From 7843af9940d80ea3d482e62b46c8cdce3f594b51 Mon Sep 17 00:00:00 2001 From: Wang Weiye Date: Thu, 22 Sep 2022 16:48:36 +0800 Subject: [PATCH] 20220922 evening --- 工具/关键字筛选题号.ipynb | 12 +- 工具/批量添加题库字段数据.ipynb | 406 +++++++++++---------- 工具/讲义生成.ipynb | 27 +- 工具/题号选题pdf生成.ipynb | 16 +- 题库0.3/Problems.json | 612 ++++++++++++++++++++++++-------- 5 files changed, 704 insertions(+), 369 deletions(-) diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index c021123c..5064c57e 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 2, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } @@ -21,7 +21,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"tags\":[\"第五单元\"],\"_notcontent\":[\"三角形式\"]}\n", + " {\"tags\":[\"第五单元\"],\"content\":[\"mathrm{i}\",\"复数\",\"虚数\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", @@ -85,7 +85,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.8.8 ('base')", + "display_name": "Python 3.9.7 ('base')", "language": "python", "name": "python3" }, @@ -99,12 +99,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.8" + "version": "3.9.7" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" + "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" } } }, 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 0f868383..f0b3c4c0 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 @@ -2,205 +2,225 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.895\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t1.000\t0.947\t1.000\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.737\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.605\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.947\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.947\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t1.000\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.447\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.895\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.658\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.658\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.842\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.947\t0.895\t0.974\t0.895\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.842\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.474\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三10班\t0.500\t0.526\t0.447\n", - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.913\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.913\t0.870\t0.957\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.739\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.652\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.870\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t1.000\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t1.000\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.739\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.565\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.522\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.391\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.696\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.913\t0.870\t0.870\t0.739\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.870\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.261\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三11班\t0.783\t0.565\t0.522\n", - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.917\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.875\t0.917\t0.958\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.958\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.708\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t1.000\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.833\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.917\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.958\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.917\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.875\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t1.000\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t1.000\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.833\t0.958\t0.958\t0.625\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.958\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.292\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三12班\t0.667\t0.625\t0.750\n", - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t1.000\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t1.000\t0.926\t0.963\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.852\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.963\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.926\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.815\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t1.000\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.889\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.926\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.963\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.963\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.852\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t1.000\t0.963\t0.963\t0.963\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.926\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.741\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三01班\t0.963\t0.889\t0.889\n", - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.933\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t1.000\t0.933\t0.967\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.833\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.800\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.900\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.767\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.967\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t1.000\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.800\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.767\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.967\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.900\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.833\t0.867\t0.900\t0.800\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.967\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.567\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三02班\t0.867\t0.867\t0.867\n", - "题号: 008215 , 字段: usages 中已添加数据: 20220922\t2023届高三03班\t0.931\n", - "题号: 010210 , 字段: usages 中已添加数据: 20220922\t2023届高三03班\t1.000\t1.000\t0.897\n", - "题号: 003056 , 字段: usages 中已添加数据: 20220922\t2023届高三03班\t0.897\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三03班\t0.690\n", - "题号: 008106 , 字段: usages 中已添加数据: 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20220922\t2023届高三04班\t0.938\n", - "题号: 010226 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.844\n", - "题号: 008106 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.906\n", - "题号: 005904 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.812\n", - "题号: 009547 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.969\n", - "题号: 010214 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.969\n", - "题号: 010232 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.781\n", - "题号: 005956 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.844\n", - "题号: 030016 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.812\n", - "题号: 004642 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.781\n", - "题号: 010220 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.875\t0.781\t0.938\t0.719\n", - "题号: 008312 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.969\n", - "题号: 006579 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.469\n", - "题号: 010268 , 字段: usages 中已添加数据: 20220922\t2023届高三04班\t0.844\t0.531\t0.781\n", - "题号: 008215 , 字段: usages 中已添加数据: 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, 字段: objs 中已添加数据: K0504003B\n", + "题号: 003334 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 003339 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 001878 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 001878 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 001897 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 001897 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001914 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 001917 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 003330 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 003330 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 000973 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 000947 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 000947 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 000153 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 000154 , 字段: objs 中已有该数据: K0504006B\n", + "题号: 000154 , 字段: objs 中已添加数据: K0505004B\n", + "题号: 000871 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 