From 817726bdf52f8dbe917975e1b312ac5102ac6e44 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 5 Nov 2022 10:22:25 +0800 Subject: [PATCH] 20221105 morning --- 工具/关键字筛选题号.ipynb | 6 +- 工具/批量添加题库字段数据.ipynb | 311 +++-- 工具/批量题号选题pdf生成.ipynb | 140 +- 工具/文本文件/metadata.txt | 1161 ++++++++++------- 工具/文本文件/题号筛选.txt | 2 +- 工具/根据目标列表批量生成对应题目的字典.ipynb | 48 +- 工具/模板文件/题目编辑.aux | 3 + 工具/模板文件/题目编辑.log | 888 +++++++++++++ 工具/模板文件/题目编辑.synctex.gz | Bin 0 -> 2262 bytes 工具/讲义生成.ipynb | 14 +- 工具/题号选题pdf生成.ipynb | 10 +- 题库0.3/Problems.json | 535 ++++++-- 12 files changed, 2245 insertions(+), 873 deletions(-) create mode 100644 工具/模板文件/题目编辑.aux create mode 100644 工具/模板文件/题目编辑.log create mode 100644 工具/模板文件/题目编辑.synctex.gz 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 449d9244..fb447156 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": 3, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 2, + "execution_count": 3, "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", - " {\"tags\":[\"第六单元\"],\"content_not\":[\"体积\",\"面积\"],\"usages_not\":[\"2023届\"]}\n", + " {\"tags\":[\"空间向量\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 8ce01b73..b17d88ba 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,142 +2,193 @@ "cells": [ { "cell_type": "code", - "execution_count": 9, + "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 000291 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000292 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000293 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000294 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000296 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000297 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000299 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000301 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000302 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000304 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000305 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 000781 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001944 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001947 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001948 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001949 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001950 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001951 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001952 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001953 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001954 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001955 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001956 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001957 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001958 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001959 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001960 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001961 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001962 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001963 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001964 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001965 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001966 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001968 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001969 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001971 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001972 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001973 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001974 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001975 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001976 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001977 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001978 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001979 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001980 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001981 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001985 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001987 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 001991 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 003624 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 003647 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 003679 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004096 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004243 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004348 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004656 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004698 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 004740 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009855 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009856 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009857 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009858 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009859 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009860 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009861 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009862 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009863 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009864 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009865 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009867 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009868 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009870 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009871 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009872 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 009873 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 010706 , 字段: tags 中已添加数据: 空间向量\n", - "题号: 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K0625003X\n", + "题号: 010714 , 字段: objs 中已添加数据: K0625003X\n", + "题号: 010716 , 字段: objs 中已添加数据: K0625003X\n", + "题号: 001949 , 字段: objs 中已添加数据: K0625004X\n", + "题号: 010711 , 字段: objs 中已添加数据: K0625004X\n", + "题号: 010736 , 字段: objs 中已添加数据: K0625004X\n", + "题号: 000301 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 001953 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 001959 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 001979 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 004698 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 010719 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 030452 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 030454 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 030458 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 030459 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 030464 , 字段: objs 中已添加数据: K0626001X\n", + "题号: 001955 , 字段: objs 中已添加数据: K0626003X\n", + "题号: 001957 , 字段: objs 中已添加数据: K0626003X\n", + "题号: 010717 , 字段: objs 中已添加数据: K0626003X\n", + "题号: 000781 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 001961 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 003624 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 003679 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 009862 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 010722 , 字段: objs 中已添加数据: K0627002X\n", + "题号: 001962 , 字段: objs 中已添加数据: K0627003X\n", + "题号: 030463 , 字段: objs 中已添加数据: K0627003X\n", + "题号: 000291 , 字段: objs 中已添加数据: K0627004X\n", + "题号: 000781 , 字段: objs 中已添加数据: K0627004X\n", + "题号: 010723 , 字段: objs 中已添加数据: K0627004X\n", + "题号: 030457 , 字段: objs 中已添加数据: K0627004X\n", + "题号: 000293 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 001952 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 001963 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 001964 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 001965 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 001980 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 009861 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 009863 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010718 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010721 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010724 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010725 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010726 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 010729 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 030456 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 030457 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 030463 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 030469 , 字段: objs 中已添加数据: K0627005X\n", + "题号: 000302 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 001951 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 001966 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 001968 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 001969 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 001977 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 009861 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 009864 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 030453 , 字段: objs 中已添加数据: K0627006X\n", + "题号: 000297 , 字段: objs 中已有该数据: K0627007X\n", + "题号: 000304 , 字段: objs 中已有该数据: K0627007X\n", + "题号: 009871 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 010721 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 010729 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 010737 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 030460 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 030472 , 字段: objs 中已添加数据: K0627007X\n", + "题号: 001971 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 001972 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 001974 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 001975 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 001976 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 003624 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 009865 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 030467 , 字段: objs 中已添加数据: K0628001X\n", + "题号: 001985 , 字段: objs 中已添加数据: K0628002X\n", + "题号: 010713 , 字段: objs 中已添加数据: K0628002X\n", + "题号: 010724 , 字段: objs 中已添加数据: K0628002X\n", + "题号: 010727 , 字段: objs 中已添加数据: K0628002X\n", + "题号: 000299 , 字段: objs 中已有该数据: K0628004X\n", + "题号: 000305 , 字段: objs 中已有该数据: K0628004X\n", + "题号: 001973 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 001978 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 004656 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 009867 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 010714 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 010718 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 010720 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 030459 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 030460 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 030462 , 字段: objs 中已添加数据: K0628004X\n", + "题号: 000294 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 001975 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 001979 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 004740 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 010731 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 010733 , 字段: objs 中已添加数据: K0628005X\n", + "题号: 001960 , 字段: objs 中已添加数据: KNONE\n", + "题号: 030471 , 字段: objs 中已添加数据: KNONE\n", + "题号: 000293 , 字段: objs 中已有该数据: K0629001X\n", + "题号: 000296 , 字段: objs 中已有该数据: K0629001X\n", + "题号: 000297 , 字段: objs 中已有该数据: K0629001X\n", + "题号: 003647 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 004348 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 009868 