diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb
index f0dce079..0c5f8a5c 100644
--- a/工具/寻找阶段末尾空闲题号.ipynb
+++ b/工具/寻找阶段末尾空闲题号.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -11,7 +11,7 @@
      "text": [
       "首个空闲id: 12739 , 直至 020000\n",
       "首个空闲id: 21441 , 直至 030000\n",
-      "首个空闲id: 30757 , 直至 999999\n"
+      "首个空闲id: 31057 , 直至 999999\n"
      ]
     }
    ],
diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb
index 543c7f62..4275a3b8 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
@@ -2,94 +2,114 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "题号: 030757 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030758 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030759 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030760 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030761 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030762 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030763 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030764 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030765 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030766 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030767 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030768 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030769 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030770 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030771 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030772 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030773 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030774 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030775 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030776 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030777 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030778 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030779 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030780 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030781 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030782 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030783 , 字段: tags 中已添加数据: a\n",
-      "题号: 030783 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030784 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030785 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030786 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030787 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030788 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030789 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030790 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030791 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030792 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030793 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030794 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030795 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030796 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030797 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030798 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030799 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030800 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030801 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030802 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030803 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030804 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030805 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030806 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030807 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030808 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030809 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030810 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030811 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030812 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030813 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030814 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030815 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030816 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030817 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030818 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030819 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030820 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030821 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030822 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030823 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030824 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030825 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030826 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030827 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030828 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030829 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030830 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030831 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030832 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030833 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030834 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030835 , 字段: tags 中已添加数据: 第三单元\n",
-      "题号: 030836 , 字段: tags 中已添加数据: 第三单元\n"
+      "题号: 031057 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031058 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031059 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031060 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031061 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031062 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031063 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031064 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031065 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031066 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031067 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031068 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031069 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031070 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031071 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031072 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031073 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031074 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031075 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031076 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031077 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031078 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031079 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031080 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031081 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031082 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031083 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031084 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031085 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031086 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031087 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031088 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031089 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031090 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031091 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031092 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031093 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031094 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031095 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031096 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031097 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031098 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031099 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031100 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031101 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031102 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031103 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031104 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031105 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031106 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031107 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031108 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031109 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031110 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031111 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031112 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031113 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031114 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031115 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031116 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031117 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031118 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031119 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031120 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031121 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031122 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031123 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031124 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031125 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031126 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031127 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031128 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031129 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031130 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031131 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031132 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031133 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031134 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031135 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031136 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031137 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031138 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031139 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031140 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031141 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031142 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031143 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031144 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031145 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031146 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031147 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031148 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031149 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031150 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031151 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031152 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031153 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031154 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031155 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031156 , 字段: tags 中已添加数据: 第七单元\n",
+      "题号: 031157 , 字段: tags 中已添加数据: 第七单元\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 081c397a..12b29604 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
@@ -9,9 +9,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "开始编译教师版本pdf文件:  临时文件/高三上末尾作业_教师用_20230105.tex\n",
+      "开始编译教师版本pdf文件:  临时文件/高三上末尾作业_教师用_20230108.tex\n",
       "0\n",
-      "开始编译学生版本pdf文件:  临时文件/高三上末尾作业_学生用_20230105.tex\n",
+      "开始编译学生版本pdf文件:  临时文件/高三上末尾作业_学生用_20230108.tex\n",
       "0\n"
      ]
     }
@@ -26,13 +26,13 @@
     "\"\"\"---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",
-    "\"7.3.3-正态分布-done\":\"30515,30516,10903,30519,30534,30517,30518,30520,30535,30538,30540,30552\",\n",
-    "\"8.1.1-成对数据间的关系-done\":\"030554,030521,030522,030523,030524\",\n",
-    "\"8.1.2-相关系数-done\":\"10905,10906,10908,30525,30526,30591\",\n",
-    "\"8.2.1-一元线性回归分析的基本思想-done\":\"30527,30528,30567,10911,10912,30573\",\n",
-    "\"8.2.2-一元线性回归分析的应用举例\":\"10913,10915,10916,10917,10918,30529,30570\",\n",
-    "\"8.3.1-2乘2列联表独立性检验-done\":\"10920,10922,30530,30531,30532,30578,30579,30580,30593\",\n",
-    "\"8.3.2-独立性检验的具体应用\":\"10919,10921,30533,30582,30583,30588,30596\"\n",
+    "\"7.3.3-正态分布\":\"30515,30516,10903,30519,30534,30517,30518,30520,30535,30538,30540,30552\",\n",
+    "\"8.1.1-成对数据间的关系\":\"030554,030521,030522,030523,030524\",\n",
+    "\"8.1.2-相关系数\":\"10905,10906,10908,30525,30526,30591\",\n",
+    "\"8.2.1-一元线性回归分析的基本思想\":\"30527,30528,30567,10911,10912,30573\",\n",
+    "\"8.2.2-一元线性回归分析的应用举例\":\"10913,10915,10916,10917,10918\",\n",
+    "\"8.3.1-2乘2列联表独立性检验\":\"10920,10922,30530,30531,30532,30578,30579,30580,30593\",\n",
+    "\"8.3.2-独立性检验的具体应用\":\"30582,30588\"\n",
     "\n",
     "}\n",
     "\n",
@@ -186,7 +186,7 @@
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index f8e649c7..296a4b6e 100644
--- a/工具/模板文件/复习课目标模板.tex
+++ b/工具/模板文件/复习课目标模板.tex
@@ -5,6 +5,7 @@
 \usepackage{ifthen,indentfirst,enumerate,color,lastpage}
 \usepackage{tikz}
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+\usepackage{multirow}
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 \usepackage{longtable}
 \usepackage[top=1in, bottom=1in,left=0.8in,right=0.8in]{geometry}
diff --git a/工具/模板文件/日常选题讲义模板.tex b/工具/模板文件/日常选题讲义模板.tex
index 45ca0db0..a680b271 100644
--- a/工具/模板文件/日常选题讲义模板.tex
+++ b/工具/模板文件/日常选题讲义模板.tex
@@ -5,6 +5,7 @@
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diff --git a/工具/模板文件/测验周末卷模板.tex b/工具/模板文件/测验周末卷模板.tex
index 6869b8fa..fbae8ea6 100644
--- a/工具/模板文件/测验周末卷模板.tex
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@@ -5,6 +5,7 @@
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diff --git a/工具/模板文件/第一轮复习讲义模板.tex b/工具/模板文件/第一轮复习讲义模板.tex
index 5813b0eb..a291a721 100644
--- a/工具/模板文件/第一轮复习讲义模板.tex
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@@ -5,6 +5,7 @@
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diff --git a/工具/模板文件/题目清单.tex b/工具/模板文件/题目清单.tex
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 \usepackage{ifthen,indentfirst,enumerate,color,titletoc}
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diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb
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-    "starting_id = 30757\n",
+    "starting_id = 31057\n",
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-      "030534 填空题\n",
-      "030535 解答题\n",
-      "030536 解答题\n",
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-      "030545 解答题\n",
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-      "030548 解答题\n",
-      "030549 解答题\n",
-      "030550 解答题\n",
-      "030551 解答题\n",
-      "030552 解答题\n",
-      "030553 解答题\n",
-      "030554 解答题\n",
-      "030555 解答题\n",
-      "030556 解答题\n",
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-      "030558 解答题\n",
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-      "030585 解答题\n",
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-      "030587 解答题\n",
-      "030588 解答题\n",
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-      "030590 解答题\n",
-      "030591 选择题\n",
-      "030592 选择题\n",
-      "030593 选择题\n",
-      "030594 解答题\n",
-      "030595 解答题\n",
-      "030596 解答题\n"
+      "030597 填空题\n",
+      "030598 填空题\n",
+      "030599 填空题\n",
+      "030600 填空题\n",
+      "030601 填空题\n",
+      "030602 填空题\n",
+      "030603 填空题\n",
+      "030604 填空题\n",
+      "030605 填空题\n",
+      "030606 填空题\n",
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+      "030608 填空题\n",
+      "030609 填空题\n",
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+      "030611 填空题\n",
+      "030612 填空题\n",
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+      "030625 选择题\n",
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+      "030640 选择题\n",
+      "030641 填空题\n",
+      "030642 填空题\n",
+      "030643 填空题\n",
+      "030644 选择题\n",
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+      "030646 填空题\n",
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+      "030651 选择题\n",
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+      "030655 填空题\n",
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+      "030667 选择题\n",
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+      "030669 解答题\n",
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+      "030678 填空题\n",
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+      "030689 选择题\n",
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+      "030716 填空题\n",
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+      "030719 填空题\n",
+      "030720 填空题\n",
+      "030721 填空题\n",
+      "030722 选择题\n",
+      "030723 选择题\n",
+      "030724 解答题\n",
+      "030725 解答题\n",
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+      "030728 选择题\n",
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+      "030731 解答题\n",
+      "030732 解答题\n",
+      "030733 解答题\n",
+      "030734 解答题\n",
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+      "030736 解答题\n",
+      "030737 解答题\n",
+      "030738 解答题\n",
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+      "030742 填空题\n",
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+      "030744 填空题\n",
+      "030745 填空题\n",
+      "030746 填空题\n",
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+      "030752 填空题\n",
+      "030753 填空题\n",
+      "030754 填空题\n",
+      "030755 选择题\n",
+      "030756 选择题\n",
+      "030757 填空题\n",
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+      "030759 填空题\n",
+      "030760 填空题\n",
+      "030761 填空题\n",
+      "030762 填空题\n",
+      "030763 填空题\n",
+      "030764 填空题\n",
+      "030765 选择题\n",
+      "030766 选择题\n",
+      "030767 选择题\n",
+      "030768 选择题\n",
+      "030769 解答题\n",
+      "030770 填空题\n",
+      "030771 填空题\n",
+      "030772 填空题\n",
+      "030773 填空题\n",
+      "030774 填空题\n",
+      "030775 填空题\n",
+      "030776 填空题\n",
+      "030777 填空题\n",
+      "030778 填空题\n",
+      "030779 填空题\n",
+      "030780 填空题\n",
+      "030781 填空题\n",
+      "030782 填空题\n",
+      "030783 选择题\n",
+      "030784 选择题\n",
+      "030785 选择题\n",
+      "030786 选择题\n",
+      "030787 选择题\n",
+      "030788 选择题\n",
+      "030789 选择题\n",
+      "030790 选择题\n",
+      "030791 选择题\n",
+      "030792 选择题\n",
+      "030793 解答题\n",
+      "030794 解答题\n",
+      "030795 填空题\n",
+      "030796 填空题\n",
+      "030797 填空题\n",
+      "030798 选择题\n",
+      "030799 解答题\n",
+      "030800 解答题\n",
+      "030801 解答题\n",
+      "030802 解答题\n",
+      "030803 解答题\n",
+      "030804 解答题\n",
+      "030805 解答题\n",
+      "030806 解答题\n",
+      "030807 解答题\n",
+      "030808 解答题\n",
+      "030809 解答题\n",
+      "030810 解答题\n",
+      "030811 解答题\n",
+      "030812 解答题\n",
+      "030813 解答题\n",
+      "030814 解答题\n",
+      "030815 解答题\n",
+      "030816 解答题\n",
+      "030817 解答题\n",
+      "030818 解答题\n",
+      "030819 填空题\n",
+      "030820 填空题\n",
+      "030821 填空题\n",
+      "030822 填空题\n",
+      "030823 填空题\n",
+      "030824 填空题\n",
+      "030825 选择题\n",
+      "030826 解答题\n",
+      "030827 解答题\n",
+      "030828 解答题\n",
+      "030829 解答题\n",
+      "030830 解答题\n",
+      "030831 解答题\n",
+      "030832 解答题\n",
+      "030833 解答题\n",
+      "030834 解答题\n",
+      "030835 解答题\n",
+      "030836 解答题\n",
+      "030837 填空题\n",
+      "030838 填空题\n",
+      "030839 填空题\n",
+      "030840 填空题\n",
+      "030841 填空题\n",
+      "030842 填空题\n",
+      "030843 填空题\n",
+      "030844 填空题\n",
+      "030845 填空题\n",
+      "030846 选择题\n",
+      "030847 选择题\n",
+      "030848 选择题\n",
