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 4c55e61a..8ab9e352 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": 19, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 1, + "execution_count": 19, "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 = \"30426\"\n", + "problems = \"31353\"\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": 2, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ 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 4a31eb4b..f715e656 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": 5, + "execution_count": 8, "metadata": {}, "outputs": [ { @@ -18,11 +18,11 @@ "#在 模板文件 目录中保留 赋能template.txt.\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21\"\n", + "filename = \"赋能30\"\n", "\n", "\"\"\"---设置题目列表---\"\"\"\n", "problems = r\"\"\"\n", - "536:542,31243,544,545\n", + "626:630,13307,632:635\n", "\n", "\"\"\"\n", "#完成后将含有 filename 的文件移至其它目录\n", @@ -85,7 +85,7 @@ "for id in problem_list:\n", " problemset = pro_dict[id]\n", " problem = problemset[\"content\"]\n", - " data_problems += \"\\\\item \" + \"{\\\\tiny (\"+id+\")}\"+problem + \"\\n\\n\"\n", + " data_problems += \"\\\\item \" + \"{\\\\tiny (\"+id+\")} \"+problem + \"\\n\\n\"\n", "texdata = texdata.replace(\"题目待替换\",data_problems)\n", "\n", "with open(\"临时文件/赋能答题纸/output/\"+filename+\".tex\",\"w\",encoding=\"utf8\") as f:\n", @@ -108,7 +108,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "pythontest", "language": "python", "name": "python3" }, @@ -127,7 +127,7 @@ "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 b276758a..1ab60d10 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 @@ -11,7 +11,7 @@ "text": [ "首个空闲id: 14764 , 直至 020000\n", "首个空闲id: 22106 , 直至 030000\n", - "首个空闲id: 31357 , 直至 040000\n", + "首个空闲id: 31358 , 直至 040000\n", "首个空闲id: 40246 , 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 74431c57..31531e06 100644 --- a/工具/批量题号选题pdf生成.ipynb +++ b/工具/批量题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/本学期高一高二年级新试卷_教师用_20230319.tex\n", + "开始编译教师版本pdf文件: 临时文件/赋能31to42_教师用_20230323.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/本学期高一高二年级新试卷_学生用_20230319.tex\n", + "开始编译学生版本pdf文件: 临时文件/赋能31to42_学生用_20230323.tex\n", "0\n" ] } @@ -23,26 +23,41 @@ "模板文件目录下 题目清单.tex 文件不能缺失\n", "\"\"\"\n", "\n", + "\"\"\"---设置是否在学生版中提供答案---\"\"\"\n", + "answered = True\n", + "\n", "\"\"\"---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", - "\"2024届高二下学期周末卷01\":\"40001:40017\",\n", - "\"2025届高一下学期周末卷01\":\"40018:40036\",\n", - "\"2024届高二下学期周末卷02\":\"40037:40056\",\n", - "\"2025届高一下学期周末卷02\":\"40057:40082\",\n", - "\"2025届高一下学期周末卷03\":\"40083:40104\",\n", - "\"2025届高一下学期周末卷03小测\":\"40105:40112\",\n", - "\"2025届高一下学期周末卷04旧版\":\"40113:40130\",\n", - "\"2025届高一下学期周末卷04小测\":\"40131:40139\",\n", - "\"2024届高二下学期周末卷03\":\"40140:40160\",\n", - "\"2024届高二上学期期末考试\":\"31267:31287\",\n", - "\"2025届高一上学期期末考试\":\"31288:31308\",\n", - "\"2024届高二下学期周末卷04\":\"40161:40180\",\n", - "\"2025届高一下学期周末卷04\":\"40181:40201\",\n", - "\"2024届高二下学期周末卷05\":\"40202:40225\",\n", - "\"2025届高一下学期周末卷05\":\"40226:40245\",\n", - "\"2024届空间向量校本作业\":\"22048:22083\",\n", - "\"2024届二项式定理校本作业\":\"22084:22105\"\n", + "# \"2024届高二下学期周末卷01\":\"40001:40017\",\n", + "# \"2025届高一下学期周末卷01\":\"40018:40036\",\n", + "# \"2024届高二下学期周末卷02\":\"40037:40056\",\n", + "# \"2025届高一下学期周末卷02\":\"40057:40082\",\n", + "# \"2025届高一下学期周末卷03\":\"40083:40104\",\n", + "# \"2025届高一下学期周末卷03小测\":\"40105:40112\",\n", + "# \"2025届高一下学期周末卷04旧版\":\"40113:40130\",\n", + "# \"2025届高一下学期周末卷04小测\":\"40131:40139\",\n", + "# \"2024届高二下学期周末卷03\":\"40140:40160\",\n", + "# \"2024届高二上学期期末考试\":\"31267:31287\",\n", + "# \"2025届高一上学期期末考试\":\"31288:31308\",\n", + "# \"2024届高二下学期周末卷04\":\"40161:40180\",\n", + "# \"2025届高一下学期周末卷04\":\"40181:40201\",\n", + "# \"2024届高二下学期周末卷05\":\"40202:40225\",\n", + "# \"2025届高一下学期周末卷05\":\"40226:40245\",\n", + "# \"2024届空间向量校本作业\":\"22048:22083\",\n", + "# \"2024届二项式定理校本作业\":\"22084:22105\"\n", + "\"赋能31\":\"636:645\",\n", + "\"赋能32\":\"646:655\",\n", + "\"赋能33\":\"656:665\",\n", + "\"赋能34\":\"666:675\",\n", + "\"赋能35\":\"676:685\",\n", + "\"赋能36\":\"686:695\",\n", + "\"赋能37\":\"696:705\",\n", + "\"赋能38\":\"706:715\",\n", + "\"赋能39\":\"716:725\",\n", + "\"赋能40\":\"726:735\",\n", + "\"赋能41\":\"736:745\",\n", + "\"赋能42\":\"746:755\"\n", "\n", "}\n", "\n", @@ -51,7 +66,7 @@ "\"\"\"---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 = \"本学期高一高二年级新试卷\"\n", + "filename = \"赋能31to42\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "if directory[-1] != \"/\":\n", " directory += \"/\"\n", @@ -159,6 +174,8 @@ " raw_string = \"\\\\item \" + \"{\\\\tiny (\"+id+\")} \"+problem\n", " teachers_string = raw_string.replace(\"\\\\tiny\",\"\")+\"\\n\\n关联目标:\\n\\n\"+ objects + \"\\n\\n标签: \" + tags + \"\\n\\n答案: \"+answer + \"\\n\\n\" + \"解答或提示: \" + solution + \"\\n\\n使用记录:\\n\\n\"+ usage + \"\\n\" + \"\\n\\n出处: \"+origin + \"\\n\\n\"\n", " students_string = raw_string + space + \"\\n\\n\"\n", + " if answered:\n", + " students_string += \"答案: \\\\textcolor{red}{\"+answer + \"}\\n\\n\"\n", " data_teachers += teachers_string\n", " data_students += students_string\n", "\n", diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index f9a32290..c397f4ce 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id列表, 新id的数目不能小于旧id的数目---\"\"\"\n", - "old_ids = \"21416,21433,21434,30409\"\n", - "new_ids = \"31353:31356\"\n", + "old_ids = \"603\"\n", + "new_ids = \"31360\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\"\"\"---完成编辑后记得运行第二个单元格---\"\"\"\n", "\n", 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 8e3b3f1c..0a70e22f 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 @@ -7,80 +7,24 @@ "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 22048\n", - "raworigin = \"\"\n", + "starting_id = 31358\n", + "raworigin = \"自拟题目\"\n", "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目9.tex\"\n", - "editor = \"20230319\\t王伟叶\"\n", + "editor = \"20230323\\t王伟叶\"\n", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "添加题号022048, 来源: 2024届空间向量校本作业\n", - "添加题号022049, 来源: 2024届空间向量校本作业\n", - "添加题号022050, 来源: 2024届空间向量校本作业\n", - "添加题号022051, 来源: 2024届空间向量校本作业\n", - "添加题号022052, 来源: 2024届空间向量校本作业\n", - "添加题号022053, 来源: 2024届空间向量校本作业\n", - "添加题号022054, 来源: 2024届空间向量校本作业\n", - "添加题号022055, 来源: 2024届空间向量校本作业\n", - "添加题号022056, 来源: 2024届空间向量校本作业\n", - "添加题号022057, 来源: 2024届空间向量校本作业\n", - "添加题号022058, 来源: 2024届空间向量校本作业\n", - "添加题号022059, 来源: 2024届空间向量校本作业\n", - "添加题号022060, 来源: 2024届空间向量校本作业\n", - "添加题号022061, 来源: 2024届空间向量校本作业\n", - "添加题号022062, 来源: 2024届空间向量校本作业\n", - "添加题号022063, 来源: 2024届空间向量校本作业\n", - "添加题号022064, 来源: 2024届空间向量校本作业\n", - "添加题号022065, 