diff --git a/工具/单元课时目标题目数据清点.ipynb b/工具/单元课时目标题目数据清点.ipynb
index 355fd1a9..9e97a3b7 100644
--- a/工具/单元课时目标题目数据清点.ipynb
+++ b/工具/单元课时目标题目数据清点.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 16,
+ "execution_count": null,
"metadata": {},
"outputs": [],
"source": [
@@ -98,7 +98,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Python 3.8.8 ('base')",
+ "display_name": "Python 3.9.7 ('base')",
"language": "python",
"name": "python3"
},
@@ -112,12 +112,12 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.8.8"
+ "version": "3.9.7"
},
"orig_nbformat": 4,
"vscode": {
"interpreter": {
- "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
}
}
},
diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb
index 0220a816..096db698 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
@@ -2,193 +2,679 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.943\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.457\t0.171\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.771\t0.743\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.829\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.886\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.914\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.429\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.800\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.600\t0.914\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.543\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.029\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.914\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.943\t0.457\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.429\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三10班\t0.886\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t1.000\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.350\t0.000\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.800\t0.750\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.900\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.700\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.850\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.800\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.800\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.550\t0.950\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.800\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.650\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.850\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.900\t0.650\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.350\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三11班\t0.700\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.909\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.500\t0.227\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.864\t0.591\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.955\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.818\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.864\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.591\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.773\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.727\t0.909\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.727\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.909\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.955\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t1.000\t0.636\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.409\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三12班\t0.864\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.885\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.538\t0.692\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.885\t0.769\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.962\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.885\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.885\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.962\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.692\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.769\t0.962\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.615\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.885\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.923\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.962\t0.462\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.462\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三01班\t0.808\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.933\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.633\t0.433\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.767\t0.867\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.900\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.567\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.833\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.800\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.833\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.867\t0.900\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.467\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.733\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.867\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.967\t0.800\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.467\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三02班\t0.800\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t1.000\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.500\t0.467\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.800\t0.733\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.800\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.567\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.800\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t1.000\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.800\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.833\t0.967\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.367\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.633\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.900\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.933\t0.633\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.500\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三04班\t0.967\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t1.000\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.407\t0.556\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.815\t0.926\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.889\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.630\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t1.000\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.778\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.741\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.926\t0.926\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.667\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.111\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.889\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.889\t0.593\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.667\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三03班\t0.963\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.923\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.333\t0.462\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.846\t0.769\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.923\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.744\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.923\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.897\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.821\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.897\t0.949\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.462\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.308\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.897\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.974\t0.590\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.538\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三05班\t0.872\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.947\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.368\t0.500\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.763\t0.868\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.895\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.737\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.895\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.921\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.842\n",
- "题号: 003341 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.553\t0.868\n",
- "题号: 001892 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.579\n",
- "题号: 001896 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.816\n",
- "题号: 001898 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.974\n",
- "题号: 001916 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.974\t0.711\n",
- "题号: 001905 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.395\n",
- "题号: 001902 , 字段: usages 中已添加数据: 20221010\t2023届高三06班\t0.895\n",
- "题号: 000481 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.962\n",
- "题号: 003347 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.654\t0.308\n",
- "题号: 003330 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.923\t0.808\n",
- "题号: 000153 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.962\n",
- "题号: 003356 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.615\n",
- "题号: 000414 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.769\n",
- "题号: 001882 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.769\n",
- "题号: 000141 , 字段: usages 中已添加数据: 20221010\t2023届高三07班\t0.769\n",
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+ "题号: 009892 , 字段: objs 中已添加数据: K0406005X\n",
+ "题号: 009893 , 字段: objs 中已添加数据: KNONE\n",
+ "题号: 009894 , 字段: objs 中已添加数据: K0407002X\n",
+ "题号: 009895 , 字段: objs 中已添加数据: K0407002X\n",
+ "题号: 009895 , 字段: objs 中已添加数据: K0403003X\n",
+ "题号: 009896 , 字段: objs 中已添加数据: K0408003X\n",
+ "题号: 009897 , 字段: objs 中已添加数据: K0408003X\n",
+ "题号: 009898 , 字段: objs 中已添加数据: K0408003X\n",
+ "题号: 009899 , 字段: objs 中已添加数据: K0409001X\n",
+ "题号: 009900 , 字段: objs 中已添加数据: K0409001X\n",
+ "题号: 009901 , 字段: objs 中已添加数据: K0409002X\n"
]
}
],
diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb
index cb7b9316..17390e02 100644
--- a/工具/批量题号选题pdf生成.ipynb
+++ b/工具/批量题号选题pdf生成.ipynb
@@ -2,20 +2,72 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "开始编译教师版本pdf文件: 临时文件/批量生成题目/test1_教师用_20221007.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0616003B_教师用_20221012.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/批量生成题目/test1_学生用_20221007.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0616003B_学生用_20221012.tex\n",
"0\n",
- "开始编译教师版本pdf文件: 临时文件/批量生成题目/test2_教师用_20221007.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0616004B_教师用_20221012.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/批量生成题目/test2_学生用_20221007.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0616004B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0617002B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0617002B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0617004B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0617004B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0617006B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0617006B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0617007B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0617007B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0619003B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0619003B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0619004B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0619004B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0619005B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0619005B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0620002B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0620002B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0620004B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0620004B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0620005B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0620005B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0623001B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0623001B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0623002B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0623002B_学生用_20221012.tex\n",
+ "0\n",
+ "开始编译教师版本pdf文件: 临时文件/批量生成题目/K0623004B_教师用_20221012.tex\n",
+ "0\n",
+ "开始编译学生版本pdf文件: 临时文件/批量生成题目/K0623004B_学生用_20221012.tex\n",
"0\n"
]
}
@@ -30,8 +82,21 @@
"\"\"\"---设置题目列表---\"\"\"\n",
"#字典字段为文件名, 之后为内容的题号\n",
"problems_dict = {\n",
- "\"test1\":\"1:100\",\n",
- "\"test2\":\"101:110\"\n",
+ "\"K0616003B\":\"000204,004061,009211,009212,010496,010498,010505\",\n",
+ "\"K0616004B\":\"009710,009711\",\n",
+ "\"K0617002B\":\"000210,000690,004084\",\n",
+ "\"K0617004B\":\"010499,010500\",\n",
+ "\"K0617006B\":\"003475,004994,009209,009210,009988,010497\",\n",
+ "\"K0617007B\":\"009712,010521,010522\",\n",
+ "\"K0619003B\":\"000199,000202,000211,000212,000216,000364,000411,005136,009214,009218,009400,010507,010514,010519,010524\",\n",
+ "\"K0619004B\":\"009868,010515,010520\",\n",
+ "\"K0619005B\":\"009720\",\n",
+ "\"K0620002B\":\"000352,004669,004995,010513\",\n",
+ "\"K0620004B\":\"000217,000372,000402,009207,010511,010516,010518\",\n",
+ "\"K0620005B\":\"010517\",\n",
+ "\"K0623001B\":\"004196\",\n",
+ "\"K0623002B\":\"000215,000394,009242,009731,010814,011332\",\n",
+ "\"K0623004B\":\"000200,000201,000205,000213,000419,009399,009417,009732,010532\"\n",
"\n",
"}\n",
"\n",
@@ -174,7 +239,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Python 3.8.8 ('base')",
+ "display_name": "Python 3.9.7 ('base')",
"language": "python",
"name": "python3"
},
@@ -188,12 +253,12 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.8.8"
+ "version": "3.9.7"
},
"orig_nbformat": 4,
"vscode": {
"interpreter": {
- "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
}
}
},
diff --git a/工具/根据目标列表批量生成对应题目的字典.ipynb b/工具/根据目标列表批量生成对应题目的字典.ipynb
new file mode 100644
index 00000000..f11b0119
--- /dev/null
+++ b/工具/根据目标列表批量生成对应题目的字典.ipynb
@@ -0,0 +1,135 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "with open(\"../题库0.3/Problems.json\",\"r\",encoding = \"utf8\") as f:\n",
+ " database = f.read()\n",
+ "import json\n",
+ "pro_dict = json.loads(database)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# 输入目标列表\n",
+ "t = \"\"\"K0616001B\n",
+ "K0616002B\n",
+ "K0616003B\n",
+ "K0616004B\n",
+ "K0617001B\n",
+ "K0617002B\n",
+ "K0617003B\n",
+ "K0617004B\n",
+ "K0617005B\n",
+ "K0617006B\n",
+ "K0617007B\n",
+ "K0619001B\n",
+ "K0619002B\n",
+ "K0619003B\n",
+ "K0619004B\n",
+ "K0619005B\n",
+ "K0620001B\n",
+ "K0620002B\n",
+ "K0620003B\n",
+ "K0620004B\n",
+ "K0620005B\n",
+ "K0623001B\n",
+ "K0623002B\n",
+ "K0623003B\n",
+ "K0623004B\n",
+ "\"\"\""
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\"K0616003B\":\"000204,004061,009211,009212,010496,010498,010505\",\n",
+ "\"K0616004B\":\"009710,009711\",\n",
+ "\"K0617002B\":\"000210,000690,004084\",\n",
+ "\"K0617004B\":\"010499,010500\",\n",
+ "\"K0617006B\":\"003475,004994,009209,009210,009988,010497\",\n",
+ "\"K0617007B\":\"009712,010521,010522\",\n",
+ "\"K0619003B\":\"000199,000202,000211,000212,000216,000364,000411,005136,009214,009218,009400,010507,010514,010519,010524\",\n",
+ "\"K0619004B\":\"009868,010515,010520\",\n",
+ "\"K0619005B\":\"009720\",\n",
+ "\"K0620002B\":\"000352,004669,004995,010513\",\n",
+ "\"K0620004B\":\"000217,000372,000402,009207,010511,010516,010518\",\n",
+ "\"K0620005B\":\"010517\",\n",
+ "\"K0623001B\":\"004196\",\n",
+ "\"K0623002B\":\"000215,000394,009242,009731,010814,011332\",\n",
+ "\"K0623004B\":\"000200,000201,000205,000213,000419,009399,009417,009732,010532\",\n"
+ ]
+ }
+ ],
+ "source": [
+ "dict1 = {}\n",
+ "for o in [l for l in t.split(\"\\n\") if len(l.strip())>0]:\n",
+ " dict1[o] = []\n",
+ "for id in pro_dict:\n",
+ " for o in dict1:\n",
+ " objs = pro_dict[id][\"objs\"]\n",
+ " flag = True\n",
+ " if not o in objs:\n",
+ " flag = False\n",
+ " for obj in objs:\n",
+ " if obj > o:\n",
+ " flag = False\n",
+ " break\n",
+ " if flag:\n",
+ " dict1[o].append(id)\n",
+ "for o in dict1:\n",
+ " if not dict1[o] == []:\n",
+ " print('\"'+o+'\":\"'+\",\".join(dict1[o])+'\",')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3.9.7 ('base')",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.9.7"
+ },
+ "orig_nbformat": 4,
+ "vscode": {
+ "interpreter": {
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/工具/根据范围提取课时目标.ipynb b/工具/根据范围提取课时目标.ipynb
index b95fa550..f94cb6d0 100644
--- a/工具/根据范围提取课时目标.ipynb
+++ b/工具/根据范围提取课时目标.ipynb
@@ -57,7 +57,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Python 3.8.8 ('base')",
+ "display_name": "Python 3.9.7 ('base')",
"language": "python",
"name": "python3"
},
@@ -71,12 +71,12 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.8.8"
+ "version": "3.9.7"
},
"orig_nbformat": 4,
"vscode": {
"interpreter": {
- "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
}
}
},
diff --git a/工具/目标挂钩简要清点.ipynb b/工具/目标挂钩简要清点.ipynb
new file mode 100644
index 00000000..6122b003
--- /dev/null
+++ b/工具/目标挂钩简要清点.ipynb
@@ -0,0 +1,86 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import json,os\n",
+ "with open(r\"题库0.3\\problems.json\",\"r\",encoding = \"u8\") as f:\n",
+ " database = f.read()\n",
+ "pro_dict = json.loads(database)\n",
+ "units = [\"第一单元\",\"第二单元\",\"第三单元\",\"第四单元\",\"第五单元\",\"第六单元\",\"第七单元\",\"第八单元\",\"第九单元\"]\n",
+ "count1 = [0]*9\n",
+ "count2 = [0]*9\n",
+ "for id in pro_dict:\n",
+ " for u in range(9):\n",
+ " unit = units[u]\n",
+ " if unit in \"\".join(pro_dict[id][\"tags\"]):\n",
+ " count1[u] += 1\n",
+ " if len(pro_dict[id][\"objs\"]) > 0:\n",
+ " count2[u] += 1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "第一单元 . 总题数: 1411 , 完成对应题数: 1397\n",
+ "第二单元 . 总题数: 1987 , 完成对应题数: 1985\n",
+ "第三单元 . 总题数: 1946 , 完成对应题数: 390\n",
+ "第四单元 . 总题数: 1023 , 完成对应题数: 594\n",
+ "第五单元 . 总题数: 1235 , 完成对应题数: 299\n",
+ "第六单元 . 总题数: 882 , 完成对应题数: 254\n",
+ "第七单元 . 总题数: 1219 , 完成对应题数: 59\n",
+ "第八单元 . 总题数: 1124 , 完成对应题数: 58\n",
+ "第九单元 . 总题数: 154 , 完成对应题数: 23\n"
+ ]
+ }
+ ],
+ "source": [
+ "for u in range(len(units)):\n",
+ " print(units[u],\". 总题数:\",count1[u],\", 完成对应题数:\",count2[u])\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3.9.7 ('base')",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.9.7"
+ },
+ "orig_nbformat": 4,
+ "vscode": {
+ "interpreter": {
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb
index b4671188..9ac0fcda 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": 4,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
@@ -13,9 +13,11 @@
"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/22_空间平面与平面的位置关系_教师_20221011.tex\n",
+ "正在处理题块 3 .\n",
+ "题块 3 处理完毕.\n",
+ "开始编译教师版本pdf文件: 临时文件/测验04_教师_20221012.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/22_空间平面与平面的位置关系_学生_20221011.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/测验04_学生_20221012.tex\n",
"0\n"
]
}
@@ -28,19 +30,19 @@
"\"\"\"---设置模式结束---\"\"\"\n",
"\n",
"\"\"\"---设置模板文件名---\"\"\"\n",
- "template_file = \"模板文件/第一轮复习讲义模板.tex\"\n",
- "# template_file = \"模板文件/测验周末卷模板.tex\"\n",
+ "# template_file = \"模板文件/第一轮复习讲义模板.tex\"\n",
+ "template_file = \"模板文件/测验周末卷模板.tex\"\n",
"# template_file = \"模板文件/日常选题讲义模板.tex\"\n",
"\"\"\"---设置模板文件名结束---\"\"\"\n",
"\n",
"\"\"\"---设置其他预处理替换命令---\"\"\"\n",
"#2023届第一轮讲义更换标题\n",
- "exec_list = [(\"标题数字待处理\",\"21\"),(\"标题文字待处理\",\"空间平面与平面的位置关系\")] \n",
- "enumi_mode = 0\n",
+ "# exec_list = [(\"标题数字待处理\",\"21\"),(\"标题文字待处理\",\"空间平面与平面的位置关系\")] \n",
+ "# enumi_mode = 0\n",
"\n",
"#2023届测验卷与周末卷\n",
- "# exec_list = [(\"标题替换\",\"周末卷04\")]\n",
- "# enumi_mode = 1\n",
+ "exec_list = [(\"标题替换\",\"测验04预选\")]\n",
+ "enumi_mode = 1\n",
"\n",
"#日常选题讲义\n",
"# exec_list = [(\"标题文字待处理\",\"2022年国庆卷(易错题订正)\")] \n",
@@ -49,14 +51,15 @@
"\"\"\"---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/22_空间平面与平面的位置关系\"\n",
+ "destination_file = \"临时文件/测验04\"\n",
"\"\"\"---设置目标文件名结束---\"\"\"\n",
"\n",
"\n",
"\"\"\"---设置题号数据---\"\"\"\n",
"problems = [\n",
- "'1649,30095,30144,30096,30100,9698,3499,1665,1659,303,30097,30145,188,9700',\n",
- "\"9697,9158,1645,189,9154,1704,1670,294\"\n",
+ "'1506,4125,2027,30152,4414,1013,1253,1510,1515,1880,4111,1868',\n",
+ "\"1993,4240,3645,4116\",\n",
+ "\"4636,1494,4098,4424,4509\"\n",
"]\n",
"\"\"\"---设置题号数据结束---\"\"\"\n",
"\n",
diff --git a/工具/课时目标及课时划分信息汇总.ipynb b/工具/课时目标及课时划分信息汇总.ipynb
index 969c090d..47d669c6 100644
--- a/工具/课时目标及课时划分信息汇总.ipynb
+++ b/工具/课时目标及课时划分信息汇总.ipynb
@@ -9,8 +9,8 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标表_20221007.tex\n",
- "开始编译课时划分信息pdf文件: 临时文件/按课时分类目标及题目清单_20221007.tex\n"
+ "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标表_20221012.tex\n",
+ "开始编译课时划分信息pdf文件: 临时文件/按课时分类目标及题目清单_20221012.tex\n"
]
},
{
@@ -172,7 +172,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Python 3.8.8 ('base')",
+ "display_name": "Python 3.9.7 ('base')",
"language": "python",
"name": "python3"
},
@@ -186,12 +186,12 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.8.8"
+ "version": "3.9.7"
},
"orig_nbformat": 4,
"vscode": {
"interpreter": {
- "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
+ "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
}
}
},
diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb
index f2e41d1a..58239068 100644
--- a/工具/题号选题pdf生成.ipynb
+++ b/工具/题号选题pdf生成.ipynb
@@ -2,26 +2,18 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "开始编译教师版本pdf文件: 临时文件/测验预选_教师用_20221011.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/测验预选_教师用_20221012.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/测验预选_学生用_20221011.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/测验预选_学生用_20221012.tex\n",
"0\n"
]
- },
- {
- "ename": "",
- "evalue": "",
- "output_type": "error",
- "traceback": [
- "\u001b[1;31mThe Kernel crashed while executing code in the the current cell or a previous cell. Please review the code in the cell(s) to identify a possible cause of the failure. Click here for more info. View Jupyter log for further details."
- ]
}
],
"source": [
@@ -34,8 +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",
- "000330,000355,000358,000366,000367,000368,000374,000377,000378,000380,000381,000382,000386,000388,000390,000396,000397,000401,000405,000406,000407,000409,000413,000415,000416,000422,000423,000425,000426,000427,000428,000433,000434,000437,000438,000441,000445,000446,000447,000448,000449,000450,000456,000458,000459,000460,000461,000463,000466,000468,000469,000471,000472,000476,000479,000487,000489,000490,000494,000495,000496,000497,000502,000504,000505,000506,000507,000509,000520,000525,000526,000529,000535,000536,000537,000538,000541,000547,000549,000550,000552,000554,000556,000559,000566,000567,000571,000572,000576,000577,000582,000583,000585,000586,000587,000590,000592,000594,000596,000598,000604,000607,000608,000609,000612,000614,000616,000617,000618,000626,000627,000628,000634,000636,000639,000642,000644,000646,000647,000648,000649,000650,000653,000660,000662,000664,000665,000666,000673,000675,000676,000677,000678,000684,000686,000687,000693,000697,000699,000700,000701,000702,000706,000709,000715,000716,000718,000726,000730,000732,000736,000742,000746,000748,000750,000756,000758,000759,000761,000762,000763,000766,000767,000768,000769,000776,000777,000782,000785,000789,000797,000799,000803,000807,000808,000810,000816,000817,000818,000819,000826,000831,000834,000836,000838,000845,000846,000847,000850,000851,000855,000857,000859,000862,000868,000872,000879,000881,000890,000891,000896,000898,000899,000900,000904,000910,000911,000912,000913,000915,000918,000919,000920,000921,000926,000930,000931,000932,000933,000936,000937,000942,000944,000945,000947,000949,000953,000955,000961,000964,000965,000967,000970,000973,000975,000976,000979,000980,000982,000983,000984,000985,000987,000988,000989,000990,000991,000992,000993,000994,000995,000996,000997,000998,000999,001000,001001,001002,001004,001005,001006,001007,001008,001009,001010,001011,001012,001013,001017,001029,001040,001041,001042,001043,001044,001045,001046,001047,001048,001051,001052,001053,001054,001055,001056,001057,001058,001059,001060,001061,001062,001063,001064,001065,001066,001067,001068,001070,001071,001075,001076,001077,001078,001079,001080,001081,001082,001083,001084,001087,001088,001089,001090,001091,001092,001093,001094,001095,001097,001098,001099,001100,001101,001102,001103,001104,001105,001106,001107,001108,001109,001110,001111,001112,001113,001114,001115,001116,001118,001119,001121,001122,001124,001125,001126,001129,001131,001133,001135,001136,001137,001140,001141,001142,001143,001144,001145,001146,001147,001148,001149,001150,001151,001152,001153,001154,001155,001156,001157,001158,001159,001160,001161,001162,001163,001165,001166,001167,001169,001170,001172,001173,001174,001175,001176,001177,001178,001179,001180,001181,001182,001183,001184,001185,001186,001187,001188,001189,001190,001191,001192,001193,001194,001195,001196,001197,001198,001199,001200,001201,001202,001203,001204,001205,001206,001207,001208,001209,001210,001212,001213,001214,001215,001216,001217,001219,001220,001222,001223,001224,001225,001228,001229,001230,001232,001233,001234,001235,001236,001237,001240,001241,001243,001246,001247,001248,001249,001250,001251,001252,001253,001254,001255,001256,001257,001258,001259,001260,001261,001263,001264,001265,001266,001267,001268,001269,001271,001272,001273,001274,001275,001276,001278,001279,001280,001281,001282,001283,001284,001285,001288,001289,001290,001291,001293,001294,001295,001297,001298,001299,001301,001302,001303,001304,001306,001307,001310,001311,001313,001315,001317,001318,001319,001320,001321,001322,001323,001327,001332,001333,001334,001335,001336,001337,001338,001341,001342,001344,001346,001347,001348,001349,001350,001354,001364,001366,001368,001369,001370,001372,001373,001374,001375,001376,001377,001378,001379,001380,001381,001382,001383,001384,001385,001386,001387,001388,001390,001391,001392,001393,001394,001395,001396,001397,001398,001400,001401,001402,001403,001404,001405,001406,001407,001408,001409,001410,001411,001413,001414,001415,001416,001417,001418,001419,001420,001421,001422,001423,001424,001425,001426,001427,001428,001429,001430,001431,001432,001433,001434,001435,001436,001437,001438,001439,001440,001441,001442,001443,001444,001445,001446,001447,001448,001449,001450,001451,001452,001453,001454,001455,001456,001457,001458,001459,001460,001461,001462,001463,001464,001465,001466,001467,001468,001469,001470,001471,001473,001474,001475,001476,001477,001478,001479,001482,001483,001485,001486,001487,001488,001489,001490,001491,001493,001494,001499,001500,001501,001502,001503,001504,001505,001506,001507,001508,001509,001510,001511,001512,001514,001515,001516,001517,001518,001519,001520,001521,001522,001523,001524,001525,001526,001527,001528,001529,001530,001531,001532,001533,001534,001536,001539,001540,001541,001542,001543,001544,001545,001546,001547,001548,001549,001550,001551,001552,001553,001555,001556,001558,001559,001560,001561,001562,001563,001564,001565,001566,001567,001568,001569,001570,001571,001572,001573,001574,001575,001576,001577,001578,001579,001580,001581,001582,001583,001584,001585,001587,001588,001589,001590,001591,001595,001598,001605,001610,001629,001642,001682,001695,001832,001849,001851,001854,001855,001859,001861,001862,001865,001866,001867,001868,001872,001873,001874,001875,001876,001878,001879,001880,001881,001883,001884,001885,001887,001890,001891,001893,001897,001899,001900,001901,001903,001904,001908,001909,001915,001917,001993,001994,001995,001996,001997,001998,002002,002003,002005,002006,002009,002011,002014,002015,002019,002021,002022,002023,002024,002027,002028,002029,002030,002031,002033,002034,002035,002036,002037,002039,002040,002041,002042,002043,002044,002045,002046,002047,002048,002049,002050,002051,002052,002053,002054,002055,002056,002058,002059,002060,002061,002062,002063,002064,002065,002066,002067,002068,002069,002070,002071,002072,002073,002074,002075,002076,002077,002078,002079,002080,002081,002082,002083,002084,002087,002089,002090,002091,002092,002093,002094,003603,003604,003620,003625,003642,003645,003646,003667,004059,004066,004067,004069,004070,004072,004074,004076,004079,004081,004086,004089,004090,004092,004094,004097,004098,004111,004113,004115,004116,004119,004123,004125,004126,004130,004132,004135,004136,004137,004139,004143,004144,004145,004149,004151,004153,004154,004155,004160,004162,004165,004168,004172,004174,004175,004176,004185,004186,004191,004194,004198,004199,004202,004203,004205,004206,004208,004209,004214,004215,004217,004219,004220,004222,004227,004228,004229,004233,004234,004235,004236,004238,004240,004244,004249,004253,004256,004258,004259,004261,004266,004269,004270,004271,004272,004273,004274,004277,004278,004279,004284,004285,004286,004287,004289,004290,004291,004292,004293,004301,004302,004305,004307,004311,004313,004314,004315,004316,004319,004323,004325,004326,004329,004332,004335,004336,004337,004339,004344,004347,004349,004350,004353,004354,004355,004357,004358,004360,004362,004364,004365,004366,004368,004371,004373,004374,004375,004376,004377,004378,004379,004380,004381,004382,004383,004384,004386,004387,004388,004389,004390,004392,004395,004396,004397,004399,004401,004403,004405,004406,004407,004408,004410,004411,004413,004414,004415,004416,004417,004418,004419,004420,004421,004422,004423,004424,004425,004427,004429,004431,004432,004433,004435,004436,004438,004442,004443,004445,004446,004447,004449,004452,004455,004456,004457,004458,004461,004463,004464,004465,004466,004468,004469,004471,004477,004481,004483,004485,004486,004489,004490,004496,004498,004499,004500,004502,004503,004506,004509,004510,004512,004515,004516,004518,004522,004523,004525,004527,004531,004532,004533,004540,004541,004542,004543,004544,004545,004546,004549,004552,004554,004555,004556,004559,004562,004563,004564,004569,004571,004619,004620,004621,004623,004630,004631,004634,004636,004637,004640,004641,004646,004649,004650,004657,004658,004745,004754,004755,004756,004757,004759,004760,004762,004763\n",
- "\n",
+ "30119,1682,30120,30115,30125,30127,30121,1695,207,9191,9200,30117,30122,7379,1697,1700,1704,3984,9227,198,214,9716,10510,30128,197,9202,1712,9717,30124,30139,10523,30116,30142,203,9239,10531,30130,30131,3996,9412,10529,3982,30126,30137,\n",
"\n",
"\"\"\"\n",
"\"\"\"---设置题目列表结束---\"\"\"\n",
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index e9a71384..0f40db6e 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -5024,7 +5024,8 @@
"id": "000200",
"content": "已知长方体一个顶点上的三条棱长分别是$3$、$4$、$5$, 且它的$8$个顶点都在同一球面上. 求这个球的表面积.",
"objs": [
- "K0623004B"
+ "K0623004B",
+ "K0623002B"
],
"tags": [
"第六单元"
@@ -5048,7 +5049,9 @@
"content": "在等边圆柱(底面直径等于高的圆柱)、球、正方体的体积相等的情况下, 讨论它们的表面积的大小关系.",
"objs": [
"K0623004B",
- "K0617006B"
+ "K0617006B",
+ "K0616003B",
+ "K0623002B"
],
"tags": [
"第六单元"
@@ -5071,7 +5074,8 @@
