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20221127 night

This commit is contained in:
weiye.wang 2022-11-27 23:46:55 +08:00
parent ef1d05b5fa
commit a7bb571e70
6 changed files with 241 additions and 83 deletions
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6
工具/关键字筛选题号.ipynb
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@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
"execution_count": 4,
"execution_count": 5,
"metadata": {},
"outputs": [
{
@ -11,7 +11,7 @@
"0"
]
},
"execution_count": 4,
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@ -21,7 +21,7 @@
"\n",
"\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n",
"keywords_dict_table = [\n",
" {\"tags\":[\"二项式定理\"]}\n",
" {\"tags\":[\"概率\"]}\n",
"]\n",
"\"\"\"---关键字设置完毕---\"\"\"\n",
"# 示例： keywords_dict_table = [\n",

2
工具/文本文件/题号筛选.txt
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@ -1 +1 @@
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000218,000219,000220,000221,000222,000223,000224,000225,000226,000227,000228,000229,000230,000231,000332,000384,000391,000512,000564,000581,000601,000611,000624,000659,000672,000685,000695,000704,000714,000744,000765,000773,000779,000812,000829,000844,000889,000969,002640,002641,002642,002643,002644,002645,002646,002647,002648,002649,002650,002651,002652,002653,002654,002655,002656,002657,002658,002659,002660,002661,002662,002663,002664,002665,003574,003575,003585,003586,003598,003640,003660,003727,003734,003751,003787,003806,003825,003873,003887,003914,003947,003979,003983,003998,004031,004032,004033,004034,004035,004036,004037,004038,004039,004040,004041,004042,004043,004044,004045,004046,004087,004110,004150,004193,004212,004232,004257,004297,004324,004341,004535,004572,004573,004574,004575,004576,004577,004578,004579,004580,004581,004582,004583,004584,004585,004586,004587,004588,004589,004590,004591,004592,004593,004594,004595,004596,004597,004598,004599,004600,004601,004602,004603,004604,004605,004606,004607,004608,004609,004610,004611,004612,004613,004614,004615,004616,004617,004618,004647,004712,004750,009345,009346,009347,009348,009349,009350,009351,009352,009353,009354,009355,009356,009357,009359,009360,009361,009362,009363,009364,009365,009366,009391,009409,009410,009423,009425,009733,009734,009735,009736,009737,009738,009739,009740,009741,009742,009743,009744,009745,009746,009747,009748,009749,009750,009751,009752,009753,009754,009755,009943,009944,009949,009950,009951,009952,009953,009954,009955,009956,009957,009958,009959,009960,009961,009962,009963,009964,009965,009966,009967,009968,009969,009992,010005,010006,010007,010008,010009,010010,010011,010012,010013,010014,010015,010016,010534,010535,010536,010537,010538,010539,010540,010541,010542,010543,010544,010545,010546,010547,010548,010549,010550,010551,010552,010553,010554,010555,010556,010557,010558,010559,010560,010870,010871,010872,010873,010874,010884,010885,010886,010887,010888,010889,010890,010891,010892,010893,010894,010895,010896,010897,010898,010899,010900,010901,010902,010903,010950,011053,011117,011141,011250,011289,011312,011351,011354,011376,011397,011439,011480,011523,011583,011586,011609,011612,011634,011640,011654,011679,011682,012036,012046,030170,030171,030172,030173,030174,030175,030176,030177,030178,030179,030180,030181,030182,030183,030184,030185,030186,030187,030188,030189,030190,030191,030192,030193,030194,030276,030277,030433

