,14 +336711,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013843": { "id": "013843", "content": "已知函数$f(x)=|2 x+a|+|2 x-1|$, $g(x)=\\dfrac{6 x-5}{2 x-1}$.\\\\\n(1) 当$a=3$时, 解不等式$f(x) \\leq 6$;\\\\\n(2) 若对任意$x_1 \\in[1, \\dfrac{5}{2}]$都存在$x_2 \\in \\mathbf{R}$, 使得$g(x_1)=f(x_2)$成立, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336724,14 +336730,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013844": { "id": "013844", "content": "已知函数$f(x)=x^2+(x-1)|x-a|$, 是否存在实数$a$, 使不等式$f(x) \\geq 2 x-3$对一切实数$x$恒成立? 若存在, 求出$a$的取值范围, 若不存在, 请说明理由.\n%10", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336743,14 +336749,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013845": { "id": "013845", "content": "已知扇形$AOB$的面积是$2 \\text{cm}^2$, 该扇形的圆心角是$1$弧度, 则该扇形的半径长为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336769,7 +336775,7 @@ "content": "已知$\\tan (\\alpha+\\beta)=\\dfrac{2}{5}$, $\\tan (\\beta-\\dfrac{\\pi}{4})=\\dfrac{1}{4}$, 则$\\tan (\\alpha+\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336788,7 +336794,7 @@ "content": "若$\\sin ^2A=\\sin ^2B+\\sin ^2C$, 则$\\triangle ABC$是\\bracket{20}\n\\twoch{一定为直角三角形}{一定为钝角三角形}{一定为锐角三角形}{不能确定为何种三角形}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -336807,7 +336813,7 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分别为$a, b, c$.\\\\\n(1) 若$\\sin C+\\sin (B-A)=\\sin 2A$, 试判断$\\triangle ABC$的形状;\n(2) 若$c=2$, $C=\\dfrac{\\pi}{3}$, 且$S_{\\triangle ABC}=\\sqrt{3}$, 求$a, b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336819,14 +336825,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013849": { "id": "013849", "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边分别为$a, b, c$, 若$b \\cos C+c \\cos B=a \\sin A$, 则$\\triangle ABC$的形状为\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{不确定}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -336845,7 +336851,7 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$的对边分别为$a, b, c$, 若$a^2+b^2+4 \\sqrt{2}=c^2$, $a b=4$, 则$\\dfrac{\\sin C}{\\tan ^2A \\sin 2B}$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336864,7 +336870,7 @@ "content": "如图, 某公司要在$A, B$两地连线上的定点$C$处建造广告牌$CD$, 其中$D$为顶端, $AC$长$35$米, $CB$长$80$米, 设$A, B$在同一水平面上, 从$A$和$B$看$D$的仰角分别为$\\alpha$和$\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2.5,0) node [right] {$B$} coordinate (B);\n\\draw (1,1.2) node [above] {$D$} coordinate (D);\n\\draw (1,0) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(D)--cycle (C)--(D);\n\\draw (A) pic [draw, \"$\\alpha$\", angle eccentricity = 1.5] {angle = C--A--D};\n\\draw (B) pic [draw, \"$\\beta$\", angle eccentricity = 1.5] {angle = D--B--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 设计中$CD$是铅垂方向, 若要求$\\alpha \\geq 2 \\beta$, 问$CD$的长至多为多少(结果精确到$0.01$米)?\\\\\n(2) 施工完成后, $CD$与铅垂方向有偏差, 现在实测得$\\alpha=38.12^{\\circ}, \\beta=18.45^{\\circ}$, 求$CD$的长(结果精确到$0.01$米)?", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336876,14 +336882,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013852": { "id": "013852", "content": "已知$f(x)=\\sqrt{\\dfrac{1-x}{1+x}}$, 若$\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, 则$f(\\cos \\alpha)+f(-\\cos \\alpha)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336902,7 +336908,7 @@ "content": "若$\\dfrac{1+\\sin \\theta+\\cos \\theta}{1+\\sin \\theta-\\cos \\theta}=\\dfrac{1}{2}$, 则$\\cos \\theta$的值等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336921,7 +336927,7 @@ "content": "在$\\triangle ABC$中, 已知$(a^2+b^2) \\sin (A-B)=(a^2-b^2) \\sin (A+B)$, 则$\\triangle ABC$的形状为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336940,7 +336946,7 @@ "content": "已知锐角$\\alpha$终边上一点$P$的坐标为$(2 \\sin 3,-2 \\cos 3)$, 若一扇形的中心角为$\\alpha$且半径为$2$, 则该扇形的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -336959,7 +336965,7 @@ "content": "已知$\\cos (\\alpha+\\beta)=\\dfrac{4}{5}$, $\\cos (\\alpha-\\beta)=-\\dfrac{4}{5}$, 且$\\dfrac{3}{2} \\pi<\\alpha+\\beta<2 \\pi$, $\\dfrac{\\pi}{2}<\\alpha-\\beta<\\pi$, 分别求$\\cos 2 \\alpha$和$\\cos 2 \\beta$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336971,14 +336977,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013857": { "id": "013857", "content": "如图, 某住宅小区的平面图呈圆心角为$120^{\\circ}$的扇形$AOB$. 小区的两个出入口设置在点$A$及点$C$处, 且小区里有一条平行于$BO$的小路$CD$. 已知某人从$C$沿$CD$走到$D$用了$10$分钟, 从$D$沿$DA$走到$A$用了$6$分钟. 若此人步行的速度为每分钟$50$米, 求该扇形的弧$AB$的长 (精确到$1$米).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (150:2) node [left] {$A$} coordinate (A);\n\\draw (30:2) node [right] {$B$} coordinate (B);\n\\draw (150:0.7) node [below left] {$D$} coordinate (D);\n\\path [draw, name path = AB] (B) arc (30:150:2);\n\\path [name path = CD] (D) --++ (30:2.5);\n\\path [name intersections = {of = AB and CD, by = C}];\n\\draw (C) node [above right] {$C$}--(D);\n\\draw (A)--(O)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -336990,14 +336996,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013858": { "id": "013858", "content": "关于$x$的方程$x^2-x \\cos A \\cos B-\\cos ^2 \\dfrac{\\mathrm{C}}{2}=0$有一个根为$1$, 则$\\triangle ABC$一定是\\bracket{20}.\n\\fourch{等腰三角形}{直角三角形}{锐角三角形}{钝角三角形}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337016,7 +337022,7 @@ "content": "如图, 位于$A$处的信息中心获悉: 在其正东方向相距$40$海里的$B$处有一艘渔船遭遇不测, 在原地等待营救. 信息中心立即把消息告知在其南偏西$30^{\\circ}$、相距$20$海里的$C$处的搜救船只, 现搜救船只朝北偏东$\\theta$的方向即沿直线$CB$前往$B$处救援, 求$\\cos \\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\draw (-120:2) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\draw [dashed] (0,-2) coordinate (S) -- (0,2);\n\\draw [dashed] (0,0) -- (-1,0);\n\\draw (A) pic [draw, \"$30^\\circ$\", angle eccentricity = 1.8] {angle = C--A--S};\n\\draw (A)--(C) node [midway, above left] {$20$};\n\\draw (A)--(B) node [midway, above] {$40$};\n\\draw [->] (0.7,1) -- (1.5,1) node [right] {东};\n\\draw [->] (1,0.7) -- (1,1.5) node [above] {北};\n\\end{tikzpicture}\n\\end{center}\n%11", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337028,14 +337034,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013860": { "id": "013860", "content": "设$a>0$, 对于函数$f(x)=\\dfrac{\\sin x+a}{\\sin x}$($00$, 函数$y=\\sin (\\omega x+\\dfrac{\\pi}{3})+2$的图像向右平移$\\dfrac{4 \\pi}{3}$个单位后与原图像重合, 则$\\omega$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337263,7 +337269,7 @@ "content": "已知函数$f(x)=\\sin x \\cos x-\\sin ^2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 若函数$f(x)$在区间$[a, \\dfrac{\\pi}{16}]$上递增, 求实数$a$的取值范围;\\\\\n(2) 若函数$f(x)$的图像关于点$Q(x_1, y_1)$对称, 且$x_1 \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$, 求点$Q$的坐标.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337275,14 +337281,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013873": { "id": "013873", "content": "已知$M=\\dfrac{a^2-a \\sin \\theta+1}{a^2-a \\cos \\theta+1}$($a, \\theta \\in \\mathbf{R}$, $a \\neq 0$), 则$M$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337301,7 +337307,7 @@ "content": "已知定义在区间$[-\\dfrac{\\pi}{2}, \\pi]$上的函数$y=f(x)$的图像关于直线$x=\\dfrac{\\pi}{4}$对称, 当$x \\geq \\dfrac{\\pi}{4}$时, 函数解析式为$f(x)=\\sin x$.\\\\\n(1) 求$y=f(x)$的函数表达式;\\\\\n(2) 如果关于$x$的方程$f(x)=a$有解, 那么将方程在$a$取某一确定值时所求得的所有解的和记为$M_a$, 求$M_a$的所有可能取值及相应的$a$的取值范围.\n%12", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337313,14 +337319,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013875": { "id": "013875", "content": "$\\triangle ABC$中, $B=60^{\\circ}, AC=\\sqrt{3}$, 则$AB+2BC$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337339,7 +337345,7 @@ "content": "函数$y=x+\\sqrt{1-x^2}$的最大值与最小值分别为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337358,7 +337364,7 @@ "content": "已知$\\sin \\theta+\\cos \\theta=\\sin \\theta \\cos \\theta$, 则角$\\theta$所在的区间可能是\\bracket{20}.\n\\fourch{$(\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2})$}{$(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{4})$}{$(-\\dfrac{\\pi}{2},-\\dfrac{\\pi}{4})$}{$(\\pi, \\dfrac{5 \\pi}{4})$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337377,7 +337383,7 @@ "content": "已知向量$\\overrightarrow {m}=(\\cos \\dfrac{x}{2},-1)$, $\\overrightarrow {n}=(\\sqrt{3} \\sin \\dfrac{x}{2}, \\cos ^2 \\dfrac{x}{2})$, 设函数$f(x)=\\overrightarrow {m} \\cdot \\overrightarrow {n}+1$.\\\\\n(1) 若$x \\in[0, \\dfrac{\\pi}{2}]$, $f(x)=\\dfrac{11}{10}$, 求$x$的值;\\\\\n(2) 在$\\triangle ABC$中, 角$A, B, C$的对边分别是$a, b, c$且满足$2 b \\cos A \\leq 2 c-\\sqrt{3} a$, 求$f(B)$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337389,14 +337395,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013879": { "id": "013879", "content": "某动物园要为刚入园的小动物建造一间两面靠墙的三角形露天活动室, 地面形状如图所示, 已知已有两面墙的夹角为$\\dfrac{\\pi}{3}$(即$\\angle ACB=\\dfrac{\\pi}{3}$), 墙$AB$的长度为$6$米 (已有两面墙的可利用长度足够大), 记$\\angle ABC=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0) node [above right] {$B$} coordinate (B);\n\\draw (60:3) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (60:0.5) coordinate (A1);\n\\draw (B) ++ (0.5,0) coordinate (B1);\n\\draw (A1) ++ (150:0.1) coordinate (A2);\n\\draw (B1) ++ (0,-0.1) coordinate (B2);\n\\draw (C) ++ (-150:0.2) coordinate (C2);\n\\fill [gray] (B2) -- (C2) -- (A2) -- (A1) -- (C) -- (B1) -- cycle;\n\\draw (A1) -- (C) -- (B1);\n\\draw (A)--(B);\n\\draw pic (B) [draw, \"$\\theta$\", angle eccentricity = 1.5] {angle = A--B--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta=\\dfrac{\\pi}{4}$, 求$\\triangle ABC$的周长(结果精确到$0.01$米);\\\\\n(2) 为了使小动物能健康成长, 要求所建造的三角形露天活动室面积即$\\triangle ABC$的面积尽可能大. 问当$\\theta$为何值时, 该活动室面积最大? 并求出最大面积.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337408,14 +337414,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013880": { "id": "013880", "content": "设$\\alpha_1$、$\\alpha_2 \\in \\mathbf{R}$, 且$\\dfrac{1}{2+\\sin \\alpha_1}+\\dfrac{1}{2+\\sin (2 \\alpha_2)}=2$, 则$|10 \\pi-\\alpha_1-\\alpha_2|$的最小值等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337434,7 +337440,7 @@ "content": "若关于$x$的方程$\\sin x \\cos x=2 a-1$有解, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337453,7 +337459,7 @@ "content": "在$\\triangle ABC$中, 角$A, B, C$所对的边长分别为$a, b, c$, 若$\\angle C=120^{\\circ}, c=\\sqrt{2} a$, 则\\bracket{20}.