From 3a6eb715c51c236a6bd8b0f89313e668639f36fd Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Tue, 6 Jun 2023 11:24:41 +0800 Subject: [PATCH] =?UTF-8?q?=E6=94=B6=E5=BD=95=E7=82=B9=E8=A6=81=E5=8D=95?= =?UTF-8?q?=E5=85=83=E6=B5=8B=E8=AF=9520=E4=BB=BD=E9=A2=98=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/批量收录题目.py | 10 +- 题库0.3/Problems.json | 6280 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 6285 insertions(+), 5 deletions(-) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index ceb0b55e..f7246108 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,9 +1,9 @@ #修改起始id,出处,文件名 -starting_id = 40862 -raworigin = "" -filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目13.tex" -editor = "20230605\t王伟叶" -indexed = False +starting_id = 17551 +raworigin = "高中数学质量测试与监控单元知识测试" +filename = r"C:\Users\weiye\Documents\wwy sync\待整理word题目\质量测试与监控.tex" +editor = "20230606\t王伟叶" +indexed = True IndexDescription = "试题" import os,re,json diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 2a7e9a2d..da08e8b5 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -453018,6 +453018,6286 @@ "space": "4em", "unrelated": [] }, + "017551": { + "id": "017551", + "content": "已知集合$A=\\{x | x<-1$或$2 \\leq x<3\\}$, $B=\\{x |-2 \\leq x<4\\}$, 则$A \\cup B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017552": { + "id": "017552", + "content": "若集合$A=\\{x | x \\leq 2\\}$与集合$B=\\{x | x \\geq a\\}$满足$A \\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017553": { + "id": "017553", + "content": "集合$A=\\{0,2, a\\}$, $B=\\{1, a^2\\}$, 若$A \\cup B=\\{0,1,2,4,16\\}$, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017554": { + "id": "017554", + "content": "满足条件$M \\cup\\{1\\}=\\{1,2,3\\}$的集合$M$的个数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017555": { + "id": "017555", + "content": "若全集$U=\\mathbf{R}$, $f(x)$、$g(x)$均为$x$的二次函数, $P=\\{x | f(x)<0\\}$, $Q=\\{x|g(x)| \\geq 0\\}$, 则不等式组$\\begin{cases}f(x)<0, \\\\ g(x)<0\\end{cases}$的解集可用$P$、$Q$表示为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017556": { + "id": "017556", + "content": "对于集合$A$、$B$, 定义两种运算``$-$''和``$\\oplus$'': $A-B=\\{x | x \\in A$且$x \\notin B\\}$, $A \\oplus B=(A-B) \\cup(B-A)$, 设$A=\\{y | y=x^2-3 x,\\ x \\in \\mathbf{R}\\}$, $B=\\{y | y=-2^x,\\ x \\in \\mathbf{R}\\}$, 则$A \\oplus B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017557": { + "id": "017557", + "content": "设曲线$C_1$和$C_2$的方程分别为$F_1(x, y)=0$和$F_2(x, y)=0$, 则点$P(a, b) \\notin C_1 \\cap C_2$的一个充分条件为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017558": { + "id": "017558", + "content": "已知函数$y=f(x)$, $x \\in[a, b]$, 那么集合$\\{(x, y) | y=f(x), x \\in[a, b]\\} \\cap\\{(x, y) | x=2\\}$中元素的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017559": { + "id": "017559", + "content": "若集合$A=\\{x | y=\\sqrt{2 x-1}\\}$, $B=\\{x|| x \\leq 1\\}$, 则$A \\cap B$是\\bracket{20}.\n\\fourch{$\\{x | \\dfrac{1}{2} \\leq x \\leq 1\\}$}{$\\{x | x \\leq-1\\}$}{$\\{x |-1 \\leq x \\leq \\dfrac{1}{2}\\}$}{$\\{x | x \\geq 1\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017560": { + "id": "017560", + "content": "函数$f(x)=x^2+m x+1$的图像关于直线$x=1$对称的充要条件是\\bracket{20}.\n\\fourch{$m=-2$}{$m=2$}{$m=-1$}{$m=1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017561": { + "id": "017561", + "content": "定义集合的一种运算``$\\ast$'': $A \\ast B=\\{z | z=x y,\\ x \\in A,\\ y \\in B\\}$. 设$A=\\{1,2\\}$, $B=\\{0,2\\}$, 则集合$A \\ast B$的所有元素之和为\\bracket{20}.\n\\fourch{$0$}{$2$}{$3$}{$6$}", 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"edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017566": { + "id": "017566", + "content": "设集合$A=\\{x | x=m^2-n^2,\\ m \\in \\mathbf{Z},\\ n \\in \\mathbf{Z}\\}$.\\\\\n(1) 求证: $11 \\in A$, $12 \\in A$, $2 k+1 \\in A$($k \\in \\mathbf{Z}$);\\\\\n(2) 用反证法证明: $10 \\notin \\mathbf{A}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试01集合与逻辑单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017567": { + "id": "017567", + "content": "设非空数集$S$满足: 若$a$、$b \\in S$, 则$a+b \\in S$, $a-b \\in S$, 则称集合$S$为闭集合. 如整数集$\\mathbf{Z}$, 有理数集$\\mathbf{Q}$等都是闭集合.\\\\\n(1) 写出一个闭集合$S$, 要求满足$S \\subset \\mathbf{R}$, 且$S \\neq \\mathbf{Z}$, $S \\neq \\mathbf{Q}$. 请加以证明;\\\\\n(2) 求证: 对于任意两个满足$S_1 \\subset \\mathbf{R}$, $S_2 \\subset \\mathbf{R}$的闭集合$S_1$、$S_2$, 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"same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017570": { + "id": "017570", + "content": "不等式$\\dfrac{x-1}{x+2}>1$的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017571": { + "id": "017571", + "content": "已知集合$A=\\{x | \\dfrac{x-7}{3-x}>0\\}$, 函数$y=\\lg (-x^2+6 x-8)$的定义域为集合$B$, 那么集合$A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017572": { + "id": "017572", + "content": "已知$a, b \\in \\mathbf{R}$, $a>b$且$a b=1$, 则$\\dfrac{a^2+b^2}{a-b}$的最小值等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017573": { + "id": "017573", + "content": "若$a>0$, 则不等式$-a<\\dfrac{1}{x}<3$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017574": { + "id": "017574", + "content": "若对于任意的$x>0$, 不等式$\\dfrac{x}{x^2+3 x+1} \\leq a$恒成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017575": { + "id": "017575", + "content": "已知实数$x>0$, $y>0$, 且$x+y=2$, 则$\\dfrac{2}{x}+\\dfrac{x}{2 y}$的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017576": { + "id": "017576", + "content": "设$a>b>0$, $m>0$, 记$x=\\dfrac{b}{a}$, $y=\\dfrac{b+m}{a+m}$, 则\\bracket{20}.\n\\fourch{$x>y$}{$x \\geq y$}{$x1$($a>0$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017579": { + "id": "017579", + "content": "如图所示, 某公园要在一块绿地的中央修建两个相同的矩形的池塘, 每个面积为$10000$平方米, 池塘前方要留$4$米宽的走道, 其余各方为$2$米宽的走道, 问每个池塘的长宽各为多少米时, 池塘的占地总面积最少?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) rectangle (2,2.5) (3,2.5) rectangle (5,0);\n\\draw (-1,-2) rectangle (6,3.5);\n\\draw (1,3) node {走道$2$米} (4,3) node {走道$2$米};\n\\draw (1,-1) node {走道$4$米} (4,-1) node {走道$4$米};\n\\draw (1,1.25) node {池塘} (4,1.25) node {池塘};\n\\draw (-0.5,1.25) node [rotate = 90] {走道$2$米} (2.5,1.25) node [rotate = 90] {走道$2$米} (5.5,1.25) node [rotate = 90] {走道$2$米};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017580": { + "id": "017580", + "content": "若$a>0, b>0$, 且$\\dfrac{1}{a}+\\dfrac{1}{b}=\\sqrt{a b}$.\\\\\n(1) 求$a^3+b^3$的最小值;\\\\\n(2) 是否存在$a, b$, 使得$2 a+3 b=6$成立, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017581": { + "id": "017581", + "content": "已知关于$x$的不等式$|x^2-4 x+a|+|x-3| \\leq 5$的解集为$M$, 且$M \\subseteq(-\\infty, 3]$. 求整数$a$的值, 并解此不等式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017582": { + "id": "017582", + "content": "已知函数$y=\\dfrac{x^2}{a x+b}(a$、$b$为常数), 且关于$x$的方程$y-x+12=0$有两个实根$x_1=3$、$x_2=4$.\\\\\n(1) 求函数$y$的表达式;\\\\\n(2) 设$k>1$, 解关于$x$的不等式$y<\\dfrac{(k+1) x-k}{2-x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试02等式与不等式单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017583": { + "id": "017583", + "content": "若$f(x)=\\log _{(\\sqrt{2}-1)} x$, 则$f(3+2 \\sqrt{2})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017584": { + "id": "017584", + "content": "如果$\\lg 108=a$, $\\lg 72=b$, 则$\\lg 48=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017585": { + "id": "017585", + "content": "函数$y=4 \\times 2^x$的图像可以由函数$y=2^x$的图像向\\blank{50}平移\\blank{50}个单位得到.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017586": { + "id": "017586", + "content": "函数$y=(a^2-1)^x$是单调减函数, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017587": { + "id": "017587", + "content": "函数$f(x)=a^x$($a>0$, $a \\neq 1$)在$[1,2]$中的最大值比最小值大$\\dfrac{a}{2}$, 则$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017588": { + "id": "017588", + "content": "已知$x^2+y^2-4 x-2 y+5=0$, 则$\\log _x(y^x)$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017589": { + "id": "017589", + "content": "关于$x$的方程$5^x=\\dfrac{a+3}{5-a}$有负根, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017590": { + "id": "017590", + "content": "当$a>0$且$a \\neq 1$时, 不论$a$为何值, 函数$y=a^{x-1}+1$的图像都通过定点\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017591": { + "id": "017591", + "content": "已知函数$f(x)=\\begin{cases}3^x, & x \\leq 0, \\\\ \\log _2 x,& x>0,\\end{cases}$那么$f(f(\\dfrac{1}{4}))$的值为\\bracket{20}.\n\\fourch{$9$}{$\\dfrac{1}{9}$}{$-9$}{$-\\dfrac{1}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017592": { + "id": "017592", + "content": "方程$\\log _2(x+4)=2^x$的根的情况是\\bracket{20}.