From 2da034fa38ea74329f56c7dc4573d5094fc67eb0 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Wed, 18 Jan 2023 22:56:50 +0800 Subject: [PATCH] 20230118 night --- 工具/文本文件/题号筛选.txt | 2 +- 工具/生成文件夹下的题号清单.ipynb | 2 +- 工具/识别题库中尚未标注的题目类型.ipynb | 577 ++++++++-- 工具/题号选题pdf生成.ipynb | 8 +- 文本处理工具/剪贴板文本整理_mathpix.ipynb | 21 +- 题库0.3/Problems.json | 1240 ++++++++++----------- 6 files changed, 1153 insertions(+), 697 deletions(-) diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 13707806..7b3de845 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ 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\ No newline at end of file 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\ No newline at end of file diff --git a/工具/生成文件夹下的题号清单.ipynb b/工具/生成文件夹下的题号清单.ipynb index 96b9ccfb..b0a66fa9 100644 --- a/工具/生成文件夹下的题号清单.ipynb +++ b/工具/生成文件夹下的题号清单.ipynb @@ -198,7 +198,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15" + "version": "3.9.15 (main, Nov 24 2022, 14:39:17) [MSC v.1916 64 bit (AMD64)]" }, "orig_nbformat": 4, "vscode": { diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index 7faf6a75..566b7023 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -9,73 +9,512 @@ "name": "stdout", "output_type": "stream", "text": [ - "012760 填空题\n", - "012761 填空题\n", - "012762 填空题\n", - "012763 填空题\n", - "012764 填空题\n", - "012765 填空题\n", - "012766 填空题\n", - "012767 填空题\n", - "012768 填空题\n", - "012769 填空题\n", - "012770 填空题\n", - "012771 填空题\n", - "012772 选择题\n", - "012773 选择题\n", - "012774 选择题\n", - "012775 选择题\n", - "012776 解答题\n", - "012777 解答题\n", - "012778 解答题\n", - "012779 解答题\n", - "012780 解答题\n", - "031158 填空题\n", - "031159 填空题\n", - "031160 选择题\n", - "031161 解答题\n", - "031162 选择题\n", - "031163 选择题\n", - "031164 选择题\n", - "031165 解答题\n", - "031166 解答题\n", - "031167 解答题\n", - "031168 解答题\n", - "031169 解答题\n", - "031170 解答题\n", - "031171 解答题\n", - "031172 解答题\n", - "031173 解答题\n", - "031174 填空题\n", - "031175 解答题\n", - "031176 解答题\n", - "031177 解答题\n", - "031178 解答题\n", - "031179 解答题\n", - "031180 解答题\n", - "031181 解答题\n", - "031182 解答题\n", - "031183 解答题\n", - "031184 解答题\n", - "031185 解答题\n", - "031186 解答题\n", - "031187 解答题\n", - "031188 解答题\n", - "031189 解答题\n", - "031190 解答题\n", - "031191 填空题\n", - "031192 填空题\n", - "031193 填空题\n", - "031194 填空题\n", - "031195 填空题\n", - "031196 填空题\n", - "031197 解答题\n", - "031198 解答题\n", - "031199 解答题\n", - "031200 解答题\n", - "031201 选择题\n", - "031202 选择题\n", - "031203 选择题\n" + "012781 选择题\n", + "012782 填空题\n", + "012783 选择题\n", + "012784 填空题\n", + "012785 填空题\n", + "012786 填空题\n", + "012787 解答题\n", + "012788 解答题\n", + "012789 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"metadata": { "kernelspec": { - "display_name": "Python 3.9.15 ('pythontest')", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -136,7 +575,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, 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 40f12a0d..2c929dd4 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": 5, + "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/概率统计_教师用_20230116.tex\n", + "开始编译教师版本pdf文件: 临时文件/待标注单元_教师用_20230118.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/概率统计_学生用_20230116.tex\n", + "开始编译学生版本pdf文件: 临时文件/待标注单元_学生用_20230118.tex\n", "0\n" ] } @@ -33,7 +33,7 @@ "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/概率统计\"\n", + "filename = \"临时文件/待标注单元\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/文本处理工具/剪贴板文本整理_mathpix.ipynb b/文本处理工具/剪贴板文本整理_mathpix.ipynb index e8865cd7..efcf8624 100644 --- a/文本处理工具/剪贴板文本整理_mathpix.ipynb +++ b/文本处理工具/剪贴板文本整理_mathpix.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 30, + "execution_count": 15, "metadata": {}, "outputs": [], "source": [ @@ -214,6 +214,9 @@ "data = re.sub(\"\\\\\\\\\\]\",r\"$\",data)\n", "data = re.sub(\"\\$\\$\",\"\",data)\n", "\n", + "#标点和$符号分开\n", + "data = re.sub(r\"([,.:;])\\$\",lambda x:x.group(1)+\" $\",data)\n", + "\n", "#选择题替换成标准格式\n", "data = re.sub(\"A\\.([\\s\\S]*?)B\\.([\\s\\S]*?)C\\.([\\s\\S]*?)D\\.([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", "data = re.sub(\"\\(A\\)([\\s\\S]*?)\\(B\\)([\\s\\S]*?)\\(C\\)([\\s\\S]*?)\\(D\\)([\\s\\S]*?)\\\\n\",multiple_choice,data)\n", @@ -229,6 +232,11 @@ "for i in range(20):\n", " data = re.sub(\"\\n[\\t ]*\\n\",\"\\n\",data)\n", "\n", + "#删除\\quad\n", + "data = re.sub(r\"\\\\q+uad\",\"\",data)\n", + "\n", + "\n", + "\n", "data1 = data #替换后暂存data1\n", "\n", "#分离文字和公式\n", @@ -266,8 +274,11 @@ " text1 = re.sub(r\"\\n\\d{1,3}\\.\",r\"\\n\\\\item \",text1)\n", " # text1 = re.sub(r\"\\s{2,}\\.\",r\"\\\\blank{50}.\",text1)\n", " # text1 = re.sub(r\"\\s{2,}\\,\",r\"\\\\blank{50},\",text1)\n", - " text1 = re.sub(r\"\\\\bracket\\{20\\}\\n\",r\"\\\\bracket{20}.\\n\",text1)\n", + " text1 = re.sub(r\"\\s*\\\\bracket\\{20\\}\\s*\\n\",r\"\\\\bracket{20}.\\n\",text1)\n", + " #改非规范选择题\n", + " text1 = re.sub(r\"[\\.;]\\}\",\"}\",text1)\n", " modified_texts.append(text1)\n", + " \n", "\n", "for equation in raw_equations:\n", " equation1 = equation\n", @@ -330,6 +341,12 @@ "modified_data = re.sub(r\"\\\\{\\\\begin\\{array\\}\\{[rcl]*\\}\",r\"\\\\begin{cases}\",modified_data)\n", "modified_data = re.sub(r\"\\\\end{array}\",r\"\\\\end{cases}\",modified_data)\n", "\n", + "#识别填空题加空格\n", + "modified_data = re.sub(r\"([\\u4e00-\\u9fa5\\$])[\\s]*\\n\\\\item\",lambda x: x.group(1)+\"\\\\blank{50}.\\n\\\\item\",modified_data)\n", + "\n", + "#识别选择题加括号\n", + "modified_data = re.sub(r\"\\$\\(\\s*\\)\\$\",r\"\\\\bracket{20}\",modified_data)\n", + "modified_data = re.sub(r\"([\\u4e00-\\u9fa5\\$])[\\s]*\\n\\\\fourch\",lambda x: x.group(1)+\"\\\\bracket{20}.\\n\\\\fourch\",modified_data)\n", "\n", "setCopy(modified_data)\n", "\n", diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 4717f542..ce8f3d35 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -315036,7 +315036,7 @@ "content": "设$a, b \\in \\mathbf{R}$, 则``$a+b \\geq 4$''是``$a>2$且$b>2$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315055,7 +315055,7 @@ "content": "设常数$a \\in \\mathbf{R}$. 若$\\{x | 10\\}$, $B=\\{x|| 1-x | \\geq 1-x\\}$, 则$A \\otimes B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315150,7 +315150,7 @@ "content": "设集合$A=\\{0, x, y\\}, B=\\{2, x^2\\}$. 若$B \\subseteq A$, 求$x,y$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315162,14 +315162,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012788": { "id": "012788", "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别是$a_n=3 n$, $b_n=2 n+1$($n \\in \\mathbf{N}$, $n\\ge 1$). 将集合$\\{x | x=a_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\} \\cup\\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$中的元素从小到大依次排列, 构成数列$c_1, c_2, \\cdots, c_n, \\cdots$.\\\\\n(1) 写出$c_1, c_2, c_3, c_4$;\\\\\n(2) 求证:在数列$\\{c_n\\}$中, 但不在数列$\\{b_n\\}$中的项恰为$a_2, a_4, \\cdots, a_{2 n}, \\cdots$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315181,14 +315181,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012789": { "id": "012789", "content": "已知集合$A=\\{(x, y) | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | y-1=2^{2018} \\cdot(x-1),\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B$的元素个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无限}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315207,7 +315207,7 @@ "content": "设$U$是全集, 集合$A$, $B$满足$A\\subset B$, 给出下列四个命题: \\textcircled{1} $A \\cap \\overline B=\\varnothing$; \\textcircled{2} $B \\cap \\overline A=\\overline A$; \\textcircled{3} $B \\cup \\overline A=U$; \\textcircled{4} $\\overline A \\cap \\overline B=\\overline B$. 四个命题中, 正确命题的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315226,7 +315226,7 @@ "content": "设$U$是全集, $A, B$是两个集合, 则``存在集合$C$使得$A \\subseteq C, B \\subseteq \\overline C$''是``$A \\cap B=\\varnothing$''的 \\bracket{20}\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315245,7 +315245,7 @@ "content": "已知集合$A=\\{y | y=x^2,\\ x \\in \\mathbf{R}\\}$, $B=\\{y | y=2^x,\\ x \\in \\mathbf{R}\\}$. 则$A \\cap B=$\\bracket{20}.\n\\onech{$\\{y | y>0\\}$}{$\\{y | y \\geq 0\\}$}{$\\{y | y=2,4, u\\}$, 其中常数$u<0$且$u^2=2^u$}{$\\{(2,4),(4,16),(u, u^2)\\}$, 其中常数$u<0$且$u^2=2^u$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315264,7 +315264,7 @@ "content": "已知集合$A=\\{x, 1\\}$, $B=\\{y, 1,2\\}$, 其中$x, y \\in\\{1,2,3,4,5\\}$, 且$A \\subseteq B$. 如果把满足上述条件的一对有序整数$(x, y)$作为一个点, 则这样的点的个数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315283,7 +315283,7 @@ "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| x-1 |<2,\\ x \\in \\mathbf{Z}\\}, B=\\{x | x \\geq a\\}$. 若$A \\cap B=\\{1,2\\}$, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315302,7 +315302,7 @@ "content": "已知集合$S=\\{1,2,3,4,5\\}$, $A$是$S$的一个子集. 当$x \\in A$时, 若有$x-1 \\not\\in A$, 且$x+1 \\not\\in A$, 则称$x$为$A$的一个``孤立元素'', 那么$S$中无``孤立元素''且恰有$4$个元素的子集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315321,7 +315321,7 @@ "content": "已知集合$M$是满足下列性质的函数$f(x)$的全体: 在在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立.\\\\\n(1) 求证: 函数$f(x)=x$不属于集合$M$;\\\\\n(2) 写出命题``存在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立''的否定形式.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315333,14 +315333,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012797": { "id": "012797", "content": "已知集合$M$是满足下列性质的函数$f(x)$的全体: 在在非零常数$T$, 对任意$x \\in \\mathbf{R}$, 有$f(x+T)=T f(x)$成立.\\\\\n(1) 求证: 函数$f(x)=x$不属于集合$M$;\\\\\n(2) 求证: 函数$f(x)=(\\dfrac{1}{2})^x$属于集合$M$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315352,14 +315352,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012798": { "id": "012798", "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-3 a x+2 a^2<0\\}$. 若$A \\cap B=B$, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315371,14 +315371,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012799": { "id": "012799", "content": "设常数$m \\in \\mathbf{R}$, 集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-2 x+2 m<0\\}$. 若$A \\cup B=A$, 求$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315390,14 +315390,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012800": { "id": "012800", "content": "不等式$|x-1| \\geq 2$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315416,7 +315416,7 @@ "content": "不等式$\\dfrac{5-x}{2 x-4} \\leq 1$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315435,7 +315435,7 @@ "content": "不等式$|x-1|f(x)$时$x$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315492,7 +315492,7 @@ "content": "设$a, b$为非零实数, 且$ab\\end{cases}$的一个解, 则$a, b$满足的条件为\\bracket{20}.\n\\fourch{$a<1b c$}{$a d$与$b c$的大小不确定}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315720,7 +315720,7 @@ "content": "设常数$a, b \\in \\mathbf{R}$. 若$x=1$是不等式组$\\begin{cases}xb\\end{cases}$的唯一整数解, 则$a, b$满足的条件为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315739,7 +315739,7 @@ "content": "设$a, b \\in \\mathbf{Z}$, 且$a+b=2023$, 则$2^a+2^b$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315758,7 +315758,7 @@ "content": "设常数$k \\in \\mathbf{R}$. 已知关于$x$的不等式$(1+k^2) x \\leq k^6+2$的解集是$M$. 有下面四个命题: \\textcircled{1} 对任意实数$k$, 总有$0 \\in M$; \\textcircled{2} 存在实数$k$, 使得$1 \\notin M$; \\textcircled{3} 存在实数$k$, 使得$2 \\in M$; \\textcircled{4} 对任意实数$k$, 总有$3 \\notin M$. 上述四个命题中, 正确的命题的序号为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315777,7 +315777,7 @@ "content": "设$a_1, b_1, a_2, b_2$均为非零实数, 不等式$a_1 x+b_1>0$和$a_2 x+b_2>0$的解集分别为集合$M$和$N$, 那么``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}$''是``$M=N$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -315796,7 +315796,7 @@ "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|| 2-x |<5, x \\in \\mathbf{R}\\}$, $B=\\{x|| x+a | \\geq 4, \\ x \\in \\mathbf{R}\\}$. 若$A \\cup B=\\mathbf{R}$, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315808,14 +315808,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012822": { "id": "012822", "content": "已知数列$\\{a_n\\}$,$ a_n=2 n-10$, $S_n$是数列$\\{a_n\\}$的前$n$项和. 设$c_n=a_n S_n$, 在数列$\\{c_n\\}$中,\\\\\n(1) 是否存在小于零的项? 若存在, 求出这些项的序数; 若不存在, 说明理由;\\\\\n(2) 求小于$10^4$的$\\{c_n\\}$的项的个数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315827,14 +315827,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012823": { "id": "012823", "content": "不等式$\\log _{\\frac{1}{2}}(3-x) \\geq 1$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315853,7 +315853,7 @@ "content": "函数$d(x)=\\begin{cases}0, & x \\in \\mathbf{Q}, \\\\ 1, & x \\notin \\mathrm{Q},\\end{cases}$ 则$d(d(x))=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315872,7 +315872,7 @@ "content": "设常数$a \\in \\mathbf{R}$. 若关于$x$的方程$\\dfrac{x-1}{3 x-2}=a$有实数解, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315891,7 +315891,7 @@ "content": "已知函数$f(x)=\\begin{cases}x^2+4 x, & x \\geq 0,\\\\ 4 x-x^2, & x<0.\\end{cases}$ 若$f(2-a^2)>f(a)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315910,7 +315910,7 @@ "content": "设常数$a \\in \\mathbf{R}$. 若$y=\\log _{\\frac{1}{2}}(x^2-a x+2)$在$[-1,+\\infty)$上是减函数, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315929,7 +315929,7 @@ "content": "已知函数$f(x)=\\sqrt{m x^2+(m-3) x+1}$的值域是$[0,+\\infty)$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315948,7 +315948,7 @@ "content": "定义域和值域均为$[-a, a]$(常数$a>0$) 的函数$y=f(x)$和$y=g(x)$的图像如图所示, 给出下列四个命题: \\textcircled{1} 方程$f(g(x))=0$有且仅有三个解; \\textcircled{2} 方程$g(f(x))=0$有且仅有三个解; \\textcircled{3} 方程$f(f(x))=0$有且仅有九个解; \\textcircled{4} 方程$g(g(x))=0$有且仅有一个解. 那么, 其中正确命题的序号是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,-2) .. controls +(75:1) and +(180:0.6) .. (-1,0.6) .. controls +(0:0.6) and +(225:1.5) .. (1,0) .. controls +(45:0.5) and +(255:1) .. (2,2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (-1,1) node [above] {\\small $y=f(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2) rectangle (2,2);\n\\draw (-2,2) .. controls +(-45:1) and +(165:0.6) .. (0,0.5) .. controls +(-15:0.6) and +(135:0.5) .. (1,0) .. controls +(-45:0.5) and +(105:1) .. (2,-2);\n\\filldraw (1,0) circle (0.03);\n\\draw (1,0) node [below] {\\tiny $\\dfrac a2$};\n\\draw (1,1) node [above] {\\small $y=g(x)$};\n\\draw (-2,0) node [below left] {$-a$};\n\\draw (2,0) node [below right] {$a$};\n\\draw (0,2) node [above right] {$a$};\n\\draw (0,-2) node [below right] {$-a$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315967,7 +315967,7 @@ "content": "设函数$f(x)=\\dfrac{a^x}{1+a^x}$($a>0$, $a \\neq 1$), $[m]$表示不超过实数$m$的最大整数, 则函数$g(x)=[f(x)-\\dfrac{1}{2}]+[f(-x)-\\dfrac{1}{2}]$的值域为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -315986,7 +315986,7 @@ "content": "记$\\min \\{p, q\\}=\\begin{cases}p, & p \\leq q, \\\\ q, & p>q,\\end{cases}$ 若函数$f(x)=\\min \\{3+\\log _{\\frac{1}{4}} x, \\log _2 x\\}$.\\\\\n(1) 用分段函数形式写出函数$f(x)$的解析式;\\\\\n(2) 求$f(x)<2$的解集.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -315998,14 +315998,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012832": { "id": "012832", "content": "设函数$f(x)=a x$, $g(x)=|x-a|$, 常数$a>0$.\\\\\n(1) 当$a=2$时, 解关于$x$的不等式$f(x)>g(x)$;\\\\\n(2) 记$F(x)=f(x)-g(x)$, 若$F(x)$在$(0,+\\infty)$上有最大值, 求$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316017,14 +316017,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012833": { "id": "012833", "content": "设$a$为实数, 记函数$f(x)=a \\sqrt{1-x^2}+\\sqrt{1+x}+\\sqrt{1-x}$的最大值为$g(a)$.\\\\\n(1) 设$t=\\sqrt{1+x}+\\sqrt{1-x}$, 求$t$的取值范围, 并把$f(x)$表示为$t$的函数$m(t)$;\\\\\n(2) 求$g(a)$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316036,14 +316036,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012834": { "id": "012834", "content": "函数$f(x)=\\sqrt{x+1}+\\dfrac{1}{2-x}$的定义域为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316062,7 +316062,7 @@ "content": "若集合$A=\\{x | \\lg x<1\\}$, $B=\\{y | y=\\sin x,\\ x \\in \\mathrm{R}\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316081,7 +316081,7 @@ "content": "``$a=1$''是``函数$f(x)=|x-a|$在区间$[1,+\\infty)$上为严格增函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -316100,7 +316100,7 @@ "content": "若$x>1$, 则函数$y=\\dfrac{x^2-x+1}{x-1}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316119,7 +316119,7 @@ "content": "函数$f(x)=x^2-2 x+2$($x \\leq 0$)的反函数是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316138,7 +316138,7 @@ "content": "若函数$f(x)=\\log _a(2-a x)$在$[0,1]$上单调递减, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316157,7 +316157,7 @@ "content": "设函数$f(x)=a^{x+1}-2$($a>1$)的反函数为$y=f^{-1}(x)$, 若函数$y=f^{-1}(x)$的图像不经过第二象限, 则$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316176,7 +316176,7 @@ "content": "已知$f(x)=4^x-k \\cdot 2^x+1$, 当$x \\in \\mathbf{R}$时, $f(x)$恒为正值, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316195,7 +316195,7 @@ "content": "已知定义在$\\mathbf{R}$上的奇函数$f(x)$, 满足$f(x-4)=-f(x)$, 且在区间$[0,2]$上是增函数, 若方程$f(x)=m$($m>0$)在区间$[-8,8]$上有四个不同的根$x_1, x_2, x_3, x_4$, 则$x_1+x_2+x_3+x_4=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316214,7 +316214,7 @@ "content": "设函数$f(x)=\\dfrac{2 x}{1+|x|}$($x \\in \\mathbf{R}$), 区间$M=[a, b]$($a0$, 函数$g(x)=\\dfrac{x}{x+1}$, $h(x)=\\dfrac{1}{x+a}$, 且$f(x)=g(x) \\cdot h(x)$.