\n\\twoch{$a>b$}{$a=latex,scale = 1.5]\n\\draw (0,0) node [below] {$Q$} coordinate (Q);\n\\draw (1,0) node [below] {$R$} coordinate (R);\n\\draw (0,1) node [below right] {$P$} coordinate (P);\n\\draw (1,1) node [below left] {$S$} coordinate (S);\n\\draw (Q) ++ ({-4/3},0) node [left] {$B$} coordinate (B);\n\\draw (R) ++ ({3/4},0) node [right] {$C$} coordinate (C);\n\\draw (P) ++ ({atan(3/4)}:{4/5}) node [above] {$A$} coordinate (A);\n\\draw (B)--(C)--(A)--cycle;\n\\draw (Q)--(P)--(S)--(R);\n\\draw (C) arc (0:180:{37/24});\n\\end{tikzpicture}\n\\end{center}\n(1) 用$a, \\theta$表示$S_1$和$S_2$;\\\\ \n(2) 当$a$固定, $\\theta$变化时, 求$\\dfrac{S_1}{S_2}$取最小值时的角$\\theta$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337522,14 +337528,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013886": { "id": "013886", "content": "设$\\varphi \\in[0,2 \\pi)$, 若关于$x$的方程$\\sin (2 x+\\varphi)=a$在区间$[0, \\pi]$上有三个解, 且它们的和为$\\dfrac{4 \\pi}{3}$, 则$\\varphi=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337548,7 +337554,7 @@ "content": "已知点$A(2,0)$, 点$P$是以原点$O$为圆心、$1$为半径的圆上的任意一点, 将点$P$绕点$O$逆时针旋转$90^{\\circ}$得点$Q$, 线段$AP$的中点为$M$, 求$|MQ|$的最大值与最小值.\n%13", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337560,14 +337566,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013888": { "id": "013888", "content": "$\\triangle ABC$所在平面内一点$P$, 满足$\\overrightarrow{AP}+\\overrightarrow{BP}+\\overrightarrow{CP}=\\overrightarrow{0}$, 若$D$为$\\triangle ABC$边$BC$的中点, 且$\\dfrac{|\\overrightarrow{AP}|}{|\\overrightarrow{PD}|}=\\lambda$, 则$\\lambda=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337586,7 +337592,7 @@ "content": "在直角三角形$ABC$中, $\\angle A=\\dfrac{\\pi}{2}$, $AB=1$, $AC=2$, $M$是$\\triangle ABC$内一点, 且$AM=\\dfrac{1}{2}$, 若$\\overrightarrow{AM}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 则$\\lambda+2 \\mu$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337605,7 +337611,7 @@ "content": "如图所示, 在同一个平面内, 向量$\\overrightarrow{OA}, \\overrightarrow{OB}, \\overrightarrow{OC}$的模分别为$1,1, \\sqrt{2}$, $\\overrightarrow{OA}$与$\\overrightarrow{OC}$的夹角为$\\alpha$, 且$\\tan \\alpha=7$, $\\overrightarrow{OB}$与$\\overrightarrow{OC}$的夹角为$45^{\\circ}$. 若$\\overrightarrow{OC}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$($m, n \\in \\mathbf{R}$), 则$m+n=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [below] {$A$} coordinate (A);\n\\draw ($(O)!{sqrt(2)}!{atan(7)}:(A)$) node [above] {$C$} coordinate (C);\n\\draw ($(O)!{sqrt(2)/2}!45:(C)$) node [left] {$B$} coordinate (B);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw [->] (O)--(C);\n\\draw (O) pic [draw, \"$\\alpha$\", angle eccentricity = 1.5] {angle = A--O--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337624,7 +337630,7 @@ "content": "在$\\triangle ABC$中, $D$、$E$分别是$AB$、$AC$的中点, $M$是直线$DE$上的动点. 若$\\triangle ABC$(形状可变化)的面积为$1$, 则$\\overrightarrow{MB} \\cdot \\overrightarrow{MC}+\\overrightarrow{BC}^2$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337643,7 +337649,7 @@ "content": "在正方形$ABCD$中, $AB=2$, 点$E$为$BC$的中点, 点$F$在边$CD$上. 若$\\overrightarrow{AE} \\cdot \\overrightarrow{BF}=0$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337662,7 +337668,7 @@ "content": "已知$\\triangle ABC$的三个顶点$A, B, C$及所在平面内一点$P$满足$\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}=\\overrightarrow{AB}$, 则点$P$一定在\\bracket{20}.\n\\fourch{$\\triangle ABC$内部}{$\\triangle ABC$外部}{$AB$边上}{$AC$边上}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337681,7 +337687,7 @@ "content": "已知$\\overrightarrow{AD}=\\overrightarrow{BC}$, 且$\\dfrac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|}+\\dfrac{\\overrightarrow{AD}}{|\\overrightarrow{AD}|}=\\dfrac{\\overrightarrow{AC}}{|\\overrightarrow{AC}|}$, 则关于下列判断: \\textcircled{1} $AC \\perp BD$; \\textcircled{2} $\\angle BAD=$$120^{\\circ}$, 正确的说法是\\bracket{20}.\n\\fourch{\\textcircled{1}正确, \\textcircled{2}不正确}{\\textcircled{1}不正确 , \\textcircled{2}正确}{\\textcircled{1} \\textcircled{2}都不正确}{\\textcircled{1} \\textcircled{2}都正确}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337700,7 +337706,7 @@ "content": "已知圆心为$O$, 半径为$1$的圆上有三点$A, B, C$. 若$5 \\overrightarrow{OA}+12 \\overrightarrow{OB}+13 \\overrightarrow{OC}=\\overrightarrow{0}$, 则$\\angle ACB=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337719,7 +337725,7 @@ "content": "在$\\triangle ABC$中, 已知$\\overrightarrow{CD}=2 \\overrightarrow{DB}$, $P$为线段$AD$上的一点, 且满足$\\overrightarrow{CP}=m \\overrightarrow{CA}+\\dfrac{4}{9} \\overrightarrow{CB}$, 若$\\triangle ABC$的面积为$\\sqrt{3}$, $\\angle ACB=\\dfrac{\\pi}{3}$, 则$|\\overrightarrow{CP}|$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337738,7 +337744,7 @@ "content": "已知点$C$是平面$ABD$上一点, $\\angle BAD=\\dfrac{\\pi}{3}$, $CB=1$, $CD=3$, 若$\\overrightarrow{AP}=\\overrightarrow{AB}+\\overrightarrow{AD}$则$|\\overrightarrow{AP}|$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337757,7 +337763,7 @@ "content": "正方形$ABCD$的边长为$2$, 对角线$AC$、$BD$相交于点$O$, 动点$P$满足$|\\overrightarrow{OP}|=\\dfrac{\\sqrt{2}}{2}$, 若$\\overrightarrow{AP}=m \\overrightarrow{AB}+n \\overrightarrow{AD}$, 其中$m$、$n \\in \\mathbf{R}$, 则$\\dfrac{2 m+1}{2 n+2}$的最大值是\\blank{50}.\n%14", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337776,7 +337782,7 @@ "content": "在$\\triangle ABC$中, 已知$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=\\overrightarrow{BA} \\cdot \\overrightarrow{BC}$.\\\\\n(1) 求证: $|\\overrightarrow{AC}|=|\\overrightarrow{BC}|$;\\\\\n(2) 若$|\\overrightarrow{AC}+\\overrightarrow{BC}|=|\\overrightarrow{AC}-\\overrightarrow{BC}|=\\sqrt{6}$, 求$|\\overrightarrow{BA}-t \\overrightarrow{BC}|$的最小值及相应的$t$值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -337788,14 +337794,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013900": { "id": "013900", "content": "已知$\\triangle ABC$是边长为$2$的等边三角形, $P$为平面$ABC$内一点, 则$\\overrightarrow{PA} \\cdot(\\overrightarrow{PB}+\\overrightarrow{PC})$的最小值是\\bracket{20}.\n\\fourch{$-2$}{$-\\dfrac{3}{2}$}{$-\\dfrac{4}{3}$}{$-1$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337814,7 +337820,7 @@ "content": "已知$P$是边长为$1$的正六边形$ABCDEF$的边上的任意一点.\\\\\n(1) $\\overrightarrow{AP} \\cdot \\overrightarrow{AB}$的取值范围为\\blank{50};\\\\\n(2) 若$M$是$BC$边的中点, 则$\\overrightarrow{AM} \\cdot \\overrightarrow{AP}$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337833,7 +337839,7 @@ "content": "已知实数$x, y$满足: $x^2+(y-2)^2=1$, $w=\\dfrac{x+\\sqrt{3} y}{\\sqrt{x^2+y^2}}$的取值范围是\\bracket{20}.\n\\fourch{$(\\sqrt{3}, 2]$}{$[1,2]$}{$(0,2]$}{$(\\dfrac{\\sqrt{3}}{2}, 1]$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337852,7 +337858,7 @@ "content": "已知实数$x_1$、$x_2$、$y_1$、$y_2$满足: $x_1^2+y_1^2=1$, $x_2^2+y_2^2=1$, $x_1 x_2+y_1 y_2=\\dfrac{1}{2}$, 则$\\dfrac{|x_1+y_1-1|}{\\sqrt{2}}+\\dfrac{|x_2+y_2-1|}{\\sqrt{2}}$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337871,7 +337877,7 @@ "content": "在$\\triangle ABC$中, $a=5$, $b=4$, $\\angle C=45^{\\circ}$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337890,7 +337896,7 @@ "content": "已知$\\overrightarrow {a}, \\overrightarrow {b}$均为单位向量, 它们的夹角为$60^{\\circ}$, 那么$|\\overrightarrow {a}+3 \\overrightarrow {b}|$等于\\bracket{20}.\n\\fourch{$\\sqrt{7}$}{$\\sqrt{10}$}{$\\sqrt{13}$}{$4$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337909,7 +337915,7 @@ "content": "$\\triangle ABC$的面积为$S$, $S \\in(\\dfrac{1}{2}, 2)$, $\\overrightarrow{AB} \\cdot \\overrightarrow{CA}=1$且$\\overrightarrow{AB}$与$\\overrightarrow{CA}$的夹角为$\\theta$, 则角$\\theta$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337928,7 +337934,7 @@ "content": "点$P$为椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{15}=1$上的任意一点, $EF$为圆$(x-1)^2+y^2=4$的任一条直径, 则$\\overrightarrow{PE} \\cdot \\overrightarrow{PF}$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337947,7 +337953,7 @@ "content": "如图所示, 正八边形$A_1A_2A_3A_4A_5A_6A_7A_8$的边长为$2$, 若$P$为该正八边形边上的动点, 则$\\overrightarrow{A_1A_3} \\cdot \\overrightarrow{A_1P}$的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0) coordinate (P);\n\\foreach \\i/\\j in {1/below,2/below,3/right,4/right,5/above,6/above,7/left,8/left}\n{\\draw (P) --++ ({45*(\\i-2)}:2) node [\\j] {$A_\\i$} coordinate (A\\i) coordinate (P);};\n\\draw [->] (A1)--(A3);\n\\draw ($(A5)!0.3!(A6)$) node [above] {$P$} coordinate (P);\n\\draw [->] (A1) -- (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt{2}]$}{$[-2 \\sqrt{2}, 8+6 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 2 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 8+6 \\sqrt{2}]$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -337966,7 +337972,7 @@ "content": "在平面四边形$ABCD$中, $AB \\perp BC$, $AD \\perp CD$, $\\angle BAD=120^{\\circ}$, $AB=AD=1$. 若点$E$为边$CD$上的动点, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{BE}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -337985,7 +337991,7 @@ "content": "已知平面上三个不同的单位向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}=\\dfrac{1}{2}$, 若$\\overrightarrow {e}$为平面内的任意单位向量, 则$|\\overrightarrow {a} \\cdot \\overrightarrow {e}|+2|\\overrightarrow {b} \\cdot \\overrightarrow {e}|+3|\\overrightarrow {c} \\cdot \\overrightarrow {e}|$的最大值为\\blank{50}.\n%15", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338004,7 +338010,7 @@ "content": "设函数$f(x)=\\sin (x-\\dfrac{\\pi}{6})$, 若对于任意$\\alpha \\in[-\\dfrac{5 \\pi}{6},-\\dfrac{\\pi}{2}]$, 在区间$[0, m]$上总存在唯一确定的$\\beta$, 使得$f(\\alpha)+f(\\beta)=0$, 则$m$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338023,7 +338029,7 @@ "content": "在$\\triangle ABC$中, $a, b, c$分别是角$A, B, C$所对边的边长, 若$\\cos A+\\sin A-\\dfrac{2}{\\cos B+\\sin B}=0$, 则$\\dfrac{a+b}{c}$的值是\\bracket{20}.\n\\fourch{$1$}{$\\sqrt{2}$}{$\\sqrt{3}$}{$2$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338042,7 +338048,7 @@ "content": "在$\\triangle ABC$中, 若$|\\overrightarrow{AC}|=2 \\sqrt{3}$, 且$(\\cos C) \\overrightarrow{AB}+(\\cos A) \\overrightarrow{BC}=(\\sin B) \\overrightarrow{AC}$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 求$\\triangle ABC$的面积$S$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338054,14 +338060,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013914": { "id": "013914", "content": "某锐角三角形三内角的度数成等差数列, 且最长边与最短边之比为$m$, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338080,7 +338086,7 @@ "content": "已知平面向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, $|\\overrightarrow {c}|=3$, $0 \\leq \\lambda \\leq 1$, $\\overrightarrow {b} \\cdot \\overrightarrow {c}=0$, 则$|\\overrightarrow {a}-\\lambda \\overrightarrow {b}-(1-\\lambda) \\overrightarrow {c}|$所有取不到的值所构成的集合是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338099,7 +338105,7 @@ "content": "如图, 点$P$位于两条平行直线$l_1, l_2$的下方, 它到直线$l_1, l_2$距离分别为$1$, $3$, 动点$N, M$分别在$l_1, l_2$上, 满足$|\\overrightarrow{PM}+\\overrightarrow{PN}|=8$, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的最大值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) -- (3,0) node [right] {$l_1$} (-2,2) -- (3,2) node [right] {$l_2$};\n\\draw [->] ({2-sqrt(14)},-1) node [below] {$P$} coordinate (P) -- (1,0) node [below right] {$N$};\n\\draw [->] (P) -- (2,2) node [below right] {$M$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$15$}{$12$}{$10$}{$9$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338118,7 +338124,7 @@ "content": "已知$\\triangle ABC$中, $|\\overrightarrow{BC}|=3 \\sqrt{2}$, $|\\overrightarrow{CA}|=4$, $|\\overrightarrow{AB}|=2 \\sqrt{3}$, $PQ$是以$A$为圆心, $2$为半径的圆的直径.