diff --git a/工具/分年级专用工具/赋能卷生成.ipynb b/工具/分年级专用工具/赋能卷生成.ipynb index 9c39d8eb..e54f3d35 100644 --- a/工具/分年级专用工具/赋能卷生成.ipynb +++ b/工具/分年级专用工具/赋能卷生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 8, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -17,11 +17,11 @@ "#在 临时文件/赋能答题纸 目录中保留一个pdf(赋能试卷的答题纸), 不留别的pdf文件. \n", "#在 临时文件/赋能答题纸 目录中保留 赋能template.tex.\n", "\"\"\"---设置文件名---\"\"\"\n", - "filename = \"赋能02\"\n", + "filename = \"赋能03\"\n", "\n", "\"\"\"---设置题目列表---\"\"\"\n", "problems = r\"\"\"\n", - "1750,337,2675,339:345\n", + "346,347,348,349,30071,351,352,30072,354,30073\n", "\"\"\"\n", "#完成后将含有 filename 的文件移至其它目录\n", "\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 62e71ab7..848e39d0 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 @@ -2,47 +2,37 @@ "cells": [ { "cell_type": "code", - "execution_count": 7, + "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "题号: 003148 , 字段: ans 中已修改数据: $\\{x|\\dfrac\\pi 2+2k\\pi\\le x\\le \\dfrac{3\\pi}2+2k\\pi, \\ k\\in \\mathbf{Z}\\}$\n", - "题号: 003155 , 字段: ans 中已修改数据: (1) $\\{x|2k\\pi=latex,scale = 0.5]\n", - "\\draw [->] (-1,0) -- (15,0) node [below] {$x$};\n", - "\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n", - "\\draw (0,0) node [below left] {$O$};\n", - "\\draw [domain = 0:720] plot ({\\x/180*pi},{2*sin(\\x/2)});\n", - "\\draw [dashed] (pi,0) node [below] {$\\pi$} -- (pi,2) -- (0,2) node [left] {$2$};\n", - "\\draw [dashed] ({3*pi},0) node [above] {$3\\pi$} -- ({3*pi},-2) -- (0,-2) node [left] {$-2$};\n", - "\\draw ({4*pi},0) node [above] {$4\\pi$} ({2*pi},0) node [below left] {$2\\pi$};\n", - "\\end{tikzpicture}\n", - "题号: 003169 , 字段: ans 中已修改数据: $y=\\sin(2x+\\dfrac\\pi 3)$\n", - "题号: 001554 , 字段: ans 中已修改数据: \\textcircled{1} 横坐标不变, 纵坐标变为原来的$\\dfrac 12$倍, \\textcircled{2} 向上平移$\\dfrac 54$个单位, \\textcircled{3} 向左平移$\\dfrac \\pi 6$个单位, \\textcircled{4} 纵坐标不变, 横坐标变为原来的$\\dfrac 12$倍.\n", - "题号: 003176 , 字段: ans 中已修改数据: $f(x)=-4\\sin(\\dfrac \\pi 8 x+\\dfrac \\pi 4)$\n", - "题号: 003178 , 字段: ans 中已修改数据: B\n", - "题号: 003181 , 字段: ans 中已修改数据: $-\\dfrac{\\sqrt{3}}3$\n", - "题号: 003167 , 字段: ans 中已有该数据: \n", - "题号: 003171 , 字段: ans 中已有该数据: \n", - "题号: 003166 , 字段: ans 中已有该数据: \n", - "题号: 003168 , 字段: ans 中已有该数据: \n", - "题号: 008337 , 字段: ans 中已有该数据: \n", - "题号: 003153 , 字段: ans 中已有该数据: \n", - "题号: 003160 , 字段: ans 中已有该数据: \n", - "题号: 003183 , 字段: ans 中已有该数据: \n", - "题号: 003164 , 字段: ans 中已有该数据: \n", - "题号: 003182 , 字段: ans 中已有该数据: \n", - "题号: 000131 , 字段: ans 中已有该数据: \n" + "题号: 002835 , 字段: objs 中已添加数据: K0215001B\n", + "题号: 002835 , 字段: objs 中已添加数据: K0216005B\n", + "题号: 030025 , 字段: objs 中已添加数据: K0215003B\n", + "题号: 030041 , 字段: objs 中已添加数据: K0204004B\n", + "题号: 030041 , 字段: objs 中已添加数据: K0206002B\n", + "题号: 030042 , 字段: objs 中已添加数据: K0220002B\n", + "题号: 030042 , 字段: objs 中已添加数据: K0219003B\n", + "题号: 030043 , 字段: objs 中已添加数据: K0214002B\n", + "题号: 030044 , 字段: objs 中已添加数据: K0214002B\n", + "题号: 030045 , 字段: objs 中已添加数据: K0215003B\n", + "题号: 030046 , 字段: objs 中已添加数据: K0219001B\n", + "题号: 030046 , 字段: objs 中已添加数据: K0219003B\n", + "题号: 030046 , 字段: objs 中已添加数据: K0220001B\n", + "题号: 030047 , 字段: objs 中已添加数据: K0219003B\n", + "题号: 030048 , 字段: objs 中已添加数据: K0219001B\n", + "题号: 030049 , 字段: objs 中已添加数据: K0218001B\n", + "题号: 030050 , 字段: objs 中已添加数据: K0223001B\n", + "题号: 030050 , 字段: objs 中已添加数据: K0223004B\n", + "题号: 030051 , 字段: objs 中已添加数据: K0234003X\n", + "题号: 030052 , 字段: objs 中已添加数据: K0221002B\n", + "题号: 030053 , 字段: objs 中已添加数据: K0221002B\n", + "题号: 030060 , 字段: objs 中已添加数据: K0221002B\n", + "题号: 030073 , 字段: objs 中已添加数据: K0218001B\n", + "题号: 030073 , 字段: objs 中已添加数据: K0219001B\n" ] } ], diff --git a/工具/添加关联题目.ipynb b/工具/添加关联题目.ipynb index accf7ccb..519007b6 100644 --- a/工具/添加关联题目.ipynb +++ b/工具/添加关联题目.ipynb @@ -2,15 +2,15 @@ "cells": [ { "cell_type": "code", - "execution_count": 47, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "import os,re,json,time\n", "\n", "\"\"\"---设置原题目id与新题目id---\"\"\"\n", - "old_id = \"3181\"\n", - "new_id = \"30068\"\n", + "old_id = \"355\"\n", + "new_id = \"30073\"\n", "\"\"\"---设置完毕---\"\"\"\n", "\n", "old_id = old_id.zfill(6)\n", diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb index 848a4987..6b8b1265 100644 --- a/工具/讲义生成.ipynb +++ b/工具/讲义生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [ { @@ -11,9 +11,11 @@ "text": [ "正在处理题块 1 .\n", "题块 1 处理完毕.\n", - "开始编译教师版本pdf文件: 临时文件/国庆卷_教师_20220924.tex\n", + "正在处理题块 2 .\n", + "题块 2 处理完毕.\n", + "开始编译教师版本pdf文件: 临时文件/17_平面向量的投影及数量积_教师_20220924.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/国庆卷_学生_20220924.tex\n", + "开始编译学生版本pdf文件: 临时文件/17_平面向量的投影及数量积_学生_20220924.tex\n", "0\n" ] } @@ -26,34 +28,35 @@ "\"\"\"---设置模式结束---\"\"\"\n", "\n", "\"\"\"---设置模板文件名---\"\"\"\n", - "# template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", + "template_file = \"模板文件/第一轮复习讲义模板.tex\"\n", "# template_file = \"模板文件/测验周末卷模板.tex\"\n", - "template_file = \"模板文件/日常选题讲义模板.tex\"\n", + "# template_file = \"模板文件/日常选题讲义模板.tex\"\n", "\"\"\"---设置模板文件名结束---\"\"\"\n", "\n", "\"\"\"---设置其他预处理替换命令---\"\"\"\n", "#2023届第一轮讲义更换标题\n", - "# exec_list = [(\"标题数字待处理\",\"19\"),(\"标题文字待处理\",\"复数的几何意义与实系数二次方程\")] \n", - "# enumi_mode = 0\n", + "exec_list = [(\"标题数字待处理\",\"17\"),(\"标题文字待处理\",\"平面向量的投影及数量积\")] \n", + "enumi_mode = 0\n", "\n", "#2023届测验卷与周末卷\n", "# exec_list = [(\"标题替换\",\"周末卷03\")]\n", "# enumi_mode = 1\n", "\n", "#日常选题讲义\n", - "exec_list = [(\"标题文字待处理\",\"2022年国庆卷(易错题订正)\")] \n", - "enumi_mode = 0\n", + "# exec_list = [(\"标题文字待处理\",\"2022年国庆卷(易错题订正)\")] \n", + "# enumi_mode = 0\n", "\n", "\"\"\"---其他预处理替换命令结束---\"\"\"\n", "\n", "\"\"\"---设置目标文件名---\"\"\"\n", - "destination_file = \"临时文件/国庆卷\"\n", + "destination_file = \"临时文件/17_平面向量的投影及数量积\"\n", "\"\"\"---设置目标文件名结束---\"\"\"\n", "\n", "\n", "\"\"\"---设置题号数据---\"\"\"\n", "problems = [\n", - "\"30029:30068\"\n", + "\"481,3347,3330,153,3356,414,1882,141,3341,1892,1896,1898,1916,1905,1902\",\n", + "\"894,1912,1914,871,1877,760,792,142,1906,1913,1886,3345,1911,159\"\n", "]\n", "\"\"\"---设置题号数据结束---\"\"\"\n", "\n", diff --git a/工具/课时目标及课时划分信息汇总.ipynb b/工具/课时目标及课时划分信息汇总.ipynb index fbde2f4b..0123dcf5 100644 --- a/工具/课时目标及课时划分信息汇总.ipynb +++ b/工具/课时目标及课时划分信息汇总.ipynb @@ -9,8 +9,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标_20220918.tex\n", - "开始编译课时划分信息pdf文件: 临时文件/课时划分_20220918.tex\n" + "开始编译单元与课时目标信息pdf文件: 临时文件/课时目标及单元目标_20220924.tex\n", + "开始编译课时划分信息pdf文件: 临时文件/课时划分_20220924.tex\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 6b96ba71..d0e98ca8 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": 4, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/国庆卷粗_教师用_20220924.tex\n", + "开始编译教师版本pdf文件: 临时文件/第四单元未挂钩_教师用_20220924.