20221211 evening
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"0"
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"execution_count": 3,
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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"source": [
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"import os,re,json\n",
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"\"\"\"这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块\"\"\"\n",
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"problems = \"12227\"\n",
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"problems = \"12298,12305,12307\"\n",
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"\n",
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"def generate_number_set(string,dict):\n",
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" string = re.sub(r\"[\\n\\s]\",\"\",string)\n",
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3.8.8 ('base')",
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"display_name": "mathdept",
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"language": "python",
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"name": "python3"
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.8 (default, Apr 13 2021, 15:08:03) [MSC v.1916 64 bit (AMD64)]"
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"version": "3.9.15"
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},
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"orig_nbformat": 4,
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"vscode": {
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"interpreter": {
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"hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
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"hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc"
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}
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}
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.15"
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"version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]"
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},
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"orig_nbformat": 4,
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"vscode": {
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},
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"004697": {
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"id": "004697",
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"content": "已知非空集合$A,B$满足: $A\\cup B=R$, $A\\cap B=\\varnothing$, 函数$f(x)=\\begin{cases}\nx^2, & x\\in A, \\\\ 2x-1, & x\\in B. \\end{cases}$ 对于下列两个命题: \\textcircled{1} 存在唯一的非空集合对$(A,B)$, 使得$f(x)$为偶函数; \\textcircled{2} 存在无穷多非空集合对$(A,B)$, 使得方程$f(x)=2$无解. 下面判断正确的是\\bracket{20}.\n\\fourch{\\textcircled{1} 正确, \\textcircled{2} 错误}{\\textcircled{1} 错误, \\textcircled{2} 正确}{\\textcircled{1} 、\\textcircled{2} 都正确}{\\textcircled{1} 、\\textcircled{2} 都错误}",
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"content": "已知非空集合$A,B$满足: $A\\cup B=\\mathbf{R}$, $A\\cap B=\\varnothing$, 函数$f(x)=\\begin{cases}\nx^2, & x\\in A, \\\\ 2x-1, & x\\in B. \\end{cases}$ 对于下列两个命题: \\textcircled{1} 存在唯一的非空集合对$(A,B)$, 使得$f(x)$为偶函数; \\textcircled{2} 存在无穷多非空集合对$(A,B)$, 使得方程$f(x)=2$无解. 下面判断正确的是\\bracket{20}.\n\\fourch{\\textcircled{1} 正确, \\textcircled{2} 错误}{\\textcircled{1} 错误, \\textcircled{2} 正确}{\\textcircled{1} 、\\textcircled{2} 都正确}{\\textcircled{1} 、\\textcircled{2} 都错误}",
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"objs": [
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"K0217004B",
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"K0223002B"
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@ -125409,7 +125409,8 @@
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],
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"origin": "2022届高三上一模第16题",
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"edit": [
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"20220711\t王伟叶"
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"20220711\t王伟叶",
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"20221211\t王伟叶"
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],
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"same": [],
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"related": [],
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@ -300008,7 +300009,7 @@
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},
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"012128": {
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"id": "012128",
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"content": "已知函数$y=f(x)$在定义域$\\mathbf{R}$上是单调函数, 值域为$(-\\infty ,\\ 0)$, 满足$f(-1)=-\\dfrac 13$, 且对于任意$x,\\ y\\in \\mathbf{R}$, 都有$f(x+y)=-f(x)f(y)$. $y=f(x)$的反函数为$y=f^{-1}(x)$, 若将$y=kf(x)$(其中常数$k>0$)的反函数的图像向上平移1个单位, 将得到函数$y=f^{-1}(x)$的图像, 则实数k的值为\\blank{50}.",