\\\\\n(1) 若$a=1$, 并设函数$f(x)$的定义域是$[1,2]$, 求函数$f(x)$的值域;\\\\\n(2) 对于给定的常数$a$, 是否存在实数$t$, 使得$g(t)=h(t)$成立? 若存在, 求出这样的所有$t$的值; 若不存在, 说明理由;\\\\\n(3) 若$a>1$, 问是否存在常数$a$的值, 使函数$f(x)$的定义域是$[1, a]$, 值域为$[\\dfrac{1}{2(a+1)}, \\dfrac{1}{a^2}]$? 若存在, 求出这样的$a$的值; 若不存在, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316264,14 +316264,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012846": { "id": "012846", "content": "函数$y=2^x+1$的反函数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316290,7 +316290,7 @@ "content": "函数$y=\\dfrac{x^2+5}{\\sqrt{x^2+4}}$的值域是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316309,7 +316309,7 @@ "content": "设函数$f(x)=a|x|+\\dfrac{b}{x}$($a$、$b$为常数), 且\\textcircled{1} $f(-2)=0$; \\textcircled{2} $f(x)$有且仅有两个严格增区间, 则同时满足上述条件的一个有序数对$(a, b)$为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316328,7 +316328,7 @@ "content": "已知函数$f(x)=\\log _a(2^x+b-1)$($a>0$, $a \\neq 1$)的图像如图所示, 则$a, b$满足的关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.95:2.5,samples = 100] plot (\\x,{ln(pow(2,\\x)-0.5)/ln(5)});\n\\draw (-0.2,-1) -- (0,-1) node [right] {$-1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$00$且$a \\neq 1)$在$\\mathbf{R}$上既是奇函数, 又是减函数, 则$g(x)=\\log _a(x+k)$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},\\x);\n\\end{tikzpicture}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -316366,7 +316366,7 @@ "content": "已知$f(x)=\\begin{cases}(2-a) x+1, & x<1, \\\\ a^x, & x \\geq 1\\end{cases}$满足: 对任意$x_1 \\neq x_2$, 都有$\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$成立, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316385,7 +316385,7 @@ "content": "已知$f(x)$是定义在$\\mathbf{R}$上的函数, 且$f(1)=1$, 对任意的$x \\in \\mathbf{R}$都有下列两式成立: $f(x+5) \\geq f(x)+5$; $f(x+1) \\leq f(x)+1$. 若$g(x)=f(x)+1-x$, 则$g(6)$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316404,7 +316404,7 @@ "content": "设函数$y=f(x)$在$\\mathbf{R}$上有定义, 对于任意给定正数$M$, 定义函数$f_M(x)=\\begin{cases}f(x), & f(x) \\leq M, \\\\ M, & f(x)>M,\\end{cases}$ 则称函数$f_M(x)$为$f(x)$的``孪生函数'', 若给定函数$f(x)=2-x^2$, $M=1$, 则$f_M(2)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316423,7 +316423,7 @@ "content": "设常数$a \\in \\mathbf{R}$, 已知函数$f(x)=x+\\dfrac{a}{x}$, $x \\in[1,+\\infty)$, 求$f(x)$的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316435,14 +316435,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012855": { "id": "012855", "content": "设常数$a \\in \\mathbf{R}$, 已知函数$f(x)=x^2+\\dfrac{a}{x}$.\\\\\n(1) 判断函数$f(x)$的奇偶性;\\\\\n(2) 若$f(x)$在区间$[2,+\\infty)$上是严格增函数, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316454,14 +316454,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012856": { "id": "012856", "content": "设关于$x$的方程$x^2+2 x=(a+2) x+2$的两个实根为$x_1$、$x_2$, 是否存在实数$m$, 使得不等式$m^2+t m+1 \\geq|x_1-x_2|$对任意$a \\in[-1,1]$及任意$t \\in[-1,1]$恒成立? 若存在, 求$m$的取值范围; 若不存在, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316473,14 +316473,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012857": { "id": "012857", "content": "函数$f(x)=x^2+\\dfrac{4}{x^2+3}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316499,7 +316499,7 @@ "content": "若$f(x+2)=\\begin{cases}\\tan x, & x \\geq 0, \\\\ \\log _2(-x), & x<0,\\end{cases}$ 则$f(\\dfrac{\\pi}{4}+2) \\cdot f(-2)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316518,7 +316518,7 @@ "content": "已知函数$y=f(x)$对任意实数$x$都有$f(-x)=f(x)$, $f(x)=-f(x+1)$, 且在$[0,1]$上单调递减, 则$f(\\dfrac{7}{2})$、$f(\\dfrac{7}{3})$、$f(\\dfrac{7}{5})$的大小顺序是\\blank{50}.(用``$>$''连接)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316537,7 +316537,7 @@ "content": "设函数$f(x)=\\dfrac{1}{1-\\sqrt{x}}$($0 \\leq x<1$)的反函数为$f^{-1}(x)$, 则\\bracket{20}.\n\\twoch{$f^{-1}(x)$在其定义域上是增函数且最大值为$1$}{$f^{-1}(x)$在其定义域上是减函数且最小值为$0$}{$f^{-1}(x)$在其定义域上是减函数且最大值为$1$}{$f^{-1}(x)$在其定义域上是增函数且最小值为$0$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -316556,7 +316556,7 @@ "content": "若函数$f(x)$($x \\in \\mathbf{R}$)为奇函数, 且存在反函数$f^{-1}(x)$(与$f(x)$不同), $F(x)=\\dfrac{2^{f(x)}-2^{f^{-1}(x)}}{2^{f(x)}+2^{f^{-1}(x)}}$, 则下列关于函数$F(x)$的奇偶性的说法中正确的是\\bracket{20}.\n\\twoch{$F(x)$是奇函数非偶函数}{$F(x)$是偶函数非奇函数}{$F(x)$既是奇函数又是偶函数}{$F(x)$既非奇函数又非偶函数}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -316575,7 +316575,7 @@ "content": "设函数$g(x)=x^2-2$, $f(x)=\\begin{cases}g(x)+x+4, & x=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw (0,0) -- (2,2);\n\\filldraw [fill = white] (0,0) circle (0.03);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 0.5:2,samples = 100] plot (\\x,{1/\\x});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 0.5:1,samples = 100] plot (\\x,{1/\\x}) -- (2,2);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0.1,1) -- (0,1) node [left] {$1$};\n\\draw [dashed] (1,0) -- (1,1) -- (0,1);\n\\draw [domain = 2:1,samples = 100] plot (\\x,{1/\\x}) -- (0,0);\n\\filldraw [fill = white] (0,0) circle (0.03);\n\\end{tikzpicture}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -316613,7 +316613,7 @@ "content": "设常数$a \\in \\mathbf{R}$. 若函数$f(x)=|x+a|-|x-1|$是定义在$\\mathbf{R}$上的奇函数, 但不是偶函数, 则函数$f(x)$的严格增区间为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316632,7 +316632,7 @@ "content": "设$f(x)$是定义域为$\\mathbf{R}$的函数, 且满足$f(x+2)=f(x+1)-f(x)$, 如果$f(1)=\\lg \\dfrac{3}{2}$, $f(2)=\\lg 15$, 则$f(2022)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316651,7 +316651,7 @@ "content": "函数$y=f(x)$是定义在$\\mathbf{R}$上的恒不为零的函数, 且对于任意的$x, y \\in \\mathbf{R}$, 都满足$f(x) \\cdot f(y)=f(x+y)$, 则下列四个结论中, 正确的是\\blank{50}.\\\\\n\\textcircled{1} $f(0)=0$; \\textcircled{2} 对任意$x \\in \\mathbf{R}$, 都有$f(x)>0$; \\textcircled{3} $f(0)=1$; \\textcircled{4} 若$x<0$时, 有$f(x)>f(0)$, 则$f(x)$在$\\mathbf{R}$上的单调递减.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316670,7 +316670,7 @@ "content": "设$f(x)=\\dfrac{-2^x+a}{2^{x+1}+b}$($a, b$为实常数). 若$f(x)$是奇函数, 求$a$与$b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316682,14 +316682,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012868": { "id": "012868", "content": "已知函数$f(x)=x+\\dfrac{m}{x}+2$($m$为实常数).\\\\\n(1) 若函数$y=f(x)$在区间$[2,+\\infty)$上是增函数, 求实数$m$的取值范围;\\\\\n(2) 设$m<0$, 若不等式$f(x) \\leq k x$在$x \\in[\\dfrac{1}{2}, 1]$有解, 求$k$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316701,14 +316701,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012869": { "id": "012869", "content": "若函数$f(x)=\\sqrt{x}+1$的反函数为$f^{-1}(x)$, 则$f^{-1}(1)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316727,7 +316727,7 @@ "content": "函数$y=(x-1)^{\\frac{3}{5}}$的图像不经过第\\blank{50}象限.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316746,7 +316746,7 @@ "content": "若函数$y=x^2+(a+2) x+3$, $x \\in[a, b]$的图像关于直线$x=1$对称, 则$b=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316765,7 +316765,7 @@ "content": "若偶函数$f(x)$满足: 当$x>0$时, $f(x)$为严格减函数, 且$f(\\pi)=0$, 则$\\dfrac{f(x)}{x}<0$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316784,7 +316784,7 @@ "content": "方程$9^x-6^x=2^{2 x+1}$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316803,7 +316803,7 @@ "content": "已知关于$x$的方程$9^x+(a+4) \\cdot 3^x+4=0$有实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316822,7 +316822,7 @@ "content": "已知函数$f(x)$是定义在实数集$\\mathbf{R}$上的不恒为零的偶函数, 且对任意实数$x$都有$x f(x+1)=(1+x) f(x)$, 则$f(f(\\dfrac{5}{2}))$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316841,7 +316841,7 @@ "content": "对于函数$f(x)=\\dfrac{x}{1+|x|}$($x \\in \\mathbf{R}$), 给出以下三个命题: \\textcircled{1} 函数$f(x)$的值域为$[-1,1]$; \\textcircled{2} 若$x_1 \\neq x_2$, 则一定有$f(x_1) \\neq f(x_2)$; \\textcircled{3} 若规定$f_1(x)=f(x)$, $f_n(x)=f(f_{n-1}(x))$, 则$f_n(x)=\\dfrac{x}{1+n|x|}$对任意$n \\in N$, $n\\ge 1$恒成立. 上述三个命题中正确命题序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316860,7 +316860,7 @@ "content": "已知函数$f(x)=2^x-\\dfrac{1}{2^{|x|}}$.\\\\\n(1) 若$f(x)=2$, 求$x$的值;\\\\\n(2) 若$2^t f(2 t)+m f(t) \\geq 0$对于$t \\in[1,2]$恒成立, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316872,14 +316872,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012878": { "id": "012878", "content": "求证: 函数$f(x)=x^3+x-1$在区间$(\\dfrac{1}{2}, 1)$上存在唯一的零点.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316891,14 +316891,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012879": { "id": "012879", "content": "若函数$f(x)=x^2+m x+2$在区间$[0,2]$上存在两个不同的零点, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316910,14 +316910,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012880": { "id": "012880", "content": "已知函数$f(x)$的定义域是$x \\neq 0$的一切实数, 对定义域内的任意$x_1, x_2$都有$f(x_1 \\cdot x_2)=f(x_1)+f(x_2)$, 且当$x>1$时$f(x)>0$, $f(2)=1$.\\\\\n(1) 求证:$f(x)$是偶函数;\\\\\n(2) $f(x)$在$(0,+\\infty)$上是增函数;\\\\\n(3) 解不等式$f(2 x^2-1)<2$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -316929,14 +316929,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012881": { "id": "012881", "content": "奇函数$y=f(x)$, 当$x<0$时, $f(x)=x+\\lg |x|$, 则$f(10)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316955,7 +316955,7 @@ "content": "关于$x$的方程$\\lg ^2 x+\\lg x^2=0$的解是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316974,7 +316974,7 @@ "content": "函数$y=\\log _{\\frac{1}{3}}(1-x^2)$的单调增区间为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -316993,7 +316993,7 @@ "content": "已知$x_0$是函数$f(x)=2^x+\\dfrac{1}{1-x}$的一个零点. 若$x_1 \\in(1, x_0)$, $x_2 \\in(x_0,+\\infty)$, 则\\bracket{20}.\n\\fourch{$f(x_1)<0$, $f(x_2)<0$}{$f(x_1)<0$, $f(x_2)>0$}{$f(x_1)>0$, $f(x_2)<0$}{$f(x_1)>0$, $f(x_2)>0$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317012,7 +317012,7 @@ "content": "函数$f(x)$的定义域关于原点对称, 对定义域中任一$x$值, 恒有$|f(x)|=|f(-x)|$成立, 则\\bracket{20}.\n\\twoch{$f(x)$是奇函数}{$f(x)$是偶函数}{$f(x)$不可能既非奇函数也非偶函数}{$f(x)$有可能既非奇函数也非偶函数}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317031,7 +317031,7 @@ "content": "已知$f(x)=\\log _{\\frac{1}{2}} x$的反函数为$f^{-1}(x)$, 若$f^{-1}(a) \\cdot f^{-1}(b)=\\dfrac{1}{4}$, 则$f(a+b)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317050,7 +317050,7 @@ "content": "设函数$f(x)$在$(-\\infty,+\\infty)$内有定义, 给出下列四个函数: \\textcircled{1} $y=-|f(x)|$; \\textcircled{2} $y=x f(x^2)$; \\textcircled{3} $y=-f(-x)$; \\textcircled{4} $y=f(x+1)-f(1-x)$. 其中必为奇函数的是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317069,7 +317069,7 @@ "content": "设$f(x)$是定义在$\\mathbf{R}$上且以$3$为周期的奇函数, 若$f(1) \\leq 1$, $f(2)=\\dfrac{2 a-3}{a+1}$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317088,7 +317088,7 @@ "content": "已知函数$f(x)=\\log_2(4^x+1)-x$. 若函数$F(x)=f(x)-m$的一个零点在区间$(0, \\dfrac{1}{2})$内, 则实数$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317107,7 +317107,7 @@ "content": "设$a, b, k$是实数, 二次函数$f(x)=x^2+a x+b$满足: $f(k-1)$与$f(k)$异号, $f(k+1)$与$f(k)$同号. 在以下关于$f(x)$的零点的命题中, 假命题的序号为\\bracket{20}.\\\\\n\\textcircled{1} 该二次函数的两个零点之差一定大于$2$; \\textcircled{2} 该二次函数的零点都小于$k$; \\textcircled{3} 该二次函数的零点都大于$k-1$.\n\\fourch{(1)(2)}{(2)(3)}{(1)(3)}{(1)(2)(3)}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317126,7 +317126,7 @@ "content": "已知函数$f(x)=a \\cdot 2^x+b \\cdot 3^x$, 其中常数$a, b$满足$a b \\neq 0$.\\\\\n(1) 若$ab>0$, 判断函数$y=f(x)$的单调性;\\\\\n(2) 若$ab<0$, 求$f(x+1)>f(x)$时$x$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317138,14 +317138,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012892": { "id": "012892", "content": "已知函数$f(x)=\\log _4(4^x+1)$, $g(x)=(k-1) x$, 记$F(x)=f(x)-g(x)$, 并且$F(x)$为偶函数.\\\\\n(1) 求常数$k$的值;\\\\\n(2) 若对一切$a \\in \\mathbf{R}$, 不等式$F(a)>-\\dfrac{1}{2} m$恒成立, 求实数$m$的取值范围;\\\\\n(3) 设$M(x)=\\log _4(a \\cdot 2^x-\\dfrac{4}{3} a)$, 若函数$F(x)$与$M(x)$的图像有且只有一个公共点, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317157,14 +317157,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012893": { "id": "012893", "content": "函数$y=\\log _{0.7}(x^2-3 x+2)$的单调减区间为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317183,7 +317183,7 @@ "content": "若$\\alpha \\in\\{-1,-3, \\dfrac{1}{3}, 2\\}$, 则使函数$y=x^\\alpha$的定义域为$\\mathbf{R}$且在$(-\\infty, 0)$上单调递增的$\\alpha$值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317202,7 +317202,7 @@ "content": "函数$y=\\dfrac{1}{x^2-4 x+5}$的图像关于\\bracket{20}.\n\\fourch{$y$轴对称}{原点对称}{直线$x=2$对称}{点$(2,1)$对称}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317221,7 +317221,7 @@ "content": "定义运算: $a \\otimes b=\\begin{cases}b, & a \\geq b, \\\\ a, & a0$.\\\\\n(1) 当$01$;\\\\\n(2) 是否存在实数$a$、$b$($a0$, 那么$f(x)$在$(1,+\\infty)$上是\\bracket{20}.\n\\fourch{增函数}{减函数}{非单调函数}{由$a$值决定单调性}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317449,7 +317449,7 @@ "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$满足$f(-2-x)=f(-2+x)$, 且当$x \\geq -2$时, 函数的解析式为$f(x)=x^2-1$, 则当$x<-2$时, 函数的解析式$f(x)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317468,7 +317468,7 @@ "content": "函数$y=f(x)$的反函数为$y=f^{-1}(x)$, 如果函数$y=f(x)$的图像过点$(2,-2)$, 那么函数$y=f^{-1}(-2 x)+1$的图像一定过点\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317487,7 +317487,7 @@ "content": "函数$f(x)$对于任意实数$x$满足条件$f(x+2)=\\dfrac{1}{f(x)}$, 若$f(1)=-\\dfrac{1}{5}$, 则$f(f(3))=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317506,7 +317506,7 @@ "content": "用$\\min \\{a, b\\}$表示$a$、$b$两数中的最小值. 若函数$f(x)=\\min \\{|x|,|x+t|\\}$的图像关于直线$x=-\\dfrac{1}{2}$对称, 则$t$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317525,7 +317525,7 @@ "content": "已知函数$f(x)$满足: $f(1)=\\dfrac{1}{4}$, $4 f(x) f(y)=f(x+y)+f(x-y)$($x, y \\in \\mathbf{R}$), 则$f(2022)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317544,7 +317544,7 @@ "content": "设函数$f(x)=\\sqrt{a x^2+b x+c}$($a<0$)的定义域为$D$, 若所有点$(s, f(t))$($s, t \\in D$)构成一个正方形区域, 则$a$的值为\\bracket{20}.\n\\fourch{$-2$}{$-4$}{$-8$}{不能确定}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -317563,7 +317563,7 @@ "content": "已知函数$f(x)$和$g(x)$的图像关于原点对称, 且$f(x)=x^2+x$.\\\\\n(1) 求函数$y=g(x)$的解析式;\\\\\n(2) 若$h(x)=g(x)-m \\cdot f(x)+3$在$[-1,1]$上是增函数, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317575,14 +317575,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012915": { "id": "012915", "content": "记函数$f(x)$的定义域为$D$, 若存在$x_0 \\in D$, 使$f(x_0)=x_0$成立, 则称以$(x_0, x_0)$为坐标的点为函数$f(x)$图像上的不动点.\\\\\n(1) 若函数$f(x)=\\dfrac{3 x+a}{x+b}$的图像上有两个关于原点对称的不动点, 求$a$、$b$应满足的条件;\\\\\n(2) 在 (1) 的条件下, 若$a=8$, 记$f(x)$图像上的两个不动点分别为$M$、$N$, 点$P$为函数$f(x)$图像上的另一点, 且其纵坐标$y_P>3$, 求点$P$到直线$MN$距离的最小值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317594,14 +317594,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012916": { "id": "012916", "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2 a_n-1$, $a_1=0$, 则$a_3=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317620,7 +317620,7 @@ "content": "已知等比数列$\\{a_n\\}$的通项为$a_n=\\dfrac{1}{3^n}$, 则其前$n$项和$S_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317639,7 +317639,7 @@ "content": "已知等差数列$\\{a_n\\}$的公差为$2$, 则等差数列$\\{a_{2 n-1}\\}$的公差为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317658,7 +317658,7 @@ "content": "已知数列$\\{a_n\\}$, 则$a_{99}$是数列$\\{a_{2 n-1}\\}$的第\\blank{50}项.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317677,7 +317677,7 @@ "content": "设$\\{a_n\\}$为等差数列, 公差$d=\\dfrac{1}{2}$, 前$100$项之和为$145$, 则$a_1+a_3+\\cdots+a_{99}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317696,7 +317696,7 @@ "content": "已知数列$\\{a_n\\}$满足: $a_{4 n-3}=1$, $a_{4 n-1}=0$, $a_{2 n}=a_n$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则$a_{2009}=$\\blank{50}, $a_{2014}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317715,7 +317715,7 @@ "content": "定义``等和数列'': 在一个数列中, 如果每一项与它的后一项的和都为同一个常数, 那么这个数列叫做等和数列, 这个常数叫做该数列的公和. 已知数列$\\{a_n\\}$是等和数列, 且$a_1=2$, 公和为$5$, 那么这个数列的前$n$项和$S_n$的计算公式为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317734,7 +317734,7 @@ "content": "数列$\\{a_n\\}$的通项公式$a_n=n \\cos \\dfrac{n \\pi}{2}+1$, 前$n$项和为$S_n$, 则$S_{2012}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317753,7 +317753,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_{2 n}-a_{2 n-1}=n$, $a_{2 n+1}-a_{2 n}=1$, $a_1=1$, 求通项$a_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317765,14 +317765,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012925": { "id": "012925", "content": "设$n \\in \\mathbf{N}$, $n \\geq 2$. 一个$n$行$n$列矩阵$(a_{i j})$满足$a_{11}=1$, $a_{21}=2$, 第一列从上至下构成等比数列, 每一行从左至右构成公差为$1$的等差数列.\\\\\n(1) 求该矩阵第$i$行第$j$列元素$a_{i j}$($i, j \\in[1, n] \\cap \\mathbf{N}$);\\\\\n(2) 求该矩阵主对角线上的元素之和.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317784,14 +317784,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012926": { "id": "012926", "content": "已知数列$\\{a_n\\}$与$\\{b_n\\}$满足$b_n=n a_n$($n \\in \\mathbf{N}$), 记数列$\\{a_n\\}$的前$n$项和为$S_n$, 数列$\\{b_n\\}$的前$n$项和为$T_n$.