001877 , 字段: objs 中已添加数据: K0504006B\n", + "题号: 003356 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 001901 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 000414 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 000760 , 字段: objs 中已添加数据: K0504007B\n", + "题号: 001882 , 字段: objs 中已添加数据: K0505001B\n", + "题号: 001883 , 字段: objs 中已添加数据: K0505001B\n", + "题号: 000792 , 字段: objs 中已添加数据: K0505001B\n", + "题号: 000792 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 000819 , 字段: objs 中已添加数据: K0505001B\n", + "题号: 000819 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 001849 , 字段: objs 中已添加数据: K0505001B\n", + "题号: 001849 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001885 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 000141 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 000141 , 字段: objs 中已有该数据: K0508003B\n", + "题号: 000142 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 000142 , 字段: objs 中已有该数据: K0505005B\n", + "题号: 000936 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 000936 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001853 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 001853 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001880 , 字段: objs 中已添加数据: K0505003B\n", + "题号: 003338 , 字段: objs 中已添加数据: K0505004B\n", + "题号: 001900 , 字段: objs 中已添加数据: K0505004B\n", + "题号: 001900 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001879 , 字段: objs 中已添加数据: K0505004B\n", + "题号: 003340 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 003340 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 003341 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 003342 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 003349 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 003349 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 003354 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001884 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 001906 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 000152 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 000152 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 000152 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 000699 , 字段: objs 中已添加数据: K0505005B\n", + "题号: 000699 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 003335 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001890 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001892 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001893 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001896 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001899 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001908 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 001913 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 000855 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 000554 , 字段: objs 中已添加数据: K0508003B\n", + "题号: 003360 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 003360 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 001916 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 001916 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 001886 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 001891 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 001898 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 001898 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 001903 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 000147 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 000147 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 000157 , 字段: objs 中已添加数据: K0508004B\n", + "题号: 003345 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 003350 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 003357 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 001887 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 000149 , 字段: objs 中已添加数据: K0508005B\n", + "题号: 003359 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001881 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001905 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001909 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001911 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001875 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 000158 , 字段: objs 中已有该数据: K0509004B\n", + "题号: 000160 , 字段: objs 中已有该数据: K0509004B\n", + "题号: 000161 , 字段: objs 中已有该数据: K0509004B\n", + "题号: 000955 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001876 , 字段: objs 中已添加数据: K0509004B\n", + "题号: 001902 , 字段: objs 中已添加数据: K0510001B\n", + "题号: 000159 , 字段: objs 中已有该数据: K0510003B\n" ] } ], diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index 2603f236..96077c9e 100644 --- a/工具/讲义生成.ipynb +++ b/工具/讲义生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 11, + "execution_count": 13, "metadata": {}, "outputs": [ { @@ -13,11 +13,9 @@ "题块 1 处理完毕.\n", "正在处理题块 2 .\n", "题块 2 处理完毕.\n", - "正在处理题块 3 .\n", - "题块 3 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/周末卷03_教师_20220922.tex\n", + "开始编译教师版本pdf文件: 临时文件/平面向量的投影及数量积预选_教师_20220922.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/周末卷03_学生_20220922.tex\n", + "开始编译学生版本pdf文件: 临时文件/平面向量的投影及数量积预选_学生_20220922.tex\n", "0\n" ] } @@ -30,30 +28,29 @@ "\"\"\"---设置模式结束---\"\"\"\n", "\n", "\"\"\"---设置模板文件名---\"\"\"\n", - "# template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", - "template_file = \"模板文件/测验周末卷模板.tex\"\n", + "template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", + "# template_file = \"模板文件/测验周末卷模板.tex\"\n", "\"\"\"---设置模板文件名结束---\"\"\"\n", "\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\",\"16\"),(\"标题文字待处理\",\"平面向量的概念及线性运算\")] \n", - "# enumi_mode = 0\n", + "exec_list = [(\"标题数字待处理\",\"17\"),(\"标题文字待处理\",\"平面向量的投影及数量积预选\")] \n", + "enumi_mode = 0\n", "\n", "#2023届测验卷与周末卷\n", - "exec_list = [(\"标题替换\",\"周末卷03\")]\n", - "enumi_mode = 1\n", + "# exec_list = [(\"标题替换\",\"周末卷03\")]\n", + "# enumi_mode = 1\n", "\"\"\"---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/周末卷03\"\n", + "destination_file = \"临时文件/平面向量的投影及数量积预选\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"3115,6264,3096,1472,6604,30027,30028\",\n", - "\"6305\",\n", - "\"6460,6463\"\n", + "\"481,3347,3334,3339,1897,3330,153,3356,414,1882,141,3340,3341,1890,1892,1896,1898,147,1916,1905,1902\",\n", + "\"894,1912,1878,1914,973,154,871,1877,1901,760,792,142,1906,699,1913,156,1886,157,3345,1911,159\"\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 20888652..f79c1106 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": 14, + "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/周末卷03三角专题预选_教师用_20220922.tex\n", + "开始编译教师版本pdf文件: 临时文件/复数预选_教师用_20220922.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/周末卷03三角专题预选_学生用_20220922.tex\n", + "开始编译学生版本pdf文件: 临时文件/复数预选_学生用_20220922.tex\n", "0\n" ] } @@ -26,14 +26,14 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库\n", "problems = r\"\"\"\n", - 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+ 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"\n", "\"\"\"\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/周末卷03三角专题预选\"\n", + "filename = \"临时文件/复数预选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", @@ -163,7 +163,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.8.8 ('base')", + "display_name": "Python 3.9.7 ('base')", "language": "python", "name": "python3" }, @@ -177,12 +177,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.8" + "version": "3.9.7" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" + "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" } } }, diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 7a79871c..f4889ea9 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -3399,7 +3399,10 @@ "content": "已知$\\overrightarrow a$、$\\overrightarrow b$均为非零向量, 写出$|\\overrightarrow a+\\overrightarrow b|=|\\overrightarrow a|+|\\overrightarrow b|$成立的充要条件.", "objs": [ "K0501003B", - "K0502002B" + "K0502002B", + "K0501001B", + "K0501004B", + "K0505005B" ], "tags": [ "第五单元" @@ -3422,7 +3425,8 @@ "id": "000140", "content": "已知$\\overrightarrow a$、$\\overrightarrow b$为非零向量, 且$\\overrightarrow a$、$\\overrightarrow b$、$5\\overrightarrow a-4\\overrightarrow b$在同一起点上. 求证: 它们的终点在同一条直线上.", "objs": [ - "K0501004B" + "K0501004B", + "K0503001B" ], "tags": [ "第五单元" @@ -3445,7 +3449,8 @@ "id": "000141", "content": "在矩形$ABCD$中, 边$AB$、$AD$的长分别为$2$、$1$, 若$M$、$N$分别是边$BC$、$CD$上的点, 且满足$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 则$\\overrightarrow{AM}\\cdot \\overrightarrow{AN}$的取值范围是\\blank{50}.", "objs": [ - "K0508003B" + "K0508003B", + "K0505003B" ], "tags": [ "第五单元" @@ -3468,7 +3473,8 @@ "id": "000142", "content": "已知两个向量$\\overrightarrow {e_1}$、$\\overrightarrow {e_2}$满足$|\\overrightarrow {e_1}|=2$, $|\\overrightarrow {e_2}|=1$, $\\langle \\overrightarrow {e_1}, \\overrightarrow {e_2}\\rangle=60^\\circ$, 且向量$2\\lambda \\overrightarrow {e_1}+7\\overrightarrow {e_2}$与向量$\\overrightarrow {e_1}+\\lambda \\overrightarrow {e_2}$的夹角为钝角. 