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 010735 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 010740 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 030461 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 030465 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 030472 , 字段: objs 中已添加数据: K0629001X\n", + "题号: 000302 , 字段: objs 中已有该数据: K0630002X\n", + "题号: 000304 , 字段: objs 中已有该数据: K0630002X\n", + "题号: 001948 , 字段: objs 中已添加数据: K0630002X\n", + "题号: 004348 , 字段: objs 中已添加数据: K0630002X\n", + "题号: 009870 , 字段: objs 中已添加数据: K0630002X\n", + "题号: 010721 , 字段: objs 中已添加数据: K0630002X\n", + "题号: 010730 , 字段: objs 中已添加数据: K0630002X\n", + "题号: 000294 , 字段: objs 中已有该数据: K0630004X\n", + "题号: 000296 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 003647 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 004096 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 004243 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 004656 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 004698 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 004740 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 009871 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 010732 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 010735 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 030462 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 030472 , 字段: objs 中已添加数据: K0630004X\n", + "题号: 030468 , 字段: objs 中已添加数据: K0631001X\n", + "题号: 000305 , 字段: objs 中已有该数据: K0631002X\n", + "题号: 001981 , 字段: objs 中已添加数据: K0631002X\n", + "题号: 004243 , 字段: objs 中已添加数据: K0631002X\n", + "题号: 009873 , 字段: objs 中已添加数据: K0631002X\n", + "题号: 010739 , 字段: objs 中已添加数据: K0631002X\n", + "题号: 000296 , 字段: objs 中已添加数据: K0631003X\n", + "题号: 000299 , 字段: objs 中已有该数据: K0631003X\n", + "题号: 004096 , 字段: objs 中已添加数据: K0631003X\n", + "题号: 009872 , 字段: objs 中已添加数据: K0631003X\n", + "题号: 010737 , 字段: objs 中已添加数据: K0631003X\n", + "题号: 030461 , 字段: objs 中已添加数据: K0631003X\n", + "题号: 030468 , 字段: objs 中已添加数据: K0631003X\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 132496d7..bd7a7d55 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 @@ -2,64 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0406001X_教师用_20221103.tex\n", + "开始编译教师版本pdf文件: 临时文件/批量生成题目/立体几何1批量.tex_教师用_20221105.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0406001X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0406002X_教师用_20221103.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0406002X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0406003X_教师用_20221103.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0406003X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0406004X_教师用_20221103.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0406004X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0406005X_教师用_20221103.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0406005X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 临时文件/批量生成题目/递推数列/K0407001X_教师用_20221103.tex\n", - "0\n", - "开始编译学生版本pdf文件: 临时文件/批量生成题目/递推数列/K0407001X_学生用_20221103.tex\n", - "0\n", - "开始编译教师版本pdf文件: 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临时文件/批量生成题目/递推数列/K0409002X_学生用_20221103.tex\n", + "开始编译学生版本pdf文件: 临时文件/批量生成题目/立体几何1批量.tex_学生用_20221105.tex\n", "0\n" ] } @@ -74,26 +26,19 @@ "\"\"\"---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", - "\"K0406001X\":\"001739,001740\",\n", - "\"K0406002X\":\"010770,010771\",\n", - "\"K0406003X\":\"000514,001741,001742,001743,001744,003202,008400,008401,008402,008406,008408,009890\",\n", - "\"K0406004X\":\"000403,003211,003219,003630,010778\",\n", - "\"K0406005X\":\"000575,003210,003215,003218,003226,009892,010772,010773,010777\",\n", - "\"K0407001X\":\"001821\",\n", - "\"K0407002X\":\"000307,000320,000574,000950,001795,001803,001813,001818,001823,001824,001825,003205,003206,003213,003214,003273,003309,003310,003319,004179,006968,006969,006974,006982,006983,008404,008405,008407,008409,009894,009895,010774,010775,010779,010780\",\n", - "\"K0407003X\":\"001804,001805,001806,001807,001810,001811,001814,001815,001816,001819,001820,001822,003312,003322,006973\",\n", - "\"K0407004X\":\"001809,003600,010776\",\n", - "\"K0408002X\":\"006910,006912\",\n", - "\"K0408003X\":\"000315,000322,001018,001019,001020,001021,001022,001023,001024,001025,003274,003275,003276,003277,003278,003282,003284,003285,003286,003287,003289,004981,006909,006911,006913,006915,006916,006917,006918,006919,006920,006921,006922,006923,006924,006925,006926,006927,006928,006929,006930,006931,006932,006933,006934,006935,006936,006937,006938,006939,006940,006941,006942,006943,006944,006945,006946,006947,006948,006949,006950,006951,006952,006953,006956,006988,006989,006990,008457,008458,008459,008461,008462,008463,008464,008465,008466,008467,008468,008469,008534,009896,009897,009898,010782,010783,010784,010785,010786,010787\",\n", - "\"K0409001X\":\"000316,000324,000595,001026,001027,001028,001467,001817,003279,003280,003283,003288,006914,006954,006955,006957,006958,006959,006960,008460,008470,008471,008472,008473,008522,008523,008524,008525,008535,009899,009900,010788\",\n", - "\"K0409002X\":\"000323,003281,008474,008475,009901\"\n", + "\"K0629001X\":\"000293,000297,009868,010740,030465\",\n", + "\"K0630002X\":\"000298,000302,000304,001948,004348,009870,010721,010730\",\n", + "\"K0630004X\":\"000294,000300,003647,004656,004698,004740,009871,010732,010735,030462,030472\",\n", + "\"K0631002X\":\"000303,000305,001981,004243,009873,010739\",\n", + "\"K0631003X\":\"000295,000296,000299,004096,009872,010737,030461,030468\"\n", "}\n", "\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---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/批量生成题目/递推数列/\"\n", + "directory = \"临时文件/批量生成题目/\"\n", + "filename = \"立体几何1批量.tex\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "if directory[-1] != \"/\":\n", " directory += \"/\"\n", @@ -147,17 +92,24 @@ "current_time = time.localtime()\n", "time_string = \"_\"+str(current_time.tm_year).zfill(4)+str(current_time.tm_mon).zfill(2)+str(current_time.tm_mday).zfill(2)\n", "\n", - "for filename in problems_dict:\n", - " problems = problems_dict[filename]\n", - " teachers_latex_file = directory + filename + \"_教师用\" + time_string + \".tex\"\n", - " students_latex_file = directory + filename + \"_学生用\" + time_string + \".tex\"\n", + "\n", + "data_teachers = \"\"\n", + "data_students = \"\"\n", + "\n", + "teachers_latex_file = directory + filename + \"_教师用\" + time_string + \".tex\"\n", + "students_latex_file = directory + filename + \"_学生用\" + time_string + \".tex\"\n", + "\n", + "for section_name in problems_dict:\n", + " problems = problems_dict[section_name]\n", + "\n", + " data_teachers += r\"\\newpage\" + \"\\n\\n\" + r\"\\section{\" + section_name +\"}\\n\\n\"\n", + " data_students += r\"\\newpage\" + \"\\n\\n\" + r\"\\section{\" + section_name +\"}\\n\\n\"\n", "\n", " #生成题目列表\n", " problem_list = [id for id in generate_number_set(problems.strip(),pro_dict) if id in pro_dict]\n", "\n", - " data_teachers = \"\"\n", - " data_students = \"\"\n", - " id_list = \"\"\n", + "\n", + "\n", "\n", " #生成教师题目字符串与学生题目字符串, 准备替换至latex文件\n", " for id in problem_list:\n", @@ -190,33 +142,35 @@ " space = (\"\" if problemset[\"space\"] == \"\" else r\"\\vspace*{\"+problemset[\"space\"]+\"}\\n\")\n", " tags = (\"|\".join(problemset[\"tags\"]) if len(problemset[\"origin\"])>0 else \"暂无标签\")\n", " 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", + " 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", " data_teachers += teachers_string\n", " data_students += students_string\n", "\n", + "#去除第一个newpage\n", + "data_teachers = data_teachers[10:]\n", + "data_students = data_students[10:]\n", "\n", + "#替换latex文件的内容并编译\n", + "with open(\"模板文件/题目清单.tex\",\"r\",encoding = \"utf8\") as f:\n", + " latex_raw = f.read()\n", + "#识别操作系统\n", + "if sys.platform != \"win32\":\n", + " latex_raw = re.sub(r\"fontset[\\s]*=[\\s]*windows\",\"fontset = fandol\",latex_raw)\n", + " latex_raw = re.sub(r\"\\\\setCJKmainfont\",r\"% \\\\setCJKmainfont\",latex_raw)\n", + "latex_teachers = latex_raw.replace(\"编译模板\",data_teachers)\n", + "with open(teachers_latex_file,\"w\",encoding = \"utf8\") as f:\n", + " f.write(latex_teachers)\n", + "print(\"开始编译教师版本pdf文件: \", teachers_latex_file)\n", + "os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ teachers_latex_file)\n", + "print(os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ teachers_latex_file))\n", "\n", - " #替换latex文件的内容并编译\n", - " with open(\"模板文件/题目清单.tex\",\"r\",encoding = \"utf8\") as f:\n", - " latex_raw = f.read()\n", - " #识别操作系统\n", - " if sys.platform != \"win32\":\n", - " latex_raw = re.sub(r\"fontset[\\s]*=[\\s]*windows\",\"fontset = fandol\",latex_raw)\n", - " latex_raw = re.sub(r\"\\\\setCJKmainfont\",r\"% \\\\setCJKmainfont\",latex_raw)\n", - " latex_teachers = latex_raw.replace(\"编译模板\",data_teachers)\n", - " with open(teachers_latex_file,\"w\",encoding = \"utf8\") as f:\n", - " f.write(latex_teachers)\n", - " print(\"开始编译教师版本pdf文件: \", teachers_latex_file)\n", - " os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ teachers_latex_file)\n", - " print(os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ teachers_latex_file))\n", - "\n", - " latex_students = latex_raw.replace(\"编译模板\",data_students)\n", - " with open(students_latex_file,\"w\",encoding = \"utf8\") as f:\n", - " f.write(latex_students)\n", - " print(\"开始编译学生版本pdf文件: \", students_latex_file)\n", - " os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ students_latex_file)\n", - " print(os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ students_latex_file))\n" + "latex_students = latex_raw.replace(\"编译模板\",data_students)\n", + "with open(students_latex_file,\"w\",encoding = \"utf8\") as f:\n", + " f.write(latex_students)\n", + "print(\"开始编译学生版本pdf文件: \", students_latex_file)\n", + "os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ students_latex_file)\n", + "print(os.system(\"xelatex -interaction=batchmode -output-directory=\" + directory + \" \"+ students_latex_file))\n" ] }, { diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 3a8723b9..4b4f65b5 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,517 +1,722 @@ -tags -291 -空间向量 - - -292 -空间向量 - - -293 -空间向量 - - -294 -空间向量 - - -296 -空间向量 - - -297 -空间向量 - - -299 -空间向量 - - -301 -空间向量 - - -302 -空间向量 - - -304 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-30463 -空间向量 - - -30464 -空间向量 - - -30465 -空间向量 - - -30466 -空间向量 +K0624002X 30467 -空间向量 +K0624002X -30468 -空间向量 +301 +K0624003X -30469 -空间向量 +1944 +K0624003X + + +1947 +K0624003X + + +1956 +K0624003X + + +1957 +K0624003X + + +9861 +K0624003X + + +10707 +K0624003X + + +10708 +K0624003X + + +10715 +K0624003X + + +10717 +K0624003X + + +30455 +K0624003X + + +30456 +K0624003X 30470 -空间向量 +K0624003X -30471 -空间向量 +292 +K0625002X + + +9859 +K0625002X + + +10709 +K0625002X + + +1950 +K0625003X + + +1954 +K0625003X + + +1956 +K0625003X + + +1958 +K0625003X + + +10706 +K0625003X + + +10710 +K0625003X + + +10714 +K0625003X + + +10716 +K0625003X + + +1949 +K0625004X + + +10711 +K0625004X + + +10736 +K0625004X + + +301 +K0626001X + + +1953 +K0626001X + + +1959 +K0626001X + + +1979 +K0626001X + + +4698 +K0626001X + + +10719 +K0626001X + + +30452 +K0626001X + + +30454 +K0626001X + + +30458 +K0626001X + + +30459 +K0626001X + + +30464 +K0626001X + + +1955 +K0626003X + + +1957 +K0626003X + + +10717 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\ No newline at end of file +000291,000292,000293,000294,000296,000297,000299,000301,000302,000304,000305,000781,001944,001947,001948,001949,001950,001951,001952,001953,001954,001955,001956,001957,001958,001959,001960,001961,001962,001963,001964,001965,001966,001968,001969,001971,001972,001973,001974,001975,001976,001977,001978,001979,001980,001981,001985,001987,001991,003624,003647,003679,004096,004243,004348,004656,004698,004740,009855,009856,009857,009858,009859,009860,009861,009862,009863,009864,009865,009867,009868,009870,009871,009872,009873,010706,010707,010708,010709,010710,010711,010712,010713,010714,010715,010716,010717,010718,010719,010720,010721,010722,010723,010724,010725,010726,010727,010729,010730,010731,010732,010733,010735,010736,010737,010738,010739,010740,030452,030453,030454,030455,030456,030457,030458,030459,030460,030461,030462,030463,030464,030465,030466,030467,030468,030469,030470,030471,030472 \ No newline at end of file diff --git a/工具/根据目标列表批量生成对应题目的字典.ipynb b/工具/根据目标列表批量生成对应题目的字典.ipynb index 3f5b8df5..6085d94b 100644 --- a/工具/根据目标列表批量生成对应题目的字典.ipynb +++ b/工具/根据目标列表批量生成对应题目的字典.