+      "030849 解答题\n",
+      "030850 填空题\n",
+      "030851 填空题\n",
+      "030852 填空题\n",
+      "030853 填空题\n",
+      "030854 选择题\n",
+      "030855 选择题\n",
+      "030856 选择题\n",
+      "030857 选择题\n",
+      "030858 填空题\n",
+      "030859 填空题\n",
+      "030860 填空题\n",
+      "030861 填空题\n",
+      "030862 填空题\n",
+      "030863 填空题\n",
+      "030864 填空题\n",
+      "030865 填空题\n",
+      "030866 填空题\n",
+      "030867 填空题\n",
+      "030868 填空题\n",
+      "030869 选择题\n",
+      "030870 选择题\n",
+      "030871 填空题\n",
+      "030872 填空题\n",
+      "030873 填空题\n",
+      "030874 填空题\n",
+      "030875 填空题\n",
+      "030876 填空题\n",
+      "030877 选择题\n",
+      "030878 选择题\n",
+      "030879 解答题\n",
+      "030880 解答题\n",
+      "030881 解答题\n",
+      "030882 解答题\n",
+      "030883 填空题\n",
+      "030884 填空题\n",
+      "030885 填空题\n",
+      "030886 填空题\n",
+      "030887 填空题\n",
+      "030888 填空题\n",
+      "030889 填空题\n",
+      "030890 填空题\n",
+      "030891 填空题\n",
+      "030892 填空题\n",
+      "030893 填空题\n",
+      "030894 填空题\n",
+      "030895 填空题\n",
+      "030896 填空题\n",
+      "030897 填空题\n",
+      "030898 选择题\n",
+      "030899 选择题\n",
+      "030900 解答题\n",
+      "030901 解答题\n",
+      "030902 解答题\n",
+      "030903 解答题\n",
+      "030904 解答题\n",
+      "030905 解答题\n",
+      "030906 解答题\n",
+      "030907 解答题\n",
+      "030908 解答题\n",
+      "030909 解答题\n",
+      "030910 解答题\n",
+      "030911 解答题\n",
+      "030912 填空题\n",
+      "030913 解答题\n",
+      "030914 解答题\n",
+      "030915 解答题\n",
+      "030916 解答题\n",
+      "030917 解答题\n",
+      "030918 解答题\n",
+      "030919 解答题\n",
+      "030920 解答题\n",
+      "030921 解答题\n",
+      "030922 解答题\n",
+      "030923 解答题\n",
+      "030924 解答题\n",
+      "030925 解答题\n",
+      "030926 解答题\n",
+      "030927 解答题\n",
+      "030928 填空题\n",
+      "030929 填空题\n",
+      "030930 填空题\n",
+      "030931 填空题\n",
+      "030932 填空题\n",
+      "030933 填空题\n",
+      "030934 填空题\n",
+      "030935 填空题\n",
+      "030936 填空题\n",
+      "030937 填空题\n",
+      "030938 填空题\n",
+      "030939 填空题\n",
+      "030940 填空题\n",
+      "030941 填空题\n",
+      "030942 填空题\n",
+      "030943 填空题\n",
+      "030944 选择题\n",
+      "030945 选择题\n",
+      "030946 选择题\n",
+      "030947 选择题\n",
+      "030948 选择题\n",
+      "030949 填空题\n",
+      "030950 填空题\n",
+      "030951 填空题\n",
+      "030952 填空题\n",
+      "030953 填空题\n",
+      "030954 填空题\n",
+      "030955 选择题\n",
+      "030956 选择题\n",
+      "030957 选择题\n",
+      "030958 选择题\n",
+      "030959 填空题\n",
+      "030960 填空题\n",
+      "030961 填空题\n",
+      "030962 填空题\n",
+      "030963 填空题\n",
+      "030964 填空题\n",
+      "030965 解答题\n",
+      "030966 填空题\n",
+      "030967 填空题\n",
+      "030968 填空题\n",
+      "030969 解答题\n",
+      "030970 填空题\n",
+      "030971 填空题\n",
+      "030972 填空题\n",
+      "030973 填空题\n",
+      "030974 填空题\n",
+      "030975 填空题\n",
+      "030976 填空题\n",
+      "030977 填空题\n",
+      "030978 填空题\n",
+      "030979 选择题\n",
+      "030980 选择题\n",
+      "030981 选择题\n",
+      "030982 填空题\n",
+      "030983 选择题\n",
+      "030984 填空题\n",
+      "030985 填空题\n",
+      "030986 填空题\n",
+      "030987 填空题\n",
+      "030988 填空题\n",
+      "030989 选择题\n",
+      "030990 选择题\n",
+      "030991 选择题\n",
+      "030992 选择题\n",
+      "030993 选择题\n",
+      "030994 选择题\n",
+      "030995 选择题\n",
+      "030996 选择题\n",
+      "030997 选择题\n",
+      "030998 选择题\n",
+      "030999 填空题\n",
+      "031000 填空题\n",
+      "031001 填空题\n",
+      "031002 选择题\n",
+      "031003 选择题\n",
+      "031004 选择题\n",
+      "031005 选择题\n",
+      "031006 选择题\n",
+      "031007 填空题\n",
+      "031008 填空题\n",
+      "031009 填空题\n",
+      "031010 填空题\n",
+      "031011 填空题\n",
+      "031012 填空题\n",
+      "031013 填空题\n",
+      "031014 填空题\n",
+      "031015 填空题\n",
+      "031016 填空题\n",
+      "031017 填空题\n",
+      "031018 填空题\n",
+      "031019 填空题\n",
+      "031020 填空题\n",
+      "031021 填空题\n",
+      "031022 填空题\n",
+      "031023 填空题\n",
+      "031024 填空题\n",
+      "031025 填空题\n",
+      "031026 解答题\n",
+      "031027 解答题\n",
+      "031028 解答题\n",
+      "031029 解答题\n",
+      "031030 解答题\n",
+      "031031 解答题\n",
+      "031032 解答题\n",
+      "031033 解答题\n",
+      "031034 解答题\n",
+      "031035 解答题\n",
+      "031036 解答题\n",
+      "031037 解答题\n",
+      "031038 解答题\n",
+      "031039 解答题\n",
+      "031040 解答题\n",
+      "031041 解答题\n",
+      "031042 解答题\n",
+      "031043 解答题\n",
+      "031044 解答题\n",
+      "031045 解答题\n",
+      "031046 解答题\n",
+      "031047 解答题\n",
+      "031048 解答题\n",
+      "031049 解答题\n",
+      "031050 解答题\n",
+      "031051 解答题\n",
+      "031052 解答题\n",
+      "031053 解答题\n",
+      "031054 解答题\n",
+      "031055 解答题\n",
+      "031056 解答题\n",
+      "031057 填空题\n",
+      "031058 填空题\n",
+      "031059 选择题\n",
+      "031060 填空题\n",
+      "031061 选择题\n",
+      "031062 填空题\n",
+      "031063 填空题\n",
+      "031064 填空题\n",
+      "031065 填空题\n",
+      "031066 填空题\n",
+      "031067 选择题\n",
+      "031068 选择题\n",
+      "031069 填空题\n",
+      "031070 填空题\n",
+      "031071 填空题\n",
+      "031072 填空题\n",
+      "031073 填空题\n",
+      "031074 填空题\n",
+      "031075 选择题\n",
+      "031076 选择题\n",
+      "031077 解答题\n",
+      "031078 解答题\n",
+      "031079 解答题\n",
+      "031080 解答题\n",
+      "031081 解答题\n",
+      "031082 解答题\n",
+      "031083 解答题\n",
+      "031084 解答题\n",
+      "031085 解答题\n",
+      "031086 解答题\n",
+      "031087 解答题\n",
+      "031088 解答题\n",
+      "031089 解答题\n",
+      "031090 解答题\n",
+      "031091 解答题\n",
+      "031092 解答题\n",
+      "031093 解答题\n",
+      "031094 解答题\n",
+      "031095 填空题\n",
+      "031096 填空题\n",
+      "031097 填空题\n",
+      "031098 填空题\n",
+      "031099 填空题\n",
+      "031100 填空题\n",
+      "031101 填空题\n",
+      "031102 填空题\n",
+      "031103 填空题\n",
+      "031104 填空题\n",
+      "031105 填空题\n",
+      "031106 选择题\n",
+      "031107 选择题\n",
+      "031108 选择题\n",
+      "031109 选择题\n",
+      "031110 解答题\n",
+      "031111 解答题\n",
+      "031112 解答题\n",
+      "031113 解答题\n",
+      "031114 解答题\n",
+      "031115 解答题\n",
+      "031116 解答题\n",
+      "031117 解答题\n",
+      "031118 填空题\n",
+      "031119 填空题\n",
+      "031120 填空题\n",
+      "031121 填空题\n",
+      "031122 填空题\n",
+      "031123 填空题\n",
+      "031124 填空题\n",
+      "031125 填空题\n",
+      "031126 填空题\n",
+      "031127 填空题\n",
+      "031128 填空题\n",
+      "031129 解答题\n",
+      "031130 解答题\n",
+      "031131 解答题\n",
+      "031132 解答题\n",
+      "031133 解答题\n",
+      "031134 填空题\n",
+      "031135 填空题\n",
+      "031136 填空题\n",
+      "031137 填空题\n",
+      "031138 填空题\n",
+      "031139 选择题\n",
+      "031140 选择题\n",
+      "031141 选择题\n",
+      "031142 选择题\n",
+      "031143 选择题\n",
+      "031144 选择题\n",
+      "031145 解答题\n",
+      "031146 填空题\n",
+      "031147 填空题\n",
+      "031148 填空题\n",
+      "031149 填空题\n",
+      "031150 填空题\n",
+      "031151 填空题\n",
+      "031152 填空题\n",
+      "031153 填空题\n",
+      "031154 填空题\n",
+      "031155 选择题\n",
+      "031156 选择题\n",
+      "031157 选择题\n"
      ]
     }
    ],
diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb
index cdc7e21c..cb3ac48d 100644
--- a/工具/题号选题pdf生成.ipynb
+++ b/工具/题号选题pdf生成.ipynb
@@ -2,16 +2,16 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "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/中档题_ver2_教师用_20230107.tex\n",
+      "开始编译教师版本pdf文件:  临时文件/题库_教师用_20230108.tex\n",
       "0\n",
-      "开始编译学生版本pdf文件:  临时文件/中档题_ver2_学生用_20230107.tex\n",
+      "开始编译学生版本pdf文件:  临时文件/题库_学生用_20230108.tex\n",
       "0\n"
      ]
     }
@@ -26,14 +26,13 @@
     "\"\"\"---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",
-    "a\n",
     "\n",
     "\"\"\"\n",
     "\"\"\"---设置题目列表结束---\"\"\"\n",
     "\n",
     "\"\"\"---设置文件名---\"\"\"\n",
     "#目录和文件的分隔务必用/\n",
-    "filename = \"临时文件/中档题_ver2\"\n",
+    "filename = \"临时文件/题库\"\n",
     "\"\"\"---设置文件名结束---\"\"\"\n",
     "\n",
     "\n",
diff --git a/文本处理工具/表格整理.ipynb b/文本处理工具/表格整理.ipynb
index d5b5d97b..0049b2ad 100644
--- a/文本处理工具/表格整理.ipynb
+++ b/文本处理工具/表格整理.ipynb
@@ -82,7 +82,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.15"
+   "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]"
   },
   "orig_nbformat": 4,
   "vscode": {
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 225515bf..649aec91 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -361455,7 +361455,7 @@
         "tags": [
             "第一单元"
         ],
-        "genre": "",
+        "genre": "填空题",
         "ans": "",
         "solution": "",
         "duration": -1,
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@@ -366466,7 +366466,7 @@
         "same": [],
         "related": [],
         "remark": "",
-        "space": ""
+        "space": "12ex"
     },
     "030836": {
         "id": "030836",
@@ -366475,7 +366475,7 @@
         "tags": [
             "第三单元"
         ],
-        "genre": "",
+        "genre": "解答题",
         "ans": "",
         "solution": "",
         "duration": -1,
@@ -366487,6 +366487,6747 @@
         "same": [],
         "related": [],
         "remark": "",
+        "space": "12ex"
+    },
+    "030837": {
+        "id": "030837",
+        "content": "已知数列$\\{a_n\\}$为等差数列, 数列$\\{a_n\\}$的前$5$项和$S_5=20$, $a_5=6$, 则$a_{10}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030838": {
+        "id": "030838",
+        "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_2+a_8=15-a_5$, 则$S_9$等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030839": {
+        "id": "030839",
+        "content": "若等差数列$\\{a_n\\}$满足$a_3+a_5=16$, 则$a_4=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030840": {
+        "id": "030840",
+        "content": "已知等差数列$\\{a_n\\}$($n \\in \\mathbf{N}$, $n\\ge 1$)满足$a_3+a_7=a_5^2+1$, 则$a_5=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030841": {
+        "id": "030841",
+        "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_1=3$, $a_9=27$, 则$S_{22}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030842": {
+        "id": "030842",
+        "content": "等差数列$\\{a_n\\}$满足$a_3+a_2=8$, $a_4+a_3=12$, 则数列$\\{a_n\\}$前$n$项的和为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030843": {
+        "id": "030843",
+        "content": "等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_5=S_7$, 且$a_2+a_3=8$, 则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{S_n}{n^2}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030844": {
+        "id": "030844",
+        "content": "已知公差不为$0$的等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_4$、$S_5$、$S_7 \\in\\{-10,0\\}$, 则$S_n$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030845": {
+        "id": "030845",
+        "content": "已知等差数列$\\{a_n\\}$的首项$a_1=2$, 且对任意$m$、$n \\in \\mathbf{N}$($m \\neq n$, $m,n\\ge 1$), 存在$k \\in \\mathbf{N}$, $k\\ge 1$, 使得$a_m+a_n=a_k$成立, 则$a_1+a_2+a_3+a_4+a_5$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030846": {
+        "id": "030846",
+        "content": "设$a$、$m$是实数, 则 ``$m=5$'' 是 ``$m$为$a$和$10-a$的等差中项'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030847": {
+        "id": "030847",
+        "content": "已知公差为$d$的等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 则 ``$S_n-n a_n<0$, 对$n>1$, $n \\in \\mathbf{N}$恒成立'' 是 ``$d>0$'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030848": {
+        "id": "030848",
+        "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 如果$-a_1<a_9<-a_2$, 则 \\bracket{20}.\n\\fourch{$S_9>0$且$S_{10}>0$}{$S_9>0$且$S_{10}<0$}{$S_9<0$且$S_{10}>0$}{$S_9<0$且$S_{10}<0$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030849": {
+        "id": "030849",
+        "content": "已知$\\{a_n\\}$是公差为$d$的等差数列, 前$n$项和为$S_n, a_1$、$a_2$、$a_3$、$a_4$的平均值为$4, a_5$、$a_6$、$a_7$、$a_8$的平均值为$12$.\\\\\n(1) 求证:$S_n=n^2$;\\\\\n(2) 是否存在实数$t$, 使得$|\\dfrac{a_{n+1}}{a_n}-t|<1$对任意$n \\in \\mathbf{N}$, $n\\ge 1$恒成立, 若存在, 求出$t$的取值范围, 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题18",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030850": {
+        "id": "030850",
+        "content": "已知等比数列$\\{a_n\\}$各项均为正数, 其中$a_1=1, a_2+a_3=12$, 则$\\{a_n\\}$的公比为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030851": {
+        "id": "030851",
+        "content": "已知等比数列$\\{a_n\\}$的首项为$2$, 公比为$q$($q \\in \\mathbf{R}$), 且$a_2$、$a_3+2$、$a_4$成等差数列, 则$q=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030852": {
+        "id": "030852",
+        "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2^n$, 数列$\\{b_n\\}$是首项为$1$, 公比为$q$的等比数列, 若$b_k<a_k<b_{k+1}$, 其中$k=1$、$2$、$\\cdots$、$10$, 则公比$q$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030853": {
+        "id": "030853",
+        "content": "已知无穷等比数列$\\{a_n\\}$的各项均为正整数, 且$\\dfrac{a_1 a_{n+1}}{a_{n+1}-a_1 a_n}=2022$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则满足条件的不同数列$\\{a_n\\}$的个数为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030854": {
+        "id": "030854",
+        "content": "若$a$、$b$、$c$、$d$成等比数列, 则下列三个数列: \\textcircled{1} $a+b$、$b+c$、$c+d$; \\textcircled{2} $a b$、$b c$、$c d$; \\textcircled{3} $a-b$、$b-c$、$c-d$. 其中必成等比数列的个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030855": {
+        "id": "030855",
+        "content": "记方程\\textcircled{1}: $x^2+a_1 x+1=0$, 方程\\textcircled{2}: $x^2+a_2 x+2=0$, 方程\\textcircled{3}: $x^2+a_3 x+4=0$, 其中$a_1$、$a_2$、$a_3$是正实数. 当$a_1$、$a_2$、$a_3$成等比数列时, 下列选项中, 能推出方程\\textcircled{3}有两个不相等的实根的是\\bracket{20}.\n\\twoch{方程\\textcircled{1}有实根, 且\\textcircled{2}有实根}{方程\\textcircled{1}有实根, 且\\textcircled{2}无实根}{方程\\textcircled{1}无实根, 且\\textcircled{2}有实根}{方程\\textcircled{1}无实根, 且\\textcircled{2}无实根}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030856": {
+        "id": "030856",
+        "content": "已知无穷等比数列$\\{a_n\\}$中$a_1=2$, $|a_2|<2$, 它的前$n$项和为$S_n$, 则下列命题正确的是\\bracket{20}.\n\\twoch{数列$\\{S_n\\}$是递增数列}{数列$\\{S_n\\}$是递减数列}{数列$\\{S_n\\}$存在最小项}{数列$\\{S_n\\}$存在最大项}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030857": {
+        "id": "030857",
+        "content": "设各项均为正整数的无穷等差数列$\\{a_n\\}$, 满足$a_{338}=2022$, 且存在正整数$k$, 使$a_1$、$a_{338}$、$a_k$成等比数列, 则公差$d$的所有可能取值的个数为\\bracket{20}.\n\\fourch{$1$}{$4$}{$5$}{无穷多}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030858": {
+        "id": "030858",
+        "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2-7 n$, 且满足$16<a_k+a_{k+1}<22$, 则正整数$k=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030859": {
+        "id": "030859",
+        "content": "已知数列$\\{a_n\\}$的首项$a_1=2$, 且对任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$a_{n+1}=\\dfrac{2 a_n}{a_n+2}$, 则$\\displaystyle\\lim _{n \\to +\\infty} a_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030860": {
+        "id": "030860",
+        "content": "数列$\\{a_n\\}$满足$a_1=2, a_2=\\dfrac 1{1-a_1}$, 若对于大于$2$的正整数$n$, $a_n=\\dfrac 1{1-a_{n-1}}$, 则$a_{102}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030861": {
+        "id": "030861",
+        "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$, 其中$a_n$是$\\sqrt 2=1.41421356237 \\cdots$的小数点后的第$n$位数字, (例如$a_1=4$, $a_6=3$), 若$b_1=a_\\gamma$, 且对任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 均有$b_{n+1}=a_{b_n}$, 则满足$b_n=n-2019$的所有$n$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030862": {
+        "id": "030862",
+        "content": "若各项均为正数的有穷数列$\\{y_n\\}$满足$y_{i+1} \\geq y_i+1$, $n \\geq 3$, $1 \\leq i \\leq n-1$, $i \\in \\mathbf{N}$, $n \\in \\mathbf{N}$, $y_1+y_2+y_3+\\cdots+y_n=2022$, 则满足不等式$y_n+n \\geq M$的正整数$M$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030863": {
+        "id": "030863",
+        "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 满足$2S_n=3 a_n-1$($n \\in \\mathbf{N}$, $n\\ge 1$), 函数$f(x)$定义域为$\\mathbf{R}$, 对任意$x \\in \\mathbf{R}$都有$f(x+1)=\\dfrac{1+f(x)}{1-f(x)}$. 若$f(2)=1-\\sqrt 2$, 则$f(a_{2022})$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030864": {
+        "id": "030864",
+        "content": "设项数为$4$的数列$\\{a_n\\}$满足: $a_i \\in\\{-1,0,1\\}$, $i \\in\\{1,2,3,4\\}$, 且对任意$1 \\leq k<l \\leq 4$, $k \\in \\mathbf{N}$, $l \\in \\mathbf{N}$, 都有$|a_k+a_{k+1}+\\cdots+a_l|\\leq 1$, 则这样的数列$\\{a_n\\}$共有\\blank{50}个.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030865": {
+        "id": "030865",
+        "content": "等差数列$\\{a_n\\}$满足: \\textcircled{1} $a_1<0, a_2>\\dfrac 32$;  \\textcircled{2} 在区问$(11,20)$中的项恰好比区间$[41,50]$中的项少$2$项, 则数列$\\{a_n\\}$的通项公式$a_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030866": {
+        "id": "030866",
+        "content": "已知$D=(10, t)$, 数列$\\{a_n\\}$满足$a_{n+1}^2+a_n^2=2(a_{n+1}+1)(a_n-1)+1$, $n \\in \\mathbf{N}$, $n\\ge 1$, 若对任意正实数$\\lambda$, 总存在$a_1 \\in D$和相邻两项$a_k$、$a_{k+1}$, 使厨$a_{i+1}+\\lambda a_k=0$成立, 则实数$t$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030867": {
+        "id": "030867",
+        "content": "若数列$\\{a_n\\}$满足$a_0=0$, 且$|a_k|=|a_{k-1}+3|$($k \\in \\mathbf{N}$, $k\\ge 1$), 则$|a_1+a_2+\\cdots+a_{19}+a_{20}|$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030868": {
+        "id": "030868",
+        "content": "在数列$\\{a_n\\}$中, 已知$a_1=a_5=1$, 且$a_{n+1}=\\begin{cases}\\dfrac 12 a_n, & a_n \\text{是偶数},  \\\\a_n+d, & a_n \\text{不是偶数}, \\end{cases}$ 则正整数$d=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030869": {
+        "id": "030869",
+        "content": "已知$n \\in \\mathbf{N}$, $n\\ge 1$, 记$\\max \\{x_1, \\cdots, x_n\\}$衾示$x_1, \\cdots, x_n$中的最大值, $\\min \\{y_1, \\cdots, y_n\\}$表示$y_1, \\cdots, y_n$中的最小值. 若$f(x)=x^2-3 x+2$, $g(x)=2^x-1$, 数列$\\{a_n\\}$和$\\{b_n\\}$满足$a_{n+1}=\\min \\{f(a_n), g(a_n)\\}$, $b_{n+1}=\\max \\{b_n, g(b_n)\\}$, $a_1=a$, $b_1=b$, $a, b \\in \\mathbf{R}$. 则下列说法中正确的是\\bracket{20}.\n\\twoch{若$a \\geq 4$, 则存在正整数$m$, 使得$a_{m+1}<a_m$}{若$a \\leq 2$, 则$\\displaystyle\\lim _{n \\to \\infty} a_n=0$}{若$b \\geq 2$, 则$\\displaystyle\\lim _{n \\to \\infty} b_n=0$}{若$b \\in \\mathbf{R}$, 则存在正整数$m$, 使得$b_{m+1}<b_m$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030870": {
+        "id": "030870",
+        "content": "在数列$\\{a_n\\}$中, $a_1=2$, $a_2=a$, $a_{n+2}=\\begin{cases}\\dfrac{a_{n+1}}{a_n}, & a_{n+1} \\geq a_n, \\\\\\dfrac{a_n}{a_{n+1}}, & a_{n+1}<a_n,\\end{cases}$ $(n \\in \\mathbf{N}$, $n\\ge 1$), 对于命题:\\\\\n\\textcircled{1} 存在$a \\in[2,+\\infty)$, 对于任意的正整数$n$, 都有$a_{n+3}=a_n$;\\\\\n\\textcircled{2} 对于任意$a \\in[2,+\\infty)$和任意的正整数$n$, 都有$a_n \\leq a$.\n下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}是真命题, \\textcircled{2}也是真命题}{\\textcircled{1}是真命题, \\textcircled{2}是假命题}{\\textcircled{1}是假命题, \\textcircled{2}是真命题}{\\textcircled{1}是假命题, \\textcircled{2}也是假命题}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030871": {
+        "id": "030871",
+        "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且满足$a_n=1+(n-1) d$, \n$5 a_2=a_8$, 则$S_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030872": {
+        "id": "030872",
+        "content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 公比$q>1$, 且$a_2+1$为$a_1$与$a_3$的等差中项, $S_3=14$, 若数列$\\{b_n\\}$渦足$b_n=\\log_2 a_n$, 其前$n$项和为$T_n$, 则$T_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030873": {
+        "id": "030873",
+        "content": "已知无穷数列$\\{a_n\\}$各项均为整数, 且满足$a_2=-1$, $a_{4 n-1}<a_{4 m}$($n=1,2,3, \\cdots$), $a_{m+n} \\in\\{a_m+a_n+1, a_m+a_n+2\\}$($m, n=1,2, \\cdots$), 则该数列的前$8$项和$S_8=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030874": {
+        "id": "030874",
+        "content": "已知集合$A=\\{x | x=2 n-1,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, $B=\\{x | x=2^n, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 将$A \\cup B$中的所有元素按从小到大的顺序排列构成一个数列$\\{a_n\\}$, 设数列$\\{a_n\\}$的前$n$项和为$S_n$, 则使得$S_n>1000$成立的最小的$n$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030875": {
+        "id": "030875",
+        "content": "设函数$f(x)=\\dfrac{2^{x+1}}{2^x+\\sqrt 2}$, $x \\in \\mathbf{R}$, 数列$\\{a_n\\}$中, $n \\in \\mathbf{N}$, $n\\ge 1$, $a_1=f(\\dfrac 12)$, $a_2=f(\\dfrac 13)+f(\\dfrac 23)$, 一般地, $a_k=f(\\dfrac 1{k+1})+f(\\dfrac 2{k+1})+f(\\dfrac 3{k+1})+\\cdots+f(\\dfrac k{k+1})$, (其中$k=1,2,3, \\cdots$). 则数列$\\{a_n\\}$的前$n$项和$S_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030876": {