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale =0.6]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw ({3*sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\path [name path = c1] (B) ++ ({2*sqrt(3)},0) arc (0:70:{2*sqrt(3)});\n\\path [name path = c2] (C) ++ (-4,0) arc (180:125:4);\n\\path [name intersections = {of = c1 and c2, by = A}];\n\\draw (A)--(B) (A) node [above] {$A$} --(C) (B)--(C);\n\\draw (A) circle (2);\n\\draw (A) ++ (220:2) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (40:2) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\draw [->] (B)--(P);\n\\draw [->] (C)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值、最小值, 并指出取最大值、 最小值时向量$\\overrightarrow{PQ}$的方向.\n%16", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338130,14 +338136,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013918": { "id": "013918", "content": "若等比数列$\\{a_n\\}$的前$n$项和$S_n=a \\cdot 3^n-2$, 则$a_2=$\\bracket{20}.\n\\fourch{$4$}{$12$}{$24$}{$36$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338156,7 +338162,7 @@ "content": "设$S_n$为等比数列$\\{a_n\\}$的前$n$项和. 若$a_1=1$, 且$3S_1, 2S_2, S_3$成等差数列, 则$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338175,7 +338181,7 @@ "content": "已知数列$\\{a_n\\}$中, $a_1=2, a_{n+1}-1=a_n+2 n$($n \\in \\mathbf{N}$, $n\\ge 1$), 则数列$\\{a_n\\}$的通项公式为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338194,7 +338200,7 @@ "content": "数列$\\{a_n\\}$的前$n$项和为$S_n, 2S_n-n a_n=n$($n \\in \\mathbf{N}$, $n\\ge 1$), 若$S_{20}=-360$, 则$a_2=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338213,7 +338219,7 @@ "content": "若数列$\\{a_n\\}$同时满足下列两个条件, 就称数列$\\{a_n\\}$具有``性质$m$'': \\textcircled{1} 对任意正整数$n$, $\\dfrac{a_n+a_{n+2}}{2}1$), 在第二步证明从$n=k$到$n=k+1$成立时, 左边增加的项数是\\bracket{20}.\n\\fourch{$2^k$个}{$2^k-1$个}{$2^{k-1}$个}{$2^k+1$个}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338403,7 +338409,7 @@ "content": "已知$a_n=\\begin{cases}2 n-1, & n<2020, \\\\ (-\\dfrac{1}{2})^{n-1}, & n \\geq 2020,\\end{cases}$ $S_n$是数列$\\{a_n\\}$的前$n$项和, 那么\\bracket{20}.\n\\twoch{$\\displaystyle\\lim_{n\\to\\infty} a_n$和$\\displaystyle\\lim_{n\\to\\infty} S_n$都存在}{$\\displaystyle\\lim_{n\\to\\infty} a_n$和$\\displaystyle\\lim_{n\\to\\infty} S_n$都不存在}{$\\displaystyle\\lim_{n\\to\\infty} a_n$存在, $\\displaystyle\\lim_{n\\to\\infty} S_n$不存在}{$\\displaystyle\\lim_{n\\to\\infty} a_n$不存在, $\\displaystyle\\lim_{n\\to\\infty} S_n$存在}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338422,7 +338428,7 @@ "content": "各项为正数的无穷等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{S_n}{S_{n+1}}=1$, 则其公比$q$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338441,7 +338447,7 @@ "content": "已知点$A(0, \\dfrac{2}{n})$, $B(0,-\\dfrac{2}{n})$, $C(4+\\dfrac{2}{n}, 0)$, 其中$n$为正整数, 设$S_n$表示$\\triangle ABC$外接圆的面积, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338460,7 +338466,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_n0$)的图像上, 其中点$B_n$的坐标为$(n, 0)$($n \\geq 2$, $n \\in \\mathbf{N}$), 矩形$A_n B_n P_n Q_n$的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:6, samples =100] plot (\\x,{2*\\x/(\\x*\\x+1)});\n\\draw ({1/3},0) node [below] {$A_n$} coordinate (A_n) -- ({1/3},{6/10}) node [above] {$Q_n$} coordinate (Q_n) -- (3,{6/10}) node [above] {$P_n$} coordinate (P_n) -- (3,0) node [below] {$B_n$} coordinate (B_n);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338555,7 +338561,7 @@ "content": "已知数列$\\{a_n\\}$满足$\\dfrac{a_{n+1}+a_n-1}{a_{n+1}-a_n+1}=n$($n$为正整数) 且$a_2=6$, 则数列$\\{a_n\\}$的通项公式为$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338574,7 +338580,7 @@ "content": "已知等差数列$\\{a_n\\}$的公差为$2$, 前$n$项和为$S_n$, 且$S_1, S_2, S_4$成等比数列.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=(-1)^{n-1} \\dfrac{4 n}{a_n a_{n+1}}$, 求数列$\\{b_n\\}$的前$n$项和$T_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338586,14 +338592,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013942": { "id": "013942", "content": "已知数列$\\{a_n\\}$满足: $a_1=1, a_2=x$($x \\in \\mathbf{N}$, $x\\ge 1$), $a_{n+2}=|a_{n+1}-a_n|$, 若前$2010$项中恰好含有$666$项为$0$, 则$x$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338612,7 +338618,7 @@ "content": "已知$f(x)=\\dfrac{1}{4} x^2+a x+b$, 且关于$x$的不等式$f(x)<0$的解集为$(-2,0)$. 各项均为正数的数列$\\{a_n\\}$的前$n$项和为$S_n$, 点列$(a_n, S_n)$($n \\in \\mathbf{N}$, $n\\ge 1$)在函数$y=f(x)$的图像上.\\\\\n(1) 求函数$y=f(x)$的解析式;\\\\\n(2) 若$b_n=k^{\\frac{a_n}{2}}$($k>0$), 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{2 b_n-1}{b_n+2}$的值;\\\\\n(3) 令$c_n=\\begin{cases}a_n, n \\text {为奇数}, \\\\ c_{\\frac n 2}, n\\text{为偶数},\\end{cases}$ 求数列$\\{c_n\\}$的前$2012$项中满足$c_m=6$的所有项数之和.\n%18", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338624,14 +338630,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013944": { "id": "013944", "content": "已知数列$\\{a_n\\}$满足, $a_1=1$, 且$(n+1) a_{n+1}-n a_n-3=0$, 若对任意的$a \\in[-2,2]$, 不等式$a_n \\leq 2 t^2+a t-1$恒成立, 则实数$t$的范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338650,7 +338656,7 @@ "content": "对于无穷数列$\\{a_n\\}$与$\\{b_n\\}$, 记$A=\\{x | x=a_n, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, $B=\\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 若同时满足条件: \\textcircled{1} $\\{a_n\\},\\{b_n\\}$均为严格增数列; \\textcircled{2} $A \\cap B=\\varnothing$且$A \\cup B=\\mathbf{N}\\cap [1,+\\infty)$, 则称$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列.\\\\\n(1) 若$a_n=2 n-1, b_n=4 n-2$, 判断$\\{a_n\\}$与$\\{b_n\\}$是否为无穷互补数列, 并说明理由;\\\\\n(2) 若$a_n=2^n$且$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, 求数列$\\{b_n\\}$的前$16$项的和;\\\\\n(3) 若$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, $\\{a_n\\}$为等差数列且$a_{16}=36$, 求$\\{a_n\\}$与$\\{b_n\\}$的通项公式.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338662,14 +338668,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013946": { "id": "013946", "content": "如图, 曲线$y^2=x$($y \\geq 0$)上的点$P_i$与$x$轴的正半轴上的点$Q_i$及原点$O$构成一系列正三角形$OP_1Q_1, Q_1P_2Q_2, \\cdots, Q_{n-1} P_n Q_n, \\cdots$, 记正三角形$Q_{n-1} P_n Q_n$(其中$Q_0$为$O$) 的边长为$a_n$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:5,samples = 100] plot (\\x,{pow(\\x,0.5)});\n\\draw (0,0) -- ({1/3},{sqrt(3)/3}) node [above] {$P_1$} -- ({2/3},0) node [below] {$Q_1$} -- ({4/3},{2/sqrt(3)}) node [above] {$P_2$} -- (2,0) node [below] {$Q_2$} -- (3,{sqrt(3)}) node [above] {$P_3$} -- (4,0) node [below] {$Q_3$};\n\\draw (4.5,{sqrt(3)/3}) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$a_1$的值;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式$a_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338681,14 +338687,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013947": { "id": "013947", "content": "已知数列$\\{a_n\\}$满足: \\textcircled{1} $\\{a_n\\}$是由非负整数组成的无穷数列; \\textcircled{2} $a_1=0$, $a_2=3$, $a_{n+1} a_n=(a_{n-1}+2)(a_{n-2}+2)$, $n=3,4,5, \\cdots$.\\\\\n(1) 试确定$a_3$的值, 并证明: $a_n=a_{n-2}+2$($n=3,4,5, \\cdots$);\\\\\n(2) 求$\\{a_n\\}$的通项公式及前$n$项和$S_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338700,14 +338706,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013948": { "id": "013948", "content": "已知数列$\\{a_n\\}$, 那么``对任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 点$P_n(n, a_n)$都在直线$y=x+1$上''是``$\\{a_n\\}$为等差数列''的\\bracket{20}.\n\\twoch{必要非充分条件}{充分非必要条件}{充要条件}{既不充分圤不必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -338726,7 +338732,7 @@ "content": "在等差数列$\\{a_n\\}$中, 若$a_5=3$, $a_6=-2$, 则$a_4+a_5+\\cdots+a_{10}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338745,7 +338751,7 @@ "content": "已知等差数列$\\{a_n\\}$满足$3 a_4=7 a_1$, 且$a_1>0$, $S_n$是$\\{a_n\\}$的前$n$项和, 则$S_n$取得最大值时$n$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338764,7 +338770,7 @@ "content": "定义``等和数列'': 在一个数列中, 如果每一项与它的后一项的和都为同一个常数, 那么这个数列叫做等和数列, 这个常数叫做该数列的公和. 已知数列$\\{a_n\\}$是等和数列, 且$a_1=2$, 公和为$5$, 那么$a_8$的值为\\blank{50}, 这个数列的前$n$项和$S_n$的计算公式为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338783,7 +338789,7 @@ "content": "数列$\\{a_n\\}$中, $a_1=8$, $a_4=2$且满足$a_{n+2}=2 a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$S_n=|a_1|+|a_2|+\\cdots+|a_n|$, 求$S_n$;\\\\\n(3) 设$b_n=\\dfrac{1}{n(12-a_n)}$($n \\in \\mathbf{N}$, $n\\ge 1$), $T_n=b_1+b_2+\\cdots+b_n$, 是否存在最大的整数$m$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 均有$T_n>\\dfrac{m}{32}$成立? 若存在, 求出$m$的值; 若不存在, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338795,14 +338801,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013953": { "id": "013953", "content": "设数列$\\{a_n\\}$的各项都是正数, $k$是一个给定的正整数, 若对于任意的正整数$m$, $a_m$、$a_{m+k}$、$a_{m+2 k}$都成等比数列, 则称数列$\\{a_n\\}$为``$D_k$型''数列.\\\\\n(1) 若$\\{a_n\\}$是``$D_1$型''数列, 且$a_1=1$, $a_3=\\dfrac{1}{4}$, 求$\\displaystyle\\lim_{n\\to\\infty}(a_1+a_2+\\cdots+a_n)$的值;\\\\\n(2) 若$\\{a_n\\}$是``$D_2$型''数列, 且$a_1=a_2=a_3=1$, $a_8=8$, 求$\\{a_n\\}$的前$n$项利$S_n$;\\\\\n(3) 若$\\{a_n\\}$既是``$D_2$型''数列, 又是``$D_3$型''数列, 求证: 数列$\\{a_n\\}$是等比数列.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338814,14 +338820,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013954": { "id": "013954", "content": "设数列$\\{a_n\\}$的通项公式为$a_n=p n+q$($n \\in \\mathbf{N}$, $n\\ge 1$, $p>0$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n \\geq m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$, 求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_m\\}$的前$2 m$项和;\\\\\n(3) 是否存在$(p, q)$, 使得$b_m=3 m+2$($n \\in \\mathbf{N}$)? 如果存在, 求所有的$(p, q)$所构成的集合; 如果不存在, 请说明理由.