12
工具/添加题目到数据库.ipynb
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@ -7,15 +7,15 @@
"outputs": [],
"source": [
"#修改起始id,出处,文件名\n",
"starting_id = 12075\n",
"origin = \"2023届华东师范大学一附中高三上学期期中考试\"\n",
"starting_id = 30485\n",
"origin = \"空中课堂必修第三册复习课例题\"\n",
"filename = r\"C:\\Users\\Weiye\\Documents\\wwy sync\\临时工作区\\自拟题目4.tex\"\n",
"editor = \"20221121\\t王伟叶\""
"editor = \"20221127\\t王伟叶\""
]
},
{
"cell_type": "code",
"execution_count": 4,
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
@ -70,8 +70,8 @@
" pid = str(id).zfill(6)\n",
" if pid in pro_dict:\n",
" duplicate_flag = True\n",
" # NewProblem = CreateNewProblem(id = pid, content = p, origin = origin , dict = pro_dict,editor = editor)\n",
" NewProblem = CreateNewProblem(id = pid, content = p, origin = origin + \"试题\" + str(id- starting_id+1), dict = pro_dict,editor = editor)\n",
" NewProblem = CreateNewProblem(id = pid, content = p, origin = origin , dict = pro_dict,editor = editor)\n",
" # NewProblem = CreateNewProblem(id = pid, content = p, origin = origin + \"试题\" + str(id- starting_id+1), dict = pro_dict,editor = editor)\n",
" pro_dict[pid] = NewProblem\n",
" id += 1\n",
"\n",

73
工具/识别题库中尚未标注的题目类型.ipynb
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@ -9,49 +9,36 @@
"name": "stdout",
"output_type": "stream",
"text": [
"012033 填空题\n",
"012034 填空题\n",
"012035 填空题\n",
"012036 填空题\n",
"012037 填空题\n",
"012038 填空题\n",
"012039 填空题\n",
"012040 填空题\n",
"012041 填空题\n",
"012042 填空题\n",
"012043 填空题\n",
"012044 填空题\n",
"012045 选择题\n",
"012046 选择题\n",
"012047 选择题\n",
"012048 选择题\n",
"012049 解答题\n",
"012050 解答题\n",
"012051 解答题\n",
"012052 解答题\n",
"012053 解答题\n",
"012054 填空题\n",
"012055 填空题\n",
"012056 填空题\n",
"012057 填空题\n",
"012058 填空题\n",
"012059 填空题\n",
"012060 填空题\n",
"012061 填空题\n",
"012062 填空题\n",
"012063 填空题\n",
"012064 填空题\n",
"012065 填空题\n",
"012066 选择题\n",
"012067 选择题\n",
"012068 选择题\n",
"012069 选择题\n",
"012070 解答题\n",
"012071 解答题\n",
"012072 解答题\n",
"012073 解答题\n",
"012074 解答题\n",
"030481 填空题\n"
"012075 填空题\n",
"012076 填空题\n",
"012077 填空题\n",
"012078 填空题\n",
"012079 填空题\n",
"012080 填空题\n",
"012081 填空题\n",
"012082 填空题\n",
"012083 填空题\n",
"012084 填空题\n",
"012085 填空题\n",
"012086 填空题\n",
"012087 选择题\n",
"012088 选择题\n",
"012089 选择题\n",
"012090 选择题\n",
"012091 解答题\n",
"012092 解答题\n",
"012093 解答题\n",
"012094 解答题\n",
"012095 解答题\n",
"030485 解答题\n",
"030486 解答题\n",
"030487 解答题\n",
"030488 解答题\n",
"030489 解答题\n",
"030490 解答题\n",
"030491 解答题\n",
"030492 解答题\n",
"030493 解答题\n"
]
}
],

8
工具/题号选题pdf生成.ipynb
View File

@ -2,16 +2,16 @@
"cells": [
{
"cell_type": "code",
"execution_count": 10,
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"开始编译教师版本pdf文件: 临时文件/二项式定理待赋目标_教师用_20221126.tex\n",
"开始编译教师版本pdf文件: 临时文件/概率待赋目标_教师用_20221127.tex\n",
"0\n",
"开始编译学生版本pdf文件: 临时文件/二项式定理待赋目标_学生用_20221126.tex\n",
"开始编译学生版本pdf文件: 临时文件/概率待赋目标_学生用_20221127.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/二项式定理待赋目标\"\n",
"filename = \"临时文件/概率待赋目标\"\n",
"\"\"\"---设置文件名结束---\"\"\"\n",
"\n",
"\n",

223
