diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index db0bb2a6..30ae2976 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"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", - "problems = \"31188\"\n", + "problems = \"12671\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", @@ -51,7 +51,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ diff --git a/工具/分年级专用工具/小闲平台作业测验数据导入.ipynb b/工具/分年级专用工具/小闲平台作业测验数据导入.ipynb index de8936f4..249d4ffd 100644 --- a/工具/分年级专用工具/小闲平台作业测验数据导入.ipynb +++ b/工具/分年级专用工具/小闲平台作业测验数据导入.ipynb @@ -120,7 +120,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.8.15 (default, Nov 24 2022, 14:38:14) [MSC v.1916 64 bit (AMD64)]" }, "orig_nbformat": 4, "vscode": { 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 7366d0d2..f4c3054e 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 @@ -2,14 +2,14 @@ "cells": [ { "cell_type": "code", - "execution_count": 4, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "首个空闲id: 12760 , 直至 020000\n", + "首个空闲id: 13287 , 直至 020000\n", "首个空闲id: 21441 , 直至 030000\n", "首个空闲id: 31204 , 直至 999999\n" ] diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index 5ce5ca5d..2d6d4847 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -2,21 +2,21 @@ "cells": [ { "cell_type": "code", - "execution_count": 6, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 12760\n", - "origin = \"2023届静安区一模\"\n", - "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\静安一模.tex\"\n", - "editor = \"20230110、3\\t王伟叶\"\n", - "indexed = True\n" + "starting_id = 12781\n", + "origin = \"2022届高三第二轮复习讲义\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\第二轮讲义转码.tex\"\n", + "editor = \"20230118\\t王伟叶\"\n", + "indexed = False\n" ] }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ diff --git a/文本处理工具/剪贴板文本整理_mathpix.ipynb b/文本处理工具/剪贴板文本整理_mathpix.ipynb index c909b5a1..e8865cd7 100644 --- a/文本处理工具/剪贴板文本整理_mathpix.ipynb +++ b/文本处理工具/剪贴板文本整理_mathpix.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 30, "metadata": {}, "outputs": [], "source": [ @@ -256,7 +256,7 @@ " text1 = re.sub(\"\\{[\\s]+?\",\"{\",text1)\n", " text1 = re.sub(\"[\\s]+?\\}\",\"}\",text1)\n", " #填空题的处理\n", - " text1 = re.sub(\"[ _]{2,}\",r\"\\\\blank{50}\",text1)\n", + " # text1 = re.sub(\"[ _]{2,}\",r\"\\\\blank{50}\",text1)\n", " #选择题的处理\n", " text1 = re.sub(r\"\\(\\\\blank\\{50\\}\\)\",\"\\\\\\\\bracket{20}\",text1)\n", " text1 = re.sub(r\"\\([\\s]{1,10}\\)\",\"\\\\\\\\bracket{20}\",text1)\n", @@ -264,8 +264,8 @@ " text1 = re.sub(\",[ ]*\",\", \",text1)\n", " text1 = re.sub(r\"\\.\\}\",\"}\",text1)\n", " text1 = re.sub(r\"\\n\\d{1,3}\\.\",r\"\\n\\\\item \",text1)\n", - " text1 = re.sub(r\"\\s{2,}\\.\",r\"\\\\blank{50}.\",text1)\n", - " text1 = re.sub(r\"\\s{2,}\\,\",r\"\\\\blank{50},\",text1)\n", + " # text1 = re.sub(r\"\\s{2,}\\.\",r\"\\\\blank{50}.\",text1)\n", + " # text1 = re.sub(r\"\\s{2,}\\,\",r\"\\\\blank{50},\",text1)\n", " text1 = re.sub(r\"\\\\bracket\\{20\\}\\n\",r\"\\\\bracket{20}.\\n\",text1)\n", " modified_texts.append(text1)\n", "\n", @@ -274,6 +274,9 @@ " #合并一些公式中的无效空格\n", " for i in range(2):\n", " equation1 = re.sub(r\"([0-9A-Z])\\s+([0-9A-Z])\",lambda x:x.group(1)+x.group(2),equation1)\n", + " #改变组合数和排列数\n", + " equation1 = re.sub(r\"([CP])(_[^_\\^]{,5}\\^)\",lambda x:r\"\\mathrm{\"+x.group(1)+\"}\"+x.group(2),equation1)\n", + " \n", " modified_equations.append(equation1)\n", "\n", "\n", @@ -307,7 +310,7 @@ "modified_data = modified_data.replace(r\"\\vec\",r\"\\overrightarrow \")\n", "modified_data = modified_data.replace(r\"\\bar\",r\"\\overline \")\n", "#mathpix的极限修改\n", - "modified_data = modified_data.replace(r\"\\lim[\\s]*_{n \\rightarrow \\infty}\",r\"\\displaystyle\\lim_{n\\to\\infty}\")\n", + "modified_data = re.sub(r\"\\\\lim[\\s]*_\\{n \\\\to \\\\infty\\}\",r\"\\\\displaystyle\\\\lim_{n\\\\to\\\\infty}\",modified_data)\n", "#mathpix的顿号修改\n", "modified_data = modified_data.replace(r\" 、 \",r\"$、$\")\n", "#改slant等\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 3740f68d..4717f542 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -312829,7 +312829,7 @@ }, "012665": { "id": "012665", - "content": "某人去公园郊游, 在草地上搭建了如图所示的简易遮阳篷$ABC$, 遮阳篷是一个直角边长为$6$的等腰直角三角形, 斜边$AB$朝南北方向固定在地上, 正西方向射出的太阳光线与地面成$30^{\\circ}$角, 则当遮阳篷$ABC$与地面所成的角大小为\\blank{50}时, 所遮阴影面$ABC$面积达到最大.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [right] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C'$} coordinate (C');\n\\draw (0,2,0) node [above] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (C) -- (C') -- cycle;\n\\draw [dashed] (A) -- (B) -- (C) (B) -- (C');\n\\end{tikzpicture}\n\\end{center}", + "content": "某人去公园郊游, 在草地上搭建了如图所示的简易遮阳篷$ABC$, 遮阳篷是一个直角边长为$6$的等腰直角三角形, 斜边$AB$朝南北方向固定在地上, 正西方向射出的太阳光线与地面成$30^{\\circ}$角, 则当遮阳篷$ABC$与地面所成的角大小为\\blank{50}时, 所遮阴影面$ABC'$面积达到最大.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [right] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C'$} coordinate (C');\n\\draw (0,2,0) node [above] {$C$} coordinate (C);\n\\fill [pattern = north east lines] (A) -- (B) -- (C) -- cycle;\n\\draw (A) -- (C) -- (C') -- cycle;\n\\draw [dashed] (A) -- (B) -- (C) (B) -- (C');\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], "genre": "填空题", @@ -312924,7 +312924,7 @@ }, "012670": { "id": "012670", - "content": "已知$O$为坐标原点, 点$A(1,1)$在抛物线$C: x^2=2 p y$($p>0$)上, 过点$B(0,-1)$的直线交抛物线$C$于$P$、$Q$两点: \\textcircled{1} 抛物线$C$的准线为$y=-\\dfrac 12$; \\textcircled{2} 直线$AB$与抛物线$C$相切; \\textcircled{3} $|OP|\\cdot|OQ|>|OA|^2$; \\textcircled{4} $|BP|\\cdot|BQ|=|BA|^2$, 以上结论中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", + "content": "已知$O$为坐标原点, 点$A(1,1)$在抛物线$C: x^2=2 p y$($p>0$)上, 过点$B(0,-1)$的直线交抛物线$C$于$P$、$Q$两点: \\textcircled{1} 抛物线$C$的准线为$y=-\\dfrac 12$; \\textcircled{2} 直线$AB$与抛物线$C$相切; \\textcircled{3} $|OP|\\cdot|OQ|>|OA|^2$; \\textcircled{4} $|BP|\\cdot|BQ|=|BA|^2$, 以上结论中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", "objs": [], "tags": [], "genre": "选择题", @@ -312943,7 +312943,7 @@ }, "012671": { "id": "012671", - "content": "已知函数$f(x)=\\sin 2 x+\\sqrt 3 \\cos 2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求函数$f(x)$的单调增区间;\\\\\n(2) 在锐角$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 当$f(A)=0$, $b=1$, 且三角形$ABC$的面积为$\\sqrt 3$时, 求$a$.", + "content": "已知函数$f(x)=\\sin 2 x+\\sqrt 3 \\cos 2 x$, $x \\in \\mathbf{R}$.\\\\\n(1) 求函数$f(x)$的单调增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 当$f(A)=0$, $b=1$, 且三角形$ABC$的面积为$\\sqrt 3$时, 求$a$.", "objs": [], "tags": [], "genre": "解答题", @@ -314644,7 +314644,7 @@ "usages": [], "origin": "2023届静安区一模试题1", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314663,7 +314663,7 @@ "usages": [], "origin": "2023届静安区一模试题2", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314682,7 +314682,7 @@ "usages": [], "origin": "2023届静安区一模试题3", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314701,7 +314701,7 @@ "usages": [], "origin": "2023届静安区一模试题4", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314720,7 +314720,7 @@ "usages": [], "origin": "2023届静安区一模试题5", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314739,7 +314739,7 @@ "usages": [], "origin": "2023届静安区一模试题6", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314758,7 +314758,7 @@ "usages": [], "origin": "2023届静安区一模试题7", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314777,7 +314777,7 @@ "usages": [], "origin": "2023届静安区一模试题8", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314796,7 +314796,7 @@ "usages": [], "origin": "2023届静安区一模试题9", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314815,7 +314815,7 @@ "usages": [], "origin": "2023届静安区一模试题10", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314834,7 +314834,7 @@ "usages": [], "origin": "2023届静安区一模试题11", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314853,7 +314853,7 @@ "usages": [], "origin": "2023届静安区一模试题12", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314872,7 +314872,7 @@ "usages": [], "origin": "2023届静安区一模试题13", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314891,7 +314891,7 @@ "usages": [], "origin": "2023届静安区一模试题14", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314910,7 +314910,7 @@ "usages": [], "origin": "2023届静安区一模试题15", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314929,7 +314929,7 @@ "usages": [], "origin": "2023届静安区一模试题16", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314948,7 +314948,7 @@ "usages": [], "origin": "2023届静安区一模试题17", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314967,7 +314967,7 @@ "usages": [], "origin": "2023届静安区一模试题18", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -314986,7 +314986,7 @@ "usages": [], "origin": "2023届静安区一模试题19", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -315005,7 +315005,7 @@ "usages": [], "origin": "2023届静安区一模试题20", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], @@ -315024,13 +315024,9627 @@ "usages": [], "origin": "2023届静安区一模试题21", "edit": [ - "20230110、3\t王伟叶" + "20230113\t王伟叶" ], "same": [], "related": [], "remark": "", "space": "12ex" }, + "012781": { + "id": "012781", + "content": "设$a, b \\in \\mathbf{R}$, 则``$a+b \\geq 4$''是``$a>2$且$b>2$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012782": { + "id": "012782", + "content": "设常数$a \\in \\mathbf{R}$. 若$\\{x | 10\\}$, $B=\\{x|| 1-x | \\geq 1-x\\}$, 则$A \\otimes B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012787": { + "id": "012787", + "content": "设集合$A=\\{0, x, y\\}, B=\\{2, x^2\\}$. 若$B \\subseteq A$, 求$x,y$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012788": { + "id": "012788", + "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别是$a_n=3 n$, $b_n=2 n+1$($n \\in \\mathbf{N}$, $n\\ge 1$). 将集合$\\{x | x=a_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\} \\cup\\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$中的元素从小到大依次排列, 构成数列$c_1, c_2, \\cdots, c_n, \\cdots$.\\\\\n(1) 写出$c_1, c_2, c_3, c_4$;\\\\\n(2) 求证:在数列$\\{c_n\\}$中, 但不在数列$\\{b_n\\}$中的项恰为$a_2, a_4, \\cdots, a_{2 n}, \\cdots$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012789": { + "id": "012789", + "content": "已知集合$A=\\{(x, y) | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | y-1=2^{2018} \\cdot(x-1),\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B$的元素个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无限}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012790": { + "id": "012790", + "content": "设$U$是全集, 集合$A$, $B$满足$A\\subset B$, 给出下列四个命题: \\textcircled{1} $A \\cap \\overline B=\\varnothing$; \\textcircled{2} $B \\cap \\overline A=\\overline A$; \\textcircled{3} $B \\cup \\overline A=U$; \\textcircled{4} $\\overline A \\cap \\overline B=\\overline B$. 四个命题中, 正确命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012791": { + "id": "012791", + "content": "设$U$是全集, $A, B$是两个集合, 则``存在集合$C$使得$A \\subseteq C, B \\subseteq \\overline C$''是``$A \\cap B=\\varnothing$''的 \\bracket{20}\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012792": { + "id": "012792", + "content": "已知集合$A=\\{y | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{y | y=2^x,\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B=$\\bracket{20}.\n\\onech{$\\{y | y>0\\}$}{$\\{y | y \\geq 0\\}$}{$\\{y | y=2,4, u\\}$, 其中常数$u<0$且$u^2=2^u$}{$\\{(2,4),(4,16),(u, u^2)\\}$, 其中常数$u<0$且$u^2=2^u$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012793": { + "id": "012793", + "content": "已知集合$A=\\{x, 1\\}$, $B=\\{y, 1,2\\}$, 其中$x, y \\in\\{1,2,3,4,5\\}$, 且$A \\subseteq B$. 如果把满足上述条件的一对有序整数$(x, y)$作为一个点, 则这样的点的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012794": { + "id": "012794", + "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| x-1 |<2,\\ x \\in \\mathbf{Z}\\}, B=\\{x | x \\geq a\\}$. 若$A \\cap B=\\{1,2\\}$, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012795": { + "id": "012795", + "content": "已知集合$S=\\{1,2,3,4,5\\}$, $A$是$S$的一个子集. 当$x \\in A$时, 若有$x-1 \\not\\in A$, 且$x+1 \\not\\in A$, 则称$x$为$A$的一个``孤立元素'', 那么$S$中无``孤立元素''且恰有$4$个元素的子集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012796": { + "id": "012796", + "content": "已知集合$M$是满足下列性质的函数$f(x)$的全体: 在在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立.\\\\\n(1) 求证: 函数$f(x)=x$不属于集合$M$;\\\\\n(2) 写出命题``存在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立''的否定形式.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012797": { + "id": "012797", + "content": "已知集合$M$是满足下列性质的函数$f(x)$的全体: 在在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立.\\\\\n(1) 求证: 函数$f(x)=x$不属于集合$M$;\\\\\n(2) 求证: 函数$f(x)=(\\dfrac{1}{2})^x$属于集合$M$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012798": { + "id": "012798", + "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-3 a x+2 a^2<0\\}$. 若$A \\cap B=B$, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012799": { + "id": "012799", + "content": "设常数$m \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-2 x+2 m<0\\}$. 若$A \\cup B=A$, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012800": { + "id": "012800", + "content": "不等式$|x-1| \\geq 2$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012801": { + "id": "012801", + "content": "不等式$\\dfrac{5-x}{2 x-4} \\leq 1$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012802": { + "id": "012802", + "content": "不等式$|x-1|f(x)$时$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012805": { + "id": "012805", + "content": "设$a, b$为非零实数, 且$ab\\end{cases}$的一个解, 则$a, b$满足的条件为\\bracket{20}.\n\\fourch{$a<1b c$}{$a d$与$b c$的大小不确定}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012817": { + "id": "012817", + "content": "设常数$a, b \\in \\mathbf{R}$. 若$x=1$是不等式组$\\begin{cases}xb\\end{cases}$的唯一整数解, 则$a, b$满足的条件为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012818": { + "id": "012818", + "content": "设$a, b \\in \\mathbf{Z}$, 且$a+b=2023$, 则$2^a+2^b$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012819": { + "id": "012819", + "content": "设常数$k \\in \\mathbf{R}$. 已知关于$x$的不等式$(1+k^2) x \\leq k^6+2$的解集是$M$. 有下面四个命题: \\textcircled{1} 对任意实数$k$, 总有$0 \\in M$; \\textcircled{2} 存在实数$k$, 使得$1 \\notin M$; \\textcircled{3} 存在实数$k$, 使得$2 \\in M$; \\textcircled{4} 对任意实数$k$, 总有$3 \\notin M$. 上述四个命题中, 正确的命题的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012820": { + "id": "012820", + "content": "设$a_1, b_1, a_2, b_2$均为非零实数, 不等式$a_1 x+b_1>0$和$a_2 x+b_2>0$的解集分别为集合$M$和$N$, 那么``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}$''是``$M=N$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012821": { + "id": "012821", + "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| 2-x |<5, x \\in \\mathbf{R}\\}$, $B=\\{x|| x+a | \\geq 4, \\ x \\in \\mathbf{R}\\}$. 若$A \\cup B=\\mathbf{R}$, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012822": { + "id": "012822", + "content": "已知数列$\\{a_n\\}$,$ a_n=2 n-10$, $S_n$是数列$\\{a_n\\}$的前$n$项和. 设$c_n=a_n S_n$, 在数列$\\{c_n\\}$中,\\\\\n(1) 是否存在小于零的项? 若存在, 求出这些项的序数; 若不存在, 说明理由;\\\\\n(2) 求小于$10^4$的$\\{c_n\\}$的项的个数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012823": { + "id": "012823", + "content": "不等式$\\log _{\\frac{1}{2}}(3-x) \\geq 1$的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012824": { + "id": "012824", + "content": "函数$d(x)=\\begin{cases}0, & x \\in \\mathbf{Q}, \\\\ 1, & x \\notin \\mathrm{Q},\\end{cases}$ 则$d(d(x))=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012825": { + "id": "012825", + "content": "设常数$a \\in \\mathbf{R}$. 若关于$x$的方程$\\dfrac{x-1}{3 x-2}=a$有实数解, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012826": { + "id": "012826", + "content": "已知函数$f(x)=\\begin{cases}x^2+4 x, & x \\geq 0,\\\\ 4 x-x^2, & x<0.\\end{cases}$ 若$f(2-a^2)>f(a)$, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012827": { + "id": "012827", + "content": "设常数$a \\in \\mathbf{R}$. 若$y=\\log _{\\frac{1}{2}}(x^2-a x+2)$在$[-1,+\\infty)$上是减函数, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012828": { + "id": "012828", + "content": "已知函数$f(x)=\\sqrt{m x^2+(m-3) x+1}$的值域是$[0,+\\infty)$, 则实数$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012829": { + "id": "012829", + "content": "定义域和值域均为$[-a, a]$(常数$a>0$) 的函数$y=f(x)$和$y=g(x)$的图像如图所示, 给出下列四个命题: \\textcircled{1} 方程$f(g(x))=0$有且仅有三个解; \\textcircled{2} 方程$g(f(x))=0$有且仅有三个解; \\textcircled{3} 方程$f(f(x))=0$有且仅有九个解; \\textcircled{4} 方程$g(g(x))=0$有且仅有一个解. 