\\\\\n(1) 若数列$\\{a_n\\}$是以$1$为首项, $\\dfrac{1}{3}$为公比的等比数列, 求$T_n$;\\\\\n(2) 求证: ``$\\dfrac{T_n}{S_n}=\\dfrac{2 b_n+n}{2 a_n+n}$($n \\in \\mathbf{N}$, $n\\ge 1$)''是``$a_n=n$($n \\in \\mathbf{N}$, $n\\ge 1$)''的必要条件.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -317803,14 +317803,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012927": { "id": "012927", "content": "$2$与$6$的等差中项为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317829,7 +317829,7 @@ "content": "$2$与$6$的等比中项为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317848,7 +317848,7 @@ "content": "已知递增的等差数列$\\{a_n\\}$满足$a_1=1, a_3=a_2^2-4$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317867,7 +317867,7 @@ "content": "已知等比数列$\\{a_n\\}$, 若$\\{a_{2 n-1}\\}$公比为$4$, 则$\\{a_n\\}$公比为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317886,7 +317886,7 @@ "content": "定义在全体正整数上的函数$f(n)=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$, 则$f(n+1)-f(n)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317905,7 +317905,7 @@ "content": "数列$\\{a_n\\}$中, $a_1=1$, $a_n$, $a_{n+1}$是方程$x^2-(2 n+1) x+\\dfrac{1}{b_n}=0$的两个根, 数列$\\{b_n\\}$的前$n$项和$S_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317924,7 +317924,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+2}-a_n=2^n$, $a_1=1$, 则$a_{2 n-1}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317943,7 +317943,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=2$, $a_n=a_{n-1}+(2 n-1)+\\cdot+2^n$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317962,7 +317962,7 @@ "content": "已知$10$行$10$列矩阵$\\{a_{i j}\\}$的第$i$行第$j$列元素$a_{i j}=i \\cdot j$, 则该矩阵所有元素的和为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -317981,7 +317981,7 @@ "content": "数列$\\{a_n\\}$满足$a_{n+1}+(-1)^n a_n=2 n-1$, 则$\\{a_n\\}$的前$60$项和为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318000,7 +318000,7 @@ "content": "已知数列$\\{a_n\\}$为等差数列, 公差$d \\neq 0$, $\\{a_n\\}$的部分项组成下列数列: $a_{k_1}, a_{k_2}, \\cdots, a_{k_n}$恰为等比数列, 其中$k_1=1$, $k_2=5$, $k_3=17$, 求$k_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318012,14 +318012,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012938": { "id": "012938", "content": "已知数列$\\{a_n\\}$的通项$a_n=2^n$, 删去该数列的第$2,5, \\cdots, 3 k-1, \\cdots$项, 得到一个新数列$\\{b_n\\}$.\\\\\n(1) 求$\\{b_n\\}$的通项;\\\\\n(2) 求$\\{b_n\\}$的前$n$项和$S_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318031,14 +318031,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012939": { "id": "012939", "content": "计算:$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^{n+1}+2^{n+1}}{3^n+2^n}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318057,7 +318057,7 @@ "content": "设数列$\\{a_n\\}$, $\\{b_n\\}$都是等差数列, 若$a_1+b_1=7$, $a_3+b_3=21$, 则$a_5+b_5=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318076,7 +318076,7 @@ "content": "已知数列$\\{a_n\\}$满足$\\dfrac{1}{a_{n+1}}-\\dfrac{1}{a_n}=1$, $a_1=2$, 则$a_{10}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318095,7 +318095,7 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=2^n$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318114,7 +318114,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=3 a_n+3$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318133,7 +318133,7 @@ "content": "$\\displaystyle\\lim _{n \\to \\infty}(1-2 x)^n$存在, 则$x$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318152,7 +318152,7 @@ "content": "设$a$是实数, 若极限$\\displaystyle\\lim _{n \\to \\infty}(a n-\\dfrac{n^2+1}{n+1})$存在, 则$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318171,7 +318171,7 @@ "content": "无穷等比数列$\\{a_n\\}$的各项和为$3$, 数列$\\{b_n\\}$满足$b_n=a_{2 n-1}+a_{2 n}$, 则$\\{b_n\\}$的各项和为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318190,7 +318190,7 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=2$且$3 a_{n+1}+2S_n=3$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318202,14 +318202,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012948": { "id": "012948", "content": "已知数列$\\{a_n\\}$满足: $a_1=\\dfrac{3}{2}$且$a_n=\\dfrac{3 n a_{n-1}}{2 a_{n-1}+n-1}$, $n \\geq 2$. 数列$\\{b_n\\}$满足$b_n=1-\\dfrac{n}{a_n}$.\\\\\n(1) 求证: $\\{b_n\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318221,14 +318221,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012949": { "id": "012949", "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=q a_n+2$, $a_1=2$, 其中实常数$q>0$.\\\\\n(1) 求通项$a_n$;\\\\\n(2) 计算$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{a_{n+1}}{a_n}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318240,14 +318240,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012950": { "id": "012950", "content": "数列$1, x, y, 2$是等差数列, 则$y-x=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318266,7 +318266,7 @@ "content": "已知$x$是实数, 数列$\\{a_n\\}$的通项为$a_n=\\log _x n$. 若$\\{a_n\\}$递增, 则$x$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318285,7 +318285,7 @@ "content": "计算:$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^{-n}+2^{-n}}{3^{1-n}+2^{1-n}}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318304,7 +318304,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2 a_n+3$, $a_1=1$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318323,7 +318323,7 @@ "content": "设等比数列$\\{a_n\\}$的公比$q=-\\dfrac{1}{2}$, 且$\\lim _{n \\to \\infty}(a_1+a_3+\\cdots+a_{2 n-1})=\\dfrac{8}{3}$, 则$a_1=$", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318335,14 +318335,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012955": { "id": "012955", "content": "无穷等比数列$\\{a_n\\}$的各项和为 1 , 则首项$a_1$的取值范围为", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318354,14 +318354,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012956": { "id": "012956", "content": "若对任意的正整数$n, a_1 a_2 \\cdots a_n=2^{n+1}$, 则$a_n=$", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318373,14 +318373,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012957": { "id": "012957", "content": "设$a$是一个实数, 若$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^n}{3^{n+1}+a^n}=\\dfrac{1}{3}$, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318399,7 +318399,7 @@ "content": "已知数列$\\{a_n\\}$的递推式为$a_{n+1}=2 a_n+n-1$, $a_1=1$, 利用$b_n=a_n+n$可求得$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318418,7 +318418,7 @@ "content": "数列$\\{a_n\\}$的前$n-1$项之和$S_{n-1}=a_n$($n \\geq 2$, $n \\in \\mathbf{N}$), $a_1=1$, 则通项$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318437,7 +318437,7 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n$, $a_1=1$.\\\\\n(1) 若数列$\\{a_n\\}$是公差为$d$的等差数列, 计算: $\\displaystyle\\lim_{n\\to\\infty} \\dfrac{S_{2 n}}{S_n}$;\\\\\n(2) 若数列$\\{a_n\\}$是公比为$q$的等比数列, 计算: $\\displaystyle\\lim_{n\\to\\infty} \\dfrac{S_{2 n}}{S_n}$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318449,14 +318449,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012961": { "id": "012961", "content": "已知$x$轴上有一点列$P_k(x_k, 0)$($k=1,2,3, \\cdots$), 满足$x_1=0$, $x_2=1$. 且对任意的正整数$n$, $\\overrightarrow{P_n P_{n+2}}=\\lambda \\overrightarrow{P_{n+2} P_{n+1}}$, 其中常数$\\lambda \\neq-1$.\\\\\n(1) 设$a_n=x_{n+1}-x_n$, 求证: $\\{a_n\\}$是等比数列;\\\\\n(2) 求通项$x_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318468,14 +318468,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012962": { "id": "012962", "content": "已知数列$\\{a_n\\}$中, $\\dfrac{a_{n+1}}{a_n}=2$, $a_3=8$, 则$a_1=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318494,7 +318494,7 @@ "content": "在等差数列$\\{a_n\\}$中, $a_3+a_7=37$, 则$a_2+a_8=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318513,7 +318513,7 @@ "content": "已知数列$\\{a_n\\}$的通项$a_n=10-n$, 则集合$\\{n | a_n>0\\}$共有\\blank{50}个元素.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318532,7 +318532,7 @@ "content": "已知等比数列$\\{a_n\\}$中$a_2=1$, 则其前 3 项的和$S_3$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318551,7 +318551,7 @@ "content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac{n-\\sqrt{60}}{n-\\sqrt{59}}$($1 \\leq n \\leq 100$), 则此数列中最大项为第\\blank{50}项, 最小项为第\\blank{50}项.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318570,7 +318570,7 @@ "content": "已知数列$\\{a_n\\}$是以$-2$为公差的等差数列, $S_n$是其前$n$项和, 若$S_7$是数列$\\{S_n\\}$中的唯一最大项, 则数列$\\{a_n\\}$的首项$a_1$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318589,7 +318589,7 @@ "content": "等差数列$\\{a_n\\}$中, $S_n$为前$n$项和, 且$S_6S_8$, 给出下列命题: \\textcircled{1} 数列$\\{a_n\\}$中前$7$项是递增的, 从第$8$项开始递减; \\textcircled{2} $S_9$一定小于$S_6$; \\textcircled{3} $a_1$是各项中的最大的; \\textcircled{4} $S_7$不一定是$\\{S_n\\}$中最大项. 其中正确的序号是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -318608,7 +318608,7 @@ "content": "设$\\{a_n\\}$是公比为实数$q$的等比数列, $|q|>1$, 令$b_n=a_n+1$($n=1,2, \\cdots$), 若数列$\\{b_n\\}$有连续四项在集合$\\{-53,-23,19,37,82\\}$中, 则$q=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318627,7 +318627,7 @@ "content": "设等差数列$\\{a_n\\}$的首项$a_1$及公差$d$都为整数, 前$n$项和为$S_n$. 若$a_1 \\geq 6$, $a_{11}>0$, $S_{14} \\leq 77$, 求所有可能的数列$\\{a_n\\}$的通项公式.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318639,14 +318639,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012971": { "id": "012971", "content": "已知等差数列$\\{a_n\\}$通项$a_n=1000 n$, 等比数列$\\{b_n\\}$通项$b_n=2^n$.\\\\\n(1) 求数列$\\{a_n-b_n\\}$的最大项;\\\\\n(2) 解不等式: $a_n>b_n$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318658,14 +318658,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012972": { "id": "012972", "content": "已知等差数列$\\{a_n\\}$中$a_1=2$, 公差是正整数$d$, 等比数列$\\{b_n\\}$中, $b_1=a_1$, $b_2=a_2$.\\\\\n(1) 试给出一个$d$的值, 使得$n \\geq 3$时, $b_n$都不在$\\{a_n\\}$中, 并说明理由;\\\\\n(2) 判断$d=10$时, 是否数列$\\{b_n\\}$中的所有项都是$\\{a_n\\}$中的项, 并证明你的结论.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318677,14 +318677,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012973": { "id": "012973", "content": "已知三角形$ABC$中, $A, B, C$成等差数列, 则$B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318703,7 +318703,7 @@ "content": "设公比为$q$($q>0$)的等比数列$\\{a_n\\}$的前$n$项和为$S_n$. 若$S_2=3 a_2+2$, $S_4=3 a_4+2$, 则$q=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318722,7 +318722,7 @@ "content": "已知各项非零的等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{10}-S_7=k a_9$, 则实数$k=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318741,7 +318741,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=2^n a_n$, $a_1=1$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318760,7 +318760,7 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(\\dfrac{3}{4})^{n-1}[(\\dfrac{3}{4})^{n-1}-1]$, 则数列$\\{a_n\\}$的最大项的值为\\blank{50}, 最小项的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318779,7 +318779,7 @@ "content": "数列$\\{a_n\\}$的通项公式为$a_n=6 n-3$, 数列$\\{b_n\\}$的通项公式为$b_n=5 n-4$, 若$a_n \\leq 1000$, $b_n \\leq 1000$, 由数列$\\{a_n\\}$与数列$\\{b_n\\}$中共有的项构成数列$\\{c_n\\}$, 则数列$\\{c_n\\}$中共有\\blank{50}项.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318798,7 +318798,7 @@ "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_4$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318817,7 +318817,7 @@ "content": "已知等差数列$\\{a_n\\}$的首项及公差均为正数, 令$b_n=\\sqrt{a_n}+\\sqrt{a_{2012-n}}$($n=1,2,3, \\cdots, 2011$). 当$b_k$是数列$\\{b_n\\}$的最大项时, $k=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318836,7 +318836,7 @@ "content": "在等差数列$\\{a_n\\}$中, $a_3+a_4+a_5=84$, $a_9=73$, 对任意$m \\in \\mathbf{N}$, $n\\ge 1$, 将数列$\\{a_n\\}$中落入区间$(9^m, 9^{2 m})$内的项的个数记为$b_m$, 则数列$b_m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318855,7 +318855,7 @@ "content": "已知数列$\\{a_n\\}$满足$a_1=m$, $m \\in \\mathbf{N}$, $n\\ge 1$, $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2}, & a_n \\text {是偶数}, \\\\ 3 a_n+1, & a_n \\text{是奇数}.\\end{cases}$ $a_6=1$, 则$m$的所有可能值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318874,7 +318874,7 @@ "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_2 a_n=S_2+S_n$对一切正整数$n$都成立.\\\\\n(1) 求$a_1$, $a_2$的值;\\\\\n(2) 设$a_1>0$, 数列$\\{\\lg \\dfrac{10 a_1}{a_n}\\}$的前$n$项和为$T_n$, 当$n$为何值时, $T_n$最大? 并求出$T_n$的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318886,14 +318886,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012984": { "id": "012984", "content": "已知等差数列$\\{a_n\\}$中$a_1=2$, 公差是正整数$d$, 等比数列$\\{b_n\\}$中, $b_1=a_1$, $b_2=a_2$. 是否存在$d$, 使得数列$\\{b_n\\}$中的所有项都是$\\{a_n\\}$中的项? 若存在, 求出所有这样的$d$; 若不存在, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -318905,14 +318905,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012985": { "id": "012985", "content": "已知数列$\\{a_n\\}$满足$a_{n+1}=a_n^2-2$, $a_1=2$, 则$a_5=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318931,7 +318931,7 @@ "content": "已知等比数列的前$n$项和为$3^n-1$, 则公比为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318950,7 +318950,7 @@ "content": "已知数列$\\{a_n\\}$对任意正整数$p, q$满足$a_{p+q}=a_p+a_q$, 且$a_2=-6$, 那么$a_{10}$等于\\bracket{20}.\n\\fourch{$-165$}{$-33$}{$-30$}{$-21$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -318969,7 +318969,7 @@ "content": "已知等差数列$\\{a_n\\}$的前$n$项和$S_n$满足$S_5=5$, $S_{10}=15$, 则$S_{15}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -318988,7 +318988,7 @@ "content": "已知$a_n=2^n+3^n$, $b_n=a_{n+1}+k a_n$, 若$\\{b_n\\}$是等比数列, 则$k=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319007,7 +319007,7 @@ "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2+k n+2$. 若对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 有$a_{n+1}>a_n$恒成立, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319026,7 +319026,7 @@ "content": "若$f(n)$为$n^2+1$($n \\in \\mathbf{N}$, $n\\ge 1$)的各位数字之和, 如$14^2+1=197$, $1+9+7=17$, 则$f(14)=17$; 记$f_1(n)=f(n), f_2(n)=f(f_1(n)), \\cdots, f_{k+1}(n)=f(f_k(n))$, $k \\in \\mathbf{N}$, $k\\ge 1$. 则$f_{2015}(8)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319045,7 +319045,7 @@ "content": "若在由正整数构成的无穷数列$\\{a_n\\}$中, 对任意的正整数$n$, 都有$a_n \\leq a_{n+1}$, 且对任意的正整数$k$, 该数列中恰有$2 k-1$个$k$, 则$a_{2022}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319064,7 +319064,7 @@ "content": "设$t$是实数, 数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_n=\\dfrac{1}{2^n}$.\\\\\n(1) 若对一切正整数$n$, $S_n>a_n+t$恒成立, 求$t$的取值范围;\\\\\n(2) 若存在正整数$n$, 使$S_n>a_n+t$成立, 求$t$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319076,14 +319076,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012994": { "id": "012994", "content": "数列$\\{a_n\\}$满足$a_n=3 a_{n-1}+3^n-1$($n \\geq 2$), $a_3=95$.\\\\\n(1) 求$a_1, a_2$的值;\\\\\n(2) 是否存在实数$t$, 使得数列$\\{\\dfrac{a_n+t}{3^n}\\}$是等差数列? 若有, 求出$t$的值; 若没有, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319095,14 +319095,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012995": { "id": "012995", "content": "设常数$a \\neq \\dfrac{1}{4}$, 数列$\\{a_n\\}$的首项$a_1=a$, $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2},& n \\text {为偶数}, \\\\ a_n+\\dfrac{1}{4},& n \\text{为奇数}.\\end{cases}$ 记$b_n=a_{2 n-1}-\\dfrac{1}{4}$.\\\\\n(1) 求$a_2$, $a_3$的值;\\\\\n(2) 判断$\\{b_n\\}$是否为等比数列, 并证明你的结论.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319114,14 +319114,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "012996": { "id": "012996", "content": "$1, x, y$既是等差数列, 又是等比数列, 则$x+y=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319140,7 +319140,7 @@ "content": "已知等比数列$\\{a_n\\}$中, $a_1 a_2=1$, $a_2 a_3=2$, 则$a_n a_{n+1}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319159,7 +319159,7 @@ "content": "设数列$\\{a_n\\}$, 则``$a_{n+2}-a_n$是常数''是``$a_{n+1}-a_n$是常数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -319178,7 +319178,7 @@ "content": "已知等比数列$\\{a_n\\}$为递增数列, 且$a_5^2=a_{10}$, $2(a_n+a_{n+2})=5 a_{n+1}$, 则数列的通项公式$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319197,7 +319197,7 @@ "content": "若数列$\\{a_n\\}$满足$a_1=\\dfrac{1}{3}$, $a_n-a_{n-1}=\\dfrac{1}{n(n+2)}$, 则$a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319216,7 +319216,7 @@ "content": "若数列$\\{a_n\\}$满足$a_1=1$, $a_2=2$, 且$a_n=\\dfrac{a_{n-1}}{a_{n-2}}$($n \\geq 3$), 则$a_{4015}$为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319235,7 +319235,7 @@ "content": "若$f(\\dfrac{1}{2}+x)+f(\\dfrac{1}{2}-x)=2$对任意的实数$x$成立, 则$f(\\dfrac{1}{3000})+f(\\dfrac{2}{3000})+f(\\dfrac{3}{3000})+\\cdots+f(\\dfrac{2999}{3000})=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319254,7 +319254,7 @@ "content": "对数列$\\{a_n\\}$, 定义$\\{\\Delta^1 a_n\\}$为数列$\\{a_n\\}$的一阶差分数列, 其中$\\Delta^1 a_n=a_{n+1}-a_n$, $n \\in \\mathbf{N}$, $n \\ge 1$. 对正整数$k$, 定义$\\{\\Delta^k a_n\\}$为$\\{a_n\\}$的$k$阶差分数列, 其中$\\Delta^{k+1} a_n=\\Delta^k a_{n+1}-\\Delta^k a_n=\\Delta^1(\\Delta^k a_n)$. 