题库0.3/Problems.json
View File

@ -295546,7 +295546,7 @@
"content": "设集合$A=\\{x|(x-1)(x-4)<0\\}$, 集合$B=\\mathbf{Z}$, 则$A\\cap B=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295565,7 +295565,7 @@
"content": "已知$\\mathrm{i}$为虚数单位, 则复数$z=\\dfrac{3+\\mathrm{i}}{2+\\mathrm{i}}$的模$|z|=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295584,7 +295584,7 @@
"content": "方程$\\log_2(3x+4)=3$的解为$x=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295603,7 +295603,7 @@
"content": "在二项式$(x+\\dfrac2x)^6$的展开式中, 常数项是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295622,7 +295622,7 @@
"content": "若圆锥侧面积为$20\\pi$, 且母线与底面所成角为$\\arccos \\dfrac 4\n5$, 则该圆锥的侧面积为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295641,7 +295641,7 @@
"content": "设点$H(2,3)$, 若直线$l$经过点$H$, 且与直线$OH$垂直($O$为坐标原点), 则直线$l$的方程为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295660,7 +295660,7 @@
"content": "函数$y=2\\cos\\left(x+\\dfrac{\\pi}{4}\\right)\\cos\\left(x-\\dfrac{\\pi}{4}\\right)+\\sqrt{3}\\sin 2x$的值域为\\blank{80}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295679,7 +295679,7 @@
"content": "函数$y=\\dfrac{x}{x+1}$的图像是一个中心对称图形, 其对称中心的坐标为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295698,7 +295698,7 @@
"content": "已知随机变量$X$的分布列为$\\begin{pmatrix}\n -1 & 0 & 1 \\\\\n \\dfrac 12 & \\dfrac 13 & \\dfrac 16 \n \\end{pmatrix}$, 另一个随机变量$Y$满足$X+2Y=4$, 则$Y$的期望$E[Y]=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295717,7 +295717,7 @@
"content": "在由数字$1, 2, 3, 4, 5$组成的数字不重复的五位数中, 小于$50000$的奇数有\\blank{50}个.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295736,7 +295736,7 @@
"content": "平面上的三个单位向量$\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$满足$2\\overrightarrow{c}=3\\overrightarrow{a}+4\\overrightarrow{b}$, 则$\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$两两间的夹角中, 最小的角的大小为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295755,7 +295755,7 @@
"content": "设区间$(m,n)$($m<n$)的长度为$n-m$. 若$a,b\\in (0,+\\infty)$, 且不等式$(x^2-a)(x^2-b)$的解集是若干个长度之和为$4$的区间的并集, 则$\\dfrac{a+5}{b}$的最小值为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "",
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
@ -295774,7 +295774,7 @@
"content": "已知$a,b$是实数, 则``$a>b$''是``$a^3+1>b^3+1$''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,
@ -295793,7 +295793,7 @@
"content": "演讲比赛共有$9$位评委分别给出某选手的原始评分, 评定该选手的成绩时, 从$9$个原始评分中去掉$1$个最高分、$1$个最低分, 得到$7$个有效评分. $7$个有效评分与$9$个原始评分相比, 不变的数字特征是\\bracket{20}.\n\\fourch{中位数}{平均数}{方差}{极差}",
"objs": [],
"tags": [],
"genre": "",
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
@ -295812,7 +295812,7 @@
"content": "已知$\\omega$是常数, 若函数$y=|\\sin (\\omega x+\\dfrac \\pi 3)|$图像的一条对称轴是直线$x=\\dfrac\\pi 6$. 则$\\omega$的值不可能在区间\\bracket{20}中.\n\\fourch{$(0,2]$}{$(2,4]$}{$(4,6]$}{$(6,8]$}",
"objs": [],
"tags": [],
"genre": "",
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
@ -295831,7 +295831,7 @@
"content": "对于两个定义在$\\mathbf{R}$上的函数$y=f(x)$与$y=g(x)$, 构造新的函数$y=h(x)$如下: 对任意$x_0\\in \\mathbf{R}$, $h(x_0)=f(x_0)+g(x_0)$. 现已知$y=h(x)$是严格增函数, 对于以下两个命题: \n\\textcircled{1} $y=f(x)$与$y=g(x)$中至少有一个是严格增函数;\n\\textcircled{2} $y=f(x)$与$y=g(x)$中至少有一个无最大值.\n其中\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都是真命题}{只有\\textcircled{1}是真命题}{只有\\textcircled{2}是真命题}{没有真命题}",
