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 024d31c3..449d9244 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": 1, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 1, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -21,12 +21,16 @@ "\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\":\"导数\"}\n", + " {\"tags\":[\"第六单元\"],\"content_not\":[\"体积\",\"面积\"],\"usages_not\":[\"2023届\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", "# {\"tags\": [\"第三单元\"], \"content1\": [r\"[\\d]\\alpha\",\"2x\"], \"content2\": [\"sin\"], \"content3\": [\"cos\"],\"content4\": [\"cot\"], \"content5\": [\"tan\"]},\n", "# ]\n", + "# 实例2: keywords_dict_table = [\n", + "# {\"tags\":[\"第三单元\"],\"content\":[\"f\\(\",\"y=\",\"函数\"],\"usages\":[r\"0\\.9\",r\"0\\.8[3-9]\"],\"usages_not\":[r\"0\\.[0-7]\",r\"0\\.8[0-2]\"],\"usages1\":[\"2023届\"]},\n", + "# {\"tags\":[\"第五单元\"],\"usages\":[r\"0\\.9\"],\"usages_not\":[r\"0\\.[0-7]\",r\"0\\.8[0-2]\"],\"usages1\":[\"2023届\"]}\n", + "# ]\n", "\"\"\"---设置输出文件名---\"\"\"\n", "filename = \"文本文件/题号筛选.txt\"\n", "\"\"\"---文件名设置完毕---\"\"\"\n", @@ -85,7 +89,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.9.7 ('base')", + "display_name": "Python 3.8.8 ('base')", "language": "python", "name": "python3" }, @@ -99,12 +103,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.7" + "version": "3.8.8" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba" + "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac" } } }, diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index b9183aaf..1f6a48e8 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "text": [ "首个空闲id: 12030 , 直至 020000\n", "首个空闲id: 20227 , 直至 030000\n", - "首个空闲id: 30451 , 直至 999999\n" + "首个空闲id: 30473 , 直至 999999\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 97cd755c..8ce01b73 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 @@ -2,941 +2,142 @@ "cells": [ { "cell_type": "code", - "execution_count": 6, + "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 000333 , 字段: objs 中已有该数据: K0405002X\n", - "题号: 000333 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000336 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000353 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000370 , 字段: objs 中已有该数据: K0405002X\n", - "题号: 000370 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000376 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000403 , 字段: objs 中已有该数据: K0406004X\n", - "题号: 000408 , 字段: objs 中已有该数据: K0402005X\n", - "题号: 000430 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000443 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000455 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000457 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000475 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000488 , 字段: objs 中已有该数据: K0405002X\n", - "题号: 000488 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000499 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000514 , 字段: objs 中已有该数据: K0406003X\n", - "题号: 000516 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000527 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000534 , 字段: objs 中已有该数据: K0402001X\n", - "题号: 000534 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000543 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000546 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000548 , 字段: objs 中已有该数据: K0401003X\n", - "题号: 000548 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000562 , 字段: objs 中已有该数据: K0401003X\n", - "题号: 000562 , 字段: objs 中已有该数据: K0403001X\n", - "题号: 000573 , 字段: objs 中已有该数据: K0401004X\n", - "题号: 000573 , 字段: objs 中已有该数据: K0401003X\n", - "题号: 000574 , 字段: objs 中已有该数据: K0407002X\n", - "题号: 000575 , 字段: objs 中已有该数据: K0406005X\n", - "题号: 000575 , 字段: objs 中已有该数据: K0406004X\n", - "题号: 000583 , 字段: objs 中已有该数据: K0403001X\n", - "题号: 000583 , 字段: objs 中已有该数据: K0316002B\n", - "题号: 000588 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000591 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000595 , 字段: objs 中已有该数据: K0409001X\n", - "题号: 000595 , 字段: objs 中已有该数据: K0407002X\n", - "题号: 000599 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000600 , 字段: objs 中已有该数据: K0819005X\n", - "题号: 000600 , 字段: objs 中已有该数据: K0401002X\n", - "题号: 000606 , 字段: objs 中已有该数据: K0405001X\n", - "题号: 000633 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000638 , 字段: objs 中已有该数据: K0405001X\n", - "题号: 000662 , 字段: objs 中已有该数据: K0405002X\n", - "题号: 000674 , 字段: objs 中已有该数据: K0405001X\n", - "题号: 000681 , 字段: objs 中已有该数据: K0401004X\n", - "题号: 000713 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000733 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000752 , 字段: objs 中已有该数据: K0401002X\n", - "题号: 000752 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 000784 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 000784 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 000791 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000791 , 字段: objs 中已有该数据: K0405001X\n", - "题号: 000798 , 字段: objs 中已有该数据: K0403004X\n", - "题号: 000798 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 000814 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 000814 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000820 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 000820 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 000835 , 字段: objs 中已有该数据: K0403004X\n", - "题号: 000835 , 字段: objs 中已有该数据: K0119001B\n", - "题号: 000856 , 字段: objs 中已有该数据: KNONE\n", - "题号: 000867 , 字段: objs 中已有该数据: K0402005X\n", - "题号: 000867 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000875 , 字段: objs 中已有该数据: K0403006X\n", - "题号: 000875 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 000875 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 000897 , 字段: objs 中已有该数据: K0401003X\n", - "题号: 000897 , 字段: objs 中已有该数据: K0826001X\n", - "题号: 000928 , 字段: objs 中已有该数据: K0405002X\n", - "题号: 000928 , 字段: objs 中已有该数据: K0405003X\n", - "题号: 000950 , 字段: objs 中已有该数据: K0407002X\n", - "题号: 000958 , 字段: objs 中已有该数据: K0405005X\n", - "题号: 000966 , 字段: objs 中已有该数据: K0405005X\n", - "题号: 001018 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001019 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001020 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001021 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001022 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001023 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001024 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001025 , 字段: objs 中已有该数据: K0408003X\n", - "题号: 001026 , 字段: objs 中已有该数据: K0409001X\n", - "题号: 001027 , 字段: objs 中已有该数据: K0409001X\n", - "题号: 001028 , 字段: objs 中已有该数据: K0409001X\n", - "题号: 001467 , 字段: objs 中已有该数据: K0409001X\n", - "题号: 001739 , 字段: objs 中已有该数据: K0406001X\n", - "题号: 001740 , 字段: objs 中已有该数据: K0406001X\n", - "题号: 001741 , 字段: objs 中已有该数据: K0406003X\n", - "题号: 001742 , 字段: objs 中已有该数据: K0406003X\n", - "题号: 001743 , 字段: objs 中已有该数据: K0406003X\n", - "题号: 001744 , 字段: objs 中已有该数据: K0406003X\n", - "题号: 001745 , 字段: objs 中已有该数据: K0402005X\n", - "题号: 001746 , 字段: objs 中已有该数据: K0402005X\n", - "题号: 001747 , 字段: objs 中已有该数据: K0401002X\n", - "题号: 001748 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001749 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001750 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001751 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001752 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001753 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001754 , 字段: objs 中已有该数据: K0401006X\n", - "题号: 001755 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001756 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001757 , 字段: objs 中已有该数据: K0401006X\n", - "题号: 001758 , 字段: objs 中已有该数据: K0401006X\n", - "题号: 001758 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001759 , 字段: objs 中已有该数据: K0401007X\n", - "题号: 001759 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001760 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001761 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001762 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001763 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001764 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001765 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001766 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001767 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001768 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001769 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001770 , 字段: objs 中已有该数据: K0403005X\n", - "题号: 001771 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001772 , 字段: objs 中已有该数据: K0403005X\n", - "题号: 001773 , 字段: objs 中已有该数据: K0403005X\n", - "题号: 001774 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001775 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001776 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001777 , 字段: objs 中已有该数据: K0403001X\n", - "题号: 001777 , 字段: objs 中已有该数据: K0401002X\n", - "题号: 001778 , 字段: objs 中已有该数据: K0404004X\n", - "题号: 001779 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 001779 , 字段: objs 中已有该数据: K0401003X\n", - "题号: 001780 , 字段: objs 中已有该数据: K0403002X\n", - "题号: 001781 , 字段: objs 中已有该数据: K0404002X\n", - "题号: 001781 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001782 , 字段: objs 中已有该数据: K0403006X\n", - "题号: 001782 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001783 , 字段: objs 中已有该数据: K0403003X\n", - "题号: 001784 , 字段: objs 中已有该数据: K0402002X\n", - "题号: 001785 , 字段: objs 中已有该数据: K0402002X\n", - "题号: 001786 , 字段: objs 中已有该数据: K0402002X\n", - "题号: 001787 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001787 , 字段: objs 中已有该数据: K0401002X\n", - "题号: 001788 , 字段: objs 中已有该数据: K0403003X\n", - "题号: 001789 , 字段: objs 中已有该数据: K0402001X\n", - "题号: 001790 , 字段: objs 中已有该数据: KNONE\n", - "题号: 001791 , 字段: objs 中已有该数据: KNONE\n", - "题号: 001792 , 字段: objs 中已有该数据: K0404001X\n", - "题号: 001793 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001794 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001794 , 字段: objs 中已有该数据: K0404003X\n", - "题号: 001795 , 字段: objs 中已有该数据: K0310001B\n", - "题号: 001795 , 字段: objs 中已有该数据: K0407002X\n", - "题号: 001796 , 字段: objs 中已有该数据: K0402004X\n", - "题号: 001796 , 字段: objs 中已有该数据: 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\ No newline at end of file diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index 2aa41758..a9119356 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": 4, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id---\"\"\"\n", - "old_id = \"30353\"\n", - "new_id = \"30451\"\n", + "old_id = \"8438\"\n", + "new_id = \"30473\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\n", "old_id = old_id.zfill(6)\n", diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index e7986b13..f53f224e 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -7,10 +7,10 @@ "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 30449\n", - "origin = \"自拟题目\"\n", + "starting_id = 30452\n", + "origin = \"高中数学教与学例题与习题\"\n", "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目4.tex\"\n", - "editor = \"20221103\\t王伟叶\"" + "editor = \"20221104\\t王伟叶\"" ] }, { diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index 1ecb4fa4..2210cb84 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": 8, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -13,9 +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", - "开始编译教师版本pdf文件: 临时文件/线上测验02_教师_20221103.tex\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", "0\n", - "开始编译学生版本pdf文件: 临时文件/线上测验02_学生_20221103.tex\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", "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\",\"29\"),(\"标题文字待处理\",\"导数的应用\")] \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", "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/线上测验02\"\n", + "destination_file = \"临时文件/30_等差数列与等比数列\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"30374,30372,30343,30408,30410,30411,30377,9922,30352,30451,9926\",\n", - "\"30379,30407,30415,30356,4009,30418,30419,30421,30369,30382\"\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", "]\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 ecf90d20..2abd3448 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": 1, + "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/赋能05_教师用_20221103.tex\n", + "开始编译教师版本pdf文件: 临时文件/立体几何空间向量选题_教师用_20221104.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/赋能05_学生用_20221103.tex\n", + "开始编译学生版本pdf文件: 临时文件/立体几何空间向量选题_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", - "000366,000367,000368,000369,030281,000371,000372,000373,000374,000375\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", "\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/赋能05\"\n", + "filename = \"临时文件/立体几何空间向量选题\"\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 7e6a3134..40de2ee1 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -7462,7 +7462,8 @@ "K0624001X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7485,7 +7486,8 @@ "K0625001X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7508,7 +7510,8 @@ "K0629001X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7533,7 +7536,8 @@ "K0613007B" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7582,7 +7586,8 @@ "K0631002X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7606,7 +7611,8 @@ "K0627007X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7661,7 +7667,8 @@ "K0631003X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7716,7 +7723,8 @@ "K0625004X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7740,7 +7748,8 @@ "K0630002X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7794,7 +7803,8 @@ "K0630002X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -7818,7 +7828,8 @@ "K0631002X" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -15209,7 +15220,7 @@ }, "000573": { "id": "000573", - "content": "若存在公差为$d$的等差数列$\\{a_n\\} \\ (n\\in \\mathbf{N}^*)$满足$a_3a_4+1=0$, 则公差$d$的取值范围是\\blank{50}.", + "content": "若存在公差为$d$, 各项均为实数的等差数列$\\{a_n\\} \\ (n\\in \\mathbf{N}^*)$满足$a_3a_4+1=0$, 则公差$d$的取值范围是\\blank{50}.", "objs": [ "K0401004X", "K0401003X" @@ -15226,7 +15237,8 @@ ], "origin": "赋能练习", "edit": [ - "20220624\t朱敏慧, 王伟叶" + "20220624\t朱敏慧, 王伟叶", + "20221104\t徐慧" ], "same": [], "related": [], @@ -20677,7 +20689,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "$(-4,-3,2)$", @@ -51783,7 +51796,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -51855,7 +51869,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -51879,7 +51894,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -51905,7 +51921,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -51929,7 +51946,8 @@ "content": "已知空间四边形$ABCD$中, $AB=AC$, $\\angle DAB=\\angle DAC$. 求证: $DA\\perp BC$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -51953,7 +51971,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -51977,7 +51996,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52001,7 +52021,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52025,7 +52046,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52049,7 +52071,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52073,7 +52096,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52097,7 +52121,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52121,7 +52146,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52145,7 +52171,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52169,7 +52196,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52193,7 +52221,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52217,7 +52246,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52241,7 +52271,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52265,7 +52296,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52289,7 +52321,8 @@ "content": "空间向量$(3,4,12)$的单位向量为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52315,7 +52348,8 @@ "content": "若向量$(-2,3,m)$与$(n,-9,2)$平行, 则$m-n=$\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52363,7 +52397,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52387,7 +52422,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52435,7 +52471,8 @@ "content": "平行于$y$轴的直线的一个方向向量为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52459,7 +52496,8 @@ "content": "$zOx$平面的一个法向量为\\blank{50}.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52483,7 +52521,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52507,7 +52546,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52531,7 +52571,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52555,7 +52596,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52579,7 +52621,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52603,7 +52646,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52627,7 +52671,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52651,7 +52696,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52675,7 +52721,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -52771,7 +52818,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52819,7 +52867,8 @@ "content": "在棱长为$4$的正方体$ABCD-A'B'C'D'$中, 点$E,F$分别在棱$AD,BC$上, $AE:ED=CF:FB=1:2$, 点$G,H$分别是$AA'$和$A'B'$的中点, 求异面直线$EF$与$GH$的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -52921,7 +52970,8 @@ "content": "在棱长为$1$的正四面体$ABCD$中, $G,H$分别为$BD,CD$的中点, 设$\\overrightarrow{AB}=\\overrightarrow{b}, \\overrightarrow{AC}=\\overrightarrow{c}, \\overrightarrow{AD}=\\overrightarrow{d}$. 按下列指定步骤求异面直线$AH, CG$的距离.(希望能借此题体会出坐标法与基向量法之间的异同点)\\\\ \n(1) 用$\\overrightarrow{b},\\overrightarrow{c},\\overrightarrow{d}$表示向量$\\overrightarrow{AH},\\overrightarrow{CG}$;\\\\ \n(2) 用$\\overrightarrow{b},\\overrightarrow{c},\\overrightarrow{d}$表示$AH,CG$公垂线的一个方向向量$\\overrightarrow{e}$;\\\\ \n(3) 求异面直线$AH,CG$之间的距离.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -84019,7 +84069,7 @@ }, "003240": { "id": "003240", - "content": "已知数列$\\{a_n\\}$是等比数列, 且$a_n>0$, 若$b_n=\\log_2a_n$, 则\\bracket{20}\n\\twoch{$\\{b_n\\}$一定是递增的等差数列}{$\\{b_n\\}$不可能是等比数列}{$\\{b_n+1\\}$一定是等差数列}{$\\{3^b_n\\}$不是等比数列}", + "content": "已知数列$\\{a_n\\}$是等比数列, 且$a_n>0$, 若$b_n=\\log_2a_n$, 则\\bracket{20}\n\\twoch{$\\{b_n\\}$一定是递增的等差数列}{$\\{b_n\\}$不可能是等比数列}{$\\{b_n+1\\}$一定是等差数列}{$\\{3^{b_n}\\}$不是等比数列}", "objs": [ "K0403006X", "K0403003X" @@ -84028,13 +84078,14 @@ "第四单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], "origin": "2022届高三第一轮复习讲义", "edit": [ - "20220701\t王伟叶" + "20220701\t王伟叶", + "20221104\t徐慧" ], "same": [], "related": [], @@ -84321,7 +84372,7 @@ }, "003253": { "id": "003253", - "content": "设$\\{a_n\\}$是各项为正数的无穷数列, $A_i$是边长为$a_i$、$a_{i+1}$的矩形的面积($i=1,2,\\cdots$), 则$\\{a_n\\}$为等比数列的充要条件是\\bracket{20}.\n\\onech{$\\{a_n\\}$是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$或$a_2,a_4,\\cdots,a_{2n},\\cdots$是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots,a_{2n},\\cdots$均是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots,a_{2n},\\cdots$均是等比数列, 且公比相同}", + "content": "设$\\{a_n\\}$是各项为正数的无穷数列, $A_i$是边长为$a_i$、$a_{i+1}$的矩形的面积($i=1,2,\\cdots$), 则$\\{A_n\\}$为等比数列的充要条件是\\bracket{20}.\n\\onech{$\\{a_n\\}$是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$或$a_2,a_4,\\cdots,a_{2n},\\cdots$是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots,a_{2n},\\cdots$均是等比数列}{$a_1,a_3,\\cdots,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots,a_{2n},\\cdots$均是等比数列, 且公比相同}", "objs": [ "K0403003X" ], @@ -84329,13 +84380,14 @@ "第四单元" ], "genre": "选择题", - "ans": "", + "ans": "D", "solution": "", "duration": -1, "usages": [], "origin": "2022届高三第一轮复习讲义", "edit": [ - "20220701\t王伟叶" + "20220701\t王伟叶", + "20221104\t徐慧" ], "same": [ "011618" @@ -93235,7 +93287,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "选择题", "ans": "", @@ -93807,7 +93860,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -93854,7 +93908,7 @@ }, "003649": { "id": "003649", - "content": "如图, $A-B-C$为海岸线, $AB$为线段, $\\overset{\\LARGE{\\frown}}{BC}$为四分之一圆弧. $BD=39.2$km, $\\angle BDC=22^\\circ$, $\\angle CBD=68^\\circ$, $\\angle BDA=58^\\circ$.\\\\\n(1) 求$\\overset{\\LARGE{\\frown}}{BC}$的长度;\\\\\n(2) 若$AB=40$km, 求$D$到海岸线$A-B-C$的最短距离(精确到$0.001$km).\n\\begin{center}\n \\begin{tikzpicture}\n \\path (0,0) coordinate (D) node [below right] {$D$};\n \\path (112:3.92) coordinate (B) node [above left] {$B$};\n \\path (0,3.63456) coordinate (C) node [above right] {$C$};\n \\path (-4.2365,0.747) coordinate (A) node [left] {$A$};\n \\draw (D) -- (C) -- (B) (A) -- (D) -- (B);\n \\draw [very thick] (C) arc (45:135:1.0383565) (A) -- (B);\n \\end{tikzpicture}\n\\end{center}", + "content": "如图, $A-B-C$为海岸线, $AB$为线段, $\\overset{\\frown}{BC}$为四分之一圆弧. $BD=39.2$km, $\\angle BDC=22^\\circ$, $\\angle CBD=68^\\circ$, $\\angle BDA=58^\\circ$.\\\\\n(1) 求$\\overset{frown}{BC}$的长度;\\\\\n(2) 若$AB=40$km, 求$D$到海岸线$A-B-C$的最短距离(精确到$0.001$km).\n\\begin{center}\n \\begin{tikzpicture}\n \\path (0,0) coordinate (D) node [below right] {$D$};\n \\path (112:3.92) coordinate (B) node [above left] {$B$};\n \\path (0,3.63456) coordinate (C) node [above right] {$C$};\n \\path (-4.2365,0.747) coordinate (A) node [left] {$A$};\n \\draw (D) -- (C) -- (B) (A) -- (D) -- (B);\n \\draw [very thick] (C) arc (45:135:1.0383565) (A) -- (B);\n \\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ "第三单元" @@ -94595,7 +94649,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "填空题", "ans": "", @@ -104249,7 +104304,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -108055,7 +108111,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -110762,7 +110819,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -118772,7 +118830,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -119849,7 +119908,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -120894,7 +120954,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -167332,7 +167393,7 @@ "第四单元" ], "genre": "选择题", - "ans": "", + "ans": "A", "solution": "", "duration": -1, "usages": [], @@ -167831,7 +167892,7 @@ "第四单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], @@ -170765,7 +170826,7 @@ }, "006892": { "id": "006892", - "content": "连接三角形三边中点得第二个三角形, 再连接第二个三角形三边中点得第三个二角形, 如此不断地作下去, 则所得的一切三角形(不包括第一个三角形)的而积之和与第一个三角形面积之比为\\bracket{20}.\n\\fourch{$1$}{$\\dfrac 12$}{$\\dfrac 13$}{$\\dfrac 14$}", + "content": "连接三角形三边中点得第二个三角形, 再连接第二个三角形三边中点得第三个二角形, 如此不断地作下去, 则所得的一切三角形(不包括第一个三角形)的面积之和与第一个三角形面积之比为\\bracket{20}.\n\\fourch{$1$}{$\\dfrac 12$}{$\\dfrac 13$}{$\\dfrac 14$}", "objs": [ "K0405005X" ], @@ -170773,13 +170834,14 @@ "第四单元" ], "genre": "选择题", - "ans": "", + "ans": "C", "solution": "", "duration": -1, "usages": [], "origin": "代数精编第七章数列", "edit": [ - "20220720\t王伟叶" + "20220720\t王伟叶", + "20221104\t徐慧" ], "same": [], "related": [], @@ -205949,7 +206011,9 @@ "20220726\t王伟叶" ], "same": [], - "related": [], + "related": [ + "030473" + ], "remark": "", "space": "12ex" }, @@ -237342,7 +237406,8 @@ "content": "空间中有异面向量的概念吗? 为什么?", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237363,7 +237428,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237384,7 +237450,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237405,7 +237472,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237426,7 +237494,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237447,7 +237516,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237468,7 +237538,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237489,7 +237560,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237510,7 +237582,8 @@ "content": "求向量$\\overrightarrow a=(0,1,0)$与$\\overrightarrow b=(1,-1,0)$的夹角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237531,7 +237604,8 @@ "content": "已知向量$\\overrightarrow a=(-m,1,3)$平行于向量$\\overrightarrow b=(2,n,1)$, 求$m$、$n$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237552,7 +237626,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237594,7 +237669,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237617,7 +237693,8 @@ "K0619004B" ], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "$\\dfrac 67$", @@ -237665,7 +237742,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237686,7 +237764,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237707,7 +237786,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -237728,7 +237808,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256746,7 +256827,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256767,7 +256849,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256788,7 +256871,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256809,7 +256893,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256830,7 +256915,8 @@ "content": "已知空间四边形$ABCD$中, $AB\\perp CD, AC\\perp BD$. 求证: $AD\\perp BC$.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256851,7 +256937,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256872,7 +256959,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256893,7 +256981,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256914,7 +257003,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256935,7 +257025,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256956,7 +257047,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256977,7 +257069,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -256998,7 +257091,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257016,10 +257110,11 @@ }, "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", + "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257040,7 +257135,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257061,7 +257157,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257082,7 +257179,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257103,7 +257201,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257124,7 +257223,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257145,7 +257245,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257166,7 +257267,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257187,7 +257289,8 @@ "content": "在$\\triangle ABC$中, 已知$\\overrightarrow{AB}=(2, 4, 0)$, $\\overrightarrow{BC}=(-1, 3, 0)$. 求$\\angle ABC$的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257229,7 +257332,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257250,7 +257354,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257271,7 +257376,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257292,7 +257398,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257313,7 +257420,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257355,7 +257463,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257376,7 +257485,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257397,7 +257507,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257418,7 +257529,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 设$AB$、$DD_1$的中点分别为$M$、$N$. 求直线$B_1M$与$CN$所成角的大小.", "objs": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257439,7 +257551,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -257460,7 +257573,8 @@ "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": [], "tags": [ - "第六单元" + "第六单元", + "空间向量" ], "genre": "解答题", "ans": "", @@ -261576,7 +261690,8 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{2,4,6\\}$, 则$A\\cup B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\{1,2,3,4,6\\}$", @@ -261614,7 +261729,8 @@ "content": "不等式$\\dfrac{x-2}{x+1}<0$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$(-1,2)$", @@ -261654,7 +261770,8 @@ "content": "函数$y=\\lg (x-1)+\\dfrac 1{\\sqrt {2-x}}$的定义域是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$(1,2)$", @@ -261690,7 +261807,8 @@ "content": "函数$y=\\sin( \\omega x-\\dfrac{\\pi}{3})$($\\omega >0$)的最小正周期是$\\pi$, 则$\\omega =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$2$", @@ -261727,7 +261845,8 @@ "content": "若函数$f(x)=\\log_2(x+1)+a$的反函数的图像经过点$(4, 1)$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -261750,7 +261869,8 @@ "content": "已知幂函数$f(x)={x^\\alpha}$的图像过点$(2,\\dfrac{\\sqrt 2}2)$, 则$f(x)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$(0,+\\infty)$", @@ -261783,7 +261903,8 @@ "content": "甲、乙两人从$5$门不同的选修课中各选修$2$门, 则甲、乙所选的课程中恰有$1$门相同的选法有\\blank{50}种.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$60$", @@ -261818,7 +261939,8 @@ "content": "设集合$M=\\{x|x^2\\le 1\\}$, $N=\\{b\\}$, 若$M\\cup N=M$, 则实数$b$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$[-1,1]$", @@ -261851,7 +261973,8 @@ "content": "将函数$f(x)=\\begin{vmatrix}\n\\sqrt 3 & \\cos 2x \\\\ 1 & \\sin 2x \\end{vmatrix}$的图像向左平移$m$($m>0$)个单位, 所得图像对应的函数为偶函数, 则$m$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -261872,7 +261995,8 @@ "content": "如图, 在$\\triangle ABC$中, $\\angle B=45^\\circ$, $D$是$BC$边上的一点, $AD=5$, $AC=7$, $DC=3$, 则$AB$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.25]\n\\draw (0,0) node [below] {$D$} coordinate (D);\n\\draw (3,0) node [below] {$C$} coordinate (C);\n\\draw (-2.5,{5*sqrt(3)/2}) node [above] {$A$} coordinate (A);\n\\draw ({-2.5-5*sqrt(3)/2},0) node [below] {$B$} coordinate (B);\n\\draw (B) -- (C) -- (A) -- cycle (A) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\dfrac{5\\sqrt{6}}2$", @@ -261905,7 +262029,8 @@ "content": "若函数$f(x)$满足: \\textcircled{1} 在定义域$D$内是严格单调函数; \\textcircled{2} 存在$[a, b]\\subseteq D$($a=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C);\n\\draw (A) --++ (0,5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) --++ (0,5) node [below right] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,5) node [right] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (A) -- ($(B)!{1/6}!(B1)$) -- ($(C)!{1/3}!(C1)$) ($(A)!0.5!(A1)$) -- ($(B)!{2/3}!(B1)$) -- ($(C)!{5/6}!(C1)$);\n\\draw [dashed] ($(C)!{1/3}!(C1)$) -- ($(A)!0.5!(A1)$) ($(C)!{5/6}!(C1)$) -- (A1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$13$", @@ -261971,7 +262097,8 @@ "content": "``$x<2$''是``$x^2<4$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "B", @@ -262004,7 +262131,8 @@ "content": "对任意向量$\\overrightarrow a$、$\\overrightarrow b$, 下列关系式中不恒成立的是\\bracket{20}.\n\\twoch{$(\\overrightarrow a+\\overrightarrow b)^2=|\\overrightarrow a+\\overrightarrow b|^2$}{$(\\overrightarrow a+\\overrightarrow b)\\cdot (\\overrightarrow a-\\overrightarrow b)=\\overrightarrow a^2-\\overrightarrow b^2$}{$|\\overrightarrow a\\cdot \\overrightarrow b|\\le|\\overrightarrow a|\\cdot|\\overrightarrow b|$}{$|\\overrightarrow a-\\overrightarrow b|\\le||\\overrightarrow a|-|\\overrightarrow b||$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "D", @@ -262037,7 +262165,8 @@ "content": "设$m$、$n$为两条直线, $\\alpha$、$\\beta$为两个平面, 则下列命题中假命题是\\bracket{20}.\n\\twoch{若$m\\perp n$, $m\\perp \\alpha$, $n\\perp \\beta$, 则$\\alpha \\perp \\beta$}{若$m\\parallel n$, $m\\perp \\alpha$, $n\\parallel \\beta$, 则$\\alpha \\perp \\beta$}{若$m\\perp n$, $m\\parallel \\alpha$, $n\\parallel \\beta$, 则$\\alpha \\parallel \\beta$}{若$m\\parallel n$, $m\\perp \\alpha$, $n\\perp \\beta$, 则$\\alpha \\parallel \\beta$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "C", @@ -262070,7 +262199,8 @@ "content": "已知函数$y=f(x)$的定义域为$D$, $x_1,x_2\\in D$. 关于$y=f(x)$ 的两个命题:\\\\\n命题\\textcircled{1}: 若当$f(x_1)+f(x_2)=0$时, 都有$x_1+x_2=0$, 则函数$y=f(x)$是$D$上的奇函数.\\\\\n命题\\textcircled{2}: 若当$f(x_1)=latex,scale = 0.8]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$C$} coordinate (C);\n\\draw ({2*sqrt(2)},0,2) node [right] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (C) -- (B) -- (D) (A) -- (B) (C) -- (M);\n\\draw (A) -- (C) -- (D) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $\\dfrac{4\\sqrt{2}}3$; (2) $\\arccos\\dfrac{\\sqrt{3}}6$.", @@ -262136,7 +262267,8 @@ "content": "已知$x\\in \\mathbf{R}$, 设$\\overrightarrow m=(2\\cos x, \\sin x+\\cos x)$, $\\overrightarrow n=(\\sqrt 3\\sin x, \\sin x-\\cos x)$, 记函数$f(x)=\\overrightarrow m\\cdot \\overrightarrow n$.\\\\\n(1) 求函数$f(x)$取最小值时$x$的取值范围;\\\\\n(2) 设$\\triangle ABC$的角$A$, $B$, $C$所对的边分别为$a$, $b$, $c$, 若$f(C)=2$, $c=\\sqrt 3$, 求$\\triangle ABC$的面积$S$的最大值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $\\{x|x=k\\pi-\\dfrac\\pi 6, \\ k\\in \\mathbf{Z}\\}$; (2) $\\dfrac{3\\sqrt{3}}4$.", @@ -262169,7 +262301,8 @@ "content": "如图, 某城市有一矩形街心广场$ABCD$, 如图. 其中$AB=4$百米, $BC=3$百米.现将在其内部挖掘一个三角形水池$DMN$种植荷花, 其中点$M$在边$BC$上, 点$N$在边$AB$上, 要求$\\angle MDN=\\dfrac{\\pi}4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (4,0) node [below] {$B$} coordinate (B) -- (4,3) node [above] {$C$} coordinate (C) -- (0,3) node [above] {$D$} coordinate (D) -- cycle;\n\\draw (2,0) node [below] {$N$} coordinate (N) -- (4,1) node [right] {$M$} coordinate (M) -- (D) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AN=CM=2$百米, 判断$\\triangle DMN$是否符合要求, 并说明理由;\\\\\n(2) 设$\\angle CDM=\\theta$, 写出$\\triangle DMN$面积的$S$关于$\\theta$的表达式, 并求$S$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) 不符合要求, 证明略; (2) $S=\\dfrac{12}{1+\\sin(2\\theta+\\dfrac \\pi 4)}, \\ \\theta\\in [0,\\arctan\\dfrac 34]$, $S$的最小值为$12\\sqrt{2}-12$.", @@ -262202,7 +262335,8 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $a_2=a$.\\\\\n(1) 若数列$\\{a_n\\}$是等差数列, 且$a_8=15$, 求实数$a$的值;\\\\\n(2) 若数列$\\{a_n\\}$满足$a_{n+2}-a_n=2$($n\\in \\mathbf{N}$且$n\\ge 1$), 且$S_{19}=19a_{10}$, 求证: 数列$\\{a_n\\}$是等差数列;", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $3$; (2) 证明略.", @@ -262235,7 +262369,8 @@ "content": "已知函数$f(x)=2^x+k\\cdot 2^{-x}$($x\\in \\mathbf{R}$).\\\\\n(1) 判断函数$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 设$k>0$, 问函数$f(x)$的图像是否关于某直线$x=m$成轴对称图形, 如果是, 求出$m$的值; 如果不是, 请说明理由; (可利用真命题: ``函数$g(x)$的图像关于某直线$x=m$成轴对称图形''的充要条件为``函数$g(m+x)$是偶函数'')\\\\\n(3) 设$k=-1$, 函数$h(x)=a\\cdot 2^x-2^{1-x}-\\dfrac 43a$, 若函数$f(x)$与$h(x)$的图像有且只有一个公共点, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) 当$k=1$时, $y=f(x)$是偶函数; 当$k=-1$时, $y=f(x)$是奇函数; 当$k\\ne 1$且$k \\ne -1$时, $y=f(x)$既不是奇函数, 又不是偶函数; (2) $m=\\log_4 k$; (3) $\\{-3\\}\\cup (1,+\\infty)$.", @@ -262268,7 +262403,8 @@ "content": "设集合$A=\\{x\\in\\mathbf{R}|0\\le x\\le 1\\}$, $B=\\{x\\in \\mathbf{R}|(x-1)(x-2)\\le 0\\}$, 则$A\\cup B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$[0,2]$", @@ -262302,7 +262438,8 @@ "content": "函数$y=x^2$($x\\ge 0$)的反函数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -262330,7 +262467,8 @@ "content": "若$0<\\alpha<\\pi$, $\\cos\\alpha=-\\dfrac 13$, 则$\\tan\\alpha=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$-2\\sqrt{2}$", @@ -262364,7 +262502,8 @@ "content": "复数$\\dfrac{2+4\\mathrm{i}}{1+\\mathrm{i}}$的虚部为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$1$", @@ -262398,7 +262537,8 @@ "content": "若正方体的棱长为$1$, 则其外接球的体积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\dfrac{\\sqrt{3}}2\\pi$", @@ -262432,7 +262572,8 @@ "content": "已知函数$f(x)=\\sin(3x+\\varphi)$($-\\dfrac\\pi 2<\\varphi<\\dfrac\\pi 2$)的图像关于直线$x=\\dfrac \\pi 4$对称, 则$\\varphi=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$-\\dfrac\\pi 4$", @@ -262466,7 +262607,8 @@ "content": "一个袋中装有同样大小、质量的$10$个球, 其中$2$个红球、$3$个蓝球、$5$个黑球. 经过充分混合后, 若从此袋中任意取出$4$个球, 则三种颜色的球均取到的概率为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\dfrac 12$", @@ -262500,7 +262642,8 @@ "content": "若抛物线$y^2=8x$的准线与曲线$\\dfrac{x^2}a+\\dfrac{y^2}4=1$($a>0$)只有一个公共点, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\{4\\}$", @@ -262534,7 +262677,8 @@ "content": "设函数$f(x)=\\dfrac 1x-\\lg x$, 则不等式$f(\\dfrac 1x-1)<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$(0,\\dfrac 12)$", @@ -262568,7 +262712,8 @@ "content": "若$\\ln x$与$\\ln y$的算术平均值为$1$, 则$\\mathrm{e}^x$与$\\mathrm{e}^y$的几何平均值的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\mathrm{e}^{\\mathrm{e}}$", @@ -262602,7 +262747,8 @@ "content": "正方形$ABCD$的边长为$4$, $O$是正方形$ABCD$的中心, 过中心$O$的直线$l$与边$AB$交于点$M$, 与边$CD$交于点$N$. $P$为平面上一点, 满足$2\n\\overrightarrow{OP}=\\lambda \\overrightarrow{OB}+(1-\\lambda)\\overrightarrow{OC}$, 则$\\overrightarrow{PM}\\cdot \\overrightarrow{PN}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$-7$", @@ -262638,7 +262784,8 @@ "content": "已知常数$b,c\\in \\mathbf{R}$, 若函数$f(x)=x+\\dfrac bx+c$在区间$[1,+\\infty)$上存在零点, 则$b^2+c^2$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$[\\dfrac 12,+\\infty)$", @@ -262672,7 +262819,8 @@ "content": "曲线$y^2=8x$的准线方程是\\bracket{20}.\n\\fourch{$x=4$}{$x=2$}{$x=-2$}{$x=-4$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "C", @@ -262706,7 +262854,8 @@ "content": "设$x,y$均为实数, 且$\\begin{vmatrix}x & 3 \\\\ 6 & 2 \\end{vmatrix} - \\begin{vmatrix} 1 & 4 \\\\ 5 & 7\\end{vmatrix} = 7$, 则在以下各项中$(x,y)$的可能取值只能是\\bracket{20}.\n\\fourch{$(2,1)$}{$(2,-1)$}{$(-1, 2)$}{$(-1, -2)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -262727,7 +262876,8 @@ "content": "已知垂直竖在水平地面上相距$20\\text{m}$的两根旗杆的高分别为$10\\text{m}$、$15\\text{m}$, 地面上的动点$P$到两旗杆顶点的仰角相等, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆}{圆}{双曲线}{抛物线}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "B", @@ -262761,7 +262911,8 @@ "content": "已知常数$b,c\\in \\mathbf{R}$, 关于$x$的方程$x^2+b|x|+c=0$在复数集$\\mathbf{C}$上给出下列两个结论: \\textcircled{1} 存在$b,c$, 使得该方程有且只有两个根, 且这两个根互为共轭虚根; \\textcircled{2} 存在$b,c$, 使得该方程有且只有$6$个互不相等的根, 则\\bracket{20}.\n\\fourch{\\textcircled{1}与\\textcircled{2}均正确}{\\textcircled{1}正确, \\textcircled{2}不正确}{\\textcircled{1}不正确, \\textcircled{2}正确}{\\textcircled{1}与\\textcircled{2}均不正确}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "A", @@ -262795,7 +262946,8 @@ "content": "设常数$a\\in \\mathbf{R}$, 函数$f(x)=a\\sin 2x+\\cos(2\\pi-2x)+1$.\\\\\n(1) 若$a=\\sqrt{3}$, 求$f(x)$的单调递增区间;\\\\\n(2) 若$f(x)$为偶函数, 求$f(x)$的值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $[k\\pi-\\dfrac \\pi 3,k\\pi+\\dfrac \\pi 6], \\ k\\in \\mathbf{Z}$; (2) $[0,2]$.", @@ -262829,7 +262981,8 @@ "content": "设$\\triangle ABC$的内角$A,B,C$所对的边分别为$a,b,c$, 且满足$\\sin A=\\sqrt{3}\\sin B$, $C=\\dfrac\\pi 6$.\\\\\n(1) 若$ac=\\sqrt{3}$, 求$\\triangle ABC$的面积;\\\\\n(2) 能否将$\\triangle ABC$的边长按某种顺序排列为一个等比数列? 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $\\dfrac{\\sqrt{3}}4$; (2) 不能, 证明略.", @@ -262863,7 +263016,8 @@ "content": "某商场共有三层楼, 在其圆柱形空间内安装两部等长的扶梯I和II供顾客乘用. 如图, 一顾客自一楼点$A$处乘I到达二楼的点$B$处后, 沿着二楼面上的圆弧$BM$逆时针步行至点$C$处, 且$C$为圆弧$BM$的中点, 再乘II到达三楼的点$D$处. 设圆柱形空间三个楼面圆的中心分别为$O,O_1,O_2$, 半径为$8\\text{m}$, 相邻楼层的间距$AM=4\\text{m}$, 两部扶梯与楼面所成角的大小均为$\\arcsin\\dfrac 13$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\filldraw (0,8) circle (0.1) node [left] {$O_2$};\n\\filldraw (0,4) circle (0.1) node [left] {$O_1$};\n\\filldraw (0,0) circle (0.1) node [left] {$O$};\n\\draw (0,8) ellipse (8 and 2);\n\\draw (8,4) arc (0:-180:8 and 2) (8,0) arc (0:-180:8 and 2);\n\\draw [dashed] (8,4) arc (0:180:8 and 2) (8,0) arc (0:180:8 and 2);\n\\draw (8,0) node [right] {$A$} coordinate (A) -- (8,4) node [right] {$M$} coordinate (M) -- (8,8);\n\\draw (-8,0) -- (-8,8);\n\\draw ({8*cos(-120)},{4+2*sin(-120)}) node [below] {$B$} coordinate (B);\n\\draw ({8*cos(-55)},{4+2*sin(-55)}) node [below] {$C$} coordinate (C);\n\\draw ({8*cos(60)},{8+2*sin(60)}) node [above] {$D$} coordinate (D);\n\\draw [dashed] (A) -- (B) node [midway, below] {I};\n\\draw [dashed] (C) -- (D) node [midway, below left] {II};\n\\end{tikzpicture}\n\\end{center}\n(1) 求此顾客在二楼面上步行的路程;\\\\\n(2) 求异面直线$AB$与$CD$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $2\\pi$米; (2) $\\arccos\\dfrac{4\\sqrt{2}-1}9$.", @@ -262897,7 +263051,8 @@ "content": "已知曲线$\\Gamma:x^2-y|y|=1$与$x$轴分别相交于$A,B$两点($A$在$B$的左侧), $\\Gamma$与$y$轴相交于点$C$. 已知$F_1(-c,0)$, $F_2(c,0)$, $c>0$, $\\triangle BCF_1$的面积为$\\dfrac{1+\\sqrt{2}}{2}$.