已知数列$\\{a_n\\}$的通项公式$a_n=n^2$, 则$\\Delta^1 a_n=$\\blank{50}, $\\Delta^2 a_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319273,7 +319273,7 @@ "content": "在两个不等的正数$a, b$之间插入$n$个正数$x_1, x_2, \\cdots, x_n$, 若$a, x_1, x_2, \\cdots, x_n, b$成等差数列, 则$\\dfrac{a+b}{2}=\\dfrac{x_1+x_2+\\cdots+x_n}{n}$; 相应地: 若$a, x_1, x_2, \\cdots, x_n, b$成等比数列, 则可得到的结论是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319292,7 +319292,7 @@ "content": "设$a_1=2$, $a_{n+1}=\\dfrac{2}{a_n+1}$, $b_n=|\\dfrac{a_n+2}{a_n-1}|$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则数列$\\{b_n\\}$的通项公式为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319311,7 +319311,7 @@ "content": "设$S_n=1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{n}$, $f(n)=S_{2 n+1}-S_{n+1}$.\\\\\n(1) 判断数列$f(n)$的单调性;\\\\\n(2) 试确定实数$t$的范围, 使得对于$n \\in \\mathbf{N}$, $n>1$, 不等式$f(n)>t^2-\\dfrac{11}{20} t^{-2}$恒成立.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319323,14 +319323,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013007": { "id": "013007", "content": "已知数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{2}$, $2 a_{n+1}-a_n=n$.\\\\\n(1) 令$b_n=a_{n+1}-a_n-1$, 求证:$\\{b_n\\}$是等比数列.\\\\\n(2) 求数列$\\{a_n\\}$的通项公式;\\\\\n(3) 是否存在实数$\\lambda$, 使得数列$\\{a_{n+1}-\\lambda a_n\\}$为等差数列? 若存在, 求出$\\lambda$的值; 若不存在, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319342,14 +319342,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013008": { "id": "013008", "content": "终边落在$y$轴的正半轴上的所有角构成的集合为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319368,7 +319368,7 @@ "content": "已知$\\sin \\alpha=\\dfrac{1}{2}+\\cos \\alpha$, 则$\\sin 2 \\alpha=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319387,7 +319387,7 @@ "content": "若$\\tan \\theta+\\dfrac{1}{\\tan \\theta}=4$, 则$\\sin 2 \\theta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319406,7 +319406,7 @@ "content": "若扇形的圆心角为$120^{\\circ}$, 半径为$5$, 则扇形的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319425,7 +319425,7 @@ "content": "若$0 \\leq \\alpha \\leq 2 \\pi$, $\\sin \\alpha>\\sqrt{3} \\cos \\alpha$, 则$\\alpha$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319444,7 +319444,7 @@ "content": "满足$\\arcsin 2 x>\\arcsin (1-x)$的$x$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319463,7 +319463,7 @@ "content": "$\\triangle ABC$中, $A=60^{\\circ}$, $b=1$, $S_{\\triangle ABC}=\\sqrt{3}$, 则$\\dfrac{a+b+c}{\\sin A+\\sin B+\\sin C}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319482,7 +319482,7 @@ "content": "若$\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, 0)$, $\\cos (\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{1}{3}$, $\\cos (\\dfrac{\\pi}{4}-\\dfrac{\\beta}{2})=\\dfrac{\\sqrt{3}}{3}$, 则$\\cos (\\alpha+\\dfrac{\\beta}{2})=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319501,7 +319501,7 @@ "content": "满足方程$\\sin (2 x+\\dfrac{\\pi}{4})=\\cos (\\dfrac{\\pi}{6}-x)$的最小的正角是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319520,7 +319520,7 @@ "content": "设$\\triangle ABC$的内角$A, B, C$所对的边为$a, b, c$, 则下列命题正确的是\\blank{50}.\\\\\n\\textcircled{1} 若$a b>c^2$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{2} 若$a+b>2 c$, 则$C<\\dfrac{\\pi}{3}$; \\textcircled{3} 若$a^3+b^3=c^3$, 则$C<\\dfrac{\\pi}{2}$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319539,7 +319539,7 @@ "content": "已知$\\dfrac{3 \\pi}{4}<\\alpha<\\pi$, $\\tan \\alpha+\\cot \\alpha=-\\dfrac{10}{3}$.\\\\\n(1) 求$\\tan \\alpha$的值;\\\\\n(2) 求$\\dfrac{5 \\sin ^2 \\dfrac{\\alpha}{2}+8 \\sin \\dfrac{\\alpha}{2} \\cos \\dfrac{\\alpha}{2}+11 \\cos ^2 \\dfrac{\\alpha}{2}-8}{\\sqrt{2} \\sin (\\alpha-\\dfrac{\\pi}{2})}$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319551,14 +319551,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013019": { "id": "013019", "content": "已知$x_1, x_2$是方程$x^2-x \\sin \\theta+\\cos \\theta=0$的两个根, $0<\\theta<\\pi$, 求$\\arctan x_1+\\arctan x_2$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319570,14 +319570,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013020": { "id": "013020", "content": "已知函数$f(x)=2 \\cos (\\omega x+\\dfrac{\\pi}{6})$(其中$\\omega>0$, $x \\in \\mathbf{R}$)的最小正周期为$10 \\pi$.\\\\\n(1) 求$\\omega$的值;\\\\\n(2) 设$\\alpha$、$\\beta \\in[0, \\dfrac{\\pi}{2}]$, $f(5 \\alpha+\\dfrac{5}{3} \\pi)=-\\dfrac{6}{5}$, $f(5 \\beta-\\dfrac{5}{6} \\pi)=\\dfrac{16}{17}$, 求$\\cos (\\alpha+\\beta)$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319589,14 +319589,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013021": { "id": "013021", "content": "若$\\tan (\\dfrac{\\pi}{4}-\\alpha)=3$, 则$\\cot \\alpha$等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319615,7 +319615,7 @@ "content": "已知$\\cos 2 \\alpha=\\dfrac{1}{5}$, 则$\\sin ^4 \\alpha-\\cos ^4 \\alpha$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319634,7 +319634,7 @@ "content": "若$\\tan \\theta=2$, 则$\\sin 2 \\theta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319653,7 +319653,7 @@ "content": "若$\\alpha, \\beta$为锐角, 且$\\cos \\alpha=\\dfrac{4}{5}$, $\\cos (\\alpha+\\beta)=\\dfrac{3}{5}$, 则$\\sin \\beta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319672,7 +319672,7 @@ "content": "若$\\alpha \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, $\\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 且$\\tan \\alpha$、$\\tan \\beta$是方程$x^2+3 \\sqrt{3} x+4=0$的两个相异实根, 则$\\alpha+\\beta=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319691,7 +319691,7 @@ "content": "设$a, b, c \\in \\mathbf{R}$, 且$\\cos 2 x=a \\cos ^2 x+b \\cos x+c$对任意$x \\in \\mathbf{R}$恒成立, 则$a^2+b^2+c^2=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319710,7 +319710,7 @@ "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$, 若$A=30^{\\circ}$, $C=135^{\\circ}$, $a=1$, 则$c=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319729,7 +319729,7 @@ "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$, 若$a=\\dfrac{\\sqrt{5}}{2} b$, $A=2B$, 则$\\cos B=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319748,7 +319748,7 @@ "content": "若$0\\dfrac 3\\pi x$}{$\\sin x<\\dfrac{4}{\\pi^2} x^2$}{$\\sin x>\\dfrac{4}{\\pi^2} x^2$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -319767,7 +319767,7 @@ "content": "有一道解三角形的问题, 缺少一个条件, 具体如下: ``在$\\triangle ABC$中, 已知$a=\\sqrt{3}$, $B=45^{\\circ}$, 求角$A$的大小.''\n经推断, 缺少的条件为三角形一边的长度, 且答案提示$A=60^{\\circ}$, 则所缺条件为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319786,7 +319786,7 @@ "content": "已知$\\sin \\alpha$、$\\cos \\alpha$是方程$8 x^2+6 k x+2 k+1=0$的两个相异实根, 求实数$k$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319798,14 +319798,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013032": { "id": "013032", "content": "已知函数$f(x)=-\\sqrt{3} \\sin ^2 x+\\sin x \\cos x$.\\\\\n(1) 求$f(\\dfrac{25 \\pi}{6})$的值;\\\\\n(2) 设$\\alpha \\in(0, \\dfrac{\\pi}{2})$, 且$f(\\alpha)=\\dfrac{1}{4}-\\dfrac{\\sqrt{3}}{2}$, 求$\\sin 2 \\alpha$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319817,14 +319817,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013033": { "id": "013033", "content": "某兴趣小组测量电视塔$AE$的高度$H$(单位: $\\text{m}$), 如示意图, 垂直放置的标杆$BC$的高度$h=4 \\text{m}$, 仰角$\\angle ABE=\\alpha$, $\\angle ADE=\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (1,0) node [above left] {$B$} coordinate (B);\n\\draw (1,0.6) node [above] {$C$} coordinate (C);\n\\draw ($(D)!2.5!(B)$) node [below right] {$A$} coordinate (A);\n\\draw ($(D)!2.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw (D) -- (A) (B) -- (C) (A) -- (E);\n\\draw [dashed] (D) -- (E) (B) -- (E);\n\\draw (B) ++ (0,-0.3) --++ (0,0.2) (A) ++ (0,-0.3) --++ (0,0.2);\n\\draw [<->] (B) ++ (0,-0.2) --++ (1.5,0) node [midway, fill = white] {$d$};\n\\draw (B) pic [draw, \"$\\alpha$\", angle eccentricity = 1.3] {angle = A--B--E};\n\\draw (D) pic [draw, \"$\\beta$\", angle eccentricity = 1.3] {angle = A--D--E};\n\\end{tikzpicture}\n\\end{center}\n(1) 该小组已经测得一组$\\alpha$、$\\beta$的值, $\\tan \\alpha=1.24$, $\\tan \\beta=1.20$, 请据此算出$H$的值;\\\\\n(2) 该小组分析若干测得的数据后, 认为适当调整标杆到电视塔的距离$d$(单位: $\\text{m}$), 使$\\alpha$与$\\beta$之差较大, 可以提高测量精确度. 若电视塔的实际高度为$125 \\text{m}$, 试问$d$为多少时, $\\alpha-\\beta$最大?", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -319836,14 +319836,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013034": { "id": "013034", "content": "\"$x=2 k \\pi+\\dfrac{\\pi}{4}$($k \\in \\mathbf{Z}$)''是``$\\tan x=1$''成立的\\bracket{20}.\n\\twoch{必要非充分条件}{充分非必要条件}{充要条件}{既非充分又非必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -319862,7 +319862,7 @@ "content": "函数$y=\\sin (x+\\dfrac{\\pi}{3}) \\sin (x+\\dfrac{\\pi}{2})$的最小正周期$T=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319881,7 +319881,7 @@ "content": "函数$y=2 \\sin (\\dfrac{\\pi}{3}-2 x)$的单调递增区间为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319900,7 +319900,7 @@ "content": "函数$2 \\sin x=1$的解集为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319919,7 +319919,7 @@ "content": "函数$y=2 \\arccos (x-2)$($1 \\leq x<2$)的反函数是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319938,7 +319938,7 @@ "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0 \\leq \\varphi \\leq \\pi$)是$\\mathbf{R}$上的偶函数, 其图像关于点$(\\dfrac{3 \\pi}{4}, 0)$对称, 则$\\varphi$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319957,7 +319957,7 @@ "content": "已知函数$y=f(x)$的周期为$2 \\pi$, 当$x \\in[0,2 \\pi)$时, $f(x)=\\sin \\dfrac{x}{2}$, 那么方程$f(x)=\\dfrac{1}{2}$的解集是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319976,7 +319976,7 @@ "content": "对函数$y=\\sin x+\\cos x$, 有下列命题: \\textcircled{1} 若$x \\in[0, \\dfrac{\\pi}{2}]$, 则$y \\in[0, \\sqrt{2}]$; \\textcircled{2} 与函数$y=\\sin x-\\cos x$的图像关于直线$x=k \\pi+\\dfrac{\\pi}{2}$($k \\in \\mathbf{Z}$)对称; \\textcircled{3} 在区间$[\\dfrac{\\pi}{4}, \\dfrac{5 \\pi}{4}]$上单调递减; \\textcircled{4} 其图像可由$y=\\sqrt{2} \\sin 2 x$的图像纵坐标不变横坐标变为原来的$\\dfrac{1}{2}$后再向左平移$\\dfrac{\\pi}{4}$个单位得到. 正确的命题是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -319995,7 +319995,7 @@ "content": "$\\triangle ABC$的三内角$A, B, C$的对边边长分别为$a, b, c$. 若$C=60^{\\circ}$, $c=2$, 且$\\triangle ABC$的面积为$\\sqrt{3}$, 求$a, b$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320007,14 +320007,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013043": { "id": "013043", "content": "已知函数$f(x)=2 \\sin ^2(\\dfrac{\\pi}{4}+x)-\\sqrt{3} \\cos 2 x$, $x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$. 若不等式$|f(x)-m|<2$在$x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$上恒成立, 求实数$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320026,14 +320026,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013044": { "id": "013044", "content": "已知函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{6})+\\sin (\\omega x-\\dfrac{\\pi}{6})-2 \\cos ^2 \\dfrac{\\omega x}{2}$, $x \\in \\mathbf{R}$(其中$\\omega>0$).\\\\\n(1) 求函数$f(x)$的值域;\\\\\n(2) 若对任意的$a \\in \\mathbf{R}$, 函数$y=f(x)$, $x \\in(a, a+\\pi]$的图像与直线$y=-1$有且仅有两个不同的交点, 试确定$\\omega$的值(不必证明), 并求函数$y=f(x)$, $x \\in \\mathbf{R}$的单调增区间.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320045,14 +320045,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013045": { "id": "013045", "content": "``$\\theta=\\dfrac{2 \\pi}{3}$''是``$\\tan \\theta=2 \\cos (\\dfrac{\\pi}{2}+\\theta)$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -320071,7 +320071,7 @@ "content": "函数$y=\\sin (2 x+\\dfrac{\\pi}{6})-\\cos (2 x+\\dfrac{\\pi}{3})$的最小正周期和最大值分别为\\bracket{20}.\n\\fourch{$\\pi, \\sqrt{3}$}{$\\pi, \\sqrt{2}$}{$2 \\pi, \\sqrt{3}$}{$2 \\pi, \\sqrt{2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -320090,7 +320090,7 @@ "content": "已知函数$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$)的最小正周期为$\\pi$, 则该函数的图像\\bracket{20}.\n\\twoch{关于点$(\\dfrac{\\pi}{3}, 0)$对称}{关于直线$x=\\dfrac{\\pi}{4}$对称}{关于点$(\\dfrac{\\pi}{4}, 0)$对称}{关于直线$x=\\dfrac{\\pi}{3}$对称}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -320109,7 +320109,7 @@ "content": "函数$f(x)=3 \\sin (2 x-\\dfrac{\\pi}{3})$的图像为$C$. \\textcircled{1} 图像$C$关于直线$x=\\dfrac{11}{12} \\pi$对称; \\textcircled{2} 函数$f(x)$在区间$(-\\dfrac{\\pi}{12}, \\dfrac{5 \\pi}{12})$内是增函数; \\textcircled{3} 由$y=3 \\sin 2 x$的图像向右平移$\\dfrac{\\pi}{3}$个单位长度可以得到图像$C$. 以上三个论断中, 正确论断的个数是\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$3$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -320128,7 +320128,7 @@ "content": "函数$f(x)=\\sqrt{3} \\sin x+\\sin (\\dfrac{\\pi}{2}+x)$的最大值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320147,7 +320147,7 @@ "content": "由函数$y=2 \\sin 3 x$, $x \\in[\\dfrac{\\pi}{6}, \\dfrac{5 \\pi}{6}]$与$y=2$($x \\in \\mathbf{R}$)的图像围成一个封闭图形, 这个封闭图形的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320166,7 +320166,7 @@ "content": "若函数$f(x)=a^2 \\sin 2 x+(a-2) \\cos 2 x$的图像关于直线$x=-\\dfrac{\\pi}{8}$对称, 则实数$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320185,7 +320185,7 @@ "content": "以下四个命题中, 正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\cos \\alpha=\\cos \\beta$, 则$\\alpha-\\beta=2 k \\pi$($k$是某个整数); \\textcircled{2} 函数$y=2 \\cos (2 x+\\dfrac{\\pi}{3})$的图像关于点$(\\dfrac{\\pi}{12}, 0)$对称; \\textcircled{3} 函数$y=\\sin |x|$是周期函数, 且$2 \\pi$是它的一个周期; \\textcircled{4} 函数$y=\\cos (\\sin x)$是偶函数.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320204,7 +320204,7 @@ "content": "已知$-\\dfrac{\\pi}{2}0$.\\\\\n(1) 求函数$y=f(x)$的值域;\\\\\n(2) 若$f(x)$在区间$[-\\dfrac{3 \\pi}{2}, \\dfrac{\\pi}{2}]$上为增函数, 求$\\omega$的最大值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320235,14 +320235,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013055": { "id": "013055", "content": "函数$f(x)=6 \\cos ^2 \\dfrac{\\omega x}{2}+\\sqrt{3} \\sin \\omega x-3$($\\omega>0$)在一个周期内的图像如图所示, $A$为图像的最高点, $B$、$C$为图像与$x$轴的交点, 且$\\triangle ABC$为正三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-2,0) -- (7.5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:7, samples = 100] plot (\\x,{2*sqrt(3)*sin(45*\\x+60)});\n\\draw (-4/3,0) node [above left] {$B$} coordinate (B);\n\\draw (2/3,{2*sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw (8/3,0) node [above right] {$C$} coordinate (C);\n\\draw (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\omega$的值及函数$f(x)$的值域;\\\\\n(2) 若$f(x_0)=\\dfrac{8 \\sqrt{3}}{5}$, 且$x_0 \\in(-\\dfrac{10}{3}, \\dfrac{2}{3})$, 求$f(x_0+1)$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320254,14 +320254,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013056": { "id": "013056", "content": "若直线$l$过点$(3,4)$, 且$(1,2)$是它的一个法向量, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320280,7 +320280,7 @@ "content": "若$\\overrightarrow {d}=(2,-1)$是直线$l$的一个方向向量, 则直线$l$的倾斜角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320299,7 +320299,7 @@ "content": "已知直线$l_1: x+a y+2=0$和$l_2:(a-2) x+3 y+6 a=0$, 则$l_1\\parallel l_2$的充要条件是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320318,7 +320318,7 @@ "content": "直线$a x+b y-a b=0$($a>0$, $b>0$)的倾斜角是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320337,7 +320337,7 @@ "content": "直线$l$上有两点$M(t-1,2)$, $N(t-3, t^2-4 t+4)$, 则直线$l$的倾斜角的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320356,7 +320356,7 @@ "content": "光线从点$M(-2,1)$出发, 先经$x$轴反射, 然后再经$y$轴反射后到达$N(-1,2)$, 则光线从点$M$到点$N$所经过的路程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320375,7 +320375,7 @@ "content": "将直线$l_1: n x+y-n=0$, $l_2: x+n y-n=0$($n \\in \\mathbf{N}$, $n \\geq 2$), $x$轴及$y$轴围成的封闭区域的面积记为$S_n$, 则$\\displaystyle\\lim_{n\\to\\infty} S_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320394,7 +320394,7 @@ "content": "若实数$a, b, c$成等差数列, 点$P(-1,0)$在动直线$l: a x+b y+c=0$上的射影为$M$, 点$N(0,3)$, 则线段$MN$的长度的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320413,7 +320413,7 @@ "content": "已知一直线$l$被两条平行直线$l_1: 3 x+4 y-7=0$和$l_2: 3 x+4 y+8=0$所截得的线段长为$\\dfrac{15}{4}$, 且直线$l$经过点$(2,3)$, 求直线$l$的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320425,14 +320425,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013065": { "id": "013065", "content": "已知抛物线$C: y^2=4 x$的焦点为$F$, 过点$K(-1,0)$的直线$l$与$C$相交于$A, B$两点, 点$A$关于$x$轴的对称点为$D$.\\\\\n(1) 证明: 点$F$在直线$BD$上;\\\\\n(2) 设$\\overrightarrow{FA} \\cdot \\overrightarrow{FB}=\\dfrac{8}{9}$, 求直线$l$的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320444,14 +320444,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013066": { "id": "013066", "content": "已知集合$M$是平面直角坐标系中方程为$x-2 k y+k^2=0$($k \\in \\mathbf{R}$)的直线的集合, 集合$S$是满足以下条件的点的集合: 对于集合$S$中的每一个点, 集合$M$中有且仅有一条直线经过该点.\\\\\n(1) 判断下列直线是否为集合$M$中的直线: $l_1: x-y+1=0$, $l_2: x-2 y+1=0$;\\\\\n(2) 判断下列各点是否为集合$S$中的点:$D(2,1)$, $E(1,1)$;\\\\\n(3) 求集合$S$中的点的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320463,14 +320463,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013067": { "id": "013067", "content": "直线$l$过点$P(1,1)$, 且其一个方向向量与向量$(2,3)$垂直, 则$l$的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320489,7 +320489,7 @@ "content": "设$\\alpha$是直线的倾斜角, 且$\\cos \\alpha=-\\dfrac{1}{5}$, 则$\\alpha$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320508,7 +320508,7 @@ "content": "已知三条直线$y=3 x+2$, $2 x+y+3=0$, $k x+y=0$. 