"objs": [],
"tags": [],
"genre": "",
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
@ -295850,7 +295850,7 @@
"content": "如图, 设$P-ABCD$是底面为矩形的四棱锥, $PA\\perp$平面$ABCD$. $PA=AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [left] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- (P) (P) -- (C);\n\\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PC\\perp BD$, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若直线$PD$与平面$PAB$所成的角的大小为$\\arctan 2$, 求直线$PC$与平面$ABCD$所成的角的大小.",
"objs": [],
"tags": [],
"genre": "",
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
@ -295862,14 +295862,14 @@
"same": [],
"related": [],
"remark": "",
"space": ""
"space": "12ex"
},
"012092": {
"id": "012092",
"content": "设$a$是实常数, 并记$f(x)=x^3+ax^2+2x$.\\\\\n(1) 当$a=-\\dfrac{5}{2}$时, 求函数$y=f(x)$的单调减区间;\\\\ \n(2) 是否存在$a$, 使得函数$y=f(x)$在实数范围内有且仅有三个零点, 且三个零点可按某种顺序排列后成等差数列? 若存在, 求所有满足条件的$a$的值; 若不存在, 说明理由.",
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"genre": "解答题",
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@ -295881,14 +295881,14 @@
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"space": "12ex"
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"012093": {
"id": "012093",
"content": "如图, 某市郊外景区内一条笔直的公路$a$经过三个景点$A$、$B$、$C$. 景区管委会又开发了风景优美的景点$D$.经测量景点$D$位于景点$A$的北偏东$30^\\circ$方向$8$千米处, 且位于景点$B$的正北方向, 还位于景点$C$的北偏西$75^\\circ$方向上.已知$AB=5$千米.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.25]\n \\draw [->] (6,8) -- (10,8) node [right] {东};\n \\draw [->] (6,8) -- (6,12) node [above] {北};\n \\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {$N$} coordinate (N);\n \\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n \\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n \\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n \\path [name path = DC] (D) --++ (-15:4);\n \\path [name intersections = {of = linea and DC, by = C}];\n \\draw (D) -- (C) node [below] {$C$};\n \\draw (60:2) arc (60:90:2);\n \\draw (75:4) node {$30^\\circ$};\n \\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点$D$向景点$B$修一条笔直的公路, 不考虑其他因素, 求出这条公路的长(结果精确到$0.1$千米);\\\\\n(2) 求景点$C$与景点$D$之间的距离(结果精确到$0.1$千米).",
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"genre": "解答题",
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@ -295900,14 +295900,14 @@
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"012094": {
"id": "012094",
"content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2^n+\\lambda n$, 其中常数$\\lambda\\in \\mathbf{R}$.\\\\\n(1) 若$a_3=4a_2$, 求$\\lambda$的值;\\\\\n(2) 若$\\{a_n\\}$前$10$项的和为$1551$, 试分析$\\{a_n\\}$的单调性;\\\\\n(3) 对于常数$t$, 记集合$C_t=\\{n|a_n=t\\}$, 试求当$\\lambda$与$t$变化时, 集合$C_t$中元素个数的最大值.",
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"genre": "解答题",
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@ -295919,14 +295919,14 @@
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"space": "12ex"
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"012095": {
"id": "012095",
"content": "已知椭圆$E$的方程为$\\dfrac{x^2}{12}+\\dfrac{y^2}{4}=1$, $F_1(-2\\sqrt{2},0)$与$F_2(2\\sqrt{2},0)$是$E$的两个焦点, $A(0,-2)$是$E$的下顶点.\\\\\n(1) 设斜率为$1$的直线$l$过点$F_1$, 且与$E$交于$M,N$两点, 求弦$MN$的长;\\\\\n(2) 若$E$上一点$P$满足$|F_1P|=3|F_2P|$, 求$\\triangle F_1F_2P$的面积;\\\\\n(3) 是否存在椭圆$E$上, 且位于第一象限的点$Q$, 使得射线$QA$平分$\\angle F_1QF_2$? 若存在, 请写出一个满足条件的点$Q$的坐标并加以验证; 若不存在, 说明理由.",