\\\\\n(1) 若过$F_2$的直线$l$与$\\Gamma$有且仅有一个公共点, 直接写出$l$倾斜角的取值范围;\\\\\n(2) 过点$B$作斜率存在的直线$m$交$\\Gamma$于$P,Q$两点(异于点$B$), 且点$P$在第一象限, 求证: $P,Q$的横坐标之积为定值, 并求该定值;\\\\\n(3) 在(2)的条件下, 当$\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_1Q}=3+2\\sqrt{2}$时, 求$\\dfrac{|AP|}{|AQ|}$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $[\\dfrac\\pi 4,\\dfrac{3\\pi}4]$; (2) 定值为$1$; (3) $3+2\\sqrt{2}$.", @@ -262931,7 +263086,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_n\\ne 0$恒成立.\\\\\n(1) 若$a_na_{n+2}=ka_{n+1}^2$且$a_n>0$, 当$\\{\\lg a_n\\}$成等差数列时, 求$k$的值;\\\\\n(2) 若$a_na_{n+2}=2a_{n+1}^2$且$a_n>0$, 当$a_1=1$, $a_4=16\\sqrt{2}$时, 求$\\{a_n\\}$的通项公式;\\\\\n(3) 若$a_na_{n+2}=-\\dfrac 12a_{n+1}a_{n+3}$, $a_1=-1$, $a_3\\in [4, 8]$, $a_{2020}<0$, 求$a_1+a_2+\\cdots+a_{2020}$的最大值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $1$; (2) $a_n=2^{\\frac 12(n-1)^2}$; (3) $\\dfrac{1-4^{505}}{3}$.", @@ -262965,7 +263121,8 @@ "content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x||x-1|>1\\}$, $B=\\{x|\\dfrac{x-3}{x+1}<0\\}$, 则$\\overline{A}\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -262988,7 +263145,8 @@ "content": "已知幂函数的图像过点$(2,\\dfrac 14)$, 则该幂函数的单调递增区间是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263011,7 +263169,8 @@ "content": "若$S_n$是等差数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$): $-1,2,5,8,\\cdots$的前$n$项和, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{S_n}{n^2+1}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263036,7 +263195,8 @@ "content": "某圆锥体的底面圆的半径长为$\\sqrt 2$, 其侧面展开图是圆心角为$\\dfrac 23\\pi$的扇形, 则该圆锥体的体积是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263059,7 +263219,8 @@ "content": "过点$P(-2,1)$作圆$x^2+y^2=5$的切线, 则该切线的点法向式方程是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263082,7 +263243,8 @@ "content": "函数$f(x)=\\sqrt 3\\sin x\\cos x+\\cos ^2x$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263107,7 +263269,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263130,7 +263293,8 @@ "content": "某高级中学欲从本校的$7$位古诗词爱好者(其中男生$2$人、女生$5$人)中随机选取3名同学作为学校诗词朗读比赛的主持人, 若要求主持人中至少有一位是男同学, 则不同选取方法的种数是\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263153,7 +263317,8 @@ "content": "已知数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$), 若$a_1=1$, $a_{n+1}+a_n=(\\dfrac 12)^n$, 则$\\displaystyle \\lim_{n\\to\\infty} a_{2n}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263176,7 +263341,8 @@ "content": "已知函数$f(x)=\\begin{cases} \\log_2(x+a), & -a0 \\end{cases}$有三个不同的零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263199,7 +263365,8 @@ "content": "在边长为$1$的正六边形$ABCDEF$中, 记以$A$为起点, 其余顶点为终点的向量分别为$\\overrightarrow{a_1}$, $\\overrightarrow{a_2}$, $\\overrightarrow{a_3}$, $\\overrightarrow{a_4}$, $\\overrightarrow{a_5}$, 若$\\overrightarrow{a_i}$与$\\overrightarrow{a_j}$的夹角记为$\\theta _{ij}$, 其中$i,j\\in \\{1,2,3,4,5\\}$, 且$i\\ne j$, 则$|\\overrightarrow{a_i}|\\cdot \\cos \\theta _{ij}$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263220,7 +263387,8 @@ "content": "设$l_1$、$l_2$是平面上过点$M$, 夹角为$\\dfrac{\\pi }3$的两条直线, 且与圆心为$O$, 半径长为$1$的圆均相切(圆心在两直线所夹的锐角中), 设圆周上一点$P$到$l_1$、$l_2$的距离分别为$d_1$、$d_2$, 那么$2d_1+d_2$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263241,7 +263409,8 @@ "content": "设函数$y=f(x)$, ``该函数的图像过点$(1,1)$''是``该函数为幂函数''的\t\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263262,7 +263431,8 @@ "content": "下列关于函数$y=\\sin x$与$y=\\arcsin x$的命题中, 正确的是\\bracket{20}.\n\\fourch{它们互为反函数}{都是增函数}{都是周期函数}{都是奇函数}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263283,7 +263453,8 @@ "content": "如图, 平面直角坐标系中, 曲线(实线部分)的方程可以是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -{sqrt(3)}:{sqrt(3)},samples = 100] plot ({sqrt(\\x*\\x+1)},\\x);\n\\draw [domain = -{sqrt(3)}:{sqrt(3)},samples = 100] plot ({-sqrt(\\x*\\x+1)},\\x);\n\\draw (-1,0) node [below left] {$-1$} -- (0,-1) node [below left] {$-1$} -- (1,0) node [below right] {$1$};\n\\draw [dashed] (-2,1) -- (-1,0) (1,0) -- (2,1);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$(|x|-y-1)\\cdot (1-x^2+y^2)=0$}{$\\sqrt {|x|-y-1}\\cdot (1-x^2+y^2)=0$}{$(|x|-y-1)\\cdot \\sqrt {1-x^2+y^2}=0$}{$\\sqrt {|x|-y-1}\\cdot \\sqrt {1-x^2+y^2}=0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263304,7 +263475,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$的八个顶点中任取两个点作直线, 与直线$A_1B$异面且夹角成$60^{\\circ}$的直线的条数为\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263325,7 +263497,8 @@ "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 点$E$、$F$分别是所在棱$A_1B_1$、$AB$的中点, 点$O_1$是面$A_1B_1C_1D_1$的中心, 如图所示.\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 (A1) -- ($(A)!0.5!(B)$) node [below] {$F$};\n\\draw [dashed] (C) -- ($(A1)!0.5!(B1)$) node [above] {$E$};\n\\filldraw ($(A1)!0.5!(C1)$) circle (0.03) node [right] {$O_1$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$O_1-FBC$的体积$V_{O_1-FBC}$;\\\\\n(2) 求异面直线$A_1F$与$CE$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263346,7 +263519,8 @@ "content": "已知函数$f(x)=\\dfrac a{2^x-1}+b$, 其中$a$、$b\\in \\mathbf{R}$.\\\\\n(1) 当$a=6$, $b=0$时, 求满足$f(|x|)=2^x$的$x$的值;\\\\\n(2) 若$f(x)$为奇函数且非偶函数, 求$a$与$b$的关系式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263367,7 +263541,8 @@ "content": "如图, 某大型厂区有三个值班室$A$、$B$、$C$, 值班室$A$在值班室$B$的正北方向$2$千米处, 值班室$C$在值班室$B$的正东方向$2\\sqrt 3$千米处.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0) node [below] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw ($(C)!0.25!(A)$) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) -- cycle (B) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 保安甲沿$CA$从值班室$C$出发行至点$P$处, 此时$PC=1$, 求$PB$的距离;\\\\\n(2) 保安甲沿$CA$从值班室$C$出发前往值班室$A$, 保安乙沿$AB$从值班室$A$出发前往值班室$B$, 甲乙同时出发, 甲的速度为$1$千米/小时, 乙的速度为$2$千米/小时, 若甲乙两人通过对讲机联系, 对讲机在厂区的最大通话距离为$3$千米(含$3$千米), 试问有多长时间两人不能通话?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263388,7 +263563,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}9+\\dfrac{y^2}4=1$.\\\\\n(1) 若抛物线$C$的焦点与$\\Gamma$的焦点重合, 求$C$的标准方程;\\\\\n(2) 若$\\Gamma$的上顶点$A$、右焦点$F$及$x$轴上一点$M$构成直角三角形, 求点$M$的坐标;\\\\\n(3) 若$O$为$\\Gamma$的中心, $P$为$\\Gamma$上一点(非$\\Gamma$的顶点), 过$\\Gamma$的左顶点$B$, 作$BQ\\parallel OP$, $BQ$交$y$轴于点$Q$, 交$\\Gamma$于点$N$, 求证: $\\overrightarrow{BN}\\cdot \\overrightarrow{BQ}=2\\overrightarrow{OP}^2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263409,7 +263585,8 @@ "content": "给定整数$n$($n\\ge 4$), 设集合$A=\\{a_1,a_2,\\cdots ,a_n\\}$, $B=\\{a_i+a_j|a_i,a_j\\in A,\\ 1\\le i\\le j\\le n\\}$.\\\\\n(1) 若$A=\\{-3,0,1,2\\}$, 求集合$B$;\\\\\n(2) 若$a_1,a_2,\\cdots ,a_n$构成以$a_1$为首项, $d$($d>0$)为公差的等差数列, 求证: 集合$B$中的元素个数为$2n-1$;\\\\\n(3) 若$a_1,a_2,\\cdots ,a_n$构成以$3$为首项, $3$为公比的等比数列, 求集合$B$中元素的个数及所有元素之和.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263430,7 +263607,8 @@ "content": "已知集合$A=\\mathbf{N}^*$, $B=\\{x||2x-1|<5\\}$, 则$A\\cap B=$\\blank{50}. (用列举法表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263451,7 +263629,8 @@ "content": "已知复数$z$满足$z\\mathrm{i}=2+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263475,7 +263654,8 @@ "content": "若函数$f(x)=2^x+1$的图像与$g(x)$的图像关于直线$y=x$对称, 则$g(9)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263498,7 +263678,8 @@ "content": "若$\\tan (\\alpha +\\dfrac{\\pi }4)=-3$, 则$\\tan \\alpha =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263521,7 +263702,8 @@ "content": "在$(1-2x)^6$的二项展开式中, $x^3$项的系数为\\blank{50}. (用数字作答)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263549,7 +263731,8 @@ "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 高为$3$, 则异面直线$AA_1$与$BD_1$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\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--++ (0,3) 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) ++ (2,3) 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) --++ (0,3);\n\\draw [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263572,7 +263755,8 @@ "content": "新冠病毒爆发初期, 全国支援武汉的活动中, 需要从$A$医院某科室的$6$名男医生(含一名主任医师)、$4$名女医生(含一名主任医师)中分别选派$3$名男医生和$2$名女医生, 要求至少有一名主任医师参加, 则不同的选派方案共有\\blank{50}种. (用数字作答)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263595,7 +263779,8 @@ "content": "设$k\\in \\{-2,-1,\\dfrac 13,\\dfrac 23,2\\}$, 若对任意$x\\in (-1,0)\\cup (0,1)$, 都成立$x^k>|x|$, 则$k$取值的集合是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263618,7 +263803,8 @@ "content": "某校开设$9$门选修课程, 其中$A$, $B$, $C$三门课程由于上课时间相同, 至多选一门, 若规定每位学生选修$4$门, 则一共有\\blank{50}种不同的选修方案.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263639,7 +263825,8 @@ "content": "如图所示, 在平面直角坐标系$xOy$中, 动点$P$以每秒$\\dfrac{\\pi }2$的角速度从点$A$出发, 沿半径为$2$的上半圆逆时针移动到$B$, 再以每秒$\\dfrac{\\pi }3$的角速度从点$B$沿半径为1的下半圆逆时针移动到坐标原点$O$, 则上述过程中动点$P$的纵坐标$y$关于时间$t$的函数表达式为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [right] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [very thick] (1.6,0) node [below right] {$A$} arc (0:180:1.6) node [below left] {$B$} arc (180:360:0.8);\n\\draw [dashed] (1.6,0) arc (0:-180:1.6) arc (-180:-360:0.8);\n\\draw (0,0) -- (45:1.6) node [above right] {$P$};\n\\draw (-0.8,0) --++ (-50:0.8) node [below] {$P$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263660,7 +263847,8 @@ "content": "设$a>0$, $a\\ne 1$, $M>0$, $N>0$, 我们可以证明对数的运算性质如下:\\\\\n因为$a^{\\log_aM+\\log_aN}=a^{\\log_aM}a^{\\log_aN}=MN$, \\textcircled{1}\\\\\n所以$\\log_a(MN)=\\log_aM+\\log_aN$.\\\\\n我们将\\textcircled{1}式称为证明的``关键步骤''. 则证明$\\log_a(M^r)=r\\log_aM$(其中$M>0$, $r\\in \\mathbf{R}$)的``关键步骤''为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263681,7 +263869,8 @@ "content": "已知函数$f(x)=|x+\\dfrac 1x|$, 给出下列命题:\\\\\n\\textcircled{1} 存在实数a, 使得函数$y=f(x)+f(x-a)$为奇函数;\\\\\n\\textcircled{2} 对任意实数a, 均存在实数m, 使得函数$y=f(x)+f(x-a)$关于$x=m$对称;\\\\\n\\textcircled{3} 若对任意非零实数a, $f(x)+f(x-a)\\ge k$都成立, 则实数$k$的取值范围为$(-\\infty ,4]$;\\\\\n\\textcircled{4} 存在实数k, 使得函数$y=f(x)+f(x-a)-k$对任意非零实数a均存在6个零点.\\\\\n其中的真命题是\\blank{50}. (写出所有真命题的序号)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263702,7 +263891,8 @@ "content": "若$a$为实数, 则``$a<1$''是``$\\dfrac 1a>1$''的 \\bracket{20}\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263723,7 +263913,8 @@ "content": "若$\\lg2=a$, $\\lg3=b$, 则$\\log_512$等于\\bracket{20}\n\\fourch{$\\dfrac{2a+b}{1+a}$}{$\\dfrac{a^2b}{1+a}$}{$\\dfrac{2a+b}{1-a}$}{$\\dfrac{a^2b}{1-a}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263744,7 +263935,8 @@ "content": "已知点$P$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)右支上一点, 点$F_1$, $F_2$分别为双曲线的左右焦点, 点$I$是$\\triangle PF_1F_2$的内心(三角形内切圆的圆心), 若恒有$S_{\\triangle IPF_1}-S_{\\triangle IPF_2}=\\dfrac{\\sqrt 3}2S_{\\triangle IF_1F_2}$, 则双曲线的渐近线方程是\\bracket{20}.\n\\fourch{$y=\\pm x$}{$y=\\pm \\dfrac{\\sqrt 2}2x$}{$y=\\pm \\sqrt 3x$}{$y=\\pm \\dfrac{\\sqrt 3}3x$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263765,7 +263957,8 @@ "content": "如图, 正四棱锥$P-ABCD$的底面边长和高均为$2$, $M$是侧棱$PC$的中点, 若过$AM$作该正四棱锥的截面, 分别交棱$PB$、$PD$于点$E$、$F$(可与端点重合), 则四棱锥$P-AEMF$的体积的取值范围是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C) (A) -- (B) -- (C);\n\\draw [dashed] (P) -- (D) (A) -- (D) -- (C);\n\\draw ($(P)!0.5!(C)$) node [right] {$M$} coordinate (M);\n\\def\\l{1.6}\n\\draw ($(P)!{1/\\l}!(B)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!{1/(3-\\l)}!(D)$) node [left] {$F$} coordinate (F);\n\\draw (A) -- (E) -- (M);\n\\draw [dashed] (A) -- (F) -- (M) (A) -- (M);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[\\dfrac 12,1]$}{$[\\dfrac 12,\\dfrac 43]$}{$[1,\\dfrac 43]$}{$[\\dfrac 89,1]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -263786,7 +263979,8 @@ "content": "如图, 在圆柱$OO_1$ 中, $AB$是圆柱的母线, $BC$是圆柱的底面$\\odot O$的直径, $D$是底面圆周上异于$B$、$C$的点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0) arc (0:-180:2 and 0.5);\n\\draw (2,2) arc (0:360:2 and 0.5);\n\\draw [dashed] (2,0) arc (0:180:2 and 0.5);\n\\draw (-2,2) node [left] {$A$} coordinate (A) -- (-2,0) node [left] {$B$} coordinate (B);\n\\draw (2,2) -- (2,0) node [right] {$C$} coordinate (C);\n\\draw ({2*cos(-110)},{0.5*sin(-110)}) node [below] {$D$} coordinate (D);\n\\draw [dashed] (B) -- (D) -- (C) (A) -- (C) (B) -- (C) (A) -- (D);\n\\draw (-2,2) -- (2,2);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\filldraw (0,2) circle (0.03) node [above] {$O_1$} coordinate (O1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CD\\perp$平面$ABD$;\\\\\n(2) 若$BD=2$, $CD=4$, $AC=6$, 求圆柱$OO_1$的侧面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263807,7 +264001,8 @@ "content": "已知函数$f(x)=2\\sqrt 2\\sin \\dfrac x2\\cos \\dfrac x2+2\\sqrt 2\\cos ^2\\dfrac x2-\\sqrt 2$.\\\\\n(1) 求函数$f(x)$在区间$[0,\\pi]$上的值域;\\\\\n(2) 若方程$f(\\omega x)=\\sqrt 3(\\omega >0)$在区间$[0,\\pi]$上至少有两个不同的解, 求$\\omega$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263828,7 +264023,8 @@ "content": "大数据时代对于数据分析能力的要求越来越高, 数据拟合是一种把现有数据通过数学方法来代入某种算式的表示方式. 比如$A_i(a_i,b_i)$($i=1,2,3,\\cdots,n$)是平面直角坐标系上的一系列点, 其中$n$是不小于$2$的正整数, 用函数$y=f(x)$来拟合该组数据, 尽可能使得函数图像与点列$(a_i,b_i)$比较接近. 其中一种衡量接近程度的指标是函数的拟合误差, 拟合误差越小越好, 定义函数$y=f(x)$的拟合误差为:\n$\\Delta (f(x))=\\dfrac 1n[(f(a_1)-b_1)^2+(f(a_2)-b_2)^2+\\cdots +(f(a_n)-b_n)^2]$.\\\\\n已知在平面直角坐标系上, 有$5$个点的坐标数据如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$x$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\\ \\hline\n$y$ & $2.2$ & $1$ & $2$ & $4.6$ & $7$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1)若用函数$f_1(x)=x^2-4x+5$来拟合上述表格中的数据, 求$\\Delta (f_1(x))$;\\\\\n(2)若用函数$f_2(x)=2^{|x-2|}+m$来拟合上述表格中的数据.\\\\\n\\textcircled{1} 求该函数的拟合误差$\\Delta (f_2(x))$的最小值, 并求出此时的函数解析式$y=f_2(x)$;\\\\\n\\textcircled{2} 指出用$f_1(x),f_2(x)$(指\\textcircled{1}中使$\\Delta(f_2(x))$最小的函数)中的哪一个函数来拟合上述表格中的数据更好?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263849,7 +264045,8 @@ "content": "对于函数$y=f(x)$, 若函数$F(x)=f(x+1)-f(x)$是增函数, 则称函数$y=f(x)$具有性质$A$.\\\\\n(1) 若$f(x)=x^2+2^x$, 求$F(x)$的解析式, 并判断$f(x)$是否具有性质$A$;\\\\\n(2) 判断命题``减函数不具有性质$A$''是否真命题, 并说明理由;\\\\\n(3) 若函数$f(x)=kx^2+x^3(x\\ge 0)$具有性质$A$, 求实数$k$的取值范围, 并讨论此时函数$g(x)=f(\\sin x)-\\sin x$在区间$[0,\\pi]$上零点的个数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263870,7 +264067,8 @@ "content": "现定义: 设$a$是非零实常数, 若对于任意的$x\\in D$, 都有$f(a-x)=f(a+x)$, 则称函数$y=f(x)$为``关于$a$的偶型函数''.\\\\\n(1) 请以三角函数为例, 写出一个``关于$2$的偶型函数''的解析式, 并给予证明;\\\\\n(2) 设定义域为$\\mathbf{R}$的``关于$a$的偶型函数''$y=f(x)$在区间$(-\\infty ,a)$上单调递增, 求证: $y=f(x)$在区间$(a,+\\infty)$上单调递减;\\\\\n(3) 设定义域为$\\mathbf{R}$的``关于$\\dfrac 12$的偶型函数''$y=f(x)$是奇函数. 若$n\\in \\mathbf{N}^*$, 请猜测$f(n)$的值, 并用数学归纳法证明你的结论.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -263891,7 +264089,8 @@ "content": "若集合$A=\\{x|1\\le x\\}$, $B=\\{-1,1,2,3\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263915,7 +264114,8 @@ "content": "已知复数$z$满足$z\\cdot (1-\\mathrm{i})=1+3\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263942,7 +264142,8 @@ "content": "若$\\sin \\alpha =\\dfrac 13$, 则$\\sin (\\dfrac \\pi 2-2\\alpha)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263965,7 +264166,8 @@ "content": "已知圆锥的母线$l=2$, 母线与旋转轴的夹角$\\alpha =30^{\\circ}$, 则圆锥的侧面积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -263986,7 +264188,8 @@ "content": "已知函数$f(x)$图像与函数$g(x)=2^x$的图像关于$y=x$对称, 则$f(4)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264007,7 +264210,8 @@ "content": "若关于$x$、$y$的方程组$\\begin{cases} 2x+3y=1 \\\\ ax-y=2 \\end{cases}$无解, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264031,7 +264235,8 @@ "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 且$\\begin{vmatrix} \\sqrt 3b+2c & 2a \\\\\\cos B & 1 \\end{vmatrix}=0$, 则角$A=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264054,7 +264259,8 @@ "content": "已知$A,B$分别是函数$f(x)=2\\sin \\omega x$($\\omega >0$)在$y$轴右侧图像上的第一个最高点和第一个最低点, 且$\\angle AOB=\\dfrac\\pi 2$, 则该函数的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264077,7 +264283,8 @@ "content": "疫情期间家长会, 我校要从$5$名男生, $3$名女生中选派$4$名志愿者担任家长入校测量体温、查看行程码、健康码、登记信息四项不同的工作, 若其中女生不能从事测量体温, 则不同的选派方案共有\\blank{50}种.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264098,7 +264305,8 @@ "content": "正方形$ABCD$的边长为$4$, $O$是正方形$ABCD$的中心, 过中心$O$的直线$l$与边$AB$交于点$M$, 与边$CD$交于点$N$, $P$为平面上一点, 满足: 存在$\\lambda\\in \\mathbf{R}$, 使得$2\\overrightarrow{OP}=\\lambda \\overrightarrow{OB}+(1-\\lambda)\\overrightarrow{OC}$, 则$\\overrightarrow{PM}\\cdot \\overrightarrow{PN}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264121,7 +264329,8 @@ "content": "若函数$f(x)=\\begin{cases}|\\lg (x-1)|, & x>1, \\\\ \\sin x, & x<0, \\end{cases}$ 则$y=f(x)$图像上关于原点$O$对称的点共有\\blank{50}对.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264142,7 +264351,8 @@ "content": "已知函数$y=f(x)$, 对任意$x\\in \\mathbf{R}$, 都有$f(x+2)\\cdot f(x)=k$($k$为常数), 且当$x\\in [0,2]$时, $f(x)=x^2+1$, 则$f(2022)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264163,7 +264373,8 @@ "content": "已知$l$是平面$\\alpha$的一条斜线, 直线$m\\subseteq \\alpha$, 则\\bracket{20}.\n\\twoch{存在唯一的一条直线$m$, 使得$l\\perp m$}{存在无限多条直线$m$, 使得$l\\perp m$}{存在唯一的一条直线$m$, 使得$l \\parallel m$}{存在无限多条直线$m$, 使得$l \\parallel m$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264184,7 +264395,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$ 中, 下列四个结论中错误的是\\bracket{20}.\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 (C) -- (B1);\n\\draw [dashed] (A) -- (C) -- (D1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$B_1C$与直线$AC$所成的角为$60^\\circ$}{直线$B_1C$与平面$AD_1C$所成的角为$60^\\circ$}{直线$B_1C$与直线$AD_1$所成的角为$90^\\circ$}{直线$B_1C$与直线$AB$所成的角为$90^\\circ$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264207,7 +264419,8 @@ "content": "若$ab$}{$a^2>b^2$}{$\\sqrt[3]a<\\sqrt[3]b$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264228,7 +264441,8 @@ "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 已知点$(n,a_n)$在直线$y=10-2x$上, 若有且只有四个正整数$n$满足$S_n\\ge k$, 则实数$k$的取值范围是\\bracket{20}.\n\\fourch{$(8,14]$}{$(14,18]$}{$(18,20]$}{$(18,\\dfrac{81}4]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264249,7 +264463,8 @@ "content": "在直三棱柱$ABC-A_1B_1C_1$中, $\\angle ABC=90^\\circ$, $AB=BC=1$, $BB_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [above right] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\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 [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C1);\n\\draw (A) -- (C) -- (C1) -- (A1) -- cycle (A1) -- (B1) -- (C1) (A1) -- (C);\n\\draw [dashed] (A) -- (B) -- (C) (B) -- (B1) (B) -- (A1) (B) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$B_1C_1$与$A_1C$所成角的大小;\\\\\n(2) 求点$C_1$与平面$A_1BC$的距离.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264270,7 +264485,8 @@ "content": "已知函数$f(x)=\\sqrt 3\\sin x\\cos x-\\sin ^2x+2$.\\\\\n(1) 求$f(x)$的最小正周期和值域;\\\\\n(2) 若对任意的$x\\in \\mathbf{R}$, $f^2(x)-k\\cdot f(x)+1\\le 0$恒成立, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264293,7 +264509,8 @@ "content": "如图, 上海天马山上的``护珠塔''因其倾斜度超过意大利的比萨斜塔而号称``世界第一斜塔'', 兴趣小组同学实施如下方案来测量塔的倾斜度和塔高, 如图, 记$O$点为塔基、$P$点为塔尖、点$P$在地面上的射影为点$H$, 在塔身$OP$射影所在直线上选点$A$, 使仰角$\\angle HAP=45^{^\\circ }$, 过$O$点与$OA$成$120^{^\\circ }$的地面上选$B$点, 使仰角$\\angle HBP=45^\\circ$(点$A$、$B$、$O$都在同一水平面上), 此时测得$\\angle OAB=27^\\circ$, $A$与$B$之间距离为$33.6$米, 试求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw (O) ++ (0.1,0) coordinate (O1);\n\\draw (O) ++ (-0.1,0) coordinate (O2);\n\\draw (0.3,0) node [above right] {$H$} coordinate (H);\n\\draw (2.8,0) node [right] {$A$} coordinate (A);\n\\draw (0.3,2.5) node [above] {$P$} coordinate (P);\n\\draw (-0.8,-1.3) node [below] {$B$} coordinate (B);\n\\filldraw [fill = gray!50, draw = black] (P) -- (O1) -- (O2) -- cycle; \n\\draw (B) -- (O) (B) -- (H) (B) -- (A) (O) -- (P) (H) -- (P) (A) -- (P) (B) -- (P);\n\\draw (A) -- ($(A)!1.1!(O)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 塔高(即线段$PH$的长, 精确到$0.1$米);\\\\\n(2) 塔的倾斜度(即$\\angle OPH$的大小, 精确到$0.1^\\circ$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264314,7 +264531,8 @@ "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AD=AA_1=1$, $AB\\text=2$, 点$E$在棱$AB$上移动.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{1/2}) node [right] {$C$} coordinate (C)\n--++ (0,1) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{1/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{1/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{1/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw ($(A)!{2-sqrt(3)}!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D1) -- (E) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $D_1E\\perp A_1D$;\\\\\n(2) 当$E$为$AB$的中点时, 求直线$A_1E$与面$ACD_1$所成角的正弦值;\\\\\n(3) 棱$AB$上是否存在点$E$, 使得二面角$D_1-EC-D$的大小为$\\dfrac{\\pi }4$, 若存在求出$AE$的长; 若不存在说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264335,7 +264553,8 @@ "content": "设$f(x)=\\dfrac{-2^x+a}{2^{x+1}+b}$, $a,b$为实常数.\\\\\n(1) 当$a=b=1$时, 证明: $f(x)$不是奇函数;\\\\\n(2) 若$f(x)$是奇函数, 求$a$与$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264356,7 +264575,8 @@ "content": "已知集合$A=\\{1,3,5,6,7\\}$, $B=\\{2,4,5,6,8\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264379,7 +264599,8 @@ "content": "不等式$|3x-2|<1$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264407,7 +264628,8 @@ "content": "已知$(2x^2-\\dfrac 1x)^n$($n\\in \\mathbf{N}^*)$的展开式中各项的二项式系数之和为$128$, 则其展开式中含$\\dfrac 1x$项的系数是\\blank{50}.(结果用数值表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264428,7 +264650,8 @@ "content": "已知函数$f(x)$是以$2$为周期的偶函数, 当$0\\le x\\le 1$时, $f(x)=\\lg (x+1)$, 令函数$g(x)=f(x)(x\\in [1,2])$, 则$g(x)$的反函数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264449,7 +264672,8 @@ "content": "若$x>0$, $y>0$, 且$4x+y=xy$, 则$x+y-m\\ge 0$恒成立的实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264470,7 +264694,8 @@ "content": "已知函数$f(x)=A\\sin (\\omega x+\\varphi)$($\\omega >0$, $0<\\varphi <\\dfrac{\\pi }2$)的部分图像如图所示, 则函数$f(x)$的解析式为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3,samples = 100] plot (\\x,{2*sin(2*\\x/pi*180+30)});\n\\draw (0,1) node [left] {$1$};\n\\draw ({5*pi/12},0) node [below left] {$\\dfrac{5\\pi}{12}$};\n\\draw ({11*pi/12},0) node [below right] {$\\dfrac{11\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264493,7 +264718,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264514,7 +264740,8 @@ "content": "设函数$f(x)=\\lg (1+|x|)-\\dfrac 1{1+x^2}$, 则使得$f(2x)1$)在区间$(-2,6]$上恰有$3$个不同的零点, 则实数$a$的取值范是\\blank{50}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264556,7 +264784,8 @@ "content": "已知函数$f(x)=x^2-a|x|+\\dfrac 1{x^2+1}+a$有且只有一个零点, 若方程$f(x)=k$无解, 则实数$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264577,7 +264806,8 @@ "content": "设函数$f(x)$的定义域是$(0,1)$, 满足: \\textcircled{1} 对任意的$x\\in (0,1)$, $f(x)>0$; \\textcircled{2} 对任意的$x_1$、$x_2\\in (0,1)$, 都有$\\dfrac{f(x_1)}{f(x_2)}+\\dfrac{f(1-x_1)}{f(1-x_2)}\\le 2$; \\textcircled{3} $f(\\dfrac 12)=2$. 则函数$g(x)=xf(x)+\\dfrac 1x$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264598,7 +264828,8 @@ "content": "用$M_I$表示函数$y=\\sin x$在闭区间$I$上的最大值, 若正数$a$满足$M_{[0,a]}\\ge 2M_{[a,2a]}$, 则$a$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264619,7 +264850,8 @@ "content": "魏晋时期数学家刘徽在他的著作《九章算术注》中, 称一个正方体内两个互相垂直的内切圆柱所围成的几何体为``牟合方盖''. 刘徽通过计算得知正方体的内切球的体积与``牟合方盖''的体积之比应为$\\pi:4$. 若正方体的棱长为$2$, 则``牟合方盖''的体积为\\bracket{20}.\n\\fourch{$16$}{$16\\sqrt 3$}{$\\dfrac{16}3$}{$\\dfrac{128}3$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264640,7 +264872,8 @@ "content": "若$f(x)$是$\\mathbf{R}$上的奇函数, 且$f(x)$在$[0,+\\infty)$上单调递增, 则下列结论:\\\\\n\\textcircled{1} $y=|f(x)|$是偶函数;\\\\\n\\textcircled{2} 对任意$x\\in \\mathbf{R}$都有$f(-x)+|f(x)|=0$;\\\\\n\\textcircled{3} $y=f(x)f(-x)$在$(-\\infty ,0]$上单调递增;\\\\\n\\textcircled{4} 反函数$y=f^{-1}(x)$存在且在$(-\\infty ,0]$上单调递增.\\\\\n其中正确结论的个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264661,7 +264894,8 @@ "content": "函数$f(x)$的定义域为$D$, 若$f(x)$存在反函数, 且$f(x)$的反函数就是它本身, 则称$f(x)$为自反函数, 有下列四个命题:\\\\\n\\textcircled{1} 函数$f(x)=-\\dfrac x{x+1}$是自反函数;\\\\\n\\textcircled{2} 若$f(x)$为自反函数, 则对任意的$x\\in D$, 成立$f(f(x))=x$;\n\\textcircled{3} 若函数$f(x)=\\sqrt {1-x^2}$($a\\le x\\le b$)为自反函数, 则$b-a$的最大值为$1$;\\\\\n\\textcircled{4} 若$f(x)$是定义在$\\mathbf{R}$上的自反函数, 则方程$f(x)=x$有解;\\\\\n其中正确命题的序号为\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{3}\\textcircled{4}}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264682,7 +264916,8 @@ "content": "设函数$f(x)$的定义域为$\\mathbf{R}$, 满足$f(x+1)=2f(x)$, 且当$x\\in (0,1]$时, $f(x)=x(x-1)$.\n若对任意$x\\in (-\\infty ,m]$, 都有$f(x)\\ge -\\dfrac 89$, 则$m$的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty ,\\dfrac 94]$}{$(-\\infty ,\\dfrac 73]$}{$(-\\infty ,\\dfrac 52]$}{$(-\\infty ,\\dfrac 83]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -264703,7 +264938,8 @@ "content": "如图, 已知正方体$ABCD-A'B'C'D'$的棱长为$1$.\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] {$C'$} coordinate (C1)\n--++ (-2,0) node [above left] {$D'$} coordinate (D1) --++ (225:{2/2}) node [left] {$A'$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B'$} 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 (A1) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 正方体$ABCD-A'B'C'D'$中哪些棱所在的直线与直线$A'B$是异面直线?\\\\\n(2) 若$M,N$分别是$A'B,BC'$的中点, 求异面直线$MN$与$BC$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264724,7 +264960,8 @@ "content": "已知函数$f(x)=\\dfrac{ax-2}{x+2}$其中$a\\in \\mathbf{R}$.\\\\\n(1) 解关于$x$的不等式$f(x)\\le -1$;\\\\\n(2) 求$a$的取值范围, 使$f(x)$在区间$(0,+\\infty)$上是单调减函数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264745,7 +264982,8 @@ "content": "我国的``洋垃圾禁止入境''政策已实施一年多. 某沿海地区的海岸线为一段圆弧$\\overset\\frown{AB}$, 对应的圆心角$\\angle AOB=\\dfrac\\pi 3$. 该地区为打击洋垃圾走私, 在海岸线外侧$20$海里内的海域$ABCD$对不明船只进行识别查证(如图: 其中海域与陆地近似看作在同一平面内).在圆弧的两端点$A,B$分别建有监测站, $A$与$B$之间的直线距离为$100$海里.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (50:2) node [right] {$B$} coordinate (B);\n\\draw (50:2.4) node [right] {$C$} coordinate (C);\n\\draw (110:2) node [left] {$A$} coordinate (A);\n\\draw (110:2.4) node [left] {$D$} coordinate (D);\n\\draw (B) arc (50:110:2) -- (D) arc (110:50:2.4) -- cycle;\n\\draw [dashed] (O) -- (A) (O) -- (B);\n\\draw (80:1.33) node {\\rotatebox{-10}{陆地}};\n\\draw (80:2.2) node {\\rotatebox{-10}{海域}};\n\\end{tikzpicture}\n\\end{center}\n(1) 求海域$ABCD$的面积;\\\\\n(2) 现海上$P$点处有一艘不明船只, 在$A$点测得其距$A$点$40$海里, 在$B$点测得其距$B$点$20\\sqrt {19}$海里. 判断这艘不明船只是否进入了海域$ABCD$? 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264766,7 +265004,8 @@ "content": "已知函数$f(x)$, 若存在非零常数$k$, 对于任意实数$x$, 都有$f(x+k)+f(x)=x$成立, 则称函数$f(x)$是``$M_k$类函数''.\\\\\n(1) 若函数$f(x)=ax+b$是``$M_1$类函数'', 求实数$a$、$b$的值;\\\\\n(2) 若函数$g(x)$是``$M_2$类函数'', 且当$x\\in [0,2]$时, $g(x)=x(2-x)$, 求函数$g(x)$在$x\\in [2,6]$时的最大值和最小值;\\\\\n(3) 已知函数$f(x)$是``$M_k$类函数'', 是否存在一次函数$h(x)=Ax+B$(常数$A$、$B\\in \\mathbf{R}$, $A\\ne 0$), 使得函数$F(x)=f(x)+h(x)$是周期函数, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264787,7 +265026,8 @@ "content": "设$n$为正整数, 集合$A=\\{\\alpha|\\alpha =(t_1,t_2,\\cdots ,t_n),\\ t_k\\in \\{0,1\\}, \\ k=1,2,\\cdots,n\\}$, 对于集合$A$中的任意元素$\\alpha =(x_1,x_2,\\cdots ,x_n)$和$\\beta =(y_1,y_2,\\cdots ,y_n)$, 记\n$M(\\alpha ,\\beta)=\\dfrac 12[(x_1+y_1-|x_1-y_1|)+(x_2+y_2-|x_2-y_2|)+\\cdots +(x_n+y_n-|x_n-y_n|)]$.\\\\\n(1) 当$n=3$时, 若$\\alpha =(1,1,0)$, $\\beta =(0,1,1)$, 求$M(\\alpha ,\\alpha)$和$M(\\alpha ,\\beta)$的值;\\\\\n(2) 当$n=4$时, 设$B$是$A$的子集, 且满足: 对于$B$中的任意元素$\\alpha$、$\\beta$, 当$\\alpha$、$\\beta$相同时, $M(\\alpha ,\\beta)$是奇数; 当$\\alpha$、$\\beta$不同时, $M(\\alpha ,\\beta)$是偶数. 求集合$B$中元素个数的最大值;\\\\\n(3) 给定不小于$2$的$n$, 设$B$是$A$的子集, 且满足: 对于$B$中的任意两个不同的元素$\\alpha$、$\\beta$, $M(\\alpha ,\\beta)=0$, 写出一个集合$B$, 使其元素个数最多, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -264808,7 +265048,8 @@ "content": "已知集合$A=(-\\infty ,-3)$, $B=(-4,+\\infty)$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264836,7 +265077,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264857,7 +265099,8 @@ "content": "已知复数$z$满足$\\dfrac 1{z-1}=\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264880,7 +265123,8 @@ "content": "函数$f(x)=\\log_2(2x+4)$的反函数为$f^{-1}(x)$, 则$f^{-1}(4)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264903,7 +265147,8 @@ "content": "从甲、乙、丙、丁$4$名同学中随机选$2$名同学参加志愿者服务, 则甲、乙两人都没有被选到的概率为\\blank{50}(用数字作答).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264924,7 +265169,8 @@ "content": "已知二项式$(2x+\\dfrac 1x)^6$, 则其展开式中的常数项为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264949,7 +265195,8 @@ "content": "计算: $\\displaystyle\\lim_{n\\to \\infty} \\dfrac{|4n-23|}{2n}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -264984,7 +265231,8 @@ "content": "已知圆锥的底面半径为$1$, 高为$\\sqrt 3$, 则该圆锥的侧面展开图的圆心角$\\theta$的大小为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265007,7 +265255,8 @@ "content": "已知$\\alpha \\in (0,\\pi)$, 且有$1-2\\sin 2\\alpha =\\cos 2\\alpha$, 则$\\cos \\alpha =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265028,7 +265277,8 @@ "content": "设$F_1,F_2$分别是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, 点$P$在双曲线右支上且满足$|PF_2|=|F_1F_2|$, 双曲线的渐近线方程为$4x\\pm 3y=0$, 则$\\cos \\angle PF_1F_2=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265051,7 +265301,8 @@ "content": "若$a,b$分别是正数$p$, $q$的算术平均数和几何平均数, 且$a,b,-2$ 这三个数可适当排序后成等差数列, 也可适当排序后成等比数列, 则$p+q+pq$的值形成的集合是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265074,7 +265325,8 @@ "content": "设$f(x)=x^2+2a\\cdot x+b\\cdot 2^x$, 其中$a,b\\in \\mathbf{N}$, $x\\in \\mathbf{R}$, 如果函数$y=f(x)$与函数$y=f(f(x))$都有零点且它们的零点完全相同, 则有序数对$(a,b)$为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265095,7 +265347,8 @@ "content": "直线$x+3y-1=0$的一个法向量可以是\\bracket{20}.\n\\fourch{$(3,-1)$}{$(3,1)$}{$(1,3)$}{$(-1,3)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265118,7 +265371,8 @@ "content": "在$\\triangle ABC$中, 若$\\overrightarrow{AB}\\cdot \\overrightarrow{BC}+\\overrightarrow{AB}^2=0$, 则$\\triangle ABC$的形状一定是\\bracket{20}.\n\\fourch{等边三角形}{直角三角形}{等腰三角形}{等腰直角三角形}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265139,7 +265393,8 @@ "content": "已知函数$f(x)=A\\sin (\\omega x+\\varphi)$($A>0$, $\\omega >0$)的图像与直线$y=b$($09$成立}{存在实数$x$、$y$满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 并使得$4(x+1)(y+1)>7$成立}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)=-9$的实数$x$、$y$不存在}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)<-9$的实数$x$、$y$不存在}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265185,7 +265441,8 @@ "content": "如图在三棱锥$P-ABC$中, 棱$AB$、$AC$、$AP$两两垂直, $AB=AC=AP=3$, 点$M$在$AP$上, 且$AM=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [right] {$M$} coordinate (M);\n\\draw (P) -- (B) -- (C) -- cycle;\n\\draw [dashed] (A) -- (B) (A) -- (C) (A) -- (P) (M) -- (B) (M) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BM$和$PC$所成的角的大小;\\\\\n(2) 求三棱锥$P-BMC$的体积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265208,7 +265465,8 @@ "content": "已知函数$f(x)=\\sin x\\cos (\\dfrac{\\pi }2+x)+\\sqrt 3\\sin x\\cos x$.\\\\\n(1) 求函数$f(x)$的最小正周期及对称中心;\\\\\n(2) 若$f(x)=a$在区间$[0,\\dfrac{\\pi }2]$上有两个解$x_1,x_2$, 求$a$的取值范围及$x_1+x_2$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265229,7 +265487,8 @@ "content": "某企业接到生产$3000$台某产品的甲、乙、丙$3$种部件的订单, 每台产品需要这$3$种部件的数量分别为$2, 2, 1$(单位: 件). 已知每个工人每天可生产甲部件$6$件, 或乙部件$3$件, 或丙部件$2$件. 该企业计划安排$200$名工人分成三组分别生产这$3$种部件, 生产乙部件的人数与生产甲部件的人数成正比例, 比例系数为$k$($k\\ge 2$为正整数).\\\\\n(1) 设生产甲部件的人数为$x$, 分别写出完成甲、乙、丙$3$种部件生产需要的时间;\\\\\n(2) 假设这$3$种部件的生产同时开工, 试确定正整数$k$的值, 使完成订单任务的时间最短, 并给出时间最短时具体的人数分组方案.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265250,7 +265509,8 @@ "content": "已知$F_1$、$F_2$分别为椭圆$\\Gamma :\\dfrac{x^2}4+y^2=1$的左、右焦点, $M$为$\\Gamma$上的一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\r{0.75}\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = ellipse] (0,0) ellipse (2 and 1);\n\\filldraw ({2*cos(75)},{sin(75)}) circle (0.03) node [above] {$M$} coordinate (M);\n\\draw (M) circle (\\r);\n\\draw [name path = tangent1] ($(O)!2.2!{-asin(\\r/sqrt(4*pow(cos(75),2)+pow(sin(75),2)))}:(M)$) -- (O);\n\\draw [name path = tangent2] ($(O)!1.5!{asin(\\r/sqrt(4*pow(cos(75),2)+pow(sin(75),2)))}:(M)$) -- (O);\n\\draw [name intersections = {of = tangent1 and ellipse, by = P}];\n\\draw [name intersections = {of = tangent2 and ellipse, by = Q}];\n\\draw (P) node [above] {$P$};\n\\draw (Q) node [below left] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$M$的坐标为$(1,m)$($m>0$), 求$\\triangle F_1MF_2$的面积;\\\\\n(2) 若点$M$的坐标为$(0,1)$, 且直线$y=kx-\\dfrac 35$($k\\in \\mathbf{R}$)与$\\Gamma$交于两不同点$A$、$B$, 求证: $\\overrightarrow{MA}\\cdot \\overrightarrow{MB}$为定值, 并求出该定值;\\\\\n(3)如图, 设点$M$的坐标为$(s,t)$, 过坐标原点$O$作圆$M:(x-s)^2+(y-t)^2=r^2$(其中$r$为定值, $00, \\end{cases}$ 当$x\\in [a,a+1]$时, 不等式$f(x+a)\\ge f(2a-x)$恒成立, 则实数$a$的最大值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265513,7 +265784,8 @@ "content": "已知$\\omega =-\\dfrac 12+\\dfrac{\\sqrt 3}2\\mathrm{i}$, 集合$A=\\{z|z=1+\\omega +\\omega ^2+\\cdots +\\omega ^n, \\ n\\in \\mathbf{N}^*\\}$,\n集合$B=\\{x|x=z_1\\cdot z_2,\\ z_1, z_2\\in A\\}$($z_1$可以等于$z_2$), 则集合$B$的子集个数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265534,7 +265806,8 @@ "content": "如图所示, 已知函数$y=\\dfrac{1+x}x$($x>0$)图像上的点$A$, 和函数$y=\\dfrac{1-x}x$($x>0$)上的两点$B$、$C$, 且线段$AB$平行于$y$轴, 当三角形$ABC$为正三角形时, 点$C$的坐标为$(p,q)$, 则$\\dfrac pq$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {2/7}:5, samples = 100] plot (\\x,{(1-\\x)/\\x}) node [right] {$y=\\dfrac{1-x}{x}$($x>0$)};\n\\draw [domain = {2/3}:5, samples = 100] plot (\\x,{(1+\\x)/\\x}) node [right] {$y=\\dfrac{1+x}{x}$($x>0$)};\n\\draw ({1/2*(sqrt(3)+sqrt(3+4*sqrt(3)))},{1/(1/2*(sqrt(3)+sqrt(3+4*sqrt(3))))}) ++ (0,1) node [above] {$A$} coordinate (A);\n\\draw (A) ++ (0,-2) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (210:2) node [left] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265555,7 +265828,8 @@ "content": "若$\\overrightarrow a$与$\\overrightarrow b-\\overrightarrow c$都是非零向量, 则``$\\overrightarrow a\\cdot \\overrightarrow b=\\overrightarrow a\\cdot \\overrightarrow c$''是``$\\overrightarrow a\\perp (\\overrightarrow b-\\overrightarrow c)$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充分必要}{既不充分也不必要}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265576,7 +265850,8 @@ "content": "一个公司有$8$名员工, 其中$6$位员工的月工资分别为$5200$、$5300$、$5500$、$6100$、$6500$、$6600$, 另两位员工数据不清楚, 那么$8$位员工月工资的中位数不可能是\\bracket{20}.\n\\fourch{$5800$}{$6000$}{$6200$}{$6400$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265597,7 +265872,8 @@ "content": "设$z_1,z_2$为复数, 则下列命题中一定成立的是\\bracket{20}.\n\\twoch{如果$z_1-z_2>0$, 那么$z_1>z_2$}{如果$|z_1|=|z_2|$, 那么$z_1=\\pm z_2$}{如果$|\\dfrac{z_1}{z_2}|>1$, 那么$|z_1|>|z_2|$}{如果$z_1^2+z_2^2=0$, 那么$z_1=z_2=0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265618,7 +265894,8 @@ "content": "对数列$\\{a_n\\}$, $\\{b_n\\}$, 若区间$[a_n,b_n]$满足下列条件: \\textcircled{1} $[a_{n+1},b_{n+1}]\\subseteq [a_n,b_n]$($n\\in \\mathbf{N}^*$); \\textcircled{2} $\\displaystyle\\lim_{n\\to\\infty} (b_n-a_n)=0$, \n则称$\\{[a_n,b_n]\\}$为区间套, 并称$\\{a_n\\}$, $\\{b_n\\}$为区间套生成数列. 下列选项中, 是区间套生成数列的是\\bracket{20}.\n\\twoch{$a_n=(\\dfrac 12)^n$, $b_n=(\\dfrac 23)^n$}{$a_n=(\\dfrac 13)^n$, $b_n=\\dfrac n{n^2+1}$}{$a_n=\\dfrac{n-1}n$, $b_n=1+(\\dfrac 13)^n$}{$a_n=\\dfrac{n+3}{n+2}$, $b_n=\\dfrac{n+2}{n+1}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -265639,7 +265916,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 异面直线$AD$与$BC_1$所成角的大小为$60^\\circ$, 求:\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*sqrt(3)}) 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*sqrt(3)}) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [below] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,{2*sqrt(3)});\n\\draw (B) -- (C1) (A) -- (B1) -- (D1);\n\\draw [dashed] (A) -- (D1) (D) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 线段$A_1B_1$到底面$ABCD$的距离;\\\\\n(2) 三棱椎$B_1-ABC_1$的体积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265660,7 +265938,8 @@ "content": "已知函数$f(x)=2^x+\\dfrac a{2^x}$, 其中$a$为实常数.\\\\\n(1) 若$f(0)=7$, 解关于$x$的方程$f(x)=5$;\\\\\n(2) 若对于任意的$x\\in \\mathbf{R}$, 恒有$f(x)\\ge a$成立, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265681,7 +265960,8 @@ "content": "在上海世博会期间, 某工厂生产$A,B,C$三种世博纪念品, 每种纪念品均有精品型和普通型两种. 某一天产量如下表(单位:个):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n & 纪念品$A$ & 纪念品$B$ & 纪念品$C$ \\\\ \\hline\n精品型 & $100$ & $150$ & $n$ \\\\ \\hline\n普通型 & $300$ & $450$ & $600$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n现采用分层抽样的方法在这一天生产的纪念品中抽取$200$个, 其中有$A$种纪念品$40$个.\\\\\n(1)\t求$n$的值;\\\\\n(2)\t从$B$种精品型纪念品中抽取$5$个, 其某种指标的数据分别如下: $x,y,10,11,9$. 把这$5$个数据看作一个总体, 其均值为$10$、方差为$2$, 求$|x-y|$的值;\\\\\n(3)\t用分层抽样的方法在$C$种纪念品中抽取一个容量为$5$的样本. 将该样本看成一个总体, 从中任取$2$个纪念品, 求至少有$1$个精品型纪念品的概率.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265702,7 +265982,8 @@ "content": "如图, 在平面直角坐标系$xOy$中, 已知抛物线$C:y^2=4x$的焦点为$F$, 点$A$是第一象限内抛物线$C$上的一点, 点$D$的坐标为$(t,0)$($t>0$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2, samples = 100] plot ({pow(\\x,2)/2},\\x);\n\\filldraw (0.5,0) circle (0.03) node [below] {$F$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$|OA|=\\sqrt 5$, 求点$A$的坐标;\\\\\n(2) 若$\\triangle AFD$为等腰直角三角形, 且$\\angle FAD=90^\\circ$, 求点$D$的坐标;\\\\\n(3) 弦$AB$经过点$D$, 过弦$AB$上一点$P$作直线$x=-t$的垂线, 垂足为点$Q$, 求证: ``直线$QA$与抛物线相切''的一个充要条件是``$P$为弦$AB$的中点''.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265723,7 +266004,8 @@ "content": "已知无穷数列$\\{a_n\\}$的前$n$项和为$S_n$, 若对于任意的正整数$n$, 均有$S_{2n-1}\\ge 0$, $S_{2n}\\le 0$, 则称数列$\\{a_n\\}$具有性质$P$.\\\\\n(1) 判断首项为$1$, 公比为$-2$的无穷等比数列$\\{a_n\\}$是否具有性质$P$, 并说明理由;\\\\\n(2) 已知无穷数列$\\{a_n\\}$具有性质$P$, 且任意相邻四项之和都相等, 求证: $S_4=0$;\\\\\n(3) 已知$b_n=2n-1$($n\\in \\mathbf{N}^*$), 数列$\\{c_n\\}$是等差数列, $a_n=\\begin{cases} b_{\\frac{n+1}2}, & n\\text{为奇数}, \\\\c_{\\frac n2}, & n\\text{为偶数}. \\end{cases}$ 若无穷数列$\\{a_n\\}$具有性质$P$, 求$c_{2021}$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -265744,7 +266026,8 @@ "content": "不等式$\\dfrac 1x<1$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265775,7 +266058,8 @@ "content": "抛物线$y^2=2x$的焦点坐标为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265806,7 +266090,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265831,7 +266116,8 @@ "content": "已知向量$\\overrightarrow a=(1,-2)$, $\\overrightarrow b=(3,4)$, 则向量$\\overrightarrow a$在向量$\\overrightarrow b$的方向上的投影为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265854,7 +266140,8 @@ "content": "已知数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.若$S_9=36$, 则$a_3+a_4+a_8=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265875,7 +266162,8 @@ "content": "已知直线$l:x-y+b=0$被圆$C:x^2+y^2=25$所截得的弦长为$6$, 则$b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265898,7 +266186,8 @@ "content": "已知函数$y=f(x)$是定义在$\\mathbf{R}$上的偶函数, 且在$[0,+\\infty)$上是增函数, 若$f(a+1)\\le f(4)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265919,7 +266208,8 @@ "content": "函数$f(x)=(\\sqrt 3\\sin x+\\cos x)(\\sqrt 3\\cos x-\\sin x)$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265942,7 +266232,8 @@ "content": "过双曲线$C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}4=1$的右焦点$F$作一条垂直于$x$轴的垂线交双曲线$C$的两条渐近线于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265965,7 +266256,8 @@ "content": "若关于$x$的不等式$|2^x-m|-\\dfrac 1{2^x}<0$在区间$[0,1]$内恒成立, 则实数$m$的范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -265988,7 +266280,8 @@ "content": "已知数列$\\{a_n\\}$满足: $na_{n+2}=1007(n-1)a_{n+1}+2018(n+1)a_n(n\\in \\mathbf{N}^*)$, 且$a_1=1,a_2=2,$若$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{a_{n+1}}{a_n}=A,$则$A=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266009,7 +266302,8 @@ "content": "已知函数$f(x)=\\begin{cases} \\dfrac x{4x^2+16}, & x\\ge 2, \\\\ (\\dfrac 12)^{|x-a|}, & x<2, \\end{cases}$ 若对任意的$x_1\\in [2,+\\infty)$, 都存在唯一的$x_2\\in (-\\infty ,2)$, 满足$f(x_1)=f(x_2)$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266030,7 +266324,8 @@ "content": "若实数$x,y\\in \\mathbf{R}$, 则陈述句甲``$\\begin{cases} x+y>4, \\\\ xy>4 \\end{cases}$''是陈述句乙``$\\begin{cases} x>2, \\\\ y>2 \\end{cases}$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分又非必要}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266051,7 +266346,8 @@ "content": "已知$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi }2$, $AB=AC=1$, 点$P$是$AB$边上的动点, 点$Q$是$AC$边上的动点, 则$\\overrightarrow{BQ}\\cdot \\overrightarrow{CP}$的最小值为\\bracket{20}.