From c4cf616b474bc3e64444fd9a97c363d9db3c93b0 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 19 Mar 2023 22:58:47 +0800 Subject: [PATCH] 20230319 night --- 工具/关键字筛选题号.ipynb | 6 +- 工具/寻找阶段末尾空闲题号.ipynb | 2 +- 工具/批量题号选题pdf生成.ipynb | 10 +- 工具/文本文件/题号筛选.txt | 2 +- 工具/添加题目到数据库.ipynb | 132 ++-- 题库0.3/Problems.json | 1102 +++++++++++++++++++++++++++++++ 6 files changed, 1196 insertions(+), 58 deletions(-) diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index 79c9cf11..8390d263 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 2, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } @@ -21,7 +21,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"origin\":[\"空中课堂\"],\"origin2\":[\"2023\"]},\n", + " {\"origin\":[\"校本作业\"]},\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index 77e1c347..ac346d78 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -12,7 +12,7 @@ "首个空闲id: 14764 , 直至 020000\n", "首个空闲id: 22048 , 直至 030000\n", "首个空闲id: 31353 , 直至 040000\n", - "首个空闲id: 40202 , 直至 999999\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 5e12783f..74431c57 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": 2, + "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/本学期高一高二年级新试卷_教师用_20230318.tex\n", + "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", "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_20230318.tex\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", "0\n" ] } @@ -40,7 +40,9 @@ "\"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", + "\"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", "\n", "}\n", "\n", diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 76b02e4d..95dea3dc 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ 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\ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index 6375c6df..8e3b3f1c 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 @@ -7,66 +7,80 @@ "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 40202\n", + "starting_id = 22048\n", "raworigin = \"\"\n", - "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目10.tex\"\n", - "editor = \"20230318\\t王伟叶\"\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", "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "添加题号040202, 来源: 2024届高二下学期周末卷05\n", - "添加题号040203, 来源: 2024届高二下学期周末卷05\n", - "添加题号040204, 来源: 2024届高二下学期周末卷05\n", - "添加题号040205, 来源: 2024届高二下学期周末卷05\n", - "添加题号040206, 来源: 2024届高二下学期周末卷05\n", - "添加题号040207, 来源: 2024届高二下学期周末卷05\n", - "添加题号040208, 来源: 2024届高二下学期周末卷05\n", - "添加题号040209, 来源: 2024届高二下学期周末卷05\n", - "添加题号040210, 来源: 2024届高二下学期周末卷05\n", - "添加题号040211, 来源: 2024届高二下学期周末卷05\n", - "添加题号040212, 来源: 2024届高二下学期周末卷05\n", - "添加题号040213, 来源: 2024届高二下学期周末卷05\n", - "添加题号040214, 来源: 2024届高二下学期周末卷05\n", - "添加题号040215, 来源: 2024届高二下学期周末卷05\n", - "添加题号040216, 来源: 2024届高二下学期周末卷05\n", - "添加题号040217, 来源: 2024届高二下学期周末卷05\n", - "添加题号040218, 来源: 2024届高二下学期周末卷05\n", - "添加题号040219, 来源: 2024届高二下学期周末卷05\n", - "添加题号040220, 来源: 2024届高二下学期周末卷05\n", - "添加题号040221, 来源: 2024届高二下学期周末卷05\n", - "添加题号040222, 来源: 2024届高二下学期周末卷05\n", - "添加题号040223, 来源: 2024届高二下学期周末卷05\n", - "添加题号040224, 来源: 2024届高二下学期周末卷05\n", - "添加题号040225, 来源: 2024届高二下学期周末卷05\n", - "添加题号040226, 来源: 2025届高一下学期周末卷05\n", - "添加题号040227, 来源: 2025届高一下学期周末卷05\n", - "添加题号040228, 来源: 2025届高一下学期周末卷05\n", - "添加题号040229, 来源: 2025届高一下学期周末卷05\n", - "添加题号040230, 来源: 2025届高一下学期周末卷05\n", - "添加题号040231, 来源: 2025届高一下学期周末卷05\n", - "添加题号040232, 来源: 2025届高一下学期周末卷05\n", - "添加题号040233, 来源: 2025届高一下学期周末卷05\n", - "添加题号040234, 来源: 2025届高一下学期周末卷05\n", - "添加题号040235, 来源: 2025届高一下学期周末卷05\n", - "添加题号040236, 来源: 2025届高一下学期周末卷05\n", - "添加题号040237, 来源: 2025届高一下学期周末卷05\n", - "添加题号040238, 来源: 2025届高一下学期周末卷05\n", - "添加题号040239, 来源: 2025届高一下学期周末卷05\n", - "添加题号040240, 来源: 2025届高一下学期周末卷05\n", - "添加题号040241, 来源: 2025届高一下学期周末卷05\n", - "添加题号040242, 来源: 2025届高一下学期周末卷05\n", - "添加题号040243, 来源: 2025届高一下学期周末卷05\n", - "添加题号040244, 来源: 2025届高一下学期周末卷05\n", - "添加题号040245, 来源: 2025届高一下学期周末卷05\n" + "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" ] } ], @@ -88,7 +102,7 @@ " for p in ProblemList_raw:\n", " startpos = data.index(p)\n", " tempdata = data[:startpos]\n", - " suflist = re.findall(r\"\\n(\\%[\\S]+)\\n\",tempdata)\n", + " suflist = re.findall(r\"\\n(\\%\\s{0,1}[\\S]+)\\n\",tempdata)\n", " if len(suflist) > 0:\n", " suffix = suflist[-1].replace(\"%\",\"\").strip()\n", " else:\n", @@ -161,6 +175,26 @@ " 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/题库0.3/Problems.json b/题库0.3/Problems.json index 8e3ea943..9d6b8990 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -408527,6 +408527,1108 @@ "remark": "", "space": "12ex" }, + "022048": { + "id": "022048", + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022059": { + "id": "022059", + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届空间向量校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022089": { + "id": "022089", + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022094": { + "id": "022094", + "content": "求$(1+2 x)^3(1-x)^4$展开式中$x^6$的系数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022096": { + "id": "022096", + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022097": { + "id": "022097", + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022099": { + "id": "022099", + "content": "用二项式定理证明: $99^{10}-1$能被$1000$整除.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "022100": { + "id": "022100", + "content": "求$77^{77}-15$除以$19$的余数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届二项式定理校本作业", + "edit": [ + "20230319\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, "030001": { "id": "030001", "content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0