那么, 其中正确命题的序号是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,-2) .. controls +(75:1) and +(180:0.6) .. (-1,0.6) .. controls +(0:0.6) and +(225:1.5) .. (1,0) .. controls +(45:0.5) and +(255:1) .. (2,2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (-1,1) node [above] {\\small $y=f(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,2) .. controls +(-45:1) and +(165:0.6) .. (0,0.5) .. controls +(-15:0.6) and +(135:0.5) .. (1,0) .. controls +(-45:0.5) and +(105:1) .. (2,-2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (1,1) node [above] {\\small $y=g(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012830": { + "id": "012830", + "content": "设函数$f(x)=\\dfrac{a^x}{1+a^x}$($a>0$, $a \\neq 1$), $[m]$表示不超过实数$m$的最大整数, 则函数$g(x)=[f(x)-\\dfrac{1}{2}]+[f(-x)-\\dfrac{1}{2}]$的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012831": { + "id": "012831", + "content": "记$\\min \\{p, q\\}=\\begin{cases}p, & p \\leq q, \\\\ q, & p>q,\\end{cases}$ 若函数$f(x)=\\min \\{3+\\log _{\\frac{1}{4}} x, \\log _2 x\\}$.\\\\\n(1) 用分段函数形式写出函数$f(x)$的解析式;\\\\\n(2) 求$f(x)<2$的解集.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012832": { + "id": "012832", + "content": "设函数$f(x)=a x$, $g(x)=|x-a|$, 常数$a>0$.\\\\\n(1) 当$a=2$时, 解关于$x$的不等式$f(x)>g(x)$;\\\\\n(2) 记$F(x)=f(x)-g(x)$, 若$F(x)$在$(0,+\\infty)$上有最大值, 求$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012833": { + "id": "012833", + "content": "设$a$为实数, 记函数$f(x)=a \\sqrt{1-x^2}+\\sqrt{1+x}+\\sqrt{1-x}$的最大值为$g(a)$.\\\\\n(1) 设$t=\\sqrt{1+x}+\\sqrt{1-x}$, 求$t$的取值范围, 并把$f(x)$表示为$t$的函数$m(t)$;\\\\\n(2) 求$g(a)$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012834": { + "id": "012834", + "content": "函数$f(x)=\\sqrt{x+1}+\\dfrac{1}{2-x}$的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012835": { + "id": "012835", + "content": "若集合$A=\\{x | \\lg x<1\\}$, $B=\\{y | y=\\sin x,\\ x \\in \\mathrm{R}\\}$, 则$A \\cup B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012836": { + "id": "012836", + "content": "``$a=1$''是``函数$f(x)=|x-a|$在区间$[1,+\\infty)$上为严格增函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012837": { + "id": "012837", + "content": "若$x>1$, 则函数$y=\\dfrac{x^2-x+1}{x-1}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012838": { + "id": "012838", + "content": "函数$f(x)=x^2-2 x+2$($x \\leq 0$)的反函数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012839": { + "id": "012839", + "content": "若函数$f(x)=\\log _a(2-a x)$在$[0,1]$上单调递减, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012840": { + "id": "012840", + "content": "设函数$f(x)=a^{x+1}-2$($a>1$)的反函数为$y=f^{-1}(x)$, 若函数$y=f^{-1}(x)$的图像不经过第二象限, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012841": { + "id": "012841", + "content": "已知$f(x)=4^x-k \\cdot 2^x+1$, 当$x \\in \\mathbf{R}$时, $f(x)$恒为正值, 则$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012842": { + "id": "012842", + "content": "已知定义在$\\mathbf{R}$上的奇函数$f(x)$, 满足$f(x-4)=-f(x)$, 且在区间$[0,2]$上是增函数, 若方程$f(x)=m$($m>0$)在区间$[-8,8]$上有四个不同的根$x_1, x_2, x_3, x_4$, 则$x_1+x_2+x_3+x_4=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012843": { + "id": "012843", + "content": "设函数$f(x)=\\dfrac{2 x}{1+|x|}$($x \\in \\mathbf{R}$), 区间$M=[a, b]$($a0$, 函数$g(x)=\\dfrac{x}{x+1}$, $h(x)=\\dfrac{1}{x+a}$, 且$f(x)=g(x) \\cdot h(x)$.\\\\\n(1) 若$a=1$, 并设函数$f(x)$的定义域是$[1,2]$, 求函数$f(x)$的值域;\\\\\n(2) 对于给定的常数$a$, 是否存在实数$t$, 使得$g(t)=h(t)$成立? 若存在, 求出这样的所有$t$的值; 若不存在, 说明理由;\\\\\n(3) 若$a>1$, 问是否存在常数$a$的值, 使函数$f(x)$的定义域是$[1, a]$, 值域为$[\\dfrac{1}{2(a+1)}, \\dfrac{1}{a^2}]$? 若存在, 求出这样的$a$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012846": { + "id": "012846", + "content": "函数$y=2^x+1$的反函数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012847": { + "id": "012847", + "content": "函数$y=\\dfrac{x^2+5}{\\sqrt{x^2+4}}$的值域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012848": { + "id": "012848", + "content": "设函数$f(x)=a|x|+\\dfrac{b}{x}$($a$、$b$为常数), 且\\textcircled{1} $f(-2)=0$; \\textcircled{2} $f(x)$有且仅有两个严格增区间, 则同时满足上述条件的一个有序数对$(a, b)$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012849": { + "id": "012849", + "content": "已知函数$f(x)=\\log _a(2^x+b-1)$($a>0$, $a \\neq 1$)的图像如图所示, 则$a, b$满足的关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.95:2.5,samples = 100] plot (\\x,{ln(pow(2,\\x)-0.5)/ln(5)});\n\\draw (-0.2,-1) -- (0,-1) node [right] {$-1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$00$且$a \\neq 1)$在$\\mathbf{R}$上既是奇函数, 又是减函数, 则$g(x)=\\log _a(x+k)$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\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\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\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\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},\\x);\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012851": { + "id": "012851", + "content": "已知$f(x)=\\begin{cases}(2-a) x+1, & x<1, \\\\ a^x, & x \\geq 1\\end{cases}$满足: 对任意$x_1 \\neq x_2$, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012852": { + "id": "012852", + "content": "已知$f(x)$是定义在$\\mathbf{R}$上的函数, 且$f(1)=1$, 对任意的$x \\in \\mathbf{R}$都有下列两式成立: $f(x+5) \\geq f(x)+5$; $f(x+1) \\leq f(x)+1$. 若$g(x)=f(x)+1-x$, 则$g(6)$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012853": { + "id": "012853", + "content": "设函数$y=f(x)$在$\\mathbf{R}$上有定义, 对于任意给定正数$M$, 定义函数$f_M(x)=\\begin{cases}f(x), & f(x) \\leq M, \\\\ M, & f(x)>M,\\end{cases}$ 则称函数$f_M(x)$为$f(x)$的``孪生函数'', 若给定函数$f(x)=2-x^2$, $M=1$, 则$f_M(2)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012854": { + "id": "012854", + "content": "设常数$a \\in \\mathbf{R}$, 已知函数$f(x)=x+\\dfrac{a}{x}$, $x \\in[1,+\\infty)$, 求$f(x)$的最小值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012855": { + "id": "012855", + "content": "设常数$a \\in \\mathbf{R}$, 已知函数$f(x)=x^2+\\dfrac{a}{x}$.\\\\\n(1) 判断函数$f(x)$的奇偶性;\\\\\n(2) 若$f(x)$在区间$[2,+\\infty)$上是严格增函数, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012856": { + "id": "012856", + "content": "设关于$x$的方程$x^2+2 x=(a+2) x+2$的两个实根为$x_1$、$x_2$, 是否存在实数$m$, 使得不等式$m^2+t m+1 \\geq|x_1-x_2|$对任意$a \\in[-1,1]$及任意$t \\in[-1,1]$恒成立? 若存在, 求$m$的取值范围; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012857": { + "id": "012857", + "content": "函数$f(x)=x^2+\\dfrac{4}{x^2+3}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012858": { + "id": "012858", + "content": "若$f(x+2)=\\begin{cases}\\tan x, & x \\geq 0, \\\\ \\log _2(-x), & x<0,\\end{cases}$ 则$f(\\dfrac{\\pi}{4}+2) \\cdot f(-2)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012859": { + "id": "012859", + "content": "已知函数$y=f(x)$对任意实数$x$都有$f(-x)=f(x)$, $f(x)=-f(x+1)$, 且在$[0,1]$上单调递减, 则$f(\\dfrac{7}{2})$、$f(\\dfrac{7}{3})$、$f(\\dfrac{7}{5})$的大小顺序是\\blank{50}.(用``$>$''连接)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012860": { + "id": "012860", + "content": "设函数$f(x)=\\dfrac{1}{1-\\sqrt{x}}$($0 \\leq x<1$)的反函数为$f^{-1}(x)$, 则\\bracket{20}.\n\\twoch{$f^{-1}(x)$在其定义域上是增函数且最大值为$1$}{$f^{-1}(x)$在其定义域上是减函数且最小值为$0$}{$f^{-1}(x)$在其定义域上是减函数且最大值为$1$}{$f^{-1}(x)$在其定义域上是增函数且最小值为$0$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012861": { + "id": "012861", + "content": "若函数$f(x)$($x \\in \\mathbf{R}$)为奇函数, 且存在反函数$f^{-1}(x)$(与$f(x)$不同), $F(x)=\\dfrac{2^{f(x)}-2^{f^{-1}(x)}}{2^{f(x)}+2^{f^{-1}(x)}}$, 则下列关于函数$F(x)$的奇偶性的说法中正确的是\\bracket{20}.\n\\twoch{$F(x)$是奇函数非偶函数}{$F(x)$是偶函数非奇函数}{$F(x)$既是奇函数又是偶函数}{$F(x)$既非奇函数又非偶函数}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012862": { + "id": "012862", + "content": "设函数$g(x)=x^2-2$, $f(x)=\\begin{cases}g(x)+x+4, & x=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw (0,0) -- (2,2);\n\\filldraw [fill = white] (0,0) circle (0.03);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 0.5:2,samples = 100] plot (\\x,{1/\\x});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 0.5:1,samples = 100] plot (\\x,{1/\\x}) -- (2,2);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 2:1,samples = 100] plot (\\x,{1/\\x}) -- (0,0);\n\\filldraw [fill = white] (0,0) circle (0.03);\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012864": { + "id": "012864", + "content": "设常数$a \\in \\mathbf{R}$. 若函数$f(x)=|x+a|-|x-1|$是定义在$\\mathbf{R}$上的奇函数, 但不是偶函数, 则函数$f(x)$的严格增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012865": { + "id": "012865", + "content": "设$f(x)$是定义域为$\\mathbf{R}$的函数, 且满足$f(x+2)=f(x+1)-f(x)$, 如果$f(1)=\\lg \\dfrac{3}{2}$, $f(2)=\\lg 15$, 则$f(2022)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012866": { + "id": "012866", + "content": "函数$y=f(x)$是定义在$\\mathbf{R}$上的恒不为零的函数, 且对于任意的$x, y \\in \\mathbf{R}$, 都满足$f(x) \\cdot f(y)=f(x+y)$, 则下列四个结论中, 正确的是\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} 对任意$x \\in \\mathbf{R}$, 都有$f(x)>0$; \\textcircled{3} $f(0)=1$; \\textcircled{4} 若$x<0$时, 有$f(x)>f(0)$, 则$f(x)$在$\\mathbf{R}$上的单调递减.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012867": { + "id": "012867", + "content": "设$f(x)=\\dfrac{-2^x+a}{2^{x+1}+b}$($a, b$为实常数). 若$f(x)$是奇函数, 求$a$与$b$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012868": { + "id": "012868", + "content": "已知函数$f(x)=x+\\dfrac{m}{x}+2$($m$为实常数).\\\\\n(1) 若函数$y=f(x)$在区间$[2,+\\infty)$上是增函数, 求实数$m$的取值范围;\\\\\n(2) 设$m<0$, 若不等式$f(x) \\leq k x$在$x \\in[\\dfrac{1}{2}, 1]$有解, 求$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012869": { + "id": "012869", + "content": "若函数$f(x)=\\sqrt{x}+1$的反函数为$f^{-1}(x)$, 则$f^{-1}(1)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012870": { + "id": "012870", + "content": "函数$y=(x-1)^{\\frac{3}{5}}$的图像不经过第\\blank{50}象限.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012871": { + "id": "012871", + "content": "若函数$y=x^2+(a+2) x+3$, $x \\in[a, b]$的图像关于直线$x=1$对称, 则$b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012872": { + "id": "012872", + "content": "若偶函数$f(x)$满足: 当$x>0$时, $f(x)$为严格减函数, 且$f(\\pi)=0$, 则$\\dfrac{f(x)}{x}<0$的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012873": { + "id": "012873", + "content": "方程$9^x-6^x=2^{2 x+1}$的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012874": { + "id": "012874", + "content": "已知关于$x$的方程$9^x+(a+4) \\cdot 3^x+4=0$有实数解, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012875": { + "id": "012875", + "content": "已知函数$f(x)$是定义在实数集$\\mathbf{R}$上的不恒为零的偶函数, 且对任意实数$x$都有$x f(x+1)=(1+x) f(x)$, 则$f(f(\\dfrac{5}{2}))$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012876": { + "id": "012876", + "content": "对于函数$f(x)=\\dfrac{x}{1+|x|}$($x \\in \\mathbf{R}$), 给出以下三个命题: \\textcircled{1} 函数$f(x)$的值域为$[-1,1]$; \\textcircled{2} 若$x_1 \\neq x_2$, 则一定有$f(x_1) \\neq f(x_2)$; \\textcircled{3} 若规定$f_1(x)=f(x)$, $f_n(x)=f(f_{n-1}(x))$, 则$f_n(x)=\\dfrac{x}{1+n|x|}$对任意$n \\in N$, $n\\ge 1$恒成立. 上述三个命题中正确命题序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012877": { + "id": "012877", + "content": "已知函数$f(x)=2^x-\\dfrac{1}{2^{|x|}}$.\\\\\n(1) 若$f(x)=2$, 求$x$的值;\\\\\n(2) 若$2^t f(2 t)+m f(t) \\geq 0$对于$t \\in[1,2]$恒成立, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012878": { + "id": "012878", + "content": "求证: 函数$f(x)=x^3+x-1$在区间$(\\dfrac{1}{2}, 1)$上存在唯一的零点.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012879": { + "id": "012879", + "content": "若函数$f(x)=x^2+m x+2$在区间$[0,2]$上存在两个不同的零点, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012880": { + "id": "012880", + "content": "已知函数$f(x)$的定义域是$x \\neq 0$的一切实数, 对定义域内的任意$x_1, x_2$都有$f(x_1 \\cdot x_2)=f(x_1)+f(x_2)$, 且当$x>1$时$f(x)>0$, $f(2)=1$.\\\\\n(1) 求证:$f(x)$是偶函数;\\\\\n(2) $f(x)$在$(0,+\\infty)$上是增函数;\\\\\n(3) 解不等式$f(2 x^2-1)<2$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012881": { + "id": "012881", + "content": "奇函数$y=f(x)$, 当$x<0$时, $f(x)=x+\\lg |x|$, 则$f(10)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012882": { + "id": "012882", + "content": "关于$x$的方程$\\lg ^2 x+\\lg x^2=0$的解是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012883": { + "id": "012883", + "content": "函数$y=\\log _{\\frac{1}{3}}(1-x^2)$的单调增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012884": { + "id": "012884", + "content": "已知$x_0$是函数$f(x)=2^x+\\dfrac{1}{1-x}$的一个零点. 若$x_1 \\in(1, x_0)$, $x_2 \\in(x_0,+\\infty)$, 则\\bracket{20}.\n\\fourch{$f(x_1)<0$, $f(x_2)<0$}{$f(x_1)<0$, $f(x_2)>0$}{$f(x_1)>0$, $f(x_2)<0$}{$f(x_1)>0$, $f(x_2)>0$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012885": { + "id": "012885", + "content": "函数$f(x)$的定义域关于原点对称, 对定义域中任一$x$值, 恒有$|f(x)|=|f(-x)|$成立, 则\\bracket{20}.\n\\twoch{$f(x)$是奇函数}{$f(x)$是偶函数}{$f(x)$不可能既非奇函数也非偶函数}{$f(x)$有可能既非奇函数也非偶函数}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012886": { + "id": "012886", + "content": "已知$f(x)=\\log _{\\frac{1}{2}} x$的反函数为$f^{-1}(x)$, 若$f^{-1}(a) \\cdot f^{-1}(b)=\\dfrac{1}{4}$, 则$f(a+b)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012887": { + "id": "012887", + "content": "设函数$f(x)$在$(-\\infty,+\\infty)$内有定义, 给出下列四个函数: \\textcircled{1} $y=-|f(x)|$; \\textcircled{2} $y=x f(x^2)$; \\textcircled{3} $y=-f(-x)$; \\textcircled{4} $y=f(x+1)-f(1-x)$. 其中必为奇函数的是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012888": { + "id": "012888", + "content": "设$f(x)$是定义在$\\mathbf{R}$上且以$3$为周期的奇函数, 若$f(1) \\leq 1$, $f(2)=\\dfrac{2 a-3}{a+1}$, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012889": { + "id": "012889", + "content": "已知函数$f(x)=\\log_2(4^x+1)-x$. 若函数$F(x)=f(x)-m$的一个零点在区间$(0, \\dfrac{1}{2})$内, 则实数$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012890": { + "id": "012890", + "content": "设$a, b, k$是实数, 二次函数$f(x)=x^2+a x+b$满足: $f(k-1)$与$f(k)$异号, $f(k+1)$与$f(k)$同号. 在以下关于$f(x)$的零点的命题中, 假命题的序号为\\bracket{20}.\\\\\n\\textcircled{1} 该二次函数的两个零点之差一定大于$2$; \\textcircled{2} 该二次函数的零点都小于$k$; \\textcircled{3} 该二次函数的零点都大于$k-1$.\n\\fourch{(1)(2)}{(2)(3)}{(1)(3)}{(1)(2)(3)}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012891": { + "id": "012891", + "content": "已知函数$f(x)=a \\cdot 2^x+b \\cdot 3^x$, 其中常数$a, b$满足$a b \\neq 0$.\\\\\n(1) 若$ab>0$, 判断函数$y=f(x)$的单调性;\\\\\n(2) 若$ab<0$, 求$f(x+1)>f(x)$时$x$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012892": { + "id": "012892", + "content": "已知函数$f(x)=\\log _4(4^x+1)$, $g(x)=(k-1) x$, 记$F(x)=f(x)-g(x)$, 并且$F(x)$为偶函数.\\\\\n(1) 求常数$k$的值;\\\\\n(2) 若对一切$a \\in \\mathbf{R}$, 不等式$F(a)>-\\dfrac{1}{2} m$恒成立, 求实数$m$的取值范围;\\\\\n(3) 设$M(x)=\\log _4(a \\cdot 2^x-\\dfrac{4}{3} a)$, 若函数$F(x)$与$M(x)$的图像有且只有一个公共点, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012893": { + "id": "012893", + "content": "函数$y=\\log _{0.7}(x^2-3 x+2)$的单调减区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012894": { + "id": "012894", + "content": "若$\\alpha \\in\\{-1,-3, \\dfrac{1}{3}, 2\\}$, 则使函数$y=x^\\alpha$的定义域为$\\mathbf{R}$且在$(-\\infty, 0)$上单调递增的$\\alpha$值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012895": { + "id": "012895", + "content": "函数$y=\\dfrac{1}{x^2-4 x+5}$的图像关于\\bracket{20}.\n\\fourch{$y$轴对称}{原点对称}{直线$x=2$对称}{点$(2,1)$对称}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012896": { + "id": "012896", + "content": "定义运算: $a \\otimes b=\\begin{cases}b, & a \\geq b, \\\\ a, & a0$.\\\\\n(1) 当$01$;\\\\\n(2) 是否存在实数$a$、$b$($a0$, 那么$f(x)$在$(1,+\\infty)$上是\\bracket{20}.\n\\fourch{增函数}{减函数}{非单调函数}{由$a$值决定单调性}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012908": { + "id": "012908", + "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$满足$f(-2-x)=f(-2+x)$, 且当$x \\geq -2$时, 函数的解析式为$f(x)=x^2-1$, 则当$x<-2$时, 函数的解析式$f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012909": { + "id": "012909", + "content": "函数$y=f(x)$的反函数为$y=f^{-1}(x)$, 如果函数$y=f(x)$的图像过点$(2,-2)$, 那么函数$y=f^{-1}(-2 x)+1$的图像一定过点\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012910": { + "id": "012910", + "content": "函数$f(x)$对于任意实数$x$满足条件$f(x+2)=\\dfrac{1}{f(x)}$, 若$f(1)=-\\dfrac{1}{5}$, 则$f(f(3))=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012911": { + "id": "012911", + "content": "用$\\min \\{a, b\\}$表示$a$、$b$两数中的最小值. 若函数$f(x)=\\min \\{|x|,|x+t|\\}$的图像关于直线$x=-\\dfrac{1}{2}$对称, 则$t$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012912": { + "id": "012912", + "content": "已知函数$f(x)$满足: $f(1)=\\dfrac{1}{4}$, $4 f(x) f(y)=f(x+y)+f(x-y)$($x, y \\in \\mathbf{R}$), 则$f(2022)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012913": { + "id": "012913", + "content": "设函数$f(x)=\\sqrt{a x^2+b x+c}$($a<0$)的定义域为$D$, 若所有点$(s, f(t))$($s, t \\in D$)构成一个正方形区域, 则$a$的值为\\bracket{20}.\n\\fourch{$-2$}{$-4$}{$-8$}{不能确定}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012914": { + "id": "012914", + "content": "已知函数$f(x)$和$g(x)$的图像关于原点对称, 且$f(x)=x^2+x$.