From c11fb31fd997eaf0c5e0cd2b091259014e08567e Mon Sep 17 00:00:00 2001 From: Wang Weiye Date: Fri, 16 Sep 2022 20:49:27 +0800 Subject: [PATCH] 20220916 evening --- 工具/关键字筛选题号.ipynb | 4 +- 工具/批量添加题库字段数据.ipynb | 39 ++- 工具/题号选题pdf生成.ipynb | 11 +- 题库0.3/Problems.json | 412 ++++++++++++++++++++++++-------- 4 files changed, 351 insertions(+), 115 deletions(-) diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index 23377c2b..af4ee9d1 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": null, + "execution_count": 10, "metadata": {}, "outputs": [], "source": [ @@ -10,7 +10,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"tags\":[\"第三单元\"],\"objs\":[r\"K030[\\d]\",r\"K031[0123]\"]}\n", + " {\"tags\":[\"第三单元\"],\"content\":[r\"A[\\s\\S]*B\"],\"content2\":[r\"c\"],}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 0dce22fb..d4fbdbf4 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -2,17 +2,44 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 004184 , 字段: objs 中已添加数据: K0203002B\n", - "题号: 004184 , 字段: objs 中已添加数据: K0205002B\n", - "题号: 004184 , 字段: objs 中已添加数据: K0215005B\n", - "题号: 004184 , 字段: objs 中已添加数据: K0221002B\n" + "题号: 000136 , 字段: objs 中已有该数据: K0319002B\n", + "题号: 001492 , 字段: objs 中已添加数据: K0319002B\n", + "题号: 009597 , 字段: objs 中已添加数据: K0319002B\n", + "题号: 000136 , 字段: objs 中已有该数据: K0319003B\n", + "题号: 001492 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001493 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001495 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001496 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001497 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001498 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 001513 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 003154 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 006096 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 006097 , 字段: objs 中已添加数据: K0319003B\n", + "题号: 003179 , 字段: objs 中已添加数据: K0324003B\n", + "题号: 001534 , 字段: objs 中已添加数据: K0324004B\n", + "题号: 001535 , 字段: objs 中已添加数据: K0324004B\n", + "题号: 001538 , 字段: objs 中已添加数据: K0324004B\n", + "题号: 006058 , 字段: objs 中已添加数据: K0324004B\n", + "题号: 010305 , 字段: objs 中已添加数据: K0324004B\n", + "题号: 003152 , 字段: objs 中已添加数据: K0324005B\n", + "题号: 003172 , 字段: objs 中已添加数据: K0324005B\n", + "题号: 003177 , 字段: objs 中已添加数据: K0324005B\n", + "题号: 001538 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 001537 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 001536 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 003172 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 006062 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 006225 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 009612 , 字段: objs 中已添加数据: K0324006B\n", + "题号: 010109 , 字段: objs 中已添加数据: K0324006B\n" ] } ], @@ -20,7 +47,7 @@ "import os,re,json\n", "\n", "\"\"\"---明确数据文件位置---\"\"\"\n", - "datafile = \"临时文件/字段数据.txt\"\n", + "datafile = \"临时文件/l15objs.txt\"\n", "# 双回车分隔，记录内单回车分隔列表，首行为字段名\n", "\"\"\"---文件位置结束---\"\"\"\n", "\n", diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 2740ec69..394f1abb 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/周末卷3初稿_教师用_20220915.tex\n", + "开始编译教师版本pdf文件: 临时文件/第15讲_周期性与其他三角函数预选_教师用_20220916.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/周末卷3初稿_学生用_20220915.tex\n", + "开始编译学生版本pdf文件: 临时文件/第15讲_周期性与其他三角函数预选_学生用_20220916.tex\n", "0\n" ] } @@ -26,14 +26,15 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库\n", "problems = r\"\"\"\n", - "10965:10985\n", + "136,1492,9597,1493,1495,1496,1497,1498,1513,3154,6096,6097,3179,1534,1535,1538,6058,10305,3152,3172,3177,1537,1536,6062,6225,9612,10109\n", + "\n", "\n", "\"\"\"\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/周末卷3初稿\"\n", + "filename = \"临时文件/第15讲_周期性与其他三角函数预选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 5fd8a1bf..e07ab677 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -2460,7 +2460,8 @@ "content": "在$\\triangle ABC$中, 已知$a=5$, $b=4$, $A=2B$. 