若它们不能围成一个三角形, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320527,7 +320527,7 @@ "content": "点$P(-2,0)$关于直线$x-y-2=0$的对称点的坐标为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320546,7 +320546,7 @@ "content": "已知直线$l_1$的斜率是$\\dfrac{1}{2}$, 直线$l_2$的倾斜角是$l_1$的倾斜角的$2$倍, 则$l_2$的斜率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320565,7 +320565,7 @@ "content": "过直线$l_1: 2 x+3 y-5=0$与直线$l_2: 3 x-2 y-3=0$的交点$P$, 且平行于直线$2 x+y-3=0$的直线的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320584,7 +320584,7 @@ "content": "直线$l$被两直线$l_1: x-3 y+10=0$和$l_2: 2 x+y+8=0$所截得的线段的中点为$P(0,1)$, 则直线$l$的方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320603,7 +320603,7 @@ "content": "过直线$l: 3 x+4 y-5=0$上的一点$P$向圆$(x-3)^2+(y-4)^2=4$作两条切线$l_1, l_2$. 设$l_1$与$l_2$的夹角为$\\theta$, 则$\\theta$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320622,7 +320622,7 @@ "content": "已知两条直线$l_1: y=x$和$l_2: a x-y=0$, 其中$a$为实数, 当这两条直线的夹角在$(0, \\dfrac{\\pi}{12})$内变动时, 实数$a$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320641,7 +320641,7 @@ "content": "设$f(x, y)=A x+B y+C$, 这里$A, B, C$是常数, $A, B$不全为零. 若点$P(x_0, y_0)$不在直线$f(x, y)=0$上, 则曲线$f(x, y)-f(x_0, y_0)=0$表示\\bracket{20}.\n\\twoch{不过点$P$但平行于$l$的直线}{过点$P$且垂直于$l$的直线}{过点$P$且平行于$l$的直线}{不过点$P$但垂直于$l$的直线}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -320660,7 +320660,7 @@ "content": "已知直线$l: 5 x+2 y+3=0$.\\\\\n(1) 求直线$l_1: 3 x+7 y-13=0$与$l$所成的角的大小;\\\\\n(2) 若$l_2$经过点$P(2,1)$, 且与$l$的夹角等于$45^{\\circ}$, 求直线$l_2$的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320672,14 +320672,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013078": { "id": "013078", "content": "若$A, B$是抛物线$y^2=4 x$上的不同两点, 弦$AB$(不平行于$y$轴)的垂直平分线与$x$轴相交于点$P$, 则称弦$AB$是点$P$的一条``相关弦''. 已知当$x>2$时, 点$P(x, 0)$存在无穷多条``相关弦''. 现给定$x_0>2$.\\\\\n(1) 证明: 点$P(x_0, 0)$的所有``相关弦''的中点的橫坐标相同;\\\\\n(2) 当$x_0$取定时, 点$P(x_0, 0)$的所有``相关弦''的弦长是否存在最大值? 若存在, 求其最大值(用$x_0$表示); 若不存在, 请说明理由.(提示: 是否存在和$x_0$的取值有关)", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320691,14 +320691,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013079": { "id": "013079", "content": "设$m$是常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{9}=1$的一个焦点, 则$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320717,7 +320717,7 @@ "content": "方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示焦点在$y$轴上的椭圆, 则实数$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320736,7 +320736,7 @@ "content": "已知双曲线$C$与椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{8}=1$有相同的焦点, 直线$y=\\sqrt{3} x$为$C$的一条渐近线, 则双曲线$C$的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320755,7 +320755,7 @@ "content": "已知$F_1, F_2$是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$的两个焦点, 过$F_2$的直线交椭圆于$A, B$两点, 若$|AB|=5$, 则$|AF_1|+|BF_1|=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320774,7 +320774,7 @@ "content": "已知抛物线方程为$y^2=2 p x$($p>0$), 过焦点$F$的直线与抛物线交于$A, B$两点, 以$AB$为直径的圆$M$与抛物线的准线$l$的位置关系为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320793,7 +320793,7 @@ "content": "已知点$A$的坐标是$(1, \\dfrac{1}{2}), P$是椭圆$x^2+4 y^2=1$上的动点, 则线段$PA$中点的轨迹方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320812,7 +320812,7 @@ "content": "某海域内有一孤岛, 岛四周的海平面(视为平面)上有一浅水区(含边界), 其边界是长轴长为$2 a$, 短轴长为$2 b$的椭圆. 已知岛上甲、乙导航灯的海拔高度分别为$h_1, h_2$, 且两个导航灯在海平面上的投影恰好落在椭圆的两个焦点上, 现有船只经过该海域(船只的大小忽略不计), 在船上测得甲、乙导航灯的仰角分别为$\\theta_1, \\theta_2$, 那么船只进入该浅水区的判断条件是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:360,samples = 100] plot ({2*cos(\\x)},0,{sqrt(3)*sin(\\x)});\n\\draw (-1,0,0,0) coordinate (A) -- (-1,1.2,0) coordinate (A1) node [midway, left] {$h_1$};\n\\draw (1,0,0) coordinate (B) -- (1,1,0) coordinate (B1) node [midway, right] {$h_2$};\n\\draw (0,0,1.2) coordinate (C);\n\\draw [dashed] (A) -- (C) (A1) -- (C) (B) -- (C) (B1) -- (C);\n\\draw (C) pic [draw, \"$\\theta_1$\", angle eccentricity = 1.3] {angle = A1--C--A};\n\\draw (C) pic [draw, \"$\\theta_2$\", angle eccentricity = 1.5] {angle = B--C--B1};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320831,7 +320831,7 @@ "content": "在平面直角坐标系中, 设$P_1(x_1, y_1), P_2(x_2, y_2)$为不同的两点, 直线$l$的方程为$a x+b y+c=0$, 定义$\\delta_1=\\dfrac{a x_1+b y_1+c}{\\sqrt{a^2+b^2}}$和$\\delta_2=\\dfrac{a x_2+b y_2+c}{\\sqrt{a^2+b^2}}$为点$P_1, P_2$到直线$l$的有向距离. 以下正确命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 若$\\delta_1-\\delta_2=0$, 则直线$P_1P_2$与$l$平行;\\\\\n\\textcircled{2} 若$\\delta_1+\\delta_2=0$, 则直线$P_1P_2$与$l$平行;\\\\\n\\textcircled{3} 若$\\delta_1+\\delta_2=0$, 则直线$P_1P_2$与$l$垂直;\\\\\n\\textcircled{4} 若$\\delta_1 \\delta_2<0$, 则直线$P_1P_2$与$l$相交.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320850,7 +320850,7 @@ "content": "已知点$A(-2,0), B(2,0)$, 点$C$在双曲线$x^2-y^2=1$上运动, 求以$AB$、$BC$为邻边的平行四边形$ABCP$的顶点$P$的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320862,14 +320862,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013088": { "id": "013088", "content": "过定点$F(4,0)$作直线$l$交$y$轴于$Q$点, 过$Q$点作$QT \\perp FQ$交$x$轴于$T$点, 延长$TQ$至$P$点, 使$|PQ|=|TQ|$, 求动点$P$的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320881,14 +320881,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013089": { "id": "013089", "content": "已知过点$A(-3,1)$且倾斜角为$45^{\\circ}$的直线$l$与焦点为$(-\\sqrt{6}, 0),(\\sqrt{6}, 0)$的椭圆交于$B, C$两点, 若线段$BC$在$A$点被平分, 则这样的椭圆是否存在? 若存在, 求出椭圆的方程; 若不存在, 说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -320900,14 +320900,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013090": { "id": "013090", "content": "抛物线$y=8 x^2$的准线方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320926,7 +320926,7 @@ "content": "设双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0, b>0)$的虚轴长为$2$, 焦距为$2 \\sqrt{3}$, 则双曲线的渐近线方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320945,7 +320945,7 @@ "content": "设点$O$为坐标原点, 点$M$是曲线$y=\\dfrac{1}{2} x^2+1$上的一个动点, 且点$M$为线段$OP$的中点, 则动点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320964,7 +320964,7 @@ "content": "若双曲线$8 m x^2-m y^2=8$的一个焦点是$(0,3)$, 则实数$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -320983,7 +320983,7 @@ "content": "若抛物线$y^2=2 x$的焦点弦$AB$的两端点为$A(x_1, y_1), B(x_2, y_2)$, 则$\\dfrac{y_1 y_2}{x_1 x_2}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321002,7 +321002,7 @@ "content": "已知$F_1, F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的焦点. 过$F_2$作垂直于$x$轴的直线交双曲线于点$P$, 且$\\angle PF_1F_2=30^{\\circ}$, 则双曲线的渐近线方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321021,7 +321021,7 @@ "content": "已知定圆$C_1:(x-7)^2+y^2=4$, $C_2:(x+7)^2+y^2=25$, 动圆$M$与两定圆外切, 则动圆圆心$M$的轨迹方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321040,7 +321040,7 @@ "content": "若$F$是双曲线$x^2-y^2=1$的左焦点, 点$P$在第三象限的双曲线上, 则直线$FP$的倾斜角的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321059,7 +321059,7 @@ "content": "在平面直角坐标系中, 设$M(x_1, y_1)$, $N(x_2, y_2)$为不同的两点, 直线$l$的方程为$a x+b y+c=0$, 设$\\lambda=\\dfrac{a x_1+b y_1+c}{a x_2+b y_2+c}$, 有以下四个命题:\\\\\n\\textcircled{1} 存在实数$\\lambda$, 使点$N$在直线$l$上;\\\\\n\\textcircled{2} 若$\\lambda=1$, 则过$M, N$两点的直线于直线$l$平行;\\\\\n\\textcircled{3} 若$\\lambda=-1$, 则直线$l$经过线段$MN$的中点;\\\\\n\\textcircled{4} 若$\\lambda>1$, 则点$M, N$在直线$l$的同侧, 且直线$l$与线段$MN$的延长线相交.\\\\\n上述命题中, 真命题的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321078,7 +321078,7 @@ "content": "记椭圆$E_n: \\dfrac{x^2}{4}+\\dfrac{n y^2}{4 n+1}=1$, 其中$n=1,2, \\cdots$. 当点$(x, y)$分别在$E_1, E_2, \\cdots$上时, $x+y$的最大值分别是$M_1, M_2, \\cdots$, 则$\\displaystyle\\lim_{n\\to\\infty} M_n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321097,7 +321097,7 @@ "content": "抛物线$x^2=4 y$的焦点$F$, 过点$(0,-1)$作直线交抛物线于不同的两点$A, B$, 以$AF, BF$为邻边作平行四边形$FARB$, 求顶点$R$的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321109,14 +321109,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013101": { "id": "013101", "content": "已知圆$O: x^2+y^2=4$.\\\\\n(1) 直线$l_1: \\sqrt{3} x+y-2 \\sqrt{3}=0$与圆$O$相交于$A, B$两点, 求弦$AB$的长;\\\\\n(2) 设$M(x_1, y_1)$, $P(x_2, y_2)$是圆$O$上的两个动点, 点$M$关于原点的对称点为$M_1$, 点$M$关于$x$轴的对称点为$M_2$. 如果直线$P$和$M_1, M_2$均不重合, 且$PM_1, PM_2$都和$y$轴相交, 且分别交于$S(0, m)$和$T(0, n)$, 问$m \\cdot n$是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321128,14 +321128,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013102": { "id": "013102", "content": "若动点$P$到点$F(2,0)$的距离与它到直线$x+2=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321154,7 +321154,7 @@ "content": "抛物线$y=-\\dfrac{x^2}{8}$的准线方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321173,7 +321173,7 @@ "content": "以原点为圆心, 且截直线$3 x+4 y+15=0$所得弦长为$8$的圆的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321192,7 +321192,7 @@ "content": "动直线$(2 k-1) x-(k+3) y-(k-11)=0$($k \\in \\mathbf{R}$)所过的定点是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321211,7 +321211,7 @@ "content": "设抛物线$y^2=8 x$的准线与$x$轴交于点$Q$, 若过点$Q$的直线$l$与抛物线有公共点, 则直线$l$的斜率的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321230,7 +321230,7 @@ "content": "若双曲线$\\dfrac{3 x^2}{2}-\\dfrac{y^2}{2}=1$的右支上有一点$P$到两坐标轴的距离相等, 则点$P$的坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321249,7 +321249,7 @@ "content": "直线$y=x+3$与曲线$\\dfrac{y^2}{9}-\\dfrac{x|x|}{4}=1$的公共点的个数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321268,7 +321268,7 @@ "content": "若命题``$F(x, y)=0$的解为坐标的点都是曲线$C$上的点''是真命题, 则下列命题正确的有\\blank{50}.\\\\\n\\textcircled{1} 曲线$C$上的点的坐标都是方程$F(x, y)=0$的解;\\\\\n\\textcircled{2} 坐标不满足方程$F(x, y)=0$的点不在曲线$C$上;\\\\\n\\textcircled{3} 曲线$C$是方程$F(x, y)=0$的曲线;\\\\\n\\textcircled{4} 不是曲线$C$上的点的坐标, 一定不满足方程$F(x, y)=0$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321287,7 +321287,7 @@ "content": "设双曲线与椭圆$\\dfrac{x^2}{27}+\\dfrac{y^2}{36}=1$有共同的焦点, 且与椭圆相交的一个交点的纵坐标为$4$, 求这个双曲线的方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321299,14 +321299,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013111": { "id": "013111", "content": "若抛物线$y=a x^2-1$上存在关于直线$x+y=0$对称的不同两点, 求实数$a$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321318,14 +321318,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013112": { "id": "013112", "content": "$P$是双曲线$\\dfrac{x^2}{4}-y^2=1$的右顶点, 过点$P$的两条互相垂直的直线分别与双曲线的右支交于点$A, B$, 问直线$AB$是否一定过$x$轴上一定点? 如果不存在这样的定点, 请说明理由; 如果存在这样的定点, 试求出这个定点的坐标.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321337,14 +321337,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013113": { "id": "013113", "content": "已知方程$\\dfrac{x^2}{2-k}+\\dfrac{y^2}{k-1}=1$表示的曲线是双曲线, 则$k$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321363,7 +321363,7 @@ "content": "双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{39}=1$上一点$P$到双曲线一个焦点的距离为$12$, 则$P$到另一个焦点的距离为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321382,7 +321382,7 @@ "content": "若方程$2 x^2+2 y^2+a x+1=0$表示圆, 则实数$a$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321401,7 +321401,7 @@ "content": "双曲线$x^2-y^2=1$, 点$F_1, F_2$为其两个焦点, 点$P$为双曲线上一点, 若$PF_1 \\perp PF_2$, 则$|PF_1|+|PF_2|$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321420,7 +321420,7 @@ "content": "点$(4,-1)$关于直线$y=-x+1$对称点的坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321439,7 +321439,7 @@ "content": "若直线$y=x+k$与曲线$y=\\sqrt{2-x^2}$相交于两点, 则实数$k$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321458,7 +321458,7 @@ "content": "过抛物线$y^2=4 x$的焦点作倾斜角为$\\dfrac{\\pi}{3}$的弦$AB$, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321477,7 +321477,7 @@ "content": "设$P$是抛物线$y^2=4 x$上一动点, $F$是抛物线的焦点, 定点$B(3,2)$, 则$|PB|+|PF|$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321496,7 +321496,7 @@ "content": "已知两点$M(-5,0), N(5,0)$, 若直线上存在点$P$, 使$|PM|-|PN|=8$, 则称该直线为``$B$型直线'', 现给出下列直线: \\textcircled{1} $y=x+2$; \\textcircled{2} $y=2$; \\textcircled{3} $y=\\dfrac{2}{3} x$; \\textcircled{4} $y=2 x+3$. 其中是``$B$型直线''的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -321515,7 +321515,7 @@ "content": "已知双曲线$C: \\dfrac{x^2}{2}-y^2=1$, 点$M(0,1)$, 点$P$是双曲线上任意一点, 若点$Q$是点$P$关于原点的对称点, 且$\\lambda=\\overrightarrow{MP} \\cdot \\overrightarrow{MQ}$, 则$\\lambda$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321534,7 +321534,7 @@ "content": "已知两点$A(-1,0)$、$B(1,0)$, 点$P(x, y)$是直角坐标平面上的动点, 若将点$P$的横坐标保持不变、纵坐标扩大到$\\sqrt{2}$倍后得到点$Q(x, \\sqrt{2} y)$满足$\\overrightarrow{AQ} \\cdot \\overrightarrow{BQ}=1$. 求动点$P$所在曲线$C$的轨迹方程.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321546,14 +321546,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013124": { "id": "013124", "content": "已知直线$l: x-a y+a=0$与双曲线$x^2-y^2=1$的左支交于$A, B$两点, 过弦$AB$的中点$Q$与点$P(-2,1)$的直线交$y$轴于$(0, b)$点. 当$a$变化时, 求实数$b$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321565,14 +321565,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013125": { "id": "013125", "content": "点$P(-3,0)$是椭圆$x^2+2 y^2-k=0$上的点, 则椭圆的焦点坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321591,7 +321591,7 @@ "content": "双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}{25}=1$的渐近线方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321610,7 +321610,7 @@ "content": "与椭圆$4 x^2+5 y^2=20$有相同的焦点, 且顶点在原点的抛物线方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321629,7 +321629,7 @@ "content": "设$AB$是椭圆$\\Gamma$的长轴, 点$C$在$\\Gamma$上, 且$\\angle CBA=\\dfrac{\\pi}{4}$, 若$AB=4, BC=\\sqrt{2}$, 则$\\Gamma$的两个焦点之间的距离为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321648,7 +321648,7 @@ "content": "设双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的右顶点为$A$, 右焦点为$F$, 过点$F$且与双曲线的一条渐近线平行的直线与另\n一条渐近线交于点$B$, 则$\\triangle AFB$的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321667,7 +321667,7 @@ "content": "中心在原点, 对称轴为坐标轴, 椭圆的短轴的一个顶点$B$与两个焦点$F_1, F_2$组成的三角形的周长为$4+2 \\sqrt{3}$, 且$\\angle F_1BF_2=\\dfrac{2 \\pi}{3}$, 则椭圆的方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321686,7 +321686,7 @@ "content": "若动点$(x, y)$在曲线$\\dfrac{x^2}{4}+\\dfrac{y^2}{b^2}=1(01$), 点$P$是$C$上的动点, $M$是右顶点, 定点$A$的坐标为$(2,0)$.\\\\\n(1) 若$M$与$A$重合, 求$C$的焦点坐标;\\\\\n(2) 若$m=3$, 求$|PA|$的最大值与最小值;\\\\\n(3) 若$|PA|$的最小值为$|MA|$, 求$m$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321755,14 +321755,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013135": { "id": "013135", "content": "已知曲线$C_1: \\dfrac{x^2}{2}-y^2=1$, 曲线$C_2:|y|=|x|+1, P$是平面上一点, 若存在过点$P$的直线与$C_1, C_2$都有公共点, 则称$P$为``$C_1-C_2$型点''.\\\\\n(1) 在正确证明$C_1$的左焦点是``$C_1-C_2$型点''时, 要使用一条过该焦点的直线, 试写出一条这样的直线的方程(不要求验证);\\\\\n(2) 设直线$y=k x$与$C_2$有公共点, 求证$|k|>1$, 进而证明原点不是``$C_1-C_2$型点''.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321774,14 +321774,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013136": { "id": "013136", "content": "直线$x-y-3=0$被双曲线$\\dfrac{x^2}{4}-y^2=1$所截得的弦长为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321800,7 +321800,7 @@ "content": "已知点$M(x, y)$到点$F_1(-5,0)$和$F_2(5,0)$的距离差是$8$, 则点$M$的轨迹方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321819,7 +321819,7 @@ "content": "双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{9}=1$上一点$P$到双曲线一个焦点的距离为$12$, 则$P$到另一个焦点的距离为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321838,7 +321838,7 @@ "content": "设$F_1, F_2$为双曲线$\\dfrac{x^2}{4}-y^2=1$的两焦点, 点$P$在双曲线上且满足$\\angle F_1PF_2=90^{\\circ}$, 则$\\triangle F_1PF_2$的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321857,7 +321857,7 @@ "content": "若抛物线$x^2=a y$经过点$A(1, \\dfrac{1}{4})$, 则点$A$到此抛物线的焦点的距离为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321876,7 +321876,7 @@ "content": "已知抛物线$y^2=2 x$及点$A(\\dfrac{2}{3}, 0)$, 则抛物线上距点$A$最近的点$P$的坐标为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321895,7 +321895,7 @@ "content": "设斜率为$2$的直线$l$过抛物线$y^2=a x$($a \\neq 0$)的焦点$F$, 且和$y$轴交于点$A$, 若$\\triangle OAF(O$为坐标原点)的面积为$4$, 则抛物线方程为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321914,7 +321914,7 @@ "content": "抛物线$x^2=2 p y$($p>0$)的焦点为$F$, 其准线与双曲线$\\dfrac{x^2}{3}-\\dfrac{y^2}{3}=1$相交于$A, B$两点, 若$\\triangle ABF$为等边三角形, 则$p=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321933,7 +321933,7 @@ "content": "已知直线$y=a$交抛物线$y=x^2$于$A, B$两点. 若该抛物线上存在点$C$, 使$\\angle ACB$为直角, 则$a$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -321952,7 +321952,7 @@ "content": "已知平面上的线段$l$及点$P$, 在$l$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$, 求点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离$d(P, l)$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321964,14 +321964,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013146": { "id": "013146", "content": "已知双曲线$H$的中心在原点, 抛物线$y^2=8 x$的焦点是双曲线$H$的一个焦点, 且$H$经过点$(\\sqrt{2}, \\sqrt{3})$.\\\\\n(1) 求双曲线$H$的方程;\\\\\n(2) 设双曲线$H$的实轴左顶点为$A$, 右焦点为$F$, 在第一象限内任取双曲线$H$上一点$P$, 试问是否存在常数$\\lambda$, 使得$\\angle PFA=\\lambda \\angle PAF$恒成立? 证明你的结论.