\n\\fourch{$-4$}{$-2$}{$-1$}{$0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266072,7 +266368,8 @@ "content": "设$\\{a_n\\}$是等差数列, 下列命题中正确的是\\bracket{20}.\n\\twoch{若$a_1+a_2>0$, 则$a_2+a_3>0$}{若$a_1+a_3<0$, 则$a_1+a_2<0$}{若$0\\sqrt {a_1a_3}$}{若$a_1<0$, 则$(a_2-a_1)(a_2-a_3)>0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266093,7 +266390,8 @@ "content": "已知点$A(1,-2)$, $B(2,0)$, $P$为曲线$y=\\sqrt {3-\\dfrac 34x^2}$上任意一点, 则$\\overrightarrow{AP}\\cdot \\overrightarrow{AB}$的取值范围为\\bracket{20}.\n\\fourch{$[1,7]$}{$[-1,7]$}{$[1,3+2\\sqrt 3]$}{$[-1,3+2\\sqrt 3]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266114,7 +266412,8 @@ "content": "如图, 四棱锥$S-ABCD$的底面是正方形, $SD \\perp$平面$ABCD$, $SD=AD=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,2) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$S$} coordinate (S);\n\\draw (S) -- (A) -- (B) -- (C) -- cycle (S) -- (B);\n\\draw [dashed] (A) -- (D) -- (C) (S) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AC\\perp SB$;\\\\\n(2) 求二面角$C-SA-D$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266135,7 +266434,8 @@ "content": "已知函数$f(x)=2\\sqrt 3\\sin x\\cos x-2\\sin ^2x$.\\\\\n(1) 若角$\\alpha$的终边与单位圆交于点$P(\\dfrac 35,\\dfrac 45)$, 求$f(\\alpha)$的值;\\\\\n(2) 当$x\\in [-\\dfrac{\\pi }6, \\dfrac{\\pi }3]$时, 求$f(x)$的单调递增区间和值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266156,7 +266456,8 @@ "content": "设数列$\\{a_n\\}$满足$a_{n+1}=2a_n+n^2-4n+1$, $b_n=a_n+n^2-2n$.\\\\\n(1) 若$a_1=2$, 求证: 数列$\\{b_n\\}$为等比数列;\\\\\n(2) 在(1)的条件下, 对于正整数$2$、$q$、$r$($20$, $b>0$)的左、右焦点分别是 $F_1$、$F_2$, 左、右两顶点分别是 $A_1$、$A_2$, 弦$AB$和$CD$所在直线分别平行于$x$轴与$y$轴, 线段$BA$的延长线与线段$CD$相交于点$P$(如图).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,{-2*sqrt(3)}) -- (0,{2*sqrt(3)}) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot ({2*sqrt(pow(\\x,2)/3+1)},\\x);\n\\draw [domain = -3:3] plot ({-2*sqrt(pow(\\x,2)/3+1)},\\x);\n\\draw [dashed] (-4,{-2*sqrt(3)}) -- (4,{2*sqrt(3)}) (-4,{2*sqrt(3)}) -- (4,{-2*sqrt(3)});\n\\draw (1,{-2*sqrt(3)}) -- (1,{2*sqrt(3)}) node [right] {$l$};\n\\filldraw ({-sqrt(7)},0) circle (0.03) node [below] {$F_1$} ({sqrt(7)},0) circle (0.03) node [below] {$F_2$};\n\\filldraw (-2,0) circle (0.03) node [below right] {$A_1$} coordinate (A1) (2,0) circle (0.03) node [above left] {$A_2$} coordinate (A2);\n\\draw ({-4/sqrt(3)},1) node [left] {$B$} coordinate (B) -- ({4/sqrt(3)},1) node [above left] {$A$} coordinate (A);\n\\draw [dashed] (A) -- (2.5,1) node [right] {$P$} coordinate (P);\n\\draw (2.5,{-3*sqrt(3)/4}) node [right] {$D$} coordinate (D) -- (2.5,{3*sqrt(3)/4}) node [right] {$C$} coordinate (C);\n\\draw (C) -- (A1) (C) -- ($(C)!3!(A2)$) node [left] {$N$} coordinate (N) ($(C)!{1.5/4.5}!(A1)$) node [above left] {$M$} coordinate (M);\n\\filldraw (C) circle (0.03) (D) circle (0.03) (M) circle (0.03) (N) circle (0.03) (A) circle (0.03) (B) circle (0.03) (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\overrightarrow d=(2,\\sqrt 3)$是$\\Gamma$的一条渐近线的一个方向向量, 试求$\\Gamma$的两渐近线的夹角$\\theta$;\\\\\n(2) 若$|PA|=1$, $|PB|=5$ , $|PC|=2$, $|PD|=4$, 试求双曲线$\\Gamma$的方程;\\\\\n(3) 在(1)的条件下, 且$|A_1A_2|=4$, 点$C$与双曲线的顶点不重合, 直线$CA_1$和直线$CA_2$与直线$l:x=1$分别相交于点$M$和$N$, 试问: 以线段$MN$为直径的圆是否恒经过定点? 若是, 请求出定点的坐标; 若不是, 试说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266198,7 +266500,8 @@ "content": "已知平面直角坐标系$xOy$, 在$x$轴的正半轴上, 依次取点$A_1,A_2,A_3,\\cdots,A_n$($n\\in \\mathbf{N}^*$), 并在第一象限内的抛物线$y^2=\\dfrac 32x$上依次取点$B_1,B_2,B_3,\\cdots ,B_n$($n\\in \\mathbf{N}^*$), 使得$\\triangle A_{k-1}B_kA_k(k\\in \\mathbf{N}^*)$都为等边三角形, 其中$A_0$为坐标原点, 设第$n$个三角形的边长为$f(n)$.\\\\\n(1) 求$f(1),f(2)$, 并猜想$f(n)$(不要求证明);\\\\\n(2) 令$a_n=9f(n)-8$, 记$t_m$为数列$\\{a_n\\}$中落在区间$(9^m,9^{2m})$内的项的个数, 设数列$\\{t_m\\}$的前$m$项和为$S_m$, 试问是否存在实数$\\lambda$, 使得$2^{\\lambda }\\le S_m$对任意$m\\in \\mathbf{N}^*$恒成立? 若存在, 求出$\\lambda$的取值范围; 若不存在, 说明理由;\\\\\n(3) 已知数列$\\{b_n\\}$满足: $b_1=\\dfrac{\\sqrt 2}2$, $b_{n+1}=\\dfrac{\\sqrt 2}2\\sqrt {1-\\sqrt {1-b_n^2}}$, 数列$\\{c_n\\}$满足: $c_1=1$, $c_{n+1}=\\dfrac{\\sqrt {1+c_n^2}-1}{c_n}$, 求证: $b_n<\\dfrac{\\pi }{2^{n+1}}0$''是``$\\dfrac ab+\\dfrac ba>2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266529,7 +266846,8 @@ "K0216006B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266550,7 +266868,8 @@ "content": "如图所示, 已知$PA\\perp$平面$ABC$, $AD\\perp BC$于$D$, $BC=CD=AD=1$. 令$PD=x$, $\\angle BPC=\\theta$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw ({-sqrt(5)},0,0) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.2!(B)$) ++ (0,0,{2/sqrt(5)}) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [below left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (B) -- (D) -- (A) -- (P) -- cycle (P) -- (C) (P) -- (D);\n\\draw [dashed] (B) -- (A) (C) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan\\theta = \\dfrac{x}{x^2+2}$}{$\\tan\\theta = \\dfrac{x}{x^2+1}$}{$\\tan\\theta=\\dfrac{1}{x^2+2}$}{$\\tan\\theta=\\dfrac{1}{x^2+1}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266571,7 +266890,8 @@ "content": "在$\\triangle ABC$中, 角$A,B,C$所对的边分别为$a,b,c$. $b=\\sqrt{5}$, $B=\\dfrac{\\pi}{4}$.\\\\\n(1) 若$a=3$, 求$\\sin A$的值;\\\\\n(2) 若$\\triangle ABC$的面积等于$1$, 求$a$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266592,7 +266912,8 @@ "content": "如图, 圆锥的顶点是$P$, 底面中心是$O$, 已知$OP=\\sqrt{2}$, 圆$O$的直径是$AB=2$, 点$C$在弧$AB$上, 且$\\angle CAB=30^\\circ$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw [dashed] (-1,0) node [left] {$A$} coordinate (A) -- (1,0) node [right] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) (P) -- (B);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw (B) [dashed] arc (0:180:1 and 0.25) (P) -- (O);\n\\draw ({cos(-50)},{0.25*sin(-50)}) node [below] {$C$} coordinate (C);\n\\draw (C) -- (P);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积;\\\\\n(2) 求$O$到平面$APC$的距离.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266613,7 +266934,8 @@ "content": "科学家发现某种特别物质的温度$y$(单位: 摄氏度)随时间$=$x$($时间:分钟)的变化规律满足关系式: $y=m2^x+2^{1-x}$, ($0\\le x\\le 4$, $m>0$).\\\\\n(1) 若$m=2$, 求经过多少分钟, 该物质的温度为$5$摄氏度;\\\\\n(2) 如果该物质温度总不低于$2$摄氏度, 求$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266634,7 +266956,8 @@ "content": "已知实数$a$是常数, 函数$f(x)=\\log_2(a x^2+2x-a)$.\\\\\n(1) 当$a=-1$时, 求该函数的定义域;\\\\\n(2) 当$a\\le 0$时, 如果$f(x)\\ge 1$对任何$x\\in [2,3]$都成立, 求实数$a$的取值范围;\\\\\n(3) 若$a<0$, 将函数$f(x)$的图像沿$x$轴或其相反方向平移, 得到一个偶函数$g(x)$的图像, 设函数$g(x)$的最大值为$h(a)$, 求$h(a)$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $(1-\\sqrt{2},1+\\sqrt{2})$; (2) $[-\\dfrac 12,+\\infty)$; (3) $1$", @@ -266668,7 +266991,8 @@ "content": "记$f_k(x)=x^k$($x>0$, $k\\in \\mathbf{Z}$).\\\\\n(1) 求函数$F(x)=f_2(x-1)-1$的零点;\\\\\n(2) 设$\\xi,\\eta,\\mu$均为正整数, 且$\\sqrt{\\mu}$为最简根式, 若存在$n_0\\in \\mathbf{N}^*$, 使得$f_{n_0}(\\xi+\\eta\\sqrt{\\mu})$可唯一表示为$\\sqrt{\\tau}+\\sqrt{\\tau-1}$的形式($\\tau\\in\\mathbf{N}^*$). 求证: $|\\xi^2-\\eta^2\\mu|=1$;\\\\\n(3) 已知$f_{-1}(t)+f_{-1}(s)=1$, 是否存在$n_1\\in \\mathbf{N}^*$, 使得$\\dfrac{f_{n_1}(t+2)-f_{n_1}(t)-f_{n_1}(s)+f_{n_1}(2)}{f_{n_1}(4)-f_{n_1}(2)}\\ge 1$成立. 若存在, 试求出$n_1$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -266689,7 +267013,8 @@ "content": "函数$f(x)=\\log_2(x-1)$的定义域为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266714,7 +267039,8 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, 集合$B=\\{4,5\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266741,7 +267067,8 @@ "content": "函数$y=2\\cos ^2x-1$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266767,7 +267094,8 @@ "content": "已知球的体积为$36\\pi$, 则该球大圆的面积等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266791,7 +267119,8 @@ "content": "二项式$(x-\\dfrac 1x)^6$的展开式中的常数项为\\blank{50}. (用数字作答)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266815,7 +267144,8 @@ "content": "若圆锥的母线长$l=5(\\text{cm})$, 高$h=4(\\text{cm})$, 则这个圆锥的体积为\\blank{50}$(\\text{cm}^3)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266836,7 +267166,8 @@ "content": "已知函数$f(x)=a^{x+1}-2$($a>0$且$a\\ne 1$), 设$f^{-1}(x)$是$f(x)$的反函数. 若$y=f^{-1}(x)$的图像不经过第二象限, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266857,7 +267188,8 @@ "content": "函数$f(x)=2\\sin (\\omega x+\\varphi)$($\\omega >0$)的部分图像, 如图所示, 若$|AB|=5$, 则$\\omega$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:3] plot (\\x,{2*sin(pi/3*\\x/pi*180+150)});\n\\draw [dashed] (-1,0) -- (-1,2) node [above] {$A$} coordinate (A) -- (0,2) node [right] {$2$};\n\\draw [dashed] (2,0) -- (2,-2) node [below] {$B$} coordinate (B) -- (0,-2) node [left] {$-2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266878,7 +267210,8 @@ "content": "在$100$件产品中有$90$件一等品, $10$件二等品, 从中随机取出$4$件产品.\n则恰含$1$件二等品的概率是\\blank{50}. (结果精确到$0.01$)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266899,7 +267232,8 @@ "content": "已知函数$f(x)=\\log_a(x+b)$($a>0$, $a\\ne 1$, $b\\in \\mathbf{R}$)的图像, 如图所示, 则$a+b$的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3.5:4, samples = 100] plot (\\x,{ln(\\x+4)/ln(0.5)});\n\\draw (-3,0) node [below left] {$-3$} (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266920,7 +267254,8 @@ "content": "函数$F(x)=\\lg x-\\sin x$零点的个数是\\blank{50}个.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266941,7 +267276,8 @@ "content": "设函数$f(x)$和$g(x)$都是定义在集合$M$上的函数, 若对于任意$x\\in M$, 都有$f(g(x))=g(f(x))$成立, 就称函数$f(x)$与$g(x)$在$M$上互为``互换函数''.若存在非空集合$M$, 使得函数$f(x)=a^x$($a>0$, $a\\ne 1$)与$g(x)=x+1$在集合$M$上互为``互换函数'', 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -266962,7 +267298,8 @@ "content": "函数$f(x)=2^x-\\dfrac 1{2^x}$的图像关于\\bracket{20}.\n\\fourch{原点对称}{直线$y=x$对称}{直线$y=-x$对称}{$y$轴对称}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -266983,7 +267320,8 @@ "content": "三国时期赵爽在《勾股方圆图注》中, 对勾股定理的证明可用现代数学表述为如图所示, 可以利用该图作为几何解释的不等式性质是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) -- (2,0) node [right] {$E$};\n\\draw (1,0) node [below] {$H$} -- (1,2) node [above] {$D$} coordinate (D);\n\\draw (1,1) node [left] {$G$} -- (3,1) node [right] {$C$} coordinate (C);\n\\draw (2,1) node [above] {$F$} -- (2,-1) node [below] {$B$} coordinate (B);\n\\draw (A) -- (D) -- (C) -- (B) -- cycle;\n\\draw (0.5,0) node [above] {$b$} (1,1) node [below right] {$a$} ($(A)!0.5!(D)$) node [above left] {$c$};\n\\end{tikzpicture}\n\\end{center}\n\\onech{如果$a>b$, $b>c$, 那么$a>c$}{如果$a>b>0$, 那么$a^2>b^2$}{对任意正实数$a$和$b$, 有$a^2+b^2\\ge 2ab$, 当且仅当$a=b$时等号成立}{如果$a>b$, $c>0$, 那么$ac>bc$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267004,7 +267342,8 @@ "content": "若函数$f(x)=\\begin{cases} \\log_2x, & x\\ge 1, \\\\ x+c, & x<1. \\end{cases}$ 则``$c=-1$''是``$y=f(x)$是$\\mathbf{R}$上的单调增函数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267025,7 +267364,8 @@ "content": "若实数$x,y$满足$x^2+2\\cos y=1$, 则$x-\\cos y$的取值范围是\\bracket{20}.\n\\fourch{$[-1,\\sqrt{3}+1)$}{$[-1,\\sqrt 3]$}{$[1,\\sqrt 3+1)$}{$[-1,\\sqrt 3+1]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267046,7 +267386,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长$AB=2$, 异面直线$A_1A$与$B_1C$所成角的大小为$\\arctan \\dfrac 12$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\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 [above] {$B_1$} 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 (C) -- (B1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$BD_1$与底面$ABCD$所成角的正切值;\\\\\n(2) 求正四棱柱$ABCD-A_1B_1C_1D_1$的体积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267067,7 +267408,8 @@ "content": "在$\\triangle ABC$中, 内角$A,B,C$所对的边长分别是$a,b,c$.\\\\\n(1) 若$c=2$, $C=\\dfrac{\\pi }3$, 且$\\triangle ABC$的面积$S=\\sqrt 3$, 求$a,b$的值;\\\\\n(2) 若$\\sin (A+B)+\\sin (B-A)=\\sin 2A$, 试判断$\\triangle ABC$的形状.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267088,7 +267430,8 @@ "content": "如图, 某班级墙上有一壁画, 最高点$A$离地面$4$米, 最低点$B$离地面$2$米, 某同学从距离墙$x$($x>1$)米, 离地面高$a$($1\\le a\\le 2$)米的$C$处观赏该壁画, 设观赏视角$\\angle ACB=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) -- (0,1.5) node [above left] {$C$} coordinate (C) -- (4,2) node [above left] {$B$} coordinate (B) ++ (0,2) node [above] {$A$} coordinate (A);\n\\draw [very thick] (B) -- (A);\n\\draw (C) -- (A) (B) --++ (0,-2);\n\\draw (C) pic [\"$\\theta$\", draw, angle eccentricity = 1.5] {angle = B--C--A};\n\\filldraw [gray!50] (-1,0) rectangle (5.5,-0.2);\n\\draw (-0.1,1.5) -- (-0.3,1.5);\n\\draw [->] (-0.2,1.1) -- (-0.2,1.5);\n\\draw [->] (-0.2,0.4) -- (-0.2,0);\n\\draw (-0.2,0.75) node {$a$};\n\\draw [->] (1.6,0.2) -- (0,0.2);\n\\draw [->] (2.4,0.2) -- (4,0.2);\n\\draw (2,0.2) node {$x$};\n\\draw (4.1,2) -- (4.3,2);\n\\draw [->] (4.2,0.6) -- (4.2,0);\n\\draw [->] (4.2,1.4) -- (4.2,2);\n\\draw (4.2,1) node {$2$};\n\\draw (4.1,4) -- (4.9,4);\n\\draw [->] (4.6,1.6) -- (4.6,0);\n\\draw [->] (4.6,2.4) -- (4.6,4);\n\\draw (4.6,2) node {$4$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$a=1.5$米, 问该同学离墙多远时, 视角$\\theta$最大;\\\\\n(2) 若$\\tan \\theta =\\dfrac 12$, 当$a$变化时, 求$x$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267109,7 +267452,8 @@ "content": "已知函数$f(x)$的定义域是$\\{x|x\\in \\mathbf{R},\\ x\\ne \\dfrac k2, \\ k\\in \\mathbf{Z}\\}$, 且$f(x)+f(2-x)=0$, $f(x+1)=-\\dfrac 1{f(x)}$, 当$0x^2-k-1$有解? 证明你的结论.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267130,7 +267474,8 @@ "content": "已知函数$f(x)=\\ln (x^{-1}+a)$.\\\\\n(1) 设$f^{-1}(x)$是$f(x)$的反函数. 当$a=1$时, 解不等式$f^{-1}(x)>0$;\\\\\n(2) 若关于$x$的方程$f(x)+\\ln (x^2)=0$的解集中恰好有一个元素, 求实数$a$的值;\\\\\n(3) 设$a>0$, 若对任意$t\\in [\\dfrac 12,1]$, 函数$f(x)$在区间$[t,t+1]$上的最大值与最小值的差不超过$\\ln 2$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267151,7 +267496,8 @@ "content": "对于集合$A=\\{x|y=\\dfrac{|x-1|}{x-1}, \\ y\\in \\mathbf{R}\\}$, $B=\\{y|y=\\dfrac{|x-1|}{x-1},\\ x\\in \\mathbf{R},\\ x\\ne 1\\}$, 下列说法中, 正确的说法的序号为\\blank{50}.\n\\textcircled{1} $A\\subset B$; \\textcircled{2} $B\\subset A$; \\textcircled{3} $A\\cap B\\ne \\varnothing$; \\textcircled{4} $A\\cap B=\\varnothing$; \\textcircled{5} $A\\cap B=\\mathbf{R}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267172,7 +267518,8 @@ "content": "对于集合$C=\\{(x,y)|y=x^2,\\ x\\in \\mathbf{R}\\}$, $D=\\{(y,x)|y=x^2,\\ x\\in \\mathbf{R}\\}$, 则集合$C\\cap D$的元素个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无限}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267193,7 +267540,8 @@ "content": "设常数$m\\in \\mathbf{R}$. 已知集合$A=(2, 4)$, $B=(m, m^2)$. 若$A\\subset B$, 求$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267214,7 +267562,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 已知集合$A=[2, 3]$, $B=[a, 2a-\\dfrac 52]$, 且$A\\cap B\\ne \\varnothing$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267235,7 +267584,8 @@ "content": "设常数$a\\in \\mathbf{R}$.已知集合$A=\\{x|x^2-(a+1)x+a=0\\}$, $B=\\{x|x^2-4x+2a=0\\}$.\\\\\n(1) 根据$a$的值, 写出集合$A$;\\\\\n(2) 若$B\\subseteq A$, 求$a$的值的集合.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267256,7 +267606,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 若$\\{x|x^2-ax+a=0,\\ x\\in \\mathbf{R}\\}\\cap (-\\infty , 0]\\ne \\varnothing$, 则$a$的取值范围为为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267277,7 +267628,8 @@ "content": "集合$A=\\{(x,y)|x^2+y^2=25\\}$, $B=\\{(x,y)|(x-3)^2+(y-4)^2=100\\}$, 则集合$A\\cap B$的元素个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无限}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267298,7 +267650,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 已知集合$A=\\{x|y=\\sqrt {-1-\\dfrac 1{1+x}}$且$x\\in \\mathbf{R}\\}$, $B=\\{x| ax0$和$a_2x^2+b_2x+c_2>0$的解集分别为集合$M$和$N$, 那么``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''的\\bracket{20}条件.\n\\fourch{充分而不必要}{必要而不充分}{充分必要}{既不充分也不必要}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -267424,7 +267782,8 @@ "content": "下列函数中, 最小值为$2$的函数的序号是\\blank{50}.\\\\\n\\textcircled{1} $y=\\cos x+\\sec x$, $x\\in (-\\dfrac{\\pi}2, \\dfrac{\\pi}2)$; \\textcircled{2} $y=\\cos x+\\sec x$, $x\\in (0,\\dfrac{\\pi}{2})$; \\textcircled{3} $y=\\ln x^2+\\log_{x^2}\\mathrm{e}$; \\textcircled{4} $y=2^x+2^{-x}$, $x>0$; \\textcircled{5} $y=\\dfrac x{\\sqrt {x-1}}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267445,7 +267804,8 @@ "content": "设$a>0$, $a\\ne 1$, $t>0$, 比较$a^{2t}$和 $a^{t^2-3}$的大小, 并证明你的结论.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267466,7 +267826,8 @@ "content": "已知$x,y\\in \\mathbf{R}^+$, $x-y=1+xy$, 求$x-4y$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267489,7 +267850,8 @@ "content": "设常数$a\\in (0, 1)$.若$00$恒成立, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267552,7 +267916,8 @@ "content": "设实数$m, n, p, q$满足不等式组$\\begin{cases} (p-m)(p-n)>0, \\\\ (q-m)(q-n)<0, \\\\ (p-m)(q-m)>0, \\\\ m+n>p+q, \\end{cases}$则$m, n, p, q$的大小顺序用``$<$''连接为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267573,7 +267938,8 @@ "content": "不等式$\\ln x^2<\\ln (4-3x)$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267594,7 +267960,8 @@ "content": "不等式$\\log_2(1-x)>1+\\log_4x$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267617,7 +267984,8 @@ "content": "设常数$a>0$. 已知关于$x$的不等式$(x-a)(x-2a)<0$的解集为$A$. 若$A\\cap \\mathbf{Z}$的元素个数为$3$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267638,7 +268006,8 @@ "content": "不等式$\\dfrac{3|x|-2}{|x|}\\le 1$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267661,7 +268030,8 @@ "content": "设常数$a>0$且$a\\ne 1$. 若关于$x$的不等式$\\log_ax<0$的解集是$(1,+\\infty)$, 则关于$x$的不等式$\\log_a(x-\\dfrac 4x)\\ge 0$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267682,7 +268052,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 已知关于$x$的不等式$\\dfrac{ax-2}{x^2-a}<0$的解集为$M$. 若$1\\in M$且$3\\notin M$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267703,7 +268074,8 @@ "content": "设$a,b,c$是互不相等的正数, 则下列不等式中正确的不等式的序号是\\blank{50}.\\\\\n\\textcircled{1} $|a-c|\\le |a-b|+|c-b|$; \\textcircled{2} $a+\\dfrac 1a\\le a^3+\\dfrac 1{a^3}$; \\textcircled{3} $a-\\dfrac 1a\\le (a+b)^2-\\dfrac 1{(a+b)^2}$; \\textcircled{4} $\\sqrt {a+4}+\\sqrt {a+1}\\le \\sqrt {a+3}+\\sqrt {a+2}$; \\textcircled{5} $\\sqrt {a+b+c}<\\sqrt {abc}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267724,7 +268096,8 @@ "content": "函数$y=f(x)$满足对于任意$x\\ne 0$, 恒有$f(x-\\dfrac 1x)=x^2+\\dfrac 1{x^2}$. 若存在$x_0$使得$f(x_0)-x_0=2$成立, 则$x_0=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267747,7 +268120,8 @@ "content": "已知半径为$r$的扇形的面积为$1$, 试将扇形的周长$C$表示成$r$的函数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267768,7 +268142,8 @@ "content": "设常数$a>1$. 已知函数$f(x)=\\log_{\\frac 12}(x^2-ax+2)$($10$时, $f(x)=2^{\\frac 1x}-3$. 求不等式$f(x)>-1$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267896,7 +268276,8 @@ "content": "设函数$y=f(x)$为$\\mathbf{R}$上的奇函数, 且对于任意$x\\in \\mathbf{R}$, 都有$f(x)=f(2-x)$. 当$1\\le x<2$时, $f(x)=-2x+4$.\\\\\n(1) 求函数$y=f(x)$在$-1\\le x<1$时的解析式;\\\\\n(2) 求函数$y=f(x)-1$在$-100\\le x\\le 100$时的所有零点的个数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -267917,7 +268298,8 @@ "content": "已知定义在$\\mathbf{R}$上的函数$y=f(x)$是奇函数, 且$y=f(x)$也是以2为周期的一个周期函数.若$f(\\dfrac 32)=0$, 则在区间$[-2,2]$上的零点的个数的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267938,7 +268320,8 @@ "content": "函数$y=\\dfrac 1{\\sqrt {x^2-5x-6}}$的递增区间是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267961,7 +268344,8 @@ "content": "设常数$a\\in \\mathbf{R}$. 已知函数$y=f(x)$为$\\mathbf{R}$上的奇函数, 且满足对于$(-\\infty , +\\infty)$内的任意$x_1$、$x_2$, 当$x_10$时, $f(x)=(x-a)^2-1$, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -267982,7 +268366,8 @@ "content": "判断函数$f(x)=2^x+2^{-x}$在$[0,+\\infty)$上的单调性, 并证明你的结论.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268003,7 +268388,8 @@ "content": "函数$f(x)=\\ln (ax^2-4x+3)$在$(-\\infty , 1]$上为减函数, 求实常数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268024,7 +268410,8 @@ "content": "下列命题中, 正确的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 一个幂函数或是奇函数, 或是偶函数;\\\\\n\\textcircled{2} 当$\\alpha =0$时, 函数$y=x^{\\alpha }$的图像是一条射线;\\\\\n\\textcircled{3} 当$\\alpha <0$且$y=x^{\\alpha }$是奇函数时, 它的图像总是过$(-1,-1)$;\\\\\n\\textcircled{4} 若一个幂函数的图像经过第二象限的点, 则这个幂函数是偶函数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268045,7 +268432,8 @@ "content": "若集合$A=\\{y|y=x^3, \\ -1\\le x\\le 1\\}$, $B=\\{y|y=x^{-1}\\}$, 则$A\\cap B$等于\\bracket{20}.\n\\fourch{$\\{(-1,1)\\}$}{$\\{(1,1),(-1,-1)\\}$}{$\\{-1,1\\}$}{$[-1,0)\\cup (0,1]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268066,7 +268454,8 @@ "content": "设常数$n\\in \\mathbf{Z}$. 若$f(x)=x^{n^2+2n-3}$是偶函数, 且图像与两条坐标轴都无公共点, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268087,7 +268476,8 @@ "content": "若函数$f(x)=1-\\sqrt {-x^2-2x}$($-2\\le x\\le -1$), 请在空白处画出函数$y=f^{-1}(x)$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268108,7 +268498,8 @@ "content": "设常数$a,b\\in \\mathbf{R}$, $a\\ne b$, 是否存在$a, b$使得函数$f(x)=\\dfrac{ax+1}{bx+1}$存在反函数, 且反函数就是$y=f(x)$? 若存在, 求出$a, b$满足的条件; 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268129,7 +268520,8 @@ "content": "设集合$A=\\{5, \\log_2(a+3)\\}$, $B=\\{a,b\\}$, 若$A\\cap B=\\{2\\}$, 则$A\\cup B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268152,7 +268544,8 @@ "content": "已知函数$f(x)=\\log_3(\\dfrac 4x+2)$, 则方程$f^{-1}(x)=4$的解为$x=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268175,7 +268568,8 @@ "content": "方程$\\sin 2x=\\sin 3x$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268196,7 +268590,8 @@ "content": "函数$y=\\ln (-x^2+2x+3)$的单调递减区间是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268217,7 +268612,8 @@ "content": "若函数$y=f(x)$的图像与$y=x+\\dfrac 1x$的图像关于$x=1$轴对称, 则$f(x)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268238,7 +268634,8 @@ "content": "已知等差数列$\\{a_n\\}$中, $a_1=10$, 当且仅当$n=5$时, 前$n$项和$S_n$取得最大值, 则公差$d$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268259,7 +268656,8 @@ "content": "已知函数$f(x)=a\\sin x+b\\cos x$($x\\in [a^2-2,a]$)是奇函数, 则$a+b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268280,7 +268678,8 @@ "content": "不等式$x^2-3>ax-a$对一切$3\\le x\\le 4$恒成立, 则符合要求的自然数$a$有\\blank{50}个.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268301,7 +268700,8 @@ "content": "在$\\triangle ABC$中, 锐角$\\angle B$所对的边$b=10$. $\\triangle ABC$的面积$S_{\\triangle ABC}=10$, 外接圆半径$R=13$, 则$\\triangle ABC$的周长为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268322,7 +268722,8 @@ "content": "若函数$f(x)=|x-3|-\\log_ax+1$无零点, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268343,7 +268744,8 @@ "content": "已知$\\log_ax=\\log_by=-2$.若$a+b=2$, 则$x+y$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268364,7 +268766,8 @@ "content": "已知函数$f(x)$满足: \\textcircled{1} 对任意$x\\in (0, +\\infty)$, 恒有$f(2x)=2f(x)$成立; \\textcircled{2} 当$x\\in (1,2]$时, $f(x)=2-x$. 若$f(a)=f(2015)$, 则满足条件的最小的正实数$a$是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268387,7 +268790,8 @@ "content": "已知函数$f(x)$定义域为$[a,b]$. 则``函数$f(x)$在$[a,b]$上为单调函数''是``函数$f(x)$在$[a,b]$上有最大值和最小值''的\t\\bracket{20}.\n\\twoch{充分但非必要条件}{必要但非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268408,7 +268812,8 @@ "content": "若$\\dfrac 1a<\\dfrac 1b<0$, 有下面四个不等式: \\textcircled{1} $|a|>|b|$; \\textcircled{2} $ab^3$. 其中, 不正确的不等式有\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{$3$个}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268429,7 +268834,8 @@ "content": "已知: 数列$\\{a_n\\}$满足$a_1=16$, $a_{n+1}-a_n=2n$. 则$\\{\\dfrac{a_n}n\\}$的最小值为\\bracket{20}.\n\\fourch{$8$}{$7$}{$6$}{$5$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268450,7 +268856,8 @@ "content": "设函数$f_1(x)=\\log_4x-(\\dfrac 14)^x$、$f_2(x)=\\log_{\\frac 14}x-(\\dfrac 14)^x$的零点分别为$x_1$与$x_2$, 则\\bracket{20}.\n\\fourch{$00$的解集$P$, 不等式$\\log_2(x^2-1)\\le 1$的解集为$Q$. 若$Q\\subseteq P$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268492,7 +268900,8 @@ "content": "已知: 函数$f(x)=p\\sin \\omega x\\cdot \\cos \\omega x-\\cos ^2\\omega x$($p>0$, $\\omega >0$)的最大值为$\\dfrac 12$, 最小正周期为$\\dfrac{\\pi}2$.\\\\\n(1) 求: $p$, $\\omega$的值与$f(x)$的解析式;\\\\\n(2) 若$\\triangle ABC$的三条边为$a$, $b$, $c$, 满足$a^2=bc$, $a$边所对的角为$A$. 求: 角$A$的取值范围及函数$f(A)$的值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268513,7 +268922,8 @@ "content": "市场上有一种新型的强力洗衣液, 特点是去污速度快. 已知每投放$a$($1\\le a\\le 4$, 且$a\\in \\mathbf{R}$)个单位的洗衣液在一定量水的洗衣机中, 它在水中释放的浓度$y$(克/升)随着时间$x$(分钟)变化的函数关系式近似为$y=a\\cdot f(x)$, 其中$f(x)=\\begin{cases} \\dfrac{16}{8-x}-1, & 0\\le x\\le 4 \\\\ 5-\\dfrac 12x, & 40$, $t>0$, 函数$f(x)=\\begin{vmatrix}\n\\sqrt 3 & \\sin \\omega x \\\\ 1 & \\cos \\omega x \\end{vmatrix}$的最小正周期为$2\\pi$, 将$f(x)$的图像向左平移$t$个单位, 所得图像对应的函数为偶函数, 则$t$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268749,7 +269169,8 @@ "content": "两个三口之家, 共$4$个大人, $2$个小孩, 约定星期日乘红色、白色两辆轿车结伴郊游, 每辆车最多乘坐$4$人, 其中两个小孩不能独坐一辆车, 则不同的乘车方法种数是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268770,7 +269191,8 @@ "content": "向量$\\overrightarrow a$, $\\overrightarrow b$满足$|\\overrightarrow a|=1$, $|\\overrightarrow a-\\overrightarrow b|=\\dfrac{\\sqrt 3}2$, $\\overrightarrow a$与$\\overrightarrow b$的夹角为$60^\\circ$, 则$|\\overrightarrow b|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268791,7 +269213,8 @@ "content": "数列$1,\\dfrac 12,\\dfrac 21,\\dfrac 13,\\dfrac 22,\\dfrac 31,\\dfrac 14,\\dfrac 23,\\dfrac 32,\\dfrac 41,\\cdots$, 则$\\dfrac 89$是该数列的第\\blank{50}项.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268812,7 +269235,8 @@ "content": "已知直线$(1-a)x+(a+1)y-4(a+1)=0$(其中$a$为实数)过定点$P$, 点$Q$在函数$y=x+\\dfrac 1x$的图像上, 则$PQ$连线的斜率的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -268833,7 +269257,8 @@ "content": "下列函数中, 是奇函数, 且在$(0,+\\infty)$上递减的是\\bracket{20}.\n\\fourch{$y=x^2$}{$y=x^3$}{$y=x^{-\\frac 12}$}{$y=x^{-\\frac 13}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268854,7 +269279,8 @@ "content": "设$P$是$\\triangle ABC$所在平面内一点. 若$\\overrightarrow{CB}=\\lambda \\overrightarrow{PA}+\\overrightarrow{PB}$, $\\lambda \\in \\mathbf{R}$, 则点$P$一定在\\bracket{20}.\n\\fourch{$\\triangle ABC$内部}{$AC$边所在直线上}{$AB$边所在直线上}{$BC$边所在直线上}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268875,7 +269301,8 @@ "content": "若$a,b$是异面直线, 则下列命题中的假命题为\\bracket{20}.\n\\onech{过直线$a$可以作一个平面并且只可以作一个平面$\\alpha$与直线$b$平行}{过直线$a$至多可以作一个平面$\\alpha$与直线$b$垂直}{唯一存在一个平面$\\alpha$与直线$a,b$等距}{可能存在平面$\\alpha$与直线$a,b$都垂直}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268896,7 +269323,8 @@ "content": "王先生购买了一部手机, 欲使用中国移动``神州行''卡或加入联通的$130$网, 经调查其收费标准见下表: (注: 本地电话费以分为计费单位, 长途话费以秒为计费单位)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n网络 & 月租费 & 本地话费 & 长途话费 \\\\ \\hline\n甲: 联通$130$ & $12$元 & $0.36$元/分 & $0.06$元/秒 \\\\ \\hline\n乙: 移动``神州行'' & 无 & $0.60$元/分 & $0.07$元/秒 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n若王先生每月拨打本地电话的时间是拨打长途电话时间的$5$倍, 若要用联通$130$应最少打多长时间的长途电话才合算? 答: \\bracket{20}.\n\\fourch{$300$秒}{$400$秒}{$500$秒}{$600$秒}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -268917,7 +269345,8 @@ "content": "在三棱锥$P-ABC$中, $PA$, $PB$, $PC$两两垂直, $PB=5$, $PC=6$. 若三棱锥$P-ABC$的体积为$20$, $Q$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$P$} coordinate (P);\n\\draw (6,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,5) node [below] {$B$} coordinate (B);\n\\draw (0,4,0) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw (A) -- (B) -- (C) -- cycle (A) -- (Q);\n\\draw [dashed] (P) -- (A) (P) -- (B) (P) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$PA$;\\\\\n(2) 求异面直线$PB$, $AQ$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268938,7 +269367,8 @@ "content": "已知角$ABC$是$\\triangle ABC$的三个内角, $a,b,c$分别是角$A,B,C$的对边. 若向量$\\overrightarrow m=(1-\\cos(A+B),\\cos\\dfrac{A-B}2)$, $\\overrightarrow n=(\\dfrac 58,\\cos\\dfrac{A-B}2)$, 且$\\overrightarrow m\\cdot \\overrightarrow n=\\dfrac 98$.\\\\\n(1) 求$\\tan A\\cdot \\tan B$的值;\\\\\n(2) 求$\\dfrac{ab\\sin C}{a^2+b^2-c^2}$的最大值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268959,7 +269389,8 @@ "content": "某市$2013$年发放汽车牌照$12$万张, 其中燃油型汽车牌照$10$万张, 电动型汽车$2$万张. 为了节能减排和控制总量, 从$2013$年开始, 每年电动型汽车牌照按$50\\%$增长, 而燃油型汽车牌照每一年比上一年减少$0.5$万张, 同时规定一旦某年发放的牌照超过$15$万张, 以后每一年发放的电动车的牌照的数量维持在这一年的水平不变.\n(1) 记$2013$年为第一年, 每年发放的燃油型汽车牌照数构成数列$\\{a_n\\}$, 每年发放的电动型汽车牌照数构成数列$\\{b_n\\}$, 完成下列表格, 并写出这两个数列的通项公式;\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$a_1=10$ & $a_2=9.5$ & $a_3=$\\blank{30} & $a_4=$\\blank{30} & $\\cdots$ \\\\ \\hline\n$b_1=2$ & $b_2=3$ & $b_3=$\\blank{30} & $b_4=$\\blank{30} & $\\cdots$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(2) 从$2013$年算起, 累计各年发放的牌照数, 哪一年开始超过$200$万张?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -268980,7 +269411,8 @@ "content": "设常数$m\\ne 0$. 已知椭圆$\\dfrac{x^2}2+y^2=1$上两个不同的点$A,B$关于直线$y=mx+\\dfrac 12$对称.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [domain = -0.5:0.9] plot (\\x, {-2*\\x+1/2});\n\\draw ({1/2-sqrt(5/6)},{-1/2-sqrt(5/6)/2}) node [below] {$A$} coordinate (A) ({1/2+sqrt(5/6)},{-1/2+sqrt(5/6)/2}) node [below right] {$B$} coordinate (B);\n\\draw (0,0) -- (A) -- (B) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 若已知$C(0,\\dfrac 12)$, $M$为椭圆上动点, 证明: $|MC|\\le \\dfrac{\\sqrt {10}}2$;\\\\\n(2) 求实数$m$的取值范围;\\\\\n(3) 求$\\triangle AOB$面积的最大值($O$为坐标原点).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269001,7 +269433,8 @@ "content": "已知函数$f(x)=\\log_k x$($k$为常数, $k>0$且$k\\ne 1$), 且数列$\\{f(a_n)\\}$是首项为$4$, 公差为$2$的等差数列.\\\\\n(1) 求证: 数列$\\{a_n\\}$是等比数列;\\\\\n(2) 若$b_n=a_n+f(a_n)$, 当$k=\\dfrac 1{\\sqrt 2}$时, 求数列$\\{b_n\\}$的前$n$项和$S_n$的最小值;\\\\\n(3) 若$c_n=a_n\\lg a_n$, 问是否存在实数$k$, 使得$\\{c_n\\}$是递增数列? 若存在, 求出$k$的范围; 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269022,7 +269455,8 @@ "content": "已知集合$A=\\{y| y=\\sin x , \\ x\\in \\mathbf{R}\\}$, 集合$B=\\{y| y=\\sqrt {x} , \\ x\\in \\mathbf{R}\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269043,7 +269477,8 @@ "content": "已知$1+\\mathrm{i}$是实系数一元二次方程$x^2+ax+b=0$的根($i$为虚数单位), 则$2a+b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269066,7 +269501,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269094,7 +269530,8 @@ "content": "已知球的主视图的面积为$\\dfrac{\\pi}4$, 则该球的体积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269115,7 +269552,8 @@ "content": "在平面直角坐标系$xOy$中, 直线$l$的参数方程为$\\begin{cases} x=t-1, \\\\ y=t, \\end{cases}$($t$为参数) 圆$O$的参数方程为$\\begin{cases} x=\\cos \\theta, \\\\ y=\\sin \\theta, \\end{cases}$($\\theta$为参数) 则直线$l$与圆$O$的位置关系是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269136,7 +269574,8 @@ "content": "已知实数$x$、$y$满足条件$\\begin{cases} x-y\\ge 0, \\\\ y\\ge 0, \\\\ x+y\\le 1, \\end{cases}$ 则目标函数$z=2x-y$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269159,7 +269598,8 @@ "content": "方程$(\\log_3x)^2+\\log_93x=2$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269187,7 +269627,8 @@ "content": "某校高一、高二、高三共有$200$名学生, 为调查他们的体育锻炼情况, 通过分层抽样获得了20名学生一周的锻炼时间, 数据如下表(单位: 小时):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n高一 & $6$ & $6.5$ & $7$ & $7.5$ & $8$ & $$ & $$ & $$ \\\\ \\hline\n高二 & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$ & $$ \\\\ \\hline\n高三 & $3$ & $4.5$ & $6$ & $7.5$ & $9$ & $10.5$ & $12$ & $13.5$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n则根据上述样本数据估计该校学生一周的锻炼时间不小于$7$小时的人数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269208,7 +269649,8 @@ "content": "从$m$($m\\in \\mathbf{N}^*$, $m\\ge 4$)个男生、$6$个女生中任选$2$个人当发言人, 假设事件$A$表示选出的$2$个人性别相同, 事件$B$表示选出的$2$个人性别不同. 如果$A$的概率和$B$的概率相等, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269229,7 +269671,8 @@ "content": "将函数$f(x)=2\\sin 2x$的图像向左平移$\\dfrac{\\pi }6$个单位, 再向下平移$1$个单位, 得到函数的$y=g(x)$图像. 若$y=g(x)$在$[0,b]$($b>0$)上至少含有$2021$个零点, 则$b$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269250,7 +269693,8 @@ "content": "如图, 在$\\triangle ABC$中, $\\angle BAC=\\dfrac{\\pi}3$, $D$为$AB$中点, $P$为$CD$上一点, 且满足$\\overrightarrow{AP}=t\\overrightarrow{AC}+\\dfrac 13\\overrightarrow{AB}$, 若$\\triangle ABC$的面积为$\\dfrac{3\\sqrt 3}2$, 则$|\\overrightarrow{AP}|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) -- (1.5,0) node [below] {$D$} coordinate (D) -- (3,0) node [right] {$B$} coordinate (B);\n\\draw (60:2) node [above] {$C$} coordinate (C);\n\\draw ($(C)!{2/3}!(D)$) node [right] {$P$} coordinate (P);\n\\draw (A) -- (C) -- (B) (A) -- (P) (C) -- (D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269271,7 +269715,8 @@ "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$a_1=b_1=1$, 对任何正整数$n$均有$a_{n+1}=a_n+b_n+\\sqrt {a_n^2+b_n^2}$, $b_{n+1}=a_n+b_n-\\sqrt {a_n^2+b_n^2}$, 设$c_n=3^n(\\dfrac 1{a_n}+\\dfrac 1{b_n})$, 则数列$\\{c_n\\}$的前$2020$项之和为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269292,7 +269737,8 @@ "content": "已知实数$a\\ne 0$, 则``$a<1$''是``$\\dfrac 1a>1$''的\\blank{50}\\bracket{20}\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269313,7 +269759,8 @@ "content": "如图, 正方体$A_1B_1C_1D_1-ABCD$中, $E$、$F$分别为棱$A_1A$、$BC$上的点, 在平面$ADD_1A_1$内且与平面$DEF$平行的直线\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2.5,0) node [below right] {$B$} coordinate (B) --++ (45:{2.5/2}) node [right] {$C$} coordinate (C)\n--++ (0,2.5) node [above right] {$C_1$} coordinate (C1)\n--++ (-2.5,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2.5/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2.5,2.5) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2.5/2}) (B1) --++ (-2.5,0);\n\\draw [dashed] (A) --++ (45:{2.5/2}) node [left] {$D$} coordinate (D) --++ (2.5,0) (D) --++ (0,2.5);\n\\draw ($(B)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(A)!0.7!(A1)$) node [left] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (F) -- (D) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{有一条}{有二条}{有无数条}{不存在}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269334,7 +269781,8 @@ "content": "已知函数$f(x)$($x\\in D$), 若对任意的$x\\in D$, 都存在$t\\in D$, 使$f(t)=-f(x)$成立, 称$f(x)$是``拟奇函数''. 下列函数是``拟奇函数''的个数是\\bracket{20}.\n\\textcircled{1} $f(x)=x^2$; \\textcircled{2} $f(x)=\\ln x$; \\textcircled{3} $f(x)=x+\\dfrac 1x$; \\textcircled{4} $f(x)=\\cos x$\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269355,7 +269803,8 @@ "content": "设集合$S=\\{1,2,3,\\cdots,2020\\}$, 设集合$A$是集合$S$的非空子集, $A$中的最大元素和最小元素之差称为集合$A$的直径. 那么集合$S$所有直径为$71$的子集的元素个数之和为\\bracket{20}.\n\\fourch{$71\\cdot 1949$}{$2^{70}\\cdot 1949$}{$2^{70}\\cdot 37\\cdot 1949$}{$2^{70}\\cdot 72\\cdot 1949$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269376,7 +269825,8 @@ "content": "如图所示的几何体是圆柱的一部分, 它是由边长为$2$的正方形$ABCD$(及其内部)以$AB$边所在直线为旋转轴顺时针旋转$120^{\\circ}$得到的.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (2,2) node [right] {$D$} coordinate (D);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw ({2*cos(-140)},{0.5*sin(-140)}) node [left] {$E$} coordinate (E);\n\\draw (E) ++ (0,2) node [left] {$F$} coordinate (F);\n\\draw ({2*cos(-40)},{0.5*sin(-40)}) node [below] {$P$} coordinate (P);\n\\draw (E) -- (F) -- (A) -- (D) -- (C) arc (0:-140:2 and 0.5);\n\\draw (D) arc (0:-140:2 and 0.5);\n\\draw [dashed] (B) -- (A) (B) -- (E) (B) -- (C) (A) -- (C) (F) -- (P) (B) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求此几何体的体积;\\\\\n(2) 设$P$是弧$EC$上的一点, 且$BP\\perp BE$, 求异面直线$FP$与$CA$所成角的大小. (结果用反三角函数值表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269397,7 +269847,8 @@ "content": "已知锐角$\\alpha$、$\\beta$的顶点与坐标原点重合, 始边与$x$轴正方向重合, 终边与单位圆分别交于$P$、$Q$两点, 若$P$、$Q$两点的横坐标分别为$\\dfrac{3\\sqrt {10}}{10}$、$\\dfrac{2\\sqrt 5}5$.\\\\\n(1) 求$\\cos (\\alpha +\\beta)$的大小;\\\\\n(2) 在$\\triangle ABC$中, $a,b,c$为三个内角$A,B,C$对应的边长, 若已知角$C=\\alpha +\\beta$, $\\tan A=\\dfrac 34$, 且$a^2=\\lambda bc+c^2$, 求$\\lambda$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269418,7 +269869,8 @@ "content": "疫情后, 为了支持企业复工复产, 某地政府决定向当地企业发放补助款, 其中对纳税额在$3$万元至$6$万元(包括$3$万元和$6$万元)的小微企业做统一方案. 方案要求同时具备下列两个条件: \\textcircled{1} 补助款$f(x)$(万元)随企业原纳税额$x$(万元)的增加而增加; \\textcircled{2} 补助款不低于原纳税额$x$(万元)的$50\\%$. 经测算政府决定采用函数模型$f(x)=\\dfrac x4-\\dfrac bx+4$(其中$b$为参数)作为补助款发放方案.\\\\\n(1) 判断使用参数$b=12$是否满足条件, 并说明理由;\\\\\n(2) 求同时满足条件\\textcircled{1}、\\textcircled{2}的参数$b$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269439,7 +269891,8 @@ "content": "在平面直角坐标系$xOy$中, $F_1$, $F_2$分别是椭圆$\\Gamma :\\dfrac{x^2}{a^2}+y^2=1$($a>0$)的左、右焦点, 直线$l$与椭圆交于不同的两点$A$, $B$, 且$|AF_1|+|AF_2|=2\\sqrt 2$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 已知直线$l$经过椭圆的右焦点$F_2$, $P,Q$是椭圆上两点, 四边形$ABPQ$是菱形, 求直线$l$的方程;\\\\\n(3) 已知直线$l$不经过椭圆的右焦点$F_2$, 直线$AF_2$, $l$, $BF_2$的斜率依次成等差数列, 求直线$l$在$y$轴上截距的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269460,7 +269913,8 @@ "content": "若数列$\\{a_n\\}$对任意连续三项$a_i,a_{i+1},a_{i+2}$, 均有$(a_i-a_{i+2})(a_{i+2}-a_{i+1})>0$, 则称该数列为``跳跃数列''.\\\\\n(1) 判断下列两个数列是否是跳跃数列:\\\\\n\\textcircled{1} 等差数列: $1,2,3,4,5,\\cdots$; \\textcircled{2} 等比数列: $1,-\\dfrac 12,\\dfrac 14,-\\dfrac 18,\\dfrac 1{16},\\cdots$;\\\\\n(2) 若数列$\\{a_n\\}$满足对任何正整数$n$, 均有$a_{n+1}=a_1^{a_n}$($a_1>0$). 证明: 数列$\\{a_n\\}$是跳跃数列的充分必要条件是$00)$上一点, 点$P$到抛物线$C$的焦点的距离为$7$, 到$y$轴的距离为$5$, 则$p=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269596,7 +270055,8 @@ "content": "设复数$z=\\begin{vmatrix}\\cos \\alpha & \\mathrm{i} \\\\ \\sin \\alpha & \\sqrt 2+\\mathrm{i} \\end{vmatrix}$($\\mathrm{i}$为虚数单位), 若$|z|=\\sqrt 2$, 则$\\tan 2\\alpha =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269619,7 +270079,8 @@ "content": "若$(ax^2+\\dfrac 1{\\sqrt x})^{5}$的展开式中的常数项为$-\\dfrac 5{2}$, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269640,7 +270101,8 @@ "content": "设函数$f(x)$的定义域为$D$. 若对于$D$内的任意$x_1$, $x_2$($x_1\\ne x_2$), 都有$(x_2-x_1)[f(x_2)-f(x_1)]>0$, 则称函数$f(x)$为``Z函数''.有下列函数: \\textcircled{1} $f(x)=1$; \\textcircled{2} $f(x)=-2x+1$; \\textcircled{3} $f(x)=x^3$; \\textcircled{4} $f(x)=\\lg x$. 其中``Z函数''的序号是\\blank{50}(写出所有的正确序号).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269661,7 +270123,8 @@ "content": "已知直三棱柱的各棱长都相等, 体积等于$18$($\\text{cm}^3$). 若该三棱柱的所有顶点都在球$O$的表面上, 则球$O$的体积等于\\blank{50}($\\text{cm}^3$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269682,7 +270145,8 @@ "content": "已知$F_1,F_2$是椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}3=1$($a>\\sqrt 3$)的左、右焦点, 过原点$O$且倾斜角为$60^\\circ$的直线与椭圆$C$的一个交点为$M$. 若$|\\overrightarrow{MF_1}+\\overrightarrow{MF_2}|=|\\overrightarrow{MF_1}-\\overrightarrow{MF_2}|$, 则椭圆$C$的长轴长为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269703,7 +270167,8 @@ "content": "已知无穷等比数列$a_1,a_2,a_3,\\cdots$各项的和为$\\dfrac 92$, 且$a_2=-2$, 若$|S_n-\\dfrac 92|<10^{-4}$, 则$n$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269724,7 +270189,8 @@ "content": "若同一平面上不共线的四个点$P,Q,R,S$满足: $mn\\overrightarrow{RP}=n(1-3m)\\overrightarrow{QP}+m(n-1)\\overrightarrow{SP}$($m>0$、$n>0$), 则当$\\triangle PRS$的面积是$\\triangle PQR$的面积的$\\dfrac 13$倍时, $\\dfrac1{m+n}$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269745,7 +270211,8 @@ "content": "设$x\\in \\mathbf{R}$, 则``$x>3$''是``$x^2>9$'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分条件又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269769,7 +270236,8 @@ "content": "某班有学生$40$人, 将这$40$人编上$1$到$40$的号码, 用系统抽样的方法抽取一个容量为$4$的样本. 已知编号为$3$、$23$、$33$的学生在样本中, 则另一学生在样本中的编号为\\bracket{20}.\n\\fourch{$12$}{$13$}{$14$}{$15$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269790,7 +270258,8 @@ "content": "已知函数$f(x)=\\sin(\\omega x+\\dfrac{\\pi}6)+\\dfrac 12$($\\omega >0)$在区间$(0,\\dfrac{\\pi}2)$上有且仅有两个零点, 则实数$\\omega$的取值范围为\\bracket{20}.\n\\fourch{$(2, \\dfrac{14}{3}]$}{$[2,\\dfrac{14}{3})$}{$[\\dfrac{10}{3}, 4)$}{$(\\dfrac{10}{3}, 6]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269811,7 +270280,8 @@ "content": "如果数列$u_1, u_2, \\cdots, u_{10}$同时满足以下四个条件: \\textcircled{1} $u_i\\in \\mathbf{Z}$($i=1, 2, \\cdots, 10$); \\textcircled{2} 点$(u_5, 2^{u_2+u_8})$在函数$y=4^x$的图像上; \\textcircled{3} 向量$\\overrightarrow a=(1, u_1)$与$\\overrightarrow b=(3, u_{10})$互相平行;\n\\textcircled{4} $u_{i+1}-u_i$与$\\dfrac2 {u_{i+1}-u_i}$的等差中项为$\\dfrac 32$($i=1, 2, \\cdots, 9$). 那么, 这样的数列$u_1, u_2, \\cdots, u_{10}$的个数为\\bracket{20}.