来源: 2024届空间向量校本作业\n", - "添加题号022066, 来源: 2024届空间向量校本作业\n", - "添加题号022067, 来源: 2024届空间向量校本作业\n", - "添加题号022068, 来源: 2024届空间向量校本作业\n", - "添加题号022069, 来源: 2024届空间向量校本作业\n", - "添加题号022070, 来源: 2024届空间向量校本作业\n", - "添加题号022071, 来源: 2024届空间向量校本作业\n", - "添加题号022072, 来源: 2024届空间向量校本作业\n", - "添加题号022073, 来源: 2024届空间向量校本作业\n", - "添加题号022074, 来源: 2024届空间向量校本作业\n", - "添加题号022075, 来源: 2024届空间向量校本作业\n", - "添加题号022076, 来源: 2024届空间向量校本作业\n", - "添加题号022077, 来源: 2024届空间向量校本作业\n", - "添加题号022078, 来源: 2024届空间向量校本作业\n", - "添加题号022079, 来源: 2024届空间向量校本作业\n", - "添加题号022080, 来源: 2024届空间向量校本作业\n", - "添加题号022081, 来源: 2024届空间向量校本作业\n", - "添加题号022082, 来源: 2024届空间向量校本作业\n", - "添加题号022083, 来源: 2024届空间向量校本作业\n", - "添加题号022084, 来源: 2024届二项式定理校本作业\n", - "添加题号022085, 来源: 2024届二项式定理校本作业\n", - "添加题号022086, 来源: 2024届二项式定理校本作业\n", - "添加题号022087, 来源: 2024届二项式定理校本作业\n", - "添加题号022088, 来源: 2024届二项式定理校本作业\n", - "添加题号022089, 来源: 2024届二项式定理校本作业\n", - "添加题号022090, 来源: 2024届二项式定理校本作业\n", - "添加题号022091, 来源: 2024届二项式定理校本作业\n", - "添加题号022092, 来源: 2024届二项式定理校本作业\n", - "添加题号022093, 来源: 2024届二项式定理校本作业\n", - "添加题号022094, 来源: 2024届二项式定理校本作业\n", - "添加题号022095, 来源: 2024届二项式定理校本作业\n", - "添加题号022096, 来源: 2024届二项式定理校本作业\n", - "添加题号022097, 来源: 2024届二项式定理校本作业\n", - "添加题号022098, 来源: 2024届二项式定理校本作业\n", - "添加题号022099, 来源: 2024届二项式定理校本作业\n", - "添加题号022100, 来源: 2024届二项式定理校本作业\n", - "添加题号022101, 来源: 2024届二项式定理校本作业\n", - "添加题号022102, 来源: 2024届二项式定理校本作业\n", - "添加题号022103, 来源: 2024届二项式定理校本作业\n", - "添加题号022104, 来源: 2024届二项式定理校本作业\n", - "添加题号022105, 来源: 2024届二项式定理校本作业\n" + "添加题号031358, 来源: 自拟题目\n", + "添加题号031359, 来源: 自拟题目\n" ] } ], @@ -175,26 +119,6 @@ " print(\"题号有重复, 请检查.\\n\"*5)" ] }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"\\\\documentclass[10pt,a4paper]{article}\\n\\\\usepackage[UTF8,fontset = windows]{ctex}\\n\\\\setCJKmainfont[BoldFont=黑体,ItalicFont=楷体]{华文中宋}\\n\\\\usepackage{amssymb,amsmath,amsfonts,amsthm,mathrsfs,dsfont,graphicx}\\n\\\\usepackage{ifthen,indentfirst,enumerate,color,titletoc}\\n\\\\usepackage{tikz}\\n\\\\usepackage{multicol}\\n\\\\usepackage{multirow}\\n\\\\usepackage{makecell}\\n\\\\usepackage{longtable}\\n\\\\usepackage{diagbox}\\n\\\\usepackage{picinpar}\\n\\\\usetikzlibrary{arrows,calc,intersections,patterns,decorations.pathreplacing,3d,angles,quotes,positioning,shapes.geometric}\\n\\\\usepackage[bf,small,indentafter,pagestyles]{titlesec}\\n\\\\usepackage[top=1in, bottom=1in,left=0.8in,right=0.8in]{geometry}\\n\\\\renewcommand{\\\\baselinestretch}{1.65}\\n\\\\newtheorem{defi}{定义~}\\n\\\\newtheorem{eg}{例~}\\n\\\\newtheorem{ex}{~}\\n\\\\newtheorem{rem}{注~}\\n\\\\newtheorem{thm}{定理~}\\n\\\\newtheorem{coro}{推论~}\\n\\\\newtheorem{axiom}{公理~}\\n\\\\newtheorem{prop}{性质~}\\n\\\\newcommand{\\\\blank}[1]{\\\\underline{\\\\hbox to #1pt{}}}\\n\\\\newcommand{\\\\bracket}[1]{(\\\\hbox to #1pt{})}\\n\\\\newcommand{\\\\onech}[4]{\\\\par\\\\begin{tabular}{p{.9\\\\linewidth}}\\nA.~#1\\\\\\\\\\nB.~#2\\\\\\\\\\nC.~#3\\\\\\\\\\nD.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\twoch}[4]{\\\\par\\\\begin{tabular}{p{.46\\\\linewidth}p{.46\\\\linewidth}}\\nA.~#1& B.~#2\\\\\\\\\\nC.~#3& D.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\vartwoch}[4]{\\\\par\\\\begin{tabular}{p{.46\\\\linewidth}p{.46\\\\linewidth}}\\n(1)~#1& (2)~#2\\\\\\\\\\n(3)~#3& (4)~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\fourch}[4]{\\\\par\\\\begin{tabular}{p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}}\\nA.~#1 &B.~#2& C.~#3& D.~#4\\n\\\\end{tabular}}\\n\\\\newcommand{\\\\varfourch}[4]{\\\\par\\\\begin{tabular}{p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}p{.23\\\\linewidth}}\\n(1)~#1 &(2)~#2& (3)~#3& (4)~#4\\n\\\\end{tabular}}\\n\\\\begin{document}\\n\\n\\n\\\\begin{enumerate}\\n\\n% 2024届空间向量校本作业\\n\\n\\\\item 如图, 已知平行六面体$ABCD-A_1B_1C_1D_1$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [below] {$D_1$} coordinate (D_1);\\n\\\\draw (2,0,0) node [below] {$A_1$} coordinate (A_1);\\n\\\\draw (0.5,0,-2) node [below] {$C_1$} coordinate (C_1);\\n\\\\draw ($(A_1)+(C_1)-(D_1)$) node [right] {$B_1$} coordinate (B_1);\\n\\\\draw (1,2,0) node [left] {$D$} coordinate (D);\\n\\\\draw ($(D)+(A_1)-(D_1)$) node [above] {$A$} coordinate (A);\\n\\\\draw ($(A)+(C_1)-(D_1)$) node [above] {$B$} coordinate (B);\\n\\\\draw ($(D)+(B)-(A)$) node [above] {$C$} coordinate (C);\\n\\\\draw ($(D_1)!0.5!(A_1)$) node [below] {$M$} coordinate (M);\\n\\\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\\n\\\\draw (D_1)--(A_1)--(B_1)--(B)--(C)--(D)--cycle(A)--(B)(A)--(A_1)(A)--(D);\\n\\\\draw [dashed] (C_1)--(C)(C_1)--(B_1)(C_1)--(D_1)(M)--(N);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 写出以该平行六面体的顶点为起点与终点, 且与$\\\\overrightarrow{AB}$相等的向量;\\\\\\\\\\n(2) 写出以该平行六面体的顶点为起点与终点的$\\\\overrightarrow{AA_1}$的负向量;\\\\\\\\\\n(3) 写出以该平行六面体的顶点为起点与终点, 且与$\\\\overrightarrow{AD}$平行的向量;\\\\\\\\\\n(4) 设$M$、$N$分别是$A_1D_1$和$DC$的中点, 用$\\\\overrightarrow{AB}$、$\\\\overrightarrow{AA_1}$、$\\\\overrightarrow{AD}$表示向量$\\\\overrightarrow{MN}$.\\n\\\\item 对于平行六面体$ABCD-A_1B_1C_1D_1$, 求证: $\\\\overrightarrow{AB_1}+\\\\overrightarrow{AC}+\\\\overrightarrow{AD_1}=2 \\\\overrightarrow{AC_1}$.\\n\\\\item 在三棱锥$O-ABC$中, $G$是三角形$ABC$的重心, 用向量$\\\\overrightarrow{OA}$、$\\\\overrightarrow{OB}$、$\\\\overrightarrow{OC}$表示向量$\\\\overrightarrow{OG}$.\\n\\\\item 已知向量$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$、$\\\\overrightarrow {c}$两两垂直, 且$|\\\\overrightarrow {a}|=1$, $|\\\\overrightarrow {b}|=2$, $|\\\\overrightarrow {c}|=3$, $\\\\overrightarrow {m}=\\\\overrightarrow {a}+\\\\overrightarrow {b}+\\\\overrightarrow {c}$.\\\\\\\\\\n(1) 求$|\\\\overrightarrow {m}|$;\\\\\\\\\\n(2) 分别求$\\\\overrightarrow {m}$与$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$、$\\\\overrightarrow {c}$的夹角.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$分别是$A_1B_1$、$CD$的中点, $R$、$S$分别是棱$AA_1$、棱$CC_1$上的点, 且$AR=2RA_1$, $C_1S=2SC$, 求证: $PS\\\\parallel RQ$.\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$的边长为$1$. 求:\\\\\\\\\\n(1) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{AA_1}$;\\\\\\\\\\n(2) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{A_1C_1}$;\\\\\\\\\\n(3) $\\\\overrightarrow{AC} \\\\cdot \\\\overrightarrow{AC_1}$.\\n\\\\item 在长方体$ABCD-A' B' C' D'$中, $A' C'$和$B' D'$相交于$O'$, 求证$DO'\\\\parallel$平面$ACB'$.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, $G$是三角形$ACD_1$的重心. 求证: $3 \\\\overrightarrow{DG}=\\\\overrightarrow{DB_1}$.\\n\\\\item 在长方体$ABCD-A_1B_1C_1D_1$中, 已知$AB=6$, $AD=2$, $AA_1=1$, $P$是棱$AB$上的点且$PB=2AP$, $M$是棱$DC$上的点, 且$DM=2MC$, $N$是$B_1C_1$的中点, 求直线$PD_1$与$MN$所成的角$\\\\theta$的大小.\\n\\\\item 已知棱长为$1$的正四面体$A-BCD$中, $E$、$F$分别在$AB$、$CD$上, 且$\\\\overrightarrow{AE}=\\\\dfrac{1}{4} \\\\overrightarrow{AB}$, \\n$\\\\overrightarrow{CF}=\\\\dfrac{1}{3} \\\\overrightarrow{CD}$.