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/国庆卷粗_学生用_20220924.tex\n", + "开始编译学生版本pdf文件: 临时文件/第四单元未挂钩_学生用_20220924.tex\n", "0\n" ] } @@ -26,14 +26,16 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库\n", "problems = r\"\"\"\n", - "30029:30068\n", + 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+ "\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/国庆卷粗\"\n", + "filename = \"临时文件/第四单元未挂钩\"\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 00711da8..9874cbfc 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -8547,7 +8547,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "030071" + ], "remark": "", "space": "" }, @@ -8618,7 +8620,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "030072" + ], "remark": "", "space": "" }, @@ -8671,7 +8675,9 @@ "20220624\t朱敏慧, 王伟叶" ], "same": [], - "related": [], + "related": [ + "030073" + ], "remark": "", "space": "" }, @@ -12747,7 +12753,7 @@ }, "000515": { "id": "000515", - "content": "向量$\\overrightarrow{i}$、$\\overrightarrow{j}$是平面直角坐标系$x$轴、$y$轴的基本单位向量, 且$|\\overrightarrow a-\\overrightarrow i|+|\\overrightarrow a-2\\overrightarrow j|=\\sqrt5$, 则$|\\overrightarrow a+2 \\overrightarrow i|$的取值范围为\\blank{50}.", + "content": "向量$\\overrightarrow{i}$、$\\overrightarrow{j}$是平面直角坐标系$x$轴、$y$轴正方向上的单位向量, 且$|\\overrightarrow a-\\overrightarrow i|+|\\overrightarrow a-2\\overrightarrow j|=\\sqrt5$, 则$|\\overrightarrow a+2 \\overrightarrow i|$的取值范围为\\blank{50}.", "objs": [ "K0501003B", "K0502007B" @@ -12761,7 +12767,7 @@ "duration": -1, "usages": [ "20220302\t2022届高三01班\t0.581", - "20220622\t2022届高三01班 0.884" + "20220622\t2022届高三01班\t0.884" ], "origin": "赋能练习", "edit": [ @@ -46829,14 +46835,14 @@ }, "001850": { "id": "001850", - "content": "若$A,B,C,D$是平面上任意四点, 则下列命题中正确的有\\bracket{20}.\n\\twoch{$\\overrightarrow{AB}+\\overrightarrow{CD}=\\overrightarrow{BC}+\\overrightarrow{DA}$;}{$\\overrightarrow{AC}+\\overrightarrow{BD}=\\overrightarrow{BC}+\\overrightarrow{AD}$;}\n{$\\overrightarrow{AC}-\\overrightarrow{BD}=\\overrightarrow{DC}+\\overrightarrow{AB}$;}{$\\overrightarrow{AB}+\\overrightarrow{BC}=\\overrightarrow{CD}+\\overrightarrow{DA}$.}", + "content": "若$A,B,C,D$是平面上任意四点, 则下列命题中正确的有\\blank{50}(填写序号).\\\\\n\\textcircled{1} $\\overrightarrow{AB}+\\overrightarrow{CD}=\\overrightarrow{BC}+\\overrightarrow{DA}$; \\textcircled{2} $\\overrightarrow{AC}+\\overrightarrow{BD}=\\overrightarrow{BC}+\\overrightarrow{AD}$; \\textcircled{3} $\\overrightarrow{AC}-\\overrightarrow{BD}=\\overrightarrow{DC}+\\overrightarrow{AB}$; \\textcircled{4} $\\overrightarrow{AB}+\\overrightarrow{BC}=\\overrightarrow{CD}+\\overrightarrow{DA}$.", "objs": [ "K0502004B" ], "tags": [ "第五单元" ], - "genre": "选择题", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -46987,7 +46993,7 @@ }, "001856": { "id": "001856", - "content": "判断下列命题的真假, 如果是假命题则在命题前的横线上写上``F'', 如果是真命题则写上``T''.\\\\ \n\\begin{enumerate}[\\blank{30}(1)]\n\\item 与非零向量$\\overrightarrow{a}$平行的单位向量一定是$\\dfrac{1}{|\\overrightarrow{a}|}\\overrightarrow{a}$.\\\\ \n\\item 若两个非零向量互相平行, 则这两个向量所在的直线平行或重合.\\\\ \n\\item 若非零向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$满足$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 则$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$可以依次首尾相接构成三角形.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$.\\\\ \n\\item 若存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\item 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\item 若存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.\\\\ \n\\end{enumerate}", + "content": "判断下列命题的真假, 如果是假命题则在命题前的横线上写上``F'', 如果是真命题则写上``T''.\\\\ \n\\blank{30}(1) 与非零向量$\\overrightarrow{a}$平行的单位向量一定是$\\dfrac{1}{|\\overrightarrow{a}|}\\overrightarrow{a}$.\\\\ \n\\blank{30}(2) 若两个非零向量互相平行, 则这两个向量所在的直线平行或重合.\\\\ \n\\blank{30}(3) 若非零向量$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$满足$\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, 则$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$可以依次首尾相接构成三角形.\\\\ \n\\blank{30}(4) 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$.\\\\ \n\\blank{30}(5) 若存在实数$\\lambda$, 使得$\\overrightarrow{b}=\\lambda\\overrightarrow{a}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.\\\\ \n\\blank{30}(6) 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\blank{30}(7) 若$\\overrightarrow{a}$与$\\overrightarrow{b}$平行, 则存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$.\\\\ \n\\blank{30}(8) 若存在不全为零的实数$\\lambda,\\mu$, 使得$\\lambda\\overrightarrow{a}+\\mu\\overrightarrow{b}=\\overrightarrow{0}$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$平行.", "objs": [ "K0501004B", "K0502002B", @@ -47358,7 +47364,7 @@ }, "001870": { "id": "001870", - "content": "在$\\triangle ABC$中, 点$D$在$AB$上, $CD$平分$\\angle ACB$, 若$\\overrightarrow{CB}=\\overrightarrow{a}$, $\\overrightarrow{CA}=\\overrightarrow{b}$, $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=4$, 则在$\\overrightarrow{a},\\overrightarrow{b}$这组基下, $\\overrightarrow{CD}$的坐标为\\blank{80}.", + "content": "在$\\triangle ABC$中, 点$D$在$AB$上, $CD$平分$\\angle ACB$, 若$\\overrightarrow{CB}=\\overrightarrow{a}$, $\\overrightarrow{CA}=\\overrightarrow{b}$, $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=4$, 则在$\\overrightarrow{a},\\overrightarrow{b}$这组基下, 设$\\overrightarrow{CD}=x\\overrightarrow{a}+y\\overrightarrow{b}$, 则$(x,y)=$\\blank{50}.", "objs": [ "K0506003B" ], @@ -48420,7 +48426,7 @@ }, "001910": { "id": "001910", - "content": "已知$\\overrightarrow{a}=(1,-2),\\overrightarrow{b}=(2,3),\\overrightarrow{c}=(1,1)$, 将$\\overrightarrow{a}$表示为$\\overrightarrow{b_1}+\\overrightarrow{c_1}$的形式,\n其中$\\overrightarrow{b_1}\\parallel\\overrightarrow{b},\\overrightarrow{c_1}\\parallel\\overrightarrow{c}$, 结果为$\\overrightarrow{a}=$\\blank{30}$+$\\blank{30}.(在横线上填入$\\overrightarrow{b_1},\\overrightarrow{c_1}$的坐标.)", + "content": "已知$\\overrightarrow{a}=(1,-2),\\overrightarrow{b}=(2,3),\\overrightarrow{c}=(1,1)$, 将$\\overrightarrow{a}$表示为$\\overrightarrow{b_1}+\\overrightarrow{c_1}$的形式,\n其中$\\overrightarrow{b_1},\\overrightarrow{c_1}$分别为$\\overrightarrow{b},\\overrightarrow{c}$的单位向量, 结果为$\\overrightarrow{a}=$\\blank{30}$\\overrightarrow{b_1}+$\\blank{30}$\\overrightarrow{c_1}$.", "objs": [ "K0506003B" ], @@ -48494,7 +48500,9 @@ "20220625\t王伟叶" ], "same": [], - "related": [], + "related": [ + "030070" + ], "remark": "", "space": "" }, @@ -70858,7 +70866,10 @@ "002835": { "id": "002835", "content": "已知$f(x)=\\begin{cases} x-2, & x>8, \\\\ f(x+3), & x\\le 8, \\end{cases}$ 则$f(2)=$\\blank{50}.", - "objs": [], + "objs": [ + "K0215001B", + "K0216005B" + ], "tags": [ "第二单元" ], @@ -74409,7 +74420,9 @@ "002981": { "id": "002981", "content": "已知函数$f(x)=|x-\\dfrac 1x|, \\ x>0$.