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -14,48 +14,38 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# 输入目标列表\n", - "t = \"\"\"K0624001X\n", - "K0624002X\n", - "K0624003X\n", - "K0625001X\n", - "K0625002X\n", - "K0625003X\n", - "K0625004X\n", - "K0626001X\n", - "K0626002X\n", - "K0626003X\n", - "K0626004X\n", - "K0627001X\n", - "K0627002X\n", - "K0627003X\n", - "K0627004X\n", - "K0627005X\n", - "K0627006X\n", - "K0627007X\n", - "K0628001X\n", - "K0628002X\n", - "K0628003X\n", - "K0628004X\n", - "K0628005X\"\"\"" + "t = \"\"\"K0629001X\n", + "K0629002X\n", + "K0629003X\n", + "K0630001X\n", + "K0630002X\n", + "K0630003X\n", + "K0630004X\n", + "K0631001X\n", + "K0631002X\n", + "K0631003X\n", + "\"\"\"" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "\"K0624001X\":\"000291\",\n", - "\"K0625001X\":\"000292\",\n", - "\"K0625004X\":\"000301\",\n" + "\"K0629001X\":\"000293,000297,009868,010740,030465\",\n", + "\"K0630002X\":\"000298,000302,000304,001948,004348,009870,010721,010730\",\n", + "\"K0630004X\":\"000294,000300,003647,004656,004698,004740,009871,010732,010735,030462,030472\",\n", + "\"K0631002X\":\"000303,000305,001981,004243,009873,010739\",\n", + "\"K0631003X\":\"000295,000296,000299,004096,009872,010737,030461,030468\",\n" ] } ], diff --git a/工具/模板文件/题目编辑.aux b/工具/模板文件/题目编辑.aux new file mode 100644 index 00000000..cc04fc55 --- /dev/null +++ b/工具/模板文件/题目编辑.aux @@ -0,0 +1,3 @@ +\relax +\ttl@finishall +\gdef \@abspage@last{1} diff --git a/工具/模板文件/题目编辑.log b/工具/模板文件/题目编辑.log new file mode 100644 index 00000000..f4b65633 --- /dev/null +++ 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.\n", "题块 2 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/30_等差数列与等比数列_教师_20221104.tex\n", + "开始编译教师版本pdf文件: 临时文件/33_立体几何中的定量计算_预选_教师_20221105.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/30_等差数列与等比数列_学生_20221104.tex\n", + "开始编译学生版本pdf文件: 临时文件/33_立体几何中的定量计算_预选_学生_20221105.tex\n", "0\n" ] } @@ -35,7 +35,7 @@ "\n", "\"\"\"---设置其他预处理替换命令---\"\"\"\n", "#2023届第一轮讲义更换标题\n", - "exec_list = [(\"标题数字待处理\",\"30\"),(\"标题文字待处理\",\"等差数列与等比数列\")] \n", + "exec_list = [(\"标题数字待处理\",\"33\"),(\"标题文字待处理\",\"立体几何中的定量计算\")] \n", "enumi_mode = 0\n", "\n", "#2023届测验卷与周末卷\n", @@ -49,14 +49,14 @@ "\"\"\"---其他预处理替换命令结束---\"\"\"\n", "\n", "\"\"\"---设置目标文件名---\"\"\"\n", - "destination_file = \"临时文件/30_等差数列与等比数列\"\n", + "destination_file = \"临时文件/33_立体几何中的定量计算_预选\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"3225,9876,1749,1767,1789,8417,1785,321,312,3238,30473,1771,3244,3243,3253,1781,6793,8456,887,1839,6892,3216,1783\",\n", - "\"3616,573,3207,3251,3250,3239,6737,6760,3240,5851,6732,3317,3298,6717,1769,1788\"\n", + "\"293,10740,304,10721,294,3647,30462,305,9873,299,4096\",\n", + "\"30465,4348,10730,10735,4698,30472,4243,10739,296,30468\"\n", "]\n", "\"\"\"---设置题号数据结束---\"\"\"\n", "\n", diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 2abd3448..ecb39c14 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/立体几何空间向量选题_教师用_20221104.tex\n", + "开始编译教师版本pdf文件: 临时文件/立体几何空间向量_教师用_20221104.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/立体几何空间向量选题_学生用_20221104.tex\n", + "开始编译学生版本pdf文件: 临时文件/立体几何空间向量_学生用_20221104.tex\n", "0\n" ] } @@ -26,7 +26,7 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n", "problems = r\"\"\"\n", - "291,292,293,294,296,297,299,301,302,304,305,781,1944,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1968,1969,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1985,1987,1991,3624,3647,3679,4096,4243,4348,4656,4698,4740,9855,9856,9857,9858,9859,9860,9861,9862,9863,9864,9865,9867,9868,9870,9871,9872,9873,10706,10707,10708,10709,10710,10711,10712,10713,10714,10715,10716,10717,10718,10719,10720,10721,10722,10723,10724,10725,10726,10727,10729,10730,10731,10732,10733,10735,10736,10737,10738,10739,10740,30452,30453,30454,30455,30456,30457,30458,30459,30460,30461,30462,30463,30464,30465,30466,30467,30468,30469,30470,30471,30472\n", + "a\n", "\n", "\n", "\n", @@ -35,7 +35,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/立体几何空间向量选题\"\n", + "filename = \"临时文件/立体几何空间向量\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 40de2ee1..baa97bc2 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -7459,7 +7459,9 @@ "id": "000291", "content": "求连接点$A(x, y, z)$与点$B(x', y', z')$的线段$AB$的中点$M$的坐标.", "objs": [ - "K0624001X" + "K0624001X", + "K0624002X", + "K0627004X" ], "tags": [ "第六单元", @@ -7483,7 +7485,8 @@ "id": "000292", "content": "设正四面体$ABCD$的棱长为$a$, $E$为$BC$的中点, $F$为$CD$的中点. 求$\\overrightarrow{BF}\\cdot\\overrightarrow{AE}$.", "objs": [ - "K0625001X" + "K0625001X", + "K0625002X" ], "tags": [ "第六单元", @@ -7507,7 +7510,9 @@ "id": "000293", "content": "给定点$A(1, 0, 0)$、$B(3, 1, 1)$、$C(2, 0, 1)$与点$D(5, -4, 3)$.\\\\\n(1) 求$\\overrightarrow{AD}$在$\\overrightarrow{AB}$、$\\overrightarrow{BC}$、$\\overrightarrow{CA}$方向上的投影向量;\\\\\n(2) 求点$D$到平面$ABC$的距离.", "objs": [ - "K0629001X" + "K0629001X", + "K0624002X", + "K0627005X" ], "tags": [ "第六单元", @@ -7533,7 +7538,8 @@ "objs": [ "K0628004X", "K0630004X", - "K0613007B" + "K0613007B", + "K0628005X" ], "tags": [ "第六单元", @@ -7583,7 +7589,9 @@ "content": "如图, 已知$ABCD-A_1B_1C_1D_1$是底面边长为$1$的正四棱柱, $O_1$是$A_1C_1$和$B_1D_1$的交点.\n\\begin{center}\n \\begin{tikzpicture}[thick,scale = 2]\n \\draw (0,0) node [below left] {$B_1$} coordinate (B1) --++ (1,0) node [below right] {$C_1$} coordinate (C1) --++ (45:{1/2}) node [right] {$D_1$} coordinate (D1)\n --++ (0,1.6) node [above right] {$D$} coordinate (D)\n --++ (-1,0) node [above left] {$A$} coordinate (A) --++ (225:{1/2}) node [left] {$B$} coordinate (B) -- cycle;\n \\draw (B1) ++ (1,1.6) node [right] {$C$} coordinate (C) -- (C1) (C) --++ (45:{1/2}) (D) (C) --++ (-1,0);\n \\draw [dashed] (B1) --++ (45:{1/2}) node [left] {$A_1$} coordinate (A1) --++ (1,0) (A1) --++ (0,1.6);\n \\draw [dashed] (A) -- (B1) -- (D1) -- cycle (A1) -- (C1);\n \\end{tikzpicture}\n\\end{center}\n(1) 设$AB_1$与底面$A_1B_1C_1D_1$所成角的大小为$\\alpha$, 二面角$A-B_1D_1-A_1$的大小为$\\beta$. 求证: $\\tan \\beta=\\sqrt 2\\tan \\alpha$;\\\\\n(2) 若点$C$到平面$AB_1D_1$的距离为$\\dfrac 43$, 求此正四棱柱的高.", "objs": [ "K0629001X", - "K0631002X" + "K0631002X", + "K0630004X", + "K0631003X" ], "tags": [ "第六单元", @@ -7720,7 +7728,9 @@ "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 点$E$、$F$分别在$B_1B$和$D_1D$上, 且$|BE|=\\dfrac 13|BB_1|$, $|DF|=\\dfrac 23|DD_1|$.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{1.8/2}) node [right] {$C$} coordinate (C)\n --++ (0.2,2.6) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{1.8/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2.2,2.6) node [below left] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{1.8/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{1.8/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0.2,2.6);\n \\draw ($(B)!{1/3}!(B1)$) node [right] {$E$} coordinate (E) -- (C1) (E) -- (A);\n \\draw [dashed] ($(D1)!{1/3}!(D)$) node [left] {$F$} coordinate (F) -- (E) (F) -- (C1) (F) -- (A);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $A$、$E$、$C_1$、$F$四点共面;\\\\\n(2) 若$\\overrightarrow{EF} =\\lambda \\overrightarrow{AB}+ \\mu \\overrightarrow{AD}+ \\nu \\overrightarrow{AA_1}$, 求$\\lambda+\\mu+\\nu$的值.", "objs": [ "K0625003X", - "K0625004X" + "K0625004X", + "K0624003X", + "K0626001X" ], "tags": [ "第六单元", @@ -7745,7 +7755,8 @@ "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$BC$、$A_1D_1$的中点.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [below left] {$B$} coordinate (B) --++ (2,0) node [below right] {$C$} coordinate (C) --++ (45:{2/2}) node [right] {$D$} coordinate (D)\n --++ (0,2) node [above right] {$D_1$} coordinate (D1)\n --++ (-2,0) node [above left] {$A_1$} coordinate (A1) --++ (225:{2/2}) node [left] {$B_1$} coordinate (B1) -- cycle;\n \\draw (B) ++ (2,2) node [right] {$C_1$} coordinate (C1) -- (C) (C1) --++ (45:{2/2}) (C1) --++ (-2,0);\n \\draw [dashed] (B) --++ (45:{2/2}) node [left] {$A$} coordinate (A) --++ (2,0) (A) --++ (0,2);\n \\draw [dashed] ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E) -- (D) -- ($(D1)!0.5!(A1)$) node [above] {$F$} coordinate (F) (A1) -- (C);\n \\draw (E) -- (B1) -- (F);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: 四边形$B_1EDF$是菱形;\\\\\n(2) 求异面直线$A_1C$与$DE$所成角的大小.", "objs": [ "K0625003X", - "K0630002X" + "K0630002X", + "K0627006X" ], "tags": [ "第六单元", @@ -20687,7 +20698,10 @@ "000781": { "id": "000781", "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{BD_1}$的坐标为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.6, >=latex]\n \\draw (0,0) node [above left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{4/2}) node [above right] {$C$} coordinate (C)\n --++ (0,2) node [right] {$C_1$} coordinate (C1)\n --++ (-3,0) node [left] {$D_1$} coordinate (D1) --++ (225:{4/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (3,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{4/2}) (B1) --++ (-3,0);\n \\draw [dashed] (A) --++ (45:{4/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,2);\n \\draw [->] (A) --++ (225:1) node [right] {$x$};\n \\draw [->] (C) --++ (1,0) node [below] {$y$};\n \\draw [->] (D1) --++ (0,1) node [right] {$z$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0627002X", + "K0627004X" + ], "tags": [ "第六单元", "空间向量" @@ -51794,7 +51808,9 @@ "001944": { "id": "001944", "content": "已知斜三棱柱$ABC-A_1B_1C_1$中, $AC$的中点为$M$, $\\overrightarrow{A_1B_1}=\\overrightarrow{a}$, $\\overrightarrow{B_1C_1}=\\overrightarrow{b}$, $\\overrightarrow{A_1A}=\\overrightarrow{c}$. 用$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$表示$\\overrightarrow{B_1M}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -51867,7 +51883,9 @@ "001947": { "id": "001947", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 棱长为$a$, $M$分$AC_1$为$1:2$, $N$为$BB_1$的中点, 则$|MN|$为\\blank{80}.", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -51892,7 +51910,9 @@ "001948": { "id": "001948", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AB=5$, $AD=2$, $AA_1=4$, 则异面直线$A_1C$与$BC_1$所成角的大小为\\blank{80}(尽量用向量法).", - "objs": [], + "objs": [ + "K0630002X" + ], "tags": [ "第六单元", "空间向量" @@ -51919,7 +51939,9 @@ "001949": { "id": "001949", "content": "平行六面体$ABCD-A_1B_1C_1D_1$中, $AB=1$, $AD=2$, $AA_1=3$, 且$\\angle BAD=\\angle DAA_1=\\angle BAA_1=60^\\circ$.\\\\ \n(1) 求$AC_1$;\\\\ \n(2) 求$\\angle CAC_1$.", - "objs": [], + "objs": [ + "K0625004X" + ], "tags": [ "第六单元", "空间向量" @@ -51944,7 +51966,9 @@ "001950": { "id": "001950", "content": "已知空间四边形$ABCD$中, $AB=AC$, $\\angle DAB=\\angle DAC$. 求证: $DA\\perp BC$.", - "objs": [], + "objs": [ + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -51969,7 +51993,9 @@ "001951": { "id": "001951", "content": "在一组基$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$下, 已知向量$\\overrightarrow{v_1}$的坐标为$(2,1,3)$, 又向量$\\overrightarrow{v_2}$与\n$\\overrightarrow{v_1}$平行, 其坐标为$(x,y,z)$, 则$x,y,z$应满足的关系为\\blank{80}.