\\\\\\\\\\n(1) 求直线$DE$和$BF$所成的角的大小;\\\\\\\\\\n(2) 求$|\\\\overrightarrow{EF}|$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的高为$h$, 上、下底面是边长为$a$的正方形, 坐标原点$O$设在下底面的中心, $x$轴、$y$轴分别与下底面的对角线重合, $z$轴垂直于底面(如图). 写出下列点的坐标以及向量的坐标:\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\def\\\\l{2}\\n\\\\def\\\\m{2}\\n\\\\def\\\\n{2.5}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} coordinate (B);\\n\\\\draw (A) ++ (\\\\l,0,-\\\\m) node [right] {$C$} coordinate (C);\\n\\\\draw (A) ++ (0,0,-\\\\m) node [left] {$D$} coordinate (D);\\n\\\\draw (A) -- (B) -- (C);\\n\\\\draw [dashed] (A) -- (D) -- (C);\\n\\\\draw (A) ++ (0,\\\\n,0) node [left] {$A_1$} coordinate (A1);\\n\\\\draw (B) ++ (0,\\\\n,0) node [right] {$B_1$} coordinate (B1);\\n\\\\draw (C) ++ (0,\\\\n,0) node [above right] {$C_1$} coordinate (C1);\\n\\\\draw (D) ++ (0,\\\\n,0) node [above left] {$D_1$} coordinate (D1);\\n\\\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\\n\\\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\\n\\\\draw [dashed] (D) -- (D1);\\n\\\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $A$的坐标: \\\\blank{50}; (2) $D$的坐标: \\\\blank{50}; (3) $B_1$的坐标: \\\\blank{50};\\\\\\\\\\n(4) $\\\\overrightarrow{OA}$的坐标: \\\\blank{50}; (5) $\\\\overrightarrow{D_1A_1}$的坐标: \\\\blank{50}; (6) $\\\\overrightarrow{B_1D}$的坐标: \\\\blank{50}.\\n\\\\item 已知$\\\\overrightarrow {a}=(1,-5,4)$, $\\\\overrightarrow {b}=(2,1,7)$.\\\\\\\\\\n(1) 求$3 \\\\overrightarrow {a}+2 \\\\overrightarrow {b}$的坐标;\\\\\\\\\\n(2) 求$|\\\\overrightarrow {a}+\\\\overrightarrow {b}|$.\\n\\\\item 已知$\\\\overrightarrow {a}=(2,1,-2)$, $\\\\overrightarrow {b}=(5,-4,3)$, $\\\\overrightarrow {c}=(-8,4,1)$.\\\\\\\\\\n(1) 求证: $\\\\overrightarrow {a} \\\\perp \\\\overrightarrow {b}$;;\\n(2) 设$\\\\overrightarrow {a}$与$\\\\overrightarrow {c}$的夹角为$\\\\theta$, 求$\\\\cos \\\\theta$.\\n\\\\item 已知$P_1(2,5,4)$, $P_2(6,4,7)$, 设$\\\\overrightarrow {a}=\\\\overrightarrow{P_1P_2}$, 求$\\\\overrightarrow {a}$、$-\\\\overrightarrow {a}$和单位向量$\\\\overrightarrow{a_0}$的坐标.\\n\\\\item 已知$P_1(2,5,-6)$, 在$y$轴上求一点$P_2$, 使$|P_1P_2|=7$.\\n\\\\item 已知$P_1(1,2,3), P_2(5,4,7)$, 在$y$轴上求一点$Q$, 使$|P_1Q|=|P_2Q|$.\\n\\\\item 已知向量$\\\\overrightarrow {a}=(1,-3,2)$, $\\\\overrightarrow {b}=(2,0,-8)$, 求单位向量$\\\\overrightarrow {c}$, 使$\\\\overrightarrow {c}$与向量$\\\\overrightarrow {a}$、$\\\\overrightarrow {b}$都垂直. \\n\\\\item 已知平面$\\\\alpha$经过点$A(3,1,-1)$、$B(1,-1,0)$, 且平行于向量$\\\\overrightarrow {a}=(-1,0,2)$, 求平面$\\\\alpha$的一个法向量.\\n\\\\item 已知点$A$、$B$、$C$的坐标分别为$(x_1, y_1, z_1)$、$(x_2, y_2, z_2)$、$(x_3, y_3, z_3)$, $G$是$\\\\triangle ABC$的重心, 求点$G$的坐标.\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$, 求证: $BD_1 \\\\perp C_1D$.\\n\\\\item 正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\\\dfrac{\\\\sqrt{6}}{2}$.\\\\\\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\\\circ}$角?\\n\\\\item 如图, 平面$ABEF \\\\perp$平面$ABCD$, 四边形$ABEF$与$ABCD$都是直角梯形, \\n$\\\\angle BAD=\\\\angle FAB=90^{\\\\circ}$, $BC =\\\\dfrac 12 AD$且$BC\\\\parallel AD$, $BE = \\\\dfrac{1}{2} AF$且$BE\\\\parallel AF$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [above right] {$A$} coordinate (A);\\n\\\\draw (3,0,0) node [right] {$D$} coordinate (D);\\n\\\\draw (0,3,0) node [above] {$F$} coordinate (F);\\n\\\\draw (0,0,1.5) node [below] {$B$} coordinate (B);\\n\\\\draw (B) ++ (1.5,0,0) node [below] {$C$} coordinate (C);\\n\\\\draw (B) ++ (0,1.5,0) node [left] {$E$} coordinate (E);\\n\\\\draw (B)--(C)--(D)--(F)--(E)--cycle(E)--(C);\\n\\\\draw [dashed] (A)--(D)(A)--(F)(A)--(B);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 证明: $C, D, F, E$四点共面;\\\\\\\\\\n(2) 设$AB=BC=BE$, 求二面角$A-ED-B$的大小; \\n\\\\item 如图, 已知四棱锥$P-ABCD$的底面$ABCD$为等腰梯形, $AB\\\\parallel DC$, $AC \\\\perp BD$, $AC$与$BD$相交于点$O$, 且顶点$P$在底面上的射影恰为$O$点, 又$BO=2$, $PO=\\\\sqrt{2}$, $PB \\\\perp PD$.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex]\\n\\\\draw (0,0,0) node [left] {$O$} coordinate (O);\\n\\\\draw (-2,0,2) node [left] {$A$} coordinate (A);\\n\\\\draw (2,0,2) node [right] {$B$} coordinate (B);\\n\\\\draw ($(O)!-0.5!(B)$) node [left] {$D$} coordinate (D);\\n\\\\draw ($(O)!-0.5!(A)$) node [right] {$C$} coordinate (C);\\n\\\\draw (O) ++ (0,2,0) node [above] {$P$} coordinate (P);\\n\\\\draw (A)--(B)--(C)--(P)--cycle(B)--(P);\\n\\\\draw [dashed] (P)--(O)(P)--(D)--(B)(A)--(C)(A)--(D)--(C);\\n\\\\draw (O) pic [draw, scale = 0.3] {right angle = B--O--A};\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 求异面直线$PD$与$BC$所成角的余弦值;\\\\\\\\\\n(2) 求二面角$P-AB-C$的大小;\\\\\\\\\\n(3) 设点$M$在棱$PC$上, 且$\\\\dfrac{PM}{MC}=\\\\lambda$, 问$\\\\lambda$为何值时, $PC \\\\perp$平面$BMD$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列直线的一个方向向量.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} coordinate (B);\\n\\\\draw (A) ++ (\\\\l,0,-\\\\m) node [right] {$C$} coordinate (C);\\n\\\\draw (A) ++ (0,0,-\\\\m) node [left] {$D$} coordinate (D);\\n\\\\draw (A) -- (B) -- (C);\\n\\\\draw [dashed] (A) -- (D) -- (C);\\n\\\\draw (A) ++ (0,\\\\n,0) node [left] {$A_1$} coordinate (A1);\\n\\\\draw (B) ++ (0,\\\\n,0) node [right] {$B_1$} coordinate (B1);\\n\\\\draw (C) ++ (0,\\\\n,0) node [above right] {$C_1$} coordinate (C1);\\n\\\\draw (D) ++ (0,\\\\n,0) node [above left] {$D_1$} coordinate (D1);\\n\\\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\\n\\\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\\n\\\\draw [dashed] (D) -- (D1);\\n\\\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $AD_1$;\\\\\\\\\\n(2) $AA_1$;\\\\\\\\\\n(3) $AC_1$;\\\\\\\\\\n(4) $AB_1$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列向量是图中哪些经过两个顶点的直线的一个方向向量?\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} coordinate (B);\\n\\\\draw (A) ++ (\\\\l,0,-\\\\m) node [right] {$C$} coordinate (C);\\n\\\\draw (A) ++ (0,0,-\\\\m) node [left] {$D$} coordinate (D);\\n\\\\draw (A) -- (B) -- (C);\\n\\\\draw [dashed] (A) -- (D) -- (C);\\n\\\\draw (A) ++ (0,\\\\n,0) node [left] {$A_1$} coordinate (A1);\\n\\\\draw (B) ++ (0,\\\\n,0) node [right] {$B_1$} coordinate (B1);\\n\\\\draw (C) ++ (0,\\\\n,0) node [above right] {$C_1$} coordinate (C1);\\n\\\\draw (D) ++ (0,\\\\n,0) node [above left] {$D_1$} coordinate (D1);\\n\\\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\\n\\\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\\n\\\\draw [dashed] (D) -- (D1);\\n\\\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $\\\\overrightarrow {a}=(1,0,0)$;\\\\\\\\\\n(2) $\\\\overrightarrow {b}=(0,1,0)$;\\\\\\\\\\n(3) $\\\\overrightarrow {c}=(3 \\\\sqrt{2}, 0,4)$;\\\\\\\\\\n(4) $\\\\overrightarrow {d}=(0,3 \\\\sqrt{2}, 8)$.