"id": "000202",
"content": "如图, 在三棱柱的侧棱$A_1A$和$B_1B$上分别取$P$、$Q$两点, 使$PQ$平分侧面$ABB_1A_1$的面积. 求平面$PQC$把棱柱所分成的两部分的体积之比.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [left] {$C$} coordinate (C) -- (3,0) node [right] {$A$} coordinate (A);\n \\draw [dashed] (1,1) node [above right] {$B$} coordinate (B) -- (A) (B) -- (C) (B) --++ (0,2) node [above] {$B_1$} coordinate (B1);\n \\draw (A) --++ (0,2) node [right] {$A_1$} coordinate (A1) -- (B1) -- (0,2) node [left] {$C_1$} coordinate (C1); \n \\draw (C) -- (C1) -- (A1);\n \\draw ($(A)!0.7!(A1)$) node [right] {$P$} coordinate (P) -- (C);\n \\draw [dashed] (P) -- ($(B)!0.3!(B1)$) node [left] {$Q$} -- (C);\n \\end{tikzpicture}\n\\end{center}",
"objs": [
- "K0616003B"
+ "K0616003B",
+ "K0619003B"
],
"tags": [
"第六单元"
@@ -5141,7 +5145,8 @@
"id": "000205",
"content": "如果两个球的体积之比为$8:27$, 求这两个球的表面积之比.",
"objs": [
- "K0623004B"
+ "K0623004B",
+ "K0623002B"
],
"tags": [
"第六单元"
@@ -5304,7 +5309,9 @@
"content": "如图, 在圆柱中, 底面直径$AB$等于母线$AD$, 点$E$在底面的圆周上, 且$AF\\perp DE$, $F$是垂足.\n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw (0,0) node [left] {$A$} -- (0,3) node [left] {$D$} coordinate (D) (3,3) node [right] {$C$} -- (3,0) node [right] {$B$};\n \\draw (0,0) arc (180:360:1.5 and 0.5) (0,3) arc (180:-180:1.5 and 0.5);\n \\draw [dashed] (0,0) arc (180:0:1.5 and 0.5);\n \\draw [dashed] ({1.5+1.5*cos(-105)},{0.5*sin(-105)}) node [below] {$E$} coordinate (E) -- (0,0) (E) -- (3,0) (E) -- (0,3) -- (3,0) -- (0,0) -- ($(E)!0.3!(D)$) node [left] {$F$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求证: $AF\\perp DB$;\\\\\n(2) 若圆柱与三棱锥$D-ABE$的体积的比等于$3\\pi$ , 求直线$DE$与平面$ABD$所成角的大小.",
"objs": [
"K0609003B",
- "K0610004B"
+ "K0610004B",
+ "K0616003B",
+ "K0619003B"
],
"tags": [
"第六单元"
@@ -5328,7 +5335,8 @@
"content": "如图, 半球内有一内接正方体(即正方体的一个面在半球的底面圆上, 其余顶点在半球面上). 若正方体的棱长为$\\sqrt 6$, 求半球的表面积和体积. \n\\begin{center}\n \\begin{tikzpicture}[thick]\n \\draw [dashed] (0,0) coordinate (A) --++ (2,0) coordinate (B) --++ (45:{2/2}) coordinate (C)\n --++ (0,2) coordinate (C1) --++ (-2,0) coordinate (D1) --++ (225:{2/2}) coordinate (A1) -- cycle;\n \\draw [dashed] (A) ++ (2,2) coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2/2}) coordinate (D) --++ (2,0) (D) --++ (0,2); \n \\draw ($(A)!0.5!(C)$) ++ ({-sqrt(6)},0) coordinate (L) arc (180:0:{sqrt(6)});\n \\draw (L) arc (-180:0:{sqrt(6)} and {sqrt(6)/3});\n \\draw [dashed] (L) arc (180:0:{sqrt(6)} and {sqrt(6)/3});\n \\end{tikzpicture}\n\\end{center}",
"objs": [
"K0623004B",
- "K0623002B"
+ "K0623002B",
+ "K0622002B"
],
"tags": [
"第六单元"
@@ -7288,7 +7296,8 @@
"content": "已知正四棱锥的体积为$12$, 底面对角线的长为$2\\sqrt 6$. 求侧面与底面所成二面角的大小.",
"objs": [
"K0631002X",
- "K0631003X"
+ "K0631003X",
+ "K0619003B"
],
"tags": [
"第六单元"
@@ -8302,7 +8311,10 @@
"000333": {
"id": "000333",
"content": "设常数$a>0$, $(x+\\dfrac{a}{\\sqrt{x}})^9$展开式中$x^6$的系数为$4$, 则$\\displaystyle\\lim_{n\\to \\infty}(a+a^2+\\cdots+a^n)=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405002X",
+ "K0405003X"
+ ],
"tags": [
"第四单元",
"第八单元"
@@ -8389,7 +8401,9 @@
"000336": {
"id": "000336",
"content": "$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{2n-5}{n+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -8448,7 +8462,9 @@
"000338": {
"id": "000338",
"content": "若线性方程组的增广矩阵为$\\begin{pmatrix} a & 0 & 2 \\\\ 0 & 1 & b\\end{pmatrix}$, 解为$\\begin{cases} x=2, \\\\ y=1.\\end{cases}$ 则$a+b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -8803,7 +8819,10 @@
"000352": {
"id": "000352",
"content": "若圆锥的侧面展开图是半径为2$\\text{cm}$, 圆心角为$270^\\circ$的扇形, 则这个圆锥的体积为\\blank{50}$\\text{cm}^3$.",
- "objs": [],
+ "objs": [
+ "K0619003B",
+ "K0620002B"
+ ],
"tags": [
"第六单元"
],
@@ -8826,7 +8845,9 @@
"000353": {
"id": "000353",
"content": "若数列$\\{a_n\\}$的所有项都是正数, 且$\\sqrt{a_1}+\\sqrt{a_2}+\\cdots +\\sqrt{a_n}=n^2+3n$($n\\in \\mathbf{N}^*$), 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{1}{n^2}(\\dfrac{a_1}{2}+\\dfrac{a_2}{3}+\\cdots +\\dfrac{a_n}{n+1})=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -9111,7 +9132,9 @@
"000364": {
"id": "000364",
"content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $\\angle ABC=90^\\circ$, $AB=BC=1$, 若$A_1C$与平面$B_1BCC_1$所成的角为$\\dfrac{\\pi}{6}$, 则三棱锥$A_1-ABC$的体积为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 1.5]\n \\draw [dashed] (0,0) -- (1,0) node [right] {$C$} coordinate (C) (0,0) -- (225:0.5) node [below left] {$A$} coordinate (A) (0,0) node [left] {$B$} coordinate (B) -- (0,{sqrt(2)}) node [above left] {$B_1$} coordinate (B1);\n \\draw (A) --+ (0,{sqrt(2)}) node [left] {$A_1$} coordinate (A1);\n \\draw (C) --+ (0,{sqrt(2)}) node [above right] {$C_1$} coordinate (C1);\n \\draw (A1) -- (B1) -- (C1) -- (A1) -- (C) -- (A);\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -9135,7 +9158,9 @@
"000365": {
"id": "000365",
"content": "设地球半径为$R$, 若$A$、$B$两地均位于北纬$45^\\circ$, 且两地所在纬度圈上的弧长为$\\dfrac{\\sqrt{2}}{4}\\pi R$, 则$A$、$B$之间的球面距离是\\blank{50}(结果用含有$R$的代数式表示).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -9263,7 +9288,10 @@
"000370": {
"id": "000370",
"content": "已知无穷数列$\\{a_n\\}$满足$a_{n+1}=\\dfrac12{a_n}$($n\\in \\mathbf{N}^*$), 且$a_2=1$, 记$S_n$为数列$\\{a_n\\}$的前$n$项和, 则$\\displaystyle\\lim_{n\\to \\infty}S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405002X",
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -9315,7 +9343,9 @@
"000372": {
"id": "000372",
"content": "已知圆锥的母线$l=10$, 母线与旋转轴的夹角$\\alpha =30^\\circ$, 则圆锥的表面积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0620004B"
+ ],
"tags": [
"第六单元"
],
@@ -9409,7 +9439,9 @@
"000376": {
"id": "000376",
"content": "$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2n+3}{n+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -9881,7 +9913,11 @@
"000394": {
"id": "000394",
"content": "已知圆锥底面半径与球的半径都是$1\\text{cm}$, 如果圆锥的体积与球的体积恰好也相等, 那么这个圆锥的侧面积是\\blank{50}$\\text{cm}^2$.",
- "objs": [],
+ "objs": [
+ "K0619003B",
+ "K0620004B",
+ "K0623002B"
+ ],
"tags": [
"第六单元"
],
@@ -10041,7 +10077,9 @@
"000400": {
"id": "000400",
"content": "若由矩阵$\\begin{pmatrix}a & 2 \\\\ 2 & a\\end{pmatrix}\\begin{pmatrix}x \\\\ y\\end{pmatrix}=\\begin{pmatrix}a+2 \\\\ 2a\\end{pmatrix}$表示$x$、$y$的二元一次方程组无解, 则实数$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -10088,7 +10126,10 @@
"000402": {
"id": "000402",
"content": "若圆锥侧面积为$20\\pi$, 且母线与底面所成角为$\\arccos \\dfrac45$, 则该圆锥的体积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0619003B",
+ "K0620004B"
+ ],
"tags": [
"第六单元"
],
@@ -10113,7 +10154,9 @@
"000403": {
"id": "000403",
"content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+bn$, 若数列$\\{a_n\\}$是单调递增数列, 则实数$b$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0406004X"
+ ],
"tags": [
"第四单元"
],
@@ -10236,7 +10279,9 @@
"000408": {
"id": "000408",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=2^n-1$, 则此数列的通项公式为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -10312,7 +10357,9 @@
"000411": {
"id": "000411",
"content": "如图, 已知正方形$ABCD-A_1B_1C_1D_1$, $AA_1=2$, $E$为棱$CC_1$的中点, 则三棱锥$D_1-ADE$的体积为\\blank{50}.\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 ($ (C)!0.5!(C1) $) node [right] {$E$} coordinate (E);\n \\draw [dashed] (E) -- (D1) -- (A) -- (E) -- (D);\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -10470,7 +10517,9 @@
"000417": {
"id": "000417",
"content": "三阶行列式$\\begin{vmatrix} 3 & -5 & 1 \\\\ 2 & 3 & -6 \\\\ -7 & 2 & 4 \\\\ \\end{vmatrix}$中元素$-5$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -10520,7 +10569,10 @@
"000419": {
"id": "000419",
"content": "已知一个球的表面积为$16\\pi$, 则它的体积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0623002B",
+ "K0623004B"
+ ],
"tags": [
"第六单元"
],
@@ -10779,7 +10831,9 @@
"000429": {
"id": "000429",
"content": "已知二元一次方程$\\begin{cases} {a_1}x+{b_1}y={c_1}, \\\\ {a_2}x+{b_2}y={c_2} \\end{cases}$的增广矩阵是$\\begin{pmatrix} 1 & -1 & 1 \\\\ 1 & 1 & 3 \\\\ \\end{pmatrix}$, 则此方程组的解是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -10804,7 +10858,9 @@
"000430": {
"id": "000430",
"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": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -11133,7 +11189,9 @@
"000443": {
"id": "000443",
"content": "在无穷等比数列$\\{a_n\\}$中, $\\displaystyle\\lim_{n\\to\\infty}(a_1+a_2+\\cdots+a_n)=\\dfrac12$, 则$a_1$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -11440,7 +11498,9 @@
"000455": {
"id": "000455",
"content": "若$a_n$是$(2+x)^n$($n\\in \\mathbf{N}^*$, $n\\ge 2$, $x\\in \\mathbf{R}$)展开式中$x^2$项的二项式系数, 则$\\displaystyle\\lim_{n\\to\\infty}(\\dfrac 1{a_2}+\\dfrac 1{a_3}+\\cdots+\\dfrac1{a_n})=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元",
"第八单元"
@@ -11493,7 +11553,9 @@
"000457": {
"id": "000457",
"content": "$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{5^n-7^n}{5^n+7^n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -11974,7 +12036,9 @@
"000475": {
"id": "000475",
"content": "若无穷等比数列$\\{a_n\\}$的各项和为$S_n$, 首项$a_1=1$, 公比为$a-\\dfrac32$, 且$\\displaystyle\\lim_{n\\to\\infty}S_n=a$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -12315,7 +12379,10 @@
"000488": {
"id": "000488",
"content": "首项和公比均为$\\dfrac12$的等比数列$\\{a_n\\}$, $S_n$是它的前$n$项和, 则$\\displaystyle\\lim_{n\\to\\infty}S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405002X",
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -12594,7 +12661,9 @@
"000499": {
"id": "000499",
"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": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -12830,7 +12899,9 @@
"000508": {
"id": "000508",
"content": "方程组$\\begin{cases} 3x-2y=1, \\\\ 2x+3y=5 \\end{cases}$的增广矩阵是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -12980,7 +13051,9 @@
"000514": {
"id": "000514",
"content": "数列$\\{a_n\\}$的通项公式是$a_n=2n-1\\ (n\\in \\mathbf{N}^*)$, 数列$\\{b_n\\}$的通项公式是$b_n=3n \\ (n\\in \\mathbf{N}^*)$, 令集合$A=\\{a_1,a_2,\\cdots,a_n,\\cdots\\}$, $B=\\{b_1,b_2,\\cdots,b_n,\\cdots\\}$, $n\\in \\mathbf{N}^*$. 将集合$A\\cup B$中的所有元素按从小到大的顺序排列, 构成的数列记为$\\{c_n\\}$. 则数列$\\{c_n\\}$的前$28$项的和$S_{28}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -13042,7 +13115,9 @@
"000516": {
"id": "000516",
"content": "计算: $\\displaystyle\\lim_{n\\to\\infty}(1-\\dfrac n{n+1})=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -13081,7 +13156,9 @@
"000517": {
"id": "000517",
"content": "计算行列式$\\begin{vmatrix} 1-\\mathrm{i} & 2 \\\\ 3\\mathrm{i}+1 & 1+\\mathrm{i}\\end{vmatrix}$的结果是\\blank{50}(其中$\\mathrm{i}$为虚数单位).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -13329,7 +13406,9 @@
"000527": {
"id": "000527",
"content": "计算$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{\\mathrm{C}_n^2}{n^2+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -13499,7 +13578,10 @@
"000534": {
"id": "000534",
"content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$b_n=\\ln a_n$, $n\\in \\mathbf{N}^*$, 其中$\\{b_n\\}$是等差数列, 且$a_3\\cdot a_{1007}=\\mathrm{e}^4$, 则$b_1+b_2+\\cdots +b_{1009}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402001X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -13724,7 +13806,9 @@
"000543": {
"id": "000543",
"content": "若数列$\\{a_n\\}$的前$n$项和$S_n=-3n^2+2n+1 \\ (n\\in \\mathbf{N}^*)$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{a_n}{3n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -13793,7 +13877,9 @@
"000546": {
"id": "000546",
"content": "计算: $\\displaystyle\\lim_{n\\to\\infty}\\dfrac{2n}{3n-1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -13864,7 +13950,10 @@
"000548": {
"id": "000548",
"content": "已知$\\{a_n\\}$为等差数列, $S_n$为其前$n$项和, 若$a_1+a_9=18$, $a_4=7 $, 则$S_{10}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -14208,7 +14297,10 @@
"000562": {
"id": "000562",
"content": "设等差数列$\\{a_n\\}$的公差$d$不为$0$, $a_1=9d$. 若$a_k$是$a_1$与$a_{2k}$的等比中项, 则$k=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X",
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -14484,7 +14576,10 @@
"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}.",
- "objs": [],
+ "objs": [
+ "K0401004X",
+ "K0401003X"
+ ],
"tags": [
"第四单元"
],
@@ -14507,7 +14602,9 @@
"000574": {
"id": "000574",
"content": "著名的斐波那契数列$\\{a_n\\}:1,1,2,3,5,8,\\cdots$, 满足$a_1=a_2=1,a_{n+2}=a_{n+1}+a_n \\ (n\\in \\mathbf{N}^*)$, 那么$1+a_3+a_5+a_7+a_9+\\cdots+a_{2017}$是斐波那契数列中的第\\blank{50}项.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -14530,7 +14627,10 @@
"000575": {
"id": "000575",
"content": "若不等式$(-1)^n\\cdot a<3+\\dfrac{(-1)^{n+1}}{n+1}$对任意正整数$n$恒成立, 则实数$a$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0406005X",
+ "K0406004X"
+ ],
"tags": [
"第四单元"
],
@@ -14606,7 +14706,9 @@
"000578": {
"id": "000578",
"content": "若行列式$\\begin{vmatrix} 2^{x-1} & 4 \\\\ 1 & 2 \\end{vmatrix}$, 则$x=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -14631,7 +14733,9 @@
"000579": {
"id": "000579",
"content": "已知一个关于$x$, $y$的二元一次方程组的增广矩阵是$\\begin{pmatrix} 1 & -1 & 2 \\\\ 0 & 1 & 2 \\end{pmatrix}$, 则$x+y=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -14741,7 +14845,10 @@
"000583": {
"id": "000583",
"content": "在$\\triangle ABC$中, 若$\\sin A,\\sin B,\\sin C$成等比数列, 则角$B$的最大值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X",
+ "K0316002B"
+ ],
"tags": [
"第三单元",
"第四单元"
@@ -14869,7 +14976,9 @@
"000588": {
"id": "000588",
"content": "$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{3^n-1}{3^{n+1}+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -14955,7 +15064,9 @@
"000591": {
"id": "000591",
"content": "若数列$\\{a_n\\}$为等比数列, 且$a_5=3$, 则$\\begin{vmatrix} a_2 & -a_7 \\\\ a_3 & a_8 \\end{vmatrix}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -15049,7 +15160,10 @@
"000595": {
"id": "000595",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=1$, $2S_n=a_na_{n+1}$($n\\in \\mathbf{N}^*$), 若$b_n=(-1)^n\\dfrac{2n+1}{{a_n}{a_{n+1}}}$, 则数列$\\{b_n\\}$的前$n$项和$T_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0409001X",
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -15146,7 +15260,9 @@
"000599": {
"id": "000599",
"content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_n=1-\\dfrac23{a_n} \\ (n\\in \\mathbf{N}^*)$, 则$\\displaystyle\\lim_{n\\to\\infty}S_n$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -15169,7 +15285,10 @@
"000600": {
"id": "000600",
"content": "若$(x+\\dfrac1{2x})^n \\ (n\\ge 4, \\ n\\in \\mathbf{N}^*)$的二项展开式中前三项的系数依次成等差数列, 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0819005X",
+ "K0401002X"
+ ],
"tags": [
"第四单元",
"第八单元"
@@ -15216,7 +15335,9 @@
"000602": {
"id": "000602",
"content": "若行列式$\\begin{vmatrix} 1 & 2 & 4 \\\\ \\cos \\dfrac x2 & \\sin \\dfrac x2 & 0 \\\\ \\sin \\dfrac x2 & \\cos \\dfrac x2 & 8 \\end{vmatrix}$中元素$4$的代数余子式的值为$\\dfrac12$, 则实数$x$的取值集合为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -15315,7 +15436,9 @@
"000606": {
"id": "000606",
"content": "计算: $\\displaystyle\\lim_{n\\to\\infty}(1+\\dfrac1n)^3=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405001X"
+ ],
"tags": [
"第四单元"
],
@@ -15703,7 +15826,9 @@
"000621": {
"id": "000621",
"content": "某空间几何体的三视图如图所示, 则该几何体的侧面积是\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) circle (1);\n \\draw (-1,-1.1) -- (-1,-1.3) (1,-1.1) -- (1,-1.3);\n \\draw [->] (-0.2,-1.2) -- (-1,-1.2);\n \\draw [->] (0.2,-1.2) -- (1,-1.2);\n \\draw (0,-1.2) node {$4$};\n \\draw (0,-1.6) node {俯视图};\n \\draw (-1,2) -- (1,2) -- (0,5) -- cycle;\n \\draw (0,1.4) node {主视图};\n \\draw (-1,1.9) -- (-1,1.7) (1,1.9) -- (1,1.7);\n \\draw [->] (-0.2,1.8) -- (-1,1.8);\n \\draw [->] (0.2,1.8) -- (1,1.8);\n \\draw (0,1.8) node {$4$};\n \\draw (1.1,2) -- (1.3,2) (1.1,5) -- (1.3,5);\n \\draw [->] (1.2,3.2) -- (1.2,2);\n \\draw [->] (1.2,3.8) -- (1.2,5);\n \\draw (1.2,3.5) node {$6$};\n \\draw (2,2) -- (4,2) -- (3,5) -- cycle;\n \\draw (3,1.4) node {左视图};\n \\draw (2,1.9) -- (2,1.7) (4,1.9) -- (4,1.7);\n \\draw [->] (2.8,1.8) -- (2,1.8);\n \\draw [->] (3.2,1.8) -- (4,1.8);\n \\draw (3,1.8) node {$4$};\n \\draw (4.1,2) -- (4.3,2) (4.1,5) -- (4.3,5);\n \\draw [->] (4.2,3.2) -- (4.2,2);\n \\draw [->] (4.2,3.8) -- (4.2,5);\n \\draw (4.2,3.5) node {$6$};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -16013,7 +16138,9 @@
"000633": {
"id": "000633",
"content": "数列$\\{a_n\\}$是等比数列, 前n项和为$S_n$, 若$a_1+a_2=2$, $a_2+a_3=-1$, 则$\\displaystyle\\lim_{n\\to\\infty}{S_n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -16145,7 +16272,9 @@
"000638": {
"id": "000638",
"content": "已知首项为$1$公差为$2$的等差数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{a_n^2}{S_n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405001X"
+ ],
"tags": [
"第四单元"
],
@@ -16746,7 +16875,9 @@
"000662": {
"id": "000662",
"content": "各项均不为零的数列$\\{a_n\\}$的前$n$项和为$S_n$. 对任意$n\\in \\mathbf{N}^*$, $\\overrightarrow{m_n}=(a_{n+1}-a_n,2a_{n+1})$\n都是直线$y=kx$的法向量. 若$\\displaystyle\\lim_{n\\to\\infty}S_n$存在, 则实数$k$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405002X"
+ ],
"tags": [
"第四单元",
"第五单元"
@@ -16945,7 +17076,9 @@
"000670": {
"id": "000670",
"content": "已知关于$x,y$的二元一次方程组的增广矩阵为$\\begin{pmatrix} 2 & 1 & 5 \\\\ 1 & -2 & 0 \\end{pmatrix}$, 则$3x-y$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -17044,7 +17177,9 @@
"000674": {
"id": "000674",
"content": "已知等差数列$\\{a_n\\}$的公差为$2$, 前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{S_n}{{a_n}{a_{n+1}}}$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405001X"
+ ],
"tags": [
"第四单元"
],
@@ -17238,7 +17373,9 @@
"000681": {
"id": "000681",
"content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$\\dfrac{a_5}{a_3}=\\dfrac53$, 则$\\dfrac{S_5}{S_3}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -17470,7 +17607,10 @@
"000690": {
"id": "000690",
"content": "若圆柱的侧面展开图是边长为$4\\text{cm}$的正方形, 则圆柱的体积为\\blank{50}$\\text{cm}^3$(结果精确到$0.1\\text{cm}^3$).",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0617002B"
+ ],
"tags": [
"第六单元"
],
@@ -17624,7 +17764,9 @@
"000696": {
"id": "000696",
"content": "行列式$\\begin{vmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{vmatrix}$中, 元素$5$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -17999,7 +18141,9 @@
"000711": {
"id": "000711",
"content": "若线性方程组的增广矩阵为$\\begin{pmatrix} 1 & 2 & c_1 \\\\ 2 & 0 & c_2\\end{pmatrix}$、解为$\\begin{cases}x=1, \\\\ y=3,\\end{cases}$ 则$c_1+c_2=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -18054,7 +18198,9 @@
"000713": {
"id": "000713",
"content": "设无穷等比数列$\\{a_n\\}$的公比为$q$, 若$a_2=\\displaystyle\\lim_{n\\to\\infty}(a_4+a_5+\\cdots+a_n)$, 则$q=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -18160,7 +18306,9 @@
"000717": {
"id": "000717",
"content": "已知一个关于$x,y$的二元一次方程组的增广矩阵是$\\begin{pmatrix} 1 & -1 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix}$, 则$x+y=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -18519,7 +18667,9 @@
"000731": {
"id": "000731",
"content": "三阶行列式$\\begin{vmatrix}-5 & 6 & 7 \\\\ 4 & 2^x & 1 \\\\ 0 & 3 & 1 \\end{vmatrix}$中元素$-5$的代数余子式为$f(x)$, 则方程$f(x)=0$的解为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -18568,7 +18718,9 @@
"000733": {
"id": "000733",
"content": "无穷等比数列$\\{a_n\\}$的通项公式$a_n=(\\sin x)^n$, 前$n$项的和为$S_n$, 若$\\displaystyle\\lim_{n\\to\\infty}S_n=1$, $x\\in (0,\\pi)$, 则$x=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -19042,7 +19194,10 @@
"000752": {
"id": "000752",
"content": "已知数列$\\{a_n\\}$是公比为$q$的等比数列, 且$a_2,a_4,a_3$成等差数列, 则$q=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -19088,7 +19243,9 @@
"000754": {
"id": "000754",
"content": "如图, 长方体$ABCD-A_1B_1C_1D_1$的边长$AB=AA_1=1$ ,$AD=\\sqrt2$ , 它的外接球是球$O$, 则$A$, $A_1$这两点的球面距离等于\\blank{50}.\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*sqrt(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*sqrt(2)/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2*sqrt(2)/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (45:{2*sqrt(2)/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw [dashed] (A1) -- (C) (A) --(C1);\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -19567,7 +19724,9 @@
"000772": {
"id": "000772",
"content": "若某线性方程组对应的增广矩阵是$\\begin{pmatrix} m & 4 & 2 \\\\ 1 & m & m \\end{pmatrix}$, 且此方程组有唯一一组解, 则实数$m$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -19775,7 +19934,9 @@
"000780": {
"id": "000780",
"content": "如图的三个直角三角形是一个体积为$20\\text{cm}^3$的几何体的三视图, 则$h=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) -- (2.5,0) -- (0,2) -- cycle;\n \\draw (0,-0.1) -- (0,-0.3) (2.5,-0.1) -- (2.5,-0.3);\n \\draw (1.25,-0.2) node {$5$};\n \\draw [->] (1.05,-0.2) -- (0,-0.2);\n \\draw [->] (1.45,-0.2) -- (2.5,-0.2);\n \\draw (1.25,-0.5) node {主视图};\n \\draw (-0.1,0) -- (-0.3,0) (-0.1,2) -- (-0.3,2);\n \\draw (-0.2,1) node {$h$};\n \\draw [->] (-0.2,0.8) -- (-0.2,0);\n \\draw [->] (-0.2,1.2) -- (-0.2,2);\n \\draw (3.5,0) -- (6.5,0) -- (3.5,2) -- cycle;\n \\draw (3.5,-0.1) -- (3.5,-0.3) (6.5,-0.1) -- (6.5,-0.3);\n \\draw (5,-0.2) node {$6$};\n \\draw [->] (4.8,-0.2) -- (3.5,-0.2);\n \\draw [->] (5.2,-0.2) -- (6.5,-0.2);\n \\draw (5,-0.5) node {左视图};\n \\draw (0,-1) -- (2.5,-1) -- (0,-4) -- cycle;\n \\draw (1.25,-4.5) node {俯视图};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -19872,7 +20033,10 @@
"000784": {
"id": "000784",
"content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$($n\\in \\mathbf{N}*$), 且$\\dfrac{S_6}{S_3}=-\\dfrac{19}8$,$a_4-a_2=-\\dfrac{15}8$, 则$a_3$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -19946,7 +20110,9 @@
"000787": {
"id": "000787",
"content": "若二元一次方程组的增广矩阵是$\\begin{pmatrix} 1 & 2 & c_1 \\\\ 3 & 4 & c_2 \\end{pmatrix}$, 其解为$\\begin{cases} x=10, \\\\ y=0, \\end{cases}$ 则$c_1+c_2=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -20046,7 +20212,10 @@
"000791": {
"id": "000791",
"content": "已知数列$\\{a_n\\}$, 其通项公式为$a_n=3n+1$, $n\\in \\mathbf{N}^*$, $\\{a_n\\}$的前$n$项和为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{S_n}{n\\cdot {a_n}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X",
+ "K0405001X"
+ ],
"tags": [
"第四单元"
],
@@ -20244,7 +20413,10 @@
"000798": {
"id": "000798",
"content": "已知$\\{a_n\\}$是等比数列, 它的前$n$项和为$S_n$, 且$a_3=4$, $a_4=-8$, 则$S_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403004X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -20636,7 +20808,10 @@
"000814": {
"id": "000814",
"content": "设函数$f(x)=\\log_m x$($m>0$且$m\\ne 1$), 若$m$是等比数列$\\{a_n\\}$($n\\in \\mathbf{N}^*$)的公比, 且$f(a_2a_4a_6\\cdots a_{2018})=7$, 则$f(a_1^2)+f(a_2^2)+f(a_3^2)+\\cdots+f(a_{2018}^2)$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -20798,7 +20973,10 @@
"000820": {
"id": "000820",
"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": [],
+ "objs": [
+ "K0403002X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -21193,7 +21371,10 @@
"000835": {
"id": "000835",
"content": "若$\\{a_n\\}$为等比数列, $a_n>0$, 且$a_{2018}=\\dfrac{\\sqrt2}2$, 则$\\dfrac1{a_{2017}}+\\dfrac2{a_{2019}}$的最小值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403004X",
+ "K0119001B"
+ ],
"tags": [
"第四单元"
],
@@ -21401,7 +21582,9 @@
"000843": {
"id": "000843",
"content": "三棱锥$P-ABC$及其三视图中的主视图和左视图如图所示, 则棱$PB$的长为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) node [right] {$C$} coordinate (C) -- (0,2) node [above] {$P$} coordinate (P) -- (-2,0) node [left] {$A$} coordinate (A);\n \\draw (-1,0) ++ (-135:{sqrt(3)/2}) node [below] {$B$} coordinate (B) -- (A) (B) -- (C) (B) -- (P);\n \\draw [dashed] (A) -- (C);\n \\draw (2,0) -- (4,0) -- (4,2) -- cycle;\n \\draw (3,0) -- (4,2);\n \\draw (5,0) -- ({5+sqrt(3)},0) coordinate (D) -- (5,2) -- cycle;\n \\draw (2,-0.1) -- (2,-0.3) (3,-0.1) -- (3,-0.3) (4,-0.1) -- (4,-0.3);\n \\draw [->] (2.2,-0.2) -- (2,-0.2);\n \\draw [->] (2.8,-0.2) -- (3,-0.2);\n \\draw [->] (3.2,-0.2) -- (3,-0.2);\n \\draw [->] (3.8,-0.2) -- (4,-0.2);\n \\draw (2.5,-0.2) node {$2$} (3.5,-0.2) node {$2$};\n \\draw (3,-0.7) node {主视图};\n \\draw (5,-0.1) -- (5,-0.3) (D) ++ (0,-0.1) --++ (0,-0.2);\n \\draw [->] ({5+sqrt(3)/2-0.4},-0.2) -- (5,-0.2);\n \\draw [->] ({5+sqrt(3)/2+0.4},-0.2) -- ({5+sqrt(3)},-0.2);\n \\draw ({5+sqrt(3)/2},-0.2) node {$2\\sqrt{3}$};\n \\draw ({5+sqrt(3)/2},-0.7) node {主视图};\n \\draw (4.9,0) -- (4.7,0) (4.9,2) -- (4.7,2);\n \\draw [->] (4.8,0.8) -- (4.8,0);\n \\draw [->] (4.8,1.2) -- (4.8,2);\n \\draw (4.8,1) node {$4$};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -21658,7 +21841,9 @@
"000853": {
"id": "000853",
"content": "若某几何体的三视图(单位:$\\text{cm}$)如图所示, 则此几何体的体积是\\blank{50}$\\text{cm}^3$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.7]\n \\draw (1,0) -- (2,0) -- (2,3) -- (3,3) -- (3,4) -- (0,4) -- (0,3) -- (1,3) -- cycle;\n \\draw (5,0) -- (8,0) -- (8,4) -- (5,4) -- cycle (5,3) -- (8,3);\n \\draw (0,-1) -- (0,-4) -- (3,-4) -- (3,-1) -- cycle;\n \\draw [dashed] (1,-1) -- (1,-4) (2,-1) -- (2,-4);\n \\draw (1.5,-4.5) node {俯视图} (1.5,-0.5) node {主视图} (6.5,-0.5) node {俯视图};\n \\draw (3.1,-4) -- (3.3,-4) (3.1,-1) -- (3.3,-1);\n \\draw [->] (3.2,-2.1) -- (3.2,-1);\n \\draw [->] (3.2,-2.9) -- (3.2,-4);\n \\draw (3.2,-2.5) node {$3$};\n \\draw (5,4.1) -- (5,4.3) (8,4.1) -- (8,4.3);\n \\draw [->] (6.1,4.2) -- (5,4.2);\n \\draw [->] (6.9,4.2) -- (8,4.2);\n \\draw (6.5,4.2) node {$3$};\n \\draw (0,4.1) -- (0,4.3) (1,4.1) -- (1,4.3) (2,4.1) -- (2,4.3) (3,4.1) -- (3,4.3);\n \\draw [->] (0.3,4.2) -- (0,4.2);\n \\draw [->] (0.7,4.2) -- (1,4.2);\n \\draw (0.5,4.2) node {$1$};\n \\draw [->] (1.3,4.2) -- (1,4.2);\n \\draw [->] (1.7,4.2) -- (2,4.2);\n \\draw (1.5,4.2) node {$1$};\n \\draw [->] (2.3,4.2) -- (2,4.2);\n \\draw [->] (2.7,4.2) -- (3,4.2);\n \\draw (2.5,4.2) node {$1$};\n \\draw (3.1,4) -- (3.3,4) (3.1,3) -- (3.3,3) (3.1,0) -- (3.3,0);\n \\draw [->] (3.2,1.1) -- (3.2,0);\n \\draw [->] (3.2,1.9) -- (3.2,3);\n \\draw (3.2,1.5) node {$3$};\n \\draw [->] (3.2,3.2) -- (3.2,3);\n \\draw [->] (3.2,3.8) -- (3.2,4);\n \\draw (3.2,3.5) node {$1$};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -21730,7 +21915,9 @@
"000856": {
"id": "000856",
"content": "已知数列$\\{a_n\\}$的通项公式为$a_n={(-1)}^n\\cdot n+2^n, \\ n\\in \\mathbf{N}^*$, 则这个数列的前$2n$项和$S_{2n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -21999,7 +22186,10 @@
"000867": {
"id": "000867",
"content": "已知各项均为正数的数列$\\{a_n\\}$满足$\\sqrt{a_1}+\\sqrt{a_2}+\\cdots+\\sqrt{a_n}=n^2+3n$($n\\in \\mathbf{N}^*$), 则$\\dfrac{a_1}2+\\dfrac{a_2}3+\\cdots \\dfrac{a_n}{n+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -22047,7 +22237,9 @@
"000869": {
"id": "000869",
"content": "已知线性方程组的增广矩阵为$\\begin{pmatrix} 1 & -1 & 3 \\\\ a & 3 & 4 \\end{pmatrix}$, 若该线性方程组的解为$\\begin{pmatrix} -1 \\\\ 2\\end{pmatrix}$, 则实数$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -22206,7 +22398,11 @@
"000875": {
"id": "000875",
"content": "已知等比数列$\\{a_n\\}$的各项均为正数, 且满足:$a_1a_7=4$, 则数列$\\{\\log_2a_n\\}$的前$7$项之和为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403006X",
+ "K0402004X",
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -22762,7 +22958,10 @@
"000897": {
"id": "000897",
"content": "设等差数列$\\{a_n\\}$的公差为$d$, 若$a_1,a_2,a_3,a_4,a_5,a_6,a_7$的方差为$1$, 则$d$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X",
+ "K0826001X"
+ ],
"tags": [
"第四单元"
],
@@ -22963,7 +23162,9 @@
"000905": {
"id": "000905",
"content": "若行列式$\\begin{vmatrix} 1 & 2 & 4 \\\\ \\cos (\\pi +x) & 2 & 0 \\\\ -1 & 1 & 6 \\end{vmatrix}$中的元素$4$的代数余子式的值等于$\\dfrac32$, 则实数$x$的取值集合为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -23526,7 +23727,10 @@
"000928": {
"id": "000928",
"content": "若数列$\\{a_n\\}$是首项为$1$, 公比为$a-\\dfrac32$的无穷等比数列, 且$\\{a_n\\}$各项的和为$a$, 则$a$的值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405002X",
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -24100,7 +24304,9 @@
"000950": {
"id": "000950",
"content": "数列$\\{a_n\\}$中, 若$a_1=3$, $\\sqrt{a_{n+1}}=a_n$($n\\in \\mathbf{N}^*$), 则数列$\\{a_n\\}$的通项公式$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -24300,7 +24506,9 @@
"000958": {
"id": "000958",
"content": "无穷等比数列首项为$1$,公比为$q \\ (q>0)$, 前$n$项和为$S_n$, 若$\\displaystyle\\lim_{n\\to\\infty}S_n=2$, 则$q=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -24494,7 +24702,9 @@
"000966": {
"id": "000966",
"content": "已知无穷等比数列$\\{a_n\\}$的首项$a_1=18$, 公比$q=-\\dfrac12$, 则无穷等比数列$\\{a_n\\}$各项的和是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -24611,7 +24821,9 @@
"000971": {
"id": "000971",
"content": "如图所示, 是一个由圆柱和球组成的几何体的三视图, 若$a=2$, $b=3$, 则该几何体的体积等于\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.6]\n \\draw (0,1) circle (1);\n \\draw (-1,-0.1) -- (-1,-0.3) (1,-0.1) -- (1,-0.3);\n \\draw [->] (-0.2,-0.2) -- (-1,-0.2);\n \\draw [->] (0.2,-0.2) -- (1,-0.2);\n \\draw (0,-0.2) node {$a$};\n \\draw (1.1,0) -- (1.3,0) (1.1,2) -- (1.3,2);\n \\draw [->] (1.2,0.8) -- (1.2,0);\n \\draw [->] (1.2,1.2) -- (1.2,2);\n \\draw (1.2,1) node {$a$};\n \\draw (0,-1) node {俯视图};\n \\draw (-1,3.5) rectangle (1,6.5) (0,7.5) circle (1);\n \\draw (-1,3.4) -- (-1,3.2) (1,3.4) -- (1,3.2);\n \\draw [->] (-0.2,3.3) -- (-1,3.3);\n \\draw [->] (0.2,3.3) -- (1,3.3);\n \\draw (0,3.3) node {$a$};\n \\draw (1.1,3.5) -- (1.3,3.5) (1.1,6.5) -- (1.3,6.5);\n \\draw [->] (1.2,4.6) -- (1.2,3.5);\n \\draw [->] (1.2,5.4) -- (1.2,6.5);\n \\draw (1.2,5) node {$b$};\n \\draw (0,2.5) node {主视图};\n \\draw (3,3.5) rectangle (5,6.5) (4,7.5) circle (1);\n \\draw (3,3.4) -- (3,3.2) (5,3.4) -- (5,3.2);\n \\draw [->] (3.8,3.3) -- (3,3.3);\n \\draw [->] (4.2,3.3) -- (5,3.3);\n \\draw (4,3.3) node {$a$};\n \\draw (5.1,3.5) -- (5.3,3.5) (5.1,6.5) -- (5.3,6.5);\n \\draw [->] (5.2,4.6) -- (5.2,3.5);\n \\draw [->] (5.2,5.4) -- (5.2,6.5);\n \\draw (5.2,5) node {$b$};\n \\draw (4,2.5) node {左视图};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -25880,7 +26092,9 @@
"001018": {
"id": "001018",
"content": "用数学归纳法证明``$(n+1)(n+2)\\cdots(n+n)=2^n\\cdot 1\\cdot 3\\cdot 5\\cdots(2n-1)$''时, 从``$n=k$''到``$n=k+1$''的过程中, 左边应多乘的因式是\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -25904,7 +26118,9 @@
"001019": {
"id": "001019",
"content": "用数学归纳法证明: 对一切正整数$n$, $1^3+2^3+\\cdots+n^3=\\dfrac{n^2(n+1)^2}{4}$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -25928,7 +26144,9 @@
"001020": {
"id": "001020",
"content": "用数学归纳法证明: 对一切正整数$n$, $1-\\dfrac{1}{2}+\\dfrac{1}{3}-\\dfrac{1}{4}+\\cdots+\\dfrac{1}{2n-1}-\\dfrac{1}{2n}=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2n}$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -25955,7 +26173,9 @@
"001021": {
"id": "001021",
"content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=2a_n+1(n\\in\\mathbf{N}^*)$. 求证: $a_n=2^n-1(n\\in\\mathbf{N}^*)$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -25981,7 +26201,9 @@
"001022": {
"id": "001022",
"content": "用数学归纳法证明: 对一切正整数$n$, $\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{3n+1}>1$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -26005,7 +26227,9 @@
"001023": {
"id": "001023",
"content": "求证: 对任意的正整数$n$, $64$能够整除$3^{2n+1}+40n-3$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -26029,7 +26253,9 @@
"001024": {
"id": "001024",
"content": "设$a=x+\\dfrac{1}{x}$. 证明: 对任意$n \\in \\mathbf{Z}^+$, $x^n+\\dfrac{1}{x^n}$均可以表示成$a$的整系数多项式.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -26053,7 +26279,9 @@
"001025": {
"id": "001025",
"content": "设数列$\\{a_n\\}$满足下列条件: $a_1=2,\\ a_2=3$, 且对任何自然数$k$有$a_{k+2}=3a_{k+1}-2a_k$, 求证: $a_n=1+2^{n-1}, \\ (n \\in \\mathbf{Z}^+)$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -26077,7 +26305,9 @@
"001026": {
"id": "001026",
"content": "已知$a_n=\\dfrac{1}{1\\times 4}+\\dfrac{1}{4\\times 7}+\\cdots+\\dfrac{1}{(3n-2)\\times(3n+1)}$.\\\\ \n(1) 计算$a_1,a_2,a_3,a_4$, 并猜测$a_n$的一般形式;\\\\ \n(2) 用数学归纳法证明你的猜想.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -26101,7 +26331,9 @@