\\\\\n(1)\t画出函数$y=f(x)$的草图;\\\\\n(2) 当$0c)$. 若$a\\ne 0$, 关于$x$的不等式$x^2-\\left(2a+\\dfrac 1a\\right)x-1<0$的非空解集(用区间表示)记为$I(a)$, 则当区间$I(a)$的长度取得最小值时, 实数$a$的值为\\blank{50}.", "objs": [ - "K0113002B" + "K0113002B", + "K0109004B", + "K0119001B" ], "tags": [ "第一单元", @@ -92343,7 +92360,9 @@ "003769": { "id": "003769", "content": "$f(x)$是定义在$\\mathbf{R}$上且周期为$2$的函数, 在区间$[-1,1]$上, $f(x)=\\begin{cases}ax+1, & -1\\le x<0,\\\\\\dfrac{bx+2}{x+1}, & 0\\le x\\le 1,\\end{cases}$ 其中$a,b\\in \\mathbf{R}$. 若$f\\left(\\dfrac 12\\right)=f\\left(\\dfrac 32\\right)$, 则$a+3b$的值为\\blank{50}.", - "objs": [], + "objs": [ + "K0319003B" + ], "tags": [ "第二单元" ], @@ -92786,7 +92805,11 @@ "003789": { "id": "003789", "content": "设函数$f(x)=\\log_\\frac 12 x$, $g(x)=f^{-1}(|x|)$.\\\\\n(1) 求函数$g(x)$的解析式, 并画出大致图像;\\\\\n(2) 若不等式$g(x)+g(2x)\\le k$对任意$x\\in \\mathbf{R}$恒成立, 求实数$k$的取值范围.", - "objs": [], + "objs": [ + "K0213006B", + "K0210005B", + "K0211001B" + ], "tags": [ "第二单元" ], @@ -93006,7 +93029,9 @@ "003799": { "id": "003799", "content": "已知$f(x)=4-\\dfrac 1x$, 若存在区间$[a,b]\\subseteq \\left(\\dfrac 13,+\\infty\\right)$, 使得$\\{y|y=f(x), \\ x\\in [a,b]\\}=[ma,mb]$, 则实数$m$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -93857,7 +93882,9 @@ "003838": { "id": "003838", "content": "已知函数$f(x)=\\begin{cases}\n\\dfrac 3x, & x\\ge 3,\\\\ \\log_3 x, & 01, \\ b<-1)$, 则$y=f^{-1}(x)$的图像一定不经过第\\blank{50}象限.", - "objs": [], + "objs": [ + "K0210005B", + "K0208005B" + ], "tags": [ "第二单元" ], @@ -96346,7 +96385,10 @@ "003953": { "id": "003953", "content": "已知集合$M$是满足下列性质的函数$f(x)$的全体, 存在非零常数$T$, 对任意$x\\in \\mathbf{R}$, 有$f(x+T)=Tf(x)$成立.\\\\\n(1) 函数$f(x)=x$是否属于集合$M$? 说明理由;\\\\\n(2) 设$f(x)\\in M$, 且$T=2$, 已知当$10$, 则$f(a_1)+f(a_2)+f(a_3)+\\cdots+f(a_{2012})+f(a_{2013})$的值\\bracket{20}.\n\\fourch{恒为正数}{恒为负数}{恒为$0$}{可正可负}", - "objs": [], + "objs": [ + "K0401004X", + "K0220003B" + ], "tags": [ "第二单元" ], @@ -96974,7 +97019,10 @@ "003981": { "id": "003981", "content": "(理科)在方程为$\\begin{cases}\nx=\\sin 2\\theta,\\\\ y=\\sin\\theta+\\cos\\theta\n\\end{cases}$的曲线上的点是\\bracket{20}.\n\\fourch{$(2,\\sqrt{3})$}{$(1,\\sqrt{3})$}{$\\left(-\\dfrac 34,\\dfrac 12\\right)$}{$\\left(\\dfrac 12,-\\sqrt{2}\\right)$}\\\\\n(文科)若函数$y=f(x)$存在反函数, 则方程$f(x)=c$($c$为常数)\\bracket{20}.\n\\fourch{有且只有一个实根}{至少有一个实根}{至多有一个实根}{没有实数根}", - "objs": [], + "objs": [ + "K0722002X", + "K0225002B" + ], "tags": [ "第二单元", "第七单元" @@ -98987,7 +99035,10 @@ "004067": { "id": "004067", "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$满足: \\textcircled{1} $f(x)+f(2-x)=0$; \\textcircled{2} $f(x)-f(-2-x)=0$; \\textcircled{3} 在$[-1,1]$上表达式为$f(x)=\\begin{cases} \\sqrt{1-x^2}, & x\\in [-1,0], \\\\ 1-x, & x\\in (0,1], \\end{cases}$ 则函数$f(x)$与$g(x)=\\begin{cases} {2^x}, & x\\le 0 \\\\ \\log_\\frac 12x, & x>0 \\end{cases}$的图像在区间$[-3,3]$上的交点的个数为\\blank{50}.", - "objs": [], + "objs": [ + "K0218001B", + "K0223004B" + ], "tags": [ "第二单元" ], @@ -99058,7 +99109,10 @@ "004070": { "id": "004070", "content": "已知$f(x)=2x^2+2x+b$是定义在$[-1,0]$上的函数, 若$f[f(x)]\\le 0$在定义域上恒成立, 而且存在实数$x_0$满足: $f[f(x_0)]=x_0$且$f(x_0)\\ne x_0$, 则实数$b$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B", + "K0223005B" + ], "tags": [ "第二单元" ], @@ -99559,7 +99613,9 @@ "004090": { "id": "004090", "content": "在直角$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi}2$, $AB=1$, $AC=2$, $M$是$\\triangle ABC$内一点, 且$AM=\\dfrac 12$, 若$\\overrightarrow{AM}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AC}$, 则$\\lambda +2\\mu$的最大值为\\blank{50}.", - "objs": [], + "objs": [ + "K0506003B" + ], "tags": [ "第二单元", "第五单元" @@ -101686,7 +101742,9 @@ "004175": { "id": "004175", "content": "已知实数$a$、$b$使得不等式$|ax^2+bx+a|\\le x$对任意$x\\in [1,2]$都成立, 在平面直角坐标系$xOy$中, 点$(a,b)$形成的区域记为$\\Omega$, 若圆$x^2+y^2=r^2$上的任一点都在$\\Omega$中, 则$r$的最大值为\\blank{50}", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元", "第七单元" @@ -101779,9 +101837,11 @@ "004179": { "id": "004179", "content": "已知定义在实数集$\\mathbf{R}$上的函数$f(x)$满足$f(x+1)=\\dfrac 12+\\sqrt{f(x)-f^2(x)}$, 则$f(0)+f(2021)$的最小值与最大值的和为\\bracket{20}.\n\\fourch{$2$}{$3$}{$\\dfrac 32+\\dfrac{\\sqrt 2}2$}{$\\dfrac 52+\\dfrac{\\sqrt 2}2$}", - "objs": [], + "objs": [ + "K0407002X" + ], "tags": [ - "第二单元" + "第四单元" ], "genre": "选择题", "ans": "", @@ -102749,7 +102809,9 @@ "004220": { "id": "004220", "content": "已知函数\\textcircled{1} $f(x)=3\\ln x$; \\textcircled{2} $f(x)=3\\mathrm{e}^{\\cos x}$; \\textcircled{3} $f(x)=3\\mathrm{e}^x$; \\textcircled{4} $f(x)=3\\cos x$; 其中对于$f(x)$定义域内的任意一个自变量$x_1$都存在唯一一个自变量$x_2$, 使$\\sqrt{f(x_1)f(x_2)}=3$成立的函数是\\bracket{20}.\n\\fourch{\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{4}}", - "objs": [], + "objs": [ + "K0215001B" + ], "tags": [ "第二单元", "第三单元" @@ -104435,7 +104497,11 @@ "004289": { "id": "004289", "content": "已知函数$f(x)$的定义域为$D$, 若存在实常数$\\lambda$及$a$($a\\ne 0$), 对任意$x\\in D$, 当$x+a\\in D$且$x-a\\in D$时, 都有$f(x+a)+f(x-a)=\\lambda f(x)$成立, 则称函数$f(x)$具有性质$M(\\lambda,a)$.\\\\\n(1) 判断函数$f(x)=x^2$是否具有性质$M(\\lambda,a)$, 并说明理由;\\\\\n(2) 若函数$g(x)=\\sin 2x+\\sin x$具有性质$M(\\lambda,a)$, 求$\\lambda$及$a$应满足的条件;\\\\\n(3) 已知定义域为$\\mathbf{R}$的函数$y=h(x)$不存在零点, 且具有性质$M(t+\\dfrac{1}{t},t)$(其中$t>0$, $t\\ne 1$), 记$a_n=h(n)$($n\\in \\mathbf{N}^*$), 求证: 数列$\\{a_n\\}$为等比数列的充要条件是$\\dfrac{a_2}{a_1}=t$或$\\dfrac{a_2}{a_1}=\\dfrac{1}{t}$.", - "objs": [], + "objs": [ + "K0215001B", + "K0319005B", + "K0403004X" + ], "tags": [ "第二单元" ], @@ -106834,7 +106900,10 @@ "004385": { "id": "004385", "content": "设函数$f(x)$的定义域为$\\mathbf{R}$, $f(x)$满足对任意$x_1,x_2\\in \\mathbf{R}$, 当$x_1\\ne x_2$时, 恒有$|f(x_1)-f(x_2)|>2|x_1-x_2|$. 对于命题: \\textcircled{1} $f(x)$的解析式可以是$f(x)=x^3+2021x$; \\textcircled{2} $f(x)$的解析式可以是$f(x)=2021^{-x}$, 下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}、\\textcircled{2}均为真命题}{\\textcircled{1}、\\textcircled{2}均为假命题}{\\textcircled{1}为真命题、\\textcircled{2}为假命题}{\\textcircled{1}为假命题、\\textcircled{2}为真命题}", - "objs": [], + "objs": [ + "K0701001X", + "K0210005B" + ], "tags": [ "第二单元" ], @@ -106857,7 +106926,10 @@ "004386": { "id": "004386", "content": "已知常数$a\\in \\mathbf{R}$, 函数$f(x)=ax^2+\\lg \\dfrac{1+x}{1-x}$.\\\\\n(1) 若$a=0$, 判断$f(x)$的单调性并证明;\\\\\n(2) 问: 是否存在$a$, 使得$f(x)$为奇函数? 若存在, 求出所有$a$的值; 若不存在, 说明理由.", - "objs": [], + "objs": [ + "K0219003B", + "K0218001B" + ], "tags": [ "第二单元" ], @@ -106880,7 +106952,9 @@ "004387": { "id": "004387", "content": "设函数$f(x)$的定义域为$(0,+\\infty)$, 若对任意$x\\in (0,+\\infty)$, 恒有$f(2x)=2f(x)$, 则称$f(x)$为``$2$阶缩放函数''.\\\\\n(1) 已知函数$f(x)$为``$2$阶缩放函数'', 当$x\\in (1,2]$时, $f(x)=1-\\log_2 x$, 求$f(2\\sqrt{2})$的值;\\\\\n(2) 已知函数$f(x)$为``$2$阶缩放函数'', 当$x\\in (1,2]$时, $f(x)=\\sqrt{2x-x^2}$, 求证: 函数$y=f(x)-x$在$(1,+\\infty)$上无零点.