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -321983,14 +321983,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013147": { "id": "013147", "content": "已知向量$\\overrightarrow{OM}=(3,-2)$, $\\overrightarrow{ON}=(-5,-1)$, 则$\\dfrac{1}{2} \\overrightarrow{MN}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322009,7 +322009,7 @@ "content": "已知向量$\\overrightarrow {a}=(\\cos 75^{\\circ}, \\sin 75^{\\circ})$, $\\overrightarrow {b}=(\\cos 15^{\\circ}, \\sin 15^{\\circ})$, 则$|\\overrightarrow {a}-\\overrightarrow {b}|=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322028,7 +322028,7 @@ "content": "若向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$\\dfrac{\\pi}{3}$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322047,7 +322047,7 @@ "content": "已知$\\overrightarrow {m}, \\overrightarrow {n}$是夹角为$60^{\\circ}$的单位向量, 则$\\overrightarrow {a}=2 \\overrightarrow {m}+\\overrightarrow {n}$与$\\overrightarrow {b}=-3 \\overrightarrow {m}+2 \\overrightarrow {n}$的夹角是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322066,7 +322066,7 @@ "content": "在四面体$O-ABC$中, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, $D$为$BC$的中点, $E$为$AD$的中点, 则用$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$表示$\\overrightarrow{OE}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322085,7 +322085,7 @@ "content": "设$\\overrightarrow {a}, \\overrightarrow {b}$是非零向量, 若函数$f(x)=(x \\overrightarrow {a}+\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-x \\overrightarrow {b})$的图像是一条直线, 则必有\\bracket{20}.\n\\fourch{$\\overrightarrow {a} \\perp \\overrightarrow {b}$}{$\\overrightarrow {a}\\parallel \\overrightarrow {b}$}{$|\\overrightarrow {a}|=|\\overrightarrow {b}|$}{$|\\overrightarrow {a}| \\neq|\\overrightarrow {b}|$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322104,7 +322104,7 @@ "content": "在$\\triangle ABC$中, 已知$|\\overrightarrow{AB}|=4$, $|\\overrightarrow{AC}|=1$, $S_{\\triangle ABC}=\\sqrt{3}$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$的值为\\bracket{20}.\n\\fourch{$-2$}{$2$}{$\\pm 4$}{$\\pm 2$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322123,7 +322123,7 @@ "content": "在直角$\\triangle ABC$中, $CD$是斜边$AB$上的高, 则下列等式不成立的是\\bracket{20}.\n\\twoch{$|\\overrightarrow{AC}|^2=\\overrightarrow{AC} \\cdot \\overrightarrow{AB}$}{$|\\overrightarrow{BC}|^2=\\overrightarrow{BC} \\cdot \\overrightarrow{BA}$}{$|\\overrightarrow{AB}|^2=\\overrightarrow{AC} \\cdot \\overrightarrow{CD}$}{$|\\overrightarrow{CD}|^2=\\dfrac{(\\overrightarrow{AC} \\cdot \\overrightarrow{AB}) \\cdot(\\overrightarrow{BA} \\cdot \\overrightarrow{BC})}{|\\overrightarrow{AB}|^2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322142,7 +322142,7 @@ "content": "设$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$为平面向量, 已知命题: \\textcircled{1} $\\overrightarrow {a} \\cdot(\\overrightarrow {b}-\\overrightarrow {c})=\\overrightarrow {a} \\cdot \\overrightarrow {b}-\\overrightarrow {a} \\cdot \\overrightarrow {c}$; \\textcircled{2} $(\\overrightarrow {a} \\cdot \\overrightarrow {b}) \\cdot \\overrightarrow {c}=\\overrightarrow {a} \\cdot(\\overrightarrow {b} \\cdot \\overrightarrow {c})$; \\textcircled{3} $(\\overrightarrow {a}-\\overrightarrow {b})^2=|\\overrightarrow {a}|^2+2|\\overrightarrow {a}||\\overrightarrow {b}|+|\\overrightarrow {b}|^2$; \\textcircled{4} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$. 以上命题中正确的有\\blank{50}个.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322161,7 +322161,7 @@ "content": "定义平面向量之间的一种运算``$\\otimes$''如下: 对任意的$\\overrightarrow {a}=(m, n) , \\overrightarrow {b}=(p, q)$, 令$\\overrightarrow {a} \\otimes \\overrightarrow {b}=m q-n p$. 给出以下四个命题: \\textcircled{1} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$\\overrightarrow {a} \\otimes \\overrightarrow {b}=0$; \\textcircled{2} $\\overrightarrow {a} \\otimes \\overrightarrow {b}=\\overrightarrow {b} \\otimes \\overrightarrow {a}$; \\textcircled{3} 对任意的$\\lambda \\in \\mathbf{R}$, 有$(\\lambda \\overrightarrow {a}) \\otimes \\overrightarrow {b}=\\lambda(\\overrightarrow {a} \\otimes \\overrightarrow {b})$; \n\\textcircled{4} $(\\overrightarrow {a} \\otimes \\overrightarrow {b})^2+(\\overrightarrow {a} \\cdot \\overrightarrow {b})^2=|\\overrightarrow {a}|^2 \\cdot|\\overrightarrow {b}|^2$. 则其中所有真命题的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322180,7 +322180,7 @@ "content": "若$|\\overrightarrow {a}|=|\\overrightarrow {b}|=1$, $\\overrightarrow {a} \\perp \\overrightarrow {b}$, 且$2 \\overrightarrow {a}+3 \\overrightarrow {b}$与$k \\overrightarrow {a}-4 \\overrightarrow {b}$也互相垂直, 求实数$k$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322192,14 +322192,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013158": { "id": "013158", "content": "平面直角坐标系内有点$P(1, \\cos x)$, $Q(\\cos x, 1)$,$x \\in[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$.\\\\\n(1) 求向量$\\overrightarrow{OP}$和$\\overrightarrow{OQ}$的夹角$\\theta$的余弦表达式$f(x)$;\\\\\n(2) 求$\\theta$的最值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322211,14 +322211,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013159": { "id": "013159", "content": "在正三棱柱$ABC-A_1B_1C_1$中, 底面是边长为$2$的正三角形, 且$AB_1 \\perp BC_1$.\\\\\n(1) 求$AA_1$的长;\\\\\n(2) 对于$n$个向量$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\cdots, \\overrightarrow{a_n}$, 如果存在不全为零的$n$个实数$\\lambda_1, \\lambda_2, \\cdots, \\lambda_n$, 使得$\\lambda_1 \\overrightarrow{a_1}+\\lambda_2 \\overrightarrow{a_2}+\\cdots+\\lambda_n \\overrightarrow{a_n}=\\overrightarrow{0}$, 则称这$n$个向量是线性相关的; 如果$n$个向量$\\overrightarrow{a_1}, \\overrightarrow{a_2}, \\cdots, \\overrightarrow{a_n}$不是线性相关的, 则称这$n$个向量是线性无关的. 求证: $\\overrightarrow{AB_1}, \\overrightarrow{BC_1}, \\overrightarrow{AC}$这三个向量是线性无关的.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322230,14 +322230,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013160": { "id": "013160", "content": "已知$\\overrightarrow{e_1}, \\overrightarrow{e_2}$是两个不共线的平面向量, 向量$\\overrightarrow {a}=2 \\overrightarrow{e_1}-\\overrightarrow{e_2}$, $\\overrightarrow {b}=\\overrightarrow{e_1}+\\lambda \\overrightarrow{e_2}$($\\lambda \\in \\mathbf{R}$), 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$\\lambda=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322256,7 +322256,7 @@ "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(x, 6)$, 且$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$x=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322275,7 +322275,7 @@ "content": "已知$\\overrightarrow {a}=(-1, m)$, $\\overrightarrow {b}=(2 m, 4)$, 若$|\\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}|=3$, 则$m=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322294,7 +322294,7 @@ "content": "已知向量$\\overrightarrow {a}=(2,3)$, $\\overrightarrow {b}=(-1,4)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$方向上的数量投影为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322313,7 +322313,7 @@ "content": "已知$\\overrightarrow {a}=(2,0)$, $\\overrightarrow{OB}=(2,2)$, $\\overrightarrow{OC}=(\\sqrt{2} \\cos \\alpha, \\sqrt{2} \\sin \\alpha)$, $\\overrightarrow{BC}$与$\\overrightarrow {a}$所成的角为$\\theta$, 则$\\theta$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322332,7 +322332,7 @@ "content": "在$\\triangle ABC$中, $M$是$BC$的中点, $AM=3$, $BC=10$, 则$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322351,7 +322351,7 @@ "content": "设$O$是直线$AB$外一点, $\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, 点$A_1, A_2, \\cdots, A_{n-1}$是线段$AB$的$n$($n \\geq 2$)等分点, 则$\\overrightarrow{OA_1}+\\overrightarrow{OA_2}+\\cdots+\\overrightarrow{OA_{n-1}}=$\\blank{50}.(用$\\overrightarrow {a}$, $\\overrightarrow {b}$, $n$表示)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322370,7 +322370,7 @@ "content": "已知$\\triangle ABC$中的三个顶点$A, B, C$及平面内一点$P$满足$\\overrightarrow{PA}+\\overrightarrow{PB}+\\overrightarrow{PC}=\\overrightarrow{AB}$, 则点$P$与$\\triangle ABC$的关系为\\bracket{20}.\n\\twoch{$P$在$\\triangle ABC$内部}{$P$在$\\triangle ABC$外部}{$P$在$AB$边所在的直线上}{$P$是$AC$边的一个三等分点}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322389,7 +322389,7 @@ "content": "设平面上有四个互异的点$A, B, C, D$, 已知$(\\overrightarrow{DB}+\\overrightarrow{DC}-2 \\overrightarrow{DA}) \\cdot(\\overrightarrow{AB}-\\overrightarrow{AC})=0$, 则$\\triangle ABC$的形状是\\bracket{20}.\n\\fourch{直角三角形}{等腰三角形}{等腰直角三角形}{等边三角形}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322408,7 +322408,7 @@ "content": "给出下列四个命题: \\textcircled{1} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$; \\textcircled{2} 若$|\\overrightarrow {a} \\cdot \\overrightarrow {b}|=|\\overrightarrow {a}||\\overrightarrow {b}|$, 则$\\overrightarrow {a}\\parallel \\overrightarrow {b}$; \\textcircled{3} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$|\\overrightarrow {a}+\\overrightarrow {b}|=|\\overrightarrow {a}-\\overrightarrow {b}|$; \\textcircled{4} 在$\\triangle ABC$中, 三边长$BC=5$, $AC=8$, $AB=7$, 则$\\overrightarrow{BC} \\cdot \\overrightarrow{CA}=20$. 其中真命题的序号是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322427,7 +322427,7 @@ "content": "设$A$是平面向量的集合, $\\overrightarrow {a}$是定向量, 对$\\overrightarrow {x} \\in A$, 定义$f(\\overrightarrow {x})=\\overrightarrow {x}-2(\\overrightarrow {a} \\cdot \\overrightarrow {x}) \\cdot \\overrightarrow {a}$. 现给出如下四个向量: \\textcircled{1} $\\overrightarrow {a}=(0,0)$; \\textcircled{2} $\\overrightarrow {a}=(\\dfrac{\\sqrt{2}}{4}, \\dfrac{\\sqrt{2}}{4})$; \\textcircled{3} $\\overrightarrow {a}=(\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2})$; \\textcircled{4} $\\overrightarrow {a}=(-\\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2})$. 那么对于任意$\\overrightarrow {x}$、$\\overrightarrow {y} \\in A$, 使$f(\\overrightarrow {x}) \\cdot f(\\overrightarrow {y})=\\overrightarrow {x} \\cdot \\overrightarrow {y}$恒成立的向量$\\overrightarrow {a}$的序号是\\blank{50}.(写出满足条件的所有向量$\\overrightarrow {a}$的序号)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322446,7 +322446,7 @@ "content": "设$\\overrightarrow {a}=(\\cos \\theta, \\sin \\theta)$, $\\overrightarrow {b}=(2,1)$, 求$2|\\overrightarrow {a}+\\overrightarrow {b}|$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322458,14 +322458,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013172": { "id": "013172", "content": "已知平面上三个向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的模均为$1$, 它们互相之间的夹角均为$120^{\\circ}$.\\\\\n(1) 求证: $(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {c}$;\\\\\n(2) 若$|k \\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}|>1$($k \\in \\mathbf{R}$), 求$k$的取值范围.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322477,14 +322477,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013173": { "id": "013173", "content": "一个圆锥底面直径为$2$, 高为$4$, 则其母线与底面所成角的大小是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322503,7 +322503,7 @@ "content": "我国古代数学名著《九章算术》中``开立圆术''曰: 置积尺数, 以十六乘之, 九而一, 所得开立方除之, 即立圆径. ``开立圆术''相当于给出了已知球的体积$V$, 求其直径$d$的一个近似公式$d \\approx \\sqrt[3]{\\dfrac{16}{9} V}$. 人们还用过一些类似的近似公式. 根据$\\pi=3.14159 \\cdots$判断, 下列近似公式中最精确的一个是\\bracket{20}.\n\\fourch{$d \\approx \\sqrt[3]{\\dfrac{16}{9} V}$}{$d \\approx \\sqrt[3]{2 V}$}{$d \\approx \\sqrt[3]{\\dfrac{300}{157} V}$}{$d \\approx \\sqrt[3]{\\dfrac{21}{11} V}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322522,7 +322522,7 @@ "content": "已知某圆锥体的底面半径$r=3$, 沿圆锥体的母线把侧面展开后得到一个圆心角为$\\dfrac{2}{3} \\pi$的扇形, 则该圆锥体的表面积是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322541,7 +322541,7 @@ "content": "正三棱柱$ABC-A_1B_1C_1$的所有棱的长度都为$4$, 则异面直线$AB_1$与$BC_1$所成的角是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322560,7 +322560,7 @@ "content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$底面$ABCD, PA=1$, 底面$ABCD$是正方形, $PC$与底面$ABCD$所成角的大小为$\\dfrac{\\pi}{6}$, 则该四棱锥的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw (0,{2*sqrt(2)/sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle (P)--(C);\n\\draw [dashed] (A)--(P) (A)--(C) (A)--(B) (A)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322579,7 +322579,7 @@ "content": "给出下列命题, 其中正确命题的所有序号是\\blank{50}.\\\\\n\\textcircled{1} 直线上有两点到平面的距离相等, 则此直线与平面平行;\\\\\n\\textcircled{2} 夹在两个平行平面间的两条异面线段的中点连线平行于这两个平面;\\\\\n\\textcircled{3} $\\alpha$内存在不共线的三点到$\\beta$的距离相等, 则平面$\\alpha$与$\\beta$平行;\\\\\n\\textcircled{4} 垂直于同一个平面的两条直线是平行直线;\\\\\n\\textcircled{5} $l$、$m$是两条异面直线, $\\alpha$、$\\beta$是两个平面, 且$l\\parallel \\alpha$, $m\\parallel \\alpha$, $l\\parallel \\beta$, $m\\parallel \\beta$, 则平面$\\alpha$与$\\beta$平行.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322598,7 +322598,7 @@ "content": "如图, 半径为$R$的半球$O$的底面圆$O$在平面$\\alpha$内, 过点$O$作平面$\\alpha$的垂线交半球面于点$A$, 过圆$O$的直径$CD$作平面$\\alpha$成$45^{\\circ}$角的平面与半球面相交, 所得交线上到平面$\\alpha$的距离最大的点为$B$, 该交线上的一点$P$满足$\\angle BOP=60^{\\circ}$, 则$A$、$P$两点间的球面距离为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [domain = 0:180,dashed,samples=100] plot ({2*cos(\\x)},0,{-2*sin(\\x)});\n\\draw [domain = 0:180,samples=100] plot ({2* cos(\\x)},0,{2* sin(\\x)});\n\\draw [dashed] (0,0,2) node [below left] {$C$} coordinate (C) -- (0,0,-2) node [above right] {$D$} coordinate (D);\n\\draw [dashed] (0,0,0) node [below right] {$O$} coordinate (O) -- (0,2,0) node [above] {$A$} coordinate (A);\n\\draw [dashed] (O) -- ({-sqrt(2)},{sqrt(2)},0) node [above left] {$B$} coordinate (B);\n\\path [draw,domain = 0:180,samples=100,name path = semi] plot ({2*cos (\\x)},{2*sin(\\x)},0);\n\\draw [domain = 0:90,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw [domain = 90:180,dashed,samples=100] plot ({-sqrt(2)*sin(\\x)},{sqrt(2)*sin(\\x)},{2*cos(\\x)});\n\\draw ({-sqrt(2)*sin(30)},{sqrt(2)*sin(30)},{2*cos(30)}) node [left] {$P$} coordinate (P);\n\\draw [dashed] (O)--(P);\n\\draw (O) pic [draw,\"$60^\\circ$\",angle eccentricity=1.5] {angle = B--O--P};\n\\path [name path = outline] (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (-3,0,-3) -- cycle;\n\\path [name intersections = {of = semi and outline, by = {S,T}}];\n\\draw (T) -- (-3,0,-3) -- (-3,0,3) -- (3,0,3) -- (3,0,-3) -- (S);\n\\draw [dashed] (S) -- (T);\n\\draw (-2.6,0,2.6) node {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322617,7 +322617,7 @@ "content": "如图, $AD, BC$是四面体$ABCD$中互相垂直的棱, $BC=2$. 若$AD=2 c$, $AB=BD=AC=CD=a$, 其中$a, c$为常数, 则四面体$ABCD$的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,-0.5,0) node [below] {$A$} coordinate (A);\n\\draw (-1,0.5,0) node [above] {$D$} coordinate (D);\n\\draw (D)--(A)--(B)--(C)--cycle (D)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322636,7 +322636,7 @@ "content": "如图, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=6$, 异面直线$BC_1$与$AA_1$所成角的大小为$\\dfrac{\\pi}{6}$, 求该三棱柱的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{2}\n\\def\\h{{2*sqrt(3)}}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw (B) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322648,14 +322648,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013182": { "id": "013182", "content": "如图, 在长方体$AC_1$中, $AB=2$, $AD=1$, $AA_1=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B)--(C1);\n\\draw [dashed] (A)--(D1)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 直线$BC_1$平行于平面$D_1AC$;\\\\\n(2) 求直线$BC_1$到平面$D_1AC$的距离.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322667,14 +322667,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013183": { "id": "013183", "content": "如图, 圆锥顶点为$P$, 底面圆心为$O$, 其母线与底面所成的角为$22.5^{\\circ}$. $AB$和$CD$是底面圆$O$上的两条平行的弦, 轴$OP$与平面$PCD$所成的角为$60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above] {$O$} coordinate (O) ellipse (2 and 0.5);\n\\draw (0,-2) node [below] {$P$} coordinate (P);\n\\draw (P)--({sqrt(15)/2},-1/8) (P)--({-sqrt(15)/2},-1/8);\n\\draw ({2*cos(-50)},{sin(-50)/2}) node [above right] {$C$} coordinate (C);\n\\draw ({2*cos(50)},{sin(50)/2}) node [above right] {$D$} coordinate (D);\n\\draw ({2*cos(-140)},{sin(-140)/2}) node [above right] {$A$} coordinate (A);\n\\draw ({2*cos(140)},{sin(140)/2}) node [above right] {$B$} coordinate (B);\n\\draw (O)--(C)--(D)--cycle (P)--(C) (P)--(A)--(B);\n\\draw [dashed] (P)--(O) (P)--(D) (P)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$PAB$与平面$PCD$的交线平行于底面;\\\\ \n(2) 求$\\cos \\angle COD$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322686,14 +322686,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013184": { "id": "013184", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 异面直线$A_1B$与$B_1C$所成角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322712,7 +322712,7 @@ "content": "圆锥和圆柱的底面半径和高都是$R$, 则圆锥的表面积与圆柱的表面积之比为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322731,7 +322731,7 @@ "content": "对于平面$\\alpha$、$\\beta$、$\\gamma$和直线$a$、$b$、$m$、$n$, 下列命题中真命题是\\bracket{20}.