(只需写一个)", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -51994,7 +52020,9 @@ "001952": { "id": "001952", "content": "设空间向量$\\overrightarrow{u},\\overrightarrow{v},\\overrightarrow{w}$在一组基$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$下的坐标分别为$(1,2,3),\n(2,3,5),(3,5,7)$. 那么在基$\\overrightarrow{u},\\overrightarrow{v},\\overrightarrow{w}$下, 向量$\\overrightarrow{a}+\\overrightarrow{b}+2\\overrightarrow{c}$的坐标\n为\\blank{80}.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -52019,7 +52047,9 @@ "001953": { "id": "001953", "content": "设有三个空间向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$, 已知$\\overrightarrow{a}$与$\\overrightarrow{b}$不平行, $\\lambda,\\mu$是\n两个非零常数, 则$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$三个向量共面是$\\overrightarrow{c}=\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}$ 的\n\\blank{80}条件.", - "objs": [], + "objs": [ + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -52044,7 +52074,9 @@ "001954": { "id": "001954", "content": "已知$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$是三个不共面的向量, 向量$\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AC}=\\overrightarrow{b}$, $\\overrightarrow{AD}=\\overrightarrow{c}$, 若$D$点在平面$ABC$内的射影为$P$, 且$\\overrightarrow{AP}=x\\overrightarrow{a}+y\\overrightarrow{b}$, 则$x=$\\blank{180}.(用$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$之间的内积来表示)", - "objs": [], + "objs": [ + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -52069,7 +52101,9 @@ "001955": { "id": "001955", "content": "已知$\\overrightarrow{i},\\overrightarrow{j},\\overrightarrow{k}$是空间中两两夹角为$90^\\circ$的三个单位向量, $\\overrightarrow{a}=\\overrightarrow{i}+\\overrightarrow{j}$, $\\overrightarrow{b}=\\overrightarrow{i}-\\overrightarrow{j}$. 则与$\\overrightarrow{a},\\overrightarrow{b}$的夹角都为$60^\\circ$的单位向量为\\blank{180}.(用$\\overrightarrow{i},\\overrightarrow{j},\\overrightarrow{k}$表示)", - "objs": [], + "objs": [ + "K0626003X" + ], "tags": [ "第六单元", "空间向量" @@ -52094,7 +52128,10 @@ "001956": { "id": "001956", "content": "若$A,B,C,D$是四面体的四个顶点, $G$是底面$BCD$的重心, 若$\\overrightarrow{AB}=\\overrightarrow{b},\\overrightarrow{AC}=\\overrightarrow{c},\\overrightarrow{AD}=\\overrightarrow{d}$. $F$是$CD$中点.\\\\ \n(1) 用$\\overrightarrow{b},\\overrightarrow{c},\\overrightarrow{d}$表示$\\overrightarrow{AG}$;\\\\ \n(2) 若$E$是$AB$中点, $M$使得$\\overrightarrow{AM}=\\dfrac{3}{4}\\overrightarrow{AG}$. 求证: $E,F,M$ 共线.", - "objs": [], + "objs": [ + "K0624003X", + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -52119,7 +52156,10 @@ "001957": { "id": "001957", "content": "$O$是空间任一点, 若$\\overrightarrow{OG}=\\dfrac{1}{4}(\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}+\\overrightarrow{OD})$, 则称$G$是\n四面体$ABCD$的重心. 已知$G$是四面体$ABCD$ 的重心, $AG$与平面$BCD$交于点$P$, 求$AG:GP$.", - "objs": [], + "objs": [ + "K0624003X", + "K0626003X" + ], "tags": [ "第六单元", "空间向量" @@ -52144,7 +52184,9 @@ "001958": { "id": "001958", "content": "用向量法证明: 若空间四点$A,B,C,D$满足$\\angle ABC=\\angle BCD=\\angle CDA=\\angle DAB=90^\\circ$, 则$ABCD$是一个矩形.", - "objs": [], + "objs": [ + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -52169,7 +52211,9 @@ "001959": { "id": "001959", "content": "[选做]\n设$O,A,B$是不共线的三个点, 我们已证明过, 点$P$在直线$AB$上当且仅当存在和为$1$的两数$x,y$使得$\\overrightarrow{OP}=x\\overrightarrow{OA}+y\\overrightarrow{OB}$. 在空间中, 已知不共面的四点$O,A,B,C$. 类比上述命题, 提出一个结论, 并证明它.\\\\ \n你的结论为:\\\\ \n证明:", - "objs": [], + "objs": [ + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -52194,7 +52238,9 @@ "001960": { "id": "001960", "content": "已知空间四点$A(1,-2,1)$, $B(2,-1,2)$, $C(3,2,-1)$, $D(1,1,-1)$依次在第\\blank{20},\n\\blank{20},\\blank{20},\\blank{20}卦限.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第六单元", "空间向量" @@ -52219,7 +52265,9 @@ "001961": { "id": "001961", "content": "已知空间直角坐标系中点$P(a,b,c)$, 在后面的横线上依次写出下列点的坐标:\n$P$在$xOy$平面上的射影, 在$yOz$平面上的射影, 在$zOx$平面上的射影, 在$x$轴上的射影,\n在$y$轴上的射影, 在$z$轴上的射影.\\\\ \n\\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}.", - "objs": [], + "objs": [ + "K0627002X" + ], "tags": [ "第六单元", "空间向量" @@ -52244,7 +52292,9 @@ "001962": { "id": "001962", "content": "已知空间直角坐标系中点$P(a,b,c)$, 在后面的横线上依次写出下列点的坐标:\n$P$关于$xOy$的对称点, 关于$yOz$平面的对称点, 关于$zOx$平面的对称点, 关于$x$轴的对称点,\n关于$y$轴的对称点, 关于$z$轴的对称点, 关于原点的对称点.\n\\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}, \\blank{40}.", - "objs": [], + "objs": [ + "K0627003X" + ], "tags": [ "第六单元", "空间向量" @@ -52269,7 +52319,9 @@ "001963": { "id": "001963", "content": "%2-**\n已知向量$\\overrightarrow{a}=(1,2,3)$, $\\overrightarrow{b}=(3,0,-1)$, $\\overrightarrow{c}=(-\\dfrac{1}{5},1,-\\dfrac{3}{5})$, 下述结论\\\\ \n(1) $|\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}|=|\\overrightarrow{a}-\\overrightarrow{b}-\\overrightarrow{c}|$; (2) $(\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c})^2=\\overrightarrow{a}^2+\\overrightarrow{b}^2+\\overrightarrow{c}^2$;\\\\ \n(3) $(\\overrightarrow{a}\\cdot\\overrightarrow{b})\\overrightarrow{c}=(\\overrightarrow{b}\\cdot \\overrightarrow{c})\\overrightarrow{a}$; (4) $(\\overrightarrow{a}+\\overrightarrow{b})\\cdot \\overrightarrow{c}=\\overrightarrow{a}\\cdot (\\overrightarrow{b}-\\overrightarrow{c})$\\\\ \n中, 真命题有\\blank{50}.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -52294,7 +52346,9 @@ "001964": { "id": "001964", "content": "已知空间三点$A(1,2,3)$, $B(2,-1,5)$, $C(3,2,-4)$, 若四边形$ABCD$为平行四边形, 则$D$的坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -52319,7 +52373,10 @@ "001965": { "id": "001965", "content": "空间向量$(3,4,12)$的单位向量为\\blank{50}.", - "objs": [], + "objs": [ + "K0624002X", + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -52346,7 +52403,9 @@ "001966": { "id": "001966", "content": "若向量$(-2,3,m)$与$(n,-9,2)$平行, 则$m-n=$\\blank{50}.", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -52395,7 +52454,9 @@ "001968": { "id": "001968", "content": "%1-**\n已知空间四点$A(1,-2,1)$, $B(2,-1,2)$, $C(3,2,-1)$, $D(1,1,-1)$, 有一点$E$, 使$\\overrightarrow{DE}\\perp \\overrightarrow\n{AB}$, $\\overrightarrow{DE}\\perp \\overrightarrow{AC}$, 且$|\\overrightarrow{DE}|=\\sqrt{14}$同时成立. 则$E$点的坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -52420,7 +52481,9 @@ "001969": { "id": "001969", "content": "已知向量$\\overrightarrow{a}=(1,-3,2),\\overrightarrow{b}=(2,0,-8)$. 求单位向量$\\overrightarrow{c}$使得$\\overrightarrow{c}$与$\\overrightarrow{a},\\overrightarrow{b}$\n都垂直.", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -52469,7 +52532,9 @@ "001971": { "id": "001971", "content": "平行于$y$轴的直线的一个方向向量为\\blank{50}.", - "objs": [], + "objs": [ + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -52494,7 +52559,9 @@ "001972": { "id": "001972", "content": "$zOx$平面的一个法向量为\\blank{50}.", - "objs": [], + "objs": [ + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -52519,7 +52586,9 @@ "001973": { "id": "001973", "content": "已知直线$l$的一个方向向量为$\\overrightarrow{d}=(4,-8,6)$, 平面$\\alpha$的一个法向量为$\\overrightarrow{n}=(m,n,6)$, 若$l\\perp \\alpha$, 则$(m,n)=$\\blank{80}.", - "objs": [], + "objs": [ + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -52544,7 +52613,9 @@ "001974": { "id": "001974", "content": "已知平面内有两个向量$(-4,6,-1)$和$(4,3,2)$, 那么平面的单位法向量为\\blank{80}.", - "objs": [], + "objs": [ + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -52569,7 +52640,10 @@ "001975": { "id": "001975", "content": "已知点$A(5,1,3)$, $B(1,6,2)$, $C(5,0,4)$, $D(4,0,6)$, 则过$AD$且垂直于平面$ABC$的平面的一个法向量为\n\\blank{50}.", - "objs": [], + "objs": [ + "K0628001X", + "K0628005X" + ], "tags": [ "第六单元", "空间向量" @@ -52594,7 +52668,9 @@ "001976": { "id": "001976", "content": "在平面$\\alpha$上有三个点$(0,0,0)$, $(1,0,0)$和$(5,0,2)$, 在平面$\\beta$上有三个点$(4,1,5)$, $(2,2,3)$和$(1,-2,0)$. 已知平面$\\alpha$和平面$\\beta$的交线为$l$, 那么$l$的一个方向向量为\\blank{50}.", - "objs": [], + "objs": [ + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -52619,7 +52695,9 @@ "001977": { "id": "001977", "content": "已知直三棱柱$ABC-A_1B_1C_1$中, $\\angle ACB=90^\\circ$, $L$是$A_1C_1$的中点, $M$是$A_1B_1$的中点,\n$N$是$BC$的中点, 求证: $LN\\parallel MB$. (限定坐标法, 要求说明建系方法, 并作草图)", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -52644,7 +52722,9 @@ "001978": { "id": "001978", "content": "已知直三棱柱$ABC-A_1B_1C_1$中, $M$是$A_1B$的中点, $N$是$CC_1$的中点,\n求证: $MN$平行于平面$ABC$. (限定坐标法, 要求说明建系方法, 并作草图)", - "objs": [], + "objs": [ + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -52669,7 +52749,10 @@ "001979": { "id": "001979", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $E,F,G,H$分别是$DC,BC,A_1D_1,A_1B_1$的中点, $AB=2,AD=AA_1=1$.\\\\ \n(1) 求证: 平面$BDGH$平行于平面$EFB_1D_1$;\\\\ \n(2) 过$C_1$点作平面$BDGH$的平行平面, 分别交直线$BC$与$CD$于$P,Q$, 求线段$PQ$的长. (限定坐标法)", - "objs": [], + "objs": [ + "K0626001X", + "K0628005X" + ], "tags": [ "第六单元", "空间向量" @@ -52694,7 +52777,10 @@ "001980": { "id": "001980", "content": "已知向量$\\overrightarrow{a}=(3,5,2)$, $\\overrightarrow{b}=(m-1,m+1,m-2)$, 若向量$\\overrightarrow{a},\\overrightarrow{b}$所成角为锐角, 则$m$的取值范围为\\blank{80}.", - "objs": [], + "objs": [ + "K0624002X", + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -52719,7 +52805,9 @@ "001981": { "id": "001981", "content": "已知平面$\\alpha$过点$A(0,0,1),B(3,0,0)$, 且与平面$xOy$所成的二面角为$60^\\circ$, 则该平面的一个法向量为\\blank{80}.", - "objs": [], + "objs": [ + "K0631002X" + ], "tags": [ "第六单元", "空间向量" @@ -52816,7 +52904,9 @@ "001985": { "id": "001985", "content": "在正四面体$ABCD$中, $G$是三角形$ABC$的中心, $H$在线段$CD$上, $CH:HD=1:2$, $I$在线段$BD$上, $BI:ID=2:1$. 求$\\angle IGH$. (要求作草图, 给出建系过程, 用坐标法)", - "objs": [], + "objs": [ + "K0628002X" + ], "tags": [ "第六单元", "空间向量" @@ -93285,7 +93375,10 @@ "003624": { "id": "003624", "content": "在棱长为$10$的正方体$ABCD-A_1B_1C_1D_1$中,$P$为左侧面$ADD_1A_1$上一点, 已知点$P$到$A_1D_1$的距离为$3$, $P$到$AA_1$的距离为$2$, 则过点$P$且与$A_1C$平行的直线相交的正方体的面是\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,2) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw [dashed] (A1) -- (C);\n \\draw [dashed] (A1) ++ (0,-0.6) --++ (45:0.4) coordinate (P) --++ (0,0.6);\n \\draw (P) node [below] {$P$};\n \\end{tikzpicture}\n\\end{center}\n\\fourch{$ABCD$}{$BB_1C_1C$}{$CC_1D_1D$}{$AA_1B_1B$}", - "objs": [], + "objs": [ + "K0627002X", + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -93858,7 +93951,10 @@ "003647": { "id": "003647", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $M$为$BB_1$上一点, 已知$BM=2$, $CD=3$, $AD=4$, $AA_1=5$. \\\\\n(1) 求直线$A_1C$与平面$ABCD{}$的夹角;\\\\\n(2) 求点$A$到平面$A_1MC$的距离.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.5]\n \\draw [dashed] (0,0) coordinate (A) node [below] {$A$} -- (225:1.5) coordinate (B) node [below left] {$B$} (0,0) -- (4,0) coordinate (D) node [right] {$D$} (0,0) -- (0,5) coordinate (A1) node [above] {$A_1$};\n \\draw (B) --++ (4,0) node [below right] {$C$} coordinate (C) -- (D) --++ (0,5) node [above right] {$D_1$} coordinate (D1) -- (A1) --++ (225:1.5) node [left] {$B_1$} coordinate (B1) -- cycle;\n \\draw (D1) --++ (225:1.5) coordinate (C1) node [right] {$C_1$} (C1) -- (B1) (C) -- (C1);\n \\draw [dashed] ($(B)!0.4!(B1)$) node [left] {$M$} -- (A1) -- (C) -- cycle; \n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0629001X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -94647,7 +94743,9 @@ "003679": { "id": "003679", "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系. 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{AC_1}$的坐标是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth]\n\\draw [dashed] (0,0) node [left] {$D$} -- (3,0) node [above right] {$C$} (0,0) -- (225:2) node [below] {$A$} (0,0) -- (0,2) node [above left] {$D_1$};\n\\draw (3,0) -- (3,2) node [above] {$C_1$} -- (0,2) --++ (225:2) node [left] {$A_1$} --++ (0,-2) --++ (3,0) node [below right] {$B$} -- cycle;\n\\draw (0,2) ++ (225:2) --++ (3,0) node [right] {$B_1$} --+ (45:2) (3,0) ++ (225:2) --+ (0,2);\n\\draw [->] (3,0) -- (3.5,0) node [below] {$y$};\n\\draw [->] (0,2) -- (0,2.5) node [right] {$z$};\n\\draw [->] (225:2) -- (225:2.5) node [left] {$x$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0627002X" + ], "tags": [ "第六单元", "空间向量" @@ -104302,7 +104400,10 @@ "004096": { "id": "004096", "content": "如图, 在多面体$ABC-A_1B_1C_1$中, $AA_1$、$BB_1$、$CC_1$均垂直于平面$ABC$, $AA_1=4$, $CC_1=3$, $BB_1=AB=AC=2$, $\\angle BAC=120^\\circ$.