求实数$\\lambda$的取值范围.", "objs": [ - "K0505005B" + "K0505005B", + "K0505003B" ], "tags": [ "第五单元" @@ -3493,7 +3499,8 @@ "objs": [ "K0507007B", "K0507009B", - "K0508005B" + "K0508005B", + "K0509001B" ], "tags": [ "第五单元" @@ -3541,7 +3548,8 @@ "id": "000145", "content": "已知向量$\\overrightarrow a=(3, -2)$, $\\overrightarrow b=(-2, 1)$, $\\overrightarrow c=(7, -4)$, 求$\\lambda,\\mu$, 使得$\\overrightarrow c=\\lambda \\overrightarrow a+\\mu \\overrightarrow b$.", "objs": [ - "K0507007B" + "K0507007B", + "K0506003B" ], "tags": [ "第五单元" @@ -3589,7 +3597,9 @@ "objs": [ "K0505001B", "K0502001B", - "K0502002B" + "K0502002B", + "K0508004B", + "K0509004B" ], "tags": [ "第五单元" @@ -3612,7 +3622,8 @@ "id": "000148", "content": "如图, 在四边形$ABCD$中, $G$为对角线$AC$与$BD$中点连线$MN$的中点, $P$为平面上任意给定的一点. 求证: $4\\overrightarrow{PG}=\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}+\\overrightarrow{PD}$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw [thick] (0,0) node [below left] {$B$} coordinate (B) -- (2,0) node [below right] {$C$} coordinate (C) -- (2.4,2.5) node [above right] {$D$} coordinate (D) -- (-0.1,1.5) node [above left] {$A$} coordinate (A) -- cycle;\n \\draw [thick] (A) -- (C) (B) -- (D);\n \\filldraw ($(B)!0.5!(D)$) circle (0.03) node [above] {$M$} coordinate (M);\n \\filldraw ($(A)!0.5!(C)$) circle (0.03) node [below] {$N$} coordinate (N);\n \\filldraw ($(M)!0.5!(N)$) circle (0.03) node [right] {$G$} coordinate (G);\n \\filldraw (3,1) circle (0.03) node [right] {$P$};\n \\draw [thick] (M) -- (N);\n \\end{tikzpicture}\n\\end{center}", "objs": [ - "K0502001B" + "K0502001B", + "K0509002B" ], "tags": [ "第五单元" @@ -3637,7 +3648,8 @@ "content": "在四边形$ABCD$中, 向量$\\overrightarrow{AB}=\\overrightarrow i+2\\overrightarrow j$, $\\overrightarrow{BC}=-4\\overrightarrow i-\\overrightarrow j$, $\\overrightarrow{CD}=-5\\overrightarrow i-3\\overrightarrow j$. 求证: $ABCD$为梯形.", "objs": [ "K0503005B", - "K0501004B" + "K0501004B", + "K0508005B" ], "tags": [ "第五单元" @@ -3661,7 +3673,8 @@ "id": "000150", "content": "已知$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$均为非零向量, 其中的任意两个向量都不平行, 且$\\overrightarrow a+\\overrightarrow b$与$\\overrightarrow c$是平行向量, $\\overrightarrow a+\\overrightarrow c$与$\\overrightarrow b$是平行向量. 求证: $\\overrightarrow b+\\overrightarrow c$与$\\overrightarrow a$是平行向量.", "objs": [ - "K0501004B" + "K0501004B", + "K0509001B" ], "tags": [ "第五单元" @@ -3686,7 +3699,8 @@ "id": "000151", "content": "如图, 点$A$、$M$、$B$在同一条直线上, 点$O$不在该直线上, 且$\\overrightarrow{AM}=\\dfrac 13 \\overrightarrow{AB}$. 设\n$\\overrightarrow{OA}=\\overrightarrow a$, $\\overrightarrow{OB}=\\overrightarrow b$, $\\overrightarrow{OM}=\\overrightarrow c$, 试用向量$\\overrightarrow a$、$\\overrightarrow b$表示$\\overrightarrow c$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [thick] (-0.5,0) -- (3.5,0);\n \\draw [->,thick] (1.5,-1.5) node [below] {$O$}-- (0,0) node [above] {$A$};\n \\draw [->,thick] (1.5,-1.5) -- (3,0);\n \\draw [->,thick] (1.5,-1.5) -- (1,0);\n \\draw [->,thick] (0,0) -- (1,0) node [above] {$M$};\n \\draw [->,thick] (1,0) -- (3,0) node [above] {$B$};\n \\end{tikzpicture}\n\\end{center}", "objs": [ - "K0503005B" + "K0503005B", + "K0509002B" ], "tags": [ "第五单元" @@ -3713,7 +3727,10 @@ "objs": [ "K0505001B", "K0507007B", - "K0507009B" + "K0507009B", + "K0505005B", + "K0508003B", + "K0508004B" ], "tags": [ "第五单元" @@ -3736,7 +3753,8 @@ "id": "000153", "content": "如图, 在矩形$ABCD$中, $AB=\\sqrt 2$, $BC=2$, $E$为$BC$的中点, 点$F$在边$CD$上且$\\overrightarrow{AB}\\cdot \\overrightarrow{AF}=\\sqrt{2}$. 求$\\overrightarrow{AE}\\cdot \\overrightarrow{BF}$的值.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,thick,scale = 1.3]\n \\draw (0,0) node [below left] {$A$} rectangle ({sqrt(2)},2) node [above right] {$C$};\n \\draw [->,thick] (0,0) -- ({sqrt(2)},1) node [right] {$E$};\n \\draw [->] (0,0) -- (1,2) node [above] {$F$};\n \\draw [->] ({sqrt(2)},0) node [below right] {$B$} -- (1,2);\n \\draw (0,2) node [above left] {$D$};\n \\end{tikzpicture}\n\\end{center}", "objs": [ - "K0508003B" + "K0508003B", + "K0504006B" ], "tags": [ "第五单元" @@ -3760,7 +3778,8 @@ "content": "已知等边三角形$ABC$的边长为$1$, $\\overrightarrow{BC}=\\overrightarrow a$, $\\overrightarrow{CA}=\\overrightarrow b$, $\\overrightarrow{AB}=\\overrightarrow c$. 求$\\overrightarrow a\\cdot \\overrightarrow b+\\overrightarrow b\\cdot \\overrightarrow c+\\overrightarrow c\\cdot \\overrightarrow a$.", "objs": [ "K0504006B", - "K0505005B" + "K0505005B", + "K0505004B" ], "tags": [ "第五单元" @@ -3785,7 +3804,8 @@ "id": "000155", "content": "已知向量$\\overrightarrow{OA}=(k, 12)$, $\\overrightarrow{OB}=(4, 5)$, $\\overrightarrow{OC}=(-k, 10)$, 且$A$、$B$、$C$三点共线. 求实数$k$的值.", "objs": [ - "K0509001B" + "K0509001B", + "K0507007B" ], "tags": [ "第五单元" @@ -3809,7 +3829,8 @@ "content": "已知向量$\\overrightarrow{OA}=(1, 7)$, $\\overrightarrow{OB}=(5, 1)$, $\\overrightarrow{OP}=(2, 1)$, $K$为直线$OP$上的一个动点, 当$\\overrightarrow{KA}\\cdot\\overrightarrow{KB}$取最小值时, 求向量$\\overrightarrow{OK}$的坐标.", "objs": [ "K0508005B", - "K0508003B" + "K0508003B", + "K0507007B" ], "tags": [ "第五单元" @@ -3832,7 +3853,8 @@ "id": "000157", "content": "如图, 在正方形$ABCD$中, $P$是对角线$AC$上一点, $PE$垂直$AB$于点$E$, $PF$垂直$BC$于点$F$. 求证: $PD\\perp EF$.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [below left] {$A$} rectangle (2,2) node [above right] {$C$};\n \\draw (0,2) node [above left] {$D$} -- (1.5,1.5) node [above] {$P$};\n \\draw (1.5,1.5) -- (1.5,0) node [below] {$E$} -- (2,1.5) node [right] {$F$} -- cycle;\n \\draw (2,0) node [below right] {$B$};\n \\draw (0,0) -- (2,2);\n \\end{tikzpicture}\n\\end{center}", "objs": [ - "K0509001B" + "K0509001B", + "K0508004B" ], "tags": [ "第五单元" @@ -10095,7 +10117,9 @@ "000414": { "id": "000414", "content": "如图, 已知半径为1的扇形$AOB$, $\\angle AOB=60^\\circ$, $P$为弧$\\overset\\frown{AB}$上的一个动点, 则$\\overrightarrow{OP}\\cdot \\overrightarrow{AB}$取值范围是\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 1.3]\n \\draw (0,0) node [below left] {$O$} -- (2,0) node [below right] {$B$} (0,0) -- (1,{sqrt(3)}) node [above] {$A$} -- cycle;\n \\draw (2,0) arc (0:60:2);\n \\draw [->] (0,0) -- (25:2) node [above right] {$P$};\n \\draw [->] (1,{sqrt(3)}) -- (2,0);\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0504007B" + ], "tags": [ "第五单元" ], @@ -11802,8 +11826,11 @@ }, "000481": { "id": "000481", - "content": "已知向量$\\overrightarrow{a}=(1,\\sqrt{3})$, $\\overrightarrow{b}=(3,m)$. 若向量$\\overrightarrow{b}$在$\\overrightarrow{a}$方向上的投影为$3$, 则实数$m=$\\blank{50}.", - "objs": [], + "content": "已知向量$\\overrightarrow{a}=(1,\\sqrt{3})$, $\\overrightarrow{b}=(3,m)$. 若向量$\\overrightarrow{b}$在$\\overrightarrow{a}$方向上的数量投影为$3$, 则实数$m=$\\blank{50}.", + "objs": [ + "K0504002B", + "K0504007B" + ], "tags": [ "第五单元" ], @@ -12667,7 +12694,10 @@ "000515": { "id": "000515", "content": "向量$\\overrightarrow{i}$、$\\overrightarrow{j}$是平面直角坐标系$x$轴、$y$轴的基本单位向量, 且$|\\overrightarrow a-\\overrightarrow i|+|\\overrightarrow a-2\\overrightarrow j|=\\sqrt5$, 则$|\\overrightarrow a+2 \\overrightarrow i|$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0501003B", + "K0502007B" + ], "tags": [ "第五单元" ], @@ -13171,7 +13201,9 @@ "000535": { "id": "000535", "content": "如图, 向量$\\overrightarrow{OA}$与$\\overrightarrow{OB}$的夹角为$120^\\circ$, $|\\overrightarrow{OA}|=2$, $|\\overrightarrow{OB}|=1$, $P$是以$O$为圆心、$|\\overrightarrow{OB}|$为半径的弧$\\overset\\frown{BC}$上的动点, 若$\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$, 则$\\lambda \\mu$的最大值是\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.5, >=latex]\n \\draw [->] (0,0) node [below left] {$O$} -- (2,0) node [below] {$A$};\n \\draw [->] (0,0) --+ (120:1) node [above left] {$B$};\n \\draw [->] (0,0) --+ (40:1) node [above right] {$P$};\n \\draw (1,0) node [below] {$C$} arc (0:120:1);\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -13650,7 +13682,9 @@ "000554": { "id": "000554", "content": "在$\\triangle ABC$中, $\\angle A=90^\\circ $, $\\triangle ABC$的面积为$1$. 若$\\overrightarrow{BM}=\\overrightarrow{MC}$, $\\overrightarrow{BN}=4 \\overrightarrow{NC}$, 则$\\overrightarrow{AM}\\cdot \\overrightarrow{AN}$的最小值为\\blank{50}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -17310,7 +17344,10 @@ "000699": { "id": "000699", "content": "设向量$\\overrightarrow{a}=(2,3)$, 向量$\\overrightarrow{b}=(6,t)$. 若$\\overrightarrow{a}$与$\\overrightarrow{b}$的夹角为钝角, 则实数$t$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0505005B", + "K0508003B" + ], "tags": [ "第五单元" ], @@ -18847,8 +18884,10 @@ }, "000760": { "id": "000760", - "content": "已知向量$\\overrightarrow a$在向量$\\overrightarrow b$方向上的投影为$-2$, 且$|\\overrightarrow b|=3$, 则$\\overrightarrow a\\cdot \\overrightarrow b$=\\blank{50}(结果用数值表示).", - "objs": [], + "content": "已知向量$\\overrightarrow a$在向量$\\overrightarrow b$方向上的投影为$-\\dfrac 23\\overrightarrow b$, 且$|\\overrightarrow b|=3$, 则$\\overrightarrow a\\cdot \\overrightarrow b$=\\blank{50}(结果用数值表示).", + "objs": [ + "K0504007B" + ], "tags": [ "第五单元" ], @@ -19489,7 +19528,9 @@ "000785": { "id": "000785", "content": "在直角三角形$ABC$中, $\\angle A=\\dfrac{\\pi}2$, $AB=3$, $AC=4$, $E$为三角形$ABC$内一点, 且$AE=\\dfrac{\\sqrt2}2$. 