"001027": {
"id": "001027",
"content": "已知$a_1=1$, $a_2=5$. 当$n \\ge 2$时, $a_{n+1}=a_n+2a_{n-1}$.\\\\ \n(1) 求$a_1,a_2,a_3,a_4,a_5,a_6$;\\\\ \n(2) 猜测并用第二数学归纳法证明$a_n$的表达式.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -26125,7 +26357,9 @@
"001028": {
"id": "001028",
"content": "是否存在常数$a,b$, 使得\n$$1^2+3^2+\\cdots+(2n-1)^2=\\dfrac{1}{3}an(n^2+b)$$\n对任意正整数$n$均成立? 证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -37713,7 +37947,9 @@
"001467": {
"id": "001467",
"content": "[选做]\n用$\\cos\\alpha$的多项式表示$\\cos 2\\alpha$和$\\cos 4\\alpha$, 并用数学归纳法证明: 当$n \\in \\mathbf{N}^*$时, $\\cos(2^n \\alpha)$可以表示成$\\cos \\alpha$的一个$2^n$次多项式.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第三单元",
"第四单元"
@@ -43633,7 +43869,9 @@
"id": "001697",
"content": "高为$2$, 底面边长为$3$的正三棱锥底面中心到侧面的距离为\\blank{80}.",
"objs": [
- "K0609007B"
+ "K0609007B",
+ "K0618002B",
+ "K0618004B"
],
"tags": [
"第六单元"
@@ -43658,7 +43896,9 @@
"001698": {
"id": "001698",
"content": "一个棱锥被平行于底面的平面所截, 如果截面面积和底面面积之比为$3:4$, 则侧棱被分成的上下两段长度之比为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -43706,7 +43946,9 @@
"001700": {
"id": "001700",
"content": "在底面边长为$1$的正三棱锥$P-ABC$中, 二面角$P-AB-C$为$\\dfrac{\\pi}{3}$, $G$是侧面$PAB$的重心, $H$是$AC$的中点, 则$GH$的长为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -43754,7 +43996,9 @@
"001702": {
"id": "001702",
"content": "正三棱锥$S-ABC$中, 二面角$S-AB-C$的大小为$60^\\circ$，求棱锥的侧棱与底面所成角的正切值.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -43803,7 +44047,9 @@
"id": "001704",
"content": "已知正四棱锥$S-ABCD$, 求证: 二面角$A-SB-C$的平面角一定为钝角.",
"objs": [
- "K0613003B"
+ "K0613003B",
+ "K0618002B",
+ "K0618004B"
],
"tags": [
"第六单元"
@@ -43996,7 +44242,9 @@
"001712": {
"id": "001712",
"content": "棱台的上, 下底面的面积分别为$16$和$49$, 则其中截面(过每条侧棱的中点, 平行于底面的截面)的面积为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0618007B"
+ ],
"tags": [
"第六单元"
],
@@ -44265,7 +44513,9 @@
"001723": {
"id": "001723",
"content": "圆锥的母线长为$1$, 那么其底面周长的取值范围为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0618005B"
+ ],
"tags": [
"第六单元"
],
@@ -44435,7 +44685,9 @@
"001730": {
"id": "001730",
"content": "半径为$1$的球的球面上两点之间的距离为$\\sqrt{2}$, 则这两点之间的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44459,7 +44711,9 @@
"001731": {
"id": "001731",
"content": "长方体$ABCD-A'B'C'D'$的八个顶点在同一球面上, 且$AB=2,AD=\\sqrt{3},AA'=1$, 则顶点$A,B$间的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44507,7 +44761,9 @@
"001733": {
"id": "001733",
"content": "已知地球的半径为$1$, 在东经$120^\\circ$线上, 南纬$30^\\circ$的点记为$A$, 北纬$15^\\circ$的点记为$B$. 则$A,B$两地的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44531,7 +44787,9 @@
"001734": {
"id": "001734",
"content": "已知地球的半径为$1$, 在南纬$45^\\circ$线上, 东经$90^\\circ$的点记为$A$, 东经$60^\\circ$的点记为$B$. 则$A,B$两地的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44555,7 +44813,9 @@
"001735": {
"id": "001735",
"content": "已知地球的半径为$1$, $A$点在东经$120^\\circ$, 北纬$30^\\circ$的位置上, $B$点在西经$60^\\circ$, 南纬$30^\\circ$的位置上, 则$A,B$ 两地的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44581,7 +44841,9 @@
"001736": {
"id": "001736",
"content": "[选做]\n已知地球的半径为$1$, $A$点在东经$120^\\circ$, 北纬$30^\\circ$的位置上, $B$点在东经$90^\\circ$, 北纬$60^\\circ$的位置上, 则$A,B$ 两地的球面距离为\\blank{50}.(精确到$0.1$)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44607,7 +44869,9 @@
"001737": {
"id": "001737",
"content": "已知地球的半径约为$6371$千米, 大连的位置约为东经$121^\\circ$, 北纬$39^\\circ$, 里斯本的位置约为西经$10^\\circ$, 北纬$39^\\circ$.\n计算大连到里斯本的球面距离(精确到$1$千米);",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -44652,7 +44916,9 @@
"001739": {
"id": "001739",
"content": "已知数列$\\{a_n\\}$中, $a_n=\\dfrac{1}{4}n^2-\\dfrac{17}{12}n+\\dfrac{13}6, \\ (n \\in \\mathbf{N}^*)$, 下列各数中, 是这数列的某一项的是\\bracket{20}.\n\\fourch{$\\dfrac{1}{10}$}{$\\dfrac{1}{5}$}{$\\dfrac{1}{2}$}{$0$}",
- "objs": [],
+ "objs": [
+ "K0406001X"
+ ],
"tags": [
"第四单元"
],
@@ -44676,7 +44942,9 @@
"001740": {
"id": "001740",
"content": "根据已知条件, 写出下列各数列$\\{a_n\\}$的前$5$ 项:\\\\ \n(1) $a_n=\\dfrac{n+1}{n+2}$, \\blank{150};\\\\ \n(2) $a_n=\\dfrac{1+(-1)^{n+1}}{2}$, \\blank{150};\\\\ \n(3) $a_n=n\\cos n\\pi$, \\blank{150};\\\\ \n(4) $a_n=\\dfrac{8}{9}(10^n-1)$, \\blank{150};\\\\ \n(5) $a_1=1$, $a_n=a_{n-1}+4$, $n \\in \\mathbf{N}^*, \\ n>1$, \\blank{150};\\\\ \n(6) $a_6=16$, $a_n=-2a_{n-1}$, $n\\in \\mathbf{N}^*, \\ n>1$, \\blank{150}.",
- "objs": [],
+ "objs": [
+ "K0406001X"
+ ],
"tags": [
"第四单元"
],
@@ -44700,7 +44968,9 @@
"001741": {
"id": "001741",
"content": "根据下列数列的前几项, 写出它的一个通项公式:\\\\ \n(1) $1,8,15,22,29,\\cdots$, \\blank{150};\\\\ \n(2) $5,4,3,2,1,\\cdots$, \\blank{150};\\\\ \n(3) $\\dfrac{1}{2},\\dfrac{3}{4},\\dfrac{5}{6},\\dfrac{7}{8},\\cdots$, \\blank{150};\\\\ \n(4) $2,0,2,0,2,\\cdots$, \\blank{150};\\\\ \n(5) $1,1.1,1.01,1.001,1.0001,\\cdots$, \\blank{150};\\\\ \n(6) $\\dfrac{2^2-1}{3},-\\dfrac{3^2-1}{5},\\dfrac{4^2-1}{7},-\\dfrac{5^2-1}{9},\\cdots$, \\blank{150};\\\\ \n(7) $1,2,3,4,5,8,7,16,9,32,11,64,\\cdots$, \\blank{150}.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -44724,7 +44994,9 @@
"001742": {
"id": "001742",
"content": "(1) 在数列$\\{a_n\\}$中, 已知$a_1=2$, $a_n=2a_{n-1}+n, \\ (n \\ge 2, \\ n\\in \\mathbf{N})$, 则$a_5=$\\blank{100};\\\\ \n(2) 在数列$\\{b_n\\}$中, 已知$b_1=1$, $b_2=5$, $b_{n+2}=b_{n+1}-b_n, n \\in \\mathbf{N}^*$, 则$b_{2014}=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -44748,7 +45020,9 @@
"001743": {
"id": "001743",
"content": "若数列$\\{a_n\\}$的前$4$项的值两两不同, 且对任意正整数$n$均成立$a_{n+4}=a_n$. 则下列该数列的子列中, 可取遍数列$\\{a_n\\}$的前$4$项值的有\\bracket{20}.\n\\varfourch{$\\{a_{2n}\\}$}{$\\{a_{3n+2}\\}$}{$\\{a_{5n+3}\\}$}{$\\{a_{6n+3}\\}$}",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -44772,7 +45046,9 @@
"001744": {
"id": "001744",
"content": "已知数列$\\{a_n\\}$的通项$a_n=\\sin n$.\\\\ \n(1) 取出数列$\\{a_n\\}$的第$1,4,7,\\cdots,3n-2,\\cdots$ 项, 得到的新数列$\\{b_n\\}$, 则通项$b_n=$\\blank{100};\\\\ \n(2) 去除数列$\\{a_n\\}$的第$1,4,7,\\cdots,3n-2,\\cdots$ 项, 得到的新数列$\\{c_n\\}$, 则通项$c_n=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -44796,7 +45072,9 @@
"001745": {
"id": "001745",
"content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=2^n$, 则通项$a_n=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -44820,7 +45098,9 @@
"001746": {
"id": "001746",
"content": "设$a,b$是常数, 已知数列$\\{a_n\\}$的前$n$项和$S_n=an^2+bn$, 求通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -44844,7 +45124,9 @@
"001747": {
"id": "001747",
"content": "(1) 数$\\lg 2$与$\\lg 8$的等差中项为\\blank{50}.\\\\ \n(2) 数$\\dfrac{8-\\sqrt{2}}{2}$与$\\dfrac{8+\\sqrt{2}}{2}$的等差中项为\\blank{50}.\\\\ \n(3) 数$(a+b)^2$与$(a-b)^2$的等差中项为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -44868,7 +45150,9 @@
"001748": {
"id": "001748",
"content": "在等差数列$\\{a_n\\}$中, 若$a_5=12$, $a_9=21$, 则$a_{10}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -44892,7 +45176,9 @@
"001749": {
"id": "001749",
"content": "在等差数列$\\{a_n\\}$中, 若$a_1+a_2=30,a_3+a_4=40$, 则$a_5+a_6=$\\blank{50},$a_7+a_8=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -44916,7 +45202,9 @@
"001750": {
"id": "001750",
"content": "在等差数列$\\{a_n\\}$中, 若$a_3+a_4+a_5+a_6+a_7=450$, 则$a_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -44940,7 +45228,9 @@
"001751": {
"id": "001751",
"content": "已知数列$\\{a_n\\}$,$\\{b_n\\}$都是等差数列, 且$a_1=10,b_1=20,a_2+b_2=40$, 则$a_5+b_5$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -44964,7 +45254,9 @@
"001752": {
"id": "001752",
"content": "等差数列$81,78,75,\\cdots$首次出现负值是在第\\blank{50}项, 这个数列的前\\blank{50}项的和最大.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -44988,7 +45280,9 @@
"001753": {
"id": "001753",
"content": "若关于$x$的方程$x^2-x+a=0$和$x^2-x+b=0\\ (a\\ne b)$的四个根可以组成首项为$\\dfrac{1}{4}$的等差数列, 则$a+b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -45012,7 +45306,9 @@
"001754": {
"id": "001754",
"content": "下列条件中, 能确定数列$\\{a_n\\}$是等差数列的条件为\\bracket{20}.\n\\vartwoch{$2a_n=a_{n+1}+a_{n-1}(n\\geq2)$}{$\\{a_{2n-1}\\}$与$\\{a_{2n}\\}$都是等差数列}{$a_n=pn+q$, $p,q$是常数}{$\\{2a_{n}+1\\}$是等差数列}",
- "objs": [],
+ "objs": [
+ "K0401006X"
+ ],
"tags": [
"第四单元"
],
@@ -45036,7 +45332,9 @@
"001755": {
"id": "001755",
"content": "等差数列的首项为$\\dfrac{1}{5}$, 若从第$10$项起各项均大于$1$, 则此数列的公差$d$的取值范围为\\bracket{20}.\n\\fourch{$\\dfrac{4}{45}\\le d<\\dfrac{1}{10}$}{$\\dfrac{4}{45}\\le d\\le \\dfrac{1}{10}$}{$\\dfrac{4}{45}0,S_{16}<0$.\\\\ \n(1) 求公差$d$的取值范围;\\\\ \n(2) $n$为何值时, $S_n$最大? 为什么?",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -45378,7 +45704,9 @@
"001769": {
"id": "001769",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=an^2+bn+c$, 其中$a,b,c$为常数. 判断数列$\\{a_n\\}$是否是等差数列, 并说明理由.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -45402,7 +45730,9 @@
"001770": {
"id": "001770",
"content": "在等比数列$\\{a_n\\}$中, 若$a_9=-2$, $a_{13}=-32$, 则通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -45428,7 +45758,9 @@
"001771": {
"id": "001771",
"content": "在等比数列$\\{a_n\\}$中, 若$a_8=\\dfrac{1}{16}$, $q=\\dfrac{1}{2}$, 则前$8$项的和为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -45452,7 +45784,9 @@
"001772": {
"id": "001772",
"content": "在等比数列$\\{a_n\\}$中, 若前$3$项的和为$14$, $a_1=2$, 则公比为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -45476,7 +45810,9 @@
"001773": {
"id": "001773",
"content": "方程$3x^2-15x+1=0$的两根的等比中项为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -45500,7 +45836,9 @@
"001774": {
"id": "001774",
"content": "已知等比数列$\\{a_n\\}$满足$a_1+a_2+a_3+\\cdots+a_{10}=1$, $a_{1}+a_{2}+a_{3}+\\cdots+a_{20}=3$, 则$a_{1}+a_{2}+\\cdots+a_{30}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -45524,7 +45862,9 @@
"001775": {
"id": "001775",
"content": "求值: $1-2+4-8+\\cdots+(-1)^{n-1}\\cdot 2^{n-1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -45548,7 +45888,9 @@
"001776": {
"id": "001776",
"content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n=2^n-1$, 则数列$\\{a_n^2+1\\}$的前$n$项和等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -45572,7 +45914,10 @@
"001777": {
"id": "001777",
"content": "已知$a,b,c$成等比数列, 如果$a,x,b$和$b,y,c$都成等差数列, 则$\\dfrac{a}{x}+\\dfrac{c}{y}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X",
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -45596,7 +45941,9 @@
"001778": {
"id": "001778",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=3^n+k$, 则\\bracket{20}.\n\\twoch{无论$k$取何值, $\\{a_n\\}$都不是等比数列}{有且仅有一个实数$k$, 使得$\\{a_n\\}$是等比数列}\n{有多于一个(有限个)实数$k$, 使得$\\{a_n\\}$是等比数列}{无论$k$取何值, $\\{a_n\\}$都是等比数列}",
- "objs": [],
+ "objs": [
+ "K0404004X"
+ ],
"tags": [
"第四单元"
],
@@ -45620,7 +45967,10 @@
"001779": {
"id": "001779",
"content": "对于数列$\\{a_n\\}$, 已知存在$s\\ne t$, 使得$a_s=a_t$.\\\\ \n(1) 若$\\{a_n\\}$是等差数列, 证明$\\{a_n\\}$是常数列;\\\\ \n(2) 若$\\{a_n\\}$是等比数列, 证明或否定: $\\{a_n\\}$是常数列.",
- "objs": [],
+ "objs": [
+ "K0403002X",
+ "K0401003X"
+ ],
"tags": [
"第四单元"
],
@@ -45644,7 +45994,9 @@
"001780": {
"id": "001780",
"content": "设$\\{a_n\\}$是由正数组成的等比数列, 且公比$q\\ne 1$, 比较$a_1+a_8$和$a_4+a_5$的大小关系.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -45668,7 +46020,10 @@
"001781": {
"id": "001781",
"content": "设$a>0$, 求$a+a^3+a^5+\\cdots+a^{2n-1}$.",
- "objs": [],
+ "objs": [
+ "K0404002X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -45694,7 +46049,10 @@
"001782": {
"id": "001782",
"content": "已知数列$\\{a_n\\}$是一个以正数$q$为公比, 以正数$a$为首项的等比数列,\n求$\\lg a_1+\\lg a_2+\\cdots+\\lg a_n$.",
- "objs": [],
+ "objs": [
+ "K0403006X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -45718,7 +46076,9 @@
"001783": {
"id": "001783",
"content": "数列$\\{a_n\\}$与$\\{b_n\\}$的通项公式分别为$a_n=2^n$, $b_n=3n+2$, 它们的公共项由小到大排成的数列记为$\\{c_n\\}$.\\\\ \n(1) 写出$\\{c_n\\}$的前$5$项;\\\\ \n(2) 证明: $\\{c_n\\}$是等比数列.",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -45742,7 +46102,9 @@
"001784": {
"id": "001784",
"content": "已知非零实数$a,b,c$不全相等. 如果$a,b,c$成等差数列, 那么, $\\dfrac{1}{a},\\dfrac{1}{b},\\dfrac{1}{c}$是否可能成等差数列? 为什么?",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -45768,7 +46130,9 @@
"001785": {
"id": "001785",
"content": "已知$a,b,c$中任意两数之和不为零, $a^2,b^2,c^2$成等差数列, 求证: $\\dfrac{1}{b+c},\\dfrac{1}{c+a},\\dfrac{1}{a+b}$成等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -45792,7 +46156,9 @@
"001786": {
"id": "001786",
"content": "已知数列$\\{a_n\\}$是等差数列, 数列$\\{b_n\\}$的通项$b_n=a_n^2-a_{n+1}^2$, 求证: 数列$\\{b_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -45816,7 +46182,10 @@
"001787": {
"id": "001787",
"content": "[选做]\n已知等差数列$\\{a_n\\}$与$\\{b_n\\}$的前$n$项和分别为$S_n,T_n$, 且$\\dfrac{S_n}{T_n}=\\dfrac{2n+1}{3n+2}$对一切$n\\in \\mathbf{N}^*$成立. 求$\\dfrac{a_4}{b_3}$.",
- "objs": [],
+ "objs": [
+ "K0402004X",
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -45840,7 +46209,9 @@
"001788": {
"id": "001788",
"content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$是公比不相等的两个等比数列, $c_n=a_n+b_n$, 证明: 数列$\\{c_n\\}$不是等比数列.",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -45864,7 +46235,9 @@
"001789": {
"id": "001789",
"content": "求和: $\\sin^21^\\circ+\\sin^22^\\circ+\\cdots+\\sin^290^\\circ=$\\blank{40}.",
- "objs": [],
+ "objs": [
+ "K0402001X"
+ ],
"tags": [
"第四单元"
],
@@ -45890,7 +46263,9 @@
"001790": {
"id": "001790",
"content": "求和: $1\\cdot 1!+2\\cdot 2!+3\\cdot 3!+\\cdots+n\\cdot n!=$\\blank{40}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -45914,7 +46289,9 @@
"001791": {
"id": "001791",
"content": "分别写出下列数列的前$n$项和$S_n$:\\\\ \n(1) $a_n=\\dfrac{1}{(2n-1)(2n+1)}$;\\\\ \n(2) $a_n=\\dfrac{1}{n(n+3)}$;\\\\ \n(3) $a_n=\\dfrac{1}{(2n-1)(2n+1)(2n+3)}$;\\\\ \n(4) $a_n=(2n-1)(2n+1)(2n+3)$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -45938,7 +46315,9 @@
"001792": {
"id": "001792",
"content": "若数列$a_n=\\dfrac{3n-2}{3^n}$, 则数列$\\{a_n\\}$ 的前$n$项和$S_n=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0404001X"
+ ],
"tags": [
"第四单元"
],
@@ -45962,7 +46341,9 @@
"001793": {
"id": "001793",
"content": "已知数列$a_n=14-3n$, 求数列$\\{|a_n|\\}$的前$n$项和$T_n$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -45986,7 +46367,10 @@
"001794": {
"id": "001794",
"content": "已知数列$a_n=\\left\\{\\begin{array}{ll}2^n,& n\\mbox{是奇数},\\\\2n-1,& n\\mbox{是偶数}.\\end{array}\\right.$求该数列的前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "K0402004X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -46010,7 +46394,10 @@
"001795": {
"id": "001795",
"content": "[选做]\n已知$a_n=\\tan n\\cdot \\tan (n-1)$, 求$\\{a_n\\}$的前$n$项之和$S_n$.",
- "objs": [],
+ "objs": [
+ "K0310001B",
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46034,7 +46421,10 @@
"001796": {
"id": "001796",
"content": "已知数列$\\{a_n\\}$的通项$a_n=1+2+\\cdots+n$, 则其前$n$项$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X",
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46058,7 +46448,10 @@
"001797": {
"id": "001797",
"content": "已知数列$\\{a_n\\}$的通项$a_n=1+2+4+\\cdots+2^{n-1}$, 则其前$n$项$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X",
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46084,7 +46477,10 @@
"001798": {
"id": "001798",
"content": "已知数列$\\{a_n\\}$的通项$a_n=1+2+3+\\cdots+2^{n-1}$, 则其前$n$项$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -46108,7 +46504,10 @@
"001799": {
"id": "001799",
"content": "已知数列$\\{a_n\\}$的前$n$项之和为$S_n=10n-n^2$.\\\\ \n(1) 求$a_n$;\\\\ \n(2) 设$b_n=|a_n|$, 求$b_n$的前$n$项之和.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -46132,7 +46531,9 @@
"001800": {
"id": "001800",
"content": "已知数列$a_n=33-2^n$, 求数列$\\{|a_n|\\}$的前$n$项和$T_n$.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -46156,7 +46557,9 @@
"001801": {
"id": "001801",
"content": "已知数列$a_n=\\left\\{\\begin{array}{ll}n,& n=3k-2;\\\\2n,& n=3k-1\\\\1,& n=3k.\\end{array}\\right.k\\in\\mathbf{N}^*$ 求该数列的前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -46180,7 +46583,10 @@
"001802": {
"id": "001802",
"content": "定义在$\\mathbf{R}$上的函数$f(x)=\\dfrac{4^x}{4^x +2}$, $A_n=f\\left(\\dfrac{1}{n}\\right)+f\\left(\\dfrac{2}{n}\\right)+\\cdots+f\\left(\\dfrac{n-1}{n}\\right), \\ n=2,3,\\cdots$.\\\\ \n(1) 求$A_n$;\\\\ \n(2) (选做) 是否存在常数$M>0$, 使得对一切整数$n\\ge 2$, 成立$\\dfrac{1}{A_2}+\\dfrac{1}{A_3}+\\cdots+\\dfrac{1}{A_n}\\le M$.",
- "objs": [],
+ "objs": [
+ "K0217001B",
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46204,7 +46610,9 @@
"001803": {
"id": "001803",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=\\dfrac{n}{n+2}a_n \\ (n\\ge 1)$. 则数列的通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46228,7 +46636,10 @@
"001804": {
"id": "001804",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=a_n+2n-1\\ (n\\ge 1)$. 求数列的通项. (限定逐差法)",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46254,7 +46665,10 @@
"001805": {
"id": "001805",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=2a_n-3\\cdot2^n \\ (n\\ge 1)$. 求数列的通项. (限定变形的逐差法)",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46280,7 +46694,10 @@
"001806": {
"id": "001806",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=\\pi a_n+1 \\ (n\\ge 1)$. 求数列的通项.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46304,7 +46721,10 @@
"001807": {
"id": "001807",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=-1$, $a_{n+1}=3a_n+2n-1\\ (n\\ge 1)$. 求数列的通项.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46331,7 +46751,9 @@
"001808": {
"id": "001808",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=3$, $a_{n}a_{n+1}=\\dfrac{1}{2^n}\\ (n\\ge 1)$. 求数列的通项.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -46355,7 +46777,10 @@
"001809": {
"id": "001809",
"content": "[选做]\n五只猴子得到了一堆桃子, 它们发现那堆桃子不能被均分成$5$份, 于是猴子们决定先去睡觉, 明天再讨论如何分配. 夜里猴子甲偷偷起来, 吃掉了一个桃子后, 它发现余下的桃子正好可以平均分成$5$份, 于是它拿走了一份; 接着猴子乙也起来先偷吃了一个, 结果它也发现余下的桃子恰好可以被平均分成$5$份, 于是它也拿走了一份; 后面的猴子丙, 丁, 戊如法炮制, 先偷吃一个, 然后将余下的桃子平均分成$5$份并拿出了自己的一份, 问: 这一堆桃子至少有几个?",
- "objs": [],
+ "objs": [
+ "K0407004X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46379,7 +46804,9 @@
"001810": {
"id": "001810",
"content": "在数列$\\{a_n\\}$中, 若$a_1=2$, $a_{n+1}=3a_n^2\\ (n \\ge 1)$, 则数列的通项$a_n=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46403,7 +46830,9 @@
"001811": {
"id": "001811",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=2$, $a_{n+1}=3a_n+n\\ (n \\ge 1)$. 则数列的通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46429,7 +46858,9 @@
"001812": {
"id": "001812",
"content": "数列$\\{a_n\\}$满足$a_1=\\dfrac{3}{5}$, $a_n=2-\\dfrac{1}{a_{n-1}}\\ (n\\ge 2)$, 数列$\\{b_n\\}$满足$b_n=\\dfrac{1}{a_n-1}$.\\\\ \n(1) 求证: 数列$\\{b_n\\}$是等差数列;\\\\ \n(2) 求数列$\\{a_n\\}$的通项.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -46455,7 +46886,9 @@
"001813": {
"id": "001813",
"content": "(1) 在数列$\\{a_n\\}$中, 已知$a_1=0$, 且$a_{n+1}=a_n+n^2 \\ (n\\ge 1)$, 求数列的通项$a_n$;\\\\ \n(2) 利用上一小题的结论, 求$1^2+2^2+3^2+\\cdots+n^2$.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46479,7 +46912,10 @@
"001814": {
"id": "001814",
"content": "(1) 在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=2a_n+3^n\\ (n \\ge 1)$. 求数列的通项$a_n$;\\\\ \n(2) 在数列$\\{a_n\\}$中, 已知$a_1=0$, $a_{n+1}=2a_n+3^n+1\\ (n\\ge 1)$. 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46503,7 +46939,9 @@
"001815": {
"id": "001815",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=2a_n+2^n\\ (n \\ge 1)$. 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46529,7 +46967,10 @@
"001816": {
"id": "001816",
"content": "[选做]\n在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=2a_n+3\\cdot 5^{n-1}-3^n+1\\ (n\\ge 1)$, 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46553,7 +46994,9 @@
"001817": {
"id": "001817",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=\\dfrac{4}{3}$, $a_{n+1}=\\dfrac{2}{3-a_n} \\ (n\\ge 1)$, 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -46577,7 +47020,9 @@
"001818": {
"id": "001818",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_{n+1}=2a_n+\\dfrac{n+2}{n(n+1)} \\ (n\\ge 1)$, 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46601,7 +47046,9 @@
"001819": {
"id": "001819",
"content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_2=2$, 且$a_{n+2}=4a_{n+1}-4a_n\\ (n\\ge 1)$, 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46625,7 +47072,10 @@
"001820": {
"id": "001820",
"content": "(1) 在数列$\\{a_n\\}$中, 若$a_1=1$, $a_2=2$, 且$a_{n+2}=-3a_{n+1}+4a_n\\ (n\\ge 1)$, 求数列的通项$a_n$;\\\\ \n(2) 在数列$\\{a_n\\}$中, 已知$a_1=10$, $a_2=100$, 且$a_{n+2}=\\dfrac{a_n^4}{a_{n+1}^{3}}\\ (n\\ge 1)$, 求数列的通项$a_n$;\\\\ \n(3)(选做) 在数列$\\{a_n\\}$中, 若$a_1=1$, $a_2=2$, 且$a_{n+2}=-3a_{n+1}+4a_n+1\\ (n\\ge 1)$, 求数列的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46649,7 +47099,9 @@
"001821": {
"id": "001821",
"content": "已知无穷数列$\\{a_n\\}$满足$(a_{n+1}+a_n)(a_{n+1}-a_n-1)=0 \\ (n\\ge 1)$, $a_1=0$.\n这样的数列的前$10$项之和的所有可能值为\\blank{150}.",
- "objs": [],
+ "objs": [
+ "K0407001X"
+ ],
"tags": [
"第四单元"
],
@@ -46673,7 +47125,10 @@
"001822": {
"id": "001822",
"content": "若数列$\\{a_n\\}$的前$n$项之和为$S_n$, $S_n=2a_n-2n, \\ n\\ge 1$, 求$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0407003X"
+ ],
"tags": [
"第四单元"
],
@@ -46697,7 +47152,10 @@
"001823": {
"id": "001823",
"content": "已知$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=1$, $a_{n+1}=2S_n$, 求$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46723,7 +47181,10 @@
"001824": {
"id": "001824",
"content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$S_n+a_n=\\dfrac{n^2+3n-2}{2}$, 求$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46747,7 +47208,10 @@
"001825": {
"id": "001825",
"content": "整数数列$\\{a_n\\}$满足$a_1a_2+a_2a_3+\\cdots+a_{n-1}a_n=\\dfrac{(n-1)n(n+1)}{3}, \\ n=2,3,\\cdots$,\\\\ \n(1) 若$a_1=1$, 求通项$a_n$.\\\\ \n(2) (选做)求所有满足条件的数列.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -46771,7 +47235,9 @@
"001826": {
"id": "001826",
"content": "判断下列数列是否有极限。 若有, 在横线上写出极限值; 若没有, 在横线上写``没有极限''.\\\\ \n(1) $a_n=(-\\dfrac{1}{2})^n$, \\blank{50};\\\\ \n(2) $a_n=\\dfrac{n+2}{2n+1}$, \\blank{50};\\\\ \n(3) $a_n=\\left\\{\\begin{array}{ll}\\dfrac{2}{n},& n\\mbox{是奇数};\\\\\\dfrac{1}{n},& n\\mbox{是偶数},\\end{array}\\right.$ \\blank{50};\\\\ \n(4) $a_n=\\left\\{\\begin{array}{ll}1,& n\\mbox{是奇数};\\\\\\dfrac{1}{n},& n\\mbox{是偶数},\\end{array}\\right.$ \\blank{50};\\\\ \n(5) $a_n=\\left\\{\\begin{array}{ll}n,& n\\leq 100;\\\\\\dfrac{1}{n},& n>100,\\end{array}\\right.$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46795,7 +47261,9 @@
"001827": {
"id": "001827",
"content": "[选做]\n参考讲义上极限的定义, 证明: 数列$a_n=\\dfrac{(-1)^n}{n}$的极限为$0$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46819,7 +47287,9 @@
"001828": {
"id": "001828",
"content": "[选做]\n参考讲义上极限的定义, 证明: 数列$a_n=2^n$没有极限.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46843,7 +47313,9 @@
"001829": {
"id": "001829",
"content": "[选做, 难]\n证明: 若数列$\\{a_n\\}$的极限为$A$, 则任意交换$\\{a_n\\}$中元素的顺序之后, 所得的新数列的极限也为$A$.\\\\ \n(注: 这里的交换可以是无限次, 如变成$a_2,a_1,a_4,a_3,a_6,a_5,\\cdots$这样一个新数列等等)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46867,7 +47339,9 @@
"001830": {
"id": "001830",
"content": "求下列极限:\\\\ \n(1) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\dfrac{2n+1}{3n+2}=$\\blank{60};\\\\ \n(2) $\\displaystyle\\lim_{n\\rightarrow \\infty} (-1)^{n+1}\\dfrac{2n^2+1}{2n^3+3n^2-n+5}=$\\blank{60};\\\\ \n(3) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\left(\\dfrac{1}{n^2+1}+\\dfrac{4}{n^2+1}+\\cdots+\\dfrac{3n-2}{n^2+1}\\right)=$\\blank{60};\\\\ \n(4) 已知$a_n=\\mathrm{sgn} (2011-n)\\cdot \\dfrac{n+3}{2n+5}$, 则$\\displaystyle\\lim_{n\\rightarrow \\infty} a_n=$\\blank{60};($\\mathrm{sgn}(x)$是符号函数, $x>0$时函数值是$1$, $x<0$是函数值是$-1$, $x=0$时函数值是$0$)\\\\ \n(5) 已知$a_n=(1+2+3+\\cdots+n)\\left[\\left(1-\\dfrac 12\\right)\\left(1-\\dfrac 13\\right)\\cdots\\left(1-\\dfrac 1n\\right)\\right]^2$, 则$\\displaystyle\\lim_{n\\rightarrow \\infty} a_n=$\\blank{60};\\\\ \n(6) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\left(\\sqrt{5}+\\left(\\dfrac{\\sqrt{3}}2\\right)^n\\right)=$\\blank{60};\\\\ \n(7) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\dfrac{1+\\dfrac 13+\\dfrac 19+\\cdots+\\dfrac 1{3^n}}{1-\\dfrac 14+\\dfrac 1{16}-\\cdots +\\left(-\\dfrac{1}{4}\\right)^n}=$\\blank{60};\\\\ \n(8) $\\displaystyle\\lim_{n\\rightarrow \\infty} (\\sqrt{n+2}-\\sqrt{n-1})=$\\blank{60};\\\\ \n(9) $\\displaystyle\\lim_{n\\rightarrow \\infty}\\dfrac{\\sqrt{n^2+n}}{n+1}=$\\blank{60}.\\\\ \n(10) $\\displaystyle\\lim_{n\\rightarrow \\infty}\\dfrac{\\sqrt{n+1}-\\sqrt{n}}{\\sqrt{n+2}-\\sqrt{n}}=$\\blank{60};\\\\ \n(11) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\left(\\dfrac{n^3-1}{3n^2+n}-\\dfrac{n^2+1}{3n+4}\\right)=$\\blank{60};\\\\ \n(12) $\\displaystyle\\lim_{n\\rightarrow \\infty} \\dfrac{2^n+3^n}{2^n-3^n}=$\\blank{60}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46891,7 +47365,9 @@
"001831": {
"id": "001831",
"content": "已知$\\displaystyle\\lim_{n\\rightarrow \\infty} [(2n-1)a_n]=1$, 则$\\displaystyle\\lim_{n\\rightarrow \\infty} (na_n)=$\\blank{60}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46942,7 +47418,9 @@
"001833": {
"id": "001833",
"content": "设$a$是常数, 数列$\\{a_n\\}$的通项公式为$a_n=(a^2+2a)^n$, 若$\\displaystyle\\lim_{n\\rightarrow \\infty} a_n$不存在, 则$a$的取值范围为\\blank{60}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46966,7 +47444,9 @@
"001834": {
"id": "001834",
"content": "设$a$是常数, 若极限$\\displaystyle\\lim_{n\\rightarrow \\infty} \\left(\\dfrac{n^2}{n+1}-an\\right)$存在, 则$a$的取值范围为\\blank{60}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -46990,7 +47470,9 @@
"001835": {
"id": "001835",
"content": "已知实数$a,b\\in \\mathbf{R}^+$, 求$\\displaystyle\\lim_{n\\rightarrow \\infty} \\dfrac{a^{n+1}}{a^n+b^n}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47014,7 +47496,9 @@
"001836": {
"id": "001836",
"content": "已知$a\\in \\mathbf{R}$, 求$\\displaystyle\\lim_{n\\rightarrow \\infty}\\dfrac{2a}{a+(1-a)n}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47038,7 +47522,9 @@
"001837": {
"id": "001837",
"content": "已知对于数列$\\{a_n\\}$, 极限$\\displaystyle\\lim_{n\\rightarrow \\infty} \\dfrac{a_n-3}{a_n+2}=\\dfrac{4}{9}$, 求$\\displaystyle\\lim_{n\\rightarrow \\infty} a_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47062,7 +47548,9 @@
"001838": {
"id": "001838",
"content": "判断下列命题的真假, 其中假命题用``F''表示, 真命题用``T''表示.\\\\ \n\\begin{enumerate}[\\blank{20}(1)]\n\\item 所有无限循环小数都可以表示成分数.\\\\ \n\\item 如果数列$\\{a_n\\}$有极限, 那么其前$n$ 项和$S_n$也有极限;\\\\ \n\\item 如果数列$\\{a_n\\}$的前$n$项和$S_n$ 有极限, 那么$\\{a_n\\}$的极限为$0$;\\\\ \n\\item 如果正数数列$\\{a_n\\}$的极限为零, 那么其前$n$项和$S_n$必定有极限.\\\\ \n\\end{enumerate}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47086,7 +47574,9 @@
"001839": {
"id": "001839",
"content": "用最简分数表示下列循环小数:\\\\ \n(1) $0.\\dot{2}\\dot{6}=$\\blank{60};\\\\ \n(2) $3.141\\dot{5}\\dot{9}\\dot{2}=$\\blank{60}.",
- "objs": [],
+ "objs": [
+ "K0405004X"
+ ],
"tags": [
"第四单元"
],
@@ -47110,7 +47600,9 @@
"001840": {
"id": "001840",
"content": "若$a_n=\\dfrac{1}{5}+\\dfrac{2}{5^2}+\\dfrac{1}{5^3}+\\dfrac{2}{5^4}+\\cdots+\\dfrac{1}{5^{2n-1}}+\\dfrac{2}{5^{2n}}$, 则$\\displaystyle\\lim_{n\\rightarrow +\\infty} a_n=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47136,7 +47628,9 @@
"001841": {
"id": "001841",
"content": "若某无穷等比数列$\\{a_n\\}$各项和是$4$, 各项的平方和是$6$, 则$\\{a_n\\}$的公比$q=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -47160,7 +47654,9 @@
"001842": {
"id": "001842",
"content": "若$\\{a_n\\}$为无穷等比数列, $\\{a_n\\}$中每一项都是它后面所有项之和的$4$倍, 则公比$q$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -47184,7 +47680,9 @@
"001843": {
"id": "001843",
"content": "已知无穷等比数列$\\{a_n\\}$的各项和为$1$, 求其首项的取值范围.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -47208,7 +47706,9 @@
"001844": {
"id": "001844",
"content": "已知无穷等比数列$\\{a_n\\}$的首项为$a$, 公比为正数$q$. 记$\\{a_n\\}$的前$n$项和为$S_n$, $\\{a_n^2\\}$的前$n$项和为$G_n$, 求$\\displaystyle\\lim_{n\\rightarrow +\\infty} \\dfrac{S_n}{G_n}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -47232,7 +47732,9 @@
"001845": {
"id": "001845",
"content": "对于数列$\\dfrac{1}{2},\\dfrac{1}{4},\\cdots,\\dfrac{1}{2^n},\\cdots$, 试从其中找出无限项构成一个新的等比数列, 使新数列的各项和为$\\dfrac{1}{7}$.\\\\ \n(1) 写出一个满足条件的新数列的首项与公比;\\\\ \n(2) (选做) 证明满足条件的新数列是唯一的.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -49344,7 +49846,9 @@
"001918": {
"id": "001918",
"content": "[选做]\n设$A,B,C$三点坐标依次为$A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, 证明: $A, B, C$三点共线的充要条件为\n$$\\left|\\begin{array}{ccc}1&x_1&y_1\\\\1&x_2&y_2\\\\1&x_3&y_3\\end{array}\\right|=0$$.\\\\ \n(注: $\\left|\\begin{array}{ccc}1&x_1&y_1\\\\1&x_2&y_2\\\\1&x_3&y_3\\end{array}\\right|$称为行列式, 可预习讲义上行列式的内容进行理解)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49368,7 +49872,9 @@
"001919": {
"id": "001919",
"content": "对于三阶行列式\n$$\\left|\\begin{array}{ccc}1 & 2& 3\\\\4& 5&6\\\\7&8&9\\end{array}\\right|,$$\n其元素$2$的代数余子式的值为\\blank{50}, 元素$4$的余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49392,7 +49898,9 @@
"001920": {
"id": "001920",
"content": "行列式$\\left|\\begin{array}{cc}a & b \\\\ -b & a\\end{array}\\right|$的值为\\blank{50};\n行列式$\\left|\\begin{array}{ccc}1 & 4 & 9\\\\1 & 5 & 25\\\\1 & 7 & 49\\end{array}\\right|$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49416,7 +49924,9 @@
"001921": {
"id": "001921",
"content": "行列式$\\left|\\begin{array}{ccc}x & 0 & 0\\\\a & y & 0\\\\b & c & z\\end{array}\\right|$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49442,7 +49952,9 @@
"001922": {
"id": "001922",
"content": "行列式$\\left|\\begin{array}{ccc}0 & n& m\\\\-n & 0 & l\\\\ -m & -l & 0\\end{array}\\right|$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49468,7 +49980,9 @@
"001923": {
"id": "001923",
"content": "行列式$\\left|\\begin{array}{ccc}b+c&a-c&a-b\\\\b-c&c+a&b-a\\\\c-b&c-a&a+b\\end{array}\\right|$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49516,7 +50030,9 @@
"001925": {
"id": "001925",
"content": "设数列$\\{a_n\\}$,$\\{b_n\\}$,$\\{c_n\\}$及三阶行列式\n$D=\\left|\\begin{array}{ccc}a_1 & b_1& c_1\\\\a_2& b_2&c_2\\\\a_3&b_3&c_3\\end{array}\\right|$.\\\\ \n(1) 若三个数列均为等差数列, 证明: $D=0$;\\\\ \n(2) 若三个数列均为等比数列, 且公比各不相同, 证明: $D\\neq0$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49564,7 +50080,9 @@
"001927": {
"id": "001927",
"content": "用行列式解方程组(写出系数行列式$D$, $D_x$, $D_y$等等的值, 并给出方程的解):\\\\ \n(1) $\\left\\{\\begin{array}{l}3x-6y-1=0,\\\\-4y+2x=2.\\end{array}\\right.$ \\\\ \n$D=$\\blank{50}, $D_x=$\\blank{50}, $D_y=$\\blank{50}, 解为$(x,y)=$\\blank{100}.\\\\ \n(2) $\\left\\{\\begin{array}{l}3x-2y+z=0,\\\\x+y+2z=5,\\\\5x-7y+8z=-1.\\end{array}\\right.$ \\\\ \n$D=$\\blank{50}, $D_x=$\\blank{50}, $D_y=$\\blank{50}, $D_z=$\\blank{50}, 解为$(x,y,z)=$\\blank{100}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49588,7 +50106,9 @@
"001928": {
"id": "001928",
"content": "已知$a$是实数, 用行列式解方程组: $\\left\\{\\begin{array}{l}ax+3y=a+3,\\\\x+(a-2)y=2,\\end{array}\\right.$ 并叙述解的个数的不同情况.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49638,7 +50158,9 @@
"001930": {
"id": "001930",
"content": "已知$a$是实数, 用行列式解方程组: $\\left\\{\\begin{array}{l}2x+y-3z=-1,\\\\x-2y+az=-3,\\\\ay-z=1,\\end{array}\\right.$并叙述解的个数的不同情况.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49688,7 +50210,9 @@
"001932": {
"id": "001932",
"content": "写出以下矩阵计算的结果:\\\\ \n(1) $\\left(\\begin{array}{ccc}1 &3 &5\\\\ 2& 4& 6\\end{array}\\right)-3\\left(\\begin{array}{ccc}4 & 2 & -1\\\\ -3 & 2 & 0\\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.\\\\ \n(2) $\\left(\\begin{array}{ccc}7&1&5\\end{array}\\right) \\left(\\begin{array}{cc}3&1\\\\6&4\\\\2&5\\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.\\\\ \n(3) $\\left(\\begin{array}{ccc}3&0&1\\\\0&5&4\\\\2&1&5\\end{array}\\right)\\left(\\begin{array}{ccc}6&0&1\\\\2&3&4\\\\3&2&1\\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.\\\\ \n(4) $\\left(\\begin{array}{ccc}2&1&0\\\\3&2&1\\end{array}\\right)\\left(\\begin{array}{cc}2&1\\\\3&0\\\\4&1\\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.\\\\ \n(5) $\\left(\\begin{array}{ccc}2&0&0\\\\0&2&0\\\\0&0&2\\end{array}\\right)\\left(\\begin{array}{cc}a& b\\\\c & d\\\\e&f \\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.\\\\ \n(6) $\\left(\\begin{array}{ccc}a&0&0\\\\0&b&0\\\\0&0&c\\end{array}\\right)\\left(\\begin{array}{ccc}d&0&0\\\\0&e&0\\\\0&0&f\\end{array}\\right)=$\\underline{\\phantom{$\\left(\\begin{array}{ccc}a_1& b_1 & c_1\\\\a_2 & b_2 & c_2 \\end{array}\\right)$}}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49712,7 +50236,9 @@