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"content": "已知函数$y=f(x)$在定义域$\\mathbf{R}$上是单调函数, 值域为$(-\\infty ,\\ 0)$, 满足$f(-1)=-\\dfrac 13$, 且对于任意$x,\\ y\\in \\mathbf{R}$, 都有$f(x+y)=-f(x)f(y)$. $y=f(x)$的反函数为$y=f^{-1}(x)$, 若将$y=kf(x)$(其中常数$k>0$)的反函数的图像向上平移1个单位, 将得到函数$y=f^{-1}(x)$的图像, 则实数$k$的值为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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@ -300018,7 +300019,8 @@
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"usages": [],
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"origin": "2021届杨浦区一模试题12",
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"edit": [
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"20221206\t王伟叶"
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"20221206\t王伟叶",
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"20221211\t王伟叶"
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],
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"same": [],
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"related": [],
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@ -303238,7 +303240,7 @@
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},
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"012298": {
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"id": "012298",
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"content": "已知数列$\\{a_n\\}$的各项都是正数, $a_{n+1}^2-a_{n+1}=a_n$($n \\in \\mathbf{N}^*$), 若数列$\\{a_n\\}$为严格增数列, 则首项$a_1$的取值范围是\\blank{50}; 当$a_1=\\dfrac 23$时, 记$b_n=\\dfrac{(-1)^{n-1}}{a_n-1}$, 若$k<b_1+b_2+\\cdots+b_{2022}<k+1$, 则整数$k=$\\blank{50}.",
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"content": "已知数列$\\{a_n\\}$的各项都是正数, $a_{n+1}^2-a_{n+1}=a_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 若数列$\\{a_n\\}$为严格增数列, 则首项$a_1$的取值范围是\\blank{50}; 当$a_1=\\dfrac 23$时, 记$b_n=\\dfrac{(-1)^{n-1}}{a_n-1}$, 若$k<b_1+b_2+\\cdots+b_{2022}<k+1$, 则整数$k=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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@ -303248,7 +303250,8 @@
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"usages": [],
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"origin": "2023届松江区一模试题12",
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"edit": [
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"20221210\t王伟叶"
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"20221210\t王伟叶",
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"20221211\t周双"
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],
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"same": [],
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"related": [],
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@ -303371,7 +303374,7 @@
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},
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"012305": {
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"id": "012305",
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"content": "某地准备在山谷中建一座桥梁, 桥址位置的坚直截面图如图所示, 谷底$O$在水平线$MN$上、桥$AB$与$MN$平行, $OO'$为铅垂线($O'$在$AB$上). 经测量, 山谷左侧的轮廓曲线$AO$上任一点$D$到$MN$的距离$h_1$(米)与$D$到$OO'$的距离$a$(米) 之间满足关系式$h_1=\\dfrac 1{40} a^2$, 山谷右侧的轮廓曲线$BO$上任一点$F$到$MN$的距离$h_2$(米)与$F$到$OO'$的距离$b$(米)之间满足关系式$h_2=-\\dfrac 1{800} b^3+6 b$. 已知点$B$到$OO'$的距离为$40$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-8,16) node [left] {$A$} coordinate (A);\n\\draw (4,16) node [right] {$B$} coordinate (B);\n\\draw (-10,0) node [below] {$M$} coordinate (M);\n\\draw (6,0) node [below] {$N$} coordinate (N);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,16) node [above] {$O'$} coordinate (O');\n\\draw (-6,16) node [above] {$C$} coordinate (C);\n\\draw (-6,9) node [left] {$D$} coordinate (D);\n\\draw (2,16) node [above] {$E$} coordinate (E);\n\\draw (2,12) node [right] {$F$} coordinate (F);\n\\draw [ultra thick] (A) -- (B) (C) -- (D) (E) -- (F);\n\\draw (M) -- (N);\n\\draw [dashed] (O) -- (O');\n\\draw [domain = -8:0] plot (\\x,{0.25*pow(\\x,2)});\n\\draw [domain = 0:4.2] plot (\\x,{16-pow(\\x-4,2)});\n\\draw [dashed] (D) --++ (0,-9) node [midway,left] {$h_1$} coordinate (h_1) (D) --++ (6,0) node [midway,above] {$a$} coordinate (a);\n\\draw [dashed] (F) --+ (0,-12) node [midway,right] {$h_2$} coordinate (h_2) (F) --++ (-2,0) node [midway,above] {$b$} coordinate (b);\n\\end{tikzpicture}\n\\end{center}\n(1) 求谷底$O$到桥面$AB$的距离和桥$AB$的长度;\\\\\n(2) 计划在谷底两侧建造平行于$OO'$的桥墩$CD$和$EF$, 且$CE$为$80$米, 其中$C$、$E$在$AB$上(不包括端点), 桥墩$EF$每米造价为$k$(万元)、桥墩$CD$每米造价为$\\dfrac 32 k$(万元)($k>0$). 问$O'E$为多少米时, 桥墩$CD$与$EF$的总造价最低?",