求$\\cos B$.", "objs": [ "K0315004B", - "K0312003B" + "K0312003B", + "K0314004B" ], "tags": [ "第三单元" @@ -2908,7 +2909,8 @@ "id": "000119", "content": "求下列函数的最小正周期:\\\\\n(1) $y=\\sin \\dfrac x2$;\\\\\n(2) $y=2\\cos (3x-\\dfrac \\pi 4)$.", "objs": [ - "K0319005B" + "K0319005B", + "K0322003B" ], "tags": [ "第三单元" @@ -3106,7 +3108,8 @@ "id": "000127", "content": "求函数$y=\\sin (2x-\\dfrac \\pi 4)-2\\sqrt 2\\sin^2x$的最小正周期.", "objs": [ - "K0311002B" + "K0311002B", + "K0319005B" ], "tags": [ "第三单元" @@ -3204,7 +3207,8 @@ "objs": [ "K0323003B", "K0321003B", - "K0308005B" + "K0308005B", + "K0323002B" ], "tags": [ "第三单元" @@ -28788,7 +28792,7 @@ ], "same": [], "related": [], - "remark": "", + "remark": "定义域有学生写了小于$\\dfrac{4000}{3}$的要求.", "space": "12ex" }, "001148": { @@ -34598,7 +34602,9 @@ "001365": { "id": "001365", "content": "(1) 在三角形$ABC$中, $a=3$, $b=5$, $c=7$, 则$\\cos C=$\\blank{50}, $C=$\\blank{50};\\\\ \n(2) 在三角形$ABC$中, $a=4$, $b=1$, $C=30^\\circ$, 则$c=$\\blank{50}.", - "objs": [], + "objs": [ + "K0315003B" + ], "tags": [ "第三单元" ], @@ -34646,7 +34652,9 @@ "001367": { "id": "001367", "content": "已知$a,b,c$是$\\triangle ABC$的三边. 证明: \\\\ \n(1) 若$\\triangle ABC$是锐角三角形, 则$(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2)>0$;\\\\ \n(2) 若$(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2)>0$, 则$\\triangle ABC$是锐角三角形.", - "objs": [], + "objs": [ + "K0316001B" + ], "tags": [ "第三单元" ], @@ -34696,7 +34704,9 @@ "001369": { "id": "001369", "content": "[Stewart定理]\n在三角形$ABC$中, 在$BC$边上取一点$D$. 记$AC=b$, $AB=c$, $BD=u$, $DC=v$, $AD=t$. 利用$\\angle ADB$和$\\angle ADC$互补以及余弦定理, 证明:\n$$t^2=\\dfrac{b^2u+c^2v}{u+v}-uv.$$", - "objs": [], + "objs": [ + "K0316001B" + ], "tags": [ "第三单元" ], @@ -34744,7 +34754,9 @@ "001371": { "id": "001371", "content": "在三角形$ABC$中, 如果$B=45^\\circ, C=15^\\circ, b=2$, 那么该三角形的最长边长等于\\blank{80}.", - "objs": [], + "objs": [ + "K0314003B" + ], "tags": [ "第三单元" ], @@ -34916,7 +34928,9 @@ "001378": { "id": "001378", "content": "[选做]\n在三角形$ABC$中, 已知三条边上的高$h_a,h_b,h_c$分别为$1/3,1/4,1/5$, 解这个三角形.", - "objs": [], + "objs": [ + "K0315003B" + ], "tags": [ "第三单元" ], @@ -37407,7 +37421,9 @@ "001480": { "id": "001480", "content": "在$\\triangle ABC$中, 若$\\dfrac{a}{\\cos A}=\\dfrac{b}{\\cos B}=\\dfrac{c}{\\cos C}$, 则$\\triangle ABC$的形状是\\blank{100}.", - "objs": [], + "objs": [ + "K0316002B" + ], "tags": [ "第三单元" ], @@ -37433,7 +37449,9 @@ "001481": { "id": "001481", "content": "在$\\triangle ABC$中, 若$\\dfrac{a}{\\cos A}=\\dfrac{b}{\\cos B}=\\dfrac{c}{\\sin C}$, 则$\\triangle ABC$的形状是\\blank{100}.", - "objs": [], + "objs": [ + "K0316002B" + ], "tags": [ "第三单元" ], @@ -37511,7 +37529,9 @@ "001484": { "id": "001484", "content": "在$\\triangle ABC$中, 若$\\dfrac{a^2+b^2}{c^2}=\\dfrac{\\sin^2A+\\sin^2B}{\\sin 2C}$, 则$\\tan C=$\\blank{50}.", - "objs": [], + "objs": [ + "K0316002B" + ], "tags": [ "第三单元" ], @@ -37714,7 +37734,10 @@ "001492": { "id": "001492", "content": "在横线上写出下列函数的一个周期(若周期不存在则写``不存在'').\\\\ \n(1) $y=\\cos\\left(\\dfrac x\\pi+1\\right)$;\\blank{50}\\\\ \n(2) $y=\\sin(\\omega x)\\ \\omega>0$;\\blank{50}\\\\ \n(3) $y=|x|$;\\blank{50}\\\\ \n(4) $y=[x]$;\\blank{50}(这里$[x]$表示$x$的整数部分)\\\\ \n(5) $y=x-[x]$;\\blank{50}\\\\ \n(6) $y=\\sin\\dfrac x3+\\sin\\dfrac x5$;\\blank{50}\\\\ \n(7) $y=[\\sqrt{x}]$;\\blank{50}\\\\ \n(8) $y=\\sqrt{[x]}$.\\blank{50}", - "objs": [], + "objs": [ + "K0319002B", + "K0319003B" + ], "tags": [ "第三单元" ], @@ -37738,7 +37761,9 @@ "001493": { "id": "001493", "content": "设$y=f(x), \\ x\\in\\mathbf{R}$是周期为$2$的函数, 若$x\\in [0,2)$时, $f(x)=x$, 求$x\\in [98,100)$时$f(x)$的解析式.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -37786,7 +37811,9 @@ "001495": { "id": "001495", "content": "下列假命题经常被误以为是正确的, 请对每个命题举出一个反例(不需要论证):\\\\ \n(1) 若$f(x)$与$g(x)$的最小正周期均为$T$, 则$f(x)g(x)$的最小正周期为$T$;\\\\ \n(2) 若$f(x)$与$g(x)$的最小正周期均为$T$, 则$f(x)+g(x)$的最小正周期为$T$.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -37810,7 +37837,9 @@ "001496": { "id": "001496", "content": "写出下列函数的最小正周期:\\\\ \n(1) $f(x)=\\sin x+\\cos x$;\\blank{50}\\\\ \n(2) $f(x)=\\sin x\\cos x$;\\blank{50}\\\\ \n(3) $f(x)=\\sin^2 x$;\\blank{50}\\\\ \n(4) $f(x)=|\\sin x|$;\\blank{50}\\\\ \n(5) $f(x)=\\sin^6 x+\\cos^6 x$;\\blank{50}\\\\ \n(6) $f(x)=|2\\sin x+1|$;\\blank{50}\\\\ \n(7) $f(x)=\\sin x+\\sin 2x$;\\blank{50}\\\\ \n(8) $f(x)=x-[x]$.\\blank{50}", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -37834,7 +37863,9 @@ "001497": { "id": "001497", "content": "求函数$f(x)=\\cos x$的最小正周期, 并证明你的结论.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -37860,7 +37891,9 @@ "001498": { "id": "001498", "content": "求函数$f(x)=|\\cos 2x|$的最小正周期, 并证明你的结论.