\n\\onech{若$a \\perp m$, $a \\perp n$, $m\\parallel \\alpha$, $n\\parallel \\alpha$, 则$a \\perp \\alpha$}{若$a\\parallel b$, $b \\perp \\alpha$, 则$a\\parallel \\alpha$}{若$a \\perp \\beta$, $b \\perp \\beta$, $a\\parallel \\alpha$, $b\\parallel \\alpha$, 则$\\alpha\\parallel \\beta$}{若$\\alpha\\parallel \\beta$, $\\alpha \\bigcap \\gamma=a$, $\\beta \\cap \\gamma=b$, 则$a\\parallel b$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -322750,7 +322750,7 @@ "content": "下列四个命题中不正确的命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 三个点确定一个平面; \\textcircled{2} 圆锥的侧面展开图可以是一个圆面; \\textcircled{3} 底面是等边三角形, 三个侧面都是等腰三角形的三棱锥是正三棱锥; \\textcircled{4} 过球面上任意两不同点的大圆有且只有一个.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322769,7 +322769,7 @@ "content": "已知三棱柱$ABC-A_1B_1C_1$的体积为$30 \\text{cm}^3, P$为其侧棱$BB_1$上的任意一点, 则四棱锥$P-A_1CC_1A$的体积为\\blank{50}$\\mathrm{cm}^3$.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322788,7 +322788,7 @@ "content": "若一个底面边长为$\\dfrac{\\sqrt{3}}{2}$, 侧棱长为$\\sqrt{6}$的正四棱柱的所有顶点都在一个球面上, 则此球的体积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322807,7 +322807,7 @@ "content": "已知正四棱柱$ABCD-A_1B_1C_1D_1$中$AA_1=2AB$, 则$CD$与平面$BDC_1$所成角的正弦值等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322826,7 +322826,7 @@ "content": "圆锥母线长为$3$, 底面半径为$1 \\text{cm}$, 底面圆周上有一点$A$, 由$A$点出发绕圆锥一周回到$A$点的最短路线长等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322845,7 +322845,7 @@ "content": "已知三棱柱$ABC-A_1B_1C_1$的侧棱与底面垂直, 体积为$\\dfrac{9}{4}$, 底面是边长为$\\sqrt{3}$的正三角形. 若$P$为底面$A_1B_1C_1$的中心, 则$PA$与平面$ABC$所成角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322864,7 +322864,7 @@ "content": "半径为$1$的球面上的四点$A$、$B$、$C$、$D$是正四面体的顶点, 则$A$与$B$两点间的球面距离为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322883,7 +322883,7 @@ "content": "如图, 在直角梯形$ABCD$中, $\\angle B=\\angle C=90^{\\circ}$, $AB=\\sqrt{2}$, $CD=\\dfrac{\\sqrt{2}}{2}$, $BC=1$. 将$ABCD$(及其内部)绕$AB$所在的直线旋转一周, 形成一个几何体.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)}) node [above] {$A$} coordinate (A);\n\\draw (-1,{sqrt(2)/2}) node [left] {$D$} coordinate (D);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(2)}) node [above] {$A$} coordinate (A);\n\\draw (-1,{sqrt(2)/2}) node [left] {$D$} coordinate (D);\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (C) arc (180:360:1 and 0.25) (D) arc (180:360:1 and 0.25) -- (A) --(D);\n\\draw [dashed] (C) arc (180:0:1 and 0.25) (D) arc (180:0:1 and 0.25);\n\\draw (C)--(D) (1,0) -- (1,{sqrt(2)/2});\n\\draw ({cos(-50)},{0.25*sin(-50)}) node [below] {$C'$} coordinate (C');\n\\draw (C') ++ (0,{sqrt(2)/2}) node [below right] {$D'$} coordinate (D');\n\\draw (C')--(D')--(A);\n\\draw [dashed] (A)--(B) (D)--(C') (C)--(B)--(C');\n\\end{tikzpicture}\n\\end{center}\n(1) 求该几何体的体积$V$;\\\\\n(2) 设直角梯形$ABCD$绕底边$AB$所在的直线旋转角$\\theta$($\\angle CBC=\\theta \\in(0, \\pi)$)至$ABC' D'$, 问: 是否存在$\\theta$, 使得$AD' \\perp DC'$. 若存在, 求角$\\theta$的值, 若不存在, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322895,14 +322895,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013195": { "id": "013195", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中$AA_1=AD=1$, $E$为$CD$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z= {(210:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A1);\n\\draw (B1) -- (C1) -- (D1) -- (A1) -- cycle;\n\\draw (D) -- (D1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (A) -- (A1) (A)--(D1) (A)--(B1)--($(C)!0.5!(D)$) node [below right] {$E$} coordinate (E)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $B_1E \\perp AD_1$;\\\\\n(2) 在棱$AA_1$上是否存在一点$P$, 使得$DP\\parallel$平面$B_1AE$? 若存在, 求$AP$的长; 若不存在, 说明理由;\\\\\n(3) 若二面角$A-B_1E-A_1$的大小为$30^{\\circ}$, 求$AB$的长.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -322914,14 +322914,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013196": { "id": "013196", "content": "关于$x, y$的二元一次方程组$\\begin{cases}a x+y+a=0, \\\\ 4 x+a y+2=0\\end{cases}$无解, 则实数$a$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322940,7 +322940,7 @@ "content": "若矩阵$A=\\begin{pmatrix}1 & 3 \\\\ 2 & 1\\end{pmatrix}$, 则向量$(2,3)$经过矩阵$A$变换后所得的向量为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322959,7 +322959,7 @@ "content": "设关于$x$的实系数一元二次方程$x^2-2 a x+a^2-4 a+4=0$($a \\in \\mathbf{R}$)的两虚根为$x_1, x_2$且$|x_1|+|x_2|=3$, 则$a=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322978,7 +322978,7 @@ "content": "若$z \\in \\mathbf{C}$且$|z+2-2 \\mathrm{i}|=1$, 则$|z-2-2 \\mathrm{i}|$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -322997,7 +322997,7 @@ "content": "$z_1, z_2 \\in \\mathbf{C}$, $z_1^2-2 z_1 z_2+4 z_2^2=0$, $|z_2|=2$, 那么以$|z_1|$为直径的圆的面积为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323016,7 +323016,7 @@ "content": "已知复数$z$满足$|z|=|z-1|=1$, $\\text{Im} z>0$, 且$z^n=-z$, 则最小正整数$n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323035,7 +323035,7 @@ "content": "已知$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位), $z_1, z_2 \\in \\mathbf{C}$, 定义: $D(z)=|a|+|b|$, $D(z_1, z_2)=D(z_1-z_2)$. 给出下列命题:\\\\\n\\textcircled{1} 对任意$z \\in \\mathbf{C}$, 都有$D(z)>0$;\\\\\n\\textcircled{2} 若$\\overline {z}$是复数$z$的共轭复数, 则$D(\\overline {z})=D(z)$恒成立;\\\\\n\\textcircled{3} 若$D(z_1)=D(z_2)$($z_1$、$z_2 \\in \\mathbf{C})$, 则$z_1=z_2$;\\\\\n\\textcircled{4} 对任意$z_1$、$z_2$、$z_3 \\in \\mathbf{C}$, 结论$D(z_1, z_3) \\leq D(z_1, z_2)+D(z_2, z_3)$恒成立;\\\\\n\\textcircled{5} 对任意$z_1$、$z_2 \\in \\mathbf{C}$, 结论$D(z_1, z_2)=D(z_2, z_1)$恒成立.\\\\\n则其中真命题是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323054,7 +323054,7 @@ "content": "关于$x$的方程$x^2-(\\tan \\theta+\\mathrm{i}) x-(\\mathrm{i}+2)=0$($\\theta \\in \\mathbf{R}$, $x \\in \\mathbf{C}$).\\\\\n(1) 若此方程有实数根, 求锐角$\\theta$的值;\\\\\n(2) 求证: 对任意的实数$\\theta$($\\theta \\neq \\dfrac{\\pi}{2}+k \\pi$, $k \\in \\mathbf{Z}$), 原方程不可能有纯虚根.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323066,14 +323066,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013204": { "id": "013204", "content": "已知复平面上三点$P_1$、$P_2$、$P_3$分别对应复数$z$、$2 z$、$3 z$, 且$|z|=4$, 点$A$对应复数$1$, 若$\\triangle P_1AP_2$与$\\triangle$$P_2AP_3$的面积和为$2$, 求复数$z$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323085,14 +323085,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013205": { "id": "013205", "content": "对于任意的复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义$z$ ``经运算$P$''为$P(z)=x^2[\\cos (y \\pi)+\\mathrm{i} \\sin (y \\pi)]$.\\\\\n(1) 集合$A=\\{\\omega|\\omega=P(z), \\ | z |\\leq 1, \\ \\text{Re} z, \\text{Im} z\\text{均为整数}\\}$, 试用列举法写出集合$A$;\\\\\n(2) 若$z=2+y\\mathrm{i}$($y \\in \\mathbf{R}$), $P(z)$为纯虚数, 求$|z|$的最小值;\\\\\n(3) 直线$l: y=x-9$上是否存在整点$(x, y)$(坐标$x, y$均为整数), 使复数$z=x+y \\mathrm{i}$ ``经运算$P$''后, $P(z)$对应的点也在直线$l$上? 若存在, 求出所有的点; 若不存在, 请说明理由.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323104,14 +323104,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013206": { "id": "013206", "content": "关于$x$的方程$x^2+4 x+k=0$有一个根为$-2+3\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$k=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323130,7 +323130,7 @@ "content": "已知关于$x, y$的二元一次方程组$\\begin{pmatrix}m & 1 \\\\ 1 & m\\end{pmatrix}\\begin{pmatrix}x \\\\ y\\end{pmatrix}=\\begin{pmatrix}m+1 \\\\ 2 m\\end{pmatrix}$. 若方程组至多有一解, 则$m$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323149,7 +323149,7 @@ "content": "函数$y=\\begin{vmatrix}1 & 2 & 3 \\\\ x & 4 & 9 \\\\ x^2 & 8 & 27\\end{vmatrix}$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323168,7 +323168,7 @@ "content": "在复平面内, $O$是原点, $\\overrightarrow{OA}, \\overrightarrow{OC}, \\overrightarrow{AB}$表示的复数分别为$-2+\\mathrm{i}, 3+2\\mathrm{i}, 1+5\\mathrm{i}$, 那么$\\overrightarrow{BC}$表示的复数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323187,7 +323187,7 @@ "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R}$, $b \\neq 0$), 若$z^2-4 b z$是实数, 则有序数对$(a, b)$可以是\\blank{50}. (只需填一个)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323206,7 +323206,7 @@ "content": "复数$z$满足$|z-1|^2=(z-1)^2$, 则复数$z$对应点的轨迹是\\bracket{20}.\n\\fourch{一条直线}{一条双曲线}{一条抛物线}{一个圆}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -323225,7 +323225,7 @@ "content": "复数$z$满足$|z-2|+|z+\\mathrm{i}|=\\sqrt{5}$, 那么$|z|$的取值范围是\\bracket{20}.\n\\fourch{$[1, \\sqrt{5}]$}{$[1,2]$}{$[\\dfrac{2 \\sqrt{5}}{5}, 2]$}{$[\\dfrac{2 \\sqrt{5}}{5}, \\sqrt{5}]$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -323244,7 +323244,7 @@ "content": "已知$z=\\dfrac{2}{1-\\sqrt{3} \\mathrm{i}}$, 则$1+z+z^2+\\cdots+z^{2018}$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323263,7 +323263,7 @@ "content": "方程$x^2-2 x+p=0$的两根在复平面上对应的点之间的距离为$\\sqrt{3}$, 则实数$p$的值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323282,7 +323282,7 @@ "content": "设复数$\\alpha=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$, $y>0$), $\\dfrac{\\alpha}{1+\\alpha^2}$是实数.\\\\\n(1) 求证: $|\\alpha|=1$;\\\\\n(2) 若$\\dfrac{\\alpha^2}{1+\\alpha}$也是实数, 求$\\alpha$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323294,14 +323294,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013216": { "id": "013216", "content": "已知复数$z_0=1-m \\mathrm{i}$($m>0$), $z=x+y \\mathrm{i}$, $w=x'+y' \\mathrm{i}$, $x, y, x', y'$均为实数, $\\mathrm{i}$为虚数单位, 且对于任意复数$w=\\overline{z_0} \\cdot \\overline {z}$, $|w|=2|z|$.\\\\\n(1) 求$m$的值, 并分别写出$x', y'$用$x, y$表示的关系式;\\\\\n(2) 将$(x, y)$看作点$P$的坐标, $(x', y')$看作点$Q$的坐标, 上述关系可以看作是坐标平面上点的一个变换: 它将平面上的点$P$变换到这一平面上的点$Q$; 已知点$P$经该点的变换后得到的点$Q$的坐标是$(\\sqrt{3}, 2)$, 试求点$P$的坐标;\\\\\n(3) 若直线$y=k x$上任一点经上述变换后得到的点仍在该直线上, 求$k$的值.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323313,14 +323313,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013217": { "id": "013217", "content": "复数$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R})$, 将一颗骰子连续抛郑两次, 第一次点数记为$a$, 第二次点数记为$b$, 则使复数\n$z^2$为纯虚数的概率为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323339,7 +323339,7 @@ "content": "某学校组织学生参加英语测试, 成绩的频率分布直方图如图, 数据的分组一次为$[20,40),[40,60)$, $[60,80),[80,100)$若低于$60$分的人数是$15$人, 则该班的学生人数是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (6,0) node [below] {成绩/分};\n\\draw [->] (0,0) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {1/1/20,2/2/40,3/4/60,4/3/80}\n{\\draw (\\i,0) node [below] {$\\k$} --++ (0,\\j/2) --++ (1,0) --++ (0,-\\j/2);};\n\\draw (5,0) node [below] {$100$};\n\\foreach \\i/\\j/\\k in {1/1/0.005,2/2/0.01,3/4/0.015,4/3/0.02}\n{\\draw [dashed] (\\j,{\\i/2}) -- (0,{\\i/2}) node [left] {$\\k$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323358,7 +323358,7 @@ "content": "某区有$300$名学生参加数学竞赛, 随机抽取$11$名学生成绩如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 成绩 & 40 & 50 & 60 & 70 & 80 & 90 \\\\\n\\hline 人数 & 1 & 2 & 2 & 3 & 2 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则总体标准差的点估计值是\\blank{50}.(精确到$0.01$)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323377,7 +323377,7 @@ "content": "样本$(x_1, x_2, \\cdots, x_n)$的平均数为$\\overline {x}$, 样本$(y_1, y_2, \\cdots, y_m)$的平均数为$\\overline {y}$($\\overline {x} \\neq \\overline {y}$). 若样本$(x_1, x_2, \\cdots, x_n, y_1, y_2, \\cdots, y_m)$的平均数$\\overline {z}=\\alpha \\overline {x}+(1-\\alpha) \\overline {y}$, 其中$0<\\alpha<\\dfrac{1}{2}$, 则$n, m$的大小关系为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323396,7 +323396,7 @@ "content": "将$2$名教师, $4$名学生分成$2$个小组, 分别安排到甲、乙两地参加社会实践活动, 每个小组由$1$名教师和$2$名学生组成, 不同的安排方案共有\\blank{50}种.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323415,7 +323415,7 @@ "content": "若从$1,2, \\cdots \\cdots, 9$这$9$个整数中同时取$4$个不同的数, 其和为偶数, 则不同的取法共有\\blank{50}种.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323434,7 +323434,7 @@ "content": "若将函数$f(x)=x^5$表示为$f(x)=a_0+a_1(1+x)+a_2(1+x)^2+\\cdots+a_5(1+x)^5$其中$a_0, a_1, a_2, \\cdots, a_5$为实数, 则$a_3=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323453,7 +323453,7 @@ "content": "设$a \\in \\mathbf{Z}$, 且$0 \\leq a<13$, 若$51^{2012}+a$能被$13$整除, 则$a=$\\blank{50}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323472,7 +323472,7 @@ "content": "为了了解《中华人民共和国知识产权法》在学生中的普及情况, 某咨询调查机构对某校的$6$名学生进行问卷调查, $6$人得分分别为$5,6,7,8,9,10$. 把这$6$名学生看成一个总体.\\\\\n(1) 求该总体的均值和标准差(精确到$0.01$);\\\\\n(2) 用简单随机抽样的方法从这个总体中抽取一个容量为$2$的样本.\\\\\n\\textcircled{1} 在所有样本中, 写出所有样本标准差最大的样本和所有样本标准差最小的样本;\\\\ \n\\textcircled{2} 求在所有容量为$2$的样本中, 样本均值与总体均值之差的绝对值不超过$\\dfrac{1}{2}$的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323484,14 +323484,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013226": { "id": "013226", "content": "已知: $(x^{\\frac{2}{3}}+3 x^2)^n$的展开式中, 各项系数和比它的二项式系数和大$992$.\\\\\n(1) 求展开式中二项式系数最大的项;\\\\\n(2) 求展开式中系数最大的项.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323503,14 +323503,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013227": { "id": "013227", "content": "某单位组织$4$个部门的职工旅游, 规定每个部门只能在韶山、衡山、华山$3$个景区中任选一个, 假设各部门选择每个景区是独立且等可能的.\\\\\n(1) 求$3$个景区都有部门选择的概率;\\\\\n(2) 求恰有$2$个景区有部门选择的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323522,14 +323522,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013228": { "id": "013228", "content": "将$a, b, c, d, e, f$字母排成三行两列, 则不同的排列方法共有\\blank{50}种.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323548,7 +323548,7 @@ "content": "$(x-\\dfrac{1}{x})^8$的展开式中常数项为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323567,7 +323567,7 @@ "content": "在$(1+x)^n$($n \\in \\mathbf{N}$, $n\\ge 1$)的二项展开式中, 若只有$x^3$的系数最大, 则$n=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323586,7 +323586,7 @@ "content": "在$(\\sqrt{x}+\\dfrac{3}{\\sqrt[3]{x}})^n$展开式中, 各项系数的和与其各项二项式系数的和之比值为$64$, 则$n$等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323605,7 +323605,7 @@ "content": "$6$位同学互通电话, 任意两位同学之间最多通电话一次, 已知$6$位同学之间共通了$13$次电话, 则通了$4$次电话的同学人数为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323624,7 +323624,7 @@ "content": "若$\\mathrm{C}_n^1 x+\\mathrm{C}_n^2 x^2+\\cdots+\\mathrm{C}_n^n x^n$能被 7 整除, 则$x, n$的值可能为\\bracket{20}.\n\\fourch{$x=4$, $n=3$}{$x=4$, $n=4$}{$x=5$, $n=4$}{$x=6$, $n=5$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -323643,7 +323643,7 @@ "content": "某同学到银行取款时忘记了帐户密码, 但他记得: \\textcircled{1} 密码是四位数字, 如:$0235,1330,2351$等; \\textcircled{2} 四位数字中有$6,8,9$; \\textcircled{3} 四位数字各不相同. 于是他就用$6,8,9$这三个数字再随意加上一个与这三个数字不同的数字, 排成四位数输入取款机尝试, 那么他只试一次就成功的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323662,7 +323662,7 @@ "content": "甲、乙两人在一次射击比赛中各射靶$5$次, 两人成绩的条形统计图如图所示, 则下列命题正确的有\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {4/1,5/1,6/1,7/1,8/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(甲)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\draw [->] (0,0) -- (0.1,0) -- (0.3,0.5) -- (0.7,-0.5) -- (0.9,0) -- (9,0) node [below right] {环数};\n\\draw [->] (0,0) -- (0,4) node [left] {频数};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {3,4,...,10}\n{\\draw ({\\i-2},0.3) -- ({\\i-2},0) node [below] {$\\i$};};\n\\foreach \\i in {1,2,3}\n{\\draw (0.3,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {5/3,6/1,9/1}\n{\\filldraw [pattern = north east lines] ({\\i-2.3},0) --++ (0,\\j) --++ (0.6,0) --++ (0,-\\j);};\n\\draw (5,-2) node {(乙)};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 甲的成绩的平均数小于乙的成绩的平均数; \\textcircled{2} 甲的成绩的中位数等于乙的成绩的中位数; \\textcircled{3} 甲的成绩的方差小于乙的成绩的方差.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323681,7 +323681,7 @@ "content": "将序号分别为$1,2,3,4,5$的$5$张参观券全部分给$4$人, 每人至少$1$张, 如果分给同一人的$2$张参观券连号, 那么不同的分法种数是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323700,7 +323700,7 @@ "content": "为了考察某校各班参加课外书法小组的人数, 在全校随机抽取$5$个班级, 把每个班级参加该小组的人数作为样本数据. 已知样本平均数为$7$, 样本方差为$4$, 则样本数据中的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323719,7 +323719,7 @@ "content": "已知数列$\\{a_n\\}$($n$为正整数)是首项是$a_1$, 公比为$q$的等比数列.\\\\\n(1) 求和: $a_1\\mathrm{C}_2^0-a_2\\mathrm{C}_2^1+a_3\\mathrm{C}_2^2$, $a_1\\mathrm{C}_3^0-a_2\\mathrm{C}_3^1+a_3\\mathrm{C}_3^2-a_4\\mathrm{C}_3^3$;\\\\\n(2) 由(1)的结果归纳概括出关于正整数$n$的一个结论, 并加以证明.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323731,14 +323731,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013239": { "id": "013239", "content": "为了研究某高校大学新生学生的视力情况, 随机地抽查了该校$100$名进校学生的视力情况, 得到频率分布直方图, 如图. 已知前$4$组的频数从左到右依次是等比数列$\\{a_n\\}$的前四项, 后$6$组的频数从左到右依次是等差数列$\\{b_n\\}$的前六项.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,yscale = 1.4]\n\\draw [->] (-0.5,0) -- (6,0) node [below] {视力};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j/\\k in {1/0.1/4.3,2/0.3/4.4,3/0.9/4.5,4/2.7/4.6,5/2.2/4.7,6/1.7/4.8,7/1.2/4.9,8/0.7/5.0,9/0.2/5.1}\n{\\draw ({\\i/2},0) node [below] {$\\k$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\draw (5,0) node [below] {$5.2$};\n\\draw [dashed] (0.5,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw [dashed] (1,0.3) -- (0,0.3) node [left] {$0.3$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求等比数列$\\{a_n\\}$的通项公式;\\\\\n(2) 求等差数列$\\{b_n\\}$的通项公式;\\\\\n(3) 若规定视力低于$5.0$的学生属于近视学生, 试估计该校新生的近视率$p$的大小.