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.6]\n \\draw [dashed] (0,0) node [above right] {$A$} coordinate (A) -- (2,0) node [right] {$C$} coordinate (C) (0,0) -- (0,4) node [above] {$A_1$} coordinate (A1) (0,0) --({-1-sqrt(6)/4},{-sqrt(6)/4}) node [left] {$B$} coordinate (B);\n \\draw (B) --++ (0,2) node [left] {$B_1$} coordinate (B1) (C) --++ (0,3) node [right] {$C_1$} coordinate (C1);\n \\draw (B) -- (C) (A1) -- (B1) -- (C1) -- cycle;\n \\end{tikzpicture}\n\\end{center}\n(1) 求$AB_1$与$A_1B_1C_1$所成角的大小;\\\\\n(2) 求二面角$A-A_1B_1-C_1$的大小.", - "objs": [], + "objs": [ + "K0630004X", + "K0631003X" + ], "tags": [ "第六单元", "空间向量" @@ -108109,7 +108210,10 @@ "004243": { "id": "004243", "content": "如图, 四棱柱$ABCD-A_1B_1C_1D_1$中, 侧棱$AA_1\\perp$底面$ABCD$, $AB\\parallel CD$, $AB\\perp AD$, $AD=DC=1$, $AA_1=AB=2$, $E$为棱$AA_1$的中点.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.6]\n \\draw [dashed] (0,0) node [left] {$A$} -- (2,0) node [right] {$B$} coordinate (B) (0,0) -- (-135:0.5) node [below left] {$D$} coordinate (D) (0,0) -- (0,2) node [above] {$A_1$};\n \\draw (D) --++ (1,0) node [below] {$C$} coordinate (C) -- (B) --++ (0,2) coordinate (B1) node [right] {$B_1$} --++ (-2,0) --++ (225:0.5) node [left] {$D_1$} -- (D);\n \\draw (C) --++ (0,2) node [above] {$C_1$}coordinate (C1) (B1) -- (C1) --++ (-1,0) (B1) -- (C);\n \\draw [dashed] (B1) -- (0,1) node [left] {$E$} -- (C);\n \\end{tikzpicture}\n\\end{center}\n(1) 求二面角$B_1-CE-C_1$的正弦值;\\\\\n(2) 设点$M$为线段$C_1E$上, 且直线$AM$与平面\n$AD{D_1}{A_1}$所成角正弦值为$\\dfrac{\\sqrt 2}6$, 求线段$AM$的长.", - "objs": [], + "objs": [ + "K0630004X", + "K0631002X" + ], "tags": [ "第六单元", "空间向量" @@ -110817,7 +110921,10 @@ "004348": { "id": "004348", "content": "如图, 空间几何体由两部分构成, 上部是一个底面半径为$1$, 高为$2$的圆锥, 下部是一个底面半径为$1$, 高为$2$的圆柱. 圆锥和圆柱的轴在同一直线上, 圆锥的下底面与圆柱的上底面重合. 点$P$是圆锥的顶点, $AB$是圆柱下底面的一条直径, $AA_1$、$BB_1$是圆柱的两条母线. $C$是弧$AB$的中点.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [left] {$A$} arc (180:360:1 and 0.3) node [right] {$B$} (0,0) -- (0,2) node [left] {$A_1$} (2,0) -- (2,2) node [right] {$B_1$} (0,2) -- (1,4) node [above] {$P$} -- (2,2) arc (360:180:1 and 0.3);\n \\draw [dashed] (0,0) arc (180:0:1 and 0.3) (0,2) arc (180:0:1 and 0.3);\n \\draw [dashed] (1,0) node [above right] {$O$} -- (1,4) (2,0) -- (0,0) -- (1,4);\n \\draw ({1+cos(-120)},{0.3*sin(-120)}) coordinate (C) node [below] {$C$};\n \\draw [dashed] (0,0) -- (C) -- (2,0) (1,0) -- (C) -- (1,4); \n \\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$PA_1$与$BC$所成的角的大小;\\\\\n(2) 求点$B_1$到平面$PAC$的距离.", - "objs": [], + "objs": [ + "K0629001X", + "K0630002X" + ], "tags": [ "第六单元", "空间向量" @@ -118828,7 +118935,10 @@ "004656": { "id": "004656", "content": "如左图, 在$\\text{Rt}\\triangle ABC$中, $\\angle C=90^\\circ$, $BC=3$, $AC=6$, $D$、$E$分别为$AC$、$AB$上的点, 且$DE\\parallel BC$, $DE=2$, 将$\\triangle ADE$沿$DE$折起到$\\triangle A_1DE$的位置, 使$A_1C\\perp CD$, 如右图.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.8]\n \\draw (0,0) node [left] {$C$} coordinate (C) -- (3,0) node [right] {$B$} coordinate (B) -- (0,6) node [above] {$A$} coordinate (A) -- cycle;\n \\draw ($(A)!{2/3}!(C)$) node [left] {$D$} coordinate (D) -- ($(A)!{2/3}!(B)$) node [right] {$E$} coordinate (E);\n \\end{tikzpicture}\n \\begin{tikzpicture}[scale = 0.8]\n \\draw (0,0) node [left] {$C$} coordinate (C) -- (3,0) node [right] {$B$} coordinate (B) -- (0,{2*sqrt(3)}) node [above] {$A_1$} coordinate (A1) -- cycle;\n \\draw (45:1) node [left] {$D$} coordinate (D) ++ (2,0) node [right] {$E$} coordinate (E);\n \\draw (B) -- (E) -- (A1);\n \\draw [dashed] (D) -- (E) (D) -- (C) (D) -- (A1) ($(A1)!0.5!(D)$) node [right] {$M$} -- (C);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C\\perp$平面$BCDE$;\\\\\n(2) 若$M$是$A_1D$的中点, 求$CM$与平面$A_1BE$所成角的大小.", - "objs": [], + "objs": [ + "K0628004X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -119906,7 +120016,10 @@ "004698": { "id": "004698", "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的底面为直角三角形且$\\angle ACB=90^\\circ$, 直角边$CA$、$CB$的长分别为$3$、$4$, 侧棱$AA_1$的长为$4$, 点$M$、$N$分别为线段$A_1B_1$、$C_1B_1$的中点.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.8]\n \\draw (0,0) node [left] {$C$} coordinate (C);\n \\draw (4,0) node [right] {$B$} coordinate (B);\n \\draw (-45:1.5) node [below] {$A$} coordinate (A);\n \\draw (A) ++ (0,4) node [above] {$A_1$} coordinate (A1);\n \\draw (B) ++ (0,4) node [right] {$B_1$} coordinate (B1);\n \\draw (C) ++ (0,4) node [left] {$C_1$} coordinate (C1);\n \\draw ($(A1)!0.5!(B1)$) node [below right] {$M$} coordinate (M);\n \\draw ($(C1)!0.5!(B1)$) node [above] {$N$} coordinate (N);\n \\draw (A) -- (B) -- (B1) -- (C1) -- (C) -- cycle;\n \\draw (C1) -- (A) -- (M) -- (N) (A1) -- (A) (A1) -- (C1) (A1) -- (B1);\n \\draw [dashed] (N) -- (C) -- (B);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $A,C,N,M$四点共面;\\\\\n(2) 求直线$AC_1$与平面$ACNM$所成角的大小.", - "objs": [], + "objs": [ + "K0626001X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -120952,7 +121065,10 @@ "004740": { "id": "004740", "content": "如图所示, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长$1$, 侧棱长$4$, $AA_1$中点为$E$, $CC_1$中点为$F$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n --++ (0,4) node [above right] {$C_1$} coordinate (C1)\n --++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,4) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,4);\n \\draw ($(A)!0.5!(A1)$) node [left] {$E$} coordinate (E); \n \\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n \\draw (B) -- (E) (D1) -- (B1) -- (F);\n \\draw [dashed] (D1) -- (F) (B) -- (D) -- (E) (B1) -- (D);\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $\\text{平面}BDE \\parallel \\text{平面}B_1D_1F$;\\\\\n(2) 连结$B_1D$, 求直线$B_1D$与平面$BDE$所成的角的大小.", - "objs": [], + "objs": [ + "K0628005X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -167457,7 +167573,7 @@ "content": "某商品欲分两次提价, 提价方案有三种: 方案甲是先提价$a\\%$, 再提价$b\\%$; 方案乙是先提价$b\\%$, 再提价$a\\%$; 方案丙是两次均提价$\\dfrac{a+b}2\\%$($a>b>0$), 则提价最多的方案是\\bracket{20}.\n\\fourch{甲}{乙}{丙}{三种方案一样}", "objs": [], "tags": [ - "第四单元" + "第一单元" ], "genre": "选择题", "ans": "", @@ -237404,7 +237520,9 @@ "009855": { "id": "009855", "content": "空间中有异面向量的概念吗? 为什么?", - "objs": [], + "objs": [ + "K0624001X" + ], "tags": [ "第六单元", "空间向量" @@ -237426,7 +237544,9 @@ "009856": { "id": "009856", "content": "如图, 请在图中找出三个不共面的向量.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,-0.2) node [below] {$C$} coordinate (C);\n\\draw (1,1.5) node [above] {$A$} coordinate (A);\n\\draw (A) ++ (3,1) node [above] {$A_1$} coordinate (A1);\n\\draw (B) ++ (3,1) node [below] {$B_1$} coordinate (B1);\n\\draw (C) ++ (3,1) node [below] {$C_1$} coordinate (C1);\n\\draw (B) -- (C) -- (A) -- cycle (C) -- (C1) -- (A1) -- (A);\n\\draw [dashed] (B) -- (B1) -- (C1) (A1) -- (B1);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624001X" + ], "tags": [ "第六单元", "空间向量" @@ -237448,7 +237568,9 @@ "009857": { "id": "009857", "content": "化简下列算式:\\\\\n(1) $3(2\\overrightarrow a-\\overrightarrow b-4\\overrightarrow c)-4(\\overrightarrow a-2\\overrightarrow b+3\\overrightarrow c)$;\\\\\n(2) $\\overrightarrow{OA}-[\\overrightarrow{OB}-(\\overrightarrow{AB}-\\overrightarrow{AC})]$.", - "objs": [], + "objs": [ + "K0624002X" + ], "tags": [ "第六单元", "空间向量" @@ -237470,7 +237592,9 @@ "009858": { "id": "009858", "content": "如图, 棱长为$a$的正四面体$ABCD$中, $E$为棱$AB$的中点. 求$\\overrightarrow{DC}\\cdot \\overrightarrow{DE}$与$\\overrightarrow{BC}\\cdot \\overrightarrow{DE}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw ({sqrt(3)/2},0,-0.5) node [right] {$C$} coordinate (C);\n\\draw ({-sqrt(3)/2},0,-0.5) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) node [above] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw ($(A)!0.5!(B)$) node [below left] {$E$} -- (D) (B) -- (D);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624002X" + ], "tags": [ "第六单元", "空间向量" @@ -237492,7 +237616,9 @@ "009859": { "id": "009859", "content": "设$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$是三个空间向量, 求证: $\\overrightarrow a\\cdot (\\overrightarrow b+\\overrightarrow c)=\\overrightarrow a\\cdot \\overrightarrow b+\n\\overrightarrow a\\cdot \\overrightarrow c$.", - "objs": [], + "objs": [ + "K0625002X" + ], "tags": [ "第六单元", "空间向量" @@ -237514,7 +237640,9 @@ "009860": { "id": "009860", "content": "下列命题是否为真命题? 如果是, 请说明理由; 如果不是, 请举出反例.\\\\\n(1) 设$A$、$B$、$C$、$D$是空间中的四个不同的点, 直线$AB$与$CD$是异面直线, 则向量$\\overrightarrow{AB}$与$\\overrightarrow{CD}$不共面;\\\\\n(2) 如果$\\overrightarrow a$、$\\overrightarrow b$是平面$\\alpha$上的互不平行的向量, 点$C$、$D$不在平面$\\alpha$上, 那么向量$\\overrightarrow{CD}$与向量$\\overrightarrow a$、$\\overrightarrow b$不共面;\\\\\n(3) 如果$\\overrightarrow a$、$\\overrightarrow b$是平面$\\alpha$上的互不平行的向量, 点$C$在平面$\\alpha$上, 点$D$不在平面$\\alpha$上, 那么向量$\\overrightarrow{CD}$与向量$\\overrightarrow a$、$\\overrightarrow b$不共面.", - "objs": [], + "objs": [ + "K0624001X" + ], "tags": [ "第六单元", "空间向量" @@ -237536,7 +237664,11 @@ "009861": { "id": "009861", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB:AA_1:AD=\n2: 1: 1$, $E$与$F$分别是棱$AB$与$DC$的中点. 设$\\overrightarrow{AA_1}=\\overrightarrow a,\\overrightarrow{AB}=\\overrightarrow b$,\n$\\overrightarrow{AD}=\\overrightarrow c$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (4,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-4,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (4,2) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-4,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (4,0) (D) --++ (0,2);\n\\draw [dashed] ($(A)!0.5!(B)$) node [below] {$E$} -- (D) ($(C)!0.5!(D)$) node [below] {$F$} -- (A1) (B) -- (D1);\n\\draw (B1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$表示$\\overrightarrow{BD_1}$、$\\overrightarrow{A_1F}$;\\\\\n(2) 求$\\overrightarrow{A_1F}\\cdot \\overrightarrow{B_1C}$;\\\\\n(3) 判断$\\overrightarrow{A_1F}$与$\\overrightarrow{DE}$是否垂直.", - "objs": [], + "objs": [ + "K0624003X", + "K0627005X", + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -237558,7 +237690,9 @@ "009862": { "id": "009862", "content": "讨论满足下列条件的点$P$的坐标$(x,y,z)$的特征:\n(1) 点$P$在坐标平面上;\n(2) 点$P$在坐标轴上.", - "objs": [], + "objs": [ + "K0627002X" + ], "tags": [ "第六单元", "空间向量" @@ -237580,7 +237714,9 @@ "009863": { "id": "009863", "content": "求向量$\\overrightarrow a=(0,1,0)$与$\\overrightarrow b=(1,-1,0)$的夹角的大小.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -237602,7 +237738,9 @@ "009864": { "id": "009864", "content": "已知向量$\\overrightarrow a=(-m,1,3)$平行于向量$\\overrightarrow b=(2,n,1)$, 求$m$、$n$.", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -237624,7 +237762,9 @@ "009865": { "id": "009865", "content": "试证明:\\\\\n(1) 两个平面垂直的充要条件是它们的法向量垂直;\\\\\n(2) 两个平面平行的充要条件是它们的法向量平行.", - "objs": [], + "objs": [ + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -237667,7 +237807,9 @@ "009867": { "id": "009867", "content": "如图, $\\triangle ABC$中, $AC=BC=\\dfrac{\\sqrt 2}2AB$, 平面$ABED\\perp$平面$ABC$, $ABED$是边长为$1$的正方形, $G$、$F$分别是$EC$、$BD$的中点. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2) node [left] {$D$} coordinate (D);\n\\draw (B) ++ (0,2) node [right] {$E$} coordinate (E);\n\\draw (A) -- (C) -- (B) -- (E) -- (D) -- cycle;\n\\draw (C) -- (D) (C) -- (E);\n\\draw [dashed] (A) -- (B) (B) -- (D);\n\\draw [dashed] ($(B)!0.5!(D)$) node [left] {$F$} -- ($(C)!0.5!(E)$) node [right] {$G$};\n\\end{tikzpicture}\n\\end{center}\n(1) $FG\\parallel$平面$ABC$;\\\\\n(2) $AC\\perp$平面$EBC$.", - "objs": [], + "objs": [ + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -237690,7 +237832,8 @@ "id": "009868", "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": [ - "K0619004B" + "K0619004B", + "K0629001X" ], "tags": [ "第六单元", @@ -237740,7 +237883,9 @@ "009870": { "id": "009870", "content": "如图, 四边形$ABCD$是矩形, $PA\\perp$平面$ABCD$, $E$是线段$PA$的中点. 已知$|PA|=2$, $|AB|=\\sqrt 3$, $|BC|=1$. 