\n%19", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338833,14 +338839,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013955": { "id": "013955", "content": "若实数$a, b$是函数$f(x)=x^2-p x+q$($p>0$, $q>0$)的两个不同的零点, 且$a, b,-2$这三个数可适当排序后成等差数列, 也可适当排序后成等比数列, 则$p+q$的值等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338859,7 +338865,7 @@ "content": "已知数列$\\{a_n\\}$的通项公式$a_n=\\dfrac{1}{(n+1)^2}$($n \\in \\mathbf{N}$, $n\\ge 1$), 记$f(n)=(1-a_1)(1-a_2) \\cdots(1-a_n)$, 则$f(n)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338878,7 +338884,7 @@ "content": "已知函数$f(x)=\\sin x+\\tan x$. 项数为$27$的等差数列$\\{a_n\\}$满足$a_n \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且公差$d \\neq 0$. 若$f(a_1)+f(a_2)+\\ldots+f(a_{27})=0$, 则当$k=$\\blank{50}时, $f(a_k)=0$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338897,7 +338903,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_2=3$, 若$|a_{n+1}-a_n|=2^n$($n \\in \\mathbf{N}$, $n\\ge 1$), 且$\\{a_{2 n-1}\\}$是递增数列. $\\{a_{2 n}\\}$是递减数列, 则$\\displaystyle\\lim _{n \\to\\infty} \\dfrac{a_{2 n-1}}{a_{2 n}}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338916,7 +338922,7 @@ "content": "已知点$P_1(a_1, b_1)$, $P_2(a_2, b_2)$, $\\cdots$, $P_n(a_n, b_n)$($n$为正整数) 都在函数$y=(\\dfrac{1}{2})^x$的图像上, 且数列$\\{a_n\\}$是$a_1=1$, 公差为$d$的等差数列.\\\\\n(1) 证明: 数列$\\{b_n\\}$是等比数列;\\\\\n(2) 若公差$d=1$, 以点$P_n$的横、纵坐标为边长的矩形面积为$c_n$, 求最大的实数$t$, 使$c_n \\leq \\dfrac{1}{t}$($t \\in \\mathbf{R}$, $t \\neq 0$)对一切正整数$n$恒成立;\\\\\n(3) 对 (2) 中的数列$\\{a_n\\}$, 对每个正整数$k$, 在$a_k$与$a_{k+1}$之间插入$3^{k-1}$个$3$(如在$a_1$与$a_2$之间插入$3^0$个$3$, $a_2$与$a_3$之间插入$3^1$个$3$, $a_3$与$a_4$之间插入$3^2$个$3$, $\\cdots$), 得到一个新的数列$\\{d_n\\}$, 设$S_n$是数列$\\{d_n\\}$的前$n$项和. 试探究$2020$是否为数列$\\{S_n\\}$中的某一项, 写出你探究得到的结论并说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338928,14 +338934,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013960": { "id": "013960", "content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_2=5$, $\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+S_{n-2}=2S_{n-1}+2^{n-1}$($n \\geq 3$).\\\\\n(1) 试求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=\\dfrac{2^{n-1}}{a_n \\cdot a_{n+1}}$, $T_n$是数列$\\{b_n\\}$的前$n$项和, 证明: $T_n<\\dfrac{1}{6}$;\\\\ \n(3) 证明: 对任意给定的$m \\in(0, \\dfrac{1}{6})$, 均存在$n_0 \\in \\mathbf{N}$, 使得当$n>n_0$时, (2)中的$T_n>m$恒成立.\n%20", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -338947,14 +338953,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013961": { "id": "013961", "content": "直线$x-a y+2=0$($a<0$)的倾斜角是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338973,7 +338979,7 @@ "content": "已知$\\overrightarrow {d}=(a,-4)$是直线$l: 2 x+y-3=0$的方向向量, 则实数$a$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -338992,7 +338998,7 @@ "content": "已知方程:\n\\textcircled{1} $x^2+y^2-2 x-4 y+6=0$; \\textcircled{2} $x^2+y^2-2 x-4 y=0$; \\textcircled{3} $x^2+y^2-2 x-4 y+5=0$, 其中表示圆的方程序号是\\blank{50}. (请写出所有满足要求的序号)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339011,7 +339017,7 @@ "content": "设实数$a$、$b$随机取自集合$\\{1,2,3\\}$, 则直线$a x+b y+3=0$与圆$x^2+y^2=1$有公共点的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339030,7 +339036,7 @@ "content": "在$\\triangle ABC$中, $D$为$AB$的中点, $G$为线段$CD$上一点, $CG=\\dfrac{3}{4} CD$且$GA \\perp GB$, 则$\\tan C$的最大值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339049,7 +339055,7 @@ "content": "已知$\\triangle ABC$的顶点$A(3,-1)$, $AB$边上的中线所在直线方程为$3 x+4 y-8=0$, $\\angle ABC$的平分线所在直线方程为$x-y+1=0$, 求$BC$边所在直线的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339061,14 +339067,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013967": { "id": "013967", "content": "已知圆$M: 2 x^2+2 y^2-8 x-8 y-1=0$, 直线$l: x+y-9=0$, 过直线$l$上点$A$作三角形$ABC$, 使得$\\angle BAC=45^{\\circ}$, $B$、$C$在圆$M$上, $M$在边$AB$上. 求点$A$的横坐标的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339080,14 +339086,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013968": { "id": "013968", "content": "某社区计划在两条互相垂直的道路一角上, 建设一个直角三角形绿化园地$\\triangle ABC$, 点$A$在直线$l_1$上, 点$B$在直线$l_2$上, 且都可根据实际需要进行选择, 如图. 现在园地中建造一个圆形花坛$M$, 要求花坛$M$与$\\triangle ABC$三边都相切, 且花坛$M$的半径为$1$个单位长度. 求此绿化园地占地面积的最小值(精确到$0.01$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) node [below left] {$C$} coordinate (C) -- (4,0) node [above] {$l_2$};\n\\draw (0,0) -- (0,5) node [right] {$l_1$};\n\\draw (3,0) node [above] {$B$} coordinate (B) -- (0,4) node [left] {$A$} coordinate (A);\n\\draw (1,1) circle (1);\n\\filldraw (1,1) circle (0.03) node [below] {$M$} coordinate (M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339099,14 +339105,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013969": { "id": "013969", "content": "若$k,-1, b$三个实数成等差数列, 则直线$y=k x+b$必经过定点\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339125,7 +339131,7 @@ "content": "已知实数$x, y$满足$2 x+y+5=0$, 那么$x^2+y^2$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339144,7 +339150,7 @@ "content": "已知直线$l_1: x-y-1=0$, 直线$l_2: x-2 y+2=0$, 则直线$l_2$关于直线$l_1$的对称直线的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339163,7 +339169,7 @@ "content": "平面直角坐标系$xOy$中, 以点$(1,0)$为圆心且与直线$m x-y-2 m-1=0$($m \\in \\mathbf{R}$)相切的所有圆中, 半径最大的圆的标准方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339182,7 +339188,7 @@ "content": "设直线$l: 3 x+4 y+4=0$, 圆$C: (x-2)^2+y^2=r^2(r>0)$, 若圆$C$上存在两点$P, Q$, 直线$l$上存在一点$M$, 使得$\\angle PMQ=90^{\\circ}$, 求$r$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339194,14 +339200,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013974": { "id": "013974", "content": "如图, 正方形$ABCD$的边长为$20$米, 圆$O$的半径为$1$米, 圆心是正方形的中心, 点$P$、$Q$分别在线段$AD$、$CB$上, 若线段$PQ$与圆$O$有公共点, 则称点$Q$在点$P$的``盲区''中, 已知点$P$以$1.5$米/秒的速度从$A$出发向$D$移动, 同时, 点$Q$以$1$米/秒的速度从$C$出发向$B$移动, 则在点$P$$A$移动到$D$的过程中, 点$Q$在点$P$的盲区中的时长约为\\blank{50}秒. (精确到$0.1$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (2,2) node [right] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (A) rectangle (C);\n\\draw (1,1) circle (0.1) node [below] {$O$};\n\\draw (0,0.45) node [left] {$P$} coordinate (P);\n\\draw (2,1.7) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339220,7 +339226,7 @@ "content": "在平面直角坐标系$x O y$中, $A(-12,0)$, $B(0,6)$, 点$P$在圆$O: x^2+y^2=50$上. 若$\\overrightarrow{PA} \\cdot \\overrightarrow{PB} \\leq 20$, 求点$P$的横坐标的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339232,14 +339238,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013976": { "id": "013976", "content": "如图, 为了保护河上古桥$OA$, 规划建一座新桥$BC$, 同时设立一个圆形保护区. 规划要求: 新桥$BC$与河岸$AB$垂直; 保护区的边界为圆心$M$在线段$OA$上并与$BC$相切的圆. 且古桥两端$O$和$A$到该圆上任意一点的距离均不少于$80 \\text{m}$. 经测量, 点$A$位于点$O$正北方向$60 \\text{m}$处, 点$C$位于点$O$正东方向$170 \\text{m}$处($OC$为河岸), $\\tan \\angle BCO=\\dfrac{4}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (3.4,0) node [below right] {$C$} coordinate (C);\n\\draw (0,1.2) node [above left] {$A$} coordinate (A);\n\\draw (C) ++ ({90+atan(3/4)}:3) coordinate (T);\n\\draw ($(C)!(A)!(T)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(B)--(C) (A)--(O)--(C);\\\n\\draw (O) --++ (0,-0.5) coordinate (O1) (C) --++ (0,-0.5) coordinate (C1);\n\\draw [<->] ($(O)!0.5!(O1)$) -- ($(C)!0.5!(C1)$) node [midway, fill = white] {$170\\text{m}$};\n\\draw (O) --++ (-0.5,0) coordinate (O2) (A) --++ (-0.5,0) coordinate (A2);\n\\draw [<->] ($(O)!0.5!(O2)$) -- ($(A)!0.5!(A2)$) node [midway, fill = white, rotate = 90] {$60\\text{m}$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {东};\n\\draw [->] (A) -- ($(O)!1.6!(A)$) node [left] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 求新桥$BC$的长;\\\\\n(2) 当$OM$多长时, 圆形保护区的面积最大?\n%21", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339251,14 +339257,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013977": { "id": "013977", "content": "若动点$P$到点$F_1(-4,0)$、$F_2(4,0)$的距离之和为 8 , 则动点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339277,7 +339283,7 @@ "content": "抛物线$x=a y^2$($a \\neq 0$)的焦点坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339296,7 +339302,7 @@ "content": "若点$A(4,3)$在椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)上, 则在点$B(4,-3)$、$C(3,-4)$、$D(-4,-3)$、$F(-4,3)$中, 不在椭圆$\\Gamma$上的点有\\blank{50}.(请写出所有满足要求的点)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339315,7 +339321,7 @@ "content": "已知直线$l_1$、$l_2$的方程分别为$x-y+3=0$、$x+1=0$, $F$为抛物线$y^2=4 x$的焦点, 若点$P$在抛物线上移动, 点$P$到直线$l_1$、$l_2$的距离分别为$d_1$、$d_2$, 则$d_1+d_2$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339334,7 +339340,7 @@ "content": "已知一个酒杯的轴截面是抛物线的一部分, 酒杯杯口直径和酒杯深都是$8 \\text{cm}$. 现准备在酒杯内放入一个玻璃球, 使得玻璃球触及酒杯底部, 求玻璃球半径的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339346,14 +339352,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013982": { "id": "013982", "content": "如图, 从双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左焦点$F_1$引圆$x^2+y^2=a^2$的切线, 切点为$T$, 延长$F_1T$交双曲线右支于点$P$, 若$M$是线段$F_1P$的中点, 且$M$在线段$PT$上, $O$为坐标原点, 则$|MT|-|MO|$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\path [domain = -4:4, samples = 100, name path = cr, draw] plot ({sqrt(1+\\x*\\x/3)},\\x);\n\\draw [domain = -4:4, samples = 100] plot ({-sqrt(1+\\x*\\x/3)},\\x);\n\\draw (-2,0) node [below] {$F_1$} coordinate (F_1);\n\\draw (2,0) node [below] {$F_2$} coordinate (F_2);\n\\path [name path = F1P] (F_1) --++ (30:5);\n\\path [name intersections = {of = F1P and cr, by = P}];\n\\draw (P) node [right] {$P$} -- (F_1) (P) -- (F_2);\n\\draw ($(F_1)!0.5!(P)$) node [above] {$M$} coordinate (M) -- (0,0);\n\\draw ($(F_1)!(0,0)!(P)$) node [above left] {$T$} coordinate (T) -- (0,0);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339372,7 +339378,7 @@ "content": "双曲线$\\Gamma: x^2-\\dfrac{y^2}{b^2}=1(b>0)$的右顶点为$A$, 两焦点分别为$F_1$、$F_2$, 点$P$是双曲线$\\Gamma$右支上一点, $PF_1$交双曲线$\\Gamma$左支于点$Q$, 交双曲线$\\Gamma$的渐近线$y=b x$于点$R$, $M$是线段$PQ$的中点, 且$F_2R \\perp F_1P$,$ F_2R\\parallel AM$, 求实数$b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339384,14 +339390,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013984": { "id": "013984", "content": "已知抛物线$\\Gamma$经过点$P(-2,3)$, 则抛物线$\\Gamma$的标准方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339410,7 +339416,7 @@ "content": "设方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339429,7 +339435,7 @@ "content": "已知双曲线$\\Gamma$与双曲线$x^2-\\dfrac{y^2}{4}=1$有共同渐近线, 且过点$M(2,2)$, 则双曲线$\\Gamma$的标准方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339448,7 +339454,7 @@ "content": "已知抛物线的顶点在原点, 焦点在$y$轴上, 抛物线上一点$M(a,-4)$到其焦点$F$的距离为$5$, 则实数$a$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339467,7 +339473,7 @@ "content": "已知点$A(3,0)$和圆$B: (x+3)^2+y^2=16$, 动圆$C$与圆$B$外切, 且过点$A$, 则动圆圆心$C$的轨迹方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339486,7 +339492,7 @@ "content": "已知椭圆$C$的焦点为$F_1(-1,0)$, $F_2(1,0)$, 过$F_2$的直线与椭圆$C$交于$A$、$B$两点. 