若$\\overrightarrow{AE}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 则$3\\lambda +4 \\mu$的最大值等于\\blank{50}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -19657,7 +19698,10 @@ "000792": { "id": "000792", "content": "已知向量$\\overrightarrow a$、$\\overrightarrow b$的夹角为$60^{\\circ}$, $|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$, 若$(\\overrightarrow a+2 \\overrightarrow b)\\perp (x\\overrightarrow a-\\overrightarrow b)$, 则实数$x$的值为\\blank{50}.", - "objs": [], + "objs": [ + "K0505001B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -20350,7 +20394,10 @@ "000819": { "id": "000819", "content": "已知两个不同向量$\\overrightarrow{OA}=(1,m)$, $\\overrightarrow{OB}=(m-1,2)$, 若$\\overrightarrow{OA}\\perp \\overrightarrow{AB}$, 则实数$m=$\\blank{50}.", - "objs": [], + "objs": [ + "K0505001B", + "K0508004B" + ], "tags": [ "第五单元" ], @@ -21277,7 +21324,9 @@ "000855": { "id": "000855", "content": "设$m,n$分别为连续两次投掷骰子得到的点数, 且向量$\\overrightarrow a=(m,n)$,$\\overrightarrow b=(1,-1)$, 则$\\overrightarrow a$与$\\overrightarrow b$的夹角为锐角的概率是\\blank{50}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -21671,7 +21720,9 @@ "000871": { "id": "000871", "content": "若向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$, 且$\\overrightarrow a$与$\\overrightarrow b$的夹角为$\\dfrac{\\pi}3$, 则$|\\overrightarrow a+\\overrightarrow b|=$\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -21968,7 +22019,9 @@ "000883": { "id": "000883", "content": "已知$\\overrightarrow{AB}=3\\overrightarrow{AP}$, 设$\\overrightarrow{BP}=\\lambda \\overrightarrow{PA}$, 则实数$\\lambda=$\\blank{50}.", - "objs": [], + "objs": [ + "K0503001B" + ], "tags": [ "第五单元" ], @@ -22229,8 +22282,11 @@ }, "000894": { "id": "000894", - "content": "在直角坐标系$xOy$中, 已知三点$A(a,1),B(2,b),C(3,4)$, 若向量$\\overrightarrow{OA}$, $\\overrightarrow{OB}$在向量$\\overrightarrow{OC}$方向上的投影相同, 则$3a-4b$的值是\\blank{50}.", - "objs": [], + "content": "在直角坐标系$xOy$中, 已知三点$A(a,1),B(2,b),C(3,4)$, 若向量$\\overrightarrow{OA}$, $\\overrightarrow{OB}$在向量$\\overrightarrow{OC}$方向上的数量投影相同, 则$3a-4b$的值是\\blank{50}.", + "objs": [ + "K0504002B", + "K0504007B" + ], "tags": [ "第五单元" ], @@ -22831,7 +22887,11 @@ "000919": { "id": "000919", "content": "如图所示, 三个边长为$2$的等边三角形有一条边在同一直线上, 边$B_3C_3$上有$10$个不同的点$P_1,P_2,\\cdots,P_{10}$, 记$M_i=\\overrightarrow{AB_2}\\cdot \\overrightarrow{AP_i}$($i=1,2,\\cdots,10$), 则$M_1+M_2+\\cdots+M_{10}=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 3]\n \\draw (0,0) node [below] {$A$} -- (1,0) node [below] {$C_1$} -- (2,0) node [below] {$C_2$} -- (3,0) node [below] {$C_3$};\n \\draw (3,0) -- (2.5,{sqrt(3)/2}) node [above] {$B_3$} -- (2,0) -- (1.5,{sqrt(3)/2}) node [above] {$B_2$} -- (1,0) -- (0.5,{sqrt(3)/2}) node [above] {$B_1$} -- (0,0);\n \\draw ($(3,0)!0.2!(2.5,{sqrt(3)/2})$) coordinate (P1) node [right] {$P_1$} -- (0,0);\n \\draw ($(3,0)!0.4!(2.5,{sqrt(3)/2})$) coordinate (P2) node [right] {$P_2$} -- (0,0);\n \\draw ($(3,0)!0.8!(2.5,{sqrt(3)/2})$) coordinate (P3) node [right] {$P_{10}$} -- (0,0);\n \\draw ($(3,0)!0.6!(2.5,{sqrt(3)/2})$) node [right] {\\rotatebox{120}{$\\cdots$}};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0502001B", + "K0505003B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -23099,7 +23159,9 @@ "000930": { "id": "000930", "content": "矩形$ABCD$中, $AB=2$, $AD=1$, $P$为矩形内部一点, 且$AP=1$. 若$\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD} \\ (\\lambda,\\ \\mu \\in \\mathbf{R})$, 则$2 \\lambda +\\sqrt3\\mu$的最大值是\\blank{50}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -23258,7 +23320,10 @@ "000936": { "id": "000936", "content": "平面向量$\\overrightarrow a$与$\\overrightarrow b$的夹角为$60^\\circ$, $|\\overrightarrow a|=1$, $\\overrightarrow b=(3,0)$, 则$|2 \\overrightarrow a+\\overrightarrow b|=$\\blank{50}.", - "objs": [], + "objs": [ + "K0505003B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -23544,7 +23609,10 @@ "000947": { "id": "000947", "content": "已知菱形$ABCD$, 若$|\\overrightarrow{AB}|=1$, $A=\\dfrac\\pi 3$, 则向量$\\overrightarrow{AC}$在$\\overrightarrow{AB}$上的投影为\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B", + "K0504007B" + ], "tags": [ "第五单元" ], @@ -23739,7 +23807,9 @@ "000955": { "id": "000955", "content": "已知$\\triangle ABC$中, $|\\overrightarrow{AB}|=2 $, $|\\overrightarrow{AC}|=3 $, $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}<0$, 且$\\triangle ABC$的面积为$\\dfrac32$, 则$\\angle BAC=$\\blank{50}.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -24167,7 +24237,9 @@ "000973": { "id": "000973", "content": "已知$\\triangle ABC$外接圆的半径为$2$, 圆心为$O$, 且$\\overrightarrow{AB}+\\overrightarrow{AC}=2 \\overrightarrow{AO}$, $|\\overrightarrow{AB}|=|\\overrightarrow{AO}|$, 则$\\overrightarrow{CA}\\cdot \\overrightarrow{CB}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -46488,7 +46560,9 @@ "001846": { "id": "001846", "content": "``$\\overrightarrow{a}=\\overrightarrow{b}$''是``$\\overrightarrow{a}\\parallel\\overrightarrow{b}$''的\\blank{30}.\n\\twoch{充分必要条件}{充分非必要条件}{必要非充分条件}{既非充分又非必要条件}", - "objs": [], + "objs": [ + "K0501004B" + ], "tags": [ "第五单元" ], @@ -46514,7 +46588,10 @@ "001847": { "id": "001847", "content": "``$|\\overrightarrow{a}|=|\\overrightarrow{b}|$''是``$\\overrightarrow{a}=\\overrightarrow{b}$或$\\overrightarrow{a}=-\\overrightarrow{b}$''的\\blank{30}.\n\\twoch{充分必要条件}{充分非必要条件}{必要非充分条件}{既非充分又非必要条件}", - "objs": [], + "objs": [ + "K0501003B", + "K0501006B" + ], "tags": [ "第五单元" ], @@ -46540,7 +46617,9 @@ "001848": { "id": "001848", "content": "``$|\\overrightarrow{a}|=0$''是``$\\overrightarrow{a}=\\overrightarrow{0}$''的\\blank{30}.\n\\twoch{充分必要条件}{充分非必要条件}{必要非充分条件}{既非充分又非必要条件}", - "objs": [], + "objs": [ + "K0501001B" + ], "tags": [ "第五单元" ], @@ -46564,7 +46643,10 @@ "001849": { "id": "001849", "content": "已知非零向量$\\overrightarrow{a}$和$\\overrightarrow{b}$所在的直线互相垂直, 则下列各式中正确的是\\blank{30}.\n\\twoch{$|\\overrightarrow{a}|+|\\overrightarrow{b}|=|\\overrightarrow{a}-\\overrightarrow{b}|$}{$|\\overrightarrow{a}|+|\\overrightarrow{b}|=|\\overrightarrow{a}+\\overrightarrow{b}|$}{$|\\overrightarrow{a}|-|\\overrightarrow{b}|=|\\overrightarrow{a}-\\overrightarrow{b}|$}{$|\\overrightarrow{a}+\\overrightarrow{b}|=|\\overrightarrow{a}-\\overrightarrow{b}|$}", - "objs": [], + "objs": [ + "K0505001B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -46588,7 +46670,9 @@ "001850": { "id": "001850", "content": "若$A,B,C,D$是平面上任意四点, 则下列命题中正确的有\\blank{30}.\n\\twoch{$\\overrightarrow{AB}+\\overrightarrow{CD}=\\overrightarrow{BC}+\\overrightarrow{DA}$;}{$\\overrightarrow{AC}+\\overrightarrow{BD}=\\overrightarrow{BC}+\\overrightarrow{AD}$;}\n{$\\overrightarrow{AC}-\\overrightarrow{BD}=\\overrightarrow{DC}+\\overrightarrow{AB}$;}{$\\overrightarrow{AB}+\\overrightarrow{BC}=\\overrightarrow{CD}+\\overrightarrow{DA}$.}", - "objs": [], + "objs": [ + "K0502004B" + ], "tags": [ "第五单元" ], @@ -46612,7 +46696,11 @@ "001851": { "id": "001851", "content": "判断下列命题的真假, 如果是假命题则在命题前的横线上写上``F'', 如果是真命题则写上``T''.\\\\ \n\\begin{enumerate}[\\blank{30}(1)]\n\\item 向量的模一定是一个正实数.\\\\ \n\\item 零向量与任何非零向量平行.\\\\ \n\\item 长度相等的向量都相等.\\\\ \n\\item $-(-\\overrightarrow{a})=\\overrightarrow{a}$.\\\\ \n\\item $\\overrightarrow{a}+(-\\overrightarrow{a})=0$.\\\\ \n\\item 若$\\overrightarrow{a}=\\overrightarrow{b}$,$\\overrightarrow{b}=\\overrightarrow{c}$, 则$\\overrightarrow{a}=\\overrightarrow{c}$.\\\\ \n\\item 若四边形$ABCD$是平行四边形, 则$\\overrightarrow{AB}=\\overrightarrow{CD}$.\\\\ \n\\item 若$\\overrightarrow{AB}=\\overrightarrow{DC}$, 则$|\\overrightarrow{AB}|=|\\overrightarrow{CD}|$且直线$AB\\parallel CD$.\\\\ \n\\end{enumerate}", - "objs": [], + "objs": [ + "K0501001B", + "K0501004B", + "K0501006B" + ], "tags": [ "第五单元" ], @@ -46636,7 +46724,10 @@ "001852": { "id": "001852", "content": "化简: $\\overrightarrow{AB}+\\overrightarrow{MB}+\\overrightarrow{BO}-\\overrightarrow{CB}+\\overrightarrow{OM}=$\\blank{40}.", - "objs": [], + "objs": [ + "K0502004B", + "K0502006B" + ], "tags": [ "第五单元" ], @@ -46660,7 +46751,10 @@ "001853": { "id": "001853", "content": "已知$|\\overrightarrow{OA}|=12,|\\overrightarrow{OB}|=4,\\angle AOB=60^\\circ$, 则$|\\overrightarrow{OA}+\\overrightarrow{OB}|=$\\blank{40}.", - "objs": [], + "objs": [ + "K0505003B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -46684,7 +46778,9 @@ "001854": { "id": "001854", "content": "如图, 在$2\\times 4$的矩形中, 起点和终点都在小方格顶点且模与$|\\overrightarrow{AB}|$相等的向量共有\\blank{40}个(包括$\\overrightarrow{AB}$本身).