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -38231,7 +38264,8 @@ "objs": [ "K0220002B", "K0215005B", - "K0208005B" + "K0208005B", + "K0319003B" ], "tags": [ "第二单元", @@ -38753,7 +38787,9 @@ "001534": { "id": "001534", "content": "函数$y=\\tan x-\\cot x$的最小正周期为\\blank{50}.", - "objs": [], + "objs": [ + "K0324004B" + ], "tags": [ "第三单元" ], @@ -38777,7 +38813,9 @@ "001535": { "id": "001535", "content": "函数$y=\\dfrac{2\\tan x}{1-\\tan^2 x}$的最小正周期为\\blank{50}.", - "objs": [], + "objs": [ + "K0324004B" + ], "tags": [ "第三单元" ], @@ -38801,7 +38839,9 @@ "001536": { "id": "001536", "content": "写出函数$y=|\\tan x|$的最小正周期, 单调区间.", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第三单元" ], @@ -38825,7 +38865,9 @@ "001537": { "id": "001537", "content": "写出函数$y=\\tan(\\dfrac{\\pi}{3}-\\dfrac{x}{2})$的单调区间.", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第三单元" ], @@ -38851,7 +38893,10 @@ "001538": { "id": "001538", "content": "写出函数$y=\\tan x+\\cot x$的定义域, 值域, 最小正周期, 单调区间.", - "objs": [], + "objs": [ + "K0324004B", + "K0324006B" + ], "tags": [ "第三单元" ], @@ -39242,7 +39287,9 @@ "001554": { "id": "001554", "content": "已知函数$y=\\dfrac{1}{2}\\cos^2 x+\\dfrac{\\sqrt{3}}{2}\\sin x \\cos x+1$, $x \\in \\mathbf{R}$.\n该函数的图像可由$y=\\sin x$的图像经过怎样的变换得到?", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -39314,7 +39361,9 @@ "001557": { "id": "001557", "content": "函数$y=A\\sin(\\omega x+\\varphi)(A\\neq 0)$的图像相邻最高点与最低点的坐标分别为$\\left(\\dfrac{5\\pi}{12},3\\right)$, $\\left(\\dfrac{11\\pi}{12},-3\\right)$.\\\\ \n(1) 若$A>0, \\omega>0, \\varphi\\in [0,2\\pi)$, 求$A, \\omega, \\varphi$的值;\\\\ \n(2) 若$A>0, \\omega>0$, 求$\\varphi$的值;\\\\ \n(3) 若$A>0, \\varphi\\in [0,2\\pi)$, 求$\\omega, \\varphi$的值;\\\\ \n(4) 若$\\omega>0, \\varphi\\in [0,2\\pi)$, 求$A, \\omega, \\varphi$的值.", - "objs": [], + "objs": [ + "K0321004B" + ], "tags": [ "第三单元" ], @@ -77105,7 +77154,10 @@ "003128": { "id": "003128", "content": "在三角形$ABC$中,\n(1) 用三个角$A,B,C$及外接圆半径$R$表示三角形的面积$S$, 得$S=$\\blank{50};\\\\\n(2) 用三条边$a,b,c$及外接圆半径$R$表示三角形的面积$S$, 得$S=$\\blank{50};\\\\\n(3) 用内切圆半径$r$, 周长$2p$表示三角形面积$S$, 得$S=$\\blank{50}.", - "objs": [], + "objs": [ + "K0314004B", + "K0314006B" + ], "tags": [ "第三单元" ], @@ -77147,7 +77199,9 @@ "003130": { "id": "003130", "content": "在三角形$ABC$中, 若$a^2+c^2-b^2=\\dfrac 12ac$, 则角$B=$\\blank{50}.", - "objs": [], + "objs": [ + "K0315003B" + ], "tags": [ "第三单元" ], @@ -77189,7 +77243,9 @@ "003132": { "id": "003132", "content": "在三角形$ABC$中, $a=3$, $b=2$, $\\sin B=\\dfrac 13$.\\\\\n(1) 若$A$是钝角, 则角$A=$\\blank{50};\\\\\n(2) 若三角形$ABC$是钝角三角形, 则角$A=$\\blank{50}.", - "objs": [], + "objs": [ + "K0314004B" + ], "tags": [ "第三单元" ], @@ -77231,7 +77287,11 @@ "003134": { "id": "003134", "content": "在三角形$ABC$中, 若$\\sin A=\\sqrt 3\\sin C$, $B=\\dfrac{\\pi}6$, $b=2$, 则三角形$ABC$的面积为\\blank{50}.", - "objs": [], + "objs": [ + "K0314001B", + "K0314004B", + "K0315002B" + ], "tags": [ "第三单元" ], @@ -77252,7 +77312,10 @@ "003135": { "id": "003135", "content": "在锐角三角形$ABC$中, 已知$a=1$, $b=2$, 则$c$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0315003B", + "K0315004B" + ], "tags": [ "第三单元" ], @@ -77294,7 +77357,9 @@ "003137": { "id": "003137", "content": "判断下列三角形的形状:\\\\\n(1) $2\\sin A\\sin B=1+\\cos C$;\\\\\n(2) $a\\sin A=b\\cos C+c\\cos B$.", - "objs": [], + "objs": [ + "K0316002B" + ], "tags": [ "第三单元" ], @@ -77315,7 +77380,9 @@ "003138": { "id": "003138", "content": "如图, 某居民小区的平面图呈扇形$AOC$. 小区的两个出入口设置在点$A$及点$C$处. 小区里有两条笔直的小路$AD,DC$, 且$\\angle ADC$的大小为$120^\\circ$. 已知某人从$C$沿$CD$走到$D$用了$10$分钟, 从$D$沿$DA$走到$A$用了$6$分钟. 若此人步行的速度为每分钟$50$米, 求该扇形的半径$OA$的长(精确到$1$米).\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) coordinate (O) node [below] {$O$};\n \\draw ({90-51.79}:3) coordinate (C) node [right] {$C$};\n \\draw ({90+51.79}:3) coordinate (A) node [left] {$A$};\n \\draw ({90+51.79}:{48/49}) coordinate (D) node [left] {$D$};\n \\draw (A) -- (O) -- (C) arc ({90-51.79}:{90+51.79}:3) (C) -- (D);\n \\end{tikzpicture}\n \\end{center}", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -77336,7 +77403,9 @@ "003139": { "id": "003139", "content": "在三角形$ABC$中, $A=120^\\circ$, $c=5$, $a=7$, 则$b=$\\blank{50}.", - "objs": [], + "objs": [ + "K0315004B" + ], "tags": [ "第三单元" ], @@ -77357,7 +77426,9 @@ "003140": { "id": "003140", "content": "在三角形$ABC$中, $A=60^\\circ$, $a=1$, 则$\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}=$\\blank{50}.", - "objs": [], + "objs": [ + "K0314004B" + ], "tags": [ "第三单元" ], @@ -77378,7 +77449,10 @@ "003141": { "id": "003141", "content": "在三角形$ABC$中, $(a+b)^2-c^2=4$, $C=\\dfrac{\\pi}3$, 则面积$S=$\\blank{50}.", - "objs": [], + "objs": [ + "K0314001B", + "K0315003B" + ], "tags": [ "第三单元" ], @@ -77399,7 +77473,9 @@ "003142": { "id": "003142", "content": "在三角形$ABC$中, $\\sin^2 A=\\sin(B+C)\\sin(B-C)$, 则\\bracket{20}.