\n\\fourch{仅有一根}{有两个正根}{有一正根和一个负根}{有两个负根}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017593": { + "id": "017593", + "content": "解方程$\\lg (2 x^2-2 x)=\\lg (x^2-5 x+4)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017594": { + "id": "017594", + "content": "设函数$f(x)=2^{|x+1|-|x-1|}$, 求使$f(x) \\geq 2 \\sqrt{2}$的$x$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": 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+ "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试03幂函数指数函数和对数函数单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017597": { + "id": "017597", + "content": "在\\textcircled{1} $y=x$与$y=\\sqrt{x^2}$; \\textcircled{2} $y=(\\sqrt{x})^2$与$y=\\sqrt{x^2}$; \\textcircled{3} $y=|x|$与$y=\\dfrac{x^2}{x}$; \\textcircled{4} $y=|x|$与$y=\\sqrt{x^2}$; \\textcircled{5} $y=x^0$与$y=1$五组函数中, 表示相同函数的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017598": { + "id": "017598", + "content": "函数$y=\\lg (-6 x^2+13 x-6)$的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017599": { + "id": "017599", + "content": "已知偶函数$f(x)$, 当$x>0$时, 解析式为$f(x)=x^2-x$, 则当$x<0$时, $f(x)$的解析式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017600": { + "id": "017600", + "content": "函数$y=\\lg (x^2+x-a)$的定义域为$\\mathbf{R}$, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017601": { + "id": "017601", + "content": "已知函数$f(x)=x^2-(a+2) x+2-a$, 若集合$A=\\{x | f(x)<0, x \\in \\mathbf{Z}\\}$有且只有一个元素, 则正实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017602": { + "id": "017602", + "content": "函数$f(x)=\\lg \\dfrac{1-x}{1+x}$是\\bracket{20}.\n\\twoch{奇函数}{偶函数}{既是奇函数又是偶函数}{非奇非偶函数}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017603": { + "id": "017603", + "content": "若函数$y=f(x)$在$[a, b]$上是单调函数, 则使得$y=f(x+3)$必为单调函数的区间是\\bracket{20}.\n\\fourch{$[a, b+3]$}{$[a+3, b+3]$}{$[a-3, b-3]$}{$[a+3, b]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017604": { + "id": "017604", + "content": "若定义在闭区间$[-a, a]$($a>0$)上的函数$y=f(x)$为奇函数, 则下列各图中可以成为它的图像的是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,0.5) -- (1,1.5) (0,-0.5) -- (-1,-1.5);\n\\filldraw (0,-0.5) circle (0.05);\n\\filldraw [fill = white] (0,0.5) circle (0.05);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw 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方程$f(x)-x=0$的两个根$x_1, x_2$满足: $00$恒成立, 求$m$的取值范围;\\\\\n(3) 讨论函数$y=f(x)$的零点个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试04函数的基本性质单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017611": { + "id": "017611", + "content": "若$\\cos (\\theta+\\dfrac{\\pi}{3})=1$, 则$\\cos \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017612": { + "id": "017612", + "content": "若$\\sin \\theta=k \\cos \\theta$, 则$\\sin \\theta \\cdot \\cos \\theta$的值等于 (用$k$表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], 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则该弧所对的圆心角与原圆心角的比值为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$2$}{$\\dfrac{1}{3}$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017618": { + "id": "017618", + "content": "若$\\tan \\theta$和$\\tan (\\dfrac{\\pi}{4}-\\theta)$是方程$x^2+p x+q=0$的两个根, 则$p$与$q$之间的关系是\\bracket{20}.\n\\fourch{$p+q+1=0$}{$p-q+1=0$}{$p+q-1=0$}{$p-1-1=0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017619": { + "id": "017619", + "content": "若$\\triangle ABC$的三个内角的余弦值分别等于$\\triangle A' B' C'$的三个内角的正弦值, 则可确定\\bracket{20}.\n\\onech{$\\triangle ABC$和$\\triangle A' B' C'$都是锐角三角形}{$\\triangle ABC$和$\\triangle A' B' C'$都是钝角三角形}{$\\triangle ABC$是锐角三角形, $\\triangle A' B' C'$为钝角三角形}{$\\triangle ABC$是钝角三角形, $\\triangle A' B' C'$为锐角三角形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017620": { + "id": "017620", + "content": "设函数$f(x)=\\alpha \\sin x+b \\cos x$, 其中$a>0$, $b>0$, 若$f(x) \\leq f(\\dfrac{\\pi}{4})$对任意的$x \\in \\mathbf{R}$恒成立, 则下列说法正确的是\\bracket{20}.\n\\onech{$f(\\dfrac{\\pi}{2})>f(\\dfrac{\\pi}{6})$}{$f(x)$的图像关于直线$x=\\dfrac{3 \\pi}{4}$对称}{$f(x)$在$[\\dfrac{\\pi}{4}, \\dfrac{5 \\pi}{4}]$上单调增}{过点$(a, b)$的直线与函数$f(x)$的图像必有公共点}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017621": { + "id": "017621", + "content": "已知$\\sin (\\dfrac{\\pi}{4}-x)=\\dfrac{5}{13}$, $0=latex, scale = 0.2]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (16,0) node [below] {$B$} coordinate (B);\n\\path [name path = arca] (A) ++ (15,0) arc (0:35:15);\n\\path [name path = arcb] (B) ++ (-9,0) arc (180:110:9);\n\\path [name intersections = {of = arca and arcb, by = P}] (P) node [above] {$P$};\n\\draw (A)--(B)--(P)--cycle;\n\\draw ($(A)!0.5!(B)$) node [below] {居民生活区};\n\\draw [->] (B)++(0,5) --++ (0,3) node [midway, right] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 当$AP=15 \\text{km}$时, 求$\\angle APB$的值;\\\\\n(2) 发电厂尽量远离居民区, 要求$\\triangle PAB$的面积最大, 问此时发电厂与两个垃圾中转站的距离各为多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试05三角单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017625": { + "id": "017625", + "content": "函数$y=\\lg (\\sin x-\\cos x)$的定义域是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017626": { + "id": "017626", + "content": "函数$f(x)=\\cos ^2 x+\\dfrac{\\sqrt{3}}{2} \\sin 2 x$, $x \\in \\mathbf{R}$的单调增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017627": { + "id": "017627", + "content": "已知函数$y=\\sin (a x+2)$($x \\in \\mathbf{R}$)的最小正周期是$\\pi$, 则实数$a$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017628": { + "id": "017628", + "content": "已知函数$f(x)=\\sin x+a \\cos ^2 \\dfrac{x}{2}$($a$为常数, $a \\in \\mathbf{R}$), 且$x=\\dfrac{\\pi}{2}$是方程$f(x)=0$的解. 当$x \\in[0, \\pi]$时, 函数$f(x)$值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017629": { + "id": "017629", + "content": "已知函数$y=\\tan (\\omega x+\\dfrac{\\pi}{6})$的图像关于点$(\\dfrac{\\pi}{3}, 0)$对称, 且$|\\omega| \\leq 1$, 则实数$\\omega$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017630": { + "id": "017630", + "content": "函数$f(x)=\\sin x$, 对于$x_10$)在区间$[0, \\pi]$上至少有两个不同的解, 求$\\omega$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017636": { + "id": "017636", + "content": "如图, 某公园有一块直角三角形$ABC$的空地, 其中$\\angle ACB=\\dfrac{\\pi}{2}$, $\\angle ABC=\\dfrac{\\pi}{6}$, $AC$长$\\alpha$千米, 现要在空地上围出一块正角形区域$DEF$建文化景观区, 其中$D$、$E$、$F$分别在$BC$、$AC$、$AB$上, 设$\\angle DEC=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$E$} coordinate (E);\n\\draw (E) ++ (70:1) node [right] {$F$} coordinate (F);\n\\draw (E) ++ (130:1) node [left] {$D$} coordinate (D);\n\\draw (D)--(E)--(F)--cycle;\n\\path [name path = BC] (D) ++ (0,1.9) --++ (0,-2.8);\n\\path [name path = AC] (E) ++ (1,0) --++ (-1.7,0);\n\\path [name path = AB] (F) ++ (120:2) --++ (120:-3.1);\n\\path [name intersections = {of = BC and AC, by = C}] (C) node [below left] {$C$};\n\\path [name intersections = {of = AB and AC, by = A}] (A) node [below right] {$A$};\n\\path [name intersections = {of = AB and BC, by = B}] (B) node [above] {$B$};\n\\draw (A)--(B)--(C)--cycle;\n\\draw (E) pic [draw, scale = 0.4] {angle = D--E--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta=\\dfrac{\\pi}{3}$, 求$\\triangle DEF$的边长;\\\\\n(2) 当$\\theta$多大时, $\\triangle DEF$的边长最小? 并求出最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017637": { + "id": "017637", + "content": "已知函数$f(x)=2 \\sqrt{3} \\sin x \\cos x+2 \\cos ^2 x-1$($x \\in \\mathbf{R}$).\\\\\n(1) 求函数$f(x)$的最小正周期及在区间$[0, \\dfrac{\\pi}{2}]$上的最大值和最小值;\\\\\n(2) 若$f(x_0)=\\dfrac{6}{5}$, $x_0 \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$, 求$\\cos 2 x_0$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017638": { + "id": "017638", + "content": "对于三个数$a$、$b$、$c$能构成三角形的三边, 则称这三个数为``三角形数''. 对于``三角形数''$a$、$b$、$c$, 如果函数$y=f(x)$使得三个数$f(a)$、$f(b)$、$f(c)$仍为``三角形数'', 则称$y=f(x)$为这三个数的``保三角形函数''.\\\\\n(1) 对于``三角形数''$\\alpha$、$2 \\alpha$、$\\dfrac{\\pi}{4}+\\alpha$, 其中$\\dfrac{\\pi}{8}<\\alpha<\\dfrac{\\pi}{4}$且$\\tan \\alpha=p$, 判断函数$f(x)=\\tan x$是否是这三个数的``保三角形函数'', 并说明理由;\\\\\n(2) 对于``三角形数''$\\alpha$、$\\alpha+\\dfrac{\\pi}{6}$、$\\alpha+\\dfrac{\\pi}{3}$, 其中$\\dfrac{\\pi}{6}<\\alpha<\\dfrac{7 \\pi}{12}$, 判断函数$f(x)=\\sin x$是否是这三个数的``保三角形函数'', 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试06三角函数单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017639": { + "id": "017639", + "content": "已知$\\overrightarrow {a}=3 \\overrightarrow {i}+\\overrightarrow {j}$, $\\overrightarrow {b}=x \\overrightarrow {i}-6 \\overrightarrow {j}$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则实数$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017640": { + "id": "017640", + "content": "已知向量$|\\overrightarrow {a}|=3$, $\\overrightarrow {b}=(1,2)$, 且$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$\\overrightarrow {a}$的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017641": { + "id": "017641", + "content": "已知$\\triangle ABC$是边长为$1$的等边三角形, 点$D$、$E$分别是边$AB$、$BC$的中点, 联接$DE$并延长到点$F$, 使得$DE=2EF$, 则$\\overrightarrow{AF} \\cdot \\overrightarrow{BC}$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017642": { + "id": "017642", + "content": "已知平行四边形$ABCD$的三个顶点$A(0,0)$, $B(3,1)$, $C(4,1)$, 则$D$点的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017643": { + "id": "017643", + "content": "已知向量$\\overrightarrow {a}=(\\cos \\theta, \\sin \\theta)$, 向量$\\overrightarrow {b}=(\\sqrt{3},-1)$, 则$|\\overrightarrow {a}-\\overrightarrow {b}|$的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017644": { + "id": "017644", + "content": "已知点$A(-1,1)$和坐标原点$O$, 若点$M(x, y)$为平面区域$\\begin{cases}x+y \\geq 2, \\\\ x \\leq 1, \\\\ y \\leq 2\\end{cases}$上的一个动点, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{OM}$的取值范围是\\bracket{20}.