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n \\foreach \\i in {0,1,2,3,4} {\\draw (\\i,0) -- (\\i,2);};\n \\foreach \\i in {0,1,2} {\\draw (0,\\i) -- (4,\\i);};\n \\draw [->] (0,0) node [below left] {$A$} -- (1,2) node [above] {$B$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0501003B" + ], "tags": [ "第五单元" ], @@ -46734,7 +46830,10 @@ "content": "判断下列命题的真假, 如果是假命题则在命题前的横线上写上``F'', 如果是真命题则写上``T''.\\\\ \n\\begin{enumerate}[\\blank{30}(1)]\n\\item 与非零向量$\\overrightarrow{a}$平行的单位向量一定是$\\dfrac{1}{|\\overrightarrow{a}|}\\overrightarrow{a}$.\\\\ \n\\item 若两个非零向量互相平行, 则这两个向量所在的直线平行或重合.\\\\ \n\\item 若非零向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$满足$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 则$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$可以依次首尾相接构成三角形.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$.\\\\ \n\\item 若存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\item 若存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.\\\\ \n\\end{enumerate}", "objs": [ "K0501004B", - "K0502002B" + "K0502002B", + "K0502004B", + "K0502007B", + "K0503001B" ], "tags": [ "第一单元", @@ -46760,7 +46859,11 @@ "001857": { "id": "001857", "content": "在四边形$ABCD$中, $\\overrightarrow{AB}=\\overrightarrow{a}+2\\overrightarrow{b}$, $\\overrightarrow{BC}=-4\\overrightarrow{a}-\\overrightarrow{b}$, $\\overrightarrow{CD}=-5\\overrightarrow{a}-3\\overrightarrow{b}$, 则四边形$ABCD$一定是\\blank{30}.\n\\fourch{矩形}{平行四边形}{菱形}{梯形}", - "objs": [], + "objs": [ + "K0501004B", + "K0501005B", + "K0503002B" + ], "tags": [ "第五单元" ], @@ -46784,7 +46887,9 @@ "001858": { "id": "001858", "content": "已知$\\overrightarrow{AB}=-\\dfrac{3}{4}\\overrightarrow{BC}$, 则$\\overrightarrow{AC}=\\blank{30}\\overrightarrow{BA}$.(尝试不用画图的代数方法)", - "objs": [], + "objs": [ + "K0503003B" + ], "tags": [ "第五单元" ], @@ -46808,7 +46913,9 @@ "001859": { "id": "001859", "content": "已知三点$A,B,M$满足$\\overrightarrow{AM}=\\dfrac{1}{3}\\overrightarrow{AB}$, 设$O$是任意一点, 若用$\\overrightarrow{OA},\\overrightarrow{OB}$\n表示$\\overrightarrow{OM}$, 则在此表示下$\\overrightarrow{OM}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0509002B" + ], "tags": [ "第五单元" ], @@ -46832,7 +46939,9 @@ "001860": { "id": "001860", "content": "已知$\\overrightarrow{m}=3\\overrightarrow{a}-5\\overrightarrow{b}$, $\\overrightarrow{n}=2\\overrightarrow{b}-\\overrightarrow{a}$, 试用$\\overrightarrow{m},\\overrightarrow{n}$表示$\\overrightarrow{a}$与$\\overrightarrow{b}$为$\\overrightarrow{a}=$\\blank{80}, $\\overrightarrow{b}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -46856,7 +46965,9 @@ "001861": { "id": "001861", "content": "已知$x,y$是实数, 向量$\\overrightarrow{a},\\overrightarrow{b}$不平行, 若$x\\overrightarrow{a}+y\\overrightarrow{b}=\\overrightarrow{0}$, 求证: $x=y=0$.", - "objs": [], + "objs": [ + "K0506001B" + ], "tags": [ "第五单元" ], @@ -46880,7 +46991,9 @@ "001862": { "id": "001862", "content": "已知三角形$ABC$中, 三条中线$AD,BE,CF$交于点$G$, 求证: $\\overrightarrow{GA}+\\overrightarrow{GB}+\\overrightarrow{GC}=\\overrightarrow{0}$.", - "objs": [], + "objs": [ + "K0509003B" + ], "tags": [ "第五单元" ], @@ -46904,7 +47017,9 @@ "001863": { "id": "001863", "content": "已知三角形$ABC$中, 若点$G$使得$\\overrightarrow{GA}+\\overrightarrow{GB}+\\overrightarrow{GC}=\\overrightarrow{0}$, 求证: $G$为三角形$ABC$的重心.(提示: 设重心为$P$, 通过证明$\\overrightarrow{GP}=\\overrightarrow{0}$来证明$G$是重心)", - "objs": [], + "objs": [ + "K0509003B" + ], "tags": [ "第五单元" ], @@ -46928,7 +47043,9 @@ "001864": { "id": "001864", "content": "已知平行四边形$ABCD$的对角线交于$E$点, 设$O$是任意一点, 求证: $\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}+\\overrightarrow{OD}=4\\overrightarrow{OE}$.", - "objs": [], + "objs": [ + "K0509002B" + ], "tags": [ "第五单元" ], @@ -46952,7 +47069,9 @@ "001865": { "id": "001865", "content": "[选做]\n已知$A,B,C,D$是平面上给定四点, 是否一定存在平面上一点$P$, 使得$\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}+\\overrightarrow{PD}=\\overrightarrow{0}$? 如果有的话, 有几个? 为什么?", - "objs": [], + "objs": [ + "K0509001B" + ], "tags": [ "第五单元" ], @@ -46976,7 +47095,9 @@ "001866": { "id": "001866", "content": "已知向量$\\overrightarrow{a}$与向量$\\overrightarrow{b}$不平行, $\\overrightarrow{x}=-\\overrightarrow{a}+3\\overrightarrow{b},\\overrightarrow{y}=4\\overrightarrow{a}+2\\overrightarrow{b},\\overrightarrow{z}=-3\\overrightarrow{a}+12\\overrightarrow{b}$. 在$\\overrightarrow{x},\\overrightarrow{y}$ 这组基下, $\\overrightarrow{z}$的坐标为\\blank{80}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -47000,7 +47121,9 @@ "001867": { "id": "001867", "content": "已知$\\triangle ABC$和点$M$满足$\\overrightarrow{MA}+\\overrightarrow{MB}+\\overrightarrow{MC}=\\overrightarrow{0}$. 若存在实数$m$使得$\\overrightarrow{CA}+\\overrightarrow{CB}=m\\overrightarrow{CM}$成立, 则$m=$\\blank{80}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -47024,7 +47147,9 @@ "001868": { "id": "001868", "content": "已知平面上四点$O,A,B,C$满足$OA=OB=1, OC=2\\sqrt3, \\angle AOC=25^\\circ, \\angle BOC=95^\\circ,\\angle AOB=120^\\circ$, 则用向量$\\overrightarrow{OA},\\overrightarrow{OB}$表示向量$\\overrightarrow{OC}$为$\\overrightarrow{OC}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -47048,7 +47173,9 @@ "001869": { "id": "001869", "content": "已知$\\triangle ABC$, 点$D$满足$\\overrightarrow{AD}=\\dfrac{1}{4}\\overrightarrow{AB}+\\dfrac{3}{4}\\overrightarrow{AC}$. 则$S_{\\triangle ADB}:S_{\\triangle ADC}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0503005B" + ], "tags": [ "第五单元" ], @@ -47072,7 +47199,9 @@ "001870": { "id": "001870", "content": "在$\\triangle ABC$中, 点$D$在$AB$上, $CD$平分$\\angle ACB$, 若$\\overrightarrow{CB}=\\overrightarrow{a}$, $\\overrightarrow{CA}=\\overrightarrow{b}$, $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=4$, 则在$\\overrightarrow{a},\\overrightarrow{b}$这组基下, $\\overrightarrow{CD}$的坐标为\\blank{80}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -47096,7 +47225,9 @@ "001871": { "id": "001871", "content": "已知$\\overrightarrow{a},\\overrightarrow{b}$不平行, 设$x$是一个实数, 若向量$(2\\overrightarrow{a}-\\overrightarrow{b})\\parallel (x\\overrightarrow{a}+3\\overrightarrow{b})$, 则实数$x$的值为\\blank{80}.", - "objs": [], + "objs": [ + "K0501004B" + ], "tags": [ "第五单元" ], @@ -47120,7 +47251,9 @@ "001872": { "id": "001872", "content": "在$\\triangle ABC$的重心为$G$, $O$是其所在平面内任意一点, 试用$\\overrightarrow{OA},\\overrightarrow{OB},\\overrightarrow{OC}$表示$\\overrightarrow{OG}$.", - "objs": [], + "objs": [ + "K0509003B" + ], "tags": [ "第五单元" ], @@ -47144,7 +47277,9 @@ "001873": { "id": "001873", "content": "已知$\\triangle OAB$为一个给定的三角形. 设$t\\in\\mathbf{R}$, 向量$\\overrightarrow{OP}=t\\overrightarrow{OA}+(\\dfrac{1}{2}-t)\\overrightarrow{OB}$,\n当$t$取遍一切实数时, 点$P$的轨迹是什么? 说明理由.", - "objs": [], + "objs": [ + "K0503005B" + ], "tags": [ "第五单元" ], @@ -47168,7 +47303,9 @@ "001874": { "id": "001874", "content": "[选做]\n已知$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$是三个非零向量, 其中任意两个向量均不平行, 若$\\overrightarrow{a}+\\overrightarrow{b}$与$\\overrightarrow{c}$平行, $\\overrightarrow{b}+\\overrightarrow{c}$与$\\overrightarrow{a}$平行, 证明: $\\overrightarrow{c}+\\overrightarrow{a}$与$\\overrightarrow{b}$平行.", - "objs": [], + "objs": [ + "K0509001B" + ], "tags": [ "第五单元" ], @@ -47192,7 +47329,9 @@ "001875": { "id": "001875", "content": "[选做]\n已知$P$是平面上一点, $\\overrightarrow{PA},\\overrightarrow{PB},\\overrightarrow{PC}$两两不平行, 且$\\overrightarrow{PA}+2\\overrightarrow{PB}=3\\overrightarrow{PC}$. 求$S_{\\triangle APB}:S_{\\triangle BPC}:S_{\\triangle CPA}$.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -47216,7 +47355,9 @@ "001876": { "id": "001876", "content": "[选做]\n已知$\\triangle ABC$的三边长分别为$a,b,c$, $\\overrightarrow{OA}=\\overrightarrow{x}$, $\\overrightarrow{OB}=\\overrightarrow{y}$, $\\overrightarrow{OC}=\\overrightarrow{z}$. $I$ 是$\\triangle ABC$的内心, 试用$a,b,c,\\overrightarrow{x},\\overrightarrow{y},\\overrightarrow{z}$表示$\\overrightarrow{OI}$.\n(尽管可以用$\\overrightarrow{x},\\overrightarrow{y},\\overrightarrow{z}$来表示$a,b,c$, 但加入$a,b,c$后答案更漂亮.)", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -47240,7 +47381,9 @@ "001877": { "id": "001877", "content": "已知$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$是三个平面向量, 那么$\\overrightarrow{a}=\\overrightarrow{b}$是$\\overrightarrow{a}\\cdot\\overrightarrow{c}=\\overrightarrow{b}\\cdot\\overrightarrow{c}$的\\blank{30}条件.\n\\twoch{充分非必要}{必要非充分}{充分必要}{既不充分又不必要}", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -47264,7 +47407,10 @@ "001878": { "id": "001878", "content": "(1) 设$\\overrightarrow{a},\\overrightarrow{b}$的夹角为$60^\\circ$, $|\\overrightarrow{a}|=2,|\\overrightarrow{b}|=\\dfrac{1}{2}$,\n则$\\overrightarrow{a}\\cdot\\overrightarrow{b}=$\\blank{50}, $\\overrightarrow{b}$在$\\overrightarrow{a}$方向上的投影为\\blank{50}.\\\\ \n(2) 已知$|\\overrightarrow{b}|=3$, $\\overrightarrow{a}$在$\\overrightarrow{b}$方向上的投影为$\\dfrac{3}{2}$, 则$\\overrightarrow{a}\\cdot\\overrightarrow{b}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B", + "K0504007B" + ], "tags": [ "第五单元" ], @@ -47288,7 +47434,9 @@ "001879": { "id": "001879", "content": "已知等边三角形$ABC$中, $\\overrightarrow{BC}=\\overrightarrow{a}$, $\\overrightarrow{CA}=\\overrightarrow{b}$, $\\overrightarrow{AB}=\\overrightarrow{c}$. 