若$|AF_2|=2|F_2B|$, $|AB|=|BF_1|$, 则椭圆$C$的方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339505,7 +339511,7 @@ "content": "设$F_1$、$F_2$分别是椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点, 过点$F_1$的直线交椭圆$C$于$A$、$B$两点, 且$|AF_1|=3|BF_1|$.\\\\\n(1) 若$|AB|=4$, $\\triangle ABF_2$的周长为$16$, 求$|AF_2|$;\\\\\n(2) 若$\\cos \\angle AF_2B=\\dfrac{3}{5}$, $a=\\lambda b$, 求实数$\\lambda$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339517,14 +339523,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013991": { "id": "013991", "content": "如图, 已知椭圆$C_1$与$C_2$的中心是坐标原点$O$, 长轴均为$MN$且在$x$轴上, 短轴长分别为$2 m$, $2 n$($m>n$), 过原点且不与$x$轴重合的直线$l$与$C_1, C_2$的四个交点按纵坐标从大到小依次为$A$、$B$、$C$、$D$. 记$\\lambda=\\dfrac{m}{n}$, $\\triangle BDM$和$\\triangle ABN$的面积分别为$S_1$和$S_2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\path [name path = elli1, draw] (0,0) ellipse (3 and 2);\n\\path [name path = elli2, draw] (0,0) ellipse (3 and 1);\n\\draw (-3,0) node [below left] {$M$} coordinate (M);\n\\draw (3,0) node [below right] {$N$} coordinate (N);\n\\path [name path = line, draw] (-2,-2.5) -- (2,2.5);\n\\path [name intersections = {of = line and elli1, by = {A,D}}];\n\\path [name intersections = {of = line and elli2, by = {B,C}}];\n\\draw (D) node [below] {$D$}--(M)--(B) node [above] {$B$};\n\\draw (C) node [below] {$C$} (A) node [above] {$A$} -- (N)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 设直线$l: y=k x$($k>0$), 若$S_1=3S_2$, 证明: $B$、$C$是线段$AD$的四等分点;\\\\\n(2) 当直线$l$与$y$轴重合时, 若$S_1=\\lambda S_2$, 求$\\lambda$的值;\\\\\n(3) 当$\\lambda$变化时, 是否存在与坐标轴不重合的直线$l$, 使得$S_1=\\lambda S_2$? 并说明理由.\n%22", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339536,14 +339542,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013992": { "id": "013992", "content": "若经过点$F_2(2,0)$的直线$l$与双曲线$x^2-\\dfrac{y^2}{3}=1$相交于$A$、$B$两点, 且$|AB|=6$, 则满足条件的直线$l$共有\\blank{50}条.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339562,7 +339568,7 @@ "content": "设过点$A(0,-1)$的直线$l$与曲线$y^2=2 x$只有一个公共点, 求直线$l$的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339574,14 +339580,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013994": { "id": "013994", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个顶点为$A(2,0)$, 它的两个焦点和短轴的两个端点恰好是一个正方形的四个顶点, 直线$l: y=k(x-1)$与椭圆$C$交于不同的两点$M$、$N$.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 当$\\triangle AMN$的面积为$\\dfrac{\\sqrt{10}}{3}$时, 求$k$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339593,14 +339599,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013995": { "id": "013995", "content": "已知直线$l_1: y=m x+1$与双曲线$C: x^2-y^2=1$的左支相交于$A$、$B$两点, 直线$l_2$经过点$M(-2,0)$和线段$AB$的中点$N$, 若直线$l_2$与$y$轴相交于点$Q(0, b)$, 求实数$b$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339612,14 +339618,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013996": { "id": "013996", "content": "已知直线$l: x-2 y=0$, 是否存在实数$a$, 使得抛物线$y=a x^2-1$上有关于直线$l$对称的两点? 若不存在, 请说明理由; 若存在, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339631,14 +339637,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013997": { "id": "013997", "content": "过抛物线$y^2=4 x$的焦点作直线交抛物线于$A(x_1, y_1)$、$B(x_2, y_2)$两点, 且$x_1+x_2=6$, 则$|AB|$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339657,7 +339663,7 @@ "content": "椭圆$x^2+4 y^2=4$的长轴上的一个顶点为$A$, 以$A$为直角顶点作一个内接于此椭圆的等腰直角三角形, 则此三角形的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339676,7 +339682,7 @@ "content": "若经过点$M(1, m)$且与双曲线$x^2-\\dfrac{y^2}{4}=1$恰有一个公共点的直线有且仅有$2$条, 则实数$m$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339695,7 +339701,7 @@ "content": "已知抛物线$y^2=8 x$的动弦$AB$的长为$6$, 则弦$AB$的中点$M$到$y$轴的最短距离是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339714,7 +339720,7 @@ "content": "已知点$P(0,1)$, 椭圆$\\dfrac{x^2}{4}+y^2=m$($m>1$)上两点$A, B$满足$\\overrightarrow{AP}=2 \\overrightarrow{PB}$, 则当$m=$时, 点$B$横坐标的绝对值最大.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339726,14 +339732,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014002": { "id": "014002", "content": "设直线$x-3 y+m=0$($m \\neq 0$)与双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{b}=1$($b>0$)的两条渐近线分别交于点$A$、$B$. 若点$P(m, 0)$满足$|PA|=|PB|$, 则实数$b$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339752,7 +339758,7 @@ "content": "设抛物线$C: y^2=4 x$的焦点为$F$, 过$F$且斜率为$k(k>0)$的直线$l$与$C$交于$A$、$B$两点, $|AB|=8$.\\\\\n(1) 求直线$l$的方程;\\\\\n(2) 求过点$A, B$且与抛物线$C$的准线相切的圆的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339764,14 +339770,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014004": { "id": "014004", "content": "设椭圆$\\Gamma$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 点$O$为坐标原点, 点$A$的坐标为$(a, 0)$, 点$B$的坐标为$(0, b)$, 点$M$在线段$AB$上, 满足$|BM|=2|MA|$, 直线$OM$的斜率为$\\dfrac{\\sqrt{5}}{10}$.\\\\\n(1) 若$a=\\lambda b$, 求$\\lambda$的值;\\\\\n(2) 设点$C$的坐标为$(0,-b)$, $N$为线段$AC$的中点, 点$N$关于直线$AB$的对称点的纵坐标为$\\dfrac{7}{2}$, 求椭圆$\\Gamma$的方程.\n%23", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339783,14 +339789,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014005": { "id": "014005", "content": "已知抛物线$C: y^2=2 x$的焦点为$F$, 平行于$x$轴的两条直线$l_1$、$l_2$分别交抛物线$C$于$A$、$B$两点, 交抛物线$C$的准线于$P$、$Q$两点. 若$\\triangle PQF$的面积是$\\triangle ABF$的面积的两倍, 求线段$AB$中点$E$的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339802,14 +339808,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014006": { "id": "014006", "content": "已知过原点$O$的直线与椭圆$C: \\dfrac{x^2}{4}+y^2=1$交于$A$、$B$两点, 点$A(x_1, y_1)$到$y$轴的距离$|x_1| \\in[1,2)$, 点$D$在椭圆$C$上, 且$AD \\perp AB$, 直线$BD$与$x$轴、$y$轴分别交于$M$、$N$两点, 记直线$BD$、$AM$的斜率分别为$k_1$、$k_2$.\\\\\n(1) 试用$x_1$、$y_1$表示$k_1$;\\\\\n(2) 求$k_1 \\cdot k_2$的取值范围;\\\\\n(3) 求$\\triangle OMN$面积的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339821,14 +339827,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014007": { "id": "014007", "content": "已知$A$是双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右顶点, 直线$l$与双曲线$C$交于$P$、$Q$两点$(P$、$Q$都不在$x$轴上).\\\\\n(1) 若$a=1$, $b=\\sqrt{2}$, 直线$l$经过点$M(-3,0)$, 求证: $\\overrightarrow{AP} \\cdot \\overrightarrow{AQ}$为定值;\\\\\n(2) 若$AP \\perp AQ$, $a \\neq b$, 试判断直线$l$是否过定点? 若是, 请求出此定点的坐标; 若不是, 请说明理由;\\\\\n(3) 请对椭圆$C_1: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)及其一个顶点, 写出与(2)类似的结论(不要求写求解或证明过程).", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339840,14 +339846,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014008": { "id": "014008", "content": "动点$P$到点$F(2,0)$的距离与它到直线$x+2=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339866,7 +339872,7 @@ "content": "过点$M(1,2)$作直线交$y$轴于点$B$, 过点$N(-1,-1)$作直线与直线$MB$垂直, 且交$x$轴于点$A$, 则线段$AB$的中点的轨迹方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339885,7 +339891,7 @@ "content": "点$M$是曲线$y=\\dfrac{1}{2} x^2+1$上的一个动点, 且点$M$为线段$OP$的中点($O$为坐标原点), 则动点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339904,7 +339910,7 @@ "content": "设$F$是双曲线$C: \\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$的一个焦点, 点$P$在$C$上, $O$为坐标原点, 若$|OP|=|OF|$, 则$\\triangle OPF$的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339923,7 +339929,7 @@ "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$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 则直线$l$过定点\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339942,7 +339948,7 @@ "content": "设抛物线$y^2=4 x$的焦点为$F$, 不经过焦点的直线上有三个不同的点$A$、$B$、$C$, 其中点$A$、$B$在抛物线上, 点$C$在$y$轴上, 若$|AF|=a$, $|BF|=b$, 则$\\triangle BCF$与$\\triangle ACF$的面积之比是\\blank{50}(用$a$、$b$的关系式表示).", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -339961,7 +339967,7 @@ "content": "(1) 已知直线$l: 4 x-y-1=0$与抛物线$x^2=2 y$交于$A(x_A, y_A)$、$B(x_B, y_B)$两点, 直线$l$与$x$轴相交于点$C(x_C, 0)$, 求证: $\\dfrac{1}{x_A}+\\dfrac{1}{x_B}=\\dfrac{1}{x_C}$;\\\\\n(2) 试将第(1)题中的命题加以推广, 使得第(1)题中的命题是推广后得到的命题的特例, 并证明推广后得到的命题正确.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -339973,14 +339979,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014015": { "id": "014015", "content": "(1) 设$P$是圆$F_1: x^2+y^2+2 x-15=0$上的动点, $F(1,0)$, 且线段$PF$的垂直平分线交直线$PF_1$于点$Q$, 求点$Q$的轨迹$C$的方程$f(x, y)=0$;\\\\\n(2) 我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(I) 已知``盾圆$D$''的方程为$y^2= \\begin{cases}4 x, & 0 \\leq x \\leq 3, \\\\ -12(x-4), & 30$)的焦点为$F$, 过点$F$的直线$l$与抛物线$C$的准线交于点$Q$, 若满足$\\overrightarrow{QF}=3\\overrightarrow{FP}$的点$P$在抛物线$C$上, 则直线$l$的斜率为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340018,7 +340024,7 @@ "content": "如图, 线段$AB$与平面$\\alpha$斜交于点$B$, 且直线$AB$与平面$\\alpha$所成的角为$60^{\\circ}$, 平面$\\alpha$上的动点$P$满足$\\angle PAB=60^{\\circ}$, 则点$P$的轨迹是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0,2) -- (2,0,2) -- (2,0,-2) -- (-2,0,-2) -- cycle;\n\\draw ({sqrt(1/3)},0,0) node [below] {$B$} coordinate (B) -- (0,1,0) node [above] {$A$} coordinate (A);\n\\draw (0,0,{sqrt(2)}) node [left] {$P$} coordinate (P) -- (A);\n\\draw [dashed] (A) -- (0,0,0) -- (B);\n\\draw (-1.5,0,1.5) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{直线}{抛物线}{椭圆}{双曲线}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340037,7 +340043,7 @@ "content": "数学中有许多形状优美、寓意美好的曲线, 曲线$C: x^2+y^2=1+|x| y$就是其中之一(如图). 给出下列三个结论:\\\\\n\\textcircled{1} 曲线$C$恰好经过$6$个整点(即横、纵坐标均为整数的点);\\\\\n\\textcircled{2} 曲线$C$上任意一点到原点的距离都不超过$\\sqrt{2}$;\\\\\n\\textcircled{3} 曲线$C$所围成的``心形''区域的面积小于$3$. 其中, 所有正确结论的序号是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -90:90, samples = 100] plot ({cos(\\x)/sqrt(1-cos(\\x)*sin(\\x))},{sin(\\x)/sqrt(1-cos(\\x)*sin(\\x))});\n\\draw [domain = -90:90, samples = 100] plot ({-cos(\\x)/sqrt(1-cos(\\x)*sin(\\x))},{sin(\\x)/sqrt(1-cos(\\x)*sin(\\x))});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}}{\\textcircled{2}}{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{2}\\textcircled{3}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340056,7 +340062,7 @@ "content": "设椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左焦点为$F$, 上顶点为$B$, 焦距为$2 \\sqrt{5}$. 已知点$A$的坐标为$(b, 0)$, 且$|FB| \\cdot|AB|=6 \\sqrt{2}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 设直线$l: y=k x$($k>0$)与椭圆在第一象限的交点为$P$, 且$l$与直线$AB$交于点$Q$. 