\\n\\\\item 已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列平面的一个法向量.\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 0.6]\\n\\\\def\\\\l{3}\\n\\\\def\\\\m{3}\\n\\\\def\\\\n{4}\\n\\\\draw (0,0,0) node [below left] {$A$} coordinate (A);\\n\\\\draw (A) ++ (\\\\l,0,0) node [below ] {$B$} coordinate (B);\\n\\\\draw (A) ++ (\\\\l,0,-\\\\m) node [right] {$C$} coordinate (C);\\n\\\\draw (A) ++ (0,0,-\\\\m) node [left] {$D$} coordinate (D);\\n\\\\draw (A) -- (B) -- (C);\\n\\\\draw [dashed] (A) -- (D) -- (C);\\n\\\\draw (A) ++ (0,\\\\n,0) node [left] {$A_1$} coordinate (A1);\\n\\\\draw (B) ++ (0,\\\\n,0) node [right] {$B_1$} coordinate (B1);\\n\\\\draw (C) ++ (0,\\\\n,0) node [above right] {$C_1$} coordinate (C1);\\n\\\\draw (D) ++ (0,\\\\n,0) node [above left] {$D_1$} coordinate (D1);\\n\\\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\\n\\\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\\n\\\\draw [dashed] (D) -- (D1);\\n\\\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\\n\\\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\\n\\\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\\n\\\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) 平面$AA_1D_1D$;\\\\\\\\\\n(2) 平面$BB_1D_1D$.\\n\\\\item 已知点$A(0,-7,0)$、$B(2,-1,1)$、$C(2,2,2)$, 求平面$ABC$的一个法向量.\\n\\\\item 已知点$S(0,6,4)$、$A(3,5,3)$、$B(-2,11,-5)$、$C(1,-1,4)$, 求点$S$到平面$ABC$的距离.\\n\\\\item 已知平面$\\\\alpha$的一个法向量$\\\\overrightarrow {n}=(3,-2,6)$, 且经过点$A(0,7,0)$, 求原点到平面$\\\\alpha$的距离.\\n\\\\item 已知三棱锥$A-BCD$的三条侧棱$AB$、$AC$、$AD$两两垂直, 且$AB=1$, $AC=2$, $AD=3$, 求顶点$A$到平面$BCD$的距离.\\n\\\\item 正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\\\dfrac{\\\\sqrt{6}}{2}$.\\\\\\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\\\circ}$角?\\n\\\\item 已知正方体$ABCD-A_1B_1C_1D_1$, 求二面角$B-AC-D_1$的大小.\\n\\\\item 已知$ABCD-A_1B_1C_1D_1$为正方体.\\\\\\\\\\n(1) 求直线$AC$与$B_1D$所成的角的大小;\\\\\\\\\\n(2) 求直线$B_1D$与平面$ACD_1$所成的角的大小;\\\\\\\\\\n(3) 求平面$ACD_1$与平面$B_1CD_1$所成的二面角的大小.\\n\\\\item 已知正三棱锥的底面边长和高都为$a$. 求侧面与底面所成的二面角的大小.\\n\\\\item 在三棱锥$P-ABC$中, 已知底面$ABC$是以$C$为直角的直角三角形, $PC \\\\perp$平面$ABC$, $AC=18$, $PC=6$, $BC=9$, $G$是$\\\\triangle PAB$的重心, $M$是棱$AC$的中点, 求直线$CG$与直线$BM$所成的角$\\\\theta$的大小.\\n\\\\item 已知矩形$ABCD$, 且$PD \\\\perp$平面$ABCD$, 若$PB=2$, $PB$与平面$PCD$所成的角为$45^{\\\\circ}$. $PB$与平面$ABD$所成的角为$30^{\\\\circ}$, 求:\\n\\\\begin{center}\\n\\\\begin{tikzpicture}[>=latex, scale = 1.3]\\n\\\\draw (0,0,0) node [below] {$D$} coordinate (D);\\n\\\\draw (2,0,0) node [right] {$C$} coordinate (C);\\n\\\\draw (2,0,2) node [below] {$B$} coordinate (B);\\n\\\\draw (0,0,2) node [left] {$A$} coordinate (A);\\n\\\\draw (0,2,0) node [above] {$P$} coordinate (P);\\n\\\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\\n\\\\draw [dashed] (A)--(D)--(C)(B)--(D)--(P);\\n\\\\end{tikzpicture}\\n\\\\end{center}\\n(1) $CD$的长;\\\\\\\\\\n(2) 求$PB$与$CD$所成的角;\\\\\\\\\\n(3) 求二面角$C-PB-D$的余弦值.\\n\\n% 2024届二项式定理校本作业\\n\\n\\\\item 用二项式定理展开下列两式:\\\\\\\\\\n(1) $(a+2 b)^6$;\\\\\\\\\\n(2) $(1-\\\\dfrac{1}{x})^5$.\\n\\\\item 化简:\\\\\\\\ \\n(1) $(1+\\\\sqrt{x})^5+(1-\\\\sqrt{x})^5$;\\\\\\\\\\n(2) $(2 x+y)^4-(2 x-y)^4$.\\n\\\\item 分别写出$(x-1)^{15}$的二项展开式中的前$4$项.\\n\\\\item 求$(2 a^3-3 b^2)^{10}$的二项展开式中的第$8$项.\\n\\\\item $(x-1)^n$的二项展开式中第$m$项($1 \\\\leq m \\\\leq n$且$m$、$n \\\\in \\\\mathbf{N}$)的二项式的系数是\\\\bracket{20}.\\n\\\\fourch{$\\\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1} \\\\mathrm{C}_n^m$}{$\\\\mathrm{C}_n^m$}{$(-1)^m \\\\mathrm{C}_n^m$}\\n\\\\item 求$(3 x^3-\\\\dfrac{1}{3 x^3})^{10}$的二项展开式中的常数项.\\n\\\\item 已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的系数相等, 求这两项的系数.\\n\\\\item 在$(\\\\sqrt[3]{x}-\\\\dfrac{2}{\\\\sqrt{x}})^{11}$的二项展开式中,\\\\\\\\\\n(1) 求含$x^2$项的二项式系数;\\\\\\\\\\n(2) 含$x^{\\\\frac{1}{3}}$的项是第几项? 并写出这项的系数.\\n\\\\item 已知$(x \\\\sin \\\\theta+1)^6$的二项展开式$x^2$项的系数与$(x-\\\\dfrac{15}{2} \\\\cos \\\\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\\\cos \\\\theta$的值.\\n\\\\item 求证: 当$n$为正整数时, $2^n-\\\\mathrm{C}_n^1 \\\\cdot 2^{n-1}+\\\\mathrm{C}_n^2 \\\\cdot 2^{n-2}+\\\\cdots+\\\\mathrm{C}_n^{n-1} \\\\cdot 2+(-1)^n=1$.\\n\\\\item 求$(1+2 x)^3(1-x)^4$展开式中$x^6$的系数.\\n\\\\item 在$(3 x-2 y)^9$的展开式中, 二项式系数的和是\\\\blank{50}, 各项系数的和是各项系数的绝对值之和是\\\\blank{50}.\\n\\\\item $\\\\mathrm{C}_n^1+3\\\\mathrm{C}_n^2+9\\\\mathrm{C}_n^3+\\\\cdots+3^{n-1} \\\\mathrm{C}_n^n$等于\\\\bracket{20}.\\n\\\\fourch{$4^n$}{$\\\\dfrac{4^n}{3}$}{$\\\\dfrac{4^n}{3}-1$}{$\\\\dfrac{4^n-1}{3}$}\\n\\\\item 求$(\\\\dfrac{\\\\sqrt{x}}{2}-\\\\dfrac{2}{\\\\sqrt{x}})^{10}$的二项展开式的正中间一项.\\n\\\\item 求$(x \\\\sqrt{y}-y \\\\sqrt{x})^{11}$的二项展开式的正中间两项.\\n\\\\item 用二项式定理证明: $99^{10}-1$能被$1000$整除.\\n\\\\item 求$77^{77}-15$除以$19$的余数.\\n\\\\item 求$(1+2 x+x^2)^{10}(1-x)^6$的展开式中各项系数之和.\\n\\\\item 在$(x^2-\\\\dfrac{3}{x})^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求:\\n$\\\\mathrm{C}_n^0-\\\\dfrac{1}{2} \\\\mathrm{C}_n^1+\\\\dfrac{1}{4} \\\\mathrm{C}_n^2-\\\\cdots+(-1)^n \\\\cdot \\\\dfrac{1}{2^n} \\\\mathrm{C}_n^n$.\\n\\\\item 在$(1+3 x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\\\\\\n(1) 求二项展开式中二项式系数最大的项;\\\\\\\\\\n(2) 求二项展开式中系数最大的项.\\n\\\\item 已知$(x+1)^n=x^n+\\\\cdots+a x^3+b x^2+c x+1$($n \\\\geq 1$, $n \\\\in \\\\mathrm{N}$), 且$a: b=3: 1$, 求$c$的值.\\n\\\\item 已知$n$为大于$1$的自然数, 用二项式定理证明: $(1+\\\\dfrac{1}{n})^n>2$.\\n\\n\\n\\\\end{enumerate}\\n\\n\\n\\n\\n\\n\\\\end{document}\"" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "problems_string" - ] - }, { "cell_type": "code", "execution_count": null, 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 ec0c0ea1..a5df3238 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 @@ -9,50 +9,66 @@ "name": "stdout", "output_type": "stream", "text": [ - "040202 填空题\n", - "040203 填空题\n", - "040204 填空题\n", - "040205 填空题\n", - "040206 填空题\n", - "040207 填空题\n", - "040208 填空题\n", - "040209 填空题\n", - "040210 填空题\n", - "040211 填空题\n", - "040212 填空题\n", - "040213 填空题\n", - "040214 填空题\n", - "040215 填空题\n", - "040216 填空题\n", - "040217 填空题\n", - "040218 填空题\n", - "040219 填空题\n", - "040220 填空题\n", - "040221 解答题\n", - "040222 选择题\n", - "040223 选择题\n", - "040224 解答题\n", - "040225 解答题\n", - "040226 填空题\n", - "040227 填空题\n", - "040228 填空题\n", - "040229 填空题\n", - "040230 填空题\n", - "040231 填空题\n", - "040232 填空题\n", - "040233 填空题\n", - "040234 填空题\n", - "040235 填空题\n", - "040236 填空题\n", - "040237 填空题\n", - "040238 填空题\n", - "040239 填空题\n", - "040240 填空题\n", - "040241 填空题\n", - "040242 解答题\n", - "040243 解答题\n", - "040244 解答题\n", - "040245 解答题\n" + "022048 解答题\n", + "022049 解答题\n", + "022050 解答题\n", + "022051 解答题\n", + "022052 解答题\n", + "022053 解答题\n", + "022054 解答题\n", + "022055 解答题\n", + "022056 解答题\n", + "022057 解答题\n", + "022058 填空题\n", + "022059 解答题\n", + "022060 解答题\n", + "022061 解答题\n", + "022062 解答题\n", + "022063 