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -135597,7 +135671,9 @@ "005561": { "id": "005561", "content": "填写下表:\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n $x$\t & $f(x)=x^2$ & $f(x)-f(x-1)$ & $g(x)=2^x$ & $g(x)-g(x-1)$ \\\\ \\hline\n $0$ & & & & \\\\ \\hline\n $1$ & & & & \\\\ \\hline\n $2$ & & & & \\\\ \\hline\n $3$ & & & & \\\\ \\hline\n $4$ & & & & \\\\ \\hline\n $5$ & & & & \\\\ \\hline\n $6$ & & & & \\\\ \\hline\n $7$ & & & & \\\\ \\hline\n $8$ & & & & \\\\ \\hline\n $9$ & & & & \\\\ \\hline\n $10$ & & & & \\\\ \\hline\n \\end{tabular}\n\\end{center}\n(1) 比较$f(x)=x^2$与$g(x)=2^x$的函数值的大小;\\\\\n(2) 比较$f(x)=x^2$与$g(x)=2^x$的函数值递增的快慢.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -135618,7 +135694,9 @@ "005562": { "id": "005562", "content": "已知函数$f(x)=2x+1$, $g(x)=1.5^x$, $h(x)=x^{1.5}$, 试用数值计算比较三个函数在$[0,+\\infty)$上的函数值的大小、图像递增的快慢. 并说明在函数图像上的表现.\n解 列表并计算得:\n\\begin{center}\n \\begin{longtable}{|c|c|c|c|c|c|c|}\n \\hline\n $x$\t & $f(x)=2x+1$ & $f(x)-f(x-1)$ & $g(x)=1.5^x$ & $g(x)-g(x-1)$ & $h(x)=x^{1.5}$ & $h(x)-h(x-1)$ \\\\ \\hline\n \\endhead\n $0$ & $1$ & & $1$ & & $0$ & \\\\ \\hline\n $1$ & $3$ & $2$ & $1.5$ & $0.5$ & $1$ & $1$\\\\ \\hline\n $2$ & $5$ & $2$ & $2.25$ & $0.75$ & $2.82842712$ & $1.82842712$\\\\ \\hline\n $3$ & $7$ & $2$ & $3.375$ & $1.125$ & $5.19615242$ & $2.3677253$\\\\ \\hline\n $4$ & $9$ & $2$ & $5.0625$ & $1.6875$ & $8$ & $2.80384758$\\\\ \\hline\n $5$ & $11$ & $2$ & $7.59375$ & $2.53125$ & $11.1803399$ & $3.18033989$\\\\ \\hline\n $6$ & $13$ & $2$ & $11.390625$ & $3.796875$ & $14.6969385$ & $3.51659857$\\\\ \\hline\n $7$ & $15$ & $2$ & $17.085938$ & $5.6953125$ & $18.5202592$ & $3.82332072$\\\\ \\hline\n $8$ & $17$ & $2$ & $25.628906$ & $8.5429688$ & $22.627417$ & $4.10715782$\\\\ \\hline\n $9$ & $19$ & $2$ & $38.443359$ & $12.814453$ & $27$ & $4.372583$\\\\ \\hline\n $10$ & $21$ & $2$ & $57.665039$ & $19.22168$ & $31.6227766$ & $4.6227766$\\\\ \\hline\n $11$ & $23$ & $2$ & $86.497559$ & $28.83252$ & $36.4828727$ & $4.86009609$\\\\ \\hline\n $12$ & $25$ & $2$ & $129.74634$ & $43.248779$ & $41.5692194$ & $5.08634669$\\\\ \\hline\n $13$ & $27$ & $2$ & $194.61951$ & $64.873169$ & $46.8721666$ & $5.3029472$\\\\ \\hline\n $14$ & $29$ & $2$ & $291.92926$ & $97.309753$ & $52.3832034$ & $5.51103683$\\\\ \\hline\n $15$ & $31$ & $2$ & $437.89389$ & $145.96463$ & $58.0947502$ & $5.71154678$\\\\ \\hline\n $16$ & $33$ & $2$ & $656.84084$ & $218.94695$ & $64$ & $5.90524981$\\\\ \\hline\n $17$ & $35$ & $2$ & $985.26125$ & $328.42042$ & $70.0927956$ & $6.09279564$\\\\ \\hline\n $18$ & $37$ & $2$ & $1477.8919$ & $492.63063$ & $76.3675324$ & $6.27473673$\\\\ \\hline\n $19$ & $39$ & $2$ & $2216.8378$ & $738.94594$ & $82.8190799$ & $6.45154756$\\\\ \\hline\n $20$ & $41$ & $2$ & $3325.2567$ & $1108.4189$ & $89.4427191$ & $6.62363917$\\\\ \\hline\n $21$ & $43$ & $2$ & $4987.8851$ & $1662.6284$ & $96.2340896$ & $6.79137049$\\\\ \\hline\n $22$ & $45$ & $2$ & $7481.8276$ & $2493.9425$ & $103.189147$ & $6.95505712$\\\\ \\hline\n $23$ & $47$ & $2$ & $11222.741$ & $3740.9138$ & $110.304125$ & $7.11497832$\\\\ \\hline\n $24$ & $49$ & $2$ & $16834.112$ & $5611.3707$ & $117.575508$ & $7.27138262$\\\\ \\hline\n $25$ & $51$ & $2$ & $25251.168$ & $8417.0561$ & $125$ & $7.42449235$\\\\ \\hline\n $26$ & $53$ & $2$ & $37876.752$ & $12625.584$ & $132.574507$ & $7.57450735$\\\\ \\hline\n $27$ & $55$ & $2$ & $56815.129$ & $18938.376$ & $140.296115$ & $7.72160806$\\\\ \\hline\n $28$ & $57$ & $2$ & $85222.693$ & $28407.564$ & $148.162073$ & $7.86595801$\\\\ \\hline\n $29$ & $59$ & $2$ & $127834.04$ & $42611.346$ & $156.169779$ & $8.00770599$\\\\ \\hline\n $30$ & $61$ & $2$ & $191751.06$ & $63917.02$ & $164.316767$ & $8.14698784$\\\\ \\hline\n $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ & $\\cdots$ \\\\ \\hline\n \\end{longtable}\n\\end{center}\n得点$A,B,C,D$的横坐标分别约为$1.5,4.8, 6.5, 7.4$, 记作$x_A,x_B,x_C,x_D$.\\\\\n(1) 三个函数的函数值的大小情况如下:\\\\\n\\textcircled{1} 当$0g(x)>h(x)$;\n\\textcircled{2} 当$x_Ah(x)>g(x)$;\n\\textcircled{3} 由$x_Bf(x)>g(x)$;\n\\textcircled{4} 当$x_Cg(x)>f(x)$;\n\\textcircled{5} 当$x_Dh(x)>f(x)$;\n\\textcircled{6} 当$x=x_A$时, $f(x)>g(x)=h(x)$;\n\\textcircled{7} 当$x=x_B$时, $f(x)=h(x)>g(x)$;\n\\textcircled{8} 当$x=x_C$时, $f(x)=g(x)=latex]\n \\draw [->] (-0.1,0) -- (5,0) node [below] {$x$};\n \\draw [->] (0,-0.1) -- (0,5.75) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = 0:9,samples = 100, name path = firstorder] plot ({\\x/2},{(2*\\x+1)/4});\n \\draw (4.5,{19/4}) node [right] {$y=f(x)$};\n \\draw [domain = 0:7.8, samples = 100, name path = exponential] plot ({\\x/2},{1.5^(\\x)/4}); \n \\draw (3.9,{1.5^7.8/4}) node [above right] {$y=g(x)$};\n \\draw [domain = 0:8, samples = 100, name path = power] plot ({\\x/2},{\\x^(3/2)/4});\n \\draw (4,{8^(3/2)/4}) node [below right] {$y=h(x)$};\n \\path [name intersections = {of = firstorder and exponential, by = {T,C}}];\n \\path [name intersections = {of = firstorder and power, by = B}];\n \\path [name intersections = {of = exponential and power, by = {A,D}}];\n \\filldraw (A) circle (0.05) node [below right] {$A$};\n \\filldraw (B) circle (0.05) node [below right] {$B$};\n \\filldraw (C) circle (0.05) node [below right] {$C$};\n \\filldraw (D) circle (0.05) node [below right] {$D$};\n \\foreach \\i in {1,2,...,9}{\\draw (\\i/2,0.1) -- (\\i/2,0) node [below] {$\\i$};};\n \\foreach \\i in {1,3,...,21}{\\draw (0.1,\\i/4) -- (0,\\i/4) node [left] {$\\i$};};\n \\end{tikzpicture}\n\\end{center}\n由表格及图像可看出, 三个函数的函数值变化及相应增量规律为: 随着$x$的增大, 直线型均匀上升, 增量恒定; 指数型急剧上升, 在区间$[0,+\\infty)$上递增增量快速增大; 幂函数型虽上升较快, 但随着$x$的不断增大上升趋势远不如指数型, 几乎微不足道, 其增量缓慢递增.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -136613,7 +136691,9 @@ "005604": { "id": "005604", "content": "求关于$x$的方程$a^x+1=-x^2+2x+2a$($a>0$且$a\\ne 1$)的实数解的个数.", - "objs": [], + "objs": [ + "K0210005B" + ], "tags": [ "第二单元" ], @@ -136634,7 +136714,9 @@ "005605": { "id": "005605", "content": "在同一个平面直角坐标系中, 作出$t(x)=0.5x$与$g(x)=0.2\\times 2^x$的图像, 并比较它们的增长情况.", - "objs": [], + "objs": [ + "K0210005B" + ], "tags": [ "第二单元" ], @@ -136655,7 +136737,9 @@ "005606": { "id": "005606", "content": "某地区不同身高的未成年男性的体重平均值如下表(身高: $\\text{cm}$; 体重: $\\text{kg}$):\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n 身高 & $60$ & $70$ & $80$ & $90$ & $100$ & $110$\\\\ \\hline\n 体重 & $6.13$ & $7.90$ & $9.99$ & $12.15$ & $15.02$ & $17.05$\\\\ \\hline\n 身高 & $120$ & $130$ & $140$ & $150$ & $160$ & $170$\\\\ \\hline\n 体重 & $20.92$ & $26.86$ & $31.11$ & $38.85$ & $47.25$ & $55.05$\\\\ \\hline\n \\end{tabular}\n\\end{center}\n为了揭示未成年男性的身高与体重的规律, 甲选择了模型$y=ax^2+bx+c$($a>0$), 乙选择了模型$y=ba^x$($a>1$), 其中$y$表示体重, $x$表示身高.