则$\\overrightarrow{a}\\cdot\\overrightarrow{b}+\\overrightarrow{b}\\cdot\\overrightarrow{c}+\\overrightarrow{c}\\cdot\\overrightarrow{a}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0505004B" + ], "tags": [ "第五单元" ], @@ -47314,7 +47462,9 @@ "001880": { "id": "001880", "content": "对于两个夹角为$\\theta$的非零向量$\\overrightarrow{a},\\overrightarrow{b}$, 我们把数$\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{b}\\right|\\sin\\theta$\n叫做向量$\\overrightarrow{a}$与$\\overrightarrow{b}$的正弦积, 记作$\\overrightarrow{a}\\diamond\\overrightarrow{b}$.\n规定零向量和任一向量的正弦积为$0$. 下列运算律\\\\ \n(1) $\\overrightarrow{a}\\diamond\\overrightarrow{b}=\\overrightarrow{b}\\diamond\\overrightarrow{a}$;\\\\ \n(2) $(\\lambda \\overrightarrow{a})\\diamond \\overrightarrow{b}=\\lambda(\\overrightarrow{a}\\diamond\\overrightarrow{b})=\\overrightarrow{a}\\diamond(\\lambda\\overrightarrow{b})$;\\\\ \n(3) $(\\overrightarrow{a}+\\overrightarrow{b})\\diamond\\overrightarrow{c}=\\overrightarrow{a}\\diamond\\overrightarrow{c}+\\overrightarrow{b}\\diamond\\overrightarrow{c}$.\n\\\\中, 成立的有\\blank{80}.", - "objs": [], + "objs": [ + "K0505003B" + ], "tags": [ "第五单元" ], @@ -47338,7 +47488,9 @@ "001881": { "id": "001881", "content": "在$\\triangle ABC$中, 设$\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AC}=\\overrightarrow{b}$, 求证: 此三角形的面积\n$$S=\\frac{1}{2}\\sqrt{(|\\overrightarrow{a}||\\overrightarrow{b}|)^2-(\\overrightarrow{a}\\cdot\\overrightarrow{b})^2}.$$", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -47362,7 +47514,9 @@ "001882": { "id": "001882", "content": "已知$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$是三个平面向量, 证明: $(\\overrightarrow{a}\\cdot\\overrightarrow{b})\\overrightarrow{c}-(\\overrightarrow{b}\\cdot\\overrightarrow{c})\\overrightarrow{a}$与$\\overrightarrow{b}$垂直.", - "objs": [], + "objs": [ + "K0505001B" + ], "tags": [ "第五单元" ], @@ -47386,7 +47540,9 @@ "001883": { "id": "001883", "content": "已知$|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=\\sqrt{2}$, $(\\overrightarrow{a}-\\overrightarrow{b})\\perp\\overrightarrow{a}$,\n求$\\overrightarrow{a}$和$\\overrightarrow{b}$的夹角.", - "objs": [], + "objs": [ + "K0505001B" + ], "tags": [ "第五单元" ], @@ -47412,7 +47568,9 @@ "001884": { "id": "001884", "content": "已知$|\\overrightarrow{a}|=4$, $|\\overrightarrow{b}|=5$, $|\\overrightarrow{a}+\\overrightarrow{b}|=6$.\\\\ \n(1) 求$\\overrightarrow{a}$和$\\overrightarrow{b}$的夹角;\\\\ \n(2) 求$|2\\overrightarrow{a}-3\\overrightarrow{b}|$的值.", - "objs": [], + "objs": [ + "K0505005B" + ], "tags": [ "第五单元" ], @@ -47436,7 +47594,9 @@ "001885": { "id": "001885", "content": "已知$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, $|\\overrightarrow{a}|=4,|\\overrightarrow{b}|=3,|\\overrightarrow{c}|=5$.\\\\ \n(1) 求$\\overrightarrow{a}\\cdot\\overrightarrow{c}$;\\\\ \n(2) 求$\\overrightarrow{a}\\cdot\\overrightarrow{b}+\\overrightarrow{b}\\cdot\\overrightarrow{c}+\\overrightarrow{c}\\cdot\\overrightarrow{a}$.", - "objs": [], + "objs": [ + "K0505003B" + ], "tags": [ "第五单元" ], @@ -47462,7 +47622,9 @@ "001886": { "id": "001886", "content": "已知$\\overrightarrow{OA}=(-1,2)$, $\\overrightarrow{OB}=(3,m)$. 若$\\overrightarrow{OA}\\perp \\overrightarrow{AB}$, 则$m=$\\blank{80}.", - "objs": [], + "objs": [ + "K0508004B" + ], "tags": [ "第五单元" ], @@ -47488,7 +47650,9 @@ "001887": { "id": "001887", "content": "已知$\\overrightarrow{OA}=(-1,2)$, $\\overrightarrow{OB}=(3,m)$. 若$\\overrightarrow{OA}\\parallel\\overrightarrow{AB}$, 则$m=$\\blank{80}.", - "objs": [], + "objs": [ + "K0508005B" + ], "tags": [ "第五单元" ], @@ -47514,7 +47678,10 @@ "001888": { "id": "001888", "content": "向量$(-3,4)$的单位向量为\\blank{80}.", - "objs": [], + "objs": [ + "K0503002B", + "K0507007B" + ], "tags": [ "第五单元" ], @@ -47541,7 +47708,11 @@ "001889": { "id": "001889", "content": "与向量$(-3,4)$垂直的单位向量为\\blank{80}.", - "objs": [], + "objs": [ + "K0503002B", + "K0507007B", + "K0508004B" + ], "tags": [ "第五单元" ], @@ -47567,7 +47738,9 @@ "001890": { "id": "001890", "content": "已知$\\overrightarrow{a}=(-1,1)$, $\\overrightarrow{b}=(m,\\sqrt{2})$, 若$\\overrightarrow{a}$与$\\overrightarrow{b}$的夹角为$120^\\circ$, 则$m=$\\blank{80}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -47591,7 +47764,9 @@ "001891": { "id": "001891", "content": "设$\\overrightarrow{i},\\overrightarrow{j}$是相互垂直的单位向量, $\\overrightarrow{a}=(m+1)\\overrightarrow{i}-3\\overrightarrow{j}$, $\\overrightarrow{b}=\\overrightarrow{i}+(m-1)\\overrightarrow{j}$, $(\\overrightarrow{a}+\\overrightarrow{b})\\perp (\\overrightarrow{a}-\\overrightarrow{b})$, 则实数$m=$\\blank{50}.", - "objs": [], + "objs": [ + "K0508004B" + ], "tags": [ "第五单元" ], @@ -47615,7 +47790,9 @@ "001892": { "id": "001892", "content": "已知$\\overrightarrow{a}=(2,-1)$, $\\overrightarrow{b}=(m,m-1)$, 若$\\overrightarrow{a}$与$\\overrightarrow{b}$的夹角是锐角, 则实数$m$的取值范围为\\blank{80}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -47639,7 +47816,9 @@ "001893": { "id": "001893", "content": "已知$\\overrightarrow{a}=(1,-5)$, $\\overrightarrow{c}=(2,-2)$, $\\overrightarrow{a}\\cdot\\overrightarrow{b}=4$, $|\\overrightarrow{b}|=4$. 则$\\overrightarrow{b}$与$\\overrightarrow{c}$的夹角为\\blank{80}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -47663,7 +47842,9 @@ "001894": { "id": "001894", "content": "设$A(4,a)$, $B(6,8)$, $C(a,b)$. 若$OABC$为平行四边形($O$为坐标原点), 则$a-b=$\\blank{80}.", - "objs": [], + "objs": [ + "K0509001B" + ], "tags": [ "第五单元" ], @@ -47687,7 +47868,9 @@ "001895": { "id": "001895", "content": "已知$M(3,4)$, $N(12,7)$, $Q$在直线$MN$上, 且$QM:MN=1:3$. 则点$Q$的坐标为\\blank{80}.", - "objs": [], + "objs": [ + "K0509002B" + ], "tags": [ "第五单元" ], @@ -47711,7 +47894,9 @@ "001896": { "id": "001896", "content": "若向量$\\overrightarrow{a}=(-1,1)$, 且$\\overrightarrow{a}$与$\\overrightarrow{a}+2\\overrightarrow{b}$的方向相同, 则$\\overrightarrow{a}\\cdot\\overrightarrow{b}$的范围是\\blank{80}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -47735,7 +47920,10 @@ "001897": { "id": "001897", "content": "若平面向量$\\overrightarrow{a},\\overrightarrow{b}$满足$|\\overrightarrow{a}|=2$, $(2\\overrightarrow{a}+\\overrightarrow{b})\\cdot \\overrightarrow{b}=12$, 则$|\\overrightarrow{b}|$的取值范围为\\blank{80}.", - "objs": [], + "objs": [ + "K0504006B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -47759,7 +47947,10 @@ "001898": { "id": "001898", "content": "设向量$\\overrightarrow{OA}=(3,1)$, $\\overrightarrow{OB}=(-1,2)$, $\\overrightarrow{OC}\\perp \\overrightarrow{OB}$, $\\overrightarrow{BC}\\parallel \\overrightarrow{OA}$. 又$\\overrightarrow{OD}+\\overrightarrow{OA}=\\overrightarrow{OC}$. 求$\\overrightarrow{OD}$的坐标.", - "objs": [], + "objs": [ + "K0508004B", + "K0508005B" + ], "tags": [ "第五单元" ], @@ -47783,7 +47974,9 @@ "001899": { "id": "001899", "content": "已知$4\\overrightarrow{m}=\\overrightarrow{a}+2\\overrightarrow{n}$, $6\\overrightarrow{b}=\\overrightarrow{a}-2\\overrightarrow{n}$. 其中$|\\overrightarrow{a}|=3$, $|\\overrightarrow{m}|=\\dfrac{\\sqrt{13}}{2}$, $|\\overrightarrow{n}|=\\dfrac{5}{2}$.\n求$\\overrightarrow{a},\\overrightarrow{b}$的夹角.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -47807,7 +48000,10 @@ "001900": { "id": "001900", "content": "(1) 是否存在三个平面向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$, 使得$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 且$\\overrightarrow{a}\\cdot\\overrightarrow{b}=-1$, $\\overrightarrow{b}\\cdot\\overrightarrow{c}=-1$, $\\overrightarrow{c}\\cdot \\overrightarrow{a}=-1$?\\\\ \n(2) 是否存在三个平面向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$, 使得$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 且$\\overrightarrow{a}\\cdot\\overrightarrow{b}=1$, $\\overrightarrow{b}\\cdot\\overrightarrow{c}=1$, $\\overrightarrow{c}\\cdot \\overrightarrow{a}=1$?\\\\ \n(3)(选做) 是否存在三个平面向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$, 使得$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 且$\\overrightarrow{a}\\cdot\\overrightarrow{b}=-1$, $\\overrightarrow{b}\\cdot\\overrightarrow{c}=-2$, $\\overrightarrow{c}\\cdot \\overrightarrow{a}=-3$?", - "objs": [], + "objs": [ + "K0505004B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -47831,7 +48027,9 @@ "001901": { "id": "001901", "content": "在直角三角形中, $\\angle C=90^\\circ$, $AC=4$, 则$\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0504007B" + ], "tags": [ "第五单元" ], @@ -47855,7 +48053,9 @@ "001902": { "id": "001902", "content": "在$\\triangle OAB$中, $\\overrightarrow{OA}=(2\\cos\\alpha,2\\sin\\alpha)$, $\\overrightarrow{OB}=(5\\cos\\beta,5\\sin\\beta)$. 若这两个向量的数量积$\\overrightarrow{OA}\\cdot\\overrightarrow{OB}=5$, 则$S_{\\triangle OAB}=$\\blank{80}.", - "objs": [], + "objs": [ + "K0510001B" + ], "tags": [ "第五单元" ], @@ -47879,7 +48079,9 @@ "001903": { "id": "001903", "content": "设点$M$是线段$BC$的中点, 点$A$在直线$BC$ 之外, $|\\overrightarrow{BC}|^2=16$, $|\\overrightarrow{AB}+\\overrightarrow{AC}|=|\\overrightarrow{AB}-\\overrightarrow{AC}|$, 则$|\\overrightarrow{AM}|=$\\blank{80}.", - "objs": [], + "objs": [ + "K0508004B" + ], "tags": [ "第五单元" ], @@ -47903,7 +48105,9 @@ "001904": { "id": "001904", "content": "已知三角形$ABC$中, $E$是边$AC$上的点且$AE=3EC$, $F$是边$AB$上的点且$AF=2FB$, 直线$BE$与$CF$相交于点$P$. 