+        "id": "030876",
+        "content": "已知数列$\\{a_n\\}$满足:  对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$|a_{n+1}-a_n|=n$, $a_n \\leq \\dfrac{n-1}2$. 设数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_1=0$, 则$S_8$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030877": {
+        "id": "030877",
+        "content": "数列$\\{a_n\\}$的前$n$项的和$S_n$满足$S_{n+1}+S_n=n$($n \\in \\mathbf{N}$, $n\\ge 1$), 则下列选项中正确的是\\bracket{20}.\n\\twoch{数列$\\{a_{n+1}+a_n\\}$是常数列}{若$a_1<\\dfrac 13$, 则$\\{a_n\\}$是递增数列}{若$a_1=-1$, 则$S_{2022}=1013$}{若$a_1=1$, 则$\\{a_n\\}$的最小项的值为$-1$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030878": {
+        "id": "030878",
+        "content": "已知因数$f(x)=\\dfrac x{1+\\sqrt x}$($x>0$), 数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=f(a_n)$, 记数列$\\{a_n\\}$的前$n$项和为$S_n$, 则 \\bracket{20}.\n\\fourch{$3<S_{2022}<4$}{$\\dfrac 32<S_{2002}<3$}{$4<S_{2022}<\\dfrac 92$}{$\\dfrac 92<S_{2022}<5$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030879": {
+        "id": "030879",
+        "content": "在等差数列$\\{a_n\\}$中, 已知$a_1+a_2=10$, $a_3+a_4+a_5=30$.\\\\\n(1) 求数列$\\{a_n\\}$的辽项公式:\\\\\n(2) 若数列$\\{a_n+b_n\\}$是首项为$1$, 公比为$3$的等比数列, 求数列$\\{b_n\\}$的剛$n$项和$S_n$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题18",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030880": {
+        "id": "030880",
+        "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$, 其中$b_n=2^{a_n}$, $n \\in \\mathbf{N}$, $n\\ge 1$, 数列$\\{a_n+b_n\\}$的前$n$项和为$S_n$.\\\\\n(1) 若$a_n=2 n$, 求$S_n$;\\\\\n(2) 若$\\{b_n\\}$是各项为正的等比数列, $S_n=3 n$, 求数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题18",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030881": {
+        "id": "030881",
+        "content": "设$S_n$是各项为正的等比数列$\\{a_n\\}$的前$n$项的和, 且$S_2=3$, $a_3=4$, $n \\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 在数列$\\{a_n\\}$的任意$a_k$与$a_{k+1}$项之问, 都插入$k$($k \\in \\mathbf{N}$, $k\\ge 1$)个相同的数$(-1)^k k$, 组成数列$\\{b_n\\}$, 记数列$\\{b_n\\}$的前$n$项的和为$T_n$, 求$T_{100}$的值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题18",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030882": {
+        "id": "030882",
+        "content": "已知数列$\\{a_n\\}$满足以下两个条件: \\textcircled{1} $a_1=1$, 当$n \\geq 2$时, $|a_n-1|=|a_{n-1}+1|$; \\textcircled{2} 若存在某一项$a_m \\leq-3$, 则存在$k \\in\\{1,2, \\cdots, m-1\\}$, 使得$a_k=a_m+2$($m \\geq 2$且$m \\in \\mathbf{N}$).\\\\\n(1) 若$a_2>0$, 求$a_2$、$a_3$、$a_4$;\\\\\n(2) 若对一切正整数$n$, $a_{n+T}=a_n$均成立向$T$的最小值为$6$, 求该数列的前$9$项之和;\\\\\n(3) 在所有的数列$\\{a_n\\}$中, 求满足$a_m=-2021$的$m$的最小值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030883": {
+        "id": "030883",
+        "content": "已知无穷等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 首项$a_1=3$, 公比为$q$, 且$\\displaystyle\\lim _{n \\to \\infty} S_n=2$, 则$q=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030884": {
+        "id": "030884",
+        "content": "首项为$1$, 公比为$-\\dfrac 12$的无穷等比数列$\\{a_n\\}$的各项和为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030885": {
+        "id": "030885",
+        "content": "已知等比数列$\\{a_n\\}$的公比为$2$, 前$n$项和为$S_n$, 则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{S_n}{a_n}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030886": {
+        "id": "030886",
+        "content": "数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases}2 n-1, & 1 \\leq n \\leq 10, \\\\2-\\dfrac 1n, & n \\geq 11,\\end{cases}$ 则$\\displaystyle\\lim _{n \\to \\infty} a_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030887": {
+        "id": "030887",
+        "content": "若数列$\\{a_n\\}$满足$\\sqrt {a_{n+1}}=a_n-6$, $n \\in \\mathbf{N}$, $n \\ge 1$, 且$\\displaystyle\\lim _{n \\to \\infty} a_n$存在, 则$\\displaystyle\\lim _{n \\to \\infty} a_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030888": {
+        "id": "030888",
+        "content": "若数列$\\{a_n\\}$是首项为$\\dfrac 12$, 公比为$a-\\dfrac 12$的无穷等比数列, 且数列$\\{a_n\\}$各项的和为$a$, 则实数$a$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030889": {
+        "id": "030889",
+        "content": "计算: $\\displaystyle\\lim _{n \\to \\infty} \\dfrac{1+2+\\cdots+2^{n-1}}{2^n}+(\\dfrac 12+\\dfrac 1{2^2}+\\cdots+\\dfrac 1{2^n}+\\cdots)=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030890": {
+        "id": "030890",
+        "content": "设无穷等比数列$\\{a_n\\}$($n \\in \\mathbf{N}$, $n\\ge 1$)的首项$a>0$, 前两项的和为$\\dfrac 13$, 若所有奇数项的和比所有偶数项的和大$3$, 则$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030891": {
+        "id": "030891",
+        "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases}1, & n=1, \\\\(\\dfrac 12)^n, & n \\geq 2, \\end{cases}$ $S_n$是数列$\\{a_n\\}$的前$n$项和, 则$\\displaystyle\\lim _{n \\to \\infty} S_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030892": {
+        "id": "030892",
+        "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=3^n$, $n \\in \\mathbf{N}$, $n\\ge 1$,  \n则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{a_1+a_2+a_3+\\cdots+a_n}{a_n}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030893": {
+        "id": "030893",
+        "content": "已知等比数列$\\{a_n\\}$, 其前$n$项和为$S_n$. 若$a_1=1$, 公比为$3$, \n则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{S_n}{a_{n+1}}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030894": {
+        "id": "030894",
+        "content": "设无穷等比数列$\\{a_n\\}$的公比为$q$, 且$a_1=q^2+1$, 则该数列的各项和的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030895": {
+        "id": "030895",
+        "content": "已知平面直角坐标系中的点$A_n(0, \\dfrac 1n)$、$B_n(0,-\\dfrac 2n)$、$C_n(2+\\dfrac 1{2^n}, 0)$, $n \\in \\mathbf{N}$, $n\\ge 1$. 记$S_n$为$\\triangle A_n B_n C_n$外接圆的面积, 则$\\displaystyle\\lim _{n \\to \\infty} S_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030896": {
+        "id": "030896",
+        "content": "设直线$l: 3 x-y-n=0$($n \\in \\mathbf{N}$, $n\\ge 1$)与函数$f(x)=(\\dfrac 34)^{x+2}$和$g(x)=(\\dfrac 34)^x+3$的图像分别交于$P_n$、$Q_n$两点, 则$\\displaystyle\\lim _{n \\to \\infty}|P_n Q_n|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030897": {
+        "id": "030897",
+        "content": "已知点$O(0,0)$、$A_0(2,3)$和$B_0(5,6)$, 记线段$A_0B_0$的中点为$P_1$, 取线段$A_0P_1$和$P_1B_0$中的一条, 记其端点为$A_1$、$B_1$, 使之满足$(|OA_1|-5)(|OB_1|-5)<0$, 记线段$A_1B_1$的中点为$P_2$, 取线段$A_1P_2$和$P_2B_1$中的一条, 记其端点为$A_2$、$B_2$, 使之满足$(|OA_2|-5)(|OB_2|-5)<0$, 依次下去, 得到点$P_1$、$P_2$、$\\cdots$、$P_n$、$\\cdots$, 则$\\displaystyle\\lim _{n \\to \\infty}|A_0P_n|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030898": {
+        "id": "030898",
+        "content": "记数列$\\{a_n\\}$的通项公式为$a_n=\\begin{cases}(-1)^n, & n \\leq 2021, \\\\\\dfrac{2 n+1}{n+1}, & n \\geq 2022, \\end{cases}$ $n \\in \\mathbf{N}$, $n\\ge 1$, 则数列$\\{a_n\\}$的极限为\\bracket{20}.\n\\fourch{$-1$}{$1$}{$2$}{不存在}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030899": {
+        "id": "030899",
+        "content": "在数列$\\{a_n\\}$中, 已知奇数项是公比为$\\dfrac 13$的等比数列, 偶数项是公比为$\\dfrac 12$的等比数列, 且$a_1=3$, $a_2=2$, 则下列各项正确的是\\bracket{20}.\n\\twoch{$a_1+a_2+\\cdots+a_{100}>9$}{$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{a_{n+1}}{a_n}=0$}{$\\dfrac{a_{10}}{a_{11}}<10$}{$\\displaystyle\\lim _{n \\to \\infty} a_n=0$}",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030900": {
+        "id": "030900",
+        "content": "保障性租赁住房, 是政府为缓解新市民、青年人住房困难, 作出的重要决策部署. 2021 年 7 月, 国务院办公厅发布《关于加快发展保障性租赁住房的意见》后, 国内多个城市陆续发布了保障性租赁住房相关政策或征求意见稿. 为了响应国家号召, 某地区计划 2021 年新建住房$40$万平方米, 其中有$25$万平方米是保障性租任住房, 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, 保障性租赁住房的面积均比上一年增加$5$万平方米.\\\\\n(1) 到哪一年底, 该市历年所建保障性租赁住房的累计面积(以 2021 年为累计的第一年) 将首次不少于$475$万平方米?\\\\\n(2) 到哪一年底, 当年建造的保障性租赁住房的面积占该年建造住房面积的比例首次大于$85 \\%$?",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030901": {
+        "id": "030901",
+        "content": "随着人们生活水平的提高, 很多家庭都购买了家用汽车, 使用汽车共需支出三笔费用: 购置费、燃油费、养护保险费. 某种型号汽车, 购置费共$20$万元, 购买后第 1 年燃油费共$2$万元, 以后每一年都比前一年增加$0.2$万元.\\\\\n(1) 若每年养护保险费均为$1$万元, 设购买该种型号汽车$n$($n \\in \\mathbf{N}$, $n \\ge 1$)年后共支出费用为$S_n$万元, 求$S_n$的表达式;\\\\\n(2) 若购买汽车后的前 6 年, 每年养护保险费均为$1$万元, 由于部件老化和事故多发, 第 7 年起, 每一年的养护保险费都比前一年增加$10 \\%$, 设使用$n$($n \\in \\mathbf{N}$, $n\\ge 1$)年后年平均费用为$c_n$, 当$n=n_0$时, $c_n$最小. 请你列出$n>6$时$c_n$的表达式, 并利用计算器确定$n_0$的值(只需写出$n_0$的值).",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030902": {
+        "id": "030902",
+        "content": "为了防止某种新冠病毒感染, 某地居民需服用一种药物预防. 规定每人每天定时服用一次, 每次服用$m$毫克. 已知人的肾脏每$24$小时可以从体内滤除这种药物的$80 \\%$, 设第$n$次服药后(滤除之前)这种药物在人体内的含量是$a_n$毫克. (即$a_1=m$)\\\\\n(1) 已知$m=12$, 求$a_2$、$a_3$;\\\\\n(2) 该药物在人体的含量超过$25$毫克会产生毒副作用, 若人需要长期服用这种药物, 求$m$的最大值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030903": {
+        "id": "030903",
+        "content": "某公司 2021 年投资$4$千万元用于新产品的研发与生产, 计划从 2022 年起, 在今后的若干年内, 每年继续投资$1$千万元用于新产品的维护与生产, 2021 年新产品带来的收入为$0.5$千万元, 并预测在相当长的年份里新产品带来的收入均在上年度收入的基础上增长$25 \\%$. 记 2021 年为第$1$年, $f(n)$为第$1$年至此后第$n$($n \\in \\mathbf{N}$, $n\\ge 1$)年的累计利润 (注: 含第$n$年, 累计利润$=$累计收入 $-$累计投入, 单位: 千万元), 且当$f(n)$为正值时, 认为新产品赢利.\\\\\n(1) 试求$f(n)$的表达式;\\\\\n(2) 根据预测, 该新产品将从哪一年开始并持续赢利? 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030904": {
+        "id": "030904",
+        "content": "某地区 2020 年产生的生活垃圾为$20$万吨, 其中$6$万吨垃圾以环保方式处理, 剩余$14$万吨垃圾以填埋方式处理, 预测显示: 在以 2020 年为第一年的末来十年内, 该地区每年产生的生活垃圾量比上一年增长$5 \\%$, 同时, 通过环保方式处理的垃圾量比上一年增加$1.5$万吨, 剩余的垃圾以填埋方式处理. 根据预测, 解答下列问题:\\\\\n(1) 求 2021 年至 2023 年, 该地区三年通过填埋方式处理的垃圾共计多少万吨?\n(结果精确到$0.1$万吨)\\\\\n(2) 该地区在哪一年通过环保方式处理的垃圾量首次超过这一年产生生活垃圾量的$50 \\%$?",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030905": {
+        "id": "030905",
+        "content": "以太阳能和风能为代表的新能源发电具有取之不尽、零碳排放等优点. 近年来我国新能源发电的装机容量快速增长, 学校新能源发电研究课题组的同学通过查阅相关资料, 整理出《2015-2020 年全国各类发电装机容量统计表 (单位: 万万千瓦)》.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline \\multirow{2}{*}{年份} & \\multicolumn{3}{|c|}{传统能源发电} & \\multicolumn{2}{|c|}{新能源发电} & \\multirow{2}{*}{总装机} \\\\\n\\cline {2 - 6} & 火力发电 & 水力发电 & 核能发电 & 太阳能发电 & 风能发电 & \\\\\n\\hline 2015 &$10.06$&$3.20$&$0.27$&$0.43$&$1.31$&$15.27$\\\\\n\\hline 2016 &$10.60$&$3.32$&$0.34$&$0.76$&$1.47$&$16.49$\\\\\n\\hline 2017 &$11.10$&$3.44$&$0.36$&$1.30$&$1.64$&$17.84$\\\\\n\\hline 2018 &$11.44$&$3.53$&$0.45$&$1.74$&$1.84$&$19.00$\\\\\n\\hline 2019 &$11.90$&$3.56$&$0.49$&$2.10$&$2.05$&$20.10$\\\\\n\\hline 2020 &$12.45$&$3.70$&$0.50$&$2.53$&$2.82$&$22.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请根据上表提供的数据, 解决课题小组的两个问题:\\\\\n(1) 2015 年至 2020 年期间, 我国发电总装机容量平均每年比上一年增加多少万万千瓦(精确到$0.01$)? 同期新能源发电装机容量的年平均增长率是多少(精确到$0.1 \\%$)?\\\\\n(2) 假设从 2021 年开始, 我国发电总装机容量平均每年比上一年增加$2$万万千瓦, 新能源发电装机容量的年平均增长率为$20 \\%$, 问从哪一年起, 我国新能源发电装机容量首次超过发电总装机容量的$60\\%$?",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030906": {
+        "id": "030906",
+        "content": "治理垃圾是改善环境的重要举措. $A$地在未进行垃圾分类前每年需要焚烧垃圾量为$200$万吨, 当地政府从 2020 年开始推进垃圾分类工作, 通过对分类垃圾进行环保处理等一系列措施, 预计从 2020 年开始的连续$5$年, 每年需要焚烧垃圾量比上一年减少$20$万吨, 从第$6$年开始, 每年需要焚烧垃圾量为上一年的$75 \\%$(记 2020 年为第 1 年).\\\\\n(1) 写出$A$地每年需要焚烧垃圾量与治理年数$n$($n \\in \\mathbf{N}$, $n \\ge 1$)的表达式;\\\\\n(2) 设$A_n$为从 2020 年开始$n$年内需要焚烧垃圾量的年平均值, 证明数列$\\{A_n\\}$为递减数列.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030907": {
+        "id": "030907",
+        "content": "甲、乙两人同时分别入职$A$、$B$两家公司, 两家公司的基础工资标准分别为: $A$公司第一年月基础工资数为$3700$元, 以后每年月基础工资比上一年月基础工资增加$300$元; $B$公司第一年月基础工资数为$4000$元, 以后每年月基础工资都是上一年的月基础工资的$1.05$倍.\\\\\n(1) 分别求甲、乙两人工作满$10$年的基础工资收入总量(精确到$1$元);\\\\\n(2) 设甲、乙两人入职第$n$年的月基础工资分别为$a_n$、$b_n$元, 记$c_n=a_n-b_n$, 讨论数列$\\{c_n\\}$的单调性, 指出哪年起到哪年止相同年份甲的月基础工资高于乙的月基础工资, 并说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题19",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030908": {
+        "id": "030908",
+        "content": "已知函数$f(x)=2-|x|$, 无穷数列$\\{a_n\\}$满足$a_{n+1}=f(a_n)$, $n \\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 若$a_1=2$, 写出数列$\\{a_n\\}$的通项公式(不必证明);\\\\\n(2) 若$a_1>0$, 且$a_1$、$a_2$、$a_3$成等比数列, 求$a_1$的值; 问$\\{a_n\\}$是否为等比数列, 并说明理由;\\\\\n(3) 证明:$a_1$、$a_2$、$\\cdots$、$a_n$、$\\cdots$成等差数列的充要条件是$a_1=1$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030909": {
+        "id": "030909",
+        "content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=-a_n$或$a_{n+1}=a_n+2$, 对一切$n \\in \\mathbf{N}$, $n \\ge 1$都成立. 记$S_n$为数列$\\{a_n\\}$的前$n$项和. 若存在一个非零常数$T \\in \\mathbf{N}$, $T\\ge 1$, 对于任意$n \\in \\mathbf{N}$, $n\\ge 1$, $a_{n+T}=a_n$成立, 则称数列$\\{a_n\\}$为周期数列, $T$是一个周期.\\\\\n(1) 求$a_2$、$a_3$所有可能的值, 并写出$a_{2022}$的最小可能值; (不需要说明理由)\\\\\n(2) 若$a_n>0$, 且存在正整数$p$、$q$($p \\neq q$), 使得$\\dfrac{a_p}q$与$\\dfrac{a_q}p$均为整数, 求$a_{p+q}$的值;\\\\\n(3) 记集合$S=\\{n | S_n=0,\\  n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 求证: 数列$\\{a_n\\}$为周期数列的必要非充分条件为``集合$S$为无穷集合''.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030910": {
+        "id": "030910",
+        "content": "已知集合$A=\\{y | y=2 x,\\  x \\in \\mathbf{N}, \\ x\\ge 1\\}$, $B=\\{y | y=3^x,\\ x \\in \\mathbf{N}, \\ x\\ge 1\\}$, $A \\cup B$中的所有元素按从小到大的顺序排列构成数列$\\{a_n\\}$, $S_n$为数列$\\{a_n\\}$的前$n$项的和.\\\\\n(1) 求$S_{10}$;\\\\\n(2) 如果$a_m=81$, $a_{2022}=t$, 求$m$和$t$的值;\\\\\n(3) 如果$n=\\dfrac{3^k-1}2+k$($k \\in \\mathbf{N}$, $k\\ge 1$), 求$S_n$(用$k$来表示).",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030911": {
+        "id": "030911",
+        "content": "已知数列$\\{a_n\\}$满足$a_n=\\begin{cases}2, & n=1, \\\\ qa_{n-1}+\\dfrac t{a_{n-1}}, & n \\ge 2,\\ n \\in \\mathbf{N}.\\end{cases}$($q\\ge 0$, $t\\ge 0$, $q^2+t^2\\ne 0$)\\\\\n(1) 当$q>1$时, 求证: 数列$\\{a_n\\}$不可能是常数列;\\\\\n(2) 若$q t=0$, 求数列$\\{a_n\\}$的前$n$项的和;\\\\\n(3) 当$q=\\dfrac 12$, $t=1$时, 令$b_n=\\dfrac 2{\\sqrt {a_n^2-2}}$($n \\geq 2$, $n \\in \\mathbf{N}$), 判断对任意$n \\geq 2$, $n \\in \\mathbf{N}$, $b_n$是否为正整数, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030912": {
+        "id": "030912",
+        "content": "若数列$\\{a_n\\}$满足$a_n+a_{n+1}+a_{n+2}+\\cdots+a_{n+k}=0$($n \\in \\mathbf{N}$, $n\\ge 1$, $k \\in \\mathbf{N}$, $k\\ge 1$), 则称数列$\\{a_n\\}$为 ``$k$阶相消数列''. 已知 ``$2$阶相消数列''$\\{b_n\\}$的通项公式为$b_n=2 \\cos \\omega n$, 记$T_n=b_1 b_2 \\cdots b_n$, $1 \\leq n \\leq 2021$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则当$n=$\\blank{50}时, $T_n$取得最小值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030913": {
+        "id": "030913",
+        "content": "如果数列$\\{a_n\\}$每一项都是正数, 且对任意不小于$2$的正整数$n$满足$a_n^2 \\leq a_{n-1} a_{n+1}$, 则称数列$\\{a_n\\}$具有性质$M$.\\\\\n(1) 若$a_n=p \\cdot q^n$, $b_n=a n+b$($p$、$q$、$a$、$b$均为正实数), 判断数列$\\{a_n\\}$、$\\{b_n\\}$是否具有性质$M$, 并说明理由;\\\\\n(2) 若数列$\\{a_n\\}$、$\\{b_n\\}$都具有性质$M$, $c_n=a_n+b_n$, 证明: 数列$\\{c_n\\}$也具有性质$M$;\\\\\n(3) 设实数$a \\geq 2$, 方程$x^2-a x+1=0$的两根为$x_1$、$x_2$, $a_n=x_1^n+x_2^n$($n \\in \\mathbf{N}$, $n \\ge 1$), 若$\\dfrac{a_1}{a_2}+\\dfrac{a_2}{a_3}+\\cdots+\\dfrac{a_n}{a_{n+1}}>n-1$对任意$n \\in \\mathbf{N}$, $n\\ge 1$恒成立, 求所有满足条件的$a$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030914": {
+        "id": "030914",
+        "content": "设有数列$\\{x_n\\}$($n \\in \\mathbf{N}$, $n\\ge 1$), 对于任意给定的$i$($i \\in \\mathbf{N}$, $i\\ge 1$), 记满足不等式$x_j-x_i \\geq t_i(j-i)$($j \\in \\mathbf{N}$, $j\\ge 1$, $j \\neq i$)的$t_i$构成的集合为$T(i)$, 并称数列$\\{x_n\\}$具有性质$X$.\\\\\n(1) 若$t_i=1$, $j>i$, 数列: $2$、$2 m+2$、$m^2$具有性质$X$, 求实数$m$的取值范围;\\\\\n(2) 若$t_i=2$, $j>i$, 数列$\\{a_n\\}$是各项均为正整数且公比大于$1$的等比数列, 且数列$\\{a_n\\}$不具有性质$X$. 设$b_n=\\dfrac{a_{n+1}}{n+1}$($n \\in \\mathbf{N}$), 试判断数列$\\{b_n\\}$是否具有性质$X$, 并说明理由;\\\\\n(3) 若数列$\\{c_n\\}$具有性质$X$, 当$i>1$时, $T(i)$都为单元素集合, 求证: 数列$\\{c_n\\}$是等差数列.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030915": {
+        "id": "030915",
+        "content": "设$q$、$d$为常数, 若存在大于$1$的整数$k$, 使得无穷数列$\\{a_n\\}$满足$a_{n+1}=\\begin{cases}a_n+d, & \\dfrac nk \\notin \\mathbf{N}, \\\\q a_n, &\\dfrac nk \\in \\mathbf{N},\\end{cases}$ 则称数列$\\{a_n\\}$($n \\in \\mathbf{N}$)为``$M(k)$数列''.\\\\\n(1) 设$d=3$, $q=0$, 若首项为$1$的数列$\\{a_n\\}$为``$M(3)$数列'', 求$a_{2021}$;\\\\\n(2) 若首项为$1$的等比数列$\\{b_n\\}$为``$M(k)$数列'', 求数列$\\{b_n\\}$的通项公式, 并指出相应的$k$、$d$、$q$的值;\\\\\n(3) 设$d=1$, $q=2$, 若首项为$1$的数列$\\{c_n\\}$为``$M(10)$数列'', 求数列$\\{c_n\\}$的前$10 n$项和$S_{10 n}$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030916": {
+        "id": "030916",
+        "content": "将有穷数列$\\{a_n\\}$中部分项按原顺序构成的新数列$\\{b_n\\}$称为$\\{a_n\\}$的一个``子列'' , 剩余项按原顺序构成``子列''$\\{c_n\\}$. 若$\\{b_n\\}$各项的和与$\\{c_n\\}$各项的和相等, 则称$\\{b_n\\}$和$\\{c_n\\}$为数列$\\{a_n\\}$的一对``完美互补子列''.\\\\\n(1) 若数列$\\{a_n\\}$为$2$、$3$、$5$、$6$、$8$、$9$, 请问$\\{a_n\\}$是否存在 ``完美互补子列''? 并说明理由;\\\\\n(2) 已知共$100$项的等比数列$\\{a_n\\}$为递减数列, 且$a_1>0$, 公比为$q$. 若$\\{a_n\\}$存在``完美互补子列'', 求证: $\\dfrac 12<q<1$;\\\\\n(3) 数列$\\{a_n\\}$满足$a_n=n$, $1 \\leq n \\leq m$($m$、$n \\in \\mathbf{N}$, $m,n\\ge 1$), 设$\\{a_n\\}$共有$f(m)$对``完美互补子列'', 求证: 当$m=4 k$和$m=4 k+3$($k \\in \\mathbf{N}$, $k\\ge 1$)时, $\\{a_n\\}$都存在``完美互补子列''且$f(4 k+3) \\geq 3 f(4 k)$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030917": {
+        "id": "030917",
+        "content": "已知数列$\\{a_n\\}$, 若存在$A \\in \\mathbf{R}$使得数列$\\{|a_n-A|\\}$是递减数列, 则称数列$\\{a_n\\}$是 ``$A$型数列''.\\\\\n(1) 判断数列$\\pi$、$-\\sqrt 3$、$-1$、$\\dfrac 12$是否为``$0$型数列'';\\\\\n(2) 若等比数列$\\{a_n\\}$的通项公式为$a_n=q^n$($n \\in \\mathbf{N}$, $n\\ge 1$), $q>0$, 其前$n$项和为$S_n$, 且$\\{S_n\\}$是``$A$型数列'', 求$A$和$q$的取值范围;\\\\\n(3) 已知$k>0$, 数列$\\{a_n\\}$满足$a_1=0$, $a_{n+1}=k|a_n|-1$($n \\in \\mathbf{N}$), 若存在$A \\in \\mathbf{R}$, 使得$\\{a_n\\}$是``$A$型数列'', 求$k$的取值范围, 并求出所有满足条件的$A$(用$k$表示).",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030918": {
+        "id": "030918",
+        "content": "已知有穷数列$\\{a_n\\}$的各项均不相等, 将$\\{a_n\\}$的项从大到小重新排序后相应的项数构成新数列$\\{p_n\\}$, 称$\\{p_n\\}$为$\\{a_n\\}$的``序数列''. 例如, 数列$a_1$、$a_2$、$a_3$满足$a_1>a_3>a_2$, 则其 ``序数列''$\\{p_n\\}$为$1$、$3$、$2$, 若两个不同数列的 ``序数列'' 相同, 则称这两个数列互为 ``保序数列''.\\\\\n(1) 若数列$3-2 x$、$5 x+6$、$x^2$的 ``序数列'' 为$2$、$3$、$1$, 求实数$x$的取值范围;\\\\\n(2) 若项数均为$2021$的数列$\\{x_n\\}$、$\\{y_n\\}$互为``保序数列'', 其通项公式分别为$x_n=(n+\\dfrac 12) \\cdot(\\dfrac 23)^n$, $y_n=-n^2+t n$($t$为常数), 求实数$t$的取值范围;\\\\\n(3) 设$a_n=q^{n-1}+p$, 其中$p$、$q$是实常数, 且$q>-1$, 记数列$\\{a_n\\}$的前$n$项和为$S_n$, 若当正整数$k \\geq 3$时, 数列$\\{a_n\\}$的前$k$项与数列$\\{S_n\\}$的前$k$项(都按原来的顺序)总是互为``保序数列'', 求$p$、$q$满足的条件.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030919": {
+        "id": "030919",
+        "content": "对于数列$\\{a_n\\}$: 若存在正整数$n_0$, 使得当$n \\geq n_0$时, $a_n$恒为常数, 则称数列$\\{a_n\\}$是准常数数列. 现已知数列$\\{a_n\\}$的首项$a_1=a$, 且$a_{n+1}=|a_n-1|$, $n \\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 若$a=\\dfrac 32$, 试判断数列$\\{a_n\\}$是否是准常数数列;\\\\\n(2) 当$a$与$n_0$满足什么条件时, 数列$\\{a_n\\}$是准常数数列? 写出符合条件的$a$与$n_0$的关系;\\\\\n(3) 若$a \\in(k, k+1)$($k \\in \\mathbf{N}$), 求$\\{a_n\\}$的前$3 k$项的和$S_{3 k}$(结果用$k$、$a$表示).",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030920": {
+        "id": "030920",
+        "content": "设数列$\\{a_n\\}$、$\\{b_n\\}$的项数相同, 对任意不相等的正整数$s$、$t$都有$(a_s-a_t)(b_s-b_t)>0$($<0$), 则称数列$\\{a_n\\}$、$\\{b_n\\}$成同序(反序).\\\\\n(1) 若$a_n=\\dfrac 1{2^n}$, $b_n=\\log_a n$, 且$\\{a_n\\}$、$\\{b_n\\}$成反序, 求$a$的取值范围;\\\\\n(2) 记等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 公差为$d$, 求证: $\\{a_n\\}$和$\\{S_n\\}$同序的充要条件是$d(a_1+d)>0$;\\\\\n(3) 若数列$\\{a_n\\}$的通项公式为$a_n=q^{n-1}$($q \\neq 1$, $q>0$), 其前$n$项的和为$S_n$, 令$b_n=\\dfrac{S_n}n$. 研究$\\{a_n\\}$、$\\{b_n\\}$是成同序, 反序, 还是其它情况? 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030921": {
+        "id": "030921",
+        "content": "记实数$a$、$b$中较小者为$\\min \\{a, b\\}$, 例如$\\min \\{1,2\\}=1$, $\\min \\{1,1\\}=1$, 对于无穷数列$\\{a_n\\}$, 记$h_k=\\min \\{a_{2 k-1}$, $a_{2 k}\\}$. 若对任意$k \\in \\mathbf{N}$, $k\\ge 1$均有$h_k<h_{k+1}$, 则称数列$\\{a_n\\}$为 ``趋向递增数列''.\\\\\n(1) 已知数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式分别为$a_n=\\cos \\dfrac{n \\pi}2$, $b_n=(-\\dfrac 12)^n$, 判断数列$\\{a_n\\}$、$\\{b_n\\}$是否为 ``趋向递增数列''? 并说明理由;\\\\\n(2) 已知首项为$1$, 公比为$q$的等比数列$\\{c_n\\}$是 ``趋向递增数列'', 求公比$q$的取值范围;\\\\\n(3) 若数列$\\{d_n\\}$满足$d_1$、$d_2$为正实数, 且$d_n=|d_{n+2}-d_{n+1}|$, 求证: 数列$\\{d_n\\}$为 ``趋向递增数列'' 的必要非充分条件是$\\{d_n\\}$中没有$0$.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030922": {
+        "id": "030922",