\n\\fourch{$[-1,0]$}{$[0,1]$}{$[0,2]$}{$[-1,2]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017645": { + "id": "017645", + "content": "将函数$y=\\log _3(2 x+1)-4$的图像按向量$\\overrightarrow {a}$平移后得到的是函数$y=\\log _3^{2 x}$的图像, 则$\\overrightarrow {a}$的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017646": { + "id": "017646", + "content": "在等腰梯形$ABCD$中, 已知$AB\\parallel DC$, $AB=2$, $BC=1$, $\\angle ABC=60^{\\circ}$, 动点$E$和$F$分别在线段$BC$和$DC$上, 且$\\overrightarrow{BE}=\\lambda \\overrightarrow{BC}$, $\\overrightarrow{DF}=\\dfrac{1}{9 \\lambda} \\overrightarrow{DC}$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{AF}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2.5]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (60:1) node [above] {$D$} coordinate (D);\n\\draw (B) ++ (120:1) node [above] {$C$} coordinate (C);\n\\draw ($(B)!0.8!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(D)!{0.8/9}!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw [->] (A)--(E);\n\\draw [->] (A)--(F);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017647": { + "id": "017647", + "content": "设$\\overrightarrow {a}$与$\\overrightarrow {b}$为非零向量, 给出下列命题:\\\\\n\\textcircled{1} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$平行, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$向量的方向相同或相反;\\\\\n\\textcircled{2} 若$\\overline{AB}=\\overrightarrow {a}, \\overline{CD}=\\overrightarrow {b}, \\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$A$、$B$、$C$、$D$四点必在一条直线上;\\\\\n\\textcircled{3} 若$\\overrightarrow {a}$与$\\overrightarrow {b}$共线, 则$|\\overrightarrow {a}|+|\\overrightarrow {b}|=|\\overrightarrow {a}+\\overrightarrow {b}|$;\\\\\n\\textcircled{4}若$\\overrightarrow {a}$与$\\overrightarrow {b}$反向, 则$\\overrightarrow {a}=-\\dfrac{|\\overrightarrow {a}|}{|\\overrightarrow {b}|} \\cdot \\overrightarrow {b}$.\\\\\n其中正确命题的个数有\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017648": { + "id": "017648", + "content": "设$\\overrightarrow{x_0}, \\overrightarrow{x_1}, \\overrightarrow{x_2}$为平面上的三个向量, 且满足$\\overrightarrow{x_0}\\cdot \\overrightarrow{x_1}=\\dfrac{1}{k}$, $\\overrightarrow{x_1}\\cdot\\overrightarrow{x_k}=\\dfrac{1}{k+1}$, $\\overrightarrow{x_2} \\cdot\\overrightarrow{x_k}=\\dfrac{1}{k+2}$($k=1,2$), 则能使$a \\overrightarrow{x_1}+b \\overrightarrow{x_2}=\\overrightarrow{x_0}$成立的常数$a$、$b$的值是\\bracket{20}.\n\\fourch{$a=6$, $b=6$}{$a=-6$, $b=6$}{$a=6$, $b=-6$}{$a=-6$, $b=-6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017649": { + "id": "017649", + "content": "已知非零向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {c}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''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}{既不充分又不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017650": { + "id": "017650", + "content": "定义: 如果一个向量列从第二项起, 每一项与它的前一项的差都等于同一个常向量, 那么这个向量列叫做等差向量列, 这个常向量叫做等差向量列的公差.\n已知向量列$\\{\\overrightarrow {a}_n\\}$是以$\\overrightarrow{a_1}=(1,3)$为首项, 公差$\\overrightarrow {d}=(1,0)$的等差向量列. 若向量$\\overrightarrow{a_n}$与非零向量$\\overrightarrow{b_n}=(x_n, x_{n+1})$($n \\in \\mathbf{N}$, $n \\geq 1$)垂直, 则$\\dfrac{x_{10}}{x_1}=$\\bracket{20}.\n\\fourch{$\\dfrac{44800}{729}$}{$\\dfrac{4480}{243}$}{$-\\dfrac{44800}{729}$}{$-\\dfrac{4480}{243}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017651": { + "id": "017651", + "content": "已知$\\overrightarrow {p}=(2,-3)$, $\\overrightarrow {q}=(1,2), \\overrightarrow {a}=(9,4)$, 且$\\overrightarrow {a}=m \\cdot \\overrightarrow {p}+n \\cdot \\overrightarrow {q}$, 求实数$m$、$n$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017652": { + "id": "017652", + "content": "平面内有向量$\\overrightarrow{OA}=(1,7)$, $\\overrightarrow{OB}=(5,1)$, $\\overrightarrow{OP}=(2,1)$, 点$X$为直线$OP$上的一个动点.\\\\\n(1) 当$\\overrightarrow{XA} \\cdot \\overrightarrow{XB}$取最小值时, 求$\\overrightarrow{OX}$的坐标;\\\\\n(2) 当点$X$满足 (1) 的条件和结论时, 求$\\cos \\angle AXB$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017653": { + "id": "017653", + "content": "已知向量$\\overrightarrow {a}=(\\cos x, \\sin x)$, $\\overrightarrow {b}=(3,-\\sqrt{3})$, $x \\in[0, \\pi]$.\\\\\n(1) 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 求$x$的值;\\\\\n(2) 记$f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$, 求函数$y=f(x)$的最大值和最小值及对应的$x$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017654": { + "id": "017654", + "content": "在$\\triangle ABC$中, 内角$A, B, C$的对边分别为$a, b, c$, 且$a>c$, 已知$\\overrightarrow{BA} \\cdot \\overrightarrow{BC}=2$, $\\cos B=\\dfrac{1}{3}$, $b=3$.\\\\\n(1) 求$a$和$c$的值;\\\\\n(2) 求$\\cos (B-C)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017655": { + "id": "017655", + "content": "已知$x, y \\in \\mathbf{R}$, $\\overrightarrow {i}$, $\\overrightarrow {j}$分别为直角坐标系中$x, y$轴正方向上的单位向量. 若向量$\\overrightarrow {a}=x \\overrightarrow {i}+(y+\\sqrt{3}) \\overrightarrow {j}$, $\\overrightarrow {b}=x \\overrightarrow {i}+(y-\\sqrt{3}) \\overrightarrow {j}$, 且$|\\overrightarrow {a}|+|\\overrightarrow {b}|=4$.\\\\\n(1) 求点$M(x, y)$的轨迹$C$的方程;\\\\\n(2) 过点$Q(-2,0)$作直线$l$与曲线$C$交于$A, B$两点, 设$P$是过点$(-\\dfrac{5}{17}, 0)$且以$\\overrightarrow {j}$为方向向量的直线上一动点, 满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\overrightarrow{OB}$($O$为坐标原点), 问是否存在这样的直线$l$, 使得四边形$OAPB$为矩形? 若存在, 求出直线$l$的方程; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试07平面向量单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017656": { + "id": "017656", + "content": "设$z_1=2-\\mathrm{i}$, $z_2=1+3 \\mathrm{i}$, 则复数$z=\\dfrac{z_1}{\\mathrm{i}}+\\dfrac{\\overline {z}_2}{5}$的虚部为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017657": { + "id": "017657", + "content": "已知$\\dfrac{a+b \\mathrm{i}}{2-\\mathrm{i}}=3+\\mathrm{i}$($a, b \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位), 则$a+b$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017658": { + "id": "017658", + "content": "复数$\\dfrac{(1-\\mathrm{i})(1+\\mathrm{i})}{\\mathrm{i}}$在复平面中所对应的点到原点的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017659": { + "id": "017659", + "content": "已知复数$z_1=x+2 \\mathrm{i}$, $z_2=-2+\\mathrm{i}$且$|z_1|<|z_2|$, 则实数$x$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017660": { + "id": "017660", + "content": "在复数范围内分解因式: $2 x^2+3 x+2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017661": { + "id": "017661", + "content": "已知$OACB$是复平面上的平行四边形, $O$是原点, 点$A$对应的复数是$3+\\mathrm{i}$, 向量$\\overrightarrow{OB}$对应的复数是$2+4 \\mathrm{i}$, 则点$C$对应的复数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017662": { + "id": "017662", + "content": "已知复数$z$满足$|z|=1$, 则$|z+\\mathrm{i}+1|$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017663": { + "id": "017663", + "content": "已知$z_1$、$z_2$是复数, 定义复数的一种运算``$\\otimes$''为: $z_1 \\otimes z_2=\\begin{cases}z_1 z_2, & |z_1|>|z_2|, \\\\ z_1+z_2 & |z_1| \\leq|z_2|,\\end{cases}$ 若$z_1=2+\\mathrm{i}$且$z_1 \\otimes z_2=3+4 \\mathrm{i}$, 则复数$z_2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017664": { + "id": "017664", + "content": "为求解方程$x^5-1=0$的虚根, 可以把原方程变形为$(x-1)(x^4+x^3+x^2+x+1)=0$, 再变形为$(x-1)(x^2+a x+1)(x^2+b x+1)=0$, 由此可得原方程的一个虚数根为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017665": { + "id": "017665", + "content": "设$C$是由全体复数组成的集合, $A$是由全体实数组成的集合, $B$是由全体纯虚数组成的集合, 全集$U=C$, 则下列结论正确的是\\bracket{20}.\n\\fourch{$A \\cup B=C$}{$\\overline {A} \\supset B$}{$A \\cap \\overline {B}=\\varnothing$}{$B \\cap \\overline {B}=C$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017666": { + "id": "017666", + "content": "已知复数$z$满足$|z|=2$, 且$(z-a)^2=a$, 则实数$a$不可能取值\\bracket{20}.\n\\fourch{$\\dfrac{1+\\sqrt{3}}{2}$}{$\\dfrac{1-\\sqrt{17}}{2}$}{$1$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017667": { + "id": "017667", + "content": "下列类比推理命题(其中$\\mathbf{Q}$为有理数集, $\\mathbf{R}$为实数集, $\\mathbf{C}$为复数集):\\\\\n\\textcircled{1} ``若$a, b \\in \\mathbf{R}$, 则$a-b=0 \\Rightarrow a=b$''类比推出``若$a, b \\in \\mathbf{C}$, 则$a-b=0 \\Rightarrow a=b$'';\\\\\n\\textcircled{2} ``若$a, b, c, d \\in \\mathbf{R}$, 则复数$a+b \\mathrm{i}=c+d \\mathrm{i} \\Rightarrow a=c, b=d$''类比推出``若$a, b, c, d \\in \\mathbf{Q}$, 则$a+b \\sqrt{2}=c+d \\sqrt{2} \\Rightarrow a=c, b=d$'';\\\\\n\\textcircled{3} ``若$a, b \\in \\mathbf{R}$, 则$a-b>0 \\Rightarrow a>b$''类比推出``若$a, b \\in \\mathbf{C}$, 则$a-b>0 \\Rightarrow a>b$''.