设$\\overrightarrow{AB}=\\overrightarrow{b}$, $\\overrightarrow{AC}=\\overrightarrow{c}$, 试用$\\overrightarrow{b},\\overrightarrow{c}$表示$\\overrightarrow{AP}$.\n(不可使用平面几何的方法)", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -47927,7 +48131,9 @@ "001905": { "id": "001905", "content": "在平行四边形$ABCD$中, $AC^2\\cdot BD^2=AB^4+AD^4$, 求$\\angle DAB$.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -47951,7 +48157,9 @@ "001906": { "id": "001906", "content": "已知四边形$ABCD$中, $AB=3$, $BC=4$, $CD=5$, $DA=6$, 求$\\overrightarrow{AC}\\cdot \\overrightarrow{BD}$.", - "objs": [], + "objs": [ + "K0505005B" + ], "tags": [ "第五单元" ], @@ -47975,7 +48183,9 @@ "001907": { "id": "001907", "content": "一条直线经过三角形$OAB$的重心$G$, 分别交边$OA,OB$于点$P,Q$,\n设$\\overrightarrow{OP}=x\\overrightarrow{OA},\\overrightarrow{OQ}=y\\overrightarrow{OB}(x,y\\neq 0)$, 求证: $\\dfrac{1}{x}+\\dfrac{1}{y}=3$.", - "objs": [], + "objs": [ + "K0509001B" + ], "tags": [ "第五单元" ], @@ -47999,7 +48209,9 @@ "001908": { "id": "001908", "content": "在直角三角形$ABC$中, 已知斜边$BC=a$, 若长为$2a$的线段$PQ$以点$A$为中点, 问: $\\overrightarrow{PQ}$与$\\overrightarrow{BC}$ 的夹角$\\theta$为何值时, $\\overrightarrow{BP}\\cdot\\overrightarrow{CQ}$的值取到最大, 并求出这个最大值.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -48023,7 +48235,9 @@ "001909": { "id": "001909", "content": "[选做]\n已知$O$在三角形$ABC$内部,\n且$\\overrightarrow{OA}+2\\overrightarrow{OB}+3\\overrightarrow{OC}=\\overrightarrow{0}$. 求$S_{\\triangle ABC}:S_{\\triangle AOC}$.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -48047,7 +48261,9 @@ "001910": { "id": "001910", "content": "已知$\\overrightarrow{a}=(1,-2),\\overrightarrow{b}=(2,3),\\overrightarrow{c}=(1,1)$, 将$\\overrightarrow{a}$表示为$\\overrightarrow{b_1}+\\overrightarrow{c_1}$的形式,\n其中$\\overrightarrow{b_1}\\parallel\\overrightarrow{b},\\overrightarrow{c_1}\\parallel\\overrightarrow{c}$, 结果为$\\overrightarrow{a}=$\\blank{30}$+$\\blank{30}.(在横线上填入$\\overrightarrow{b_1},\\overrightarrow{c_1}$的坐标.)", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -48073,7 +48289,9 @@ "001911": { "id": "001911", "content": "已知$\\triangle ABC$的三个顶点分别为$A(2,-1)$, $B(3,2)$, $C(-3,-1)$. 则三角形$ABC$的面积为\\blank{50}.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -48097,7 +48315,9 @@ "001912": { "id": "001912", "content": "向量$\\overrightarrow{b}=(11,12)$在$\\overrightarrow{a}=(3,4)$方向上的投影为\\blank{80}.", - "objs": [], + "objs": [ + "K0504003B" + ], "tags": [ "第五单元" ], @@ -48121,7 +48341,9 @@ "001913": { "id": "001913", "content": "已知$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$是单位向量, 且$\\overrightarrow{a}\\cdot\\overrightarrow{b}=0$, 则$(\\overrightarrow{a}-\\overrightarrow{c})\\cdot(\\overrightarrow{b}-\\overrightarrow{c})$的最小值为\\blank{80}.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -48145,7 +48367,9 @@ "001914": { "id": "001914", "content": "已知圆$O$的半径为$1$, $PA,PB$为该圆的两条切线, $A,B$为两切点. $\\overrightarrow{PA}\\cdot\\overrightarrow{PB}$的最小值为\\blank{80}.", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -48171,7 +48395,9 @@ "001915": { "id": "001915", "content": "已知$\\triangle ABC$中的两边$AB,AC$的中点分别为$M,N$, 在$BN$的延长线上取点$P$, 使$NP=BN$, 在$CM$的延长线上取点$Q$, 使$MQ=CM$. 利用向量法证明: $P,A,Q$三点共线.", - "objs": [], + "objs": [ + "K0501004B" + ], "tags": [ "第五单元" ], @@ -48195,7 +48421,10 @@ "001916": { "id": "001916", "content": "已知$\\overrightarrow{OA}=(3,-4)$, $\\overrightarrow{OB}=(6,-3)$, $\\overrightarrow{OC}=(5-m,-3-m)$.\\\\ \n(1) 若$A,B,C$共线, 求实数$m$的值;\\\\ \n(2) 若$\\triangle ABC$是直角三角形, 求实数$m$的值.", - "objs": [], + "objs": [ + "K0508004B", + "K0508005B" + ], "tags": [ "第五单元" ], @@ -48219,7 +48448,9 @@ "001917": { "id": "001917", "content": "在$\\triangle ABC$中, $O$为中线$AM$上的一个动点, 若$AM=2$, 求$\\overrightarrow{OA}\\cdot(\\overrightarrow{OB}+\\overrightarrow{OC})$的最小值, 以及取最小值时$O$的位置.", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -81820,7 +82051,10 @@ "003326": { "id": "003326", "content": "若$OEF$是不共线的任意三点, 则以下各式中成立的是\\bracket{20}.\n\\twoch{$\\overrightarrow{EF}=\\overrightarrow{OF}+\\overrightarrow{OE}$\t}{$\\overrightarrow{EF}=\\overrightarrow{OF}-\\overrightarrow{OE}$\t }{$\\overrightarrow{EF}=-\\overrightarrow{OF}+\\overrightarrow{OE}$\t }{$\\overrightarrow{EF}=-\\overrightarrow{OF}-\\overrightarrow{OE}$}", - "objs": [], + "objs": [ + "K0502002B", + "K0502007B" + ], "tags": [ "第五单元" ], @@ -81841,7 +82075,11 @@ "003327": { "id": "003327", "content": "已知$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$为非零向量, 下列命题中假命题是\\blank{50}.\\\\\n\\textcircled{1} $\\overrightarrow a+(-\\overrightarrow a)=0$;\\\\\n\\textcircled{2} 若$| \\overrightarrow a|=|\\overrightarrow b|$, 则$\\overrightarrow a=\\overrightarrow b$或$\\overrightarrow a=-\\overrightarrow b$;\\\\\n\\textcircled{3} $\\overrightarrow a\\parallel \\overrightarrow b$是$| \\overrightarrow a+\\overrightarrow b|=|\\overrightarrow a|+|\\overrightarrow b|$成立的充分非必要条件;\\\\\n\\textcircled{4} $\\overrightarrow a+\\overrightarrow b+\\overrightarrow c=\\overrightarrow 0$是$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$可以首尾相接构成三角形的必要非充分条件.", - "objs": [], + "objs": [ + "K0501001B", + "K0501004B", + "K0501006B" + ], "tags": [ "第五单元" ], @@ -81862,7 +82100,10 @@ "003328": { "id": "003328", "content": "设$\\overrightarrow m$、$\\overrightarrow n$为非零向量, 则``存在负数$\\lambda$, 使得$\\overrightarrow m=\\lambda \\overrightarrow n$''是``$\\overrightarrow m\\cdot \\overrightarrow n<0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", - "objs": [], + "objs": [ + "K0503001B", + "K0504006B" + ], "tags": [ "第五单元" ], @@ -81883,7 +82124,9 @@ "003329": { "id": "003329", "content": "若$\\overrightarrow{P_1O}=-3\\overrightarrow{OP_2}$, 则$\\overrightarrow{P_1P_2}=$\\blank{50}$\\overrightarrow{P_2O}$.", - "objs": [], + "objs": [ + "K0503003B" + ], "tags": [ "第五单元" ], @@ -81904,7 +82147,10 @@ "003330": { "id": "003330", "content": "已知$\\triangle ABC$中, $AB=2$, $AC=3$, $\\angle A=120^\\circ$, 设$\\overrightarrow a=\\overrightarrow{AB},\\overrightarrow b=\\overrightarrow{AC}$, 用$\\overrightarrow a,\\overrightarrow b$表示$\\overrightarrow{BC}$的单位向量为\\blank{50};$|\\overrightarrow a+\\overrightarrow b|=$\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -81925,7 +82171,9 @@ "003331": { "id": "003331", "content": "若$|\\overrightarrow{AB}|=8$, $|\\overrightarrow{AC}|=9$, 则$|\\overrightarrow{BC}|$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0502002B" + ], "tags": [ "第五单元" ], @@ -81946,7 +82194,9 @@ "003332": { "id": "003332", "content": "已知向量$\\overrightarrow a$、$\\overrightarrow b$是单位向量, $\\overrightarrow a\\cdot \\overrightarrow b=0$, 且向量$\\overrightarrow c$满足$|\\overrightarrow c-\\overrightarrow a-\\overrightarrow b|=1$, 则$|\\overrightarrow c|$的取值范围是\\bracket{20}.\n\\fourch{$[\\sqrt 2-1,\\sqrt 2+1]$}{$[\\sqrt 2-1,\\sqrt 2]$}{$[\\sqrt 2,\\sqrt 2+1]$}{$[2-\\sqrt 2,2+\\sqrt 2]$}", - "objs": [], + "objs": [ + "K0502002B" + ], "tags": [ "第五单元" ], @@ -81967,7 +82217,9 @@ "003333": { "id": "003333", "content": "若平面上三点$A,B,C$共线, $O$是直线$AB$外一点, 且$\\overrightarrow{OC}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$($\\lambda,\\mu \\in \\mathbf{R}$), 求$\\lambda +\\mu$的值.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -81988,7 +82240,10 @@ "003334": { "id": "003334", "content": "已知$|\\overrightarrow a+\\overrightarrow b|=2| \\overrightarrow a-\\overrightarrow b|$, $|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$. 求:\\\\\n(1) $|3\\overrightarrow a-2\\overrightarrow b|$;\\\\\n(2) $\\overrightarrow a$与$\\overrightarrow a+\\overrightarrow b$的夹角;\\\\\n(3) $\\overrightarrow a$在$\\overrightarrow a+\\overrightarrow b$方向上的投影.", - "objs": [], + "objs": [ + "K0504003B", + "K0505005B" + ], "tags": [ "第五单元" ], @@ -82009,7 +82264,9 @@ "003335": { "id": "003335", "content": "已知$|\\overrightarrow{a}|$=$\\sqrt 2$, $|\\overrightarrow{b}|=3$, $\\overrightarrow{a}$和$\\overrightarrow{b}$的夹角为$45^\\circ$, 求当向量$\\overrightarrow{a}+\\lambda \\overrightarrow{b}$与$\\lambda \\overrightarrow{a}+\\overrightarrow{b}$夹角为锐角时, 求 $\\lambda$的取值范围.", - "objs": [], + "objs": [ + "K0508003B" + ], "tags": [ "第五单元" ], @@ -82030,7 +82287,9 @@ "003336": { "id": "003336", "content": "若点$O$是$\\triangle ABC$内一点, $\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow 0$, 则点$O$是$\\triangle ABC$的\\blank{50}心.", - "objs": [], + "objs": [ + "K0509003B" + ], "tags": [ "第五单元" ], @@ -82051,7 +82310,9 @@ "003337": { "id": "003337", "content": "在平行四边形$ABCD$中, $AC$与$BD$交于点$O$, $E$是线段$OD$的中点, $AE$的延长线与$CD$交于点$F$. 