你认为谁选择的模型较好?", - "objs": [], + "objs": [ + "K0215001B" + ], "tags": [ "第二单元" ], @@ -136676,7 +136760,9 @@ "005607": { "id": "005607", "content": "用计算器计算并填写下表:\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n $x$\t& $f(x)=x^{\\frac 12}$ & $g(x)=x^{0.6}$ & $h(x)=2.1^x$ & $s(x)=2.2^x$ \\\\ \\hline\n $0$ & & & & \\\\ \\hline\n $1$ & & & & \\\\ \\hline\n $2$ & & & & \\\\ \\hline\n $3$ & & & & \\\\ \\hline\n $4$ & & & & \\\\ \\hline\n $5$ & & & & \\\\ \\hline\n $6$ & & & & \\\\ \\hline\n $7$ & & & & \\\\ \\hline\n $8$ & & & & \\\\ \\hline\n $9$ & & & & \\\\ \\hline\n $10$ & & & & \\\\ \\hline\n \\end{tabular}\n\\end{center}\n从表中变化的现象可以归纳出哪些函数递增的规律?\\\\\n(1) 幂函数$f(x)$与$g(x)$之间比较得出的规律;\n(2) 指数函数$h(x)$与$s(x)$之间比较得出的规律;\n(3) 幂函数$f(x)=x^{\\frac 12}$与指数函数$h(x)$之间比较得出的规律", - "objs": [], + "objs": [ + "K0215001B" + ], "tags": [ "第二单元" ], @@ -138479,7 +138565,9 @@ "005683": { "id": "005683", "content": "若$00$, $a\\ne 1$, 比较$p=|\\log_a(1-x)|$和$q=|\\log_a(1+x)|$的大小.", - "objs": [], + "objs": [ + "K0206002B" + ], "tags": [ "第二单元" ], @@ -138546,7 +138634,9 @@ "005686": { "id": "005686", "content": "已知关于$x$的方程$ax^2-4ax+1=0$的两个实数根$\\alpha ,\\beta$满足不等式$|\\lg \\alpha -\\lg \\beta|\\le 1$, 求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0206002B" + ], "tags": [ "第二单元" ], @@ -138659,7 +138749,10 @@ "005691": { "id": "005691", "content": "设$f(x)$是定义在$(-\\infty ,+\\infty)$上的偶函数, 且它在$[0,+\\infty)$上是增函数, 记$a=f(-\\log_{\\sqrt 2}\\sqrt 3)$, $b=f(-\\log_{\\sqrt 3}\\sqrt 2)$, $c=f(-2)$, 则$a,b,c$的大小关系是\\bracket{20}.\n\\fourch{$a>b>c$}{$b>c>a$}{$c>a>b$}{$c>b>a$}", - "objs": [], + "objs": [ + "K0206002B", + "K0218001B" + ], "tags": [ "第二单元" ], @@ -138680,7 +138773,9 @@ "005692": { "id": "005692", "content": "下列函数图像中, 不正确的是\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw [domain = -1.4:2.5] plot ({sqrt((1/3)^\\x)},\\x);\n \\draw [domain = -1.4:2.5] plot ({-sqrt((1/3)^\\x)},\\x);\n \\draw (0,-1.5) node [below] {(A)};\n \\end{tikzpicture}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw [domain = -0.9:2.5] plot ({-(1/3)^\\x},\\x);\n \\draw (0,-1.5) node [below] {(B)};\n \\end{tikzpicture}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw [domain = -0.9:2.5] plot ({(1/3)^\\x},{abs(\\x)});\n \\draw (0,-1.5) node [below] {(C)};\n \\end{tikzpicture}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-1.5) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw [domain = 0.1:2.5] plot (\\x,{\\x^(-1/3)});\n \\draw (0,-1.5) node [below] {(D)};\n \\end{tikzpicture}\n\\end{center}\n\\fourch{$y=\\log_{\\frac 13}x^2$}{$y=\\log_{\\frac 13}(-x)$}{$y=|\\log_3x|$}{$y=|x^{-\\frac 13}|$}", - "objs": [], + "objs": [ + "K0213007B" + ], "tags": [ "第二单元" ], @@ -138701,7 +138796,9 @@ "005693": { "id": "005693", "content": "在同一平面直角坐标系中画出函数$y=x+a$与$y=\\log_ax$的图像, 可能是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[scale = 0.5, >=latex]\n \\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw (0.1,1) -- (0,1) node [left] {$1$};\n \\draw [domain = -1.8:1.2] plot (\\x,{\\x+1.5});\n \\draw [domain = -2.5:2] plot ({1.5^\\x},{-\\x}); \n\\end{tikzpicture}}\n{\\begin{tikzpicture}[scale = 0.5, >=latex]\n \\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw (0.1,1) -- (0,1) node [left] {$1$};\n \\draw [domain = -1.8:2.2] plot (\\x,{\\x+0.7}); \n \\draw [domain = -1.9:2.5] plot ({1.5^\\x},{\\x}); \n\\end{tikzpicture}}\n{\\begin{tikzpicture}[scale = 0.5, >=latex]\n \\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw (0.1,1) -- (0,1) node [left] {$1$};\n \\draw [domain = -1.8:2.2] plot (\\x,{\\x+0.7}); \n \\draw [domain = -1.9:2.5] plot ({0.7^\\x},{\\x}); \n\\end{tikzpicture}}\n{\\begin{tikzpicture}[scale = 0.5, >=latex]\n \\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0) node [below] {$1$};\n \\draw (0.1,1) -- (0,1) node [left] {$1$};\n \\draw [domain = -1.8:2.8] plot (\\x,{\\x-0.2}); \n \\draw [domain = -1.9:2.5] plot ({1.4^\\x},{\\x}); \n\\end{tikzpicture}}", - "objs": [], + "objs": [ + "K0213007B" + ], "tags": [ "第二单元" ], @@ -138722,7 +138819,9 @@ "005694": { "id": "005694", "content": "函数$y=f(x)$的图像如图所示, 则$y=\\log_{0.7}f(x)$的示意图是\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0.1) -- (1,0) node [below] {$1$} (2,0.1) -- (2,0) node [below] {$2$} (0.1,1) -- (0,1) node [left] {$1$};\n \\draw [dashed] (2,0) -- (2,3);\n \\draw [dashed] (1,0) -- (1,1) -- (0,1);\n \\draw [domain = 0.3:1.7] plot (\\x,{3*(\\x-1)^2+1});\n \\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0.1) -- (1,0) node [below] {$1$} (2,0.1) -- (2,0) node [below right] {$2$};\n \\draw [dashed] (2,-3) -- (2,3);\n \\draw [domain = 0.3:1] plot (\\x,{ln(3*(\\x-1)^2+1)/ln(0.7)});\n \\draw [domain = 1:1.7] plot (\\x,{-ln(3*(\\x-1)^2+1)/ln(0.7)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0.1) -- (1,0) node [below] {$1$} (2,0.1) -- (2,0) node [below right] {$2$};\n \\draw [dashed] (2,-3) -- (2,3);\n \\draw [domain = 0.3:1] plot (\\x,{-ln(3*(\\x-1)^2+1)/ln(0.7)});\n \\draw [domain = 1:1.7] plot (\\x,{ln(3*(\\x-1)^2+1)/ln(0.7)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0.1) -- (1,0) node [below] {$1$} (2,0.1) -- (2,0) node [below right] {$2$};\n \\draw [dashed] (2,-3) -- (2,3);\n \\draw [domain = 0.3:1.7] plot (\\x,{ln(3*(\\x-1)^2+1)/ln(0.7)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n \\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n \\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw (1,0.1) -- (1,0) node [below] {$1$} (2,0.1) -- (2,0) node [below right] {$2$};\n \\draw [dashed] (2,-3) -- (2,3);\n \\draw [domain = 0.3:1.7] plot (\\x,{-ln(3*(\\x-1)^2+1)/ln(0.7)});\n\\end{tikzpicture}}", - "objs": [], + "objs": [ + "K0213007B" + ], "tags": [ "第二单元" ], @@ -139180,7 +139279,10 @@ "005713": { "id": "005713", "content": "函数$y=\\lg \\dfrac{1-x}{1+x}$\\bracket{20}.\n\\twoch{是奇函数, 且在$(-1, 1)$是增函数}{是奇函数, 且在$(-1, 1)$上是减函数}{是偶函数, 且在$(-1, 1)$是增函数}{是偶函数, 且在$(-1, 1)$上是减函数}", - "objs": [], + "objs": [ + "K0218001B", + "K0219003B" + ], "tags": [ "第二单元" ], @@ -140023,7 +140125,10 @@ "005749": { "id": "005749", "content": "已知函数$f(x)=\\log_{\\frac 12}(x^2-2x)$.\\\\\n(1) 求它的单调区间;\\\\\n(2) 求$f(x)$为增函数时的反函数.", - "objs": [], + "objs": [ + "K0219003B", + "K0225005B" + ], "tags": [ "第二单元" ], @@ -140044,7 +140149,12 @@ "005750": { "id": "005750", "content": "已知函数$f(x)=\\log_a\\dfrac{x+b}{x-b}$($a>0$, $b>0$且$a\\ne 1$).\\\\\n(1) 求$f(x)$的定义域;\\\\\n(2) 讨论$f(x)$的奇偶性;\\\\\n(3) 讨论$f(x)$的单调性;\\\\\n(4) 求$f(x)$的反函数$f^{-1}(x)$.", - "objs": [], + "objs": [ + "K0218001B", + "K0219003B", + "K0225005B", + "K0212002B" + ], "tags": [ "第二单元" ], @@ -140067,7 +140177,9 @@ "005751": { "id": "005751", "content": "已知函数$f(x)=\\lg \\dfrac{x+1}{x-1}+\\lg (x-1)+\\lg (a-x)$($a>1$).