求异面直线$BE$与$PC$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,-1) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-1) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$E$} coordinate (E);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C) (A) -- (B) -- (C) (E) -- (B);\n\\draw [dashed] (A) -- (D) -- (C) (P) -- (D);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0630002X" + ], "tags": [ "第六单元", "空间向量" @@ -237762,7 +237907,10 @@ "009871": { "id": "009871", "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$是棱$AB$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (D1) -- (C) -- cycle (A1) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DA_1\\perp ED_1$;\\\\\n(2) 确定点$E$的位置, 使得直线$DA_1$与平面$CED_1$所成的角是$45^\\circ$.", - "objs": [], + "objs": [ + "K0627007X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -237784,7 +237932,9 @@ "009872": { "id": "009872", "content": "在正方体$ABCD-A'B'C'D'$中, $E$、$F$分别是$BC$、$CD$的中点. 求二面角$B-B'E-F$的大小.", - "objs": [], + "objs": [ + "K0631003X" + ], "tags": [ "第六单元", "空间向量" @@ -237806,7 +237956,9 @@ "009873": { "id": "009873", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 求平面$DA_1B$与平面$A_1B_1C_1D_1$所成二面角的正弦值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,3) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{3/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,3) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (45:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw (A1) -- (B);\n\\draw [dashed] (A1) -- (D) -- (B);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0631002X" + ], "tags": [ "第六单元", "空间向量" @@ -256825,7 +256977,10 @@ "010706": { "id": "010706", "content": "在长方体$ABCD-A'B'C'D'$中, $|AB|=4$, $|BC|=3$, $|AA'|=5$. 写出:\\\\(1) 与$\\overrightarrow{AC'}$有相等模的向量;\\\\\n(2) $\\overrightarrow{AB}$的相等向量;\\\\\n(3) 与$\\overrightarrow{AA'}$垂直的向量.", - "objs": [], + "objs": [ + "K0624002X", + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -256847,7 +257002,9 @@ "010707": { "id": "010707", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $\\overrightarrow{CA}=\\overrightarrow a$, $\\overrightarrow{CB}=\\overrightarrow b$, $\\overrightarrow{CC_1}=\\overrightarrow c$. 将向量$\\overrightarrow{A_1B}$表示为$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$的线性组合.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,0) ++ (0,0,{-1.5*sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) --++ (0,3) node [left] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,3) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,3) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed] (C) -- (A) (C) -- (B) (C) -- (C1);\n\\draw (A) -- (B) (A1) -- (B1) (C1) -- (A1) (C1) -- (B1) (A1) -- (B);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -256869,7 +257026,9 @@ "010708": { "id": "010708", "content": "如图, 在正方体$ABCD-A'B'C'D'$中, $E$是$A'C'$的中点, 点$F$在$AE$上, 且$|AF|=\\dfrac 12|EF|$. 试用向量$\\overrightarrow{AA'}$、$\\overrightarrow{AB}$与$\\overrightarrow{AD}$的线性组合表示$\\overrightarrow{AF}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (45:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,3) node [above right] {$C'$} coordinate (C1)\n--++ (-3,0) node [above left] {$D'$} coordinate (D1) --++ (225:{3/2}) node [left] {$A'$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,3) node [right] {$B'$} coordinate (B1) -- (B) (B1) --++ (45:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (45:{3/2}) node [below right] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw [dashed] (A1) -- (C1);\n\\draw ($(A1)!0.5!(C1)$) node [above] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (A);\n\\filldraw ($(A)!{1/3}!(E)$) circle (0.03) node [left] {$F$} coordinate (F); \n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -256891,7 +257050,10 @@ "010709": { "id": "010709", "content": "已知$\\overrightarrow a\\perp \\overrightarrow b$, $\\overrightarrow c$与$\\overrightarrow a$、$\\overrightarrow b$的夹角都是$60^\\circ$, 且$|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$, $|\\overrightarrow c|=3$. 计算:\\\\\n(1) $(3\\overrightarrow a-2\\overrightarrow b)\\cdot (\\overrightarrow b-3\\overrightarrow c)$;\\\\\n(2) $|\\overrightarrow a+2\\overrightarrow b-\\overrightarrow c|$.", - "objs": [], + "objs": [ + "K0624002X", + "K0625002X" + ], "tags": [ "第六单元", "空间向量" @@ -256913,7 +257075,9 @@ "010710": { "id": "010710", "content": "已知空间四边形$ABCD$中, $AB\\perp CD, AC\\perp BD$. 求证: $AD\\perp BC$.", - "objs": [], + "objs": [ + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -256935,7 +257099,9 @@ "010711": { "id": "010711", "content": "如图, 在四面体$ABCD$中, $E$、$M$、$N$分别是棱$AB$、$AC$、$AD$的中点, $E_1$、$M_1$、$N_1$分别是棱$CD$、$BD$、$BC$的中点, $G$是线段$EE_1$的中点. 试判断下列各组中的三点是否共线:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.4]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$D$} coordinate (D);\n\\draw (1.4,2.3) node [above] {$A$} coordinate (A);\n\\draw (2,-0.8) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(C)$) node [right] {$E_1$} coordinate (E1);\n\\filldraw ($(A)!0.5!(C)$) circle (0.03) node [above right] {$M$} coordinate (M);\n\\filldraw ($(A)!0.5!(D)$) circle (0.03) node [above right] {$N$} coordinate (N);\n\\filldraw ($(B)!0.5!(C)$) circle (0.03) node [below left] {$N_1$} coordinate (N1);\n\\filldraw ($(B)!0.5!(D)$) circle (0.03) node [below] {$M_1$} coordinate (M1);\n\\filldraw ($(E)!0.5!(E1)$) circle (0.03) node [above] {$G$} coordinate (G);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A) -- (C);\n\\draw [dashed] (E) -- (E1) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) $G$、$M$、$M_1$;\\\\\n(2) $G$、$N$、$N_1$.", - "objs": [], + "objs": [ + "K0625004X" + ], "tags": [ "第六单元", "空间向量" @@ -256957,7 +257123,9 @@ "010712": { "id": "010712", "content": "如图, $A$是$\\triangle BCD$所在平面外一点, $G$是$\\triangle BCD$的重心.求证: $\\overrightarrow{AG}=\\dfrac 13(\\overrightarrow{AB}+\\overrightarrow{AC}+\\overrightarrow{AD})$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw (1,1) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(D)$) node [below] {$E$} coordinate (E);\n\\draw ($(E)!{1/3}!(C)$) node [above right] {$G$} coordinate (G);\n\\draw (G) ++ (0.1,2.5) node [above] {$A$} coordinate (A);\n\\draw (A) -- (D) -- (C) -- cycle;\n\\draw [dashed] (A) -- (G) (D) -- (B) -- (A) (B) -- (C) (E) -- (C);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624002X" + ], "tags": [ "第六单元", "空间向量" @@ -256979,7 +257147,9 @@ "010713": { "id": "010713", "content": "如图, 在三棱锥$D-ABC$中, $\\angle DAC=\\angle BAC=60^\\circ$, $AC=1$, $AB=2$, $AD=3$.\\\\\n(1) 求$\\overrightarrow{AC}\\cdot \\overrightarrow{BD}$, 并说明异面直线$AC$与$BD$所成的角$\\theta$的大小在棱$BD$长度增大时是怎样变化的;\\\\\n(2) 若$AC\\perp BC$, 判断点$D$在平面$ABC$上的射影是否可能在直线$BC$上, 给出你的结论并加以证明.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0.5,0,{-0.5*sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ({3/2},{9/4},{-3*sqrt(3)/4}) node [above] {$D$} coordinate (D);\n\\draw (A) -- (B) (A) -- (D) (B) -- (D);\n\\draw [dashed] (C) -- (A) (C) -- (B) (C) -- (D);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0628002X" + ], "tags": [ "第六单元", "空间向量" @@ -257001,7 +257171,10 @@ "010714": { "id": "010714", "content": "在空间中还可以讨论一个向量$\\overrightarrow{AB}$在一个平面$\\alpha$上的投影. 如图, 若$\\overrightarrow a=\\overrightarrow{AB}$, 点$A$与点$B$在平面$\\alpha$上的投影分别是点$A'$与点$B'$, 则$\\overrightarrow a=\\overrightarrow{AB}$在平面$\\alpha$上的投影就是向量$\\overrightarrow{A'B'}$. 现在给定向量$\\overrightarrow a$、平面$\\alpha$以及平面$\\alpha$上的非零向量$\\overrightarrow b$. 设向量$\\overrightarrow a$在平面$\\alpha$上的投影是向量$\\overrightarrow{a'}$, 向量$\\overrightarrow{a'}$在向量$\\overrightarrow b$方向上的投影是向量$\\overrightarrow{a''}$. 求证:向量$\\overrightarrow{a''}$是向量$\\overrightarrow a$在向量$\\overrightarrow b$方向上的投影.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (3,0) --++ (1.5,1.5) --++ (-3,0) -- (0,0) ++ (0.5,0) node [above] {$\\alpha$};\n\\draw [->] (2,0.7) node [left] {$A'$} -- (3.2,1) node [right] {$B'$};\n\\draw [->] (2,1.9) node [left] {$A$} -- (3.2,1.7) node [right] {$B$};\n\\draw [dashed] (2,0.7) -- (2,1.9) (3.2,1) -- (3.2,1.7);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0625003X", + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -257023,7 +257196,9 @@ "010715": { "id": "010715", "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 设$\\overrightarrow{D_1A}=\\overrightarrow a$, $\\overrightarrow{D_1B_1}=\\overrightarrow b$, $\\overrightarrow{D_1C}=\\overrightarrow c$. 试用$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$表示$\\overrightarrow{D_1B}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0.2,1.5) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2.2,1.5) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0.2,1.5);\n\\draw [dashed] (D1) -- (A) (D1) -- (B) (D1) -- (C) (D1) -- (B1);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -257045,7 +257220,9 @@ "010716": { "id": "010716", "content": "已知$\\overrightarrow a$、$\\overrightarrow b$是空间的非零向量, 分析$\\overrightarrow a\\cdot \\overrightarrow b=|\\overrightarrow a|\\cdot|\\overrightarrow b|$与$\\overrightarrow a\\parallel \\overrightarrow b$的关系.", - "objs": [], + "objs": [ + "K0625003X" + ], "tags": [ "第六单元", "空间向量" @@ -257067,7 +257244,10 @@ "010717": { "id": "010717", "content": "在正方体$ABCD-A'B'C'D'$中, $E$是面$A'B'C'D'$的中心. 求下列各式中实数$\\lambda$、$\\mu$、$\\nu$的值:\\\\\n(1) $\\overrightarrow{BD'}=\\lambda \\overrightarrow{AD}+\\mu \\overrightarrow{AB}+ \\nu \\overrightarrow{AA'}$;\\\\\n(2) $\\overrightarrow{AE}=\\lambda \\overrightarrow{AD}+\\mu \\overrightarrow{AB}+\\nu \\overrightarrow{AA'}$.", - "objs": [], + "objs": [ + "K0624003X", + "K0626003X" + ], "tags": [ "第六单元", "空间向量" @@ -257089,7 +257269,10 @@ "010718": { "id": "010718", "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $BD_1$交平面$ACB_1$于点$E$. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (55:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,3) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (235:{3/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,3) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (55:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (55:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw (A) -- (B1) -- (C);\n\\draw [dashed] (A) -- (C) (B) -- (D1) ($(A)!0.5!(C)$) coordinate (O) -- (B1);\n\\filldraw ($(O)!{1/3}!(B1)$) circle (0.03) node [right] {$E$} coordinate (E);\n\\end{tikzpicture}\n\\end{center}\n(1) $BD_1\\perp$平面$ACB_1$;\\\\\n(2) $|BE|=\\dfrac 12|ED_1|$.", - "objs": [], + "objs": [ + "K0627005X", + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -257111,7 +257294,9 @@ "010719": { "id": "010719", "content": "在平面上有如下命题: ``若$O$为直线$AB$外的一点, 则点$P$在直线$AB$上的充要条件是: 存在实数$\\lambda$、$\\mu$, 满足$\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$, 且$\\lambda +\\mu=1$.'' 类比此命题, 给出空间某点在某一平面上的充要条件并加以证明.", - "objs": [], + "objs": [ + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -257133,7 +257318,9 @@ "010720": { "id": "010720", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$BB_1$、$D_1B_1$的中点. 求证: $EF\\perp$平面$B_1AC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (3,0) node [below right] {$B$} coordinate (B) --++ (55:{3/2}) node [right] {$C$} coordinate (C)\n--++ (0,3) node [above right] {$C_1$} coordinate (C1)\n--++ (-3,0) node [above left] {$D_1$} coordinate (D1) --++ (235:{3/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (3,3) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (55:{3/2}) (B1) --++ (-3,0);\n\\draw [dashed] (A) --++ (55:{3/2}) node [left] {$D$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw (A) -- (B1) -- (C) (B1) -- (D1);\n\\draw [dashed] (A) -- (C);\n\\filldraw ($(B)!{1/2}!(B1)$) circle (0.03) node [right] {$E$} coordinate (E);\n\\filldraw ($(B1)!{1/2}!(D1)$) circle (0.03) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -257155,7 +257342,11 @@ "010721": { "id": "010721", "content": "在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$DD_1$、$DB$的中点, 点$G$在棱$CD$上, $|CG|=\\dfrac 14|CD|$, $H$是$C_1G$的中点.