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"content": "某地准备在山谷中建一座桥梁, 桥址位置的竖直截面图如图所示, 谷底$O$在水平线$MN$上、桥$AB$与$MN$平行, $OO'$为铅垂线($O'$在$AB$上). 经测量, 山谷左侧的轮廓曲线$AO$上任一点$D$到$MN$的距离$h_1$(米)与$D$到$OO'$的距离$a$(米) 之间满足关系式$h_1=\\dfrac 1{40} a^2$, 山谷右侧的轮廓曲线$BO$上任一点$F$到$MN$的距离$h_2$(米)与$F$到$OO'$的距离$b$(米)之间满足关系式$h_2=-\\dfrac 1{800} b^3+6 b$. 已知点$B$到$OO'$的距离为$40$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-8,16) node [left] {$A$} coordinate (A);\n\\draw (4,16) node [right] {$B$} coordinate (B);\n\\draw (-10,0) node [below] {$M$} coordinate (M);\n\\draw (6,0) node [below] {$N$} coordinate (N);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,16) node [above] {$O'$} coordinate (O');\n\\draw (-6,16) node [above] {$C$} coordinate (C);\n\\draw (-6,9) node [left] {$D$} coordinate (D);\n\\draw (2,16) node [above] {$E$} coordinate (E);\n\\draw (2,12) node [right] {$F$} coordinate (F);\n\\draw [ultra thick] (A) -- (B) (C) -- (D) (E) -- (F);\n\\draw (M) -- (N);\n\\draw [dashed] (O) -- (O');\n\\draw [domain = -8:0] plot (\\x,{0.25*pow(\\x,2)});\n\\draw [domain = 0:4.2] plot (\\x,{16-pow(\\x-4,2)});\n\\draw [dashed] (D) --++ (0,-9) node [midway,left] {$h_1$} coordinate (h_1) (D) --++ (6,0) node [midway,above] {$a$} coordinate (a);\n\\draw [dashed] (F) --+ (0,-12) node [midway,right] {$h_2$} coordinate (h_2) (F) --++ (-2,0) node [midway,above] {$b$} coordinate (b);\n\\end{tikzpicture}\n\\end{center}\n(1) 求谷底$O$到桥面$AB$的距离和桥$AB$的长度;\\\\\n(2) 计划在谷底两侧建造平行于$OO'$的桥墩$CD$和$EF$, 且$CE$为$80$米, 其中$C$、$E$在$AB$上(不包括端点), 桥墩$EF$每米造价为$k$(万元)、桥墩$CD$每米造价为$\\dfrac 32 k$(万元)($k>0$). 问$O'E$为多少米时, 桥墩$CD$与$EF$的总造价最低?",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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@ -303381,7 +303384,8 @@
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"usages": [],
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"origin": "2023届松江区一模试题19",
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"edit": [
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"20221210\t王伟叶"
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"20221210\t王伟叶",
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"20221211\t周双"
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],
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"same": [],
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"related": [],
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@ -303409,7 +303413,7 @@
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},
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"012307": {
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"id": "012307",
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"content": "己知定义在$\\mathbf{R}$上的函数$f(x)=\\mathrm{e}^{k x+b}$($\\mathrm{e}$是自然对数的底数) 满足$f(x)=f'(x)$且$f(-1)=1$, 删除无穷数列$f(1)$、$f(2)$、$f(3)$、$\\cdots$、$f(n)$、$\\cdots$中的第$3$项、第$6$项、$\\cdots$、第$3n$项, $\\cdots$, ($n \\in \\mathbf{N}$, $n\\ge 1$), 余下的项按原来顺序组成一个新数列$\\{t_n\\}$, 记数列$\\{t_n\\}$前$n$项和为$T_n$.\\\\\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 已知数列$\\{t_n\\}$的通项公式是$t_n=f(g(n))$, $n \\in \\mathbf{N}$, $n\\ge 1$, 求函数$g(n)$的解析式;\n(3) 设集合$X$是实数集$\\mathbf{R}$的非空子集, 如果正实数$a$满足: 对任意$x_1$、$x_2 \\in X$, 都有$|x_1-x_2|\\leq a$, 则称$a$为集合$X$的一个``阈度'', 记集合$H=\\{w | w=\\dfrac{T_n}{f(\\dfrac{3 n}2-\\dfrac{1+3(-1)^n}4)}, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 试问集合$H$存在``阈度''吗? 若存在, 求出集合$H$``阈度''的取值范围, 若不存在, 试说明理由.",
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"content": "己知定义在$\\mathbf{R}$上的函数$f(x)=\\mathrm{e}^{k x+b}$($\\mathrm{e}$是自然对数的底数) 满足$f(x)=f'(x)$且$f(-1)=1$, 删除无穷数列$f(1)$、$f(2)$、$f(3)$、$\\cdots$、$f(n)$、$\\cdots$中的第$3$项、第$6$项、$\\cdots$、第$3n$项, $\\cdots$, ($n \\in \\mathbf{N}$, $n\\ge 1$), 余下的项按原来顺序组成一个新数列$\\{t_n\\}$, 记数列$\\{t_n\\}$前$n$项和为$T_n$.\\\\\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 已知数列$\\{t_n\\}$的通项公式是$t_n=f(g(n))$, $n \\in \\mathbf{N}$, $n\\ge 1$, 求函数$g(n)$的解析式;\\\\\n(3) 设集合$X$是实数集$\\mathbf{R}$的非空子集, 如果正实数$a$满足: 对任意$x_1$、$x_2 \\in X$, 都有$|x_1-x_2|\\leq a$, 则称$a$为集合$X$的一个``阈度'', 记集合$H=\\{w | w=\\dfrac{T_n}{f(\\dfrac{3 n}2-\\dfrac{1+3(-1)^n}4)}, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 试问集合$H$存在``阈度''吗? 若存在, 求出集合$H$``阈度''的取值范围, 若不存在, 试说明理由.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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@ -303419,7 +303423,8 @@
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"usages": [],
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"origin": "2023届松江区一模试题21",
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"edit": [
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"20221210\t王伟叶"
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"20221210\t王伟叶",
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"20221211\t周双"
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],
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"same": [],
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"related": [],
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@ -303466,7 +303471,7 @@
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},
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"012310": {
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"id": "012310",
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"content": "已知复数$z_1=2+a \\mathrm{i}$, $z_2=3+\\mathrm{i}$, 若$z_1,z_2$是纯虚数, 则实数$a=$\\blank{50}.",
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"content": "已知复数$z_1=2+a \\mathrm{i}$, $z_2=3+\\mathrm{i}$, 若$z_1\\cdot z_2$是纯虚数, 则实数$a=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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