"001933": {
"id": "001933",
"content": "填入适当的矩阵, 使得二元一次方程组$\\left\\{\\begin{array}{l}a_1x+b_1y=c_1,\\\\a_2x+b_2y=c_2\\end{array}\\right.$表示为矩阵的乘积:\\\\ \n\\underline{\\phantom{$\\left(\\begin{array}{cc}a_1& b_1\\\\a_2 & b_2\\end{array}\\right)$}}$\\left(\\begin{array}{c}x\\\\y\\end{array}\\right)=\\left(\\begin{array}{c}c_1\\\\c_2\\end{array}\\right)$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49736,7 +50262,9 @@
"001934": {
"id": "001934",
"content": "判断下列命题的真假, 用``T''或``F''分别表示真命题与假命题(其中$\\det(A)$表示方阵$A$的行列式).\\\\ \n\\begin{enumerate}[\\blank{30}1.]\n\\item 对任意两个同阶方阵, $\\det(A+B)=\\det A+\\det B$.\\\\ \n\\item 对于任意的方阵$A$和实数$k$, $\\det (kA)=k\\det A$.\\\\ \n\\item 已知$B$是一个$3$阶方阵, 最多存在一个$3$阶方阵$A$, 使得$A^2=B$.\\\\ \n\\item 对任意同阶方阵$A,B$, 一定有且仅有一个矩阵$C$, 使得$AC=B$.\\\\ \n\\item 对任意两个同阶方阵, $A\\cdot A+2A\\cdot B+B\\cdot B=(A+B)^2$.\\\\ \n\\item 已知$B$是一个$3$阶方阵, 一定存在$3$阶方阵$A$, 使得$A^2=B$. (注: $A^2=A\\cdot A$)\\\\ \n\\end{enumerate}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49760,7 +50288,9 @@
"001935": {
"id": "001935",
"content": "已知$A=\\left(\\begin{array}{cc}3&1\\\\1&3\\end{array}\\right)$, $B=\\left(\\begin{array}{cc}1&2\\\\2&1\\end{array}\\right)$. 写出一个$2$行$2$列的矩阵$C$, 使得$AC=B$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49784,7 +50314,9 @@
"001936": {
"id": "001936",
"content": "(1) 已知$A$是一个$1$行$3$列的矩阵, $B$是一个$3$行$1$列的矩阵, 问$\\det(AB)$是否一定为零? 并说明理由.\\\\ \n(2) 已知$A$是一个$1$行$3$列的矩阵, $B$是一个$3$行$1$列的矩阵, 问$\\det(BA)$是否一定为零? 并说明理由.\\\\ \n(3) 已知$A$是一个$2$行$3$列的矩阵, $B$是一个$3$行$2$列的矩阵, 问$\\det(AB)$是否一定为零? 并说明理由.\\\\ \n(4) (选做) 已知$A$是一个$2$行$3$列的矩阵, $B$是一个$3$行$2$列的矩阵, 问$\\det(BA)$是否一定为零? 并说明理由.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49808,7 +50340,9 @@
"001937": {
"id": "001937",
"content": "[选做]\n我们知道二阶单位矩阵$I_2=\\left(\\begin{array}{cc}1&0\\\\0&1\\end{array}\\right)$和所有二阶方阵的乘法都可以交换, 即任取一个二阶方阵$A$, $AI_2=I_2 A(=A)$. 试找出所有这样的二阶方阵$\\left(\\begin{array}{cc}a&b\\\\c&d\\end{array}\\right)$, 使得它们和任何一个二阶方阵的乘法都可以交换(提示: 在证明必要性时可以用一些矩阵来作乘法试试看).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49832,7 +50366,9 @@
"001938": {
"id": "001938",
"content": "写出下列方程组的系数矩阵, 并用行初等变换的方法解方程组(要有过程):\\\\ \n(1) $\\left\\{\\begin{array}{ll}2x+y=5,\\\\3x-2y=4.\\end{array}\\right.$\\\\ \n(2) $\\left\\{\\begin{array}{l}x+y+z=6,\\\\3x+y-z=2,\\\\5x-2y+3z=10.\\end{array}\\right.$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -49928,7 +50464,9 @@
"001942": {
"id": "001942",
"content": "[选做, 可另附纸]\n已知关于$x,y,z$的三元一次方程组$$\\left\\{\\begin{array}{l}a_1x+b_1y+c_1z=0,\\\\a_2x+b_2y+c_2z=0,\\\\a_3x+b_3y+c_3z=0,\\end{array}\\right.$$\n求证: 该方程组有非零解(即$x,y,z$不全等于零的解)当且仅当方程组系数矩阵的行列式的值为零.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -51036,7 +51574,9 @@
"001988": {
"id": "001988",
"content": "设地球的半径为$R$, 那么地球表面任意两个不同地点的球面距离的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -51060,7 +51600,9 @@
"001989": {
"id": "001989",
"content": "设地球半径为$6400\\mathrm{km}$, 地球上的两点$A(30^\\circ\\mathrm{N}, 60^\\circ\\mathrm{E})$与$B(50^\\circ\\mathrm{N}, 110^\\circ\\mathrm{E})$之间的球面距离约为\\blank{50}.(精确到$10\\mathrm{km}$)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -51084,7 +51626,9 @@
"001990": {
"id": "001990",
"content": "设地球半径为$6400\\mathrm{km}$, 地球上的两点$$A(39^\\circ54'23.54''\\mathrm{N}, 116^\\circ23'28.16''\\mathrm{E}), \\ B(39^\\circ59'28.66''\\mathrm{N}, 116^\\circ23'24.84''\\mathrm{E})$$之间的球面距离约为\\blank{50}.(精确到$0.1\\mathrm{km}$)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -80983,7 +81527,9 @@
"003202": {
"id": "003202",
"content": "写出下列数列的一个通项公式:\\\\\n(1) $-3,1,5,9,13,\\cdots$: $a_n$=\\blank{50}; (2)$\\dfrac 27,\\dfrac 4{11},\\dfrac 12,\\dfrac 45,2$: $a_n$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -81004,7 +81550,9 @@
"003203": {
"id": "003203",
"content": "已知数列$\\{a_n\\}$满足: $a_n=n+\\dfrac 6n$, 则数列$\\{a_n\\}$中最小项为第\\blank{50}项.",
- "objs": [],
+ "objs": [
+ "K0401001X"
+ ],
"tags": [
"第四单元"
],
@@ -81025,7 +81573,9 @@
"003204": {
"id": "003204",
"content": "(1) 数列$\\{a_n\\}$满足: $a_1+a_2+a_3+\\cdots +a_n=8$, 则$a_n=$\\blank{50};\\\\\n(2) 数列$\\{a_n\\}$满足: $a_1\\cdot a_2\\cdot a_3\\cdots a_n=8$, 则$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -81046,7 +81596,9 @@
"003205": {
"id": "003205",
"content": "已知$a_1=1$, $a_2=3$, $a_{n+2}=a_{n+1}-a_n$, 则$a_{2030}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -81067,7 +81619,9 @@
"003206": {
"id": "003206",
"content": "数列$\\{a_n\\}$满足$a_{n+1}=\\begin{cases}2a_n,& 0\\le a_n<\\dfrac 12,\\\\ 2a_n-1, &\\dfrac 12\\le a_n<1. \\end{cases}$ 若$a_1=\\dfrac 67$, 则$a_2=$\\blank{50}; $a_3=$\\blank{50}; $a_{2021}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -81088,7 +81642,9 @@
"003207": {
"id": "003207",
"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": [],
+ "objs": [
+ "K0401001X"
+ ],
"tags": [
"第四单元"
],
@@ -81112,7 +81668,9 @@
"003208": {
"id": "003208",
"content": "已知数列$\\{a_n\\}$的通项$a_n=n+\\mathrm{e}^n$.\\\\\n(1) 把该数列的前$10$项去掉, 得到新数列$\\{b_n\\}$, 则通项$b_n=$\\blank{50};\\\\\n(2) 将该数列的奇数项按原来的先后顺序排列, 得到新数列$\\{c_n\\}$, 则通项${c_n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401001X"
+ ],
"tags": [
"第四单元"
],
@@ -81133,7 +81691,9 @@
"003209": {
"id": "003209",
"content": "已知数列$\\{a_n\\}$的前$n$项和是$S_n=2\\cdot 3^n+3$, 求数列$\\{a_n\\}$的通项$a_n$.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -81154,7 +81714,9 @@
"003210": {
"id": "003210",
"content": "已知数列$\\{a_n\\}$的通项$a_n=(n+1)(\\dfrac{10}{11})^n$, 试问该数列有没有最大项? 若有, 求出最大项; 若没有, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0406005X"
+ ],
"tags": [
"第四单元"
],
@@ -81175,7 +81737,9 @@
"003211": {
"id": "003211",
"content": "已知$\\{a_n\\}$是递增数列, 且$a_n=n^2+\\lambda n$, 求实数$\\lambda$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0406004X"
+ ],
"tags": [
"第四单元"
],
@@ -81196,7 +81760,9 @@
"003212": {
"id": "003212",
"content": "已知数列$\\{a_n\\}$的通项$a_n=2^n$. 对任意的$k\\in \\mathbf{N}^*$, 在$a_{2k}$与$a_{2k+1}$中间插入一项$k$, 构成新数列$\\{b_n\\}:2,4,1,8,16,2,32,64,3,128,\\cdots$. 求数列$\\{b_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0401001X"
+ ],
"tags": [
"第四单元"
],
@@ -81217,7 +81783,9 @@
"003213": {
"id": "003213",
"content": "已知数列$\\{a_n\\}$满足$a_{n+2}=a_n$, ${a_1}=1$, ${a_2}=2$, 则通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -81238,7 +81806,9 @@
"003214": {
"id": "003214",
"content": "已知数列$\\{a_n\\}$满足$a_{n+1}=a_n^2-k$, $a_1=1$, $a_3=-1$, 则常数$k=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -81259,7 +81829,9 @@
"003215": {
"id": "003215",
"content": "已知数列$\\{a_n\\}$满足: $a_n=\\dfrac 1{n-5.5}$, 则此数列中最大项的值为\\blank{50}, 最小项的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0406005X"
+ ],
"tags": [
"第四单元"
],
@@ -81280,7 +81852,9 @@
"003216": {
"id": "003216",
"content": "已知数列$\\{a_n\\}$满足: $a_n=2^n$, 删去数列中第$1,4,\\cdots,3n-2,\\cdots$项, 得到新数列的通项$b_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401001X"
+ ],
"tags": [
"第四单元"
],
@@ -81301,7 +81875,9 @@
"003217": {
"id": "003217",
"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": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -81324,7 +81900,10 @@
"003218": {
"id": "003218",
"content": "设$\\lambda$是实常数, 数列$\\{a_n\\}$的通项$a_n=n+\\dfrac{\\lambda}n$.\\\\\n(1) 若数列$\\{a_n\\}$递增, 求$\\lambda$的取值范围;\\\\\n(2) 若数列$\\{a_n\\}$中, 唯一最小项为$a_4$, 求$\\lambda$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0406004X",
+ "K0406005X"
+ ],
"tags": [
"第四单元"
],
@@ -81345,7 +81924,9 @@
"003219": {
"id": "003219",
"content": "已知正项数列$\\{a_n\\}$满足$a_n-\\dfrac 1a_n=-2n$, 求证: 数列$\\{a_n\\}$是递减数列.",
- "objs": [],
+ "objs": [
+ "K0406004X"
+ ],
"tags": [
"第四单元"
],
@@ -81366,7 +81947,10 @@
"003220": {
"id": "003220",
"content": "等差数列$\\{a_n\\}$中, 已知$a_1=3$, $d=2$, 则通项$a_n=$\\blank{50}, 前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81387,7 +81971,9 @@
"003221": {
"id": "003221",
"content": "等差数列$\\{a_n\\}$中, 已知$a_1=3$, $a_2+a_5=-4$, $a_n=-11$, 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X"
+ ],
"tags": [
"第四单元"
],
@@ -81408,7 +81994,9 @@
"003222": {
"id": "003222",
"content": "记等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_3=0$, $a_7+a_8=0$, 则$S_7=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81431,7 +82019,9 @@
"003223": {
"id": "003223",
"content": "等差数列$\\{a_n\\}$中, 已知$a_1=1$, $a_1+a_2+a_5=13$, 则前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81452,7 +82042,9 @@
"003224": {
"id": "003224",
"content": "已知等差数列$\\{a_n\\}$的前$n$项之和为$S_n$, 若$S_{15}$为一确定常数, 则下列各式也为确定常数的是\\bracket{20}.\n\\fourch{$a_2+a_{13}$}{$a_2\\cdot a_{13}$}{$a_1+a_8+a_{15}$}{$a_1\\cdot a_8\\cdot a_{15}$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81473,7 +82065,9 @@
"003225": {
"id": "003225",
"content": "在$a$和$b$($a0$, $S_7<0$.\\\\\n(1) 求公差$d$的取值范围;\\\\\n(2) 数列$\\{S_n\\}$是否有最大项? 若有, 求出该项为第几项; 若无, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0402006X"
+ ],
"tags": [
"第四单元"
],
@@ -81599,7 +82205,9 @@
"003231": {
"id": "003231",
"content": "等差数列$\\{a_n\\}$中, $a_1+a_4+a_7=9$, $a_2+a_5+a_8=3$, 则$a_3+a_6+a_9=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -81620,7 +82228,9 @@
"003232": {
"id": "003232",
"content": "设$S_n$为等差数列$\\{a_n\\}$的前n项和, 若$S_5=10$, $S_{10}=-5$, 则$S_{15}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -81641,7 +82251,11 @@
"003233": {
"id": "003233",
"content": "设$a$是实数, 若等差数列$\\{a_n\\}$的前$n$项和$S_n=n+a$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402002X",
+ "K0402005X",
+ "K0402006X"
+ ],
"tags": [
"第四单元"
],
@@ -81662,7 +82276,10 @@
"003234": {
"id": "003234",
"content": "已知等差数列$\\{a_n\\}$, $\\{b_n\\}$的前$n$项和分别为$S_n,T_n$, 若$\\dfrac{S_n}{T_n}=\\dfrac{n-1}{n+1}$, 则$\\dfrac{a_8}{b_8}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81683,7 +82300,9 @@
"003235": {
"id": "003235",
"content": "等差数列$\\{a_n\\}$中, $S_n$为前$n$项和, 且$S_6S_8$, 给出下列命题:\\\\\n(1) 数列$\\{a_n\\}$中前$7$项是递增的, 从第$8$项开始递减;\n(2) $S_9$一定小于$S_6$;\n(3) $a_1$是$\\{a_n\\}$各项中的最大的;\n(4) $S_7$不一定是$\\{S_n\\}$中最大项. 其中正确的序号是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -81704,7 +82323,9 @@
"003236": {
"id": "003236",
"content": "设等比数列$\\{b_n\\}$各项为正, 数列$\\{a_n\\}$满足: $a_n=\\dfrac{\\lg b_1+\\lg b_2+\\cdots+\\lg b_n}n$, 证明: 数列 $\\{a_n\\}$为等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -81725,7 +82346,9 @@
"003237": {
"id": "003237",
"content": "设数列$\\{a_n\\}$的通项公式为$a_n=pn+q$($n\\in \\mathbf{N}^*, \\ p>0$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n>m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_n\\}$的前$2m$项和公式.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -81746,7 +82369,9 @@
"003238": {
"id": "003238",
"content": "实数组成的等比数列$\\{a_n\\}$中, 已知$a_1=2$, $a_4=54$, 则通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -81767,7 +82392,9 @@
"003239": {
"id": "003239",
"content": "等比数列$\\{a_n\\}$中, $a_1=4$, $a_2=2$, 则$a_1a_2+a_2a_3+\\cdots +a_na_{n+1}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -81788,7 +82415,10 @@
"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}",
- "objs": [],
+ "objs": [
+ "K0403006X",
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -81809,7 +82439,9 @@
"003241": {
"id": "003241",
"content": "等比数列$\\{a_n\\}$满足$a_1=1$, $a_3=81$, 则$a_2=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -81830,7 +82462,9 @@
"003242": {
"id": "003242",
"content": "若实数$a$、$b$、$c$、$d$、$e$依次构成等比数列, 且$a=-1$, $e=-81$, 则$c$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -81851,7 +82485,9 @@
"003243": {
"id": "003243",
"content": "若等比数列$\\{a_n\\}$的前$n$项和为$S_n=3^n+a$, 则实数$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404004X"
+ ],
"tags": [
"第四单元"
],
@@ -81872,7 +82508,9 @@
"003244": {
"id": "003244",
"content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 则$S_4,S_8-S_4,S_{12}-S_8,S_{16}-S_{12}$成等差数列. 类比以上结论有: 设等比数列$\\{b_n\\}$的前$n$项积为$T_n$, 则$T_4$,\\blank{50}, \\blank{50}, $\\dfrac{T_{16}}{T_{12}}$成等比数列.",
- "objs": [],
+ "objs": [
+ "K0403006X"
+ ],
"tags": [
"第四单元"
],
@@ -81893,7 +82531,9 @@
"003245": {
"id": "003245",
"content": "几位大学生响应国家的创业号召, 开发了一款应用软件. 为激发大家学习数学的兴趣, 他们推出了``解数学题获取软件激活码''的活动. 这款软件的激活码为下面数学问题的答案: 已知数列$1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, \\cdots$, 其中第一项是$2^0$, 接下来的两项是$2^0,2^1$, 再接下来的三项是$2^0,2^1,2^2$, 依此类推.求满足如下条件的最小整数$N$($N>100$), 且该数列的前$N$项和为2的整数幂.那么该款软件的激活码是\\bracket{20}.\n\\fourch{$440$}{$330$}{$220$}{$110$}",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -81914,7 +82554,9 @@
"003246": {
"id": "003246",
"content": "已知由实数组成的数列$\\{a_n\\}$, 前$n$项和记为$S_n$, 若数列$\\{a_n\\}$为等比数列, $S_{100}=100S_{50}$, 求$\\dfrac{a_{100}}{a_{50}}$的值.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -81935,7 +82577,9 @@
"003247": {
"id": "003247",
"content": "已知数列$\\{c_n\\}$, 其中$c_n=2^n+3^n$, 是否存在实数$p$使得数列$\\{c_{n+1}-p{c_n}\\}$为等比数列, 若存在, 求出$p$; 若不存在, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -81956,7 +82600,9 @@
"003248": {
"id": "003248",
"content": "已知等比数列$\\{a_n\\}$中每一项均为实数, 设数列$\\{a_n\\}$的前$n$项和为$S_n$.\\\\\n(1) 证明: $(S_{2n}-S_n)^2=S_n(S_{3n}-S_{2n})$;\\\\\n(2) 试给出一个例子使得$S_n,S_{2n}-S_n,S_{3n}-S_{2n}$依次不构成等比数列;\\\\\n(3) 若$S_{10}=2$, $S_{30}=14$, 求$S_{20}$.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -81977,7 +82623,9 @@
"003249": {
"id": "003249",
"content": "等比数列$\\{a_n\\}$满足$a_1=2$, $a_2=1$, 则通项$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -82000,7 +82648,9 @@
"003250": {
"id": "003250",
"content": "若等比数列$\\{a_n\\}$的公比为$3$, 则等比数列$\\{a_n\\cdot a_{n+3}\\}$的公比为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -82021,7 +82671,9 @@
"003251": {
"id": "003251",
"content": "若实数$a$使得$a,a^2,a$依次构成等比数列, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -82042,7 +82694,9 @@
"003252": {
"id": "003252",
"content": "若数列$\\{a_n\\}$为等差数列, 则$a_9=4a_3-3a_1$. 类比以上结论有: 若数列$\\{b_n\\}$为等比数列, 则$b_9=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403006X"
+ ],
"tags": [
"第四单元"
],
@@ -82063,7 +82717,9 @@
"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}",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -82086,7 +82742,9 @@
"003254": {
"id": "003254",
"content": "设$p\\in \\mathbf{R}$, 已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=a_n^2-p$, 是否存在$p$使得$\\{a_n\\}$是等比数列? 若存在, 求出$p$的值; 若不存在, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -82107,7 +82765,10 @@
"003255": {
"id": "003255",
"content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$a_1=1$, $S_{n+1}=4a_n+2$.\\\\\n(1) 设$b_n=a_{n+1}-2a_n$, 证明数列$\\{b_n\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0403003X",
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -82128,7 +82789,9 @@
"003256": {
"id": "003256",
"content": "求和: $\\sin^21^\\circ +\\sin^22^\\circ+\\sin^23^\\circ+\\cdots+\\sin^288^\\circ+\\sin^289^\\circ=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82149,7 +82812,9 @@
"003257": {
"id": "003257",
"content": "设$f(x)=\\dfrac 1{3^x+\\sqrt 3}$, 利用课本中推导等差数列前$n$项和的公式的方法, 可求得$f(-5)+f(-4)+\\cdots+f(0)+\\cdots+f(5)+f(6)$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82170,7 +82835,9 @@
"003258": {
"id": "003258",
"content": "已知数列$\\{a_n\\}$的通项$a_n=1+2+2^2+\\cdots+2^n$, 则其前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82193,7 +82860,9 @@
"003259": {
"id": "003259",
"content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac 1{(2n-1)(2n+1)}$, 则其前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82217,7 +82886,9 @@
"003260": {
"id": "003260",
"content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac 3{n(n+3)}$, 则其前$n$项和$S_n$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82241,7 +82912,10 @@
"003261": {
"id": "003261",
"content": "等比数列$\\{a_n\\}$中前$n$项和为$S_n$, $n\\in \\mathbf{N}^*$, 若$S_n=48$, $S_{2n}=60,$则$S_{4n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X",
+ "K0404002X"
+ ],
"tags": [
"第四单元"
],
@@ -82262,7 +82936,9 @@
"003262": {
"id": "003262",
"content": "在等差数列$\\{a_n\\}$中, 满足$3a_4=7a_7$, 且$a_1>0$, $S_n$是数列$\\{a_n\\}$前$n$项的和, 若$S_n$取得最大值, 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -82283,7 +82959,9 @@
"003263": {
"id": "003263",
"content": "已知数列$\\{a_n\\}$的通项$a_n=n\\cdot 2^n$, 求其前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82304,7 +82982,9 @@
"003264": {
"id": "003264",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2-20n$, 求数列$\\{|a_n|\\}$的前$n$项和$T_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82325,7 +83005,9 @@
"003265": {
"id": "003265",
"content": "求数列$\\{\\dfrac{(n+1)^2+1}{(n+1)^2-1}\\}$的前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82346,7 +83028,9 @@
"003266": {
"id": "003266",
"content": "(1) 设$n$为正整数, 求和: $1-3+5-7+9+\\cdots +(-1)^{n-1}\\cdot (2n-1)$;\\\\\n(2) 已知数列$\\{a_n\\}$的通项$a_n=\\begin{cases}3n+1, & n\\text{为奇数}, \\\\ 2^{\\frac n2}, & n\\text{为偶数}, \\end{cases}$ 求其前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82367,7 +83051,9 @@
"003267": {
"id": "003267",
"content": "数列$\\{a_n\\}$的通项$a_n=2^n\\cdot 3^n$, 则其前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -82388,7 +83074,9 @@
"003268": {
"id": "003268",
"content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac 2{\\sqrt{n+2}+\\sqrt n}$, 则其前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82412,7 +83100,9 @@
"003269": {
"id": "003269",
"content": "等差数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_3=3$, $S_4=10$, 则数列$\\{S_n\\}$的前$n$项和为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -82433,7 +83123,9 @@
"003270": {
"id": "003270",
"content": "求数列$\\{\\dfrac n{2^n}\\}$的前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82454,7 +83146,9 @@
"003271": {
"id": "003271",
"content": "已知数列$\\{a_n\\}$满足$a_n=\\begin{cases}n, & n\n\\text{是奇数}, \\\\ 2^n, & n\\text{是偶数}. \\end{cases}$ 试求数列$\\{a_n\\}$的前$n$项和$S_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -82475,7 +83169,10 @@
"003272": {
"id": "003272",
"content": "如果有穷数列$a_1,a_2,a_3,\\cdots,a_m$($m$为正整数)满足条件$a_1=a_m$, $a_2=a_{m-1}$, $\\cdots$, $a_m=a_1$, 即$a_i=a_{m-i+1}$($i=1,2,\\cdots,m$), 我们称其为``对称数列''. 例如数列$1,2,5,2,1$与数列$8,4,2,2,4,8$都是``对称数列''.\\\\\n(1) 设$\\{c_n\\}$是$49$项的``对称数列'', 其中$c_{25},c_{26},\\cdots,c_{49}$是首项为$1$, 公比为$2$的等比数列, 求$\\{c_n\\}$各项的和$S$;\\\\\n(2) 设$\\{d_n\\}$是$100$项的``对称数列'', 其中$d_{51},d_{52}\\cdots,d_{100}$是首项为$2$, 公差为$3$的等差数列. 求$\\{d_n\\}$前$n$项的和$S_n$($n=1,2,\\cdots,100$).",
- "objs": [],
+ "objs": [
+ "K0404003X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -82496,7 +83193,9 @@
"003273": {
"id": "003273",
"content": "设数列$\\{a_n\\}$满足$a_1=0$且$\\dfrac 1{1-a_{n+1}}-\\dfrac 1{1-a_n}=1$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\dfrac{1-\\sqrt{a_{n+1}}}{\\sqrt n}$, 记$S_n={b_1}+{b_2}+\\cdots +b_n$, 求$\\{S_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -82517,7 +83216,9 @@
"003274": {
"id": "003274",
"content": "数学归纳法证明$1+a+a^2+\\cdots+a^{n+1}=\\dfrac{1-a^{n+2}}{1-a}\\ (a\\ne 1)$, 在验证$n=1$时, 左边计算所得项为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82538,7 +83239,9 @@
"003275": {
"id": "003275",
"content": "用数学归纳法证明``对于任意正偶数$n$, $a^n-b^n$能被$a+b$整除''时, 其第二步论证应该是\\bracket{20}.\n\\onech{假设$n=k$, $k\\in \\mathbf{N}^*$时命题成立, 证明$n=k+1$时, 命题也成立}{假设$n=2k$, $k\\in \\mathbf{N}^*$时命题成立, 证明$n=2k+1$时, 命题也成立}{假设$n=k$, $k\\in \\mathbf{N}^*$时命题成立, 证明$n=k+2$时, 命题也成立}{假设$n=2k$, $k\\in \\mathbf{N}^*$时命题成立, 证明$n=2k+2$时, 命题也成立}",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82559,7 +83262,9 @@
"003276": {
"id": "003276",
"content": "用数学归纳法证明: $1^2-2^2+3^2-4^2+\\cdots+(2n-1)^2-(2n)^2=-n(2n+1)$, $n$从$k$到$k+1$时, 等式左边增加的项为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82580,7 +83285,9 @@
"003277": {
"id": "003277",
"content": "根据$1=1$, $1-4=-(1+2)$, $1-4+9=1+2+3$, $1-4+9-16=-(1+2+3+4)$, $\\cdots$, 请写一个能体现其一般规律的数学表达式:\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82601,7 +83308,9 @@
"003278": {
"id": "003278",
"content": "设$f(x)$是定义在正整数集上的函数, 且$f(x)$满足: ``当$f(k)\\ge k^2$成立时, 总可推出$f(k+1)\\ge (k+1)^2$成立''. 那么, 下列说法中正确的是\\bracket{20}.\n\\onech{若$f(3)\\ge 9$成立, 则当$k\\ge 1$时, 均有$f(k)\\ge k^2$成立}{若$f(5)\\ge 25$成立, 则当$k\\le 5$时, 均有$f(k)\\ge k^2$成立}{若$f(7)<49$成立, 则当$k\\ge 8$时, 均有$f(k)1+\\dfrac n2\\ (n\\ge 2)$, $n$从$k$到$k+1$时, 不等式左边增加的项为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82773,7 +83496,9 @@
"003286": {
"id": "003286",
"content": "根据 $1=1$, $2+3+4=9$, $3+4+5+6+7=25$, $\\cdots$, 请写一个能体现其一般规律的数学表达式:\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -82794,7 +83519,9 @@
"003287": {
"id": "003287",
"content": "(1) 已知数列$\\{a_n\\}$满足$a_1=3$, $a_{n+1}=a_n^2-2\\ (n\\in \\mathbf{N}^*)$. 求证: 当$n\\in \\mathbf{N}^*$时, $a_n\\ge 3$;\\\\\n(2) *已知数列$\\{a_n\\}$满足$a_n\\ge 0$, $a_1=0$, $a_{n+1}^2+a_{n+1}-1=a_n^2 \\ (n\\in \\mathbf{N}^*)$. 求证: 当$n\\in \\mathbf{N}^*$时, $a_n=latex, line cap = round, line join = round]\n \\draw [dashed] (0,0) rectangle (4,4);\n \\draw (0,2) -- (2,4) -- (4,2) -- (2,0) -- cycle;\n \\draw (1,3) -- (3,3) -- (3,1) -- (1,1) -- cycle;\n \\draw (2,3) -- (3,2) -- (2,1) -- (1,2) -- cycle;\n \\draw (0,4) node [above left] {$A$} (4,4) node [above right] {$B$} (4,0) node [below right] {$C$} (0,0) node [below left] {$D$};\n \\draw (2,4) node [above] {$A_1$} (4,2) node [right] {$B_1$} (2,0) node [below] {$C_1$} (0,2) node [left] {$D_1$};\n \\draw (3,3) node [above right] {$A_2$} (3,1) node [below right] {$B_2$} (1,1) node [below left] {$C_2$} (1,3) node [above left] {$D_2$};\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -83006,7 +83751,9 @@
"003297": {
"id": "003297",
"content": "已知公比为$q(00)$的等比数列, 前$n$项和为$S_n$, 若$G_n=a_1^2+a_2^2+\\cdots+a_n^2$, 求$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{S_n}{G_n}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -83090,7 +83843,9 @@
"003301": {
"id": "003301",
"content": "设无穷等比数列$\\{a_n\\}$满足$\\displaystyle\\lim_{n\\to \\infty}(a_1+a_3+a_5+\\cdots +a_{2n-1})=\\dfrac 83$, 则首项$a_1$的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -83111,7 +83866,9 @@
"003302": {
"id": "003302",
"content": "(1) $\\displaystyle\\lim_{n\\to \\infty}\\dfrac{(-2)^n+1}{(-2)^{n+1}+1}$=\\blank{50};\\\\\n(2) $\\displaystyle\\lim_{n\\to \\infty}\\dfrac{6-2+4-8+\\cdots+(-2)^{n+1}}{4+3+9+27+\\cdots+3^n}$=\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -83132,7 +83889,9 @@
"003303": {
"id": "003303",
"content": "(1)若$\\displaystyle\\lim_{n\\to \\infty}(\\dfrac{n^3-1}{2n^2+n}-an-b)=0$, 则$a$=\\blank{50}, $b$=\\blank{50};\\\\\n(2) 若$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{5^n}{5^{n+1}+(a+1)^n}=\\dfrac 15$, 则实数$a$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -83153,7 +83912,9 @@
"003304": {
"id": "003304",
"content": "设$\\{a_n\\}$为无穷等比数列, 若$\\{a_n\\}$的任意一项都是它后面所有项和的$4$倍, 则公比为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405003X"
+ ],
"tags": [
"第四单元"
],
@@ -83174,7 +83935,9 @@
"003305": {
"id": "003305",
"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$2S_n0,\\ 0.60, \\ 0.70$, 求通项$a_n$;\n(2) (不需要理由)试写出所有可能的数列$\\{a_n\\}$的前三项.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -83602,7 +84403,9 @@
"003325": {
"id": "003325",
"content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$满足: $a_1=\\lambda$, $a_{n+1}=\\dfrac 23a_n+n-4$, $b_n=(-1)^n(a_n-3n+21)$, 其中$\\lambda$为实数.\\\\\n(1) 对任意实数$\\lambda$, 证明数列$\\{a_n\\}$不是等比数列;\\\\\n(2) *若数列$\\{b_n\\}$是等比数列, 求$\\lambda$的取值范围;\\\\\n(3) *若$a_n<3n$对一切$n\\in \\mathbf{N}^*$成立, 求$\\lambda$的取值范围.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -86998,7 +87801,10 @@
"003475": {
"id": "003475",
"content": "若一个圆柱的侧面展开图是一个正方形, 则这个圆柱的表面积与侧面积的比是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0617002B",
+ "K0617006B"
+ ],
"tags": [
"第六单元"
],
@@ -87084,7 +87890,9 @@
"003479": {
"id": "003479",
"content": "取地球的半径为$6370$千米, 在北纬$45^\\circ$线上, 求相隔$30^\\circ$的两条经线之间的球面距离(精确到$0.1$千米).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -87233,7 +88041,9 @@
"003486": {
"id": "003486",
"content": "在四棱锥$A-BCDE$中, 已知$AD\\perp\\text{底面}BCDE$, $AC\\perp BC$, $AE\\perp BE$, 若$\\angle CBE=90^\\circ$, $CE=\\sqrt 3$, $AD=1$, 求$B,D$两点在棱锥$A-BCDE$外接球表面的球面距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 3]\n \\draw (0,0) node [below left] {$E$} -- (1,0) node [below right] {$B$} --++ (45:{sqrt(2)/2}) node [right] {$C$} coordinate (C);\n \\draw [dashed] (C) --++ (-1,0) node [left] {$D$} coordinate (D) -- (0,0) (D) --++ (0,1) node [above] {$A$} coordinate (A);\n \\draw (A) -- (0,0) (A) -- (C) (A) -- (1,0);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -88831,7 +89641,9 @@
"003554": {
"id": "003554",
"content": "若矩阵$\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix}$为$n$阶单位阵, 则$a-b-c+d+n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -88852,7 +89664,9 @@
"003555": {
"id": "003555",
"content": "方程组$\\begin{cases} 2x+y=7, \\\\ x-y=2 \\end{cases}$的系数行列式的值为\\blank{50}, 系数矩阵的行向量为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -88873,7 +89687,9 @@
"003556": {
"id": "003556",
"content": "行列式$\\begin{vmatrix}1 & 2 & 0 \\\\ 1 & 5 & 1 \\\\ 1 & 8 & 0 \\end{vmatrix}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -88896,7 +89712,9 @@
"003557": {
"id": "003557",
"content": "在三阶行列式$\\begin{vmatrix}1 & 2 & 3 \\\\4 & 5 & 6 \\\\7 & 8 & -9 \\end{vmatrix}$中, 元素$6$的余子式为\\blank{50}, 元素$8$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -88964,7 +89782,9 @@
"003560": {
"id": "003560",
"content": "已知关于$x,y,z$的方程组$\\begin{cases} x+y+z=1, \\\\ x+y+az=1, \\\\ x+ay+a^2z=2, \\end{cases}$其中$a\\in \\mathbf{R}$.\\\\\n(1) 若关于$x$、$y$的方程组$\\begin{cases} a_1x+b_1y=c_1, \\\\ a_2x+b_2y=c_2 \\end{cases}$可以用矩阵记号$\\begin{pmatrix}\n a_1 & b_1 \\\\ a_2 & b_2 \\end{pmatrix}\\begin{pmatrix} x \\\\ y \\end{pmatrix}=\\begin{pmatrix} c_1 \\\\ c_2 \\end{pmatrix}$来表示, 请试给出上述三元一次方程组的矩阵表示;\\\\\n(2) 用行列式的方法解此方程组.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -89006,7 +89826,9 @@
"003562": {
"id": "003562",
"content": "行列式$\\begin{vmatrix} 1 & 3 & 0 \\\\ 4 & 5 & 1 \\\\ 7 & 8 & 2 \\end{vmatrix}$中, 元素$4$的余子式的值为\\blank{50}, 元素$3$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -89029,7 +89851,9 @@
"003563": {
"id": "003563",
"content": "将 $a\\begin{vmatrix} 1 & 2 \\\\0 & 4 \\end{vmatrix}+b \\begin{vmatrix}\n-1 & 3 \\\\ 0 & 4 \\end{vmatrix}+c \\begin{vmatrix}\n-1 & 3 \\\\ 1 & 2 \\end{vmatrix}$化为一个三阶行列式: $\\begin{vmatrix}a & -1 & 3 \\\\ \\blank{10} & \\blank{10} & \\blank{10} \\\\ \\blank{10} & \\blank{10} & \\blank{10}\\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -89073,7 +89897,9 @@
"003565": {
"id": "003565",
"content": "关于$x,y,z$的方程组$\\begin{cases} x+y+az=1, \\\\ x+ay+z=a, \\\\ x-y+z=3 \\end{cases}$的增广矩阵是\\blank{50}; 若此方程组有唯一解, 则实数a的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -89094,7 +89920,9 @@
"003566": {
"id": "003566",
"content": "设$m$是实数, 用行列式的方法解关于$x$、$y$的方程组$\\begin{cases} (m+1)x-(2m-1)y=3m, \\\\ (3m+1)x-(4m-1)y=5m+4. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -89745,7 +90573,9 @@
"003596": {
"id": "003596",
"content": "已知无穷等比数列$\\{a_n\\}$和$\\{b_n\\}$, 满足$a_1=3$, $b_n=a_{2n}$, $a_n$的各项和为$9$, 则数列$\\{b_n\\}$的各项和为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -89833,7 +90663,9 @@
"003600": {
"id": "003600",
"content": "已知$a_i\\in \\mathbf{N}^* \\ (i=1,2,\\cdots,9)$, 若对任意的$k\\in \\mathbf{N}^* \\ (2\\le k\\le 8)$, $a_k=a_{k-1}+1$或$a_k=a_{k+1}-1$中有且仅有一个成立, 且$a_1=6$, $a_9=9$, 则$a_1+a_2+\\cdots+a_9$的最小值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0407004X"
+ ],
"tags": [
"第四单元"
],
@@ -90220,7 +91052,9 @@
"003615": {
"id": "003615",
"content": "已知行列式$\\begin{vmatrix}1 & a & b \\\\ 2 & c & d \\\\ 3 & 0 & 0 \\end{vmatrix}=6$, 则行列式$\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -90241,7 +91075,9 @@
"003616": {
"id": "003616",
"content": "已知等差数列$\\{a_n\\}$的首项$a_1\\ne 0$, 且满足$a_1+a_{10}=a_9$, 则$\\dfrac{a_1+a_2+\\cdots+a_9}{a_{10}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401003X"
+ ],
"tags": [
"第四单元"
],
@@ -90487,7 +91323,7 @@
},
"003627": {
"id": "003627",
- "content": "已知$f(x)=\\sin\\omega x$($\\omega>0$).\\\\\n(1) $f(x)$的周期是$4\\pi$, 求$\\omega$, 并求此时$f(x)=\\dfrac12$的解集;\\\\\n(2) 已知$\\omega=1$, $g(x)=f^2(x)+\\sqrt3f(-x)f(\\dfrac{\\pi}2-x)$, $x\\in [0,\\dfrac{\\pi}4]$, 求$g(x)$的值域.",
+ "content": "已知$f(x)=\\sin\\omega x$($\\omega>0$).\\\\\n(1) $f(x)$的最小正周期是$4\\pi$, 求$\\omega$, 并求此时$f(x)=\\dfrac12$的解集;\\\\\n(2) 已知$\\omega=1$, $g(x)=f^2(x)+\\sqrt3f(-x)f(\\dfrac{\\pi}2-x)$, $x\\in [0,\\dfrac{\\pi}4]$, 求$g(x)$的值域.",
"objs": [],
"tags": [
"第三单元"
@@ -90554,7 +91390,10 @@
"003630": {
"id": "003630",
"content": "已知有限数列$\\{a_n\\}$, 若满足$|a_1-a_2|\\le |a_1-a_3|\\le \\cdots \\le |a_1-a_m|$, $m$是项数, 则称$\\{a_n\\}$满足性质$P$.\\\\\n(1) 判断数列$3,2,5,1$和$4,3,2,5,1$是否具有性质$P$, 请说明理由;\\\\\n(2) 若首项$a_1=1$, 公比为$q$的等比数列, 项数为$10$, 具有性质$P$, 求$q$的取值范围;\\\\\n(3) 若$\\{a_n\\}$是$1,2,\\cdots,m$的一个排列($m\\ge 4$), $\\{b_n\\}$符合$b_k=a_{k+1}$($k=1,2,\\cdots,m-1$), $\\{a_n\\}$, $\\{b_n\\}$都具有性质$P$, 求所有满足条件的$\\{a_n\\}$.",
- "objs": [],
+ "objs": [
+ "K0403002X",
+ "K0406004X"
+ ],
"tags": [
"第四单元"
],
@@ -90742,7 +91581,9 @@
"003638": {
"id": "003638",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+a_n=2$, 则$S_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -91056,7 +91897,9 @@
"003652": {
"id": "003652",
"content": "行列式$\\begin{vmatrix}\n4 & 1 \\\\ 2 & 5\n\\end{vmatrix}$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -91858,7 +92701,9 @@
"003685": {
"id": "003685",
"content": "关于$x$、$y$的二元一次方程组$\\begin{cases}\nx+5y=0, \\\\ 2x+3y=4\\end{cases}$的系数行列式$D$为\\bracket{15}.\n\\fourch{$\\begin{vmatrix}\n0 & 5 \\\\ 4 & 3\n\\end{vmatrix}$}{$\\begin{vmatrix}\n1 & 0 \\\\ 2 & 4\n\\end{vmatrix}$}{$\\begin{vmatrix}\n1 & 5 \\\\ 2 & 3\n\\end{vmatrix}$}{$\\begin{vmatrix}\n6 & 0 \\\\ 5 & 4\n\\end{vmatrix}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -92588,7 +93433,9 @@
"003717": {
"id": "003717",
"content": "设地球半径为$R$, 甲地位于北纬$45^\\circ$东经$105^\\circ$, 乙地位于南纬$30^\\circ$东经$105^\\circ$, 则甲乙两地间的球面距离是\\bracket{20}.\n\\fourch{$\\dfrac{5\\pi}{12}R$}{$\\dfrac{7\\pi}{12}R$}{$\\dfrac{\\sqrt{2}}2R$}{$\\dfrac{\\sqrt{3}}2R$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -94276,7 +95123,9 @@
"003792": {
"id": "003792",
"content": "若关于$x,y$的二元线性方程组的增广矩阵为$\\begin{pmatrix}1 & 3 & 5\\\\2 & 4 & 6\\end{pmatrix}$, 则$x-y=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -94955,7 +95804,9 @@
"003823": {
"id": "003823",
"content": "已知关于$x,y$的二元一次方程组的增广矩阵为$\\begin{pmatrix}2 & 3 & 1 \\\\1 & 1 & 2\\end{pmatrix}$, 则$D_x=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -96597,7 +97448,9 @@
"003898": {
"id": "003898",
"content": "已知球$O$的半径为$4$, $A,B$是球面上两点, $\\angle AOB=45^\\circ$, 则$A,B$两点的球面距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -98461,7 +99314,9 @@
"003982": {
"id": "003982",
"content": "设$M$是球$O$半径$OP$的中点, 分别过$M,O$作垂直于$OP$的平面, 截球面得两个圆, 则这两个圆的面积比值为\\bracket{20}.\n\\fourch{$\\dfrac 14$}{$\\dfrac 12$}{$\\dfrac 23$}{$\\dfrac 34$}",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -98503,7 +99358,10 @@
"003984": {
"id": "003984",
"content": "把边长为$a$的正方形减去图中的阴影部分, 沿图中所画折线折成一个正三棱锥, 求这个正三棱锥的高.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t\\coordinate (P) at ({3/2},{3*sqrt(3)/2});\n\t\\coordinate (Q) at ({3-3*sqrt(3)/2},{3/2});\n\t\\filldraw [gray] (0,0)--(Q)--(0,3)--cycle;\n\t\\filldraw [gray] (0,3)--(P)--(3,3)--cycle;\n\t\\draw (0,0) rectangle (3,3);\n\t\n\t\\draw (0,0)--(Q)--(P)--(3,3);\n\t\\draw (0,3)--(Q)--(3,0)--(P)--cycle;\n\t\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0618002B",