+        "content": "对于项数为$m$的数列$\\{a_n\\}$, 若满足: $1 \\leq a_1<a_2<\\cdots<a_m$, 且对任意$1 \\leq i \\leq j \\leq m$, $a_i \\cdot a_j$与$\\dfrac{a_j}{a_i}$中至少有一个是$\\{a_n\\}$中的项, 则称$\\{a_n\\}$具有性质$P$.\\\\\n(1) 分别判断数列$1$、$3$、$9$和数列$2$、$4$、$8$是否具有性质$P$, 并说明理由;\\\\\n(2) 如果数列$a_1$、$a_2$、$a_3$、$a_4$具有性质$P$, 求证:$a_1=1$, $a_4=a_2 \\cdot a_3$;\\\\\n(3) 如果数列$\\{a_n\\}$具有性质$P$, 且项数为大于等于$5$的奇数. 判断$\\{a_n\\}$是否为等比数列? 并说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030923": {
+        "id": "030923",
+        "content": "若数列$\\{a_n\\}$同时满足下列两个条件, 则称数列$\\{a_n\\}$具有 ``性质$A$''.\\\\\n\\textcircled{1} $\\dfrac{a_n+a_{n+2}}2>a_{n+1}$($n \\in \\mathbf{N}$, $n\\ge 1$);\\\\ \\textcircled{2} 存在实数$A$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 有$a_n \\geq A$成立.\\\\\n(1) 设$a_n=n^2-4 n+5$, $b_n=\\sin \\dfrac{n \\pi}4$, 试判断$\\{a_n\\}$、$\\{b_n\\}$是否具有``性质$A$'';\\\\\n(2) 设递增的等比数列$\\{c_n\\}$的前$n$项和为$S_n$, 若$c_2=-1$, $S_3=-\\dfrac 72$, 证明: 数列$\\{S_n\\}$具有 ``性质$A$'', 并求出$A$的取值范围;\\\\\n(3) 设数列$\\{d_n\\}$的通项公式$d_n=\\dfrac{2^{n+1} t+n^2+2 n t+t^2}{2^n}$($n \\in \\mathbf{N}$, $n\\ge 1$), 若数列$\\{d_n\\}$具有 ``性质$A$'', 其满足条件的$A$的最大值$A_0=10$, 求$t$的值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030924": {
+        "id": "030924",
+        "content": "若项数为$k$($k \\in \\mathbf{N}$且$k \\geq 3$)的有穷数列$\\{a_n\\}$满足: $|a_1-a_2|\\leq|a_2-a_3|\\leq \\cdots \\leq|a_{k-1}-a_k|$, 则称数列$\\{a_n\\}$具有 ``性质$M$''.\\\\\n(1) 判断下列数列是否具有 ``性质$M$'', 并说明理由;\\\\\n\\textcircled{1} $1$、$2$、$4$、$3$; \\textcircled{2} $2$、$4$、$8$、$16$.\\\\\n(2) 设$b_m=|a_m-a_{m+1}|$($m=1,2, \\cdots, k-1$), 若数列$\\{a_n\\}$具有 ``性质$M$'', 且各项互不相同. 求证: ``数列$\\{a_n\\}$为等差数列'' 的充要条件是 ``数列$\\{a_n\\}$为常数列'';\\\\\n(3) 已知数列$\\{a_n\\}$具有 ``性质$M$'' . 若存在数列$\\{a_n\\}$, 使得数列$\\{a_n\\}$是连续$k$个正整数$1,2, \\cdots, k$的一个排列, 且$|a_1-a_2|+|a_2-a_3|+\\cdots+|a_{k-1}-a_k|=k+2$, 求$k$的所有可能值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030925": {
+        "id": "030925",
+        "content": "已知数列$\\{x_n\\}$. 若存在$B \\in \\mathbf{R}$, 使得$\\{|x_n-B|\\}$为递减数列, 则$\\{x_n\\}$称为 ``$B$型数列''.\\\\\n(1) 是否存在$B \\in \\mathbf{R}$使得有穷数列$1$、$\\sqrt 3$、$2$为$B$型数列? 若是, 写出$B$的一个值; 否则, 说明理由;\\\\\n(2) 已知$2022$项的数列$\\{u_n\\}$中, $u_n=(-1)^n \\cdot(2022-n)$($n \\in \\mathbf{N}$, $1 \\leq n \\leq 2022$), 求使得$\\{u_n\\}$为$B$型数列的实数$B$的取值范围;\\\\\n(3) 已知存在唯一的$B \\in \\mathbf{R}$, 使得无穷数列$\\{a_n\\}$是$B$型数列. 证明: 存在递增的无穷正整数列$n_1<n_2<\\cdots<n_k<\\cdots$, 使得$\\{a_{n_{2 k-1}}\\}$为递增数列, $\\{a_{n_{2 k}}\\}$为递减数列.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030926": {
+        "id": "030926",
+        "content": "对于集合$A=\\{a_1, a_2, a_3, \\cdots, a_n\\}$, $n \\geq 2$且$n \\in \\mathbf{N}$, 定义$A+A=\\{x+y | x \\in A, y \\in A$且$x \\neq y\\}$. 集合$A$中的元素个数记为$|A|$, 当$|A+A|=\\dfrac{n(n-1)}2$时, 称集合$A$具有性质$\\Gamma$.\\\\\n(1) 判断集合$A_1=\\{1,2,3\\}$, $A_2=\\{1,2,4,5\\}$是否具有性质$\\Gamma$, 并说明理由;\\\\\n(2) 设集合$B=\\{1,3, p, q\\}$($p, q \\in \\mathbf{N}$, 且$3<p<q$)具有性质$\\Gamma$, 若$B+B$中的所有元素能构成等差数列, 求$p$、$q$的值;\\\\\n(3) 若集合$A$具有性质$\\Gamma$, 且$A+A$中的所有元素能构成等差数列, 问: 集合$A$中的元素个数是否存在最大值? 若存在, 求出该最大值; 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030927": {
+        "id": "030927",
+        "content": "已知集合$M=\\{x | 1 \\leq x \\leq m,\\  x \\in \\mathbf{Z}\\}$($\\mathbf{Z}$是整数集, $m$是大于$3$的正整数). 若含有$m$项的数列$\\{a_n\\}$满足: 任意的$i$、$j \\in M$, 都有$a_i \\in M$, 且当$i \\neq j$时有$a_i \\neq a_j$, 当$i<m$时有$|a_{i+1}-a_i|=2$或$|a_{i+1}-a_i|=3$, 则称该数列为$P$数列.\\\\\n(1) 写出所有满足$m=5$且$a_1=1$的$P$数列;\\\\\n(2) 若数列$\\{a_n\\}$为$P$数列, 证明:$\\{a_n\\}$不可能是等差数列;\\\\\n(3) 已知含有$100$项的$P$数列$\\{a_n\\}$满足$a_5$、$a_{10}$、$\\cdots$、$a_{5 k}$、$\\cdots$、$a_{100}$($k=1,2,3, \\cdots, 20$)是公差为$d$($d>0$)的等差数列, 求$d$所有可能的值.",
+        "objs": [],
+        "tags": [
+            "第四单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030928": {
+        "id": "030928",
+        "content": "已知向量$\\overrightarrow a=(-k, 1)$, $\\overrightarrow b=(5,3 k-4)$, 若$\\overrightarrow a \\perp \\overrightarrow b$, 则实数$k=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030929": {
+        "id": "030929",
+        "content": "已知$\\overrightarrow{OA} \\perp \\overrightarrow{AB}$, 若$\\overrightarrow{OA}=(1,1,0)$, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030930": {
+        "id": "030930",
+        "content": "已知向量$\\overrightarrow a$、$\\overrightarrow b$满足$|\\overrightarrow a|=2$, $|\\overrightarrow b|=1$, $|\\overrightarrow a+\\overrightarrow b|=\\sqrt 3$, 则$|\\overrightarrow a-\\overrightarrow b|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030931": {
+        "id": "030931",
+        "content": "设向量$\\overrightarrow a$与$\\overrightarrow b$的夹角为$\\theta$, 定义$\\overrightarrow a$与$\\overrightarrow b$的 ``向量积'': $\\overrightarrow a \\times \\overrightarrow b$是一个向量, 它的模$|\\overrightarrow a \\times \\overrightarrow b|=|\\overrightarrow a|\\cdot|\\overrightarrow b|\\cdot \\sin \\theta$, 若$\\overrightarrow a=(-\\dfrac{\\sqrt 3}2,-\\dfrac 12)$, $\\overrightarrow b=(\\dfrac 12, \\dfrac{\\sqrt 3}2)$, 则$|\\overrightarrow a \\times \\overrightarrow b|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030932": {
+        "id": "030932",
+        "content": "若$O$为$\\triangle ABC$内一点, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{BC}+\\overrightarrow{OB} \\cdot \\overrightarrow{CA}+\\overrightarrow{OC} \\cdot \\overrightarrow{AB}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030933": {
+        "id": "030933",
+        "content": "已知向量$\\overrightarrow a$、$\\overrightarrow b$, 若$|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$, 则向量$\\overrightarrow a+2 \\overrightarrow b$在$\\overrightarrow a$方向上的数量投影的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030934": {
+        "id": "030934",
+        "content": "已知$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$是夹角为$60^{\\circ}$的两个单位向量, 若$\\overrightarrow{e_1}+k \\overrightarrow{e_2}$和$k \\overrightarrow{e_1}+\\overrightarrow{e_2}$垂直, 则实数$k=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030935": {
+        "id": "030935",
+        "content": "在$\\triangle ABC$中, $D$是$BC$中点, $AB=2$, $AC=4$, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{CB}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030936": {
+        "id": "030936",
+        "content": "已知菱形$ABCD$的边长为$1$, $\\angle DAB=\\dfrac{\\pi}3$, 点$E$为该菱形边上任意一点, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AE}$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030937": {
+        "id": "030937",
+        "content": "已知双曲线$\\dfrac{x^2}4-\\dfrac{y^2}{b^2}=1$($b>0$)的右焦点为$F$, 若双曲线上存在关于原点$O$对称的两点$P$、$Q$, 使$\\overrightarrow{FP} \\cdot \\overrightarrow{FQ}=4$, 则$b$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030938": {
+        "id": "030938",
+        "content": "已知$A(-1,0)$、$B(1,0)$、$P(1, \\sqrt 3)$, 点$C$是圆$x^2+y^2=1$上的动点, 则$\\overrightarrow{PC} \\cdot \\overrightarrow{PB}+\\overrightarrow{PC} \\cdot \\overrightarrow{PA}$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030939": {
+        "id": "030939",
+        "content": "在平面直角坐标系中, 已知点$A(-1,0)$、$B(0,3)$, $E$、$F$为圆$x^2+y^2=4$上两个动点, 且$|\\overrightarrow{EF}|=4$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{BF}$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030940": {
+        "id": "030940",
+        "content": "已知$\\overrightarrow a$、$\\overrightarrow b$是空间相互垂直的单位向量, 且$|\\overrightarrow c|=5$, $\\overrightarrow c \\cdot \\overrightarrow a=\\overrightarrow c \\cdot \\overrightarrow b=2 \\sqrt 2$, 则$|\\overrightarrow c-m \\overrightarrow a-n \\overrightarrow b|$的最小值是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030941": {
+        "id": "030941",
+        "content": "如图, 动点$C$在以$AB$为直径的半圆$O$上(异于$A$、$B$), $\\angle DCB=\\dfrac{\\pi}2$, 且$DC=CB$, 若$|AB|=2$, 则$\\overrightarrow{OC} \\cdot \\overrightarrow{OD}$的取值范围为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (120:1) node [above left] {$C$} coordinate (C);\n\\draw ($(C)!1!90:(B)$) node [above] {$D$} coordinate (D);\n\\draw (A) arc (180:0:1) -- (A) (B) -- (C) -- (D) (O) -- (D) (O) -- (C); \n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030942": {
+        "id": "030942",
+        "content": "设平面上的向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow x$、$\\overrightarrow y$满足关系$\\overrightarrow a=\\overrightarrow y-\\overrightarrow x$, $\\overrightarrow b=m \\overrightarrow x-\\overrightarrow y$($m \\geq 2$), 又设$\\overrightarrow a$与$\\overrightarrow b$的模均为$1$且互相垂直, 则$\\overrightarrow x$与$\\overrightarrow y$的夹角取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030943": {
+        "id": "030943",
+        "content": "已知平面向量$\\overrightarrow a, \\overrightarrow b, \\overrightarrow c$满足$|\\overrightarrow a|=1$, $|\\overrightarrow b|=2$, $\\overrightarrow a^2=\\overrightarrow a \\cdot \\overrightarrow b$, $2 \\overrightarrow c^2=\\overrightarrow b \\cdot \\overrightarrow c$, 则$|\\overrightarrow c-\\overrightarrow a|^2+|\\overrightarrow c-\\overrightarrow b|^2$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030944": {
+        "id": "030944",
+        "content": "设$\\overrightarrow a=(x, y)$, $\\overrightarrow b=(m, n)$, 且$\\overrightarrow a$、$\\overrightarrow b$均为非零向量, 则 ``$\\dfrac xm=\\dfrac yn$'' 是 ``$\\overrightarrow a\\parallel \\overrightarrow b$'' 的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030945": {
+        "id": "030945",
+        "content": "在$\\triangle ABC$中, $AB=AC=3$, $\\overrightarrow{BD}=2 \\overrightarrow{DC}$. 若$\\overrightarrow{AD} \\cdot \\overrightarrow{BC}=4$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=$\\bracket{20}.\n\\fourch{$3$}{$-3$}{$2$}{$-2$}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030946": {
+        "id": "030946",
+        "content": "已知正方形$ABCD$的边长为$4$, 点$M$、$N$分别在边$AD$、$BC$上, 且$AM=1$, $BN=2$, 若点$P$在正方形$ABCD$的边上, 则$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$的取值范围是\\bracket{20}.\n\\fourch{$[-6,6]$}{$[-6,2]$}{$[-2,6]$}{$[-2,2]$}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030947": {
+        "id": "030947",
+        "content": "已知平面向量$\\overrightarrow a$、$\\overrightarrow m$、$\\overrightarrow n$, 满足$|\\overrightarrow a|=4$, $\\begin{cases}\\overrightarrow m^2-\\overrightarrow a \\cdot \\overrightarrow m+1=0, \\\\\\overrightarrow n^2-\\overrightarrow a \\cdot \\overrightarrow n+1=0,\\end{cases}$ 则当$\\overrightarrow m$与$\\overrightarrow n$的夹角最大时, $|\\overrightarrow m-\\overrightarrow n|$的值为\\bracket{20}.\n\\fourch{$4$}{$2$}{$\\sqrt 3$}{$1$}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030948": {
+        "id": "030948",
+        "content": "已知向量$\\overrightarrow a$与$\\overrightarrow b$的夹角为$120^{\\circ}$, 且$\\overrightarrow a \\cdot \\overrightarrow b=-2$, 向量$\\overrightarrow c$满足$\\overrightarrow c=\\lambda \\overrightarrow a+(1-\\lambda) \\overrightarrow b$($0<\\lambda<1$), 且$\\overrightarrow a \\cdot \\overrightarrow c=\\overrightarrow b \\cdot \\overrightarrow c$, 记向量$\\overrightarrow c$在向量$\\overrightarrow a$与$\\overrightarrow b$方向上的数量投影分别为$x$、$y$. 现有两个结论: \\textcircled{1} 若$\\lambda=\\dfrac 13$, 则$|\\overrightarrow a|=2|\\overrightarrow b|$; \\textcircled{2} $x^2+y^2+x y$的最大值为$\\dfrac 34$. 则正确的判断是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030949": {
+        "id": "030949",
+        "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 设$\\overrightarrow{AB}=\\overrightarrow a$, $\\overrightarrow{AD}=\\overrightarrow b$, $\\overrightarrow{AA_1}=\\overrightarrow c$, 若用向量$\\overrightarrow a$、$\\overrightarrow b$、$\\overrightarrow c$表示向量$\\overrightarrow{AC_1}$, 则$\\overrightarrow{AC_1}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030950": {
+        "id": "030950",
+        "content": "已知抛物线$y^2=2 p x$($p>0$)的焦点为$F, A$、$B$为此抛物线上的异于坐标原点$O$的两个不同的点, 满足$|\\overrightarrow{FA}|+|\\overrightarrow{FB}|+|\\overrightarrow{FO}|=12$, 且$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FO}=\\overrightarrow 0$, 则$p=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030951": {
+        "id": "030951",
+        "content": "已知点$P$为正$\\triangle ABC$边上或内部的一点, 且实数$x$、$y$满足$\\overrightarrow{AP}=x \\overrightarrow{AB}+2 y \\overrightarrow{AC}$, 则$x-y$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030952": {
+        "id": "030952",
+        "content": "已知平面向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$满足$|\\overrightarrow{OA}|=|\\overrightarrow{OB}|=1$, 若关于$x$的方程$|x \\cdot \\overrightarrow{OA}-\\overrightarrow{OB}|=\\dfrac 14$有实数解, 则$\\triangle AOB$面积的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030953": {
+        "id": "030953",
+        "content": "设$O$为坐标原点, $P$是以$F$为焦点的抛物线$y^2=2 p x(p>0)$上任意一点, $M$是线段$PF$上的点, 且$\\overrightarrow{PM}=4 \\overrightarrow{MF}$, 则直线$OM$斜率的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030954": {
+        "id": "030954",
+        "content": "若向量$\\overrightarrow{MA}$、$\\overrightarrow{MB}$的夹角为$\\dfrac{\\pi}6$, 且$|\\overrightarrow{MA}-\\overrightarrow{MB}|=2$, 则$|2 \\overrightarrow{MA}+\\overrightarrow{MB}|$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030955": {
+        "id": "030955",
+        "content": "``$\\overrightarrow a=\\overrightarrow b$ ''$是 ``$|$\\overrightarrow a|=|\\overrightarrow b |$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030956": {
+        "id": "030956",
+        "content": "设$O$为$\\triangle ABC$所在平面上一点, 若实数$x$、$y$、$z$满足$x \\overrightarrow{OA}+y \\overrightarrow{OB}+z \\overrightarrow{OC}=\\overrightarrow0(x^2+y^2+z^2 \\neq 0)$, 则 ``$x y z=0$'' 是 ``点$O$在$\\triangle ABC$的边所在直线上'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030957": {
+        "id": "030957",
+        "content": "已知$A$、$B$、$C$是平面内不共线的三点, 点$O$满足\n$\\overrightarrow{OA}+2 \\overrightarrow{OB}+\\lambda \\overrightarrow{OC}=\\overrightarrow 0$, $\\lambda$为实常数, 现有下述两个命题: \\textcircled{1} 当$\\lambda \\neq-3$时, 满足条件的点$O$存在且是唯一的; \\textcircled{2} 当$\\lambda=-3$时, 满足条件的点$O$不存在.\n则下列说法正确的一项是\\bracket{20}.\n\\twoch{命题\\textcircled{1}和\\textcircled{2}均为真命题}{命题\\textcircled{1}为真命题, 命题\\textcircled{2}为假命题}{命题\\textcircled{1}和\\textcircled{2}均为假命题}{命题\\textcircled{1}为假命题, 命题\\textcircled{2}为真命题}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030958": {
+        "id": "030958",
+        "content": "已知正六边形$ABCDEF$的边长为$2$, 当$\\lambda_i \\in\\{-1,1\\}$($i=1,2,3,4,5$)时, $|\\lambda_1 \\overrightarrow{AB}+\\lambda_2 \\overrightarrow{AC}+\\lambda_3 \\overrightarrow{AD}+\\lambda_4 \\overrightarrow{AE}+\\lambda_5 \\overrightarrow{AF}|$的最大值为\\bracket{20}.\n\\fourch{$6$}{$12$}{$18$}{$8+4 \\sqrt 3$}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030959": {
+        "id": "030959",
+        "content": "已知复数$z=1+2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030960": {
+        "id": "030960",
+        "content": "已知复数$z$满足$z \\cdot \\mathrm{i}=1+\\mathrm{i}$($\\mathrm{i}$是虚数单位), 则复数$z$的模等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030961": {
+        "id": "030961",
+        "content": "已知复数$z$满足$\\mathrm{i} \\cdot z=1+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$|\\overline z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030962": {
+        "id": "030962",
+        "content": "已知$\\mathrm{i}$是虚数单位, 若复数$z=\\mathrm{i} \\cdot(1+\\mathrm{i})$, 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030963": {
+        "id": "030963",
+        "content": "若复数$z$满足$\\mathrm{i} z=\\sqrt 3-\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030964": {
+        "id": "030964",
+        "content": "已知$\\mathrm{i}$为虚数单位, 复数$z=\\mathrm{i}(1+3 \\mathrm{i})$, 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030965": {
+        "id": "030965",
+        "content": "复数$z$满足$z(2+\\mathrm{i})=5(\\mathrm{i}$为虚数单位), 则$|z|=$",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030966": {
+        "id": "030966",
+        "content": "若复数$z=\\dfrac{4+3 \\mathrm{i}}{1-2 \\mathrm{i}}$, 其中$\\mathrm{i}$为虚数单位, 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030967": {
+        "id": "030967",
+        "content": "复数$z=2-\\mathrm{i}$, 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030968": {
+        "id": "030968",
+        "content": "已知复数$z$满足: $\\mathrm{i}+\\dfrac{2+\\mathrm{i}}{\\overline z}=0$($\\mathrm{i}$为虚数单位), 则$|z|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030969": {
+        "id": "030969",
+        "content": "已知$z_1=1+\\mathrm{i}, z_2=2+3 \\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$z_1+\\overline{z_2}=$",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "030970": {
+        "id": "030970",
+        "content": "已知$(1+\\mathrm{i}) z=2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030971": {
+        "id": "030971",
+        "content": "若$\\dfrac{m+\\mathrm{i}}{1+\\mathrm{i}}$为纯虚数 ($\\mathrm{i}$为虚数单位), 则实数$m=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030972": {
+        "id": "030972",
+        "content": "已知复数$z_1=1+\\mathrm{i}$, $z_2=\\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$z_1 \\cdot z_2=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030973": {
+        "id": "030973",
+        "content": "已知复数$z$满足$1+z=(1-z) \\mathrm{i}$, 其中$\\mathrm{i}$是虚数单位, 则$z$的虚部为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030974": {
+        "id": "030974",
+        "content": "已知复数$z$满足: $\\overline z=\\dfrac 2{1+\\mathrm{i}}$($\\mathrm{i}$为虚数单位), 则$\\mathrm{Im} z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030975": {
+        "id": "030975",
+        "content": "已知复数$z=1+\\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 则$z^2+z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030976": {
+        "id": "030976",
+        "content": "设$\\mathrm{i}$为虚数单位, 若复数$z=(1+2 \\mathrm{i})(2-\\mathrm{i})$, 则$z$的实部与虚部的和为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030977": {
+        "id": "030977",
+        "content": "若复数$z=\\dfrac 1{1+\\mathrm{i}}$($\\mathrm{i}$为虚数单位), 则$z \\cdot \\overline z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030978": {
+        "id": "030978",
+        "content": "已知集合$S=\\{x | x=a+b \\mathrm{i}, \\ a, b \\in \\mathbf{Z}\\}$, $\\mathrm{i}$是虛数单位, 对任意$x_1$、$x_2 \\in S$($x_1$、$x_2$可以相等) 均有$\\dfrac{x_1}{x_2} \\in S$, 则符合条件的元素个数最多的集合$S=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030979": {
+        "id": "030979",
+        "content": "已知复数$z=(2 \\sin \\alpha-1)+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则 ``$z$为纯虚数'' 是 ``$\\alpha=\\dfrac{\\pi}6$''的\\bracket{20}条件.\n\\twoch{充分不必要}{必要不充分}{充要}{既不充分也不必要}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030980": {
+        "id": "030980",
+        "content": "若$z_1$、$z_2 \\in \\mathbf{C}$, 则 ``$z_1$、$z_2$均为实数'' 是 ``$z_1-z_2$是实数'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030981": {
+        "id": "030981",
+        "content": "已知$z$为复数, 则下列命题不正确的是\\bracket{20}.\n\\twoch{若$z=\\overline z$, 则$z$为实数}{若$z^2<0$, 则$z$为纯虚数}{若$|z+1|=|z-1|$, 则$z$为纯虚数}{若$z^3=1$, 则$\\overline z=z^2$}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030982": {
+        "id": "030982",
+        "content": "若关于$x$的实系数一元二次方程$x^2-m x+3 m-8=0$有两个共轭虚数根, 则$m$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030983": {
+        "id": "030983",
+        "content": "``实系数一元二次方程$a x^2+b x+5 a=0$的根是$-1 \\pm 2 \\mathrm{i}$'' 是``$b=2 a \\neq 0$''($\\mathrm{i}$是虚数单位)的\\bracket{20}条件.\n\\fourch{充要}{必要不充分}{充分不必要}{既不充分也不必要}",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030984": {
+        "id": "030984",
+        "content": "在复平面内, 复数$z$对应的点的坐标是$(1,2)$, 则$\\mathrm{i} \\cdot z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030985": {
+        "id": "030985",
+        "content": "若复数$z$在复平面内对应的点为$(1,-1)$, 则$\\dfrac 2z=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030986": {
+        "id": "030986",
+        "content": "已知复数$z$的虚部为$1$, 且$|z|=2$, 则$z$在复平面内所对应的点$Z$到虚轴的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030987": {
+        "id": "030987",
+        "content": "若复数$z$满足$|z|=1$, 则$|z-2|$的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030988": {
+        "id": "030988",
+        "content": "在复平面$xOy$内, 复数$z_1$、$z_2$所对应的点分别为$Z_1$、$Z_2$, 对于\n下列四个式子: \\textcircled{1} $z_1^2=|z_1|^2$; \\textcircled{2} $|z_1 \\cdot z_2|=|z_1|\\cdot|z_2|$; \\textcircled{3} $\\overline{OZ_1}^2=|\\overrightarrow{OZ_1}|^2$; \\textcircled{4} $|\\overrightarrow{OZ_1} \\cdot \\overrightarrow{OZ_2}|=|\\overrightarrow{OZ_1}|\\cdot|\\overrightarrow{OZ_2}|$. 其中恒成立的是\\blank{50}(写出所有恒成立式子的序号).",