\\\\\n其中类比结论正确的个数是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017668": { + "id": "017668", + "content": "已知复数$z=(m^2+5 m+6)+(m^2-2 m-15) \\mathrm{i}$, 当实数$m$为何值时:\\\\\n(1) $z$为实数;\\\\\n(2) $z$为虚数;\\\\\n(3) $z$为纯虚数;\\\\\n(4) 复数$z$对应的点$Z$在第四象限.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017669": { + "id": "017669", + "content": "设复数$z$的共轭复数为$\\overline {z}$, 已知$(1+2 \\mathrm{i}) \\overline {z}=4+3 \\mathrm{i}$.\\\\\n(1) 求复数$z$及$\\dfrac{z}{\\overline {z}}$;\\\\\n(2) 求满足$|z_1-1|=|z|$的复数$z_1$对应的点的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017670": { + "id": "017670", + "content": "已知关于$x$的实系数一元二次方程$a x^2+b x+c=0$有两个虚数根$x_1$、$x_2$, 且$(1-3 a \\mathrm{i}) \\mathrm{i}=c-\\dfrac{a}{\\mathrm{i}}$($\\mathrm{i}$为虚数单位), $|x_1-x_2|=1$, 求实数$b$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017671": { + "id": "017671", + "content": "对于任意的复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义运算$P(z)=x^2[\\cos (y \\pi)+\\mathrm{i}\\sin(y \\pi)]$.\\\\\n(1) 集合$A=\\{\\omega|\\omega=P(z), \\ | z | \\leq 1, \\ \\mathrm{Re} z$、$\\mathrm{Im} z \\in \\mathbf{Z}\\}$, 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$l: y=x-9$上是否存在格点$(x, y)$(坐标$x, y$均为整数的点), 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": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试08复数单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017672": { + "id": "017672", + "content": "已知$\\alpha$、$\\beta$是不同的两个平面, 直线$a \\subset \\alpha$, 直线$b \\subset \\beta$, 命题$p: a$与$b$没有公共点; 命题$q: \\alpha\\parallel \\beta$, 则$p$是$q$的\\blank{50}条件.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017673": { + "id": "017673", + "content": "设直线$m$与平面$\\alpha$相交但不垂直, 给出以下说法:\\\\\n\\textcircled{1} 在平面$\\alpha$内有且只有一条直线与直线$m$垂直;\\\\\n\\textcircled{2} 过直线$m$有且只有一个平面与平面$\\alpha$垂直;\\\\\n\\textcircled{3} 与直线$m$垂直的直线不可能与平面$\\alpha$平行;\\\\\n\\textcircled{4} 与直线$m$平行的平面不可能与平面$\\alpha$垂直.\\\\\n其中错误的是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017674": { + "id": "017674", + "content": "在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, 点$M$和$N$分别是矩形$ABCD$和$BB_1C_1C$的中心, 则过点$A$、$M$、$N$的平面截正方体所得截面的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017675": { + "id": "017675", + "content": "在$30^{\\circ}$的二面角的一个面内有一点$P$, 点$P$到另一个面的距离是$10$, 则点$P$到二面角棱的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017676": { + "id": "017676", + "content": "若将一个$45^{\\circ}$的直角三角板的一直角边放在一桌面上, 另一直角边与桌面所成角为$45^{\\circ}$, 则此时该三角板的斜边与桌面所成角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017677": { + "id": "017677", + "content": "如图所示, 在四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$, 且底面各边都相等, $M$是$PC$上的一动点, 当点$M$满足\\blank{50}时, 平面$MBD$$\\perp$平面$PCD$. (只要填写一个你认为是正确的条件即可)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (0,0,-2) node [left] {$D$} coordinate (D);\n\\draw (0,2.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(C)!0.45!(P)$) node [above] {$M$} coordinate (M);\n\\draw (P)--(A)--(B)--(C)--cycle(B)--(M)(B)--(P);\n\\draw [dashed] (A)--(D)--(M)(P)--(D)--(B)(D)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017678": { + "id": "017678", + "content": "设$a$、$b$是两条不同的直线, $\\alpha$、$\\beta$是两个不同的平面, 则下面四个命题中错误的是\\bracket{20}.\n\\twoch{若$a \\perp b$, $a \\perp \\alpha$, $b \\not \\subset \\alpha$, 则$b\\parallel \\alpha$}{若$a \\perp b$, $a \\perp \\alpha, b \\perp \\beta$, 则$\\alpha \\perp \\beta$}{若$a \\perp \\beta$, $\\alpha \\perp \\beta$, 则$a\\parallel \\alpha$或$a \\subset \\alpha$}{若$a \\perp \\alpha$, $\\alpha \\perp \\beta$, 则$a \\perp \\beta$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017679": { + "id": "017679", + "content": "如图, $\\alpha \\perp \\beta$, $\\alpha \\cap \\beta=l$, $A \\in \\alpha$, $B \\in \\beta$, $A, B$到$l$的距离分别是$a$和$b$, $AB$与$\\alpha, \\beta$所成的角分别是$\\theta$和$\\varphi$, $AB$在$\\alpha, \\beta$内的射影长分别是$m$和$n$. 若$a>b$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (S) (0,0,-4) coordinate (T);\n\\draw ($(S)!0.4!(T)$) coordinate (P) ($(S)!0.75!(T)$) coordinate (Q);\n\\draw (S) --++ (1.5,0) node [above] {$\\beta$} --++ (0,0,-4) -- (T) (S) --++ (0,1.5) node [midway, right] {$l$} node [right] {$\\alpha$} --++ (0,0,-4) -- (T) -- (S);\n\\draw (P) --++ (0,1.2) node [midway, right] {$a$} node [above] {$A$} coordinate (A);\n\\draw (Q) --++ (0.8,0) node [midway, below] {$b$} node [right] {$B$} coordinate (B) -- (A);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\theta>\\varphi$, $m>n$}{$\\theta>\\varphi$, $mn$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017680": { + "id": "017680", + "content": "如图, $P$为$\\triangle ABC$所在平面外一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$P$} coordinate (P);\n\\draw ($(C)!0.5!(P)$) node [above right] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(D)(B)--(P);\n\\draw [dashed] (A)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AP=AC$, $BP=BC$, $D$为$PC$的中点, 证明: $PC \\perp$平面$ABD$;\\\\\n(2) 若$AP=BP$, $AC=BC$, 证明: $PC \\perp AB$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017681": { + "id": "017681", + "content": "正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别为$A_1B_1$、$A_1D_1$的中点, $E$、$F$分别是$B_1C_1$、$C_1D_1$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 (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [below left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(B_1)$) node [below] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(D_1)$) node [left] {$N$} coordinate (N);\n\\draw ($(B_1)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$F$} coordinate (F);\n\\draw (A)--(M)--(N)(B)--(E)--(F);\n\\draw [dashed] (B)--(D)--(F)(A)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $E$、$F$、$B$、$D$共面;\\\\\n(2) 求证: 平面$AMN\\parallel$平面$EFDB$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017682": { + "id": "017682", + "content": "如图所示, $PA \\perp$平面$ABCD$, 矩形$ABCD$的边长$AB=1$, $BC=2$, $E$为$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [right] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(E)--(D);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $PE \\perp DE$;\\\\\n(2) 如果$PA=2$, 求异面直线$AE$与$PD$所成的角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017683": { + "id": "017683", + "content": "如图, 已知正方形$ABCD$和矩形$ACEF$所在的平面互相垂直, 且$AB=1$, $AF=1$, 点$M$是线段$EF$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\l,0) node [above] {$F$} coordinate (F);\n\\draw (C) ++ (0,\\l,0) node [above] {$E$} coordinate (E);\n\\draw ($(E)!0.5!(F)$) node [above] {$M$} coordinate (M);\n\\draw (D)--(A)--(B)--(F)--(E)--cycle(D)--(F)(A)--(F);\n\\draw [dashed] (A)--(C)(B)--(D)(D)--(C)--(B)(E)--(C)(A)--(M)(B)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AM\\parallel$平面$BDE$;\\\\\n(2) 求二面角$A-DF-B$的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017684": { + "id": "017684", + "content": "如图, 已知$ABCD$为正方形, $PD \\perp$平面$ABCD$, $PD=AD=2$.\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,\\l,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)(P)--(B)(P)--(C);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C)(A)--(C)(B)--(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$PC$与平面$PBD$所成角的大小;\\\\\n(2) 求异面直线$PC$与$BD$所成角的大小;\\\\\n(3) 在线段$PB$上是否存在一点$E$, 使得$PC \\perp$平面$ADE$. 若存在, 确定点$E$的位置; 不存在, 则说明理由?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试09空间直线与平面单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017685": { + "id": "017685", + "content": "一个圆锥的底面圆的半径和高分别为$2$和$6$, 则该圆锥的侧面积是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017686": { + "id": "017686", + "content": "正三棱锥底面边长为$a$, 侧棱与底面所成的角的大小为$45^{\\circ}$, 则它的斜高等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017687": { + "id": "017687", + "content": "某地球仪上一点$A$位于北伟$30^{\\circ}$的纬线上, 纬线的长主为$12 \\pi \\text{cm}$, 则该地球仪的表面积是\\blank{50}$\\text{cm}^2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017688": { + "id": "017688", + "content": "如图所示, 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}$, 该三棱柱的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{1}\n\\def\\h{{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 [dashed] (A) -- (C);\n\\draw (B)--(C_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017689": { + "id": "017689", + "content": "如图所示, 边长为$2$的正方形$ABCD$中, $E$为$AB$的中点, 将它沿$EC$、$ED$折起, 使$EA, EB$重合, 组成一个四面体, 这个四面体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw (A) rectangle (C) (D)--(E)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017690": { + "id": "017690", + "content": "在三棱柱$ABC-A_1B_1C_1$中, $AB=BC=2$, $\\angle ABC=120^{\\circ}$, 侧面$A_1ACC_1 \\perp$底面$ABC$, 侧棱$AA_1$与底面$ABC$成$60^{\\circ}$角, $D$为$AC$的中点, $\\angle A_1D\\mathrm{C}_1=90^{\\circ}$, 则三棱柱$ABC-A_1B_1C_1$的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017691": { + "id": "017691", + "content": "一个圆锥的侧面积是其底面积的$2$倍, 则该圆锥的母线与底面所成的角为\\bracket{20}.