解答题\n", + "022064 解答题\n", + "022065 解答题\n", + "022066 解答题\n", + "022067 解答题\n", + "022068 解答题\n", + "022069 解答题\n", + "022070 解答题\n", + "022071 解答题\n", + "022072 解答题\n", + "022073 解答题\n", + "022074 解答题\n", + "022075 解答题\n", + "022076 解答题\n", + "022077 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-16330,7 +16330,7 @@ }, "000556": { "id": "000556", - "content": "设全集$U=\\mathbf{Z}$, 集合$M=\\{1,2\\}$, $P=\\{-2,-1,0,1,2\\}$, 则$P\\cap \\complement_U M$=\\blank{50}.", + "content": "设全集$U=\\mathbf{Z}$, 集合$M=\\{1,2\\}$, $P=\\{-2,-1,0,1,2\\}$, 则$P\\cap \\overline{M}$=\\blank{50}.", "objs": [ "K0104001B", "K0104006B" @@ -16543,7 +16543,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "031357" + ], "remark": "", "space": "" }, @@ -16689,7 +16691,7 @@ }, "000569": { "id": "000569", - "content": "若$\\begin{vmatrix} 4^x & 2 \\\\ 2^x & 1 \\end{vmatrix}=0$, 则$x=$\\blank{50}.", + "content": "若$4^x-2\\cdot 2^x=0$, 则$x=$\\blank{50}.", "objs": [], "tags": [ "暂无对应" @@ -16789,7 +16791,7 @@ }, "000573": { "id": "000573", - "content": "若存在公差为$d$, 各项均为实数的等差数列$\\{a_n\\} \\ (n\\in \\mathbf{N}^*)$满足$a_3a_4+1=0$, 则公差$d$的取值范围是\\blank{50}.", + "content": "若存在公差为$d$, 各项均为实数的等差数列$\\{a_n\\}$($n\\in \\mathbf{N}$, $n\\ge 1$)满足$a_3a_4+1=0$, 则公差$d$的取值范围是\\blank{50}.", "objs": [ "K0401004X", "K0401003X" @@ -16827,7 +16829,7 @@ }, "000574": { "id": "000574", - "content": "著名的斐波那契数列$\\{a_n\\}:1,1,2,3,5,8,\\cdots$, 满足$a_1=a_2=1,a_{n+2}=a_{n+1}+a_n \\ (n\\in \\mathbf{N}^*)$, 那么$1+a_3+a_5+a_7+a_9+\\cdots+a_{2017}$是斐波那契数列中的第\\blank{50}项.", + "content": "著名的斐波那契数列$\\{a_n\\}:1,1,2,3,5,8,\\cdots$, 满足$a_1=a_2=1$, $a_{n+2}=a_{n+1}+a_n$($n\\in \\mathbf{N}$, $n\\ge 1$), 那么$1+a_3+a_5+a_7+a_9+\\cdots+a_{2017}$是斐波那契数列中的第\\blank{50}项.", "objs": [ "K0407002X" ], @@ -16887,7 +16889,7 @@ "第一单元" ], "genre": "填空题", - "ans": "$-\\frac 35$", + "ans": "$3$", "solution": "", "duration": -1, "usages": [ @@ -16912,7 +16914,7 @@ "第三单元" ], "genre": "填空题", - "ans": "$2$", + "ans": "$-\\frac 35$", "solution": "", "duration": -1, "usages": [ @@ -16966,7 +16968,7 @@ "暂无对应" ], "genre": "填空题", - "ans": "$-160$", + "ans": "$6$", "solution": "", "duration": -1, "usages": [ @@ -16998,7 +17000,7 @@ "二项式定理" ], "genre": "填空题", - "ans": "$\\dfrac 1{12}$", + "ans": "$-160$", "solution": "", "duration": -1, "usages": [ @@ -17033,7 +17035,7 @@ "概率" ], "genre": "填空题", - "ans": "$1$", + "ans": "$\\dfrac 1{12}$", "solution": "", "duration": -1, "usages": [ @@ -17050,7 +17052,7 @@ }, "000582": { "id": "000582", - "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$, 若点$(n,S_n) \\ (n\\in \\mathbf{N}^*)$在函数$y=\\log_2 (x+1)$的反函数的图像上, 则$a_n$=\\blank{50}.", + "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$, 若点$(n,S_n)$($n\\in \\mathbf{N}$, $n\\ge 1$)在函数$y=\\log_2 (x+1)$的反函数的图像上, 则$a_n$=\\blank{50}.", "objs": [ "K0226004B", "K0402005X" @@ -17299,7 +17301,7 @@ }, "000591": { "id": "000591", - "content": "若数列$\\{a_n\\}$为等比数列, 且$a_5=3$, 则$\\begin{vmatrix} a_2 & -a_7 \\\\ a_3 & a_8 \\end{vmatrix}=$\\blank{50}.", + "content": "若数列$\\{a_n\\}$为等比数列, 且$a_5=3$, 则$a_2a_8+a_3a_7=$\\blank{50}.", "objs": [ "KNONE" ], @@ -17347,7 +17349,7 @@ }, "000593": { "id": "000593", - "content": "若$(2x+\\dfrac 1x)^n$的二项展开式中的所有二项式系数之和等于$256$, 则该展开式中常数项的值为\\blank{50}.", + "content": "若$(x+\\dfrac 1x)^n$的二项展开式中的所有系数之和等于$6561$, 则该展开式中常数项的值为\\blank{50}.", "objs": [ "KNONE" ], @@ -17398,7 +17400,7 @@ }, "000595": { "id": "000595", - "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $2S_n=a_na_{n+1}$($n\\in \\mathbf{N}^*$), 若$b_n=(-1)^n\\dfrac{2n+1}{{a_n}{a_{n+1}}}$, 则数列$\\{b_n\\}$的前$n$项和$T_n=$\\blank{50}.", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $2S_n=a_na_{n+1}$($n\\in \\mathbf{N}$, $n\\ge 1$), 若$b_n=(-1)^n\\dfrac{2n+1}{{a_n}{a_{n+1}}}$, 则数列$\\{b_n\\}$的前$n$项和$T_n=$\\blank{50}.", "objs": [ "K0409001X", "K0407002X" @@ -17424,7 +17426,7 @@ }, "000596": { "id": "000596", - "content": "设全集$U=\\{1,2,3,4\\}$, 集合$A=\\{x|x^2-5x+4<0,x\\in \\mathbf{Z}\\}$, 则$\\complement_U A$=\\blank{50}.", + "content": "设全集$U=\\{1,2,3,4\\}$, 集合$A=\\{x|x^2-5x+4<0,x\\in \\mathbf{Z}\\}$, 则$\\overline{A}$=\\blank{50}.", "objs": [ "K0114001B", "K0104006B" @@ -17499,7 +17501,7 @@ }, "000599": { "id": "000599", - "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_n=1-\\dfrac23{a_n} \\ (n\\in \\mathbf{N}^*)$, 则$\\displaystyle\\lim_{n\\to\\infty}S_n$=\\blank{50}.", + "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_n=1-\\dfrac23{a_n}$($n\\in \\mathbf{N}$, $n \\ge 1$), 则$\\displaystyle\\lim_{n\\to\\infty}S_n$=\\blank{50}.", "objs": [ "KNONE" ], @@ -17524,7 +17526,7 @@ }, "000600": { "id": "000600", - "content": "若$(x+\\dfrac1{2x})^n \\ (n\\ge 4, \\ n\\in \\mathbf{N}^*)$的二项展开式中前三项的系数依次成等差数列, 则$n=$\\blank{50}.", + "content": "若$(x+\\dfrac1{2x})^n$($n\\ge 4$, $n\\in \\mathbf{N}$)的二项展开式中前三项的系数依次成等差数列, 则$n=$\\blank{50}.", "objs": [ "K0819005X", "K0401002X" @@ -17623,7 +17625,8 @@ "000952" ], "related": [ - "000952" + "000952", + "031360" ], "remark": "", "space": "" @@ -18049,7 +18052,7 @@ }, "000620": { "id": "000620", - "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots+bx+c \\ (n\\in \\mathbf{N}^*, \\ n\\ge 3)$, 且$b=4c$, 则$a$的值为\\blank{50}.", + "content": "若$(x+2)^n=x^n+ax^{n-1}+\\cdots+bx+c$($n\\in \\mathbf{N}$, $n\\ge 3$), 且$b=4c$, 则$a$的值为\\blank{50}.", "objs": [ "K0819005X" ], @@ -18256,7 +18259,7 @@ }, "000627": { "id": "000627", - "content": "若全集$U=\\mathbf{R}$, 集合$A=\\{x|x\\ge 1\\}\\cup\\{x|x<0\\}$, 则$\\complement_U A=$\\blank{50}.", + "content": "若全集$U=\\mathbf{R}$, 集合$A=\\{x|x\\ge 1\\}\\cup\\{x|x<0\\}$, 则$\\overline{A}=$\\blank{50}.", "objs": [ "K0104003B", "K0104006B" @@ -18390,7 +18393,7 @@ }, "000632": { "id": "000632", - "content": "若$(\\sqrt x-\\dfrac1x)^n$的二项展开式中各项的二项式系数的和是$64$, 则展开式中的常数项的值为\\blank{50}.", + "content": "若$(\\sqrt x+\\dfrac1x)^n$的二项展开式中各项系数的和是$64$, 则展开式中的常数项的值为\\blank{50}.", "objs": [ "KNONE" ], @@ -18416,7 +18419,7 @@ }, "000633": { "id": "000633", - "content": "数列$\\{a_n\\}$是等比数列, 前n项和为$S_n$, 若$a_1+a_2=2$, $a_2+a_3=-1$, 则$\\displaystyle\\lim_{n\\to\\infty}{S_n}=$\\blank{50}.", + "content": "数列$\\{a_n\\}$是等比数列, 前$n$项和为$S_n$, 若$a_1+a_2=2$, $a_2+a_3=-1$, 则$\\displaystyle\\lim_{n\\to\\infty}{S_n}=$\\blank{50}.", "objs": [ "K0405003X" ], @@ -409056,7 +409059,7 @@ "content": "如图, 已知平行六面体$ABCD-A_1B_1C_1D_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D_1$} coordinate (D_1);\n\\draw (2,0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (0.5,0,-2) node [below] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)+(C_1)-(D_1)$) node [right] {$B_1$} coordinate (B_1);\n\\draw (1,2,0) node [left] {$D$} coordinate (D);\n\\draw ($(D)+(A_1)-(D_1)$) node [above] {$A$} coordinate (A);\n\\draw ($(A)+(C_1)-(D_1)$) node [above] {$B$} coordinate (B);\n\\draw ($(D)+(B)-(A)$) node [above] {$C$} coordinate (C);\n\\draw ($(D_1)!0.5!