若$\\overrightarrow{AC}=\\overrightarrow a$, $\\overrightarrow{BD}=\\overrightarrow b$, 则$\\overrightarrow{AF}=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=stealth, line cap = round, line join = round, scale = 2]\n \\draw (0,0) node [below left] {$A$} -- (2,0) node [below right] {$B$}-- (3,1) node [above right] {$C$} -- (1,1) node [above left] {$D$}-- cycle;\n \\draw (0,0) -- (3,1) (2,0) -- (1,1);\n \\draw (0,0) -- (5/3,1) node [above] {$F$};\n \\draw (1.5,0.5) node [below] {$O$};\n \\draw (1.25,0.75) node [below] {$E$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -82072,7 +82333,9 @@ "003338": { "id": "003338", "content": "平面上点$ABC$满足$|\\overrightarrow{AB}|=3$, $|\\overrightarrow{BC}|=4$, $|\\overrightarrow{CA}|=5$, 则$\\overrightarrow{AB}\\cdot \\overrightarrow{BC}+\\overrightarrow{BC}\\cdot \\overrightarrow{CA}+\\overrightarrow{CA}\\cdot \\overrightarrow{AB}$=\\blank{50}.", - "objs": [], + "objs": [ + "K0505004B" + ], "tags": [ "第五单元" ], @@ -82093,7 +82356,9 @@ "003339": { "id": "003339", "content": "在$\\triangle ABC$中, $\\angle A=60^\\circ$, $AB=3$, $AC=2$. 若$\\overrightarrow{BD}=2\\overrightarrow{DC}$, $\\overrightarrow{AE}=\\lambda \\overrightarrow{AC}-\\overrightarrow{AB}$, 且$\\overrightarrow{AD}\\cdot \\overrightarrow{AE}=-4$, 则$\\lambda$的值为\\blank{50}.", - "objs": [], + "objs": [ + "K0504006B" + ], "tags": [ "第五单元" ], @@ -82114,7 +82379,10 @@ "003340": { "id": "003340", "content": "$\\overrightarrow a$、$\\overrightarrow b$是非零向量且满足$(\\overrightarrow a-2\\overrightarrow b)\\perp \\overrightarrow a$, $(\\overrightarrow b-2\\overrightarrow a)\\perp \\overrightarrow b$, 则$\\overrightarrow a$与$\\overrightarrow b$的夹角是\\blank{50}.", - "objs": [], + "objs": [ + "K0505005B", + "K0508004B" + ], "tags": [ "第五单元" ], @@ -82135,7 +82403,9 @@ "003341": { "id": "003341", "content": "(1) 已知$\\overrightarrow a$与$\\overrightarrow b$都是非零向量, 且$|\\overrightarrow a|=|\\overrightarrow b|=|\\overrightarrow a-\\overrightarrow b|$, 求$\\overrightarrow a$与$\\overrightarrow a+\\overrightarrow b$的夹角;\\\\\n(2) 已知向量$\\overrightarrow a$与$\\overrightarrow b$的夹角为$120^\\circ$, $|\\overrightarrow a|=3,|\\overrightarrow a+\\overrightarrow b|=\\sqrt{13}$, 求$|\\overrightarrow b|$的值.", - "objs": [], + "objs": [ + "K0505005B" + ], "tags": [ "第五单元" ], @@ -82156,7 +82426,9 @@ "003342": { "id": "003342", "content": "*已知向量$\\overrightarrow a$、$\\overrightarrow b$满足$| \\overrightarrow a|=1$, $|\\overrightarrow{b}|=2$, 求$|\\overrightarrow a+\\overrightarrow b|+|\\overrightarrow a-\\overrightarrow b|$的最小值、最大值.", - "objs": [], + "objs": [ + "K0505005B" + ], "tags": [ "第五单元" ], @@ -82177,7 +82449,11 @@ "003343": { "id": "003343", "content": "若$\\overrightarrow{AB}=(2,4)$, $\\overrightarrow{AC}=(1,3)$, 则$\\overrightarrow{BC}$方向相反的单位向量是\\blank{50}.", - "objs": [], + "objs": [ + "K0502007B", + "K0503002B", + "K0507007B" + ], "tags": [ "第五单元" ], @@ -82198,7 +82474,9 @@ "003344": { "id": "003344", "content": "已知点$P_1(2,-1)$、$P_2(0,5)$, 若点$P$在直线$P_1P_2$上, 且满足$|\\overrightarrow{P_1P_2}|=2|\\overrightarrow{PP_2}|$, 则点$P$的坐标为\\blank{50}.", - "objs": [], + "objs": [ + "K0509002B" + ], "tags": [ "第五单元" ], @@ -82219,7 +82497,9 @@ "003345": { "id": "003345", "content": "若三点$A(2,2)$、$B(a,0)$、$C(0,4)$共线, 则$a$的值等于\\blank{50}.", - "objs": [], + "objs": [ + "K0508005B" + ], "tags": [ "第五单元" ], @@ -82240,7 +82520,10 @@ "003346": { "id": "003346", "content": "已知$\\overrightarrow{e_1},\\overrightarrow{e_2}$是不平行的向量, 设$\\overrightarrow a=\\overrightarrow {e_1}+k\\overrightarrow {e_2}$, $\\overrightarrow b=k\\overrightarrow {e_1}+\\overrightarrow {e_2}$, 则$\\overrightarrow a$与$\\overrightarrow b$平行的充要条件是实数$k$等于\\blank{50}.", - "objs": [], + "objs": [ + "K0501004B", + "K0503001B" + ], "tags": [ "第五单元" ], @@ -82261,7 +82544,11 @@ "003347": { "id": "003347", "content": "已知向量$\\overrightarrow a=(1,2)$, $\\overrightarrow b=(0,3)$, 则与$\\overrightarrow a$垂直的单位向量的坐标为\\blank{50}; $\\overrightarrow b$在$\\overrightarrow a$的方向上的投影为\\blank{50}.", - "objs": [], + "objs": [ + "K0503002B", + "K0504003B", + "K0508004B" + ], "tags": [ "第五单元" ], @@ -82282,7 +82569,9 @@ "003348": { "id": "003348", "content": "已知$\\triangle ABC$的三个顶点分别是$A(1,\\dfrac 32)$, $B(4,-2)$, $C(1,y)$, 其重心坐标为$G(x,-1)$, 则$x,y$的值分别是\n\\blank{50}.", - "objs": [], + "objs": [ + "K0509003B" + ], "tags": [ "第五单元" ], @@ -82303,7 +82592,10 @@ "003349": { "id": "003349", "content": "若$\\overrightarrow a=(x,1)$, $\\overrightarrow b=(2,3x)$, 那么$\\dfrac{\\overrightarrow a\\cdot \\overrightarrow b}{| \\overrightarrow a|^2}+{|\\overrightarrow b |^2}$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0505005B", + "K0508003B" + ], "tags": [ "第五单元" ], @@ -82324,7 +82616,9 @@ "003350": { "id": "003350", "content": "设向量$\\overrightarrow{OA}=(1,-2)$, $\\overrightarrow{OB}=(a,-1)$, $\\overrightarrow{OC}=(-b,0)$, 其中点$O$为坐标原点, $a>0$, $b>0$, 若$A$、$B$、$C$三点共线, 则$\\dfrac 1a+\\dfrac 2b$的最小值为\\blank{50}.", - "objs": [], + "objs": [ + "K0508005B" + ], "tags": [ "第五单元" ], @@ -82345,7 +82639,9 @@ "003351": { "id": "003351", "content": "已知直线$l$上两个点$A(0,3)$、$C(3,0)$, $O$为坐标原点.\\\\\n(1) 若$\\overrightarrow{OD}=-\\dfrac 13\\overrightarrow{OA}+\\dfrac 43\\overrightarrow{OC}$, 试确定点$D$与直线$l$的位置关系;\\\\\n(2) 已知点$B(1,2)$是直线$l$上的一点, 求证: 若存在实数$m,n$使向量$\\overrightarrow{OB}=m\\cdot \\overrightarrow{OA}+n\\cdot \\overrightarrow{OC}$, 则 $m+n=1$;\\\\\n(3) 若存在实数$m,n$使向量$\\overrightarrow{OB}=m\\overrightarrow{OA}+n\\overrightarrow{OC}$, 且$m+n=2$, 写出满足条件的所有点$B$的轨迹.", - "objs": [], + "objs": [ + "K0509001B" + ], "tags": [ "第五单元" ], @@ -82366,7 +82662,10 @@ "003352": { "id": "003352", "content": "已知$\\overrightarrow m=(2\\sqrt 3,1)$, $\\overrightarrow n=(\\cos^2\\dfrac A2,\\sin A)$, $A$、$B$、$C$是$\\triangle ABC$的内角.\\\\\n(1) 当$A=\\dfrac{\\pi}2$时, 求$|\\overrightarrow n|$的值;\\\\\n(2) 若$C=\\dfrac{2\\pi}3$, $|AB|=3$, 当$\\overrightarrow m\\cdot\\overrightarrow n$取最大值时, 求$A$的大小及边$BC$的长.", - "objs": [], + "objs": [ + "K0501003B", + "K0504006B" + ], "tags": [ "第五单元" ], @@ -82387,7 +82686,9 @@ "003353": { "id": "003353", "content": "已知$\\triangle ABC$的顶点坐标分别为$A(1,0)$, $B(5,8)$, $C(7,-4)$, 在边$AB$上有一点$P$, 其横坐标为$4$, 在边$AC$上求一点$Q$, 使线段$PQ$把$\\triangle ABC$分成面积相等的两个部分.", - "objs": [], + "objs": [ + "K0509002B" + ], "tags": [ "第五单元" ], @@ -82409,7 +82710,8 @@ "id": "003354", "content": "给出下列命题:\\\\\n\\textcircled{1} 非零向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=|\\overrightarrow b|=|\\overrightarrow a-\\overrightarrow b|$, 则$\\overrightarrow a$与$\\overrightarrow a+\\overrightarrow b$的夹角为$30^\\circ$;\\\\\n\\textcircled{2} $\\overrightarrow a\\cdot \\overrightarrow b>0$, 是$\\overrightarrow a$、$\\overrightarrow b$的夹角为锐角的充要条件;\\\\\n\\textcircled{3} 将函数$y=|x-1|$的图像按向量$\\overrightarrow a=(-1,0)$平移, 得到的图像对应的函数表达式为$y=|x|$;\\\\\n\\textcircled{4} 在$\\triangle ABC$中, 若$(\\overrightarrow{AB}+\\overrightarrow{AC})\\cdot (\\overrightarrow{AB}-\\overrightarrow{AC})=0$, 则$\\triangle ABC$为等腰三角形.\\\\\n以上命题正确的是\\blank{50}(注: 把你认为正确的命题的序号都填上).", "objs": [ - "K0208005B" + "K0208005B", + "K0505005B" ], "tags": [ "第五单元" @@ -82431,7 +82733,9 @@ "003355": { "id": "003355", "content": "若$\\overrightarrow a$和$\\overrightarrow b$夹角为$120^\\circ$, 且$|\\overrightarrow a|=|\\overrightarrow b|=1$, $| \\overrightarrow c|=2$, $\\overrightarrow c$与$\\overrightarrow a$、$\\overrightarrow b$夹角均为$60^\\circ$, 用$\\overrightarrow a$和$\\overrightarrow b$表示$\\overrightarrow c$为\\blank{50}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ], @@ -82452,7 +82756,9 @@ "003356": { "id": "003356", "content": "在平面直角坐标系中, 已知$A(1,0)$、$B(0,-1)$, $P$是曲线$y=\\sqrt{1-x^2}$上一个动点, 则$\\overrightarrow{BP}\\cdot \\overrightarrow{BA}$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0504007B" + ], "tags": [ "第五单元" ], @@ -82475,7 +82781,9 @@ "003357": { "id": "003357", "content": "设$\\overrightarrow{PA}=(k,12)$, $\\overrightarrow{PB}=(4,5)$, $\\overrightarrow{PC}=(10,k)$, 则$k=$\\blank{50}时, 点$A$、$B$、$C$共线.", - "objs": [], + "objs": [ + "K0508005B" + ], "tags": [ "第五单元" ], @@ -82496,7 +82804,10 @@ "003358": { "id": "003358", "content": "已知直角梯形$ABCD$ , $AD\\parallel BC$, $\\angle BAD=90^\\circ$. $AD=2$, $BC=1$, $P$是腰$AB$上的动点, 则$|\\overrightarrow{PC}+\\overrightarrow{PD}|$的最小值为\\blank{50}.", - "objs": [], + "objs": [ + "K0509002B", + "K0509004B" + ], "tags": [ "第五单元" ], @@ -82517,7 +82828,9 @@ "003359": { "id": "003359", "content": "已知三角形$ABC$, $\\overrightarrow{AB}=(k-1,2)$, $\\overrightarrow{AC}=(-1,2)$.\\\\\n(1) 若$k=4$, 求$S_{\\triangle ABC}$; \\\\\n(2)若三角形为直角三角形, 求$S_{\\triangle ABC}$.", - "objs": [], + "objs": [ + "K0509004B" + ], "tags": [ "第五单元" ], @@ -82538,7 +82851,10 @@ "003360": { "id": "003360", "content": "已知平面内三点$P(-2,0)$, $Q(-1,1)$和$R(-3,0)$, 设$\\overrightarrow m=\\overrightarrow{PQ}$, $\\overrightarrow n=\\overrightarrow{PR}$, 当实数$k$为何值时, 向量$k\\overrightarrow m+\\overrightarrow n$与向量$k\\overrightarrow m-2\\overrightarrow n$互相垂直、平行?", - "objs": [], + "objs": [ + "K0508004B", + "K0508005B" + ], "tags": [ "第五单元" ], @@ -82559,7 +82875,9 @@ "003361": { "id": "003361", "content": "在矩形$ABCD$中, $AB=1$, $AD=2$, 动点$P$在以点$C$为圆心且与$BD$相切的圆上. 若$\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD}$, 求$\\lambda +\\mu$的最大值.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第五单元" ],