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -98589,7 +99447,9 @@
"003988": {
"id": "003988",
"content": "已知正三棱锥的侧棱长是底面边长的$2$倍, 则侧棱与底面所成角的余弦值等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -98610,7 +99470,9 @@
"003989": {
"id": "003989",
"content": "正三棱锥$P-ABC$的高为$2$, 侧棱与底面$ABC$成$45^\\circ$角, 则点$A$到侧面$PBC$的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -98759,7 +99621,9 @@
"003996": {
"id": "003996",
"content": "过球面上两点作球的大圆, 可能的个数是\\bracket{20}.\n\\fourch{有且只有一个}{一个或无穷多个}{无数个}{以上结论都不正确}",
- "objs": [],
+ "objs": [
+ "K0622004B"
+ ],
"tags": [
"第六单元"
],
@@ -100310,7 +101174,9 @@
"004061": {
"id": "004061",
"content": "如果一个圆柱的高不变, 要使它的体积扩大为原来的$5$倍, 那么它的底面半径应该扩大为原来的\\blank{50}倍.",
- "objs": [],
+ "objs": [
+ "K0616003B"
+ ],
"tags": [
"第六单元"
],
@@ -100356,7 +101222,9 @@
"004063": {
"id": "004063",
"content": "把三阶行列式$\\begin{vmatrix} 2^x & 0 & 3 \\\\x & 4 & 0 \\\\1 & x-3 & -1 \\end{vmatrix}$中第$1$行第$3$列元素的代数余子式记为$f(x)$, 则关于$x$的不等式$f(x)<0$的解集为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -100889,7 +101757,10 @@
"004084": {
"id": "004084",
"content": "若圆柱的侧面展开图是边长为$4$的正方形, 则圆柱的体积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0617002B"
+ ],
"tags": [
"第六单元"
],
@@ -100991,7 +101862,9 @@
"004088": {
"id": "004088",
"content": "如图, 长方体$ABCD-A_1B_1C_1D_1$的边长$AB=AA_1=1$, $AD=\\sqrt 2$, 它的外接球是球$O$, 则$A$、$A_1$这两点的球面距离等于\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (30:{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) --++ (210:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n \\draw (A) ++ (2,2) node [above] {$B_1$} coordinate (B1) -- (B) (B1) --++ (30:{2/2}) (B1) --++ (-2,0);\n \\draw [dashed] (A) --++ (30:{2/2}) node [below] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n \\draw [dashed] (A) -- (C1) (C) -- (A1);\n \\draw ($(A)!0.5!(C1)$) node [below] {$O$};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -101375,7 +102248,9 @@
"004103": {
"id": "004103",
"content": "在行列式$D=\\begin{vmatrix}1&3&7\\\\2&5&-2\\\\1&2&3\\end{vmatrix}$中, 第二行第三列的元素$3$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -102166,7 +103041,9 @@
"004134": {
"id": "004134",
"content": "行列式$\\begin{vmatrix}1 & 2 \\\\ 3 & 4\\end{vmatrix}=$\\bracket{20}.\n\\fourch{$-4$}{$-2$}{$2$}{$4$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -103124,7 +104001,9 @@
"004171": {
"id": "004171",
"content": "已知正四棱柱$ABCD-A_1B_1C_1D_1$的八个顶点都在同一球面上, 若$AB=1$, $AA_1=\\sqrt 2$, 则$A$、$C$两点间的球面距离是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -103523,7 +104402,9 @@
"004187": {
"id": "004187",
"content": "计算行列式的值, $\\begin{vmatrix}0 & 1 \\\\2 & 3 \\end{vmatrix}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -103732,7 +104613,10 @@
"004196": {
"id": "004196",
"content": "课本中介绍了应用祖暅原理推导棱锥体积公式的做法. 祖暅原理也可用来求旋转体的体积. 现介绍用祖暅原理求球体体积公式的做法: 可构造一个底面半径和高都与球半径相等的圆柱, 然后在圆柱内挖去一个以圆柱下底面圆心为顶点, 圆柱上底面为底面的圆锥, 用这样一个几何体与半球应用祖暅原理(左图), 即可求得球的体积公式. 请研究和理解球的体积公式求法的基础上, 解答以下问题: 已知椭圆的标准方程为$\\dfrac{x^2}4+\\dfrac{y^2}{25}=1$, 将此椭圆绕$y$轴旋转一周后, 得一橄榄状的几何体(右图), 其体积等于\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) arc (180:0:2) arc (0:-180:2 and 0.5);\n \\draw [dashed] (0,0) arc (180:0:2 and 0.5) -- (0,0);\n \\fill [color = gray!30] (2,1) ellipse ({sqrt(3)} and {sqrt(3)/4});\n \\draw ({2-sqrt(3)},{1}) arc (180:360:{sqrt(3)} and {sqrt(3)/4});\n \\draw [dashed] ({2-sqrt(3)},{1}) arc (180:0:{sqrt(3)} and {sqrt(3)/4});\n \\draw [dashed] (2,0) -- (2,1) (2,0.2) node [left] {$h$};\n \\draw [dashed] (2,0) -- ({2+sqrt(3)},1) (3,0) node [below] {$R$};\n \\filldraw [even odd rule, gray!30] (7,1) ellipse (2 and 0.5) (7,1) ellipse (1 and 0.25);\n \\draw (5,0) arc (180:360:2 and 0.5) (5,2) arc (180:-180:2 and 0.5) (5,0) -- (5,2) (9,0) -- (9,2);\n \\draw [dashed] (5,0) -- (9,0) (7,0) -- (7,1) (7,0) -- (5,2) (7,0) -- (9,2) (8,0) node [below] {$R$} (7,0.4) node [left] {$h$};\n \\draw (5,1) arc (180:360:2 and 0.5);\n \\draw [dashed] (5,1) arc (180:0:2 and 0.5) (6,1) arc (180:-180:1 and 0.25);\n \\end{tikzpicture}\n \\begin{tikzpicture}{>=latex}\n \\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (0,0) ellipse (1 and 1.5);\n \\draw [dashed] (0,0) ellipse (0.5 and 1.5);\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0616001B",
+ "K0623001B"
+ ],
"tags": [
"第六单元",
"第七单元"
@@ -104566,7 +105450,9 @@
"004230": {
"id": "004230",
"content": "如果圆锥的底面积为$\\pi$, 母线长为$2$, 那么该圆锥的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618005B"
+ ],
"tags": [
"第六单元"
],
@@ -104781,7 +105667,9 @@
"004239": {
"id": "004239",
"content": "关于$x$、$y$的二元一次方程组$\\begin{cases} 3x+4y=1,\\\\ x-3y=10 \\end{cases}$的增广矩阵为\\bracket{20}.\n\\fourch{$\\begin{pmatrix}3&4&-1\\\\1&-3&10\\end{pmatrix}$}{$\\begin{pmatrix}3&4&-1\\\\1&-3&-10\\end{pmatrix}$}{$\\begin{pmatrix}3&4&1\\\\1&-3&10\\end{pmatrix}$}{$\\begin{pmatrix}3&4&1\\\\1&3&10\\end{pmatrix}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -106220,7 +107108,9 @@
"004296": {
"id": "004296",
"content": "若某线性方程组对应的增广矩阵是$\\begin{pmatrix} m & 4 & 2 \\\\1 & m & m \\end{pmatrix}$, 且此方程组有唯一的一组解, 则实数m的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -106414,7 +107304,9 @@
"004304": {
"id": "004304",
"content": "如图几何体是由五个相同正方体叠成的, 其三视图中的左视图序号是\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\filldraw [gray!50] (2,0) --++ (45:0.5) coordinate (T) --++ (0,1) --++ (225:0.5) -- cycle;\n \\filldraw [gray!50] (T) --++ (45:0.5) --++ (0,1) --++ (225:0.5) -- cycle;\n \\draw (1,0) ++ (0,1) ++ (45:0.5) coordinate (S);\n \\filldraw [gray!50] (S) --++ (45:0.5) --++ (0,1) --++ (225:0.5) -- cycle;\n \\draw (T) --++ (45:0.5) --++ (0,1) --++ (225:0.5) -- cycle;\n \\draw (T) --++ (225:0.5) --++ (0,1) --++ (45:0.5);\n \\draw (S) --++ (45:0.5) --++ (0,1) --++ (225:0.5) -- cycle;\n \\draw (0,0) -- (2,0) (0,0) -- (0,1) (1,0) -- (1,1) (0,1) -- (2,1) (1,1) --++ (45:0.5) (0,1) --++ (45:0.5) coordinate (P) --++ (2,0);\n \\draw (P) --++ (0,1) --++ (1,0) (P) ++ (0,1) --++ (45:0.5) --++ (1,0) (S) ++ (45:0.5) --++ (1,0);\n \\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}\n \\draw (0,0) rectangle (2,1) (1,0) -- (1,2) -- (0,2) -- (0,1);\n\\end{tikzpicture}}{\\begin{tikzpicture}\n \\draw (0,0) rectangle (2,1) (1,0) -- (1,2) -- (2,2) -- (2,1);\n\\end{tikzpicture}}{\\begin{tikzpicture}\n \\draw (0,0) rectangle (1,2) (0,1) -- (2,1) -- (2,0) -- (1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}\n \\draw (0,0) rectangle (2,2) (1,0) -- (1,2) (0,1) -- (2,1);\n\\end{tikzpicture}}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -111490,7 +112382,9 @@
"004501": {
"id": "004501",
"content": "展开式为$ad-bc$的行列式是\\bracket{20}.\n\\fourch{$\\begin{vmatrix}a & b \\\\ d & c \\end{vmatrix}$}{$\\begin{vmatrix} a & c \\\\ b & d \\end{vmatrix}$}{$\\begin{vmatrix}\n a & d \\\\ b & c \\end{vmatrix}$}{$\\begin{vmatrix} b & a \\\\ d & c \\end{vmatrix}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -111734,7 +112628,9 @@
"004511": {
"id": "004511",
"content": "若关于$x,y$的方程组为$\\begin{cases} x+y=1, \\\\ x-y=2, \\end{cases}$ 则该方程组的增广矩阵为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -115105,7 +116001,9 @@
"004644": {
"id": "004644",
"content": "已知线性方程组的增广矩阵为$\\begin{pmatrix}\n2 & 0 & m \\\\ 1 & n & 2 \\end{pmatrix}$, 解为$\\begin{cases}\nx=1, \\\\y=1, \\end{cases}$ 则$m+n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -115737,7 +116635,9 @@
"004669": {
"id": "004669",
"content": "已知圆锥的底面半径为$1$, 其侧面展开图为一个半圆, 则该圆锥的母线长为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0620002B"
+ ],
"tags": [
"第六单元"
],
@@ -116373,7 +117273,9 @@
"004694": {
"id": "004694",
"content": "关于$x$、$y$的二元一次方程组$\\begin{cases}x+2y=3, \\\\ 3x+4y=-1 \\end{cases}$的增广矩阵为\\bracket{20}.\n\\fourch{$\\begin{pmatrix}1 & 2 \\\\3 & 4 \\end{pmatrix}$}{$\\begin{vmatrix} 1 & 2 \\\\3 & 4 \\end{vmatrix}$}{$\\begin{pmatrix} 1 & 2 & -3 \\\\ 3 & 4 & 1\\end{pmatrix}$}{$\\begin{pmatrix}1 & 2 & 3 \\\\ 3 & 4 & -1\\end{pmatrix}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -116694,7 +117596,9 @@
"004707": {
"id": "004707",
"content": "已知$\\sin x=\\dfrac{3}{5}$, $x\\in (\\dfrac \\pi 2,\\pi)$, 则行列式$\\begin{vmatrix} \\sin x & -1 \\\\ 1 & \\sec x \\end{vmatrix}$的值等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第三单元"
],
@@ -116816,7 +117720,9 @@
"004712": {
"id": "004712",
"content": "三阶矩阵$\\begin{pmatrix}\n a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33}\n\\end{pmatrix}$中有$9$个不同的数$a_{ij}$($i=1,2,3$, $j=1,2,3$), 从中任取三个, 则至少有两个数位于同行或同列的概率是\\blank{50}(结果用分数表示).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第八单元"
],
@@ -117262,7 +118168,9 @@
"004730": {
"id": "004730",
"content": "设$a,b,c,d\\in \\mathbf{R}$, 若行列式$\\begin{vmatrix} a & b & 1 \\\\ c & d & 2 \\\\ 0 & 0 & 3 \\end{vmatrix}=9$, 则行列式$\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix}$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -117800,7 +118708,9 @@
"004752": {
"id": "004752",
"content": "三棱锥$P-ABC$中, 底面$ABC$是锐角三角形, $PC$垂直平面$ABC$, 若其三视图中主视图和左视图如图所示, 则棱$PB$的长为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.7,>=latex]\n \\draw (0,0) node [right] {$C$} coordinate (C);\n \\draw (-4,0) node [left] {$A$} coordinate (A);\n \\draw (-2,0) ++ (-135:{sqrt(3)}) node [below] {$B$} coordinate (B);\n \\draw (A) -- (B) -- (C) --++ (0,4) node [above] {$P$} coordinate (P);\n \\draw (A) -- (P) -- (B);\n \\draw [dashed] (A) -- (C);\n \\draw (2,0) -- (6,0) -- (6,4) -- cycle;\n \\draw (4,0) -- (6,4);\n \\foreach \\i in {2,4,6} {\\draw (\\i,-0.1) -- (\\i,-0.5);};\n \\draw [->] (2.6,-0.3) -- (2,-0.3);\n \\draw [->] (3.4,-0.3) -- (4,-0.3);\n \\draw [->] (4.6,-0.3) -- (4,-0.3);\n \\draw [->] (5.4,-0.3) -- (6,-0.3);\n \\draw (3,-0.3) node {$2$} (5,-0.3) node {$2$};\n \\draw (4,-1.3) node {主视图};\n \\draw (8,0) --++ ({2*sqrt(3)},0) -- (8,4) -- cycle;\n \\draw (8,-0.1) -- (8,-0.5) ({8+2*sqrt(3)},-0.1) --++ (0,-0.4) (7.9,0) --++ (-0.4,0) (7.9,4) --++ (-0.4,0);\n \\draw [->] (7.7,1.6) -- (7.7,0);\n \\draw [->] (7.7,2.4) -- (7.7,4);\n \\draw [->] ({8+sqrt(3)-1},-0.3) -- (8,-0.3);\n \\draw [->] ({8+sqrt(3)+1},-0.3) -- ({8+2*sqrt(3)},-0.3);\n \\draw ({8+sqrt(3)},-0.3) node {$2\\sqrt{3}$};\n \\draw (7.7,2) node {$4$};\n \\draw (10,-1.3) node {左视图};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -123597,7 +124507,9 @@
"id": "004994",
"content": "已知圆柱的体积为定值$V$, 求圆柱全面积的最小值.",
"objs": [
- "K0119001B"
+ "K0119001B",
+ "K0616003B",
+ "K0617006B"
],
"tags": [
"第一单元",
@@ -123622,7 +124534,8 @@
"content": "从半径为$R$的圆形铁片里剪去一个扇形, 然后把剩下部分卷成一个圆锥形漏斗, 要使漏斗有最大容量, 剪去扇形的圆心角$\\theta$应是多少弧度?",
"objs": [
"K0119001B",
- "K0619003B"
+ "K0619003B",
+ "K0620002B"
],
"tags": [
"第一单元",
@@ -127022,7 +127935,8 @@
"K0119002B",
"K0619003B",
"K0222002B",
- "K0221002B"
+ "K0221002B",
+ "K0616004B"
],
"tags": [
"第一单元",
@@ -130037,7 +130951,8 @@
"id": "005260",
"content": "已知三棱锥的三条侧棱两两互相垂直, 且六条棱之和为定值$m$, 求证: 它的体积$V\\le \\dfrac{5\\sqrt 2-7}{162}m^3$.",
"objs": [
- "KNONE"
+ "KNONE",
+ "K0619003B"
],
"tags": [
"第一单元",
@@ -162213,7 +163128,9 @@
"006666": {
"id": "006666",
"content": "若$m,a_1,a_2,n$和$m,b_1,b_2,n$($m\\ne n$)分别是两个等差数列, 则$\\dfrac{{a_2}-{a_1}}{{b_2}-{b_1}}$的值为\\bracket{20}.\n\\fourch{$\\dfrac 23$}{$\\dfrac 34$}{$\\dfrac 32$}{$\\dfrac 43$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162234,7 +163151,9 @@
"006667": {
"id": "006667",
"content": "若等差数列$\\{a_n\\}$的前三项依次为$a-1,a+1,2a+3$, 则此数列的通项$a_n$等于\\bracket{20}.\n\\fourch{$2n-5$}{$2n-3$}{$2n-1$}{$2n+1$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162255,7 +163174,9 @@
"006668": {
"id": "006668",
"content": "在等差数列$\\{a_n\\}$中, 若$a_3+a_4+a_5+a_6+a_7=450$, 则$a_2+a_8$等于\\bracket{20}.\n\\fourch{$45$}{$75$}{$180$}{$320$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162276,7 +163197,9 @@
"006669": {
"id": "006669",
"content": "在等差数列$\\{a_n\\}$中, 已知$a_1+a_4+a_7=39$, $a_2+a_5+a_8=33$, 则$a_3+a_6+a_9$的值是\\bracket{20}.\n\\fourch{$30$}{$27$}{$24$}{$21$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162297,7 +163220,9 @@
"006670": {
"id": "006670",
"content": "在递增的等差数列$\\{a_n\\}$中, 已知$a_3+a_6+a_9=12$, $a_3a_6a_9=28$, 则通项$a_n$等于\\bracket{20}.\n\\fourch{$n-2$}{$16-n$}{$n-2$或$16-n$}{$2-n$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162318,7 +163243,9 @@
"006671": {
"id": "006671",
"content": "若等差数列$\\{a_n\\}$的公差$d$不为零, 且$a_1\\ne d$, 前$20$项之和$S_{20}=10M$, 则$M$等于\\bracket{20}.\n\\fourch{$a_6+a_5$}{$a_2+2a_{10}$}{$2a_{10}+d$}{$10a_2+d$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162339,7 +163266,9 @@
"006672": {
"id": "006672",
"content": "在等差数列$\\{a_n\\}$中, 已知前$4$项和是$1$, 前$8$项和是$4$, 则$a_{17}+a_{18}+a_{19}+a_{20}$的值等于\\bracket{20}.\n\\fourch{$7$}{$8$}{$9$}{$10$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162360,7 +163289,9 @@
"006673": {
"id": "006673",
"content": "在等差数列$\\{a_n\\}$中, 若前$15$项的和$S_{15}=90$, 则$a_8$等于\\bracket{20}.\n\\fourch{$6$}{$\\dfrac{45}4$}{$12$}{$\\dfrac{45}2$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162381,7 +163312,9 @@
"006674": {
"id": "006674",
"content": "若数列$\\{a_n\\}$满足$a_{n+1}=\\dfrac{3a_n+2}3$, 且$a_1=0$, 则$a_7=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162402,7 +163335,9 @@
"006675": {
"id": "006675",
"content": "若等差数列$\\{a_n\\}$满足$a_7=p$, $a_{14}=q$($p\\ne q$), 则$a_{21}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162423,7 +163358,9 @@
"006676": {
"id": "006676",
"content": "首项为$-24$的等差数列从第$10$项开始为正数, 则公差$d$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162444,7 +163381,9 @@
"006677": {
"id": "006677",
"content": "若等差数列$\\{a_n\\}$的公差$d\\ne 0$, 且$a_1,a_2$为关于$x$的方程$x^2-a_3x+a_4=0$的两根, 则$\\{a_n\\}$的通项公式$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162465,7 +163404,9 @@
"006678": {
"id": "006678",
"content": "若$a,x,b,2x$依次成等差数列, 则$a:b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162486,7 +163427,9 @@
"006679": {
"id": "006679",
"content": "若$a,b,\\lg 6,2\\lg 2+\\lg 3$依次成等差数列, 则$a=$\\blank{50}, $b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162507,7 +163450,9 @@
"006680": {
"id": "006680",
"content": "等差数列$\\{a_n\\}$中, 若$a_1+a_3+a_5=-1$, 则$a_1+a_2+a_3+a_4+a_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162528,7 +163473,9 @@
"006681": {
"id": "006681",
"content": "等差数列$\\{a_n\\}$中, 若$a_3+a_{11}=10$, 则$a_2+a_4+a_{15}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162549,7 +163496,9 @@
"006682": {
"id": "006682",
"content": "等差数列$\\{a_n\\}$中, 若$a_2+a_3+a_4+a_5=34$, $a_2a_5=52$, 且$a_4>a_2$, 则$a_5$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162570,7 +163519,9 @@
"006683": {
"id": "006683",
"content": "等差数列$\\{a_n\\}$中, 若$a_1-a_4-a_8-a_{12}+a_{15}=2$, 则$a_3+a_{13}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162591,7 +163542,9 @@
"006684": {
"id": "006684",
"content": "等差数列$\\{a_n\\}$中, 若$a_1+a_2+a_3+a_4+a_5=30$, $a_6+a_7+a_8+a_9+a_{10}=80$, 则$a_{11}+a_{12}+a_{13}+a_{14}+a_{15}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162612,7 +163565,9 @@
"006685": {
"id": "006685",
"content": "等差数列$\\{a_n\\}$中, 若$a_2+a_7+a_{12}=21$, 则前$13$项和$S_{13}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162633,7 +163588,9 @@
"006686": {
"id": "006686",
"content": "等差数列$\\{a_n\\}$中, 若前$10$项和$S_{10}=100$, 前$20$项和$S_{20}=400$, 则前$30$项和$S_{30}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162654,7 +163611,9 @@
"006687": {
"id": "006687",
"content": "等差数列$\\{a_n\\}$中, 若$a_{11}=20$, 则前$21$项和$S_{21}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162675,7 +163634,9 @@
"006688": {
"id": "006688",
"content": "若一个等差数列的前$10$项和是前$5$项和的$4$倍, 则其首项与公差之比等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162696,7 +163657,9 @@
"006689": {
"id": "006689",
"content": "等差数列$\\{a_n\\}$中, 若前$100$项之和等于前$10$项和的$100$倍, 则$\\dfrac{a_{100}}{a_{10}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162717,7 +163680,9 @@
"006690": {
"id": "006690",
"content": "若$100$个连续整数之和在$13400$与$13500$之间, 则此连续整数中最小的一个等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162738,7 +163703,9 @@
"006691": {
"id": "006691",
"content": "在等差数列$\\{a_n\\}$中, 已知$a_m=p$, $a_n=q$($m\\ne n$), 求$a_{m+n}$.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162759,7 +163726,9 @@
"006692": {
"id": "006692",
"content": "若$\\{a_n\\}$是等差数列, 数列$\\{b_n\\}$满足: $b_n=(\\dfrac 12)^{a_n}$, $b_1+b_2+b_3=\\dfrac{21}8$, $b_1b_2b_3=\\dfrac 18$, 求通项公式$a_n$.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162780,7 +163749,9 @@
"006693": {
"id": "006693",
"content": "已知等差数列的第$1$项和第$4$项之和为$10$, 且第$2$项减去第$3$项的差为$2$, 求此数列的前$n$项之和.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162801,7 +163772,9 @@
"006694": {
"id": "006694",
"content": "求所有能被$7$整除且被$11$除余$2$的三位数之和.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162822,7 +163795,9 @@
"006695": {
"id": "006695",
"content": "首项$a_1\\ne 0$的等差数列$\\{a_n\\}$中, 已知前$9$项和与前$4$项和之比$S_9:S_4=81:16$, 求$a_9:a_4$的值.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162843,7 +163818,9 @@
"006696": {
"id": "006696",
"content": "在等差数列$\\{a_n\\}$中, 已知公差$d=1$, 前$98$项和$S_{98}=137$, 求$a_2+a_4+a_6+a_8+\\cdots +a_{94}+a_{96}+a_{98}$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162864,7 +163841,9 @@
"006697": {
"id": "006697",
"content": "若等差数列$\\{a_n\\}$满足$a_1+a_3+a_5+a_7+a_9=\\dfrac{25}2$, $a_2+a_4+a_6+a_8+a_{10}=15$, 求前$20$项之和$S_{20}$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162885,7 +163864,9 @@
"006698": {
"id": "006698",
"content": "若三角形三边长成等差数列, 周长为$36$, 内切圆周长为$6\\pi$, 则此三角形是\\bracket{20}.\n\\twoch{正三角形}{等腰三角形, 但不是直角三角形}{直角三角形, 但不是等腰三角形}{等腰直角三角形}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162906,7 +163887,9 @@
"006699": {
"id": "006699",
"content": "若$a,b,c$的倒数依次成等差数列, 且$a,b,c$互不相等, 则$\\dfrac{a-b}{b-c}$等于\\bracket{20}.\n\\fourch{$\\dfrac ca$}{$\\dfrac ab$}{$\\dfrac ac$}{$\\dfrac bc$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162927,7 +163910,9 @@
"006700": {
"id": "006700",
"content": "若等差数列$\\{a_n\\}$的公差$d=\\dfrac 12$, $a_1+a_3+a_5+a_7+a_9+\\cdots +a_{95}+a_{97}+a_{99}=60$, 则前$100$项之和$S_{100}$等于\\bracket{20}.\n\\fourch{$120$}{$145$}{$150$}{$170$}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -162948,7 +163933,9 @@
"006701": {
"id": "006701",
"content": "在等差数列$\\{a_n\\}$中, $S_m=S_n=l$($m\\ne n$), 则$a_1+a_{m+n}$等于\\bracket{20}.\n\\fourch{$mnl$}{$(m+n)l$}{$0$}{$(m+n-1)l$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162969,7 +163956,9 @@
"006702": {
"id": "006702",
"content": "若等差数列$\\{a_n\\}$满足$3a_8=5a_{13}$, 且$a_1>0$, 则前$n$项之和$S_n$的最大值是\\bracket{20}.\n\\fourch{$S_{10}$}{$S_{11}$}{$S_{20}$}{$S_{21}$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -162990,7 +163979,9 @@
"006703": {
"id": "006703",
"content": "若一个等差数列共有$2n+1$项, 其中奇数项之和为$290$, 偶数项之和为$261$, 则第$n+1$项为\\bracket{20}.\n\\fourch{$30$}{$29$}{$28$}{$27$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163011,7 +164002,9 @@
"006704": {
"id": "006704",
"content": "记两个等差数列$\\{a_n\\}$和$\\{b_n\\}$的前$n$项和分别为$S_n$和$T_n$, 且$\\dfrac{S_n}{T_n}=\\dfrac{7n+1}{4n+27}$($n\\in \\mathbf{N}^*$), 则$\\dfrac{a_{11}}{b_{11}}$等于\\bracket{20}.\n\\fourch{$\\dfrac 74$}{$\\dfrac 32$}{$\\dfrac 43$}{$\\dfrac{78}{71}$}",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163032,7 +164025,9 @@
"006705": {
"id": "006705",
"content": "在等差数列$\\{a_n\\}$中, 若前三项之和为$12$, 最后三项之和为$75$, 各项之和为$145$, 则$n=$\\blank{50}, $a_1=$\\blank{50}, 公差$d=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163053,7 +164048,9 @@
"006706": {
"id": "006706",
"content": "在等差数列$\\{a_n\\}$中, 若前四项之和为$21$, 末四项之和为$67$, 前$n$项之和为$286$, 则该数列的项数为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163074,7 +164071,9 @@
"006707": {
"id": "006707",
"content": "在等差数列$\\{a_n\\}$中, 若$a_1+a_2+a_3+\\cdots +a_{15}=a$, $a_{n-11}+a_{n-13}+\\cdots +a_n=b$, 则$\\{a_n\\}$的前$n$项和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163095,7 +164094,9 @@
"006708": {
"id": "006708",
"content": "在等差数列$\\{a_n\\}$中, 若前$9$项和为$18$, 前$n$项和为$240$, 且$a_{n-4}=30$, $n>9$, 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163116,7 +164117,9 @@
"006709": {
"id": "006709",
"content": "若等差数列$18, 15, 12, \\cdots$的前$n$项和最大, 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163137,7 +164140,9 @@
"006710": {
"id": "006710",
"content": "若等差数列$-21, -19, -17, \\cdots$的前$n$项和最小. 则$n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163158,7 +164163,9 @@
"006711": {
"id": "006711",
"content": "在等差数列$\\{a_n\\}$中, 弱$a_9+a_{10}=a$, $a_{29}+a_{30}=b$, 则$a_{99}+a_{100}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -163179,7 +164186,9 @@
"006712": {
"id": "006712",
"content": "两个等差数列: $2, 5, 8, \\cdot, 197$和$2, 7, 12, \\cdots, 197$中,\\\\\n(1) 有多少相同的项?\\\\\n(2) 求这些相同项之和.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -163200,7 +164209,10 @@
"006713": {
"id": "006713",
"content": "求和: $100^2-99^2+98^2-97^2+\\cdots +4^2-3^2+2^2-1^2$.",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163221,7 +164233,10 @@
"006714": {
"id": "006714",
"content": "若$\\{a_n\\}$是等差数列, 求证: $a_1^2-a_2^2+a_3^2-a_4^2+\\cdots +a_{2n-1}^2-a_{2n}^2=\\dfrac n{2n-1}(a_1^2-a_{2n}^2)$.",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163242,7 +164257,9 @@
"006715": {
"id": "006715",
"content": "若四个数依次成等差数列, 且四个数的平方和为$94$, 首尾两数之积比中间两数之积少$18$, 求此四数.",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -163263,7 +164280,9 @@
"006716": {
"id": "006716",
"content": "已知$\\lg a,\\lg b,\\lg c$与$\\lg a-\\lg 2b,\\lg 2b-\\lg 3c,\\lg 3c-\\lg a$都是等差数列, 试求$a,b,c$之比.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -163284,7 +164303,9 @@
"006717": {
"id": "006717",
"content": "已知$\\triangle ABC$的三边成等差数列, 且最大角与最小角之差为$90^\\circ$, 求证: 其三边之比为$(\\sqrt 7+1):\\sqrt 7:(\\sqrt 7-1)$.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -163305,7 +164326,10 @@
"006718": {
"id": "006718",
"content": "在$\\triangle ABC$中. 已知$\\lg \\tan A,\\lg \\tan B,\\lg \\tan C$依次成等差数列, 求$\\angle B$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0310002B"
+ ],
"tags": [
"第四单元"
],
@@ -163326,7 +164350,11 @@
"006719": {
"id": "006719",
"content": "若等差数列的第$p$项是$q$, 第$q$项是$p$($p\\ne q$), 求它的第$p+q$项及前$p+q$项的和.",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0401003X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163347,7 +164375,9 @@
"006720": {
"id": "006720",
"content": "在等差数列中, 若前$p$项的和与前$q$项的和相等求前$p+q$项的和.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163368,7 +164398,9 @@
"006721": {
"id": "006721",
"content": "一等差数列共有奇数项, 且奇数项之和为$80$, 偶数项之和为$75$, 求此数列的中间项与项数.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163389,7 +164421,9 @@
"006722": {
"id": "006722",
"content": "已知一个等差数列的项数$n$为奇数, 求其奇数项之和与偶数项之和的比.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163410,7 +164444,9 @@
"006723": {
"id": "006723",
"content": "已知等差数列$\\{a_n\\}$满足$a_1=-60$, $a_{17}=-12$, 记$b_n=|a_n|$, 求数列$\\{b_n\\}$前$30$项之和.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163431,7 +164467,9 @@
"006724": {
"id": "006724",
"content": "若等差数列$\\{a_n\\}$的通项为$a_n=10-3n$, 求$|a_1|+|a_2|+\\cdots +|a_n|$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163536,7 +164574,9 @@
"006729": {
"id": "006729",
"content": "已知一个数列$\\{a_n\\}$的前$n$项和$S_n=2n(n+1)$, 求此数列的第$100$项.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -163557,7 +164597,9 @@
"006730": {
"id": "006730",
"content": "已知数列$\\{a_n\\}$前$n$项和$S_n=na_n-n^2+n$, 求$a_{100}-a_{99}$.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -163578,7 +164620,9 @@
"006731": {
"id": "006731",
"content": "已知数列$\\{a_n\\}$前$n$项和$S_n=2n^2-3n-1$, 求此数列的通项公式.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -163599,7 +164643,9 @@
"006732": {
"id": "006732",
"content": "已知$\\{a_n\\}$是首项为$a$的等差数列, 记$b_n=\\dfrac{a_1+a_2+\\cdots +a_n}n$, 求证: 数列$\\{b_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -163620,7 +164666,10 @@
"006733": {
"id": "006733",
"content": "已知等差数列$\\{a_n\\}$及关于$x$的方程$a_ix^2+2a_{i+1}x+a_{i+2}=0$($i=1,2,\\cdots ,n,n\\in \\mathbf{N}^*$), 其中$a_1$及公差$d$均为非零实数.\\\\\n(1) 求证: 这些方程有公共根;\\\\\n(2) 若方程的另一根为$a_i$, 求证: $\\dfrac 1{a_1+1},\\dfrac 1{a_2+1},\\cdots,\\dfrac 1{a_n+1}$依次成等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X",
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -163641,7 +164690,10 @@
"006734": {
"id": "006734",
"content": "已知$a_{n+1}=\\dfrac{2{a_n}}{{a_n}+2}$, $a_1=2$.\\\\\n(1) 求证: 数列$\\{\\dfrac 1{a_n}\\}$是等差数列;\\\\\n(2) 求$a_5$;\\\\\n(3) 求$\\{a_n\\}$.",
- "objs": [],
+ "objs": [
+ "K0402002X",
+ "K0401003X"
+ ],
"tags": [
"第四单元"
],
@@ -163662,7 +164714,9 @@
"006735": {
"id": "006735",
"content": "若一个首项为$1$的等差数列$\\{a_n\\}$的前$n$项和与其后的$2n$项和之比是与$n$无关的定值, 试求此数列的通项公式.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -163683,7 +164737,10 @@
"006736": {
"id": "006736",
"content": "若公差不为零的等差数列的第$2, 3, 6$项依次是一等比数列的连续三项, 则这个等比数列的公比等于\\bracket{20}.\n\\fourch{$\\dfrac 34$}{$-\\dfrac 13$}{$\\dfrac 13$}{$3$}",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -163704,7 +164761,9 @@
"006737": {
"id": "006737",
"content": "若自然数$m,n,p,r$满足$m+n=p+r$, 则等比数列$\\{a_n\\}$必定满足\\bracket{20}.\n\\fourch{$\\dfrac{a_m}{a_p}=\\dfrac{a_r}{a_n}$}{$\\dfrac{a_m}{a_n}=\\dfrac{a_r}{a_p}$}{$a_m+a_n=a_p+a_r$}{$a_m-a_n=a_p-a_r$}",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -163725,7 +164784,9 @@
"006738": {
"id": "006738",
"content": "在等比数列$\\{a_n\\}$中, 已知$a_9=-2$, 则此数列前$17$项之积等于\\bracket{20}.\n\\fourch{$2^{16}$}{$-2^{16}$}{$2^{17}$}{$-2^{17}$}",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -163746,7 +164807,9 @@
"006739": {
"id": "006739",
"content": "已知数列$\\{a_n\\}$是公比$q\\ne 1$的等比数列, 则在\\textcircled{1} $\\{a_na_{n+1}\\}$, \\textcircled{2} $\\{a_{n+1}-a_n\\}$, \\textcircled{3} $\\{a_n^3\\}$, \\textcircled{4} $\\{na_n\\}$这四个数列中, 成等比数列的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -163788,7 +164851,9 @@
"006741": {
"id": "006741",
"content": "在等比数列$\\{a_n\\}$中, 若公比为$q$, $a_n=a_m\\cdot x$, 则$x=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -163809,7 +164874,9 @@
"006742": {
"id": "006742",
"content": "在等比数列$\\{a_n\\}$中, 若$a_5=2$, $a_{10}=10$, 则$a_{15}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -163830,7 +164897,9 @@
"006743": {
"id": "006743",
"content": "在等比数列$\\{a_n\\}$中, 若$a_4=5$, $a_8=6$, 则$a_2a_{10}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -163851,7 +164920,9 @@
"006744": {
"id": "006744",
"content": "在等比数列$\\{a_n\\}$中, 若$a_1a_2\\cdots a_9=512$, 则$a_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -163872,7 +164943,9 @@
"006745": {
"id": "006745",
"content": "若$\\{a_n\\}$是等比数列, 且$a_n>0$, $a_2\\cdot a_4+2a_3\\cdot a_5+a_4\\cdot a_6=25$, 则$a_3+a_5=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403001X"
+ ],
"tags": [
"第四单元"
],
@@ -164187,7 +165260,9 @@
"006760": {
"id": "006760",
"content": "某厂去年产值为$a$, 计划在今后五年内每年比上年产值增长$10\\%$, 则从今年起到第五年, 这个厂的总产值为\\bracket{20}.\n\\fourch{$1.1^4a$}{$1.1^5a$}{$11(1.1^5-1)a$}{$10(1.1^6-1)a$}",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -164208,7 +165283,9 @@
"006761": {
"id": "006761",
"content": "某人从$2006$年起, 每年$1$月$1$日到银行新存人$a$元(一年定期), 若年利率为$r$保持不变, 且每平到期存款均自动转为新的一年定期, 到$2010$年$1$月$1$日将所有存款及利息全部取回, 他可取回的钱数(单位为元)为\\bracket{20}.\n\\fourch{$a(1+r)^5$}{$\\dfrac ar[(1+r)^5-(1+r)]$}{$a(1+r)^6$}{$\\dfrac ar[(1+r)^6-(1+r)]$}",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -164229,7 +165306,9 @@
"006762": {
"id": "006762",
"content": "若数列前$n$项的和$S_n=2^n-1$, 则此数列奇数项的前$n$项的和是\\bracket{20}.\n\\fourch{$\\dfrac 13(2^{n+1}-1)$}{$\\dfrac 13(2^{n+1}-2)$}{$\\dfrac 13(2^{2n}-1)$}{$\\dfrac 13(2^{2n}-2)$}",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164250,7 +165329,9 @@
"006763": {
"id": "006763",
"content": "若等比数列的前$n$项和$S_n=4^n+a$, 则$a$的值等于\\bracket{20}.\n\\fourch{$-4$}{$-1$}{$0$}{$1$}",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164271,7 +165352,9 @@
"006764": {
"id": "006764",
"content": "在等比数列$\\{a_n\\}$中, 已知$a_1+a_2+a_3=6$, $a_2+a_3+a_4=-3$, 则$a_3+a_4+a_5+a_6+a_7+a_8$等于\\bracket{20}.\n\\fourch{$\\dfrac{21}{16}$}{$\\dfrac{19}{16}$}{$\\dfrac 98$}{$\\dfrac 34$}",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164292,7 +165375,9 @@
"006765": {
"id": "006765",
"content": "在等比数列$\\{a_n\\}$中, 已知对任意自然数$n$, $a_1+a_2+a_3+\\cdots +a_n=2^n-1$, 则$a_1^2+a_2^2+a_3^2+\\cdots +a_n^2$等于\\bracket{20}.\n\\fourch{$(2^n-1)^2$}{$\\dfrac 13(2^n-1)$}{$4^n-1$}{$\\dfrac 13(4^n-1)$}",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164313,7 +165398,9 @@
"006766": {
"id": "006766",
"content": "在等比数列$\\{a_n\\}$中, 若前$n$项和为$S_n$, 且$a_3=3S_2+2$, $a_4=3S_3+2$, 则公比等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164334,7 +165421,9 @@
"006767": {
"id": "006767",
"content": "在等比数列$\\{a_n\\}$中, 若公比等于$2$, 且前$4$项之和等于$1$, 那么前$8$项之和等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164355,7 +165444,10 @@
"006768": {
"id": "006768",
"content": "在等比数列$\\{a_n\\}$中, 若第一、二、三这三项之和为$168$, 第四、五、六这三项之和为$21$, 则公比$q=$\\blank{50}, 首项$a_1=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -164376,7 +165468,10 @@
"006769": {
"id": "006769",
"content": "在等比数列$\\{a_n\\}$中, 若$a_1+a_2+a_3+a_4+a_5=31$, $a_2+a_3+a_4+a_5+a_6=62$, 则其通项公式$a_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0403005X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -165860,7 +166955,9 @@
"006839": {
"id": "006839",
"content": "若数列$\\{a_n\\}$满足$a_1=\\sqrt 6$, $a_{n+1}=\\sqrt {a_n+6}$($n\\in \\mathbf{N}^*$), 且$\\displaystyle \\lim_{n\\to \\infty} a_n$存在, 求$\\displaystyle \\lim_{n\\to \\infty} a_n$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165881,7 +166978,9 @@
"006840": {
"id": "006840",
"content": "用极限定义证明: 数列$\\{\\dfrac n{2n+1}\\}$的极限为$\\dfrac 12$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165902,7 +167001,9 @@
"006841": {
"id": "006841",
"content": "用极限定义证明: $\\displaystyle \\lim_{n\\to \\infty} (1-\\dfrac 1{2^n})=1$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165923,7 +167024,9 @@
"006842": {
"id": "006842",
"content": "用极限定义证明: $\\displaystyle \\lim_{n\\to \\infty} (\\sqrt {n+1}-\\sqrt n)=0$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165946,7 +167049,9 @@
"006843": {
"id": "006843",
"content": "$\\displaystyle \\lim_{n\\to \\infty} (\\dfrac{{n^2}+1}{n^3}+\\dfrac{{n^2}+2}{n^3}+\\cdots +\\dfrac{{n^2}+n}{n^3})$的值为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{不存在}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165967,7 +167072,9 @@
"006844": {
"id": "006844",
"content": "若$f(n)=1+2+\\cdots +n$($n\\in \\mathbf{N}^*$), 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{f(n^2)}{[f(n)]^2}$值是\\bracket{20}.\n\\fourch{$2$}{$0$}{$1$}{$\\dfrac 12$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -165988,7 +167095,9 @@
"006845": {
"id": "006845",
"content": "若$S_n$是无穷等差数列$1, 3, 5, \\cdots$的前$n$项之和, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{S_n}{S_{2n}}$的值等于\\bracket{20}.\n\\fourch{$\\dfrac 14$}{$1$}{$2$}{$4$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166009,7 +167118,9 @@
"006846": {
"id": "006846",
"content": "若数列$\\{a_n\\}$满足$a_n=(1+\\dfrac 12)(1+\\dfrac 13)(1+\\dfrac 14)\\cdots (1+\\dfrac 1{n+1})$, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{a^n}n$的值等于\\bracket{20}.\n\\fourch{$0$}{$\\dfrac 12$}{$1$}{不存在}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166030,7 +167141,9 @@
"006847": {
"id": "006847",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{(k-2){n^2}+4n}{2({n^2}+7)}=2$, 则实数$k$的值等于\\bracket{20}.\n\\fourch{$4$}{$6$}{$8$}{$0$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166051,7 +167164,9 @@
"006848": {