(A_1)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(D)$) node [above left] {$N$} coordinate (N);\n\\draw (D_1)--(A_1)--(B_1)--(B)--(C)--(D)--cycle(A)--(B)(A)--(A_1)(A)--(D);\n\\draw [dashed] (C_1)--(C)(C_1)--(B_1)(C_1)--(D_1)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 写出以该平行六面体的顶点为起点与终点, 且与$\\overrightarrow{AB}$相等的向量;\\\\\n(2) 写出以该平行六面体的顶点为起点与终点的$\\overrightarrow{AA_1}$的负向量;\\\\\n(3) 写出以该平行六面体的顶点为起点与终点, 且与$\\overrightarrow{AD}$平行的向量;\\\\\n(4) 设$M$、$N$分别是$A_1D_1$和$DC$的中点, 用$\\overrightarrow{AB}$、$\\overrightarrow{AA_1}$、$\\overrightarrow{AD}$表示向量$\\overrightarrow{MN}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409068,14 +409071,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022049": { "id": "022049", "content": "对于平行六面体$ABCD-A_1B_1C_1D_1$, 求证: $\\overrightarrow{AB_1}+\\overrightarrow{AC}+\\overrightarrow{AD_1}=2 \\overrightarrow{AC_1}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409087,14 +409090,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022050": { "id": "022050", "content": "在三棱锥$O-ABC$中, $G$是三角形$ABC$的重心, 用向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$表示向量$\\overrightarrow{OG}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409106,14 +409109,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022051": { "id": "022051", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$两两垂直, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, $|\\overrightarrow {c}|=3$, $\\overrightarrow {m}=\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}$.\\\\\n(1) 求$|\\overrightarrow {m}|$;\\\\\n(2) 分别求$\\overrightarrow {m}$与$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的夹角.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409125,14 +409128,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022052": { "id": "022052", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$分别是$A_1B_1$、$CD$的中点, $R$、$S$分别是棱$AA_1$、棱$CC_1$上的点, 且$AR=2RA_1$, $C_1S=2SC$, 求证: $PS\\parallel RQ$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409144,14 +409147,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022053": { "id": "022053", "content": "已知正方体$ABCD-A_1B_1C_1D_1$的边长为$1$. 求:\\\\\n(1) $\\overrightarrow{AC} \\cdot \\overrightarrow{AA_1}$;\\\\\n(2) $\\overrightarrow{AC} \\cdot \\overrightarrow{A_1C_1}$;\\\\\n(3) $\\overrightarrow{AC} \\cdot \\overrightarrow{AC_1}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409163,14 +409166,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022054": { "id": "022054", "content": "在长方体$ABCD-A' B' C' D'$中, $A' C'$和$B' D'$相交于$O'$, 求证$DO'\\parallel$平面$ACB'$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409182,14 +409185,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022055": { "id": "022055", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $G$是三角形$ACD_1$的重心. 求证: $3 \\overrightarrow{DG}=\\overrightarrow{DB_1}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409201,14 +409204,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022056": { "id": "022056", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 已知$AB=6$, $AD=2$, $AA_1=1$, $P$是棱$AB$上的点且$PB=2AP$, $M$是棱$DC$上的点, 且$DM=2MC$, $N$是$B_1C_1$的中点, 求直线$PD_1$与$MN$所成的角$\\theta$的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409220,14 +409223,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022057": { "id": "022057", "content": "已知棱长为$1$的正四面体$A-BCD$中, $E$、$F$分别在$AB$、$CD$上, 且$\\overrightarrow{AE}=\\dfrac{1}{4} \\overrightarrow{AB}$, \n$\\overrightarrow{CF}=\\dfrac{1}{3} \\overrightarrow{CD}$.\\\\\n(1) 求直线$DE$和$BF$所成的角的大小;\\\\\n(2) 求$|\\overrightarrow{EF}|$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409239,14 +409242,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022058": { "id": "022058", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的高为$h$, 上、下底面是边长为$a$的正方形, 坐标原点$O$设在下底面的中心, $x$轴、$y$轴分别与下底面的对角线重合, $z$轴垂直于底面(如图). 写出下列点的坐标以及向量的坐标:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $A$的坐标: \\blank{50}; (2) $D$的坐标: \\blank{50}; (3) $B_1$的坐标: \\blank{50};\\\\\n(4) $\\overrightarrow{OA}$的坐标: \\blank{50}; (5) $\\overrightarrow{D_1A_1}$的坐标: \\blank{50}; (6) $\\overrightarrow{B_1D}$的坐标: \\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -409265,7 +409268,7 @@ "content": "已知$\\overrightarrow {a}=(1,-5,4)$, $\\overrightarrow {b}=(2,1,7)$.\\\\\n(1) 求$3 \\overrightarrow {a}+2 \\overrightarrow {b}$的坐标;\\\\\n(2) 求$|\\overrightarrow {a}+\\overrightarrow {b}|$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409277,14 +409280,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022060": { "id": "022060", "content": "已知$\\overrightarrow {a}=(2,1,-2)$, $\\overrightarrow {b}=(5,-4,3)$, $\\overrightarrow {c}=(-8,4,1)$.\\\\\n(1) 求证: $\\overrightarrow {a} \\perp \\overrightarrow {b}$;;\n(2) 设$\\overrightarrow {a}$与$\\overrightarrow {c}$的夹角为$\\theta$, 求$\\cos \\theta$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409296,14 +409299,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022061": { "id": "022061", "content": "已知$P_1(2,5,4)$, $P_2(6,4,7)$, 设$\\overrightarrow {a}=\\overrightarrow{P_1P_2}$, 求$\\overrightarrow {a}$、$-\\overrightarrow {a}$和单位向量$\\overrightarrow{a_0}$的坐标.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409315,14 +409318,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022062": { "id": "022062", "content": "已知$P_1(2,5,-6)$, 在$y$轴上求一点$P_2$, 使$|P_1P_2|=7$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409334,14 +409337,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022063": { "id": "022063", "content": "已知$P_1(1,2,3), P_2(5,4,7)$, 在$y$轴上求一点$Q$, 使$|P_1Q|=|P_2Q|$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409353,14 +409356,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022064": { "id": "022064", "content": "已知向量$\\overrightarrow {a}=(1,-3,2)$, $\\overrightarrow {b}=(2,0,-8)$, 求单位向量$\\overrightarrow {c}$, 使$\\overrightarrow {c}$与向量$\\overrightarrow {a}$、$\\overrightarrow {b}$都垂直.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409372,14 +409375,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022065": { "id": "022065", "content": "已知平面$\\alpha$经过点$A(3,1,-1)$、$B(1,-1,0)$, 且平行于向量$\\overrightarrow {a}=(-1,0,2)$, 求平面$\\alpha$的一个法向量.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409391,14 +409394,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022066": { "id": "022066", "content": "已知点$A$、$B$、$C$的坐标分别为$(x_1, y_1, z_1)$、$(x_2, y_2, z_2)$、$(x_3, y_3, z_3)$, $G$是$\\triangle ABC$的重心, 求点$G$的坐标.