\n\\fourch{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}{$75^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017692": { + "id": "017692", + "content": "如图所示, 三棱锥的四个顶点$P$、$A$、$B$、$C$在同一个球面上, 顶点$P$在平面$ABC$上的射影是$H$, 若球心在直线$PH$上, 则点$H$一定是$\\triangle ABC$的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,1.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(C)$) node [above right] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(P)--(H);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{重心}{垂心}{内心}{外心}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017693": { + "id": "017693", + "content": "三棱锥$P-ABC$的四个顶点都在球$O$的球面上, 已知$PA$、$PB$、$PC$两两垂直, $PA=1$, $PB+PC=4$, 当三棱锥的体积最大时, 球$O$的体积为\\bracket{20}.\n\\fourch{$36 \\pi$}{$9 \\pi$}{$\\dfrac{9}{2} \\pi$}{$\\dfrac{9}{4} \\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017694": { + "id": "017694", + "content": "如图所示, 在正方体$ABCD-A_1B_1C_1D_1$中, 给出下列四个命题:\\\\\n\\textcircled{1} 点$P$在直线$BC_1$上运动时, 三棱锥$A-D_1PC$的体积不变;\\\\\n\\textcircled{2} 点$P$在直线$BC_1$上运动时, 直线$AP$与平面$ACD_1$所成角的大小不变;\\\\\n\\textcircled{3} 点$P$在直线$BC_1$上运动时, 二面角$P-AD_1-C$的大小不变;\\\\\n\\textcircled{4} 点$P$是平面$ABCD$上到点$D$和$C_1$距离相等的动点, 则$P$的轨迹是过点$B$的直线.\\\\\n其中的真命题是\\bracket{20}.\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (B)--(C_1);\n\\draw [dashed] (A)--(C)--(D_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017695": { + "id": "017695", + "content": "如图所示, 在正三棱柱$ABC-A_1B_1C_1$中, $AA_1=A_1B_1=4$, 点$D$、$E$分别为棱$AA_1$、$A_1B_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-125:0.5cm)}]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,\\h) node [below right] {$C_1$} coordinate (C_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (A) -- (C) -- (B) (A) -- (A_1) (C) -- (C_1) (B) -- (B_1) (A_1) -- (C_1) -- (B_1) (A_1) -- (B_1);\n\\draw [dashed] (A) -- (B);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$D$} coordinate (D);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$E$} coordinate (E);\n\\draw (E)--(C_1)--(B)(C_1)--(D);\n\\draw [dashed] (E)--(D)--(B)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$BC_1$与平面$ABB_1A_1$所成的角的大小 (结果用反三角函数值表示);\\\\\n(2) 求四面体$BDEC_1$的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017696": { + "id": "017696", + "content": "如图所示, 已知圆柱$OO_1$的轴截面$ABCD$为正方形, $E$为上底面圆周上一点, 且$2CE=\\sqrt{3} CD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\filldraw (0,0) node [below] {$O$} coordinate (O) circle (0.03);\n\\filldraw (0,2) node [below] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O) ++ (-1,0) node [left] {$A$} coordinate (A);\n\\draw ($(O)!-1!(A)$) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,2) node [left] {$D$} coordinate (D);\n\\draw (B) ++ (0,2) node [right] {$C$} coordinate (C);\n\\draw (O_1) ++ (120:1 and 0.25) node [above] {$E$} coordinate (E);\n\\draw (A) arc (180:360:1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25) (A)--(C)(A)--(E)(A)--(B);\n\\draw (C) arc (0:360:1 and 0.25) -- (C) -- (E) (A)--(D)(B)--(C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AE \\perp CE$;\\\\\n(2) 求平面$ACE$与圆$O$所在平面所成的锐二面角的余弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017697": { + "id": "017697", + "content": "如图$1$所示, 四边形$PBCD$是直角梯形, $\\angle PBC=90^{\\circ}$, $CD=1$, $PB=BC=2$, 点$A$是$PB$的中点, $E$是$BC$的中点, 现沿$AD$将平面$PAD$折起, 设$\\angle PAB=\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (0,1) node [left] {$A$} coordinate (A);\n\\draw (0,2) node [left] {$P$} coordinate (P);\n\\draw (2,1) node [right] {$D$} coordinate (D);\n\\filldraw (1,0) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (P)--(B)--(C)--(D)--cycle(A)--(D);\n\\draw (1,-0.5) node [below] {图1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\filldraw (1,0) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (B) ++ (0,0,-1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (A) ++ (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (B)--(C)--(D)--(P)--(A)--cycle(A)--(E)(P)--(C);\n\\draw [dashed] (A)--(D);\n\\draw (1,-0.5) node [below] {图2};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (B) ++ (0,0,-1) node [above right] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(2)/2},{-sqrt(2)/2}) node [above] {$P$} coordinate (P);\n\\draw (B)--(C)--(D)--(P)--cycle(B)--(D);\n\\draw [dashed] (A)--(D)(B)--(A)--(P);\n\\draw (1,-0.5) node [below] {图3};\n\\end{tikzpicture}\n\\end{center}\n(1) 如图 2, 当$\\theta=90^{\\circ}$时, 求异面直线$PC$与$AE$所成的角的大小;\\\\\n(2) 如图 3, 当$\\theta$为多大时, 三棱锥$P-ABD$的体积为$\\dfrac{\\sqrt{2}}{6}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试10简单几何体单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017698": { + "id": "017698", + "content": "从自动打包机包装的食盐中, 随机抽取$20$袋, 测得各袋的质量分别为 (单位: 克): $494$, $496$, $494$, $495$, $498$, $497$, $501$,$502$, $504$, $496$, $497$, $503$, $506$, $508$, $507$, $492$, $496$, $500$, $501$, $499$. 则该自动包装机包装的袋装食盐质量在 $497.5$到$501.5$之间的概率约是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017699": { + "id": "017699", + "content": "从一副混合后的扑克牌($52$张)中, 随机抽取$1$张. 事件$A$为``抽得红桃K'', 事件$B$为``抽得黑桃'', 则概率$P(A \\cup B)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017700": { + "id": "017700", + "content": "连续$4$次抛掷一枚硬币, 至少出现$1$次正面的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017701": { + "id": "017701", + "content": "甲、乙两人进行三局比赛, 并规定如果有一人胜满两局, 则比赛结束. 若每局比赛甲获胜的概率为$\\dfrac{2}{3}$, 则比赛以甲$2$胜$1$负而结束的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017702": { + "id": "017702", + "content": "某水产试验基地实行某种鱼的人工孵化, $10000$个鱼卵能孵出$8513$尾鱼苗. 要孵出$50000$尾鱼苗, 大概需要准备\\blank{50}个鱼卵.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017703": { + "id": "017703", + "content": "已知事件$A$、$B$是独立的, 如果$P(A)=0.3$, $P(B)=0.8$, 那么$P(A \\cap B)=$\\blank{50}, $P(A \\cap \\overline {B})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017704": { + "id": "017704", + "content": "甲、乙两人进行$5$局$3$胜制的比赛, 每局两人获胜的可能性相等. 若已知第一局甲胜, 则甲最终获胜的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017705": { + "id": "017705", + "content": "某保险公司把被保险人分为$3$类: ``谨慎的''、``一般的''、``冒失的''. 统计资料表明, 这$3$类被保险人在一年内发生事故的概率依次为$0.05$、$0.15$和$0.30$. 若``谨慎的''被保险人占$20 \\%$, , ``一般的''被保险人占$50 \\%$, ``冒失的''被保险人占$30 \\%$, 随机抽取该保险公司的一位被保险人, 此人在一年内出事故的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017706": { + "id": "017706", + "content": "在进行抛掷一枚均匀硬币的试验时, 以下说法正确的是\\bracket{20}.\n\\onech{若前一次抛掷得正面朝上, 则后一次抛掷一定得反面朝上}\n{在$1000$次抛掷中, 正面朝上应有$5000$次}{随着抛掷次数的逐渐增加, 正面朝上出现的频率的近似值为$\\dfrac{1}{2}$}{随着抛掷次数的逐渐增加, 正面朝上出现的频率越来越趋于稳定}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017707": { + "id": "017707", + "content": "如果事件$A, B$互斥, 那么\\bracket{20}.\n\\fourch{$A \\cup B$是必然事件}{$\\overline {A} \\cup \\overline {B}$是必然事件}{$\\overline {A}$与$\\overline {B}$互斥}{$\\overline {A}$与$\\overline {B}$一定不互斥}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017708": { + "id": "017708", + "content": "若独立地重复一个伯努利试验$n$次, 记成功的频率为$f(n)$, 则随着$n$的逐渐增大, 有$\\cdots$\\bracket{20}.\n\\twoch{$f(n)$与某个常数越来越接近}{$f(n)$与某个常数差的绝对值逐渐减少}{$f(n)$无限趋近于某个常数}{$f(n)$的值趋于稳定}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017709": { + "id": "017709", + "content": "$6$张奖券中有$2$张是有奖的, 先后由甲、乙两人各抽一张. 对于以下两种情况, 分别研究``甲中奖''和``乙中奖''这两个事件是否独立? \\textcircled{1} 抽后返回奖券; \\textcircled{2} 抽后不返回奖券. \\bracket{20}.\n\\fourch{\\textcircled{1} 独立; \\textcircled{2} 不独立}{\\textcircled{1} 不独立;$\\textcircled{2}$独立}{\\textcircled{1} 不独立; \\textcircled{2} 不独立}{\\textcircled{1} 独立; \\textcircled{2}独立}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017710": { + "id": "017710", + "content": "掷两颗骰子, 求:\\\\\n(1) 它们的点数都是偶数的概率;\\\\\n(2) 它们的点数之和是偶数的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017711": { + "id": "017711", + "content": "甲、乙两名实习生加工零件为一等品的概率分别是$0.75$和$0.8$. 甲、乙每人独立地加工一个零件, 求:\\\\\n(1) 只有甲生产的零件为一等品的概率;\\\\\n(2) 至少有一个零件为一等品的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017712": { + "id": "017712", + "content": "已知关于$x$的一元二次方程$x^2-b x+c=0$. 其中$b$、$c$是分别掷两颗骰子得到的点数. 求下列事件的概率:\\\\\n(1) 方程有两个不相等的实根;\\\\\n(2) $x=2$是方程的根.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017713": { + "id": "017713", + "content": "罐子中有$b$个黑球, $r$个红球. 从中随机摸出一个球, 观察其颜色后放回, 并在罐子中加人同色球$c$个. 再从罐子中第二次摸出一个球, 求第二次摸出的是黑球的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017714": { + "id": "017714", + "content": "抛掷一枚均匀的硬币两次.