\n\\fourch{$78$}{$80$}{$82$}{$90$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -269832,7 +270302,8 @@ "content": "在三棱锥$P-ABC$中, $PA=PB=PC=AC=2\\sqrt 2$, $BA=BC=2$, $O$是线段$AC$的中点, $M$是线段$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$O$} coordinate (O);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below right] {$M$} coordinate (M);\n\\draw (A) -- (P) -- (C) -- (B) -- cycle (P) -- (B) (P) -- (M);\n\\draw [dashed] (A) -- (C) (P) -- (O) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PO\\perp$平面$ABC$;\\\\\n(2) 求直线$PM$与平面$PBO$所成的角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269853,7 +270324,8 @@ "content": "将关于$x$的函数$y=\\dfrac{m(x+2)^2}x$($m\\in \\mathbf{R}$)的图像向右平移$2$个单位后得到的函数图像记为$C$, 并设$C$所对应的函数为$f(x)$.\\\\\n(1) 当$m>0$时, 试直接写出函数$f(x)$的单调递减区间;\\\\\n(2) 设$f(4)=8$, 若函数$g(x)=x^2-2ax+5$($a>1$)对于任意$t_1\\in [0, 1]$, 总存在$t_2\\in [0, 1]$, 使得$g(t_2)=f (t_1)$成立, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269874,7 +270346,8 @@ "content": "某工厂制作如图所示的一种标识, 在半径为$R$的圆内作一个关于圆心对称的 ``H''型图形, ``H''型图形由两竖一横三个等宽的矩形组成, 两个竖直的矩形全等且它们的长边是横向矩形长边的$\\dfrac{3}{2}$倍, 设$O$为圆心, $\\angle AOB=2\\alpha$, 记``H''型图形的面积为$S$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (2) node [left] {$O$} coordinate (O);\n\\filldraw (0,0) circle (0.03);\n\\draw (45:2) node [above right] {$A$} coordinate (A);\n\\draw (-45:2) node [below right] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [right] {$M$} coordinate (M);\n\\draw [dashed] (O) -- (M) (O) -- (A) (O) -- (B);\n\\draw (O) pic [\"$\\alpha$\",draw,angle eccentricity = 1.5, scale = 0.6] {angle = M--O--A};\n\\draw (A) -- (B);\n\\draw (A) ++ ({-sqrt(2)/3},0) node [left] {$D$} coordinate (D);\n\\draw (B) ++ ({-sqrt(2)/3},0) node [left] {$C$} coordinate (C);\n\\draw (A) -- (D) -- (C) -- (B);\n\\draw (M) pic [draw,scale = 0.2] {right angle = O--M--B};\n\\draw ({sqrt(2)*2/3},{sqrt(2)/6}) -- ({-sqrt(2)*2/3},{sqrt(2)/6});\n\\draw ({sqrt(2)*2/3},{-sqrt(2)/6}) -- ({-sqrt(2)*2/3},{-sqrt(2)/6});\n\\draw (135:2) --++ ({sqrt(2)/3},0) --++ (0,{-2*sqrt(2)}) --++ ({-sqrt(2)/3},0) --cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 将$AB,AD$用$R,\\alpha$表示, 并将$S$表示成$\\alpha$的函数;\\\\\n(2) 为了突出``H''型图形, 设计时应使$S$尽可能大, 则当$\\alpha$为何值时, $S$最大? 并求出$S$的最大值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269895,7 +270368,8 @@ "content": "已知椭圆$C$的方程为$\\dfrac{x^2}2+y^2=1$.、、\n(1) 设$M(x_M,y_M)$是椭圆$C$上的点, 证明: 直线$\\dfrac{x_Mx}2+y_My=1$与椭圆$C$有且只有一个公共点;\\\\\n(2) 过点$N(1,\\sqrt 2)$作两条与椭圆只有一个公共点的直线, 公共点分别记为$A$、$B$, 点$N$在直线$AB$上的射影为点$Q$, 求点$Q$的坐标;\\\\\n(3) 互相垂直的两条直线$l_1$与$l_2$相交于点$P$, 且$l_1$、$l_2$都与椭圆$C$只有一个公共点, 求证点$P$落在$x^2+y^2=3$上.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269916,7 +270390,8 @@ "content": "若数列$\\{a_n\\}$满足``对任意正整数$i$, $j$, $i\\ne j$, 都存在正整数$k$, 使得$a_k=a_i\\cdot a_j$'', 则称数列$\\{a_n\\}$具有``性质$P$''.\\\\\n(1) 判断各项均等于$a$的常数列是否具有``性质$P$'', 并说明理由;\\\\\n(2) 若公比为$2$的无穷等比数列$\\{a_n\\}$具有``性质$P$'', 求首项$a_1$的值;\\\\\n(3) 若首项$a_1=2$的无穷等差数列$\\{a_n\\}$具有``性质$P$'', 求公差$d$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -269937,7 +270412,8 @@ "content": "设集合$A=\\{1,2,3,4\\}$, 集合$B=\\{1,3,5,7\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269963,7 +270439,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -269986,7 +270463,8 @@ "content": "函数$y=\\dfrac{\\cos ^2x+1}2$的最小正周期为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270010,7 +270488,8 @@ "content": "设$\\mathrm{i}$是虚数单位, 复数$z$满足$(1+2\\mathrm{i})z=|4+3\\mathrm{i}|$, 则$z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270031,7 +270510,8 @@ "content": "若$\\{a_n\\}$是无穷等比数列, 首项$a_1=1$, 公比$q=\\dfrac 13$, 则$\\{a_n\\}$各项的和$S=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270052,7 +270532,8 @@ "content": "在$4$名男生, $4$名女生中随机选出$3$名学生参加某次活动, 则选出的学生中至多$1$名女生的概率为\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270073,7 +270554,8 @@ "content": "实数$x,y$满足约束条件$\\begin{cases}x+2y\\le 4, \\\\ 2x+y\\le 3, \\\\ x\\ge 0, \\\\ y\\ge 0, \\end{cases}$ 目标函数$z=x+y$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270096,7 +270578,8 @@ "content": "已知曲线$C_1$的参数方程为$\\begin{cases}\nx=\\sqrt 3t-1, \\\\ y=t+\\sqrt 3, \\end{cases}$($t$是参数) 曲线$C_2$的参数方程为$\\begin{cases} x=-2+\\sqrt 5\\cos \\theta, \\\\ y=\\sqrt 5\\sin \\theta, \\end{cases}$($\\theta$是参数) 则$C_1$和$C_2$的两个交点之间的距离为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270117,7 +270600,8 @@ "content": "数列$\\{a_n\\}$满足$a_1=1$, 且$a_n+a_{n+1}=3n+2$对任意$n\\in \\mathbf{N}^*$均成立, 则$a_{2022}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270138,7 +270622,8 @@ "content": "设$n\\in \\mathbf{N}^*$, 若$(2+\\sqrt x)^n$的二项展开式中, 有理项的系数之和为$365$, 则$n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270159,7 +270644,8 @@ "content": "设$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$是同一平面上的三个两两不同的单位向量. 若$(\\overrightarrow a\\cdot \\overrightarrow b):(\\overrightarrow b\\cdot \\overrightarrow c):(\\overrightarrow c\\cdot \\overrightarrow a)=1:1:3$, 则$\\overrightarrow a\\cdot\\overrightarrow b$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270180,7 +270666,8 @@ "content": "设$n\\in \\mathbf{N}^*$, 圆$C_n:x^2+y^2=R_n^2$($R_n>0$)与$y$轴正半轴的交点为$M$, 与曲线$y=\\sqrt x$的交点为$N(x_n,y_n)$, 直线$MN$与$x$轴的交点为$A(a_n,0)$.若数列$\\{x_n\\}$满足: $x_{n+1}=4x_n+3$, $x_1=3$. 则常数$p=$\\blank{50}使数列$\\{a_{n+1}-p\\cdot a_n\\}$成等比数列.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270201,7 +270688,8 @@ "content": "不等式$\\dfrac{x-1}{x-2}\\le 0$的解集为\\bracket{20}.\n\\fourch{$[1,2]$}{$[1,2)$}{$(-\\infty,1]\\cup [2,+\\infty)$}{$(-\\infty,1)\\cup (2,+\\infty)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270222,7 +270710,8 @@ "content": "设$z$是复数, 则``$z$是虚数''是``$z^3$是虚数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270243,7 +270732,8 @@ "content": "设$F_1,F_2$是椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的两焦点, $A$与$B$分别是该椭圆的右顶点与上顶点. $P$是该椭圆上的一个动点, $O$是坐标原点. 记$s=\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_2P}-2\\overrightarrow{OP}^2$. 在动点$P$在第一象限内从$A$沿椭圆向左上方运动到$B$的过程中, $s$的大小的变化情况为\\bracket{20}.\n\\fourch{逐渐变大}{逐渐变小}{先变大后变小}{先变小后变大}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270264,7 +270754,8 @@ "content": "设$\\{a_n\\}$是$2022$项的实数数列, $\\{a_n\\}$中的每一项都不为零. $\\{a_n\\}$中任意连续$11$项$a_n,a_{n+1},\\cdots ,a_{n+10}$的乘积是定值($n=1,2,3,\\cdots ,2012$). 命题\\\\\n\\textcircled{1} 不存在满足条件的数列, 使得其中恰有$365$个$1$;\\\\\n\\textcircled{2} 存在满足条件的数列, 使得其中恰有$550$个$1$;\\\\\n的真假情况为\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}都是真命题}{\\textcircled{1}是真命题, \\textcircled{2} 是假命题}{\\textcircled{2}是真命题, \\textcircled{1} 是假命题}{\\textcircled{1}和\\textcircled{2}都是假命题}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270285,7 +270776,8 @@ "content": "如图, 线段$OA$和$OB$是以$P$为顶点的圆锥的底面上的两条相互垂直的半径. 点$M$是母线$BP$的中点. 已知$OA=OM=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below right] {$O$} coordinate (O);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw (B) arc (0:-180:2 and 0.5);\n\\draw [dashed] (B) arc (0:180:2 and 0.5);\n\\draw [dashed] (O) --++ (0,{2*sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [above right] {$M$} coordinate (M);\n\\draw ({2*cos(-120)},{0.5*sin(-120)}) node [below left] {$A$} coordinate (A);\n\\draw (P) -- (-2,0) (P) -- (A) (P) -- (B);\n\\draw [dashed] (A) -- (O) -- (B) (O) -- (M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆锥的体积;\\\\\n(2) 求异面直线$OM$与$AP$所成的角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270306,7 +270798,8 @@ "content": "已知三角形$ABC$中, 三个内角$ABC$的对应边分别为$a$、$b$、$c$, 且$a=5$, $b=7$.\\\\\n(1) 若$B=\\dfrac{\\pi}3$, 求$c$;\\\\\n(2) 设点$M$是边$AB$的中点, 若$CM=3$, 求三角形$ABC$的面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270327,7 +270820,8 @@ "content": "某地出现了虫害. 农业科学家引入了``虫害指数''数列$\\{I_n\\}$, $\\{I_n\\}$表示第$n$周的虫害的严重程度, 虫害指数越大, 严重程度越高. 为了治理虫害, 需要环境整治、杀灭害虫, 然而由于人力资源有限, 每周只能采取以下两个策略之一:\\\\\n策略A: 环境整治, ``虫害指数''数列满足$I_{n+1}=1.02I_n-0.20$;\\\\\n策略B: 杀灭害虫, ``虫害指数''数列满足$I_{n+1}=1.08I_n-0.46$;\\\\\n当某周``虫害指数''小于$1$时, 危机就在这周解除.\\\\\n(1) 设第一周的虫害指数$I_1\\in [1,8]$, 用哪一个策略将使第二周的虫害的严重程度更小?\\\\\n(2) 设第一周的虫害指数$I_1=3$, 如果每周都采用最优的策略, 虫害的危机最快在第几周解除?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270348,7 +270842,8 @@ "content": "已知双曲线$H:x^2-\\dfrac{y^2}{b^2}=1$($b>0$), 经过点$D(2,0)$的直线$l$与该双曲线交于$M$、$N$两点.\\\\\n(1) 若$l$与$x$轴垂直, 且$|MN|=6$, 求$b$的值;\\\\\n(2) 若$b=\\sqrt 2$, 且$M$、$N$的横坐标之和为$-4$, 证明: $\\angle MON=90^\\circ$;\\\\\n(3) 设直线$l$与$y$轴交于点$E$, $\\overrightarrow{EM}=\\lambda \\cdot \\overrightarrow{MD}$, $\\overrightarrow{EN}=\\mu \\cdot \\overrightarrow{ND}$, 求证: $\\lambda +\\mu$为定值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270369,7 +270864,8 @@ "content": "$f(x)=2^{x+m}+m x+1$, 其中$m$是实常数.\\\\\n(1) 若$f(\\dfrac 1m)>18$, 求$m$的取值范围;\\\\\n(2) 若$m>0$, 求证: 函数$f(x)$的零点有且仅有一个;\\\\\n(3) 若$m>0$, 设函数$y=f(x)$的反函数为$y=f^{-1}(x)$, 若$a_1,a_2,a_3,a_4$是公差$d>0$的等差数列且均在函数$f(x)$的值域中, 求证: $f^{-1}(a_1)+f^{-1}(a_4)0$)的左右焦点分别为$F_1$、$F_2$, 抛物线$y^2=2x$的焦点为$F$, 若$\\overrightarrow{F_1F}=3\\overrightarrow{FF_2}$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270552,7 +271055,8 @@ "content": "甲、乙、丙、丁$4$名同学参加志愿者服务, 分别到三个路口疏导交通, 每个路口有$1$名或$2$名志愿者. 若每种分配方案的可能性相等, 则甲、乙两人在同一路口的概率为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270573,7 +271077,8 @@ "content": "将函数$f(x)=\\sin x$的图像向右平移$\\varphi$($\\varphi >0$)个单位后得到函数$y=g(x)$的图像. 若对满足$|f(x_1)-g(x_2)|=2$的任意$x_1$、$x_2$, $|x_1-x_2|$的最小值是$\\dfrac{\\pi }3$, 则$\\varphi$的最小值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270594,7 +271099,8 @@ "content": "如图, 已知$AB$是边长为$1$的正六边形的一条边,\n$P$在正六边形内(含边界), 则$\\overrightarrow{AP}\\cdot \\overrightarrow{BP}$的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A) --++ (1,0) node [below] {$B$} coordinate (B) --++ (60:1) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1);\n\\filldraw (0.7,0.8) circle (0.03) node [right] {$P$} coordinate (P);\n\\draw [->] (A) -- (P);\n\\draw [->] (B) -- (P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270615,7 +271121,8 @@ "content": "已知曲线$C:y=\\dfrac 2x$($1\\le x\\le 2$), 若对于曲线$C$上的任意一点$P(x,y)$, 都有$(x+y+c_1)(x+y+c_2)\\le 0$, 则$|c_1-c_2|$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270636,7 +271143,8 @@ "content": "在数列$\\{a_n\\}$中, $a_1=3$, $a_{n+1}=1+a_1\\cdot a_2\\cdot a_3\\cdots a_n$, 记$T_n$为数列$\\{\\dfrac 1{a_n}\\}$的前$n$项和, 则$\\displaystyle\\lim_{n\\to\\infty} T_n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270659,7 +271167,8 @@ "content": "设$\\overrightarrow a$、$\\overrightarrow b$分别是两条异面直线$l_1$、$l_2$的方向向量, 向量$\\overrightarrow a$、$\\overrightarrow b$夹角的取值范围为$A$, $l_1$、$l_2$所成角的取值范围为$B$, 则``$\\alpha \\in A$''是``$\\alpha \\in B$''的\\bracket{20}.\n\\twoch{充要条件}{充分不必要条件}{必要不充分条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270680,7 +271189,8 @@ "content": "若等比数列$\\{a_n\\}$的公比为$q$, 则关于$x$、$y$的二元一次方程组$\\begin{cases} a_1x+a_5y=2, \\\\ a_2x+a_6y=1 \\end{cases}$的解的情况下列说法正确的是\\bracket{20}.\n\\twoch{对任意$q\\in \\mathbf{R}$($q\\ne 0$), 方程组有唯一解}{对任意$q\\in \\mathbf{R}$($q\\ne 0$), 方程组都无解}{当且仅当$q=\\dfrac 12$时, 方程组有无穷多解}{当且仅当$q=\\dfrac 12$时, 方程组无解}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270701,7 +271211,8 @@ "content": "已知实数$a$、$b$满足$(a+2)(b+1)=8$, 有结论:\\\\\n\\textcircled{1} 当$a>0$, $b>0$时, $ab$存在最大值;\\\\\n\\textcircled{2} 当$a<0$, $b<0$时, $ab$存在最小值. 正确的判断是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270722,7 +271233,8 @@ "content": "已知函数$f(x)=\\dfrac 1x+|2x-a|$. 若存在相异的实数$x_1,x_2\\in (-\\infty ,0)$, 使得$f(x_1)=f(x_2)$成立, 则实数$a$的取值范围为\\bracket{20}.\n\\fourch{$(-\\infty,-\\dfrac{\\sqrt{2}}2)$}{$(-\\infty,-\\sqrt{2})$}{$(\\dfrac{\\sqrt{2}}2,+\\infty)$}{$(\\sqrt 2,+\\infty)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -270743,7 +271255,8 @@ "content": "已知在直三棱柱$ABC-A_1B_1C_1$中, $\\angle BAC=90^\\circ$, $AB=BB_1=1$, 直线$B_1C$与平面$ABC$成$30^\\circ$的角.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw ($(B)!{1/3}!(C)$) ++ (0,0,{sqrt(2)/sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw (B) -- (A) -- (C) --++ (0,1) node [right] {$C_1$} coordinate (C1) --++ ({-sqrt(3)},0) node [left] {$B_1$} coordinate (B1) -- cycle;\n\\draw (A) --++ (0,1) node [below right] {$A_1$} coordinate (A1) -- (B1) (A1) -- (C1);\n\\draw [dashed] (B1) -- (C) (B) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1)\t求三棱锥$C_1-AB_1C$的体积;\\\\\n(2) 求二面角$B-B_1C-A$的余弦值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270764,7 +271277,8 @@ "content": "已知复数$z$满足$|z|=\\sqrt 2$, $z^2$的虚部为$2$.\\\\\n(1) 求复数$z$;\\\\\n(2) 设复数$z,z^2,z-z^2$在复平面上对应的点分别为$A,B,C$, 求: $(\\overrightarrow{OA}+\\overrightarrow{OB})\\cdot \\overrightarrow{OC}$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270785,7 +271299,8 @@ "content": "某公园内有一块以$O$为圆心半径为$20$米的圆形区域. 为丰富市民的业余文化生活, 现提出如下设计方案: 如图, 在圆形区域内搭建露天舞台, 舞台为扇形$OAB$区域, 其中两个端点$A$, $B$分别在圆周上; 观众席为等腰梯形$ABQP$内且在圆$O$外的区域, 其中$AP=AB=BQ$, $\\angle PAB=\\angle QBA=\\dfrac{2\\pi}3$, 且$AB,PQ$在点$O$的同侧. 为保证视听效果, 要求观众席内每一个观众到舞台中心$O$处的距离都不超过$60$米(即要求$PO\\le 60$). 设$\\angle OAB=\\alpha$, $\\alpha \\in (0, \\dfrac{\\pi}3)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (1) node [above] {$O$} coordinate (O);\n\\filldraw (0,0) circle (0.03);\n\\draw (O) --++ (-30:1) node [right] {$B$} coordinate (B);\n\\draw (O) --++ (-150:1) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B);\n\\draw (A) --++ (-120:{sqrt(3)}) node [left] {$P$} coordinate (P);\n\\draw (B) --++ (-60:{sqrt(3)}) node [right] {$Q$} coordinate (Q);\n\\draw (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\alpha =\\dfrac{\\pi}6$时, 求舞台表演区域的面积;\\\\\n(2) 对于任意$\\alpha$, 上述设计方案是否均能符合要求?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270806,7 +271321,8 @@ "content": "双曲线$C:x^2-\\dfrac{y^2}{b^2}=1$($b>0$)的左顶点为$A$, 右焦点为$F$, 点$B$是双曲线$C$上一点.\\\\\n(1) 当$b=2$时, 求双曲线两条渐近线的夹角;\\\\\n(2) 若直线$BF$的倾斜角为$\\dfrac{\\pi}4$, 与双曲线$C$的另一交点为$D$, 且$|BD|=8$, 求$b$的值;\\\\\n(3) 若$\\overrightarrow{AF}\\cdot \\overrightarrow{BF}=0$, 且$|\\overrightarrow{AF}|=|\\overrightarrow{BF}|$, 点$E$是双曲线$C$上位于第一象限的动点, 求证: $\\angle EFA=2\\angle EAF$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270827,7 +271343,8 @@ "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$\\dfrac 12\\le \\dfrac{a_{n+1}}{a_n}\\le 2$($n\\in \\mathbf{N}^*$), 则称$\\{a_n\\}$是``紧密数列''.\\\\\n(1) 已知数列$\\{a_n\\}$是``紧密数列'', 其前$5$项依次为$1,\\dfrac 32,\\dfrac 94,x,\\dfrac{81}{16}$, 求$x$的取值范围;\\\\\n(2) 若数列$\\{a_n\\}$的前$n$项和为$S_n=\\dfrac 14(n^2+3n)$($n\\in \\mathbf{N}^*$), 判断$\\{a_n\\}$是否是``紧密数列'', 并说明理由;\\\\\n(3) 设$\\{a_n\\}$是公比为$q$的等比数列, 若$\\{a_n\\}$与$\\{S_n\\}$都是``紧密数列'', 求$q$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -270850,7 +271367,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270873,7 +271391,8 @@ "content": "设集合$A=\\{x|y=\\sqrt {x-1}, \\ x\\in \\mathbf{R}\\}$, $B=\\{y|y=\\sqrt {1-x^2}, \\ x\\in \\mathbf{R}\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270896,7 +271415,8 @@ "content": "若函数$f(x)=1+\\dfrac 1x$($x>0$)的反函数为$f^{-1}(x)$, 则不等式$f^{-1}(x)>2$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270919,7 +271439,8 @@ "content": "已知常数$a>0$, 双曲线$4x^2-y^2=1$的一条渐近线与直线$ax+y+1=0$垂直, 则\t$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270942,7 +271463,8 @@ "content": "以抛物线$y^2=4x$的焦点为圆心, 且与该抛物线的准线相切的圆的标准方程\t为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270963,7 +271485,8 @@ "content": "若$\\log_ab=-2$, 则$a^2+b$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -270987,7 +271510,8 @@ "K0623002B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\dfrac 32$", @@ -271015,7 +271539,8 @@ "content": "若$\\alpha \\in (\\dfrac{\\pi}4,\\dfrac{\\pi}2)$, $\\sin 2\\alpha =\\dfrac 1{16}$, 则$\\cos \\alpha -\\sin \\alpha =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271036,7 +271561,8 @@ "content": "在平面直角坐标系$xOy$中, 将点$A(2,1)$绕原点$O$逆时针旋转$\\dfrac{\\pi}4$到点$B$. 设直线$OB$的倾\t斜角为$\\alpha$, 则$\\cos \\alpha =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271059,7 +271585,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+1}+a_n=4n+2$. 若$\\{a_n\\}$单调递增, 则$a_1$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271080,7 +271607,8 @@ "content": "已知复数$z_1,z_2$满足$|z_1|\\le 1$, $-1\\le \\text{Re}z_2\\le 1$, $-1\\le \\mathrm{Im}z_2\\le 1$. 设$z=z_1+z_2$, 则$z$在复平面\t上对应的点组成的图形的面积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271101,7 +271629,8 @@ "content": "已知椭圆$\\Gamma:\\dfrac{x^2}2+y^2=1$的左、右焦点分别为$F_1,F_2$, 点$A$为椭圆$\\Gamma$的上顶点, 坐标平面内的动点$P$满足$|AP|^2=2\\overrightarrow{PF_1}\\cdot \\overrightarrow{PF_2}$, 则$|\\overrightarrow{PF_1}+2\\overrightarrow{PF_2}|$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271122,7 +271651,8 @@ "content": "若$\\cos \\alpha >0$, $\\sin 2\\alpha <0$, 则角$\\alpha$的终边所在象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271143,7 +271673,8 @@ "content": "过抛物线$y^2=8x$的焦点作直线与抛物线相交于两点, 且这两点的横坐标之和$9$, 则满足\t条件的直线\\bracket{20}.\n\\fourch{有$1$条}{有$2$条}{有无穷多条}{不存在}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271164,7 +271695,8 @@ "content": "在$(\\sqrt x+\\dfrac 2{x^2})^n$($n\\in \\mathbf{N}^*$)的展开式中, 设含有$(\\sqrt x)^{n-r}(\\dfrac 2{x^2})^r$的项为第$r+1$($0\\le r\\le n$, $n\\in \\mathbf{N}$)项. 若第$3$项与第$5$项的系数之比为$3:56$, 则展开式中的常数项为\\bracket{20}.\n\\fourch{$180$}{$160$}{$120$}{$100$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271185,7 +271717,8 @@ "content": "已知等差数列$\\{a_n\\}$与等比数列$\\{b_n\\}$都不是常数列. 设集合$S=\\{k|a_k=b_k, \\ k\\in \\mathbf{N}^*\\}$, 则$S$\t元素个数的最大值为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271206,7 +271739,8 @@ "content": "如图, 在$\\mathbf{R}t\\triangle ABC$中, $AB\\perp AC$. $PA\\perp$平面$ABC$, $D$为$AC$的中点. 过点$D$作$DE\\perp$平面$ABC$, $DE=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (1,0,0) node [above right] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,1,0) node [above right] {$E$} coordinate (E);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw (B) -- (D) -- (C) -- cycle;\n\\draw (D) -- (E) -- (P) -- (B) -- (E);\n\\draw [dashed] (A) -- (B) (A) -- (D) (A) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 作出平面$PBE$与平面$ABC$的交线(保留作图痕迹);\\\\\n(2) 若$PA=AC=2$, $DE=1$, 四棱锥$B-ADEP$的体积为$\\dfrac{\\sqrt 3}3$, 求平面$PBE$与平面$ABC$所成锐二面角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271227,7 +271761,8 @@ "content": "设函数$f(x)=2\\sin (x+\\dfrac{\\pi }3)\\cos x$.\\\\\n(1) 求$f(x)$在$[0,\\dfrac{\\pi }2]$上的值域;\\\\\n(2) 记$\\triangle ABC$三个内角$A,B,C$的对边分别为$a,b,c$. 若$A$为锐角, $f(A)=\\dfrac{\\sqrt 3}2$, $b=2$, $c=3$, 求$\\cos (A-B)$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271248,7 +271783,8 @@ "content": "某企业参加$A$项目生产的工人为$1000$人, 平均每人每年创造利润$10$万元. 根据现实的需要, 从$A$项目中调出$x$人参与$B$项目的售后服务工作, 每人每年可以创造利润$10(a-\\dfrac{3x}{500})$万元(其中常数$a>0$), $A$项目余下的工人每人每年创造利润需要提高$0.2x\\%$.\\\\\n(1) 若要保证$A$项目余下的工人创造的年总利润不低于原来$1000$名工人创造的年总利润, 则最多调出多少人参加$B$项目从事售后服务工作?\\\\\n(2) 当从$A$项目调出的人数不超过总人数的$40\\%$时, 总能使得$A$项目中留岗工人创造的年总利润不低于调出的工人所创造的年总利润, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271269,7 +271805,8 @@ "content": "已知常数$k>0$, 在平面直角坐标系$xOy$中, $A,B$两点分别在第一象限、第四象限, 直线$OA,OB$的一个方向向量分别为$\\overrightarrow{d_1}=(1,k)$, $\\overrightarrow{d_2}=(1,-k)$, 动点$P$在$\\angle AOB$内, 过点$P$分别作$PM\\perp OA$于$M$, $PN\\perp OB$于$N$, $M,N$分别在射线$OA,OB$上.\\\\\n(1) 若$k=1$, $P(\\dfrac 32,\\dfrac 12)$, 求$|OM|$的值;\\\\\n(2) 若$P(2,1)$, $\\triangle OMP$的面积为$\\dfrac 65$, 求$k$的值;\\\\\n(3) 设线段$MN$的中点为$T$. 当$\\triangle MON$的面积为$\\dfrac 1k$时, 求证: $|OT|\\ge \\dfrac 1k$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271290,7 +271827,8 @@ "content": "若无穷数列$\\{a_n\\}$满足: \\textcircled{1} 存在互异的$p,q\\in \\mathbf{N}^*$使得$a_p=a_q$; \\textcircled{2} 对任意$n\\in \\mathbf{N}^*$, 当$n\\ne p$且$n\\ne q$时, 恒有$a_n>a_p$, 则称$\\{a_n\\}$具有性质$P$.\\\\\n(1) 已知常数$k\\in \\mathbf{N}^*$, 无穷数列$\\{a_n\\}$的通项公式为$a_n=n+\\dfrac kn$. 若$\\{a_n\\}$具有性质$P$, 求$k$的值;\\\\\n(2) 已知常数$\\lambda \\in \\mathbf{R}$, 无穷数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases} -2n+11, & 1\\le n\\le 5, \\\\ 2^{n-5}+\\lambda, & n\\ge 6. \\end{cases}$若$\\{a_n\\}$具有性质$P$, 求$\\lambda$的值以及$\\{a_n\\}$的前$n$项和$S_n$;\\\\\n(3) 已知常数$t\\in \\mathbf{Z}$, 无穷数列$\\{a_n\\}$的通项公式为$a_n=(tn+3)(\\dfrac 9{10})^n$. 问: 是否存在$t$, 使得$\\{a_n\\}$具有性质$P$? 若存在, 求出所有$t$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271311,7 +271849,8 @@ "content": "$(x^2+\\dfrac 1x)^8$的展开式中$x^4$项的系数是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271334,7 +271873,8 @@ "content": "设变量$x, y$满足约束条件$\\begin{cases} 0\\le x\\le 1, \\\\ y\\le 2, \\\\ x\\le y.\\end{cases}$则$z=x+y$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271355,7 +271895,8 @@ "content": "已知奇函数$y=f(x)$的周期为$2$, 且当$x\\in (0,1]$时, $f(x)=\\log_2 x$. 则$f(7.5)$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271378,7 +271919,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271399,7 +271941,8 @@ "content": "投掷两颗六个面上分别刻有$1$到$6$的点数的均匀的骰子, 得到其向上的点数分别为$m$和$n$, 则复数$\\dfrac{m+n\\mathrm{i}}{n+m\\mathrm{i}}$为虚数的概率为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271420,7 +271963,8 @@ "content": "某茶农打算在自己的茶园建造一个容积为$500$立方米的长方体无盖蓄水池, 要求池底面的长和宽之和为$20$米. 若每平方米的池底面造价是池侧壁的两倍, 则为了使蓄水池的造价最低, 蓄水池的高应该为\\blank{50}米.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271441,7 +271985,8 @@ "content": "如图, 在直角梯形$ABCD$中, $AD\\parallel BC$, $\\angle DC=90^\\circ$, $AD=2$, $BC=1$, $P$为梯形的腰$DC$上的动点, 则\n$|\\overrightarrow{PA}+3\\overrightarrow{PB}|$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A) --++ (2,0) node [right] {$D$} coordinate (D) --++ (0,1.5) node [right] {$C$} coordinate (C) --++ (-1,0) node [left] {$B$} coordinate (B) -- cycle;\n\\draw (A) -- ($(C)!0.5!(D)$) node [right] {$P$} coordinate (P) (P) -- (B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271464,7 +272009,8 @@ "K0615004B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271485,7 +272031,8 @@ "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的两焦点分别为$F_1,F_2$, $P$为双曲线上一点, $PF_2\\perp x$轴, 且$|PF_2|$是$|PF_1|$与$|F_1F_2|$的等差中项, 则双曲线的渐近线方程为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271506,7 +272053,8 @@ "content": "若四边形$ABCD$是边长为$4$的菱形, $P$为其所在平面上的任意点, 则$|\\overrightarrow{PA}\\cdot \\overrightarrow{PC}-\\overrightarrow{PB}\\cdot \\overrightarrow{PD}|$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271527,7 +272075,8 @@ "content": "已知函数$f(x)=\\begin{cases}\n\\tan x, & x\\in (-\\dfrac{\\pi }2,\\dfrac{\\pi }3]\\cup (\\dfrac{2\\pi }3,\\dfrac{3\\pi }2), \\\\ -\\dfrac{6\\sqrt 3}{\\pi }x+3\\sqrt 3, & x\\in (\\dfrac{\\pi }3,\\dfrac{2\\pi }3]. \\end{cases}$若$f(x)$在区间$D$上的最大值存在, 记该最大值为$K\\{D\\}$, 则满足等式$K\\{[0,a)\\}=3\\cdot K\\{[a,2a]\\}$的实数$a$的取值集合是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271548,7 +272097,8 @@ "content": "已知数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)满足$a_{n+1}=|a_2-a_1|+|a_3-a_2|+\\cdots +|a_n-a_{n-1}|$($n\\ge 2$), 且$a_1=1$, $a_2=a$($a>1$), 则$a_1+a_2+a_3+\\cdots +a_{24}=$\\blank{50}. (结果用含$a$的式子表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271569,7 +272119,8 @@ "content": "为客观了解上海市民家庭存书量, 上海市统计局社情民意调查中心通过电话调查系统开展专项调查, 成功访问了$2007$位市民. 在这项调查中, 总体、样本及样本的容量分别是\\bracket{20}.\n\\onech{总体是上海市民家庭总数量, 样本是$2007$位市民家庭的存书量, 样本的容量是$2007$}{总体是上海市民家庭的存书量, 样本是$2007$位市民家庭的存书量, 样本的容量是$2007$}{总体是上海市民家庭的存书量, 样本是$2007$位市民, 样本的容量是$2007$}{总体是上海市民家庭总数量, 样本是$2007$位市民, 样本的容量是$2007$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271590,7 +272141,8 @@ "content": "某县共有$300$个村, 现采取系统抽样方法, 抽取15个村作为样本, 调查农民的生活和生产状况, 将$300$个村编上$1$到$300$的号码, 求得间隔数$k=\\dfrac{300}{15}=20$, 即每$20$个村抽取一个村, 在$1$到$20$中随机抽取一个数字$7$, 则在$41$到$60$这$20$个数中应取得号码是\\bracket{20}.\n\\fourch{$45$}{$46$}{$47$}{$48$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271611,7 +272163,8 @@ "content": "已知抛物线的方程为$y^2=4x$, 过其焦点$F$的直线交抛物线于$M,N$两点, 交$y$轴于点$E$, 若$\\overrightarrow{EM}=\\lambda _1\\overrightarrow{MF},\\overrightarrow{EN}=\\lambda _2\\overrightarrow{NF}$, 则$\\lambda _1+\\lambda _2=$\\bracket{20}.\n\\fourch{$2$}{$-2$}{$1$}{$-1$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271632,7 +272185,8 @@ "content": "已知关于$x$的实系数方程: $x^2-4x+5=0$和$x^2+2mx+m=0$的四个不同的根, 若这四个根在复平面上对应的点共圆, 则$m$取值范围是\\bracket{20}.\n\\fourch{$\\{5\\}$}{$\\{-1\\}$}{$(0,1)$}{$(0,1)\\cup \\{-1\\}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -271653,7 +272207,8 @@ "content": "如图, 在直三棱柱$ABC-A_1BC_1$中, $AB\\perp BC$, $AB=BC=2$, $AA_1=2\\sqrt 3$, $M$是侧棱$C_1C$上一点, 设$MC=h$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B) -- (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,{2*sqrt(3)},0) node [above] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,{2*sqrt(3)},0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,{2*sqrt(3)},0) node [right] {$C_1$} coordinate (C1);\n\\draw (B) -- (B1) -- (A1) -- (C1) -- (C) (B1) -- (C1);\n\\draw [dashed] (A) -- (A1) (A) -- (B) (A) -- (C);\n\\draw ($(C)!0.5!(C1)$) node [right] {$M$} coordinate (M);\n\\draw (M) -- (B);\n\\draw [dashed] (A) -- (M);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$h=\\sqrt 3$, 求多面体$ABM-A_1B_1C_1$的体积;\\\\\n(2) 若异面直线$BM$与$A_1C_1$所成的角为$60^{^\\circ}$, 求$h$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271674,7 +272229,8 @@ "content": "已知函数$f(x)=3\\cos ^2\\omega x+\\sqrt 3\\sin \\omega x\\cos \\omega x$($\\omega >0$).\\\\\n(1) 当$f(x)$的最小正周期为$2\\pi$时, 求$\\omega$的值;\\\\\n(2) 当$\\omega =1$时, 设三角形$\\triangle ABC$的内角$A,B,C$对应的边分别为$a,b,c$, 已知$f(\\dfrac A2)=3$, 且$a=2\\sqrt 7$, $b=6$, 求$\\triangle ABC$的面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271695,7 +272251,8 @@ "content": "如图, $AB$两地相距$100$公里, 两地政府为提升城市的抗疫能力, 决定在$AB$之间选址$P$点建造储备仓库, 共享民生物资, 当点$P$在线段$AB$的中点$C$时, 建造费用为$2000$万元; 若点$P$在线段$AC$上(不含点$A$), 则建造费用与$PA$之间的距离成反比; 若点$P$在线段$CB$上(不含点$B$), 则建造费用与$PB$之间的距离成反比, 现假设$PA$之间的距离为$x$千米$(0=latex]\n\\filldraw (0,0) node [left] {$A$} coordinate (A) circle (0.03);\n\\filldraw (4,0) node [below] {$C$} coordinate (C) circle (0.03);\n\\filldraw (8,0) node [right] {$B$} coordinate (B) circle (0.03);\n\\draw (2,0) node [below] {$50$千米} (6,0) node [below] {$50$千米};\n\\filldraw (2.5,0) circle (0.03) node [above] {$P$};\n\\draw (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 若规划仓库使用的年数为$n$($n\\in \\mathbf{N}^*)$,$H(x)=f(x)+ng(x)$, 求$H(x)$的最小值, 并解释其何处取得最小值的实际意义.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271716,7 +272273,8 @@ "content": "已知椭圆$\\Gamma :\\dfrac{x^2}4+y^2=1$的左右顶点分别为$AB$, $P$是椭圆$\\Gamma$上异于$AB$的一点, 直线$l:x=4$, 直线$AP,BP$分别交直线$l$于两点$C,D$, 线段$CD$的中点为$E$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (4,2) -- (4,-2);\n\\draw (0,0) ellipse (2 and 1);\n\\draw (-2,0) node [below left] {$A$} coordinate (A) (2,0) node [below] {$B$} coordinate (B);\n\\draw (1.2,0.8) node [above] {$P$} coordinate (P);\n\\draw ($(A)!{6/3.2}!(P)$) node [right] {$C$} coordinate (C);\n\\draw ($(P)!{2.8/0.8}!(B)$) node [right] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(D)$) node [right] {$E$} coordinate (E);\n\\draw (1,0) node [below] {$N$} coordinate (N);\n\\draw (P) -- (N) -- (E) (A) -- (C) (P) -- (D);\n\\draw (B) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 设直线$AP,BP$的斜率分别为$k_{AP},k_{BP}$, 求$k_{AP}\\cdot k_{BP}$的值;\\\\\n(2) 设$\\triangle ABP, \\triangle ABC$的面积分别为$S_1S_2$, 如果$S_2=2S_1$, 求直线$AP$的方程;\\\\\n(3) 在$x$轴上是否存在定点$N(n,0)$, 使得当直线$NP,NE$的斜率$k_{NP},k_{NE}$存在时, $k_{NP}\\cdot k_{NE}$为定值? 若存在, 求出$k_{NP}\\cdot k_{NE}$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -271737,7 +272295,8 @@ "content": "对于有限集$S=\\{a_1,a_2,a_3,\\cdots ,a_{m-1},a_m\\}$($m\\in \\mathbf{N}^*$, $m\\ge 3$), 如果存在函数$f(x)$($f(x)=x$除外), 其图像在区间$D$($S\\subseteq D$)上是一段连续曲线, 且满足$f(S)=S$, 其中$f(S)=\\{f(x)|x \\in S\\}$, 那么称这个函数$f(x)$是$P$变换, 集合$S$是$P$集合, 数列$a_1,a_2,a_3,\\cdots ,a_{m-1},a_m$是$P$数列.\\\\\n例如, $S=\\{1,2,3\\}$是$P$集合, 此时函数$f(x)=4-x$是$P$变换, 数列$1,2,3$或$3,2,1$等都是$P$数列.\\\\\n(1) 判断数列$1,2,5,8,9$是否是$P$数列? 说明理由;\\\\\n(2) 若各项均为正数的递增数列$\\{a_n\\}$($1\\le n\\le 2021$, $n\\in \\mathbf{N}^*$)是$P$数列, 若$P$变换$f(x)=\\dfrac 9x$, 求$a_1\\cdot a_2\\cdot \\cdots \\cdot a_{2021}$的值;\\\\\n(3) 元素都是正数的有限集$S=\\{a_1,a_2,a_3,\\cdots ,a_{m-1},a_m\\}$($m\\in \\mathbf{N}^*$, $m\\ge 3$), 若$a_i0$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271959,7 +272526,8 @@ "content": "某商场举行购物抽奖促销活动, 规定每位顾客从装有编号为$0$、$1$、$2$、$3$的四个相同小球的抽奖箱中, 每次取出一球记下编号后放回, 连续取两次, 若取出的两个小球编号相加之和等于$6$, 则中一等奖, 等于$5$中二等奖, 等于$4$或$3$中三等奖. 则顾客抽奖中三等奖的概率为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -271982,7 +272550,8 @@ "content": "已知函数$f(x)=\\lg (\\sqrt {x^2+1}+ax)$的定义域为$\\mathbf{R}$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272005,7 +272574,8 @@ "content": "在$\\triangle ABC$中, $M$是$BC$的中点, $\\angle A=120^\\circ$, $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=-\\dfrac 12$, 则线段$AM$长的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272026,7 +272596,8 @@ "content": "若实数$x$、$y$满足$4^x+4^y=2^{x+1}+2^{y+1}$, 则$S=2^x+2^y$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272047,7 +272618,8 @@ "content": "命题``若$x=1$, 则$x^2-3x+2=0$''的逆否命题是\\bracket{20}.\n\\twoch{若$x\\ne 1$, 则$x^2-3x+2\\ne 0$}{若$x^2-3x+2=0$, 则$x=1$}{若$x^2-3x+2=0$, 则$x\\ne 1$}{若$x^2-3x+2\\ne 0$, 则$x\\ne 1$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272068,7 +272640,8 @@ "content": "某单位现有职工$52$人, 将所有职工编号, 用系统抽样的方法抽取一个容量为$4$的样本. 已知$6$号、$32$号、$45$号职工在样本中, 则另一个在样本中的职工编号为\\bracket{20}.\n\\fourch{$18$}{$19$}{$20$}{$21$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272091,7 +272664,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272114,7 +272688,8 @@ "K0216006B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272135,7 +272710,8 @@ "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为直角梯形, $\\angle BAD=90^\\circ$, $AD\\parallel BC$, $AB=2$, $AD=1$, $PA=BC=4$, $PA\\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,4,0) node [left] {$P$} coordinate (P);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [below] {$D$} coordinate (D);\n\\draw (4,0,2) node [right] {$C$} coordinate (C);\n\\draw (B) -- (C) -- (P) -- cycle;\n\\draw [dashed] (A) -- (B) (A) -- (D) (A) -- (P) (B) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BD$与$PC$所成角的大小;\\\\\n(2) 求二面角$A-PC-D$的余弦值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272156,7 +272732,8 @@ "content": "在$\\triangle ABC$中, 内角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 已知$a-b=2$, $c=4$, $\\sin A=2\\sin B$.\\\\\n(1) 求$\\triangle ABC$的面积$S$;\\\\\n(2) 求$\\sin (2A-B)$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272177,7 +272754,8 @@ "content": "某创新团队拟开发一种新产品, 根据市场调查估计能获得$200$万元到$1000$万元的收益. 现准备制定一个奖励方案: 奖金$y$(单位: 万元)随收益$x$(单位: 万元)的增加而增加, 且奖金不超过$9$万元, 同时奖金不超过收益的$20\\%$.\\\\\n(1) 若建立函数$y=f(x)$模型制定奖励方案, 试用数学语言表述该团队对奖励函数$f(x)$模型的基本要求, 并分析函数$y=\\dfrac x{150}+2$是否符合团队要求的奖励函数模型, 并说明原因;\\\\\n(2) 若该团队采用模型函数$f(x)=\\dfrac{10x-3a}{x+2}$作为奖励函数模型, 试求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272198,7 +272776,8 @@ "content": "已知椭圆$\\Gamma$: $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的焦距为$2\\sqrt 3$, 点$P(0, 2)$关于直线$y=-x$的对称点在椭圆$\\Gamma$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and 1);\n\\draw [domain = -1.5:0.2, name path = line] plot (\\x,{2*\\x+2});\n\\draw [name intersections = {of = elli and line, by = {C,D}}];\n\\draw (0,0) -- (C) node [above] {$C$} (0,0) -- (D) node [below left] {$D$};\n\\filldraw (1,0) circle (0.03) node [below] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 如图, 过点$P$的直线$l$与椭圆$\\Gamma$交于两个不同的点$C$、$D$(点$C$在点$D$的上方), 试求$\\triangle COD$面积的最大值;\\\\\n(3) 若直线$m$经过点$M(1, 0)$, 且与椭圆$\\Gamma$交于两个不同的点$A$、$B$, 是否存在直线$l_0$: $x=x_0$(其中$x_0>2$), 使得$A$、$B$到直线$l_0$的距离$d_A$、$d_B$满足$\\dfrac{d_A}{d_B}=\\dfrac{|MA|}{|MB|}$恒成立? 若存在, 求出$x_0$的值; 若不存在, 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272219,7 +272798,8 @@ "content": "给定数列$\\{a_n\\}$, 若满足$a_1=a$($a>0$且$a\\ne 1$), 对于任意的$n, m\\in \\mathbf{N}^*$, 都有$a_{n+m}=a_n\\cdot a_m$, 则称数列$\\{a_n\\}$为指数数列.\\\\\n(1) 已知数列$\\{a_n\\}$, $\\{b_n\\}$的通项公式分别为$a_n=3\\cdot 2^{n-1}$, $b_n=3^n$, 试判断$\\{a_n\\}$, $\\{b_n\\}$是不是指数数列(需说明理由);\\\\\n(2) 若数列$\\{a_n\\}$满足: $a_1=2$, $a_2=4$, $a_{n+2}=3a_{n+1}-2a_n$, 证明: $\\{a_n\\}$是指数数列;\\\\\n(3) 若数列$\\{a_n\\}$是指数数列, $a_1=\\dfrac{t+3}{t+4}$($t\\in \\mathbf{N}^*$), 证明: 数列$\\{a_n\\}$中任意三项都不能构成等差数列.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272240,7 +272820,8 @@ "content": "不等式$|1-x|>1$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272268,7 +272849,8 @@ "content": "若函数$f(x)=\\sqrt {8-ax-2x^2}$是偶函数, 则该函数的定义域是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272291,7 +272873,8 @@ "content": "若$\\sin \\alpha =\\dfrac 13$, 则$\\cos (\\alpha -\\dfrac{\\pi }2)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272317,7 +272900,8 @@ "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": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272340,7 +272924,8 @@ "content": "在等比数列$\\{a_n\\}$中, 公比$q=2$, 前$n$项和为$S_n$, 若$S_5=1$, 则$S_{10}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272363,7 +272948,8 @@ "content": "若$x$、$y$满足$\\begin{cases} x\\le 2, \\\\ x-y+1\\ge 0, \\\\ x+y-2\\ge 0, \\end{cases}$ 则$z=2x-y$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272388,7 +272974,8 @@ "content": "已知圆$C:(x-4)^2+(y-3)^2=4$和两点$A(-m,0)$, $B(m,0)$, $m>0$, 若圆$C$上至少存在一点$P$, 使得$\\angle APB=90^\\circ$, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272411,7 +272998,8 @@ "content": "$(1+\\dfrac 1{x^2})(1+x)^6$展开式中$x^2$项的系数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272434,7 +273022,8 @@ "content": "高三某位同学参加物理、化学、政治科目的等级考, 已知这位同学在物理、化学、政治科目考试中达$A^+$的概率分别为$\\dfrac 78$、$\\dfrac 34$、$\\dfrac 5{12}$, 这三门科目考试成绩的结果互不影响, 则这位考生至少得$2$个$A^+$的概率是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272455,7 +273044,8 @@ "content": "已知$f(x)$是定义在$[-2,2]$上的奇函数, 当$x\\in (0,2]$时, $f(x)=2^x-1$, 函数$g(x)=x^2-2x+m$, 如果对于任意的$x_1\\in [-2,2]$, 总存在$x_2\\in [-2,2]$, 使得$f(x_1)\\le g(x_2)$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272478,7 +273068,8 @@ "content": "已知曲线$C:y=-\\sqrt {9-x^2}$, 直线$l:y=2$, 若对于点$A(0,m)$, 存在$C$上的点$P$和$l$上的点$Q$, 使得$\\overrightarrow{AP}+\\overrightarrow{AQ}=\\overrightarrow 0$, 则$m$取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272503,7 +273094,8 @@ "content": "如图所示, $\\angle BAC=\\dfrac{2\\pi}3$, 圆$M$与$AB$、$AC$分别相切于点$D$、$E$, 点$P$是圆$M$及其内部任意一点, 且$\\overrightarrow{AP}=x\\overrightarrow{AD}+y\\overrightarrow{AE}$($x,y\\in \\mathbf{R}$), 则$x+y$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272524,7 +273116,8 @@ "content": "下列函数中, 周期为$\\pi$, 且在$[\\dfrac{\\pi }4,\\dfrac{\\pi }2]$上为减函数的是\\bracket{20}.\n\\fourch{$y=\\sin (2x+\\dfrac{\\pi }2)$}{$y=\\cos (2x+\\dfrac{\\pi }2)$}{$y=\\sin (x+\\dfrac{\\pi }2)$}{$y=\\cos (x+\\dfrac{\\pi }2)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272545,7 +273138,8 @@ "content": "设$\\alpha$、$\\beta$是两个不同的平面, $b$是直线且$b\\subset\\beta$, 则``$b\\perp \\alpha$''是``$\\alpha \\perp \\beta$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272566,7 +273160,8 @@ "content": "若已知极限$\\displaystyle\\lim_{n\\to \\infty} \\dfrac{\\sin n}n=0$, 则$\\displaystyle\\lim_{n\\to \\infty} \\dfrac{n-3\\sin n}{\\sin n-2n}$的值为\\bracket{20}.\n\\fourch{$-3$}{$-\\dfrac 32$}{$-1$}{$-\\dfrac 12$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272587,7 +273182,8 @@ "content": "若函数$f(x)=\\lg [\\sin (\\pi x)\\cdot \\sin (2\\pi x)\\cdot \\sin (3\\pi x)\\cdot \\sin (4\\pi x)]$的定义域与区间$[0,1]$的交集是$n$个两两不交的开区间的并集, 则$n$的值为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -272610,7 +273206,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272631,7 +273228,8 @@ "content": "已知椭圆$C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个顶点坐标为$A(2,0)$, 且长轴长是短轴长的两倍.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 过点$D(1,0)$且斜率存在的直线交椭圆于$G$、$H$, $G$关于$x$轴的对称点为$G'$, 求证: 直线$G'H$恒过定点$(4,0)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272652,7 +273250,8 @@ "content": "某高科技企业研制出一种型号为$A$的精密数控车床, $A$型车床为企业创造的价值逐年减少(以投产一年的年初到下一年的年初为$A$型车床所创造价值的第一年). 若第$1$年$A$型车床创造的价值是$250$万元, 且第$1$年至第$6$年, 每年$A$型车床创造的价值减少$30$万元; 从第$7$年开始, 每年$A$型车床创造的价值是上一年价值的$50\\%$. 现用$a_n$($n\\in \\mathbf{N}^*$)表示$A$型车床在第$n$年创造的价值.\\\\\n(1) 求数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)的通项公式$a_n$;\\\\\n(2) 记$S_n$为数列$\\{a_n\\}$的前$n$项和, $T_n=\\dfrac{S_n}n$. 企业经过成本核算, 若$T_n>100$万元, 则继续使用$A$型车床, 否则在第$n+1$年年初更换$A$型车床. 试问该企业须在第几年年初更换$A$型车床?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272673,7 +273272,8 @@ "content": "设函数$f(x)=|\\dfrac 2x-ax+5|$($a\\in \\mathbf{R}$).\\\\\n(1) 求函数的零点;\\\\\n(2) 当$a=3$时, 求证: $f(x)$在区间$(-\\infty ,-1)$上单调递减;\\\\\n(3) 若对任意的正实数$a$, 总存在$x_0\\in [1,2]$, 使得$f(x_0)\\ge m$, 求实数$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272694,7 +273294,8 @@ "content": "给定数列$\\{a_n\\}$, 若数列$\\{a_n\\}$中任意(不同)两项之和仍是该数列中的一项, 则称该数列是``封闭数列''.\\\\\n(1) 已知数列$\\{a_n\\}$的通项公式为$a_n=3^n$, 试判断$\\{a_n\\}$是否为封闭数列, 并说明理由;\\\\\n(2) 证明: 公差为$d$的等差数列$\\{a_n\\}$成为``封闭数列''的一个充要条件是: 存在整数$m\\ge -1$, 使$a_1=md$;\\\\\n(3) 已知数列$\\{a_n\\}$满足$a_{n+2}+a_n=2a_{n+1}$且$a_2-a_1=2$, 设$S_n$是该数列$\\{a_n\\}$的前$n$项和, 试问: 是否存在这样的``封闭数列'' $\\{a_n\\}$, 使得对任意$n\\in \\mathbf{N}^*$都有$S_n\\ne 0$, 且$\\dfrac 18<\\dfrac 1{S_1}+\\dfrac 1{S_2}+\\cdots +\\dfrac 1{S_n}<\\dfrac{11}{18}$, 若存在, 求数列$\\{a_n\\}$的首项$a_1$的所有取值, 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -272715,7 +273316,8 @@ "content": "设$a\\in \\mathbf{R}$. 已知集合$A=\\{a,a^2\\}$. 若$1\\in A$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272736,7 +273338,8 @@ "content": "已知复数$z$满足$z\\mathrm{i}=2+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$\\mathrm{Im}z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272759,7 +273362,8 @@ "content": "若函数$f(x)=2^x+1$的图像与$y=g(x)$的图像关于直线$y=x$对称, 则$g(3)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272783,7 +273387,8 @@ "content": "若$\\tan (\\alpha -\\dfrac{\\pi}4)=-3$, 则$\\tan (\\pi-\\alpha)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272806,7 +273411,8 @@ "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$3$, 高为$4$, 则异面直线$AA_1$与$BD_1$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\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--++ (0,4) 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) ++ (3,4) 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) --++ (0,4);\n\\draw [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272829,7 +273435,8 @@ "content": "在$(1-2x)^{6}$的二项展开式中, $x^{3}$项的系数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272856,7 +273463,8 @@ "content": "新冠病毒爆发初期, 全国支援武汉的活动中, 需要从$A$医院某科室的$7$名男医生(含一名主任医师)、 $5$名女医生(含一名主任医师)中分别选派$3$名男医生和 $2$名女医生, 要求至少有一名主任医师参加, 则不同的选派方案共有\\blank{50}种.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272879,7 +273487,8 @@ "content": "设$k\\in \\{-2,-1,\\dfrac 13,\\dfrac 23, 2\\}$, 若对任意$x\\in (-1,0)\\cup (0,1)$, 都成立$x^k>\\|x|$, 则$k$取值的集合是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272904,7 +273513,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272927,7 +273537,8 @@ "content": "已知定义在$[0,+\\infty)$上的函数$f(x)=\\begin{cases} 13-|x-1|, & 0\\le x<2, \\\\ f(x-2)-2, & x\\ge 2. \\end{cases}$设函数$y=f(x)$在$[2n-2,2n)$($n\\in \\mathbf{N}^*$)上的最大值记作$a_n$. 若用正整数$n$表示$a_n$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272948,7 +273559,8 @@ "content": "已知平面向量$\\overrightarrow a,\\overrightarrow b,\\overrightarrow c$, 对任意实数t, 都有$|\\overrightarrow c-t\\overrightarrow a|\\ge|\\overrightarrow c-\\overrightarrow a|$, $|\\overrightarrow c-t\\overrightarrow b|\\ge|\\overrightarrow c-\\overrightarrow b|$成立. 若$\\overrightarrow a\\cdot \\overrightarrow a=3$, $\\overrightarrow b\\cdot \\overrightarrow b=2$, $\\overrightarrow a\\cdot \\overrightarrow b=1$, 则$|\\overrightarrow c|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272969,7 +273581,8 @@ "content": "已知函数$f(x)=|x+\\dfrac 1x|$, 给出下列命题:\\\\\n\\textcircled{1} 存在实数$a<0$, 函数$y=f(x)+f(x-a)$是偶函数;\\\\\n\\textcircled{2} 存在实数$a>0$, 使得函数$y=f(x)+f(x-a)$关于直线$x=1$对称;\\\\\n\\textcircled{3} 对于任意实数$a$, 关于$x$的不等式$f(x)+f(x-a)\\le 8$总有解.