\\\\\n(1) 求函数$y=g(x)$的解析式;\\\\\n(2) 若$h(x)=g(x)-m \\cdot f(x)+3$在$[-1,1]$上是增函数, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012915": { + "id": "012915", + "content": "记函数$f(x)$的定义域为$D$, 若存在$x_0 \\in D$, 使$f(x_0)=x_0$成立, 则称以$(x_0, x_0)$为坐标的点为函数$f(x)$图像上的不动点.\\\\\n(1) 若函数$f(x)=\\dfrac{3 x+a}{x+b}$的图像上有两个关于原点对称的不动点, 求$a$、$b$应满足的条件;\\\\\n(2) 在 (1) 的条件下, 若$a=8$, 记$f(x)$图像上的两个不动点分别为$M$、$N$, 点$P$为函数$f(x)$图像上的另一点, 且其纵坐标$y_P>3$, 求点$P$到直线$MN$距离的最小值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012916": { + "id": "012916", + "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2 a_n-1$, $a_1=0$, 则$a_3=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012917": { + "id": "012917", + "content": "已知等比数列$\\{a_n\\}$的通项为$a_n=\\dfrac{1}{3^n}$, 则其前$n$项和$S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012918": { + "id": "012918", + "content": "已知等差数列$\\{a_n\\}$的公差为$2$, 则等差数列$\\{a_{2 n-1}\\}$的公差为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012919": { + "id": "012919", + "content": "已知数列$\\{a_n\\}$, 则$a_{99}$是数列$\\{a_{2 n-1}\\}$的第\\blank{50}项.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012920": { + "id": "012920", + "content": "设$\\{a_n\\}$为等差数列, 公差$d=\\dfrac{1}{2}$, 前$100$项之和为$145$, 则$a_1+a_3+\\cdots+a_{99}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012921": { + "id": "012921", + "content": "已知数列$\\{a_n\\}$满足: $a_{4 n-3}=1$, $a_{4 n-1}=0$, $a_{2 n}=a_n$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则$a_{2009}=$\\blank{50}, $a_{2014}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012922": { + "id": "012922", + "content": "定义``等和数列'': 在一个数列中, 如果每一项与它的后一项的和都为同一个常数, 那么这个数列叫做等和数列, 这个常数叫做该数列的公和. 已知数列$\\{a_n\\}$是等和数列, 且$a_1=2$, 公和为$5$, 那么这个数列的前$n$项和$S_n$的计算公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012923": { + "id": "012923", + "content": "数列$\\{a_n\\}$的通项公式$a_n=n \\cos \\dfrac{n \\pi}{2}+1$, 前$n$项和为$S_n$, 则$S_{2012}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012924": { + "id": "012924", + "content": "已知数列$\\{a_n\\}$满足$a_{2 n}-a_{2 n-1}=n$, $a_{2 n+1}-a_{2 n}=1$, $a_1=1$, 求通项$a_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012925": { + "id": "012925", + "content": "设$n \\in \\mathbf{N}$, $n \\geq 2$. 一个$n$行$n$列矩阵$(a_{i j})$满足$a_{11}=1$, $a_{21}=2$, 第一列从上至下构成等比数列, 每一行从左至右构成公差为$1$的等差数列.\\\\\n(1) 求该矩阵第$i$行第$j$列元素$a_{i j}$($i, j \\in[1, n] \\cap \\mathbf{N}$);\\\\\n(2) 求该矩阵主对角线上的元素之和.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012926": { + "id": "012926", + "content": "已知数列$\\{a_n\\}$与$\\{b_n\\}$满足$b_n=n a_n$($n \\in \\mathbf{N}$), 记数列$\\{a_n\\}$的前$n$项和为$S_n$, 数列$\\{b_n\\}$的前$n$项和为$T_n$.\\\\\n(1) 若数列$\\{a_n\\}$是以$1$为首项, $\\dfrac{1}{3}$为公比的等比数列, 求$T_n$;\\\\\n(2) 求证: ``$\\dfrac{T_n}{S_n}=\\dfrac{2 b_n+n}{2 a_n+n}$($n \\in \\mathbf{N}$, $n\\ge 1$)''是``$a_n=n$($n \\in \\mathbf{N}$, $n\\ge 1$)''的必要条件.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012927": { + "id": "012927", + "content": "$2$与$6$的等差中项为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012928": { + "id": "012928", + "content": "$2$与$6$的等比中项为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012929": { + "id": "012929", + "content": "已知递增的等差数列$\\{a_n\\}$满足$a_1=1, a_3=a_2^2-4$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012930": { + "id": "012930", + "content": "已知等比数列$\\{a_n\\}$, 若$\\{a_{2 n-1}\\}$公比为$4$, 则$\\{a_n\\}$公比为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012931": { + "id": "012931", + "content": "定义在全体正整数上的函数$f(n)=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$, 则$f(n+1)-f(n)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012932": { + "id": "012932", + "content": "数列$\\{a_n\\}$中, $a_1=1$, $a_n$, $a_{n+1}$是方程$x^2-(2 n+1) x+\\dfrac{1}{b_n}=0$的两个根, 数列$\\{b_n\\}$的前$n$项和$S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012933": { + "id": "012933", + "content": "已知数列$\\{a_n\\}$满足$a_{n+2}-a_n=2^n$, $a_1=1$, 则$a_{2 n-1}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012934": { + "id": "012934", + "content": "已知数列$\\{a_n\\}$满足$a_1=2$, $a_n=a_{n-1}+(2 n-1)+\\cdot+2^n$, 则通项$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012935": { + "id": "012935", + "content": "已知$10$行$10$列矩阵$\\{a_{i j}\\}$的第$i$行第$j$列元素$a_{i j}=i \\cdot j$, 则该矩阵所有元素的和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012936": { + "id": "012936", + "content": "数列$\\{a_n\\}$满足$a_{n+1}+(-1)^n a_n=2 n-1$, 则$\\{a_n\\}$的前$60$项和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012937": { + "id": "012937", + "content": "已知数列$\\{a_n\\}$为等差数列, 公差$d \\neq 0$, $\\{a_n\\}$的部分项组成下列数列: $a_{k_1}, a_{k_2}, \\cdots, a_{k_n}$恰为等比数列, 其中$k_1=1$, $k_2=5$, $k_3=17$, 求$k_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012938": { + "id": "012938", + "content": "已知数列$\\{a_n\\}$的通项$a_n=2^n$, 删去该数列的第$2,5, \\cdots, 3 k-1, \\cdots$项, 得到一个新数列$\\{b_n\\}$.\\\\\n(1) 求$\\{b_n\\}$的通项;\\\\\n(2) 求$\\{b_n\\}$的前$n$项和$S_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012939": { + "id": "012939", + "content": "计算:$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^{n+1}+2^{n+1}}{3^n+2^n}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012940": { + "id": "012940", + "content": "设数列$\\{a_n\\}$, $\\{b_n\\}$都是等差数列, 若$a_1+b_1=7$, $a_3+b_3=21$, 则$a_5+b_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012941": { + "id": "012941", + "content": "已知数列$\\{a_n\\}$满足$\\dfrac{1}{a_{n+1}}-\\dfrac{1}{a_n}=1$, $a_1=2$, 则$a_{10}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012942": { + "id": "012942", + "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=2^n$, 则通项$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012943": { + "id": "012943", + "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=3 a_n+3$, 则通项$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012944": { + "id": "012944", + "content": "$\\displaystyle\\lim _{n \\to \\infty}(1-2 x)^n$存在, 则$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012945": { + "id": "012945", + "content": "设$a$是实数, 若极限$\\displaystyle\\lim _{n \\to \\infty}(a n-\\dfrac{n^2+1}{n+1})$存在, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012946": { + "id": "012946", + "content": "无穷等比数列$\\{a_n\\}$的各项和为$3$, 数列$\\{b_n\\}$满足$b_n=a_{2 n-1}+a_{2 n}$, 则$\\{b_n\\}$的各项和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012947": { + "id": "012947", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=2$且$3 a_{n+1}+2S_n=3$, 求数列$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012948": { + "id": "012948", + "content": "已知数列$\\{a_n\\}$满足: $a_1=\\dfrac{3}{2}$且$a_n=\\dfrac{3 n a_{n-1}}{2 a_{n-1}+n-1}$, $n \\geq 2$. 数列$\\{b_n\\}$满足$b_n=1-\\dfrac{n}{a_n}$.\\\\\n(1) 求证: $\\{b_n\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012949": { + "id": "012949", + "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=q a_n+2$, $a_1=2$, 其中实常数$q>0$.\\\\\n(1) 求通项$a_n$;\\\\\n(2) 计算$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{a_{n+1}}{a_n}$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012950": { + "id": "012950", + "content": "数列$1, x, y, 2$是等差数列, 则$y-x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012951": { + "id": "012951", + "content": "已知$x$是实数, 数列$\\{a_n\\}$的通项为$a_n=\\log _x n$. 若$\\{a_n\\}$递增, 则$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012952": { + "id": "012952", + "content": "计算:$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^{-n}+2^{-n}}{3^{1-n}+2^{1-n}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012953": { + "id": "012953", + "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2 a_n+3$, $a_1=1$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012954": { + "id": "012954", + "content": "设等比数列$\\{a_n\\}$的公比$q=-\\dfrac{1}{2}$, 且$\\lim _{n \\to \\infty}(a_1+a_3+\\cdots+a_{2 n-1})=\\dfrac{8}{3}$, 则$a_1=$", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012955": { + "id": "012955", + "content": "无穷等比数列$\\{a_n\\}$的各项和为 1 , 则首项$a_1$的取值范围为", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012956": { + "id": "012956", + "content": "若对任意的正整数$n, a_1 a_2 \\cdots a_n=2^{n+1}$, 则$a_n=$", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012957": { + "id": "012957", + "content": "设$a$是一个实数, 若$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^n}{3^{n+1}+a^n}=\\dfrac{1}{3}$, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012958": { + "id": "012958", + "content": "已知数列$\\{a_n\\}$的递推式为$a_{n+1}=2 a_n+n-1$, $a_1=1$, 利用$b_n=a_n+n$可求得$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012959": { + "id": "012959", + "content": "数列$\\{a_n\\}$的前$n-1$项之和$S_{n-1}=a_n$($n \\geq 2$, $n \\in \\mathbf{N}$), $a_1=1$, 则通项$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012960": { + "id": "012960", + "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n$, $a_1=1$.\\\\\n(1) 若数列$\\{a_n\\}$是公差为$d$的等差数列, 计算: $\\displaystyle\\lim_{n\\to\\infty} \\dfrac{S_{2 n}}{S_n}$;\\\\\n(2) 若数列$\\{a_n\\}$是公比为$q$的等比数列, 计算: $\\displaystyle\\lim_{n\\to\\infty} \\dfrac{S_{2 n}}{S_n}$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012961": { + "id": "012961", + "content": "已知$x$轴上有一点列$P_k(x_k, 0)$($k=1,2,3, \\cdots$), 满足$x_1=0$, $x_2=1$. 且对任意的正整数$n$, $\\overrightarrow{P_n P_{n+2}}=\\lambda \\overrightarrow{P_{n+2} P_{n+1}}$, 其中常数$\\lambda \\neq-1$.\\\\\n(1) 设$a_n=x_{n+1}-x_n$, 求证: $\\{a_n\\}$是等比数列;\\\\\n(2) 求通项$x_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012962": { + "id": "012962", + "content": "已知数列$\\{a_n\\}$中, $\\dfrac{a_{n+1}}{a_n}=2$, $a_3=8$, 则$a_1=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012963": { + "id": "012963", + "content": "在等差数列$\\{a_n\\}$中, $a_3+a_7=37$, 则$a_2+a_8=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012964": { + "id": "012964", + "content": "已知数列$\\{a_n\\}$的通项$a_n=10-n$, 则集合$\\{n | a_n>0\\}$共有\\blank{50}个元素.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012965": { + "id": "012965", + "content": "已知等比数列$\\{a_n\\}$中$a_2=1$, 则其前 3 项的和$S_3$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012966": { + "id": "012966", + "content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac{n-\\sqrt{60}}{n-\\sqrt{59}}$($1 \\leq n \\leq 100$), 则此数列中最大项为第\\blank{50}项, 最小项为第\\blank{50}项.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012967": { + "id": "012967", + "content": "已知数列$\\{a_n\\}$是以$-2$为公差的等差数列, $S_n$是其前$n$项和, 若$S_7$是数列$\\{S_n\\}$中的唯一最大项, 则数列$\\{a_n\\}$的首项$a_1$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012968": { + "id": "012968", + "content": "等差数列$\\{a_n\\}$中, $S_n$为前$n$项和, 且$S_6S_8$, 给出下列命题: \\textcircled{1} 数列$\\{a_n\\}$中前$7$项是递增的, 从第$8$项开始递减; \\textcircled{2} $S_9$一定小于$S_6$; \\textcircled{3} $a_1$是各项中的最大的; \\textcircled{4} $S_7$不一定是$\\{S_n\\}$中最大项. 其中正确的序号是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012969": { + "id": "012969", + "content": "设$\\{a_n\\}$是公比为实数$q$的等比数列, $|q|>1$, 令$b_n=a_n+1$($n=1,2, \\cdots$), 若数列$\\{b_n\\}$有连续四项在集合$\\{-53,-23,19,37,82\\}$中, 则$q=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012970": { + "id": "012970", + "content": "设等差数列$\\{a_n\\}$的首项$a_1$及公差$d$都为整数, 前$n$项和为$S_n$. 若$a_1 \\geq 6$, $a_{11}>0$, $S_{14} \\leq 77$, 求所有可能的数列$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012971": { + "id": "012971", + "content": "已知等差数列$\\{a_n\\}$通项$a_n=1000 n$, 等比数列$\\{b_n\\}$通项$b_n=2^n$.\\\\\n(1) 求数列$\\{a_n-b_n\\}$的最大项;\\\\\n(2) 解不等式: $a_n>b_n$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012972": { + "id": "012972", + "content": "已知等差数列$\\{a_n\\}$中$a_1=2$, 公差是正整数$d$, 等比数列$\\{b_n\\}$中, $b_1=a_1$, $b_2=a_2$.\\\\\n(1) 试给出一个$d$的值, 使得$n \\geq 3$时, $b_n$都不在$\\{a_n\\}$中, 并说明理由;\\\\\n(2) 判断$d=10$时, 是否数列$\\{b_n\\}$中的所有项都是$\\{a_n\\}$中的项, 并证明你的结论.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012973": { + "id": "012973", + "content": "已知三角形$ABC$中, $A, B, C$成等差数列, 则$B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012974": { + "id": "012974", + "content": "设公比为$q$($q>0$)的等比数列$\\{a_n\\}$的前$n$项和为$S_n$. 若$S_2=3 a_2+2$, $S_4=3 a_4+2$, 则$q=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012975": { + "id": "012975", + "content": "已知各项非零的等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{10}-S_7=k a_9$, 则实数$k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012976": { + "id": "012976", + "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2^n a_n$, $a_1=1$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012977": { + "id": "012977", + "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(\\dfrac{3}{4})^{n-1}[(\\dfrac{3}{4})^{n-1}-1]$, 则数列$\\{a_n\\}$的最大项的值为\\blank{50}, 最小项的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012978": { + "id": "012978", + "content": "数列$\\{a_n\\}$的通项公式为$a_n=6 n-3$, 数列$\\{b_n\\}$的通项公式为$b_n=5 n-4$, 若$a_n \\leq 1000$, $b_n \\leq 1000$, 由数列$\\{a_n\\}$与数列$\\{b_n\\}$中共有的项构成数列$\\{c_n\\}$, 则数列$\\{c_n\\}$中共有\\blank{50}项.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012979": { + "id": "012979", + "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_4$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012980": { + "id": "012980", + "content": "已知等差数列$\\{a_n\\}$的首项及公差均为正数, 令$b_n=\\sqrt{a_n}+\\sqrt{a_{2012-n}}$($n=1,2,3, \\cdots, 2011$). 当$b_k$是数列$\\{b_n\\}$的最大项时, $k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012981": { + "id": "012981", + "content": "在等差数列$\\{a_n\\}$中, $a_3+a_4+a_5=84$, $a_9=73$, 对任意$m \\in \\mathbf{N}$, $n\\ge 1$, 将数列$\\{a_n\\}$中落入区间$(9^m, 9^{2 m})$内的项的个数记为$b_m$, 则数列$b_m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012982": { + "id": "012982", + "content": "已知数列$\\{a_n\\}$满足$a_1=m$, $m \\in \\mathbf{N}$, $n\\ge 1$, $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2}, & a_n \\text {是偶数}, \\\\ 3 a_n+1, & a_n \\text{是奇数}.\\end{cases}$ $a_6=1$, 则$m$的所有可能值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012983": { + "id": "012983", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_2 a_n=S_2+S_n$对一切正整数$n$都成立.\\\\\n(1) 求$a_1$, $a_2$的值;\\\\\n(2) 设$a_1>0$, 数列$\\{\\lg \\dfrac{10 a_1}{a_n}\\}$的前$n$项和为$T_n$, 当$n$为何值时, $T_n$最大? 并求出$T_n$的最大值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012984": { + "id": "012984", + "content": "已知等差数列$\\{a_n\\}$中$a_1=2$, 公差是正整数$d$, 等比数列$\\{b_n\\}$中, $b_1=a_1$, $b_2=a_2$. 是否存在$d$, 使得数列$\\{b_n\\}$中的所有项都是$\\{a_n\\}$中的项? 若存在, 求出所有这样的$d$; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012985": { + "id": "012985", + "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=a_n^2-2$, $a_1=2$, 则$a_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012986": { + "id": "012986", + "content": "已知等比数列的前$n$项和为$3^n-1$, 则公比为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012987": { + "id": "012987", + "content": "已知数列$\\{a_n\\}$对任意正整数$p, q$满足$a_{p+q}=a_p+a_q$, 且$a_2=-6$, 那么$a_{10}$等于\\bracket{20}.\n\\fourch{$-165$}{$-33$}{$-30$}{$-21$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012988": { + "id": "012988", + "content": "已知等差数列$\\{a_n\\}$的前$n$项和$S_n$满足$S_5=5$, $S_{10}=15$, 则$S_{15}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012989": { + "id": "012989", + "content": "已知$a_n=2^n+3^n$, $b_n=a_{n+1}+k a_n$, 若$\\{b_n\\}$是等比数列, 则$k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012990": { + "id": "012990", + "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+k n+2$. 若对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 有$a_{n+1}>a_n$恒成立, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012991": { + "id": "012991", + "content": "若$f(n)$为$n^2+1$($n \\in \\mathbf{N}$, $n\\ge 1$)的各位数字之和, 如$14^2+1=197$, $1+9+7=17$, 则$f(14)=17$; 记$f_1(n)=f(n), f_2(n)=f(f_1(n)), \\cdots, f_{k+1}(n)=f(f_k(n))$, $k \\in \\mathbf{N}$, $k\\ge 1$. 则$f_{2015}(8)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012992": { + "id": "012992", + "content": "若在由正整数构成的无穷数列$\\{a_n\\}$中, 对任意的正整数$n$, 都有$a_n \\leq a_{n+1}$, 且对任意的正整数$k$, 该数列中恰有$2 k-1$个$k$, 则$a_{2022}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012993": { + "id": "012993", + "content": "设$t$是实数, 数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_n=\\dfrac{1}{2^n}$.\\\\\n(1) 若对一切正整数$n$, $S_n>a_n+t$恒成立, 求$t$的取值范围;\\\\\n(2) 若存在正整数$n$, 使$S_n>a_n+t$成立, 求$t$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012994": { + "id": "012994", + "content": "数列$\\{a_n\\}$满足$a_n=3 a_{n-1}+3^n-1$($n \\geq 2$), $a_3=95$.\\\\\n(1) 求$a_1, a_2$的值;\\\\\n(2) 是否存在实数$t$, 使得数列$\\{\\dfrac{a_n+t}{3^n}\\}$是等差数列? 若有, 求出$t$的值; 若没有, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012995": { + "id": "012995", + "content": "设常数$a \\neq \\dfrac{1}{4}$, 数列$\\{a_n\\}$的首项$a_1=a$, $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2},& n \\text {为偶数}, \\\\ a_n+\\dfrac{1}{4},& n \\text{为奇数}.