\\\\\n(1) 已知第一次出现正面, 求第二次也出现正面的概率;\\\\\n(2) 已知其中有一次出现正面, 求另一次也出现正面的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017715": { + "id": "017715", + "content": "把$1$、$2$、$3$、$4$、$5$、$6$、$7$、$8$、$9$、$10$分别写在$10$个形状大小相同的卡片上, 从中抽取一张卡片, 设事件$A$表示``卡片上是偶数'', 事件$B$表示``卡片上是$5$的倍数'', 事件$C$表示``卡片上是合数''.\\\\\n(1) 事件$A$与事件$B$是否独立?\\\\\n(2) 事件$A$与事件$C$是否独立?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试11概率初步单元测试试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017716": { + "id": "017716", + "content": "在五个数字$1,2,3,4,5$中, 若随机取出三个数字, 则剩下两个数字都是奇数的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017717": { + "id": "017717", + "content": "今年我市约有$50000$名高一学生参加地理高中学业水平考, 为了了解这$50000$名学生的地理成绩, 准备从中随机抽取$2500$名学生的地理成绩进行统计分析, 那么某位高一学生地理成绩被抽中的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017718": { + "id": "017718", + "content": "$10$件产品中有$3$件次品, 从中随机取出$5$件, 则恰含$1$件次品的概率是\\blank{50}(结果用数值表示).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017719": { + "id": "017719", + "content": "某林场有树苗$30000$棵, 其中松树苗$4000$棵. 为调查树苗的生长情况, 采用分层抽样的方法抽取一个容量为$150$的样本, 则样本中松树苗的数量为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017720": { + "id": "017720", + "content": "已知一组数据: $125,121,123,125,127,129,125,128,130,129,126,124,125,127,126$. 则这组数据的第$25$百分位数和第$80$百分位数分别是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017721": { + "id": "017721", + "content": "为了了解某地区高三学生的身体发育情况, 抽查了该地区$100$名年龄为$17.5$岁至$18$岁的男生体重$(\\text{kg})$, 得到频率分布直方图如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.45, yscale = 50]\n\\draw [->] (53.5,0) -- (53.6,0) -- (53.7,0.003) -- (53.9,-0.003) -- (54,0) -- (80.5,0) node [below] {体重(kg)};\n\\draw [->] (53.5,0) -- (53.5,0.09) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i/\\j in {54.5/0.01,56.5/0.03,58.5/0.05,60.5/0.05,62.5/0.07,64.5/0.08,66.5/0.07,68.5/0.06,70.5/0.04,72.5/0.03,74.5/0.01}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (2,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {56.5/0.03,58.5/0.05,62.5/0.07}\n{\\draw [dashed] (\\i,\\j) -- (53.5,\\j) node [left] {$\\k$};};\n\\draw (76.5,0) node [below] {$76.5$};\n\\end{tikzpicture}\n\\end{center}\n根据上图可得这$100$名学生中体重在$[56.5,64.5)$的学生人数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017722": { + "id": "017722", + "content": "从$4$名男同学和$6$名女同学中随机选取$3$人参加某社团活动, 选出的$3$人中男女同学都有的概率为\\blank{50}(结果用数值表示).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017723": { + "id": "017723", + "content": "同时抛掷两枚质地均匀的骰子(一种各面上分别标有$1,2,3,4,5,6$个点的正方体玩具), 观察向上的点数, 则两个点数之积不小于$4$的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017724": { + "id": "017724", + "content": "随机抽取某果园一批橙子中的$10$个, 经称重分别为$245$、$260$、$235$、$240$、$245$、$255$、$250$、$225 、$$240$、$255$克, 试用这些橙子去估计该批次橙子, 则该批次橙子的重量的标准差是\\blank{50}克(精确到小数点后两位).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017725": { + "id": "017725", + "content": "总体由编号为$01$, $02$, $\\cdots$, $29$, $30$的$30$个个体组成. 利用所给的随机数表选取$6$个个体, 选取的方法是从随机数表第$1$行的第$3$列和第$4$列数字开始, 由左到右一次选取两个数字, 则选出来的第$5$个个体的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cccccccc}\n1712 & 1340 & 3320 & 3826 & 1389 & 5103 & 7417 & 7637 \\\\ \n1304 & 0774 & 2119 & 3056 & 6218 & 3735 & 9683 & 5087\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017726": { + "id": "017726", + "content": "从个位数与十位数之和为奇数的两位数中任取一个, 其个位数为$0$的概率是\\bracket{20}.\n\\fourch{$\\dfrac{1}{9}$}{$\\dfrac{2}{9}$}{$\\dfrac{1}{3}$}{$\\dfrac{4}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017727": { + "id": "017727", + "content": "某校为了了解学生的课外阅读情况, 随机调查了$50$名学生, 得到他们在某一天各自课外阅读所用时间的数据, 结果用条形图表示. 根据条形图要得这$50$名学生这一天平均每人的课外阅读时间为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,xscale = 1.5, yscale = 0.1]\n\\draw [->] (0,0) -- (3.5,0) node [below] {时间(小时)};\n\\draw [->] (0,0) -- (0,25) node [left] {人数(人)};\n\\draw (0,0) rectangle (0.3,5) (0,0) ++ (0.15,0) node [below] {$0$};\n\\draw (0.5,0) rectangle (0.8,20) (0.5,0) ++ (0.15,0) node [below] {$0.5$};\n\\draw (1,0) rectangle (1.3,10) (1,0) ++ (0.15,0) node [below] {$1.0$};\n\\draw (1.5,0) rectangle (1.8,10) (1.5,0) ++ (0.15,0) node [below] {$1.5$};\n\\draw (2,0) rectangle (2.3,5) (2,0) ++ (0.15,0) node [below] {$2.0$};\n\\foreach \\i/\\j in {2/5,1.5/10,0.5/20}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\j$};};\n\\draw (0,15) node [left] {$15$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0.6$小时}{$0.9$小时}{$1.0$小时}{$1.5$小时}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017728": { + "id": "017728", + "content": "一个容量为$20$的样本数据, 分组后, 组距与频数如下:\n$[10,20), 2$; $[20,30), 3$; $[30,40), 4$; $[40,50), 5$; $[50,60), 4$; $[60,70), 2$, 则样本在$[0,50)$上的频率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{20}$}{$\\dfrac{1}{4}$}{$\\dfrac{1}{2}$}{$\\dfrac{7}{10}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017729": { + "id": "017729", + "content": "某社区安置了$15$个体温检测点, 每个检测点每天检测的人数都是随机的, 不受位置等因素影响, 如图是由某天检测人数绘制的茎叶图, 则某个检测点某天检测人数达$145$及以上的概率是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{c|ccccccc}\n13 & 0 & 2 & 4 & 6 \\\\\n14 & 0 & 5 & 5 & 5 & 6 & 8 & 8\\\\\n15 & 2 & 3 & 3 & 4 \n\\end{tabular}\n\\end{center}\n\\fourch{$\\dfrac{7}{15}$}{$\\dfrac{8}{15}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017730": { + "id": "017730", + "content": "某居民小区所有$263$户家庭人口数分组列表如下, 求:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline 家庭人口数 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\n\\hline 家庭数 & 20 & 29 & 59 & 50 & 46 & 36 & 19 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 总体平均数、众数、中位数;\\\\\n(2) 求总体标准差.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017731": { + "id": "017731", + "content": "某中学从甲、乙两个班中各选出$7$名学生参加数学竞赛, 将他们的成绩 (满分$100$分) 进行统计分析, 绘制成如图所示的茎叶图, 已知甲班学生成绩的众数是$83$, 乙班学生成绩的平均数是$86$.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{r|}{甲} & & \\multicolumn{3}{l}{乙}\\\\\n8 & 9 & 7 & 6\\\\\n$x$ & 3 & 8 & 1 & 2 & $y$\\\\\n6 & 2 & 9 & 1 & 1 & 6\n\\end{tabular}\n\\end{center}\n(1) 求$x$、$y$的值;\\\\\n(2) 设成绩在$85$分以上 (含$85$分) 的学生为优秀学生, 从甲、乙两班的优秀学生中各取$1$人, 记甲班选取的学生成绩不低于乙班选取的学生成绩为事件$A$, 求事件$A$发生的概率$P(A)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017732": { + "id": "017732", + "content": "有$10$张卡片, 其号码分别为$1,2,3, \\cdots, 10$. 从中任取$3$张.\\\\\n(1) 求恰有一张号码为$3$的倍数的概率;\\\\\n(2) 求至少有一张号码为$3$的倍数的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017733": { + "id": "017733", + "content": "我国是世界上严重缺水的国家之一, 城市缺水问题较为突出. 某市为了节约生活用水, 计划在本市试行居民生活用水定额管理(即确定一个居民月均用水量标准, 用水量不超过$a$的部分按照平价收费, 超过$a$的部分按照议价收费). 为了较为合理地确定出这个标准, 通过抽样获得了$100$位居民某年的月均用水量(单位: $\\text{t}$), 制作了频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1, yscale = 4]\n\\draw [->] (0,0) -- (5.2,0) node [below] {月均用水量/t};\n\\draw [->] (0,0) -- (0,0.75) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\foreach \\i in {0.1,0.2,0.3,0.4,0.5,0.6}\n{\\draw [dashed] (3.5,\\i) -- (0,\\i) node [left] {$\\i$};};\n\\foreach \\i/\\j in {0/0.1,0.5/0.2,1/0.3,1.5/0,2/0.6,2.5/0.3,3/0.1\n}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.5,0) --++ (0,-\\j);};\n\\draw (3.5,0) node [below] {$3.5$};\n\\end{tikzpicture}\n\\end{center}\n(1) 由于某种原因频率分布直方图部分数据丢失, 请在图中将其补充完整;\\\\\n(2) 用样本估计总体, 如果希望$80 \\%$的居民每月的用水量不超过标准, 则月均用水量的最低标准定为多少吨? 并说明理由;\\\\\n(3) 从频率分布直方图中估计该$100$位居民月均用水量的平均数. (同一组中的数据用该区间的中点值代表)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试12概率与统计初步单元测试试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017734": { + "id": "017734", + "content": "直线$l_1: x+m y+1=0$与直线$l_2: y=2 x-1$垂直, 则$m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017735": { + "id": "017735", + "content": "已知过点$(0,1)$的直线$l: x \\tan \\alpha-y-3 \\tan \\beta=0$的一个法向量为$(2,-1)$, 则$\\tan (\\alpha+\\beta)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017736": { + "id": "017736", + "content": "点$P(1,1)$到直线$x \\cos \\theta+y \\sin \\theta=2$的距离为$d$, 则$d$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017737": { + "id": "017737", + "content": "已知点$M(2,4)$, $N(5,-4)$, 点$P$在$y$轴上, 且$\\angle MPN=90^{\\circ}$, 则点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017738": { + "id": "017738", + "content": "已知直线$l: y=k x-\\sqrt{3}$与直线$2 x+3 y-6=0$的交点位于第一象限, 则直线$l$的倾斜角的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017739": { + "id": "017739", + "content": "若直线$l$过点$P(0,1)$, 且被两平行直线$l_1: 2 x+y-6=0$和$l_2: 4 x+2 y-5=0$截得长为$\\dfrac{7}{2}$的线段, 则直线$l$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017740": { + "id": "017740", + "content": "设$m$、$n \\in \\mathbf{R}$, 若直线$(m+1) x+(n+1) y-2=0$与圆$(x-1)^2+(y-1)^2=1$相切, 则$m+n$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017741": { + "id": "017741", + "content": "设$m \\in \\mathbf{R}$, 过定点$A$的动直线$x+m y=0$与过定点$B$的动直线$m x-y-2 m+4=0$交于点$P(x, y)$, 则$|PA| \\cdot|PB|$的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017742": { + "id": "017742", + "content": "若两直线$l_1: (a-1) x-3 y-2=0$与$l_2: x-(a+1) y+2=0$平行, 则$a$的值为\\bracket{20}.