+        "objs": [],
+        "tags": [
+            "第五单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030989": {
+        "id": "030989",
+        "content": "已知$l_1$、$l_2$是平面$\\alpha$内的两条直线, $l$是空间的一条直线, 则 ``$l \\perp \\alpha ''$是 ``$l \\perp l_1$且$l \\perp l_2$'' 的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030990": {
+        "id": "030990",
+        "content": "已知空间三条直线$a$、$b$、$m$及平面$\\beta$, 且$a$、$b \\subset \\beta$, 条件甲:$m \\perp a, m \\perp b$; 条件乙:$m \\perp \\beta$. 则 ``条件乙'' 是 ``条件甲'' 的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030991": {
+        "id": "030991",
+        "content": "设$m$、$n$是两条不同的直线, $\\alpha$、$\\beta$是两个不同的平面, 则下列命题中的真命题为\\bracket{20}.\n\\twoch{若$m\\parallel \\alpha, n\\parallel \\alpha$, 则$m\\parallel n$}{若$m \\perp \\alpha, n \\perp \\alpha$, 则$m\\parallel n$}{若$m\\parallel \\alpha, m\\parallel \\beta$, 则$\\alpha\\parallel \\beta$}{若$m \\perp \\alpha, \\alpha \\perp \\beta$, 则$m\\parallel \\beta$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030992": {
+        "id": "030992",
+        "content": "下列命题中, 正确的是\\bracket{20}.\n\\onech{三点确定一个平面}{垂直于同一直线的两条直线平行}{若直线$l$与平面$\\alpha$上的无数条直线都垂直, 则$l \\perp \\alpha$}{若$a$、$b$、$c$是三条直线, $a\\parallel b$且与$c$都相交, 则直线$a$、$b$、$c$在同一平面上}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030993": {
+        "id": "030993",
+        "content": "如图, 已知$A$、$B$、$C$、$D$、$E$、$F$分别是正方体所在棱的中点, 则下列直线中与直线$EF$相交的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0)  coordinate (A);\n\\draw (A) ++ (\\l,0,0)  coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l)  coordinate (C);\n\\draw (A) ++ (0,0,-\\l)  coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0)  coordinate (A1);\n\\draw (B) ++ (0,\\l,0)  coordinate (B1);\n\\draw (C) ++ (0,\\l,0)  coordinate (C1);\n\\draw (D) ++ (0,\\l,0)  coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(B)!0.5!(C)$) circle (0.03) node [right] {$E$};\n\\filldraw ($(C)!0.5!(C1)$) circle (0.03) node [right] {$F$};\n\\filldraw ($(D1)!0.5!(C1)$) circle (0.03) node [above] {$A$};\n\\filldraw ($(D1)!0.5!(A1)$) circle (0.03) node [left] {$B$};\n\\filldraw ($(A1)!0.5!(A)$) circle (0.03) node [left] {$C$};\n\\filldraw ($(D1)!0.5!(D)$) circle (0.03) node [right] {$D$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{直线$AB$}{直线$BC$}{直线$CD$}{直线$DA$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030994": {
+        "id": "030994",
+        "content": "如图, 已知点$A \\in$平面$\\alpha$, 点$O \\in \\alpha$, 直线$a \\subset \\alpha$, 点$P \\notin \\alpha$且$PO \\perp \\alpha$, 则 ``直线$a \\perp$直线$OA$'' 是 ``直线$a \\perp$直线$PA$'' 的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw (1.5,0) node [above right] {$A$} coordinate (A);\n\\draw (0,1.3) node [left] {$P$} coordinate (P);\n\\path [name path = OP] (O) -- ($(P)!2!(O)$);\n\\path [name path = PA] (A) -- ($(P)!1.8!(A)$);\n\\draw ($(O)!-0.2!(A)$) -- ($(O)!1.5!(A)$);\n\\draw ($(P)!-0.3!(O)$) -- (O);\n\\draw ($(P)!-0.2!(A)$) -- (A);\n\\path [name path = para, draw]  (O) ++ (-2,-0.7) --++ (1.4,1.4) --++ (4,0) --++ (-1.4,-1.4) -- cycle;\n\\draw (O) ++ (-1.5,-0.7) node [above] {$\\alpha$};\n\\draw (A) ++ (0,-0.4) --++ (0.8,0.8) node [right] {$a$};\n\\path [name intersections = {of = OP and para, by = S}];\n\\path [name intersections = {of = PA and para, by = T}];\n\\draw [dashed] (O) -- (S) (A) -- (T);\n\\draw (S) -- ($(O)!1.5!(S)$) (T) -- ($(A)!1.5!(T)$);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030995": {
+        "id": "030995",
+        "content": "已知平面$\\alpha$经过圆柱$OO_1$的旋转轴, 点$A$、$B$是在圆柱$OO_1$的侧面上, 但不在平面$\\alpha$上, 则下列 4 个命题中真命题的个数是\\bracket{20}.\\\\\n\\textcircled{1} 总存在直线$l$, $l \\subset \\alpha$且$l$与$AB$异面;\\\\\n\\textcircled{2} 总存在直线$l$, $l \\subset \\alpha$且$l \\perp AB$;\n\\textcircled{3} 总存在平面$\\beta$, $AB \\subset \\beta$且$\\beta \\perp \\alpha$;\\\\\n\\textcircled{4} 总存在平面$\\beta$, $AB \\subset \\beta$且$\\beta\\parallel \\alpha$.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030996": {
+        "id": "030996",
+        "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 点$M$、$N$分别在棱$AA_1$、$CC_1$上, 则 ``直线$MN \\perp$直线$C_1B$'' 是 ``直线$MN \\perp$平面$C_1BD$'' 的\\bracket{20}条件.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(240:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.2!(C1)$) node [right] {$N$} coordinate (N);\n\\draw ($(A)!0.8!(A1)$) node [left] {$M$} coordinate (M);\n\\draw [dashed] (M) -- (N) (B) -- (D) (D) -- (C1);\n\\draw (B) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030997": {
+        "id": "030997",
+        "content": "在下列判断两个不同平面$\\alpha$与$\\beta$平行的$4$个命题中, 真命题的个数是\\bracket{20}.\\\\\n\\textcircled{1} $\\alpha$、$\\beta$都垂直于平面$\\gamma$, 那么$\\alpha\\parallel \\beta$;\\\\\n\\textcircled{2} $\\alpha$、$\\beta$都平行于平面$\\gamma$, 那么$\\alpha\\parallel \\beta$;\\\\\n\\textcircled{3} $\\alpha$、$\\beta$都垂直于直线$l$, 那么$\\alpha\\parallel \\beta$;\\\\\n\\textcircled{4} 如果$l, m$是两条异面直线, 且$l\\parallel \\alpha$, $m\\parallel \\alpha$, $l\\parallel \\beta$, $m\\parallel \\beta$, 那么$\\alpha\\parallel \\beta$.\n\\fourch{$0$}{$1$}{$2$}{$3$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030998": {
+        "id": "030998",
+        "content": "在空间中, 直线$AB$平行于直线$EF$, 直线$BC$、$EF$为异面直线, 若$\\angle ABC=120^{\\circ}$, 则异面直线$BC$、$EF$所成角的大小为\\bracket{20}.\n\\fourch{$30^{\\circ}$}{$60^{\\circ}$}{$120^{\\circ}$}{$150^{\\circ}$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "030999": {
+        "id": "030999",
+        "content": "若正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 则顶点$A$到平面$BB_1D_1D$的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031000": {
+        "id": "031000",
+        "content": "如图所示, 在正方体$ABCD-A_1B_1C_1D_1$中, 若$E$是$D_1C_1$的中点, 则异面直线$A_1C_1$与$DE$所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C1)!0.5!(D1)$) node [above] {$E$} coordinate (E);\n\\draw [dashed] (D) -- (E);\n\\draw (A1) -- (C1);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031001": {
+        "id": "031001",
+        "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $P$为底面$ABCD$内 (包括边界)的动点, 满足: 直线$D_1P$与直线$CC_1$所成角的大小为$\\dfrac{\\pi}6$, 则线段$DP$扫过的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (D) ++ ({2/sqrt(6)},0,{2/sqrt(6)}) node [right] {$P$} coordinate (P);\n\\draw [dashed] (D1) -- (P);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031002": {
+        "id": "031002",
+        "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, 点$E$、$F$分别是棱$A_1C_1, BC$的中点, 则下列结论不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\filldraw ($(B)!0.5!(C)$) circle (0.03) node [below right] {$F$} coordinate (F);\n\\filldraw ($(A_1)!0.5!(C_1)$) circle (0.03) node [above] {$E$} coordinate (E);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$CC_1\\parallel$平面$A_1ABB_1$}{$AF\\parallel $平面$A_1B_1C_1$}{$EF\\parallel$平面$A_1ABB_1$}{$AE\\parallel$平面$B_1BCC_1$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031003": {
+        "id": "031003",
+        "content": "如图, 已知$P$、$Q$、$R$分别是正方体$ABCD-A_1B_1C_1D_1$的棱$AB$、$BC$和$C_1D_1$的中点, 由点$P$、$Q$、$R$确定的平面$\\beta$截该正方体所得截面为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(A)!0.5!(B)$) circle (0.03) node [below] {$P$} coordinate (P);\n\\filldraw ($(B)!0.5!(C)$) circle (0.03) node [right] {$Q$} coordinate (Q);\n\\filldraw ($(C1)!0.5!(D1)$) circle (0.03) node [above] {$R$} coordinate (R);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{三角形}{四边形}{五边形}{六边形}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031004": {
+        "id": "031004",
+        "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$, $M$、$N$分别是$A_1D$、$D_1B$的中点, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(D)$) node [below] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(D1)$) node [right] {$N$} coordinate (N);\n\\draw [dashed] (M) -- (N) (A1) -- (D) (B) -- (D1) (B) -- (D);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n\\onech{直线$A_1D$与直线$D_1B$相交, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$平行, 直线$MN \\perp$平面$BDD_1B_1$}{直线$A_1D$与直线$D_1B$垂直, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$异面, 直线$MN \\perp$平面$BDD_1B_1$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031005": {
+        "id": "031005",
+        "content": "如图, 在棱长为$1$的正方体\n$ABCD-A_1B_1C_1D_1$中, $P$、$Q$、$R$分别是棱$AB$、$BC$、$BB_1$的中点, 以$\\triangle PQR$为底面作一个直三棱柱, 使其另一个底面的三个顶点也都在正方体$ABCD-A_1B_1C_1D_1$的表面上, 则这个直三棱柱的体积为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(B)!0.5!(B1)$) node [left] {$R$} coordinate (R);\n\\draw (P) -- (R) -- (Q);\n\\draw [dashed] (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac 38$}{$\\dfrac{\\sqrt 3}8$}{$\\dfrac 3{16}$}{$\\dfrac{\\sqrt 3}{16}$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031006": {
+        "id": "031006",
+        "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是线段$AB$、$BD_1$上的动点, 且直线$EF$与$AA_1$所成的角为$\\arctan \\sqrt 2$, 则下列直线中与$EF$所成的角必为$\\arctan \\dfrac{\\sqrt 2}2$的是\\bracket{20}.\n\\fourch{$CD$}{$BD$}{$BC_1$}{$DC_1$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031007": {
+        "id": "031007",
+        "content": "若圆柱的高、底面半径均为$1$, 则其表面积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031008": {
+        "id": "031008",
+        "content": "底面半径长为$2$, 母线长为$3$的圆柱的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031009": {
+        "id": "031009",
+        "content": "已知圆柱的母线长$4 \\text{cm}$, 底面半径$2 \\text{cm}$, 则该圆柱的侧面积为\\blank{50}$\\mathrm{cm}^2$.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031010": {
+        "id": "031010",
+        "content": "已知一个圆柱的高不变, 它的体积扩大为原来的$4$倍, 则它的侧面积扩大为原来的\\blank{50}倍.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031011": {
+        "id": "031011",
+        "content": "一块边长为$10 \\text{cm}$的正方形铁片按如图所示的阴影部分裁下, 然后用余下的四个全等的等腰三角形作侧面, 以它们的公共顶点$P$为顶点, 加工成一个如图所示的正四棱锥形容器, 当$x=6 \\text{cm}$时, 该容器的容积为\\blank{50}$\\text{cm}^3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\fill [pattern = north east lines] (0.4,0) -- (1,1) -- (0,0.4) -- (0,0) -- cycle;\n\\fill [pattern = north east lines] (1.6,0) -- (1,1) -- (2,0.4) -- (2,0) -- cycle;\n\\fill [pattern = north east lines] (1.6,2) -- (1,1) -- (2,1.6) -- (2,2) -- cycle;\n\\fill [pattern = north east lines] (0.4,2) -- (1,1) -- (0,1.6) -- (0,2) -- cycle;\n\\draw (0.4,0) -- (1.6,2) (0,0.4) -- (2,1.6) (1.6,0) -- (0.4,2) (0,1.6) -- (2,0.4);\n\\draw (0,0) rectangle (2,2);\n\\draw (1,1) node [above = 0.2] {$P$} coordinate (P);\n\\draw [dashed] (1,1) -- (1,0) node [midway, left] {$5$};\n\\draw (2,1) node [right] {$10$};\n\\draw (0.4,0) --++ (0,-0.3) (1.6,0) --++ (0,-0.3);\n\\draw [<->] (0.4,-0.15) -- (1.6,-0.15) node [midway, fill = white] {$x$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (-3,0,3) node [left] {$A$} coordinate (A);\n\\draw (3,0,3) node [right] {$B$} coordinate (B);\n\\draw (3,0,-3) node [right] {$C$} coordinate (C);\n\\draw (-3,0,-3) node [below right] {$D$} coordinate (D);\n\\draw (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031012": {
+        "id": "031012",
+        "content": "若圆锥的侧面积是底面积的$2$倍, 则该圆锥的母线与底面所成角的大小等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031013": {
+        "id": "031013",
+        "content": "一个圆锥的侧面展开图是圆心角为$\\dfrac{4 \\pi}3$, 半径为$18 \\text{cm}$的扇形, 则圆锥的母线与底面所成角的余弦值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031014": {
+        "id": "031014",
+        "content": "已知某圆锥的底面圆的半径为$\\sqrt 2$, 若其侧面展开图为一个半圆, 则该圆锥的侧面积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031015": {
+        "id": "031015",
+        "content": "已知圆锥的侧面积为$\\dfrac{2 \\pi}9$, 若其过轴的截面为正三角形, 则该圆锥的母线的长为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031016": {
+        "id": "031016",
+        "content": "若圆锥的底面面积为$\\pi$, 母线长为$2$, 则该圆锥的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031017": {
+        "id": "031017",
+        "content": "已知圆锥的底面半径为$1$, 母线长为$3$, 则圆锥的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031018": {
+        "id": "031018",
+        "content": "已知一个圆锥的底面半径为$1 \\text{cm}$, 侧面积为$2 \\pi \\text{cm}^2$, 则该圆锥的母线与底面所成的角的大小为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031019": {
+        "id": "031019",
+        "content": "已知圆锥的轴截面是边长为$2$的等边三角形, 则这个圆锥的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031020": {
+        "id": "031020",
+        "content": "如图, 倒置圆锥形容器装有$2$升水, 水平高度正好是圆锥高的一半, 那么, 这个容器的容积是\\blank{50}升.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\fill [gray!20] (-0.5,1) -- (0,0) -- (0.5,1) arc (0:180:0.5 and 0.125);\n\\draw (0,0) -- (1,2) (0,0) -- (-1,2);\n\\draw (0,2) ellipse (1 and 0.25);\n\\draw (0.5,1) arc (360:180:0.5 and 0.125);\n\\draw [dashed] (0.5,1) arc (0:180:0.5 and 0.125);\n\\draw [dashed] (-0.5,1) -- (0.5,1) (0,0) -- (0,2) -- (1,2);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031021": {
+        "id": "031021",
+        "content": "若圆锥的母线长为$5$, 底面半径为$3$, 则该圆锥的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031022": {
+        "id": "031022",
+        "content": "已知圆锥的母线长等于$2$, 侧面积等于$2 \\pi$, 则该圆锥的体积等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031023": {
+        "id": "031023",
+        "content": "设圆锥底面圆周上两点$A$、$B$间的距离为$2$, 圆锥顶点到直线$AB$的距离为$\\sqrt 3$, $AB$和圆锥的轴的距离为$1$, 则该圆锥的侧面积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031024": {
+        "id": "031024",
+        "content": "如果一个圆锥的底面积和侧面积分别为$9 \\pi$和$15 \\pi$, 则该圆锥母线与底面所成角的大小为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031025": {
+        "id": "031025",
+        "content": "在空间直角坐标系$O-x y z$中, 若平面$OMQ$的一个法向量$\\overrightarrow n=(2,1,-2)$, 则点$P(-1,1,4)$到平面$OMQ$的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031026": {
+        "id": "031026",
+        "content": "如图, 直三棱柱$ABC-A_1B_1C_1$中, $\\angle ACB=90^{\\circ}$, $CA=CB=CC_1=2$, 点$D$是线段$A_1B_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,0,0) node [below right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (A) --++ (0,2) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2) node [left] {$B_1$} coordinate (B_1);\n\\draw [dashed] (C) --++ (0,2) node [above left] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (B) -- (C) (A) -- (C);\n\\draw ($(A_1)!0.5!(B_1)$) node [below right] {$D$} coordinate (D);\n\\draw (B) -- (D);\n\\draw [dashed] (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱柱$ABC-A_1B_1C_1$的体积;\\\\\n(2) 已知$P$为侧棱$BB_1$的中点, 求点$P$到平面$BCD$的距离.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031027": {
+        "id": "031027",
+        "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=4$, 异面直线$BC_1$与$AA_1$所成角的大小为$\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,{2/sqrt(3)}) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,{2/sqrt(3)}) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,{2/sqrt(3)}) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (B) -- (C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求正三棱柱$ABC-A_1B_1C_1$的体积;\\\\\n(2) 求直线$BC_1$与平面$AA_1C_1C$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031028": {
+        "id": "031028",
+        "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $AB=2$, $AA_1=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,4) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,4) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,4) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (B) -- (C_1);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$M$} coordinate (M);\n\\draw (M) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求正三棱柱$ABC-A_1B_1C_1$的体积;\\\\\n(2) 若点$M$是侧棱$AA_1$的中点, 求异面直线$BM$与$B_1C_1$所成角的余弦值.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031029": {
+        "id": "031029",
+        "content": "如图, 直三棱柱$ABC-A_1B_1C_1$中, 已知$AB=BC=BB_1=2$, $AB \\perp BC, D$为$AB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (A) -- (C);\n\\draw (B) -- (C_1);\n\\draw ($(A)!0.5!(B)$) node [below left] {$D$} coordinate (D);\n\\draw (A_1) -- (D);\n\\draw [dashed] (D) -- (C) -- (A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BC_1$与$DC$所成角的大小;\\\\\n(2) 求证:$BC_1\\parallel$平面$A_1CD$.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031030": {
+        "id": "031030",
+        "content": "在直三棱柱$ABC-A_1B_1C_1$中, $\\angle ABC=\\dfrac{\\pi}2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,0) node [below right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (A) --++ (0,{2*sqrt(2)}) node [left] {$A_1$} coordinate (A_1);\n\\draw [dashed] (B) --++ (0,{2*sqrt(2)}) node [above left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,{2*sqrt(2)}) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle (A_1) -- (C);\n\\draw [dashed] (B) -- (C) (A) -- (C) (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$B_1C_1$与$AC$所成角的大小;\\\\\n(2) 若$A_1C$与平面$ABC$所成角为$\\dfrac{\\pi}4$, 求三棱锥$A_1-ABC$的体积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031031": {
+        "id": "031031",
+        "content": "如图, 直三棱柱$ABC-A_1B_1C_1$中, $AB \\perp AC, AB=AC=AA_1=2$, 点$D$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw [dashed] (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) -- (C) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (B) -- (A) -- (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (D) -- (C_1);\n\\draw [dashed] (D) -- (A) -- (C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$C_1-ACD$的体积;\\\\\n(2) 求异面直线$AC$与$C_1D$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031032": {
+        "id": "031032",
+        "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的底面为直角三角形且$\\angle ACB=90^{\\circ}$, 直角边$CA$、$CB$的长分别为$3$、$4$, 侧棱$AA_1$的长为$4$, 点$M$、$N$分别为线段$A_1B_1$、$C_1B_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-45:0.5cm)}]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,1.5) node [below] {$A$} coordinate (A);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw (C) -- (A) -- (B) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (B) -- (C);\n\\draw ($(A_1)!0.5!(B_1)$) node [below right] {$M$} coordinate (M);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$N$} coordinate (N);\n\\draw (M) -- (N) (M) -- (A);\n\\draw [dashed] (N) -- (C);\n\\draw (A) -- (C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$A$、$C$、$N$、$M$四点共面;\\\\\n(2) 求直线$AC_1$与平面$ACNM$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031033": {
+        "id": "031033",
+        "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, 已知$AC=BC=4$, $AA_1=3$, $AB=4 \\sqrt 2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,0) node [below right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (B) --++ (0,1.5) node [right] {$B_1$} coordinate (B_1);\n\\draw (A) --++ (0,1.5) node [left] {$A_1$} coordinate (A_1);\n\\draw [dashed] (C) --++ (0,1.5) node [above left] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (B) -- (C) (A) -- (C);\n\\draw (A) -- (B_1);\n\\draw [dashed] (A) -- (C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A-BCC_1B_1$的体积;\\\\\n(2) 求直线$AC_1$与平面$ABB_1A_1$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031034": {
+        "id": "031034",
+        "content": "在直三棱柱$ABC-A_1B_1C_1$中, $AC \\perp BC$, $AC=BC=CC_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [right] {$B$} coordinate (B);\n\\draw ({-sqrt(2)},0,{sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw [dashed] (C) --++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw [dashed] (A) -- (C) (B) -- (C);\n\\draw (A) -- (B_1);\n\\draw [dashed] (A) -- (C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A-BCC_1B_1$的体积$V$;\\\\\n(2) 求直线$AB_1$与平面$ACC_1A_1$所成角的正切值.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031035": {
+        "id": "031035",
+        "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, \n$M$为线段$AB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$C_1-MBC$的体积;\\\\\n(2) 求异面直线$CD$与$MC_1$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031036": {
+        "id": "031036",