"id": "006848",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{a{n^2}+cn}{b{n^2}+c}=2$, $\\displaystyle \\lim_{n\\to \\infty} \\dfrac{bn+c}{cn+a}=3$, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{a{n^2}+bn+c}{c{n^2}+an+b}=$\\bracket{20}.\n\\fourch{$\\dfrac 16$}{$\\dfrac 23$}{$\\dfrac 32$}{$6$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166072,7 +167187,9 @@
"006849": {
"id": "006849",
"content": "数列极限$\\displaystyle \\lim_{n\\to \\infty} (n+1-\\sqrt {n^2+n})$是\\bracket{20}.\n\\fourch{不存在}{$\\dfrac 12$}{$1$}{$\\dfrac 32$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166093,7 +167210,9 @@
"006850": {
"id": "006850",
"content": "以下各式中, 当$n\\to \\infty$时, 极限值为$\\dfrac 12$的是\\bracket{20}.\n\\fourch{$\\dfrac{n-2}{2n(n+1)}$}{$\\dfrac{2n+1}{3n+2}$}{$(\\sqrt {n+1}-\\sqrt n)\\sqrt n$}{$\\dfrac{1+4+7+\\cdots +(3n-2)}{2{n^2}}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166114,7 +167233,9 @@
"006851": {
"id": "006851",
"content": "$\\displaystyle \\lim_{n\\to \\infty} (\\dfrac{2{n^2}+5n-1}{3{n^3}-2{n^2}}+\\dfrac{3+5n}{3n-1})=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166135,7 +167256,9 @@
"006852": {
"id": "006852",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{1+3+5+7+\\cdots +(2n-1)}{1+4+7+11+\\cdots +(3n-2)}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166156,7 +167279,9 @@
"006853": {
"id": "006853",
"content": "若$\\{a_n\\}$是公差不为零的等差数列, $S_n$是它的前$n$项之和, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{n{a_n}}{S_n}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166177,7 +167302,9 @@
"006854": {
"id": "006854",
"content": "$\\displaystyle \\lim_{n\\to \\infty} [\\dfrac 1{(3n+1)(2n-1)}+\\dfrac 5{(3n+1)(2n-1)}+\\dfrac 9{(3n+1)(2n-1)}+\\cdots +\\dfrac{4n-3}{(3n+1)(2n-1)}]=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166198,7 +167325,9 @@
"006855": {
"id": "006855",
"content": "$\\displaystyle \\lim_{n\\to \\infty} (1-\\dfrac 12)(1-\\dfrac 13)(1-\\dfrac 14)\\cdots (1-\\dfrac 1n)=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166221,7 +167350,9 @@
"006856": {
"id": "006856",
"content": "$\\displaystyle \\lim_{n\\to \\infty} (1-\\dfrac 1{2^2})(1-\\dfrac 1{3^2})(1-\\dfrac 1{4^2})\\cdots (1-\\dfrac 1{n^2})=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166244,7 +167375,9 @@
"006857": {
"id": "006857",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} (2n-\\sqrt {4n^2+an+3})=1$, 则$a$等于\\bracket{20}.\n\\fourch{$-7$}{$-4$}{$0$}{$4$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166265,7 +167398,9 @@
"006858": {
"id": "006858",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac 1n[1^2+(1+\\dfrac 1n)^2+(1+\\dfrac 2n)^2+(1+\\dfrac 3n)^2+\\cdots (1+\\dfrac{n+1}n)^2]$的值为\\bracket{20}.\n\\fourch{$2$}{$\\dfrac 73$}{$\\dfrac 95$}{$\\dfrac 37$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166286,7 +167421,9 @@
"006859": {
"id": "006859",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} (\\dfrac{n^2+1}{n+1}-an-b)=0$, 则$a=$\\blank{50}, $b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166307,7 +167444,9 @@
"006860": {
"id": "006860",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{3{a^n}+p{b^n}+c}{7{a^n}-3{b^n}+{c^2}}=-5$($11$)的等比数列前$n$项之和为$S_n$, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{S_n}{{S_{n+1}}}$的值为\\bracket{20}.\n\\fourch{$1$}{$q$}{$\\dfrac 1q$}{不存在}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166605,7 +167770,9 @@
"006874": {
"id": "006874",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} (\\dfrac{1-a}{2a})^n=0$, 则$a$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166626,7 +167793,9 @@
"006875": {
"id": "006875",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} [2-(\\dfrac q{1-q})^n]=2$, 则$q$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166647,7 +167816,9 @@
"006876": {
"id": "006876",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{x^{2n+1}}}{1+{x^{2n}}}=x$($x\\ne 0$), 则$x$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166668,7 +167839,9 @@
"006877": {
"id": "006877",
"content": "若$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{3^n}+{a^n}}{{3^{n+1}}+{a^{n+1}}}=\\dfrac 13$, 则$a$的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166691,7 +167864,9 @@
"006878": {
"id": "006878",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{2^{2n-1}}+1}{{4^n}-{3^n}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166722,7 +167897,9 @@
"006879": {
"id": "006879",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{5^{n+1}}-{{10}^{n-1}}}{{{10}^{n+1}}-{5^{n-1}}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166743,7 +167920,9 @@
"006880": {
"id": "006880",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{{(-2)}^{n+1}}}{1-2+4-\\cdots +{{(-2)}^{n-1}}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166766,7 +167945,9 @@
"006881": {
"id": "006881",
"content": "$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{1+2+{2^2}+\\cdots +{2^{n-1}}}{1-{2^{n-1}}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166789,7 +167970,9 @@
"006882": {
"id": "006882",
"content": "若$|p|<3$, 则$\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{p^n}+{3^n}}{1+3+{3^2}+\\cdots +{3^n}}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166810,7 +167993,9 @@
"006883": {
"id": "006883",
"content": "若$|a|<1$, 则$\\displaystyle \\lim_{n\\to \\infty} [(1+a)(1+a^2)(1+a^4)\\cdots (1+a^{2^n})]=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166831,7 +168016,9 @@
"006884": {
"id": "006884",
"content": "在正数数列$\\{a_n\\}$中, 已知$a_1=2$, $a_{n-1}$与$a_n$满足关系式$\\lg a_n=\\lg a_{n-1}+\\lg t$, 其中$t$为大于零的常数.求:\\\\\n(1) 数列$\\{a_n\\}$的通项公式;\\\\\n(2) $\\displaystyle \\lim_{n\\to \\infty} \\dfrac{{a_n}+1}{{a_n}-1}$的值.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166852,7 +168039,9 @@
"006885": {
"id": "006885",
"content": "已知数列$\\{a_n\\}$的前$n$项之和$S_n$满足$S_n=1+ra_n$($r\\ne 1$).\\\\\n(1) 求证: $\\{s_n-1\\}$是公比为$\\dfrac r{r-1}$的等比数列;\\\\\n(2) 求适合$\\displaystyle \\lim_{n\\to \\infty} S_n=1$的$r$的取值范围.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166873,7 +168062,9 @@
"006886": {
"id": "006886",
"content": "已知等差数列$\\{a_n\\}$的首项为$1$, 公差为$d$, 前$n$项和为$A_n$; 等比数列$\\{b_n\\}$的首项为$1$, 公比为$q$($|q|<1$), 前$n$项和为$B_n$.记$S_n=B_1+B_2+\\cdots +B_n$, 若$\\displaystyle \\lim_{n\\to \\infty} (\\dfrac{A_n}n-S_n)=1$, 求$d$和$q$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -166896,7 +168087,9 @@
"006887": {
"id": "006887",
"content": "无穷数列$\\dfrac 12\\sin \\dfrac{\\pi }2,\\dfrac 1{2^2}\\sin \\dfrac{2\\pi }2,\\dfrac 1{2^3}\\sin \\dfrac{3\\pi }2,\\cdots,\\dfrac 1{2^n}\\sin \\dfrac{n\\pi }2, \\cdots$的各项之和为\\bracket{20}.\n\\fourch{$\\dfrac 13$}{$\\dfrac 27$}{$\\dfrac 25$}{不存在}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -166917,7 +168110,9 @@
"006888": {
"id": "006888",
"content": "将循环小数$0.\\dot3\\dot6$化成最简分数后, 分子与分母的和等于\\bracket{20}.\n\\fourch{15}{45}{126}{135}",
- "objs": [],
+ "objs": [
+ "K0405004X"
+ ],
"tags": [
"第四单元"
],
@@ -166938,7 +168133,9 @@
"006889": {
"id": "006889",
"content": "记$b=\\cos 30^\\circ$, 又无穷数列$\\{a_n\\}$满足$a_1=2$, $\\log _ba_{n+1}=\\log _ba_n+2$, 则$\\displaystyle \\lim_{n\\to \\infty} (a_2+a_3+\\cdots +a_n)$等于\\bracket{20}.\n\\fourch{$8$}{$6$}{$\\dfrac 83$}{$2$}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -166959,7 +168156,9 @@
"006890": {
"id": "006890",
"content": "无穷等比数列(公比$q$满足$|q|<1$)中, 若任何一项都等于该项后所有项的和, 则等比数列的公比是\\bracket{20}.\n\\fourch{$\\dfrac 14$}{$\\dfrac 12$}{$-\\dfrac 12$}{$-\\dfrac 14$}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -166980,7 +168179,9 @@
"006891": {
"id": "006891",
"content": "一个公比的绝对值小于$1$的无穷等比数列中, 已知各项的和为$15$, 各项的平方和为$45$, 则此数列的首项为\\bracket{20}.\n\\fourch{$6$}{$5$}{$3$}{$2$}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167001,7 +168202,9 @@
"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{$1$}{$\\dfrac 12$}{$\\dfrac 13$}{$\\dfrac 14$}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167022,7 +168225,9 @@
"006893": {
"id": "006893",
"content": "设$a$是方程$\\log _2x+\\log _2(x+\\dfrac 34)+\\log _24=0$的根, 则无穷数列$a,a^2,a^3,\\cdots$的各项之和等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167043,7 +168248,9 @@
"006894": {
"id": "006894",
"content": "已知$S_n=\\dfrac 15+\\dfrac 2{5^2}+\\dfrac 1{5^3}+\\dfrac 2{5^4}+\\cdots +\\dfrac 1{5^{2n-1}}+\\dfrac 2{5^{2n}}$, 则$\\displaystyle \\lim_{n\\to \\infty} S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167067,7 +168274,9 @@
"006895": {
"id": "006895",
"content": "无穷数列$0.\\dot1\\dot5,0.0\\dot1\\dot5,0.00\\dot1\\dot5, \\cdots$所有项的和等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405004X"
+ ],
"tags": [
"第四单元"
],
@@ -167088,7 +168297,9 @@
"006896": {
"id": "006896",
"content": "若$\\theta$是一个定锐角, $\\theta _1$是$\\dfrac{\\theta }2$的余角, $\\theta _2$是$\\dfrac{\\theta _1}2$的余角, $\\theta _3$是$\\dfrac{\\theta _2}2$的余角, $\\cdots$, $\\theta _n$是$\\dfrac{\\theta _{n-1}}2$的余角, 则$\\displaystyle \\lim_{n\\to \\infty} \\theta _n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167109,7 +168320,9 @@
"006897": {
"id": "006897",
"content": "$\\dfrac 13+\\dfrac 3{3^2}+\\dfrac 7{3^3}+\\cdots +\\dfrac{{2^n}-1}{3^n}+\\cdots =$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167130,7 +168343,9 @@
"006898": {
"id": "006898",
"content": "记$S_n=1-\\dfrac 12-\\dfrac 14+\\dfrac 18-\\dfrac 1{16}-\\dfrac 1{32}+\\dfrac 1{64}-\\dfrac 1{128}-\\dfrac 1{256}+\\cdots +\\dfrac 1{2^{3n-3}}-\\dfrac 1{2^{3n-2}}-\\dfrac 1{2^{3n-1}}$, 则$\\displaystyle \\lim_{n\\to \\infty} S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167151,7 +168366,9 @@
"006899": {
"id": "006899",
"content": "在等比数列$\\{a_n\\}$中, 已知$\\displaystyle \\lim_{n\\to \\infty} (a_1+a_2+\\cdots +a_n)=\\dfrac 12$, 求$a_1$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167175,7 +168392,9 @@
"006900": {
"id": "006900",
"content": "在等比数列$\\{a_n\\}$中, 已知$a_1+a_2+a_3=18$, $a_2+a_3+a_4=-9$, 且$S_n=a_1+2a_2+3(a_3+a_4+\\cdots +a_n)$($n\\ge 3$), 求$\\displaystyle \\lim_{n\\to \\infty} S_n$.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167196,7 +168415,9 @@
"006901": {
"id": "006901",
"content": "已知$\\{a_n\\}$是公比为正数的等比数列, 且$\\dfrac 1{a_2}+\\dfrac 1{a_3}+\\dfrac 1{a_4}=117$, $a_1\\cdot a_2\\cdot a_3=\\dfrac 1{3^6}$, 求$\\displaystyle \\lim_{n\\to \\infty} (a_1+a_2+\\cdots a_n)$.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167217,7 +168438,9 @@
"006902": {
"id": "006902",
"content": "已知数列$\\{a_n\\}$的前$n$项之和$S_n$满足$S_n=1-\\dfrac 23a_n$($n\\in \\mathbf{N}^*$).\\\\\n(1) 求$\\displaystyle \\lim_{n\\to \\infty} S_n$;\\\\\n(2) 若记数列$\\{a_nS_n\\}$的前$n$项之和为$U_n$, 求$\\displaystyle \\lim_{n\\to \\infty} U_n$.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167238,7 +168461,9 @@
"006903": {
"id": "006903",
"content": "已知数列$\\{a_n\\}$的首项$a_1=b$($b\\ne 0$), 它的前$n$项和$S_n$组成的数列$\\{S_n\\}$($n\\in \\mathbf{N}^*$)是一个公比为$q$($q\\ne 0$, $|q|<1$)的等比数列.\\\\\n(1) 求证: $a_2,a_3,a_4,\\cdots ,a_n,\\cdots$是一个等比数列;\\\\\n(2) 设$W_n=a_1S_1+a_2S_2+\\cdots +a_nS_n$, 求$\\displaystyle \\lim_{n\\to \\infty} W_n$(用$b,q$表示).",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -167259,7 +168484,9 @@
"006904": {
"id": "006904",
"content": "在$45^\\circ$角的一边上, 取距离顶点为$a$的一点, 由这点向另一边作垂线, 然后再由这个垂线的垂足向另一边作垂线, $\\cdots$, 如此无限地继续下去, 求所有这些垂线长的和.",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -177484,7 +178711,10 @@
"007379": {
"id": "007379",
"content": "在六棱锥各棱所在的$12$条直线中, 异面直线共有\\bracket{20}.\n\\fourch{$12$对}{$24$对}{$36$对}{$48$对}",
- "objs": [],
+ "objs": [
+ "K0618001B",
+ "K0618003B"
+ ],
"tags": [
"第八单元"
],
@@ -179082,7 +180312,11 @@
"007454": {
"id": "007454",
"content": "直线$l_1\\parallel l_2$, $l_1$上有$4$个点, $l_2$上有$6$个点, 以这些点为端点连接成线段, 则它们在$l_1$与$l_2$之间的交点最多有\\blank{50}个.",
- "objs": [],
+ "objs": [
+ "K0615001B",
+ "K0615002B",
+ "K0615005B"
+ ],
"tags": [
"第八单元"
],
@@ -180092,7 +181326,10 @@
"007501": {
"id": "007501",
"content": "以四棱台的顶点为顶点, 能组成\\blank{50}个四面体.",
- "objs": [],
+ "objs": [
+ "K0615001B",
+ "K0618007B"
+ ],
"tags": [
"第八单元"
],
@@ -200996,7 +202233,9 @@
"008400": {
"id": "008400",
"content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2(1+3n)$, 填写下表:\n\\begin{center}\n \\begin{tabular}{|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|p{.05\\textwidth}<{\\centering}|}\n \\hline\n $n$ & $1$ & $2$ & $3$ & $\\cdots$ & $11$ & $\\cdots$ & & $\\cdots$ & \\\\ \\hline\n $a_n$ & & & & $\\cdots$ & & $\\cdots$ & $128$ & $\\cdots$ & $602$ \\\\ \\hline \n \\end{tabular}\n\\end{center}\n在数列$-1,0,\\dfrac 19,\\dfrac 18,\\cdots ,\\dfrac{n-2}{n^2},\\cdots$中, $\\dfrac 2{25}$是它的第\\blank{50}项.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -201017,7 +202256,9 @@
"008401": {
"id": "008401",
"content": "已知无穷数列$1\\times 2,2\\times 3,3\\times 4,\\cdots ,n(n+1),\\cdots$.\\\\\n(1) 求这个数列的第$10$项、第$31$项及第$48$项;\\\\\n(2) $420$是不是这个数列的项? 如果是, 那么是第几项?",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -201038,7 +202279,9 @@
"008402": {
"id": "008402",
"content": "写出下列数列的一个通项公式, 使它的前$4$项分别是下列各数:\\\\\n(1) $4, 8, 12, 16$;\\\\\n(2) $\\dfrac 12,\\dfrac 23,\\dfrac 34,\\dfrac 45$;\\\\\n(3) $-\\dfrac 1{2\\times 1},\\dfrac 1{2\\times 2},-\\dfrac 1{2\\times 3},\\dfrac 1{2\\times 4}$;\\\\\n(4) $1,-\\sqrt[3]2,\\sqrt[3]3,-\\sqrt[3]4$.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -201059,7 +202302,9 @@
"008403": {
"id": "008403",
"content": "根据下面的图形及相应的点数在横线上画出适当的图形.\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.5]\n \\filldraw (0,0) circle (0.05);\n \\filldraw (2,0) circle (0.05) (3,0) circle (0.05) (2,0) ++ (60:1) circle (0.05);\n \\filldraw (5,0) circle (0.05) (6,0) circle (0.05) (7,0) circle (0.05) (5,0) ++ (60:1) circle (0.05) ++ (1,0) circle (0.05) ++ (120:1) circle (0.05);\n \\filldraw (9,0) circle (0.05) (10,0) circle (0.05) (11,0) circle (0.05) (12,0) circle (0.05) (9,0) ++ (60:1) circle (0.05) ++ (1,0) circle (0.05) ++ (1,0) circle (0.05) (9,0) ++ (60:2) circle (0.05) ++ (1,0) circle (0.05) ++ (120:1) circle (0.05);\n \\end{tikzpicture}\n \\blank{100}.\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -201080,7 +202325,9 @@
"008404": {
"id": "008404",
"content": "根据下列条件, 分别写出数列$\\{a_n\\}$的前$5$项.\\\\\n(1) $a_1=5$, $a_n=a_{n-1}+4$($n\\ge 2$);\\\\\n(2) $a_1=2$, $a_n=3a_{n-1}$($n\\ge 2$);\\\\\n(3) $a_1=1$, $a_n=a_{n-1}+\\dfrac 1{a_{n-1}}$($n\\ge 2$).",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -201101,7 +202348,9 @@
"008405": {
"id": "008405",
"content": "根据下列条件, 分别写出数列$\\{a_n\\}$的第$3$项至第$5$项.\\\\\n(1) $a_1=1$, $a_n=2a_{n-1}+3$;\\\\\n(2) $a_1=1$, $a_n=2a_{n-1}+2^n$.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -201122,7 +202371,9 @@
"008406": {
"id": "008406",
"content": "写出数列: $\\dfrac 12,\\dfrac 38,\\dfrac 5{18},\\dfrac 7{32},\\cdots$的一个通项公式.",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -201143,7 +202394,9 @@
"008407": {
"id": "008407",
"content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=a_n+n+1$.求$a_6$.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -201166,7 +202419,9 @@
"008408": {
"id": "008408",
"content": "已知数列$\\{a_n\\}$满足$a_n=\\dfrac{n^2+n+1}3$.\\\\\n(1) 写出$a_{10}$与$a_{n+1}$;\\\\\n(2) $79\\dfrac 23$是不是数列$\\{a_n\\}$的项? 如果是, 那么是第几项?",
- "objs": [],
+ "objs": [
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -201187,7 +202442,9 @@
"008409": {
"id": "008409",
"content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_2=6$, $a_{n+2}=-a_n$.\\\\\n(1) 写出这个数列的前$8$项;\\\\\n(2) 根据第(1)题的结论, 猜想这个数列的项所具有的特征.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -201208,7 +202465,9 @@
"008410": {
"id": "008410",
"content": "判断下列数列中, 哪些是等差数列.是等差数列的, 请写出等差数列的公差$d$.\\\\\n(1) $1, 11, 121$;\\\\\n(2) $1, 2, 1$;\\\\\n(3) $\\lg 2,\\lg 4,\\lg 8$;\\\\\n(4) $2, 2, 2$.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201229,7 +202488,9 @@
"008411": {
"id": "008411",
"content": "已知数列$\\{a_n\\}$是等差数列, 请在下表中填入适当的数:\n\\begin{center}\n \\begin{tabular}{|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|}\n \\hline\n $a_1$ & $a_2$ & $a_3$ & 公差$d$ & $a_5$ \\\\ \\hline\n $-3$ & & $6$ & & \\\\ \\hline\n & $-5$ & & $2$ & \\\\ \\hline\n \\end{tabular}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201252,7 +202513,9 @@
"008412": {
"id": "008412",
"content": "根据所给的条件填写下表:\n\\begin{center}\n \\begin{tabular}{|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|}\n \\hline\n & $a_1$ & $d$ & $n$ & $a_n$ \\\\ \\hline\n 等差数列$\\{a_n\\}$ & $5$ & $10$ & $12$ & \\\\ \\hline\n 等差数列$\\{a_n\\}$ & $-5$ & $6$ & & $61$ \\\\ \\hline\n \\end{tabular}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201275,7 +202538,9 @@
"008413": {
"id": "008413",
"content": "已知数列$\\{a_n\\}$是等差数列, 且$a_1+a_6=12$, $a_4=7$, 求这个数列的通项公式.",
- "objs": [],
+ "objs": [
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -201296,7 +202561,9 @@
"008414": {
"id": "008414",
"content": "已知数列$\\{a_n\\}$是等差数列, 且$a_7=2$, $a_8=-4$, 求$a_1$与$a_{10}$.",
- "objs": [],
+ "objs": [
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -201317,7 +202584,9 @@
"008415": {
"id": "008415",
"content": "已知数列$\\{a_n\\}$是等差数列, 且$a_6=4$, $a_{14}=64$.设$a_6$与$a_{14}$的等差中项为$x$, $a_6$与$x$的等差中项为$y$, $x$与$a_{14}$的等差中项为$z$, 求$x+y+z$.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201338,7 +202607,9 @@
"008416": {
"id": "008416",
"content": "分别求下面两题中两数的等差中项:\\\\\n(1) $\\dfrac{8-\\sqrt 2}2$与$\\dfrac{8+\\sqrt 2}2$;\\\\\n(2) $(a+b)^2$与$(a-b)^2$.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201361,7 +202632,9 @@
"008417": {
"id": "008417",
"content": "已知数列$\\{a_n\\}$是等差数列, 且$b_n=a_n+a_{n+1}$.求证: 数列$\\{b_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0401006X"
+ ],
"tags": [
"第四单元"
],
@@ -201382,7 +202655,9 @@
"008418": {
"id": "008418",
"content": "已知三个数成等差数列, 首末两项之积为中间项的$5$倍, 后两项的和为第一项的$8$倍, 求这三个数.",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201403,7 +202678,9 @@
"008419": {
"id": "008419",
"content": "已知非零实数$a,b,c$不全相等.如果$a,b,c$成等差数列, 那么$\\dfrac 1a,\\dfrac 1b,\\dfrac 1c$能不能构成等差数列? 为什么?",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201426,7 +202703,9 @@
"008420": {
"id": "008420",
"content": "已知数列$\\{a_n\\}$是等差数列.\\\\\n(1) $2a_5=a_4+a_6$是否成立? 为什么?\\\\\n(2) 求证: $2a_n=a_{n-1}+a_{n+1}$($n\\ge 2$);\\\\\n(3) 由第(2)题你可以推广出怎样的结论?",
- "objs": [],
+ "objs": [
+ "K0401002X"
+ ],
"tags": [
"第四单元"
],
@@ -201447,7 +202726,10 @@
"008421": {
"id": "008421",
"content": "夏季高山上的温度从山脚起每升高$100$米降低$0.7^\\circ\\text{C}$.已知山脚的温度是$34.9^\\circ\\text{C}$, 山顶的温度是$23^\\circ\\text{C}$, 求山的相对高度.",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -201468,7 +202750,10 @@
"008422": {
"id": "008422",
"content": "某产品按质量分成$10$个档次, 生产最低档次的利润是$8$元/件.每提高一个档次, 利润每件增加$2$元, 产量减少$3$件.如果在某段时间内, 最低档的产品可生产$60$件, 那么在相同时间内, 生产第几档次的产品可获得最大利润? (最低档次为第$1$档)",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -201489,7 +202774,9 @@
"008423": {
"id": "008423",
"content": "已知等差数列$\\{a_n\\}$分别满足下列条件, 求解相应问题.\\\\\n(1) $d=\\dfrac 13$, $n=37$, $S_n=629$, 求$a_1$;\\\\\n(2) $d=2$, $n=15$, $a_n=-10$, 求$S_n$;\\\\\n(3) $a_1=20$, $a_n=54$, $S_n=999$, 求$d$;\\\\\n(4) $a_1=\\dfrac 56$, $d=-\\dfrac 16$, $S_n=-5$, 求$a_n$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -201510,7 +202797,10 @@
"008424": {
"id": "008424",
"content": "已知等差数列$\\{a_n\\}$的第$6$项是$5$, 第$3$项与第$8$项的和也是$5$, 求这个数列的前$9$项和.",
- "objs": [],
+ "objs": [
+ "K0402002X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -201531,7 +202821,10 @@
"008425": {
"id": "008425",
"content": "求$100$以内能被$7$整除的所有正整数的和.",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -201554,7 +202847,10 @@
"008426": {
"id": "008426",
"content": "某礼堂有$18$排座位, 第$1$排有$26$个座位, 以后每一排都比前一排多$2$个座位.这个礼堂共能坐多少人?",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -201575,7 +202871,10 @@
"008427": {
"id": "008427",
"content": "某单位开发了一个受政府扶持的新项目, 得到政府无息贷款$50$万元用于购买设备.已知该设备在使用过程中第一天使用费是$101$元, $\\cdots$, 第$n$天的使用费是$(100+n)$元.如果总费用$=$购置费$+$使用费, 那么使用多少天后, 平均每天的费用最低?",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -201596,7 +202895,9 @@
"008428": {
"id": "008428",
"content": "已知$a,b,c,d$成等差数列, 求证: $2a-3b$, $2b-3c$, $2c-3d$成等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -201617,7 +202918,9 @@
"008429": {
"id": "008429",
"content": "如果数列$\\{a_n\\}$、$\\{b_n\\}$是项数相同的两个等差数列, $p,q$是常数, 那么数列$\\{pa_n+qb_n\\}$是等差数列吗? 为什么?",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -201638,7 +202941,9 @@
"008430": {
"id": "008430",
"content": "已知数列$\\{a_n\\}$的各项均不为零, 且$a_n=\\dfrac{3a_{n-1}}{a_{n-1}+3}$($n\\ge 2$), $b_n=\\dfrac 1{a_n}$.求证: 数列$\\{b_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -201659,7 +202964,9 @@
"008431": {
"id": "008431",
"content": "已知等差数列$\\{a_n\\}$的前$15$项和为$135$, 求这个数列的第$8$项.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -201680,7 +202987,9 @@
"008432": {
"id": "008432",
"content": "已知等差数列$\\{a_n\\}$的前$5$项和为$0$, 前$10$项和为$-100$, 求这个数列的前$20$项和.",
- "objs": [],
+ "objs": [
+ "K0402005X"
+ ],
"tags": [
"第四单元"
],
@@ -201701,7 +203010,10 @@
"008433": {
"id": "008433",
"content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2-3n$, 求证: 数列$\\{a_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -201724,7 +203036,10 @@
"008434": {
"id": "008434",
"content": "已知数列$\\{b_n\\}$的前$n$项和为$T_n=an^2+bn+c$($a\\ne 0$).判断数列$\\{b_n\\}$是否是等差数列, 并说明理由.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -201745,7 +203060,9 @@
"008435": {
"id": "008435",
"content": "已知数列$\\{a_n\\}$是等比数列, 请在下表中填入适当的数:\n\\begin{center}\n \\begin{tabular}{|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|}\n \\hline\n $a_1$ & $a_2$ & $a_3$ & 公比$q$ & $a_5$ \\\\ \\hline\n & $-1$ & $3$ & & \\\\ \\hline\n & $4$ & & $2$ & \\\\ \\hline\n \\end{tabular}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -201768,7 +203085,10 @@
"008436": {
"id": "008436",
"content": "根据所给的条件填写下表:\n\\begin{center}\n \\begin{tabular}{|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|p{.15\\textwidth}<{\\centering}|}\n \\hline\n & $a_1$ & $q$ & $n$ & $a_n$ \\\\ \\hline\n 等比数列$\\{a_n\\}$ & $9$ & & $4$ & $243$ \\\\ \\hline\n 等比数列$\\{a_n\\}$ & & $-2$ & $7$ & $32$ \\\\ \\hline\n \\end{tabular}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0401002X",
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -201791,7 +203111,9 @@
"008437": {
"id": "008437",
"content": "已知数列$\\{a_n\\}$是等比数列, 且$a_4=-27$, $a_1=1$, 求$a_5a_8$.",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -201812,7 +203134,9 @@
"008438": {
"id": "008438",
"content": "已知数列$\\{a_n\\}$是等比数列, 且$a_9=-2$, $a_{13}=-32$, 求这个数列的通项公式.",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -201833,7 +203157,9 @@
"008439": {
"id": "008439",
"content": "已知直角三角形的斜边长为$c$, 两条直角边长分别为$a,b$($a0$, 求$a+a^3+a^5+\\cdots +a^{2n-1}$.",
- "objs": [],
+ "objs": [
+ "K0403003X",
+ "K0404002X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -202156,7 +203519,9 @@
"008454": {
"id": "008454",
"content": "已知等比数列$\\{a_n\\}$的前$5$项和为$10$, 前$10$项和为$50$, 求这个数列的前$15$项和.",
- "objs": [],
+ "objs": [
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -202179,7 +203544,10 @@
"008455": {
"id": "008455",
"content": "已知数列$\\{a_n\\}$是一个以$q$($q>0$)为公比、以$a_1$($a_1>0$)为首项的等比数列, 求$\\lg a_1+\\lg a_2+\\cdots +\\lg a_n$.",
- "objs": [],
+ "objs": [
+ "K0404003X",
+ "K0403006X"
+ ],
"tags": [
"第四单元"
],
@@ -202200,7 +203568,10 @@
"008456": {
"id": "008456",
"content": "某水库到$2006$年底浮萍面积达$1$万亩, 侵占大量湖面, 还造成水质富氧化, 估计今后浮萍面积将平均每年增加$0.08$万亩.政府投入资金研究对策将浮萍变成饲料, 估计$2007$年能处理$0.05$万亩, 今后每年将提高$10\\%$的处理能力.到哪一年底浮萍面积最大?",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -202221,7 +203592,9 @@
"008457": {
"id": "008457",
"content": "用数学归纳法证明: $1+a+a^2+\\cdots +a^{n+1}=\\dfrac{1-a^{n+2}}{1-a}$($a\\ne 1$, $n\\in \\mathbf{N}^*$). 在验证$n=1$时, 等式左边为\\bracket{20}.\n\\fourch{$1$}{$1+a$}{$1+a+a^2$}{$1+a+a^2+a^3$}",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202245,7 +203618,9 @@
"008458": {
"id": "008458",
"content": "用数学归纳法证明: $(n+1)(n+2)\\cdots (n+n)=2^n\\cdot 1\\cdot 3\\cdot \\cdots \\cdot (2n-1)$($n\\in \\mathbf{N}^*$), 从$k$到$k+1$时, 等式左边需添加的代数式是\\bracket{20}.\n\\fourch{$2k+1$}{$\\dfrac{2k+1}{k+1}$}{$2(2k+1)$}{$\\dfrac{2k+3}{k+1}$}",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202266,7 +203641,9 @@
"008459": {
"id": "008459",
"content": "用数学归纳法证明: $1\\times 2+2\\times 5+\\cdots +n(3n-1)=n^2(n+1)$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202289,7 +203666,9 @@
"008460": {
"id": "008460",
"content": "观察以下等式:\\\\\n\\begin{align*}\n 2 &= 2,\\\\ \n 2+4 &= 6,\\\\\n 2+4+6 & = 12,\\\\\n 2+4+6+8 & = 20,\\\\\n \\cdots &\n\\end{align*}\n试写出数列$\\{2n\\}$的前$n$项和公式, 并用数学归纳法证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -202310,7 +203689,9 @@
"008461": {
"id": "008461",
"content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=2a_n+1$($n\\in \\mathbf{N}^*$).用数学归纳法证明: $a_n=2^n-1$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202333,7 +203714,9 @@
"008462": {
"id": "008462",
"content": "用数学归纳法证明: $1-2^2+3^2-4^2+\\cdots +(-1)^{n-1}n^2=(-1)^{n-1}\\dfrac{n(n+1)}2$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202356,7 +203739,9 @@
"008463": {
"id": "008463",
"content": "用数学归纳法证明: $2^{3n}-1$($n\\in \\mathbf{N}^*$)能被$7$整除.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202379,7 +203764,9 @@
"008464": {
"id": "008464",
"content": "用数学归纳法证明: $-1+3-5+\\cdots +(-1)^n(2n-1)=(-1)^nn$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202402,7 +203789,9 @@
"008465": {
"id": "008465",
"content": "用数学归纳法证明: $1^3+2^3+3^3+\\cdots +n^3=[\\dfrac 12n(n+1)]^2$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202425,7 +203814,9 @@
"008466": {
"id": "008466",
"content": "用数学归纳法证明: $1\\times 2+2\\times 3+3\\times 4+\\cdots +n(n+1)=\\dfrac{n(n+1)(n+2)}3$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202448,7 +203839,9 @@
"008467": {
"id": "008467",
"content": "已知数列$\\{a_n\\}$满足$a_1=1$, 设该数列的前$n$项和为$S_n$, 且$S_n,S_{n+1},2a_1$成等差数列, 用数学归纳法证明: $S_n=\\dfrac{2^n-1}{2^{n-1}}$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202471,7 +203864,9 @@
"008468": {
"id": "008468",
"content": "用数学归纳法证明:\n$(a_1+a_2+\\cdots +a_n)^2=a_1^2+a_2^2+\\cdots +a_n^2+2(a_1a_2+a_1a_3+\\cdots +a_{n-1}a_n)$($n\\ge 2$, $n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202492,7 +203887,9 @@
"008469": {
"id": "008469",
"content": "已知数列$\\{a_n\\}$满足$a_1=\\dfrac 12$, $a_1+a_2+\\cdots +a_n=n^2a_n$($n\\in \\mathbf{N}^*$), 试用数学归纳法证明: $a_n=\\dfrac 1{n(n+1)}$.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -202513,7 +203910,9 @@
"008470": {
"id": "008470",
"content": "观察下列数字:\n\\begin{center}\n \\begin{tabular}{ccccccc}\n $1$ \\\\\n $2$ & $3$ & $4$ \\\\\n $3$ & $4$ & $5$ & $6$ & $7$ \\\\\n $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\\\\n $\\cdots$ \n \\end{tabular}\n\\end{center}\n猜想第$n$行的各数之和$S_n=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -202534,7 +203933,9 @@
"008471": {
"id": "008471",
"content": "依次计算$(1-\\dfrac 12)$, $(1-\\dfrac 12)(1-\\dfrac 13)$, $(1-\\dfrac 12)(1-\\dfrac 13)(1-\\dfrac 14)$, $\\cdots$的值; 根据计算的结果, 猜想$T_n=(1-\\dfrac 12)(1-\\dfrac 13)(1-\\dfrac 14)\\cdots (1-\\dfrac 1{n+1})$($n\\in \\mathbf{N}^*$)的表达式, 并用数学归纳法加以证明.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -202555,7 +203956,9 @@
"008472": {
"id": "008472",
"content": "已知数列: $\\dfrac 1{1\\times 2}$, $\\dfrac 1{2\\times 3}$, $\\dfrac 1{3\\times 4}$, …, $\\dfrac 1{n\\times (n+1)}$, $\\cdots$, 设$S_n$为该数列的前$n$项和.计算$S_1,S_2,S_3,S_4$的值; 根据计算的结果, 猜想$S_n=\\dfrac 1{1\\times 2}+\\dfrac 1{2\\times 3}+\\dfrac 1{3\\times 4}+\\cdots +\\dfrac 1{n(n+1)}$($n\\in \\mathbf{N}^*$)的表达式, 并用数学归纳法加以证明.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -202576,7 +203979,9 @@
"008473": {
"id": "008473",
"content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=\\dfrac{3a_n}{a_n+3}$, $a_n\\ne 0$($n\\in \\mathbf{N}^*$).\\\\\n(1) 求$a_2,a_3,a_4$;\\\\\n(2) 猜想$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -202599,7 +204004,9 @@
"008474": {
"id": "008474",
"content": "是否存在常数$a,b,c$, 使等式$1\\cdot (n^2-1^2)+2\\cdot (n^2-2^2)+\\cdots +n\\cdot (n^2-n^2)=an^4+bn^2+c$对一切正整数$n$都成立? 证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0409002X"
+ ],
"tags": [
"第四单元"
],
@@ -202622,7 +204029,9 @@
"008475": {
"id": "008475",
"content": "是否存在大于$1$的正整数$m$, 使得$f(n)=(2n+7)\\cdot 3^n+9$对任意正整数$n$都能被$m$整除? 若存在, 求出$m$的最大值, 并证明你的结论; 若不存在, 请说明理由.",
- "objs": [],
+ "objs": [
+ "K0409002X"
+ ],
"tags": [
"第四单元"
],
@@ -203660,7 +205069,10 @@
"008519": {
"id": "008519",
"content": "设$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 求证: 数列$\\{\\dfrac{S_n}n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402002X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -203683,7 +205095,9 @@
"008520": {
"id": "008520",
"content": "某区域环境噪声平均值如下表:\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n 年份 & $2001$ & $2002$ & $2003$ & $2004$ \\\\ \\hline\n 分贝 & $57.8$ & $57.2$ & $56.6$ & $56.0$ \\\\ \\hline\n \\end{tabular}\n\\end{center}\n如果噪声平均值按照表中规律依次逐年减小, 那么从哪一年起平均值将小于$42$分贝?",
- "objs": [],
+ "objs": [
+ "K0401007X"
+ ],
"tags": [
"第四单元"
],
@@ -203704,7 +205118,11 @@
"008521": {
"id": "008521",
"content": "假设某市$2004$年新建住房面积$400$万平方米, 其中有$250$万平方米是中低价房.预计在今后的若干年内, 该市每年新建住房面积平均比上一年增长$8\\%$.另外, 每年新建住房中, 中低价房的面积均比上一年增加$50$万平方米.\\\\\n(1) 到哪一年底该市历年所建中低价房的累计面积(以$2004$年为累计的第一年)将首次不少于$4750$万平方米?\\\\\n(2) 到哪一年底当年建造的中低价房的面积占该年建造住房面积的比例首次大于$85\\%$?",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0403005X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -203725,7 +205143,9 @@
"008522": {
"id": "008522",
"content": "在用数学归纳法证明等式: $1^2+2^2+\\cdots +n^2+\\cdots +2^2+1^2=\\dfrac{n(2n^2+1)}3$($n\\in \\mathbf{N}^*$)的过程中, 假设当$n=k$时等式成立后, 在证明当$n=k+1$时等式也成立时, 等式的左边应添加哪些项?",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -203746,7 +205166,9 @@
"008523": {
"id": "008523",
"content": "某个命题与正整数有关, 如果当$n=k(k\\in \\mathbf{N}^*)$时命题成立, 那么可以推得当$n=k+1$时命题也成立.现在已知当$n=5$时该命题不成立, 所以该命题在\\bracket{20}.\n\\fourch{$n=6$时成立}{$n=6$时不成立}{$n=4$时成立}{$n=4$时不成立}",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -203767,7 +205189,9 @@
"008524": {
"id": "008524",
"content": "用数学归纳法证明: $\\dfrac 12+\\dfrac 2{2^2}+\\dfrac 3{2^3}+\\cdots +\\dfrac n{2^n}=2-\\dfrac{n+2}{2^n}$($n\\in \\mathbf{N}^*$).",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -203790,7 +205214,9 @@
"008525": {
"id": "008525",
"content": "(1) 依次计算下列各式的值:\n$\\dfrac 11,\\dfrac 11+\\dfrac 1{1+2},\\dfrac 11+\\dfrac 1{1+2}+\\dfrac 1{1+2+3},\\dfrac 11+\\dfrac 1{1+2}+\\dfrac 1{1+2+3}+\\dfrac 1{1+2+3+4}$;\\\\\n(2) 根据第(1)题的计算结果, 猜想$S_n=\\dfrac 11+\\dfrac 1{1+2}+\\dfrac 1{1+2+3}+\\cdots +\\dfrac 1{1+2+3+\\cdots +n}$ ($n\\in \\mathbf{N}^*$)的表达式, 并用数学归纳法证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -205567,7 +206993,9 @@
"008608": {
"id": "008608",
"content": "假设某班的男生中, 步行上学、骑自行车上学和乘车上学的人数分别为$12$、$5$、$3$, 而在这个班的女生中, 采取相应方式上学的人数分别为$10$、$3$、$7$, 试设计一个矩阵来表示这些数据.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205588,7 +207016,9 @@
"008609": {
"id": "008609",
"content": "已知矩阵$A=\\begin{pmatrix}1 & x \\\\y & -1 \\end{pmatrix}$和$B=\\begin{pmatrix}m & 0 \\\\-1 & n \\end{pmatrix}$. 当$A=B$时, 求实数$x,y,m,n$的值.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205609,7 +207039,9 @@
"008610": {
"id": "008610",
"content": "写出下列线性方程组的系数矩阵和增广矩阵, 并用矩阵变换的方法求解.\\\\\n(1) $\\begin{cases} 2x+y=5, \\\\3x-2y=4; \\end{cases}$\\\\\n(2) $\\begin{cases} x+y+z=6, \\\\3x+y-z=2, \\\\5x-2y+3z=10. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205632,7 +207064,9 @@
"008611": {
"id": "008611",
"content": "利用网络, 查出北京、天津、上海和重庆四个直辖市之间的距离(单位: 千米), 并用矩阵表示.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205653,7 +207087,9 @@
"008612": {
"id": "008612",
"content": "写出下列线性方程组的系数矩阵和增广矩阵, 并用矩阵变换的方法求解.\\\\\n(1) $\\begin{cases} x-2y=3, \\\\2x+y=11; \\end{cases}$\\\\(2) $\\begin{cases} x-2z=1, \\\\y+4z=6, \\\\2x-y+z=5. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205739,7 +207175,9 @@