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"genre": "解答题",
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"duration": -1,
@ -295938,7 +295938,7 @@
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"space": "12ex"
},
"020001": {
"id": "020001",
@ -315090,5 +315090,176 @@
],
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"space": "12ex"
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"030485": {
"id": "030485",
"content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 哪些棱所在的直线与直线$BD_1$是异面直线?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\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 [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 [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}",
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"030486": {
"id": "030486",
"content": "如图, 点$P$是矩形$ABCD$所在平面外的一点, $M$、$N$分别是$AB$、$PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\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 (D) ++ (0.2,1.5,0) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (P) -- (D);\n\\draw ($(P)!0.5!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (M) -- (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$平面$PAD$;\\\\ \n(2) 若$\\triangle PAD$是等边三角形, 求异面直线$MN$与$BC$所成的角的大小;\\\\ \n(3) 设$Q$是线段$DC$上的一点, 若平面$PAD\\parallel$平面$MNQ$, 求点$Q$的位置.",
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"030487": {
"id": "030487",
"content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $E$为$DD_1$上一点. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\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 ($(D)!0.5!(D1)$) node [left] {$E$} coordinate (E);\n\\draw [dashed] (A) -- (C) (B) -- (D);\n\\draw [dashed] (A) -- (E) -- (C);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$D_1DBB_1\\perp$平面$EAC$;\\\\\n(2) 若$DE=2$, 求$AE$与平面$D_1DBB_1$所成的角的大小;\\\\\n(3) 若$E$为$DD_1$的中点, 且$B_1D\\perp$平面$EAC$, 求$DD_1$的长度.",
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"030488": {
"id": "030488",
"content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=3$, $BC=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{4}\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\\end{tikzpicture}\n\\end{center}\n(1) 求点$A_1$到平面$B_1BDD_1$的距离;\\\\\n(2) 若$A_1B$和平面$B_1BDD_1$所成的角的大小为$\\arcsin \\dfrac{12}{25}$, 求$AA_1$的长度.",
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"030489": {
"id": "030489",
"content": "如图, 在斜三棱柱$ABC-A_1B_1C_1$中, $\\angle A_1AC=\\angle ACB=\\dfrac{\\pi }2$, $\\angle AA_1C=\\dfrac{\\pi }6$, 侧棱$BB_1$与底面所成的角为$\\dfrac{\\pi }3$, $AA_1=4\\sqrt 3$, $BC=4$. 求斜三棱柱$ABC-A_1B_1C_1$的体积$V$. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw ({-2*sqrt(2)},0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-2*sqrt(2)}) node [above right] {$C$} coordinate (C);\n\\draw (A) -- (B);\n\\draw [dashed] (A) -- (C) -- (B);\n\\draw (A) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed] (C) -- (C1) (A1) -- (C);\n\\draw (A) -- (A1) (B) -- (B1) (A1) -- (B1) (A1) -- (C1) -- (B1); \n\\end{tikzpicture}\n\\end{center}",
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"space": "12ex"