\\\\\n(1) 是否存在一个实数$a$使得函数$y=f(x)$的图像关于某一条垂直于$x$轴的直线对称? 若存在, 求出这个实数$a$; 若不存在, 说明理由;\\\\\n(2) 当$f(x)$的最大值为2时, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0206002B" + ], "tags": [ "第二单元" ], @@ -140254,7 +140366,9 @@ "005759": { "id": "005759", "content": "方程$2^{|x+1|}=3$的解集是\\bracket{20}.\n\\fourch{$\\{\\log_{\\frac 12}\\dfrac 23\\}$}{$\\{\\log_2\\dfrac 23\\}$}{$\\{\\log_2\\dfrac 32,\\log_2\\dfrac 16\\}$}{$\\{\\log_2\\dfrac 13,-\\log_{\\frac 12}6\\}$}", - "objs": [], + "objs": [ + "K0204004B" + ], "tags": [ "第二单元" ], @@ -140275,7 +140389,9 @@ "005760": { "id": "005760", "content": "方程$2x^2+2^x-3=0$的实数根有\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{无数个}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140296,7 +140412,9 @@ "005761": { "id": "005761", "content": "满足$(x-2)^{5-|x|}=1$的实数根存\\bracket{20}.\n\\fourch{$4$个}{$3$个}{$2$个}{无数个}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140317,7 +140435,9 @@ "005762": { "id": "005762", "content": "方程$6\\cdot 7^{|x|}-7^{-x}=1$的解集是\\bracket{20}.\n\\fourch{$\\{\\log_7\\dfrac 12\\}$}{$\\{\\log_75\\}$}{$\\{\\log_7\\dfrac 12,\\log_75\\}$}{$\\varnothing$}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140338,7 +140458,9 @@ "005763": { "id": "005763", "content": "若对于任意实数$p$, 函数$y=(p-1)2^x-\\dfrac p2$的图像恒过一定点, 则这个点的坐标是\\bracket{20}.\n\\fourch{$(1,-\\dfrac 12)$}{$(0, -1)$}{$(-1,-\\dfrac 12)$}{$(-2,-\\dfrac 14)$}", - "objs": [], + "objs": [ + "K0210002B" + ], "tags": [ "第二单元" ], @@ -140359,7 +140481,9 @@ "005764": { "id": "005764", "content": "方程$2^{2x+1}-33\\cdot 2^{x-2}+1=0$的解是\\bracket{20}.\n\\fourch{$\\{-2,-3\\}$}{$\\{2,-3\\}$}{$\\{2,3\\}$}{$\\{-2,3\\}$}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140380,7 +140504,9 @@ "005765": { "id": "005765", "content": "方程$3^{x^2}=(3^x)^2$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140401,7 +140527,9 @@ "005766": { "id": "005766", "content": "方程$3^x=2^x$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140425,7 +140553,9 @@ "005767": { "id": "005767", "content": "方程$\\dfrac{3^{x^2+1}}{3^{x-1}}=81$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140446,7 +140576,9 @@ "005768": { "id": "005768", "content": "方程$5^{x-1}\\cdot 10^{3x}=8^x$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140467,7 +140599,9 @@ "005769": { "id": "005769", "content": "方程$2^{x-1}=3^{2x}$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140490,7 +140624,9 @@ "005770": { "id": "005770", "content": "方程$2\\cdot 4^x-7\\cdot 2^x+3=0$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140511,7 +140647,9 @@ "005771": { "id": "005771", "content": "方程$9^x-3^{x+2}-10=0$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140532,7 +140670,9 @@ "005772": { "id": "005772", "content": "方程$3^{x+1}-3^{-x}=2$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140555,7 +140695,9 @@ "005773": { "id": "005773", "content": "已知$a>0$且$a\\ne 1$, 则方程$a(a^x+1)=a^{-x}+1$的解为\\blank{50}.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140576,7 +140718,9 @@ "005774": { "id": "005774", "content": "解方程: $3\\times 16^x+36^x=2\\times 81^x$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140597,7 +140741,9 @@ "005775": { "id": "005775", "content": "解方程: $(\\sqrt {5+2\\sqrt 6})^x+(\\sqrt {5-2\\sqrt 6})^x=10$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140618,7 +140764,9 @@ "005776": { "id": "005776", "content": "解方程: $\\sqrt[x]9-\\sqrt[x]6=\\sqrt[x]4$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140639,7 +140787,9 @@ "005777": { "id": "005777", "content": "解方程: $4^{x+\\sqrt {x^2-2}}-5\\times 2^{x-1+\\sqrt {x^2-2}}=6$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140660,7 +140810,9 @@ "005778": { "id": "005778", "content": "已知关于$x$的方程$2a^{2x-2}-7a^{x-1}+3=0$有一个根是$2$, 求实数$a$的值, 并求方程其余的根.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140683,7 +140835,9 @@ "005779": { "id": "005779", "content": "解关于$x$的方程$\\dfrac{a^x-a^{-x}}{a^x+a^{-x}}=b$(实数$a>0$, $a\\ne 1$, $b\\in \\mathbf{R}$).", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140704,7 +140858,9 @@ "005780": { "id": "005780", "content": "若关于$x$的指数方程$9^x+(a+4)3^x+4=0$有实数解, 试求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140725,7 +140881,9 @@ "005781": { "id": "005781", "content": "若关于$x$的方程$2a\\cdot 3^{-|x-1|}-3^{-2|x-1|}-2a-1=0$有实数解, 求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140746,7 +140904,10 @@ "005782": { "id": "005782", "content": "方程$\\lg (x-1)^2=2$的解集是\\bracket{20}.\n\\fourch{$\\{11\\}$}{$\\{-9\\}$}{$\\{11,-9\\}$}{$\\{-11,9\\}$}", - "objs": [], + "objs": [ + "K0204004B", + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140767,7 +140928,10 @@ "005783": { "id": "005783", "content": "关于$x$的方程$\\log_ax^2=\\log_a(\\sqrt {a+1}-\\sqrt a)-\\log_a(\\sqrt {a+1}+\\sqrt a)$($a>0$且$a\\ne 1$)的解为\\bracket{20}.\n\\fourch{$\\sqrt {a+1}+\\sqrt a$}{$\\sqrt {a+1}-\\sqrt a$}{$\\pm (\\sqrt {a+1}+\\sqrt a)$}{$\\pm (\\sqrt {a+1}-\\sqrt a)$}", - "objs": [], + "objs": [ + "K0204004B", + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140788,7 +140952,9 @@ "005784": { "id": "005784", "content": "若$f(x)=1+\\lg x$, $g(x)=x^2$, 则使$2f[g(x)]=g[f(x)]$成立的$x$值等于\\bracket{20}.\n\\fourch{$10^{1+\\sqrt 2}$或$10^{1-\\sqrt 2}$}{$1+\\sqrt 2$或$1-\\sqrt 2$}{$10^{1+\\sqrt 3}$或$10^{1-\\sqrt 3}$}{$1+\\sqrt 3$或$1-\\sqrt 3$}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140809,7 +140975,10 @@ "005785": { "id": "005785", "content": "方程$\\log_5(x-8)^2=2+\\log_5(x-2)$的解是\\bracket{20}.\n\\fourch{3或$\\dfrac 12$}{$\\dfrac 12$}{$3$或$38$}{$2$}", - "objs": [], + "objs": [ + "K0204004B", + "K0223004B" + ], "tags": [ "第二单元" ], @@ -140830,7 +140999,9 @@ "005786": { "id": "005786", "content": "方程$\\sqrt {\\lg x-4}=4-\\lg x$的解集是\\bracket{20}.