\\end{cases}$ 记$b_n=a_{2 n-1}-\\dfrac{1}{4}$.\\\\\n(1) 求$a_2$, $a_3$的值;\\\\\n(2) 判断$\\{b_n\\}$是否为等比数列, 并证明你的结论.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012996": { + "id": "012996", + "content": "$1, x, y$既是等差数列, 又是等比数列, 则$x+y=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012997": { + "id": "012997", + "content": "已知等比数列$\\{a_n\\}$中, $a_1 a_2=1$, $a_2 a_3=2$, 则$a_n a_{n+1}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012998": { + "id": "012998", + "content": "设数列$\\{a_n\\}$, 则``$a_{n+2}-a_n$是常数''是``$a_{n+1}-a_n$是常数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "012999": { + "id": "012999", + "content": "已知等比数列$\\{a_n\\}$为递增数列, 且$a_5^2=a_{10}$, $2(a_n+a_{n+2})=5 a_{n+1}$, 则数列的通项公式$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013000": { + "id": "013000", + "content": "若数列$\\{a_n\\}$满足$a_1=\\dfrac{1}{3}$, $a_n-a_{n-1}=\\dfrac{1}{n(n+2)}$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013001": { + "id": "013001", + "content": "若数列$\\{a_n\\}$满足$a_1=1$, $a_2=2$, 且$a_n=\\dfrac{a_{n-1}}{a_{n-2}}$($n \\geq 3$), 则$a_{4015}$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013002": { + "id": "013002", + "content": "若$f(\\dfrac{1}{2}+x)+f(\\dfrac{1}{2}-x)=2$对任意的实数$x$成立, 则$f(\\dfrac{1}{3000})+f(\\dfrac{2}{3000})+f(\\dfrac{3}{3000})+\\cdots+f(\\dfrac{2999}{3000})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013003": { + "id": "013003", + "content": "对数列$\\{a_n\\}$, 定义$\\{\\Delta^1 a_n\\}$为数列$\\{a_n\\}$的一阶差分数列, 其中$\\Delta^1 a_n=a_{n+1}-a_n$, $n \\in \\mathbf{N}$, $n \\ge 1$. 对正整数$k$, 定义$\\{\\Delta^k a_n\\}$为$\\{a_n\\}$的$k$阶差分数列, 其中$\\Delta^{k+1} a_n=\\Delta^k a_{n+1}-\\Delta^k a_n=\\Delta^1(\\Delta^k a_n)$. 已知数列$\\{a_n\\}$的通项公式$a_n=n^2$, 则$\\Delta^1 a_n=$\\blank{50}, $\\Delta^2 a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013004": { + "id": "013004", + "content": "在两个不等的正数$a, b$之间插入$n$个正数$x_1, x_2, \\cdots, x_n$, 若$a, x_1, x_2, \\cdots, x_n, b$成等差数列, 则$\\dfrac{a+b}{2}=\\dfrac{x_1+x_2+\\cdots+x_n}{n}$; 相应地: 若$a, x_1, x_2, \\cdots, x_n, b$成等比数列, 则可得到的结论是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013005": { + "id": "013005", + "content": "设$a_1=2$, $a_{n+1}=\\dfrac{2}{a_n+1}$, $b_n=|\\dfrac{a_n+2}{a_n-1}|$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则数列$\\{b_n\\}$的通项公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013006": { + "id": "013006", + "content": "设$S_n=1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{n}$, $f(n)=S_{2 n+1}-S_{n+1}$.\\\\\n(1) 判断数列$f(n)$的单调性;\\\\\n(2) 试确定实数$t$的范围, 使得对于$n \\in \\mathbf{N}$, $n>1$, 不等式$f(n)>t^2-\\dfrac{11}{20} t^{-2}$恒成立.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013007": { + "id": "013007", + "content": "已知数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{2}$, $2 a_{n+1}-a_n=n$.\\\\\n(1) 令$b_n=a_{n+1}-a_n-1$, 求证:$\\{b_n\\}$是等比数列.\\\\\n(2) 求数列$\\{a_n\\}$的通项公式;\\\\\n(3) 是否存在实数$\\lambda$, 使得数列$\\{a_{n+1}-\\lambda a_n\\}$为等差数列? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013008": { + "id": "013008", + "content": "终边落在$y$轴的正半轴上的所有角构成的集合为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013009": { + "id": "013009", + "content": "已知$\\sin \\alpha=\\dfrac{1}{2}+\\cos \\alpha$, 则$\\sin 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013010": { + "id": "013010", + "content": "若$\\tan \\theta+\\dfrac{1}{\\tan \\theta}=4$, 则$\\sin 2 \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013011": { + "id": "013011", + "content": "若扇形的圆心角为$120^{\\circ}$, 半径为$5$, 则扇形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013012": { + "id": "013012", + "content": "若$0 \\leq \\alpha \\leq 2 \\pi$, $\\sin \\alpha>\\sqrt{3} \\cos \\alpha$, 则$\\alpha$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013013": { + "id": "013013", + "content": "满足$\\arcsin 2 x>\\arcsin (1-x)$的$x$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013014": { + "id": "013014", + "content": "$\\triangle ABC$中, $A=60^{\\circ}$, $b=1$, $S_{\\triangle ABC}=\\sqrt{3}$, 则$\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013015": { + "id": "013015", + "content": "若$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, 0)$, $\\cos (\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{1}{3}$, $\\cos (\\dfrac{\\pi}{4}-\\dfrac{\\beta}{2})=\\dfrac{\\sqrt{3}}{3}$, 则$\\cos (\\alpha+\\dfrac{\\beta}{2})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013016": { + "id": "013016", + "content": "满足方程$\\sin (2 x+\\dfrac{\\pi}{4})=\\cos (\\dfrac{\\pi}{6}-x)$的最小的正角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013017": { + "id": "013017", + "content": "设$\\triangle ABC$的内角$A, B, C$所对的边为$a, b, c$, 则下列命题正确的是\\blank{50}.\\\\\n\\textcircled{1} 若$a b>c^2$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{2} 若$a+b>2 c$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{3} 若$a^3+b^3=c^3$, 则$C<\\dfrac{\\pi}{2}$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013018": { + "id": "013018", + "content": "已知$\\dfrac{3 \\pi}{4}<\\alpha<\\pi$, $\\tan \\alpha+\\cot \\alpha=-\\dfrac{10}{3}$.\\\\\n(1) 求$\\tan \\alpha$的值;\\\\\n(2) 求$\\dfrac{5 \\sin ^2 \\dfrac{\\alpha}{2}+8 \\sin \\dfrac{\\alpha}{2} \\cos \\dfrac{\\alpha}{2}+11 \\cos ^2 \\dfrac{\\alpha}{2}-8}{\\sqrt{2} \\sin (\\alpha-\\dfrac{\\pi}{2})}$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013019": { + "id": "013019", + "content": "已知$x_1, x_2$是方程$x^2-x \\sin \\theta+\\cos \\theta=0$的两个根, $0<\\theta<\\pi$, 求$\\arctan x_1+\\arctan x_2$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013020": { + "id": "013020", + "content": "已知函数$f(x)=2 \\cos (\\omega x+\\dfrac{\\pi}{6})$(其中$\\omega>0$, $x \\in \\mathbf{R}$)的最小正周期为$10 \\pi$.\\\\\n(1) 求$\\omega$的值;\\\\\n(2) 设$\\alpha$、$\\beta \\in[0, \\dfrac{\\pi}{2}]$, $f(5 \\alpha+\\dfrac{5}{3} \\pi)=-\\dfrac{6}{5}$, $f(5 \\beta-\\dfrac{5}{6} \\pi)=\\dfrac{16}{17}$, 求$\\cos (\\alpha+\\beta)$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013021": { + "id": "013021", + "content": "若$\\tan (\\dfrac{\\pi}{4}-\\alpha)=3$, 则$\\cot \\alpha$等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013022": { + "id": "013022", + "content": "已知$\\cos 2 \\alpha=\\dfrac{1}{5}$, 则$\\sin ^4 \\alpha-\\cos ^4 \\alpha$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013023": { + "id": "013023", + "content": "若$\\tan \\theta=2$, 则$\\sin 2 \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013024": { + "id": "013024", + "content": "若$\\alpha, \\beta$为锐角, 且$\\cos \\alpha=\\dfrac{4}{5}$, $\\cos (\\alpha+\\beta)=\\dfrac{3}{5}$, 则$\\sin \\beta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013025": { + "id": "013025", + "content": "若$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且$\\tan \\alpha$、$\\tan \\beta$是方程$x^2+3 \\sqrt{3} x+4=0$的两个相异实根, 则$\\alpha+\\beta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013026": { + "id": "013026", + "content": "设$a, b, c \\in \\mathbf{R}$, 且$\\cos 2 x=a \\cos ^2 x+b \\cos x+c$对任意$x \\in \\mathbf{R}$恒成立, 则$a^2+b^2+c^2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013027": { + "id": "013027", + "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$, 若$A=30^{\\circ}$, $C=135^{\\circ}$, $a=1$, 则$c=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013028": { + "id": "013028", + "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$, 若$a=\\dfrac{\\sqrt{5}}{2} b$, $A=2B$, 则$\\cos B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013029": { + "id": "013029", + "content": "若$0\\dfrac 3\\pi x$}{$\\sin x<\\dfrac{4}{\\pi^2} x^2$}{$\\sin x>\\dfrac{4}{\\pi^2} x^2$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013030": { + "id": "013030", + "content": "有一道解三角形的问题, 缺少一个条件, 具体如下: ``在$\\triangle ABC$中, 已知$a=\\sqrt{3}$, $B=45^{\\circ}$, 求角$A$的大小.''\n经推断, 缺少的条件为三角形一边的长度, 且答案提示$A=60^{\\circ}$, 则所缺条件为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013031": { + "id": "013031", + "content": "已知$\\sin \\alpha$、$\\cos \\alpha$是方程$8 x^2+6 k x+2 k+1=0$的两个相异实根, 求实数$k$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013032": { + "id": "013032", + "content": "已知函数$f(x)=-\\sqrt{3} \\sin ^2 x+\\sin x \\cos x$.\\\\\n(1) 求$f(\\dfrac{25 \\pi}{6})$的值;\\\\\n(2) 设$\\alpha \\in(0, \\dfrac{\\pi}{2})$, 且$f(\\alpha)=\\dfrac{1}{4}-\\dfrac{\\sqrt{3}}{2}$, 求$\\sin 2 \\alpha$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013033": { + "id": "013033", + "content": "某兴趣小组测量电视塔$AE$的高度$H$(单位: $\\text{m}$), 如示意图, 垂直放置的标杆$BC$的高度$h=4 \\text{m}$, 仰角$\\angle ABE=\\alpha$, $\\angle ADE=\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (1,0) node [above left] {$B$} coordinate (B);\n\\draw (1,0.6) node [above] {$C$} coordinate (C);\n\\draw ($(D)!2.5!(B)$) node [below right] {$A$} coordinate (A);\n\\draw ($(D)!2.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw (D) -- (A) (B) -- (C) (A) -- (E);\n\\draw [dashed] (D) -- (E) (B) -- (E);\n\\draw (B) ++ (0,-0.3) --++ (0,0.2) (A) ++ (0,-0.3) --++ (0,0.2);\n\\draw [<->] (B) ++ (0,-0.2) --++ (1.5,0) node [midway, fill = white] {$d$};\n\\draw (B) pic [draw, \"$\\alpha$\", angle eccentricity = 1.3] {angle = A--B--E};\n\\draw (D) pic [draw, \"$\\beta$\", angle eccentricity = 1.3] {angle = A--D--E};\n\\end{tikzpicture}\n\\end{center}\n(1) 该小组已经测得一组$\\alpha$、$\\beta$的值, $\\tan \\alpha=1.24$, $\\tan \\beta=1.20$, 请据此算出$H$的值;\\\\\n(2) 该小组分析若干测得的数据后, 认为适当调整标杆到电视塔的距离$d$(单位: $\\text{m}$), 使$\\alpha$与$\\beta$之差较大, 可以提高测量精确度. 若电视塔的实际高度为$125 \\text{m}$, 试问$d$为多少时, $\\alpha-\\beta$最大?", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013034": { + "id": "013034", + "content": "\"$x=2 k \\pi+\\dfrac{\\pi}{4}$($k \\in \\mathbf{Z}$)''是``$\\tan x=1$''成立的\\bracket{20}.\n\\twoch{必要非充分条件}{充分非必要条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013035": { + "id": "013035", + "content": "函数$y=\\sin (x+\\dfrac{\\pi}{3}) \\sin (x+\\dfrac{\\pi}{2})$的最小正周期$T=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013036": { + "id": "013036", + "content": "函数$y=2 \\sin (\\dfrac{\\pi}{3}-2 x)$的单调递增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013037": { + "id": "013037", + "content": "函数$2 \\sin x=1$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013038": { + "id": "013038", + "content": "函数$y=2 \\arccos (x-2)$($1 \\leq x<2$)的反函数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013039": { + "id": "013039", + "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0 \\leq \\varphi \\leq \\pi$)是$\\mathbf{R}$上的偶函数, 其图像关于点$(\\dfrac{3 \\pi}{4}, 0)$对称, 则$\\varphi$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013040": { + "id": "013040", + "content": "已知函数$y=f(x)$的周期为$2 \\pi$, 当$x \\in[0,2 \\pi)$时, $f(x)=\\sin \\dfrac{x}{2}$, 那么方程$f(x)=\\dfrac{1}{2}$的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013041": { + "id": "013041", + "content": "对函数$y=\\sin x+\\cos x$, 有下列命题: \\textcircled{1} 若$x \\in[0, \\dfrac{\\pi}{2}]$, 则$y \\in[0, \\sqrt{2}]$; \\textcircled{2} 与函数$y=\\sin x-\\cos x$的图像关于直线$x=k \\pi+\\dfrac{\\pi}{2}$($k \\in \\mathbf{Z}$)对称; \\textcircled{3} 在区间$[\\dfrac{\\pi}{4}, \\dfrac{5 \\pi}{4}]$上单调递减; \\textcircled{4} 其图像可由$y=\\sqrt{2} \\sin 2 x$的图像纵坐标不变横坐标变为原来的$\\dfrac{1}{2}$后再向左平移$\\dfrac{\\pi}{4}$个单位得到. 正确的命题是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013042": { + "id": "013042", + "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$. 若$C=60^{\\circ}$, $c=2$, 且$\\triangle ABC$的面积为$\\sqrt{3}$, 求$a, b$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013043": { + "id": "013043", + "content": "已知函数$f(x)=2 \\sin ^2(\\dfrac{\\pi}{4}+x)-\\sqrt{3} \\cos 2 x$, $x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$. 若不等式$|f(x)-m|<2$在$x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$上恒成立, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013044": { + "id": "013044", + "content": "已知函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{6})+\\sin (\\omega x-\\dfrac{\\pi}{6})-2 \\cos ^2 \\dfrac{\\omega x}{2}$, $x \\in \\mathbf{R}$(其中$\\omega>0$).\\\\\n(1) 求函数$f(x)$的值域;\\\\\n(2) 若对任意的$a \\in \\mathbf{R}$, 函数$y=f(x)$, $x \\in(a, a+\\pi]$的图像与直线$y=-1$有且仅有两个不同的交点, 试确定$\\omega$的值(不必证明), 并求函数$y=f(x)$, $x \\in \\mathbf{R}$的单调增区间.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013045": { + "id": "013045", + "content": "``$\\theta=\\dfrac{2 \\pi}{3}$''是``$\\tan \\theta=2 \\cos (\\dfrac{\\pi}{2}+\\theta)$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013046": { + "id": "013046", + "content": "函数$y=\\sin (2 x+\\dfrac{\\pi}{6})-\\cos (2 x+\\dfrac{\\pi}{3})$的最小正周期和最大值分别为\\bracket{20}.\n\\fourch{$\\pi, \\sqrt{3}$}{$\\pi, \\sqrt{2}$}{$2 \\pi, \\sqrt{3}$}{$2 \\pi, \\sqrt{2}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013047": { + "id": "013047", + "content": "已知函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$)的最小正周期为$\\pi$, 则该函数的图像\\bracket{20}.\n\\twoch{关于点$(\\dfrac{\\pi}{3}, 0)$对称}{关于直线$x=\\dfrac{\\pi}{4}$对称}{关于点$(\\dfrac{\\pi}{4}, 0)$对称}{关于直线$x=\\dfrac{\\pi}{3}$对称}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013048": { + "id": "013048", + "content": "函数$f(x)=3 \\sin (2 x-\\dfrac{\\pi}{3})$的图像为$C$. \\textcircled{1} 图像$C$关于直线$x=\\dfrac{11}{12} \\pi$对称; \\textcircled{2} 函数$f(x)$在区间$(-\\dfrac{\\pi}{12}, \\dfrac{5 \\pi}{12})$内是增函数; \\textcircled{3} 由$y=3 \\sin 2 x$的图像向右平移$\\dfrac{\\pi}{3}$个单位长度可以得到图像$C$. 以上三个论断中, 正确论断的个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013049": { + "id": "013049", + "content": "函数$f(x)=\\sqrt{3} \\sin x+\\sin (\\dfrac{\\pi}{2}+x)$的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013050": { + "id": "013050", + "content": "由函数$y=2 \\sin 3 x$, $x \\in[\\dfrac{\\pi}{6}, \\dfrac{5 \\pi}{6}]$与$y=2$($x \\in \\mathbf{R}$)的图像围成一个封闭图形, 这个封闭图形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013051": { + "id": "013051", + "content": "若函数$f(x)=a^2 \\sin 2 x+(a-2) \\cos 2 x$的图像关于直线$x=-\\dfrac{\\pi}{8}$对称, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013052": { + "id": "013052", + "content": "以下四个命题中, 正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\cos \\alpha=\\cos \\beta$, 则$\\alpha-\\beta=2 k \\pi$($k$是某个整数); \\textcircled{2} 函数$y=2 \\cos (2 x+\\dfrac{\\pi}{3})$的图像关于点$(\\dfrac{\\pi}{12}, 0)$对称; \\textcircled{3} 函数$y=\\sin |x|$是周期函数, 且$2 \\pi$是它的一个周期; \\textcircled{4} 函数$y=\\cos (\\sin x)$是偶函数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013053": { + "id": "013053", + "content": "已知$-\\dfrac{\\pi}{2}0$.\\\\\n(1) 求函数$y=f(x)$的值域;\\\\\n(2) 若$f(x)$在区间$[-\\dfrac{3 \\pi}{2}, \\dfrac{\\pi}{2}]$上为增函数, 求$\\omega$的最大值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013055": { + "id": "013055", + "content": "函数$f(x)=6 \\cos ^2 \\dfrac{\\omega x}{2}+\\sqrt{3} \\sin \\omega x-3$($\\omega>0$)在一个周期内的图像如图所示, $A$为图像的最高点, $B$、$C$为图像与$x$轴的交点, 且$\\triangle ABC$为正三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-2,0) -- (7.5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:7, samples = 100] plot (\\x,{2*sqrt(3)*sin(45*\\x+60)});\n\\draw (-4/3,0) node [above left] {$B$} coordinate (B);\n\\draw (2/3,{2*sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw (8/3,0) node [above right] {$C$} coordinate (C);\n\\draw (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\omega$的值及函数$f(x)$的值域;\\\\\n(2) 若$f(x_0)=\\dfrac{8 \\sqrt{3}}{5}$, 且$x_0 \\in(-\\dfrac{10}{3}, \\dfrac{2}{3})$, 求$f(x_0+1)$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013056": { + "id": "013056", + "content": "若直线$l$过点$(3,4)$, 且$(1,2)$是它的一个法向量, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013057": { + "id": "013057", + "content": "若$\\overrightarrow {d}=(2,-1)$是直线$l$的一个方向向量, 则直线$l$的倾斜角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013058": { + "id": "013058", + "content": "已知直线$l_1: x+a y+2=0$和$l_2:(a-2) x+3 y+6 a=0$, 则$l_1\\parallel l_2$的充要条件是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013059": { + "id": "013059", + "content": "直线$a x+b y-a b=0$($a>0$, $b>0$)的倾斜角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013060": { + "id": "013060", + "content": "直线$l$上有两点$M(t-1,2)$, $N(t-3, t^2-4 t+4)$, 则直线$l$的倾斜角的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013061": { + "id": "013061", + "content": "光线从点$M(-2,1)$出发, 先经$x$轴反射, 然后再经$y$轴反射后到达$N(-1,2)$, 则光线从点$M$到点$N$所经过的路程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013062": { + "id": "013062", + "content": "将直线$l_1: n x+y-n=0$, $l_2: x+n y-n=0$($n \\in \\mathbf{N}$, $n \\geq 2$), $x$轴及$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013063": { + "id": "013063", + "content": "若实数$a, b, c$成等差数列, 点$P(-1,0)$在动直线$l: a x+b y+c=0$上的射影为$M$, 点$N(0,3)$, 则线段$MN$的长度的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013064": { + "id": "013064", + "content": "已知一直线$l$被两条平行直线$l_1: 3 x+4 y-7=0$和$l_2: 3 x+4 y+8=0$所截得的线段长为$\\dfrac{15}{4}$, 且直线$l$经过点$(2,3)$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013065": { + "id": "013065", + "content": "已知抛物线$C: y^2=4 x$的焦点为$F$, 过点$K(-1,0)$的直线$l$与$C$相交于$A, B$两点, 点$A$关于$x$轴的对称点为$D$.