\n\\fourch{$A=90^\\circ$}{$B=90^\\circ$}{$C=90^\\circ$}{$A=B=C$}", - "objs": [], + "objs": [ + "K0316002B" + ], "tags": [ "第三单元" ], @@ -77420,7 +77496,9 @@ "003143": { "id": "003143", "content": "在三角形$ABC$中, $a=\\sqrt 3$, $b=\\sqrt 5$, $c=\\sqrt 7$, 则$bc\\cos A+ca \\cos B+ab \\cos C=$\\blank{50}.", - "objs": [], + "objs": [ + "K0315002B" + ], "tags": [ "第三单元" ], @@ -77462,7 +77540,9 @@ "003145": { "id": "003145", "content": "已知$D,C,B$三点在地面同一直线上, $DC=a$, 从$C,D$两点测得$A$点的仰角分别为$\\alpha,\\beta$($\\alpha>\\beta$), 则点$A$离地面的高$AB=$\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (0,0) node [below left] {$D$} -- (4,0) node [below] {$C$} -- (7,0) node [below right] {$B$} -- (7,3) node [above right] {$A$};\n \\draw (4,0) -- (7,3);\n \\draw (0,0) -- (7,3);\n \\draw (4.5,0) arc (0:atan(1):0.5);\n \\draw (5,0) node [above] {$\\alpha$};\n \\draw (0.5,0) arc(0:atan(3/7):0.5);\n \\draw (1.5,0) node [above] {$\\beta$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -77483,7 +77563,9 @@ "003146": { "id": "003146", "content": "在一个特定时段内, 以点$E$为中心的$7$海里以内海域被设为警戒水域. 点$E$正北$55$海里处有一个雷达观测站$A$. 某时刻测得一艘匀速直线行驶的船只位于点$A$北偏东$45^\\circ$且与点$A$相距$40\\sqrt 2$海里的位置$B$, 经过$40$分钟又测得该船已行驶到点$A$北偏东$45^\\circ+\\arcsin\\dfrac{\\sqrt{26}}{26}$且与点$A$相距$10\\sqrt{13}$海里的位置$C$.\n(1) 求该船的行驶速度(单位: 海里$/$小时);\n(2) 若该船不改变航行方向继续行驶, 判断它是否会进入警戒水域, 并说明理由.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, line cap = round, scale = 0.5]\n \\draw (0,0) -- (0,5.5) node [left] {$A$} coordinate (A);\n \\draw (0,5.5) -- ++ (45:{4*sqrt(2)}) coordinate (B) node [right] {$B$};\n \\draw (A) ++ ({45-asin(1/sqrt(26))}:{sqrt(13)}) coordinate (C) node [right] {$C$} -- (B);\n \\draw (C) -- (A);\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -77527,7 +77609,9 @@ "003148": { "id": "003148", "content": "函数$y=\\sqrt{-\\cos x}$的定义域为\\blank{50}.", - "objs": [], + "objs": [ + "K0322002B" + ], "tags": [ "第三单元" ], @@ -77557,7 +77641,9 @@ "003149": { "id": "003149", "content": "函数$y=\\sin x+\\sqrt 3\\cos x$ ($-\\dfrac{\\pi}2\\le x\\le \\dfrac{\\pi}2$)的值域为\\blank{50}.", - "objs": [], + "objs": [ + "K0320002B" + ], "tags": [ "第三单元" ], @@ -77620,7 +77706,9 @@ "003152": { "id": "003152", "content": "已知函数$f(x)=a\\sin 2x+b\\tan x+1$. 若实数$t$满足$f(t)=7$, 则$f(\\pi-t)=$\\blank{50}.", - "objs": [], + "objs": [ + "K0324005B" + ], "tags": [ "第三单元" ], @@ -77641,7 +77729,9 @@ "003153": { "id": "003153", "content": "若函数$f(x)=\\dfrac{\\cos^2 x}{1+\\sin x}$, 则函数$f(x)$\\bracket{20}.\n\\twoch{有最大值, 也有最小值}{有最大值, 但无最小值}{无最大值, 但有最小值}{无最大值, 也无最小值}", - "objs": [], + "objs": [ + "K0320001B" + ], "tags": [ "第三单元" ], @@ -77662,7 +77752,9 @@ "003154": { "id": "003154", "content": "已知$T>0$. 下列命题中, 能成为命题``函数$f(x)$的一个周期为$T$''的必要不充分条件的是\\bracket{20}.\n\\twoch{函数$f(x)$的一个周期是$-T$}{函数$f(x)$的一个周期是$2T$}{函数$f(x)$的一个周期是$\\dfrac T2$}{函数$f(x)$存在最小正周期}", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -77683,7 +77775,10 @@ "003155": { "id": "003155", "content": "求下列函数的定义域:\\\\\n(1) $y=\\log_{\\sin x}(1+2\\cos x)$;\\\\\n(2) $y=\\sqrt{\\sin x}+\\dfrac 1{\\sqrt{16-x^2}}$.", - "objs": [], + "objs": [ + "K0320001B", + "K0322002B" + ], "tags": [ "第三单元" ], @@ -77704,7 +77799,9 @@ "003156": { "id": "003156", "content": "求下列函数的最大值与最小值:\\\\\n(1) $y=2\\sin x(\\sin x+\\cos x)$;\\\\\n(2) $y=\\sin(\\dfrac{\\pi}4+\\dfrac x2)\\sin(\\dfrac{\\pi}4-\\dfrac x2)$, $\\dfrac{\\pi}4\\le x\\le \\dfrac{5\\pi}4$;\\\\\n(3) $y=1+\\sin x+\\cos x+\\sin x\\cos x$, $x\\in [-\\pi,0]$.", - "objs": [], + "objs": [ + "K0320002B" + ], "tags": [ "第三单元" ], @@ -77748,7 +77845,10 @@ "003158": { "id": "003158", "content": "函数$y=2\\cos x$, $\\dfrac{\\pi}3\\le x\\le \\dfrac{4\\pi}3$的值域为\\blank{50}.", - "objs": [], + "objs": [ + "K0320002B", + "K0322003B" + ], "tags": [ "第三单元" ], @@ -77769,7 +77869,10 @@ "003159": { "id": "003159", "content": "函数$y=2\\cos 2x$, $00$, $\\omega>0$, $0\\le \\varphi<2\\pi$. 如图为定义在$\\mathbf{R}$上的函数$f(x)=A\\sin (\\omega x+\\varphi)$的图像的一部分, 则$f(x)$的解析式为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, line cap = round, scale = 0.8]\n \\draw [->] (-1,0) -- (7,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below right] {$O$};\n \\draw [domain = -pi/6:25*pi/12, samples = 1000] plot (\\x, {2*sin(2*\\x/3/pi*180+20)});\n \\draw [dashed] (7*pi/12,2) -- (0,2) node [left] {$2$};\n \\draw [dashed] (25*pi/12,0) node [above] {$\\dfrac{25\\pi}{12}$} --++ (0,-2) -- (0,-2) node [left] {$-2$};\n \\draw (-pi/6,0) node [below] {$-\\dfrac{\\pi}{6}$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -77963,7 +78078,9 @@ "003168": { "id": "003168", "content": "要得到$y=\\sin(\\dfrac x2+\\dfrac{\\pi}4)$的图像, 可以将$y=\\sin\\dfrac x2$的图像\\bracket{20}.