\\\\\n其中的真命题是\\blank{50}. (写出所有真命题的序号)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -272990,7 +273603,8 @@ "content": "已知$a,b,l$是空间中的三条直线, 其中直线$a,b$在平面$\\alpha$上, 则``$l\\perp a$且$l\\perp b$ ''是``$l\\perp$平面$\\alpha$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273011,7 +273625,8 @@ "content": "为得到函数$y=\\cos x+\\sqrt 3\\sin x$的图像, 可以将$y=2\\cos x$的图像 \\bracket{20}个单位.\n\\fourch{向右平移$\\dfrac{\\pi}{6}$}{向左平移$\\dfrac{\\pi}{6}$}{向右平移$\\dfrac{\\pi}3$}{向左平移$\\dfrac{\\pi}3$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273032,7 +273647,8 @@ "content": "某企业欲做一个介绍企业发展史的铭牌, 铭牌的截面形状是如图所示的扇形环面(由扇形$OAD$挖去扇形$OBC$后构成). 已知$OA=10$, $OB=x$($0=latex,scale = 0.2]\n\\def\\t{10/15/pi*180}\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw [dashed] (0,0) --++ ({90-\\t}:5) node [below right] {$C$} coordinate (C) (0,0) --++ ({90+\\t}:5) node [below left] {$B$} coordinate (B);\n\\draw (B) -- ($(B)!-1!(O)$) node [left] {$A$} coordinate (A) (C) -- ($(C)!-1!(O)$) node [right] {$D$} coordinate (D);\n\\draw (D) arc ({90-\\t}:{90+\\t}:10) (C) arc ({90-\\t}:{90+\\t}:5);\n\\draw (O) pic [\"$\\theta$\", angle eccentricity = 1.5, scale = 0.5] {angle = C--O--B};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{10+x}{10+2x}$}{$\\dfrac{10+2x}{10+x}$}{$\\dfrac{10-x}{10+2x}$}{$\\dfrac{10-x}{10+x}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273053,7 +273669,8 @@ "content": "在坐标平面内, 点$A$的坐标为$A(5,0)$. 若对于正实数$k$, 总存在函数$f(x)=ax^2$($a>0$), 使$\\angle QOA=2\\angle POA$, 这里$P(1,f(1))$、$Q(k,f(k))$, 则$k$的取值范围是\\bracket{20}.\n\\fourch{$(2,+\\infty)$}{$(3,+\\infty)$}{$[4,+\\infty)$}{$[8,+\\infty)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273074,7 +273691,8 @@ "content": "如图, 在圆柱$OO_1$ 中, $AB$是圆柱的母线, $BC$是圆柱的底面$\\bigodot O$的直径, $D$是底面圆周上异于$B$、$C$的点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw ({-sqrt(5)/2},0) node [left] {$B$} coordinate (B) arc (180:360:{sqrt(5)/2} and {sqrt(5)/8}) node [right] {$C$} coordinate (C);\n\\draw [dashed] ({-sqrt(5)/2},0) arc (180:0:{sqrt(5)/2} and {sqrt(5)/8});\n\\draw (B) --++ (0,2) node [left] {$A$} coordinate (A) (C) --++ (0,2) -- (A);\n\\draw (A) arc (180:-180:{sqrt(5)/2} and {sqrt(5)/8});\n\\filldraw (0,0) circle (0.03) node [above] {$O$} coordinate (O);\n\\filldraw (O) ++ (0,2) circle (0.03) node [above] {$O_1$} coordinate (O1);\n\\draw ({sqrt(5)/2*cos(-120)},{sqrt(5)/8*sin(-120)}) node [below] {$D$} coordinate (D);\n\\draw [dashed] (D) -- (B) (D) -- (C) (B) -- (C) -- (A) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CD\\perp AD$.\\\\\n(2) 若$BD=1$, $CD=2$, $AC=3$, 求圆柱$OO_1$的侧面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273095,7 +273713,8 @@ "content": "已知函数$f(x)=\\sin \\dfrac x2\\cos \\dfrac x2+\\cos ^2\\dfrac x2-\\dfrac 12$.\\\\\n(1) 求函数$y=f(x)$在区间$[0,\\pi]$上的值域;\\\\\n(2) 设$\\omega >0$. 若关于$x$的方程$4f(\\omega x)-\\sqrt 2=0$在区间$[0,\\pi]$上恰有两个不同的解, 求$\\omega$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273116,7 +273735,8 @@ "content": "大数据时代对于数据分析能力的要求越来越高, 数据拟合是一种把现有数据通过数学方法来代入某种算式的表示方式. 比如$A_i(a_i,b_i)$($i=1,2,3,\\cdots,n$)是平面直角坐标系上的一系列点, 其中$n$是不小于$2$的正整数, 用函数$y=f(x)$来拟合该组数据, 尽可能使得函数图像与点列$A_i(a_i,b_i)$比较接近. 其中一种衡量接近程度的指标是函数的拟合误差, 拟合误差越小越好, 定义函数$y=f(x)$的拟合误差为:\n$$\\Delta (f(x))=\\dfrac 1n[(f(a_1)-b_1)^2+(f(a_2)-b_2)^2+\\cdots +(f(a_n)-b_n)^2].$$\n已知在平面直角坐标系上, 有$5$个点的坐标数据如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n$x$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\\ \\hline\n$y$ & $2.2$ & $1$ & $2$ & $4.6$ & $7$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1) 若用函数$f_1(x)=x^2-4x+5$来拟合上述表格中的数据, 求$\\Delta (f_1(x))$;\\\\\n(2) 设$m\\in \\mathbf{R}$. 若用函数$f_2(x)=2^{|x-2|}+m$来拟合上述表格中的数据.\\\\\n\\textcircled{1} 求该函数的拟合误差$\\Delta (f_2(x))$的最小值, 并求出此时的函数解析式$y=f_2(x)$.\\\\\n\\textcircled{2} 根据实数$m$, 讨论用$f_1(x)$, $f_2(x)$(这里$f_2(x)$指\\textcircled{1} 中取得$\\Delta(f_2(x))$的最小值的那一个)中的哪一个函数来拟合上述表格中的数据更好? 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273137,7 +273757,8 @@ "content": "已知椭圆$\\Gamma :\\dfrac{x^2}6+\\dfrac{y^2}2=1$. 直线$l$与$x$轴的正半轴和$y$轴分别交于点$Q,P$, 与椭圆$\\Gamma$相交于两点$M\\mathbf{N}$, 各点互不重合, 且满足$\\overrightarrow{PM}=\\lambda_1\\overrightarrow{MQ}$, $\\overrightarrow{PN}=\\lambda_2\\overrightarrow{NQ}$.\\\\\n(1) 求焦距, 并证明: 焦距的平方是长轴长的平方、短轴长的平方的等差中项;\\\\\n(2) 若直线$l$的方程为$y=-x+1$, 求$\\dfrac 1{\\lambda_1}+\\dfrac 1{\\lambda_2}$的值;\\\\\n(3) 若$\\lambda_1+\\lambda_2=-3$, 试证明直线$l$恒过定点, 并求此定点的坐标.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273158,7 +273779,8 @@ "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)满足$a_1=1$, $a_2=t$, $a_n=f(a_{n-1})$, $f(a_n)+kf(a_{n-1})=\\lambda (a_n+ka_{n-1})$($n\\ge 2$, $n\\in \\mathbf{N}^*)$, 实数$k,t$是非零常数.\\\\\n(1) 若数列$\\{a_n\\}(n\\in \\mathbf{N}^{*})$是常数列, 求证: $k=-1$或$\\lambda =1$;\\\\\n(2) 若$\\lambda =1$, $t<1$, 求证: ``数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)是等差数列''的一个充要条件是``$k=-1$'';\\\\\n(3) 若$k=-1$, $t>1$, 是否存在实数$\\lambda$, 使得数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)是等比数列? 若存在, 求$\\lambda$的值; 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273179,7 +273801,8 @@ "content": "集合$A=\\{x|00)$的一条渐近线方程为$2x-y=0$, 则实数$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273323,7 +273951,8 @@ "content": "已知函数$f(x)=\\lg \\dfrac{1-x}{1+x}+\\sin x+1$, 若$f(m)=4$, 则$f(-m)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273344,7 +273973,8 @@ "content": "数列$\\{a_n\\}$的通项公式$a_n=\\begin{cases}\n\\dfrac 1n, & n=1,2, \\\\ \\dfrac 1{2^n}, & n\\ge 3, \\end{cases}$($n\\in \\mathbf{N}^*$), 前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to \\infty} S_n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273365,7 +273995,8 @@ "content": "甲、乙、丙三个不同单位的医疗队里各有$3$人, 职业分别为医生、护士与化验师, 现在要从中抽取$3$人组建一支志愿者队伍, 则他们的单位与职业都不相同的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273386,7 +274017,8 @@ "content": "已知双曲线的中心是坐标原点, 它的一个顶点为$A(\\sqrt 2,0)$, 两条渐近线与以$A$为圆心$1$为半径的圆都相切, 则该双曲线的标准方程是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273407,7 +274039,8 @@ "content": "已知集合$A=\\{(x,y)|(x+y)^2+x+y-2\\le 0\\}$,\n$B=\\{(x,y)|(x-2a)^2+(y-a-1)^2\\le a^2-\\dfrac a2\\}$, 若$A\\cap B\\ne \\varnothing$, 则实数$a$取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273428,7 +274061,8 @@ "content": "设$n\\in \\mathbf{N}^*$, $a_n$为$(x+2)^n-(x+1)^n$的展开式的各项系数之和, $m=-\\dfrac 12t+6$, , $b_n=[\\dfrac{a_1}3]+[\\dfrac{2a_2}{3^2}]+...+[\\dfrac{na_n}{3^n}]$($[x]$表示不超过实数$x$的最大整数), 则当$n$和$t$变动时, $(n-t)^2+(b_n-m)^2$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273449,7 +274083,8 @@ "content": "已知直角坐标平面上两条直线的方程分别为$l_1:a_1x+b_1y+c_1=0$, $l_2:a_2x+b_2y+c_2=0$, 那么``$\\begin{vmatrix}\na_1 & b_1 \\\\a_2 & b_2 \\end{vmatrix}=0$''是``两直线$l_1$、$l_2$平行''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273470,7 +274105,8 @@ "content": "如图, 若一个水平放置的图形的斜二测直观图是一个底角为$45^\\circ$且腰和上底均为$1$的等腰梯形, 则原平面图形的面积是\\bracket{20}.\n\\fourch{$\\dfrac{2+\\sqrt 2}2$}{$\\dfrac{1+\\sqrt 2}2$}{$2+\\sqrt 2$}{$1+\\sqrt 2$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273491,7 +274127,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 下列结论错误的是\\bracket{20}.\n\\onech{$(\\overrightarrow{A_1A}+\\overrightarrow{A_1D_1}+\\overrightarrow{A_1B_1})^2=3\\overrightarrow{A_1B_1}^2$}{$\\overrightarrow{A_1C}\\cdot (\\overrightarrow{A_1B_1}-\\overrightarrow{A_1A})=0$}{向量$\\overrightarrow{AD_1}$ 与$\\overrightarrow{A_1B}$ 的夹角是$120^\\circ$}{正方体$ABCD-A_1B_1C_1D_1$的体积为$|\\overrightarrow{AB}\\cdot \\overrightarrow{AA_1}\\cdot \\overrightarrow{AD}|$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273512,7 +274149,8 @@ "content": "函数$f(x)$是定义在$\\mathbf{R}$上的奇函数, 且$f(x-1)$为偶函数, 当$x\\in [0,1]$ 时, $f(x)=\\sqrt x$.若函数$g(x)=f(x)-x-m$ 有三个零点, 则实数$m$的取值范围是\\bracket{20}.\n\\twoch{$(-\\dfrac 14,\\dfrac 14)$}{$(1-\\sqrt {2},\\sqrt {2}-1)$}{$\\{m|4k-\\dfrac 14=latex, scale = 1.8]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,{sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,{sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- cycle (P) -- (C);\n\\draw [dashed] (P) -- (A) -- (B) (A) -- (D);\n\\draw ($(P)!0.5!(D)$) node [above right] {$E$} coordinate (E);\n\\draw [dashed] (A) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的体积;\\\\\n(2) 求异面直线$AE$与$PC$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273554,7 +274193,8 @@ "content": "已知函数$f(x)=2\\cos ^2\\dfrac x2+\\sqrt 3\\sin x$.\\\\\n(1) 求函数$f(x)$在区间$[0,\\pi]$上的单调递增区间;\\\\\n(2) 当$f(\\alpha)=\\dfrac{11}{5}$, 且$-\\dfrac{2\\pi}3<\\alpha <\\dfrac{\\pi}6$, 求$\\sin (2\\alpha +\\dfrac{\\pi}3)$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273575,7 +274215,8 @@ "content": "随着疫情的有效控制, 人们的生产生活逐渐向正常秩序恢复, 位于我区的某著名赏花园区重新开放.据统计研究, 近期每天赏花的人数大致符合以下数学模型($n\\in \\mathbf{N}^*$):\\\\\n以$f(n)=\\begin{cases}\n200n+1500, & 1\\le n\\le 6, \\\\ 300\\cdot 3^{\\frac{n-6}{11}}+2400, & 7\\le n\\le 28, \\\\ 23400-650n, & 29\\le n\\le 36 \\end{cases}$表示第$n$个时刻进入园区的人数;\\\\\n以$g(n)=\\begin{cases}\n0, & 1\\le n\\le 15, \\\\ 400n-5000, & 16\\le n\\le 28, \\\\ 8200, & 29\\le n\\le 36 \\end{cases}$表示第$n$个时刻离开园区的人数.\\\\\n设定每$15$分钟为一个计算单位, 上午$8$点$15$分作为第$1$个计算人数单位, 即$n=1$; $8$点$30$分作为第$2$个计算单位, 即$n=2$; 依次类推, 把一天内从上午$8$点到下午$5$点分成$36$个计算单位(最后结果四舍五入, 精确到整数).\\\\\n(1) 试分别计算当天$12: 30$至$13: 30$这一小时内, 进入园区的游客人数$f(19)+f(20)+f(21)+f(22)$和离开园区的游客人数$g(19)+g(20)+g(21)+g(22)$;\\\\\n(2) 请问, 从$12$点(即$n=16$)开始, 园区内游客总人数何时达到最多? 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273596,7 +274237,8 @@ "content": "已知动直线$l$与椭圆C:$x^2+\\dfrac{y^2}2=1$交于$P(x_1,y_1)$、$Q(x_2,y_2)$两不同点, 且$\\triangle OPQ$的面积$S_{\\triangle OPQ}=\\dfrac{\\sqrt 2}2$, 其中$O$为坐标原点.\\\\\n(1) 若动直线$l$垂直于$x$轴, 求直线$l$的方程;\\\\\n(2) 证明$x_1^2+x_2^2$和$y_1^2+y_2^2$均为定值;\\\\\n(3) 椭圆$C$上是否存在点$D,E,G$, 使得三角形面积$S_{\\triangle ODE}=S_{\\triangle ODG}=S_{\\triangle OEG}=\\dfrac{\\sqrt 2}2$? 若存在, 判断$\\triangle DEG$的形状; 若不存在, 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -273617,7 +274259,8 @@ "content": "若无穷数列$\\{a_n\\}$满足: 存在$k,n_0\\in \\mathbf{N}^*$, 对任意的$n\\ge n_0$($n\\in \\mathbf{N}^*$), 都有$a_{n+k}-a_n=d$($d$为常数), 则称 $\\{a_n\\}$具有性质$Q(k,n_0,d)$.\\\\\n(1) 若无穷数列$\\{a_n\\}$具有性质$Q(3,1,0)$, 且$a_1=1$, $a_2=2$, $a_3=3$, 求$a_2+a_3+a_4$的值;\\\\\n(2) 若无穷数列$\\{b_n\\}$是等差数列, 无穷数列$\\{c_n\\}$是公比为正数的等比数列, $b_1=c_5=1$, $b_5=c_1=81$, $a_n=b_n+c_n$, 判断$\\{a_n\\}$是否具有性质$Q(k,n_0,0)$, 并说明理由;\\\\\n(3) 设无穷数列$\\{a_n\\}$既具有性质$Q(i,2,d_1)$, 又具有性质$Q(j,2,d_2)$, 其中$i,j\\in \\mathbf{N}^*$, $i=latex, scale = 0.3]\n\\draw [->] (-12,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -12:2.5, samples = 100] plot (\\x,{pow(2,\\x)+cos(\\x/pi*180)});\n\\draw (0,2) node [above left] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273892,7 +274545,8 @@ "content": "已知平面向量$\\overrightarrow a,\\overrightarrow b,\\overrightarrow c$满足$|\\overrightarrow a|=|\\overrightarrow b|=|\\overrightarrow c|=1$, 且$\\overrightarrow a\\cdot \\overrightarrow b=\\dfrac 12$, 若存在实数$x,y$, 使得$\\overrightarrow c=x\\overrightarrow a+y\\overrightarrow b$, 则$x+y$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273913,7 +274567,8 @@ "content": "已知函数$f(x)=\\begin{cases}|5^x-1|, & x<1, \\\\ \\dfrac{8}{x+1},& x\\ge 1, \\end{cases}$, 若方程$f(f(x))=a$恰有$5$个不同的实数根, 则实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -273934,7 +274589,8 @@ "content": "已知两条直线$l_1$、$l_2$的方程分别为$l_1:ax+y-1=0$和$l_2:x-2y+1=0$, 则``$a=2$''是``直线$l_1\\perp l_2$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273957,7 +274613,8 @@ "content": "在正方体$ABCD-A_1B_1C_1D_1$ 中, 下列四个结论中错误的是\\bracket{20}.\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 (B1) -- (C);\n\\draw [dashed] (A) -- (C) -- (D1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\onech{直线$B_1C$与直线$AC$所成的角为$60^\\circ$}{直线$B_1C$与平面$AD_1C$所成的角为$60^\\circ$}{直线$B_1C$与直线$AD_1$所成的角为$90^\\circ$}{直线$B_1C$与直线$AB$所成的角为$90^\\circ$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -273980,7 +274637,8 @@ "content": "已知实数$x>0$, $y>0$, 且$2x+\\dfrac 1y=1$, 则$\\dfrac yx$的\\bracket{20}.\n\\fourch{最小值为$8$}{最大值为$8$}{最小值为$2$}{最大值为$2$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274001,7 +274659,8 @@ "content": "在平面上, 已知定点$A(2,0)$, 动点$P(\\sin\\alpha,\\cos\\alpha)$. 当$\\alpha$在区间$[-\\dfrac \\pi 4,\\dfrac \\pi 4]$上变化时, 动线段$AP$所形成图形的面积为\\bracket{20}.\n\\fourch{$\\sqrt 2-\\dfrac \\pi 4$}{$\\sqrt 3-\\dfrac\\pi 4$}{$\\dfrac{5\\pi}{24}+\\dfrac{\\sqrt 3}2-\\dfrac{\\sqrt 2}2$}{$\\dfrac \\pi 4$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274022,7 +274681,8 @@ "content": "如图1, 在三棱柱$ABC-A_1B_1C_1$中, 已知$AB\\perp AC$, $AB=AC=1$, $AA_1=2$, 且$AA_1\\perp$平面$ABC$. 过$A_1$、$C_1$、$B$三点作平面截此三棱柱, 截得一个三棱锥和一个四棱锥(如图2).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,1) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (B) -- (C) -- (C1) -- cycle (C1) -- (B1) (C1) -- (B);\n\\draw [dashed] (A) -- (C) (A) -- (B) (A) -- (A1) (A1) -- (B);\n\\draw (0.5,-0.8) node {图1};\n\\draw (A) ++ (3,0) node [left] {$A$} coordinate (A3);\n\\draw (B) ++ (3,0) node [right] {$B$} coordinate (B3);\n\\draw (C) ++ (3,0) node [left] {$C$} coordinate (C3);\n\\draw (C3) ++ (0,2) node [left] {$C_1$} coordinate (C4);\n\\draw (A3) ++ (0,2) node [above] {$A_1$} coordinate (A4);\n\\draw (B3) -- (C3) -- (C4) -- (A4) -- cycle (B3) -- (C4);\n\\draw [dashed] (A3) -- (B3) (A3) -- (C3) (A3) -- (A4);\n\\draw (B) ++ (4.5,0) node [right] {$B$} coordinate (B5);\n\\draw (B5) ++ (0,2) node [right] {$B_1$} coordinate (B6);\n\\draw (C1) ++ (4.5,0) node [left] {$C_1$} coordinate (C6);\n\\draw (A1) ++ (4.5,0) node [above] {$A_1$} coordinate (A6);\n\\draw (C6) -- (B5) -- (B6) -- (A6) -- cycle (B6) -- (C6);\n\\draw [dashed] (A6) -- (B5);\n\\draw (4.25,-0.8) node {图2};\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BC_1$与$AA_1$所成角的大小(结果用反三角函数表示);\\\\\n(2) 求四棱锥$B-ACC_1A_1$的体积和表面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274043,7 +274703,8 @@ "content": "已知函数$f(x)=\\sqrt 3\\sin x\\cos x+\\cos ^2x+1$.\\\\\n(1) 求$f(x)$的最小正周期和值域;\\\\\n(2) 若对任意的$x\\in \\mathbf{R}$, $f^2(x)-k\\cdot f(x)-2\\le 0$恒成立, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274066,7 +274727,8 @@ "content": "某网店有$3$(万件)商品, 计划在元旦旺季售出商品$x$(万件). 经市场调查测算, 花费$t$(万元)进行促销后, 商品的剩余量$3-x$与促销费$t$之间的关系为$3-x=\\dfrac k{t+1}$(其中$k$为常数), 如果不搞促销活动, 只能售出$1$(万件)商品.\\\\\n(1) 要使促销后商品的剩余量不大于$0.1$(万件), 促销费$t$至少为多少(万元)?\\\\\n(2) 已知商品的进价为$32$(元/件), 另有固定成本$3$(万元). 定义每件售出商品的平均成本为$32+\\dfrac 3x$(元).若将商品售价定为: ``每件售出商品平均成本的$1.5$倍''与``每件售出商品平均促销费的一半''之和, 则当促销费$t$为多少(万元)时, 该网店售出商品的总利润最大? 此时商品的剩余量为多少?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274087,7 +274749,8 @@ "content": "已知椭圆$\\Gamma \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点坐标为$(2,0)$, 且长轴长为短轴长的$\\sqrt 2$倍. 直线$l$交椭圆$\\Gamma$于不同的两点$M$和$N$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse ({sqrt(8)} and 2);\n\\draw [name path = line] (1,2.5) -- (-2,-2);\n\\draw [name intersections = {of = elli and line, by = {M,N}}];\n\\draw (M) node [above] {$M$};\n\\draw (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 若直线$l$经过点$T(0,4)$, 且$\\triangle OMN$的面积为$2\\sqrt 2$, 求直线$l$的方程;\\\\\n(3) 若直线$l$的方程为$y=kx+t$($k\\ne 0$), 点$M$关于$x$轴的对称点为$M'$, 直线$MN$、$M'N$ 分别与$x$轴相交于$P$、$Q$两点, 求证: $|OP|\\cdot|OQ|$为定值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274108,7 +274771,8 @@ "content": "对于由$m$个正整数构成的有限集$M=\\{a_1,a_2,a_3,\\cdots ,a_m\\}$, 记$P(M)=a_1+a_2+\\cdots +a_m$, 特别规定$P(\\varnothing)=0$. 若集合$M$满足: 对任意的正整数$k\\le P(M)$, 都存在集合$M$的两个子集$A$、$B$, 使得$k=P(A)-P(B)$成立, 则称集合$M$为``满集''.\\\\\n(1) 分别判断集合$M_1=\\{1,2\\}$与$M_2=\\{1,4\\}$是否是``满集'', 请说明理由;\\\\\n(2) 若$a_1,a_2,\\cdots ,a_m$是公差为$d(d\\in \\mathbf{N}^*)$的等差数列, 求证: 集合$M$为``满集''的必要条件是$a_1=1$, $d=1$或$2$;\\\\\n(3) 若$a_1,a_2,\\cdots ,a_m$由小到大能排列成首项为$1$, 公比为$3$的等比数列, 求证: 集合$M$是``满集''.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274129,7 +274793,8 @@ "content": "设全集$U=\\mathbf{R}$, 若集合$A=\\{1,2,3,4\\}$, $B=\\{x|2\\le x\\le 3\\}$, 则$A\\cap \\complement _UB\\text=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274150,7 +274815,8 @@ "content": "已知点$(2,5)$在函数$f(x)=1+a^x$($a>0$且$a\\ne 1$)的图像上, 则$f(x)$的反函数$f^{-1}(x)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274171,7 +274837,8 @@ "content": "不等式$\\dfrac{x+1}x>1$的解为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274201,7 +274868,8 @@ "content": "已知球的主视图所表示图形的面积为$9\\pi$, 则该球的体积是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274222,7 +274890,8 @@ "content": "函数$f(x)=\\begin{vmatrix}\n\\cos 2x & -\\sin x \\\\\\cos x & \\dfrac{\\sqrt 3}2 \\end{vmatrix}$在区间$[0,\\dfrac{\\pi}{2}]$上的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274243,7 +274912,8 @@ "content": "若$2+\\mathrm{i}$($\\mathrm{i}$是虚数单位)是关于$x$的实系数方程$x^2+mx+n=0$的一个根, 则圆锥曲线$\\dfrac{x^2}m+\\dfrac{y^2}n=1$的焦距为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274264,7 +274934,8 @@ "content": "已知点$O(0,0)$, $A(2,0)$, $B(1,-2\\sqrt 3)$, $P$是曲线$y=\\sqrt {1-\\dfrac{x^2}4}$上一个动点, 则$\\overrightarrow{OP}\\cdot \\overrightarrow{BA}$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274285,7 +274956,8 @@ "content": "甲、乙两队进行排球决赛, 现在的情形是甲队只要再赢一局就获冠军, 乙队需要再赢两局才能得冠军.若两队在每局赢的概率都是$0.5$, 则甲队获得冠军的概率为\\blank{50}.(结果用数值表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274306,7 +274978,8 @@ "content": "已知函数$f(x)=x+\\dfrac 4x-1$, 若存在$x_1,x_2,\\cdots ,x_n\\in [\\dfrac 14,4]$使得$f(x_1)+f(x_2)+\\cdots +f(x_{n-1})=f(x_n)$, 则正整数$n$的最大值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274327,7 +275000,8 @@ "content": "在平面直角坐标系中, 设点$O(0,0)$, $A(3,\\sqrt 3)$, 点$P(x,y)$的坐标满足$\\begin{cases} \\sqrt 3x-y\\le 0, \\\\ x-\\sqrt 3y+2\\ge 0, \\\\ y\\ge 0 ,\\end{cases}$则$\\overrightarrow{OA}$在$\\overrightarrow{OP}$上的投影的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274348,7 +275022,8 @@ "content": "函数$f(x)=\\sin \\omega x$($\\omega >0$)的图像与其对称轴在$y$轴右侧的交点从左到右依次记为$A_1,A_2,A_3,\\cdots ,A_n,\\cdots$, 在点列$\\{A_n\\}$中存在三个不同的点$A_k$、$A_i$、$A_p$, 使得$\\triangle A_kA_iA_p$是等腰直角三角形, 将满足上述条件的$\\omega$值从小到大组成的数列记为$\\{\\omega _n\\}$, 则$\\omega_{2019}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274369,7 +275044,8 @@ "content": "我们把一系列向量$\\overrightarrow{a_i}$($i=1,2,\\cdots ,n)$按次序排成一列, 称为向量列, 记作$\\{\\overrightarrow{a_i}\\}$, 已知向量列$\\{\\overrightarrow{a_i}\\}$满足$\\overrightarrow{a_1}=(1,1)$, $\\overrightarrow{a_n}=(x_n,y_n)=\\dfrac 12(x_{n-1}-y_{n-1},x_{n-1}+y_{n-1})$($n\\ge 2$), 设$\\theta_n$表示向量$\\overrightarrow{a_{n-1}}$与$\\overrightarrow{a_n}$的夹角, 若$b_n=\\dfrac{n^2}{\\pi}\\theta_n$, 对任意正整数$n$, 不等式$\\sqrt{\\dfrac 1{b_{n+1}}}+\\sqrt {\\dfrac 1{b_{n+2}}}+\\cdots +\\sqrt {\\dfrac 1{b_{2n}}}>\\log_a(1-2a)$恒成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274390,7 +275066,8 @@ "content": "满足条件$|z-\\mathrm{i}|=|3+4\\mathrm{i}|$($\\mathrm{i}$是虚数单位)的复数$z$在复平面上对应的点的轨迹是\\bracket{20}.\n\\fourch{直线}{圆}{椭圆}{双曲线}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274411,7 +275088,8 @@ "content": "设$n\\in \\mathbf{N}^*$, 则``数列$\\{a_n\\}$为等比数列''是``数列$\\{a_n\\}$满足$a_n\\cdot a_{n+3}=a_{n+1}\\cdot a_{n+2}$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274432,7 +275110,8 @@ "content": "已知直线$l_1:4x-3y+6=0$和直线$l_2:x=-1$, 则抛物线$y^2=4x$上一动点$P$到直线$l_1$和直线$l_2$的距离之和的最小值是\\bracket{20}.\n\\fourch{$\\dfrac{37}{16}$}{$\\dfrac{11}{5}$}{$2$}{$\\dfrac{7}{4}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274453,7 +275132,8 @@ "content": "已知$\\{a_n\\}$是公差为$d$($d>0$)的等差数列, 若存在实数$x_1, x_2, x_3, \\cdots, x_9$满足方程组$$\\begin{cases} \\sin x_1+\\sin x_2+\\sin x_3+\\cdots +\\sin x_9=0, \\\\ a_1\\sin x_1+a_2\\sin x_2+a_3\\sin x_3+\\cdots +a_9\\sin x_9=25, \\end{cases}$$则$d$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac 98$}{$\\dfrac 89$}{$\\dfrac 54$}{$\\dfrac 45$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274474,7 +275154,8 @@ "content": "在$\\triangle ABC$中, 角$A$, $B$, $C$的对边分别是$a$, $b$, $c$, 且$2\\cos 2A+4\\cos (B+C)+3=0$.\\\\\n(1) 求角$A$的大小;\\\\\n(2) 若$a=\\sqrt 3$, $b+c=3$, 求$b$和$c$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274495,7 +275176,8 @@ "content": "如图: 正四棱柱$ABCD-A_1B_1C_1D_1$中, 底面边长为$2$, $BC_1$与底面$ABCD$所成角的大小为$\\arctan 2$, $M$是$DD_1$的中点, $N$是$BD$上的一动点, 设$\\overrightarrow{DN}=\\lambda \\overrightarrow{DB}(0<\\lambda <1)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\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 [above] {$B_1$} 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 ($(D)!0.5!(D1)$) node [right] {$M$} coordinate (M);\n\\draw ($(D)!0.5!(B)$) node [above right] {$N$} coordinate (N);\n\\draw [dashed] (B) -- (D) (M) -- (N) (D1) -- (A);\n\\draw (B) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda =\\dfrac{1}{2}$时, 证明: $MN$与平面$ABC_1D_1$平行;\\\\\n(2) 若点$N$到平面$BCM$的距离为$d$, 试用$\\lambda$表示$d$, 并求出$d$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274516,7 +275198,8 @@ "content": "$2018$年世界人工智能大会已于$2018$年$9$月在上海徐汇西岸举行, 某高校的志愿者服务小组受大会展示项目的启发, 会后决定开发一款``猫捉老鼠''的游戏.如下图: $A$、$B$两个信号源相距$10$米, $O$是$AB$的中点, 过$O$点的直线$l$与直线$AB$的夹角为$45^\\circ$.机器猫在直线$l$上运动, 机器鼠的运动轨迹始终满足: 接收到$A$点的信号比接收到$B$点的信号晚$\\dfrac 8{v_0}$秒(注: 信号每秒传播$v_0$米).在时刻$t_0$时, 测得机器鼠距离$O$点为$4$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}\n(1) 以$O$为原点, 直线$AB$为$x$轴建立平面直角坐标系(如图), 求时刻$t_0$时机器鼠所在位置的坐标;\\\\\n(2) 游戏设定: 机器鼠在距离直线$l$不超过$1.5$米的区域运动时, 有``被抓''的风险.如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''风险?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274537,7 +275220,8 @@ "content": "对于项数为$m(m\\ge 3)$的有穷数列$\\{a_n\\}$, 若存在项数为$m+1$, 公差为$d$的等差数列$\\{b_n\\}$, 使得$b_kf_2(x). \\end{cases}$\\\\\n(1) 设函数$f_1(x)=\\sqrt x$, $f_2(x)=(\\dfrac 12)^{x-1}$($x\\ge 0$), 求函数$y=f(x)$的值域;\\\\\n(2) 设函数$f_1(x)=\\lg (|p-x|+1)$($00$, 若$f(x)$的定义域与值域相同, 则非零实数$a$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -274834,7 +275530,8 @@ "content": "已知函数$f(x)=\\begin{cases} -2x, & x<0, \\\\ x^2-1, & x\\ge 0,\\end{cases}$若方程$f(x)+2\\sqrt {1-x^2}+|f(x)-2\\sqrt {1-x^2}|-2ax-4=0$有三个根$x_19$成立}{存在实数$x$、$y$满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 并使得$4(x+1)(y+1)>7$成立}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)=-9$的实数$x$、$y$不存在}{满足$\\begin{cases}|x|\\le 1, \\\\|x+y|\\le 1, \\end{cases}$ 且使得$4(x+1)(y+1)<-9$的实数$x$、$y$不存在}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -274943,7 +275644,8 @@ "content": "如图, 在长方体$ABCD=A_1B_1C_1D_1$中, $T$为$DD_1$上一点, 已知$DT=2$, $AB=4$, $BC=2$, $AA_1=6$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\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--++ (0,6) 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) ++ (4,6) 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) --++ (0,6);\n\\filldraw ($(D)!{1/3}!(D1)$) circle (0.06) node [right] {$T$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$TC$与平面$ABCD$所成角的大小(用反三角函数值表示);\\\\\n(2) 求点$C_1$到平面$A_1TC$的距离.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274964,7 +275666,8 @@ "content": "已知函数$f(x)=\\sqrt 3\\sin 2x+\\cos 2x-1$($x\\in \\mathbf{R}$).\\\\\n(1) 写出函数$f(x)$的最小正周期和单调递增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$, $B$, $C$所对的边分别为$a$, $b$, $c$, 若$f(B)=0$, $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}=\\dfrac 32$, 且$a+c=4$, 求$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -274985,7 +275688,8 @@ "content": "定义在$D$上的函数$f(x)$, 如果满足: 存在常数$M>0$, 对任意$x\\in D$, 都有$|f(x)|\\le M$成立, 则称$f(x)$是$D$上的有界函数, 其中$M$称为函数$f(x)$的上界.\\\\\n(1) 设$f(x)=\\dfrac x{x+1}$, 判断$f(x)$在$[-\\dfrac 12, \\dfrac 12]$上是否为有界函数, 若是, 请说明理由, 并写出$f(x)$的所有上界$M$的集合; 若不是, 也请说明理由;\\\\\n(2) 若函数$g(x)=1+a\\cdot (\\dfrac 12)^x+(\\dfrac 14)^x$在$[0, +\\infty)$上是以$3$为上界的有界函数, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275006,7 +275710,8 @@ "content": "已知椭圆$\\Omega:9x^2+y^2=m^2$($m>0$), 直线$l$不过原点$O$且不平行于坐标轴, $l$与$\\Omega$有两个交点$A$、$B$, 线段$AB$的中点为$M$.\\\\\n(1) 若$m=3$, 点$K$在椭圆$\\Omega$上, $F_1,F_2$分别为椭圆的两个焦点, 求$\\overrightarrow{KF_1}\\cdot \\overrightarrow{KF_2}$的范围;\\\\\n(2) 证明: 直线$OM$的斜率与$l$的斜率的乘积为定值;\\\\\n(3) 若$l$过点$(\\dfrac m3,m)$, 射线$OM$与$\\Omega$交于点$P$, 四边形$OAPB$能否为平行四边形? 若能, 求此时$l$的斜率; 若不能, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275027,7 +275732,8 @@ "content": "已知数列$\\{a_n\\}$、$\\{b_n\\}$满足: $a_1=\\dfrac 14$, $a_n+b_n=1$, $b_{n+1}=\\dfrac{b_n}{1-a_n^2}$.\\\\\n(1) 求$b_1$, $b_2$, $b_3$, $b_4$, 并求$\\{b_n\\}$的通项公式;\\\\\n(2) 设$S_n=a_1a_2+a_2a_3+\\cdots +a_na_{n+1}$, 若不等式$4aS_n0$时, $f(x)=x+\\dfrac{m^2}x-1$(这里$m$为正常数), 若$f(x)\\le m-2$对一切$x\\le 0$成立, 则\n$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275287,7 +276003,8 @@ "content": "如图, 已知$O$为矩形$P_1P_2P_3P_4$内的一点, 满足$OP_1=4$,\n$OP_3=5$, $P_1P_3=7$, 则$\\overrightarrow{OP_2}\\cdot \\overrightarrow{OP_4}$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw (0,0) node [left] {$P_1$} coordinate (P1);\n\\draw ({sqrt(20)},0) node [right] {$P_2$} coordinate (P2);\n\\draw (P1) ++ (0,{sqrt(29)}) node [left] {$P_4$} coordinate (P4);\n\\draw (P2) ++ (0,{sqrt(29)}) node [right] {$P_3$} coordinate (P3);\n\\draw (P1) -- (P2) -- (P3) -- (P4) -- cycle;\n\\path [name path = c1] (P1) circle (4);\n\\path [name path = c2] (P3) circle (5);\n\\path [name intersections = {of = c1 and c2, by = {O,T}}];\n\\draw (O) node [above left] {$O$} -- (P1) (O) -- (P3);\n\\draw [->] (O) -- (P2);\n\\draw [->] (O) -- (P4);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275308,7 +276025,8 @@ "content": "将实数$x,y,z$中的最小值记为$\\min \\{x,y,z\\}$.在锐角$\\triangle POQ$中, $\\angle POQ=60^\\circ$, $PQ=1$, 点$T$在$\\triangle POQ$的边上或内部运动, 且$TO=\\min \\{TP,TO,TQ\\}$, 由$T$所组成的图形为$M$. 设$\\triangle POQ$、$M$的面积为$S_{\\triangle POQ}$、$S_M$, 若$S_M:(S_{\\triangle POQ}-S_M)=2:5$, 则$S_M=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275329,7 +276047,8 @@ "content": "``$\\sin x=\\dfrac 12$''是``$x=\\dfrac{\\pi }6$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275350,7 +276069,8 @@ "content": "在$(\\dfrac 2x-x)^6$的二项展开式中, 常数项等于\\bracket{20}.\n\\fourch{$-160$}{$160$}{$-150$}{$150$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275371,7 +276091,8 @@ "content": "若函数$f(x)$($x\\in \\mathbf{R}$)满足$f(-1+x)$、$f(1+x)$均为奇函数, 则下列四个结论正确的是\\bracket{20}.\n\\fourch{$f(-x)$为奇函数}{$f(-x)$为偶函数}{$f(x+3)$为奇函数}{$f(x+3)$为偶函数}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275392,7 +276113,8 @@ "content": "在正三棱柱$ABC-A_1B_1C_1$中, $AB=AA_1=1$, 点$P$满足$\\overrightarrow{BP}=\\lambda \\overrightarrow{BC}+\\mu \\overrightarrow{BB_1}$, 其中$\\lambda \\in [0,1]$, $\\mu \\in [0,1]$, 则下列命题为真命题的个数为\\bracket{20}.\n\\textcircled{1} 当$\\lambda =1$时, $\\triangle AB_1P$的周长为定值;\\\\\n\\textcircled{2} 当$\\mu =1$时, 三棱锥$P-A_1BC$的体积为定值;\\\\\n\\textcircled{3} 当$\\lambda =\\dfrac 12$时, 有且仅有一个点$P$, 使得$A_1P\\perp BP$;\\\\\n\\textcircled{4} 当$\\mu =\\dfrac 12$时, 有且仅有一个点$P$, 使得$A_1B\\perp$平面$AB_1P$.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275413,7 +276135,8 @@ "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PA\\perp$底面$ABCD$, $AD=2$, $AB=4$, $PA=6$, 点$E$在侧棱$PA$上, 且$AE=1$, $F$为侧棱$PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,6,0) node [above] {$P$} coordinate (P);\n\\draw (-4,0,0) node [left] {$B$} coordinate (B);\n\\draw (-4,0,2) node [left] {$C$} coordinate (C);\n\\draw (0,0,2) node [right] {$D$} coordinate (D);\n\\filldraw ($(C)!0.5!(P)$) circle (0.03) node [left] {$F$} coordinate (F);\n\\filldraw ($(A)!{1/6}!(P)$) circle (0.03) node [right] {$E$} coordinate (E);\n\\draw (A) -- (P) -- (C) -- (D) -- cycle (P) -- (D);\n\\draw [dashed] (B) -- (P) (B) -- (C) (B) -- (A) (C) -- (E) (D) -- (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$E-ABD$的体积;\\\\\n(2) 求异面直线$CE$与$DF$所成角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275434,7 +276157,8 @@ "content": "设$z+1$为关于$x$的方程$x^2+mx+n=0$, $m,n\\in\\mathbf{R}$的虚根, $\\mathrm{i}$为虚数单位.\\\\\n(1) 当$z=-1+\\mathrm{i}$时, 求$m$、$n$的值;\\\\\n(2) 若$n=1$, 在复平面上, 设复数$z$所对应的点为$P$, 复数$2+4\\mathrm{i}$所对应的点为$Q$, 试求$|PQ|$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275455,7 +276179,8 @@ "content": "某渔业公司最近开发的一种新型淡水养虾技术具有方法简便且经济效益好的特点, 研究表明: 用该技术进行淡水养虾时, 在一定的条件下, 每尾虾的平均生长速度为$g(x)$(单位: 千克/年)养殖密度为$x$($x>0$, 单位: 尾/立方分米), 当$x$不超过$4$时, $g(x)$的值恒为$2$; 当$4\\le x\\le 20$时, $g(x)$是$x$的一次函数, 且当$x$达到$20$时, 因养殖空间受限等原因, $g(x)$的值为$0$.\\\\\n(1) 当$00$,\n$b>0$)的右顶点, 直线$x+2y+1=0$与$C$的一条渐近线平行.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw ({-sqrt(5)},0) node [below left] {$F_1$} coordinate (F1);\n\\draw ({sqrt(5)},0) node [below right] {$F_2$} coordinate (F2);\n\\draw (4,{sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (P) -- (F1) (P) -- (F2);\n\\draw (1,0) node [above] {$M$} coordinate (M);\n\\draw ($(P)!{4/3}!(M)$) node [below left] {$N$} coordinate (N);\n\\draw (P) -- (N);\n\\draw [name path = line] ($(F1)!-0.6!(N)$) -- ($(F1)!3!(N)$);\n\\draw [name path = leftcurve, domain = -2:2] plot ({-sqrt(1+pow(\\x,2))*2},\\x);\n\\draw [name path = rightcurve, domain = -2:2] plot ({sqrt(1+pow(\\x,2))*2},\\x);\n\\draw [name intersections = {of = line and leftcurve, by = D}];\n\\draw [name intersections = {of = line and rightcurve, by = E}];\n\\draw (D) node [below] {$D$} -- (E) node [below] {$E$} -- (F2) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$C$的方程;\\\\\n(2) 如图, $F_1$、$F_2$为$C$的左右焦点, 动点$P(x_0,y_0)$($y_0\\ge 1$)在$C$的右支上, 且$\\angle F_1PF_2$的平分线与$x$轴、$y$轴分别交于点$M(m,0)$($-\\sqrt 50$, $m$为正偶数, 数列$\\{x_n\\}$满足$x_1=b<0$, 且$x_{n+1}=f_{(b,a)}(\\dfrac 1{x_n^m})(n\\in \\mathbf{N}^*)$, 证明: 若$ab^{m-1}+2\\ge 0$, 则数列$\\{x_n\\}$有界.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275518,7 +276245,8 @@ "content": "已知集合$A=\\{1,2,4,6,8\\}$, $B=\\{x|x=2k, \\ k\\in A\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275541,7 +276269,8 @@ "content": "已知$\\dfrac{\\overline z}{1-\\mathrm{i}}=2+\\mathrm{i}$, 则复数$z$的虚部为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275564,7 +276293,8 @@ "content": "设函数$f(x)=\\sin x-\\cos x$, 且$f(a)=1$, 则$\\sin 2a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275589,7 +276319,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275612,7 +276343,8 @@ "content": "数列$\\{a_n\\}$是首项为$1$, 公差为$2$的等差数列, $S_n$是它前$n$项和, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{S_n}{a_n^2}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275638,7 +276370,8 @@ "content": "已知角$A$是$\\triangle ABC$的内角, 则``$\\cos A=\\dfrac 12$''是``$\\sin A=\\dfrac {\\sqrt 3}2$''的\\blank{50}条件(填``充分非必要''、``必要非充分''、``充要条件''、``既非充分又非必要''之一)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275661,7 +276394,8 @@ "content": "若双曲线$x^2-\\dfrac{y^2}{b^2}=1$的一个焦点到其渐近线距离为$2\\sqrt{2}$, 则该双曲线焦距等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275684,7 +276418,8 @@ "content": "一个底面半径为$2$的圆柱被与其底面所成角是$60^\\circ$的平面所截, 截面是一个椭圆, 则该椭圆的焦距等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (-1,-2) arc (180:360:1 and 0.25);\n\\draw (-1,2) arc (180:-180:1 and 0.25);\n\\draw [dashed] (-1,-2) arc (180:0:1 and 0.25);\n\\draw (-1,-2) -- (-1,2) (1,-2) -- (1,2);\n\\draw [domain = 180:360] plot ({cos(\\x)},{-sqrt(3)+sin(\\x)+sqrt(3)*(cos(\\x)+1)});\n\\draw [domain = 180:0, dashed] plot ({cos(\\x)},{-sqrt(3)+sin(\\x)+sqrt(3)*(cos(\\x)+1)});\n\\filldraw (0,0) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275705,7 +276440,8 @@ "content": "设$F_1,F_2$分别是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, 点$P$在双曲线右支上且满足$|PF_2|=|F_1F_2|$, 双曲线的渐近线方程为$4x\\pm 3y=0$, 则$\\cos \\angle PF_1F_2=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275728,7 +276464,8 @@ "content": "若$a,b$分别是正数$p,q$的算术平均数和几何平均数, 且$a,b,-2$这三个数可适当排序后成等差数列, 也可适当排序后成等比数列, 则$p+q+pq$的值形成的集合是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275751,7 +276488,8 @@ "content": "点$M(20,40)$, 抛物线$y^2=2px$($p>0$)的焦点为$F$, 若对于抛物线上的任意点$P$, $|PM|+|PF|$的最小值为$41$, 则$p$的值等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275772,7 +276510,8 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=-2$, 且$S_n=\\dfrac 32a_n+n$(其中$S_n$为数列$\\{a_n\\}$前$n$项和), $f(x)$是定义在$\\mathbf{R}$上的奇函数, 且满足$f(2-x)=f(x)$, 则$f(a_{2022})=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -275793,7 +276532,8 @@ "content": "若$a>b$, 则下列各式中恒正的是\\bracket{20}.\n\\fourch{$\\lg(a-b)$}{$a^3-b^3$}{$0.5^a-0.5^b$}{$|a|-|b|$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275814,7 +276554,8 @@ "content": "在空间中, $\\alpha$表示平面, $m$、$n$表示两条直线, 则下列命题中错误的是\\bracket{20}.\n\\onech{若$m\\parallel \\alpha$, $m$、$n$不平行, 则$n$与$\\alpha$不平行}{若$m\\parallel \\alpha$, $m$、$n$不垂直, 则$n$与$\\alpha$不垂直}{若$m\\perp \\alpha$, $m$、$n$不平行, 则$n$与$\\alpha$不垂直}{若$m\\perp \\alpha$, $m$、$n$不垂直, 则$n$与$\\alpha$不平行}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275835,7 +276576,8 @@ "content": "已知函数$f(x)=A\\sin(\\omega x+\\varphi)$($A>0$, $\\omega>0$)的图像与直线$y=b$($00$;\\\\\n\\textcircled{2} 当$a>0$时, $a+b$有最小值, 无最大值;\\\\\n\\textcircled{3} $a^2+b^2>1$;\\\\\n\\textcircled{4} 当$a>0$且$a\\ne 1$时, $\\dfrac{b+1}{a-1}$的取值范围是$(-\\infty,-\\dfrac 94)\\cup (\\dfrac 34,+\\infty)$;\\\\\n正确的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -275879,7 +276622,8 @@ "content": "如图在三棱锥$P-ABC$中, 棱$AB$、$AC$、$AP$两两垂直, $AB=AC=AP=3$, 点$M$在$AP$上, 且$AM=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,3) node [left] {$B$} coordinate (B);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw (0,1,0) node [above right] {$M$} coordinate (M);\n\\draw (P) -- (B) -- (C) -- cycle;\n\\draw [dashed] (A) -- (P) (B) -- (A) -- (C) (B) -- (M) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BM$和$PC$所成的角的大小;\\\\\n(2) 求三棱锥$P-BMC$的体积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275902,7 +276646,8 @@ "content": "已知函数$f(x)=(a+1)x^2+(a-1)x+(a^2-1)$, 其中$a\\in \\mathbf{R}$\\\\\n(1) 当$f(x)$是奇函数时, 求实数$a$的值;\\\\\n(2) 当函数$f(x)$在$[2,+\\infty)$上单调递增时, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275923,7 +276668,8 @@ "content": "如图所示, $A$,$B$两处各有一个垃圾中转站, $B$在$A$的正东方向$16\\text{km}$处, $AB$的南面为居民生活区. 为了妥善处理生活垃圾, 政府决定在$AB$的北面$P$处建一个发电厂, 利用垃圾发电. 要求发电厂到两个垃圾中转站的距离(单位: $\\text{km}$)与它们每天集中的生活垃圾量(单位: 吨)成反比, 现估测得$A,B$两处中转站每天集中的生活垃圾量分别约为$30$吨和$50$吨.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.05) node [left] {$A$} coordinate (A) -- (3,0) circle (0.05) node [right] {$B$} coordinate (B);\n\\path [name path = c1] (A) ++ (3,0) arc (0:40:3);\n\\path [name path = c2] (B) ++ (-1.8,0) arc (180:100:1.8);\n\\path [name intersections = {of = c1 and c2, by = P}];\n\\draw [dotted] (A) -- (P) node [above] {$P$} -- (B);\n\\filldraw (P) circle (0.03);\n\\draw [->] (3,1) -- (3,1.5) node [right] {北};\n\\draw (1.5,0) node [below] {居民生活区};\n\\end{tikzpicture}\n\\end{center}\n(1) 当$AP=15\\text{km}$时, 求$\\angle APB$的值;\\\\\n(2) 发电厂尽量远离居民区, 要求$\\triangle PAB$的面积最大. 问此时发电厂与两个垃圾中转站的距离各为多少?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275944,7 +276690,8 @@ "content": "已知点$A(-1,0)$, $B(1,0)$, 直线$l:ax+by+c=0$(其中$a,b,c\\in \\mathbf{R}$), 点$P$在直线$l$上.\\\\\n(1) 若$a,b,c$是常数列, 求$|PB|$的最小值;\\\\\n(2) 若$a,b,c$成等差数列, 且$PA\\perp l$, 求$|PB|$的最大值;\\\\\n(3) 若$a,b,c$成等比数列, 且$PA\\perp l$, 求$|PB|$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275965,7 +276712,8 @@ "content": "已知函数$f(x)=2-|x|$, 无穷数列满足$\\{a_n\\}$满足$a_{n+1}=f(a_n)$, $n\\in \\mathbf{N}^*$.\\\\\n(1) 若$a_1=2$, 写出数列的通项公式(不必证明);\\\\\n(2) 若$a_1>0$, 且$a_1,a_2,a_3$成等比数列, 求$a_1$的值; 此时$\\{a_n\\}$是否为等比数列, 说明理由;\\\\\n(3) 证明: $a_1,a_2,\\cdots,a_n,\\cdots$成等差数列的一个充要条件是$a_1=1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -275986,7 +276734,8 @@ "content": "已知集合$A=\\{1,2,3,4\\}$, $B=\\{3,4,5\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276015,7 +276764,8 @@ "content": "若排列数$\\mathrm{P}_6^m=6\\times 5\\times 4$, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276038,7 +276788,8 @@ "content": "不等式$\\dfrac{x-1}{x}>1$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276072,7 +276823,8 @@ "content": "已知球的体积为$36\\pi$, 则该球主视图的面积等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276099,7 +276851,8 @@ "content": "已知复数$z$满足$z+\\dfrac{3}{z}=0$, 则$|z|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276122,7 +276875,8 @@ "content": "设双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{b^2}=1 \\ (b>0)$的焦点为$F_1$、$F_2$, $P$为该双曲线上的一点, 若$|PF_1|=5$, 则$|PF_2|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276145,7 +276899,8 @@ "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}[>=latex, scale = 0.6]\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": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276168,7 +276923,8 @@ "content": "定义在$(0,+\\infty)$上的函数$y=f(x)$的反函数为$y=f^{-1}(x)$. 若$g(x)=\\begin{cases}3^x-1, & x\\le 0,\\\\ f(x), & x>0\\end{cases}$为奇函数, 则$f^{-1}(x)=2$的解为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276191,7 +276947,8 @@ "content": "已知四个函数: \\textcircled{1} $y=-x$, \\textcircled{2} $y=-\\dfrac{1}{x}$, \\textcircled{3} $y=x^3$, \\textcircled{4} $y=x^{\\frac{1}{2}}$. 从中任选$2$个, 则事件``所选$2$个函数的图像有且仅有一个公共点''的概率为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276214,7 +276971,8 @@ "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$, 其中$a_n=n^2, \\ n\\in \\mathbf{N}^*$, $\\{b_n\\}$的项是互不相等的正整数. 若对于任意$n\\in \\mathbf{N}^*$, $\\{b_n\\}$的第$a_n$项等于$\\{a_n\\}$的第$b_n$项, 则$\\dfrac{\\lg (b_1b_4b_9b_{16})}{\\lg(b_1b_2b_3b_4)}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276238,7 +276996,8 @@ "content": "设$\\alpha_1,\\alpha_2\\in \\mathbf{R}$, 且$\\dfrac{1}{2+\\sin\\alpha_1}+\\dfrac{1}{2+\\sin(2\\alpha_2)}=2$, 则$|10\\pi-\\alpha_1-\\alpha_2|$的最小值等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276262,7 +277021,8 @@ "content": "如图, 用$35$个单位正方形拼成一个矩形, 点$P_1,P_2,P_3,P_4$以及四个标记为``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点在正方形的顶点处, 设集合$\\Omega=\\{P_1,P_2,P_3,P_4\\}$, 点$P\\in \\Omega$. 过$P$作直线$l_P$, 使得不在$l_P$上的``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点分布在$l_P$的两侧. 用$D_1(l_P)$和$D_2(l_P)$分别表示$l_P$一侧和另一侧的``\\begin{tikzpicture}\n\\filldraw ({-0.0625*sqrt(3)},-0.0625) -- ({0.0625*sqrt(3)},-0.0625) -- (0,0.125) -- cycle;\n\\end{tikzpicture}''的点到$l_P$的距离之和. 若过$P$的直线$l_P$中有且只有一条满足$D_1(l_P)=D_2(l_P)$, 则$\\Omega$中所有这样的$P$为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[scale = 0.6]\n\\foreach \\i in {0,1,2,3,4,5,6,7}{\\draw (\\i,0) -- (\\i,5);};\n\\foreach \\i in {0,1,2,3,4,5}{\\draw (0,\\i) -- (7,\\i);};\n\\filldraw (1,0) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (7,1) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (0,3) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (4,4) ++ ({-0.0625*sqrt(3)},-0.0625) coordinate(P) --++ ({2*0.0625*sqrt(3)},0) --++ ({-0.0625*sqrt(3)},{3*0.0625}) -- (P);\n\\filldraw (0,4) circle (0.05) node [left] {$P_1$};\n\\filldraw (3,2) circle (0.05) node [below left] {$P_2$};\n\\filldraw (4,2) circle (0.05) node [below right] {$P_3$};\n\\filldraw (6,5) circle (0.05) node [above] {$P_4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276287,7 +277047,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276310,7 +277071,8 @@ "content": "在数列$\\{a_n\\}$中, $a_n=\\left(-\\dfrac{1}{2}\\right)^n, \\ n\\in \\mathbf{N}^*$, 则$\\displaystyle\\lim_{n\\to \\infty}a_n$\\bracket{15}.\n\\fourch{等于$-\\dfrac{1}{2}$}{等于$0$}{等于$\\dfrac{1}{2}$}{不存在}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276333,7 +277095,8 @@ "content": "已知$a,b,c$为实常数, 数列$\\{x_n\\}$的通项$x_n=an^2+bn+c, \\ n\\in \\mathbf{N}^*$, 则``存在$k\\in \\mathbf{N}^*$, 使得$x_{100+k},x_{200+k},x_{300+k}$成等差数列''的一个必要条件是\\bracket{15}.\n\\fourch{$a\\ge 0$}{$b\\le 0$}{$c=0$}{$a-2b+c=0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276356,7 +277119,8 @@ "content": "在平面直角坐标系$xOy$中, 已知椭圆$C_1:\\dfrac{x^2}{36}+\\dfrac{y^2}{4}=1$和$C_2:x^2+\\dfrac{y^2}{9}=1$. $P$为$C_1$上的动点, $Q$为$C_2$上的动点, $w$是$\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}$的最大值. 记$\\Omega=\\{(P,Q)|P\\text{在}C_1\\text{上}, \\ Q\\text{在}C_2\\text{上, 且}\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}=w\\}$, 则$\\Omega$中的元素有\\bracket{15}.\n\\fourch{$2$个}{$4$个}{$8$个}{无穷个}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276379,7 +277143,8 @@ "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的底面为直角三角形, 两直角边$AB$和$AC$的长分别为$4$和$2$, 侧棱$AA_1$的长为$5$.