\\\\\n(1) 证明: 点$F$在直线$BD$上;\\\\\n(2) 设$\\overrightarrow{FA} \\cdot \\overrightarrow{FB}=\\dfrac{8}{9}$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013066": { + "id": "013066", + "content": "已知集合$M$是平面直角坐标系中方程为$x-2 k y+k^2=0$($k \\in \\mathbf{R}$)的直线的集合, 集合$S$是满足以下条件的点的集合: 对于集合$S$中的每一个点, 集合$M$中有且仅有一条直线经过该点.\\\\\n(1) 判断下列直线是否为集合$M$中的直线: $l_1: x-y+1=0$, $l_2: x-2 y+1=0$;\\\\\n(2) 判断下列各点是否为集合$S$中的点:$D(2,1)$, $E(1,1)$;\\\\\n(3) 求集合$S$中的点的轨迹方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013067": { + "id": "013067", + "content": "直线$l$过点$P(1,1)$, 且其一个方向向量与向量$(2,3)$垂直, 则$l$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013068": { + "id": "013068", + "content": "设$\\alpha$是直线的倾斜角, 且$\\cos \\alpha=-\\dfrac{1}{5}$, 则$\\alpha$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013069": { + "id": "013069", + "content": "已知三条直线$y=3 x+2$, $2 x+y+3=0$, $k x+y=0$. 若它们不能围成一个三角形, 则$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013070": { + "id": "013070", + "content": "点$P(-2,0)$关于直线$x-y-2=0$的对称点的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013071": { + "id": "013071", + "content": "已知直线$l_1$的斜率是$\\dfrac{1}{2}$, 直线$l_2$的倾斜角是$l_1$的倾斜角的$2$倍, 则$l_2$的斜率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013072": { + "id": "013072", + "content": "过直线$l_1: 2 x+3 y-5=0$与直线$l_2: 3 x-2 y-3=0$的交点$P$, 且平行于直线$2 x+y-3=0$的直线的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013073": { + "id": "013073", + "content": "直线$l$被两直线$l_1: x-3 y+10=0$和$l_2: 2 x+y+8=0$所截得的线段的中点为$P(0,1)$, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013074": { + "id": "013074", + "content": "过直线$l: 3 x+4 y-5=0$上的一点$P$向圆$(x-3)^2+(y-4)^2=4$作两条切线$l_1, l_2$. 设$l_1$与$l_2$的夹角为$\\theta$, 则$\\theta$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013075": { + "id": "013075", + "content": "已知两条直线$l_1: y=x$和$l_2: a x-y=0$, 其中$a$为实数, 当这两条直线的夹角在$(0, \\dfrac{\\pi}{12})$内变动时, 实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013076": { + "id": "013076", + "content": "设$f(x, y)=A x+B y+C$, 这里$A, B, C$是常数, $A, B$不全为零. 若点$P(x_0, y_0)$不在直线$f(x, y)=0$上, 则曲线$f(x, y)-f(x_0, y_0)=0$表示\\bracket{20}.\n\\twoch{不过点$P$但平行于$l$的直线}{过点$P$且垂直于$l$的直线}{过点$P$且平行于$l$的直线}{不过点$P$但垂直于$l$的直线}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013077": { + "id": "013077", + "content": "已知直线$l: 5 x+2 y+3=0$.\\\\\n(1) 求直线$l_1: 3 x+7 y-13=0$与$l$所成的角的大小;\\\\\n(2) 若$l_2$经过点$P(2,1)$, 且与$l$的夹角等于$45^{\\circ}$, 求直线$l_2$的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013078": { + "id": "013078", + "content": "若$A, B$是抛物线$y^2=4 x$上的不同两点, 弦$AB$(不平行于$y$轴)的垂直平分线与$x$轴相交于点$P$, 则称弦$AB$是点$P$的一条``相关弦''. 已知当$x>2$时, 点$P(x, 0)$存在无穷多条``相关弦''. 现给定$x_0>2$.\\\\\n(1) 证明: 点$P(x_0, 0)$的所有``相关弦''的中点的橫坐标相同;\\\\\n(2) 当$x_0$取定时, 点$P(x_0, 0)$的所有``相关弦''的弦长是否存在最大值? 若存在, 求其最大值(用$x_0$表示); 若不存在, 请说明理由.(提示: 是否存在和$x_0$的取值有关)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013079": { + "id": "013079", + "content": "设$m$是常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013080": { + "id": "013080", + "content": "方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013081": { + "id": "013081", + "content": "已知双曲线$C$与椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{8}=1$有相同的焦点, 直线$y=\\sqrt{3} x$为$C$的一条渐近线, 则双曲线$C$的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013082": { + "id": "013082", + "content": "已知$F_1, F_2$是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$的两个焦点, 过$F_2$的直线交椭圆于$A, B$两点, 若$|AB|=5$, 则$|AF_1|+|BF_1|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013083": { + "id": "013083", + "content": "已知抛物线方程为$y^2=2 p x$($p>0$), 过焦点$F$的直线与抛物线交于$A, B$两点, 以$AB$为直径的圆$M$与抛物线的准线$l$的位置关系为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013084": { + "id": "013084", + "content": "已知点$A$的坐标是$(1, \\dfrac{1}{2}), P$是椭圆$x^2+4 y^2=1$上的动点, 则线段$PA$中点的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013085": { + "id": "013085", + "content": "某海域内有一孤岛, 岛四周的海平面(视为平面)上有一浅水区(含边界), 其边界是长轴长为$2 a$, 短轴长为$2 b$的椭圆. 已知岛上甲、乙导航灯的海拔高度分别为$h_1, h_2$, 且两个导航灯在海平面上的投影恰好落在椭圆的两个焦点上, 现有船只经过该海域(船只的大小忽略不计), 在船上测得甲、乙导航灯的仰角分别为$\\theta_1, \\theta_2$, 那么船只进入该浅水区的判断条件是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:360,samples = 100] plot ({2*cos(\\x)},0,{sqrt(3)*sin(\\x)});\n\\draw (-1,0,0,0) coordinate (A) -- (-1,1.2,0) coordinate (A1) node [midway, left] {$h_1$};\n\\draw (1,0,0) coordinate (B) -- (1,1,0) coordinate (B1) node [midway, right] {$h_2$};\n\\draw (0,0,1.2) coordinate (C);\n\\draw [dashed] (A) -- (C) (A1) -- (C) (B) -- (C) (B1) -- (C);\n\\draw (C) pic [draw, \"$\\theta_1$\", angle eccentricity = 1.3] {angle = A1--C--A};\n\\draw (C) pic [draw, \"$\\theta_2$\", angle eccentricity = 1.5] {angle = B--C--B1};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013086": { + "id": "013086", + "content": "在平面直角坐标系中, 设$P_1(x_1, y_1), P_2(x_2, y_2)$为不同的两点, 直线$l$的方程为$a x+b y+c=0$, 定义$\\delta_1=\\dfrac{a x_1+b y_1+c}{\\sqrt{a^2+b^2}}$和$\\delta_2=\\dfrac{a x_2+b y_2+c}{\\sqrt{a^2+b^2}}$为点$P_1, P_2$到直线$l$的有向距离. 以下正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\delta_1-\\delta_2=0$, 则直线$P_1P_2$与$l$平行;\\\\\n\\textcircled{2} 若$\\delta_1+\\delta_2=0$, 则直线$P_1P_2$与$l$平行;\\\\\n\\textcircled{3} 若$\\delta_1+\\delta_2=0$, 则直线$P_1P_2$与$l$垂直;\\\\\n\\textcircled{4} 若$\\delta_1 \\delta_2<0$, 则直线$P_1P_2$与$l$相交.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013087": { + "id": "013087", + "content": "已知点$A(-2,0), B(2,0)$, 点$C$在双曲线$x^2-y^2=1$上运动, 求以$AB$、$BC$为邻边的平行四边形$ABCP$的顶点$P$的轨迹方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013088": { + "id": "013088", + "content": "过定点$F(4,0)$作直线$l$交$y$轴于$Q$点, 过$Q$点作$QT \\perp FQ$交$x$轴于$T$点, 延长$TQ$至$P$点, 使$|PQ|=|TQ|$, 求动点$P$的轨迹方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013089": { + "id": "013089", + "content": "已知过点$A(-3,1)$且倾斜角为$45^{\\circ}$的直线$l$与焦点为$(-\\sqrt{6}, 0),(\\sqrt{6}, 0)$的椭圆交于$B, C$两点, 若线段$BC$在$A$点被平分, 则这样的椭圆是否存在? 若存在, 求出椭圆的方程; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013090": { + "id": "013090", + "content": "抛物线$y=8 x^2$的准线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013091": { + "id": "013091", + "content": "设双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0, b>0)$的虚轴长为$2$, 焦距为$2 \\sqrt{3}$, 则双曲线的渐近线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013092": { + "id": "013092", + "content": "设点$O$为坐标原点, 点$M$是曲线$y=\\dfrac{1}{2} x^2+1$上的一个动点, 且点$M$为线段$OP$的中点, 则动点$P$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013093": { + "id": "013093", + "content": "若双曲线$8 m x^2-m y^2=8$的一个焦点是$(0,3)$, 则实数$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013094": { + "id": "013094", + "content": "若抛物线$y^2=2 x$的焦点弦$AB$的两端点为$A(x_1, y_1), B(x_2, y_2)$, 则$\\dfrac{y_1 y_2}{x_1 x_2}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013095": { + "id": "013095", + "content": "已知$F_1, F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的焦点. 过$F_2$作垂直于$x$轴的直线交双曲线于点$P$, 且$\\angle PF_1F_2=30^{\\circ}$, 则双曲线的渐近线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013096": { + "id": "013096", + "content": "已知定圆$C_1:(x-7)^2+y^2=4$, $C_2:(x+7)^2+y^2=25$, 动圆$M$与两定圆外切, 则动圆圆心$M$的轨迹方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013097": { + "id": "013097", + "content": "若$F$是双曲线$x^2-y^2=1$的左焦点, 点$P$在第三象限的双曲线上, 则直线$FP$的倾斜角的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013098": { + "id": "013098", + "content": "在平面直角坐标系中, 设$M(x_1, y_1)$, $N(x_2, y_2)$为不同的两点, 直线$l$的方程为$a x+b y+c=0$, 设$\\lambda=\\dfrac{a x_1+b y_1+c}{a x_2+b y_2+c}$, 有以下四个命题:\\\\\n\\textcircled{1} 存在实数$\\lambda$, 使点$N$在直线$l$上;\\\\\n\\textcircled{2} 若$\\lambda=1$, 则过$M, N$两点的直线于直线$l$平行;\\\\\n\\textcircled{3} 若$\\lambda=-1$, 则直线$l$经过线段$MN$的中点;\\\\\n\\textcircled{4} 若$\\lambda>1$, 则点$M, N$在直线$l$的同侧, 且直线$l$与线段$MN$的延长线相交.\\\\\n上述命题中, 真命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013099": { + "id": "013099", + "content": "记椭圆$E_n: \\dfrac{x^2}{4}+\\dfrac{n y^2}{4 n+1}=1$, 其中$n=1,2, \\cdots$. 当点$(x, y)$分别在$E_1, E_2, \\cdots$上时, $x+y$的最大值分别是$M_1, M_2, \\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty} M_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013100": { + "id": "013100", + "content": "抛物线$x^2=4 y$的焦点$F$, 过点$(0,-1)$作直线交抛物线于不同的两点$A, B$, 以$AF, BF$为邻边作平行四边形$FARB$, 求顶点$R$的轨迹方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013101": { + "id": "013101", + "content": "已知圆$O: x^2+y^2=4$.\\\\\n(1) 直线$l_1: \\sqrt{3} x+y-2 \\sqrt{3}=0$与圆$O$相交于$A, B$两点, 求弦$AB$的长;\\\\\n(2) 设$M(x_1, y_1)$, $P(x_2, y_2)$是圆$O$上的两个动点, 点$M$关于原点的对称点为$M_1$, 点$M$关于$x$轴的对称点为$M_2$. 如果直线$P$和$M_1, M_2$均不重合, 且$PM_1, PM_2$都和$y$轴相交, 且分别交于$S(0, m)$和$T(0, n)$, 问$m \\cdot n$是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013102": { + "id": "013102", + "content": "若动点$P$到点$F(2,0)$的距离与它到直线$x+2=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013103": { + "id": "013103", + "content": "抛物线$y=-\\dfrac{x^2}{8}$的准线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013104": { + "id": "013104", + "content": "以原点为圆心, 且截直线$3 x+4 y+15=0$所得弦长为$8$的圆的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013105": { + "id": "013105", + "content": "动直线$(2 k-1) x-(k+3) y-(k-11)=0$($k \\in \\mathbf{R}$)所过的定点是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013106": { + "id": "013106", + "content": "设抛物线$y^2=8 x$的准线与$x$轴交于点$Q$, 若过点$Q$的直线$l$与抛物线有公共点, 则直线$l$的斜率的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013107": { + "id": "013107", + "content": "若双曲线$\\dfrac{3 x^2}{2}-\\dfrac{y^2}{2}=1$的右支上有一点$P$到两坐标轴的距离相等, 则点$P$的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013108": { + "id": "013108", + "content": "直线$y=x+3$与曲线$\\dfrac{y^2}{9}-\\dfrac{x|x|}{4}=1$的公共点的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013109": { + "id": "013109", + "content": "若命题``$F(x, y)=0$的解为坐标的点都是曲线$C$上的点''是真命题, 则下列命题正确的有\\blank{50}.\\\\\n\\textcircled{1} 曲线$C$上的点的坐标都是方程$F(x, y)=0$的解;\\\\\n\\textcircled{2} 坐标不满足方程$F(x, y)=0$的点不在曲线$C$上;\\\\\n\\textcircled{3} 曲线$C$是方程$F(x, y)=0$的曲线;\\\\\n\\textcircled{4} 不是曲线$C$上的点的坐标, 一定不满足方程$F(x, y)=0$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013110": { + "id": "013110", + "content": "设双曲线与椭圆$\\dfrac{x^2}{27}+\\dfrac{y^2}{36}=1$有共同的焦点, 且与椭圆相交的一个交点的纵坐标为$4$, 求这个双曲线的方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013111": { + "id": "013111", + "content": "若抛物线$y=a x^2-1$上存在关于直线$x+y=0$对称的不同两点, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013112": { + "id": "013112", + "content": "$P$是双曲线$\\dfrac{x^2}{4}-y^2=1$的右顶点, 过点$P$的两条互相垂直的直线分别与双曲线的右支交于点$A, B$, 问直线$AB$是否一定过$x$轴上一定点? 如果不存在这样的定点, 请说明理由; 如果存在这样的定点, 试求出这个定点的坐标.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013113": { + "id": "013113", + "content": "已知方程$\\dfrac{x^2}{2-k}+\\dfrac{y^2}{k-1}=1$表示的曲线是双曲线, 则$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013114": { + "id": "013114", + "content": "双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{39}=1$上一点$P$到双曲线一个焦点的距离为$12$, 则$P$到另一个焦点的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013115": { + "id": "013115", + "content": "若方程$2 x^2+2 y^2+a x+1=0$表示圆, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013116": { + "id": "013116", + "content": "双曲线$x^2-y^2=1$, 点$F_1, F_2$为其两个焦点, 点$P$为双曲线上一点, 若$PF_1 \\perp PF_2$, 则$|PF_1|+|PF_2|$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013117": { + "id": "013117", + "content": "点$(4,-1)$关于直线$y=-x+1$对称点的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013118": { + "id": "013118", + "content": "若直线$y=x+k$与曲线$y=\\sqrt{2-x^2}$相交于两点, 则实数$k$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013119": { + "id": "013119", + "content": "过抛物线$y^2=4 x$的焦点作倾斜角为$\\dfrac{\\pi}{3}$的弦$AB$, 则$|AB|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013120": { + "id": "013120", + "content": "设$P$是抛物线$y^2=4 x$上一动点, $F$是抛物线的焦点, 定点$B(3,2)$, 则$|PB|+|PF|$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013121": { + "id": "013121", + "content": "已知两点$M(-5,0), N(5,0)$, 若直线上存在点$P$, 使$|PM|-|PN|=8$, 则称该直线为``$B$型直线'', 现给出下列直线: \\textcircled{1} $y=x+2$; \\textcircled{2} $y=2$; \\textcircled{3} $y=\\dfrac{2}{3} x$; \\textcircled{4} $y=2 x+3$. 其中是``$B$型直线''的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013122": { + "id": "013122", + "content": "已知双曲线$C: \\dfrac{x^2}{2}-y^2=1$, 点$M(0,1)$, 点$P$是双曲线上任意一点, 若点$Q$是点$P$关于原点的对称点, 且$\\lambda=\\overrightarrow{MP} \\cdot \\overrightarrow{MQ}$, 则$\\lambda$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013123": { + "id": "013123", + "content": "已知两点$A(-1,0)$、$B(1,0)$, 点$P(x, y)$是直角坐标平面上的动点, 若将点$P$的横坐标保持不变、纵坐标扩大到$\\sqrt{2}$倍后得到点$Q(x, \\sqrt{2} y)$满足$\\overrightarrow{AQ} \\cdot \\overrightarrow{BQ}=1$. 求动点$P$所在曲线$C$的轨迹方程.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013124": { + "id": "013124", + "content": "已知直线$l: x-a y+a=0$与双曲线$x^2-y^2=1$的左支交于$A, B$两点, 过弦$AB$的中点$Q$与点$P(-2,1)$的直线交$y$轴于$(0, b)$点. 当$a$变化时, 求实数$b$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013125": { + "id": "013125", + "content": "点$P(-3,0)$是椭圆$x^2+2 y^2-k=0$上的点, 则椭圆的焦点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013126": { + "id": "013126", + "content": "双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}{25}=1$的渐近线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013127": { + "id": "013127", + "content": "与椭圆$4 x^2+5 y^2=20$有相同的焦点, 且顶点在原点的抛物线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013128": { + "id": "013128", + "content": "设$AB$是椭圆$\\Gamma$的长轴, 点$C$在$\\Gamma$上, 且$\\angle CBA=\\dfrac{\\pi}{4}$, 若$AB=4, BC=\\sqrt{2}$, 则$\\Gamma$的两个焦点之间的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013129": { + "id": "013129", + "content": "设双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的右顶点为$A$, 右焦点为$F$, 过点$F$且与双曲线的一条渐近线平行的直线与另\n一条渐近线交于点$B$, 则$\\triangle AFB$的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013130": { + "id": "013130", + "content": "中心在原点, 对称轴为坐标轴, 椭圆的短轴的一个顶点$B$与两个焦点$F_1, F_2$组成的三角形的周长为$4+2 \\sqrt{3}$, 且$\\angle F_1BF_2=\\dfrac{2 \\pi}{3}$, 则椭圆的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013131": { + "id": "013131", + "content": "若动点$(x, y)$在曲线$\\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1(01$), 点$P$是$C$上的动点, $M$是右顶点, 定点$A$的坐标为$(2,0)$.\\\\\n(1) 若$M$与$A$重合, 求$C$的焦点坐标;\\\\\n(2) 若$m=3$, 求$|PA|$的最大值与最小值;\\\\\n(3) 若$|PA|$的最小值为$|MA|$, 求$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013135": { + "id": "013135", + "content": "已知曲线$C_1: \\dfrac{x^2}{2}-y^2=1$, 曲线$C_2:|y|=|x|+1, P$是平面上一点, 若存在过点$P$的直线与$C_1, C_2$都有公共点, 则称$P$为``$C_1-C_2$型点''.\\\\\n(1) 在正确证明$C_1$的左焦点是``$C_1-C_2$型点''时, 要使用一条过该焦点的直线, 试写出一条这样的直线的方程(不要求验证);\\\\\n(2) 设直线$y=k x$与$C_2$有公共点, 求证$|k|>1$, 进而证明原点不是``$C_1-C_2$型点''.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013136": { + "id": "013136", + "content": "直线$x-y-3=0$被双曲线$\\dfrac{x^2}{4}-y^2=1$所截得的弦长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013137": { + "id": "013137", + "content": "已知点$M(x, y)$到点$F_1(-5,0)$和$F_2(5,0)$的距离差是$8$, 则点$M$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013138": { + "id": "013138", + "content": "双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{9}=1$上一点$P$到双曲线一个焦点的距离为$12$, 则$P$到另一个焦点的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013139": { + "id": "013139", + "content": "设$F_1, F_2$为双曲线$\\dfrac{x^2}{4}-y^2=1$的两焦点, 点$P$在双曲线上且满足$\\angle F_1PF_2=90^{\\circ}$, 则$\\triangle F_1PF_2$的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013140": { + "id": "013140", + "content": "若抛物线$x^2=a y$经过点$A(1, \\dfrac{1}{4})$, 则点$A$到此抛物线的焦点的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013141": { + "id": "013141", + "content": "已知抛物线$y^2=2 x$及点$A(\\dfrac{2}{3}, 0)$, 则抛物线上距点$A$最近的点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013142": { + "id": "013142", + "content": "设斜率为$2$的直线$l$过抛物线$y^2=a x$($a \\neq 0$)的焦点$F$, 且和$y$轴交于点$A$, 若$\\triangle OAF(O$为坐标原点)的面积为$4$, 则抛物线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013143": { + "id": "013143", + "content": "抛物线$x^2=2 p y$($p>0$)的焦点为$F$, 其准线与双曲线$\\dfrac{x^2}{3}-\\dfrac{y^2}{3}=1$相交于$A, B$两点, 若$\\triangle ABF$为等边三角形, 则$p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013144": { + "id": "013144", + "content": "已知直线$y=a$交抛物线$y=x^2$于$A, B$两点. 