\n\\fourch{$\\{100\\}$}{$\\{1000\\}$}{$\\{10000\\}$}{$\\{\\dfrac 1{10000}\\}$}", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141075,7 +141246,10 @@ "005796": { "id": "005796", "content": "已知集合$A=\\{x|x^2-ax+a^2-19=0\\}$, $B=\\{x|\\log_2(x^2-5x+8)=1\\}$, $C=\\{x|x^2+2x-8=0\\}$满足$A\\cap B\\ne \\varnothing$, $A\\cap C\\ne \\varnothing$, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0223004B", + "K0204004B" + ], "tags": [ "第二单元" ], @@ -141098,7 +141272,10 @@ "005797": { "id": "005797", "content": "已知$f(x)=\\log_a(a^x-1)$($a>0$, $a\\ne 1$), 解方程$f(2x)=f^{-1}(x)$.", - "objs": [], + "objs": [ + "K0204004B", + "K0225005B" + ], "tags": [ "第二单元" ], @@ -141119,7 +141296,9 @@ "005798": { "id": "005798", "content": "解方程$\\log_{\\frac 12}(9^{x-1}-5)=\\log_{\\frac 12}(3^{x-1}-2)-2$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141142,7 +141321,9 @@ "005799": { "id": "005799", "content": "解方程$\\log_{0.5x}2-\\log_{0.5x^3}x^2=\\log_{0.5x^3}4$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141163,7 +141344,9 @@ "005800": { "id": "005800", "content": "解方程$(\\sqrt x)^{\\log_5x-1}=5$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141184,7 +141367,9 @@ "005801": { "id": "005801", "content": "解方程$10^{\\lg ^2x}+x^{\\lg x}=20$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141205,7 +141390,9 @@ "005802": { "id": "005802", "content": "解方程$|\\log_2x|=|\\log_2(2x^2)|-2$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141228,7 +141415,9 @@ "005803": { "id": "005803", "content": "解方程组$\\begin{cases} \\log_yx-3\\log_xy=2, \\\\ (2^x)^y=(\\dfrac 12)^{-16}. \\end{cases}$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141249,7 +141438,9 @@ "005804": { "id": "005804", "content": "解关于$x$的方程: $\\lg (x+a)+1=\\lg (ax-1)$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141272,7 +141463,9 @@ "005805": { "id": "005805", "content": "解关于$x$的方程: $\\lg (ax-1)-\\lg (x-3)=1$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141295,7 +141488,9 @@ "005806": { "id": "005806", "content": "解关于$x$的方程: $2\\lg x-\\lg (x-1)=\\lg a$.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141316,7 +141511,9 @@ "005807": { "id": "005807", "content": "已知函数$f(x)=a^{x-\\dfrac 12}$满足$f(\\lg a)=\\sqrt {10}$, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0205002B" + ], "tags": [ "第二单元" ], @@ -141337,7 +141534,10 @@ "005808": { "id": "005808", "content": "已知函数$f(x)=x^2-x+k$满足$\\log_2f(a)=2$, $f(\\log_2a)=k$($a>0$且$a\\ne 1$), 求$f(\\log_2x)$在什么区间上是减函数, 并求出$a$与$k$的值.", - "objs": [], + "objs": [ + "K0219003B", + "K0220001B" + ], "tags": [ "第二单元" ], @@ -141358,7 +141558,9 @@ "005809": { "id": "005809", "content": "若关于$x$的方程$\\lg 2x\\cdot \\lg 3x=-a^2$有两个相异实根, 求实数$a$的取值范围, 并求此方程两根之积.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141379,7 +141581,9 @@ "005810": { "id": "005810", "content": "若关于$x$的方程$(\\lg ax)(\\lg ax^2)=4$所有的解都大于$1$, 求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141400,7 +141604,9 @@ "005811": { "id": "005811", "content": "若关于$x$的方程$\\lg (ax)\\cdot \\lg (ax^2)=4$有两个小于$1$的正根$\\alpha ,\\beta$, 且满足$|\\lg \\alpha -\\lg \\beta|\\le 2\\sqrt 3$, 求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141421,7 +141627,9 @@ "005812": { "id": "005812", "content": "已知函数$f(x)=x^2\\lg a+2x+4\\lg a$的最大值是$3$, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -141444,7 +141652,9 @@ "005813": { "id": "005813", "content": "若关于$x$的方程$\\log_2x+1=2\\log_2(x-a)$恰有一个实数解, 求实数$a$的取值范围.", - "objs": [], + "objs": [ + "K0223004B" + ], "tags": [ "第二单元" ], @@ -141465,7 +141675,11 @@ "005814": { "id": "005814", "content": "已知函数$f(x)=\\log_a(a-ka^x)$($a>0$, $a\\ne 1$, $k\\in \\mathbf{R}$).\n(1) 当$01$时, $f(x)$的反函数就是它自身, 求$k$的值;\\\\\n(3) 在(2)的条件下, 求$f^{-1}(x^2-2)=f(x)$的解.", - "objs": [], + "objs": [ + "K0212002B", + "K0225001B", + "K0225005B" + ], "tags": [ "第二单元" ], @@ -141509,8 +141723,10 @@ }, "005816": { "id": "005816", - "content": "已知$f(x)=x^2+ax+b$($a,b$均为实数), 集合$A=\\{x|x=f(x) ,x\\in \\mathbf{R}\\}=\\{-1,3\\}$, $B=\\{x|x=f[f(x)],x\\in \\mathbf{R}\\}$, 用列举法求集合.", - "objs": [], + "content": "已知$f(x)=x^2+ax+b$($a,b$均为实数), 集合$A=\\{x|x=f(x) ,x\\in \\mathbf{R}\\}=\\{-1,3\\}$, $B=\\{x|x=f[f(x)],x\\in \\mathbf{R}\\}$, 列举法表示集合$B$.", + "objs": [ + "K0109001B" + ], "tags": [ "第二单元" ], @@ -141796,7 +142012,9 @@ "005828": { "id": "005828", "content": "若函数$f(x)$的定义域为$\\mathbf{R}^+$, 且满足$f(xy)=f(x)+f(y)$, $f(8)=3$, 求$f(\\sqrt 2)$的值.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -141817,7 +142035,9 @@ "005829": { "id": "005829", "content": "若函数$f(x)$的定义域为$\\mathbf{R}$, 且满足$f(x)+2f(-x)=-x^3+6x^2-3x+3$, 求$f(0)$的值, 并求$f(x)$的表达式.", - "objs": [], + "objs": [ + "K0215001B" + ], "tags": [ "第二单元" ], @@ -141838,7 +142058,10 @@ "005830": { "id": "005830", "content": "已知$f(x+y)=f(x)+f(y)$对于任何实数$x,y$都成立.\\\\\n(1) 求证: $f(2x)=2f(x)$;\\\\\n(2) 求$f(0)$的值;\\\\\n(3) 求证: $f(x)$为奇函数.", - "objs": [], + "objs": [ + "K0215001B", + "K0217004B" + ], "tags": [ "第二单元" ], @@ -141859,7 +142082,9 @@ "005831": { "id": "005831", "content": "已知函数$f(x)$对任何实数$x,y$满足$f(x+y)+f(x-y)=2f(x)f(y)$, 且$f(0)\\ne 0$, 求证: $f(x)$是偶函数.", - "objs": [], + "objs": [ + "K0217004B" + ], "tags": [ "第二单元" ], @@ -141880,7 +142105,10 @@ "005832": { "id": "005832", "content": "已知函数$f(x)$($x\\ne 0$)满足$f(xy)=f(x)+f(y)$.\n(1) 求证: $f(1)=f(-1)=0$;\\\\\n(2) 求证: $f(x)$为偶函数;\\\\\n(3) 若$f(x)$在$(0,+\\infty)$上是增函数, 解不等式$f(x)+f(x-\\dfrac 12)\\le 0$.", - "objs": [], + "objs": [ + "K0217004B", + "K0223005B" + ], "tags": [ "第二单元" ], @@ -141901,7 +142129,10 @@ "005833": { "id": "005833", "content": "已知函数$f(x)$对一切实数$x,y$满足$f(0)\\ne 0$, $f(x+y)=f(x)\\cdot f(y)$, 且当$x<0$时, $f(x)>1$.求证:\n(1)当$x>0$时, $00$).", - "objs": [], + "objs": [ + "K0215001B" + ], "tags": [ "第二单元" ], @@ -142009,7 +142247,10 @@ "005838": { "id": "005838", "content": "已知函数$f(x)=x^2-2mx+m+6$.\\\\\n(1) 若对任意实数$x$都有$f(x)>0$, 求实数$m$的取值范围;\\\\\n(2) 若实数$\\alpha ,\\beta$满足$f(\\alpha)=f(\\beta)=0$, 求$\\alpha ^2+\\beta ^2$的最小值.", - "objs": [], + "objs": [ + "K0221002B", + "K0215001B" + ], "tags": [ "第二单元" ], @@ -142030,7 +142271,9 @@ "005839": { "id": "005839", "content": "已知函数$f(x)=x^2-2kx+2$在$x\\ge -1$时恒有$f(x)\\ge k$, 求实数$k$的取值范围.", - "objs": [], + "objs": [ + "K0223005B" + ], "tags": [ "第二单元" ], @@ -142051,7 +142294,9 @@ "005840": { "id": "005840", "content": "已知$f(x)=-9x^2-6ax+2a-a^2$在$-\\dfrac 13\\le x\\le \\dfrac 13$内有最大值$-3$, 求实数$a$的值.", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -142072,7 +142317,9 @@ "005841": { "id": "005841", "content": "已知$y=f(x)$在其定义域上是增函数, 求证: $y=f(x)$的反函数$y=f^{-1}(x)$在其定义域上也是增函数.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -142093,7 +142340,10 @@ "005842": { "id": "005842", "content": "已知函数$f(x)=x^3+x+1$($x\\in \\mathbf{R}$), 求证:\\\\\n(1) $f(x)$是$\\mathbf{R}$上的增函数;\\\\\n(2) 方程$x^3+x+1=0$只有一个实数解.", - "objs": [], + "objs": [ + "K0223004B", + "K0219003B" + ], "tags": [ "第二单元" ], @@ -142114,7 +142364,10 @@ "005843": { "id": "005843", "content": "已知函数$f(x)=\\dfrac x{1+x^2}$($x\\in \\mathbf{R}$).\\\\\n(1) 求$f(x)$的值域;\\\\\n(2) 讨论$f(x)$的单调性.", - "objs": [], + "objs": [ + "K0219003B", + "K0215005B" + ], "tags": [ "第二单元" ], @@ -142135,7 +142388,9 @@ "005844": { "id": "005844", "content": "若二次函数$f(x)=ax^2+bx+c$满足$f(x_1)=f(x_2)$, ($x_1\\ne x_2$)求证: 直线$x=\\dfrac{x_1+x_2}2$是该二次函数图像的对称轴.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -142156,7 +142411,9 @@ "005845": { "id": "005845", "content": "若对于任何实数$x$, 函数$y=f(x)$始终满足$f(a+x)=f(a-x)$, 求证: 函数$y=f(x)$的图像关于直线$x=a$对称.