\n\\fourch{向左平移$\\dfrac{\\pi}2$个单位}{向右平移$\\dfrac{\\pi}2$个单位}{向左平移$\\dfrac{\\pi}4$个单位}{向右平移$\\dfrac{\\pi}4$个单位}", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -77986,7 +78103,10 @@ "003169": { "id": "003169", "content": "把函数$y=\\sin x$的图像上所有点向左平移$\\dfrac{\\pi}3$个单位长度, 再把所得图像上所有点的横坐标变为原来的$\\dfrac 12$(纵坐标不变), 得到的图像是函数\\blank{50}的图像.", - "objs": [], + "objs": [ + "K0320002B", + "K0323003B" + ], "tags": [ "第三单元" ], @@ -78028,7 +78148,9 @@ "003171": { "id": "003171", "content": "设常数$\\theta\\in \\mathbf{R}$. 函数$f(x)=\\cos(x+\\theta)$是偶函数, 当且仅当$\\theta=$\\blank{50}.", - "objs": [], + "objs": [ + "K0322002B" + ], "tags": [ "第三单元" ], @@ -78049,7 +78171,10 @@ "003172": { "id": "003172", "content": "若函数$y=\\tan \\omega x$在$(-\\dfrac{\\pi}2,\\dfrac{\\pi}2)$上是减函数, 则实数$\\omega$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0324005B", + "K0324006B" + ], "tags": [ "第三单元" ], @@ -78112,7 +78237,12 @@ "003175": { "id": "003175", "content": "已知函数$f(x)=(2\\sin(x+\\dfrac{\\pi}3)+\\sin x)\\cos x-\\sqrt 3\\sin^2 x$.\\\\\n(1) 求函数$f(x)$的值域与周期;\\\\\n(2) 若$x\\in [0,\\dfrac{\\pi}2]$, 求$f(x)$的单调递减区间;\\\\\n(3) *设常数$a>0$, 若函数$y=f(x)$的图像关于直线$x=a$对称, 求$a$的最小值;\\\\\n(4) 设常数$m\\in \\mathbf{R}$, 若存在$x_0\\in [0,\\dfrac{5\\pi}{12}]$, 使得$mf(x_0)-2=0$成立, 求$m$的取值范围.", - "objs": [], + "objs": [ + "K0319005B", + "K0320002B", + "K0321003B", + "K0323003B" + ], "tags": [ "第三单元" ], @@ -78133,7 +78263,9 @@ "003176": { "id": "003176", "content": "设$A\\ne 0$, $\\omega>0$, $-\\dfrac{\\pi}2<\\varphi<\\dfrac{\\pi}2$, 函数$f(x)=A\\sin(\\omega x+\\varphi)$的部分图像如右图所示, 则$f(x)$的解析式为\\blank{50}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, line cap = round, scale = 0.4]\n \\draw [->] (-4,0) -- (11,0) node [below] {$x$};\n \\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n \\draw (0,0) node [below right] {$O$};\n \\draw [domain = -2:11, samples = 1000] plot (\\x, {-4*sin(180*\\x/8+45)});\n \\draw [dashed] (2,-4) -- (0,-4) node [left] {$-4$} (10,4) -- (0,4) node [left] {$4$};\n \\draw (-2,0) node [above] {$-2$};\n \\draw (6,0) node [below] {$6$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0321004B" + ], "tags": [ "第三单元" ], @@ -78154,7 +78286,9 @@ "003177": { "id": "003177", "content": "函数$f(x)=\\tan 2x$的图像的对称中心是\\blank{50}.", - "objs": [], + "objs": [ + "K0324005B" + ], "tags": [ "第三单元" ], @@ -78175,7 +78309,10 @@ "003178": { "id": "003178", "content": "函数$y=\\sin(2x+\\dfrac{\\pi}4)$图像的对称轴可以是 \\bracket{20}.\n\\fourch{$x=-\\dfrac{3\\pi}4$}{$x=-\\dfrac{3\\pi}8$}{$x=\\dfrac{3\\pi}8$}{$x=\\dfrac{3\\pi}4$}", - "objs": [], + "objs": [ + "K0320002B", + "K0321001B" + ], "tags": [ "第三单元" ], @@ -78196,7 +78333,9 @@ "003179": { "id": "003179", "content": "与函数$y=\\tan(2x+\\dfrac{\\pi}4)$没有公共点的直线可以是\\bracket{20}.\n\\fourch{$x=-\\dfrac{\\pi}2$}{$x=-\\dfrac{\\pi}4$}{$x=\\dfrac{\\pi}8$}{$x=\\dfrac{\\pi}4$}", - "objs": [], + "objs": [ + "K0324003B" + ], "tags": [ "第三单元" ], @@ -78217,7 +78356,10 @@ "003180": { "id": "003180", "content": "*设$\\omega>0$, $0<\\varphi <\\pi$, 若函数$f(x)=\\cos(\\omega x+\\varphi)$为奇函数, 且图像与直线$y=\\dfrac 12$的所有交点中, 距离最近的两个交点的距离为$\\pi$, 则$\\omega =$\\blank{50}, $\\varphi=$\\blank{50}.", - "objs": [], + "objs": [ + "K0321001B", + "K0323003B" + ], "tags": [ "第三单元" ], @@ -78238,7 +78380,9 @@ "003181": { "id": "003181", "content": "*设常数$a\\in \\mathbf{R}$. 若函数$y=\\sin 2x+a\\cos 2x$的图像关于直线$x=-\\dfrac{\\pi}6$对称, 则$a=$\\blank{50}.", - "objs": [], + "objs": [ + "K0321001B" + ], "tags": [ "第三单元" ], @@ -78259,7 +78403,9 @@ "003182": { "id": "003182", "content": "*设常数$a\\in \\mathbf{R}$. 若关于$x$的方程$3\\sin x+4\\cos x=a$在区间$(0,2\\pi)$内恰有两个相异实根$\\alpha,\\beta$, 求$a$的取值范围及$\\alpha+\\beta$的值.", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -78280,7 +78426,11 @@ "003183": { "id": "003183", "content": "求函数$y=\\sin^4 x+2\\sqrt 3\\sin x\\cos x-\\cos^4 x$的最小正周期和值域, 写出该函数在$[0,\\pi]$上的递增区间.", - "objs": [], + "objs": [ + "K0319005B", + "K0320001B", + "K0320002B" + ], "tags": [ "第三单元" ], @@ -78669,7 +78819,9 @@ "003201": { "id": "003201", "content": "设常数$m\\in \\mathbf{R}$, 关于$x$的方程$2-\\sin 2x=m(2+\\sin 2x), \\ x\\in [0,\\pi)$的解集为$A$.\\\\\n(1) 若$A\\ne \\varnothing$, 求$m$的取值范围;\\\\\n(2) 若$A\\subseteq (0,\\pi)$, 且$A$中至少有两个元素, 求$m$的取值范围.", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -87336,7 +87488,11 @@ "003606": { "id": "003606", "content": "已知在$\\triangle ABC$中,$A,B,C$所对边分别为$a,b,c$, 且$a=3$, $b=2c$.\\\\\n(1) 若$A=\\dfrac{2\\pi}3$, 求$\\triangle ABC$的面积;\\\\ \n(2) 若$2\\sin B-\\sin C=1$, 求$\\triangle ABC$的周长.", - "objs": [], + "objs": [ + "K0314001B", + "K0314003B", + "K0315004B" + ], "tags": [ "第三单元" ], @@ -88824,7 +88980,7 @@ ], "same": [], "related": [], - "remark": "", + "remark": "$x=32.5$处单调性的连接做得相当不好.