若该抛物线上存在点$C$, 使$\\angle ACB$为直角, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013145": { + "id": "013145", + "content": "已知平面上的线段$l$及点$P$, 在$l$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$, 求点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离$d(P, l)$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013146": { + "id": "013146", + "content": "已知双曲线$H$的中心在原点, 抛物线$y^2=8 x$的焦点是双曲线$H$的一个焦点, 且$H$经过点$(\\sqrt{2}, \\sqrt{3})$.\\\\\n(1) 求双曲线$H$的方程;\\\\\n(2) 设双曲线$H$的实轴左顶点为$A$, 右焦点为$F$, 在第一象限内任取双曲线$H$上一点$P$, 试问是否存在常数$\\lambda$, 使得$\\angle PFA=\\lambda \\angle PAF$恒成立? 证明你的结论.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013147": { + "id": "013147", + "content": "已知向量$\\overrightarrow{OM}=(3,-2)$, $\\overrightarrow{ON}=(-5,-1)$, 则$\\dfrac{1}{2} \\overrightarrow{MN}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013148": { + "id": "013148", + "content": "已知向量$\\overrightarrow {a}=(\\cos 75^{\\circ}, \\sin 75^{\\circ})$, $\\overrightarrow {b}=(\\cos 15^{\\circ}, \\sin 15^{\\circ})$, 则$|\\overrightarrow {a}-\\overrightarrow {b}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013149": { + "id": "013149", + "content": "若向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013150": { + "id": "013150", + "content": "已知$\\overrightarrow {m}, \\overrightarrow {n}$是夹角为$60^{\\circ}$的单位向量, 则$\\overrightarrow {a}=2 \\overrightarrow {m}+\\overrightarrow {n}$与$\\overrightarrow {b}=-3 \\overrightarrow {m}+2 \\overrightarrow {n}$的夹角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013151": { + "id": "013151", + "content": "在四面体$O-ABC$中, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, $D$为$BC$的中点, $E$为$AD$的中点, 则用$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$表示$\\overrightarrow{OE}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013152": { + "id": "013152", + "content": "设$\\overrightarrow {a}, \\overrightarrow {b}$是非零向量, 若函数$f(x)=(x \\overrightarrow {a}+\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-x \\overrightarrow {b})$的图像是一条直线, 则必有\\bracket{20}.\n\\fourch{$\\overrightarrow {a} \\perp \\overrightarrow {b}$}{$\\overrightarrow {a}\\parallel \\overrightarrow {b}$}{$|\\overrightarrow {a}|=|\\overrightarrow {b}|$}{$|\\overrightarrow {a}| \\neq|\\overrightarrow {b}|$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013153": { + "id": "013153", + "content": "在$\\triangle ABC$中, 已知$|\\overrightarrow{AB}|=4$, $|\\overrightarrow{AC}|=1$, $S_{\\triangle ABC}=\\sqrt{3}$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$的值为\\bracket{20}.\n\\fourch{$-2$}{$2$}{$\\pm 4$}{$\\pm 2$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013154": { + "id": "013154", + "content": "在直角$\\triangle ABC$中, $CD$是斜边$AB$上的高, 则下列等式不成立的是\\bracket{20}.\n\\twoch{$|\\overrightarrow{AC}|^2=\\overrightarrow{AC} \\cdot \\overrightarrow{AB}$}{$|\\overrightarrow{BC}|^2=\\overrightarrow{BC} \\cdot \\overrightarrow{BA}$}{$|\\overrightarrow{AB}|^2=\\overrightarrow{AC} \\cdot \\overrightarrow{CD}$}{$|\\overrightarrow{CD}|^2=\\dfrac{(\\overrightarrow{AC} \\cdot \\overrightarrow{AB}) \\cdot(\\overrightarrow{BA} \\cdot \\overrightarrow{BC})}{|\\overrightarrow{AB}|^2}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013155": { + "id": "013155", + "content": "设$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$为平面向量, 已知命题: \\textcircled{1} $\\overrightarrow {a} \\cdot(\\overrightarrow {b}-\\overrightarrow {c})=\\overrightarrow {a} \\cdot \\overrightarrow {b}-\\overrightarrow {a} \\cdot \\overrightarrow {c}$; \\textcircled{2} $(\\overrightarrow {a} \\cdot \\overrightarrow {b}) \\cdot \\overrightarrow {c}=\\overrightarrow {a} \\cdot(\\overrightarrow {b} \\cdot \\overrightarrow {c})$; \\textcircled{3} $(\\overrightarrow {a}-\\overrightarrow {b})^2=|\\overrightarrow {a}|^2+2|\\overrightarrow {a}||\\overrightarrow {b}|+|\\overrightarrow {b}|^2$; \\textcircled{4} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$. 以上命题中正确的有\\blank{50}个.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013156": { + "id": "013156", + "content": "定义平面向量之间的一种运算``$\\otimes$''如下: 对任意的$\\overrightarrow {a}=(m, n) , \\overrightarrow {b}=(p, q)$, 令$\\overrightarrow {a} \\otimes \\overrightarrow {b}=m q-n p$. 给出以下四个命题: \\textcircled{1} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$\\overrightarrow {a} \\otimes \\overrightarrow {b}=0$; \\textcircled{2} $\\overrightarrow {a} \\otimes \\overrightarrow {b}=\\overrightarrow {b} \\otimes \\overrightarrow {a}$; \\textcircled{3} 对任意的$\\lambda \\in \\mathbf{R}$, 有$(\\lambda \\overrightarrow {a}) \\otimes \\overrightarrow {b}=\\lambda(\\overrightarrow {a} \\otimes \\overrightarrow {b})$; \n\\textcircled{4} $(\\overrightarrow {a} \\otimes \\overrightarrow {b})^2+(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2=|\\overrightarrow {a}|^2 \\cdot|\\overrightarrow {b}|^2$. 则其中所有真命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013157": { + "id": "013157", + "content": "若$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, $\\overrightarrow {a} \\perp \\overrightarrow {b}$, 且$2 \\overrightarrow {a}+3 \\overrightarrow {b}$与$k \\overrightarrow {a}-4 \\overrightarrow {b}$也互相垂直, 求实数$k$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013158": { + "id": "013158", + "content": "平面直角坐标系内有点$P(1, \\cos x)$, $Q(\\cos x, 1)$,$x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$.\\\\\n(1) 求向量$\\overrightarrow{OP}$和$\\overrightarrow{OQ}$的夹角$\\theta$的余弦表达式$f(x)$;\\\\\n(2) 求$\\theta$的最值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013159": { + "id": "013159", + "content": "在正三棱柱$ABC-A_1B_1C_1$中, 底面是边长为$2$的正三角形, 且$AB_1 \\perp BC_1$.\\\\\n(1) 求$AA_1$的长;\\\\\n(2) 对于$n$个向量$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\cdots, \\overrightarrow{a_n}$, 如果存在不全为零的$n$个实数$\\lambda_1, \\lambda_2, \\cdots, \\lambda_n$, 使得$\\lambda_1 \\overrightarrow{a_1}+\\lambda_2 \\overrightarrow{a_2}+\\cdots+\\lambda_n \\overrightarrow{a_n}=\\overrightarrow{0}$, 则称这$n$个向量是线性相关的; 如果$n$个向量$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\cdots, \\overrightarrow{a_n}$不是线性相关的, 则称这$n$个向量是线性无关的. 求证: $\\overrightarrow{AB_1}, \\overrightarrow{BC_1}, \\overrightarrow{AC}$这三个向量是线性无关的.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013160": { + "id": "013160", + "content": "已知$\\overrightarrow{e_1}, \\overrightarrow{e_2}$是两个不共线的平面向量, 向量$\\overrightarrow {a}=2 \\overrightarrow{e_1}-\\overrightarrow{e_2}$, $\\overrightarrow {b}=\\overrightarrow{e_1}+\\lambda \\overrightarrow{e_2}$($\\lambda \\in \\mathbf{R}$), 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013161": { + "id": "013161", + "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(x, 6)$, 且$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013162": { + "id": "013162", + "content": "已知$\\overrightarrow {a}=(-1, m)$, $\\overrightarrow {b}=(2 m, 4)$, 若$|\\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}|=3$, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013163": { + "id": "013163", + "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(-1,4)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$方向上的数量投影为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013164": { + "id": "013164", + "content": "已知$\\overrightarrow {a}=(2,0)$, $\\overrightarrow{OB}=(2,2)$, $\\overrightarrow{OC}=(\\sqrt{2} \\cos \\alpha, \\sqrt{2} \\sin \\alpha)$, $\\overrightarrow{BC}$与$\\overrightarrow {a}$所成的角为$\\theta$, 则$\\theta$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013165": { + "id": "013165", + "content": "在$\\triangle ABC$中, $M$是$BC$的中点, $AM=3$, $BC=10$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013166": { + "id": "013166", + "content": "设$O$是直线$AB$外一点, $\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, 点$A_1, A_2, \\cdots, A_{n-1}$是线段$AB$的$n$($n \\geq 2$)等分点, 则$\\overrightarrow{OA_1}+\\overrightarrow{OA_2}+\\cdots+\\overrightarrow{OA_{n-1}}=$\\blank{50}.(用$\\overrightarrow {a}$, $\\overrightarrow {b}$, $n$表示)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013167": { + "id": "013167", + "content": "已知$\\triangle ABC$中的三个顶点$A, B, C$及平面内一点$P$满足$\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}=\\overrightarrow{AB}$, 则点$P$与$\\triangle ABC$的关系为\\bracket{20}.\n\\twoch{$P$在$\\triangle ABC$内部}{$P$在$\\triangle ABC$外部}{$P$在$AB$边所在的直线上}{$P$是$AC$边的一个三等分点}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013168": { + "id": "013168", + "content": "设平面上有四个互异的点$A, B, C, D$, 已知$(\\overrightarrow{DB}+\\overrightarrow{DC}-2 \\overrightarrow{DA}) \\cdot(\\overrightarrow{AB}-\\overrightarrow{AC})=0$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{直角三角形}{等腰三角形}{等腰直角三角形}{等边三角形}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013169": { + "id": "013169", + "content": "给出下列四个命题: \\textcircled{1} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$; \\textcircled{2} 若$|\\overrightarrow {a} \\cdot \\overrightarrow {b}|=|\\overrightarrow {a}||\\overrightarrow {b}|$, 则$\\overrightarrow {a}\\parallel \\overrightarrow {b}$; \\textcircled{3} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=|\\overrightarrow {a}-\\overrightarrow {b}|$; \\textcircled{4} 在$\\triangle ABC$中, 三边长$BC=5$, $AC=8$, $AB=7$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=20$. 其中真命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013170": { + "id": "013170", + "content": "设$A$是平面向量的集合, $\\overrightarrow {a}$是定向量, 对$\\overrightarrow {x} \\in A$, 定义$f(\\overrightarrow {x})=\\overrightarrow {x}-2(\\overrightarrow {a} \\cdot \\overrightarrow {x}) \\cdot \\overrightarrow {a}$. 现给出如下四个向量: \\textcircled{1} $\\overrightarrow {a}=(0,0)$; \\textcircled{2} $\\overrightarrow {a}=(\\dfrac{\\sqrt{2}}{4}, \\dfrac{\\sqrt{2}}{4})$; \\textcircled{3} $\\overrightarrow {a}=(\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2})$; \\textcircled{4} $\\overrightarrow {a}=(-\\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2})$. 那么对于任意$\\overrightarrow {x}$、$\\overrightarrow {y} \\in A$, 使$f(\\overrightarrow {x}) \\cdot f(\\overrightarrow {y})=\\overrightarrow {x} \\cdot \\overrightarrow {y}$恒成立的向量$\\overrightarrow {a}$的序号是\\blank{50}.(写出满足条件的所有向量$\\overrightarrow {a}$的序号)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013171": { + "id": "013171", + "content": "设$\\overrightarrow {a}=(\\cos \\theta, \\sin \\theta)$, $\\overrightarrow {b}=(2,1)$, 求$2|\\overrightarrow {a}+\\overrightarrow {b}|$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013172": { + "id": "013172", + "content": "已知平面上三个向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的模均为$1$, 它们互相之间的夹角均为$120^{\\circ}$.\\\\\n(1) 求证: $(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {c}$;\\\\\n(2) 若$|k \\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}|>1$($k \\in \\mathbf{R}$), 求$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013173": { + "id": "013173", + "content": "一个圆锥底面直径为$2$, 高为$4$, 则其母线与底面所成角的大小是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013174": { + "id": "013174", + "content": "我国古代数学名著《九章算术》中``开立圆术''曰: 置积尺数, 以十六乘之, 九而一, 所得开立方除之, 即立圆径. ``开立圆术''相当于给出了已知球的体积$V$, 求其直径$d$的一个近似公式$d \\approx \\sqrt[3]{\\dfrac{16}{9} V}$. 人们还用过一些类似的近似公式. 根据$\\pi=3.14159 \\cdots$判断, 下列近似公式中最精确的一个是\\bracket{20}.\n\\fourch{$d \\approx \\sqrt[3]{\\dfrac{16}{9} V}$}{$d \\approx \\sqrt[3]{2 V}$}{$d \\approx \\sqrt[3]{\\dfrac{300}{157} V}$}{$d \\approx \\sqrt[3]{\\dfrac{21}{11} V}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013175": { + "id": "013175", + "content": "已知某圆锥体的底面半径$r=3$, 沿圆锥体的母线把侧面展开后得到一个圆心角为$\\dfrac{2}{3} \\pi$的扇形, 则该圆锥体的表面积是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013176": { + "id": "013176", + "content": "正三棱柱$ABC-A_1B_1C_1$的所有棱的长度都为$4$, 则异面直线$AB_1$与$BC_1$所成的角是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013177": { + "id": "013177", + "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$底面$ABCD, PA=1$, 底面$ABCD$是正方形, $PC$与底面$ABCD$所成角的大小为$\\dfrac{\\pi}{6}$, 则该四棱锥的体积是\\blank{50}.\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) node [left] {$B$} coordinate (B);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw (0,{2*sqrt(2)/sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle (P)--(C);\n\\draw [dashed] (A)--(P) (A)--(C) (A)--(B) (A)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013178": { + "id": "013178", + "content": "给出下列命题, 其中正确命题的所有序号是\\blank{50}.\\\\\n\\textcircled{1} 直线上有两点到平面的距离相等, 则此直线与平面平行;\\\\\n\\textcircled{2} 夹在两个平行平面间的两条异面线段的中点连线平行于这两个平面;\\\\\n\\textcircled{3} $\\alpha$内存在不共线的三点到$\\beta$的距离相等, 则平面$\\alpha$与$\\beta$平行;\\\\\n\\textcircled{4} 垂直于同一个平面的两条直线是平行直线;\\\\\n\\textcircled{5} $l$、$m$是两条异面直线, $\\alpha$、$\\beta$是两个平面, 且$l\\parallel \\alpha$, $m\\parallel \\alpha$, $l\\parallel \\beta$, $m\\parallel \\beta$, 则平面$\\alpha$与$\\beta$平行.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013179": { + "id": "013179", + "content": "如图, 半径为$R$的半球$O$的底面圆$O$在平面$\\alpha$内, 过点$O$作平面$\\alpha$的垂线交半球面于点$A$, 过圆$O$的直径$CD$作平面$\\alpha$成$45^{\\circ}$角的平面与半球面相交, 所得交线上到平面$\\alpha$的距离最大的点为$B$, 该交线上的一点$P$满足$\\angle BOP=60^{\\circ}$, 则$A$、$P$两点间的球面距离为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:180,dashed,samples=100] plot ({2*cos(\\x)},0,{-2*sin(\\x)});\n\\draw [domain = 0:180,samples=100] plot ({2* cos(\\x)},0,{2* sin(\\x)});\n\\draw [dashed] (0,0,2) node [below left] {$C$} coordinate (C) -- (0,0,-2) node [above right] {$D$} coordinate (D);\n\\draw [dashed] (0,0,0) node [below right] {$O$} coordinate (O) -- (0,2,0) node [above] {$A$} coordinate (A);\n\\draw [dashed] (O) -- ({-sqrt(2)},{sqrt(2)},0) node [above left] {$B$} coordinate (B);\n\\path [draw,domain = 0:180,samples=100,name path = semi] plot ({2*cos (\\x)},{2*sin(\\x)},0);\n\\draw [domain = 0:90,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw [domain = 90:180,dashed,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw ({-sqrt(2)*sin(30)},{sqrt(2)*sin(30)},{2*cos(30)}) node [left] {$P$} coordinate (P);\n\\draw [dashed] (O)--(P);\n\\draw (O) pic [draw,\"$60^\\circ$\",angle eccentricity=1.5] {angle = B--O--P};\n\\path [name path = outline] (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (-3,0,-3) -- cycle;\n\\path [name intersections = {of = semi and outline, by = {S,T}}];\n\\draw (T) -- (-3,0,-3) -- (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (S);\n\\draw [dashed] (S) -- (T);\n\\draw (-2.6,0,2.6) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013180": { + "id": "013180", + "content": "如图, $AD, BC$是四面体$ABCD$中互相垂直的棱, $BC=2$. 若$AD=2 c$, $AB=BD=AC=CD=a$, 其中$a, c$为常数, 则四面体$ABCD$的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,-0.5,0) node [below] {$A$} coordinate (A);\n\\draw (-1,0.5,0) node [above] {$D$} coordinate (D);\n\\draw (D)--(A)--(B)--(C)--cycle (D)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013181": { + "id": "013181", + "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=6$, 异面直线$BC_1$与$AA_1$所成角的大小为$\\dfrac{\\pi}{6}$, 求该三棱柱的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{2}\n\\def\\h{{2*sqrt(3)}}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw (B) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013182": { + "id": "013182", + "content": "如图, 在长方体$AC_1$中, $AB=2$, $AD=1$, $AA_1=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} 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);\n\\draw [dashed] (A)--(D1)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 直线$BC_1$平行于平面$D_1AC$;\\\\\n(2) 求直线$BC_1$到平面$D_1AC$的距离.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013183": { + "id": "013183", + "content": "如图, 圆锥顶点为$P$, 底面圆心为$O$, 其母线与底面所成的角为$22.5^{\\circ}$. $AB$和$CD$是底面圆$O$上的两条平行的弦, 轴$OP$与平面$PCD$所成的角为$60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$O$} coordinate (O) ellipse (2 and 0.5);\n\\draw (0,-2) node [below] {$P$} coordinate (P);\n\\draw (P)--({sqrt(15)/2},-1/8) (P)--({-sqrt(15)/2},-1/8);\n\\draw ({2*cos(-50)},{sin(-50)/2}) node [above right] {$C$} coordinate (C);\n\\draw ({2*cos(50)},{sin(50)/2}) node [above right] {$D$} coordinate (D);\n\\draw ({2*cos(-140)},{sin(-140)/2}) node [above right] {$A$} coordinate (A);\n\\draw ({2*cos(140)},{sin(140)/2}) node [above right] {$B$} coordinate (B);\n\\draw (O)--(C)--(D)--cycle (P)--(C) (P)--(A)--(B);\n\\draw [dashed] (P)--(O) (P)--(D) (P)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$PAB$与平面$PCD$的交线平行于底面;\\\\ \n(2) 求$\\cos \\angle COD$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013184": { + "id": "013184", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 异面直线$A_1B$与$B_1C$所成角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013185": { + "id": "013185", + "content": "圆锥和圆柱的底面半径和高都是$R$, 则圆锥的表面积与圆柱的表面积之比为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013186": { + "id": "013186", + "content": "对于平面$\\alpha$、$\\beta$、$\\gamma$和直线$a$、$b$、$m$、$n$, 下列命题中真命题是\\bracket{20}.