若$\\dfrac{|AQ|}{|PQ|}=\\dfrac{5 \\sqrt{2}}{4} \\sin \\angle AOQ$($O$为原点), 求$k$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340068,14 +340074,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014020": { "id": "014020", "content": "已知点$P(1,2)$在抛物线$C: y^2=2 p x$上. 过点$Q(0,1)$的直线$l$与抛物线$C$有两个不同的交点$A, B$, 且直线$PA$交$y$轴于$M$, 直线$PB$交$y$轴于$N$.\\\\\n(1) 求抛物线$C$的方程;\\\\\n(2) 求直线$l$的斜率的取值范围;\\\\\n(3) 设$O$为原点, $\\overrightarrow{QM}=\\lambda \\overrightarrow{QO}$, \n$\\overrightarrow{QN}=\\mu \\overrightarrow{QO}$, 求证: $\\dfrac{1}{\\lambda}+\\dfrac{1}{\\mu}$为定值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340087,14 +340093,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014021": { "id": "014021", "content": "已知双曲线$C: \\dfrac{x^2}{2}-y^2=1$, 设过点$A(-3 \\sqrt{2}, 0)$的直线$l$的方向向量$\\overrightarrow {e}=(1, k)$.\\\\\n(1) 当直线$l$与双曲线$C$的一条渐近线$m$平行时, 求直线$l$的方程及$l$与$m$的距离;\\\\\n(2) 证明: 当$k>\\dfrac{\\sqrt{2}}{2}$时, 在双曲线$C$的右支上不存在点$Q$, 使之到直线$l$的距离为$\\sqrt{6}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340106,14 +340112,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014022": { "id": "014022", "content": "在平面直角坐标系$xOy$中, 已知椭圆$C: \\dfrac{x^2}{3}+y^2=1$. 斜率为$k$($k>0$)且不过原点的直线$l$交椭圆$C$于$A$、$B$两点, 线段$AB$的中点为$E$, 射线$OE$交椭圆$C$于点$G$, 交直线$x=-3$于点$D(-3, m)$.\\\\\n(1) 求$mk$的值;\\\\\n(2) 若$|OG|^2=|OD| \\cdot|OE|$, 求证: 直线$l$过定点;\\\\\n(3) 在(2)的条件下, 判断点$B$与$G$能否关于$x$轴对称? 若能, 求出此时点$B$的坐标; 若不能, 请说明理由.\n%25", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340125,14 +340131,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014023": { "id": "014023", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 异面直线$A_1B$与$B_1C$所成角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340151,7 +340157,7 @@ "content": "如图, 以$O$为圆心、$OA$长为半径作圆弧$\\overset\\frown{AB}$, 联结$AB$, 已知$\\angle AOB=90^{\\circ}, OA=OB=1$, 求图中阴影部分绕轴$OA$旋转一周得到的旋转体的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (A) -- (O) --(B);\n\\filldraw [pattern = north east lines] (A) -- (B) arc (0:90:2);\n\\draw (O) pic [draw,scale = 0.5] {right angle = B--O--A};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340163,14 +340169,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014025": { "id": "014025", "content": "如图所示, 边长为$2$的等边$\\triangle PAB$是圆锥的轴截面, $Q$是母线$PB$的中点, 若动点$M$沿圆锥侧面从点$A$运动到点$Q$, 则最短路径的长为\\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 (0,{sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (P)--(O) (A)--(B);\n\\draw (A) arc (180:360:1 and 0.3);\n\\draw [dashed] (A) arc (180:0:1 and 0.3);\n\\filldraw ($(P)!0.5!(B)$) circle (0.03) node [right] {$Q$} coordinate (Q);\n\\filldraw (0.3,0.6) circle (0.03) node [left] {$M$} coordinate (M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340189,7 +340195,7 @@ "content": "如图, 四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PA \\perp$底而$ABCD$. $AB=PA=1$, $AD=\\sqrt{3}$, $E, F$分别为棱$PD, PA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,0) node [right] {$D$} coordinate (D);\n\\draw ({sqrt(3)},0,1) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [below right] {$F$} coordinate (F);\n\\draw ($(D)!0.5!(P)$) node [above right] {$E$} coordinate (E);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D);\n\\draw [dashed] (B)--(A)--(D) (P)--(A) (F)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $B$、$C$、$E$、$F$四点共面;\\\\\n(2) 求异面直线$PB$与$AE$所成的角.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340201,14 +340207,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014027": { "id": "014027", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 动点$E$在棱$A_1B_1$上, 动点$Q$在棱$CD$上, 若$A_1E=x$, $DQ=y$($0=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.7!(B1)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!0.6!(D)$) node [below] {$Q$} coordinate (Q);\n\\draw [dashed] (E)--(Q) (A)--(D1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{与$x$无关, 与$y$有关}{与$x$有关, 与$y$无关: }{与$x, y$都有关}{与$x, y$都无关}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340227,7 +340233,7 @@ "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 动点$E, F$在棱$A_1B_1$上, 动点$P, Q$分别在棱$AD, CD$上, 若$EF=1$, $A_1E=x$, $DQ=y$, $DP=z$($x, y, z$大于零), 则四面体$PEFQ$的体积 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(A1)!0.3!(B1)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A1)!0.8!(B1)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\filldraw ($(C)!0.6!(D)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\filldraw ($(A)!0.3!(D)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与$x, y, z$都有关}{与$x$有关, 与$y, z$无关}{与$y$有关, 与$x, z$无关}{与$z$有关, 与$x, y$无关}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340246,7 +340252,7 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E, F$在棱$BC$上, $M, N$在棱$DD_1$上, 若$AB=2$, $MN=1$, $EF=1$, $DM=x$, $BE=y$, 则四面体$EFMN$的体积\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.3!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(B)!0.8!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(D)!0.25!(D1)$) node [left] {$M$} coordinate (M);\n\\draw ($(D)!0.75!(D1)$) node [left] {$N$} coordinate (N);\n\\draw [dashed] (M)--(E) (M)--(F) (N)--(E) (N)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{与$x$无关, 是关于$y$的严格减函数}{与$y$无关, 是关于$x$的严格增函数}{与$x, y$都有关}{与$x, y$都无关}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340265,7 +340271,7 @@ "content": "如图, 圆锥的底而半径$OA=2$, 高$PO=6$, 点$C$是底而直径$AB$所对弧的中点, 点$D$是母线$PA$的中点.\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 right] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (-100:1 and 0.3) node [below] {$C$} coordinate (C);\n\\filldraw ($(A)!0.5!(P)$) node [left] {$D$} coordinate (D) circle (0.03);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (A)--(B) (P)--(O)--(C);\n\\draw (A) arc (180:360:1 and 0.3);\n\\draw [dashed] (A) arc (180:0:1 and 0.3);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积与体积;\\\\\n(2) 过$D$、$C$、$O$的平面与圆锥的侧面相交得到的曲线为以$DO$为对称轴的抛物线$\\Gamma$的一部分, 求抛物线$\\Gamma$的焦点$F$到$A$的距离.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340277,14 +340283,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014031": { "id": "014031", "content": "已知$l_1$、$l_2$、$l_3$是空间三条不同的直线, 下列命题正确的是\\bracket{20}.\n\\twoch{$l_1 \\perp l_2$, $l_2 \\perp l_3 \\Rightarrow l_1\\parallel l_3$}{$l_1 \\perp l_2$, $l_2 \\perp l_3 \\Rightarrow l_1 \\perp l_3$}{$l_1\\parallel l_2$, $l_2\\parallel l_3 \\Rightarrow l_1\\parallel l_3$}{$l_1\\parallel l_2$, $l_2\\parallel l_3 \\Rightarrow l_1$、$l_2$、$l_3$共面}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340303,7 +340309,7 @@ "content": "``直线$l$上有两点到平面$\\alpha$的距离相等''是``直线$l$与平面$\\alpha$平行''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": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340322,7 +340328,7 @@ "content": "若圆锥的侧面积为$2 \\pi$, 底面积为$\\pi$, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340341,7 +340347,7 @@ "content": "设球$O$的表面积为$16 \\pi$, 圆$O_1$是球$O$的一小圆, $O_1O=\\sqrt{2}$, $A$、$B$是圆$O_1$上的两点, 若$A$、$B$两点间的球面距离为$\\dfrac{2 \\pi}{3}$, 则$\\angle AO_1B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340360,7 +340366,7 @@ "content": "若一个底而边长为$\\dfrac{\\sqrt{3}}{2}$, 侧棱长为$\\sqrt{6}$的止六棱柱的所有顶点都在一个球面上, 则此球的体积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340379,7 +340385,7 @@ "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $A_1C$与底而$ABCD$所成的角为$60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{{sqrt(6)}}\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 [dashed] (A1)--(C) (B1)--(D1);\n\\draw (A1)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求㫒而直线$A_1B$与$B_1D_1$所成角的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340391,14 +340397,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014037": { "id": "014037", "content": "如图, $AD$与$BC$是四面体$ABCD$中互相垂直的棱, $BC=2$, 若$AD=2 c$, 且$AB+BD=AC+CD=2 a$, 其中$a$、$c$为常数, 则四面体$ABCD$的体积的最大值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,-0.8,0) node [below] {$A$} coordinate (A);\n\\draw (0,1.2,0) node [above] {$D$} coordinate (D);\n\\draw (1,0,0.8) node [below] {$B$} coordinate (B);\n\\draw (1,0,-0.8) node [right] {$C$} coordinate (C);\n\\draw (A)--(D) (B)--(D) (C)--(D) (A)--(B)--(C);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340417,7 +340423,7 @@ "content": "将半径都为$1$的$4$个钢球完全装入形状为正四面体的容器里, 这个正四面体的高的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}+2 \\sqrt{6}}{3}$}{$2+\\dfrac{2 \\sqrt{6}}{3}$}{$4+\\dfrac{2 \\sqrt{6}}{3}$}{$\\dfrac{4 \\sqrt{3}+2 \\sqrt{6}}{3}$}\n%26", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340436,7 +340442,7 @@ "content": "已知平面$A_1BC_1$中, $\\overrightarrow{A_1C_1}=(0,1,0)$, $\\overrightarrow{A_1B}=(1,0,1)$, 则平面$A_1BC_1$的一个法向量为$\\overrightarrow {n}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340455,7 +340461,7 @@ "content": "已知直线$l_1$与直线$l_2$异面, 若$l_1$的一个方向向量为$\\overrightarrow {d_1}=(2,0,-2)$, $l_2$的一个方向向量为$\\overrightarrow{d_2}=(1,-1,-1)$, 则异面直线$l_1$与$l_2$所成角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340474,7 +340480,7 @@ "content": "平面$\\alpha$的一个法向量为$\\overrightarrow {m}=(a,-3,2)$, 直线$l$的一个方向向量为$\\overrightarrow {d}=(-2,-1,3)$, 若直线$l$与平面$\\alpha$所成的角为$\\dfrac{\\pi}{6}$, 则$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340493,7 +340499,7 @@ "content": "如图的几何体是由一个棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$与一个侧棱长为$2$的正四棱锥$P-A_1B_1C_1D_1$组合而成. 若点$E$是棱$BC$的中点, $F$是直线$A_1B_1$上的一个动点, 当异面直线$PE$与$AF$所成角最大时, 求线段$A_1F$的长.\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);\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1) (C1) -- (D1) -- (A1);\n\\draw ($(A1)!0.5!(C1)$) ++ (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw (A1) -- (P) -- (B1) (P) -- (C1);\n\\draw [dashed] (D1) -- (P);\n\\filldraw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A1)!0.45!(B1)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\draw (A)--(F);\n\\draw [dashed] (P)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340505,14 +340511,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014043": { "id": "014043", "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为直角梯形, $BC\\parallel AD$, $AB \\perp BC$, $\\angle ADC=45^{\\circ}$, $PA \\perp$平面$ABCD$, $AB=AP=1$, $AD=3$.