\n单调性的实际意义不会说.", "space": "12ex" }, "003671": { @@ -94866,7 +95022,9 @@ "003943": { "id": "003943", "content": "在$\\triangle ABC$中, 角$A,B,C$所对的边分别为$a,b,c$. 若$a=\\sqrt{2}$, $b=2$, $\\sin B+\\cos B=\\sqrt{2}$, 则角$A$的大小为\\blank{50}.", - "objs": [], + "objs": [ + "K0315004B" + ], "tags": [ "第三单元" ], @@ -98478,7 +98636,9 @@ "004098": { "id": "004098", "content": "某城市的棚户区改造建筑用地平面示意图如图所示, 经过调研、规划确定, 棚改规划用地区域近似为圆面, 该圆的内接四边形$ABCD$区域是原棚户区建筑用地, 测量可知边界$AB=AD=2(\\text{km}), BC=3(\\text{km}),CD=1(\\text{km})$.\\\\\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (1.5,0.288675) coordinate (O) circle ({sqrt(7/3)});\n \\draw (0,0) node [left] {$B$} -- (1,{sqrt(3)}) node [above] {$A$} coordinate (A)-- ({20/7},0.9897433) node [right] {$D$} -- (3,0) node [right] {$C$} -- cycle;\n \\draw (O) ++ (260:{sqrt(7/3)}) node [below] {$P$} coordinate (P) -- (A) (P) -- (3,0) -- (A);\n \\end{tikzpicture}\n\\end{center}\n(1) 求$AC$的长及原棚户区建筑用地$ABCD$的面积;\\\\\n(2) 因地理条件限制, 边界$AD$, $DC$不能变更, 而边界$AB$, $BC$可以调整, 为了增加棚户区建筑用地的面积, 请在弧 $\\overset\\frown{ABC}$上设计一点$P$, 使得棚户区改造后的\n新建筑用地(四边形$APCD$)的面积最大, 并求出这个面积最大值.", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -99001,7 +99161,9 @@ "004119": { "id": "004119", "content": "如图, $A,B,C$三地在以$O$为圆心的圆形区域边界上, $AB=30$公里, $AC=10$公里, $\\angle BAC=60^\\circ$, $D$是圆形区域外一景点, $\\angle DBC=90^\\circ$, $\\angle DCB=60^\\circ$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [left] {$A$} -- (3,0) node [right] {$B$} -- (60:1) node [above] {$C$} coordinate (C) -- (0,0);\n \\draw (1.5,-0.288675) node [below] {$O$} circle (1.52752523);\n \\filldraw (1.5,-0.288675) circle (0.03);\n \\draw (43.90:6.245) node [above] {$D$} coordinate (D) -- (C) (D) -- (3,0) (D) -- (0,0);\n \\end{tikzpicture}\n\\end{center}\n(1) $O$、$A$相距多少公里(精确到小数点后两位)?\n(2) 若一汽车从$A$处出发, 以每小时$50$公里的速度沿公路$AD$行驶到$D$处, 需要多少小时(精确到小数点后两位)?", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -99522,7 +99684,9 @@ "004140": { "id": "004140", "content": "某居民小区为缓解业主停车难的问题, 拟对小区内一块扇形空地$AOB$进行改建. 如图所示, 平行四边形$OMPN$区域为停车场, 其余部分建成绿地, 点$P$在围墙$\\overset\\frown{AB}$上, 点$M$和$N$分别在道路$OA$和道路$OB$上, 且$OA=60\\text{m}$, $\\angle AOB=\\dfrac\\pi 3$. 设$\\angle POB=\\theta$.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (0,0) node [left] {$B$} arc (180:140:3) node [left] {$P$} coordinate (P) arc (140:120:3) node [above] {$A$} -- (3,0) node [right] {$O$}-- cycle;\n \\draw [dashed] (3,0) -- (P);\n \\draw (P) --++ ({3/sin(60)*sin(20)},0) node [right] {$M$};\n \\draw (P) --++ (-60:{3/sin(60)*sin(40)}) node [below] {$N$};\n \\end{tikzpicture}\n\\end{center}\n(1) 求停车场面积$S$(单位: $\\text{m}^2$)关于$\\theta$的函数关系式, 并写出$\\theta$的取值范围;\\\\\n(2) 求停车场面积$S$的最大值以及相应$\\theta$的值.", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -100538,7 +100702,9 @@ "004181": { "id": "004181", "content": "随着生活水平的不断提高, 人们更加关注健康, 重视锻炼, 通过``小步道'', 走出``大健康'', 健康步道成为引领健康生活的一道亮丽风景线. 如图, $A-B-C-A$为某区的一条健康步道, $AB$、$AC$为线段, $\\overset\\frown{BC}$是以$BC$为直径的半圆, $AB=2\\sqrt 3\\text{km}$, $AC=4\\text{km}$, $\\angle BAC=\\dfrac{\\pi}6$.\n\\begin{center}\n \\begin{tikzpicture}[scale =0.7]\n \\draw (0,0) node [right] {$A$} -- (0,{2*sqrt(3)}) node [right] {$B$} arc (0:180:1) node [left] {$C$} -- cycle;\n \\draw (0,0) --++ (150:{8/sqrt(3)}) node [left] {$D$} coordinate (D) -- (-2,{2*sqrt(3)}); \n \\draw (D) ++ (-30:0.5) arc (-30:0:0.5) node [right] {$\\dfrac{\\pi}3$} arc (0:30:0.5);\n \\end{tikzpicture}\n\\end{center}\n(1) 求$\\overset\\frown{BC}$的长度;\\\\\n(2) 为满足市民健康生活需要, 提升城市品位, 改善人居环境, 现计划新建健康步道$A-D-C$($B$, $D$在$AC$两侧), 其中$AD$, $CD$为线段. 若$\\angle ADC=\\dfrac{\\pi}3$, 求新建的健康步道$A-D-C$的路程最多可比原有健康步道$A-B-C$的路程增加多少长度(精确到$0.01\\text{km}$)?", - "objs": [], + "objs": [ + "K0317002B" + ], "tags": [ "第三单元" ], @@ -112676,7 +112842,9 @@ "004672": { "id": "004672", "content": "在$\\triangle ABC$中, $b=2,c=1$, $\\angle B-\\angle C=\\dfrac{\\pi}2$, 则$\\triangle ABC$的周长为\\blank{50}.", - "objs": [], + "objs": [ + "K0315003B" + ], "tags": [ "第三单元" ], @@ -144232,7 +144400,9 @@ "006027": { "id": "006027", "content": "作出函数$y=|\\sin x|$的图像.", - "objs": [], + "objs": [ + "K0318003B" + ], "tags": [ "第三单元" ], @@ -144253,7 +144423,9 @@ "006028": { "id": "006028", "content": "作出函数$y=|\\cos x|+\\cos x$的图像.", - "objs": [], + "objs": [ + "K0318003B" + ], "tags": [ "第三单元" ], @@ -144318,7 +144490,9 @@ "006031": { "id": "006031", "content": "作出函数$y=f(\\sin x)$的图像, 其中$f(x)=\\begin{cases}\n 2, & x\\ge 0, \\\\ -1, & x<0. \\end{cases}$", - "objs": [], + "objs": [ + "K0318003B" + ], "tags": [ "第三单元" ], @@ -144891,7 +145065,9 @@ "006058": { "id": "006058", "content": "函数$y=\\tan 3\\pi x$的最小正周期为\\bracket{20}.