\n\\onech{若$a \\perp m$, $a \\perp n$, $m\\parallel \\alpha$, $n\\parallel \\alpha$, 则$a \\perp \\alpha$}{若$a\\parallel b$, $b \\perp \\alpha$, 则$a\\parallel \\alpha$}{若$a \\perp \\beta$, $b \\perp \\beta$, $a\\parallel \\alpha$, $b\\parallel \\alpha$, 则$\\alpha\\parallel \\beta$}{若$\\alpha\\parallel \\beta$, $\\alpha \\bigcap \\gamma=a$, $\\beta \\cap \\gamma=b$, 则$a\\parallel b$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013187": { + "id": "013187", + "content": "下列四个命题中不正确的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 三个点确定一个平面; \\textcircled{2} 圆锥的侧面展开图可以是一个圆面; \\textcircled{3} 底面是等边三角形, 三个侧面都是等腰三角形的三棱锥是正三棱锥; \\textcircled{4} 过球面上任意两不同点的大圆有且只有一个.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013188": { + "id": "013188", + "content": "已知三棱柱$ABC-A_1B_1C_1$的体积为$30 \\text{cm}^3, P$为其侧棱$BB_1$上的任意一点, 则四棱锥$P-A_1CC_1A$的体积为\\blank{50}$\\mathrm{cm}^3$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013189": { + "id": "013189", + "content": "若一个底面边长为$\\dfrac{\\sqrt{3}}{2}$, 侧棱长为$\\sqrt{6}$的正四棱柱的所有顶点都在一个球面上, 则此球的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013190": { + "id": "013190", + "content": "已知正四棱柱$ABCD-A_1B_1C_1D_1$中$AA_1=2AB$, 则$CD$与平面$BDC_1$所成角的正弦值等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013191": { + "id": "013191", + "content": "圆锥母线长为$3$, 底面半径为$1 \\text{cm}$, 底面圆周上有一点$A$, 由$A$点出发绕圆锥一周回到$A$点的最短路线长等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013192": { + "id": "013192", + "content": "已知三棱柱$ABC-A_1B_1C_1$的侧棱与底面垂直, 体积为$\\dfrac{9}{4}$, 底面是边长为$\\sqrt{3}$的正三角形. 若$P$为底面$A_1B_1C_1$的中心, 则$PA$与平面$ABC$所成角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013193": { + "id": "013193", + "content": "半径为$1$的球面上的四点$A$、$B$、$C$、$D$是正四面体的顶点, 则$A$与$B$两点间的球面距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013194": { + "id": "013194", + "content": "如图, 在直角梯形$ABCD$中, $\\angle B=\\angle C=90^{\\circ}$, $AB=\\sqrt{2}$, $CD=\\dfrac{\\sqrt{2}}{2}$, $BC=1$. 将$ABCD$(及其内部)绕$AB$所在的直线旋转一周, 形成一个几何体.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)}) node [above] {$A$} coordinate (A);\n\\draw (-1,{sqrt(2)/2}) node [left] {$D$} coordinate (D);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)}) node [above] {$A$} coordinate (A);\n\\draw (-1,{sqrt(2)/2}) node [left] {$D$} coordinate (D);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (C) arc (180:360:1 and 0.25) (D) arc (180:360:1 and 0.25) -- (A) --(D);\n\\draw [dashed] (C) arc (180:0:1 and 0.25) (D) arc (180:0:1 and 0.25);\n\\draw (C)--(D) (1,0) -- (1,{sqrt(2)/2});\n\\draw ({cos(-50)},{0.25*sin(-50)}) node [below] {$C'$} coordinate (C');\n\\draw (C') ++ (0,{sqrt(2)/2}) node [below right] {$D'$} coordinate (D');\n\\draw (C')--(D')--(A);\n\\draw [dashed] (A)--(B) (D)--(C') (C)--(B)--(C');\n\\end{tikzpicture}\n\\end{center}\n(1) 求该几何体的体积$V$;\\\\\n(2) 设直角梯形$ABCD$绕底边$AB$所在的直线旋转角$\\theta$($\\angle CBC=\\theta \\in(0, \\pi)$)至$ABC' D'$, 问: 是否存在$\\theta$, 使得$AD' \\perp DC'$. 若存在, 求角$\\theta$的值, 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013195": { + "id": "013195", + "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中$AA_1=AD=1$, $E$为$CD$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z= {(210:0.5cm)}]\n\\def\\l{2}\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,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\l,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) (A)--(D1) (A)--(B1)--($(C)!0.5!(D)$) node [below right] {$E$} coordinate (E)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $B_1E \\perp AD_1$;\\\\\n(2) 在棱$AA_1$上是否存在一点$P$, 使得$DP\\parallel$平面$B_1AE$? 若存在, 求$AP$的长; 若不存在, 说明理由;\\\\\n(3) 若二面角$A-B_1E-A_1$的大小为$30^{\\circ}$, 求$AB$的长.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013196": { + "id": "013196", + "content": "关于$x, y$的二元一次方程组$\\begin{cases}a x+y+a=0, \\\\ 4 x+a y+2=0\\end{cases}$无解, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013197": { + "id": "013197", + "content": "若矩阵$A=\\begin{pmatrix}1 & 3 \\\\ 2 & 1\\end{pmatrix}$, 则向量$(2,3)$经过矩阵$A$变换后所得的向量为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013198": { + "id": "013198", + "content": "设关于$x$的实系数一元二次方程$x^2-2 a x+a^2-4 a+4=0$($a \\in \\mathbf{R}$)的两虚根为$x_1, x_2$且$|x_1|+|x_2|=3$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013199": { + "id": "013199", + "content": "若$z \\in \\mathbf{C}$且$|z+2-2 \\mathrm{i}|=1$, 则$|z-2-2 \\mathrm{i}|$的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013200": { + "id": "013200", + "content": "$z_1, z_2 \\in \\mathbf{C}$, $z_1^2-2 z_1 z_2+4 z_2^2=0$, $|z_2|=2$, 那么以$|z_1|$为直径的圆的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013201": { + "id": "013201", + "content": "已知复数$z$满足$|z|=|z-1|=1$, $\\text{Im} z>0$, 且$z^n=-z$, 则最小正整数$n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013202": { + "id": "013202", + "content": "已知$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位), $z_1, z_2 \\in \\mathbf{C}$, 定义: $D(z)=|a|+|b|$, $D(z_1, z_2)=D(z_1-z_2)$. 给出下列命题:\\\\\n\\textcircled{1} 对任意$z \\in \\mathbf{C}$, 都有$D(z)>0$;\\\\\n\\textcircled{2} 若$\\overline {z}$是复数$z$的共轭复数, 则$D(\\overline {z})=D(z)$恒成立;\\\\\n\\textcircled{3} 若$D(z_1)=D(z_2)$($z_1$、$z_2 \\in \\mathbf{C})$, 则$z_1=z_2$;\\\\\n\\textcircled{4} 对任意$z_1$、$z_2$、$z_3 \\in \\mathbf{C}$, 结论$D(z_1, z_3) \\leq D(z_1, z_2)+D(z_2, z_3)$恒成立;\\\\\n\\textcircled{5} 对任意$z_1$、$z_2 \\in \\mathbf{C}$, 结论$D(z_1, z_2)=D(z_2, z_1)$恒成立.\\\\\n则其中真命题是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013203": { + "id": "013203", + "content": "关于$x$的方程$x^2-(\\tan \\theta+\\mathrm{i}) x-(\\mathrm{i}+2)=0$($\\theta \\in \\mathbf{R}$, $x \\in \\mathbf{C}$).\\\\\n(1) 若此方程有实数根, 求锐角$\\theta$的值;\\\\\n(2) 求证: 对任意的实数$\\theta$($\\theta \\neq \\dfrac{\\pi}{2}+k \\pi$, $k \\in \\mathbf{Z}$), 原方程不可能有纯虚根.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013204": { + "id": "013204", + "content": "已知复平面上三点$P_1$、$P_2$、$P_3$分别对应复数$z$、$2 z$、$3 z$, 且$|z|=4$, 点$A$对应复数$1$, 若$\\triangle P_1AP_2$与$\\triangle$$P_2AP_3$的面积和为$2$, 求复数$z$.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013205": { + "id": "013205", + "content": "对于任意的复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义$z$ ``经运算$P$''为$P(z)=x^2[\\cos (y \\pi)+\\mathrm{i} \\sin (y \\pi)]$.\\\\\n(1) 集合$A=\\{\\omega|\\omega=P(z), \\ | z |\\leq 1, \\ \\text{Re} z, \\text{Im} z\\text{均为整数}\\}$, 试用列举法写出集合$A$;\\\\\n(2) 若$z=2+y\\mathrm{i}$($y \\in \\mathbf{R}$), $P(z)$为纯虚数, 求$|z|$的最小值;\\\\\n(3) 直线$l: y=x-9$上是否存在整点$(x, y)$(坐标$x, y$均为整数), 使复数$z=x+y \\mathrm{i}$ ``经运算$P$''后, $P(z)$对应的点也在直线$l$上? 若存在, 求出所有的点; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013206": { + "id": "013206", + "content": "关于$x$的方程$x^2+4 x+k=0$有一个根为$-2+3\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013207": { + "id": "013207", + "content": "已知关于$x, y$的二元一次方程组$\\begin{pmatrix}m & 1 \\\\ 1 & m\\end{pmatrix}\\begin{pmatrix}x \\\\ y\\end{pmatrix}=\\begin{pmatrix}m+1 \\\\ 2 m\\end{pmatrix}$. 若方程组至多有一解, 则$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013208": { + "id": "013208", + "content": "函数$y=\\begin{vmatrix}1 & 2 & 3 \\\\ x & 4 & 9 \\\\ x^2 & 8 & 27\\end{vmatrix}$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013209": { + "id": "013209", + "content": "在复平面内, $O$是原点, $\\overrightarrow{OA}, \\overrightarrow{OC}, \\overrightarrow{AB}$表示的复数分别为$-2+\\mathrm{i}, 3+2\\mathrm{i}, 1+5\\mathrm{i}$, 那么$\\overrightarrow{BC}$表示的复数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013210": { + "id": "013210", + "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R}$, $b \\neq 0$), 若$z^2-4 b z$是实数, 则有序数对$(a, b)$可以是\\blank{50}. (只需填一个)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013211": { + "id": "013211", + "content": "复数$z$满足$|z-1|^2=(z-1)^2$, 则复数$z$对应点的轨迹是\\bracket{20}.\n\\fourch{一条直线}{一条双曲线}{一条抛物线}{一个圆}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013212": { + "id": "013212", + "content": "复数$z$满足$|z-2|+|z+\\mathrm{i}|=\\sqrt{5}$, 那么$|z|$的取值范围是\\bracket{20}.\n\\fourch{$[1, \\sqrt{5}]$}{$[1,2]$}{$[\\dfrac{2 \\sqrt{5}}{5}, 2]$}{$[\\dfrac{2 \\sqrt{5}}{5}, \\sqrt{5}]$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013213": { + "id": "013213", + "content": "已知$z=\\dfrac{2}{1-\\sqrt{3} \\mathrm{i}}$, 则$1+z+z^2+\\cdots+z^{2018}$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013214": { + "id": "013214", + "content": "方程$x^2-2 x+p=0$的两根在复平面上对应的点之间的距离为$\\sqrt{3}$, 则实数$p$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013215": { + "id": "013215", + "content": "设复数$\\alpha=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$, $y>0$), $\\dfrac{\\alpha}{1+\\alpha^2}$是实数.\\\\\n(1) 求证: $|\\alpha|=1$;\\\\\n(2) 若$\\dfrac{\\alpha^2}{1+\\alpha}$也是实数, 求$\\alpha$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013216": { + "id": "013216", + "content": "已知复数$z_0=1-m \\mathrm{i}$($m>0$), $z=x+y \\mathrm{i}$, $w=x'+y' \\mathrm{i}$, $x, y, x', y'$均为实数, $\\mathrm{i}$为虚数单位, 且对于任意复数$w=\\overline{z_0} \\cdot \\overline {z}$, $|w|=2|z|$.\\\\\n(1) 求$m$的值, 并分别写出$x', y'$用$x, y$表示的关系式;\\\\\n(2) 将$(x, y)$看作点$P$的坐标, $(x', y')$看作点$Q$的坐标, 上述关系可以看作是坐标平面上点的一个变换: 它将平面上的点$P$变换到这一平面上的点$Q$; 已知点$P$经该点的变换后得到的点$Q$的坐标是$(\\sqrt{3}, 2)$, 试求点$P$的坐标;\\\\\n(3) 若直线$y=k x$上任一点经上述变换后得到的点仍在该直线上, 求$k$的值.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013217": { + "id": "013217", + "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R})$, 将一颗骰子连续抛郑两次, 第一次点数记为$a$, 第二次点数记为$b$, 则使复数\n$z^2$为纯虚数的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013218": { + "id": "013218", + "content": "某学校组织学生参加英语测试, 成绩的频率分布直方图如图, 数据的分组一次为$[20,40),[40,60)$, $[60,80),[80,100)$若低于$60$分的人数是$15$人, 则该班的学生人数是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (6,0) node [below] {成绩/分};\n\\draw [->] (0,0) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {1/1/20,2/2/40,3/4/60,4/3/80}\n{\\draw (\\i,0) node [below] {$\\k$} --++ (0,\\j/2) --++ (1,0) --++ (0,-\\j/2);};\n\\draw (5,0) node [below] {$100$};\n\\foreach \\i/\\j/\\k in {1/1/0.005,2/2/0.01,3/4/0.015,4/3/0.02}\n{\\draw [dashed] (\\j,{\\i/2}) -- (0,{\\i/2}) node [left] {$\\k$};};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013219": { + "id": "013219", + "content": "某区有$300$名学生参加数学竞赛, 随机抽取$11$名学生成绩如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 成绩 & 40 & 50 & 60 & 70 & 80 & 90 \\\\\n\\hline 人数 & 1 & 2 & 2 & 3 & 2 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则总体标准差的点估计值是\\blank{50}.(精确到$0.01$)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013220": { + "id": "013220", + "content": "样本$(x_1, x_2, \\cdots, x_n)$的平均数为$\\overline {x}$, 样本$(y_1, y_2, \\cdots, y_m)$的平均数为$\\overline {y}$($\\overline {x} \\neq \\overline {y}$). 若样本$(x_1, x_2, \\cdots, x_n, y_1, y_2, \\cdots, y_m)$的平均数$\\overline {z}=\\alpha \\overline {x}+(1-\\alpha) \\overline {y}$, 其中$0<\\alpha<\\dfrac{1}{2}$, 则$n, m$的大小关系为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013221": { + "id": "013221", + "content": "将$2$名教师, $4$名学生分成$2$个小组, 分别安排到甲、乙两地参加社会实践活动, 每个小组由$1$名教师和$2$名学生组成, 不同的安排方案共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013222": { + "id": "013222", + "content": "若从$1,2, \\cdots \\cdots, 9$这$9$个整数中同时取$4$个不同的数, 其和为偶数, 则不同的取法共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013223": { + "id": "013223", + "content": "若将函数$f(x)=x^5$表示为$f(x)=a_0+a_1(1+x)+a_2(1+x)^2+\\cdots+a_5(1+x)^5$其中$a_0, a_1, a_2, \\cdots, a_5$为实数, 则$a_3=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013224": { + "id": "013224", + "content": "设$a \\in \\mathbf{Z}$, 且$0 \\leq a<13$, 若$51^{2012}+a$能被$13$整除, 则$a=$\\blank{50}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013225": { + "id": "013225", + "content": "为了了解《中华人民共和国知识产权法》在学生中的普及情况, 某咨询调查机构对某校的$6$名学生进行问卷调查, $6$人得分分别为$5,6,7,8,9,10$. 把这$6$名学生看成一个总体.\\\\\n(1) 求该总体的均值和标准差(精确到$0.01$);\\\\\n(2) 用简单随机抽样的方法从这个总体中抽取一个容量为$2$的样本.\\\\\n\\textcircled{1} 在所有样本中, 写出所有样本标准差最大的样本和所有样本标准差最小的样本;\\\\ \n\\textcircled{2} 求在所有容量为$2$的样本中, 样本均值与总体均值之差的绝对值不超过$\\dfrac{1}{2}$的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013226": { + "id": "013226", + "content": "已知: $(x^{\\frac{2}{3}}+3 x^2)^n$的展开式中, 各项系数和比它的二项式系数和大$992$.\\\\\n(1) 求展开式中二项式系数最大的项;\\\\\n(2) 求展开式中系数最大的项.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013227": { + "id": "013227", + "content": "某单位组织$4$个部门的职工旅游, 规定每个部门只能在韶山、衡山、华山$3$个景区中任选一个, 假设各部门选择每个景区是独立且等可能的.\\\\\n(1) 求$3$个景区都有部门选择的概率;\\\\\n(2) 求恰有$2$个景区有部门选择的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013228": { + "id": "013228", + "content": "将$a, b, c, d, e, f$字母排成三行两列, 则不同的排列方法共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013229": { + "id": "013229", + "content": "$(x-\\dfrac{1}{x})^8$的展开式中常数项为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013230": { + "id": "013230", + "content": "在$(1+x)^n$($n \\in \\mathbf{N}$, $n\\ge 1$)的二项展开式中, 若只有$x^3$的系数最大, 则$n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013231": { + "id": "013231", + "content": "在$(\\sqrt{x}+\\dfrac{3}{\\sqrt[3]{x}})^n$展开式中, 各项系数的和与其各项二项式系数的和之比值为$64$, 则$n$等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013232": { + "id": "013232", + "content": "$6$位同学互通电话, 任意两位同学之间最多通电话一次, 已知$6$位同学之间共通了$13$次电话, 则通了$4$次电话的同学人数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013233": { + "id": "013233", + "content": "若$\\mathrm{C}_n^1 x+\\mathrm{C}_n^2 x^2+\\cdots+\\mathrm{C}_n^n x^n$能被 7 整除, 则$x, n$的值可能为\\bracket{20}.\n\\fourch{$x=4$, $n=3$}{$x=4$, $n=4$}{$x=5$, $n=4$}{$x=6$, $n=5$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013234": { + "id": "013234", + "content": "某同学到银行取款时忘记了帐户密码, 但他记得: \\textcircled{1} 密码是四位数字, 如:$0235,1330,2351$等; \\textcircled{2} 四位数字中有$6,8,9$; \\textcircled{3} 四位数字各不相同. 于是他就用$6,8,9$这三个数字再随意加上一个与这三个数字不同的数字, 排成四位数输入取款机尝试, 那么他只试一次就成功的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013235": { + "id": "013235", + "content": "甲、乙两人在一次射击比赛中各射靶$5$次, 两人成绩的条形统计图如图所示, 则下列命题正确的有\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {4/1,5/1,6/1,7/1,8/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(甲)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {5/3,6/1,9/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(乙)};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲的成绩的平均数小于乙的成绩的平均数; \\textcircled{2} 甲的成绩的中位数等于乙的成绩的中位数; \\textcircled{3} 甲的成绩的方差小于乙的成绩的方差.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013236": { + "id": "013236", + "content": "将序号分别为$1,2,3,4,5$的$5$张参观券全部分给$4$人, 每人至少$1$张, 如果分给同一人的$2$张参观券连号, 那么不同的分法种数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013237": { + "id": "013237", + "content": "为了考察某校各班参加课外书法小组的人数, 在全校随机抽取$5$个班级, 把每个班级参加该小组的人数作为样本数据. 已知样本平均数为$7$, 样本方差为$4$, 则样本数据中的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013238": { + "id": "013238", + "content": "已知数列$\\{a_n\\}$($n$为正整数)是首项是$a_1$, 公比为$q$的等比数列.\\\\\n(1) 求和: $a_1\\mathrm{C}_2^0-a_2\\mathrm{C}_2^1+a_3\\mathrm{C}_2^2$, $a_1\\mathrm{C}_3^0-a_2\\mathrm{C}_3^1+a_3\\mathrm{C}_3^2-a_4\\mathrm{C}_3^3$;\\\\\n(2) 由(1)的结果归纳概括出关于正整数$n$的一个结论, 并加以证明.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013239": { + "id": "013239", + "content": "为了研究某高校大学新生学生的视力情况, 随机地抽查了该校$100$名进校学生的视力情况, 得到频率分布直方图, 如图. 已知前$4$组的频数从左到右依次是等比数列$\\{a_n\\}$的前四项, 后$6$组的频数从左到右依次是等差数列$\\{b_n\\}$的前六项.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,yscale = 1.4]\n\\draw [->] (-0.5,0) -- (6,0) node [below] {视力};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j/\\k in {1/0.1/4.3,2/0.3/4.4,3/0.9/4.5,4/2.7/4.6,5/2.2/4.7,6/1.7/4.8,7/1.2/4.9,8/0.7/5.0,9/0.2/5.1}\n{\\draw ({\\i/2},0) node [below] {$\\k$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\draw (5,0) node [below] {$5.2$};\n\\draw [dashed] (0.5,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw [dashed] (1,0.3) -- (0,0.3) node [left] {$0.3$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求等比数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求等差数列$\\{b_n\\}$的通项公式;\\\\\n(3) 若规定视力低于$5.0$的学生属于近视学生, 试估计该校新生的近视率$p$的大小.