\\\\\n(1) 求三棱柱$ABC-A_1B_1C_1$的体积;\\\\\n(2) 设$M$是$BC$中点, 求直线$A_1M$与平面$ABC$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) node [below right] {$A$} coordinate (A) -- (-4,0) node [left] {$B$} coordinate (B) (0,0) -- (45:1) node [right] {$C$} coordinate (C);\n\\draw [dashed] (B) -- (C);\n\\draw (A) --++ (0,5) node [right] {$A_1$} coordinate (A1) ;\n\\draw (B) --++ (0,5) node [left] {$B_1$} coordinate (B1);\n\\draw (C) --++ (0,5) node [above right] {$C_1$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\t\t\t\t\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276402,7 +277167,8 @@ "content": "已知函数$f(x)=\\cos^2 x-\\sin^2 x+\\dfrac{1}{2}, \\ x\\in (0,\\pi)$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\\n(2) 设$\\triangle ABC$为锐角三角形, 角$A$所对的边$a=\\sqrt{19}$, 角$B$所对的边$b=5$, 若$f(A)=0$, 求$\\triangle ABC$的面积.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276425,7 +277191,8 @@ "content": "根据预测, 某地第$n \\ (n\\in \\mathbf{N}^*)$个月共享单车的投放量和损失量分别为$a_n$和$b_n$(单位: 辆), 其中$a_n=\\begin{cases}\n5n^4+15, & 1\\le n\\le 3,\\\\ -10n+470, & n \\ge 4,\n\\end{cases}$ $b_n=n+5$, 第$n$个月底的共享单车的保有量是前$n$个月的累计投放量与累计损失量的差.\\\\\n(1) 求该地区第$4$个月底的共享单车的保有量;\\\\\n(2) 已知该地共享单车停放点第$n$个月底的单车容纳量$S_n=-4(n-46)^2+8800$(单位: 辆). 设在某月底, 共享单车保有量达到最大, 问该保有量是否超出了此时停放点的单车容纳量?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276448,7 +277215,8 @@ "content": "在平面直角坐标系$xOy$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{4}+y^2=1$, $A$为$\\Gamma$的上顶点, $P$为$\\Gamma$上异于上、下顶点的动点. $M$为$x$正半轴上的动点.\\\\\n(1) 若$P$在第一象限, 且$|OP|=\\sqrt{2}$, 求$P$的坐标;\\\\\n(2) 设$P\\left(\\dfrac{8}{5},\\dfrac{3}{5}\\right)$. 若以$A,P,M$为顶点的三角形是直角三角形, 求$M$的横坐标;\\\\\n(3) 若$|MA|=|MP|$, 直线$AQ$与$\\Gamma$交于另一点$C$, 且$\\overrightarrow{AQ}=2\\overrightarrow{AC}$, $\\overrightarrow{PQ}=4\\overrightarrow{PM}$, 求直线$AQ$的方程.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276472,7 +277240,8 @@ "content": "设定义在$\\mathbf{R}$上的函数$f(x)$满足: 对于任意的$x_1,x_2\\in \\mathbf{R}$, 当$x_10$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276523,7 +277293,8 @@ "content": "若复数$z=1-2\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z\\cdot \\overline z+z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276547,7 +277318,8 @@ "content": "动点$P$到点$F(2,0)$的距离与它到直线$x+2=0$的距离相等, 则$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276570,7 +277342,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276591,7 +277364,8 @@ "content": "圆$C:x^2+y^2-2x-4y+4=0$的圆心到直线$l:3x+4y+4=0$的距离$d=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276615,7 +277389,8 @@ "content": "随机变量$\\xi$的概率分布率由下表给出: \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$x$ & $7$ & $8$ & $9$ & $10$ \\\\ \\hline\n$P(\\xi=x)$ & $0.3$ & $0.35$ & $0.2$ & $0.15$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n则随机变量$\\xi$的均值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276636,7 +277411,8 @@ "content": "$2010$年上海世博会园区每天$9:00$开园, $20:00$停止入园. 在下面的框图中, $S$表示上海世博会官方网站在每个整点报道的入园总人数, $a$表示整点报道前$1$个小时内入园人数, 则空白的执行框内应填入\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, node distance = 3ex]\n\\node [draw, rounded corners] (start) {开始};\n\\node [draw, below=of start] (step1) {$T\\leftarrow 9$, $S\\leftarrow 0$};\n\\node [draw, trapezium, trapezium right angle = 120, below=of step1] (step2) {输出$T,S$};\n\\node [draw, diamond, aspect = 3, below= of step2] (step3) {$T\\le 19$};\n\\node [draw, below= of step3] (step4) {$T\\leftarrow T+1$};\n\\node [draw, trapezium, trapezium right angle = 120, below = of step4] (step5) {输入$a$};\n\\node [draw, below = of step5] (step6) {\\phantom{$S\\leftarrow S+a$}};\n\\node [right = of step3] (anchor2) {};\n\\node [below = of step6] (anchor3) {};\n\\node [draw, below = of anchor3] (end) {结束};\n\\node [left = of step3] (anchor1) {};\n\\draw[->] (start) -- (step1);\n\\draw[->] (step1) -- (step2);\n\\draw[->] (step2) -- (step3);\n\\draw[->] (step3) -- node[right] {是}(step4) ;\n\\draw[->] (step4) -- (step5);\n\\draw[->] (step5) -- (step6);\n\\draw[->] (step6) -- (anchor1|-step6) -- (anchor1|-step2) -- (step2);\n\\draw[->] (step3) --node[above] {否} (anchor2|-anchor2) -- (anchor2|-anchor3) -- (anchor3|-anchor3) -- (end);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276657,7 +277433,8 @@ "content": "对任意不等于$1$的正数$a$, 函数$f(x)=\\log_a(x+3)$的反函数的图像都经过点$P$, 则点$P$的坐标是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276678,7 +277455,8 @@ "content": "从一副混合后的扑克牌($52$张)中随机抽取$1$张, 事件$A$为``抽得红桃K'', 事件$B$为``抽得为黑桃'', 则概率$P(A\\cup B)=$\\blank{50}.(结果用最简分数表示)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276703,7 +277481,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276724,7 +277503,8 @@ "content": "将直线$l_1:nx+y-n=0$、$l_2:x+ny-n=0$($n\\in \\mathbf{N}^*$, $n\\ge 2$)$x$轴、$y$轴围成的封闭图形的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276747,7 +277527,8 @@ "content": "如图所示, 在边长为$4$的正方形纸片$ABCD$中, $AC$与$BD$相交于$O$, 剪去$\\triangle AOB$, 将剩余部分沿$OC$、$OD$折叠, 使$OA$、$OB$重合, 则以$A(B)$、$C$、$D$、$O$为顶点的四面体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,2) (0,2) -- (2,0) (0,0) rectangle (2,2);\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (2,2) node [right] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (1,1) node [below] {$O$} coordinate (O);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276768,7 +277549,8 @@ "content": "如图所示, 直线$x=2$与双曲线$\\Gamma :\\dfrac{x^2}4-y^2=1$的渐近线交于$E_1$, $E_2$两点, 记$\\overrightarrow{OE_1}=\\overrightarrow{e_1}$, $\\overrightarrow{OE_2}=\\overrightarrow{e_2}$, 任取双曲线$\\Gamma$上的点$P$, 若$\\overrightarrow{OP}=a\\overrightarrow{e_1}+b\\overrightarrow{e_2}$($a,b\\in \\mathbf{R}$), 则$a$、$b$满足的一个等式是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-4,-2) -- (4,2) (4,-2) -- (-4,2);\n\\draw [domain = {-sqrt(3)}:{sqrt(3)}] plot ({2*sqrt(1+pow(\\x,2))},\\x);\n\\draw [domain = {-sqrt(3)}:{sqrt(3)}] plot ({-2*sqrt(1+pow(\\x,2))},\\x);\n\\draw (2,{-sqrt(3)}) -- (2,{sqrt(3)});\n\\draw (2,1) node [above left] {$E_1$} coordinate (E1);\n\\draw (2,-1) node [below left] {$E_2$} coordinate (E2); \n\\draw [->] (0,0) -- node [above] {$\\overrightarrow{e_1}$} coordinate (e1) (E1);\n\\draw [->] (0,0) -- node [below] {$\\overrightarrow{e_2}$} coordinate (e1) (E2);\n\\draw (-3,{sqrt(5/4)}) node [below left] {$P$} coordinate (P);\n\\draw [->] (0,0) -- (P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276789,7 +277571,8 @@ "content": "在集合$U=\\{a,b,c,d\\}$的子集中选出$2$个不同的子集, 需同时满足以下两个条件: \\textcircled{1} $a$、$b$都要选出; \\textcircled{2} 对选出的任意两个子集$A$和$B$, 必有$A\\subseteq B$或$B\\subseteq A$, 那么共有\\blank{50}种不同的选法.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -276810,7 +277593,8 @@ "content": "``$x=2k\\pi +\\dfrac{\\pi }4$($k\\in \\mathbf{Z}$)''是``$\\tan x=1$''成立的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分条件}{既不充分也不必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276831,7 +277615,8 @@ "content": "直线$l$的参数方程是$\\begin{cases} x=1+2t, \\\\ y=2-t, \\end{cases}$($t\\in \\mathbf{R}$), 则$l$的方向向量是$\\overrightarrow d$可以是\\bracket{20}.\n\\fourch{$(1, 2)$}{$(2, 1)$}{$(-2, 1)$}{$(1, -2)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276852,7 +277637,8 @@ "content": "若$x_0$是方程$(\\dfrac 12)^x=x^{\\frac 13}$的解, 则$x_0$属于区间\\bracket{20}.\n\\fourch{$(\\dfrac 23, 1)$}{$(\\dfrac 12, \\dfrac 23)$}{$(\\dfrac 13, \\dfrac 12)$}{$(0, \\dfrac 13)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276873,7 +277659,8 @@ "content": "某人要制作一个三角形, 要求它的三条高的长度分别为$\\dfrac 1{13},\\dfrac 1{11},\\dfrac 15$, 则此人能\\bracket{20}.\n\\twoch{不能作出这样的三角形}{作出一个锐角三角形}{作出一个直角三角形}{作出一个钝角三角形}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -276894,7 +277681,8 @@ "content": "已知$0=latex,scale = 1.6]\n\\draw [domain = 0:360, samples = 100] plot ({cos(\\x)},2,{sin(\\x)}); \n\\draw ({cos(-atan(sqrt(2)/4))},0,{sin((-atan(sqrt(2)/4)))}) coordinate (R);\n\\draw [domain = {-atan(sqrt(2)/4)}:{-atan(sqrt(2)/4)+180}, samples = 50] plot ({cos(\\x)},0,{sin(\\x)}) coordinate (L);\n\\draw [domain = {-atan(sqrt(2)/4)}:{-atan(sqrt(2)/4)-180}, dashed, samples = 50] plot ({cos(\\x)},0,{sin(\\x)});\n\\draw (L) --++ (0,2) (R) --++ (0,2);\n\\draw ({cos(180)},0,{sin(180)}) node [left] {$A_1$} coordinate (A1);\n\\draw ({cos(135)},0,{sin(135)}) node [below left] {$A_2$} coordinate (A2);\n\\draw ({cos(90)},0,{sin(90)}) node [below] {$A_3$} coordinate (A3);\n\\draw ({cos(45)},0,{sin(45)}) node [below right] {$A_4$} coordinate (A4);\n\\draw ({cos(0)},0,{sin(0)}) node [right] {$A_5$} coordinate (A5);\n\\draw ({cos(-45)},0,{sin(-45)}) node [above right] {$A_6$} coordinate (A6);\n\\draw ({cos(-90)},0,{sin(-90)}) node [above left] {$A_7$} coordinate (A7);\n\\draw ({cos(-135)},0,{sin(-135)}) node [above left] {$A_8$} coordinate (A8);\n\\draw [dashed] (A1) --++ (0,2) node [above] {$B_1$} coordinate (B1);\n\\draw (A2) --++ (0,2) node [below right] {$B_2$} coordinate (B2);\n\\draw (A3) --++ (0,2) node [below right] {$B_3$} coordinate (B3);\n\\draw (A4) --++ (0,2) node [below right] {$B_4$} coordinate (B4);\n\\draw (A5) --++ (0,2) node [right] {$B_5$} coordinate (B5);\n\\draw [dashed] (A6) --++ (0,2) node [above] {$B_6$} coordinate (B6);\n\\draw [dashed] (A7) --++ (0,2) node [above] {$B_7$} coordinate (B7);\n\\draw [dashed] (A8) --++ (0,2) node [above] {$B_8$} coordinate (B8);\n\\draw (B1) -- (B5) (B2) -- (B6) (B3) -- (B7) (B4) -- (B8);\n\\draw [dashed] (A1) -- (A5) (A2) -- (A6) (A3) -- (A7) (A4) -- (A8);\n\\draw [dashed] (A1) -- (B3) (A3) -- (B5);\n\\end{tikzpicture}\n\\end{center}\n(1) 当圆柱底面半径$r$取何值时, $S$取得最大值? 并求出该最大值(结果精确到$0.01$平方米);\\\\\n(2) 在灯笼内, 以矩形骨架的顶点为点, 安装一些霓虹灯, 当灯笼的底面半径为$0.3$米时, 求图中两根直线$A_1B_3$与$A_3B_5$所在异面直线所成角的大小(结果用反三角函数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276957,7 +277747,8 @@ "content": "若实数$x$、$y$、$m$满足$|x-m|>|y-m|$, 则称$x$比$y$远离$m$.\\\\\n(1) 若$x^2-1$比$1$远离$0$, 求$x$的取值范围;\\\\\n(2) 对任意两个不相等的正数$a$、$b$, 证明: $a^3+b^3$比$a^2b+ab^2$远离$2ab\\sqrt {ab}$;\\\\\n(3) 已知函数$f(x)$的定义域$D=\\{x| x\\ne \\dfrac{k\\pi }2+\\dfrac{\\pi }4, \\ k\\in \\mathbf{Z}, \\ x\\in \\mathbf{R}\\}$.任取$x\\in D$, $f(x)$等于$\\sin x$和$\\cos x$中远离$0$的那个值.写出函数$f(x)$的解析式, 并指出它的基本性质(结论不要求证明).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276978,7 +277769,8 @@ "content": "已知椭圆$\\Gamma$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 点$P$的坐标为$(-a, b)$.\\\\\n(1) 若直角坐标平面上的点$M$、$A(0, -b)$、$B(a, 0)$满足$\\overrightarrow{PM}=\\dfrac{1}{2}(\\overrightarrow{PA}+\\overrightarrow{PB})$, 求点$M$的坐标;\\\\\n(2) 设直线$l_1:y=k_1x+p$交椭圆$\\Gamma$于$C$、$D$两点, 交直线$l_2:y=k_2x$于点$E$. 若$k_1\\cdot k_2=-\\dfrac{b^2}{a^2}$, 证明: $E$为$CD$的中点;\\\\\n(3) 对于椭圆$\\Gamma$上的点$Q(a \\cos\\theta , b \\sin\\theta )$($0<\\theta <\\pi$), 如果椭圆$\\Gamma$上存在不同的两点$P_1$、$P_2$满足$\\overrightarrow{PP_1}+\\overrightarrow{PP_2}=\\overrightarrow{PQ}$, 写出求作点$P_1$、$P_2$的步骤, 并求出使$P_1$、$P_2$存在的$\\theta$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -276999,7 +277791,8 @@ "content": "函数$f(x)=\\dfrac 1{x-2}$的反函数为$f^{-1}(x)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277020,7 +277813,8 @@ "content": "若全集$U=\\mathbf{R}$, 集合$A=\\{x|x\\ge 1\\}\\cup \\{x|x\\le 0\\}$, 则$\\complement_UA=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277043,7 +277837,8 @@ "content": "设$m$为常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}m-\\dfrac{x^2}9=1$的一个焦点, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277066,7 +277861,8 @@ "content": "不等式$\\dfrac{x+1}x<3$的解为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277098,7 +277894,8 @@ "content": "在极坐标系中, 直线$\\rho (2\\cos \\theta +\\sin \\theta)=2$与直线$\\rho \\cos \\theta =1$的夹角大小为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277119,7 +277916,8 @@ "content": "在相距$2$千米的$A$、$B$两点处测量目标$C$, 若$\\angle CAB=75^\\circ$, $\\angle CBA=60^\\circ$, 则$A$、$C$两点之间的距离是\\blank{50}千米.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277140,7 +277938,8 @@ "content": "若圆锥的侧面积为$2\\pi$, 底面积为$\\pi$, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277161,7 +277960,8 @@ "content": "函数$y=\\sin (\\dfrac{\\pi}2+x)\\cos (\\dfrac{\\pi}6-x)$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277182,7 +277982,8 @@ "content": "马老师从课本上抄录一个随机变量$\\xi$的概率分布律如下表请小牛同学计算$\\xi$的数学期望, 尽管``!''处无法完全看清, 且两个``?''处字迹模糊, 但能肯定这两个``?''处的数值相同. 据此, 小牛给出了正确答案$E\\xi=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$x$ & $1$ & $2$ & $3$ \\\\ \\hline\n$P(\\xi=x)$ & ? & ! & ? \\\\ \\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277205,7 +278006,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277226,7 +278028,8 @@ "content": "在正三角形$ABC$中, $D$是$BC$上的点, $AB=3$, $BD=1$, 则$\\overrightarrow{AB}\\cdot \\overrightarrow{AD}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277247,7 +278050,8 @@ "content": "随机抽取$9$个同学中, 至少有$2$个同学在同一月出生的概率是\\blank{50}(默认每月天数相同, 结果精确到$0.001$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277270,7 +278074,8 @@ "content": "设$g(x)$是定义在$R$上、以$1$为周期的函数, 若$f(x)=x+g(x)$在$[3,4]$上的值域为$[-2,5]$, 则$f(x)$在区间$[-10,10]$上的值域为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277291,7 +278096,8 @@ "content": "已知点$O(0,0)$、$Q_0(0,1)$和$R_0(3,1)$, 记$Q_0R_0$的中点为$P_1$, 取$Q_0P_1$和$P_1R_0$中的一条, 记其端点为$Q_1$、$R_1$, 使之满足$(|OQ_1|-2)(|OR_1|-2)<0$; 记$Q_1R_1$的中点为$P_2$, 取$Q_1P_2$和$P_2R_1$中的一条, 记其端点为$Q_2$、$R_2$, 使之满足$(|OQ_2|-2)(|OR_2|-2)<0$; 依次下去, 得到点$P_1,P_2,\\cdots ,P_n,\\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty}|Q_0P_n|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277312,7 +278118,8 @@ "content": "若$a,b\\in \\mathbf{R}$, 且$ab>0$, 则下列不等式中, 恒成立的是\\bracket{20}.\n\\fourch{$a^2+b^2>2ab$}{$a+b\\ge 2\\sqrt {ab}$}{$\\dfrac 1a+\\dfrac 1b>\\dfrac 2{\\sqrt {ab}}$}{$\\dfrac ba+\\dfrac ab\\ge 2$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277335,7 +278142,8 @@ "content": "下列函数中, 既是偶函数, 又是在区间$(0,+\\infty)$上单调递减的函数为\\bracket{20}.\n\\fourch{$y=\\ln \\dfrac 1{|x|}$}{$y=x^3$}{$y=2^{|x|}$}{$y=\\cos x$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277358,7 +278166,8 @@ "content": "设$A_1,A_2,A_3,A_4,A_5$是空间中给定的$5$个不同的点, 则使$\\overrightarrow{MA_1}+\\overrightarrow{MA_2}+\\overrightarrow{MA_3}+\\overrightarrow{MA_4}+\\overrightarrow{MA_5}=\\overrightarrow 0$成立的点$M$的个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$5$}{$10$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277379,7 +278188,8 @@ "content": "设$\\{a_n\\}$是各项为正数的无穷数列, $A_i$是边长为$a_i,a_{i+1}$的矩形面积($i=1,2,\\cdots$), 则$\\{A_n\\}$为等比数列的充要条件为\\bracket{20}.\n\\onech{$\\{a_n\\}$是等比数列}{$a_1,a_3,\\cdots ,a_{2n-1},\\cdots$或$a_2,a_4,\\cdots ,a_{2n},\\cdots$是等比数列}{$a_1,a_3,\\cdots ,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots ,a_{2n},\\cdots$均是等比数列}{$a_1,a_3,\\cdots ,a_{2n-1},\\cdots$和$a_2,a_4,\\cdots ,a_{2n},\\cdots$均是等比数列, 且公比相同}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277402,7 +278212,8 @@ "content": "已知复数$z_1$满足$(z_1-2)(1+\\mathrm{i})=1-\\mathrm{i}$($\\mathrm{i}$为虚数单位), 复数$z_2$的虚部为$2$, $z_1\\cdot z_2$是实数, 求$z_2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277423,7 +278234,8 @@ "content": "已知函数$f(x)=a\\cdot 2^x+b\\cdot 3^x$, 其中常数$a,b$满足$ab\\ne 0$.\\\\\n(1) 若$ab>0$, 判断函数$f(x)$的单调性;\\\\\n(2) 若$ab<0$, 求$f(x+1)>f(x)$时$x$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277446,7 +278258,8 @@ "content": "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}[>=latex]\n\\draw (0,0) node [below left] {$B_1$} coordinate (A) --++ (2,0) node [below right] {$C_1$} coordinate (B) --++ (45:{2/2}) node [right] {$D_1$} coordinate (C)\n--++ (0,3) node [above right] {$D$} coordinate (C1)\n--++ (-2,0) node [above left] {$A$} coordinate (D1) --++ (225:{2/2}) node [left] {$B$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,3) node [right] {$C$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$A_1$} coordinate (D) --++ (2,0) (D) --++ (0,3);\n\\draw [dashed] (A) -- (C) (B) -- (D) (D1) -- (A) (D1) -- (C);\n\\draw ($(A)!0.5!(C)$) node [below] {$O_1$} coordinate (O1);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$AB_1$与底面$A_1B_1C_1D_1$所成的角的大小为$\\alpha$, 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$\\dfrac 43$, 求正四棱柱$ABCD-A_1B_1C_1D_1$的高.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277467,7 +278280,8 @@ "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别为$a_n=3n+6$, $b_n=2n+7$($n\\in \\mathbf{N}^*$), 将集合$\\{x|x=a_n,\\ n\\in \\mathbf{N}^*\\}\\cup \\{x|x=b_n,\\ n\\in \\mathbf{N}^*\\}$中的元素从小到大依次排列, 构成数列$c_1,c_2,c_3,\\cdots ,c_n,\\cdots$.\\\\\n(1) 求$c_1,c_2,c_3,c_4$;\\\\\n(2) 求证: 在数列$\\{c_n\\}$中、但不在数列$\\{b_n\\}$中的项恰为$a_2,a_4,\\cdots ,a_{2n},\\cdots$;\\\\\n(3) 求数列$\\{c_n\\}$的通项公式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277488,7 +278302,8 @@ "content": "已知平面上的线段$l$及点$P$, 在$l$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P,l)$.\\\\\n(1) 求点$P(1,1)$到线段$l:x-y-3=0$($3\\le x\\le 5$)的距离$d(P,l)$;\\\\\n(2) 设$l$是长为$2$的线段, 求点集$D=\\{P|d(P,l)\\le 1\\}$所表示图形的面积;\\\\\n(3) 写出到两条线段$l_1,l_2$距离相等的点的集合$\\Omega =\\{P|d(P,l_1)=d(P,l_2)\\}$, 其中$l_1=AB$, $l_2=CD$, $A,B,C,D$是下列三组点中的一组. 对于下列三组点只需选做一种, 满分分别是\\textcircled{1} 2分, \\textcircled{2} 6分, \\textcircled{3} 8分; 若选择了多于一种的情形, 则按照序号较小的解答计分.\\\\\n\\textcircled{1} $A(1,3)$, $B(1,0)$, $C(-1,3)$, $D(-1,0)$;\\\\\n\\textcircled{2} $A(1,3)$, $B(1,0)$, $C(-1,3)$, $D(-1,-2)$;\\\\\n\\textcircled{3} $A(0,1)$, $B(0,0)$, $C(0,0)$, $D(2,0)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277509,7 +278324,8 @@ "content": "计算: $\\dfrac{3-\\mathrm{i}}{1+\\mathrm{i}}=$\\blank{50}($\\mathrm{i}$为虚数单位).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277530,7 +278346,8 @@ "content": "若集合$A=\\{x|2x+1>0\\}$, $B=\\{x||x-1|<2\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277554,7 +278371,8 @@ "content": "函数$f(x)=\\begin{vmatrix} 2 & \\cos x \\\\ \\sin x & -1\\end{vmatrix}$的值域是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277575,7 +278393,8 @@ "content": "若$\\overrightarrow{n}=(-2,1)$是直线$l$的一个法向量, 则$l$的倾斜角的大小为\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277598,7 +278417,8 @@ "content": "在$(x-\\dfrac 2x)^6$的二项展开式中, 常数项等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277629,7 +278449,8 @@ "content": "有一列正方体, 棱长组成以$1$为首项、$\\dfrac 12$为公比的等比数列, 体积分别记为$V_1,V_2,\\cdots,V_n,\\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty}(V_1+V_2+\\cdots+V_n)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277650,7 +278471,8 @@ "content": "已知函数$f(x)=\\mathrm{e}^{|x-a|}$($a$为常数). 若$f(x)$在区间$[1,+\\infty)$上是增函数, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277671,7 +278493,8 @@ "content": "若一个圆锥的侧面展开图是面积为$2\\pi$的半圆面, 则该圆锥的体积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277692,7 +278515,8 @@ "content": "已知$f(x)+x^2$是奇函数, 且$f(1)=1$, 若$g(x)=f(x)+2$, 则$g(-1)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277713,7 +278537,8 @@ "content": "如图, 在极坐标系中, 过点$M(2,0)$的直线$l$与极轴的夹角$\\alpha=\\dfrac \\pi 6$, 若将$l$的极坐标方程写成$\\rho = f(\\theta)$的形式, 则$f(\\theta)=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) node [below] {$O$} -- (3,0) node [below] {$x$} coordinate (x);\n\\draw (1,0) ++ (-150:1) --++ (30:3) node [above] {$l$} coordinate (l);\n\\draw (1,0) node [below] {$M$} coordinate (M);\n\\draw (M) pic [draw, \"$\\alpha$\", angle eccentricity = 1.5] {angle = x--M--l};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277734,7 +278559,8 @@ "content": "三位同学参加跳高、跳远、铅球项目的比赛, 若每人都选择其中两个项目, 则有且仅有两人选择的项目完全相同的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277755,7 +278581,8 @@ "content": "在平行四边形$ABCD$中, $\\angle A=\\dfrac \\pi 3$, 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": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277776,7 +278603,8 @@ "content": "已知函数$y=f(x)$的图像是折线段$ABC$, 其中$A(0,0)$、$B(\\dfrac 12,5)$、$C(1,0)$, 函数$y=xf(x)$($0\\le x\\le 1$)的图像与$x$轴围成的图形的面积为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277797,7 +278625,8 @@ "content": "如图, $AD$与$BC$是四面体$ABCD$中互相垂直的棱, $BC=2$, 若$AD=2c$,\n且$AB+BD=AC=CD=2a$, 其中$a$、$c$为常数, 则四面体$ABCD$的体积的最大值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,-1,0) node [below] {$A$} coordinate (A);\n\\draw (0,1.5,0) node [above] {$D$} coordinate (D);\n\\draw (2,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,-1) node [right] {$C$} coordinate (C);\n\\draw (A) -- (D) -- (C) -- (B) -- cycle (B) -- (D);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -277818,7 +278647,8 @@ "content": "若$1+2\\mathrm{i}$是关于$x$的实系数方程$x^2+bx+c=0$的一个复数根, 则\\bracket{20}.\n\\fourch{$b=2$, $c=3$}{$b=-2$, $c=3$}{$b=-2$, $c=-1$}{$b=2$, $c=-1$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277839,7 +278669,8 @@ "content": "在$\\triangle ABC$中, 若$\\sin^2 A+\\sin^2 B<\\sin^2 C$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{不能确定}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -277860,7 +278691,8 @@ "content": "设$10\\le x_1D\\xi_2$}{$D\\xi_1=D\\xi_2$}{$D\\xi_1=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- cycle (P) -- (C);\n\\draw [dashed] (A) -- (P) (B) -- (A) -- (D) (A) -- ($(P)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 三角形$PCD$的面积;\\\\\n(2) 异面直线$BC$与$AE$所成的角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277923,7 +278757,8 @@ "content": "已知函数$f(x)=\\lg (x+1)$.\\\\\n(1) 若$0=latex, scale = 0.4]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3, ->] plot (\\x,{pow(\\x,2)/3}) node [above] {$P$} coordinate (P);\n\\draw (0,-3) node [left] {$A$} coordinate (A);\n\\draw [->] (A) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$t=0.5$时, 写出失事船所在位置$P$的纵坐标. 若此时两船恰好会合, 求救援船速度的大小和方向;\\\\\n(2) 问救援船的时速至少是多少海里才能追上失事船?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277965,7 +278801,8 @@ "content": "在平面直角坐标系$xOy$中, 已知双曲线$C_1:2x^2-y^2=1$.\\\\\n(1) 过$C_1$的左顶点引$C_1$的一条渐近线的平行线, 求该直线与另一条渐近线及$x$轴围成的三角形的面积;\\\\\n(2) 设斜率为$1$的直线$l$交$C_1$于$P$、$Q$两点, 若$l$与圆$x^2+y^2=1$相切, 求证: $OP\\perp OQ$;\\\\\n(3) 设椭圆$C_2:4x^2+y^2=1$. 若$M$、$N$分别是$C_1$、$C_2$上的动点, 且$OM\\perp ON$, 求证: $O$到直线$MN$的距离是定值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -277986,7 +278823,8 @@ "content": "对于数集$X=\\{-1,x_1,x_2,\\cdots,x_n\\}$, 其中$0\\le x_12$, 且$\\{-1,1,2,x\\}$具有性质$P$, 求$x$的值;\\\\\n(2) 若$X$具有性质$P$, 求证: $1\\in X$, 且当$x_n>1$时, $x_1=1$;\\\\\n(3) 若$X$具有性质$P$, 且$x_1=1$, $x_2=q$($q$为常数), 求有穷数列$x_1,x_2,\\cdots,x_n$的通项公式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278007,7 +278845,8 @@ "content": "计算: $\\displaystyle\\lim_{n\\to\\infty} \\dfrac{n+20}{3n+13}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278046,7 +278885,8 @@ "content": "设$m\\in \\mathbf{R}$, $m^2+m-2+(m^2-1)\\mathrm{i}$是纯虚数, 其中$\\mathrm{i}$是虚数单位, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278067,7 +278907,8 @@ "content": "若$\\begin{vmatrix}\nx^2 & y^2 \\\\-1 & 1 \\end{vmatrix}=\\begin{vmatrix}\nx & x \\\\y & -y \\end{vmatrix}$, 则$x+y=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278090,7 +278931,8 @@ "content": "已知$\\triangle ABC$的内角$A$、$B$、$C$所对应边分别为$a$、$b$、$c$, 若$3a^2+2ab+3b^2-3c^2=0$, 则角$C$的大小是\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278111,7 +278953,8 @@ "content": "设常数$a\\in \\mathbf{R}$, 若$(x^2+\\dfrac ax)^5$的二项展开式中$x^7$项的系数为$-10$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278134,7 +278977,8 @@ "content": "方程$\\dfrac 3{3^x-1}+\\dfrac 13=3^{x-1}$的实数解为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278155,7 +278999,8 @@ "content": "在极坐标系中, 曲线$\\rho =\\cos \\theta +1$与$\\rho \\cos \\theta =1$的公共点到极点的距离为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278178,7 +279023,8 @@ "content": "盒子中装有编号为$1, 2, 3, 4, 5, 6, 7, 8, 9$的九个球, 从中任意取出两个, 则这两个球的编号之积为偶数的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278199,7 +279045,8 @@ "content": "设$AB$是椭圆$\\Gamma$的长轴, 点$C$在$\\Gamma$上, 且$\\angle CBA=\\dfrac{\\pi}4$, 若$AB=4$, $BC=\\sqrt 2$, 则$\\Gamma$的两个焦点之间的距离为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278220,7 +279067,8 @@ "content": "设非零常数$d$是等差数列$x_1,x_2,x_3,\\cdots ,x_{19}$的公差, 随机变量$\\xi$等可能地取值$x_1,x_2,x_3,\\cdots ,x_{19}$, 则方差$D\\xi =$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278241,7 +279089,8 @@ "content": "若$\\cos x\\cos y+\\sin x\\sin y=\\dfrac 12$, $\\sin 2x+\\sin 2y=\\dfrac 23$, 则$\\sin (x+y)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278262,7 +279111,8 @@ "content": "设$a$为实常数, $y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 当$x<0$时, $f(x)=9x+\\dfrac{a^2}x+7$, 若$f(x)\\ge a+1$对一切$x\\ge 0$成立, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278283,7 +279133,8 @@ "content": "在$xOy$平面上, 将两个半圆弧$(x-1)^2+y^2=1(x\\ge 1)$和$(x-3)^2+y^2=1(x\\ge 3)$、两条直线$y=1$和$y=-1$围成的封闭图形记为$D$, 如图中阴影部分. 记$D$绕$y$轴旋转一周而成的几何体为$\\Omega$, 过$(0,y)$($|y|\\le 1$)作$\\Omega$的水平截面, 所得截面面积为$4\\pi \\sqrt {1-y^2}+8\\pi$, 试利用祖暅原理、一个平放的圆柱和一个长方体, 得出$\\Omega$的体积值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw [pattern = north east lines] (1,1) -- (3,1) arc (90:-90:1) -- (1,-1) arc (-90:90:1);\n\\foreach \\i in {1,2,3,4}{\\draw (\\i,0.1) -- (\\i,0) node [fill = white, below] {$\\i$};};\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-1,1}{\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278304,7 +279155,8 @@ "content": "对区间$I$上有定义的函数$g(x)$, 记$g(I)=\\{y|y=g(x),x\\in I\\}$, 已知定义域为$[0,3]$的函数$y=f(x)$有反函数$y=f^{-1}(x)$, 且$f^{-1}([0,1))=[1,2)$, $f^{-1}((2,4])=[0,1)$, 若方程$f(x)-x=0$有解$x_0$, 则$x_0=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278325,7 +279177,8 @@ "content": "设常数$a\\in \\mathbf{R}$, 集合$A=\\{x|(x-1)(x-a)\\ge 0\\}$, $B=\\{x|x\\ge a-1\\}$, 若$A\\cup B=\\mathbf{R}$, 则$a$的取值范围为\\bracket{20}.\n\\fourch{$(-\\infty ,2)$}{$(-\\infty ,2]$}{$(2,+\\infty)$}{$[2,+\\infty)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278346,7 +279199,8 @@ "content": "钱大姐常说``便宜没好货'', 她这句话的意思是: ``不便宜''是``好货''的\\bracket{20}.\n\\twoch{充分条件}{必要条件}{充分必要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278369,7 +279223,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278390,7 +279245,8 @@ "content": "在边长为$1$的正六边形$ABCDEF$中, 记以$A$为起点, 其余顶点为终点的向量分别为$\\overrightarrow{a_1},\\overrightarrow{a_2},\\overrightarrow{a_3},\\overrightarrow{a_4},\\overrightarrow{a_5}$; 以$D$为起点, 其余顶点为终点的向量分别为$\\overrightarrow{d_1},\\overrightarrow{d_2},\\overrightarrow{d_3},\\overrightarrow{d_4},\\overrightarrow{d_5}$. 若$m,M$分别为$(\\overrightarrow{a_i}+\\overrightarrow{a_j}+\\overrightarrow{a_k})\\cdot (\\overrightarrow{d_r}+\\overrightarrow{d_s}+\\overrightarrow{d_t})$的最小值、最大值, 其中$\\{i,j,k\\}\\subseteq \\{1,2,3,4,5\\}$, $\\{r,s,t\\}\\subseteq \\{1,2,3,4,5\\}$, 则$m,M$满足\\bracket{20}.\n\\fourch{$m=0$, $M>0$}{$m<0$, $M>0$}{$m<0$, $M=0$}{$m<0$, $M<0$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278411,7 +279267,8 @@ "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=2$, $AD=1$, $A_1A=1$, 证明直线$BC_1$平行于平面$DA_1C$, 并求直线$BC_1$到平面$D_1AC$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{1/2}) node [right] {$C$} coordinate (C)\n--++ (0,1) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{1/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{1/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{1/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw (B) -- (C1);\n\\draw [dashed] (A) -- (C) -- (D1) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278432,7 +279289,8 @@ "content": "甲厂以$x$千克/小时的速度运输生产某种产品(生产条件要求$1\\le x\\le 10$), 每小时可获得利润是$100(5x+1-\\dfrac 3x)$元.\\\\\n(1) 要使生产该产品$2$小时获得的利润不低于$3000$元, 求$x$的取值范围;\\\\\n(2) 要使生产$900$千克该产品获得的利润最大, 问: 甲厂应该选取何种生产速度? 并求最大利润.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278453,7 +279311,8 @@ "content": "已知函数$f(x)=2\\sin (\\omega x)$, 其中常数$\\omega >0$.\\\\\n(1) 若$y=f(x)$在$[-\\dfrac{\\pi }4,\\dfrac{2\\pi }3]$上单调递增, 求$\\omega$的取值范围;\\\\\n(2) 令$\\omega =2$, 将函数$y=f(x)$的图像向左平移$\\dfrac{\\pi }6$个单位, 再向上平移$1$个单位, 得到函数$y=g(x)$的图像, 区间$[a,b]$($a,b\\in \\mathbf{R}$且$a1$, 进而证明原点不是``$C_1-C_2$型点'';\\\\\n(3) 求证: 圆$x^2+y^2=\\dfrac 12$内的点都不是``$C_1-C_2$型点''.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278495,7 +279355,8 @@ "content": "给定常数$c>0$, 定义函数$f(x)=2|x+c+4|-|x+c|$, 数列$a_1,a_2,a_3,\\cdots$满足$a_{n+1}=f(a_n)$, $n\\in \\mathbf{N}^*$.\\\\\n(1) 若$a_1=-c-2$, 求$a_2$及$a_3$;\\\\\n(2) 求证: 对任意$n\\in \\mathbf{N}^*$, $a_{n+1}-a_n\\ge c$;\\\\\n(3) 是否存在$a_1$, 使得$a_1,a_2,\\cdots a_n,\\cdots$成等差数列? 若存在, 求出所有这样的$a_1$, 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278516,7 +279377,8 @@ "content": "函数$y=1-2\\cos ^2(2x)$的最小正周期是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278542,7 +279404,8 @@ "content": "若复数$z=1+2\\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$(z+\\dfrac 1{\\overline z})\\cdot \\overline z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278563,7 +279426,8 @@ "content": "抛物线$y^2=2px$的焦点与椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{5}=1$的右焦点重合, 则该抛物线的准线方程为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278584,7 +279448,8 @@ "content": "设$f(x)=\\begin{cases}x, & x\\in (\\infty,a),\\\\ x^2, & x\\in [a,+\\infty),\\end{cases}$若$f(2)=4$, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278605,7 +279470,8 @@ "content": "若实数$x,y$满足$xy=1$, 则$x^2+2y^2$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278626,7 +279492,8 @@ "content": "若圆锥的侧面积是底面积的$3$倍, 则其母线与底面角的大小为\\blank{50}(结果用反三角函数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278647,7 +279514,8 @@ "content": "已知曲线$C$的极坐标方程为$\\rho(3\\cos\\theta-4\\sin\\theta)=1$, 则$C$与极轴的交点到极点的距离是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278668,7 +279536,8 @@ "content": "设无穷等比数列$\\{a_n\\}$的公比为$q$, 若$a_1=\\displaystyle\\lim_{n\\to \\infty}(a_3+a_4+\\cdots)$, 则$q=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278691,7 +279560,8 @@ "content": "若$f(x)=x^\\frac 23-x^\\frac 12$, 则满足$f(x)<0$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278715,7 +279585,8 @@ "content": "为强化安全意识, 某商场拟在未来的连续$10$天中随机选择$3$天进行紧急疏散演练, 则选择的$3$天恰好为连续$3$天的概率是\\blank{50}(结果用最简分数表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278736,7 +279607,8 @@ "content": "已知互异的复数$a, b$满足$ab\\ne 0$, 集合$\\{a, b\\}=\\{a^2,b^2\\}$, 则$a+b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278757,7 +279629,8 @@ "content": "设常数$a$使方程$\\sin x+\\sqrt{3}\\cos x=a$在闭区间$[0, 2\\pi]$上恰有三个解$x_1,x_2,x_3$, 则$x_1+x_2+x_3=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278778,7 +279651,8 @@ "content": "某游戏的得分为$1, 2, 3, 4, 5$, 随机变量$\\xi$表示小白玩游戏的得分. 若$E\\xi = 4.2$, 则小白得$5$分的概率至少为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278799,7 +279673,8 @@ "content": "已知曲线$C:x=-\\sqrt{4-y^2}$, 直线$l: x=6$. 若对于点$A(m, 0)$, 存在$C$上的点$P$和$l$上的点$Q$使得$\\overrightarrow{AP}+\\overrightarrow{AQ}=\\overrightarrow 0$, 则$m$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -278823,7 +279698,8 @@ "content": "设$a,b\\in \\mathbf{R}$, 则``$a+b>4$''是``$a>2$且$b>2$''的\\bracket{20}.\n\\twoch{充分条件}{必要条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278844,7 +279720,8 @@ "content": "如图, 四个棱长为$1$的正方体排成一个正四棱柱, $AB$是一条侧棱, $P_i$($i=1,2,\\cdots$)是上底面上其余的八个点, 则$\\overrightarrow{AB}\\cdot \\overrightarrow{AP_i}$($i=1,2,\\cdots$)的不同值的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) coordinate (B) --++ (45:{2/2}) coordinate (C)\n--++ (0,1) coordinate (C1) node [above left] {$P_8$}\n--++ (-2,0) coordinate (D1) node [above left] {$P_2$} --++ (225:{2/2}) node [above left] {$B$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) coordinate (B1) node [above left] {$P_6$} -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$P_1$} coordinate (P1);\n\\draw ($(B1)!0.5!(C1)$) node [above left] {$P_7$} coordinate (P7);\n\\draw ($(A1)!0.5!(B1)$) node [above left] {$P_3$} coordinate (P3);\n\\draw ($(P1)!0.5!(P7)$) node [above left] {$P_4$} coordinate (P4);\n\\draw ($(C1)!0.5!(D1)$) node [above left] {$P_5$} coordinate (P5);\n\\draw (P1) -- (P7) (P3) -- (P5) (P3) --++ (0,-1) coordinate (S) (P7) --++ (0,-1);\n\\draw [dashed] (P1) --++ (0,-1) --++ (2,0) (P5) --++ (0,-1) -- (S);\n\\draw [dashed] (P4) --++ (0,-1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$2$}{$4$}{$8$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278865,7 +279742,8 @@ "content": "已知$P_1(a_1,b_1)$与$P_2(a_2,b_2)$是直线$y=kx+1$($k$为常数)上两个不同的点, 则关于$x$和$y$的方程组$\\begin{cases} a_1x+b_1y=1, \\\\ a_2x+b_2y=1 \\end{cases}$的解的情况是\\bracket{20}.\n\\twoch{无论$k,P_1,P_2$如何, 总是无解}{无论$k,P_1,P_2$如何, 总有唯一解}{存在$k,P_1,P_2$, 使之恰有两解}{存在$k,P_1,P_2$, 使之有无穷多解}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278886,7 +279764,8 @@ "content": "设$f(x)=\\begin{cases}(x-a)^2, & x\\le 0, \\\\ x+\\dfrac 1x+a, & x>0. \\end{cases}$若$f(0)$是$f(x)$的最小值, 则$a$的取值范围为\\bracket{20}.\n\\fourch{$[-1, 2]$}{$[-1, 0]$}{$[1, 2]$}{$[0,2]$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -278907,7 +279786,8 @@ "content": "底面边长为$2$的正三棱锥$P-ABC$, 其表面展开图是三角形$P_1P_2P_3$, 如图, 求$\\triangle P_1P_2P_3$的各边长及此三棱锥的体积$V$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$C$} coordinate (C);\n\\draw (-60:1) node [below] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C) -- cycle;\n\\draw (A) ++ (60:1) node [above] {$P_2$} coordinate (P2);\n\\draw (A) ++ (-120:1) node [left] {$P_3$} coordinate (P3);\n\\draw (P3) ++ (2,0) node [right] {$P_1$} coordinate (P1);\n\\draw [dashed] (P1) -- (P2) -- (P3) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278928,7 +279808,8 @@ "content": "设常数$a\\ge 0$, 函数$f(x)=\\dfrac{2^x+a}{2^x-a}$\\\\\n(1) 若$a=4$, 求函数$y=f(x)$的反函数$y=f^{-1}(x)$;\\\\\n(2) 根据$a$的不同取值, 讨论函数$y=f(x)$的奇偶性, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278949,7 +279830,8 @@ "content": "如图, 某公司要在$A,B$两地连线上的定点$C$处建造广告牌$CD$, 其中$D$为顶端, $AC$长$35$米, $CB$长$80$米, 设$A,B$在同一水平面上, 从$A$和$B$看$D$的仰角分别为$\\alpha$和$\\beta$.\n(1) 设计中$CD$是铅垂方向, 若要求$\\alpha\\ge 2\\beta$, 问$CD$的长至多为多少(结果精确到$0.01$米)?\\\\\n(2) 施工完成后, $CD$与铅垂方向有偏差, 现在实测得$\\alpha=38.12^\\circ$, $\\beta=18.45^\\circ$, 求$CD$的长(结果精确到$0.01$米)?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0) node [below] {$B$} coordinate (B);\n\\draw (1.5,1.7) node [above] {$D$} coordinate (D);\n\\draw ($(A)!(D)!(B)$) node [below] {$C$} coordinate (C);\n\\draw (A) -- (B) -- (D) -- cycle (D) -- (C);\n\\draw (A) pic [\"$\\alpha$\",draw, angle eccentricity = 1.5] {angle = C--A--D};\n\\draw (B) pic [\"$\\beta$\",draw, angle eccentricity = 1.5] {angle = D--B--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278970,7 +279852,8 @@ "content": "在平面直角坐标系$xOy$中, 对于直线$l:ax+by+c=0$和点$P_1(x_1,y_1)$, $P_2(x_2,y_2)$, 记$\\eta=(ax_1+by_1+c)(ax_2+by_2+c)$. 若$\\eta<0$, 则称点$P_1,P_2$被直线$l$分隔. 若曲线$C$与直线$l$没有公共点, 且曲线$C$上存在点$P_1,P_2$被直线$l$分隔, 则称直线$l$为曲线$C$的一条分隔线.\\\\\n(1) 求证: 点$A(1,2)$, $B(-1,0)$被直线$x+y-1=0$分隔;\\\\\n(2) 若直线$y=kx$是曲线$x^2-4y^2=1$的分隔线, 求实数$k$的取值范围;\\\\\n(3) 动点$M$到点$Q(0,2)$的距离与到$y$轴的距离之积为$1$, 设点$M$的轨迹为$E$, 求证: 通过原点的直线中, 有且仅有一条直线是$E$的分割线.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -278991,7 +279874,8 @@ "content": "已知数列$\\{a_n\\}$满足$\\dfrac 13a_n\\le a_{n+1}\\le 3a_n$, $n\\in \\mathbf{N}^*$, $a_1=1$.\\\\\n(1) 若$a_2=2$, $a_3=x$, $a_4=9$, 求$x$的取值范围;\\\\\n(2) 若$\\{a_n\\}$是公比为$q$等比数列. 记$S_n=a_1+a_2+\\cdots+a_n$, 若$\\dfrac13 S_n\\le S_{n+1}\\le 3S_n$, $n\\in \\mathbf{N}^*$, 求$q$的取值范围;\\\\\n(3) 若$a_1,a_2,\\cdots,a_k$成等差数列, 且$a_1+a_2+\\cdots+a_k=1000$, 求正整数$k$的最大值, 以及$k$取最大值时相应数列$a_1,a_2,\\cdots,a_k$的公差.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279012,7 +279896,8 @@ "content": "设全集$U=\\mathbf{R}$. 若集合$A=\\{1,2,3,4\\}$, $B=\\{x|2\\le x\\le 3\\}$, 则$A\\cap \\overline B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279033,7 +279918,8 @@ "content": "若复数$z$满足$3z+\\overline z=1+\\mathrm{i}$, 其中$\\mathrm{i}$为虚数单位, 则$z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279058,7 +279944,8 @@ "KNONE" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279082,7 +279969,8 @@ "content": "若正三棱柱的所有棱长均为$a$, 且其体积为$16\\sqrt 3$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279103,7 +279991,8 @@ "content": "抛物线$y^2=2px$($p>0$)上的动点$Q$到焦点的距离的最小值为$1$, 则$p=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279124,7 +280013,8 @@ "content": "若圆锥的侧面积与过轴的截面面积之比为$2\\pi$, 则其母线与轴的夹角的大小为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279145,7 +280035,8 @@ "content": "方程$\\log_2(9^{x-1}-5)=\\log_2(3^{x-1}-2)+2$的解为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279169,7 +280060,8 @@ "content": "在报名的$3$名男教师和$6$名女教师中, 选取$5$人参加义务献血, 要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279194,7 +280086,8 @@ "content": "已知点$P$和$Q$的横坐标相同, $P$的纵坐标是$Q$的纵坐标的$2$倍, $P$和$Q$的轨迹分别为双曲线$C_1$和$C_2$. 若$C_1$的渐近线方程为$y=\\pm \\sqrt 3x$, 则$C_2$的渐近线方程为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279215,7 +280108,8 @@ "content": "设$f^{-1}(x)$为$f(x)=2^{x-2}+\\dfrac x2$, $x\\in [0,2]$的反函数, 则$y=f(x)+f^{-1}(x)$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279236,7 +280130,8 @@ "content": "在$(1+x+\\dfrac 1{x^{2015}})^{10}$的展开式中, $x^2$项的系数为\\blank{50}(结果用数值表示).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279257,7 +280152,8 @@ "content": "赌博有陷阱. 某种赌博每局的规则是: 赌客先在标记有$1,2,3,4,5$的卡片中随机摸取一张, 将卡片上的数字作为其赌金(单位: 元); 随后放回该卡片, 再随机摸取两张, 将这两张卡片上数字之差的绝对值的$1.4$倍作为其奖金(单位: 元). 若随机变量$\\xi _1$和$\\xi _2$分别表示赌客在一局赌博中的赌金和奖金, 则$E \\xi _1-E \\xi _2=$\\blank{50}(元).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279278,7 +280174,8 @@ "content": "已知函数$f(x)=\\sin x$. 若存在$x_1$, $x_2$, $\\cdots$, $x_m$满足$0\\le x_1=latex,scale = 1.5]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,1) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F) (C) -- (D1);\n\\draw (A1) -- (E) (C1) -- (F) (A1) -- (C1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279427,7 +280330,8 @@ "content": "如图, $A$, $B$, $C$三地有直道相通, $AB=5$千米, $AC=3$千米, $BC=4$千米. 现甲、乙两警员同时从$A$地出发匀速前往$B$地, 经过$t$小时, 他们之间的距离为$f(t)$(单位: 千米).甲的路线是$AB$, 速度为$5$千米/小时, 乙的路线是$ACB$, 速度为$8$千米/小时. 乙到达$B$地后原地等待. 设$t=t_1$时乙到达$C$地.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (A) --++ ({atan(4/3)}:3) node [above] {$C$} coordinate (C);\n\\draw (5,0) node [right] {$B$} coordinate (B);\n\\draw (A) -- (B) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$t_1$与$f(t_1)$的值;\\\\\n(2) 已知警员的对讲机的有效通话距离是$3$千米.当$t_1\\le t\\le 1$时, 求$f(t)$的表达式, 并判断$f(t)$在$[t_1,1]$上的最大值是否超过$3$? 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279448,7 +280352,8 @@ "content": "已知椭圆$x^2+2y^2=1$, 过原点的两条直线$l_1$和$l_2$分别于椭圆交于$A$、$B$和$C$、$D$, 记得到的平行四边形$ABCD$的面积为$S$.\\\\\n(1) 设$A(x_1,y_1)$, $C(x_2,y_2)$, 用$A$、$C$的坐标表示点$C$到直线$l_1$的距离, 并证明$S=2|x_1y_1-x_2y_1|$;\\\\\n(2) 设$l_1$与$l_2$的斜率之积为$-\\dfrac 12$, 求面积$S$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279469,7 +280374,8 @@ "content": "已知数列$\\{a_n\\}$与$\\{b_n\\}$满足$a_{n+1}-a_n=2(b_{n+1}-b_n)$, $n\\in \\mathbf{N}^*$.\\\\\n(1) 若$b_n=3n+5$, 且$a_1=1$, 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$\\{a_n\\}$的第$n_0$项是最大项, 即$a_{n_0}>a_n$($n\\in \\mathbf{N}^*$), 求证: 数列$\\{b_n\\}$的第$n_0$项是最大项;\\\\\n(3) 设$a_1=\\lambda <0$, $b_n=\\lambda ^n$($n\\in \\mathbf{N}^*$), 求$\\lambda$的取值范围, 使得$\\{a_n\\}$有最大值$M$与最小值$m$, 且$\\dfrac M m\\in (-2,2)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279490,7 +280396,8 @@ "content": "对于定义域为$\\mathbf{R}$的函数$g(x)$, 若存在正常数$T$, 使得$\\cos g(x)$是以$T$为周期的函数, 则称$g(x)$为余弦周期函数, 且称$T$为其余弦周期. 已知$f(x)$是以$T$为余弦周期的余弦周期函数, 其值域为$\\mathbf{R}$. 设$f(x)$单调递增, $f(0)=0$, $f(T)=4\\pi$.\\\\\n(1) 验证$h(x)=x+\\sin \\dfrac x3$是以$6\\pi$为周期的余弦周期函数;\\\\\n(2) 设$a=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (1.5,0) node [below right] {$B$} coordinate (B) --++ (45:{1.5/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-1.5,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{1.5/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (1.5,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{1.5/2}) (B1) --++ (-1.5,0);\n\\draw [dashed] (A) --++ (45:{1.5/2}) node [left] {$D$} coordinate (D) --++ (1.5,0) (D) --++ (0,2);\n\\draw [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279639,7 +280552,8 @@ "content": "方程$3\\sin x=1+\\cos 2x$在区间$[0,2\\pi]$上的解为\\blank{50}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279660,7 +280574,8 @@ "content": "在$(\\sqrt[3]x-\\dfrac 2x)^n$的二项式中, 所有项的二项式系数之和为$256$, 则常数项等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279683,7 +280598,8 @@ "content": "已知$\\triangle ABC$的三边长分别为$3, 5, 7$, 则该三角形的外接圆半径等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279704,7 +280620,8 @@ "content": "设$a>0$, $b>0$. 