+        "content": "在正四棱柱$ABCD-A_1B_1C_1D_1$中, $AB=2$, 连接$A_1C_1$、$C_1B$、$A_1B$, 得到三棱锥$B_1-A_1BC_1$的体积为$2$, 点$P$、$Q$分别为$A_1D$和$AC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B) -- (C1) -- (A1) -- cycle;\n\\draw [dashed] (A1) -- (D) (A) -- (C);\n\\filldraw ($(A1)!0.5!(D)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(A)!0.5!(C)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 求正四棱柱$ABCD-A_1B_1C_1D_1$的表面积;\\\\\n(2) 求异面直线$D_1P$与$C_1Q$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031037": {
+        "id": "031037",
+        "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$4$, $P$、$Q$分别是棱$BC$与$B_1C_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] ($(B)!0.5!(C)$) node [below right] {$P$} coordinate (P) -- (D1);\n\\draw ($(B1)!0.5!(C1)$) node [right] {$Q$} coordinate (Q) -- (A1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求以$A_1$、$D_1$、$P$、$Q$为顶点的四面体的体积;\\\\\n(2) 求异面直线$D_1P$与$A_1Q$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031038": {
+        "id": "031038",
+        "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$1, P$是$CC_1$的中点, 过$AP$的平面与$BB_1$、$DD_1$分别交于$Q$、$R$, 且$BQ=\\dfrac 14$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [above right] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.25!(B1)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(C)!0.5!(C1)$) node [right] {$P$} coordinate (P);\n\\draw ($(D)!0.25!(D1)$) node [left] {$R$} coordinate (R);\n\\draw (A) -- (Q) -- (P);\n\\draw [dashed] (A) -- (R) -- (P) (A) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$PR$与$A_1B_1$所成角的大小;\\\\\n(2) 求$C_1$到平面$AQPR$的距离.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031039": {
+        "id": "031039",
+        "content": "如图所示, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长$1$, 侧棱长$4$, $AA_1$中点为$E$, $CC_1$中点为$F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3.5}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\m) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\m) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\n,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\draw ($(A)!0.5!(A1)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C1)$) node [right] {$F$} coordinate (F);\n\\draw (B1) -- (F) -- (D1) -- cycle;\n\\draw [dashed] (B) -- (D) -- (E) -- cycle (B1) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$BDE\\parallel$平面$B_1D_1F$;\\\\\n(2) 连结$B_1D$, 求直线$B_1D$与平面$BDE$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031040": {
+        "id": "031040",
+        "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$4$, 点$E$是棱$DD_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(D)!0.5!(D1)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$P$} coordinate (P);\n\\draw [dashed] (A1) -- (E) -- (C1) (D1) -- (P);\n\\draw (A1) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$A_1E$与直线$B_1C$所成的角;\\\\\n(2) 若底面$ABCD$上的点$P$满足$PD_1 \\perp$平面$A_1EC_1$, 求线段$DP$的长度.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031041": {
+        "id": "031041",
+        "content": "如图所示, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 侧棱长为$4$, 设$\\overrightarrow{DE}=\\lambda \\overrightarrow{DD_1}$($0<\\lambda<1$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\m) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\m) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\n,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1);\n\\draw (B1) -- (C);\n\\draw ($(D)!0.25!(D1)$) node [right] {$E$} coordinate (E);\n\\draw ($(B1)!(C1)!(C)$) coordinate (T);\n\\draw ($(C)!0.25!(T)$) node [above] {$G$} coordinate (G);\n\\draw [dashed] (E) -- (G) (E) -- (B1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$\\lambda=\\dfrac 12$时, 求直线$B_1E$与平面$ABCD$所成角的大小;\\\\\n(2) 当$\\lambda=\\dfrac 14$时, 若$\\overrightarrow{B_1G}=t \\overrightarrow{B_1C}$, 且$\\overrightarrow{EG} \\cdot \\overrightarrow{B_1C}=0$, \n求正实数$t$的值.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031042": {
+        "id": "031042",
+        "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$是边长为$2$的正方形, $PA \\perp$平面$ABCD$. $PC$与平面$ABCD$所成角的大小为$\\dfrac{\\pi}3, M$为$PA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,2.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [right] {$M$} coordinate (M);\n\\draw (B) -- (C) -- (D) (P) -- (B) (P) -- (C) (P) -- (D);\n\\draw [dashed] (B) -- (A) -- (D) (P) -- (A) (B) -- (M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的体积;\\\\\n(2) 求异面直线$BM$与$PC$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031043": {
+        "id": "031043",
+        "content": "如图所示, 在三棱锥$P-ABC$中, $PA \\perp$平面$ABC$, $CD \\perp AB$于$D$点, $PA=AB=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw (1,0,0) node [above] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) -- (C) -- (B) -- cycle (P) -- (C);\n\\draw [dashed] (A) -- (B) (C) -- (D);\n\\draw pic [draw, scale = 0.3] {right angle = A--D--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$CD \\perp PB$;\\\\\n(2) 若三棱锥$P-ABC$的体积为$\\dfrac{16}3, \\angle ACB=\\dfrac{\\pi}2$, 求$PC$与平面$PAB$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031044": {
+        "id": "031044",
+        "content": "已知三棱锥$P-ABC, PA$、$BA$、$CA$两两互相垂直, 且长度均为$1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--cycle (P)--(D);\n\\draw [dashed] (B)--(A)--(C) (A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱锥$P-ABC$的表面积;\\\\\n(2) 若点$D$为$BC$的中点, 求$PD$与平面$PAC$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031045": {
+        "id": "031045",
+        "content": "如图, 四棱锥$P-ABCD$的底面是矩形, $PD \\perp$底面$ABCD, M$为$BC$的中点, $PD=DC=1$, 直线$PB$与平面$ABCD$所成的角为$\\dfrac{\\pi}6$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{2*sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [right] {$M$} coordinate (M);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(D)--(C) (D)--(P) (A)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的体积;\\\\\n(2) 求异面直线$AM$与$PC$所成的角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031046": {
+        "id": "031046",
+        "content": "在四棱锥$P-ABCD$中, 底面是边长为 2 的菱形, $\\angle DAB=60^{\\circ}$, 对角线$AC$与$BD$相交于点$O, PO \\perp$平面$ABCD, PB$与平面$ABCD$所成的角为$60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw ({sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [above right] {$D$} coordinate (D);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$E$} coordinate (E);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(C) (B)--(D) (P)--(O) (D)--(E) (P)--(D) (A)--(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若$E$是$PB$的中点, 求异面直线$DE$与$PA$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031047": {
+        "id": "031047",
+        "content": "如图, 已知四棱锥$S-ABCD$的底面$ABCD$是梯形, $AD\\parallel BC$, $\\angle BAD=90^{\\circ}$, $SA \\perp$平面$ABCD$, $SA=BC=1$, $AD=2$, $AB=\\sqrt 3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,{sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$S$} coordinate (S);\n\\draw (S)--(B)--(C)--(D)--cycle (S)--(C);\n\\draw [dashed] (B)--(A)--(D) (S)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$S-ABCD$的体积;\\\\\n(2) 求直线$BS$与平面$SCD$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031048": {
+        "id": "031048",
+        "content": "如图, 在四棱锥$P-ABCD$中, 底面$ABCD$是矩形, \n$PA \\perp$平面$ABCD$, $PA=AD=1$, $AB=\\sqrt 3$, $F$是$PD$的中点, 点$E$在棱$CD$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,{2*sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle (P)--(C);\n\\draw [dashed] (B)--(A)--(D) (P)--(A);\n\\draw ($(C)!0.5!(D)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (A)--(F) (A)--(C);\n\\draw (P)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的表面积;\\\\\n(2) 求证: $PE \\perp AF$.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031049": {
+        "id": "031049",
+        "content": "如图, 四棱锥$P-ABCD$的底面为菱形, $PD \\perp$平面$ABCD$, $\\angle BAD=60^{\\circ}$, $E$为棱$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0,{sqrt(3)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(D)--(C) (P)--(D)--(B) (D)--(E); \n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $ED \\perp$平面$PAD$;\\\\\n(2) 若$PD=AD=2$, 求点$D$到平面$PBC$的距离.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031050": {
+        "id": "031050",
+        "content": "如图, 四棱锥$P-ABCD$的底面是矩形, $PD \\perp$底面$ABCD$, $AB=1, BC=\\sqrt 2$, 四棱锥$P-ABCD$的体积为$\\dfrac{\\sqrt 2}3$, $M$为$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{2*sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(B)!0.5!(C)$) node [below right] {$M$} coordinate (M);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B) (P)--(M);\n\\draw [dashed] (A)--(D)--(C) (P)--(D)--(B) (A)--(M); \n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AM$与$PB$所成的角;\\\\\n(2) 求直线$PM$与平面$PBD$所成的角.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031051": {
+        "id": "031051",
+        "content": "如图, 在正四棱锥$P-ABCD$中, $PA=AB=2 \\sqrt 2$, $E$、$F$分别为$PB$、$PD$的中点, 平面$AEF$与棱$PC$的交点为$G$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)},0,{sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ ({2*sqrt(2)},0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,{-2*sqrt(2)}) node [right] {$C$} coordinate (C);\n\\draw (C) ++ ({-2*sqrt(2)},0,0) node [below] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\path [name path = PC] (P)--(C);\n\\path [name path = AG] (A)--($(A)!1.5!($(E)!0.5!(F)$)$);\n\\path [name intersections = {of = PC and AG, by = G}];\n\\draw (A)--(B)--(C)--(P)--cycle (P)--(B) (A)--(E)--(G) node [above right] {$G$} coordinate (G);\n\\draw [dashed] (A)--(F) (P)--(D) (A)--(D)--(C) (F)--(G);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AE$与$PF$所成角的大小;\\\\\n(2) 求平面$AEGF$与平面$ABCD$所成锐二面角的大小;\\\\\n(3) 求点$G$的位置.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031052": {
+        "id": "031052",
+        "content": "如图, 圆锥的底面直径与母线长均为$4, PO$是圆锥的高, 点$C$是底面直径$AB$所对弧的中点, 点$D$是母线$PA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (0,{sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$D$} coordinate (D);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$C$} coordinate (C);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (C)--(O)--(P) (A)--(B) (C)--(D) (D)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆锥的体积;\\\\\n(2) 求直线$CD$与平面$PAB$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031053": {
+        "id": "031053",
+        "content": "如图, 已知圆锥的底面半径$r=2$, 经过旋转轴$SO$的截面是等边三角形$SAB$, 点$Q$为半圆弧$AB$的中点, 点$P$为母线$SA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [right] {$A$} coordinate (A);\n\\draw (-1,0) node [left] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)}) node [above] {$S$} coordinate (S);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$Q$} coordinate (Q);\n\\draw ($(A)!0.5!(S)$) node [right] {$P$} coordinate (P);\n\\draw (A)--(S)--(B);\n\\draw (B) arc (180:360:1 and 0.25);\n\\draw [dashed] (B) arc (180:0:1 and 0.25);\n\\draw [dashed] (A)--(B) (S)--(O)--(Q) (Q)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求此圆锥的表面积;\\\\\n(2) 求异面直线$PQ$与$SO$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031054": {
+        "id": "031054",
+        "content": "已知圆锥的顶点为$S$, 底面圆心为$O$, 母线$SA$的长为$2 \\sqrt 2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$A$} coordinate (A);\n\\draw (0,1) node [above] {$S$} coordinate (S);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (B) arc (0:-180:1 and 0.25);\n\\draw [dashed] (B) arc (0:180:1 and 0.25);\n\\draw (S)--(-1,0) (S)--(A) (S)--(B);\n\\draw [dashed] (O)--(A) (O)--(S) (O)--(B) (A)--(B) (S)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 若圆锥的侧面积为$2 \\sqrt 2 \\pi$, 求圆锥的体积;\\\\\n(2) $A$、$B$是底面圆周上的两个点, $\\angle AOB=90^{\\circ}, M$为线段$AB$的中点, 若圆锥的底面半径为$2$, 求直线$SM$与平面$SOA$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031055": {
+        "id": "031055",
+        "content": "如图, 圆锥的底面半径$OA=2$, 高$PO=6$, 点$C$是底面直径$AB$所对弧的中点, 点$D$是母线$PA$的中点. 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$D$} coordinate (D);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$C$} coordinate (C);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (C)--(O)--(P) (A)--(B) (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 该圆锥的表面积;\\\\\n(2) 直线$CD$与平面$PAB$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031056": {
+        "id": "031056",
+        "content": "如图, 已知圆柱的轴截面$ABCD$是边长为$2$的正方形, $E$是弧$AD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [above right] {$B$} coordinate (B);\n\\draw (1,0) coordinate (T);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw ({1*cos(-120)},{0.25*sin(-120)}) node [below left] {$A_1$} coordinate (A_1);\n\\draw (0,1) node [above] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(A_1)$) node [below left] {$M$} coordinate (M);\n\\draw (1,0) arc (0:-180:1 and 0.25);\n\\draw [dashed] (1,0) arc (0:180:1 and 0.25);\n\\draw (C)--(1,0) (C)--(A) (C)--(A_1);\n\\draw [dashed] (1,0)--(A) (B)--(C) (C)--(M) (B)--(A_1)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆柱的表面积和体积;\\\\\n(2) 求异面直线$BE$与$AD$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题17",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031057": {
+        "id": "031057",
+        "content": "若直线$l$的一个法向量是$\\overrightarrow n=(1,-\\sqrt 3)$, 则直线$l$的倾斜角的大小为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题02",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031058": {
+        "id": "031058",
+        "content": "不等式组$\\begin{cases}x \\leq 3, \\\\x+y \\geq 0, \\\\x-y+4 \\geq 0\\end{cases}$表示的平面区域的面积等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031059": {
+        "id": "031059",
+        "content": "若直线$l$的一个方向向量为$(1,-3)$, 则$l$的法向量可以是\\bracket{20}.\n\\fourch{$(-3,1)$}{$(-1,-3)$}{$(3,1)$}{$(1,3)$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031060": {
+        "id": "031060",
+        "content": "直线$y-2=0$与直线$y=2 x-1$的夹角大小等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031061": {
+        "id": "031061",
+        "content": "``$a_1 b_2-a_2 b_1=0$'' 是 ``直线$a_1 x+b_1 y=1$和$a_2 x+b_2 y=1$平行'' 的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031062": {
+        "id": "031062",
+        "content": "圆$x^2+y^2-2 x-4 y+4=0$的圆心到直线$3 x+4 y+4=0$的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031063": {
+        "id": "031063",
+        "content": "设$a \\in \\mathbf{R}$, $k \\in \\mathbf{R}$, 三条直线$l_1: a x-y-2 a+5=0$, $l_2: x+a y-3 a-4=0$, $l_3: y=k x$, 则$l_1$与$l_2$的交点$M$到$l_3$的距离的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031064": {
+        "id": "031064",
+        "content": "圆$x^2+y^2+4 \\sin \\theta \\cdot x+4 \\cos \\theta \\cdot y+1=0$的半径等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031065": {
+        "id": "031065",
+        "content": "设$P_n(x_n, y_n)$是直线$2 x-y=\\dfrac n{n+1}$($n \\in \\mathbf{N}$, $n \\ge 1$)与圆$x^2+y^2=1$在第一象限的交点, 则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{y_n-1}{x_n-1}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031066": {
+        "id": "031066",
+        "content": "在平面直角坐标系$x O y$中, 已知圆$C:(x-2)^2+y^2=4$, 点$A$是直线$x-y+2=0$上的一个动点, 直线$AP$、$AQ$分别切圆$C$于$P$、$Q$两点, 则线段$PQ$长的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3, 0) -- (5, 0) node [below] {$x$};\n\\draw [->] (0, -3) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw (2, 0) node [below] {$C$} coordinate (C) circle (2);\n\\draw [domain = -3:1] plot (\\x, \\x+2);\n\\draw (-2.5, -0.5) node [below] {$A$} coordinate (A);\n\\draw ({(46-sqrt(66))/41}, {(-4+9*sqrt(66))/41}) node [above] {$Q$} coordinate (Q);\n\\draw ({(46+sqrt(66))/41}, {(-4-9*sqrt(66))/41}) node [below] {$P$} coordinate (P);\n\\draw (A)--(Q) (A)--(P) (A)--(C) (P)--(Q);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031067": {
+        "id": "031067",
+        "content": "从圆$C_1: x^2+y^2=4$上的一点向圆$C_2: x^2+y^2=1$引两条切线, 连接两切点间的线段称为切点弦, 则圆$C_2$内不与任何切点弦相交的区域面积为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}6$}{$\\dfrac{\\pi}4$}{$\\dfrac{\\pi}3$}{$\\dfrac{\\pi}2$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031068": {
+        "id": "031068",
+        "content": "已知直线$l: \\dfrac xa+\\dfrac yb=1$与圆$x^2+y^2=100$有公共点, 且公共点的横、纵坐标均为整数, 则满足$4 \\sqrt {5 a^2+5 b^2} \\geq|a b|$的$l$有 \\bracket{20}条.\n\\fourch{$40$}{$46$}{$52$}{$54$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031069": {
+        "id": "031069",
+        "content": "已知椭圆$x^2+\\dfrac{y^2}{a^2}=1$($a>0$)的一个焦点坐标为$(0,1)$, 则$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031070": {
+        "id": "031070",
+        "content": "设椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上的一点$P$到椭圆两焦点的距离的乘积为$s$, 则当$s$取得最大值时, 点$P$的坐标是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031071": {
+        "id": "031071",
+        "content": "已知椭圆$\\dfrac{(n+1) x^2}{4 n+1}+\\dfrac{(n+2) y^2}{n+1}=1$的右焦点为$F_n(c_n, 0)$, 其中$n \\in \\mathbf{N}$, $n\\ge 1$, 则$\\displaystyle\\lim _{n\\to \\infty} c_n=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031072": {
+        "id": "031072",
+        "content": "已知椭圆$C: \\dfrac{x^2}9+\\dfrac{y^2}{b^2}=1$($b>0$)的左、右两个焦点分别为$F_1$、$F_2$, 过$F_2$的直线交椭圆$C$于$A$、$B$两点, 若$\\triangle F_1AB$是等边三角形, 则$b$的值等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031073": {
+        "id": "031073",
+        "content": "设椭圆$\\Gamma: \\dfrac{x^2}8+\\dfrac{y^2}4=1$的左、右两焦点分别为$F_1$、$F_2$, $P$是$\\Gamma$上的点, 则使得$\\triangle PF_1F_2$是直角三角形的点$P$的个数为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031074": {
+        "id": "031074",
+        "content": "如果椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点$(1,0)$关于直线$y=b x$的对称点在椭圆上, 则$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031075": {
+        "id": "031075",
+        "content": "以坐标原点为中心的椭圆的长轴长等于$8$, 且以抛物线$x^2=12 y$的焦点为一个焦点, 则该椭圆的标准方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{55}+\\dfrac{y^2}{64}=1$}{$\\dfrac{x^2}{28}+\\dfrac{y^2}{64}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}7=1$}{$\\dfrac{x^2}7+\\dfrac{y^2}{16}=1$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031076": {
+        "id": "031076",
+        "content": "已知$a$、$b \\in \\mathbf{R}$, 复数$z=a+2 b \\mathrm{i}$(其中$\\mathrm{i}$为虚数单位)满足$z \\cdot \\overline z=4$, 给出下列结论:\\\\\n\\textcircled{1} $a^2+b^2$的取值范围是$[1,4]$;\\\\\n\\textcircled{2} $\\sqrt {(a-\\sqrt 3)^2+b^2}+\\sqrt {(a+\\sqrt 3)^2+b^2}=4$;\\\\\n\\textcircled{3} $\\dfrac{b-\\sqrt 5}a$的取值范围是$(-\\infty,-1] \\cup[1,+\\infty)$;\\\\\n\\textcircled{4} $\\dfrac 1{a^2}+\\dfrac 1{b^2}$的最小值为$2$. 其中正确结论的个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031077": {
+        "id": "031077",
+        "content": "设常数$m>0$且$m \\neq 1$, 椭圆$\\Gamma: \\dfrac{x^2}{m^2}+y^2=1$, 点$P$是$\\Gamma$上的动点.\\\\\n(1) 若点$P$的坐标为$(2,0)$, 求$\\Gamma$的焦点坐标;\\\\\n(2) 设$m=3$, 若定点$A$的坐标为$(2,0)$, 求$|PA|$的最大值与最小值;\\\\\n(3) 设$m=\\dfrac 12$, 若$\\Gamma$上的另一动点$Q$满足$OP \\perp OQ$($O$为坐标原点), 求证:$O$到直线$PQ$的距离是定值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031078": {
+        "id": "031078",
+        "content": "如图, 已知椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$的左焦点为$F_1$, 点$P$是椭圆$C$上位于第一象限的点, $M$、$N$是$y$轴上的两个动点 (点$M$位于$x$轴上方), 满足$PM \\perp PN$且$F_1M \\perp F_1N$, 线段$PN$交$x$轴于点$Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5, 0) -- (2.5, 0) node [below] {$x$};\n\\draw [->] (0, -2.5) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [above left] {$O$};\n\\path [name path = elli, draw] (0, 0) ellipse (2 and {sqrt(3)});\n\\draw (0, -0.4) node [below left] {$N$} coordinate (N);\n\\draw (0, 2.5) node [right] {$M$} coordinate (M);\n\\draw (-1, 0) node [below] {$F_1$} coordinate (F_1);\n\\path [name path = circ] (0, 1.05) circle (1.45);\n\\path [name intersections = {of = circ and elli, by = {P, Q}}];\n\\draw (P) node [above right] {$P$} --(N) (P)--(M)--(F_1)--(N);\n\\draw pic [draw, scale = 0.3] {right angle = N--F_1--M};\n\\draw pic [draw, scale = 0.3] {right angle = M--P--N};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$|F_1P|=\\dfrac 52$, 求点$P$的坐标;\\\\\n(2) 若四边形$F_1MPN$为矩形, 求点$M$的坐标;\\\\\n(3) 求证:$\\dfrac{|PQ|}{|QN|}$为定值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031079": {
+        "id": "031079",