\n\\fourch{$0$}{$2$}{$-2$}{$\\pm 2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017743": { + "id": "017743", + "content": "已知点$M, N$分别在直线$l_1: x+y=0$与直线$l_2: x+y-3=0$, 且$MN \\perp l_1$, 点$P(-1,-3)$, $Q(\\dfrac{7}{2}, \\dfrac{1}{2})$, 则$|PM|+|QN|$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{15}}{2}$}{$\\sqrt{15}$}{$\\sqrt{13}$}{$3 \\sqrt{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017744": { + "id": "017744", + "content": "如图, 在平面直角坐标系$x O y$中, 点$A(0,3)$, 直线$l: y=2 x-4$, 设圆$C$的半径为$1$, 圆心在$l$上. 若圆心$C$也在直线$y=x-1$上, 过点$A$作圆$C$的切线, 求切线的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.25]\n\\draw [->] (-2,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (0,3) node [right] {$A$} coordinate (A) circle (0.12);\n\\draw (-1,-6) -- (5,6) node [right] {$l$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017745": { + "id": "017745", + "content": "直线$l$被两条直线$l_1: 4 x+y+3=0$和$l_2: 3 x-5 y-5=0$截得的线段$AB$的中点为$P(-1,2)$, 求直线$l$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017746": { + "id": "017746", + "content": "直线$l$与两坐标轴构成等腰直角三角形, 且点$P(4,3)$到直线$l$的距离为$3 \\sqrt{2}$, 求此直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017747": { + "id": "017747", + "content": "在平面直角坐标系$x O y$中, $O$为坐标原点. 定义$P(x_1, y_1)$、$Q(x_2, y_2)$两点之间的``直角距离''为$d(P, Q)=|x_1-x_2|+|y_1-y_2|$. 已知$B(1,0)$, 点$M$为直线$x-y+2=0$上的动点, 求$d(B, M)$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017748": { + "id": "017748", + "content": "设集合$L=\\{l|$直线$l$与直线$y=3 x$相交, 且以交点的横坐标为斜率$\\}$.\\\\\n(1) 是否存在直线$l_0$使$l_0 \\in L$且$l_0$过点$(1,5)$, 若存在, 请写出$l_0$的方程, 若不存在, 请说明理由;\\\\\n(2) 点$P(-3,5)$与集合$L$中的哪一条直线的距离最小?\\\\\n(3) 设$a \\in(0,+\\infty)$, 点$P(-3, a)$与集合$L$中的直线的距离最小值记为$f(a)$, 求$f(a)$的解析式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017749": { + "id": "017749", + "content": "已知直线$l$过点$P(-\\dfrac{5}{2}, \\dfrac{3}{2})$, 且直线$l$的一个方向向量为$\\overrightarrow {m}=(3,3)$. 一组直线$l_1$, $l_2$, $\\cdots$, $l_n$, $\\cdots$, $l_{2 n}$($n \\in \\mathbf{N}$, $n \\geq 1$)都与直线$l$平行且与椭圆$C: \\dfrac{x^2}{10}+\\dfrac{y^2}{6}=1$均有交点, 它们到直线$l$的距离依次为$d, 2 d, \\cdots, n d, \\cdots, 2 n d$($d>0$), 直线$l_n$恰好过椭圆$C$的中心, 试用$n$表示$d$的关系式, 并求出直线$l_i$($i=1,2, \\cdots, 2 n$)的方程 (用$n$、$i$表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试13坐标平面上的直线单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017750": { + "id": "017750", + "content": "已知圆$O_1: x^2+y^2=1$, 圆$O_2: x^2+y^2+2 x-4 y+3=0$, 则两圆的圆心距是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017751": { + "id": "017751", + "content": "已知方程$2 x^2+m y^2=1$表示焦点在$y$轴上的椭圆, 则实数$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017752": { + "id": "017752", + "content": "已知双曲线$\\dfrac{x^2}{m}-\\dfrac{y^2}{4}=1$的一条渐近线方程为$y=x$, 则实数$m$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017753": { + "id": "017753", + "content": "若直线$a x-y+1=0$经过抛物线$y^2=4 x$的焦点, 则实数$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017754": { + "id": "017754", + "content": "以原点为中心的椭圆的两焦点在$x$轴上, 长轴是短轴的$2$倍, 且过点$(2,-1)$, 则该椭圆的标准方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017755": { + "id": "017755", + "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{(a+3)^2}=1$($a>0$)的一条渐近线方程为$y=2 x$, 则实数$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017756": { + "id": "017756", + "content": "设$P$是曲线$y^2=4 x$上的一个动点, 则点$P$到点$(0,1)$的距离与点$P$到直线$x=-1$的距离之和的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017757": { + "id": "017757", + "content": "设双曲线$\\dfrac{x^2}{12}-\\dfrac{y^2}{4}=1$的右焦点为$F(c, 0)$, 点$P$到$F(c, 0)$的距离与到直线$x+4=0$的距离相等, 则点$P$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017758": { + "id": "017758", + "content": "已知椭圆$\\dfrac{x^2}{a}+y^2=1$($a>1$)和双曲线$\\dfrac{x^2}{m}-y^2=1$($m>0$)有相同焦点, 则\\bracket{20}.\n\\fourch{$a=m+2$}{$m=a+2$}{$a^2=m^2+2$}{$m^2=a^2+2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017759": { + "id": "017759", + "content": "无论$m$为任何实数, 直线$l: y=x+m$与双曲线$C: \\dfrac{x^2}{2}-\\dfrac{y^2}{b^2}=1$($b>0$)恒有公共点, 则双曲线$C$的焦距的取值范围是\\bracket{20}.\n\\fourch{$(2 \\sqrt{2},+\\infty)$}{$(4,+\\infty)$}{$(2 \\sqrt{6},+\\infty)$}{$(4 \\sqrt{2},+\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017760": { + "id": "017760", + "content": "抛物线$C$的顶点为坐标原点$O$, 焦点在$x$轴上, 直线$l: x=1$交$C$于$P, Q$两点, 且$OP \\perp OQ$. 已知点$M(2,0)$, 且$\\odot M$与$l$相切. 求$C$, $\\odot M$的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017761": { + "id": "017761", + "content": "已知$\\odot M: x^2+(y-2)^2=1$, $Q$是$x$轴上的动点, $QA$、$QB$分别切$\\odot M$于$A$、$B$两点.\\\\ \n(1) 如果$|AB|=\\dfrac{4 \\sqrt{2}}{3}$, 求直线$MQ$的方程;\\\\\n(2) 求动弦$AB$的中点$P$的轨迹方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017762": { + "id": "017762", + "content": "已知抛物线$y^2=2 p x$($p>0$), 其准线方程为$x+1=0$, 直线$l$过点$T(t, 0)$($t>0$)且与抛物线交于$A$、$B$两点, $O$为坐标原点.\\\\\n(1) 求抛物线方程, 并证明: $\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值与直线$l$倾斜角的大小无关;\\\\\n(2) 若$P$为抛物线上的动点, 记$|PT|$的最小值为函数$d(t)$, 求$d(t)$的解析式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017763": { + "id": "017763", + "content": "已知离心率为$\\dfrac{1}{2}$的椭圆$E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左顶点及右焦点分别为点$A$、$F$, 且$|AF|=3$.\\\\\n(1) 求$E$的方程;\\\\\n(2) 过点$F$的直线$l$与$E$交于$M, N$两点, $P$是直线$l$上异于$F$的点, 且$|MF| \\cdot|PN|=|NF| \\cdot|PM|$, 证明: 点$P$在定直线上.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017764": { + "id": "017764", + "content": "已知$F_1(-2,0)$, $F_2(2,0)$, 点$P$满足$|PF_1|-|PF_2|=2$, 记点$P$的轨迹为$E$.\\\\\n(1) 求轨迹$E$的方程;\\\\\n(2) 若直线$l$过点$F_2$且法向量为$\\overrightarrow {n}=(a, 1)$, 直线与轨迹$E$交于$P$、$Q$两点.\\\\\n(I) 过$P$、$Q$作$y$轴的垂线$PA$、$QB$, 垂足分别为$A$、$B$, 记$|PQ|=\\lambda|AB|$, 试确定$\\lambda$的取值范围;\\\\\n(II) 在$x$轴上是否存在定点$M$, 无论直线$l$绕点$F_2$怎样转动, 使$\\overrightarrow{MP} \\cdot \\overrightarrow{MQ}=0$恒成立? 如果存在, 求出定点$M$; 如果不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试14圆锥曲线单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017765": { + "id": "017765", + "content": "若向量$\\overrightarrow {b}$与向量$\\overrightarrow {a}=(2,-1,2)$共线, 且满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}+18=0$, 则向量$\\overrightarrow {b}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017766": { + "id": "017766", + "content": "若向量$\\overrightarrow {a}=(1, \\lambda, 2)$与$\\overrightarrow {b}=(2,-1,2)$的夹角的余弦值为$\\dfrac{8}{9}$, 则$\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017767": { + "id": "017767", + "content": "如图, 已知矩形$ABCD$中, $AB=1, BC=a, PA \\perp$平面$ABCD$, 若在$BC$上只有一点$Q$满足$PQ \\perp DQ$, 则$a$的值等于\\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 (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.6!(C)$) node [below] {$Q$} coordinate (Q);\n\\draw (A) ++ (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(Q)--(D)(A)--(B)--(C)--(D)--cycle(A)--(P);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017768": { + "id": "017768", + "content": "已知空间四边形$ABCD$的每条边和对角线的长都等于$1$, 点$E$、$F$分别是$AB$、$AD$的中点, 则$\\overrightarrow{EF} \\cdot \\overrightarrow{DC}$等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017769": { + "id": "017769", + "content": "在正三棱柱$ABC-A_1B_1C_1$中, $AB=\\sqrt{2} BB_1$, 则$AB_1$与$C_1B$所成的角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017770": { + "id": "017770", + "content": "在棱长为$a$的正方体$OABC-O' A' B' C'$中, $E$、$F$分别为棱$AB$、$BC$上的动点, 且$AE=BF$, 则$A' F$与$C' E$的位置关系为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017771": { + "id": "017771", + "content": "如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 若$AB=2$, $BC=2$, $AA_1=1$, $\\angle A_1AD=\\angle A_1AB=\\angle ABD=60^{\\circ}$, 则$BD_1$的长为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw ($1/6*(B)+1/6*(D)$) ++ (0,{sqrt(6)/3},0) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)-(A)+(B)$) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)-(A)+(C)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)-(A)+(D)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle(B)--(B_1)--(C_1)(A_1)--(B_1);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017772": { + "id": "017772", + "content": "如图, 四面体$ABCD$中, $AB, BC, BD$两两垂直, 且$AB=BC=2$, $E$是$AC$中点, 异面直线$AD$与$BE$所成的角为$\\arccos \\dfrac{\\sqrt{10}}{10}$, 则四面体的体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (0,2.5,0) node [above] {$D$} coordinate (D);\n\\draw ($(A)!0.5 !