若关于$x,y$的方程组$\\begin{cases}\nax+y=1, \\\\ x+by=1 \\end{cases}$无解, 则$a+b$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279725,7 +280642,8 @@ "content": "无穷数列$\\{a_n\\}$由$k$个不同的数组成, $S_n$为$\\{a_n\\}$的前$n$项和.若对任意$n\\in \\mathbf{N}^*$, $S_n\\in \\{2,3\\}$, 则$k$的最大值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279748,7 +280666,8 @@ "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": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279771,7 +280690,8 @@ "content": "设$a,b\\in \\mathbf{R}$, $c\\in [0,2\\pi)$, 若对任意实数$x$都有$2\\sin (3x-\\dfrac{\\pi }3)=a\\sin (bx+c)$, 则满足条件的有序实数组$(a,b,c)$的组数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279792,7 +280712,8 @@ "content": "如图, 在平面直角坐标系$xOy$中, $O$为正八边形$A_1A_2\\cdots A_8$的中心, $A_1(1,0)$. 任取不同的两点$A_i,A_j$, 点$P$满足$\\overrightarrow{OP}+\\overrightarrow{OA_i}+\\overrightarrow{OA_j}=\\overrightarrow 0$, 则点$P$落在第一象限的概率是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0:1) node [above right] {$A_1$} coordinate (A1) -- (45:1) node [above right] {$A_2$} coordinate (A2) -- (90:1) node [above left] {$A_3$} coordinate (A3) -- (135:1) node [above left] {$A_4$} coordinate (A4) -- (180:1) node [below left] {$A_5$} coordinate (A5) -- (225:1) node [below left] {$A_6$} coordinate (A6) -- (270:1) node [below right] {$A_7$} coordinate (A7) -- (315:1) node [below right] {$A_8$} coordinate (A8) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -279813,7 +280734,8 @@ "content": "设$a\\in \\mathbf{R}$, 则``$a>1$''是``$a^2>1$''的\\bracket{20}.\t\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -279837,7 +280759,8 @@ "content": "下列极坐标方程中, 对应的曲线为下图的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\draw [domain = 0:360, samples = 100] plot ({(6-5*sin(\\x))*cos(\\x)},{(6-5*sin(\\x))*sin(\\x)});\n\\draw [->] (0,0) node [below] {$O$} -- (12,0) node [below] {$x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\rho=6+5\\cos\\theta$}{$\\rho =6+5\\sin\\theta$}{$\\rho =6-5\\cos\\theta$}{$\\rho =6-5\\sin \\theta$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -279858,7 +280781,8 @@ "content": "已知无穷等比数列$\\{a_n\\}$的公比为$q$, 前n项和为$S_n$, 且$\\displaystyle\\lim_{n\\to\\infty} S_n=S$.下列条件中, 使得$2S_n0$, $0.60$, $0.7=latex]\n\\draw (2,0) node [right] {$A$} coordinate (A) arc (0:-180:2 and 0.5);\n\\draw [dashed] (2,0) arc (0:180:2 and 0.5);\n\\draw (A) --++ (0,2) node [right] {$A_1$} coordinate (A1) arc (0:360:2 and 0.5);\n\\draw (-2,0) -- (-2,2);\n\\draw (0,0) node [below] {$O$} coordinate (O) (0,2) node [above] {$O_1$} coordinate (O1);\n\\draw [dashed] (A) -- (O) -- (O1) -- (A1);\n\\draw ({2*cos(-130)},{0.5*sin(-130)}) node [below] {$C$} coordinate (C);\n\\draw (0,2) ++ ({2*cos(-70)},{0.5*sin(-70)}) node [below] {$B_1$} coordinate (B1);\n\\draw [dashed] (O) -- (C) -- (B1) -- (O1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$C-O_1A_1B_1$的体积;\\\\\n(2) 求异面直线$B_1C$与$AA_1$所成的角的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279923,7 +280849,8 @@ "content": "有一块正方形菜地$EFGH$, $EH$所在直线是一条小河, 收货的蔬菜可送到$F$点或河边运走. 于是, 菜地分为两个区域$S_1$和$S_2$, 其中$S_1$中的蔬菜运到河边较近, $S_2$中的蔬菜运到$F$点较近, 而菜地内$S_1$和$S_2$的分界线$C$上的点到河边与到$F$点的距离相等, 现建立平面直角坐标系, 其中原点$O$为$EF$的中点, 点$F$的坐标为$(1, 0)$, 如图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,0) node [below] {$E$} coordinate (E) rectangle (2,4) node [above] {$G$} coordinate (G);\n\\draw (2,0) node [below] {$F$} coordinate (F);\n\\draw (-2,4) node [above] {$H$} coordinate (H);\n\\draw (-1,2.5) node {$S_1$};\n\\draw (1,1) node {$S_2$};\n\\draw [very thick, domain = 0:4, samples = 100] plot ({pow(\\x,2)/8},\\x);\n\\filldraw ({1/2},2) circle (0.05) node [right] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求菜地内的分界线$C$的方程;\\\\\n(2) 菜农从蔬菜运量估计出$S_1$面积是$S_2$面积的两倍, 由此得到$S_1$面积的``经验值''为$\\dfrac 83$. 设$M$是$C$上纵坐标为$1$的点, 请计算以$EH$为一边、另一边过点$M$的矩形的面积, 及五边形$EOMGH$的面积, 并判断哪一个更接近于$S_1$面积的经验值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279944,7 +280871,8 @@ "content": "双曲线$x^2-\\dfrac{y^2}{b^2}=1(b>0)$的左、右焦点分别为$F_1$、$F_2$, 直线$l$过$F_2$且与双曲线交于$AB$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi }2$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt 3$, 若$l$的斜率存在, 且$(\\overrightarrow{F_1A}+\\overrightarrow{F_1B})\\cdot \\overrightarrow{AB}=0$, 求$l$的斜率.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279967,7 +280895,8 @@ "content": "已知$a\\in \\mathbf{R}$, 函数$f(x)=\\log_2(\\dfrac 1x+a)$.\\\\\n(1) 当$a=5$时, 解不等式$f(x)>0$;\\\\\n(2) 若关于$x$的方程$f(x)-\\log_2[(a-4)x+2a-5]=0$的解集中恰好有一个元素, 求$a$的取值范围;\\\\\n(3) 设$a>0$, 若对任意$t\\in [\\dfrac 12,1]$, 函数$f(x)$在区间$[t,t+1]$上的最大值与最小值的差不超过$1$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -279988,7 +280917,8 @@ "content": "若无穷数列$\\{a_n\\}$满足: 只要$a_p=a_q$($p,q\\in \\mathbf{N}^*$), 必有$a_{p+1}=a_{q+1}$, 则称$\\{a_n\\}$具有性质$P$.\\\\\n(1) 若$\\{a_n\\}$具有性质$P$, 且$a_1=1,a_2=2,a_4=3,a_5=2$, $a_6+a_7+a_8=21$, 求$a_3$;\\\\\n(2) 若无穷数列$\\{b_n\\}$是等差数列, 无穷数列$\\{c_n\\}$是公比为正数的等比数列, $b_1=c_5=1$, $b_5=c_1=81$, $a_n=b_n+c_n$. 判断$\\{a_n\\}$是否具有性质$P$, 并说明理由;\\\\\n(3) 设$\\{b_n\\}$是无穷数列, 已知$a_{n+1}=b_n+\\sin a_n$($n\\in \\mathbf{N}^*)$. 求证: ``对任意$a_1$, $\\{a_n\\}$都具有性质$P$''的充要条件为``$\\{b_n\\}$是常数列''.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280009,7 +280939,8 @@ "content": "判断下列各组对象能否组成集合, 并请说明理由.\\\\\n(1) 方程$x^2+x+3=0$的所有实数解;\\\\\n(2) 比较难的数学题;\\\\\n(3) 一部分末位是$3$的自然数;\\\\\n(4) 太阳、$2$、上海市.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280030,7 +280961,8 @@ "content": "判断下列集合是有限集还是无限集, 并说明理由.\\\\\n(1) $6$的正整数倍的全体组成的集合;\\\\\n(2) $600$的正约数的全体组成的集合;\\\\\n(3) $2019$年在上海出生的所有人组成的集合;\\\\\n(4) 给定的一条长度为$1$的线段$AB$上的所有点组成的集合.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280051,7 +280983,8 @@ "content": "用符号``$\\in$''或``$\\notin$''填空:\\\\\n(1) $0$\\blank{20}$\\mathbf{N}$;\\\\\n(2) $1$\\blank{20}$\\mathbf{Z}$;\\\\\t\n(3) $\\sqrt 2$\\blank{20}$\\mathbf{Q}$;\\\\\n(4) $-\\sqrt {\\pi }$\\blank{20}$\\mathbf{R}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -280072,7 +281005,8 @@ "content": "用符号``$\\in$''或``$\\notin$''填空:\\\\\n(1) $0.\\dot1\\dot3$\\blank{20}$\\mathbf{Q}$;\\\\\n(2) $-\\sqrt 3$\\blank{20}$\\mathbf{Q}$;\\\\\n(3) $\\pi^2$\\blank{20}$\\mathbf{R}$;\\\\\n(4) $\\dfrac 1{\\sqrt 2-1}-\\sqrt 2$\\blank{20}$\\mathbf{Z}$;\\\\\n(5) $0$\\blank{20}$\\mathbf{\\varnothing}$;\\\\\n(6) $\\dfrac 1{1-\\dfrac 1{1-\\dfrac 12}}$\\blank{20}$\\mathbf{N}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -280093,7 +281027,8 @@ "content": "判断下列各组对象能否组成集合. 若能组成集合, 指出是有限集还是无限集; 若不能组成集合, 请说明理由.\\\\\n\\textcircled{1} 上海市$2020$年入学的全体高一年级新生;\\\\\n\\textcircled{2} 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n\\textcircled{3} 影响力比较大的中国数学家;\\\\\n\\textcircled{4} 不等式$3x-10<0$的所有正整数解;\\\\\n\\textcircled{5} 所有的平面四边形;\\\\\n\\textcircled{6} 函数$y=\\dfrac 1x$图像上所有的点.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280114,7 +281049,8 @@ "content": "对于一个确定的实数$x$, 由$x,-x,|x|,-\\sqrt {x^2}$中的一个值或几个值组成的所有集合中, 元素的个数最多有\\blank{50}个.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -280137,7 +281073,8 @@ "content": "已知关于$x$的方程$\\sqrt {x^2+4x+a}=x+2$, 若以该方程的所有解为元素组成的集合是无限集, 则实数$a$满足的条件为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -280160,7 +281097,8 @@ "content": "用列举法表示下列集合:\\\\\n(1) 所有不大于$10$的正整数组成的集合;\\\\\n(2) 方程$(x-1)(x-2)(x-3)(x-4)=0$的所有解组成的集合;\\\\\n(3) 集合$\\{1,2,3,4\\}$中任意两个不同元素之和组成的集合.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280181,7 +281119,8 @@ "content": "选择适当的方法表示下列集合:\\\\\n(1) 大于$0$且小于$10$的全体偶数组成的集合$A$\n(2) 全体偶数组成的集合$B$;\\\\\n(3) 被$3$除余$2$的所有自然数组成的集合$C$;\\\\\n(4) 被$3$除余$2$的所有整数组成的集合$D$;\\\\\n(5) 直角坐标平面上由第二象限与第四象限中的所有点组成的集合$E$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280202,7 +281141,8 @@ "content": "用区间表示下列集合:\\\\\n(1) $\\{x|1\\le x<2\\}$;\\\\\n(2) 不等式$2x\\le 6$的所有解组成的集合.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280225,7 +281165,8 @@ "content": "请说明下列集合的表示方法是什么, 并尝试使用其他的方法表示该集合:\\\\\n(1) 集合$A=\\{x|\\dfrac 6{6-x}\\in \\mathbf{Z}$且$x\\in \\mathbf{Z}\\}$;\\\\\n(2) 集合$B=\\{(1,1),(1,-1),(-1,1),(-1,-1)\\}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280246,7 +281187,8 @@ "content": "设集合$M=\\{a|a =x^2-y^2,\\ x,y\\in \\mathbf{Z}\\}$, 下列数中不属于$M$的为\\bracket{20}.\n\\fourch{$3$}{$6$}{$9$}{$12$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -280269,7 +281211,8 @@ "content": "已知集合$A=\\{x|x=a+\\sqrt 2b,\\ a,b\\in \\mathbf{Z}\\}$, 若$x_1,x_2\\in A$, 证明: $x_1x_2\\in A$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280292,7 +281235,8 @@ "content": "确定$x$与$y$, 使得集合$\\{2x,x+y\\}=\\{4,8\\}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280313,7 +281257,8 @@ "content": "确定下列每组中两个集合之间的关系:\\\\\n(1) $A=\\{n|n$是$12$的正约数$\\}$, $B=\\{1,2,3,6\\}$;\\\\\n(2) $C=\\{x|-1\\le x< 2\\}$, $D=\\{x|x< 2$或$x>3\\}$;\\\\\n(3) $E=\\{n|n=3k+1,\\ k\\in \\mathbf{N}\\}$, $F=\\{n|n=3m-2,\\ m\\in \\mathbf{N}\\}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280334,7 +281279,8 @@ "content": "写出集合$\\{a,b,c\\}$的所有子集, 并指出哪些是真子集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280355,7 +281301,8 @@ "content": "已知集合$A=\\{x|0=latex]\n\\draw (0,0) rectangle (3,2) node [below left] {$U$};\n\\begin{scope}[even odd rule]\n\\filldraw [pattern = north east lines] (1,1) circle (0.8) (2,1) circle (0.8);\n\\end{scope}\n\\draw (1,1) circle (0.8) (2,1) circle (0.8);\n\\draw (1,1) node {$A$} (2,1) node {$B$};\n\\end{tikzpicture}\\\\\n(2) \\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (3,3) node [below left] {$U$};\n\\filldraw [pattern = north east lines] (1,2) circle (0.8);\n\\filldraw [white] (2,2) circle (0.8) (1.5,1) circle (0.8);\n\\begin{scope}\n\\clip (2,2) circle (0.8);\n\\clip (1.5,1) circle (0.8);\n\\filldraw [pattern = north east lines] (1,2) circle (0.8);\n\\end{scope}\n\\draw (1,2) circle (0.8) (2,2) circle (0.8) (1.5,1) circle (0.8);\n\\draw (1,2) node {$A$} (2,2) node {$B$} (1.5,1) node {$C$};\n\\end{tikzpicture}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280596,7 +281553,8 @@ "content": "下列语句哪些是命题? 如果是命题, 那么它们是真命题还是假命题? 为什么?\\\\\n(1) 个位数字是$5$的自然数能被$5$整除;\\\\\n(2) 凡直角三角形都相似;\\\\\n(3) 请起立;\\\\\n(4) 若两个角互为补角, 则这两个角不相等;\\\\\n(5) 若两个三角形的三组对应边分别相等, 则这两个三角形全等;\\\\\n(6) 你是高一学生吗?\\\\\n(7) $x>3$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280617,7 +281575,8 @@ "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 若一个数是偶数, 则这个数不是素数;\\\\\n(2) 若菱形的一组邻角相等, 则这个菱形是正方形;\\\\\n(3) 如果集合$A$是集合$B$的子集, 那么$B$不是$A$的子集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280638,7 +281597,8 @@ "content": "将下列命题改写成``若$\\alpha$, 则$\\beta$''的形式, 并判断``$\\alpha \\Rightarrow \\beta$''是否成立.\\\\\n(1) 等腰三角形的两底角相等;\\\\\n(2) 凡是素数都是奇数;\\\\\n(3) 对顶角相等.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280659,7 +281619,8 @@ "content": "已知下列三组陈述句:\\\\\n(1) $\\alpha$: $a+b$是偶数, $\\beta$: $a,b$都是偶数;\\\\\n(2) $\\alpha$: $q<0$, $\\beta$: 关于$x$的方程$x^2+2x+q=0$($q\\in \\mathbf{R}$)有两个不相等的实数根;\\\\\n(3) $\\alpha$: $ab=0$, $\\beta$: $a^2+b^2=0$.\n其中满足关系``$\\alpha \\Rightarrow \\beta$''的题号是\\blank{50}. 满足关系``$\\alpha \\Leftarrow \\beta$''的题号是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -280680,7 +281641,8 @@ "content": "已知$a$是常数, 命题$p$: 存在实数$x$, 使得$|x|-a<0$. 若命题$p$是假命题, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280701,7 +281663,8 @@ "content": "已知$a$是常数, 命题$\\alpha$: $-12$, 则$x>1$或$y>1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280852,7 +281821,8 @@ "content": "证明: $\\sqrt 2$是无理数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280875,7 +281845,8 @@ "content": "已知$a_1,a_2,\\cdots,a_{10}$是实数, 且满足$a_1+a_2+\\cdots +a_{10}\\le 100$. 求证: $a_1,a_2,\\cdots,a_{10}$中至少有一个不大于$10$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280896,7 +281867,8 @@ "content": "已知$x,y$是实数, 若$x$是有理数, $y$是无理数, 求证: $x+y$是无理数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280917,7 +281889,8 @@ "content": "$a,b,c$是整数, 若$ab+bc+ca$为偶数, 求证: $a,b,c$中至少有两个是偶数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280938,7 +281911,8 @@ "content": "设$a,b,c\\in \\{1,2,3\\}$, 且$a,b,c$两两不等, 求证: $(a-1)(b-2)(c-3)$是偶数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280959,7 +281933,8 @@ "content": "设$a$、$b$、$c$、$d$是实数, 判断下列命题的真假, 并说明理由:\\\\\n(1) 如果$a=b$, 且$c=d$, 那么$a+c=b+d$;\\\\\n(2) 如果$a=b$, 且$c=d$, 那么$ac=bd$;\\\\\n(3) 如果$a=b\\ne 0$, 那么$\\dfrac 1a=\\dfrac 1b$;\\\\\n(4) 如果$a=b$, 那么$a^n=b^n$, 其中$n$是正整数;\\\\\n(5) 如果$ac=bc$, 那么$a=b$;\\\\\n(6) 如果$(a-b)^2+(b-c)^2=0$, 那么$a=b=c$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -280980,7 +281955,8 @@ "content": "设$a$、$b\\in \\mathbf{R}$, 分别求下列关于$x$的方程的解集:\\\\\n(1) $ax=1$;\\\\\n(2) $1-ax=a-x$;\\\\\n(3) $ax=b$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281001,7 +281977,8 @@ "content": "设$k\\in \\mathbf{R}$, 分别求下列关于$x$与$y$的二元一次方程组的解集:\\\\\n(1) $\\begin{cases} y=2x+1, \\\\ y=kx+3; \\end{cases}$\n(2) $\\begin{cases} y=-2x+k, \\\\ y=k^2x+3. \\end{cases}$", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281022,7 +281999,8 @@ "content": "设$a$、$b$、$c$、$d$均为实数, 判断下列命题的真假, 并说明理由:\\\\\n(1) 如果$\\dfrac ac=\\dfrac bc$, 那么$a=b$;\\\\\n(2) 如果$a^n=b^n$, 那么$a=b$, 其中$n$是正整数;\\\\\n(3) 如果$\\dfrac a{b-a}=\\dfrac c{d-c}$, 且$bd\\ne 0$, 那么$\\dfrac ab=\\dfrac cd$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281043,7 +282021,8 @@ "content": "设$m$、$k$均为实数, 求关于$x$的方程$2x+m=kx+3$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281064,7 +282043,8 @@ "content": "设$k$为实数, 求关于$x$与$y$的二元一次方程组$\\begin{cases} y=(3-2k)x-2, \\\\ y=k^2x-1 \\end{cases}$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281087,7 +282067,8 @@ "content": "设$k$为实数, 求关于$x$与$y$的二元一次方程组$\\begin{cases} y=x+k^2+2k, \\\\ y=k^2x+3 \\end{cases}$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281110,7 +282091,8 @@ "content": "设$x$、$y$均为正整数, 求关于$x$与$y$的方程$|x-2|+|y-1|=1$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281131,7 +282113,8 @@ "content": "求关于$x$的一元二次方程$ax^2+bx+c=0$($a\\ne 0$)的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281152,7 +282135,8 @@ "content": "求证: $a_1=a_2$, $b_1=b_2$, $c_1=c_2$是等式$a_1x^2+b_1x+c_1=a_2x^2+b_2x+c_2$恒成立的充要条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281173,7 +282157,8 @@ "content": "已知方程$x^2+x-3=0$的两个根为$x_1$、$x_2$, 求下列各式的值:\\\\\n(1) $x_1^2x_2+x_2^2x_1$;\\\\\n(2) $|x_1-x_2|$;\\\\\n(3)$x_1^3+x_2^3$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281194,7 +282179,8 @@ "content": "已知一元二次方程$x^2+3x-3=0$的两个实根分别为$x_1$、$x_2$, 求作二次项系数是$1$, 且分别以下列数值为根的一元二次方程:\\\\\n(1)$-x_1$, $-x_2$;\\\\\n(2)$\\dfrac 1{x_1}$, $\\dfrac 1{x_2}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281217,7 +282203,8 @@ "content": "$a$、$b$、$c$均为非零实数, 若关于$x$的一元二次方程$ax^2+bx+c=0$的解集为$\\{1,-3\\}$, 求关于$x$的一元二次方程$cx^2-bx+a=0$的解集.(要求: 用两种不同方法解答.)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281238,7 +282225,8 @@ "content": "解方程$\\sqrt [3]{35+x}+\\sqrt [3]{26-x}=1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281259,7 +282247,8 @@ "content": "证明: 如果$a+b>c$, 那么$a>c-b$; 反之亦然.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281280,7 +282269,8 @@ "content": "已知$a>b$, $c>d$. 求证: $a+c>b+d$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281303,7 +282293,8 @@ "content": "已知$a>b$, $c>d$. 求证: $a-d>b-c$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281326,7 +282317,8 @@ "content": "(1) 已知$a>b>0$, 求证: $\\dfrac 1b>\\dfrac 1a>0$;\\\\\n(2) 已知$0>a>b$, 求证: $0>\\dfrac 1b>\\dfrac 1a$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281347,7 +282339,8 @@ "content": "已知$a>b>0$, $c>d>0$. 求证: $\\dfrac ad>\\dfrac bc$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281368,7 +282361,8 @@ "content": "设$a>b>0$. 试比较$\\dfrac ab$, $\\dfrac{2a+b}{a+2b}$, $\\dfrac{3a+b}{a+3b}$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281389,7 +282383,8 @@ "content": "设$a,b,c$均为正数. 若$\\dfrac c{a+b}<\\dfrac a{b+c}<\\dfrac b{a+c}$, 则$a,b,c$的大小关系是\\bracket{20}.\n\\fourch{$cb>0$, $c>d>0$. 求证: $ac>bd$;\\\\\n(2) 已知$a>b>0$, 求证: $a^n>b^n$, 其中$n$是正整数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281431,7 +282427,8 @@ "content": "已知$ab$为正数, $n$为正整数, 求证: 如果$a^n>b^n$, 那么$a>b$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281452,7 +282449,8 @@ "content": "设$a$是实数, 比较$(a+1)^2$与$a^2-a+1$的值的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281473,7 +282471,8 @@ "content": "已知$a,b,c$为实数, 证明: $a^2+b^2+c^2\\ge bc+ca+ab$.写出等号成立的一个充要条件, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281494,7 +282493,8 @@ "content": "若$a<0$, $b<0$, 比较$\\dfrac{b^2}a+\\dfrac{a^2}b$与$a+b$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281515,7 +282515,8 @@ "content": "已知$a$是实数且$a\\ne 0$, 证明: $\\sqrt {a^4+\\dfrac 1{a^4}}-\\sqrt 2\\ge a^2+\\dfrac 1{a^2}-2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281536,7 +282537,8 @@ "content": "设$a$为实数, 求关于$x$的不等式$ax<1$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281557,7 +282559,8 @@ "content": "设$a$为实数, 解关于$x$的一元一次不等式组$\\begin{cases} 2x+a>0, \\\\ 3x-6a<0. \\end{cases}$", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281580,7 +282583,8 @@ "content": "解不等式$(x-3)(x+1)>0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281604,7 +282608,8 @@ "content": "解不等式$(x-1)(x+4)<0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281628,7 +282633,8 @@ "content": "解不等式$-2x^2+3x-\\dfrac 12\\ge 0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281649,7 +282655,8 @@ "content": "若$a<-1$, 解关于$x$的不等式: $(x-a)(\\dfrac 1a-x)<0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281670,7 +282677,8 @@ "content": "某厂计划全年完成产值$6000$万元, 前三个季度已完成$4300$万元. 如果$10$月份的产值是$500$万元, 设在最后两个月里, 月增长率是$x$($x\\ge 0$). 若要完成全年任务, 求$x$的最小值(精确到$1\\%$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281691,7 +282699,8 @@ "content": "解下列不等式:\\\\\n(1) $x^2\\le 4x-4$;\\\\\n(2) $x(x+1)\\ge 7x-9$;\\\\\n(3) $4x^2-4x+3>0$;\\\\\n(4) $x^2\\le x-2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281712,7 +282721,8 @@ "content": "写出一个形如$ax^2+bx+c>0$的一元二次不等式, 使它的解集为$(-1,3)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281733,7 +282743,8 @@ "content": "设$a\\in \\mathbf{R}$, 解关于$x$的不等式: $x^2-(a+1)x+a\\le 0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281754,7 +282765,8 @@ "content": "设$m\\in \\mathbf{R}$, 解关于$x$的不等式: $mx^2+(m-2)x-2>0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281775,7 +282787,8 @@ "content": "设$a_1,a_2,b_1,b_2,c_1,c_2$均为非零实数, 关于$x$的不等式$a_1x^2+b_1x+c_1>0$与$a_2x^2+b_2x+c_2>0$的解集分别为$M$和$N$, 那么``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -281796,7 +282809,8 @@ "content": "解下列不等式组:\\\\\n(1) $\\begin{cases} x-3>0,\\\\ x^2-3x-4>0; \\end{cases}$\\\\\n(2) $\\begin{cases} 10+7x-3x^2\\ge 0, \\\\ 2x^2-5x+2>0. \\end{cases}$", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281817,7 +282831,8 @@ "content": "若关于$x$的不等式$x^2+(k-1)x+4>0$的解集为$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281838,7 +282853,8 @@ "content": "若关于$x$的不等式$(k-5)x^2-(5-k)x-k+10>0$的解集为$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281859,7 +282875,8 @@ "content": "已知一元二次不等式$x^2+bx+c<0$的解集为$(1,2)$, 求实数$b$、$c$的值以及不等式$bx^2-5x+c\\le 0$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281880,7 +282897,8 @@ "content": "已知关于$x$的一元二次方程$4x^2+(m-2)x+m-5=0$有两个负实根, 求实数$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281901,7 +282919,8 @@ "content": "设关于$x$的一元二次不等式$ax^2+bx+c>0$的解集为$(\\alpha ,\\beta)$, 其中$0<\\alpha <\\beta$, 求关于$x$的不等式$cx^2+bx+a<0$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281922,7 +282941,8 @@ "content": "已知关于$x$的一元二次方程$x^2+2mx+6-m=0$($m\\in \\mathbf{R}$), 求分别满足下列条件的$m$的取值范围:\\\\\n(1) 一根大于$1$, 另一根小于$1$;\\\\\n(2) 两根都大于$1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281943,7 +282963,8 @@ "content": "解分式不等式$\\dfrac{x+3}{4-x}>0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281964,7 +282985,8 @@ "content": "解不等式$\\dfrac{5x+3}{x-1}\\le 3$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -281985,7 +283007,8 @@ "content": "解不等式$\\dfrac{x+5}{x^2+2x+3}\\le 1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282006,7 +283029,8 @@ "content": "某服装公司生产的衬衫每件定价$80$元, 在某城市年销售$8$万件.现该公司计划在该市招收代理商来销售衬衫, 以降低管理和营销成本.已知代理商要收取的代理费为总销售金额的$r\\%$(即每$100$元销售额收取$r$元), 为确保单件衬衫的利润保持不变, 服装公司将每件衬衫的价格提高到$\\dfrac{80}{1-r\\%}$元, 但提价后每年的销量会减少$0.62r$万件.求$r$的取值范围, 以确保代理商每年收取的代理费不少于$16$万元.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282027,7 +283051,8 @@ "content": "已知集合$A=\\{x|\\dfrac{2x-a}{x+a}<0\\}$, 若$3\\in A$且$5\\notin A$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282048,7 +283073,8 @@ "content": "设$a\\in \\mathbf{R}$, 且$a\\ne 1$, 比较$\\dfrac{a+2}{a-1}$与$-1$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282071,7 +283097,8 @@ "content": "解不等式$|x-1|<2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282092,7 +283119,8 @@ "content": "解不等式$|2x+1|\\ge 3$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282113,7 +283141,8 @@ "content": "(1) 解不等式$|1-2x|>x$;\\\\\n(2) 解不等式$|x-2|\\le \\dfrac{x+1}2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282134,7 +283163,8 @@ "content": "解不等式$|x-3|+|x-5|<4$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282157,7 +283187,8 @@ "content": "解不等式: $x-1<|2x-1|0$, 求证: $x+\\dfrac 1x\\ge 2$, 并指出等号成立的条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282222,7 +283255,8 @@ "content": "已知$ab>0$, 求证: $\\dfrac ba+\\dfrac ab\\ge 2$, 并指出等号成立的条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282243,7 +283277,8 @@ "content": "设$x\\in \\mathbf{R}$, 求二次函数$y=x(4-x)$的最大值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282266,7 +283301,8 @@ "content": "判断下列结论是否正确:\\\\\n(1) 对任意的正整数$n$, 不等式$x^n+\\dfrac 1{x^n}\\ge 2$当$x\\ne 0$时总是成立的;\\\\\n(2) 对任意的实数$x$, 不等式$\\sqrt {x^2+2}+\\dfrac 1{\\sqrt {x^2+2}}\\ge 2$恒成立.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282287,7 +283323,8 @@ "content": "若$a>0$, $b>0$, $a+b=2$, 则下列不等式对一切满足条件的$a$、$b$恒成立的是\\blank{50}(写出所有恒成立的不等式的编号).\\\\\n\\textcircled{1} $ab\\le 1$; \\textcircled{2} $\\sqrt a+\\sqrt b\\le \\sqrt 2$; \\textcircled{3} $a^2+b^2\\ge 2$; \\textcircled{4} $\\dfrac 1a+\\dfrac 1b\\ge 2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -282308,7 +283345,8 @@ "content": "设$a$、$b$为正数, 且$a+2b=1$, 比较$ab$的值与$\\dfrac 18$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282329,7 +283367,8 @@ "content": "证明: (1) 在周长为常数的所有矩形中, 正方形的面积最大;\\\\\n(2) 在面积相同的所有矩形中, 正方形的周长最小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282350,7 +283389,8 @@ "content": "某新建居民小区欲建一面积为$700\\text{m}^2$的矩形绿地, 并在绿地四周铺设人行道, 设计要求绿地外南北两侧人行道宽$3\\text{m}$, 东西两侧人行道宽$4\\text{m}$, 如图所示(图中单位: $\\text{m}$), 问如何设计绿地的边长, 才能使人行道的占地面积最小. (结果精确到$0.1\\text{m}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,-1.5) rectangle (2,1.5) (-1.2,-0.9) rectangle (1.2,0.9);\n\\draw (0,0) node {绿地};\n\\draw (0,-1.5) node [below] {南} (0,1.5) node [above] {北};\n\\draw [<->] (0,0.9) -- (0,1.5) node [midway,right] {$3$};\n\\draw [<->] (0,-0.9) -- (0,-1.5) node [midway,right] {$3$};\n\\draw [<->] (1.2,0) -- (2,0) node [midway,above] {$4$};\n\\draw [<->] (-1.2,0) -- (-2,0) node [midway,above] {$4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282371,7 +283411,8 @@ "content": "如果正数$a,b,c,d$满足$a+b=cd=4$, 那么\\bracket{20}.\n\\twoch{$ab\\le c+d$, 且等号成立时$a,b,c,d$的取值唯一}{$ab\\ge c+d$, 且等号成立时$a,b,c,d$的取值唯一}{$ab\\le c+d$, 且等号成立时$a,b,c,d$的取值不唯一}{$ab\\ge c+d$, 且等号成立时$a,b,c,d$的取值不唯一}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -282394,7 +283435,8 @@ "content": "已知$a>0$, $b>0$, 且$ab=1$, 则$\\dfrac 1{2a}+\\dfrac 1{2b}+\\dfrac 8{a+b}$的最小值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -282415,7 +283457,8 @@ "content": "已知$a$、$b$为实数, 求证: $|a+b|+|a-b|\\ge 2|a|$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282436,7 +283479,8 @@ "content": "已知为$a$、$b$为实数, 求证: $|a|-|b|\\le|a-b|$, 并指出等号成立的条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282459,7 +283503,8 @@ "content": "证明: $|x-3|+|x-5|\\ge 2$对所有实数$x$恒成立, 并求等号成立时的$x$取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282482,7 +283527,8 @@ "content": "(1) 若要将$a+2b-5$表示成$m(a-1)+n(b-2)$的形式, 试确定常数$m,n$的值;\\\\\n(2) 已知$a$、$b$为实数, 若$|a-1|\\le 1$且$|b-2|\\le 1$, 求证: $|a+2b-5|\\le 3$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282503,7 +283549,8 @@ "content": "已知$a$、$b$、$c$为实数, 求证: $|a+b+c|\\le|a|+|b|+|c|$. 你能否将该不等式进一步推广到更为一般的形式? 请你写出结论, 并给出证明.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282524,7 +283571,8 @@ "content": "(1) 求$-\\dfrac 1{32}$的$5$次方根;\\\\\n(2) 求$81$的$4$次方根.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282545,7 +283593,8 @@ "content": "求下列各根式的值:\\\\\n(1) $\\sqrt [5]{(-2)^5}$;\\\\\n(2) $\\sqrt [6]{(-8)^2}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282568,7 +283617,8 @@ "content": "已知等式$(2x+3)^{x+2023}=1$(其中$x$为整数)成立, 则$x$=\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -282589,7 +283639,8 @@ "content": "设$a,b$是实数, 化简: $\\sqrt [n]{(a-b)^n}$的值(其中$n$为大于$1$的整数).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282610,7 +283661,8 @@ "content": "求下列各式的值:\\\\\n(1) $8^{\\frac 23}$;\\\\\n(2) $(\\dfrac{81}{625})^{-\\frac 34}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282631,7 +283683,8 @@ "content": "用有理数指数幂的形式表示下列各式(其中$a>0)$:\\\\\n(1) $\\sqrt [3]{a^2}$;\\\\\n(2) $a^3\\cdot \\sqrt [4]{a^3}$;\\\\ \n(3) $\\sqrt {a\\sqrt a}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282652,7 +283705,8 @@ "content": "化简下列各式:\\\\\n(1) $(x^{\\frac{\\sqrt 3}2})^{\\sqrt 3}\\cdot \\sqrt x$(其中$x>0$);\\\\ \n(2) $\\dfrac{(a^{\\frac 23}b^{\\frac 12})\\cdot (-3a^{\\frac 12}b^{\\frac 13})}{\\dfrac 13a^{\\frac 16}b^{\\frac 56}}$(其中$a>0,b>0$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282673,7 +283727,8 @@ "content": "已知$01$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282694,7 +283749,8 @@ "content": "已知$a-a^{-1}=1$, 则$a^{12}+a^{-12}$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -282715,7 +283771,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_28$;\\\\\n(2) $\\log_2\\sqrt 2$;\\\\\n(3) $\\log_{10}0.00001$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282736,7 +283793,8 @@ "content": "求下列各式中$x$的值:\\\\\n(1) $\\log_2x=-1$;\\\\\n(2) $\\log_{\\frac 12}x=3$;\\\\ \n(3) $\\ln x=-1$;\\\\\n(4) $\\log_x8=6$;\\\\ \n(5) $-\\ln \\mathrm{e}^2=x$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282757,7 +283815,8 @@ "content": "求下列各式中$x$的取值范围:\\\\\n(1) $\\log_a(1-x^2)$($a>0$且$a\\ne 1$);\\\\\n(2) $\\log_{(1+x)}(1-x)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282780,7 +283839,8 @@ "content": "求下列各式的值:\\\\\n(1) $3^{\\log_312}$;\\\\\n(2) $4^{\\log_23}$;\\\\ \n(3) $2^{\\lg 7}\\cdot 5^{\\lg 7}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282801,7 +283861,8 @@ "content": "求下列各式中$x$的取值范围:\\\\ \n(1) $\\log_2\\dfrac{3x+2}{2x-1}$;\\\\\n(2) $\\log_{(x^2-1)}(3x+2)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282822,7 +283883,8 @@ "content": "求下列各式的值:\\\\\n(1) $2^{\\log_23}+4^{\\log_23}+8^{\\log_23}+16^{\\log_23}$;\\\\\n(2) $2^{\\log_29\\cdot \\log_32\\cdot \\log_45}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282843,7 +283905,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_3(9^4\\times 3^2)$;\\\\ \n(2) $\\log_3\\sqrt[5]9$;\\\\ \n(3) $2\\log_510+\\log_53-\\log_512$;\\\\ \n(4) $\\lg 2+\\lg 2\\times \\lg 5+(\\lg 5)^2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282864,7 +283927,8 @@ "content": "已知$\\log_52=a$, $5^b=3$, 用$a$及$b$表示$\\log_512$和$\\log_5\\dfrac{81}{100}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282885,7 +283949,8 @@ "content": "在有声世界, 声强级是表示声强度相对大小的指标. 其值$y$[单位: $\\text{dB}$(分贝)]定义为$y=10\\lg \\dfrac I{I_0}$. 其中, $I$为声场中某点的声强度, 其单位为$\\text{W}/\\text{m}^2$(瓦$/$平方米), $I_0=10^{-12}\\text{W}/\\text{m}^2$为基准值.\\\\\n(1)如果$I=10\\text{W}/\\text{m}^2$, 求相应的声强级;\\\\\n(2)声强级为$60\\text{dB}$时的声强度$I_{60}$是声强级为$50\\text{dB}$时的声强度$I_{50}$的多少倍?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282906,7 +283971,8 @@ "content": "已知$24^a=12$, 试用$a$表示下列各对数:\\\\\n(1) $\\log_{24}2$;\\\\\n(2) $\\log_{24}3$;\\\\ \t\n(3) $\\log_{24}\\dfrac 4{27}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282927,7 +283993,8 @@ "content": "已知$2\\lg (x-2y)=\\lg x+\\lg y$, 求$\\dfrac xy$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282948,7 +284015,8 @@ "content": "求下列各式的值:\\\\\n(1) $\\log_225\\times \\log_34\\times \\log_59$;\\\\\n(2) $\\dfrac 1{\\log_23}+\\dfrac{\\lg 13.5}{\\lg 3}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282969,7 +284037,8 @@ "content": "设$a>0$, $a\\ne 1$, 且$N>0$. 求证: 若$m\\ne 0$, 则$\\log_{a^m}N^n=\\dfrac nm\\log_aN$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -282990,7 +284059,8 @@ "content": "(1) 已知$\\lg 2=a$, $\\lg 6=b$, 试用$a$及$b$表示$\\log_26$与$\\log_{12}15$;\\\\\n(2) 已知$\\log_{14}2=a$, 试用$a$表示$\\log_{49}16$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283011,7 +284081,8 @@ "content": "已知正数$a$, $b$满足$2^a=3^b$, $\\dfrac 1a+\\dfrac 2b=1$, 求$a$, $b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283032,7 +284103,8 @@ "content": "已知$a$, $b$都是不等于$1$的正数, 若$\\log_ab+\\log_ba=\\dfrac{10}3$, 则$\\log_ab-\\log_ba=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283053,7 +284125,8 @@ "content": "已知非零实数$a,b,c$满足$3^a=4^b=6^c$, 求证: $\\dfrac 2a+\\dfrac 1b=\\dfrac 2c$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283074,7 +284147,8 @@ "content": "写出幂函数$y=x^{\\frac 12}$的定义域, 并作出它的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283098,7 +284172,8 @@ "content": "写出幂函数$y=x^3$的定义域, 并作出它的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283122,7 +284197,8 @@ "content": "写出幂函数$y=x^{-\\frac 23}$的定义域, 并作出它的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283146,7 +284222,8 @@ "content": "下列幂函数中, 其图像不经过原点的有\\blank{50}.(请填入全部正确的序号)\\\\\n\\textcircled{1} $y=x^4$; \\textcircled{2} $y=x^{-\\frac 14}$; \\textcircled{3} $y=x^{\\frac 52}$; \\textcircled{4} $y=x^{-3}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283167,7 +284244,8 @@ "content": "已知幂函数$y=x^{n^2-2n-3}$($n$为正整数)的图像关于原点成中心对称, 且与两坐标轴都无公共点, 求$n$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283188,7 +284266,8 @@ "content": "比较下列各题中两个数的大小:\\\\\n(1) $2.5^{-2}$与$1.8^{-2}$;\\\\\n(2) $1.32^{\\frac 45}$与$(-\\sqrt 2)^{\\frac 45}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283209,7 +284288,8 @@ "content": "已知函数$y=\\dfrac 1x$和$y=\\dfrac 1{x-2}$, 说明这两个函数图像之间的关系, 并在同一平面直角坐标系中作出它们的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283232,7 +284312,8 @@ "content": "已知函数$y=\\dfrac 1{x-2}$和$y=\\dfrac{x-1}{x-2}$, 说明这两个函数图像之间的关系, 并在同一平面直角坐标系中作出它们的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283255,7 +284336,8 @@ "content": "若函数$y=\\dfrac x{x-m}$($m$为实常数)的图像先向下平移$\\text1$个单位, 再左平移$\\text1$个单位后, 能得到函数$y=\\dfrac 1x$的图像, 则$m=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283276,7 +284358,8 @@ "content": "已知幂函数$y=x^{\\frac m3-1}$($m\\in \\mathbf{Z}$)在区间$(0,+\\infty)$上是严格增函数, 其图像经过点$(-1,-1)$, 且在$x>1$时其图像位于直线$y=x$的下方, 求$m$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283297,7 +284380,8 @@ "content": "若指数函数$y=a^x$($a>0$且$a\\ne 1$)的图像经过点$(2,9)$, 求该指数函数的表达式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283320,7 +284404,8 @@ "content": "分别作出指数函数$y=2^x$和$y=3^x$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283341,7 +284426,8 @@ "content": "作出指数函数$y=(\\dfrac 12)^x$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283362,7 +284448,8 @@ "content": "已知指数函数图像的一部分如下图所示, 求该指数函数的表达式. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -1.8:3, samples = 100] plot (\\x,{pow(1/3,\\x)});\n\\draw [dashed] (0,3) -- (-1,3) -- (-1,0);\n\\draw (-1,0) node [below] {$-1$};\n\\draw (0,3) node [right] {$3$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283383,7 +284470,8 @@ "content": "在平面直角坐标系中作出函数$y=-4^x$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283404,7 +284492,8 @@ "content": "利用指数函数的性质, 比较下列各题中两个数的大小:\\\\\n(1) $1.7^{2.5}$与$1.7^{3}$;\\\\\n(2) $(\\frac{3}{4})^{\\frac{1}{6}}$与$(\\frac{4}{3})^{-\\frac{1}{5}}$;\\\\ \n(3) $a^{\\frac 12}$与$a^{\\frac 13}$($a>0$且$a\\ne 1$).", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283425,7 +284514,8 @@ "content": "求下列关于$x$的不等式的解集:\\\\ \n(1) $3^x>\\dfrac 1{27}$;\\\\\n(2) $a^{x^2-2x+3}>a^6$($00$且$a\\ne 1$, $b\\in \\mathbf{R}$)的图像不经过第二象限, 则$a$, $b$的取值范围分别是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283467,7 +284558,8 @@ "content": "无论实数$a$为何值, 函数$y=(a-2)\\cdot 2^x-\\dfrac a2$的图像都经过一个定点, 求这个定点的坐标.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283488,7 +284580,8 @@ "content": "已知指数函数$y=a^x$($a>1$)在区间$[1,2]$上的最大值比最小值大$\\dfrac a3$, 求实数$a$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283513,7 +284606,8 @@ "content": "统计资料显示: 某外来入侵物种现有种群数量为$k$, 若有理想的外部环境条件, 该物种的年平均增长率约为$20\\%$.试建立该物种的种群数量增长模型, 并预测$30$年后该物种的种群数量约为现有种群数量的多少倍. (结果精确到个位)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283534,7 +284628,8 @@ "content": "当生物死亡后, 它机体内原有的碳$14$含量会按确定的比率衰减(称为衰减率), 大约每经过$5730$年衰减为原来的一半, 这个时间称为``半衰期''.假设死亡生物体内碳$14$的年衰减率为$p$, 将刚死亡的生物体内碳$14$含量看成$1$个单位.求$p$的值, 并按照上述变化规律, 写出生物体内碳$14$含量与死亡年数之间的关系式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283555,7 +284650,8 @@ "content": "探测某片森林知道, 蓄积的木材有$20$万$\\text{m}^3$, 如果森林蓄积木材的年平均增长率为$8\\%$, 那么经过三年, 蓄积的木材为\\blank{50}万$\\text{m}^3$.(精确到$0.01$)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283576,7 +284672,8 @@ "content": "设$a>0$且$a\\ne 1$, 若函数$y=a^{2x}+2a^x-1$在区间$[-1,1]$上的最大值为14, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283597,7 +284694,8 @@ "content": "复利是一种计算利息的方法, 即把前一期的利息和本金加在一起算作本金, 再计算下一期的利息.已知一种储蓄按复利计算利息, 本金为$a$(单位: 元), 每期利率为$r$. 设本利和(本金与利息之和)为$y$(单位: 元), 存期数为$x$($x$为正整数).\\\\\n(1) 写出本利和$y$关于存期数$x$的关系式;\\\\\n(2) 如果存入本金$1000$元, 每期利率为$2.25\\%$, 试计算$5$期后的本利和.(精确到$1$元)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283618,7 +284716,8 @@ "content": "求下列函数的定义域:\\\\\n(1) $y=\\log_2(x-1)$;\\\\\n(2) $y=\\log_a(x^2-4x-5)$, 其中常数$a>0,a\\ne 1$;\\\\\n(3) $y=\\ln \\dfrac{1-2x}{x+1}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283639,7 +284738,8 @@ "content": "在同一坐标系中, 作出函数$y=\\log_{\\frac 12}x$与$y=\\log_{\\frac 13}x$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283660,7 +284760,8 @@ "content": "观察函数$y=\\log_2x$和$y=\\log_{\\frac 12}x$的图像, 判断这两个函数的图像是否关于$x$轴对称?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283681,7 +284782,8 @@ "content": "已知集合$A,B$分别是函数$y=x^{-\\frac 12}$与$y=\\lg (1-2x)$的定义域, 则$A\\cap B=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283702,7 +284804,8 @@ "content": "$a,b,c$是图中三个对数函数的底数, 它们之间的大小关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:2.5] plot (\\x,{ln(\\x)/ln(2)}) node [right] {$y=\\log_c x$};\n\\draw [domain = 0.5:2.5] plot (\\x,{ln(\\x)/ln(0.53)}) node [right] {$y=\\log_a x$};\n\\draw [domain = 0.5:2.5] plot (\\x,{ln(\\x)/ln(0.3)}) node [right] {$y=\\log_b x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$c>a>b$}{$c>b>a$}{$a>b>c$}{$b>a>c$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -283723,7 +284826,8 @@ "content": "判断函数$y=\\ln|x|$的图像是否关于某条直线对称? 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283744,7 +284848,8 @@ "content": "利用对数函数的单调性, 比较下列各题中两个对数的大小:\\\\\n(1) $\\log_25$与$\\log_26$;\\\\\n(2) $\\log_a0.1$与$\\log_a0.2$(其中常数$a>0$, $a\\ne 1$);\\\\\n(3) $\\log_57$与$\\log_67$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283765,7 +284870,8 @@ "content": "比较$99^{89}$与$89^{99}$的大小关系.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283786,7 +284892,8 @@ "content": "已知$\\lg a+\\lg b=0$($a>0$, $a\\ne 1$, $b>0$, $b\\ne 1$), 则函数$y=a^x$与$y=-\\log_bx$的图像可能是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -3:1.5] plot (\\x,{pow(2,\\x)});\n\\draw [domain = -3:1.5] plot ({-pow(2,\\x)},\\x);\n\\draw (-1,0) node [below left] {$-1$};\n\\draw (0,1) node [above left] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:1.5] plot (\\x,{pow(2,\\x)});\n\\draw [domain = -3:1.5] plot ({pow(2,\\x)},\\x);\n\\draw (1,0) node [below right] {$1$};\n\\draw (0,1) node [above left] {$1$};\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:1.5] plot ({-\\x},{pow(2,\\x)});\n\\draw [domain = -3:1.5] plot ({pow(2,\\x)},\\x);\n\\draw (1,0) node [below right] {$1$};\n\\draw (0,1) node [above right] {$1$};\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:1.5] plot (\\x,{pow(2,\\x)});\n\\draw [domain = -3:1.5] plot ({pow(2,\\x)},{-\\x});\n\\draw (1,0) node [below left] {$1$};\n\\draw (0,1) node [above left] {$1$};\n\\end{tikzpicture}}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -283807,7 +284914,8 @@ "content": "已知$\\log_a\\dfrac 34<1$, 那么$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -283828,7 +284936,8 @@ "content": "若$0|\\lg b|$, 证明: $ab<1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283849,7 +284958,8 @@ "content": "试利用对数函数的单调性估算对数$\\log_23$的第一位小数的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283872,7 +284982,8 @@ "content": "如果不考虑空气阻力, 火箭的最大速度$v$(单位: $\\text{km}/\\text{s}$)和燃料质量$M$(单位: $\\text{kg}$)、火箭(除燃料外)的质量$m_0$(单位: $\\text{kg}$)之间的关系是$v=2\\ln (1+\\dfrac M{m_0})$. 问当燃料质量至少是火箭质量的多少倍时, 火箭的最大速度才能超过$8\\text{km}/\\text{s}$? (结果精确到$0.1$倍)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283893,7 +285004,8 @@ "content": "某人在银行存入$1$万元, 若年利率为$5\\%$, 且按年计复利的条件下, 经过多少年存款才能连本带利超过$5$万元? (结果精确到$1$年)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283914,7 +285026,8 @@ "content": "求函数$y=(\\log_2x)\\cdot (\\log_22x)$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283935,7 +285048,8 @@ "content": "已知$x>0$, $x\\ne 1$, 比较$1+\\log_x3$与$2\\log_x2$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283956,7 +285070,8 @@ "content": "某公司在销售某产品时制定了一次性奖励方案: 当销售利润$x$($x>0$, 单位: 万元)不超过$10$万元时, 按销售利润的$15\\%$进行奖励; 当销售利润超过$10$万元时, 则超出部分$m$($m>0$, 单位: 万元)按$\\log_5(m+1)$进行奖励. 如果业务员小王按此方案获得了$2.5$万元的奖金, 那么他的销售利润是多少万元?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283977,7 +285092,8 @@ "content": "求下列函数的定义域:\\\\\n(1) $y=\\log_2(x+1)$;\\\\\n(2) $y=\\dfrac{\\sqrt {x+3}}{x-1}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -283998,7 +285114,8 @@ "content": "判断下列函数与函数$y=x$是否相同, 并说明理由:\\\\\n(1) $y=(\\sqrt x)^2$;\\\\\n(2) $y=\\ln \\mathrm{e}^x$;\\\\ \n(3) $y=\\sqrt [4]{x^4}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284019,7 +285136,8 @@ "content": "求函数$y=\\dfrac 1{2^x+1}$的值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284049,7 +285167,8 @@ "content": "下列四组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\twoch{$y=\\dfrac{(x+3)(x-5)}{x+3}$与$y=x-5$}{$y=\\sqrt {x^2}-1$与$y=\\sqrt [3]{x^3}-1$}{$y=\\sqrt [3]{x^4-x^3}$与$y=x\\cdot \\sqrt [3]{x-1}$}{$y=\\lg x^2$与$y=2\\lg x$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -284070,7 +285189,8 @@ "content": "求函数$y=x^2-2x$, $x\\in \\{0,1,2,3\\}$的值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284091,7 +285211,8 @@ "content": "下列四组函数中, 同组的两个函数是相同函数的是\\bracket{20}.\n\\twoch{$y=\\sqrt {x+1}\\cdot \\sqrt {x-1}$与$y=\\sqrt {(x+1)(x-1)}$}{$y=\\dfrac{x^2}{|x|}$与$y=|x|$}{$y=x^0$与$y=\\dfrac{x^2+1}{x^2+1}$}{$y=1$, $x\\in \\{1,2\\}$与$y=(x^2-5x+5)^2$, $x\\in \\{1,2\\}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -284112,7 +285233,8 @@ "content": "求函数$y=x+\\sqrt {1-x}$的值域.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284135,7 +285257,8 @@ "content": "作出函数$y=x|x|$的大致图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284160,7 +285283,8 @@ "content": "某车辆装配车间每$2\\text{h}$装配完成一辆车.按照计划, 该车间今天生产$8\\text{h}$.用解析法和图像法分别表示从当天开始生产的时刻起所经过的时间$x$(单位: $\\text{h}$)与装配完成的车辆数$y$(单位: 辆)之间的函数$y=f(x)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284181,7 +285305,8 @@ "content": "上海的出租车价格规定: 起步费$14$元, 可行$3$千米; $3$千米以后按每千米$2.5$元计价, 可再行$12$千米; 以后每千米都按$3.75$元计价(假设每一次乘车的车费由行车里程唯一确定). 用解析法表示出租车行车里程$x$(单位: 千米)与车费$y$(单位: 元)之间的函数$y=f(x)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284202,7 +285327,8 @@ "content": "已知一次函数$y=f(x)$满足: 对任意$x\\in \\mathbf{R}$都有$3f(x+1)-2f(x-1)=2x+17$, 求函数$y=f(x)$的表达式.