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409410,14 +409413,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022067": { "id": "022067", "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 求证: $BD_1 \\perp C_1D$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409429,14 +409432,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022068": { "id": "022068", "content": "正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\dfrac{\\sqrt{6}}{2}$.\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\circ}$角?", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409448,14 +409451,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022069": { "id": "022069", "content": "如图, 平面$ABEF \\perp$平面$ABCD$, 四边形$ABEF$与$ABCD$都是直角梯形, \n$\\angle BAD=\\angle FAB=90^{\\circ}$, $BC =\\dfrac 12 AD$且$BC\\parallel AD$, $BE = \\dfrac{1}{2} AF$且$BE\\parallel AF$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,3,0) node [above] {$F$} coordinate (F);\n\\draw (0,0,1.5) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (1.5,0,0) node [below] {$C$} coordinate (C);\n\\draw (B) ++ (0,1.5,0) node [left] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(F)--(E)--cycle(E)--(C);\n\\draw [dashed] (A)--(D)(A)--(F)(A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $C, D, F, E$四点共面;\\\\\n(2) 设$AB=BC=BE$, 求二面角$A-ED-B$的大小;", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409467,14 +409470,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022070": { "id": "022070", "content": "如图, 已知四棱锥$P-ABCD$的底面$ABCD$为等腰梯形, $AB\\parallel DC$, $AC \\perp BD$, $AC$与$BD$相交于点$O$, 且顶点$P$在底面上的射影恰为$O$点, 又$BO=2$, $PO=\\sqrt{2}$, $PB \\perp PD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$O$} coordinate (O);\n\\draw (-2,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw ($(O)!-0.5!(B)$) node [left] {$D$} coordinate (D);\n\\draw ($(O)!-0.5!(A)$) node [right] {$C$} coordinate (C);\n\\draw (O) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(P);\n\\draw [dashed] (P)--(O)(P)--(D)--(B)(A)--(C)(A)--(D)--(C);\n\\draw (O) pic [draw, scale = 0.3] {right angle = B--O--A};\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$PD$与$BC$所成角的余弦值;\\\\\n(2) 求二面角$P-AB-C$的大小;\\\\\n(3) 设点$M$在棱$PC$上, 且$\\dfrac{PM}{MC}=\\lambda$, 问$\\lambda$为何值时, $PC \\perp$平面$BMD$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409486,14 +409489,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022071": { "id": "022071", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列直线的一个方向向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $AD_1$;\\\\\n(2) $AA_1$;\\\\\n(3) $AC_1$;\\\\\n(4) $AB_1$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409505,14 +409508,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022072": { "id": "022072", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系. 下列向量是图中哪些经过两个顶点的直线的一个方向向量?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow {a}=(1,0,0)$;\\\\\n(2) $\\overrightarrow {b}=(0,1,0)$;\\\\\n(3) $\\overrightarrow {c}=(3 \\sqrt{2}, 0,4)$;\\\\\n(4) $\\overrightarrow {d}=(0,3 \\sqrt{2}, 8)$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409524,14 +409527,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022073": { "id": "022073", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的上、下底面都是边长为$3$的正方形, 长方体的高为$4$, 如图建立空间直角坐标系, 求下列平面的一个法向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below ] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C)--(A)(D)--(B)($(A)!0.5!(C)$)--($(A1)!0.5!(C1)$);\n\\draw [->] (A) -- ($(A)!-0.5!(C)$) node [below] {$x$} coordinate (x);\n\\draw [->] (B) -- ($(B)!-0.5!(D)$) node [right] {$y$} coordinate (y);\n\\draw [->] ($(A1)!0.5!(C1)$) --++ (0,1.5,0) node [right] {$z$} coordinate (z);\n\\end{tikzpicture}\n\\end{center}\n(1) 平面$AA_1D_1D$;\\\\\n(2) 平面$BB_1D_1D$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409543,14 +409546,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022074": { "id": "022074", "content": "已知点$A(0,-7,0)$、$B(2,-1,1)$、$C(2,2,2)$, 求平面$ABC$的一个法向量.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409562,14 +409565,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022075": { "id": "022075", "content": "已知点$S(0,6,4)$、$A(3,5,3)$、$B(-2,11,-5)$、$C(1,-1,4)$, 求点$S$到平面$ABC$的距离.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409581,14 +409584,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022076": { "id": "022076", "content": "已知平面$\\alpha$的一个法向量$\\overrightarrow {n}=(3,-2,6)$, 且经过点$A(0,7,0)$, 求原点到平面$\\alpha$的距离.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409600,14 +409603,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022077": { "id": "022077", "content": "已知三棱锥$A-BCD$的三条侧棱$AB$、$AC$、$AD$两两垂直, 且$AB=1$, $AC=2$, $AD=3$, 求顶点$A$到平面$BCD$的距离.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409619,14 +409622,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022078": { "id": "022078", "content": "正三棱柱$ABC-A_1B_1C_1$中, $AB=2AA_1=\\dfrac{\\sqrt{6}}{2}$.\\\\\n(1) $P$点在棱$A_1B_1$上什么位置时, 异面直线$AP$与$A_1C$互相垂直?\\\\\n(2) $P$点在棱$A_1B_1$上什么位置时, 直线$AP$与平面$A_1BC$成$30^{\\circ}$角?", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409638,14 +409641,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022079": { "id": "022079", "content": "已知正方体$ABCD-A_1B_1C_1D_1$, 求二面角$B-AC-D_1$的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409657,14 +409660,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022080": { "id": "022080", "content": "已知$ABCD-A_1B_1C_1D_1$为正方体.\\\\\n(1) 求直线$AC$与$B_1D$所成的角的大小;\\\\\n(2) 求直线$B_1D$与平面$ACD_1$所成的角的大小;\\\\\n(3) 求平面$ACD_1$与平面$B_1CD_1$所成的二面角的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409676,14 +409679,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022081": { "id": "022081", "content": "已知正三棱锥的底面边长和高都为$a$. 求侧面与底面所成的二面角的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409695,14 +409698,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022082": { "id": "022082", "content": "在三棱锥$P-ABC$中, 已知底面$ABC$是以$C$为直角的直角三角形, $PC \\perp$平面$ABC$, $AC=18$, $PC=6$, $BC=9$, $G$是$\\triangle PAB$的重心, $M$是棱$AC$的中点, 求直线$CG$与直线$BM$所成的角$\\theta$的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409714,14 +409717,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022083": { "id": "022083", "content": "已知矩形$ABCD$, 且$PD \\perp$平面$ABCD$, 若$PB=2$, $PB$与平面$PCD$所成的角为$45^{\\circ}$. $PB$与平面$ABD$所成的角为$30^{\\circ}$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.3]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(B)--(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) $CD$的长;\\\\\n(2) 求$PB$与$CD$所成的角;\\\\\n(3) 求二面角$C-PB-D$的余弦值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409733,14 +409736,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022084": { "id": "022084", "content": "用二项式定理展开下列两式:\\\\\n(1) $(a+2 