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"030490": {
"id": "030490",
"content": "如图, 圆锥$P-O$的底面直径和高均是$a$, 过$PO$的中点$O'$作平行于底面的截面, 以该截面为底面挖去一个圆柱, 求剩下几何体的体积和表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (0,1) node [left] {$O'$} coordinate (O1);\n\\draw (P) -- (-1,0) (P) -- (1,0);\n\\draw [dashed] (P) --++ (0,-2) --++ (1,0);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25) (-0.5,1) arc (180:0:0.5 and 0.125);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-0.5,0) --++ (0,1) (0.5,0) --++ (0,1);\n\\draw (-0.5,1) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (O) ellipse (0.5 and 0.125);\n\\end{tikzpicture}\n\\end{center}",
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"030491": {
"id": "030491",
"content": "如图, 三棱锥$P-MNQ$中, $PM\\perp NQ$, $PM\\perp MN$, $NQ\\perp MN$. 若$MN=NQ=1$, 二面角$P-NQ-M$的大小为$\\dfrac{\\pi }4$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$M$} coordinate (M);\n\\draw ({sqrt(2)},0,0) node [right] {$Q$} coordinate (Q);\n\\draw ({sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$N$} coordinate (N);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (M) -- (N) -- (Q) (P) -- (M) (P) -- (N) (P) -- (Q);\n\\draw [dashed] (M) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥$P-MNQ$的体积;\\\\\n(2) 点$M$到平面$PNQ$的距离. \n(3) 若点$E$为棱$PN$的中点, 点$F$为棱$PQ$的中点, 那么三棱锥$M-EFN$的体积是多少?",
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"030492": {
"id": "030492",
"content": "已知四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为2, 且$\\angle ADC=\\dfrac{2\\pi }3$. 设$\\overrightarrow{DA}=\\overrightarrow{a}$, $\\overrightarrow{DC}=\\overrightarrow{b}$, $\\overrightarrow{DD_1}=\\overrightarrow{c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0.5,{sqrt(14)/2},-0.5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0.5,{sqrt(14)/2},-0.5) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0.5,{sqrt(14)/2},-0.5) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0.5,{sqrt(14)/2},-0.5) 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\\end{tikzpicture}\n\\end{center}\n(1)设$E$是棱$A_1D_1$的中点.\\\\\n\\textcircled{1} 试用$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$的线性组合表示$\\overrightarrow{EB}$;\\\\\n\\textcircled{2} 若$\\angle ADD_1=\\angle CDD_1=\\alpha$, $\\alpha \\in (\\dfrac{\\pi }3,\\dfrac{\\pi }2]$, 求$|\\overrightarrow{EB}|$的取值范围;\\\\\n(2)求证: 当且仅当$\\angle ADD_1=\\angle CDD_1$时, $AC\\perp$平面$DBB_1D_1$.",
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"030493": {
"id": "030493",
"content": "已知直四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为$2$, 且$\\angle ADC=\\dfrac{2\\pi }3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,2,0) 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 ($(A1)!0.5!(D1)$) node [above left] {$E$} coordinate (E);\n\\draw ($(A)!{2/3}!(B)$) node [below] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F);\n\\end{tikzpicture}\n\\end{center}\n(1)设$E$是$A_1D_1$中点, 点$F$满足$\\overrightarrow{AF}=2\\overrightarrow{FB}$.\\\\\n\\textcircled{1} 求异面直线$EF$与$DD_1$所成角的大小;\\\\\n\\textcircled{2} 求直线$EF$与平面$DBB_1D_1$所成角的大小;\\\\ \n(2)求平面$DBB_1D_1$与平面$BDC_1$所成锐二面角的大小;\\\\\n(3)求四面体$A_1C_1BD$的体积.",
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