\n\\fourch{$\\dfrac 13$}{$\\dfrac 23$}{$\\dfrac 6{\\pi}$}{$\\dfrac 3{\\pi}$}", - "objs": [], + "objs": [ + "K0324004B" + ], "tags": [ "第三单元" ], @@ -144975,7 +145151,9 @@ "006062": { "id": "006062", "content": "下列函数中, 同时满足条件\\textcircled{1} 在$(0,\\dfrac{\\pi}2)$为增函数, \\textcircled{2} 为奇函数, \\textcircled{3} 以$\\pi$为最小正周期的函数是\\bracket{20}.\n\\fourch{$y=\\tan x$}{$y=\\cot x$}{$y=\\tan \\dfrac x2$}{$y=|\\sin x|$}", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第三单元" ], @@ -145693,7 +145871,9 @@ "006096": { "id": "006096", "content": "已知$f(x)$为偶函数, 其图像关于直线$x=a$($a\\ne 0$)对称, 求证: $f(x)$是一个以$2a$为周期的周期函数.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -145714,7 +145894,9 @@ "006097": { "id": "006097", "content": "已知$f(x)$, $g(x)$是定义在$\\mathbf{R}$上的两个函数, 且$g(x)$为奇函数.并满足: \\textcircled{1} $f(0)=1$; \\textcircled{2} 对任何$x,y\\in \\mathbf{R}$都有$f(x-y)=f(x)f(y)+g(x)g(y)$. 求证:\\\\\n(1) 对任何$x\\in \\mathbf{R}$都有$f^2(x)+g^2(x)=1$;\\\\\n(2) $f(x)$是偶函数;\\\\\n(3) 若存在非零实数$a$满足$f(a)=1$, 则$f(x)$是周期函数.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第三单元" ], @@ -148518,7 +148700,9 @@ "006225": { "id": "006225", "content": "函数$y=\\lg \\dfrac{\\tan x}{1+\\tan x}$为增函数的区间是\\bracket{20}.\n\\fourch{$(k\\pi ,k\\pi +\\dfrac{\\pi}4]$, $k\\in \\mathbf{Z}$}{$(k\\pi ,2k\\pi +\\dfrac{\\pi}4]$, $k\\in \\mathbf{Z}$}{$(2k\\pi ,2k\\pi +\\dfrac{\\pi}2]$, $k\\in \\mathbf{Z}$}{$(2k\\pi ,k\\pi +\\dfrac{\\pi}2]$, $k\\in \\mathbf{Z}$}", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第三单元" ], @@ -152526,7 +152710,9 @@ "006414": { "id": "006414", "content": "在$\\triangle ABC$中, 求证: $a(\\sin B-\\sin C)+b(\\sin C-\\sin A)+c(\\sin A-\\sin B)=0$.", - "objs": [], + "objs": [ + "K0316001B" + ], "tags": [ "第三单元" ], @@ -154729,7 +154915,8 @@ "id": "006515", "content": "已知$\\cos x=-\\dfrac 13$, $\\pi \\le x\\le 2\\pi$则$x=$\\blank{50}.", "objs": [ - "K0308004B" + "K0308004B", + "K0316001B" ], "tags": [ "第三单元" @@ -191662,7 +191849,9 @@ "008200": { "id": "008200", "content": "在$\\triangle ABC$中, 证明: $a\\cos B+b\\cos A=c$.", - "objs": [], + "objs": [ + "K0316001B" + ], "tags": [ "第三单元" ], @@ -192547,7 +192736,10 @@ "008240": { "id": "008240", "content": "作出函数$y=-\\cos x,\\ x\\in [0,2\\pi]$的大致图像.", - "objs": [], + "objs": [ + "K0318003B", + "K0322003B" + ], "tags": [ "第三单元" ], @@ -193584,7 +193776,9 @@ "008286": { "id": "008286", "content": "作出函数$y=\\sin x\\cdot \\cos x$在长度为一个周期的闭区间上的大致图像.", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -193609,7 +193803,9 @@ "008287": { "id": "008287", "content": "作出函数$y=2\\sin \\dfrac x2$在长度为一个周期的闭区间上的大致图像.", - "objs": [], + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -193771,7 +193967,9 @@ "008294": { "id": "008294", "content": "电流强度$I$随时间$t$的变化关系是$I=A\\sin (\\omega t+\\varphi _0)$, 设$\\omega =100\\pi$(弧度/秒), $A=50$(安培), $\\varphi _0=\\dfrac{\\pi}6$.\\\\\n(1) 求电流强度$I$的变化周期和频率;\\\\\n(2) 当$t=\\dfrac 1{50}$(秒)时, 求电流强度$I$.", - "objs": [], + "objs": [ + "K0323002B" + ], "tags": [ "第三单元" ], @@ -194721,8 +194919,10 @@ }, "008337": { "id": "008337", - "content": "作出$y=2\\sin (x-\\dfrac{\\pi}4)+1$在区间$[-\\pi ,\\pi]$上的图像, 并写出它的振幅、周期、频率、初相、单词区间及值域.", - "objs": [], + "content": "作出$y=2\\sin (x-\\dfrac{\\pi}4)+1$在区间$[-\\pi ,\\pi]$上的图像, 并写出它的振幅、周期、频率、初相、单调区间及值域.", + "objs": [ + "K0323003B" + ], "tags": [ "第三单元" ], @@ -221982,7 +222182,9 @@ "009597": { "id": "009597", "content": "现实生活中常碰到类似于周期的现象. 根据图中标出的尺度估算下列心电图的周期. (其中横轴的单位是$2\\text{ms}$, $1\\text{s}=1000\\text{ms}$; 纵轴的单位是$\\text{mV}$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [very thick] (0.000,0.59) -- (0.013,0.61) -- (0.027,0.63) -- (0.040,0.65) -- (0.053,0.66) -- (0.067,0.67) -- (0.080,0.67) -- (0.093,0.68) -- (0.107,0.68) -- (0.120,0.68) -- (0.133,0.67) -- (0.147,0.67) -- (0.160,0.65) -- (0.173,0.64) -- (0.187,0.62) -- (0.200,0.60) -- (0.213,0.57) -- (0.227,0.54) -- (0.240,0.51) -- (0.253,0.48) -- (0.267,0.44) -- (0.280,0.41) -- (0.293,0.40) -- (0.307,0.54) -- (0.320,0.65) -- (0.333,0.93) -- (0.347,1.13) -- (0.360,1.26) -- (0.373,1.35) -- (0.387,1.41) -- (0.400,1.42) -- (0.413,1.37) -- (0.427,1.25) -- (0.440,1.01) -- (0.453,0.85) -- (0.467,0.66) -- (0.480,0.52) -- (0.493,0.49) -- (0.507,0.48) -- (0.520,0.48) -- (0.533,0.47) -- (0.547,0.47) -- (0.560,0.48) -- (0.573,0.48) -- (0.587,0.49) -- (0.600,0.50) -- (0.613,0.51) -- (0.627,0.52) -- (0.640,0.53) -- (0.653,0.55) -- (0.667,0.56) -- (0.680,0.58) -- (0.693,0.60) -- 