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323750,14 +323750,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013240": { "id": "013240", "content": "已知实数$x, y$满足$\\begin{cases}x+2 y \\geq 4, \\\\ 2 x+y \\geq 3, \\\\ x \\geq 0, \\\\ y \\geq 0\\end{cases}$的目标函数$f=x+y$的最小值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323776,7 +323776,7 @@ "content": "已知实数$x$、$y$满足$\\begin{cases}x+y \\leq 5, \\\\ 2 x+y \\leq 6, \\\\ x \\geq 0, \\\\ y \\geq 0,\\end{cases}$ 则$z=3 x+4 y$的最大值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323795,7 +323795,7 @@ "content": "若实数$x, y$满足$\\begin{cases}2 x-y \\geq 0, \\\\ y \\geq x, \\\\ y \\geq-x+b,\\end{cases}$且$z=2 x+y$的最小值为$3$, 则实数$b$的值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323814,7 +323814,7 @@ "content": "设实数$x, y$, 满足$\\begin{cases}x-y-2 \\leq 0, \\\\ x+2 y-4 \\geq 0, \\\\ 2 y-3 \\leq 0,\\end{cases}$ 则$\\dfrac{y}{x}$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323833,7 +323833,7 @@ "content": "已知实数$x, y$满足$\\begin{cases}x \\leq 3, \\\\ x+y-3 \\geq 0, \\\\ x-y+1 \\geq 0,\\end{cases}$ 则$x^2+y^2$的最小值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323852,7 +323852,7 @@ "content": "若正三棱柱的主视图如图所示, 则此三棱柱的体积等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (2,2);\n\\draw (1,0) -- (1,2);\n\\draw (0,0) -- (0,-0.5) (1,0) -- (1,-0.5) (2,0) -- (2,-0.5);\n\\draw (2.5,0) -- (2,0) (2.5,2) -- (2,2);\n\\draw [<->] (0,-0.25) -- (1,-0.25) node [midway,fill = white] {$1$};\n\\draw [<->] (1,-0.25) -- (2,-0.25) node [midway,fill = white] {$1$};\n\\draw [<->] (2.25,0) -- (2.25,2) node [midway, fill = white] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323871,7 +323871,7 @@ "content": "一个几何体的三视图如图所示, 则该几何体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale=0.5]\n\\draw (0,0) circle (1);\n\\draw (-1,1.5) --++ (2,0) --++ (0,2) --++ (-2,0) -- cycle;\n\\draw (-1,3.5) --++ (1,1) --++ (1,-1);\n\\draw (2,1.5) --++ (2,0) --++ (0,2) --++ (-2,0) -- cycle;\n\\draw (2,3.5) --++ (1,1) --++ (1,-1);\n\\draw [dashed] (1,1.5) -- (2,1.5) (1,3.5) -- (2,3.5);\n\\draw [<->] (1.5,1.5) -- (1.5,3.5) node [midway, rotate = 90, fill = white] {$2$};\n\\draw [dashed] (0,4.5) -- (3,4.5);\n\\draw [->] (1.5,5) -- (1.5,4.5);\n\\draw (1.5,4) node [rotate = 90] {$1$};\n\\draw [dashed] (-1,1.5) --++ (0,-3.25) (1,1.5) --++ (0,-3.25);\n\\draw [<->] (-1,-1.5) -- (1,-1.5) node [midway, fill=white] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323890,7 +323890,7 @@ "content": "一个用若干块大小相同的立方块搭成的立体图形, 主视图和俯视图是同一图形(如图), 那么搭成这样一个立体图形最少需要\\blank{50}个小立方块.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) rectangle (3,1);\n\\draw (0,1) --++ (0,1) --++ (1,0) --++ (0,-2) (2,0) --++ (0,1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323909,7 +323909,7 @@ "content": "将如图所示的一个直角三角形$ABC$($\\angle C=90^{\\circ}$)绕斜边$AB$旋转一周, 所得到的几何体的正视图是下面四个图形中的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) node [left] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (0,-0.5) node [below] {$B$} coordinate (B);\n\\filldraw [pattern = north east lines] (A) -- (B) -- (C) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,2) -- cycle;\n\\draw (1,0) arc ({atan(4/3)-90}:{-atan(4/3)-90}:{5/4});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,2) -- cycle;\n\\draw (1,0) -- (0,-0.5) -- (-1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (-1,0) -- (1,0) -- (0,-2) -- cycle;\n\\draw (1,0) -- (0,0.5) -- (-1,0);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (1);\n\\filldraw (0,0) circle (0.03);\n\\end{tikzpicture}}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -323928,7 +323928,7 @@ "content": "将大小不同的两种钢板截成$A$、$B$两种规格的成品, 每张钢板可同时截得这两种规格的成品的块数如表所示. 若现在需$A$、$B$两种规格的成品分别为$12$块和$10$块, 则至少共需这两种钢板\\blank{50}张.\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline &$A$种规格的成品 &$B$种规格的成品 \\\\\n\\hline 第一种钢板 & 2 & 1 \\\\\n\\hline 第二种钢板 & 1 & 3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323947,7 +323947,7 @@ "content": "已知几何体由正方体和直三棱柱组成, 其三视图和直观图(单位: $\\text{cm}$) 如图所示. 设两条异面直线$A_1Q$和$PD$所成的角为$\\theta$, 求$\\cos \\theta$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\draw (0,0) -- (2,0) -- (2,3) -- (0,3) -- cycle (0,2) -- (2,2);\n\\draw (1,0) node [above] {$2$} (2,1) node [left] {$2$};\n\\draw (3,0) --++ (2,0) --++ (0,2) --++ (-1,1) --++ (-1,-1) -- cycle;\n\\draw (4,0) node [above] {$2$} (3,1) node [right] {$2$} (3,2) ++ (0.5,0.5) node [above left] {$\\sqrt{2}$} ++ (1,0) node [above right] {$\\sqrt{2}$};\n\\draw (0,-3) rectangle ++ (2,2) (0,-2) --++ (2,0);\n\\draw (0,-2.5) node [right] {$1$} (0,-1.5) node [right] {$1$} (1,-3) node [above] {$2$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [below right] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1);\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1) (A1) -- (D1) -- (C1);\n\\draw ($(A1)!0.5!(D1)$) ++ (0,1) node [above] {$P$} coordinate (P);\n\\draw ($(B1)!0.5!(C1)$) ++ (0,1) node [above] {$Q$} coordinate (Q);\n\\draw (A1) -- (P) -- (Q) -- (B1) (Q) -- (C1);\n\\draw [dashed] (P) --(D1);\n\\draw [dashed] (P) -- (D);\n\\draw (A1) -- (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -323959,14 +323959,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013251": { "id": "013251", "content": "已知实数$x$、$y$满足线性约束条件$\\begin{cases}3 x-y \\geq 0, \\\\ x+y-4 \\leq 0, \\\\ x-3 y+5 \\leq 0.\\end{cases}$则目标函数$z=x-y-1$的最大值是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -323985,7 +323985,7 @@ "content": "设实数$x, y$满足$\\begin{cases}x+y \\geq 2, \\\\ 2 x-y \\leq 4, \\\\ y \\leq 4,\\end{cases}$则$x-2 y$的最大值等于\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324004,7 +324004,7 @@ "content": "若$x, y$满足$\\begin{cases}-x+y \\leq 0, \\\\ -x+2 y \\geq 2,\\end{cases}$则目标函数$C=\\log _{\\frac{1}{2}}(x+y)$的最大值为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324023,7 +324023,7 @@ "content": "已知$x, y$满足$\\begin{cases}y-2 \\leq 0, \\\\ x+3 \\geq 0, \\\\ x-y-1 \\leq 0,\\end{cases}$则$\\dfrac{y-2}{x-4}$的取值范围是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324042,7 +324042,7 @@ "content": "一个几何体的三视图如下左图所示, 其中俯视图与左视图均为半径是$2$的圆, 则这个几何体的表面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) circle (1);\n\\draw (0,-1) -- (0,1);\n\\draw (1,3) arc (0:-270:1) --++ (0,-1) --++ (1,0);\n\\draw (3,3) circle (1);\n\\draw [dashed] (2,3) -- (4,3);\n\\draw (0,-1) node [below] {俯视图};\n\\draw (0,2) node [below] {主视图};\n\\draw (3,2) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324061,7 +324061,7 @@ "content": "如图, 四棱锥$S-ABCD$的底面是矩形, 锥顶点在底面的射影是矩形对角线的交点, 四棱锥及其三视图如下($AB$平行于主视图投影平面) 则四棱锥$S-ABCD$的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$B$} coordinate (B);\n\\draw (4,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [left] {$D$} coordinate (D);\n\\draw (2,3,-1) node [above] {$S$} coordinate (S);\n\\draw (A) -- (B) -- (C) -- (S) -- (A) (S) -- (B);\n\\draw [dashed] (C) -- (D) (A) -- (D) -- (S);\n\\path (2,-4);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) -- (4,0) -- (4,2) -- (0,2) -- cycle;\n\\draw (0,0) -- (4,2) (4,0) -- (0,2);\n\\draw (0,-1) -- (0,0) (4,-1) -- (4,0);\n\\draw [<->] (0,-0.5) -- (4,-0.5) node [midway, fill = white] {$4$};\n\\draw (0,4) -- (4,4) -- (2,7) -- cycle;\n\\draw (6,4) -- (8,4) -- (7,7) -- cycle;\n\\draw [dashed] (4,4) -- (6,4) (2,7) -- (7,7);\n\\draw [<->] (5,4) -- (5,7) node [midway, rotate = 90, fill = white] {$3$};\n\\draw (6,4) -- (6,3) (8,4) -- (8,3);\n\\draw [<->] (6,3.5) -- (8,3.5) node [midway, fill = white] {$2$};\n\\draw (2,4) node [below] {主视图}; \n\\draw (2,-1) node [below] {俯视图};\n\\draw (7,3) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324080,7 +324080,7 @@ "content": "已知棱长为$1$的正方体的俯视图是一个面积为$1$的正方形, 则该正方体的主视图的面积不可能等于\\bracket{20}.\n\\fourch{1}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}-1}{2}$}{$\\dfrac{\\sqrt{2}+1}{2}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324099,7 +324099,7 @@ "content": "某四棱锥的三视图如图所示, 则最长的一条侧棱长度为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (1,0) -- (2,0) -- (0,1) -- cycle;\n\\draw [dashed] (1,0) -- (0,1);\n\\draw (0,-0.1) -- (0,-0.4) (1,-0.1) -- (1,-0.4) (2,-0.1) -- (2,-0.4);\n\\draw [<->] (0,-0.25) -- (1,-0.25) node [midway, fill = white] {$1$};\n\\draw [<->] (1,-0.25) -- (2,-0.25) node [midway, fill = white] {$1$};\n\\draw (-0.1,0) -- (-0.4,0) (-0.1,1) -- (-0.4,1);\n\\draw [<->] (-0.25,0) -- (-0.25,1) node [midway, rotate = 90, fill = white] {$1$};\n\\draw (1,-0.5) node [below] {主视图};\n\\draw (3,0) -- (4,0) -- (4,1) -- cycle;\n\\draw (3,-0.1) -- (3,-0.4) (4,-0.1) -- (4,-0.4);\n\\draw [<->] (3,-0.25) -- (4,-0.25) node [midway, fill = white] {$1$};\n\\draw (3.5,-0.5) node [below] {主视图};\n\\draw (0,-1.5) -- (1,-1.5) --++ (1,-1) --++ (-2,0) --++ (0,1);\n\\draw (0,-2.5) --++ (1,1);\n\\draw (1,-2.5) node [below] {俯视图};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\sqrt{2}$}{$\\sqrt{3}$}{$\\sqrt{5}$}{$\\sqrt{6}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324118,7 +324118,7 @@ "content": "某几何体的三视图如图所示, 则该几何体的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (1);\n\\draw (-0.5,-0.5) rectangle (0.5,0.5);\n\\draw (-1,1.5) rectangle ++ (2,2);\n\\draw (2,1.5) rectangle ++ (2,2);\n\\draw [dashed] (-0.5,1.5) --++ (0,2) (0.5,1.5) --++ (0,2);\n\\draw [dashed] (2.5,1.5) --++ (0,2) (3.5,1.5) --++ (0,2);\n\\draw (-1.1,1.5) -- (-1.5,1.5) (-1.1,3.5) -- (-1.5,3.5);\n\\draw [<->] (-1.3,1.5) -- (-1.3,3.5) node [midway, rotate = 90, fill = white] {$4$};\n\\foreach \\i in {-1,-0.5,0.5,1,2,2.5,3.5,4}\n{\\draw (\\i,3.6) --++ (0,0.3);};\n\\draw [<->] (-1,3.75) -- (-0.5,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (0.5,3.75) -- (1,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (2,3.75) -- (2.5,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (3.5,3.75) -- (4,3.75) node [midway, fill = white] {$1$};\n\\draw [<->] (-0.5,3.75) -- (0.5,3.75) node [midway, fill = white] {$2$};\n\\draw [<->] (2.5,3.75) -- (3.5,3.75) node [midway, fill = white] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324137,7 +324137,7 @@ "content": "已知某一多面体内接于一个简单组合体, 如果该组合体的主视图, 左视图, 俯视图均如上图所示, 且图中的四边形是边长为$2$的正方形, 则该球的表面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle ({sqrt(3)/2});\n\\draw [dashed] (-0.5,-0.5) rectangle (0.5,0.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324156,7 +324156,7 @@ "content": "某企业生产甲、乙两种产品, 已知生产每吨甲产品要用$A$原料$3$吨、$B$原料$2$吨; 生产每吨乙产品要用$A$原料$1$吨、$B$原料$3$吨. 销售每吨甲产品可获得利润$5$万元, 每吨乙产品可获得利润$3$万元, 该企业在一个生产周期内消耗$A$原料不超过$13$吨, $B$原料不超过$18$吨, 求该企业可获得的最大利润.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324168,14 +324168,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013262": { "id": "013262", "content": "如图所示, 给出的是某几何体的三视图, 其中主视图与左视图都是边长为$2$的正三角形, 俯视图为半径等于$1$的圆. 试求这个几何体的体积与侧面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\filldraw (0,0) circle (0.03);\n\\draw (0,0) circle (1);\n\\draw (-1,2) -- (1,2) --++ (120:2) -- cycle;\n\\draw (2,2) -- (4,2) --++ (120:2) -- cycle;\n\\draw (0,-1) node [below] {俯视图};\n\\draw (0,2) node [below] {主视图};\n\\draw (3,2) node [below] {左视图};\n\\end{tikzpicture}\n\\end{center}", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324187,14 +324187,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013263": { "id": "013263", "content": "某工厂的一位产品检验员在检验产品时, 可能把正品错误地检验为次品, 同样也会把次品错误地检验为正品, 他的各次检验是相互独立的. 已知他把正品检验为次品的概率是$0.02$, 把次品检验为正品的概率为$0.01$. 现有$3$件正品和$1$件次品, 则该检验员将这$4$件产品全部检验正确的概率是\\blank{50}.(结果保留三位小数)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324213,7 +324213,7 @@ "content": "有一技术难题, 甲单独解决的概率为$\\dfrac{1}{2}$, 乙单独解决的概率为$\\dfrac{1}{3}$, 现两人单独解决难题, 则此难题能被解决的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324232,7 +324232,7 @@ "content": "曲线$y=-\\sqrt{2-x^2}$的参数方程可以是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324251,7 +324251,7 @@ "content": "若直线$l$的参数方程为: $\\begin{cases}x=1+\\dfrac{1}{2} t, \\\\ y=2-\\dfrac{\\sqrt{3}}{2} t,\\end{cases}$($t$为参数) 则$l$的倾斜角的大小为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324270,7 +324270,7 @@ "content": "直线$x+\\sqrt{3} y-4=0$和圆$\\begin{cases}x=2 \\cos \\varphi, \\\\ y=2 \\sin \\varphi, \\end{cases}$($0 \\leq \\varphi<2 \\pi$)的位置关系是\\bracket{20}.\n\\fourch{相交但不过圆心}{相交且过圆心}{相切}{相离}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324289,7 +324289,7 @@ "content": "下列参数($t$为参数)方程中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2,\\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt{|t|},\\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sin t,\\\\ y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324308,7 +324308,7 @@ "content": "将参数方程$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta,\\end{cases}$($\\theta$为参数) 化为普通方程, 所得方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324327,7 +324327,7 @@ "content": "巳知圆$C$和圆$\\begin{cases}x=4+4 \\cos \\theta, \\\\ y=5+4 \\sin \\theta,\\end{cases}$($\\theta$为参数)关于直线$\\begin{cases}x=\\dfrac{1}{\\sqrt{10}} t, \\\\ y=3-\\dfrac{1}{\\sqrt{10}} t,\\end{cases}$($t$为参数)对称, 则圆$C$的普通方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324346,7 +324346,7 @@ "content": "已知直线$l: y=3 x$, 曲线$C$的参数方程为$\\begin{cases}x=t-\\dfrac{1}{t}, \\\\ y=t+\\dfrac{1}{t},\\end{cases}$($t$为参数) $l$与$C$相交于$A, B$两点, 则$|AB|=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324365,7 +324365,7 @@ "content": "设$A$、$B$是两个随机事件, 若$P(A)=0.34$, $P(B)=0.32$, $P(A\\cap B)=0.31$, 则$P(A \\cup B)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324384,7 +324384,7 @@ "content": "在一个正四面体四个面分别写上$1,2,3,4$. 投掷一次出现$A=\\{\\text{朝下的面出现}1\\text{或}2\\}$, $B=\\{\\text{朝下的面出现}1\\text{或}3\\}$, $C=\\{\\text{朝下的面出现}3\\text{或}4\\}$, 事件$A, B, C$中相互独立的事件是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324403,7 +324403,7 @@ "content": "袋中有大小相同的红球和白球若干个, 其中红、白球个数的比为$4: 3$. 假设从袋中任取$2$个球, 取到的都是红球的概率为$\\dfrac{4}{13}$.\n(1) 试问: 袋中的红、白球各有多少个?\\\\\n(2) 现从袋中逐次取球, 每次从袋中任取$1$个球, 若取到白球, 则停止取球, 若取到红球, 则继续下一次取球. 试求: 取球不超过$3$次便停止的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324415,14 +324415,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013275": { "id": "013275", "content": "曲线$\\begin{cases}x=2 \\sin t, \\\\ y=3 \\cos t,\\end{cases}$($t$为参数) 的焦点坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324441,7 +324441,7 @@ "content": "直线$l$的参数方程是$\\begin{cases}x=1+2 t, \\\\ y=2-t,\\end{cases}$($t \\in \\mathbf{R}$) 则$l$的方向向量$\\overrightarrow {d}$可以是\\bracket{20}.\n\\fourch{$(1,2)$}{$(2,1)$}{$(-2,1)$}{$(1,-2)$}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324460,7 +324460,7 @@ "content": "若实数$x, y$满足$y=\\sqrt{1-x^2}$, 则$y-x$的取值范围为\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324479,7 +324479,7 @@ "content": "根据下列条件研究参数方程$\\begin{cases}x=2+t \\cos \\alpha, \\\\ y=-1+t \\sin \\alpha\\end{cases}$表示何种曲线:\\\\\n(1) $\\alpha$为常数, $t$为参数;\\\\\n(2) $t$为常数, $\\alpha$为参数.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324491,14 +324491,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013279": { "id": "013279", "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta,\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324517,7 +324517,7 @@ "content": "将参数方程$\\begin{cases}x=\\sin \\theta+\\cos \\theta, \\\\ y=\\sin \\theta-\\cos \\theta,\\end{cases}$$\\theta \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$, ($\\theta$为参数) 化为普通方程是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324536,7 +324536,7 @@ "content": "已知曲线$\\Gamma$的参数方程为$\\begin{cases}x=t^3-t \\cos t, \\\\ y=\\ln (t+\\sqrt{t^2+1}),\\end{cases}$ 其中参数$t \\in \\mathbf{R}$, 则曲线$\\Gamma$\\bracket{20}.\n\\fourch{关于$x$轴对称}{关于$y$轴对称}{关于原点对称}{没有对称性}", "objs": [], "tags": [], - "genre": "", + "genre": "选择题", "ans": "", "solution": "", "duration": -1, @@ -324555,7 +324555,7 @@ "content": "某一批花生种子, 如果每$1$粒发芽的概率为$\\dfrac{4}{5}$, 那么独立地播下$4$粒种子恰有$2$粒发芽的概率是\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324574,7 +324574,7 @@ "content": "袋中有$8$个颜色不同, 其它都相同的球, 其中$1$个为黑球, $3$个为白球, $4$个为红球, 若从袋中一次摸出$2$个球, 求所摸出的$2$个球恰为异色球的概率.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324586,14 +324586,14 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "013284": { "id": "013284", "content": "已知随机事件$A$、$B$是互斥事件. 若$P(A)=0.25$, $P(A \\cup B)=0.78$, 则$P(B)=$\\blank{50}.", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324612,7 +324612,7 @@ "content": "某学生参加一次某志愿者测试. 已知在备选的$10$道试题中, 预计每道题该学生答对的概率为$\\dfrac{2}{3}$. 规定每位考生都从备选题中随机抽出$3$道题进行测试, 则该学生恰答对$2$道题的概率是\\blank{50}.(用数值表示)", "objs": [], "tags": [], - "genre": "", + "genre": "填空题", "ans": "", "solution": "", "duration": -1, @@ -324631,7 +324631,7 @@ "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$, 直线$x+2 y+18=0$, 在椭圆上求一点$P$, 使点$P$到这条直线的距离最短.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -324643,7 +324643,7 @@ "same": [], "related": [], "remark": "", - "space": "" + "space": "12ex" }, "020001": { "id": "020001",