"008616": {
"id": "008616",
"content": "计算矩阵的乘积: $\\begin{pmatrix} x & y \\\\u & v \\end{pmatrix}\\begin{pmatrix} -1 & 0 \\\\0 & 1 \\end{pmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205760,7 +207198,9 @@
"008617": {
"id": "008617",
"content": "计算矩阵的乘积: $\\begin{pmatrix} 0 & 0 & 1 \\\\1 & 0 & 0 \\\\0 & 1 & 0 \\end{pmatrix}\\begin{pmatrix} a & b & c \\\\ u & v & w \\\\ x & y & z \\end{pmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205781,7 +207221,9 @@
"008618": {
"id": "008618",
"content": "计算矩阵的乘积: $\\begin{pmatrix}\n 1 & 2 & 3 & 4 \\end{pmatrix}\\begin{pmatrix}\n1 \\\\2 \\\\3 \\\\4 \\end{pmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205802,7 +207244,9 @@
"008619": {
"id": "008619",
"content": "已知矩阵$A=\\begin{pmatrix} 1 & 1 \\\\1 & 1 \\end{pmatrix}$, 求向量$(2,3)$经过矩阵$A$变换后所得的向量.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205823,7 +207267,9 @@
"008620": {
"id": "008620",
"content": "某水果批发部向$A,B,C,D$四家水果店分别批发的苹果、橘子和香蕉的数量如下(单位: 千克):\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n & 苹果\t & 橘子\t & 香蕉 \\\\ \\hline\n 水果店$A$ &\t$100$ & $40$ & $60$ \\\\ \\hline\n 水果店$B$ &\t$60$ & $35$ & $50$ \\\\ \\hline\n 水果店$C$ &\t$60$ & $30$ & $60$ \\\\ \\hline\n 水果店$D$ & $50$ & $45$ & $30$ \\\\ \\hline\n \\end{tabular}\n\\end{center}\n已知苹果、橘子和香蕉的批发价分别为每千克$1.50$元、$1.80$元和$2.20$元.\\\\\n(1) 试用矩阵表示批发部批发苹果、橘子和香蕉各为多少千克?\\\\\n(2) 试用矩阵表示并计算, $ABCD$四家水果店应支付的金额各为多少元?",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205844,7 +207290,9 @@
"008621": {
"id": "008621",
"content": "在一次演讲比赛中, 如果规定评综合得分的指标有三项: 演说词、口才和仪态, 它们的权重依次为$50\\%$、$30\\%$和$20\\%$, 现有$5$位演讲者的得分如下:\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n & 演说词 & 口才 & 仪态\\\\ \\hline\n 演讲者$1$ & $90$ & $90$ & $90$\\\\ \\hline\n 演讲者$2$ & $90$ & $85$ & $80$\\\\ \\hline\n 演讲者$3$ & $80$ & $85$ & $90$\\\\ \\hline\n 演讲者$4$ & $80$ & $60$ & $50$\\\\ \\hline\n 演讲者$5$ & $50$ & $60$ & $80$\\\\ \\hline\n \\end{tabular}\n\\end{center}\n试根据三项指标的权重, 用矩阵表示并计算$5$位演讲者的综合得分.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205865,7 +207313,9 @@
"008622": {
"id": "008622",
"content": "已知矩阵$A=\\begin{pmatrix} 3 & 1 & 4 \\\\2 & -1 & -2 \\\\2 & 4 & 1 \\end{pmatrix}$与$B=\\begin{pmatrix}\n0 & 1 & 2 \\\\3 & 4 & -1 \\\\-2 & 1 & 1 \\end{pmatrix}$, 求$2A+3B$和$5A-2B$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205886,7 +207336,9 @@
"008623": {
"id": "008623",
"content": "已知矩阵$A=\\begin{pmatrix} 5 & 8 \\\\6 & 7 \\end{pmatrix}$, 矩阵$B=\\begin{pmatrix} 5 & 3 & 4 \\\\2 & 2 & 4 \\end{pmatrix}$, 矩阵$C=\\begin{pmatrix} 15 \\\\10 \\\\6 \\end{pmatrix}$.\\\\\n(1) 求$AB$;\\\\\n(2) 求$(AB)C$;\\\\\n(3) 求$BC$;\\\\\n(4) 求$A(BC)$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -205949,7 +207401,9 @@
"008626": {
"id": "008626",
"content": "已知向量$(x,y)$经矩阵$\\begin{pmatrix}\n0 & 1 \\\\1 & 0 \\end{pmatrix}$变换后得到矩阵$(3,2)$, 求实数$x,y$的值.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206012,7 +207466,9 @@
"008629": {
"id": "008629",
"content": "计算下列行列式的值:\\\\\n(1) $\\begin{vmatrix} -6 & -2 \\\\8 & 5 \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} \\sqrt 3-\\sqrt 2 & \\sqrt 3-2 \\\\ \\sqrt 3+2 & \\sqrt 3+\\sqrt 2 \\end{vmatrix}$;\\\\\n(3) $\\begin{vmatrix} \\pi ^{m+n} & \\pi ^m-1 \\\\\\pi ^m+1 & \\pi ^{m-n} \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206033,7 +207489,9 @@
"008630": {
"id": "008630",
"content": "用行列式解下列方程组:\\\\\n(1) $\\begin{cases} x+2y=3, \\\\4x-y=3; \\end{cases}$\\\\(2) $\\begin{cases} 7x-5y-6=0, \\\\13x+9y-2=0. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206140,7 +207598,9 @@
"008635": {
"id": "008635",
"content": "展开下列行列式并化简:\\\\\n(1) $\\begin{vmatrix} x-1 & x^2 \\\\1 & x+1 \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} \\mathrm{e}^x & \\mathrm{e}^x-1 \\\\\\mathrm{e}^x+1 & \\mathrm{e}^{-x} \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206161,7 +207621,9 @@
"008636": {
"id": "008636",
"content": "用行列式解下列方程组:\\\\\n(1) $\\begin{cases}\n 15x-7y+5=0, \\\\22x-6y-14=0; \\end{cases}$\\\\\n(2) $\\begin{cases}\n \\dfrac 5x+\\dfrac 7y=3, \\\\\\dfrac 7x+\\dfrac 9y=4. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206224,7 +207686,9 @@
"008639": {
"id": "008639",
"content": "计算下列行列式的值: $\\begin{vmatrix} a & b \\\\c & d \\end{vmatrix}$、$\\begin{vmatrix} a & a+b \\\\c & c+d \\end{vmatrix}$. 你能得到关于行列式的一个怎样的性质? 猜想行列式还可能具有哪些类似的性质, 并加以证明.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206245,7 +207709,9 @@
"008640": {
"id": "008640",
"content": "用对角线法则计算下列三阶行列式:\\\\\n(1) $\\begin{vmatrix} 2 & 1 & 2 \\\\-3 & 4 & -1 \\\\3 & 6 & 5 \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} 1 & -2 & 0 \\\\2 & 1 & 0 \\\\6 & 3 & 1 \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206266,7 +207732,9 @@
"008641": {
"id": "008641",
"content": "用对角线法则展开下列三阶行列式, 并化简:\\\\\n(1) $\\begin{vmatrix} a & b & a+b \\\\b & a+b & a \\\\a+b & a & b \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} 0 & x & y \\\\x & 0 & z \\\\y & z & 0 \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206287,7 +207755,9 @@
"008642": {
"id": "008642",
"content": "试根据三阶行列式的展开式的定义, 将三阶行列式$\\begin{vmatrix} a_1 & b_1 & c_1 \\\\a_2 & b_2 & c_2 \\\\a_3 & b_3 & c_3 \\end{vmatrix}$推导成按第二列展开的形式.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206308,7 +207778,9 @@
"008643": {
"id": "008643",
"content": "用按某一行(或某一列)展开的方式计算下列三阶行列式:\\\\\n(1) $\\begin{vmatrix} 1 & 2 & 2 \\\\0 & 3 & 1 \\\\2 & 0 & 5 \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} 1 & 0 & 2 \\\\-3 & 4 & -1 \\\\2 & 0 & 5 \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206329,7 +207801,9 @@
"008644": {
"id": "008644",
"content": "用按某一行(或某一列)展开的方式化简下列三阶行列式:\\\\\n(1) $\\begin{vmatrix} x & y & z \\\\y & x & y \\\\z & z & x \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} 0 & 0 & a \\\\0 & b & c \\\\c & a & b \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206350,7 +207824,9 @@
"008645": {
"id": "008645",
"content": "把$\\begin{vmatrix} x_2 & y_2 \\\\x_3 & y_3 \\end{vmatrix}-\\begin{vmatrix} x_1 & y_1 \\\\x_3 & y_3 \\end{vmatrix}+\\begin{vmatrix} x_1 & y_1 \\\\x_2 & y_2 \\end{vmatrix}$表示成一个三阶行列式.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206371,7 +207847,9 @@
"008646": {
"id": "008646",
"content": "用行列式解下列三元一次方程组:\\\\\n(1) $\\begin{cases} x-2y+z=7, \\\\3x-5y+z=14, \\\\2x-2y-z=3; \\end{cases}$\\\\\n(2) $\\begin{cases} x+3y+z=11, \\\\2x+3y-4z=13, \\\\-x-3y+z=-9. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206392,7 +207870,9 @@
"008647": {
"id": "008647",
"content": "用行列式解关于$x,y,z$的方程组$\\begin{cases}\n x+y=a, \\\\y+z=b, \\\\z+x=c. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206457,7 +207937,9 @@
"008650": {
"id": "008650",
"content": "把$3\\begin{vmatrix} 1 & 3 \\\\3 & 1 \\end{vmatrix}-2\\begin{vmatrix} 0 & -2 \\\\3 & 1 \\end{vmatrix}-2\\begin{vmatrix} 0 & -2 \\\\1 & 3 \\end{vmatrix}$表示成一个三阶行列式为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206499,7 +207981,9 @@
"008652": {
"id": "008652",
"content": "用行列式解关于$x,y,z$的方程组$\\begin{cases}\n x-y+z=a, \\\\x+y-z=b, \\\\-x+y+z=c. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206564,7 +208048,9 @@
"008655": {
"id": "008655",
"content": "已知某个线性方程组的增广矩阵是$\\begin{pmatrix}\n 6 & -4 & 5 \\\\8 & -3 & 2 \\end{pmatrix}$.\\\\\n(1) 试写出这个增广矩阵对应的线性方程组;\\\\\n(2) 用矩阵变换的方法解这个方程组.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206585,7 +208071,9 @@
"008656": {
"id": "008656",
"content": "已知矩阵$A=\\begin{pmatrix} 2 & 1 \\\\-4 & 0 \\end{pmatrix}$, 矩阵$B=\\begin{pmatrix} 4 & 3 \\\\-7 & 0 \\end{pmatrix}$, 矩阵$C=\\begin{pmatrix} 1 & -2 & 0 \\\\-2 & 3 & 4 \\end{pmatrix}$.\\\\\n(1) 求$B-2A$;\\\\\n(2) 求$AB$;\\\\\n(3) 求$AC$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206606,7 +208094,9 @@
"008657": {
"id": "008657",
"content": "两角差的余弦公式为: $\\cos (\\alpha -\\beta)=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta$, 右式若用行列式表示, 则$\\cos (\\alpha -\\beta)=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206648,7 +208138,9 @@
"008659": {
"id": "008659",
"content": "计算下列行列式:\\\\\n(1) $\\begin{vmatrix} 3+\\sqrt 5 & \\sqrt 3-2 \\\\\\sqrt 3+2 & 3-\\sqrt 5 \\end{vmatrix}$;\\\\\n(2) $\\begin{vmatrix} 3 & 3 & -5 \\\\0 & -2 & 1 \\\\7 & 1 & 3 \\end{vmatrix}$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206669,7 +208161,9 @@
"008660": {
"id": "008660",
"content": "利用行列式解下列方程组:\\\\\n(1) $\\begin{cases} 2x-10y=3, \\\\4x+5y=1; \\end{cases}$\\\\ \n(2) $\\begin{cases} 3x+2y+z=3, \\\\-7x+4y+5z=-10, \\\\2x+3y-z=\\dfrac 92 \\end{cases}$\\\\\n(3) $\\begin{cases} \\dfrac{11}x-\\dfrac 1y-6=0, \\\\\\dfrac 3x+\\dfrac 5y+1=0. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206755,7 +208249,9 @@
"008664": {
"id": "008664",
"content": "已知矩阵$A=\\begin{pmatrix} 2 \\\\1 \\\\-3 \\end{pmatrix}$, 矩阵$B=\\begin{pmatrix}-1 \\\\ 2 \\end{pmatrix}$, 矩阵$C=\\begin{pmatrix} -3 & 0 \\\\-1 & 2 \\end{pmatrix}$, 求$(AB)C$和$A(BC)$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206797,7 +208293,9 @@
"008666": {
"id": "008666",
"content": "已知$abc\\ne 0$, 用行列式解关于$x,y,z$的方程组:\n$\\begin{cases}\n bx-ay=-2ab, \\\\-2cy+3bz=bc, \\\\cx+az=0. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -206818,7 +208316,9 @@
"008667": {
"id": "008667",
"content": "用行列式解关于$x,y$的方程组:\n$\\begin{cases} (m-1)x-y-m-1=0, \\\\(m-1)x+(m+1)y+1=0. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -207177,7 +208677,9 @@
"008684": {
"id": "008684",
"content": "某花店包装大小两种鲜花束出售, 每束包含鲜花的品种及其数量如下表(单位: 支):\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n \\diagbox{束别}{数量}{品种} & 玫瑰 & 康乃馨 & 兰草 \\\\ \\hline\n 大束 & $3$ & $5$ & $4$ \\\\ \\hline\n 小束 & $2$ & $3$ & $3$ \\\\ \\hline\n \\end{tabular}\n\\end{center}\n如果该花店在这个周末卖出的鲜花束的数量如下表:\n\\begin{center}\n \\begin{tabular}{|c|c|c|}\n \\hline\n \\diagbox{星期}{数量}{束别} & 大束 & 小束 \\\\ \\hline\n 星期六 & $20$ & $32$ \\\\ \\hline\n 星期天 & $24$ & $28$ \\\\ \\hline\n \\end{tabular}\n\\end{center}\n试用矩阵计算星期六和星期天该花店的玫瑰、康乃馨和兰草各卖出几支.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -207665,7 +209167,9 @@
"008707": {
"id": "008707",
"content": "用行列式解方程组: $\\begin{cases} \\dfrac 2x+\\dfrac 2y+\\dfrac 1z=2, \\\\-\\dfrac 7x+\\dfrac 4y+\\dfrac 5z=-10, \\\\\\dfrac 2x+\\dfrac 3y-\\dfrac 1z=\\dfrac 92. \\end{cases}$",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -207749,7 +209253,10 @@
"008711": {
"id": "008711",
"content": "近年来, 太阳能技术在生活中应用的步伐日益加快.某地区$2006$年太阳能电池的年生产量达到$7$亿瓦, 实际安装量为$5$亿瓦.假设以后若干年内太阳能电池的年生产量逐年递增$3$亿瓦, 年安装量的增长率保持在$30\\%$.\\\\\n(1) 以$2006$年为第$1$年, 写出第$n$年太阳能电池的年产量$a_n$与年安装量$b_n$;\\\\\n(2) 到哪一年, 年安装量不少于年生产量?",
- "objs": [],
+ "objs": [
+ "K0401007X",
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -218037,7 +219544,10 @@
"009191": {
"id": "009191",
"content": "判断下列说法是否正确. 如果正确, 请说明理由; 如果不正确, 请举一个反例.\\\\\n(1) 有两个相邻的侧面是矩形的棱柱是直棱柱;\\\\\n(2) 正四棱柱是正方体;\\\\\n(3) 底面是正多边形的棱锥是正棱锥.",
- "objs": [],
+ "objs": [
+ "K0615004B",
+ "K0618002B"
+ ],
"tags": [
"第六单元"
],
@@ -218058,7 +219568,9 @@
"009192": {
"id": "009192",
"content": "四棱柱集合$A$、平行六面体集合$B$、长方体集合$C$、正方体集合$D$之间有怎样的包含关系? 用文氏图表示出来.",
- "objs": [],
+ "objs": [
+ "K0615004B"
+ ],
"tags": [
"第六单元"
],
@@ -218229,7 +219741,10 @@
"009200": {
"id": "009200",
"content": "将一个直三棱柱分割成三个三棱锥, 试将这三个三棱锥分离, 并画出这些三棱锥的直观图.",
- "objs": [],
+ "objs": [
+ "K0615002B",
+ "K0618002B"
+ ],
"tags": [
"第六单元"
],
@@ -218250,7 +219765,10 @@
"009201": {
"id": "009201",
"content": "圆柱体、圆锥体的母线和旋转轴的位置关系如何?",
- "objs": [],
+ "objs": [
+ "K0615007B",
+ "K0618005B"
+ ],
"tags": [
"第六单元"
],
@@ -218271,7 +219789,9 @@
"009202": {
"id": "009202",
"content": "从一个底面半径和高都是$R$的圆柱中, 挖去一个以圆柱的上底为底、下底面的中心为顶点的圆锥, 得到一个如图所示的几何体. 如果用一个与圆柱下底面距离等于$d$并且平行于底面的平面去截这个几何体. 求截面面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) ellipse (1.5 and 0.5);\n\\draw (-1.5,0) --++ (0,-3) (1.5,0) --++ (0,-3);\n\\draw (-1.5,-3) arc (180:360:1.5 and 0.5);\n\\draw [dashed] (1.5,-3) arc (0:180:1.5 and 0.5) (-1.5,0) -- (0,-3) -- (1.5,0);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0618006B"
+ ],
"tags": [
"第六单元"
],
@@ -218292,7 +219812,9 @@
"009203": {
"id": "009203",
"content": "如果球的大圆面积增为原来的$100$倍, 那么球的半径有什么变化?",
- "objs": [],
+ "objs": [
+ "K0622004B"
+ ],
"tags": [
"第六单元"
],
@@ -218313,7 +219835,9 @@
"009204": {
"id": "009204",
"content": "已知$OA$是球$O$的半径, $OA=5$, $O_2$是$OA$上的两点, 平面$\\alpha$、$\\beta$分别通过点$O_2$, 且垂直于$OA$, 截得圆$O_1$和圆$O_2$, 当圆$O_1$、圆$O_2$的面积分别为$9$$\\pi$、$21\\pi$时, 求$O_1O_2$的长.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\filldraw (0,0) circle (0.05) node [below] {$O$};\n\\filldraw (0,5) circle (0.05) node [above] {$A$};\n\\filldraw (0,2) circle (0.05) node [right] {$O_2$};\n\\filldraw (0,4) circle (0.05) node [right] {$O_1$};\n\\draw (0,0) circle (5);\n\\draw [dashed] ({-sqrt(21)},2) arc (180:0:{sqrt(21)} and {sqrt(21)/4});\n\\draw ({-sqrt(21)},2) arc (180:360:{sqrt(21)} and {sqrt(21)/4});\n\\draw [dashed] (-3,4) arc (180:0:3 and {3/4});\n\\draw (-3,4) arc (180:360:3 and {3/4});\n\\draw [dashed] (0,0) -- (0,5);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -218355,7 +219879,9 @@
"009206": {
"id": "009206",
"content": "经过球面上不同两点的大圆有多少个? 并说明理由.",
- "objs": [],
+ "objs": [
+ "K0622004B"
+ ],
"tags": [
"第六单元"
],
@@ -218376,7 +219902,9 @@
"009207": {
"id": "009207",
"content": "已知正三棱锥的底面边长是$2$, 高是$4$, 求该正三棱锥的表面积.",
- "objs": [],
+ "objs": [
+ "K0620004B"
+ ],
"tags": [
"第六单元"
],
@@ -218418,7 +219946,9 @@
"009209": {
"id": "009209",
"content": "已知侧面积为$27$的正三棱柱的侧棱恰好是某个圆柱的三条母线, 且这个圆柱的底面半径为$2$, 求这个圆柱的表面积.",
- "objs": [],
+ "objs": [
+ "K0617006B"
+ ],
"tags": [
"第六单元"
],
@@ -218439,7 +219969,9 @@
"009210": {
"id": "009210",
"content": "如图, 已知一个圆锥的底面半径为$2$, 高为$2$, 且在这个圆锥中有一个高为$x$的圆柱.\\\\\n(1) 写出此圆柱的侧面积表达式;\\\\\n(2) 当$x$为何值时, 此圆柱的侧面积最大?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03);\n\\draw [dashed] (0,0) ellipse (1 and 0.25);\n\\draw [dashed] (0,1) ellipse (1 and 0.25);\n\\draw [dashed] (-1,0) -- (-1,1) (1,0) -- (1,1);\n\\draw [dashed] (-2,0) arc (180:0:2 and 0.5);\n\\draw (-2,0) arc (180:360:2 and 0.5);\n\\draw (-2,0) -- (0,2) -- (2,0);\n\\draw (1.1,0) -- (2.4,0) (1.1,1) -- (2.4,1);\n\\draw [->] (2.2,0.3) -- (2.2,0);\n\\draw [->] (2.2,0.7) -- (2.2,1);\n\\draw (2.2,0.5) node {$x$};\n\\draw (0,2.1) -- (0,2.5) (-2,2.1) -- (-2,2.5);\n\\draw [->] (-1.2,2.3) -- (-2,2.3);\n\\draw [->] (-0.8,2.3) -- (0,2.3);\n\\draw (-1,2.3) node {$2$};\n\\draw (-2.1,0) -- (-2.5,0) (-2.1,2) -- (-2.5,2);\n\\draw [->] (-2.3,1.2) -- (-2.3,2);\n\\draw [->] (-2.3,0.8) -- (-2.3,0);\n\\draw (-2.3,1) node {$2$};\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0617006B"
+ ],
"tags": [
"第六单元"
],
@@ -218460,7 +219992,9 @@
"009211": {
"id": "009211",
"content": "已知三棱柱的底面是$\\triangle ABC$, $AB=13\\text{cm}$, $BC=5\\text{cm}$, $CA=12\\text{cm}$, 侧棱$AA'$的长是$20\\text{cm}$, 且侧棱$AA'$与底面所成的角为$60^\\circ$, 求这个三棱柱的体积.",
- "objs": [],
+ "objs": [
+ "K0616003B"
+ ],
"tags": [
"第六单元"
],
@@ -218481,7 +220015,9 @@
"009212": {
"id": "009212",
"content": "在万吨水压机上, 有四根圆柱形钢柱, 高$18$米, 内径$0.4$米, 外径$1$米, 求这四根钢柱的质量(结果精确到$1$吨, 钢的密度为$7.9$克/立方厘米).",
- "objs": [],
+ "objs": [
+ "K0616004B"
+ ],
"tags": [
"第六单元"
],
@@ -218523,7 +220059,9 @@
"009214": {
"id": "009214",
"content": "一块正方形薄铁板的边长是$22$厘米, 以它的一个顶点为圆心、边长为半径画弧, 沿弧剪下一个扇形, 用这块扇形铁板围成一个圆锥筒, 求它的容积(结果精确到$1$立方厘米).",
- "objs": [],
+ "objs": [
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -218609,7 +220147,10 @@
"009218": {
"id": "009218",
"content": "有一个铜制工件, 它的下部分呈正四棱柱形, 上部分呈正四棱锥形, 且这个正四棱锥以正四棱柱的上底为底, 已知正四棱柱的底面边长是$50$毫米, 高是$10$毫米, 正四棱锥的侧面呈正三角形, 这个工件的质量是多少千克(结果精确到$0.1$千克, 铜的密度是$8.9$克/立方厘米)?",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -218651,7 +220192,9 @@
"009220": {
"id": "009220",
"content": "已知香港的位置为东经$114^\\circ 10'$, 北纬$22^\\circ 18'$, 江西井冈山的位置为东经$114^\\circ 10'$, 北纬$26^\\circ 34'$, 求这两个城市之间的距离. (地球半径约为$6371$千米, 结果精确到$1$千米)",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -218672,7 +220215,9 @@
"009221": {
"id": "009221",
"content": "在北纬$60^\\circ$圈上有甲乙两地, 它们的纬度圈上的弧长等于$\\dfrac{\\pi R}2$($R$是地球的半径), 求甲乙两地的球面距离.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -218693,7 +220238,9 @@
"009222": {
"id": "009222",
"content": "在北纬$45^\\circ$圈上有甲乙两地. 经度相差$90^\\circ$, 求甲乙两地的球面距离与地球半径的比.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -218714,7 +220261,9 @@
"009223": {
"id": "009223",
"content": "纬度为$\\alpha$的纬度圈上有甲乙两地, 它们的纬度圈上的弧长等于$\\pi R\\cos \\alpha$($R$是地球的半径), 求甲乙两地的球面距离.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -218735,7 +220284,9 @@
"009224": {
"id": "009224",
"content": "地球上有甲乙两个城市, 甲在北纬$30^\\circ$, 东经$83^\\circ$, 乙在北纬$30^\\circ$, 西经$97^\\circ$, 求过这两个城市在纬度圈上的距离与它们在地球表面上的球面距离的比.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -218756,7 +220307,12 @@
"009225": {
"id": "009225",
"content": "判断下列说法是否正确, 如果正确, 请说明理由; 如果不正确, 请举一个反例.\\\\\n(1) 直四棱柱是长方体;\\\\\n(2) 侧棱长相等, 且底面是正多边形的棱锥是正棱锥;\\\\\n(3) 各侧面都是正三角形的四棱锥是正四棱锥;\\\\\n(4) ``三条侧棱两两互相垂直, 且侧棱与底面所成角都相等''是``棱锥为正三棱锥''的充要条件.",
- "objs": [],
+ "objs": [
+ "K0615002B",
+ "K0618001B",
+ "K0618002B",
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -218777,7 +220333,9 @@
"009226": {
"id": "009226",
"content": "现有以下三个命题: \\textcircled{1} 底面是平行四边形的四棱柱是平行六面体; \\textcircled{2} 底面是矩形的平行六面体是长方体; \\textcircled{3} 直四棱柱是直平行六面体. 其中真命题的序号是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0615004B"
+ ],
"tags": [
"第六单元"
],
@@ -218798,7 +220356,9 @@
"009227": {
"id": "009227",
"content": "选择题:\n如果一个三棱锥的底面是直角三角形, 那么这个三棱锥的三个侧面\\bracket{20}\n\\twoch{都不是直角三角形}{至多只能有一个是直角三角形}{至多只能有两个是直角三角形}{可能都是直角三角形}",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -218819,7 +220379,9 @@
"009228": {
"id": "009228",
"content": "已知棱锥的侧棱与底面所成的角都相等, 试说出棱锥的顶点在底面内的射影所在的位置, 并证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -218840,7 +220402,9 @@
"009229": {
"id": "009229",
"content": "已知棱锥的顶点在底面内的射影在底面的内部, 其侧面与底面所成的角都相等, 试说出棱锥的顶点在底面内的射影所在的位置, 并证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [
"第六单元"
],
@@ -219050,7 +220614,9 @@
"009239": {
"id": "009239",
"content": "海面上地球球心角$1'$所对的大圆弧长约为$1$海里, $1$海里约是多少千米? (地球的半径约是$6371$千米)",
- "objs": [],
+ "objs": [
+ "K0622002B"
+ ],
"tags": [
"第六单元"
],
@@ -219071,7 +220637,9 @@
"009240": {
"id": "009240",
"content": "在半径是$r$的球面上, 有两点$AB$, 半径$OA$和$OB$的夹角是$n^\\circ (n<180) $求$AB$的两点的球面距离.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -219113,7 +220681,11 @@
"009242": {
"id": "009242",
"content": "已知圆柱$A$和圆锥$B$的底面直径和高都与球$C$的直径相等, 求证: 圆柱$A$、球$C$、圆锥$B$的体积的比是$3: 2: 1$.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0619003B",
+ "K0623002B"
+ ],
"tags": [
"第六单元"
],
@@ -219134,7 +220706,9 @@
"009243": {
"id": "009243",
"content": "在赤道上, 东经$140^\\circ$上有点$A$, 西经$130^\\circ$上有点$B$, 求$AB$两点的球面距离(地球半径约为$6371$千米).",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -219155,7 +220729,9 @@
"009244": {
"id": "009244",
"content": "设$AB$是球$O$的直径, $AB=50$, $O_1,O_2$是$AB$上的两点, 平面$\\alpha, \\beta$分别通过点$O_1,O_2$, 且垂直于$AB$截得圆$O_1$圆$O_2$, 当圆$O_1$圆$O_2$的面积分别为$49\\pi$、$400\\pi$时, 求$O_1,O_2$两点的距离.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -222459,7 +224035,11 @@
"009399": {
"id": "009399",
"content": "在球内有相距$9\\text{cm}$的两个平行的截面. 若两截面圆为球的小圆时, 其面积分别为$49\\pi \\text{cm}^2$、$400\\pi \\text{cm}^2$, 求此球的表面积及体积.",
- "objs": [],
+ "objs": [
+ "K0622005B",
+ "K0623002B",
+ "K0623004B"
+ ],
"tags": [
"第六单元"
],
@@ -222480,7 +224060,10 @@
"009400": {
"id": "009400",
"content": "已知圆锥的母线$l$与底面成$45^\\circ$角, 这个圆锥的体积为$9\\pi \\text{cm}^3$, 求这个圆锥的高$h$及侧面积.",
- "objs": [],
+ "objs": [
+ "K0617006B",
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -222522,7 +224105,9 @@
"009402": {
"id": "009402",
"content": "设地球半径为$R$, 城市$A$位于东经$90^\\circ$, 北纬$60^\\circ$, 城市$B$位于东经$150^\\circ$、北纬$60^\\circ$, 求城市$AB$之间的距离.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第六单元"
],
@@ -222734,7 +224319,9 @@
"009412": {
"id": "009412",
"content": "设球的半径为$4$, 用一个平面截球, 使截面圆的半径为$2$, 则截面与球心的距离\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -222839,7 +224426,9 @@
"009417": {
"id": "009417",
"content": "一个球受热膨胀, 如果它的表面积增加$21\\%$. 那么这个球的半径增加多少?",
- "objs": [],
+ "objs": [
+ "K0623004B"
+ ],
"tags": [
"第六单元"
],
@@ -222902,7 +224491,10 @@
"009420": {
"id": "009420",
"content": "以一个正方体的顶点为顶点能组成多少个三棱锥?",
- "objs": [],
+ "objs": [
+ "K0615001B",
+ "K0618001B"
+ ],
"tags": [
"第八单元"
],
@@ -229404,7 +230996,10 @@
"009706": {
"id": "009706",
"content": "证明: 棱柱的所有侧面都是平行四边形.",
- "objs": [],
+ "objs": [
+ "K0615002B",
+ "K0615006B"
+ ],
"tags": [
"第六单元"
],
@@ -229425,7 +231020,9 @@
"009707": {
"id": "009707",
"content": "证明: 平行于棱柱底面的平面截这个棱柱所得到的截面是一个与底面全等的多边形.",
- "objs": [],
+ "objs": [
+ "K0615006B"
+ ],
"tags": [
"第六单元"
],
@@ -229446,7 +231043,10 @@
"009708": {
"id": "009708",
"content": "一个水平放置的封闭圆柱形容器中装了部分的水, 此时水面的形状是什么图形? 如果把圆柱沿侧面放倒在水平的面上, 那么水面的形状又会是什么图形? 请分别画出以上两种情形的示意图.",
- "objs": [],
+ "objs": [
+ "K0615007B",
+ "K0615008B"
+ ],
"tags": [
"第六单元"
],
@@ -229488,7 +231088,10 @@
"009710": {
"id": "009710",
"content": "一个圆柱形油桶的底面半径为$50\\text{cm}$, 高为$100\\text{cm}$. 求这个油桶的体积.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0616004B"
+ ],
"tags": [
"第六单元"
],
@@ -229509,7 +231112,10 @@
"009711": {
"id": "009711",
"content": "查一查六角螺帽的尺寸规格, 并说明如何计算它的体积.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0616004B"
+ ],
"tags": [
"第六单元"
],
@@ -229530,7 +231136,10 @@
"009712": {
"id": "009712",
"content": "如图(图中单位: $\\text{cm}$)是一种机器零件, 零件下部是实心的直六棱\n柱(底面是正六边形, 侧面是全等的矩形), 上部是实心的圆柱. 求此零件的体积与表面积. (结果分别精确到$0.1\\text{cm}^3$与$0.1\\text{cm}^2$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.1]\n\\draw (-6,0) -- (6,0) -- (6,5) -- (-6,5) -- cycle;\n\\draw (6,5) --++ (8,5) --++ (0,-5) -- (6,0);\n\\draw (-6,5) --++ (-8,5) coordinate (T) --++ (0,-5) -- (-6,0);\n\\draw (T) --++ (8,5) --++ (12,0) --++ (8,-5);\n\\draw (T) ++ (14,0) coordinate (O);\n\\filldraw [white] (O) ++ (3,0) rectangle++ (-6,25);\n\\draw (O) ++ (3,0) --++ (0,25);\n\\draw (O) ++ (-3,0) --++ (0,25);\n\\draw (O) ++ (3,0) arc (0:-180:3 and 1.5);\n\\draw (O) ++ (0,25) ellipse (3 and 1.5);\n\\draw (-6,-1) -- (-6,-3) (6,-1) -- (6,-3);\n\\draw [->] (-2,-2) -- (-6,-2);\n\\draw [->] (2,-2) -- (6,-2);\n\\draw (0,-2) node {$12$};\n\\draw (15,5) -- (19,5) (15,10) -- (19,10);\n\\draw (17,7.5) node {\\rotatebox{90}{$5$}};\n\\draw [->] (17,0) -- (17,5);\n\\draw [->] (17,20) -- (17,10);\n\\draw [->] (17,25) -- (17,35);\n\\draw (6,35) -- (19,35);\n\\draw (17,22.5) node {\\rotatebox{90}{$25$}};\n\\draw (-3,38) -- (-3,42) (3,38) -- (3,42);\n\\draw [->] (-8,40) -- (-3,40);\n\\draw [->] (8,40) -- (3,40);\n\\draw (0,40) node {$6$}; \n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0617007B"
+ ],
"tags": [
"第六单元"
],
@@ -229593,7 +231202,9 @@
"009715": {
"id": "009715",
"content": "用平行于棱锥底面的平面截这个棱锥, 得到一个小棱锥. 已知这两个棱锥的高分别是$h_1$、$h_2$, 求这两个棱锥的底面面积之比.",
- "objs": [],
+ "objs": [
+ "K0618006B"
+ ],
"tags": [
"第六单元"
],
@@ -229614,7 +231225,9 @@
"009716": {
"id": "009716",
"content": "(1) 过圆锥的任意两条母线作一个平面与圆锥相截, 得到的截面是什么图形? 在什么条件下, 所得到的截面面积最大? \\\\\n(2) 如果圆锥的母线与底面所成的角为$60^\\circ$, 那么经过圆锥两\n条母线的平面与圆锥底面所成的二面角有可能小于$60^\\circ$吗?",
- "objs": [],
+ "objs": [
+ "K0618005B"
+ ],
"tags": [
"第六单元"
],
@@ -229635,7 +231248,9 @@
"009717": {
"id": "009717",
"content": "显然, 通过延长圆台的任意一条母线都可以使它们交于一点, 从而得到一个圆锥. 如图, 这样的几何体是否也可以通过延长棱的方法得到一个棱锥?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) -- (3,0,0) coordinate (B) --++ (0,0,-2) coordinate (C);\n\\draw [dashed] (0,0,0) -- (0,0,-2) coordinate (D) --++ (3,0,0);\n\\draw (1,3,-1) coordinate (S) (2,3,-1) coordinate (T);\n\\draw ($(A)!0.5!(S)$) coordinate (A1) ($(D)!0.5!(S)$) coordinate (D1) ($(C)!0.5!(T)$) coordinate (C1) ($(B)!0.5!(T)$) coordinate (B1);\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": [],
+ "objs": [
+ "K0618007B"
+ ],
"tags": [
"第六单元"
],
@@ -229698,7 +231313,9 @@
"009720": {
"id": "009720",
"content": "已知圆台上、下底面的半径分别为$r_1$、$r_2$, 高为$h$. 求证: $V_{\\text{圆台}}=\\dfrac 13\\pi (r_1^2+r_1r_2+r_2^2)h$.",
- "objs": [],
+ "objs": [
+ "K0619005B"
+ ],
"tags": [
"第六单元"
],
@@ -229803,7 +231420,9 @@
"009725": {
"id": "009725",
"content": "有两个面平行, 其余各面都是平行四边形的多面体一定是柱体吗? 请给出你的理由或反例.",
- "objs": [],
+ "objs": [
+ "K0615002B"
+ ],
"tags": [
"第六单元"
],
@@ -229845,7 +231464,9 @@
"009727": {
"id": "009727",
"content": "$O$为球心, $O_1$为小圆的圆心, 用球的半径$r$和小圆的半径$r_1$表示$\nOO_1$的距离$d$.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -229866,7 +231487,9 @@
"009728": {
"id": "009728",
"content": "已知半径为$R$的球面上三点$A$、$B$、$C$满足$AB=6$, $BC=8$, $CA=10$, 球心到平面$ABC$的距离为$12$. 求球的半径$R$.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -229887,7 +231510,9 @@
"009729": {
"id": "009729",
"content": "已知上海地处东经$120^\\circ 52'$至$122^\\circ 12'$, 北纬$30^\\circ 40'$至$31^\\circ 53'$之间, 地球半径为$6371.004\\text{km}$. 求上海所辖区域:\\\\\n(1) 经线对应的两平面所成的二面角的大小;\\\\\n(2) 纬线所在两平面的距离.",
- "objs": [],
+ "objs": [
+ "K0622006B"
+ ],
"tags": [
"第六单元"
],
@@ -229929,7 +231554,9 @@
"009731": {
"id": "009731",
"content": "把一个半径为$R$的实心铁球熔化铸成两个小球, 两个小球的半径之比为$1:2$. 求其中较小球的半径.",
- "objs": [],
+ "objs": [
+ "K0623002B"
+ ],
"tags": [
"第六单元"
],
@@ -229950,7 +231577,10 @@
"009732": {
"id": "009732",
"content": "如图, 有一个水平放置的透明无盖的正方体容器, 容器高$8\\text{cm}$, 将一个球放在容器口, 再向容器注水, 当球面恰好接触水面时, 测得水深为$\n6\\text{cm}$. 若不计容器的厚度, 求球的体积. \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (A) --++ (2,0) coordinate (B) --++ (45:{2/2}) coordinate (C)\n--++ (0,2) coordinate (C1) ++ (-2,0) coordinate (D1) ($(D1)!0.5!(0,2)$)--++ (225:{1/2}) coordinate (A1) -- (A);\n\\draw [dashed] ($(D1)!0.5!(0,2)$) -- (D1);\n\\draw (A) ++ (2,2) coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\draw (0,0) ++ (1,0) ++ (45:0.5) ++ (0,2) ++ (0,0.75) coordinate (O);\n\\draw (C1) -- (2.54,2.71);\n\\draw [dashed] (2.54,2.71) -- (D1);\n\\filldraw (O) circle (0.03);\n\\draw [dashed] (O) ++ (241.99:1.25) arc (241.99:298.01:1.25);\n\\draw (O) ++ (241.99:1.25) arc (241.99:{298.01-360}:1.25);\n\\draw (O) ++ (1.25,0) arc (0:-180:1.25 and 0.5);\n\\draw [dashed] (O) ++ (1.25,0) arc (0:180:1.25 and 0.5);\n\\draw (0,1.5) --++ (2,0) --++ (45:1);\n\\draw [dashed] (0,1.5) --++ (45:1) --++ (2,0);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0622001B",
+ "K0622002B"
+ ],
"tags": [
"第六单元"
],
@@ -232819,7 +234449,9 @@
"009868": {
"id": "009868",
"content": "已知三棱锥$ABCD$的三条侧棱$AB$、$AC$、$AD$两两垂直, 且$|AB|=1$, $|AC|=2$, $|AD|=3$. 求顶点$A$到平面$BCD$的距离.",
- "objs": [],
+ "objs": [
+ "K0619004B"
+ ],
"tags": [
"第六单元"
],
@@ -232966,7 +234598,9 @@
"009875": {
"id": "009875",
"content": "下列数列中成等差数列的是\\bracket{20}.\n\\fourch{$0,1,3,5,7$}{$1,\\dfrac 13,\\dfrac 15,\\dfrac 17,\\dfrac 19$}\n{$1,\\sqrt 2,\\sqrt 3,2,\\sqrt 5$}{$1,\\dfrac 13,-\\dfrac 13,-1,-\\dfrac 53$}",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -232987,7 +234621,9 @@
"009876": {
"id": "009876",
"content": "设数列$\\{a_n\\}$为等差数列, 其公差为$d$.\\\\\n(1) 已知$a_1=-1$, $d=4$, 求$a_8$;\\\\\n(2) 已知$a_7=8$, $d=-13$, 求$a_1$;\\\\\n(3) 已知$a_1=9$, $d=-2$, $a_n=-15$, 求$n$.",
- "objs": [],
+ "objs": [
+ "K0401004X"
+ ],
"tags": [
"第四单元"
],
@@ -233008,7 +234644,9 @@
"009877": {
"id": "009877",
"content": "已知数列$\\{a_n\\}$是等差数列, 正整数$m$、$n$、$p$、$q$满足$m+n=p+q$. 求证: $a_m+a_n=a_p+a_q$.",
- "objs": [],
+ "objs": [
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -233031,7 +234669,10 @@
"009878": {
"id": "009878",
"content": "计算$\\displaystyle\\sum_{i=1}^n 2i$.",
- "objs": [],
+ "objs": [
+ "K0402003X",
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -233052,7 +234693,9 @@
"009879": {
"id": "009879",
"content": "设数列$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=-4$, $a_8=-18$, 求$S_8$;\\\\\n(2) 已知$a_1=-4$, $a_{12}=18$, 求$S_{15}$.",
- "objs": [],
+ "objs": [
+ "K0402004X"
+ ],
"tags": [
"第四单元"
],
@@ -233073,7 +234716,10 @@
"009880": {
"id": "009880",
"content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=n^2-3n$, 求证: 数列$\\{a_n\\}$是等差数列.",
- "objs": [],
+ "objs": [
+ "K0402005X",
+ "K0402002X"
+ ],
"tags": [
"第四单元"
],
@@ -233096,7 +234742,9 @@
"009881": {
"id": "009881",
"content": "下列数列中成等比数列的是\\bracket{20}.\n\\fourch{$1,\\dfrac 14,\\dfrac 19,\\dfrac1{16}$}{$1,1,-1,-1$}{$1,\\dfrac{\\sqrt 2}2,\\dfrac 12,\\dfrac{\\sqrt 2}4$}{$\\dfrac 12,2,\\dfrac 12,2$}",
- "objs": [],
+ "objs": [
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -233117,7 +234765,9 @@
"009882": {
"id": "009882",
"content": "设数列$\\{a_n\\}$为等比数列, 其公比为$q$.\\\\\n(1) 已知$a_1=-3$, $q=2$, 求$a_5$;\\\\\n(2) 已知$a_1=1$, $q=2$, $a_n=16$, 求$n$;\\\\\n(3) 已知$a_1=\\dfrac 13$, $a_7=9$, 求$q$;\\\\\n(4) 已知$q=-\\dfrac 32$, $a_4=-27$, 求$a_1$.",
- "objs": [],
+ "objs": [
+ "K0403002X"
+ ],
"tags": [
"第四单元"
],
@@ -233138,7 +234788,9 @@
"009883": {
"id": "009883",
"content": "已知数列$\\{a_n\\}$是等比数列, 正整数$m$、$n$、$s$、$t$满足$m+n=s+t$. 求证: $a_m\\cdot a_n=a_s\\cdot a_t$.",
- "objs": [],
+ "objs": [
+ "K0403004X"
+ ],
"tags": [
"第四单元"
],
@@ -233161,7 +234813,10 @@
"009884": {
"id": "009884",
"content": "设数列$\\{a_n\\}$为等比数列, 其前$n$项和为$S_n$.\\\\\n(1) 已知$a_1=3$, 公比$q=2$, 求$S_6$;\\\\\n(2) 已知$a_1=-2.7$, 公比$q=-\\dfrac 13$, $a_n=\\dfrac1{90}$, 求$S_n$.",
- "objs": [],
+ "objs": [
+ "K0403004X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -233182,7 +234837,10 @@
"009885": {
"id": "009885",
"content": "已知等比数列$\\{a_n\\}$的前$5$项和为$10$, 前$10$项和为$50$. 求这个数列的前$15$项和.",
- "objs": [],
+ "objs": [
+ "K0404002X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -233205,7 +234863,10 @@
"009886": {
"id": "009886",
"content": "中国古代数学著作《算法统宗》中有这样一个问题: ``三百七十八里关, 初行健步不为难, 次日脚痛减一半, 六朝才得到其关, 要见次日行里数, 请公仔细算相还.''其意思为: 有一个人要走$378$里路, 第一天健步行走, 从第二天起因为脚痛, 每天走的路程为前一天的一半, 走了$6$天后到达目的地. 请问第二天走了多少里.",
- "objs": [],
+ "objs": [
+ "K0405005X",
+ "K0404003X"
+ ],
"tags": [
"第四单元"
],
@@ -233226,7 +234887,9 @@
"009887": {
"id": "009887",
"content": "计算$\\displaystyle\\sum_{i=1}^{\\infty}(\\dfrac 13)^i$.",
- "objs": [],
+ "objs": [
+ "K0405002X"
+ ],
"tags": [
"第四单元"
],
@@ -233247,7 +234910,9 @@
"009888": {
"id": "009888",
"content": "化下列循环小数为分数:\\\\\n(1) $0.\\dot1 \\dot 3$;\\\\\n(2) $1.3\\dot 3\\dot 2$.",
- "objs": [],
+ "objs": [
+ "K0405004X"
+ ],
"tags": [
"第四单元"
],
@@ -233268,7 +234933,9 @@
"009889": {
"id": "009889",
"content": "如图, 已知等边三角形$ABC$的面积等于$1$, 连接这个三角形各边的中点得到一个小的三角形$A_1B_1C_1$, 又连接三角形$A_1B_1C_1$各边的中点得到一个更小的三角形$A_2B_2C_2$, 这样的过程可以无限继续下去. 求所有三角形$A_iB_iC_i$($i=1,2,3,\\cdots$)的面积的和. \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B) -- (3,0) node [below] {$C$} coordinate (C) --++ (120:3) node [above] {$A$} coordinate (A) -- cycle;\n\\draw ($(A)!0.5!(B)$) node [left] {$A_1$} coordinate (A1) -- ($(B)!0.5!(C)$) node [below] {$B_1$} coordinate (B1) -- ($(C)!0.5!(A)$) node [right] {$C_1$} coordinate (C1) -- cycle;\n\\draw ($(A1)!0.5!(B1)$) node [left] {$A_2$} coordinate (A2) -- ($(B1)!0.5!(C1)$) node [right] {$B_2$} coordinate (B2) -- ($(C1)!0.5!(A1)$) node [above] {$C_2$} coordinate (C_2) -- cycle;\n\\draw (30:{sqrt(3)}) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0405005X"
+ ],
"tags": [
"第四单元"
],
@@ -233289,7 +234956,10 @@
"009890": {
"id": "009890",
"content": "根据数列$\\{a_n\\}$的通项公式填表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n$n$ & $1$ & $2$ & $\\cdots$ & $5$ & $\\cdots$ & $$ & $\\cdots$ & $n$ & $\\cdots$ \\\\ \\hline\n$a_n$ & $$ & $$ & $\\cdots$ & $$ & $\\cdots$ & $156$ & $\\cdots$ & $n(n+1)$ & $\\cdots$ \\\\ \\hline\n\\end{tabular}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0406002X",
+ "K0406003X"
+ ],
"tags": [
"第四单元"
],
@@ -233310,7 +234980,9 @@
"009891": {
"id": "009891",