\\\\\n(1) 求证: $EF\\perp B_1C$;\\\\\n(2) 求$EF$与$C_1G$所成角的余弦值;\\\\\n(3) 求线段$FH$的长.", - "objs": [], + "objs": [ + "K0627005X", + "K0627007X", + "K0630002X" + ], "tags": [ "第六单元", "空间向量" @@ -257177,7 +257368,9 @@ "010722": { "id": "010722", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的棱长$|AB|=14$, $|AD|=6$, $|AA_1|=10$, 以这个长方体的顶点$A$为坐标原点, 分别以射线$AB$、$AD$、$AA_1$为$x$轴、$y$轴、$z$轴的正半轴, 建立空间直角坐标系. 求长方体各顶点的坐标.", - "objs": [], + "objs": [ + "K0627002X" + ], "tags": [ "第六单元", "空间向量" @@ -257199,7 +257392,9 @@ "010723": { "id": "010723", "content": "已知$PA$垂直于正方形$ABCD$所在的平面, $M$、$N$分别是$AB$、$PC$的中点, 且$|PA|=|AD|$, 分别以射线$AB$、$AD$、$AP$为$x$轴、$y$轴、$z$轴的正半轴, 建立空间直角坐标系. 求向量$\\overrightarrow{MN}$、$\\overrightarrow{DC}$的坐标表示.", - "objs": [], + "objs": [ + "K0627004X" + ], "tags": [ "第六单元", "空间向量" @@ -257221,7 +257416,10 @@ "010724": { "id": "010724", "content": "已知$\\overrightarrow a=\\overrightarrow i+\\overrightarrow j-4\\overrightarrow k, \\overrightarrow b=\\overrightarrow i-2\\overrightarrow j+2\\overrightarrow k$. 求:\\\\\n(1) 向量$\\overrightarrow a$与$\\overrightarrow b$的夹角的大小;\\\\\n(2) 向量$\\overrightarrow a$与$\\overrightarrow b$所在直线的夹角的大小.", - "objs": [], + "objs": [ + "K0627005X", + "K0628002X" + ], "tags": [ "第六单元", "空间向量" @@ -257243,7 +257441,9 @@ "010725": { "id": "010725", "content": "已知平行四边形$ABCD$中的三个顶点的坐标分别为$A(1, 2, 3)$、$B(2, -1, 5)$与$C(3, 2, -5)$, 求顶点$D$的坐标.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -257265,7 +257465,9 @@ "010726": { "id": "010726", "content": "设$\\overrightarrow a=(a_1, a_2, a_3)$, $\\overrightarrow b=(b_1, b_2, b_3)$, 且$\\overrightarrow a\\ne \\overrightarrow b$. 记$|\\overrightarrow a-\\overrightarrow b|=m$, 求$\\overrightarrow a-\\overrightarrow b$与$x$轴正方向向量夹角的余弦值.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -257287,7 +257489,9 @@ "010727": { "id": "010727", "content": "在$\\triangle ABC$中, 已知$\\overrightarrow{AB}=(2, 4, 0)$, $\\overrightarrow{BC}=(-1, 3, 0)$. 求$\\angle ABC$的大小.", - "objs": [], + "objs": [ + "K0628002X" + ], "tags": [ "第六单元", "空间向量" @@ -257330,7 +257534,10 @@ "010729": { "id": "010729", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $|CA|=|CB|=1$,\n$\\angle BCA=90^\\circ$ ,$|AA_1|=2$, $M$、$N$分别是$A_1B_1$、$A_1A$的中点. 建立适当的空间直角坐标系, 解决如下问题:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C1);\n\\draw ($(A)!0.5!(A1)$) node [left] {$N$} coordinate (N);\n\\draw ($(A1)!0.5!(B1)$) node [below] {$M$} coordinate (M);\n\\draw (C1) -- (M) (A1) -- (B) (N) -- (B);\n\\draw (A) -- (B) -- (B1) -- (C1) -- (A1) -- cycle;\n\\draw (A1) -- (B1);\n\\draw [dashed] (A) -- (C) -- (B) (B1) -- (C) (C1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{BN}$的模;\\\\\n(2) 求$\\cos \\langle \\overrightarrow{BA_1}, \\overrightarrow{CB_1}\\rangle$;\\\\\n(3) 求证: $A_1B\\perp C_1M$.", - "objs": [], + "objs": [ + "K0627005X", + "K0627007X" + ], "tags": [ "第六单元", "空间向量" @@ -257352,7 +257559,9 @@ "010730": { "id": "010730", "content": "在正四棱柱$ABCD-A_1B_1C_1D_1$中, $|AA_1|=2|AB|=2$, $E$为$AA_1$的中点. 求异面直线$BE$与$CD_1$所成角的大小.", - "objs": [], + "objs": [ + "K0630002X" + ], "tags": [ "第六单元", "空间向量" @@ -257374,7 +257583,9 @@ "010731": { "id": "010731", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$、$P$分别是$CC_1$、$B_1C_1$、$C_1D_1$的中点. 求证: 平面$MNP\\parallel$平面$A_1BD$.", - "objs": [], + "objs": [ + "K0628005X" + ], "tags": [ "第六单元", "空间向量" @@ -257396,7 +257607,9 @@ "010732": { "id": "010732", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 求$BB_1$与平面$ACD_1$所成角的大小.", - "objs": [], + "objs": [ + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -257418,7 +257631,9 @@ "010733": { "id": "010733", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$的各条棱长均为$a$, $D$是棱$CC_1$的中点. 求证: 平面$AB_1D\\perp$平面$ABB_1A_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C1);\n\\draw (B) -- (B1) -- (A1) (B1) -- (C1);\n\\draw (A) -- (B1) -- ($(C)!0.5!(C1)$) node [right] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (C1) -- (A1) -- cycle;\n\\draw [dashed] (A) -- (D) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0628005X" + ], "tags": [ "第六单元", "空间向量" @@ -257460,8 +257675,11 @@ }, "010735": { "id": "010735", - "content": "如图, 在直棱柱$ABC-A_1B_1C_1$中, $|AA_1|=|AB|=|AC|=2$, $AB\\perp AC$, $D$、$E$、$F$分别是$A_1B_1$、$CC_1$、$BC$的中点.\n(1) 求$AE$与平面$DEF$所成角的大小;\\\\\n(2) 求$A$到平面$DEF$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,-2) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C1);\n\\draw (A) -- (B) -- (B1) -- (C1) -- (A1) -- cycle (A1) -- (B1);\n\\draw [dashed] (C1) -- (C) (A) -- (C) -- (B);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(C1)$)node [above left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw [dashed] (A) -- (E) (E) -- (D) (E) -- (F) (D) -- (F);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "content": "如图, 在直棱柱$ABC-A_1B_1C_1$中, $|AA_1|=|AB|=|AC|=2$, $AB\\perp AC$, $D$、$E$、$F$分别是$A_1B_1$、$CC_1$、$BC$的中点.\\\\\n(1) 求$AE$与平面$DEF$所成角的大小;\\\\\n(2) 求$A$到平面$DEF$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,-2) node [below] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C1);\n\\draw (A) -- (B) -- (B1) -- (C1) -- (A1) -- cycle (A1) -- (B1);\n\\draw [dashed] (C1) -- (C) (A) -- (C) -- (B);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(C1)$)node [above left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw [dashed] (A) -- (E) (E) -- (D) (E) -- (F) (D) -- (F);\n\\end{tikzpicture}\n\\end{center}", + "objs": [ + "K0629001X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -257483,7 +257701,9 @@ "010736": { "id": "010736", "content": "如图, 在空间四边形$ABCD$中, $|AC|=|AD|$, $\\angle BAC=\\angle BAD$. 求证: $CD\\perp AB$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,-1) node [right] {$C$} coordinate (C);\n\\draw (2,1,0) node [above] {$D$} coordinate (D);\n\\draw (A) -- (D) -- (C);\n\\draw (1,-1,1) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C) (B) -- (D);\n\\draw [dashed] (A) -- (C); \n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0625004X" + ], "tags": [ "第六单元", "空间向量" @@ -257504,8 +257724,11 @@ }, "010737": { "id": "010737", - "content": "如图, 在三棱锥$P-ABC$中, $PA\\perp$平面$ABC$,$ AB\\perp AC$, $|PA|=|AC|=\\dfrac 12|AB|$, $M$、$S$分别为$PB$、$BC$的中点, $N$为$AB$上一点, $|BN|=3|NA|$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,4) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$S$} coordinate (S);\n\\draw ($(P)!0.5!(B)$) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.25!(B)$) node [left] {$N$} coordinate (N);\n\\draw (B) -- (P) -- (C) -- cycle (C) -- (M);\n\\draw [dashed] (B) -- (A) -- (C) (A) -- (P) (M) -- (N) -- (S) (N) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CM\\perp SN$;\\\\\n(2) 求二面角$PBCA$的大小.", - "objs": [], + "content": "如图, 在三棱锥$P-ABC$中, $PA\\perp$平面$ABC$,$ AB\\perp AC$, $|PA|=|AC|=\\dfrac 12|AB|$, $M$、$S$分别为$PB$、$BC$的中点, $N$为$AB$上一点, $|BN|=3|NA|$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,4) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$S$} coordinate (S);\n\\draw ($(P)!0.5!(B)$) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.25!(B)$) node [left] {$N$} coordinate (N);\n\\draw (B) -- (P) -- (C) -- cycle (C) -- (M);\n\\draw [dashed] (B) -- (A) -- (C) (A) -- (P) (M) -- (N) -- (S) (N) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CM\\perp SN$;\\\\\n(2) 求二面角$P-BC-A$的大小.", + "objs": [ + "K0627007X", + "K0631003X" + ], "tags": [ "第六单元", "空间向量" @@ -257549,7 +257772,9 @@ "010739": { "id": "010739", "content": "过边长为$1$的正方形$ABCD$的顶点$A$, 作长度为$1$的线段$AE\\perp$平面$ABCD$. 求平面$ADE$与平面$BCE$所成二面角的大小.", - "objs": [], + "objs": [ + "K0631002X" + ], "tags": [ "第六单元", "空间向量" @@ -257571,7 +257796,9 @@ "010740": { "id": "010740", "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别为棱$AA_1$、$BB_1$的中点, $G$为棱$A_1B_1$上的一点. 求点$G$到平面$D_1EF$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\draw ($(A)!0.5!(A1)$)node [left] {$E$} coordinate (E) -- ($(B)!0.5!(B1)$)node [right] {$F$} coordinate (F);\n\\filldraw ($(A1)!0.8!(B1)$) circle (0.03) node [above] {$G$} coordinate (G);\n\\draw [dashed] (D1) -- (E) (D1) -- (F);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0629001X" + ], "tags": [ "第六单元", "空间向量" @@ -304874,7 +305101,9 @@ "030452": { "id": "030452", "content": "已知空间向量$\\overrightarrow{a}=(4, -1, x)$, $\\overrightarrow{b}=(2, 1, 1)$, $\\overrightarrow{c}=(1,2,1)$, 若$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$共面, 求实数$\\lambda$的值.", - "objs": [], + "objs": [ + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -304896,7 +305125,9 @@ "030453": { "id": "030453", "content": "如图所示, 在长方体$ABCD-A_1B_1C_1D_1$中, $M$为$DD_1$的中点, $N$在$AC$上, 且$AN:NC=2:1$, $E$为$BM$的中点, 求证:$A_1,E,N$三点共线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{3}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(D)!0.5!(D1)$) node [right] {$M$} coordinate (M);\n\\draw ($(A)!{2/3}!(C)$) node [below] {$N$} coordinate (N);\n\\draw [dashed] (A) -- (C) (B) -- (M);\n\\draw ($(B)!0.5!(M)$) node [above] {$E$} coordinate (E);\n\\filldraw (E) circle (0.03) (N) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0627006X" + ], "tags": [ "第六单元", "空间向量" @@ -304918,7 +305149,10 @@ "030454": { "id": "030454", "content": "给出下列四个命题:\\\\\n\\textcircled{1} 若存在实数$x,y$, 使$\\overrightarrow{p}=x\\overrightarrow{a}+y\\overrightarrow{b}$, 则$\\overrightarrow{p}$与$\\overrightarrow{a},\\overrightarrow{b}$共面;\\\\\n\\textcircled{2} 若$\\overrightarrow{p}$与$\\overrightarrow{a},\\overrightarrow{b}$共面, 则存在实数$x,y$, 使$\\overrightarrow{p}=x\\overrightarrow{a}+y\\overrightarrow{b}$;\\\\\n\\textcircled{3} 若存在实数$x,y$, 使$\\overrightarrow{MP}=x\\overrightarrow{MA}+y\\overrightarrow{MB}$, 则$P,M,A,B$共面;\\\\\n\\textcircled{4} 若点$P, M, A, B$共面, 则存在实数$x,y$, 使$\\overrightarrow{MP}=x\\overrightarrow{MA}+y\\overrightarrow{MB}$.\\\\\n其中\\blank{50}是真命题.(填序号)", - "objs": [], + "objs": [ + "K0624001X", + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -304940,7 +305174,9 @@ "030455": { "id": "030455", "content": "如图所示, 在平行六面体$ABCD-A_1B_1C_1D_1$中, $AC$与$BD$的交点为$O$, 点$M$在$BC_1$上, 且$BM=2MC_1$, 则下列向量中与$\\overrightarrow{OM}$相等的向量是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,y = (80:1)]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw ($(B)!0.5!(C1)$) node [right] {$M$} coordinate (M);\n\\draw (B) -- (C1);\n\\draw [dashed] (A) -- (C) (B) -- (D) (O) -- (M);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$-\\dfrac 12 \\overrightarrow{AB}+\\dfrac 76 \\overrightarrow{AD}+\\dfrac 23 \\overrightarrow{AA_1}$}{$\\dfrac 12 \\overrightarrow{AB}+\\dfrac 16 \\overrightarrow{AD}+\\dfrac 13 \\overrightarrow{AA_1}$}{$-\\dfrac 12 \\overrightarrow{AB}+\\dfrac 56 \\overrightarrow{AD}+\\dfrac 13 \\overrightarrow{AA_1}$}{$-\\dfrac 12 \\overrightarrow{AB}+\\dfrac 16 \\overrightarrow{AD}+\\dfrac 23 \\overrightarrow{AA_1}$}", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -304962,7 +305198,10 @@ "030456": { "id": "030456", "content": "以正方体$ABCD-A_1B_1C_1D_1$的对角线的交点为坐标原点$O$建立右手系的空间直角坐标系$O-xyz$, 其中$A(1,\\sqrt{2},0)$, $B(-1,\\sqrt{2},0)$, $D(1,0,-\\sqrt{2})$, 求点$A_1$的坐标.", - "objs": [], + "objs": [ + "K0624003X", + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -304984,7 +305223,10 @@ "030457": { "id": "030457", "content": "如图所示, 在空间直角坐标系$O-xyz$中, 正方体$ABCD-A_1B_1C_1D_1$\n的棱长为$1$, 顶点$A$位于坐标原点, 若$E$是棱$B_1C_1$的中点, $F$是侧面$CDD_1C_1$的中心.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-120:0.5)]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A(O)$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (B) --++ (0.5,0,0) node [right] {$x$} coordinate (x);\n\\draw [->] (D) --++ (0,0,-0.5) node [right] {$y$} coordinate (y);\n\\draw [->] (A1) --++ (0,0.5,0) node [left] {$z$} coordinate (z);\n\\filldraw ($(B1)!0.5!(C1)$) node [above] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(C)!0.5!