\n(1) 求点$D$到平面$PBC$的距离;\\\\\n(2) 求直线$PD$与平面$PBC$所成角的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340524,14 +340530,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014044": { "id": "014044", "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$平面$ABCD$, 且四边形$ABCD$为直角梯形, $\\angle ABC=\\angle BAD=\\dfrac{\\pi}{2}$, $PA=AD=2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D) (Q)--(C);\n\\draw [dashed] (B)--(A)--(D) (A)--(P); \n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的表面积;\\\\\n(2) 若$P, A, C, D$四点在同一球面上, 求该球的体积;\\\\\n(3) 点$Q$是线段$BP$上的动点, 当直线$CQ$与$DP$所成的角最小时, 求线段$BQ$的长.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340543,14 +340549,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014045": { "id": "014045", "content": "已知$M(2,1,8)$是平面$\\alpha$外的一点, $A(-1,-2,5)$是平面$\\alpha$内的一点, $\\overrightarrow {n}=(-4,0,3)$是平面$\\alpha$的一个法向量, 则点$M$到平面$\\alpha$的距离$d=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340569,7 +340575,7 @@ "content": "在四棱锥$P-ABCD$中, $PA \\perp$平面$ABCD$, $AB \\perp AD$, $BC\\parallel AD$, $BC=1$, $CD=\\sqrt{2}$, $\\angle CDA=45^{\\circ}$. 若$PA=AB$, 求直线$PB$与平面$PCD$所成角的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340581,14 +340587,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014047": { "id": "014047", "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$E$是棱$AB$上的动点.\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 ($(A)!0.7!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D1)--(E) (A1)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DA_1 \\perp ED_1$;\\\\\n(2) 若直线$DA_1$与平面$CED_1$所成的角是$45^{\\circ}$, 请你确定点$E$的位置, 并证明你的结论.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340600,14 +340606,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014048": { "id": "014048", "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为直角梯形, $\\angle BAD=90^{\\circ}$, $AD\\parallel BC$, $AB=2$, $AD=1$, $PA=BC=4$, $PA \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (4,0,0) node [below] {$C$} coordinate (C);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--cycle;\n\\draw [dashed] (B)--(A)--(D)--cycle (A)--(P)--(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BD$与$PC$所成角的大小;\\\\\n(2) 求二面角$A-PC-B$的余弦值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340619,14 +340625,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014049": { "id": "014049", "content": "如图所示, 正四棱锥$V-ABCD$的表面积为$12$, $AB=2$, 将正四棱锥$V-ABCD$绕棱$AB$旋转, 若$AB \\subset$平面$\\alpha$, $M$、$N$分别是$AB$、$CD$的中点, 点$V$在平面$\\alpha$上的射影为点$O$, 则$ON$的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$A$} coordinate (A);\n\\draw (-1,{sqrt(3)},1) node [left] {$C$} coordinate (C);\n\\draw (-1,{sqrt(3)},-1) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5!(C)$) ++ ({3/2},{sqrt(3)/2},0) node [above] {$V$} coordinate (V);\n\\draw (1,0,0) node [below] {$O$} coordinate (O);\n\\draw (0,0,0) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\n\\draw (V)--(D)(V)--(C)(V)--(B)(V)--(A)(D)--(C)--(B)--(A)(V)--(O)--(M);\n\\draw [dashed] (M)--(N)(A)--(D);\n\\path [name path = BC] (B)--(C);\n\\path [name path = AV] (A)--(V);\n\\path [name path = line] (-3,0,-2) -- (2,0,-2);\n\\path [name intersections = {of = line and AV, by = S}];\n\\path [name intersections = {of = line and BC, by = T}];\n\\draw (-3,0,-2) -- (-3,0,2) -- (2,0,2) -- (2,0,-2);\n\\draw (-3,0,-2) -- (T) (S) -- (2,0,-2);\n\\draw (-2.5,0,1.5) node {$\\alpha$};\n\\draw [dashed] (S)--(T);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340645,7 +340651,7 @@ "content": "已知棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为侧面$BB_1C_1C$的中心, $F$在棱$AD$上运动, 正方体表面上有一点$P$满足$\\overrightarrow{D_1P}=\\lambda \\overrightarrow{D_1F}+\\mu \\overrightarrow{D_1E}$($\\lambda \\geq 0$, $\\mu \\geq 0$), 则所有满足条件的$P$点构成图形的面积为\\blank{50}.\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 ($(B)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.4!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (F)--(D1);\n\\draw (B)--(C1);\n\\filldraw (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n%27", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340664,7 +340670,7 @@ "content": "三棱柱$ABC-A_1B_1C_1$的棱长均为$2$, $\\angle AA_1C_1=\\angle AA_1B_1=60^{\\circ}$, 则该棱柱的体积等于\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2 \\sqrt{2}$}{$3 \\sqrt{2}$}{$4 \\sqrt{2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340683,7 +340689,7 @@ "content": "如图, 点$N$为正方形$ABCD$的中心, $\\triangle ECD$为正三角形, $O$为$CD$的中点, $EO \\perp$平面$ABCD$, $M$是线段$ED$的中点, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (2,0,2) node [right] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw ($(B)!0.5!(D)$) node [below] {$N$} coordinate (N);\n\\draw (O) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw (E)--(D)(E)--(C)(E)--(O)(C)--(D)--(A)--(B)--cycle;\n\\draw (E)--(N)(M)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$BM=EN$, 且直线$BM, EN$是相交直线}{$BM \\neq EN$, 且直线$BM, EN$是相交直线}{$BM=EN$, 且直线$BM, EN$是异面直线}{$BM \\neq EN$, 且直线$BM, EN$是异面直线}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340702,7 +340708,7 @@ "content": "若三棱锥$P-ABC$的四个顶点在球$O$的球面上, $PA=PB=PC$, $\\triangle ABC$是边长为$2$的正三角形, $E, F$分别是$PA, AB$的中点, $\\angle CEF=90^{\\circ}$, 求球$O$的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [above] {$C$} coordinate (C);\n\\draw (1,0,{-sqrt(3)/3}) ++ (0,{sqrt(2/3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(A)$) node [above left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw (E)--(F);\n\\draw (A)--(B)(P)--(A)(P)--(B)(P)--(C)--(B);\n\\draw [dashed] (A)--(C)(C)--(E)(C)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340714,14 +340720,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014054": { "id": "014054", "content": "如图, 几何体是圆柱的一部分, 它是由矩形$ABCD$(及其内部)以$AB$边所在直线为旋转轴旋转$120^{\\circ}$得到的, $G$是$\\overset\\frown{DF}$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw ({2*cos(120)},0,{2*sin(120)}) node [left] {$F$} coordinate (F);\n\\draw ({2*cos(60)},0,{2*sin(60)}) node [below] {$G$} coordinate (G);\n\\draw (F) ++ (0,-3,0) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (0,-3,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,-3,0) node [below] {$B$} coordinate (B);\n\\draw ({2*cos(30)},-3,{2*sin(30)}) node [below] {$P$} coordinate (P);\n\\draw [domain = 0:120, samples = 100] plot ({2*cos(\\x)},0,{2*sin(\\x)});\n\\draw [domain = 0:120, samples = 100] plot ({2*cos(\\x)},-3,{2*sin(\\x)});\n\\draw (A)--(F)--(E)(A)--(D)--(C)(A)--(G);\n\\draw [dashed] (A)--(E)(A)--(B)--(P)(A)--(P)(A)--(C)(G)--(C)(E)--(B)--(C)(E)--(G);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$P$是$\\overset\\frown{CE}$上的一点, 且$AP \\perp BE$, 求$\\angle CBP$的大小;\\\\\n(2) 当$AB=3, AD=2$, 求二面角$E-AG-C$的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340733,14 +340739,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014055": { "id": "014055", "content": "已知$a, b$为空间中两条互相垂直的直线, 等腰直角三角形$ABC$的直角边$AC$所在直线与$a, b$都垂直, 斜边$AB$以直线$AC$为旋转轴旋转, 有下列结论:\\\\\n\\textcircled{1} 当直线$AB$与$a$成$60^{\\circ}$角时, $AB$与$b$成$30^{\\circ}$角;\\\\\n\\textcircled{2} 当直线$AB$与$a$成$60^{\\circ}$角时, $AB$与$b$成$60^{\\circ}$角;\\\\\n\\textcircled{3} 直线$AB$与$a$所成角的最小值为$45^{\\circ}$;\\\\\n\\textcircled{4} 直线$AB$与$a$所成角的最大值为$60^{\\circ}$.\n其中正确的是\\blank{50}(填写所有正确结论的编号).", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340759,7 +340765,7 @@ "content": "如图, 正四棱锥$P-ABCD$中, $B_1$为$PB$的中点, $D_1$为$PD$的中点, 则棱锥$A-B_1CD_1$与$P-ABCD$的体积之比为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (1,0,-1) node [right] {$D$} coordinate (D);\n\\draw (-1,0,-1) node [below] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [above left] {$B_1$} coordinate (B_1);\n\\draw ($(P)!0.5!(D)$) node [above right] {$D_1$} coordinate (D_1);\n\\draw (P)--(B)(P)--(C)(P)--(D)(B)--(C)--(D);\n\\draw [dashed] (P)--(A)(B)--(A)--(D);\n\\draw (C)--(B_1)(C)--(D_1);\n\\draw [dashed] (C)--(A)--(B_1)(D_1)--(A)(D_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n%28", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340778,7 +340784,7 @@ "content": "设复数$z_1=1+2 a \\mathrm{i}$, $z_2=a-\\mathrm{i}$, $a \\in \\mathbf{R}$, 集合$A=\\{z|| z-z_1 | \\leq \\sqrt{2}\\}$, $B=\\{z|| z-z_2 | \\leq 2 \\sqrt{2}\\}$, 若$A \\cap B=\\varnothing$, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340790,14 +340796,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014058": { "id": "014058", "content": "方程$2 x^2+3 a x+a^2+2 a=0$至少有一个模为$1$的根, 求实数$a$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340809,14 +340815,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014059": { "id": "014059", "content": "已知复数$z=a+b \\mathrm{i}$($a, b \\in \\mathbf{R}$), $\\overline {z}$是$z$的共轭复数, 存在实数$t$, 使得$\\overline {z}=\\dfrac{2+4 \\mathrm{i}}{t}-3 a t \\mathrm{i}$成立.\\\\\n(1) 求证: $2 \\mathrm{Re} z+\\mathrm{Im}z$为定值;\\\\\n(2) 若$|z-2| \\leq a$, 求$|z|$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340828,14 +340834,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014060": { "id": "014060", "content": "已知集合$B=\\{z | z$是虚数, 且$\\dfrac{z^2}{\\overline{z}} \\in \\mathbf{R}\\}$.\\\\\n(1) 若$z \\in B$, 求复数$z$在复平面内对应的点的轨迹方程;\\\\\n(2) 若集合$M=\\{x | x^3=1\\}$, 判断集合$M_B=\\{t | z-t \\overline {z}=0,\\ z \\in B\\}$是否为集合$M$的真子集.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340847,14 +340853,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014061": { "id": "014061", "content": "已知复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$)满足$|z-4 \\mathrm{i}|=|z+2|$, 则$2^x+4^y$的最小值为\\bracket{20}.\n\\fourch{$2$}{$4$}{$4 \\sqrt{2}$}{$8 \\sqrt{2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340873,7 +340879,7 @@ "content": "设复数$z=a+b \\mathrm{i}$($a>0$, $b \\in \\mathbf{R}$, $b \\neq 0$)是实系数方程$x^2+p x+q=0$的根, 又$z^3$为实数, 则点$(p, q)$的轨迹方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340892,7 +340898,7 @@ "content": "复数$z$满足$|z-2 \\mathrm{i}|^2-|z-1|^2=5$, 则它在复平面对应的点的轨迹是\\bracket{20}.\n\\fourch{圆}{直线}{双曲线}{椭圆}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -340911,7 +340917,7 @@ "content": "若复数$z$满足$|z-\\sqrt{2}-\\sqrt{2} \\mathrm{i}|=1$, 则$|z|$的最大值是\\blank{50}, $|z|$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340930,7 +340936,7 @@ "content": "满足方程$z^2-|z|=0$的复数的个数有\\blank{50}个.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340949,7 +340955,7 @@ "content": "已知复数$z$满足$\\overline{z^2}=z$, $|z+1|=1$, 则$z=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -340968,7 +340974,7 @@ "content": "已知复数$z=1+\\mathrm{i}$.