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -142177,7 +142434,9 @@ "005846": { "id": "005846", "content": "已知函数$f(x)$满足$f(x+2)=f(2-x)$($x\\in \\mathbf{R}$), 且$f(x)$的图像与$x$轴有15个不同的交点, 求方程$f(x)=0$的所有解的和.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -142198,7 +142457,9 @@ "005847": { "id": "005847", "content": "已知函数$f(2x+1)$是偶函数, 求函数$f(2x)$的图像的对称轴.", - "objs": [], + "objs": [ + "KNONE" + ], "tags": [ "第二单元" ], @@ -142219,7 +142480,9 @@ "005848": { "id": "005848", "content": "求函数$y=\\dfrac{3x-1}{x+2}$($x\\ne -2$)的图像的对称点.", - "objs": [], + "objs": [ + "K0217001B" + ], "tags": [ "第二单元" ], @@ -142240,7 +142503,9 @@ "005849": { "id": "005849", "content": "已知函数$f(x)$满足$f(x)+f(2-x)+2=0$($x\\in \\mathbf{R}$), 求$f(x)$的图像的对称中心.", - "objs": [], + "objs": [ + "K0217001B" + ], "tags": [ "第二单元" ], @@ -142261,7 +142526,9 @@ "005850": { "id": "005850", "content": "已知函数$f(x)=\\log_3(x^2-4mx+4m^2+m+\\dfrac 1{m-1})$, 集合$M=\\{m|m>1,m\\in \\mathbf{R}\\}$.\\\\\n(1) 求证: 当$m\\in M$时, $f(x)$的定义域为$x\\in \\mathbf{R}$; 反之, 若$f(x)$对一切实数$x$都有意义, 则$m\\in M$;\\\\\n(2) 当$m\\in M$时, 求$f(x)$的最小值;\\\\\n(3) 求证: 对每一个$m\\in M$, $f(x)$的最小值都不小于1.", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -189609,8 +189876,11 @@ }, "008003": { "id": "008003", - "content": "计算: $\\log _55\\sqrt 5+\\ln e$.", - "objs": [], + "content": "计算: $\\log _55\\sqrt 5+\\ln \\mathrm{e}$.", + "objs": [ + "K0204001B", + "K0204003B" + ], "tags": [ "第二单元" ], @@ -189631,7 +189901,10 @@ "008004": { "id": "008004", "content": "计算: $\\lg \\sqrt {10}-\\lg 0.01$.", - "objs": [], + "objs": [ + "K0204001B", + "K0204003B" + ], "tags": [ "第二单元" ], @@ -190709,7 +190982,11 @@ "008049": { "id": "008049", "content": "已知$10, \\ a\\ne 1)$没有最小值, 则$a$的取值范围是\\blank{50}.", - "objs": [], + "objs": [ + "K0214002B" + ], "tags": [ "第二单元" ], @@ -283946,7 +284237,9 @@ "030044": { "id": "030044", "content": "不等式$\\log_{\\frac{1}{2}}(x^2+x+1)<\\log_{\\frac{1}{2}}(5x-2)$的解集为\\blank{80}.", - "objs": [], + "objs": [ + "K0214002B" + ], "tags": [ "第二单元" ], @@ -283970,7 +284263,9 @@ "030045": { "id": "030045", "content": "已知函数$y=f(2x+1)$的定义域为$[0,3]$, 则函数$y=f(3x-1)$的定义域为\\blank{50}.", - "objs": [], + "objs": [ + "K0215003B" + ], "tags": [ "第二单元" ], @@ -283994,7 +284289,11 @@ "030046": { "id": "030046", "content": "作出函数$y=(x^3-1)^2-1$的大致图像, 写出它的单调区间, 并证明你的结论.", - "objs": [], + "objs": [ + "K0219001B", + "K0219003B", + "K0220001B" + ], "tags": [ "第二单元" ], @@ -284018,7 +284317,9 @@ "030047": { "id": "030047", "content": "判断函数$y=(\\sqrt{x}+2)(2^x+1)$的单调性, 并证明.", - "objs": [], + "objs": [ + "K0219003B" + ], "tags": [ "第二单元" ], @@ -284042,7 +284343,9 @@ "030048": { "id": "030048", "content": "设常数$a\\in \\mathbf{R}$. 若$y=\\log_{\\frac 12}(x^2-2ax+12)$在$[-2,+\\infty)$上是严格减函数, 则$a$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0219001B" + ], "tags": [ "第二单元" ], @@ -284066,7 +284369,9 @@ "030049": { "id": "030049", "content": "判断函数$y=\\dfrac{\\sqrt{4x^2+1}+2x+1}{\\sqrt{4x^2+1}+2x-1}$的奇偶性, 并说明理由.", - "objs": [], + "objs": [ + "K0218001B" + ], "tags": [ "第二单元" ], @@ -284090,7 +284395,10 @@ "030050": { "id": "030050", "content": "已知$f(x)=ax+\\dfrac{1}{x+1}, \\ a\\in \\mathbf{R}$. 若$y=f(x)$在$x\\in [3,4]$时有零点, 则$a$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0223001B", + "K0223004B" + ], "tags": [ "第二单元" ], @@ -284114,7 +284422,9 @@ "030051": { "id": "030051", "content": "函数$y=x^3+\\dfrac{27}{x}, \\ x\\in [1,4]$的最大值为\\blank{50}, 最小值为\\blank{50}.", - "objs": [], + "objs": [ + "K0234003X" + ], "tags": [ "第二单元" ], @@ -284138,7 +284448,9 @@ "030052": { "id": "030052", "content": "已知函数$y=-x^2+2ax$, $x\\in [0, 1]$的最大值为$a+1$. 求实数$a$的值.", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -284162,7 +284474,9 @@ "030053": { "id": "030053", "content": "已知$y=f(x)$是定义在$(-2,2)$上的奇函数, 在区间$[0,2)$上是严格增函数, 且$f(1-a)+f(1-a^2)<0$, 则实数$a$的取值范围为\\blank{50}.", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -284332,7 +284646,9 @@ "030060": { "id": "030060", "content": "如图, 用长为$l$的铁丝弯成中间为矩形, 两端为半圆形的空心框架, 若矩形一边长为$2x$, 试用解析式将此框架围成的面积$y$表示为$x$的函数.\n\\begin{center}\n \\begin{tikzpicture}[>=latex]\n \\draw (2,0) -- (2,1.5) arc (0:180:1) -- (0,0) arc (180:360:1) -- cycle;\n \\draw [<->] (0,0.3) -- (2,0.3) node [midway, fill = white] {$2x$};\n \\end{tikzpicture}\n\\end{center}", - "objs": [], + "objs": [ + "K0221002B" + ], "tags": [ "第二单元" ], @@ -284544,5 +284860,128 @@ ], "remark": "", "space": "" + }, + "030069": { + "id": "030069", + "content": "已知向量$\\overrightarrow a=(1,2)$, $\\overrightarrow b=(0,3)$, 则与$\\overrightarrow a$垂直的单位向量的坐标为\\blank{50}; $\\overrightarrow b$在$\\overrightarrow a$的方向上的投影的坐标为\\blank{50}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届高三第一轮复习讲义-20220924修改", + "edit": [ + "20220701\t王伟叶", + "20220924\t王慎有" + ], + "same": [], + "related": [ + "003347" + ], + "remark": "", + "space": "" + }, + "030070": { + "id": "030070", + "content": "向量$\\overrightarrow{b}=(11,12)$在$\\overrightarrow{a}=(3,4)$方向上的投影的坐标为\\blank{80}.", + "objs": [], + "tags": [ + "第五单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2016届创新班作业\t3121-向量的应用[2]-20220924修改", + "edit": [ + "20220625\t王伟叶", + "20220924\t王慎有" + ], + "same": [], + "related": [ + "001912" + ], + "remark": "", + "space": "" + }, + "030071": { + "id": "030071", + "content": "已知$(a+3b)^n$的展开式中, 各项系数的和为$2^{n+6}$, 则$n=$\\blank{50}.", + "objs": [], + "tags": [ + "第八单元" + ], + "genre": "填空题", + "ans": "$6$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20220924修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20220924\t王伟叶" + ], + "same": [], + "related": [ + "000350" + ], + "remark": "", + "space": "" + }, + "030072": { + "id": "030072", + "content": "若数列$\\{a_n\\}$的所有项都是正数, 且$\\sqrt{a_1}+\\sqrt{a_2}+\\cdots +\\sqrt{a_n}=n^2+3n$($n\\in \\mathbf{N}^*$), 则$\\dfrac{a_1}{2}+\\dfrac{a_2}{3}+\\cdots +\\dfrac{a_n}{n+1}=$\\blank{50}.", + "objs": [], + "tags": [ + "第四单元" + ], + "genre": "填空题", + "ans": "$2n^2+6n$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20220924修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20220924\t王伟叶" + ], + "same": [], + "related": [ + "000353" + ], + "remark": "", + "space": "" + }, + "030073": { + "id": "030073", + "content": "有以下命题:\\\\\n\\textcircled{1} 若函数$f(x)$既是奇函数又是偶函数, 则$f(x)$的值域为$\\{0\\}$; \\\\\n\\textcircled{2} 若函数$f(x)$是偶函数, 则$f(|x|)=f(x)$, 其中的真命题有\\blank{50}(写出所有真命题的序号).", + "objs": [ + "K0218001B", + "K0219001B" + ], + "tags": [ + "第二单元" + ], + "genre": "填空题", + "ans": "\\textcircled{1}\\textcircled{2}", + "solution": "", + "duration": -1, + "usages": [], + "origin": "赋能练习-20220924修改", + "edit": [ + "20220624\t朱敏慧, 王伟叶", + "20220924\t王伟叶" + ], + "same": [], + "related": [ + "000355" + ], + "remark": "", + "space": "" } } \ No newline at end of file