+        "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{12}+\\dfrac{y^2}8=1$的左、右焦点分别为$F_1$、$F_2$, 过点$F_1$的直线$l$交椭圆于$A$、$B$两点, 交$y$轴于点$P(0, t)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -3.5) -- (0, 3.5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw (0, 0) ellipse ({sqrt(12)} and {sqrt(8)});\n\\filldraw (-2, 0) circle (0.06) node [below] {$F_1$} coordinate (F_1);\n\\filldraw (2, 0) circle (0.06) node [below] {$F_2$} coordinate (F_2);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$F_1P \\perp F_2P$, 求$t$的值;\\\\\n(2) 若点$A$在第一象限, 满足$\\overrightarrow{F_1A} \\cdot \\overrightarrow{F_2A}=7$, 求$t$的值;\\\\\n(3) 在平面内是否存在定点$Q$, 使得$\\overrightarrow{QA} \\cdot \\overrightarrow{QB}$是一个确定的常数? 若存在, 求出点$Q$的坐标, 若不存在, 说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031080": {
+        "id": "031080",
+        "content": "如图, 椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$的左、右焦点分别为$F_1$、$F_2$, 过右焦点$F_2$与$x$轴垂直的直线交椭圆于$M$、$N$两点; 动点$P$、$Q$分别在直线$MN$与椭圆$C$上. 已知$|F_1F_2|=2, \\triangle MNF_1$的周长为$4 \\sqrt 2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw [->] (-2, 0) -- (2, 0) node [below] {$x$};\n\\draw [->] (0, -1.5) -- (0, 1.5) node [left] {$y$};\n\\draw (0, 0) node [below right] {$O$};\n\\draw (0, 0) ellipse ({sqrt(2)} and 1);\n\\draw (-1, 0) node [below] {$F_1$} coordinate (F_1);\n\\draw (1, 0) node [below right] {$F_2$} coordinate (F_2);\n\\draw (1, {sqrt(1/2)}) node [above right] {$M$} coordinate (M);\n\\draw (1, {-sqrt(1/2)}) node [below right] {$N$} coordinate (N);\n\\draw (M)--(N)--(F_1)--cycle;\n\\draw ($(N)!0.2!(M)$) node [right] {$P$} coordinate (P);\n\\draw ({sqrt(2)*cos(120)}, {1*sin(120)}) node [above] {$Q$} coordinate (Q);\n\\draw (P)--(Q)--(F_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$C$的方程;\\\\\n(2) 若线段$PQ$的中点在$y$轴上, 求$\\triangle F_1QP$的面积;\\\\\n(3) 是否存在以$F_1Q$、$F_1P$为邻边的矩形$F_1PEQ$, 使得点$E$在椭圆$C$上?\n若存在, 求出所有满足条件的点$Q$的横坐标; 若不存在, 说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031081": {
+        "id": "031081",
+        "content": "在平面直角坐标系$x O y$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右顶点分别为$A$、$B$, 右焦点为$F$, 且椭圆$\\Gamma$过点$(0, \\sqrt 5)$、$(2, \\dfrac 53)$, 过点$F$的直线$l$与椭圆$\\Gamma$交于$P$、$Q$两点(点$P$在$x$轴的上方).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-4, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -3) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\path [name path = elli, draw] (0, 0) ellipse (3 and {sqrt(5)});\n\\draw (-3, 0) node [below left] {$A$} coordinate (A);\n\\draw (3, 0) node [above right] {$B$} coordinate (B);\n\\draw (2, 0) node [below left] {$F$} coordinate (F);\n\\path (F) --++ (-1.5, 3) coordinate (S);\n\\path [name path = line] (S) -- ($(F)!-0.6!(S)$);\n\\draw [name intersections = {of = line and elli, by = {P, Q}}];\n\\draw ($(P)!-0.2!(Q)$) -- ($(P)!1.2!(Q)$);\n\\draw (P) node [above] {$P$} (Q) node [below] {$Q$};\n\\draw (A)--(P) (B)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 若$\\overrightarrow{PF}+2 \\overrightarrow{QF}=\\overrightarrow0$, 求点$P$的坐标;\\\\\n(3) 设直线$AP$、$BQ$的斜率分别为$k_1$、$k_2$, 是否存在常数$\\lambda$, 使得$k_1+\\lambda k_2=0$? 若存在, 请求出$\\lambda$的值; 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031082": {
+        "id": "031082",
+        "content": "已知$P(0,1)$为椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$内一定点, $Q$为直线$l: y=3$上一动点, 直线$PQ$与椭圆$C$交于$A$、$B$两点 (点$B$位于$P$、$Q$两点之间), $O$为坐标原点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5, 0) -- (2.5, 0) node [below] {$x$};\n\\draw [->] (0, -2) -- (0, 3.5) node [left] {$y$};\n\\draw (0, 0) node [below right] {$O$} coordinate (O);\n\\path [name path = elli, draw] (0, 0) ellipse (2 and {sqrt(3)});\n\\draw (0, 1) node [left] {$P$} coordinate (P);\n\\draw (2, 3) node [below] {$Q$} coordinate (Q);\n\\path [name path = line] ($(Q)!2.5!(P)$)--(Q);\n\\path [name intersections = {of = line and elli, by = {B, A}}];\n\\draw (A) node [below] {$A$} --(O)--(B) node [above] {$B$};\n\\draw (-2.5, 3) -- (2.5, 3);\n\\draw ($(Q)!-0.1!(A)$)--($(Q)!1.1!(A)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 当直线$PQ$的倾斜角为$\\dfrac{\\pi}4$时, 求直线$OQ$的斜率;\\\\\n(2) 当$\\triangle AOB$的面积为$\\dfrac 32$时, 求点$Q$的横坐标;\\\\\n(3) 设$\\overrightarrow{AP}=\\lambda \\overrightarrow{PB}, \\overrightarrow{AB}=\\mu \\overrightarrow{BQ}$, 试问$\\lambda-\\mu$是否为定值? 若是, 请求出该定值; 若不是, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031083": {
+        "id": "031083",
+        "content": "如图, 已知椭圆$\\Gamma$的中心是坐标原点$O$, 焦点在$x$轴上, 点$B$是椭圆$\\Gamma$的上顶点, 椭圆$\\Gamma$上一点$A(1, \\dfrac{\\sqrt 2}2)$到两焦点距离之和为$2 \\sqrt 2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2, 0) -- (2.5, 0) node [below] {$x$};\n\\draw [->] (0, -1.5) -- (0, 1.5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\filldraw (2, 0) circle (0.03) node [above] {$T$} coordinate (T);\n\\draw (0, 0) ellipse ({sqrt(2)} and 1);\n\\draw (0, 1) node [above left] {$B$} coordinate (B);\n\\draw (1, 0) node [below left] {$F$} coordinate (F);\n\\draw ({3-sqrt(3)}, {-sqrt(3*sqrt(3)-5)}) node [below right] {$N$} coordinate (N);\n\\draw ($(N)!3!(F)$) node [above] {$M$} coordinate (M);\n\\draw [->] (N)--(F);\n\\draw [->] (F)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 若点$P$、$Q$是椭圆$\\Gamma$上异于点$B$的两点, $BP \\perp BQ$, 且满足$3 \\overrightarrow{PC}=2 \\overrightarrow{CQ}$的点$C$在$y$轴上, 求直线$BP$的方程;\\\\\n(3) 设$x$轴上点$T$坐标为$(2,0)$, 过椭圆$\\Gamma$的右焦点$F$作直线$l$(不与$x$轴重合) 与椭圆$\\Gamma$交于$M$、$N$两点, 如图, 点$M$在$x$轴上方, 点$N$在$x$轴下方, 且$\\overrightarrow{FM}=2 \\overrightarrow{NF}$, 求$|\\overrightarrow{TM}+\\overrightarrow{TN}|$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031084": {
+        "id": "031084",
+        "content": "已知椭圆$M: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的焦距为$2 \\sqrt 2$, 过点$(\\sqrt 2, \\dfrac{\\sqrt 3}3)$, 斜率为$k$的直线$l$与椭圆有两个不同的交点$A$、$B$.\\\\\n(1) 求椭圆$M$的方程;\\\\\n(2) 若$k=1$, 求$|AB|$的最大值;\\\\\n(3) 设$P(-2,0)$, 直线$PA$与椭圆$M$的另一个交点为$C$, 直线$PB$与椭圆$M$的另一个交点为$D$. 若$C$、$D$和点$Q(-\\dfrac 74, \\dfrac 12)$共线, 求实数$k$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031085": {
+        "id": "031085",
+        "content": "已知椭圆$\\Gamma: \\dfrac{x^2}4+\\dfrac{y^2}3=1$的右焦点为$F$, 过$F$的直线$l$交$\\Gamma$于$A$、$B$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5, 0) -- (2.5, 0) node [below] {$x$};\n\\draw [->] (0, -2.5) -- (0, 2.5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\path [name path = elli, draw] (0, 0) ellipse (2 and {sqrt(3)});\n\\draw (1, 0) node [below left] {$F$} coordinate (F);\n\\path [name path = line] (F) ++ (-0.5, 2) --++ (1, -4);\n\\path [name intersections = {of = line and elli, by = {B, A}}];\n\\draw (A) node [right] {$A$} coordinate (A);\n\\draw (B) node [above right] {$B$} coordinate (B);\n\\draw ($(A)!-0.1!(B)$) -- ($(A)!1.1!(B)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 若直线$l$垂直于$x$轴, 求线段$AB$的长;\\\\\n(2) 若直线$l$与$x$轴不重合, $O$为坐标原点, 求$\\triangle AOB$面积的最大值;\\\\\n(3) 若椭圆$\\Gamma$上存在点$C$使得$|AC|=|BC|$, 且$\\triangle ABC$的重心$G$在$y$轴上, 求此时直线$l$的方程.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031086": {
+        "id": "031086",
+        "content": "已知$F_1$、$F_2$分别为椭圆$E: \\dfrac{x^2}4+\\dfrac{y^2}3=1$的左、右焦点, 过$F_1$的直线$l$交椭圆$E$于$A$、$B$两点.\\\\\n(1) 当直线$l$垂直于$x$轴时, 求弦长$|AB|$;\\\\\n(2) 当$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=-2$时, 求直线$l$的方程;\\\\\n(3) 记椭圆的右顶点为$T$, 直线$AT$、$BT$分别交直线$x=6$于$C$、$D$两点, 求证: 以$CD$为直径的圆恒过定点, 并求出定点坐标.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031087": {
+        "id": "031087",
+        "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右顶点坐标为$A(2,0)$, 左、右焦点分别为$F_1$、$F_2$, 且$|F_1F_2|=2$, 直线$l$交椭圆$\\Gamma$于不同的两点$M$和$N$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 若直线$l$的斜率为 1 , 且以$MN$为直径的圆经过点$A$, 求直线$l$的方程;\\\\\n(3) 若直线$l$与椭圆$\\Gamma$相切, 求证: 点$F_1$、$F_2$到直线$l$的距离之积为定值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031088": {
+        "id": "031088",
+        "content": "已知点$F_1$、$F_2$分别为椭圆$\\Gamma: \\dfrac{x^2}2+y^2=1$的左、右焦点, 直线$l: y=k x+t$与椭圆$\\Gamma$有且仅有一个公共点, 直线$F_1M \\perp l, F_2N \\perp l$, 垂足分别为点$M$、$N$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw [->] (-2, 0) -- (2, 0) node [below] {$x$};\n\\draw [->] (0, -1.5) -- (0, 1.5) node [left] {$y$};\n\\draw (0, 0) node [below right] {$O$};\n\\draw (0, 0) ellipse ({sqrt(2)} and 1);\n\\draw (-1.5, {1.5*0.25+sqrt(9/8)}) coordinate (S) node [left] {$l$}-- (1.5, {-1.5*0.25+sqrt(9/8)}) coordinate (T);\n\\draw (-1, 0) node [below] {$F_1$} coordinate (F_1);\n\\draw (1, 0) node [below] {$F_2$} coordinate (F_2);\n\\draw ($(S)!(F_1)!(T)$) node [above] {$M$} coordinate (M);\n\\draw ($(S)!(F_2)!(T)$) node [above] {$N$} coordinate (N);\n\\draw (F_1)--(M) (F_2)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$t^2=2 k^2+1$;\\\\\n(2) 求证:$\\overrightarrow{F_1M} \\cdot \\overrightarrow{F_2N}$为定值, 并求出该定值;\\\\\n(3) 求$|\\overrightarrow{OM}+\\overrightarrow{ON}|\\cdot|\\overrightarrow{OM}-\\overrightarrow{ON}|$的最大值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031089": {
+        "id": "031089",
+        "content": "已知点$M(x, y)$与定点$F(1,0)$的距离是点$M$到直线$x-2=0$距离的$\\dfrac{\\sqrt 2}2$倍, 设点$M$的轨迹为曲线$\\Gamma$, 直线$l: x+m y+1=0$($m \\in \\mathbf{R}$)与$\\Gamma$交于$A$、$B$两点, 点$C$是线段$AB$的中点, $P$、$Q$是$\\Gamma$上关于原点$O$对称的两点, 且$\\overrightarrow{PO}=\\lambda \\overrightarrow{OC}(\\lambda>0)$.\\\\\n(1) 求曲线$\\Gamma$的方程;\\\\\n(2) 当$\\lambda=\\sqrt 3$时, 求直线$l$的方程;\\\\\n(3) 当四边形$PAQB$的面积$S=\\sqrt 6$时, 求$\\lambda$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031090": {
+        "id": "031090",
+        "content": "如图, 已知$F_1$、$F_2$是椭圆$\\Gamma: \\dfrac{x^2}4+y^2=1$的左、右焦点, $M$、$N$是其顶点, 直线$l: y=k x+m$($k>0$)与$\\Gamma$相交于$A$、$B$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5, 0) -- (2.5, 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 (2 and 1);\n\\filldraw ({-sqrt(3)}, 0) circle (0.03) node [below] {$F_1$} coordinate (F_1);\n\\filldraw ({sqrt(3)}, 0) circle (0.03) node [below] {$F_2$} coordinate (F_2);\n\\draw (-2, 0) node [below left] {$M$} coordinate (M);\n\\draw (0, 1) node [above right] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\triangle F_2MN$的面积$P$;\\\\\n(2) 若$l \\perp F_2N$, 点$A$、$M$重合, 求$B$点的坐标;\\\\\n(3) 设直线$OA$、$OB$的斜率分别$k_1$、$k_2$, 记以$OA$、$OB$为直径的圆的面积分别为$S_1$、$S_2$, $\\triangle OAB$的面积为$S$, 若$k_1$、$k$、$k_2$恰好构成等比数列, 求$S(S_1+S_2)$的最大值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031091": {
+        "id": "031091",
+        "content": "已知$A$、$B$分别为椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的上、下顶点, $F$是椭圆$\\Gamma$的右焦点, $M$是椭圆$\\Gamma$上异于$A$、$B$的点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -2) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\path [name path = line, draw] (-4, 2) -- (4, 2);\n\\path [name path = elli, draw] (0, 0) ellipse (2 and 1);\n\\filldraw ({sqrt(3)}, 0) circle (0.03) node [below] {$F$} coordinate (F);\n\\draw (0, -1) node [below left] {$B$} coordinate (B) (0, 1) node [above left] {$A$} coordinate (A);\n\\path [name path = BR] (B) --++ (-3.8, 3.1);\n\\path [name intersections = {of = BR and elli, by = M}];\n\\path [name intersections = {of = BR and line, by = R}];\n\\draw (B)--(R) node [above] {$R$};\n\\path [name path = MA] (M)--($(M)!3!(A)$);\n\\path [name intersections = {of = MA and line, by = Q}];\n\\draw (M) node [left] {$M$} -- (Q) node [above] {$Q$};\n\\draw (0, 2) node [below right] {$P$};\n\\path [name path = AD] ($(A)!1!90:(M)$) -- (A);\n\\path [name path = BD] ($(B)!1!-90:(M)$) -- (B);\n\\path [name intersections = {of = AD and BD, by = D}];\n\\draw (A) -- (D) node [right] {$D$} coordinate (D)-- (B);\n\\draw pic [draw, scale = 0.3] {right angle = M--A--D};\n\\draw pic [draw, scale = 0.3] {right angle = D--B--M};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle AFB=\\dfrac{\\pi}3$, 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 设直线$l: y=2$与$y$轴交于点$P$, 与直线$MA$交于点$Q$, 与直线$MB$交于点$R$, 求证:$|PQ|\\cdot|PR|$的值仅与$a$有关;\\\\\n(3) 如图, 在四边形$MADB$中, $MA \\perp AD, MB \\perp BD$, 若四边形$MADB$面积$S$的最大值为$\\dfrac 52$, 求$a$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031092": {
+        "id": "031092",
+        "content": "已知椭圆$\\Gamma: \\dfrac{x^2}4+\\dfrac{y^2}3=1$的左、右焦点分别为$F_1$、$F_2$, 设$P$是第一象限内椭圆$\\Gamma$上一点, $PF_1$、$PF_2$的延长线分别交椭圆$\\Gamma$于点$Q_1$、$Q_2$, 直线$Q_1F_2$与$Q_2F_1$交于点$R$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5, 0) -- (2.5, 0) node [below] {$x$};\n\\draw [->] (0, -2.5) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [above right] {$O$};\n\\path [name path = elli, draw] (0, 0) ellipse (2 and {sqrt(3)});\n\\draw ({2*cos(45)}, {sqrt(3)*sin(45)}) node [above right] {$P$} coordinate (P);\n\\draw (-1, 0) node [above] {$F_1$} coordinate (F_1);\n\\draw (1, 0) node [above right] {$F_2$} coordinate (F_2);\n\\path [name path = PQ1] (P) -- ($(P)!1.5!(F_1)$);\n\\path [name path = PQ2] (P) -- ($(P)!2.5!(F_2)$);\n\\path [name intersections = {of = PQ1 and elli, by = Q_1}];\n\\path [name intersections = {of = PQ2 and elli, by = Q_2}];\n\\draw (P) -- (Q_1) node [left] {$Q_1$};\n\\draw (P) -- (Q_2) node [below] {$Q_2$};\n\\path [name path = F1Q2, draw] (F_1)--(Q_2);\n\\path [name path = F2Q1, draw] (F_2)--(Q_1);\n\\path [name intersections = {of = F1Q2 and F2Q1, by = R}];\n\\draw (R) node [below] {$R$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\triangle PQ_1F_2$的周长;\\\\\n(2) 当$PF_2$垂直于$x$轴时, 求直线$Q_1Q_2$的方程;\\\\\n(3) 记$\\triangle F_1Q_1R$与$\\triangle F_2Q_2R$的面积分别为$S_1$、$S_2$, 求$S_2-S_1$的最大值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031093": {
+        "id": "031093",
+        "content": "如图, 中心在原点$O$的椭圆$\\Gamma$的右焦点为$F(2 \\sqrt 3, 0)$, 长轴长为$8$. 椭圆$\\Gamma$上有两点$P$、$Q$, 连结$OP$、$OQ$, 记它们的斜率为$k_{OP}$、$k_{OQ}$, 且满足$k_{OP} \\cdot k_{OQ}=-\\dfrac 14$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-5, 0) -- (5, 0) node [below] {$x$};\n\\draw [->] (0, -4) -- (0, 4) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw [name path = directrix] ({4*sqrt(3)}, -4) -- ({4*sqrt(3)}, 5.2);\n\\draw (0, 0) ellipse (4 and 2);\n\\draw ({4*cos(acos(sqrt(7)-sqrt(3)))}, {2*sin(acos(sqrt(7)-sqrt(3)))}) node [above] {$P$} coordinate (P);\n\\draw ({-4*sin(acos(sqrt(7)-sqrt(3)))}, {2*cos(acos(sqrt(7)-sqrt(3)))}) node [above] {$R$} coordinate (R);\n\\draw ({4*sin(acos(sqrt(7)-sqrt(3)))}, {-2*cos(acos(sqrt(7)-sqrt(3)))}) node [below] {$Q$} coordinate (Q);\n\\draw (0, 0) -- (P) -- (R) -- (Q) -- (P);\n\\draw [name path = line1] (Q) -- ($(Q)!2.7!(P)$);\n\\draw [name path = line2] (R) -- ($(R)!1.7!(P)$);\n\\draw [name intersections = {of = line1 and directrix, by = N}] (N) node [right] {$N$};\n\\draw [name intersections = {of = line2 and directrix, by = M}] (M) node [right] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 求证:$|OP|^2+|OQ|^2$为一定值, 并求出这个定值;\\\\\n(3) 设直线$OQ$与椭圆$\\Gamma$的另一个交点为$R$, 直线$RP$和$PQ$分别与直线$x=4 \\sqrt 3$交于点$M$、$N$, 若$\\triangle PQR$和$\\triangle PMN$的面积相等, 求点$P$的横坐标.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031094": {
+        "id": "031094",
+        "content": "椭圆$x^2+4 y^2=68$上有两点$A(8, y_A)$和$T(x_T,-4)$, $y_A>0$, $x_T<0$. 点$A$关于椭圆中心$O$的对称点为点$B$, 点$P(t,-2 t)$在椭圆内部, $t \\neq 0$. $F_1$是椭圆的左焦点, $F_2$是椭圆的右焦点.\\\\\n(1) 若点$P$在直线$AT$上, 求点$P$坐标;\\\\\n(2) 是否存在一个点$P$, 满足$|\\overrightarrow{PF_2}|-|\\overrightarrow{PF_1}|=2 \\sqrt 3$, 若满足求出点$P$坐标, 若不存在, 请说明理由;\\\\\n(3) 设$\\triangle AOP$的面积为$S_1$, $\\triangle BTP$的面积为$S_2$, 求$\\dfrac{S_1}{S_2}$的取值范围.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031095": {
+        "id": "031095",
+        "content": "双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的焦点到其渐近线的距离是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031096": {
+        "id": "031096",
+        "content": "若双曲线$x^2-\\dfrac{y^2}m=1$的渐近线方程为$y=\\pm 2 x$, 则实数$m=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦一模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031097": {
+        "id": "031097",
+        "content": "已知中心在原点的双曲线的一个焦点坐标为$F(\\sqrt 7, 0)$, 直线$y=x-1$与该双曲线相交于$M$、$N$两点, 线段$MN$中点的横坐标为$-\\dfrac 23$, 则此双曲线的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031098": {
+        "id": "031098",
+        "content": "已知双曲线$M: x^2-\\dfrac{y^2}6=1$的左、右焦点为$F_1$、$F_2$, 过$F_1$的直线$l$与双曲线$M$的左、右两支分别交于点$A$、$B$. 若$\\triangle ABF_2$为等边三角形, 则$\\triangle ABF_2$的边长为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031099": {
+        "id": "031099",
+        "content": "设$P$为直线$y=2 x$上的一点, 且位于第一象限, 若点$P$到双曲线$\\dfrac{x^2}4-y^2=1$的两条渐近线的距离之积为$27$, 则点$P$的坐标为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031100": {
+        "id": "031100",
+        "content": "已知抛物线$y^2=2 p x$($p>0$)上一点$M(1, m)$到其焦点的距离为$5$, 双曲线$C: x^2-\\dfrac{y^2}{b^2}=1$($b>0$)的左顶点为$A$, 若双曲线$C$的一条渐近线与直线$AM$垂直, 则双曲线$C$的焦距为\\blank{50}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031101": {
+        "id": "031101",
+        "content": "已知双曲线的中心是坐标原点, 它的一个顶点为$A(\\sqrt 2, 0)$, 两条渐近线与以$A$为圆心$1$为半径的圆都相切, 则该双曲线的标准方程是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031102": {
+        "id": "031102",
+        "content": "已知曲线$\\dfrac{x^2}a+\\dfrac{y^2}{16}=1$的焦距是$10$, 曲线上的点$P$到一个焦点距离是$2$, 则点$P$到另一个焦点的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031103": {
+        "id": "031103",
+        "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别为$F_1$、$F_2$, 过$F_2$且斜率为$-\\dfrac{\\sqrt 5}2$的直线与双曲线$C$的左支交于点$A$. 若$(\\overrightarrow{F_1F_2}+\\overrightarrow{F_1A}) \\cdot \\overrightarrow{F_2A}=0$, 则双曲线$C$的渐近线方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031104": {
+        "id": "031104",
+        "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的实轴为$A_1A_2$, 对于实轴$A_1A_2$上的任意点$P$, 在实轴$A_1A_2$上都存在点$Q$, 使得$|PQ|=\\sqrt 3 b$, 则双曲线$\\Gamma$的两条渐近线夹角的最大值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031105": {
+        "id": "031105",
+        "content": "已知一族双曲线$E_n: x^2-y^2=(\\dfrac n{2022})^2$($n \\in \\mathbf{N}$, $1\\le n \\leq 2022$), 设双曲线$E_n$的左、右焦点分别为$F_{n_1}$、$F_{n_2}$, $P_n$是双曲线$E_n$右支上一动点, $\\triangle P_n F_{n_1} F_{n_2}$的内切圆$G_n$与$x$轴切于点$A_n(a_n, 0)$, 则$a_1+a_2+\\cdots+a_{2022}=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031106": {
+        "id": "031106",
+        "content": "设$m$、$n \\in \\mathbf{R}$, 则 ``$m \\cdot n<0$'' 是 ``方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$表示的曲线为双曲线'' 的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031107": {
+        "id": "031107",
+        "content": "已知$x^2-y^2=3$的两条渐近线与直线$x=4$围成三角形区域, 那么, 表示该区域的不等式组是\\bracket{20}.\n\\fourch{$\\begin{cases}x-y \\geq 0,\\\\x+y \\geq 0,\\\\0 \\leq x \\leq 4\\end{cases}$}{$\\begin{cases}x-y \\geq 0,\\\\x+y \\leq 0,\\\\0 \\leq x \\leq 4\\end{cases}$}{$\\begin{cases}x-y \\leq 0,\\\\x+y \\leq 0,\\\\0 \\leq x \\leq 4\\end{cases}$}{$\\begin{cases}x-y \\leq 0,\\\\x+y \\geq 0,\\\\0 \\leq x \\leq 4\\end{cases}$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031108": {
+        "id": "031108",
+        "content": "若方程$4 x^2+k y^2=4 k$表示双曲线, 则此双曲线的虚轴长等于\\bracket{20}.\n\\fourch{$2 \\sqrt k$}{$2 \\sqrt {-k}$}{$\\sqrt k$}{$\\sqrt {-k}$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031109": {
+        "id": "031109",
+        "content": "设点$F_1$、$F_2$是双曲线$C: \\dfrac{x^2}4-y^2=1$的左、右两焦点, 点$M$是$C$的右支上的任意$-$点, 若$\\overrightarrow{F_2M} \\cdot \\overrightarrow{F_2F_1}>0$, 则$|MF_1|+|MF_2|$的值可能是\\bracket{20}.\n\\fourch{$4$}{$2 \\sqrt 6$}{$5$}{$3 \\sqrt 3$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031110": {
+        "id": "031110",
+        "content": "如图, 在平面直角坐标系中, $F_1$、$F_2$分别为双曲线$\\Gamma: x^2-y^2=2$的左、右焦点, 点$D$为线段$F_1O$的中点, 直线$MN$过点$F_2$且与双曲线右支交于$M(x_1, y_1)$、$N(x_2, y_2)$两点, 延长$MD$、$ND$, 分别与双曲线$\\Gamma$交于$P$、$Q$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-4, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -3.5) -- (0, 3.5) node [left] {$y$};\n\\draw (0, 0) node [above right] {$O$};\n\\path [domain = -3.5:3.5, name path = right, draw, samples = 100] plot ({sqrt(2+\\x*\\x)}, \\x);\n\\path [domain = -3.5:3.5, name path = left, draw, samples = 100] plot ({-sqrt(2+\\x*\\x)}, \\x);\n\\filldraw (2, 0) circle (0.06) node [above right] {$F_2$} coordinate (F_2);\n\\filldraw (-2, 0) circle (0.06) node [above left] {$F_1$} coordinate (F_1);\n\\draw (-1, 0) node [below] {$D$} coordinate (D);\n\\path [name path = MN] (F_2) ++ (1, 3) --++ (-2, -6);\n\\path [name intersections = {of = MN and right, by = {N, M}}];\n\\draw (M) node [above] {$M$} -- (N) node [below] {$N$};\n\\path [name path = MD] (M) -- ($(M)!1.2!(D)$);\n\\path [name path = ND] (N) -- ($(N)!1.2!(D)$);\n\\path [name intersections = {of = MD and left, by = P}];\n\\path [name intersections = {of = ND and left, by = Q}];\n\\draw (M) -- (P) node [left] {$P$};\n\\draw (N) -- (Q) node [above right] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 已知点$M(3, \\sqrt 7)$, 求点$D$到直线$MN$的距离;\\\\\n(2) 求证:$x_1 y_2-x_2 y_1=2(y_2-y_1)$;\\\\\n(3) 若直线$MN$、$PQ$的斜率都存在, 且依次设为$k_1$、$k_2$, 试判断$\\dfrac{k_2}{k_1}$是否为定值, 如果是, 请求出该定值; 如果不是, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031111": {