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013240": { + "id": "013240", + "content": "已知实数$x, y$满足$\\begin{cases}x+2 y \\geq 4, \\\\ 2 x+y \\geq 3, \\\\ x \\geq 0, \\\\ y \\geq 0\\end{cases}$的目标函数$f=x+y$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013241": { + "id": "013241", + "content": "已知实数$x$、$y$满足$\\begin{cases}x+y \\leq 5, \\\\ 2 x+y \\leq 6, \\\\ x \\geq 0, \\\\ y \\geq 0,\\end{cases}$ 则$z=3 x+4 y$的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013242": { + "id": "013242", + "content": "若实数$x, y$满足$\\begin{cases}2 x-y \\geq 0, \\\\ y \\geq x, \\\\ y \\geq-x+b,\\end{cases}$且$z=2 x+y$的最小值为$3$, 则实数$b$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013243": { + "id": "013243", + "content": "设实数$x, y$, 满足$\\begin{cases}x-y-2 \\leq 0, \\\\ x+2 y-4 \\geq 0, \\\\ 2 y-3 \\leq 0,\\end{cases}$ 则$\\dfrac{y}{x}$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013244": { + "id": "013244", + "content": "已知实数$x, y$满足$\\begin{cases}x \\leq 3, \\\\ x+y-3 \\geq 0, \\\\ x-y+1 \\geq 0,\\end{cases}$ 则$x^2+y^2$的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013245": { + "id": "013245", + "content": "若正三棱柱的主视图如图所示, 则此三棱柱的体积等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (2,2);\n\\draw (1,0) -- (1,2);\n\\draw (0,0) -- (0,-0.5) (1,0) -- (1,-0.5) (2,0) -- (2,-0.5);\n\\draw (2.5,0) -- (2,0) (2.5,2) -- (2,2);\n\\draw [<->] (0,-0.25) -- (1,-0.25) node [midway,fill = white] {$1$};\n\\draw [<->] (1,-0.25) -- (2,-0.25) node [midway,fill = white] {$1$};\n\\draw [<->] (2.25,0) -- (2.25,2) node [midway, fill = white] {$2$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013246": { + "id": "013246", + "content": "一个几何体的三视图如图所示, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale=0.5]\n\\draw (0,0) circle (1);\n\\draw (-1,1.5) --++ (2,0) --++ (0,2) --++ (-2,0) -- cycle;\n\\draw (-1,3.5) --++ (1,1) --++ (1,-1);\n\\draw (2,1.5) --++ (2,0) --++ (0,2) --++ (-2,0) -- cycle;\n\\draw (2,3.5) --++ (1,1) --++ (1,-1);\n\\draw [dashed] (1,1.5) -- (2,1.5) (1,3.5) -- (2,3.5);\n\\draw [<->] (1.5,1.5) -- (1.5,3.5) node [midway, rotate = 90, fill = white] {$2$};\n\\draw [dashed] (0,4.5) -- (3,4.5);\n\\draw [->] (1.5,5) -- (1.5,4.5);\n\\draw (1.5,4) node [rotate = 90] {$1$};\n\\draw [dashed] (-1,1.5) --++ (0,-3.25) (1,1.5) --++ (0,-3.25);\n\\draw [<->] (-1,-1.5) -- (1,-1.5) node [midway, fill=white] {$2$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013247": { + "id": "013247", + "content": "一个用若干块大小相同的立方块搭成的立体图形, 主视图和俯视图是同一图形(如图), 那么搭成这样一个立体图形最少需要\\blank{50}个小立方块.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) rectangle (3,1);\n\\draw (0,1) --++ (0,1) --++ (1,0) --++ (0,-2) (2,0) --++ (0,1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013248": { + "id": "013248", + "content": "将如图所示的一个直角三角形$ABC$($\\angle C=90^{\\circ}$)绕斜边$AB$旋转一周, 所得到的几何体的正视图是下面四个图形中的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (0,-0.5) node [below] {$B$} coordinate (B);\n\\filldraw [pattern = north east lines] (A) -- (B) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,2) -- cycle;\n\\draw (1,0) arc ({atan(4/3)-90}:{-atan(4/3)-90}:{5/4});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,2) -- cycle;\n\\draw (1,0) -- (0,-0.5) -- (-1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,-2) -- cycle;\n\\draw (1,0) -- (0,0.5) -- (-1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (1);\n\\filldraw (0,0) circle (0.03);\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013249": { + "id": "013249", + "content": "将大小不同的两种钢板截成$A$、$B$两种规格的成品, 每张钢板可同时截得这两种规格的成品的块数如表所示. 若现在需$A$、$B$两种规格的成品分别为$12$块和$10$块, 则至少共需这两种钢板\\blank{50}张.\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$A$种规格的成品 &$B$种规格的成品 \\\\\n\\hline 第一种钢板 & 2 & 1 \\\\\n\\hline 第二种钢板 & 1 & 3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013250": { + "id": "013250", + "content": "已知几何体由正方体和直三棱柱组成, 其三视图和直观图(单位: $\\text{cm}$) 如图所示. 设两条异面直线$A_1Q$和$PD$所成的角为$\\theta$, 求$\\cos \\theta$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw (0,0) -- (2,0) -- (2,3) -- (0,3) -- cycle (0,2) -- (2,2);\n\\draw (1,0) node [above] {$2$} (2,1) node [left] {$2$};\n\\draw (3,0) --++ (2,0) --++ (0,2) --++ (-1,1) --++ (-1,-1) -- cycle;\n\\draw (4,0) node [above] {$2$} (3,1) node [right] {$2$} (3,2) ++ (0.5,0.5) node [above left] {$\\sqrt{2}$} ++ (1,0) node [above right] {$\\sqrt{2}$};\n\\draw (0,-3) rectangle ++ (2,2) (0,-2) --++ (2,0);\n\\draw (0,-2.5) node [right] {$1$} (0,-1.5) node [right] {$1$} (1,-3) node [above] {$2$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\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 [below right] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1);\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1) (A1) -- (D1) -- (C1);\n\\draw ($(A1)!0.5!(D1)$) ++ (0,1) node [above] {$P$} coordinate (P);\n\\draw ($(B1)!0.5!(C1)$) ++ (0,1) node [above] {$Q$} coordinate (Q);\n\\draw (A1) -- (P) -- (Q) -- (B1) (Q) -- (C1);\n\\draw [dashed] (P) --(D1);\n\\draw [dashed] (P) -- (D);\n\\draw (A1) -- (Q);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013251": { + "id": "013251", + "content": "已知实数$x$、$y$满足线性约束条件$\\begin{cases}3 x-y \\geq 0, \\\\ x+y-4 \\leq 0, \\\\ x-3 y+5 \\leq 0.\\end{cases}$则目标函数$z=x-y-1$的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013252": { + "id": "013252", + "content": "设实数$x, y$满足$\\begin{cases}x+y \\geq 2, \\\\ 2 x-y \\leq 4, \\\\ y \\leq 4,\\end{cases}$则$x-2 y$的最大值等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013253": { + "id": "013253", + "content": "若$x, y$满足$\\begin{cases}-x+y \\leq 0, \\\\ -x+2 y \\geq 2,\\end{cases}$则目标函数$C=\\log _{\\frac{1}{2}}(x+y)$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013254": { + "id": "013254", + "content": "已知$x, y$满足$\\begin{cases}y-2 \\leq 0, \\\\ x+3 \\geq 0, \\\\ x-y-1 \\leq 0,\\end{cases}$则$\\dfrac{y-2}{x-4}$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013255": { + "id": "013255", + "content": "一个几何体的三视图如下左图所示, 其中俯视图与左视图均为半径是$2$的圆, 则这个几何体的表面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (1);\n\\draw (0,-1) -- (0,1);\n\\draw (1,3) arc (0:-270:1) --++ (0,-1) --++ (1,0);\n\\draw (3,3) circle (1);\n\\draw [dashed] (2,3) -- (4,3);\n\\draw (0,-1) node [below] {俯视图};\n\\draw (0,2) node [below] {主视图};\n\\draw (3,2) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013256": { + "id": "013256", + "content": "如图, 四棱锥$S-ABCD$的底面是矩形, 锥顶点在底面的射影是矩形对角线的交点, 四棱锥及其三视图如下($AB$平行于主视图投影平面) 则四棱锥$S-ABCD$的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$B$} coordinate (B);\n\\draw (4,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [left] {$D$} coordinate (D);\n\\draw (2,3,-1) node [above] {$S$} coordinate (S);\n\\draw (A) -- (B) -- (C) -- (S) -- (A) (S) -- (B);\n\\draw [dashed] (C) -- (D) (A) -- (D) -- (S);\n\\path (2,-4);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) -- (4,0) -- (4,2) -- (0,2) -- cycle;\n\\draw (0,0) -- (4,2) (4,0) -- (0,2);\n\\draw (0,-1) -- (0,0) (4,-1) -- (4,0);\n\\draw [<->] (0,-0.5) -- (4,-0.5) node [midway, fill = white] {$4$};\n\\draw (0,4) -- (4,4) -- (2,7) -- cycle;\n\\draw (6,4) -- (8,4) -- (7,7) -- cycle;\n\\draw [dashed] (4,4) -- (6,4) (2,7) -- (7,7);\n\\draw [<->] (5,4) -- (5,7) node [midway, rotate = 90, fill = white] {$3$};\n\\draw (6,4) -- (6,3) (8,4) -- (8,3);\n\\draw [<->] (6,3.5) -- (8,3.5) node [midway, fill = white] {$2$};\n\\draw (2,4) node [below] {主视图}; \n\\draw (2,-1) node [below] {俯视图};\n\\draw (7,3) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013257": { + "id": "013257", + "content": "已知棱长为$1$的正方体的俯视图是一个面积为$1$的正方形, 则该正方体的主视图的面积不可能等于\\bracket{20}.\n\\fourch{1}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}-1}{2}$}{$\\dfrac{\\sqrt{2}+1}{2}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013258": { + "id": "013258", + "content": "某四棱锥的三视图如图所示, 则最长的一条侧棱长度为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (1,0) -- (2,0) -- (0,1) -- cycle;\n\\draw [dashed] (1,0) -- (0,1);\n\\draw (0,-0.1) -- (0,-0.4) (1,-0.1) -- (1,-0.4) (2,-0.1) -- (2,-0.4);\n\\draw [<->] (0,-0.25) -- (1,-0.25) node [midway, fill = white] {$1$};\n\\draw [<->] (1,-0.25) -- (2,-0.25) node [midway, fill = white] {$1$};\n\\draw (-0.1,0) -- (-0.4,0) (-0.1,1) -- (-0.4,1);\n\\draw [<->] (-0.25,0) -- (-0.25,1) node [midway, rotate = 90, fill = white] {$1$};\n\\draw (1,-0.5) node [below] {主视图};\n\\draw (3,0) -- (4,0) -- (4,1) -- cycle;\n\\draw (3,-0.1) -- (3,-0.4) (4,-0.1) -- (4,-0.4);\n\\draw [<->] (3,-0.25) -- (4,-0.25) node [midway, fill = white] {$1$};\n\\draw (3.5,-0.5) node [below] {主视图};\n\\draw (0,-1.5) -- (1,-1.5) --++ (1,-1) --++ (-2,0) --++ (0,1);\n\\draw (0,-2.5) --++ (1,1);\n\\draw (1,-2.5) node [below] {俯视图};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\sqrt{2}$}{$\\sqrt{3}$}{$\\sqrt{5}$}{$\\sqrt{6}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013259": { + "id": "013259", + "content": "某几何体的三视图如图所示, 则该几何体的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (1);\n\\draw (-0.5,-0.5) rectangle (0.5,0.5);\n\\draw (-1,1.5) rectangle ++ (2,2);\n\\draw (2,1.5) rectangle ++ (2,2);\n\\draw [dashed] (-0.5,1.5) --++ (0,2) (0.5,1.5) --++ (0,2);\n\\draw [dashed] (2.5,1.5) --++ (0,2) (3.5,1.5) --++ (0,2);\n\\draw (-1.1,1.5) -- (-1.5,1.5) (-1.1,3.5) -- (-1.5,3.5);\n\\draw [<->] (-1.3,1.5) -- (-1.3,3.5) node [midway, rotate = 90, fill = white] {$4$};\n\\foreach \\i in {-1,-0.5,0.5,1,2,2.5,3.5,4}\n{\\draw (\\i,3.6) --++ (0,0.3);};\n\\draw [<->] (-1,3.75) -- (-0.5,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (0.5,3.75) -- (1,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (2,3.75) -- (2.5,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (3.5,3.75) -- (4,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (-0.5,3.75) -- (0.5,3.75) node [midway, fill = white] {$2$};\n\\draw [<->] (2.5,3.75) -- (3.5,3.75) node [midway, fill = white] {$2$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013260": { + "id": "013260", + "content": "已知某一多面体内接于一个简单组合体, 如果该组合体的主视图, 左视图, 俯视图均如上图所示, 且图中的四边形是边长为$2$的正方形, 则该球的表面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle ({sqrt(3)/2});\n\\draw [dashed] (-0.5,-0.5) rectangle (0.5,0.5);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013261": { + "id": "013261", + "content": "某企业生产甲、乙两种产品, 已知生产每吨甲产品要用$A$原料$3$吨、$B$原料$2$吨; 生产每吨乙产品要用$A$原料$1$吨、$B$原料$3$吨. 销售每吨甲产品可获得利润$5$万元, 每吨乙产品可获得利润$3$万元, 该企业在一个生产周期内消耗$A$原料不超过$13$吨, $B$原料不超过$18$吨, 求该企业可获得的最大利润.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013262": { + "id": "013262", + "content": "如图所示, 给出的是某几何体的三视图, 其中主视图与左视图都是边长为$2$的正三角形, 俯视图为半径等于$1$的圆. 试求这个几何体的体积与侧面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\filldraw (0,0) circle (0.03);\n\\draw (0,0) circle (1);\n\\draw (-1,2) -- (1,2) --++ (120:2) -- cycle;\n\\draw (2,2) -- (4,2) --++ (120:2) -- cycle;\n\\draw (0,-1) node [below] {俯视图};\n\\draw (0,2) node [below] {主视图};\n\\draw (3,2) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013263": { + "id": "013263", + "content": "某工厂的一位产品检验员在检验产品时, 可能把正品错误地检验为次品, 同样也会把次品错误地检验为正品, 他的各次检验是相互独立的. 已知他把正品检验为次品的概率是$0.02$, 把次品检验为正品的概率为$0.01$. 现有$3$件正品和$1$件次品, 则该检验员将这$4$件产品全部检验正确的概率是\\blank{50}.(结果保留三位小数)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013264": { + "id": "013264", + "content": "有一技术难题, 甲单独解决的概率为$\\dfrac{1}{2}$, 乙单独解决的概率为$\\dfrac{1}{3}$, 现两人单独解决难题, 则此难题能被解决的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013265": { + "id": "013265", + "content": "曲线$y=-\\sqrt{2-x^2}$的参数方程可以是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013266": { + "id": "013266", + "content": "若直线$l$的参数方程为: $\\begin{cases}x=1+\\dfrac{1}{2} t, \\\\ y=2-\\dfrac{\\sqrt{3}}{2} t,\\end{cases}$($t$为参数) 则$l$的倾斜角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013267": { + "id": "013267", + "content": "直线$x+\\sqrt{3} y-4=0$和圆$\\begin{cases}x=2 \\cos \\varphi, \\\\ y=2 \\sin \\varphi, \\end{cases}$($0 \\leq \\varphi<2 \\pi$)的位置关系是\\bracket{20}.\n\\fourch{相交但不过圆心}{相交且过圆心}{相切}{相离}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013268": { + "id": "013268", + "content": "下列参数($t$为参数)方程中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2,\\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt{|t|},\\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sin t,\\\\ y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013269": { + "id": "013269", + "content": "将参数方程$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta,\\end{cases}$($\\theta$为参数) 化为普通方程, 所得方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013270": { + "id": "013270", + "content": "巳知圆$C$和圆$\\begin{cases}x=4+4 \\cos \\theta, \\\\ y=5+4 \\sin \\theta,\\end{cases}$($\\theta$为参数)关于直线$\\begin{cases}x=\\dfrac{1}{\\sqrt{10}} t, \\\\ y=3-\\dfrac{1}{\\sqrt{10}} t,\\end{cases}$($t$为参数)对称, 则圆$C$的普通方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013271": { + "id": "013271", + "content": "已知直线$l: y=3 x$, 曲线$C$的参数方程为$\\begin{cases}x=t-\\dfrac{1}{t}, \\\\ y=t+\\dfrac{1}{t},\\end{cases}$($t$为参数) $l$与$C$相交于$A, B$两点, 则$|AB|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013272": { + "id": "013272", + "content": "设$A$、$B$是两个随机事件, 若$P(A)=0.34$, $P(B)=0.32$, $P(A\\cap B)=0.31$, 则$P(A \\cup B)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013273": { + "id": "013273", + "content": "在一个正四面体四个面分别写上$1,2,3,4$. 投掷一次出现$A=\\{\\text{朝下的面出现}1\\text{或}2\\}$, $B=\\{\\text{朝下的面出现}1\\text{或}3\\}$, $C=\\{\\text{朝下的面出现}3\\text{或}4\\}$, 事件$A, B, C$中相互独立的事件是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013274": { + "id": "013274", + "content": "袋中有大小相同的红球和白球若干个, 其中红、白球个数的比为$4: 3$. 假设从袋中任取$2$个球, 取到的都是红球的概率为$\\dfrac{4}{13}$.\n(1) 试问: 袋中的红、白球各有多少个?\\\\\n(2) 现从袋中逐次取球, 每次从袋中任取$1$个球, 若取到白球, 则停止取球, 若取到红球, 则继续下一次取球. 试求: 取球不超过$3$次便停止的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013275": { + "id": "013275", + "content": "曲线$\\begin{cases}x=2 \\sin t, \\\\ y=3 \\cos t,\\end{cases}$($t$为参数) 的焦点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013276": { + "id": "013276", + "content": "直线$l$的参数方程是$\\begin{cases}x=1+2 t, \\\\ y=2-t,\\end{cases}$($t \\in \\mathbf{R}$) 则$l$的方向向量$\\overrightarrow {d}$可以是\\bracket{20}.\n\\fourch{$(1,2)$}{$(2,1)$}{$(-2,1)$}{$(1,-2)$}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013277": { + "id": "013277", + "content": "若实数$x, y$满足$y=\\sqrt{1-x^2}$, 则$y-x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013278": { + "id": "013278", + "content": "根据下列条件研究参数方程$\\begin{cases}x=2+t \\cos \\alpha, \\\\ y=-1+t \\sin \\alpha\\end{cases}$表示何种曲线:\\\\\n(1) $\\alpha$为常数, $t$为参数;\\\\\n(2) $t$为常数, $\\alpha$为参数.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013279": { + "id": "013279", + "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta,\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013280": { + "id": "013280", + "content": "将参数方程$\\begin{cases}x=\\sin \\theta+\\cos \\theta, \\\\ y=\\sin \\theta-\\cos \\theta,\\end{cases}$$\\theta \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$, ($\\theta$为参数) 化为普通方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013281": { + "id": "013281", + "content": "已知曲线$\\Gamma$的参数方程为$\\begin{cases}x=t^3-t \\cos t, \\\\ y=\\ln (t+\\sqrt{t^2+1}),\\end{cases}$ 其中参数$t \\in \\mathbf{R}$, 则曲线$\\Gamma$\\bracket{20}.\n\\fourch{关于$x$轴对称}{关于$y$轴对称}{关于原点对称}{没有对称性}", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013282": { + "id": "013282", + "content": "某一批花生种子, 如果每$1$粒发芽的概率为$\\dfrac{4}{5}$, 那么独立地播下$4$粒种子恰有$2$粒发芽的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013283": { + "id": "013283", + "content": "袋中有$8$个颜色不同, 其它都相同的球, 其中$1$个为黑球, $3$个为白球, $4$个为红球, 若从袋中一次摸出$2$个球, 求所摸出的$2$个球恰为异色球的概率.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013284": { + "id": "013284", + "content": "已知随机事件$A$、$B$是互斥事件. 若$P(A)=0.25$, $P(A \\cup B)=0.78$, 则$P(B)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013285": { + "id": "013285", + "content": "某学生参加一次某志愿者测试. 已知在备选的$10$道试题中, 预计每道题该学生答对的概率为$\\dfrac{2}{3}$. 规定每位考生都从备选题中随机抽出$3$道题进行测试, 则该学生恰答对$2$道题的概率是\\blank{50}.(用数值表示)", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "013286": { + "id": "013286", + "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$, 直线$x+2 y+18=0$, 在椭圆上求一点$P$, 使点$P$到这条直线的距离最短.", + "objs": [], + "tags": [], + "genre": "", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第二轮复习讲义", + "edit": [ + "20230118\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",