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284223,7 +285349,8 @@ "content": "设函数$y=f(x)$的表达式为$f(x)=\\begin{cases} (x+1)^2, & x<1, \\\\ 4-x, & x\\ge 1, \\end{cases}$则不等式$f(x)\\ge 1$的解集为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -284244,7 +285371,8 @@ "content": "已知函数$y=f(x)$的表达式为$f(x)=\\dfrac{x-1}{x+1}$, 求解关于$a$的方程$f(\\dfrac{a-1}{a+1})=-a$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284265,7 +285393,8 @@ "content": "设函数$y=f(x)$的表达式为$f(x)=\\begin{cases} x^2+bx+c, & x\\le 0, \\\\ 2,& x>0 \\end{cases}$(其中$b,c\\in \\mathbf{R}$). 若$f(-4)=f(0)$, $f(-2)=-2$, 则关于$x$的方程$f(x)=x$解的个数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -284286,7 +285415,8 @@ "content": "证明: 函数$y=2x^4-3x^2$是一个偶函数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284307,7 +285437,8 @@ "content": "证明: 函数$y=x^3-\\dfrac 1x$是一个奇函数.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284328,7 +285459,8 @@ "content": "是否存在定义在$\\mathbf{R}$上的, 且既是奇函数又是偶函数的函数? 若存在, 求出所有满足此条件的函数; 若不存在, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284349,7 +285481,8 @@ "content": "已知$y=f(x)$是奇函数, $y=g(x)$是偶函数, 且$f(-1)+g(1)=3$, $f(1)+g(-1)=5$, 则$f(-1)=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -284370,7 +285503,8 @@ "content": "(1) 设$y=f(x)$为定义在$R$上的奇函数, $y=g(x)$为定义在$\\mathbf{R}$上的偶函数, 对任意实数$x$, 都有$f(x)+g(x)=2^x$, 求$y=f(x)$与$y=g(x)$的表达式;\\\\\n(2) 对于定义在$\\mathbf{R}$上的任意函数$y=H(x)$, 是否总存在$\\mathbf{R}$上的奇函数$y=f(x)$和偶函数$y=g(x)$, 使得对任意实数$x$, 都有$H(x)=f(x)+g(x)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284391,7 +285525,8 @@ "content": "判断函数$y=(x-1)^2$, $x\\in \\mathbf{R}$的奇偶性, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284412,7 +285547,8 @@ "content": "判断函数$y=\\begin{cases} x(x+1)x>0, \\\\ x(1-x)x<0 \\end{cases}$的奇偶性, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284433,7 +285569,8 @@ "content": "已知函数$y=f(x)$, $x\\in \\mathbf{R}$, 且当$x\\ge 0$时, $f(x)=2x^3+2^x-1$.函数$y=f(x)$是否可能是奇函数? 若可能, 求$f(x)$的表达式; 若不可能, 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284454,7 +285591,8 @@ "content": "已知$a,b\\in \\mathbf{R}$, 函数$y=ax^2+(b-2)x+3$是定义在$[a^2-2,a]$上的偶函数, 则$a+b$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -284475,7 +285613,8 @@ "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 当$x>0$时, $f(x)>2$, 则下列说法不正确的是\\bracket{20}.\n\\onech{若$y=f(x)$为偶函数, 则当$x<0$时, $f(x)>2$}{若$y=f(x)$为偶函数, 则不存在非零实数$x_0$, 使得$f(x_0)\\le 2$}{若$y=f(x)$为奇函数, 则当$x<0$时, $f(x)<-2$}{若$y=f(x)$为奇函数, 则不存在实数$x_0$, 使得$-2=latex]\n\\filldraw [pattern = north east lines] (0.8,0) rectangle (3,1.2);\n\\filldraw [pattern = north east lines] (0,1.2) rectangle (0.8,2);\n\\draw (0,0) rectangle (3,2);\n\\draw (-0.1,0) -- (-0.7,0) (-0.1,1.2) -- (-0.4,1.2) (-0.1,2) -- (-0.7,2);\n\\draw [<->] (-0.25,1.2) -- (-0.25,2) node [midway, fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.55,0) -- (-0.55,2) node [midway, fill = white] {\\rotatebox{90}{$a$}};\n\\draw (0,2.1) -- (0,2.7) (0.8,2.1) -- (0.8,2.4) (3,2.1) -- (3,2.7);\n\\draw [<->] (0,2.25) -- (0.8,2.25) node [midway, fill = white] {$x$};\n\\draw [<->] (0,2.55) -- (3,2.55) node [midway, fill = white] {$b$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284859,7 +286015,8 @@ "content": "如图所示, 四边形$OABC$是平面直角坐标系中边长为1的正方形. 一直线$y=-x+t$($t\\in (0,2)$)与正方形$OABC$相交, 将正方形分为两个部分, 其中包含原点$O$的部分的面积记为$S$. 试将$S$表示为$t$的函数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) node [below] {$A$} coordinate (A) -- (2,2) node [above right] {$B$} coordinate (B) -- (0,2) node [left] {$C$} coordinate (C);\n\\draw (-0.5,1.4) -- (1.4,-0.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284880,7 +286037,8 @@ "content": "要建造一面靠墙、且面积相同的两间相邻的长方形居室, 如图所示. 如果已有材料可建成的围墙总长度为$30\\text{m}$, 那么当宽$x$(单位: $\\text{m}$)为多少时, 才能使所建造的居室面积最大? 居室的最大面积是多少?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw [pattern = north east lines] (0,0) rectangle (5,0.2);\n\\draw (0.5,0) rectangle (4.5,-1.3);\n\\draw (2.5,0) -- (2.5,-1.3);\n\\draw (4.6,-1.3) -- (5,-1.3);\n\\draw [<->] (4.8,-1.3) -- (4.8,0) node [midway, fill = white] {\\rotatebox{90}{$x$}};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284901,7 +286059,8 @@ "content": "如图, 某小区要建造一个直径为$16\\text{m}$的圆形喷水池, 并在池的周边靠近水面的位置安装一圈喷水头, 使喷出的水柱在离池中心水平距离$3\\text{m}$的地方达到最高高度$4\\text{m}$. 各方向喷来的水柱在池中心上方某一点汇合, 求该点离水面的高度.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\filldraw [pattern = north east lines] (-9,0) rectangle (9,-1);\n\\draw [domain = 0:8] plot (\\x,{(\\x-8)*(\\x+2)/25*(-4)});\n\\draw [domain = 0:8] plot (-\\x,{(\\x-8)*(\\x+2)/25*(-4)});\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284922,7 +286081,8 @@ "content": "提高黄浦江上各大桥的通行能力, 可改善浦江两岸的交通状况. 在一般情况下, 大桥上的车流速度$v$(单位: 千米/小时)是车流密度$x$(单位: 辆/千米)的函数, 车流量$f(x)$(单位: 辆/小时)(单位时间内通过桥上某观测点的车辆数)满足$f(x)=xv$. 当车流密度$x\\in[20, 200]$时, 车流速度随着车流密度的增大而减小. 当桥上的车流密度达到或超过$200$辆/千米时, 车辆通行缓慢, 车流速度很小, 可将其视作为$0$.\\\\\n(1) 已知当车流密度不超过$20$辆/千米时, 车流速度为$60$千米/小时; 当$x\\in[20,200]$时, 车流速度$v$是车流密度$x$的一次函数.\\\\\n\\textcircled{1} 当$x\\in [0,200]$时, 试用解析法将$v$表示为$x$的函数;\\\\\n\\textcircled{2} 当车流密度$x$为多大时, 车流量可以达到最大, 并求出最大值(精确到$1$辆/小时);\\\\\n(2) 为减轻周边道路通行压力, 现规定当车流密度$x\\in [60,200]$时, 车流量始终不能超过$3000$辆/小时. 当车流密度$x\\in [60, 200]$时, 请参考教材例题或习题中出现过的函数, 写出一个符合要求的车流速度函数并说明其合理性.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284943,7 +286103,8 @@ "content": "方程$x^3+2x+1=100$是否有整数解? 说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284964,7 +286125,8 @@ "content": "用函数的观点在区间$(0,+\\infty)$上解不等式$x^4+x>2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -284985,7 +286147,8 @@ "content": "设方程$x+\\log_2x=4$的解为$x_1$, $x+2^x=4$的解为$x_2$, 试求出$x_1+x_2$的结果, 并说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285006,7 +286169,8 @@ "content": "已知$a$是实常数, 设关于$x$的不等式$\\dfrac 1x\\ge x^2+a$的解集为$A$, 若$A$与区间$[1,+\\infty)$的交集非空, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285027,7 +286191,8 @@ "content": "如图所示, 在一块边长为$13\\text{cm}$的正方形金属薄片的四个角上都剪去一个边长为$x\\text{cm}$的小正方形, 做成一个容积是$140\\text{cm}^3$的无盖长方体盒子. 问: $x$是多少? (结果精确到$0.1\\text{cm}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\filldraw [pattern = north east lines] (0,0) rectangle (1,1) (5,0) rectangle (4,1) (0,5) rectangle (1,4) (5,5) rectangle (4,4);\n\\draw (0,0) rectangle (5,5);\n\\draw [dashed] (1,1) rectangle (4,4);\n\\draw (-0.1,0) -- (-1.1,0) (-0.1,5) -- (-1.1,5) (-0.1,1) -- (-0.6,1) (-0.1,4) -- (-0.6,4);\n\\draw [<->] (-0.35,0) -- (-0.35,1) node [midway,fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.35,4) -- (-0.35,5) node [midway,fill = white] {\\rotatebox{90}{$x$}};\n\\draw [<->] (-0.85,0) -- (-0.85,5) node [midway,fill = white] {\\rotatebox{90}{$13$}};\n\\draw (0,-0.1) -- (0,-1.1) (5,-0.1) -- (5,-1.1) (1,-0.1) -- (1,-0.6) (4,-0.1) -- (4,-0.6);\n\\draw [<->] (0,-0.35) -- (1,-0.35) node [midway,fill = white] {$x$};\n\\draw [<->] (4,-0.35) -- (5,-0.35) node [midway,fill = white] {$x$};\n\\draw [<->] (0,-0.85) -- (5,-0.85) node [midway,fill = white] {$13$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\path (-2,-2,0);\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-3) coordinate (C);\n\\draw (0,0,-3) coordinate (D);\n\\draw (A) ++ (0,1,0) coordinate (A1);\n\\draw (B) ++ (0,1,0) coordinate (B1);\n\\draw (C) ++ (0,1,0) coordinate (C1);\n\\draw (D) ++ (0,1,0) coordinate (D1);\n\\path [name path = up] (A1) -- (B1) -- (C1);\n\\path [name path = line1] (A) -- (D);\n\\path [name intersections = {of = up and line1, by = P}];\n\\path [name path = line2] (D) -- (C);\n\\path [name intersections = {of = up and line2, by = Q}];\n\\draw (P) -- (D) -- (Q);\n\\draw (A) -- (B) -- (C) (A) -- (A1) (B) -- (B1) (C) -- (C1) (A1) -- (B1) -- (C1) -- (D1) -- cycle (D) -- (D1);\n\\draw [dashed] (A) -- (P) (C) -- (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285048,7 +286213,8 @@ "content": "已知函数$y=x^3+2x-99$在区间$(4, 5)$上有且仅有一个零点, 求该零点的近似值. (结果精确到$0.1$)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285069,7 +286235,8 @@ "content": "已知方程$x=3-\\lg x$在区间$(2, 3)$上有且仅有一个根, 求该根的近似值. (结果精确到$0.1$)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285090,7 +286257,8 @@ "content": "求方程$0.8^x-1=\\ln x$的近似解. (结果精确到$0.1$)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285111,7 +286279,8 @@ "content": "设函数$f(x)=ax^2+bx+c$($a>0$, $b$、$c\\in \\mathbf{R}$), 且$f(1)=-\\dfrac a2$, 求证: 函数在区间$(0, 2)$上至少有一个零点.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285132,7 +286301,8 @@ "content": "若$f(x)=1+\\log_3x$, 并设$y=f^{-1}(x)$是$y=f(x)$的反函数, 求$f^{-1}(2)$, $f^{-1}(a)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285153,7 +286323,8 @@ "content": "求下列函数的反函数:\\\\\n(1) $y=4x+2$;\\\\ \n(2) $y=x^2+1$, $x\\in [1,3]$;\\\\ \n(3) $y=\\dfrac{3x+1}{4x+2}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285174,7 +286345,8 @@ "content": "设$a>0$, 若函数$y=x+\\dfrac 4x$, $x\\in (0,a]$存在反函数, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -285195,7 +286367,8 @@ "content": "已知$f(x)=2^x-1$, 设$y=f^{-1}(x)$是$y=f(x)$的反函数. 若正数$a$、$b$满足$f^{-1}(a-1)+f^{-1}(b-1)=1$, 则$f(ab)$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -285216,7 +286389,8 @@ "content": "求函数$y=x^3$的反函数, 并在同一坐标系中作出函数$y=x^3$和它的反函数的图像.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285237,7 +286411,8 @@ "content": "已知函数$y=a^x+b$的图像经过点$(1,7)$, 而其反函数的图像经过点$(4,0)$, 求实数$a$、$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -285258,7 +286433,8 @@ "content": "已知常数$a>0$且$a\\ne 1$, 若函数$y=a^x$的反函数的图像经过点$(2,-1)$, 则$a=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -285279,7 +286455,8 @@ "content": "已知函数$y=f(x)$的图像经过点$(1,2)$, 设$y=f(x+4)$的反函数为$y=g(x)$, 那么函数$y=g(x)$的图像必经过点\\bracket{20}. \n\\fourch{$(2,-3)$}{$(-3,2)$}{$(2,5)$}{$(5,2)$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -285300,7 +286477,8 @@ "content": "设函数$y=\\dfrac{1-2x}{2+x}$的反函数为$y=f(x)$, 若函数$y=g(x)$的图像与$y=f(x+1)$的图像关于直线$y=x$成轴对称, 则$g(3)$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -285321,7 +286499,8 @@ "content": "设函数$y=f(x)$的图像与其反函数$y=f^{-1}(x)$的图像的交点所组成的集合为$A$. 以下结论:\\\\\n\\textcircled{1} 集合$A$可能是空集; \n\\textcircled{2} 集合$A$中最多只有一个元素; \n\\textcircled{3} 若集合$B=\\{(x,y)|y=x\\}$, 则一定有$A\\subseteq B$; \n\\textcircled{4} 集合$A$中可以有无穷多个元素.\\\\\n其中, 所有正确结论的序号是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288642,7 +289821,8 @@ "content": "设$a,b,x,y\\in \\mathbf{R}$, 判断下列命题的真假.\\\\\n\\blank{20}(1) 若$a=b$, 则$\\sqrt a=\\sqrt b$;\\\\\n\\blank{20}(2) 若$a^2=b^2$, 则$|a|=|b|$;\\\\\n\\blank{20}(3) 若$\\dfrac ab=\\dfrac xy$, 且$b-y\\ne 0$, 则$\\dfrac{a-x}{b-y}=\\dfrac ab$;\\\\\n\\blank{20}(4) 若$ax=b$, 则$x=\\dfrac ba$;\\\\\n\\blank{20}(5) 若$a^2x=a^2y$, 则$x=y$;\\\\\n\\blank{20}(6) 若$(\\sqrt a+1)x=(\\sqrt a+1)y$, 则$x=y$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288663,7 +289843,8 @@ "content": "若$(m^2-1)x^2+(m-1)x+3=0$是关于$x$的一元一次方程, 则这个方程的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288684,7 +289865,8 @@ "content": "已知$a,b\\in \\mathbf{R}$, 解关于$x$的方程$(a-2)x-1=x+b$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288705,7 +289887,8 @@ "content": "方程组 $\\begin{cases}\\dfrac{x+y}2+\\dfrac{x-y}3=6, \\\\2(x+y)-x+y=-4 \\end{cases}$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288726,7 +289909,8 @@ "content": "已知关于$x$的方程$2a(x-1)=(5-a)x+3b$的解集为$\\mathbf{R}$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288747,7 +289931,8 @@ "content": "如果$a,b$为定值, 关于$x$的方程$\\dfrac{2kx+a}3=2+\\dfrac{x-bk}6,$无论$k$为何值时, 它总是至少有一个根为$1$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288768,7 +289953,8 @@ "content": "已知$k\\in \\mathbf{R}$, 求关于$x,y$的一元二次方程组$\\begin{cases}\n 2x+ky=1, \\\\2kx+y=3 \\end{cases}$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288792,7 +289978,8 @@ "content": "已知$a\\in \\mathbf{R}$, 要使方程组$\\begin{cases} 2x+ay=16, \\\\x-2y=0 \\end{cases}$有正整数解, 求整数组成$a$的集合.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288813,7 +290000,8 @@ "content": "设$a\\in \\mathbf{R}$, 求关于$x$的方程$ax^2+2x-3=0$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288834,7 +290022,8 @@ "content": "证明: 对于任意实数$a$, 关于$x$的方程$x^2-3(a+1)x+6a+1=0$总有实数根.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288855,7 +290044,8 @@ "content": "若关于$x$的方程$2x^2-3x+1-m=0$有一个正根和一个负根, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288878,7 +290068,8 @@ "content": "若关于$x$的方程$2x^2-3x+1-m=0$有两个不同正根, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288901,7 +290092,8 @@ "content": "已知关于$x$的方程$2x^2+2x+c=0$的两个根为$x_1$, $x_2$, 并且$|x_1-x_2|=\\sqrt 3$, 则的实数$c$值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288922,7 +290114,8 @@ "content": "已知$x_1$、$x_2$是一元二次方程$ax^2-2x-a=0$的两根, 且$\\dfrac{x_2}{x_1-1}+\\dfrac{x_1}{x_2-1}=6$, 求实数$a$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288943,7 +290136,8 @@ "content": "已知实数$p$、$q$, 满足$p^2-3p-5=0$, $5q^2+3q-1=0$, 且$pq\\ne 1$, 则$p^2+\\dfrac 1{q^2}=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -288964,7 +290158,8 @@ "content": "已知关于$x$的方程$x^2+ax+\\dfrac 12=0$有两根、且两根都比$1$小, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -288985,7 +290180,8 @@ "content": "已知关于$x$的方程$2x^2+2x+6k-1=0$有两个实数根, 且一个实根比$1$小, 另一个实根比$1$大, 求$k$的最大整数值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289006,7 +290202,8 @@ "content": "已知关于$x$的方程$ax^2+(a+2)x+a-1=0$只有整数根, 求实数$a$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289027,7 +290224,8 @@ "content": "判断下列语句是否正确, 并在相应横线上填入``\\checkmark''或``$\\times$''.\\\\\n\\blank{20}(1) 若$ax>b,a\\ne 0$则$x>\\dfrac ba$;\\\\\n\\blank{20}(2) 若$a^2x>a^2y$, 则$x>y$;\\\\\n\\blank{20}(3) 若$a>b$, 则$a^2>ab$;\\\\\n\\blank{20}(4) 若$a\\dfrac 1a>\\dfrac 1b$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -289048,7 +290246,8 @@ "content": "设$x$为实数, 下列各式中, 值恒大于$x$的是\\bracket{20}.\n\\fourch{$x^2+4$}{$2x+4$}{$x^3+4$}{$\\dfrac 1{x^2+4}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289069,7 +290268,8 @@ "content": "``$a(a-b)<0$''是``$\\dfrac ba>1$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289090,7 +290290,8 @@ "content": "已知$a,b\\in \\mathbf{R}^+$, 则$\\dfrac ba<\\dfrac{b+n}{a+n}(n\\in \\mathbf{N}$, $n\\ge 1)$成立的充要条件是\\bracket{20}.\n\\fourch{$a>b$}{$a0$, $b>0$, 且$a\\ne b$, 比较$\\dfrac{a^2}b+\\dfrac{b^2}a$与$a+b$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289176,7 +290380,8 @@ "content": "已知$x$、$y\\in \\mathbf{R}$, 比较$x^2+y^2$与$2(2x-y)-5$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289199,7 +290404,8 @@ "content": "已知$a\\in \\mathbf{R}$, $a\\ne 0$, 比较$\\dfrac 1{1+a}$与$1-a$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289223,7 +290429,8 @@ "content": "如果$a^2>b^2$, 那么下列不等式中正确的是\\bracket{20}.\n\\fourch{$a>0>b$}{$a>b>0$}{$|a|>|b|$}{$a>|b|$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289247,7 +290454,8 @@ "content": "如果$aab$}{$\\dfrac 1{b^2}<\\dfrac 1{a^2}$}{$\\dfrac 1a<\\dfrac 1b$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289271,7 +290479,8 @@ "content": "如果$a<0b^2$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289295,7 +290504,8 @@ "content": "若$x\\dfrac 1x$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -289316,7 +290526,8 @@ "content": "如果$a$''或``$<$''号填空)\\\\\n(1) $\\sqrt [n]{-a}$\\blank{50}$\\sqrt [n]{-b}(n\\ge 2,n\\in \\mathbf{N})$;\\\\\n(2) $\\dfrac 1{a^{2n}}$\\blank{50}$\\dfrac 1{b^{2n}},(n\\in \\mathbf{N},n\\ge 1)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -289339,7 +290550,8 @@ "content": "``$a|b|$''是$B$: ``$a^2>b^2$''的\\blank{50}条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -289381,7 +290594,8 @@ "content": "已知$0<\\dfrac ab<\\dfrac cd$, 比较$\\dfrac b{a+b}$与$\\dfrac d{c+d}$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289404,7 +290618,8 @@ "content": "比较$\\sqrt {x+1}-\\sqrt x$与$\\sqrt {x+2}-\\sqrt {x+1}$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289425,7 +290640,8 @@ "content": "已知$a,b,m,n$都是正实数, 且$m+n=1$, 比较$\\sqrt {ma+nb}$和$m\\sqrt a+n\\sqrt b$的大小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289448,7 +290664,8 @@ "content": "已知$m$为常数. 若对于某一个确定的$c$, 关于$x$的一元二次方程$mx^2+2x+c+1=0$有实根, 求$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289469,7 +290686,8 @@ "content": "已知$m$为常数. 若对于任意的$c$, 关于$x$的一元二次方程总有实根, 求$m$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289490,7 +290708,8 @@ "content": "求下列一元一次不等式(组)的解集:\\\\ \n(1) $2(x+1)-3(x-2)>8$;\\\\\n(2) $\\begin{cases} 3x-2(5-3x)>8, \\\\ 2x\\le 2(2x+3). \\end{cases}$", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289513,7 +290732,8 @@ "content": "已知关于$x$的不等式$(4a-3b)x>2b-a$的解集为$(-\\infty ,\\dfrac 49)$, 则$ax>b$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -289534,7 +290754,8 @@ "content": "解下列关于$x$的不等式.\\\\\n(1) $ax+4<2x+a^2$, 其中$a>2$;\\\\\n(2) $mx+1b^2(1+x)+2ab$, 其中$a,b>0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289604,7 +290827,8 @@ "content": "若关于$x$的不等式组$\\begin{cases}x-2a>0, \\\\x-3b\\le 0 \\end{cases}$的整数解有且仅有$4,5$, 则$a$的取值范围是\\blank{50}, $b$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -289625,7 +290849,8 @@ "content": "设$a$为实数, 求关于$x$的一元一次不等式$\\begin{cases} ax<1, \\\\ -2x<4a \\end{cases}$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289648,7 +290873,8 @@ "content": "已知关于$x$的一元一次不等式组$\\begin{cases} x\\ge a, \\\\x\\le 2-2a \\end{cases}$的解集中, 有且仅有$3$个整数解, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289669,7 +290895,8 @@ "content": "求解下列不等式(组)的解集: \n(1) $2x^2-3x+1<0$;\\\\\n(2) $(x+1)^2-6>0$;\\\\\n(3) $x(x-1)0$;\\\\\n(5) $x^4+2x^2-8<0$;\\\\\n(6) $\\begin{cases} 4x^2-27x+18>0, \\\\ x^2-6x+4<0; \\end{cases}$\\\\\n(7) $\\begin{cases} 6-x-x^2\\le 0, \\\\ x^2+3x-4<0; \\end{cases}$\\\\\n(8) $\\begin{cases} 5x^2-4x-7\\le 0, \\\\ x^2+x+1<0, \\\\ 7x-2>0. \\end{cases}$", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289690,7 +290917,8 @@ "content": "已知$a\\in \\mathbf{R}$, 求下列关于$x$的不等式的解集:\\\\ \n(1) $(x-a)(x-1)<0(a>1)$;\\\\\n(2) $(x-a)(x-2a)<0(a>0)$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289711,7 +290939,8 @@ "content": "已知集合$U=\\mathbf{R}$, 且集合$A=\\{x|x^2-16<0\\}$, 集合$B=\\{x|x^2-4x+3\\ge 0\\}$, 求:\\\\\n(1) $A\\cap B$;\\\\\n(2) $A\\cup B$;\\\\\n(3) $\\overline{A\\cap B}$;\\\\\n(4) $\\overline{A\\cup B}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289734,7 +290963,8 @@ "content": "已知$a\\in \\mathbf{R}$, 求下列关于$x$的不等式的解集:\\\\\n(1) $x^2-(a+1)^2x+2a(a^2+1)<0$;\n(2) $x^2+(a+2)x+2a>0$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289755,7 +290985,8 @@ "content": "已知关于$x$的二次方程$2x^2+ax+1=0$无实数解, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289779,7 +291010,8 @@ "content": "当$k$取何值时, 关于$x$的不等式$2kx^2+kx-\\dfrac 38<0$对于一切实数$x$都成立?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289802,7 +291034,8 @@ "content": "若函数$y=ax^2+2ax+1$的图像与$x$轴无交点, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289823,7 +291056,8 @@ "content": "已知集合$A=\\{x|x^2+(a-3)x+a<0\\}$非空, 且$A\\subset (0,+\\infty)$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289844,7 +291078,8 @@ "content": "已知函数$y=(k^2+4k-5)x^2+4(1-k)x+3$的图像都在$x$轴的上方, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289867,7 +291102,8 @@ "content": "已知不等式$x^2+ax+b<0$的解集为$(-3,-1)$, 求实数$a$、$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289890,7 +291126,8 @@ "content": "已知关于$x$的不等式$ax^2+bx+c>0$的解集是$\\{x|x>2$或$x<\\dfrac 12\\}$, 求关于$x$的不等式$ax^2-bx+c\\le 0$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289915,7 +291152,8 @@ "content": "已知关于$x$的不等式$ax^2+bx+c>0$的解集为$(-1,2)$, 求不等式$cx^2+ax-b\\le 0$的解集.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289936,7 +291174,8 @@ "content": "已知关于$x$的不等式组$\\begin{cases} (2x-3)(3x+2)\\le 0, \\\\ x-a>0 \\end{cases}$无实数解, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289960,7 +291199,8 @@ "content": "设$a\\in \\mathbf{R}$, 已知集合$A=\\{x|x-a >0\\}$, $B=\\{x|x^2-2ax-3a^2<0\\}$, 求$A\\cap B$与$A\\cup B$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -289981,7 +291221,8 @@ "content": "若不等式组$\\begin{cases} ax^2-x-2\\le 0, \\\\ x^2-x\\ge a(1-x) \\end{cases}$的解集为$\\mathbf{R}$, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290002,7 +291243,8 @@ "content": "若关于$x$的不等式$ax^2-(a+1)x+1<0$的解集为$\\varnothing$, 求实数$a$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290023,7 +291265,8 @@ "content": "若关于$x$的不等式$(a^2-1)x^2+2ax+1>0$有实数解, 求$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290044,7 +291287,8 @@ "content": "求下列不等式的解集: \n(1) $\\dfrac 1x<1$;\\\\\n(2) $\\dfrac{4x+3}{x-1}>5$;\\\\\n(3) $\\dfrac 2x<\\dfrac 2{x-3}$;\\\\\n(4) $\\dfrac{x+4}{x^2+2x+2}\\le 1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290065,7 +291309,8 @@ "content": "若关于$x$的方程$\\dfrac{4k-3x}{k+2}=2x$的解是正数, 则$k$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290086,7 +291331,8 @@ "content": "若关于$x$的方程$ax-1=2x+a$的解不小于$1$, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290107,7 +291353,8 @@ "content": "定义运算``$\\ast$''满足如下法则: $a\\ast b=\\dfrac{a^2-1}b$, 则不等式$x\\ast (x+1)<0$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290128,7 +291375,8 @@ "content": "若$a0$的解集是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290151,7 +291399,8 @@ "content": "已知$f(x),g(x)$是定义在$\\mathbf{R}$上的函数, 若不等式$f(x)\\ge 0$的解集为$[1,2]$, 不等式$g(x)\\ge 0$的解集为$\\varnothing$, 则不等式$\\dfrac{f(x)}{g(x)}>0$的解集是\\blank{50}\\bracket{20}\n\\fourch{$\\varnothing$}{$(-\\infty ,1)\\cup (2,+\\infty)$}{$[1,2)$}{$\\mathbf{R}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290172,7 +291421,8 @@ "content": "解下列关于$x$的不等式:\\\\\n(1) $\\dfrac a{x-1}<1$;\\\\\n(2) $\\dfrac{ax}{x-1}<1$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290193,7 +291443,8 @@ "content": "求下列不等式的解集.\\\\\n(1) $|x^2-3|<2$;\\\\\n(2) $|\\dfrac 1{2-x}|\\ge 2$;\\\\\n(3) $|x^2-3x+2|\\le 0$;\\\\\n(4) $|\\dfrac x{x+1}|>\\dfrac x{x+1}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290214,7 +291465,8 @@ "content": "若$x=5$是不等式$|x-a|\\le 4$的解中的最大值, 则$a$的值等于\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290235,7 +291487,8 @@ "content": "已知集合$A=\\{x|$存在实数$y$, 使得$y=\\sqrt {x^2-2x-8}$成立$\\}$, \n集合$B=\\{x|1-|x-a|>0,\\ x\\in \\mathbf{R}\\}$, 若$A\\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290256,7 +291509,8 @@ "content": "求下列不等式的解集.\\\\\n(1) $|x+3|-|x-1|<1$;\\\\\n(2) $|x+1|<\\dfrac 1{x-1}$;\\\\\n(3)\t$x^2-2|x|-15\\le 0$;\\\\\n(4) $x^2-x-5>|2x-1|$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290277,7 +291531,8 @@ "content": "已知不等式$|ax+1|\\le b$的解集是$[-1,3]$, 求$a$、$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290300,7 +291555,8 @@ "content": "设$x\\in \\mathbf{R}$, 则$(1-|x|)(1+x)>0$成立的充要条件是\\bracket{20}.\n\\fourch{$|x|<1$}{$x<1$}{$|x|>1$}{$x<1$且$x\\ne -1$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290323,7 +291579,8 @@ "content": "若实数$a,b$满足$ab>0$, 则在``\\textcircled{1} $|a+b|>|a|$, \\textcircled{2} $|a+b|<|b|$, \\textcircled{3} $|a+b|<|a-b|$, \\textcircled{4} $|a+b|>|a-b|$''这四个式子中, 正确的是\\bracket{20}\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290346,7 +291603,8 @@ "content": "若$|a+b|<-c$, 则在``\\textcircled{1} $a<-b-c$, \\textcircled{2} $a+b>c$, \\textcircled{3} $a+c0$, 当$|x-2|0$, 那么下列不等式中正确的是\\bracket{20}.\n\\fourch{$\\dfrac 1a<\\dfrac 1b$}{$\\sqrt {-a}<\\sqrt b$}{$a^2|b|$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290411,7 +291671,8 @@ "content": "设$a,b,c,d\\in \\mathbf{R}$, 且$a>b$, $cb+d$}{$a-c>b-d$}{$ac>bd$}{$\\dfrac ad>\\dfrac bc$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290432,7 +291693,8 @@ "content": "不等式$(x-1)\\sqrt {x+2}\\ge 0$的解集是\\bracket{20}.\n\\twoch{$\\{x|x>1\\}$}{$\\{x|x\\ge 1\\}$}{$\\{x|x\\ge -2\\text{或}x\\ne 1\\}$}{$\\{x|x=-2\\text{或}x\\ge 1\\}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290455,7 +291717,8 @@ "content": "已知集合$A=\\{x||x-a|\\le 1\\}$, $B=\\{x|x^2-5x+4\\ge 0\\}$. 若$A\\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290476,7 +291739,8 @@ "content": "一元二次不等式$ax^2+bx+1>0$的解集是$\\{x|x\\ne 2\\}$, 求实数$a$与$b$的值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290497,7 +291761,8 @@ "content": "已知集合$A=\\{x|x^2-6x+5<0,\\ x\\in \\mathbf{R}\\}$, $B=\\{x|x^2-3ax+2a^2<0, \\ x\\in \\mathbf{R}\\}$.\\\\\n(1) 若$A\\cap B=\\varnothing$, 求实数$a$的取值范围;\\\\\n(2) 若$B\\subseteq A$, 求实数$a$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290518,7 +291783,8 @@ "content": "关于$x$的不等式组$\\begin{cases} x^2-x-2>0, \\\\ 2x^2+(2k+5)x+5k<0 \\end{cases}$的整数解的集合为$\\{-2\\}$, 求实数$k$的取值范围.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290539,7 +291805,8 @@ "content": "若$q<0\\dfrac 1p\\}$}{$\\{x|-\\dfrac 1p0$, $b>0$, 且$a\\ne b$, 则下列各式恒成立的是\\bracket{20}.\n\\fourch{$\\dfrac{2ab}{a+b}<\\dfrac{a+b}2<\\sqrt {ab}$}{$\\sqrt {ab}<\\dfrac{2ab}{a+b}<\\dfrac{a+b}2$}{$\\dfrac{2ab}{a+b}<\\sqrt {ab}<\\dfrac{a+b}2$}{$\\sqrt {ab}<\\dfrac{a+b}2<\\dfrac{2ab}{a+b}$}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290753,7 +292029,8 @@ "content": "已知$a,b\\in \\mathbf{R}$. 若$M=a^2+b^2+1$, $N=a+b+ab$, 则$M$、$N$的大小关系是\\bracket{20}. \n\\fourch{$M\\ge N$}{$M\\le N$}{$M=N$}{无法确定}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "选择题", "ans": "", @@ -290774,7 +292051,8 @@ "content": "设$ab\\ne 0$, 利用基本不等式有如下证明:\\\\ \n$\\dfrac ba+\\dfrac ab=\\dfrac{b^2+a^2}{ab}\\ge \\dfrac{2ab}{ab}=2$.试判断这个证明过程是否正确. 若正确, 请说明每一步的依据; 若不正确, 请说明理由.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290797,7 +292075,8 @@ "content": "若$a>0$, $b>0$, $c>0$, $d>0$, \n则$\\dfrac ba+\\dfrac ab\\ge$\\blank{50}, $\\dfrac{b+c}a+\\dfrac{c+a}b+\\dfrac{a+b}c\\ge$\\blank{50}, $(a+b)(\\dfrac 1a+\\dfrac 1b)\\ge$\\blank{50}, $(\\dfrac ba+\\dfrac dc)(\\dfrac cb+\\dfrac ad)\\ge$\\blank{50}.\n(填能使不等式成立的最大值)", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290818,7 +292097,8 @@ "content": "当$x<1$时, 有$\\dfrac{x^2-2x+a+2}{x-1}\\le -4$成立, 且当$x=x_0$时等号成立, 则$a$的值为\\blank{50}, 等号成立时$x_0$的值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290839,7 +292119,8 @@ "content": "求证:\\\\\n(1) 若$x>0$, $y>0$, 则$\\sqrt {(1+x)(1+y)}\\ge 1+\\sqrt {xy}$;\\\\\n(2) 若$a>0$, $b>0$, 则$a+b+\\dfrac 1{\\sqrt {ab}}\\ge 2\\sqrt 2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -290860,7 +292141,8 @@ "content": "若实数$x+y=4$, 则$x^2+y^2$有最\\blank{50}值, 且此最值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290881,7 +292163,8 @@ "content": "已知$00$, 则$2x+\\dfrac 8{x+1}$的最小值是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290946,7 +292231,8 @@ "content": "已知$x>-1$, 则当$x$取\\blank{50}时, $x+\\dfrac 4{x+1}$的值最小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290967,7 +292253,8 @@ "content": "已知$x>1$.比较大小: $\\dfrac{2x^2-4x+4}{x-1}$\\blank{50}", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -290988,7 +292275,8 @@ "content": "若$x<1$, 则$\\dfrac{x^2-2x+3}{x-1}$有最\\blank{50}值, 且此最值为\\blank{50}, 此时$x=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291009,7 +292297,8 @@ "content": "若$x>0$, 则$\\dfrac x{x^2+1}$有最\\blank{50}值, 且此最值为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291030,7 +292319,8 @@ "content": "用一根长为$m$的铁丝制成一个矩形框架.当长、宽分别为多少时, 框架的面积最大?", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291054,7 +292344,8 @@ "content": "已知$x,y>0$, 且$x+y=1$, 求当$x,y$分别取何值时, $\\dfrac 1x+\\dfrac 2y$的值最小.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291075,7 +292366,8 @@ "content": "已知$x>0$, $y>0$, $x+y=1$, 求证:\\\\\n(1) $(1+\\dfrac 1x)(1+\\dfrac 1y)\\ge 9$;\\\\\n(2) $(\\dfrac 1{x^2}-1)(\\dfrac 1{y^2}-1)\\ge 9$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291098,7 +292390,8 @@ "content": "已知$a,b>0$, 满足$a+b=4$.\\\\\n(1) 求$a^2+b^2$的最小值;\\\\\n(2) 求$\\dfrac 1a+\\dfrac 4b$的最小值;\\\\\n(3) 求$\\dfrac 1{a+1}+\\dfrac 3{2b+4}$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291119,7 +292412,8 @@ "content": "设$a,b,c\\in \\mathbf{R}$, 且满足$a+2b+c=1$, 求$a^2+b^2+c^2$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291140,7 +292434,8 @@ "content": "已知$a,b$是实数, 求证: $|a|-|b|\\le|a+b|$, 并指出等号成立的条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291163,7 +292458,8 @@ "content": "证明: $|x-3|-|x-5|\\le 2$对一切实数$x$恒成立, 并指出等号成立的条件.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291184,7 +292480,8 @@ "content": "已知实数$a,b,c$满足$|a-c|<1$, $|b-c|<1$, 证明: $|a-b|<2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291205,7 +292502,8 @@ "content": "若$|x-3|+|x-4|+|x-5|\\ge a$对一切实数$x$恒成立, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291226,7 +292524,8 @@ "content": "如果实数$a,b,c,d$满足$|a-2b|\\le 6$, $|b-d|\\le 7$, $|a-3b+d|\\ge 13$, 则$|a-2b|-|b-d|=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291247,7 +292546,8 @@ "content": "已知$|x|\\le 3$, $|y|\\le 1$, $|z|\\le 4$, 且$|x-2y+z| =9$, 则$2x-3y+z=$\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291268,7 +292568,8 @@ "content": "若$|2x-1|+|3x-1|+|4x-1|+\\cdots +|9x-1|+|10x-1|$是常数, 则$x$的取值范围是\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "", @@ -291289,7 +292590,8 @@ "content": "已知$a,b,c,d\\in \\mathbf{R}$, 求证:\\\\\n(1) $(ac+bd)^2\\le (a^2+b^2)(c^2+d^2)$;\\\\\n(2) $a^2+b^2+c^2\\ge ab+bc+ca$.\\\\", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291310,7 +292612,8 @@ "content": "已知实数$a\\ge 3$, 求证: $\\sqrt a-\\sqrt {a-1}<\\sqrt {a-2}-\\sqrt {a-3}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291333,7 +292636,8 @@ "content": "已知$a$、$b$、$c$是不全相等的整数, 求证: $(a^2+1)(b^2+1)(c^2+1)>8abc$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291358,7 +292662,8 @@ "content": "已知$ab>0$, 且$a+b=1$.求证: $\\sqrt {a+\\dfrac 12}+\\sqrt {b+\\dfrac 12}\\le 2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291379,7 +292684,8 @@ "content": "已知$a>0$, $b>0$, 求证: $\\dfrac a{\\sqrt b}+\\dfrac b{\\sqrt a}\\ge \\sqrt a+\\sqrt b$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291402,7 +292708,8 @@ "content": "已知$xy\\in \\mathbf{R}$.求证: $2x^2-4x+21>2y-y^2$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291423,7 +292730,8 @@ "content": "已知$x$、$y$为正实数. 求证: $\\dfrac 1{2x}+\\dfrac 1{2y}\\ge \\dfrac 2{x+y}$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291444,7 +292752,8 @@ "content": "已知正数$a,b$满足$a+b=1$, 求证:\\\\\n(1) $(a+\\dfrac 1a)^2+(b+\\dfrac 1b)^2\\ge \\dfrac{25}2$;\\\\\n(2) $(a+\\dfrac 1a)(b+\\dfrac 1b)\\ge \\dfrac{25}4$.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291610,7 +292919,8 @@ "K0222002B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $M=10+mx-x-10\\sqrt{x}, \\ x\\in \\{1,2,3,\\cdots,16\\}$; (2) $[\\dfrac 72,\\dfrac{19}4]$.", @@ -291633,7 +292943,8 @@ "K0222002B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $2197.2\\text{m}/\\text{s}$; (2) $53.6$倍.", @@ -291656,7 +292967,8 @@ "K0222002B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $19$万元; (2) 当促销费为$7$万元时, 该网店售出商品的总利润最大, 此时商品的剩余量为$0.25$万件.", @@ -291677,7 +292989,8 @@ "content": "勤俭节约是中华民族的传统美德. 为避免舌尖上的浪费, 各地各部门采取了精准供应的措施.某学校食堂经调查分析预测, 从年初开始的前$n$($n=1,2,3,\\cdots ,12$)个月对某种食材的需求总量$S_n$(公斤)近似地满足$S_n=\\begin{cases} 635n, & 1\\le n\\le 6,\\\\ -6n^2+774n-618, & 7\\le n\\le 12. \\end{cases}$ 为保证全年每一个月该食材都够用, 食堂前$n$个月的进货总量须不低于前$n$个月的需求总量.\\\\\n(1) 如果每月初进货$646$公斤, 那么前$7$个月每月该食材是否都够用?\\\\\n(2) 若每月初等量进货$p$公斤, 为保证全年每一个月该食材都够用, 求$p$的最小值.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291700,7 +293013,8 @@ "K0222002B" ], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "(1) $(0,80]$; (2) 车流量的最大值约为$3250\\text{辆}/\\text{小时}$, 此时车流密度约为$87\\text{辆}/\\text{千米}$.", @@ -291721,7 +293035,8 @@ "content": "已知角$\\alpha$终边上一点$P$与$x$轴的距离和与轴的距离之比为$4:3$, 且$\\cos \\alpha <0$.求$\\sin \\alpha$和$\\tan \\alpha$.\n%", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "解答题", "ans": "", @@ -291742,7 +293057,8 @@ "content": "若函数$f(x)=\\log_2(x+1)+a$的图像经过点$(1,4)$, 则实数$a=$\\blank{50}.\n%010927", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$3$", @@ -291777,7 +293093,8 @@ "content": "将函数$y=\\sqrt{3}\\sin 2x-\\cos 2x$的图像向左平移$m$($m>0$)个单位, 所得图像对应的函数为偶函数, 则$m$的最小值为\\blank{50}.\n%010931", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$\\dfrac{\\pi}3$", @@ -291890,7 +293207,8 @@ "content": "函数$y=\\dfrac{1}{2^x}$的反函数为\\blank{50}.", "objs": [], "tags": [ - "" + "", + "暂无对应" ], "genre": "填空题", "ans": "$y=-\\log_2 x$", @@ -302411,7 +303729,7 @@ }, "030402": { "id": "030402", - "content": "写出一个同时满足性质\\textcircled{1}, \\textcircled{2}的函数$f(x)=$\\blank{50}. ($y=f(x)$不是常值函数)\\\\\n\\textcircled{1} $y=f'(x)$为偶函数; \\textcircled{2} $f(x+\\pi)=f'(x)$.", + "content": "写出一个同时满足性质\\textcircled{1}, \\textcircled{2}的函数$f(x)=$\\blank{50}. ($y=f(x)$不是常值函数)\\\\\n\\textcircled{1} $y=f'(x)$为偶函数; \\textcircled{2} $f'(x+\\pi)=f'(x)$.", "objs": [ "K0231002X" ], @@ -302944,7 +304262,7 @@ }, "030424": { "id": "030424", - "content": "已知函数$f(x)=x^3-3ax+a$($a\\in \\mathbf{R}$).\\\\\\n(1) 若$y=f(x)$仅有一个零点, 求$a$的取值范围;\\\\\n(2) 若函数$y=f(x)$在区间$[0,3]$上的最大值与最小值之差为$g(a)$, 求$g(a)$的最小值.", + "content": "已知函数$f(x)=x^3-3ax+a$($a\\in \\mathbf{R}$).\\\\\n(1) 若$y=f(x)$仅有一个零点, 求$a$的取值范围;\\\\\n(2) 若函数$y=f(x)$在区间$[0,3]$上的最大值与最小值之差为$g(a)$, 求$g(a)$的最小值.", "objs": [ "K0234003X" ], @@ -303046,7 +304364,9 @@ "id": "030428", "content": "已知集合$A=\\{1,2,m\\}$, $B=\\{3,4\\}$. 若$A\\cap B=\\{4\\}$, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303065,7 +304385,9 @@ "id": "030429", "content": "若$\\mathrm{P}_n^2=6$, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303084,7 +304406,9 @@ "id": "030430", "content": "函数$f(x)=\\arcsin x+1$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303103,7 +304427,9 @@ "id": "030431", "content": "若球的大圆的面积为$9\\pi$, 则该球的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303122,7 +304448,9 @@ "id": "030432", "content": "函数$f(x)=\\sin^2x-\\cos^2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303141,7 +304469,9 @@ "id": "030433", "content": "若掷一颗质地均匀的骰子, 则出现向上的点数大于$4$的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303160,7 +304490,9 @@ "id": "030434", "content": "若$f(x)=\\sin x\\cos\\theta+\\cos x\\sin\\theta$是定义在$\\mathbf{R}$上的偶函数, 其中$0\\le \\theta\\le \\dfrac\\pi 2$, 则$\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303179,7 +304511,9 @@ "id": "030435", "content": "设常数$a\\in \\mathbf{R}$, 函数$f(x)=\\ln (x+a)$. 若$f(x)$的反函数图像经过点$(3,1)$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303198,7 +304532,9 @@ "id": "030436", "content": "函数$y=\\sqrt{x}-\\sqrt{1-x}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303217,7 +304553,9 @@ "id": "030437", "content": "若非零实数$a,b$满足$a^2+4b^2=1$, 则$\\dfrac{2ab}{|a|+2|b|}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303236,7 +304574,9 @@ "id": "030438", "content": "已知$f(x)$是定义域为$\\mathbf{R}$的奇函数, 满足$f(1+x)=f(1-x)$. 若$f(1)=2$, 则$f(1)+f(2)+f(3)+\\cdots+f(2018)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303255,7 +304595,9 @@ "id": "030439", "content": "已知定义域为$(0,+\\infty)$的函数$f(x)$满足: 对任何$x\\in (0,+\\infty)$, 都有$f(3x)=3f(x)$, 且当$x\\in (1,3]$时, $f(x)=3-x$. 在下列结论中, 正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 对任何$m\\in \\mathbf{Z}$, 都有$f(3^m)=0$; \\textcircled{2} 函数$f(x)$的值域是$[0,+\\infty)$; \\textcircled{3} 存在$n\\in \\mathbf{Z}$, 使得$f(3^n+1)=17$; \\textcircled{4} ``函数$f(x)$在区间$(a,b)$上单调递减''的一个充要条件是``存在$k\\in \\mathbf{Z}$, 使得$(a,b)\\subseteq (3^k,3^{k+1})$.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303274,7 +304616,9 @@ "id": "030440", "content": "为了得到函数$y=\\sin(x+\\dfrac{5\\pi}6)$的图像, 可将函数$y=\\sin x$的图像\\bracket{20}.\n\\twoch{左移$\\dfrac{5\\pi}6$个长度}{右移$\\dfrac{5\\pi}6$个长度}{左移$\\dfrac{5\\pi}{12}$个长度}{右移$\\dfrac{5\\pi}{12}$个长度}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -303293,7 +304637,9 @@ "id": "030441", "content": "已知$a,b\\in \\mathbf{R}$, 则``$ab>0$''是``$\\dfrac ab+\\dfrac ba>2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -303312,7 +304658,9 @@ "id": "030442", "content": "符号$[x]$表示不超过$x$的最大整数, 如$[\\pi]=3$, $[-1.08]=2$, 定义函数$\\{x\\}=x-[x]$, 那么下列命题中正确的序号是\\bracket{20}.\\\\\n\\textcircled{1} 函数$\\{x\\}$的定义域为$\\mathbf{R}$, 值域为$[0,1]$; \\textcircled{2} 方程$\\{x\\}=\\dfrac 12$有无数解; \\textcircled{3} 函数$\\{x\\}$是周期函数; \\textcircled{4} 函数$\\{x\\}$是增函数.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{4}\\textcircled{1}}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -303331,7 +304679,9 @@ "id": "030443", "content": "如图所示, 已知$PA\\perp$平面$ABC$, $AD\\perp BC$于$D$, $BC=CD=AD=1$. 令$PD=x$, $\\angle BPC=\\theta$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (-1,0,2) node [below] {$D$} coordinate (D);\n\\draw (-5,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(D)!0.5!(B)$) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [right] {$P$} coordinate (P);\n\\draw (B) -- (D) -- (A) -- (P) (B) -- (P) (C) -- (P) (D) -- (P);\n\\draw [dashed] (B) -- (A) (C) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan\\theta = \\dfrac{x}{x^2+2}$}{$\\tan\\theta = \\dfrac{x}{x^2+1}$}{$\\tan\\theta = \\dfrac{1}{x^2+2}$}{$\\tan\\theta = \\dfrac{1}{x^2+1}$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -303350,7 +304700,9 @@ "id": "030444", "content": "在$\\triangle ABC$中, 角$A,B,C$所对的边分别为$a,b,c$. $b=\\sqrt{5}$, $B=\\dfrac\\pi 4$.\\\\\n(1) 若$a=3$, 求$\\sin A$的值;\\\\\n(2) 若$\\triangle ABC$的面积等于$1$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303369,7 +304721,9 @@ "id": "030445", "content": "如图, 圆锥的顶点是$P$, 底面中心是$O$, 已知$OP=\\sqrt{2}$, 圆$O$的直径是$AB=2$, 点$C$在弧$AB$上, 且$\\angle CAB=30^\\circ$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) node [left] {$A$} coordinate (A) (1,0) node [right] {$B$} coordinate (B) (0,0) node [above right] {$O$} coordinate (O) (0,{sqrt(2)}) node [above] {$P$} coordinate (P);\n\\draw ({cos(-60)},{0.3*sin(-60)}) node [below] {$C$} coordinate (C);\n\\draw (A) arc (180:360:1 and 0.3);\n\\draw [dashed] (A) arc (180:0:1 and 0.3) (A) -- (B) (O) -- (P) (A) -- (C);\n\\draw (A) -- (P) (B) -- (P) (C) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆锥的侧面积;\\\\\n(2) 求$O$到平面$APC$的距离.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303388,7 +304742,9 @@ "id": "030446", "content": "科学家发现某种特别物质的温度$y$(单位: 摄氏度)随时间$x$(单位: 分钟)的变化规律满足关系式: $y=m\\cdot 2^x+2^{1-x}$($0\\le x\\le 4$, $m>0$).\\\\\n(1) 若$m=2$, 求经过多少分钟, 该物质的温度为$5$摄氏度;\\\\\n(2) 如果该物质温度总不低于$2$摄氏度, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303407,7 +304763,9 @@ "id": "030447", "content": "已知函数$f(x)=\\log_2(ax^2+2x-a)$.\\\\\n(1) 当$a=-1$时, 求该函数的定义域;\\\\\n(2) 当$a\\le 0$时, 如果$f(x)\\ge 1$对任何$x\\in [2,3]$都成立, 求实数$a$的取值范围;\\\\\n(3) 若$a<0$, 将函数$f(x)$的图像沿$x$轴或其相反方向平移, 得到一个偶函数$g(x)$的图像, 设函数$g(x)$的最大值为$h(a)$, 求$h(a)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303426,7 +304784,9 @@ "id": "030448", "content": "记$f_k(x)=x^k$($x>0$, $k\\in \\mathbf{Z}$).\\\\\n(1) 求函数$F(x)=f_2(x-1)-1$的零点;\\\\\n(2) 设$\\xi,\\eta,\\mu$均为正整数, 且$\\sqrt{\\mu}$为最简根式, 若存在$n_0\\in \\mathbf{N}^*$, 使得$f_{n_0}(\\xi+\\eta\\sqrt{\\mu})$可唯一表示为$\\sqrt{\\tau}+\\sqrt{\\tau-1}$的形式($\\tau\\in \\mathbf{N}^*$). 求证: $|\\xi^2-\\eta^2\\mu|=1$;\\\\\n(3) 已知$f_{-1}(t)+f_{-1}(s)=1$, 是否存在$n_1\\in \\mathbf{N}^*$, 使得$\\dfrac{f_{n_1}(t+s)-f_{n_1}(t)-f_{n_1}(s)+f_{n_1}(2)}{f_{n_1}(4)-f_{n_1}(2)}\\ge 1$成立. 若存在, 试求出$n_1$的值; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303445,7 +304805,9 @@ "id": "030449", "content": "从小到大排列的$9$个数据$1.23,1.35,2.14,2.55,3.67,3.89,4.21,4.43,5.51$的第$60$百分位数$\\mathrm{P}60$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303464,7 +304826,9 @@ "id": "030450", "content": "函数$y=\\sqrt{\\dfrac 1{x^2-1}}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303506,5 +304870,493 @@ ], "remark": "", "space": "12ex" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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}", + "objs": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "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": [], + "tags": [ + "第六单元", + "空间向量" + ], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学教与学例题与习题", + "edit": [ + "20221104\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "030473": { + "id": "030473", + "content": "已知数列$\\{a_n\\}$是等比数列, 且$a_9=-2$, $a_{13}=-32$, 则这个数列的通项公式为\\blank{50}.", + "objs": [ + "K0403004X" + ], + "tags": [ + "第四单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "二期课改练习册高二第一学期-20221104修改", + "edit": [ + "20220726\t王伟叶", + "20221104\t徐慧" + ], + "same": [], + "related": [ + "008438" + ], + "remark": "", + "space": "" } } \ No newline at end of file