(D1)$) node [left] {$F$} coordinate (F) circle (0.03);\n\\draw [dashed] (A) -- (E) -- (F) (A) -- (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$E, F$的坐标及$|\\overrightarrow{EF}|$;\\\\\n(2) 求向量$\\overrightarrow{EF}$在$\\overrightarrow{DC}$方向上的投影及数量投影.", - "objs": [], + "objs": [ + "K0627004X", + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -305006,7 +305248,10 @@ "030458": { "id": "030458", "content": "已知$E, F, G, H$分别是空间四边形$ABCD$的边$AB, BC, CD, DA$的中点.\\\\\n(1) 用向量法证明$E, F, G, H$四点共面;\\\\\n(2) 设$M$是$EG$和$FH$的交点, 求证; 对空间任一点$O$, 有$\\overrightarrow{OM}=\\dfrac 14(\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}+\\overrightarrow{OD})$.", - "objs": [], + "objs": [ + "K0624002X", + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -305028,7 +305273,10 @@ "030459": { "id": "030459", "content": "如图所示, 在正方体$ABCD-A_1B_1C_1D_1$中,$E, F, G, H, K, L$分别是$AB, BB_1, B_1C_1, C_1D_1, D_1D, DA$各棱的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(B1)$) node [right] {$F$} coordinate (F);\n\\draw ($(B1)!0.5!(C1)$) node [below right] {$G$} coordinate (G);\n\\draw ($(C1)!0.5!(D1)$) node [above] {$H$} coordinate (H);\n\\draw ($(D)!0.5!(D1)$) node [left] {$K$} coordinate (K);\n\\draw ($(A)!0.5!(D)$) node [left] {$L$} coordinate (L);\n\\draw (E) -- (F) -- (G) -- (H);\n\\draw [dashed] (H) -- (K) -- (L) -- (E);\n\\draw [dashed] (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $E, F, G, H, K, L$共面;\\\\ \n(2) 求证: $A_1C\\perp$平面$EFGHKL$.", - "objs": [], + "objs": [ + "K0626001X", + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -305050,7 +305298,10 @@ "030460": { "id": "030460", "content": "如图所示, 在直三棱柱$ABC-A_1B_1C_1$中, $\\angle BAC=90^\\circ$, $AB=AC=AA_1=2$, $M$为$AB$的中点, $N$为$B_1C_1$的中点, $H$是$A_1B_1$的中点, $P$是$BC_1$与$B_1C$的交点, $Q$是$A_1N$与$C_1H$的交点.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5, z = (-140:0.5)]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (-2,0,0) node [left] {$C$} coordinate (C);\n\\draw (A) --++ (0,2,0) node [right] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,2,0) node [below right] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,2,0) node [left] {$C_1$} coordinate (C1);\n\\draw (C) -- (B) -- (A) (C1) -- (B1) -- (A1) (C1) -- (A1);\n\\draw ($(A)!0.5!(B)$) node [below right] {$M$} coordinate (M);\n\\draw ($(A1)!0.5!(B1)$) node [below right] {$H$} coordinate (H);\n\\draw ($(C1)!0.5!(B1)$) node [below left] {$N$} coordinate (N);\n\\draw ($(C1)!{2/3}!(H)$) node [below] {$Q$} coordinate (Q);\n\\draw ($(B)!0.5!(C1)$) node [left] {$P$} coordinate (P);\n\\draw (A1) -- (M) (C1) -- (H) (A1) -- (N) (B) -- (C1) (C) -- (B1);\n\\draw [dashed] (C) -- (M) (C) -- (A) (A1) -- (C) (Q) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C\\perp BC_1$;\\\\\n(2) 求证: $PQ\\parallel$平面$A_1CM$.", - "objs": [], + "objs": [ + "K0627007X", + "K0628004X" + ], "tags": [ "第六单元", "空间向量" @@ -305072,7 +305323,11 @@ "030461": { "id": "030461", "content": "如图所示, 在四棱锥$P-ABCD$中, 底面$ABCD$为梯形, $AB=AD=PD=2$, $DC=4$, $AB\\parallel DC$, $\\angle ADC=\\dfrac \\pi 2$, $PD\\perp$平面$ABCD$, $E, F$分别为$PD, PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(D)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (P) -- (A) -- (B) -- (C) (P) -- (C) (P) -- (B) (F) -- (B);\n\\draw [dashed] (D) -- (A) (D) -- (P) (D) -- (C) (A) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 判断直线$AE$与$BF$的位置关系, 并说明理由;\\\\ \n(2) 求二面角$P-BC-A$的余弦值;\\\\ \n(3) 求点$E$到平面$PBC$的距离.", - "objs": [], + "objs": [ + "K0624002X", + "K0629001X", + "K0631003X" + ], "tags": [ "第六单元", "空间向量" @@ -305094,7 +305349,10 @@ "030462": { "id": "030462", "content": "如图, 已知圆柱$OO_1$, $A$是圆$O_1$上的动点, $AO_1=1$, $OO_1=2$, $P,Q$为圆$O$上的两个定点, 且满足$PQ=\\sqrt{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) arc (180:360:1 and 0.3);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.3);\n\\draw (0,2) ellipse (1 and 0.3);\n\\draw (-1,0) -- (-1,2) (1,0) -- (1,2);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,2) node [left] {$O_1$} coordinate (O1);\n\\draw ({cos(-50)},{0.3*sin(-50)}) node [below] {$P$} coordinate (P);\n\\draw ({cos(40)},{0.3*cos(40)}) node [above right] {$Q$} coordinate (Q);\n\\draw (({cos(-110)},{2+0.3*sin(-110)}) node [below left] {$A$} coordinate (A);\n\\draw [dashed] (O) -- (O1) (O) -- (A) (O1) -- (P) (O1) -- (Q) (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\overrightarrow{AO_1}=\\overrightarrow{OP}$或$\\overrightarrow{AO_1}=\\overrightarrow{OQ}$时, 求证: $AO\\parallel$平面$O_1PQ$;\\\\\n(2) 当直线$AO$与平面$O_1PQ$所成角的正弦值取最大值时, 求三棱锥$A-O_1PQ$的体积.", - "objs": [], + "objs": [ + "K0628004X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量" @@ -305116,7 +305374,10 @@ "030463": { "id": "030463", "content": "在空间直角坐标系中, 点$A(-1,,1,-2)$关于原点的对称点为点$B$, 则$|AB|=$\\blank{50}.", - "objs": [], + "objs": [ + "K0627003X", + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -305137,8 +305398,10 @@ }, "030464": { "id": "030464", - "content": "已知$O, A, B, C$为空间中不共面的四点, 且$\\overrightarrow{OP}=\\dfrac 34\\overrightarrow{OA}+\\dfrac 18\\overrightarrow{OB}+t\\overrightarrow{OC}$, 四点共面, 则实数$t=$\\blank{50}.", - "objs": [], + "content": "已知$O, A, B, C$为空间中不共面的四点, 且$\\overrightarrow{OP}=\\dfrac 34\\overrightarrow{OA}+\\dfrac 18\\overrightarrow{OB}+t\\overrightarrow{OC}$, 若$A,B,C,P$四点共面, 则实数$t=$\\blank{50}.", + "objs": [ + "K0626001X" + ], "tags": [ "第六单元", "空间向量" @@ -305160,7 +305423,9 @@ "030465": { "id": "030465", "content": "给定点$A(1,0,0)$, $B(3,1,1)$, $C(2,0,1)$与点$D(5,-4,3)$, 则点$D$到平面$ABC$的距离为\\blank{50}.", - "objs": [], + "objs": [ + "K0629001X" + ], "tags": [ "第六单元", "空间向量" @@ -305181,7 +305446,7 @@ }, "030466": { "id": "030466", - "content": "如图所示, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 底面$ABCD$是边长为$1$的正方形, $AA_1$的长度为$2$, 且$\\angle A_1AB=\\angle A_1AD=\\dfrac \\pi 3$, \n, 则\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = (65:1)]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [below right] {$D$} 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 [above] {$B_1$} 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\\end{tikzpicture}\n\\end{center}", + "content": "如图所示, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 底面$ABCD$是边长为$1$的正方形, $AA_1$的长度为$2$, 且$\\angle A_1AB=\\angle A_1AD=\\dfrac \\pi 3$, \n, 则$A_1C$的长度为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = (65:1)]\n\\def\\l{1.5}\n\\def\\m{1.5}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [below right] {$D$} 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 [above] {$B_1$} 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\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ "第六单元", @@ -305203,8 +305468,11 @@ }, "030467": { "id": "030467", - "content": "已知空间中三点$A(0, 1, 0)$, $B(2, 2, 0)$, $C-1, 3, 1)$, 则下列结论中正确的有\\bracket{20}.\n\\onech{平面$ABC$的一个法向量是$(1,-2,5)$}{$\\overrightarrow{AB}$的一个单位向量的坐标是$(1,1,0)$}{$|\\overrightarrow{AB}|=2$}{$\\overrightarrow{AB}$与$\\overrightarrow{AC}$是共线向量}", - "objs": [], + "content": "已知空间中三点$A(0, 1, 0)$, $B(2, 2, 0)$, $C(-1, 3, 1)$, 则下列结论中正确的是\\bracket{20}.\n\\onech{平面$ABC$的一个法向量是$(1,-2,5)$}{$\\overrightarrow{AB}$的一个单位向量的坐标是$(1,1,0)$}{$|\\overrightarrow{AB}|=2$}{$\\overrightarrow{AB}$与$\\overrightarrow{AC}$是共线向量}", + "objs": [ + "K0624002X", + "K0628001X" + ], "tags": [ "第六单元", "空间向量" @@ -305226,7 +305494,10 @@ "030468": { "id": "030468", "content": "如图所示, 在正三棱台$ABC-A_1B_1C_1$中, $AB=3AA_1=\\dfrac 32 A_1B_1=3$, 记侧面$ABB_1A_1$与底面$ABC$, 侧面$ABB_1A_1$与侧面$BCC_1B_1$, 以及侧面$ABB_1A_1$与截面$A_1BC$所成的锐二面角的平面角分别为$\\alpha,\\beta,\\gamma$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw ({-sqrt(3)/2},0,{-1/2}) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)/2},0,{-1/2}) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)},0) coordinate (P);\n\\draw ($(A)!{1/3}!(P)$) node [above left] {$A_1$} coordinate (A_1);\n\\draw ($(B)!{1/3}!(P)$) node [below right] {$B_1$} coordinate (B_1);\n\\draw ($(C)!{1/3}!(P)$) node [above right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1) (A_1) -- (B);\n\\draw [dashed] (A) -- (C) (A_1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\gamma<\\beta=\\alpha$}{$\\beta=\\alpha<\\gamma$}{$\\beta<\\alpha<\\gamma$}{$\\alpha<\\beta<\\gamma$}", - "objs": [], + "objs": [ + "K0631001X", + "K0631003X" + ], "tags": [ "第六单元", "空间向量" @@ -305248,7 +305519,9 @@ "030469": { "id": "030469", "content": "已知在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $D_1D=3$, $M$是$B_1C_1$的中点, $N$是$AB$的中点, 以$D$为原点, 建立如图所示的空间直角坐标系.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [above right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A) --++ (0,0,0.8) node [below] {$x$} coordinate (x);\n\\draw [->] (C) --++ (0.8,0,0) node [below] {$y$} coordinate (y);\n\\draw [->] (D1) --++ (0,0.8,0) node [left] {$z$} coordinate (z);\n\\draw ($(A)!0.5!(B)$) node [below] {$N$} coordinate (N);\n\\draw ($(B1)!0.5!(C1)$) node [left] {$M$} coordinate (M);\n\\draw ($(D)!0.5!(N)$) node [left] {$P$} coordinate (P);\n\\draw [dashed] (D) -- (N) (M) -- (P);\n\\end{tikzpicture}\n\\end{center} \n(1) 写出点$D,N,M$的坐标;\\\\\n(2) 求线段$MD, MN$的长度;\\\\ \n(3) 设$P$是线段$DN$上的动点, 求$MP$的最小值.", - "objs": [], + "objs": [ + "K0627005X" + ], "tags": [ "第六单元", "空间向量" @@ -305270,7 +305543,9 @@ "030470": { "id": "030470", "content": "《九章算术$\\cdot$商功》主要讲述了以立体几何为主的各种形体体积的计算, 其堑堵是指底面为直角三角形的直棱柱. 如图所示, 在堑堵$ABC-A_1B_1C_1$中, $M$是$A_1C_1$的中点, $AB=2AA_1=2AC$, $\\overrightarrow{BN}=\\dfrac 13\\overrightarrow{BB_1}$, $\\overrightarrow{MG}=3\\overrightarrow{GN}$, 若$\\overrightarrow{AG}=x\\overrightarrow{AA_1}+y\\overrightarrow{AB}+z\\overrightarrow{AC}$, 则$x+y+z=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$B_1$} coordinate (B1);\n\\draw (A) ++ (0,0,-\\m) node [left] {$A_1$} coordinate (A1);\n\\draw (A) ++ (0,\\n,0) node [left] {$C$} coordinate (C);\n\\draw (A1) ++ (0,\\n,0) node [above] {$C_1$} coordinate (C1);\n\\draw (A) -- (B) -- (B1) -- (C1) -- (C) -- cycle (B) -- (C);\n\\draw [dashed] (A) -- (A1) -- (B1) (A1) -- (C1);\n\\draw ($(A1)!0.5!(C1)$) node [right] {$M$} coordinate (M);\n\\draw ($(B)!{1/3}!(B1)$) node [right] {$N$} coordinate (N);\n\\draw ($(M)!0.75!(N)$) node [above] {$G$} coordinate (G);\n\\draw [dashed] (A) -- (G) (M) -- (N);\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0624003X" + ], "tags": [ "第六单元", "空间向量" @@ -305292,7 +305567,9 @@ "030471": { "id": "030471", "content": "如图所示, 正方体$ABCD-A_1B_1C_1D_1$中, $M$为$B_1C_1$边的中点, 点$P$在底面$ABCD$和侧面$CDD_1C_1$上运动并且使$\\angle MA_1C=\\angle PA_1C$, 那么点$P$的轨迹是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1) (A1) -- (C);\n\\draw ($(B1)!0.5!(C1)$) node [right] {$M$} coordinate (M);\n\\draw (A1) -- (M);\n\\filldraw ({1/3},0,{-1/12*(35 - 3*sqrt(97))}) node [right] {$P$} coordinate (P) circle (0.01);\n\\draw [dashed] (A1) -- (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{两端圆弧}{两段椭圆弧}{两段双曲线弧}{两段抛物线弧}", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第六单元", "空间向量" @@ -305314,7 +305591,11 @@ "030472": { "id": "030472", "content": "如图所示, 在四棱锥$P-ABCD$中, 底面$ABCD$是边长为$2$的菱形, $\\angle ADC=60^\\circ$ , $\\triangle PAD$为正三角形, $O$为$AD$的中点, 且平面$PAD\\perp $平面$ABCD$, $M$是线段$PC$上的点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (-1,0,0) node [below] {$A$} coordinate (A);\n\\draw (D) ++ (-1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (C) ++ (-2,0,0) node [left] {$B$} coordinate (B);\n\\draw (O) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (B) -- (C) -- (D) -- (P) -- cycle (P) -- (C);\n\\draw [dashed] (B) -- (A) -- (P) (A) -- (D) (O) -- (P);\n\\draw ($(P)!0.5!(C)$) node [right] {$M$} coordinate (M);\n\\draw [dashed] (A) -- (M) (O) -- (M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OM\\perp BC$;\\\\ \n(2) 当$M$为线段$PC$的中点时, 求点$M$到平面$PAB$的距离;\\\\\n(3) 是否存在点$M$, 使得直线$AM$与平面$PAB$的夹角的正弦值为$\\dfrac{\\sqrt{10}}{10}$. 若存在, 求出此时$\\dfrac{PM}{PC}$的值; 若不存在, 请说明理由.", - "objs": [], + "objs": [ + "K0627007X", + "K0629001X", + "K0630004X" + ], "tags": [ "第六单元", "空间向量"