\\\\\n(1) 设$w=z^2+3 \\overline {z}-4$, 求$w$;\\\\\n(2) 如果$\\dfrac{z^2+a z+b}{z^2-z+1}=1-\\mathrm{i}$, 求实数$a, b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340980,14 +340986,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014068": { "id": "014068", "content": "设复数$z=-\\sqrt{6}+\\sqrt{2} \\mathrm{i}$, $\\mu=(\\dfrac{4}{z})^3$, \n(1) 求$\\mu$的模的大小;\\\\\n(2) 是否存在实数$x, y$使得$\\dfrac{x}{z}+\\dfrac{y}{\\mu}=z+2 \\mu$成立, 若存在, 求出$x, y$的值, 若不存在, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -340999,14 +341005,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014069": { "id": "014069", "content": "已知关于$z$的方程$z^2-(4+\\mathrm{i}) z+4-m \\mathrm{i}=0$($m \\in \\mathbf{R}$)有实根$\\lambda$.\\\\\n(1) 分别求实数根$\\lambda$以及相应利$m$的值;\\\\\n(2) 在(1)的条件下, 若$M=\\{(x, y) |$存在$b, n \\in \\mathbf{R}$, 使得$(m-n \\mathrm{i})(1-b \\mathrm{i})=x+y \\mathrm{i}, \\ x, y \\in \\mathbf{R}\\}$, 是否存在$t, \\alpha \\in \\mathbf{R}$, 满足$(t \\cos \\alpha, \\sqrt{7} \\sin \\alpha) \\in M$, 若存在, 求出$t$的取值范围, 若不存在, 请说明理由.\n%29", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341018,14 +341024,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014070": { "id": "014070", "content": "在$-3,-2,-1,0,1,2,3,4$这$8$个数中, 任取$3$个不重复的数作为二次函数$f(x)=a x^2+b x+c$的系数$a, b, c$, 问:\\\\\n(1) 能组成多少个不同的二次函数;\\\\\n(2) 能组成多少条对称轴为$y$轴的抛物线?\\\\\n(3) 能组成多少经过原点且顶点在第一或第三象限的抛物线?", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341037,14 +341043,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014071": { "id": "014071", "content": "把由$1,2,3,4,5$这五个数字组成的无重复数字的五位数, 把它们按从小到大的顺序排成一列, 构成一个数列.\\\\\n(1) $43251$是这个数列的第几项?\\\\\n(2) 这个数列的第$96$项是多少?\\\\\n(3) 求这个数列的各项和.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341056,14 +341062,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014072": { "id": "014072", "content": "如果$(x^{\\lg x}-3)^n$的展开式中最后三项的二项式系数的和等于$22$, 又展开式的中间项等于$-540000$, 求$x$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341075,14 +341081,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014073": { "id": "014073", "content": "在教材中有这样一个性质: $\\mathrm{C}_n^0+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots+\\mathrm{C}_n^n=2^n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 计算$1 \\cdot \\mathrm{C}_3^0+2 \\cdot \\mathrm{C}_3^1+3 \\cdot \\mathrm{C}_3^2+4 \\cdot \\mathrm{C}_3^3$的值的方法如下: 设$S=1 \\cdot \\mathrm{C}_3^0+2 \\cdot \\mathrm{C}_3^1+3 \\cdot \\mathrm{C}_3^2+4 \\cdot \\mathrm{C}_3^3$, 又上式的倒序为$S=4 \\cdot \\mathrm{C}_3^3+3 \\cdot \\mathrm{C}_3^2+2 \\cdot \\mathrm{C}_3^1+1 \\cdot \\mathrm{C}_3^0$.\n二式相加得$2S=5(\\mathrm{C}_3^0+\\mathrm{C}_3^1+\\mathrm{C}_3^2+\\mathrm{C}_3^3)=5 \\cdot 2^3$, 所以$S=5 \\cdot 2^2=20$. 利用类似方法求:\n$T=1 \\cdot \\mathrm{C}_2^0+2 \\cdot \\mathrm{C}_2^1+3 \\cdot \\mathrm{C}_2^2$; $R=1 \\cdot \\mathrm{C}_4^0+2 \\cdot \\mathrm{C}_4^1+3 \\cdot \\mathrm{C}_4^2+4 \\cdot \\mathrm{C}_4^3+5 \\cdot \\mathrm{C}_4^4$;\\\\\n(2) 将(1)的结论推广到一般情况, 并给予证明;\n(3) 设$S_n$是首项为$a_1$、公比为$q$的等比数列$\\{a_n\\}$的前$n$项和, 求$Q=S_1 \\mathrm{C}_n^0+S_2 \\mathrm{C}_n^1+S_3 \\mathrm{C}_n^2+\\cdots+S_{n+1} \\mathrm{C}_n^n$($n \\in \\mathbf{N}$, $n \\ge 1$)", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341094,14 +341100,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014074": { "id": "014074", "content": "某餐厅供应客饭, 每位顾客可在餐厅提供的菜肴中任选$2$荤$2$素共$4$种不同品种, 现餐厅准备了$5$种不同的荤菜, 若保证每位顾客有$200$种以上的不同选择, 则餐厅至少还需准备\\blank{50}种不同的素菜品种.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341120,7 +341126,7 @@ "content": "$A, B, C, D, E$五个人并排站成一排, 若$B$必须站在$A$的右边, 那么不同的排法共有\\bracket{20}种.\n\\fourch{$24$}{$60$}{$90$}{$120$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341139,7 +341145,7 @@ "content": "在$5$双不同的手套中, 任取$4$只, 四只手套中至少有两只配成一双的可能取法种数是\\bracket{20}.\n\\fourch{$20$}{$30$}{$130$}{$140$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341158,7 +341164,7 @@ "content": "设$(3 \\sqrt[3]{x}+\\sqrt{x})^n$展开式的各项系数和为$t$, 其二项式系数和为$h$, 若$t+h=272$, 则展开式的$x^2$项的系数是\\bracket{20}.\n\\fourch{$12$}{$1$}{$6$}{$3$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341177,7 +341183,7 @@ "content": "设$l_n$为$(1+x)^n$的展开式中的$x^2$项的系数, 则$\\displaystyle\\lim_{n\\to\\infty}(\\dfrac{1}{l_2}+\\dfrac{1}{l_3}+\\cdots+\\dfrac{1}{l_n})$等于\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341196,7 +341202,7 @@ "content": "已知$a_n$是函数$f_n(x)=(1+2 x)(1+2^2 x)(1+2^3 x) \\cdots(1+2^n x)$($n \\in \\mathbf{N}$, $n\\ge 1$)的展开式中的$x^2$的系数.\\\\\n(1) 计算$a_1, a_2, a_3$;\\\\\n(2) 求证: $a_{n+1}=a_n+2^{n+1}(2+2^2+\\cdots+2^n)$;\\\\\n(3) 是否存在常数$a$、$b$, 使得对不小于$2$的自然数$n$, 下列关系式$a_n=\\dfrac{8}{3}(2^{n-1}-1)(a \\cdot 2^n+b)$恒成立? 证明你的结论.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341208,14 +341214,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014080": { "id": "014080", "content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n$、$m$是正整数, 且$m \\leq n)$的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的性质$\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$, 是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能, 写出推广的形式, 并给出证明; 若不能, 说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.\n%30", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341227,14 +341233,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014081": { "id": "014081", "content": "某同学从物理、化学、生物、政治、历史、地理六科中随机选择三科参加考试, 则物理和化学不同时被选中的概率为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341253,7 +341259,7 @@ "content": "把$4$个不同的球任意投入$4$个不同的盒子内(每盒装球数不限), 计算:\\\\\n(1) 无空盒的概率;\\\\\n(2) 恰有一个空盒的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341265,14 +341271,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014083": { "id": "014083", "content": "某三位数密码, 每位数字可在$0-9$这$10$个数字中任选一个, 则该三位数密码中, 恰有两位数字相同的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341291,7 +341297,7 @@ "content": "某班$40$人随机平均分成两组, 两组学生一次考试成绩情况见下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline \\backslashbox{组别}{统计量} & 平均成绩 & 标准差 \\\\\n\\hline 第一组 & 90 & 6 \\\\\n\\hline 第二组 & 80 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n求全班的平均成绩和标准差.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341303,14 +341309,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014085": { "id": "014085", "content": "冬奥会即将在北京召开, 某工厂生产$A$、$B$、$C$三种奥运会纪念品, 每种纪念品均有精品型和普通型两种, 某一天产量如下表: (单位: 个)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 纪念品 A & 纪念品 B & 纪念品 C \\\\\n\\hline 精品型 & 100 & 150 &$n$\\\\\n\\hline 普通型 & 300 & 450 & 600 \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n现采用分层抽样的方法在这一天生产的纪念品中抽取$200$个, 其中有$A$种纪念品$40$个.\\\\\n(1) 求$n$的值;\\\\\n(2) 从 B 种纪念品中抽取$5$个, 其某种指标的数据分别如下: $x,y, 10,11,9$. 把这$5$个数据看作一个总体, 其均值为$10$, 方差为$2$, 求$|x-y|$的值;\\\\\n(3) 用分层抽样的方法在$C$种纪念品中抽取一个样本容量为$5$的样本, 将该样本看成一个总体, 从中任取$2$个纪念品, 求至少有$1$个精品型纪念品的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341322,14 +341328,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014086": { "id": "014086", "content": "甲、乙两队进行篮球决赛, 采取七场四胜制(当一队赢得四场胜利时, 该队获胜, 决赛结束). 根据前期比赛成绩, 甲队的主客场安排依次为``主主客客主客主''. 设甲队主场取胜的概率为$0.6$, 客场取胜的概率为$0.5$, 且各场比赛结果相互独立, 则甲队以$4: 1$获胜的概率是\\blank{50};", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341348,7 +341354,7 @@ "content": "气象意义上从春季进入夏季的标志为: ``连续$5$天的日平均温度均不低于$22$($^\\circ\\text{C}$). 现有甲、乙、丙三地连续$5$天的日平均温度的记录数据(记录数据都是正整数): \\textcircled{1} 甲地: $5$个数据的中位数为$24$, 众数为$22$; \\textcircled{2} 乙地: $5$个数据的中位数为$27$, 总体均值为$24$; \\textcircled{3} 丙地: $5$个数据中有一个数据是$32$, 总体均值为$26$, 总体方差为$10.8$; 则肯定进入夏季的地区有\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{$3$个}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341367,7 +341373,7 @@ "content": "一个容量为$n$的样本, 分成若干组, 已知某数的频数和频率分别为$40$、$0.125$, 则$n$的值为\\bracket{20}.\n\\fourch{$640$}{$320$}{$240$}{$160$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341386,7 +341392,7 @@ "content": "一个单位有$500$名职工, 其中不到$35$岁的有$125$人, $35-49$岁的有$280$人, $50$岁以上的有$95$人, 要从中抽取一个容量为$100$的样本来调查职工的体育锻炼时间, 较为恰当的抽样方法是\\bracket{20}.\n\\fourch{简单随机抽样}{系统抽样}{分层抽样}{以上三种均可}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -341405,7 +341411,7 @@ "content": "从$1$、$2$、$3$、$4$这四个数中一次随机地抽取两个数, 其中一个数是另一个数的两倍的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341424,7 +341430,7 @@ "content": "袋中有$3$个五分硬币, $3$个二分硬币和$4$个一分硬帀, 从中任取三个, 则总金额超过$8$分的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341443,7 +341449,7 @@ "content": "一个总体中有$100$个个体, 随机编号为$0,1,2, \\cdots, 99$, 依编号顺序平均分成$10$个小组, 组号依次为$1,2,3, \\cdots, 10$. 现用某种抽样方法抽取一个容量为$10$的样本, 规定如果在第$1$组随机抽取的号码为$m$, 那么在第$k$小组中抽取的号码个位数字与$m+k$的个位数字相同. 若$m=6$, 则在第$7$组中抽取的号码是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341462,7 +341468,7 @@ "content": "$2019$年, 我国施行个人所得税专项附加扣除办法, 涉及子女教育、继续教育、大病医疗、住房贷款利息或者住房租金、赡养老人等六项专项附加扣除, 某单位老、中、青员工分别有$72,108,120$人, 现采用分层抽样的方法, 从该单位上述员工中抽取$25$人调查专项附加扣除的享受情况.\\\\\n(1) 应从老、中、青员工中分别抽取多少人?\\\\\n(2) 抽取的$25$人中, 享受至少两项专项附加扣除的员工有$6$人, 分别记为$A, B, C, D, E, F$. 享受情况如下表, 其中``\\checkmark''表示享受.``$\\times$''表示不享受. 现从这$6$人中随机抽取$2$人接受采访.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\backslashbox{项目}{员工} & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ \\\\ \\hline\n子女教育 & \\checkmark & \\checkmark & $\\times$ & \\checkmark & $\\times$ & \\checkmark \\\\ \\hline\n继续教育 & $\\times$ & $\\times$ & \\checkmark & $\\times$ & \\checkmark & \\checkmark \\\\ \\hline\n大病医疗 & $\\times$ & $\\times$ & $\\times$ & \\checkmark & $\\times$ & $\\times$ \\\\ \\hline\n住房贷款利息 & \\checkmark & \\checkmark & $\\times$ & $\\times$ & \\checkmark & \\checkmark \\\\ \\hline\n住房租金 & $\\times$ & $\\times$ & \\checkmark & $\\times$ & $\\times$ & $\\times$ \\\\ \\hline\n赡养老人 & \\checkmark & \\checkmark & $\\times$ & $\\times$ & $\\times$ & \\checkmark \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(I) 使用所给字母列举出所有可能的抽取结果;\\\\\n(II) 设$M$为事件``抽取的$2$人享受的专项附加扣除至少有一项相同'', 求事件$M$发生的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341474,14 +341480,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014094": { "id": "014094", "content": "某人有$5$把钥匙, 其中有$k$把是房门钥匙, 但忘记了开房门的是哪一把. 于是, 他逐把不重复地试开, 问:\\\\\n(1) 若$k=1$, 则恰好第三次打开房门锁的概率是多少?\\\\\n(2) 若$k=1$, 则三次内打开的概率是多少?\\\\\n(3) 若$k=2$, 则三次内打开的概率是多少?\n%31", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -341493,14 +341499,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "014095": { "id": "014095", "content": "已知$(x-\\dfrac{a}{x})^8$展开式中常数项为$1120$, 其中实数$a$是常数, 则展开式中各项系数的和是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -341519,7 +341525,7 @@ "content": "已知$0