+        "id": "031111",
+        "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的焦距为$2 \\sqrt 3$, 渐近线方程为$y=\\pm \\dfrac{\\sqrt 2}2 x$.\\\\\n(1) 求双曲线$\\Gamma$的方程;\\\\\n(2) 若对任意的$m \\in \\mathbf{R}$, 直线$y=k x+m$与双曲线$\\Gamma$总有公共点, 求实数$k$的取值范围;\\\\\n(3) 若过点$(1,0)$的直线$l$与双曲线$\\Gamma$交于$M$、$N$两点, 问在$x$轴上是否存在定点$P$, 使得$\\overrightarrow{PM} \\cdot \\overrightarrow{PN}$为常数? 若存在, 求出点$P$的坐标及此常数的值, 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031112": {
+        "id": "031112",
+        "content": "已知点$F_1$、$F_2$分别为双曲线$\\Gamma: \\dfrac{x^2}2-y^2=1$的左右焦点, 直线$l: y=k x+1$与$\\Gamma$有两个不同的交点$A$、$B$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-5, 0) -- (5, 0) node [below] {$x$};\n\\draw [->] (0, -3) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw [domain = -3:3, samples = 100] plot ({sqrt(2*(1+\\x*\\x))}, \\x);\n\\draw [domain = -3:3, samples = 100] plot ({-sqrt(2*(1+\\x*\\x))}, \\x);\n\\filldraw ({sqrt(3)}, 0) circle (0.06) node [below right] {$F_2$} coordinate (F_2);\n\\filldraw ({-sqrt(3)}, 0) circle (0.06) node [below left] {$F_1$} coordinate (F_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当$F_1 \\in l$时, 求$F_2$到$l$的距离;\\\\\n(2) 若$O$为原点, 直线$l$与$\\Gamma$的两条渐近线在一、二象限的交点分别为$C$、$D$, 证明: 当$\\triangle COD$的面积最小时, 直线$CD$平行于$x$轴;\\\\\n(3) 设$P$为$x$轴上一点, 是否存在实数$k$($k>0$), 使得$\\triangle PAB$是以点$P$为直角顶点的等腰直角三角形? 若存在, 求出$k$的值及点$P$的坐标; 若不存在, 说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031113": {
+        "id": "031113",
+        "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的一条渐近线的方程为$\\sqrt {13} x-y=0$, 它的右顶点与抛物线$\\Gamma: y^2=4 \\sqrt 3 x$的焦点重合, 经过点$A(-9,0)$且不垂直于$x$轴的直线与双曲线$C$交于$M$、$N$两点.\\\\\n(1) 求双曲线$C$的标准方程;\\\\\n(2) 若点$M$是线段$AN$的中点, 求点$N$的坐标;\\\\\n(3) 设$P$、$Q$是直线$x=-9$上关于$x$轴对称的两点, 求证: 直线$PM$与$QN$的交点必在直线$x=-\\dfrac 13$上.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031114": {
+        "id": "031114",
+        "content": "已知双曲线$\\Gamma: \\dfrac{x^2}4-\\dfrac{y^2}{12}=1$, $F$为左焦点, $P$为直线$x=1$上一动点, $Q$为线段$PF$与$\\Gamma$的交点. 定义: $d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$Q$的纵坐标为$\\sqrt {15}$, 求$d(P)$的值;\\\\\n(2) 设$d(P)=\\lambda$, 点$P$的纵坐标为$t$, 试将$t^2$表示成$\\lambda$的函数并求其定义域;\\\\\n(3) 证明: 存在常数$m$、$n$, 使得$m d(P)=|PF|+n$.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031115": {
+        "id": "031115",
+        "content": "已知双曲线$\\Gamma: x^2-y^2=4$, 双曲线$\\Gamma$的右焦点为$F$, 圆$C$的圆心在$y$轴正半轴上, 且经过坐标原点$O$, 圆$C$与双曲线$\\Gamma$的右支交于$A$、$B$两点.\\\\\n(1) 当$\\triangle OFA$是以$F$为直角顶点的直角三角形, 求$\\triangle OFA$的面积;\\\\\n(2) 若点$A$的坐标是$(\\sqrt 5, 1)$, 求直线$AB$的方程;\\\\\n(3) 求证: 直线$AB$与圆$x^2+y^2=2$相切.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031116": {
+        "id": "031116",
+        "content": "第一象限内的点$P$在双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$)上, 双曲线的左、右焦点分别记为$F_1$、$F_2$, 已知$PF_1 \\perp PF_2$, $|PF_1|=2|PF_2|$, $O$为坐标原点.\\\\\n(1) 求证:$b=2 a$;\\\\\n(2) 若$\\triangle OF_2P$的面积为$2$, 求点$P$的坐标.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题18",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031117": {
+        "id": "031117",
+        "content": "设$F_1$、$F_2$分别是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、 右两焦点, 过点$F_2$的直线$l: x-m y-t=0$($m, t \\in \\mathbf{R}$)与$\\Gamma$的右支交于$M$、$N$两点, $\\Gamma$过点$(-2,3)$, 且它的虚轴的端点与焦点的距离为$\\sqrt 7$.\\\\\n(1) 求双曲线$\\Gamma$的方程;\\\\\n(2) 当$|MF_1|=|F_2F_1|$时, 求实数$m$的值;\\\\\n(3) 设点$M$关于坐标原点$O$的对称点为$P$, 当$\\overrightarrow{MF_2}=\\dfrac 12 \\overrightarrow{F_2N}$时, 求$\\triangle PMN$面积$S$的值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031118": {
+        "id": "031118",
+        "content": "抛物线$y^2=16 x$的准线方程是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安一模试题01",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031119": {
+        "id": "031119",
+        "content": "抛物线$y^2=2 p x$($p>0$)上的动点$Q$到焦点的距离的最小值为$1$, 则$p=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题05",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031120": {
+        "id": "031120",
+        "content": "若抛物线$y^2=4 x$上一点$P$到$y$轴的距离是$4$, 则点$P$到该抛物线焦点的距离是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届松江一模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031121": {
+        "id": "031121",
+        "content": "已知$F$为抛物线$C: y^2=4 x$的焦点, 过点$F$的直线$l$交抛物线$C$于$A$、$B$两点, 若$|AB|=10$, 则线段$AB$的中点$M$到直线$x+1=0$的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031122": {
+        "id": "031122",
+        "content": "以双曲线$\\dfrac{x^2}4-\\dfrac{y^2}5=1$的中心为顶点, 且以该双曲线的右焦点为焦点的抛物线方程是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦一模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031123": {
+        "id": "031123",
+        "content": "《九章算术》是我国古代内容极为丰富的数学名著, 第九卷 ``勾股'' 讲述了 ``勾股定理'' 及一些应用, 其中直角三角形的三条边长分别称 ``勾'' ``股'' ``弦'', 设点$F$是抛物线$y^2=2 p x$的焦点, 直线$l$是该抛物线的准线, 过抛物线上一点$A$作准线的垂线$AB$, 垂足为$B$, 射线$AF$交准线$l$于点$C$, 若$\\text{Rt}\\triangle ABC$的``勾''$|AB|=3$, ``股''$|CB|=3 \\sqrt 3$, 则抛物线方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题07",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031124": {
+        "id": "031124",
+        "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯口所在的圆面与反射镜的轴垂直, 灯泡位于抛物线的焦点处. 已知灯口直径是$24$厘米, 灯深$10$厘米, 则灯泡与反射镜顶点的距离是\\blank{50}厘米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\filldraw (3.6, 0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10, 0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x, 2)/14.4}, \\x);\n\\draw (10, 0) ellipse (3 and 12);\n\\draw (0, 0) node [left] {$O$} coordinate (O) --++ (0, -14);\n\\draw (10, -12) --++ (0, -2);\n\\draw [<->] (0, -13) -- (10, -13) node [midway, below] {$10\\text{cm}$};\n\\draw (10, -12) --++ (5, 0) (10, 12) --++ (5, 0);\n\\draw [<->] (14, -12) -- (14, 12) node [midway, right] {\\rotatebox{90}{$24\\text{cm}$}};\n\\draw [dashed] (10, 0) --++ (0, -12);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031125": {
+        "id": "031125",
+        "content": "过抛物线$y^2=2 p x(p>0)$的焦点$F$且斜率为$1$的直线交抛物线于$A$、$B$两点, 若$|AF|\\cdot|BF|=8$, 则$p$的值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031126": {
+        "id": "031126",
+        "content": "已知$P_1$、$P_2$、$P_3$、$\\cdots$、$P_{10}$是抛物线$y^2=8 x$上不同的点, 点$F(2,0)$, 若$\\overrightarrow{FP_1}+\\overrightarrow{FP_2}+\\cdots+\\overrightarrow{FP_{10}}=\\overrightarrow 0$, 则$|\\overrightarrow{FP_1}|+|\\overrightarrow{FP_2}|+\\cdots+|\\overrightarrow{FP_{10}}|=$\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届金山一模试题10",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031127": {
+        "id": "031127",
+        "content": "已知点$A$、$B$在抛物线$\\Gamma: y^2=4 x$上, 点$M$在$\\Gamma$的准线上, 线段$MA$、$MB$的中点均在抛物线$\\Gamma$上, 设直线$AB$与$y$轴交于点$N(0, n)$, 则$|n|$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031128": {
+        "id": "031128",
+        "content": "已知双曲线$\\Gamma_1: x^2-\\dfrac{y^2}{b^2}=1$的左、右焦点分别为$F_1$、$F_2$, 以$O$为顶点$F_2$为焦点作抛物线$\\Gamma_2$, 若双曲线$\\Gamma_1$与抛物线$\\Gamma_2$交于点$P$, 且$\\angle PF_1F_2=45^{\\circ}$, 则抛物线$\\Gamma_2$的准线方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031129": {
+        "id": "031129",
+        "content": "已知抛物线$y^2=x$.\\\\\n(1) 过抛物线焦点$F$的直线交抛物线于$A$、$B$两点, 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值(其中$O$为坐标原点);\\\\\n(2) 过抛物线上一点$C(x_0, y_0)$, 分别作两条直线交抛物线于另外两点$P(x_P, y_P)$、$Q(x_Q, y_Q)$, 交直线$x=-1$于$A_1(-1,1)$、$B_1(-1,-1)$两点, 求证:$y_P \\cdot y_Q$为常数;\\\\\n(3) 己知点$D(1,1)$, 在抛物线上是否存在异于点$D$的两个不同点$M$、$N$, 使得$DM \\perp MN$? 若存在, 求$N$点纵坐标的取值范围, 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031130": {
+        "id": "031130",
+        "content": "已知斜率为$k$的直线$l$经过抛物线$C: y^2=4 x$的焦点$F$, 且与抛物线$C$交于不同的两点$A(x_1, y_1)$、$B(x_2, y_2)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -4) -- (0, 4) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot ({\\x*\\x/4}, \\x);\n\\draw (1, 0) node [below right] {$F$} coordinate (F);\n\\draw (0.5, {-sqrt(2)}) node [below] {$A$} coordinate (A);\n\\draw (2, {sqrt(8)}) node [above] {$B$} coordinate (B);\n\\draw (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$A$和$B$到抛物线准线的距离分别为$\\dfrac 32$和$3$, 求$|AB|$;\\\\\n(2) 若$|AF|+|AB|=2|BF|$, 求$k$的值;\\\\\n(3) 点$M(t, 0), t>0$, 对任意确定的实数$k$, 若$\\triangle AMB$是以$AB$为斜边的直角三角形, 判断符合条件的点$M$有几个, 并说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031131": {
+        "id": "031131",
+        "content": "已知抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 准线为$l$, 记准线$l$与$x$轴的交点为$A$, 过$A$作直线交抛物线$C$于$M(x_1, y_1)$、$N(x_2, y_2)(x_2>x_1)$两点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1.5, 0) -- (4, 0) node [below] {$x$};\n\\draw [->] (0, -4) -- (0, 4) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw [domain = -4:4, samples = 100] plot ({\\x*\\x/4}, \\x);\n\\filldraw (1, 0) circle (0.06) node [below] {$F$} coordinate (F);\n\\draw (-1, -4) -- (-1, 4) node [left] {$l$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$x_1+x_2=2 p$, 求$|MF|+|NF|$的值;\\\\\n(2) 若$M$是线段$AN$的中点, 求直线$MN$的方程;\\\\\n(3) 若$P$、$Q$是准线$l$上关于$x$轴对称的两点, 问直线$PM$与$QN$的交点是否在一条定直线上? 请说明理由.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031132": {
+        "id": "031132",
+        "content": "在平面直角坐标系$xOy$中, 一动圆经过点$A(\\dfrac 12, 0)$且与直线$x=-\\dfrac 12$相切, 设该动圆圆心的轨迹为曲线$K, P$是曲线$K$上一点.\\\\\n(1) 求曲线$K$的方程;\\\\\n(2) 过点$A$且斜率为$k$的直线$l$与曲线$K$交于$B$、$C$两点, 若$l\\parallel OP$且直线$OP$与直线$x=1$交于$Q$点, 求$\\dfrac{|AB|\\cdot|AC|}{|OP|\\cdot|OQ|}$的值;\\\\\n(3) 若点$D$、$E$在$y$轴上, $\\triangle PDE$的内切圆的方程为$(x-1)^2+y^2=1$, 求$\\triangle PDE$面积的最小值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031133": {
+        "id": "031133",
+        "content": "如图, 点$P(x_P, y_P$) 是$y$轴左侧 (不含$y$轴) 一点, 抛物线$C: y^2=4 x$上存在不同的两点$A$、$B$, 且$PA$、$PB$的中点均在抛物线$C$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-2, 0) -- (6, 0) node [below] {$x$};\n\\draw [->] (0, -5) -- (0, 5) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\path [domain = -5:5, samples = 100, name path = para, draw] plot ({\\x*\\x/4}, \\x);\n\\draw (-1, 1) node [above] {$P$} coordinate (P);\n\\filldraw ({(7-2*sqrt(10))/8}, {(2-sqrt(10))/2}) circle (0.06) coordinate (D);\n\\filldraw ({(7+2*sqrt(10))/8}, {(2+sqrt(10))/2}) circle (0.06) coordinate (C);\n\\draw ($(P)!2!(C)$) node [above] {$A$} coordinate (A)-- ($(P)!2!(D)$) node [below] {$B$} coordinate (B);\n\\draw (A)--(P)--(B);\n\\draw ($(A)!0.5!(B)$) node [above] {$M$} coordinate (M)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$P(-1,2)$, 点$A$在第一象限, 求此时点$A$的坐标;\\\\\n(2) 设$AB$中点为$M$, 求证: 直线$PM \\perp y$轴;\\\\\n(3) 若$P$是曲线$x^2+\\dfrac{y^2}4=1(x<0)$上的动点, 求$\\triangle PAB$面积的最大值.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题20",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031134": {
+        "id": "031134",
+        "content": "已知实数$x$、$y$满足: $x|x|+y|y|=1$, 则$|x+y+\\sqrt 2|$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口一模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031135": {
+        "id": "031135",
+        "content": "已知实数$x$、$y$满足$\\dfrac{x|x|}4+y|y|=1$, 则$|x+2 y-4|$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东一模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031136": {
+        "id": "031136",
+        "content": "已知实数$x$、$y$满足$\\dfrac{x|x|}4-y|y|=1$, 则$|x-2 y+\\sqrt 5|$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031137": {
+        "id": "031137",
+        "content": "设点$P$是曲线$y=\\sqrt {x^2+1}$上的动点, 点$F(0,-\\sqrt 2)$、$A(\\sqrt 2, 0)$满足$|PF|+|PA|=4$, 则点$P$的坐标为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031138": {
+        "id": "031138",
+        "content": "构造一个二元二次方程组$\\begin{cases}f(x, y)=0,\\\\g(x, y)=0,\\end{cases}$ 使得它的解恰好为$\\begin{cases}x_1=1, \\\\y_1=2,\\end{cases}$ $\\begin{cases}x_2=3, \\\\y_2=-4,\\end{cases}$ 要求$f(x, y)=0$与$g(x, y)=0$的每个方程均要出现$x$、$y$两个末知数. 答:\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤二模试题12",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031139": {
+        "id": "031139",
+        "content": "已知点$M(2,2)$, 直线$l: x-y-1=0$, 若动点$P$到$l$的距离等于$|PM|$, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆}{双曲线}{抛物线}{直线}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届普陀二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031140": {
+        "id": "031140",
+        "content": "定义曲线$\\Gamma: \\dfrac{a^2}{x^2}+\\dfrac{b^2}{y^2}=1$为椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的``倒曲线'', 给出以下三个结论: \\textcircled{1} 曲线$\\Gamma$有对称轴; \\textcircled{2} 曲线$\\Gamma$有对称中心; \\textcircled{3} 曲线$\\Gamma$与椭圆$C$有公共点. 其中正确的结论个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届青浦二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031141": {
+        "id": "031141",
+        "content": "已知双曲线$\\dfrac{x^2}2-y^2=1$, 作$x$轴的垂线交双曲线于$A$、$B$两点, 作$y$轴的垂线交双曲线于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆}{双曲线}{圆}{抛物线}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031142": {
+        "id": "031142",
+        "content": "已知曲线$C: \\dfrac{x|x|}4+\\dfrac{y|y|}3=-1$, 对于命题: \\textcircled{1} 垂直于$x$轴的直线与曲线$C$有且只有一个交点; \\textcircled{2} 若$P_1(x_1, y_1)$、$P_2(x_2, y_2)$为曲线$C$上任意两点, 则有$\\dfrac{y_1-y_2}{x_1-x_2}<0$. 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇一模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031143": {
+        "id": "031143",
+        "content": "数学中有许多形状优美、寓意美好的血线, 曲线$C: x^2+y^2=1+|x|y$就是其中之一(如图). 给出下列两个命题, 命题$q_1:$曲线$C$上任意一点到原点的距离都不超过$\\sqrt 2$; 命题$q_2$: 曲线$C$所围成的 ``心形'' 区域的面积小于$3$. 则下列说法正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2, 0) -- (2, 0) node [below] {$x$};\n\\draw [->] (0, -1.5) -- (0, 2) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$};\n\\draw [domain = -90:90, samples = 100] plot (\\x:{1/sqrt(1-cos(\\x)*sin(\\x))}); \n\\draw [domain = 90:270, samples = 100] plot (\\x:{1/sqrt(1+cos(\\x)*sin(\\x))}); \n\\end{tikzpicture}\n\\end{center}\n\\twoch{命题$q_1$是真命题, 命题$q_2$是假命题}{命题$q_1$是假命题, 命题$q_2$是真命题}{命题$q_1$、$q_2$都是真命题}{命题$q_1$、$q_2$都是假命题}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明一模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031144": {
+        "id": "031144",
+        "content": "将曲线$\\dfrac{x^2}{16}+\\dfrac{y^2}9=1$($x \\geq 0$)与曲线$\\dfrac{x^2}7+\\dfrac{y^2}9=1$($x \\leq 0$)合成的曲线记作$C$. 设$k$为实数, 斜率为$k$的直线与$C$交于$A$、$B$两点, $P$为线段$AB$的中点, 有下列两个结论: \\textcircled{1} 存在$k$, 使得点$P$的轨迹总落在某个椭圆上; \\textcircled{2} 存在$k$, 使得点$P$的轨迹总落在某条直线上. 那么\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}均正确}{\\textcircled{1}\\textcircled{2}均错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题16",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031145": {
+        "id": "031145",
+        "content": "城市道路大多是纵横交错的矩形网格状, 从甲地到乙地的最短路径往往不是直线距离, 而是沿着网格走的直角距离. 在直角坐标系$x O y$中, 定义点$A(x_1, y_1)$、$B(x_2, y_2)$的 ``直角距离''$d(A, B)$为:$d(A, B)=|x_1-x_2|+|y_1-y_2|$. 设$M(1,1)$、$N(-1,-1)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\foreach \\i in {-4, -3, -2, -1, 1, 2, 3, 4}\n{\\draw [dashed, gray] (-4, \\i) -- (4, \\i);\n\\draw [dashed, gray] (\\i, -4) -- (\\i, 4);\n\\draw (0, \\i) node [left] {\\tiny$\\i$};\n\\draw (\\i, 0) node [below] {\\tiny$\\i$};\n\\draw [->] (-5, 0) -- (5, 0) node [below] {$x$};\n\\draw [->] (0, -5) -- (0, 5) node [left] {$y$};\n\\draw (0, 0) node [below left] {\\tiny$O$};};\n\\end{tikzpicture}\n\\end{center}\n(1) 写出一个满足$d(C, M)=d(C, N)$的点$C$的坐标;\\\\\n(2) 过点$M$、$N$作斜率为$2$的直线$l_1$、$l_2$, 点$Q$、$R$分别是直线$l_1$、$l_2$上的动点, 求$d(Q, R)$的最小值;\\\\\n(3) 设$P(x, y)$, 记方程$d(P, M)+d(P, N)=8$的曲线为$\\Gamma$, 类比椭圆研究曲线$\\Gamma$的性质(结论不要求证明), 并在所给坐标系中画出该曲线.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁一模试题21",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "12ex"
+    },
+    "031146": {
+        "id": "031146",
+        "content": "直线$l:\\begin{cases}x=1+t, \\\\y=1-t\\end{cases}$($t$为参数, $t \\in \\mathbf{R}$)的斜率为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届浦东二模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031147": {
+        "id": "031147",
+        "content": "直线$l$的参数方程为$\\begin{cases}x=2+t, \\\\y=1+2 t\\end{cases}$($t \\in \\mathbf{R}$), 则直线$l$的斜率为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届杨浦二模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031148": {
+        "id": "031148",
+        "content": "设$t \\in \\mathbf{R}$, 直线$\\begin{cases}x=2+t, \\\\y=-1-t\\end{cases}$($t$为参数)的倾斜角的大小为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届黄浦二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031149": {
+        "id": "031149",
+        "content": "已知直线$l$的参数方程为$\\begin{cases}x=1+3 t, \\\\y=2+4 t\\end{cases}$($t$是参数), 则点$(1,0)$到直线$l$的距离等于\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届崇明二模试题08",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031150": {
+        "id": "031150",
+        "content": "已知圆的参数方程为$\\begin{cases}x=2 \\cos \\theta, \\\\y=2 \\sin \\theta\\end{cases}$($\\theta$为参数), 则此圆的半径是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届奉贤一模试题03",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031151": {
+        "id": "031151",
+        "content": "在直角坐标系中, 已知圆的参数方程是$\\begin{cases}x=2 \\cos \\theta,\\\\y=2 \\sin \\theta-3\\end{cases}$($\\theta$是参数, $0 \\leq \\theta<2 \\pi$), 则圆的半径是\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届宝山二模试题04",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031152": {
+        "id": "031152",
+        "content": "曲线$\\begin{cases}x=\\dfrac{1}{\\cos ^2 \\theta}-1,\\\\y=2 \\tan \\theta,\\end{cases}$ $\\theta \\in(-\\dfrac{\\pi}2, \\dfrac{\\pi}2)$的焦点坐标为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届长宁二模试题09",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031153": {
+        "id": "031153",
+        "content": "直线$l$的方向向量$\\overrightarrow d=(1,-1)$, 且经过曲线$\\begin{cases}x=2+2 \\cos \\theta, \\\\y=-4+2 \\sin \\theta\\end{cases}$的中心, 则直线$l$的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届静安二模试题06",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031154": {
+        "id": "031154",
+        "content": "已知椭圆$\\Gamma:\\begin{cases}x=a \\cos \\theta, \\\\y=b \\sin \\theta\\end{cases}$($\\theta$为参数, $a>0, b>0)$的焦点分别$F_1(-2,0)$、$F_2(2,0)$, 点$A$为椭圆$\\Gamma$的上顶点, 直线$AF_2$与椭圆$\\Gamma$的另一个交点为$B$. 若$|BF_1|=3|BF_2|$, 则椭圆$\\Gamma$的标准方程为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届嘉定二模试题11",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031155": {
+        "id": "031155",
+        "content": "参数方程$\\begin{cases}x=t^2, \\\\y=t\\end{cases}$, (其中$t \\in \\mathbf{R}$) 表示的曲线为\\bracket{20}.\n\\fourch{圆}{椭圆}{双曲线}{抛物线}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届闵行二模试题13",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031156": {
+        "id": "031156",
+        "content": "已知双曲线$C$的参数方程为$\\begin{cases}x=t+\\dfrac 1t, \\\\y=t-\\dfrac 1t\\end{cases}$($t$为参数), 则此双曲线的焦距等于\\bracket{20}.\n\\fourch{$2$}{$4$}{$2 \\sqrt 2$}{$4 \\sqrt 2$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届虹口二模试题14",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": ""
+    },
+    "031157": {
+        "id": "031157",
+        "content": "若曲线$C_1:\\begin{cases}x=1+\\cos \\theta, \\\\y=\\sin \\theta\\end{cases}$($\\theta$为参数)上的点$P$到直线$C_2:\\begin{cases}x=t \\cdot \\sin 30^{\\circ}, \\\\y=1-t \\cdot \\cos 120^{\\circ}\\end{cases}$\n($t$为参数)的距离最短, 则点$P$的坐标是\\bracket{20}.\n\\fourch{$(1-\\dfrac{\\sqrt 2}2,-\\dfrac{\\sqrt 2}2)$}{$(1-\\dfrac{\\sqrt 2}2, \\dfrac{\\sqrt 2}2)$}{$(-\\dfrac{\\sqrt 2}2, 1-\\dfrac{\\sqrt 2}2)$}{$(\\dfrac{\\sqrt 2}2, 1-\\dfrac{\\sqrt 2}2)$}",
+        "objs": [],
+        "tags": [
+            "第七单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2022届徐汇二模试题15",
+        "edit": [
+            "20230108\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
         "space": ""
     }
 }
\ No newline at end of file