"content": "图中的三角形图案称为谢宾斯基三角形. 在下图四个三角形图案中, 着色的小三角形的个数依次排列成一个数列的前四项, 请写出其前四项, 并给出这个数列的一个通项公式.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, gray]\n\\newcommand{\\firstlevel}[2]{\\filldraw #1 --++ (#2,0) --++ (120:#2) -- cycle;}\n\\firstlevel{(0,0)}{2}\n\\newcommand{\\secondlevel}[2]{\\firstlevel{#1}{#2/2} \\firstlevel{#1 ++ ({#2/2},0)}{#2/2} \\firstlevel{#1 ++ (60:{#2/2})}{#2/2}}\n\\secondlevel{(2.5,0)}{2}\n\\newcommand{\\thirdlevel}[2]{\\secondlevel{#1}{#2/2} \\secondlevel{#1 ++ ({#2/2},0)}{#2/2} \\secondlevel{#1 ++ (60:{#2/2})}{#2/2}}\n\\thirdlevel{(5,0)}{2}\n\\newcommand{\\fourthlevel}[2]{\\thirdlevel{#1}{#2/2} \\thirdlevel{#1 ++ ({#2/2},0)}{#2/2} \\thirdlevel{#1 ++ (60:{#2/2})}{#2/2}}\n\\fourthlevel{(7.5,0)}{2}\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0403005X"
+ ],
"tags": [
"第四单元"
],
@@ -233331,7 +235003,10 @@
"009892": {
"id": "009892",
"content": "已知数列$\\{a_n\\}$的通项公式是$a_n=|2n-7|$. 试问: 该数列是否有最小项? 若有, 指出第几项最小; 若没有, 试说明理由.",
- "objs": [],
+ "objs": [
+ "K0406004X",
+ "K0406005X"
+ ],
"tags": [
"第四单元"
],
@@ -233352,7 +235027,9 @@
"009893": {
"id": "009893",
"content": "已知数列$\\{a_n\\}$对任意正整数$n$, 均满足$a_1a_2\\cdots a_n=n^2$.\\\\\n(1) 写出数列$\\{a_n\\}$的前五项;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第四单元"
],
@@ -233373,7 +235050,9 @@
"009894": {
"id": "009894",
"content": "在数列$\\{a_n\\}$中, $a_1=2$, 且$a_n=a_{n-1}+\\lg\\dfrac n{n-1}$($n\\ge 2$). 求数列$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0407002X"
+ ],
"tags": [
"第四单元"
],
@@ -233394,7 +235073,10 @@
"009895": {
"id": "009895",
"content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_n=2a_{n-1}+3$($n\\ge 2$).\\\\\n(1) 求证: 数列$\\{a_n+3\\}$为等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.",
- "objs": [],
+ "objs": [
+ "K0407002X",
+ "K0403003X"
+ ],
"tags": [
"第四单元"
],
@@ -233415,7 +235097,9 @@
"009896": {
"id": "009896",
"content": "请指出下列各题用数学归纳法证明过程中的错误.\\\\\n(1) 设$n$为正整数, 求证: $2+4+6+\\cdots +2n=n^2+n+1$.\\\\\n证明: 假设当$n=k$($k$为正整数)时等式成立, 即有$2+4+6+\\cdots+2k=k^2+k+1$. 那么当$n=k+1$时, 就有$2+4+6+\\cdots+2k+2(k+1)\n=k^2+k+1+2(k+1)=(k+1)^2+(k+1)+1$. 因此, 对于任意正整数$n$等式都成立.\\\\\n(2) 设$n$为正整数, 求证: $1+2+2^2+\\cdots+2^{n-1}=2^n-1$.\\\\\n证明: \\textcircled{1} 当$n=1$时, 左边$=1$, 右边$=1$, 等式成立.\\\\\n\\textcircled{2} 假设当$n=k$($k$为正整数)时, 等式成立, 即有\n$1+2+2^2+\\cdots+2^{k-1}=2^k-1$. 那么当$n=k+1$时, 由等比数列求和公式, 就有$1+2+2^2+\\cdots+2^{k-1}+2^k=\\dfrac{1\\times(1-2^{k+1})}{1-2}=2^{k+1}-1$, 等式也成立.\\\\\n根据\\textcircled{1}和\\textcircled{2}, 由数学归纳法可以断定$1+2+2^2+\\cdots+2^{n-1}=2^n-1$对任意正整数$n$都成立.",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -233436,7 +235120,9 @@
"009897": {
"id": "009897",
"content": "用数学归纳法证明: $-1+3-5+\\cdots+(-1)^n(2n-1)=(-1)^nn$($n$为正整数).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -233459,7 +235145,9 @@
"009898": {
"id": "009898",
"content": "用数学归纳法证明: $1\\times 2+2\\times 3+3\\times 4+\\cdots+n(n+1)=\\dfrac{n(n+1)(n+2)}3$($n$为正整数).",
- "objs": [],
+ "objs": [
+ "K0408003X"
+ ],
"tags": [
"第四单元"
],
@@ -233482,7 +235170,9 @@
"009899": {
"id": "009899",
"content": "已知数列: $\\dfrac 1{1\\times 2},\\dfrac 1{2\\times 3},\\dfrac 1{3\\times 4}, \\cdots,\\dfrac 1{n(n+1)}, \\cdots$, 设$S_n$为该数列的前$n$项和. 计算$S_1,S_2,S_3,S_4$的值; 根据计算的结果, 猜想$S_n$的表达式, 并用数学归纳法加以证明.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -233503,7 +235193,9 @@
"009900": {
"id": "009900",
"content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_n+1=\\dfrac{3a_n}{a_n+3}$, $a_n\\ne 0$.\\\\\n(1) 求$a_2$, $a_3$, $a_4$;\\\\\n(2) 猜想数列$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.",
- "objs": [],
+ "objs": [
+ "K0409001X"
+ ],
"tags": [
"第四单元"
],
@@ -233526,7 +235218,9 @@
"009901": {
"id": "009901",
"content": "是否存在常数$a$、$b$, 使等式$1^2+3^2+5^2+\\cdots+(2n-1)^2=an^3+bn$对任意正整数$n$都成立? 证明你的结论.",
- "objs": [],
+ "objs": [
+ "K0409002X"
+ ],
"tags": [
"第四单元"
],
@@ -235413,7 +237107,9 @@
"009987": {
"id": "009987",
"content": "已知$a$是实数, 行列式$\\begin{vmatrix} a & 1 \\\\ 3 & 2\\end{vmatrix}$的值与行列式$\\begin{vmatrix} a & 0 \\\\ 4 & 1\\end{vmatrix}$的值相等, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"暂无对应"
],
@@ -235434,7 +237130,10 @@
"009988": {
"id": "009988",
"content": "已知圆柱的高为$4$, 底面积为$9\\pi$, 则圆柱的侧面积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0617006B"
+ ],
"tags": [
"第六单元"
],
@@ -247101,7 +248800,9 @@
"010496": {
"id": "010496",
"content": "若圆柱的底面半径是$1$, 母线长为$2$, 则这个圆柱的体积是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B"
+ ],
"tags": [
"第六单元"
],
@@ -247122,7 +248823,10 @@
"010497": {
"id": "010497",
"content": "若一个圆柱的侧面积是$4\\pi$, 高为$1$, 则这个圆柱的体积是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0617006B"
+ ],
"tags": [
"第六单元"
],
@@ -247143,7 +248847,9 @@
"010498": {
"id": "010498",
"content": "若正六棱柱的高为$4$, 底面边长为$2$, 则这个正六棱柱的体积是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B"
+ ],
"tags": [
"第六单元"
],
@@ -247164,7 +248870,9 @@
"010499": {
"id": "010499",
"content": "将一个棱长为$a$的正方体切成$27$个全等的小正方体, 其表面积增加了\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0617004B"
+ ],
"tags": [
"第六单元"
],
@@ -247185,7 +248893,9 @@
"010500": {
"id": "010500",
"content": "已知侧面都是矩形的四棱柱, 侧棱长为$5$, 底面是边长为$2$的菱形, 则这个棱柱的侧面积是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0617004B"
+ ],
"tags": [
"第六单元"
],
@@ -247247,8 +248957,10 @@
},
"010503": {
"id": "010503",
- "content": "已知三棱柱$ABCA_1B_1C_1$的三个侧面均是矩形, 求证:它的任意两个侧面的面积之和大于第三个侧面的面积.",
- "objs": [],
+ "content": "已知三棱柱$ABC-A_1B_1C_1$的三个侧面均是矩形, 求证:它的任意两个侧面的面积之和大于第三个侧面的面积.",
+ "objs": [
+ "K0615002B"
+ ],
"tags": [
"第六单元"
],
@@ -247290,7 +249002,9 @@
"010505": {
"id": "010505",
"content": "如图, 设圆柱有一个内接棱柱(即棱柱的侧棱都是圆柱的母线, 棱柱的两个底面分别在圆柱的两个底面内). 已知圆柱的体积是$4\\sqrt 3\\pi$, 棱柱的底面是边长为$2$的正三角形. 求棱柱的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) ellipse (1 and 0.5);\n\\draw (-1,-2) arc (180:360:1 and 0.5);\n\\draw [dashed] (-1,-2) arc (180:0:1 and 0.5);\n\\draw (0,-2) ++ ({cos(50)},{0.5*sin(50)}) node [above right] {$C$} coordinate (C);\n\\draw (0,-2) ++ ({cos(170)},{0.5*sin(170)}) node [left] {$A$} coordinate (A);\n\\draw (0,-2) ++ ({cos(290)},{0.5*sin(290)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2) node [left] {$A'$} coordinate (A1);\n\\draw (B) ++ (0,2) node [above left] {$B'$} coordinate (B1);\n\\draw (C) ++ (0,2) node [above] {$C'$} coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- (A1);\n\\draw (A) -- (A1) (B) -- (B1) (A) -- (B);\n\\draw [dashed] (B) -- (C) -- (A) (C) -- (C1);\n\\draw (-1,-2) -- (-1,0) (1,-2) -- (1,0);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0616003B"
+ ],
"tags": [
"第六单元"
],
@@ -247332,7 +249046,9 @@
"010507": {
"id": "010507",
"content": "已知一个正三棱锥的底面边长为$6$, 侧棱长为$\\sqrt 15$. 求这个三棱锥的体积.",
- "objs": [],
+ "objs": [
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -247395,7 +249111,9 @@
"010510": {
"id": "010510",
"content": "棱锥被平行于底面的平面所截, 当截面分别平分棱锥的高与体积时, 相应的截面面积分别为$S_1$、$S_2$. 求证: $S_1=latex]\n\\draw (0,0) node [below left] {$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$} coordinate (D) --++ (3,0) (D) --++ (0,3);\n\\draw ($(A)!0.5!(D1)$) coordinate (P);\n\\draw ($(B)!0.5!(A1)$) coordinate (Q);\n\\draw ($(C)!0.5!(B1)$) coordinate (R);\n\\draw ($(D)!0.5!(C1)$) coordinate (S);\n\\draw ($(A)!0.5!(C)$) coordinate (X);\n\\draw ($(A1)!0.5!(C1)$) coordinate (Y);\n\\draw [dashed] (P) -- (Q) -- (R) -- (S) -- (P);\n\\draw [dashed] (X) -- (P) -- (Y) (X) -- (Q) -- (Y) (X) -- (R) -- (Y) (X) -- (S) -- (Y);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -247605,7 +249343,11 @@
"010520": {
"id": "010520",
"content": "如图, 设$E$、$F$、$G$分别是正方体$ABCD-A_1B_1C_1D_1$的共点的三条棱$A_1B_1$、$B_1B$、$B_1C_1$的中点, 过这三个点的平面截正方体得到的一个``角''是四面体$B_1EFG$. 设正方体的棱长为$1$.\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 [below left] {$B_1$} 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 ($(B1)!0.5!(A1)$) node [above] {$E$} 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 (E) -- (F) -- (G) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:四面体$B_1EFG$是以$B_1$为顶点、以$EFG$为底面的正三棱锥;\\\\\n(2) 在四面体$B_1EFG$中, 求顶点$B_1$到底面$EFG$的距离;\\\\\n(3) 如果将正方体按照题设的方法截去八个``角'', 那么剩余的多面体有几个顶点、几条棱、几个面? 并求这个剩余多面体的表面积与体积.",
- "objs": [],
+ "objs": [
+ "K0615001B",
+ "K0618002B",
+ "K0619004B"
+ ],
"tags": [
"第六单元"
],
@@ -247626,7 +249368,10 @@
"010521": {
"id": "010521",
"content": "在如图所示的多面体中, 已知$ABCD$为矩形, $ABFE$和$DCFE$为全等的等腰梯形, $AB=4$, $BC=AE=EF=2$. 求此多面体的表面积与体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,-2) node [below right] {$D$} coordinate (D);\n\\draw (4,0,-2) node [right] {$C$} coordinate (C);\n\\draw (1,{sqrt(2)},-1) node [above] {$E$} coordinate (E);\n\\draw (E) --++ (2,0,0) node [above] {$F$} coordinate (F);\n\\draw (A) -- (B) -- (C) (A) -- (E) (F) -- (B) (F) -- (C);\n\\draw [dashed] (E) -- (D) (A) -- (D) -- (C);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0616004B",
+ "K0617007B"
+ ],
"tags": [
"第六单元"
],
@@ -247647,7 +249392,10 @@
"010522": {
"id": "010522",
"content": "一个直角三角形有一个$\\dfrac\\pi 6$的内角, 这个内角所对直角边的长度为$1$. 把这个三角形绕其斜边旋转一周, 求所得旋转体的表面积与体积.",
- "objs": [],
+ "objs": [
+ "K0616004B",
+ "K0617007B"
+ ],
"tags": [
"第六单元"
],
@@ -247668,7 +249416,9 @@
"010523": {
"id": "010523",
"content": "如图, 给定一个正方体形状的土豆块, 只切一刀, 可以得到下面哪些类型的多面体? \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\definecolor{frontcolor}{RGB}{249, 203, 142}\n\\filldraw [frontcolor] (0,0,0) rectangle (2,2,0);\n\\definecolor{rightcolor}{RGB}{195, 168, 127}\n\\filldraw [rightcolor] (2,0,0) -- (2,0,-2) -- (2,2,-2) -- (2,2,0) -- cycle;\n\\definecolor{abovecolor}{RGB}{251, 218, 170}\n\\filldraw [abovecolor] (0,2,0) -- (2,2,0) -- (2,2,-2) -- (0,2,-2) -- cycle;\n\\draw (0,0,0) rectangle (2,2,0) (2,0,0) -- (2,0,-2) -- (2,2,-2) -- (2,2,0) (0,2,0) -- (0,2,-2) -- (2,2,-2);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 四面体; \\textcircled{2} 四棱锥; \\textcircled{3} 四棱柱; \\textcircled{4} 五棱锥; \\textcircled{5} 五棱柱; \\textcircled{6} 六棱锥; \\textcircled{7} 七面体. \\blank{100}(找出可能的结果, 并将序号填在横线上).",
- "objs": [],
+ "objs": [
+ "K0621001B"
+ ],
"tags": [
"第六单元"
],
@@ -247689,7 +249439,10 @@
"010524": {
"id": "010524",
"content": "如图, 有两张全等的正三角形纸片, 按照下面两种方法分别将它们剪拼成一个三棱锥和一个三棱柱. 试比较这两个多面体的体积的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) -- (60:2) -- cycle;\n\\filldraw [gray!50] (60:1) --++ (1,0) -- (1,0) -- cycle;\n\\draw (60:1) --++ (1,0) -- (1,0) -- cycle;\n\\draw [->] (2,0.5) -- (3,0.5);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) (1,0,0) coordinate (B) (0.5,0,{0.5*sqrt(3)}) coordinate (C) (C) ++ (0,0,{-0.5*sqrt(3)/3*2}) ++ (0,{sqrt(6)/3},0) coordinate (D);\n\\filldraw [gray!50] (A) -- (C) -- (B) -- cycle;\n\\draw (D) -- (A) (D) -- (B) (D) -- (C) (A) -- (C) -- (B);\n\\draw [dashed] (A) -- (B);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) -- (60:2) -- cycle;\n\\draw (30:{1/sqrt(3)}) coordinate (A);\n\\draw (A) ++ (1,0) coordinate (B);\n\\draw (A) ++ (60:1) coordinate (C);\n\\draw (A) --++ (0,{-0.5/sqrt(3)}) (A) --++ (150:{0.5/sqrt(3)});\n\\draw (B) --++ (270:{0.5/sqrt(3)}) (B) --++ (30:{0.5/sqrt(3)});\n\\draw (C) --++ (30:{0.5/sqrt(3)}) (C) --++ (150:{0.5/sqrt(3)});\n\\filldraw [gray!50] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (B) -- (C) -- cycle;\n\\draw [->] (2,0.5) -- (3,0.5);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) (1,0,0) coordinate (B) (0.5,0,{0.5*sqrt(3)}) coordinate (C);\n\\filldraw [gray!50] (A) -- (C) -- (B) -- cycle;\n\\draw (A) ++ (0,{0.5/sqrt(3)}) coordinate (A1) (B) ++ (0,{0.5/sqrt(3)}) coordinate (B1) (C) ++ (0,{0.5/sqrt(3)}) coordinate (C1);\n\\draw (A1) -- (B1) -- (C1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1) (A) -- (C) -- (B);\n\\draw [dashed] (A) -- (B);\n\\draw (C1) ++ (0,0,{-0.5*sqrt(3)/3*2}) coordinate (D);\n\\draw ($(A1)!0.5!(B1)$) -- (D) ($(C1)!0.5!(B1)$) -- (D) ($(C1)!0.5!(A1)$) -- (D);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0619003B"
+ ],
"tags": [
"第六单元"
],
@@ -247798,7 +249551,10 @@
"010529": {
"id": "010529",
"content": "已知正三角形$ABC$的三个顶点都在半径为$2$的球面上, 球心$O$到平面$ABC$的距离为$1$, $E$是线段$AB$的中点, 过点$E$作球$O$的截面, 则截面面积的最小值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622004B",
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -247819,7 +249575,9 @@
"010530": {
"id": "010530",
"content": "已知过球面上$A$、$B$、$C$三点的截面和球心的距离等于球半径的一半, 且$AB=BC=CA=2$. 求截面的面积.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [
"第六单元"
],
@@ -247840,7 +249598,9 @@
"010531": {
"id": "010531",
"content": "已知三棱柱$ABC-A_1B_1C_1$的$6$个顶点都在球$O$的球面上, 且$AB=3, AC=4$, $AB\\perp AC$, $AA_1=12$. 求球$O$的半径.",
- "objs": [],
+ "objs": [
+ "K0622002B"
+ ],
"tags": [
"第六单元"
],
@@ -247861,7 +249621,9 @@
"010532": {
"id": "010532",
"content": "如图为一个用鲜花做成的花柱, 它的下面是一个直径为$1\\text{m}$、高为$3\\text{m}$的圆柱形物体, 上面是一个半球形体. 如果每平方米大约需要鲜花$120$朵, 那么装饰这个花柱大约需要多少朵鲜花?\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (-1,0) arc (180:360:1 and 0.3);\n\\draw (-1,3) arc (180:360:1 and 0.3);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.3);\n\\draw [dashed] (-1,3) arc (180:0:1 and 0.3);\n\\draw (-1,3) arc (180:0:1);\n\\draw (-1,0) -- (-1,3) (1,0) -- (1,3);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0623004B"
+ ],
"tags": [
"第六单元"
],
@@ -253950,7 +255712,8 @@
"id": "010814",
"content": "吹一个球形的气球时, 气球半径$r$将随空气容量$V$的增加而增大.\\\\\n(1) 写出气球半径$r$关于气球内空气容量$V$的函数表达式;\\\\\n(2) 求$V=1$时, 气球的瞬时膨胀率(即气球半径关于气球内空气容量的瞬时变化率).",
"objs": [
- "K0227004X"
+ "K0227004X",
+ "K0623002B"
],
"tags": [
"第二单元"
@@ -257812,7 +259575,9 @@
"010971": {
"id": "010971",
"content": "若关于$x$、$y$的二元一次线性方程组$\\begin{cases} a_1x+b_1y=c_1 \\\\ a_2x+b_2y=c_2 \\end{cases}$的增广矩阵是$\\begin{pmatrix}\nm & 1 & 3 \\\\ 0 & 2 & n \\end{pmatrix}$, 且$\\begin{cases} x=1 \\\\ y=-1 \\end{cases}$是该线性方程组的解, 则三阶行列式$\\begin{vmatrix}\n-1 & 0 & 1 \\\\ 0 & 3 & m \\\\ 2 & n & 1 \\end{vmatrix}$中第$3$行第$2$列元素的代数余子式的值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -259196,7 +260961,9 @@
"011034": {
"id": "011034",
"content": "在$120^\\circ$的二面角内放置一个半径为$6$的小球, 它与二面角的两个半平面相切于$A$、$B$两点, 则这两个点在球面上的距离是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -259537,7 +261304,9 @@
"011050": {
"id": "011050",
"content": "行列式$\\begin{vmatrix} \\sin \\alpha & \\sin \\alpha -\\cos \\alpha \\\\\\cos \\alpha & \\sin \\alpha +\\cos \\alpha \\end{vmatrix}$的值等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -260019,7 +261788,9 @@
"011071": {
"id": "011071",
"content": "关于$x,y$的方程组$\\begin{cases} 2x-y+1=0, \\\\x+3y=0 \\end{cases}$的增广矩阵为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -260503,7 +262274,9 @@
"011093": {
"id": "011093",
"content": "三阶行列式$\\begin{vmatrix}\n3 & -5 & 1 \\\\ 2 & 3 & -6 \\\\ -7 & 2 & 4 \\end{vmatrix}$中元素$-5$的代数余子式的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -263761,7 +265534,9 @@
"011244": {
"id": "011244",
"content": "已知关于$x,y$的二元一次方程组的增广矩阵为$\\begin{pmatrix}\n 2 & 1 & 5 \\\\ 1 & -2 & 0 \\end{pmatrix}$, 则$xy=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -264656,7 +266431,9 @@
"011285": {
"id": "011285",
"content": "行列式$\\begin{vmatrix}1 & 2 & 0 \\\\2 & 3 & 5 \\\\5 & 8 & 0\\end{vmatrix}$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -265106,7 +266883,9 @@
"011306": {
"id": "011306",
"content": "已知一个关于$x$、$y$的二元一次方程组的增广矩阵是$\\begin{pmatrix}\n1 & -1 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix}$, 则$x+y=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -265539,7 +267318,9 @@
"011326": {
"id": "011326",
"content": "行列式$\\begin{vmatrix} 1 & 2 \\\\ 3 & 4 \\end{vmatrix}$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -265673,7 +267454,10 @@
"011332": {
"id": "011332",
"content": "在一个水平放置的底面半径为$\\sqrt 3$的的圆柱形量杯中装有适量的水, 现放入一个半径为$R$的实心铁球, 球完全浸没入水中且无水溢出. 若水面上升高度也为$R$, 则\t$R=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0616003B",
+ "K0623002B"
+ ],
"tags": [
""
],
@@ -266057,7 +267841,9 @@
"011350": {
"id": "011350",
"content": "一个几何体的三视图如图所示, 则这个几何体的表面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) -- (2,1) -- (0,1) -- cycle;\n\\draw (0,1) -- (1,2) -- (2,1);\n\\draw (-0.1,0) -- (-0.5,0) (-0.1,1) -- (-0.5,1) (0.9,2) -- (-0.5,2);\n\\draw [->] (-0.3,0.2) -- (-0.3,0);\n\\draw [->] (-0.3,0.8) -- (-0.3,1);\n\\draw (-0.3,0.5) node {$1$};\n\\draw [->] (-0.3,1.2) -- (-0.3,1);\n\\draw [->] (-0.3,1.8) -- (-0.3,2);\n\\draw (-0.3,1.5) node {$1$};\n\\draw (0,-0.1) -- (0,-0.5) (2,-0.1) -- (2,-0.5);\n\\draw [->] (0.8,-0.3) -- (0,-0.3);\n\\draw [->] (1.2,-0.3) -- (2,-0.3);\n\\draw (1,-0.3) node {$2$};\n\\draw (1,-0.8) node {主视图};\n\\draw (3,0) -- (5,0) -- (5,1) -- (3,1) -- cycle;\n\\draw (3,1) -- (4,2) -- (5,1);\n\\draw (2.9,0) -- (2.5,0) (2.9,1) -- (2.5,1) (3.9,2) -- (2.5,2);\n\\draw [->] (2.7,0.2) -- (2.7,0);\n\\draw [->] (2.7,0.8) -- (2.7,1);\n\\draw (2.7,0.5) node {$1$};\n\\draw [->] (2.7,1.2) -- (2.7,1);\n\\draw [->] (2.7,1.8) -- (2.7,2);\n\\draw (2.7,1.5) node {$1$};\n\\draw (3,-0.1) -- (3,-0.5) (5,-0.1) -- (5,-0.5);\n\\draw [->] (3.8,-0.3) -- (3,-0.3);\n\\draw [->] (4.2,-0.3) -- (5,-0.3);\n\\draw (4,-0.3) node {$2$};\n\\draw (4,-0.8) node {左视图};\n\\draw (1,-2.5) circle (1);\n\\draw (0.9,-1.5) -- (-0.5,-1.5) (0.9,-3.5) -- (-0.5,-3.5);\n\\draw [->] (-0.3,-2.2) -- (-0.3,-1.5);\n\\draw [->] (-0.3,-2.8) -- (-0.3,-3.5);\n\\draw (-0.3,-2.5) node {$2$};\n\\draw (0,-2.6) -- (0,-4) (2,-2.6) -- (2,-4);\n\\draw [->] (0.8,-3.8) -- (0,-3.8);\n\\draw [->] (1.2,-3.8) -- (2,-3.8);\n\\draw (1,-3.8) node {$2$};\n\\draw (1,-4.2) node {俯视图};\n\\filldraw (1,-2.5) circle (0.03);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -266141,7 +267927,9 @@
"011354": {
"id": "011354",
"content": "如图, 在正六棱柱的所有棱中任取两条, 则它们所在的直线是互相垂直的异面直线的概率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) --++ (1,0,0) coordinate (B) --++ ({1/2},0,{sqrt(3)/2}) coordinate (C) --++ ({-1/2},0,{sqrt(3)/2}) coordinate (D) --++ (-1,0,0) coordinate (E) --++ ({-1/2},0,{-sqrt(3)/2}) coordinate (F) -- cycle;\n\\draw [dashed] (A) --++ (0,-2,0) coordinate (A1);\n\\draw [dashed] (B) --++ (0,-2,0) coordinate (B1);\n\\draw (C) --++ (0,-2,0) coordinate (C1);\n\\draw (D) --++ (0,-2,0) coordinate (D1);\n\\draw (E) --++ (0,-2,0) coordinate (E1);\n\\draw (F) --++ (0,-2,0) coordinate (F1);\n\\draw (F1) -- (E1) -- (D1) -- (C1);\n\\draw [dashed] (F1) -- (A1) -- (B1) -- (C1);\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0615004B"
+ ],
"tags": [
""
],
@@ -266766,7 +268554,9 @@
"011382": {
"id": "011382",
"content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $M$、$E$是$AB$的三等分点, $G$、$N$是$CD$的三等分点, $F$、$H$分别是$BC$、$MN$的中点, 则四棱锥$A_1-EFGH$的左视图是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B) --++ (2,0) node [below right] {$C$} coordinate (C) --++ (45:{2/2}) node [right] {$D$} coordinate (D)\n--++ (0,2) node [above right] {$D_1$} coordinate (D1)\n--++ (-2,0) node [above left] {$A_1$} coordinate (A1) --++ (225:{2/2}) node [left] {$B_1$} coordinate (B1) -- cycle;\n\\draw (B) ++ (2,2) node [right] {$C_1$} coordinate (C1) -- (C) (C1) --++ (45:{2/2}) (C1) --++ (-2,0);\n\\draw [dashed] (B) --++ (45:{2/2}) node [left] {$A$} coordinate (A) --++ (2,0) (A) --++ (0,2);\n\\draw [dashed] ($(A)!{1/3}!(B)$) node [left] {$M$} coordinate (M) -- ($(D)!{1/3}!(C)$) node [right] {$N$} coordinate (N);\n\\draw [dashed] ($(A)!{2/3}!(B)$) node [left] {$E$} coordinate (E) -- ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F)--($(D)!{2/3}!(C)$) node [right] {$G$} coordinate (G);\n\\draw [dashed] (E) -- ($(M)!0.5!(N)$) node [below] {$H$} coordinate (H) -- (G);\n\\draw [dashed] (A1) -- (E) (A1) -- (F) (A1) -- (G) (A1) -- (H);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) -- (0,2) -- cycle (1,0) -- (0,2);\n\\end{tikzpicture}\n\\end{center}}{\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- ({4/3},0) -- (2,2) -- cycle ({2/3},0) -- (2,2);\n\\end{tikzpicture}\n\\end{center}}{\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({2/3},0) -- (2,0) -- (0,2) -- cycle ({4/3},0) -- (0,2);\n\\end{tikzpicture}\n\\end{center}\n}{\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (2,0) -- (0,2) -- cycle ({2/3},0) -- (0,2);\n\\end{tikzpicture}\n\\end{center}}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -267283,7 +269073,9 @@
"011405": {
"id": "011405",
"content": "在四棱锥$P-ABCD$中, $PA\\perp$平面$ABCD$, $AB\\perp AD$, $BC\\parallel AD$, $BC=1$, $CD=\\sqrt 2$, $\\angle CDA=45^{^\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.8]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [below 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] (A) -- (B) (A) -- (P) (A) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 画出四棱锥$P-ABCD$的三视图;\\\\\n(2) 若$PA=BC$, 求直线$PB$与平面$PCD$所成角的大小.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -267575,7 +269367,9 @@
"011418": {
"id": "011418",
"content": "若关于$x,y$的线性方程组$\\begin{cases} a_1x+b_1y=c_1, \\\\ a_2x+b_2y=c_2 \\end{cases}$的增广矩阵是$\\begin{pmatrix}\nm & 1 & 3 \\\\ 0 & 2 & n \\end{pmatrix}$, 且$\\begin{cases} x=-1, \\\\ y=1 \\end{cases}$是该线性方程组的解, 则三阶行列式$\\begin{vmatrix}\n-1 & 0 & 1 \\\\ 0 & 3 & m \\\\ 2 & n & 1 \\end{vmatrix}$中第$3$行第$2$列的元素的代数余子式的值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -267924,7 +269718,9 @@
"011434": {
"id": "011434",
"content": "已知线性方程组的增广矩阵为$\\begin{pmatrix}\n1 & 1 & 3 \\\\ a & 0 & 2 \\end{pmatrix}$, 若该线性方程组的解为$\\begin{pmatrix}1 \\\\ 2 \\end{pmatrix}$, 则实数$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -269836,7 +271632,9 @@
"011520": {
"id": "011520",
"content": "若线性方程组的增广矩阵为$\\begin{pmatrix}\n1 & 2 & c_1 \\\\2 & 0 & c_2 \\end{pmatrix}$的解为$\\begin{cases} x=1, \\\\ y=3, \\end{cases}$则$c_1+c_2=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -270254,7 +272052,9 @@
"011539": {
"id": "011539",
"content": "已知二元一次方程$\\begin{cases} a_1x+b_1y+c_1, \\\\ a_2x+b_2y+c_2 \\end{cases}$的增广矩阵是$\\begin{pmatrix} 1 & -1 & 1 \\\\ 1 & 1 & 3\\end{pmatrix}$, 则此方程组的解是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -270950,7 +272750,9 @@
"011569": {
"id": "011569",
"content": "关于$x$、$y$的二元一次方程组$\\begin{cases}\nx+5y=0, \\\\ 2x+3y=4\\end{cases}$的系数行列式$D$为\\bracket{15}.\n\\fourch{$\\begin{vmatrix}\n0 & 5 \\\\ 4 & 3\n\\end{vmatrix}$}{$\\begin{vmatrix}\n1 & 0 \\\\ 2 & 4\n\\end{vmatrix}$}{$\\begin{vmatrix}\n1 & 5 \\\\ 2 & 3\n\\end{vmatrix}$}{$\\begin{vmatrix}\n6 & 0 \\\\ 5 & 4\n\\end{vmatrix}$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -271231,7 +273033,9 @@
"011581": {
"id": "011581",
"content": "行列式$\\begin{vmatrix} \\cos \\dfrac{\\pi }3 & \\sin \\dfrac{\\pi }6 \\\\ \\sin \\dfrac{\\pi }3 & \\cos \\dfrac{\\pi }6 \\end{vmatrix}$的值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -271362,7 +273166,9 @@
"011587": {
"id": "011587",
"content": "在$n$行$n$列矩阵$\\begin{pmatrix}\n1 & 2 & 3 & \\cdots & n-2 & n-1 & n \\\\ 2 & 3 & 4 & \\cdots & n-1 & n & 1 \\\\ 3 & 4 & 5 & \\cdots & n & 1 & 2 \\\\ \\cdots & \\cdots & \\cdots & \\cdots & \\cdots & \\cdots & \\cdots \\\\ n & 1 & 2 & \\cdots & n-3 & n-2 & n-1 \\end{pmatrix}$中, 记位于第$i$行第$j$列的数为$a_{ij}$($i,j=1,2\\cdots ,n$). 当$n=9$时, $a_{11}+a_{22}+a_{33}+\\cdots +a_{99}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -271862,7 +273668,9 @@
"011610": {
"id": "011610",
"content": "行列式$\\begin{vmatrix}\na & b \\\\ c & d \\end{vmatrix}$($a,b,c,d\\in \\{-1,1,2\\}$)的所有可能值中, 最大的是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -273024,7 +274832,9 @@
"011663": {
"id": "011663",
"content": "在数列$\\{a_n\\}$中, $a_n=2^n-1$, 若一个$7$行$12$列的矩阵的第$i$行第$j$列的元素$a_{i,j}=a_i\\cdot a_j+a_i+a_j$, ($i=1,2,\\cdots ,7$; $j=1,2,\\cdots ,12$)则该矩阵元素能取到的不同数值的个数为\\bracket{20}.\n\\fourch{$18$}{$28$}{$48$}{$63$}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -273711,7 +275521,9 @@
"011695": {
"id": "011695",
"content": "若线性方程组的增广矩阵为$\\begin{pmatrix}\n2 & 3 & c_1 \\\\0 & 1 & c_2 \\end{pmatrix}$、解为$\\begin{cases} x=3, \\\\ y=5, \\end{cases}$, 则$c_1-c_2=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
""
],
@@ -288567,7 +290379,11 @@
"030113": {
"id": "030113",
"content": "将下列各几何体之间的关系用文氏图表示: 多面体, 长方体, 棱柱, 棱锥, 棱台, 直棱柱, 四面体, 平行六面体.",
- "objs": [],
+ "objs": [
+ "K0615004B",
+ "K0618002B",
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288586,7 +290402,10 @@
"030114": {
"id": "030114",
"content": "判断下列命题是否正确, 正确的在横线上画``$\\checkmark$'', 错误的画``$\\times$''.\\\\\n\\blank{30}(1) 长方体是四棱柱, 直四棱柱是长方体;\\\\\n\\blank{30}(2) 四棱柱, 四棱台, 五棱锥都是六面体.",
- "objs": [],
+ "objs": [
+ "K0615004B",
+ "K0621001B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288605,7 +290424,10 @@
"030115": {
"id": "030115",
"content": "一个几何体由$7$个面围成, 其中两个面是互相平行且全等的五边形, 其他各面都是全等的矩形, 则这个几何体是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0615004B",
+ "K0615005B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288624,7 +290446,9 @@
"030116": {
"id": "030116",
"content": "一个多面体至少有\\blank{50}个面, 此时这个多面体是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0621002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288643,7 +290467,11 @@
"030117": {
"id": "030117",
"content": "判断下列命题是否正确, 正确的在横线上画``$\\checkmark$'', 错误的画``$\\times$''.\\\\\n\\blank{30}(1) 一个棱柱至少有$5$个面;\\\\\n\\blank{30}(2) 平行六面体中相对的两个面是全等的平行四边形;\\\\\n\\blank{30}(3) 有一个面是平行四边形的棱锥一定是四棱锥;\\\\\n\\blank{30}(4) 正棱锥的侧面是全等的等腰三角形.",
- "objs": [],
+ "objs": [
+ "K0615003B",
+ "K0615005B",
+ "K0618002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288662,7 +290490,10 @@
"030118": {
"id": "030118",
"content": "下列命题是否正确? 若正确, 请说明理由; 若错误, 请举出反例.\\\\\n(1) 有两个面平行, 其他各个面都是平行四边形的多面体是棱柱;\\\\\n(2) 有两个面平行且相似, 其他各个面都是梯形的多面体是棱台.",
- "objs": [],
+ "objs": [
+ "K0615002B",
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288681,7 +290512,9 @@
"030119": {
"id": "030119",
"content": "把直棱柱沿任意一条侧棱展开, 然后在一个平面上将所有侧面展开, 得到的是一个什么平面图形? 答\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0615002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288700,7 +290533,10 @@
"030120": {
"id": "030120",
"content": "判断下列命题是否正确, 正确的在横线上画``$\\checkmark$'', 错误的画``$\\times$''.\\\\\n\\blank{30}(1) 侧棱垂直于底面的棱柱一定是直棱柱;\\\\\n\\blank{30}(2) 底面是正多边形的棱柱一定是正棱柱;\\\\\n\\blank{30}(3) 棱柱的侧面都是平行四边形;\\\\\n\\blank{30}(4) 斜柱的侧面都不可能是矩形.",
- "objs": [],
+ "objs": [
+ "K0615002B",
+ "K0615004B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288719,7 +290555,9 @@
"030121": {
"id": "030121",
"content": "四棱锥中, 与一条侧棱异面的棱有\\blank{50}条.",
- "objs": [],
+ "objs": [
+ "K0618001B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288738,7 +290576,9 @@
"030122": {
"id": "030122",
"content": "已知正四棱锥$V-ABCD$的底面面积为$16$, 侧棱长为$2\\sqrt{11}$, 则这个棱锥的斜高为\\blank{50}, 高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288757,7 +290597,9 @@
"030123": {
"id": "030123",
"content": "一个三棱台的上, 下底面面积之比为$4:9$, 若棱台的高是$4\\text{cm}$, 则这个棱台的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288776,7 +290618,9 @@
"030124": {
"id": "030124",
"content": "设正三棱台的上底面边长为$2\\text{cm}$, 下底面边长以及侧棱长均为$5\\text{cm}$, 则这个棱台的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288795,7 +290639,9 @@
"030125": {
"id": "030125",
"content": "圆柱的任意两条母线\\blank{50}, 两个底面\\blank{50}; 圆台的任意两条母线\\blank{50}, 两个底面\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0615007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288814,7 +290660,10 @@
"030126": {
"id": "030126",
"content": "把地球看成一个半径为$6370\\text{cm}$的球, 已知我国首都北京靠近北纬$40^\\circ$. 北纬$40^\\circ$纬线的长度约为\\blank{50}$\\text{km}$(结果精确到$1\\text{km}$).",
- "objs": [],
+ "objs": [
+ "K0622005B",
+ "K0622006B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288833,7 +290682,9 @@
"030127": {
"id": "030127",
"content": "一个圆柱的母线长为$5$, 底面半径为$2$, 则该圆柱轴截面的面积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0615008B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288852,7 +290703,9 @@
"030128": {
"id": "030128",
"content": "圆锥的母线长为$20$, 母线与轴的夹角为$30^\\circ$, 则该圆锥的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618005B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288871,7 +290724,9 @@
"030129": {
"id": "030129",
"content": "已知$A,B$都是球$O$对应的球面上的点, 过$A,B$两点可以作几个大圆? 说明理由.",
- "objs": [],
+ "objs": [
+ "K0622004B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288890,7 +290745,10 @@
"030130": {
"id": "030130",
"content": "一条直线被一个半径为$5$的球截得的线段长为$8$, 则球心到直线的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622001B",
+ "K0622003B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288909,7 +290767,10 @@
"030131": {
"id": "030131",
"content": "判断下列命题是否正确, 正确的在横线上画``$\\checkmark$'', 错误的画``$\\times$''.\\\\\n\\blank{30}(1) 球面上任意一点与球心的连线都是球的半径;\\\\\n\\blank{30}(2) 球面上任意两点连成的线段都是球的直径;\\\\\n\\blank{30}(3) 用一个球面截一个球, 得到的截面是一个圆面.",
- "objs": [],
+ "objs": [
+ "K0622001B",
+ "K0622003B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288928,7 +290789,9 @@
"030132": {
"id": "030132",
"content": "一个圆台的母线长为$5$, 两底面直径分别为$2$和$8$, 则该圆台的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288947,7 +290810,9 @@
"030133": {
"id": "030133",
"content": "记$A$为所有平行六面体组成的集合, $B$为所有直平行六面体组成的集合, $C$为所有长方体组成的集合, $D$为所有正四棱柱组成的集合, $E$为所有正方体组成的集合, 写出$A, B, C, D, E$之间的关系.",
- "objs": [],
+ "objs": [
+ "K0615004B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288966,7 +290831,10 @@
"030134": {
"id": "030134",
"content": "四棱台中, 与一条侧棱异面的棱有\\blank{50}条.",
- "objs": [],
+ "objs": [
+ "K0618003B",
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -288985,7 +290853,10 @@
"030135": {
"id": "030135",
"content": "已知一个圆台的轴截面是下底为$2$且其余边长为$1$的等腰梯形, 则该圆台的高为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618005B",
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289004,7 +290875,9 @@
"030136": {
"id": "030136",
"content": "用一个平行于圆锥底面的平面截这个圆锥, 截得圆台上、下底面半径的比是$1: 4$, 截去的圆锥的母线长是$3$, 则圆台的母线长为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618006B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289023,7 +290896,10 @@
"030137": {
"id": "030137",
"content": "如果把地球看成一个球体, 则地球上的北纬$60^\\circ$纬线长与赤道长的比为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622004B",
+ "K0622006B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289042,7 +290918,9 @@
"030138": {
"id": "030138",
"content": "用一个平面截半径为$25\\text{cm}$的球, 截面面积是$49\\pi\\text{cm}^2$, 则球心到截面的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622005B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289061,7 +290939,11 @@
"030139": {
"id": "030139",
"content": "判断下列命题是否正确, 正确的在横线上画``$\\checkmark$'', 错误的画``$\\times$''.\\\\\n\\blank{30}(1) 有一个面是多边形, 其余各面都是三角形的几何体是棱锥;\\\\\n\\blank{30}(2) 底面是正多边形的棱锥一定是正棱锥;\\\\\n\\blank{30}(3) 有两个面是平行的相似多边形, 其余各面都是梯形的几何体是棱台.",
- "objs": [],
+ "objs": [
+ "K0618001B",
+ "K0618002B",
+ "K0618007B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289080,7 +290962,9 @@
"030140": {
"id": "030140",
"content": "如果平行于一个正棱锥底面的截面面积是底面面积的$\\dfrac 12$, 那么截面截一条侧棱所得两条线段的比是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0618004B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289099,7 +290983,10 @@
"030141": {
"id": "030141",
"content": "将地球视为球体, 记地球半径为$R$, 地球球心为$O$, 设$A, B$为赤道上两点, 且半径$OA$与$OB$的夹角为$\\dfrac {2\\pi}3$, 则线段$AB$的长为\\blank{50}, 赤道在$A, B$两点间的劣弧长为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0622001B",
+ "K0622002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289118,7 +291005,9 @@
"030142": {
"id": "030142",
"content": "在正方体上任意选择$4$个顶点, 然后将它们两两相连, 则可能组成的几何图形为\\blank{50}(写出所有正确结论的编号).\\\\\n\\textcircled{1} 矩形; \\textcircled{2} 不是矩形的平行四边形; \\textcircled{3} 有三个面为等腰直角三角形, 有一个面为等边三角形的四面体; \\textcircled{4} 每个面都是等边三角形的四面体; \\textcircled{5} 每个面都是直角三角形的四面体.",
- "objs": [],
+ "objs": [
+ "K0621002B"
+ ],
"tags": [],
"genre": "",
"ans": "",
@@ -289354,7 +291243,7 @@
},
"030152": {
"id": "030152",
- "content": "在复数范围内, 方程$2x^2-\\mathrm{i} x+1=0$的解为\\blank{50}.",
+ "content": "在复数范围内, 方程$2x^2-2x+1=0$的解为\\blank{50}.",
"objs": [],
"tags": [
"第五单元"
@@ -289364,10 +291253,9 @@
"solution": "",
"duration": -1,
"usages": [],
- "origin": "2016届创新班作业\t3140-复系数一元二次方程-20221011修改",
+ "origin": "自拟题目",
"edit": [
- "20220625\t王伟叶",
- "20221011\t王伟叶"
+ "20221012\t王伟叶"
],
"same": [],
"related": [