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(10.587,0.56) -- (10.600,0.57) -- (10.613,0.57) -- (10.627,0.59) -- (10.640,0.59) -- (10.653,0.61) -- (10.667,0.62) -- (10.680,0.63) -- (10.693,0.65) -- (10.707,0.67) -- (10.720,0.69) -- (10.733,0.71) -- (10.747,0.74) -- (10.760,0.76) -- (10.773,0.79) -- (10.787,0.81) -- (10.800,0.82) -- (10.813,0.83) -- (10.827,0.84) -- (10.840,0.84) -- (10.853,0.83) -- (10.867,0.81) -- (10.880,0.79) -- (10.893,0.77) -- (10.907,0.73) -- (10.920,0.69) -- (10.933,0.65) -- (10.947,0.63) -- (10.960,0.61) -- (10.973,0.59) -- (10.987,0.58) -- (11.000,0.57) -- (11.013,0.55) -- (11.027,0.55) -- (11.040,0.54) -- (11.053,0.53) -- (11.067,0.52) -- (11.080,0.52) -- (11.093,0.52) -- (11.107,0.51) -- (11.120,0.51) -- (11.133,0.51) -- (11.147,0.51) -- (11.160,0.51) -- (11.173,0.51) -- (11.187,0.51) -- (11.200,0.52) -- (11.213,0.52) -- (11.227,0.52) -- (11.240,0.53) -- (11.253,0.53) -- (11.267,0.53) -- (11.280,0.54) -- (11.293,0.54) -- (11.307,0.54) -- (11.320,0.54) -- (11.333,0.55) -- (11.347,0.55) -- (11.360,0.55) -- (11.373,0.55) -- (11.387,0.55) -- (11.400,0.56) -- (11.413,0.56) -- (11.427,0.56) -- (11.440,0.56) -- (11.453,0.56) -- (11.467,0.56) -- (11.480,0.56) -- (11.493,0.56) -- (11.507,0.55) -- (11.520,0.55) -- (11.533,0.55) -- (11.547,0.54) -- (11.560,0.54) -- (11.573,0.55) -- (11.587,0.57) -- (11.600,0.58) -- (11.613,0.60) -- (11.627,0.62) -- (11.640,0.63) -- (11.653,0.65) -- (11.667,0.65) -- (11.680,0.66) -- (11.693,0.66) -- (11.707,0.65) -- (11.720,0.65) -- (11.733,0.63) -- (11.747,0.61) -- (11.760,0.58) -- (11.773,0.56) -- (11.787,0.52) -- (11.800,0.50) -- (11.813,0.47) -- (11.827,0.45) -- (11.840,0.43) -- (11.853,0.41) -- (11.867,0.40) -- (11.880,0.40) -- (11.893,0.43) -- (11.907,0.55) -- (11.920,0.90) -- (11.933,1.12) -- (11.947,1.39) -- (11.960,1.45) -- (11.973,1.37) -- (11.987,1.20) -- (12.000,0.97) -- (12.013,0.82) -- (12.027,0.64) -- (12.040,0.61) -- (12.053,0.60) -- (12.067,0.59) -- (12.080,0.59) -- (12.093,0.58) -- (12.107,0.58) -- (12.120,0.59) -- (12.133,0.59) -- (12.147,0.60) -- (12.160,0.60) -- (12.173,0.61) -- (12.187,0.62) -- (12.200,0.63) -- (12.213,0.63) -- (12.227,0.65) -- (12.240,0.66) -- (12.253,0.68) -- (12.267,0.70) -- (12.280,0.73) -- (12.293,0.75) -- (12.307,0.78) -- (12.320,0.82) -- (12.333,0.84) -- (12.347,0.88) -- (12.360,0.91) -- (12.373,0.93) -- (12.387,0.95) -- (12.400,0.96) -- (12.413,0.95) -- (12.427,0.94) -- (12.440,0.92) -- (12.453,0.90) -- (12.467,0.87) -- (12.480,0.84) -- (12.493,0.81) -- (12.507,0.77) -- (12.520,0.74) -- (12.533,0.71) -- (12.547,0.68) -- (12.560,0.65) -- (12.573,0.62) -- (12.587,0.60) -- (12.600,0.59) -- (12.613,0.57) -- (12.627,0.56) -- (12.640,0.56) -- (12.653,0.56) -- (12.667,0.55) -- (12.680,0.56) -- (12.693,0.56) -- (12.707,0.56) -- (12.720,0.57) -- (12.733,0.57) -- (12.747,0.58) -- (12.760,0.58) -- (12.773,0.58) -- (12.787,0.59) -- (12.800,0.59) -- (12.813,0.60) -- (12.827,0.60) -- (12.840,0.60) -- (12.853,0.60) -- (12.867,0.60) -- (12.880,0.61) -- (12.893,0.61) ;\n\\draw (0,0) rectangle (12.893,2);\n\\draw (0,0) node [left] {$-500$} node [below] {$0$};\n\\draw (0.2,1.333) -- (0,1.333) node [left] {$500$};\n\\draw (0,2) node [left] {$1000$};\n\\foreach \\i in {500,1000,...,3000}\n{\\draw (\\i/3000*12.893,0.2) -- (\\i/3000*12.893,0) node [below] {$\\i$};\n\\draw (\\i/3000*12.893,1.8) -- (\\i/3000*12.893,2);};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0319002B" + ], "tags": [ "第三单元" ], @@ -222297,7 +222499,9 @@ "009612": { "id": "009612", "content": "求函数$y=\\tan (3x+\\dfrac\\pi 4)$的定义域, 并写出其单调区间.", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第三单元" ], @@ -233218,7 +233422,9 @@ "010109": { "id": "010109", "content": "求下列各式中$x$的值(其中$x>0$):\n(1) $x^3=27$;\\\\\n(2) $x^4=121$;\\\\\n(3) $x^\\frac 32=1000$;\\\\\n(4) $x^{-\\frac 43}=\\dfrac{16}{625}$.", - "objs": [], + "objs": [ + "K0324006B" + ], "tags": [ "第二单元" ], @@ -237546,7 +237752,9 @@ "010305": { "id": "010305", "content": "求下列函数的最小正周期:\\\\\n(1) $y=\\tan (-\\dfrac 12x)$;\\\\\n(2) $y=\\tan (3x+\\dfrac \\pi 3)$.", - "objs": [], + "objs": [ + "K0324004B" + ], "tags": [ "第三单元" ], @@ -279554,7 +279762,7 @@ ], "same": [], "related": [], - "remark": "", + "remark": "第(2)小题非常多学生在求取值范围时把离散的$x$的范围悄悄改为连续的$x$的范围了.", "space": "12ex" }, "030008": { @@ -279644,7 +279852,7 @@ ], "same": [], "related": [], - "remark": "", + "remark": "第(2)小题极个别学生用了导数, 但没有对$x=20$处进行说明", "space": "12ex" }, "030012": { @@ -279846,7 +280054,7 @@ }, "030019": { "id": "030019", - "content": "把$5\\sin \\alpha -12\\cos \\alpha$化成$A\\sin (\\alpha +\\varphi)$($A>0$)的形式, 所得结果为\\blank{50}.", + "content": "把$5\\sin \\alpha +12\\cos \\alpha$化成$A\\sin (\\alpha +\\varphi)$($A>0$, $\\varphi\\in (0,2\\pi)$)的形式, 则其中常数$\\varphi$的值为\\blank{50}.", "objs": [ "K0311002B" ],