b)^6$;\\\\\n(2) $(1-\\dfrac{1}{x})^5$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409752,14 +409755,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022085": { "id": "022085", "content": "化简:\\\\ \n(1) $(1+\\sqrt{x})^5+(1-\\sqrt{x})^5$;\\\\\n(2) $(2 x+y)^4-(2 x-y)^4$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409771,14 +409774,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022086": { "id": "022086", "content": "分别写出$(x-1)^{15}$的二项展开式中的前$4$项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409790,14 +409793,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022087": { "id": "022087", "content": "求$(2 a^3-3 b^2)^{10}$的二项展开式中的第$8$项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409809,14 +409812,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022088": { "id": "022088", "content": "$(x-1)^n$的二项展开式中第$m$项($1 \\leq m \\leq n$且$m$、$n \\in \\mathbf{N}$)的二项式的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1} \\mathrm{C}_n^m$}{$\\mathrm{C}_n^m$}{$(-1)^m \\mathrm{C}_n^m$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -409835,7 +409838,7 @@ "content": "求$(3 x^3-\\dfrac{1}{3 x^3})^{10}$的二项展开式中的常数项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409847,14 +409850,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022090": { "id": "022090", "content": "已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的系数相等, 求这两项的系数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409866,14 +409869,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022091": { "id": "022091", "content": "在$(\\sqrt[3]{x}-\\dfrac{2}{\\sqrt{x}})^{11}$的二项展开式中,\\\\\n(1) 求含$x^2$项的二项式系数;\\\\\n(2) 含$x^{\\frac{1}{3}}$的项是第几项? 并写出这项的系数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409885,14 +409888,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022092": { "id": "022092", "content": "已知$(x \\sin \\theta+1)^6$的二项展开式$x^2$项的系数与$(x-\\dfrac{15}{2} \\cos \\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\cos \\theta$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409904,14 +409907,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022093": { "id": "022093", "content": "求证: 当$n$为正整数时, $2^n-\\mathrm{C}_n^1 \\cdot 2^{n-1}+\\mathrm{C}_n^2 \\cdot 2^{n-2}+\\cdots+\\mathrm{C}_n^{n-1} \\cdot 2+(-1)^n=1$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409923,14 +409926,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022094": { "id": "022094", "content": "求$(1+2 x)^3(1-x)^4$展开式中$x^6$的系数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409942,14 +409945,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022095": { "id": "022095", "content": "在$(3 x-2 y)^9$的展开式中, 二项式系数的和是\\blank{50}, 各项系数的和是各项系数的绝对值之和是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -409968,7 +409971,7 @@ "content": "$\\mathrm{C}_n^1+3\\mathrm{C}_n^2+9\\mathrm{C}_n^3+\\cdots+3^{n-1} \\mathrm{C}_n^n$等于\\bracket{20}.\n\\fourch{$4^n$}{$\\dfrac{4^n}{3}$}{$\\dfrac{4^n}{3}-1$}{$\\dfrac{4^n-1}{3}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -409987,7 +409990,7 @@ "content": "求$(\\dfrac{\\sqrt{x}}{2}-\\dfrac{2}{\\sqrt{x}})^{10}$的二项展开式的正中间一项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -409999,14 +410002,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022098": { "id": "022098", "content": "求$(x \\sqrt{y}-y \\sqrt{x})^{11}$的二项展开式的正中间两项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410018,14 +410021,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022099": { "id": "022099", "content": "用二项式定理证明: $99^{10}-1$能被$1000$整除.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410037,14 +410040,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022100": { "id": "022100", "content": "求$77^{77}-15$除以$19$的余数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410056,14 +410059,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022101": { "id": "022101", "content": "求$(1+2 x+x^2)^{10}(1-x)^6$的展开式中各项系数之和.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410075,14 +410078,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022102": { "id": "022102", "content": "在$(x^2-\\dfrac{3}{x})^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求:\n$\\mathrm{C}_n^0-\\dfrac{1}{2} \\mathrm{C}_n^1+\\dfrac{1}{4} \\mathrm{C}_n^2-\\cdots+(-1)^n \\cdot \\dfrac{1}{2^n} \\mathrm{C}_n^n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410094,14 +410097,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022103": { "id": "022103", "content": "在$(1+3 x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\n(1) 求二项展开式中二项式系数最大的项;\\\\\n(2) 求二项展开式中系数最大的项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410113,14 +410116,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022104": { "id": "022104", "content": "已知$(x+1)^n=x^n+\\cdots+a x^3+b x^2+c x+1$($n \\geq 1$, $n \\in \\mathrm{N}$), 且$a: b=3: 1$, 求$c$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410132,14 +410135,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "022105": { "id": "022105", "content": "已知$n$为大于$1$的自然数, 用二项式定理证明: $(1+\\dfrac{1}{n})^n>2$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -410151,7 +410154,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "030001": { "id": "030001", @@ -444294,6 +444297,96 @@ "remark": "", "space": "" }, + "031357": { + "id": "031357", + "content": "已知$(1+x)^6$展开式的系数的最大值为$a$, $(1+2x)^6$展开式的系数的最大值为$b$, 则$\\dfrac ba=$\\blank{50}.", + "objs": [ + "K0819005X" + ], + "tags": [ + "第八单元", + "二项式定理" + ], + "genre": "填空题", + "ans": "$12$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20230323修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20230323\t徐慧" + ], + "same": [], + "related": [ + "000563" + ], + "remark": "", + "space": "" + }, + "031358": { + "id": "031358", + "content": "若直线$x+y-c=0$, $x-y+2=0$, $y+2=0$无法围成三角形, 则实数$c$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$-6$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20230323\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "031359": { + "id": "031359", + "content": "已知$2^x<3^x$, 则$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$(0,+\\infty)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20230323\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "031360": { + "id": "031360", + "content": "点集$\\{(x,y)||x|+2|y|\\le 2\\}$在平面直角坐标系中所对应图形的面积为\\blank{50}.", + "objs": [], + "tags": [ + "第七单元" + ], + "genre": "填空题", + "ans": "$4$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20230323修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20230323\t徐慧" + ], + "same": [], + "related": [ + "000952", + "000603" + ], + "remark": "", + "space": "" + }, "040001": { "id": "040001", "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}",