(C)$)node [below] {$E$} coordinate (E);\n\\draw (A)--(C)--(D)--cycle;\n\\draw [dashed] (A)--(B)--(C)(D)--(B)--(E);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017773": { + "id": "017773", + "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$中, $AB=AC=1$, $AA_1=2$, $\\angle B_1A_1C_1=90^{\\circ}$, $D$为$BB_1$的中点, 则异面直线$C_1D$与$A_1C$所成角的余弦值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(B_1)$) node [left] {$D$} coordinate (D);\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1)(C_1)--(D);\n\\draw [dashed] (A_1)--(A)--(C)(A)--(B)(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017774": { + "id": "017774", + "content": "已知点$P$在正方体$ABCD-A_1B_1C_1D_1$的对角线$BD_1$上, $\\angle PDA=60^{\\circ}$, 则$DP$与$CC_1$所成角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017775": { + "id": "017775", + "content": "已知点$O, A, B, C$为空间不共面的四点, 且向量$\\overrightarrow {a}=\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}$, 向量$\\overrightarrow {b}=\\overrightarrow{OA}+\\overrightarrow{OB}-\\overrightarrow{OC}$, 则与向量$\\overrightarrow {a}$、$\\overrightarrow {b}$不能共同构成空间基底的向量是\\bracket{20}.\n\\fourch{$\\overrightarrow{OA}$}{$\\overrightarrow{OB}$}{$\\overrightarrow{OC}$}{$\\overrightarrow{OA}$或$\\overrightarrow{OB}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017776": { + "id": "017776", + "content": "已知$\\overrightarrow{AB}=(1,5,-2)$, $\\overrightarrow{BC}=(3,1, z)$, 若$\\overrightarrow{AB} \\perp \\overrightarrow{BC}, \\overrightarrow{BP}=(x-1, y,-3)$, 且$BP \\perp$平面$ABC$, 则实数$x, y, z$分别为\\bracket{20}.\n\\fourch{$\\dfrac{33}{7},-\\dfrac{15}{7}, 4$}{$\\dfrac{40}{7},-\\dfrac{15}{7}, 4$}{$\\dfrac{40}{7},-2,4$}{$4, \\dfrac{40}{7},-15$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017777": { + "id": "017777", + "content": "空间四边形$ABCD$的各边和对角线均相等, $E$是$BC$的中点, 那么\\bracket{20}.\n\\twoch{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}<\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}=\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}>\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$}{$\\overrightarrow{AE} \\cdot \\overrightarrow{BC}$与$\\overrightarrow{AE} \\cdot \\overrightarrow{CD}$的大小不能比较}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017778": { + "id": "017778", + "content": "设动点$P$在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$的对角线$BD_1$上, 记$\\dfrac{D_1P}{D_1B}=\\lambda$. 当$\\angle APC$为钝角时, 则$\\lambda$的取值范围是\\bracket{20}.\n\\fourch{$(0, \\dfrac{1}{3})$}{$(0, \\dfrac{1}{2})$}{$(\\dfrac{1}{2}, 1)$}{$(\\dfrac{1}{3}, 1)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017779": { + "id": "017779", + "content": "如图, 在棱长为$a$的正方体$OABC-O_1A_1B_1C_1$中, $E, F$分别是棱$AB, BC$上的动点, 且$AE=BF=x$, 其中$0 \\leq x \\leq a$, 以$O$为原点建立空间直角坐标系$O x y z$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [above right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$O$} coordinate (O);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (O) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (O) ++ (0,\\l,0) node [above left] {$O_1$} coordinate (O_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (O_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (O) -- (O_1);\n\\draw ($(A)!0.75!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B)!0.75!(C)$) node [below right] {$F$} coordinate (F);\n\\draw [->] (A) -- ($(O)!1.3!(A)$) node [right] {$x$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {$y$};\n\\draw [->] (O_1) -- ($(O)!1.3!(O_1)$) node [left] {$z$};\n\\draw [dashed] (A_1)--(F)(C_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1F \\perp C_1E$;\\\\\n(2) 若$A_1, E, F, C_1$四点共面, 求证: $\\overrightarrow{A_1F}=\\dfrac{1}{2} \\overrightarrow{A_1C_1}+\\overrightarrow{A_1E}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017780": { + "id": "017780", + "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$G$分别为$AB$、$B_1C_1$、$AA_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$D$} coordinate (D);\n\\draw (D) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (D) ++ (\\l,0,-\\l) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0,\\l,0) node [left] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [above right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (D_1) -- (A_1) -- (B_1) -- (C_1) -- cycle;\n\\draw (D) -- (D_1) (A) -- (A_1) (B) -- (B_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$G$} coordinate (G);\n\\draw (D)--(G)--(B)(A)--(D_1);\n\\draw [dashed] (D)--(B)(F)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证$EF \\perp$平面$GBD$;\\\\\n(2) 求异面直线$AD_1$与$EF$所成的角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017781": { + "id": "017781", + "content": "如图, 在四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$, 底面$ABCD$是正方形, $PA=AB$, $E$、$F$、$G$分别是线段$PA, PD, CD$的中点.\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 (2,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(P)$) node [left] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) node [below right] {$G$} coordinate (G);\n\\draw ($(P)!0.25!(B)$) node [left] {$H$} coordinate (H);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C)(F)--(G);\n\\draw [dashed] (B)--(A)--(D)(E)--(F)(E)--(G)(E)--(H)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证$PB\\parallel$平面$EFG$;\\\\\n(2) 若点$H$在侧棱$PB$上, 且$PB=4PH$, 求证: $HE \\perp$平面$EFG$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017782": { + "id": "017782", + "content": "在直三棱柱$ABC-A_1B_1C_1$中, 底面是以$\\angle ABC$为直角的等腰三角形, $AC=2 a$, $BB_1=3 a$, $D, E$分别为线段$A_1C_1, B_1C$的中点.\\\\\n(1) 求直线$BE$与$A_1C$所成的角;\\\\\n(2) 在线段$AA_1$上是否存在点$F$, 使$CF \\perp$平面$B_1DF$; 若存在, 求出$|\\overrightarrow{AF}|$; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试15空间向量及其应用一单元测试试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017783": { + "id": "017783", + "content": "已知空间四边形$OABC$, 点$M$、$N$分别为$OA$、$BC$的中点, 且$\\overrightarrow{OA}=\\overrightarrow {a}$, $\\overrightarrow{OB}=\\overrightarrow {b}$, $\\overrightarrow{OC}=\\overrightarrow {c}$, 用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$表示$\\overrightarrow{MN}$, 则$\\overrightarrow{MN}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017784": { + "id": "017784", + "content": "在空间直角坐标系$O-x y z$中, 平面$OAB$的一个法向量$\\overrightarrow {n}=(2,-2,1)$, 已知$P(-1,3,2)$, 则点$P$到平面$OAB$的距离$d$等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017785": { + "id": "017785", + "content": "在四面体$OABC$中, 空间的一点$M$满足$\\overrightarrow{OM}=\\dfrac{1}{2} \\overrightarrow{OA}+\\dfrac{1}{6} \\overrightarrow{OB}+\\lambda \\cdot \\overrightarrow{OC}$, 若$\\overrightarrow{MA}$、$\\overrightarrow{MB}$、$\\overrightarrow{MC}$共面, 则$\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017786": { + "id": "017786", + "content": "如图所示, 在三棱柱$ABC-A_1B_1C_1$中, $AA_1 \\perp$底面$ABC$, $AB=BC=AA_1$, $\\angle ABC=90^{\\circ}$, 点$E$、$F$分别是棱$AB$、$BB_1$的中点, 则直线$EF$和$BC_1$所成的角是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\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 [dashed] (A) -- (C);\n\\draw ($(A)!0.5!(B)$) node [below left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$F$} coordinate (F);\n\\draw (E)--(F)(B)--(C_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017787": { + "id": "017787", + "content": "正四棱锥$S-ABCD$中, $O$为顶点在底面上的射影, $P$为侧棱$SD$的中点, 且$SO=OD$, 则直线$BC$与平面$PAC$的夹角等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017788": { + "id": "017788", + "content": "在正方形$ABCD-A_1B_1C_1D_1$中, 平面$A_1BD$与平面$C_1BD$的夹角的余弦值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017789": { + "id": "017789", + "content": "正方体$ABCD-A_1B_1C_1D_1$中, 二面角$A-BD_1-B_1$的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017790": { + "id": "017790", + "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, $O$是底面$A_1B_1C_1D_1$的中心, 则点$O$到平面$ABC_1D_1$的距离为\\blank{50}.\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 (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(A_1)!0.5!(C_1)$) node [below] {$O$} coordinate (O) circle (0.03);\n\\draw (A_1) -- (C_1)--(B);\n\\draw [dashed] (A)--(D_1); \n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017791": { + "id": "017791", + "content": "在平行六面体$ABCD-A_1B_1C_1D_1$中, 向量$\\overrightarrow{AB}$、$\\overrightarrow{AD}$、$\\overrightarrow{AA_1}$两两的夹角均为$60^{\\circ}$, 且$|\\overrightarrow{AB}|=1$, $|\\overrightarrow{AD}|=2$, $|\\overrightarrow{AA_1}|=3$, 则$|\\overrightarrow{AC_1}|$等于\\bracket{20}.\n\\fourch{$5$}{$6$}{$4$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017792": { + "id": "017792", + "content": "正$\\triangle ABC$与正$\\triangle BCD$所在平面垂直, 则二面角$A-BD-C$的正弦值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{5}}{5}$}{$\\dfrac{\\sqrt{3}}{3}$}{$\\dfrac{2 \\sqrt{5}}{5}$}{$\\dfrac{\\sqrt{6}}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017793": { + "id": "017793", + "content": "如图, 正方体的棱长为$1, C$、$D$、$M$分别为三条棱的中点, $A$、$B$是顶点, 则点$M$到截面$ABCD$的距离为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$A$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(B)!0.5!(C)$) node [right] {$M$} coordinate (M) circle (0.03);\n\\draw ($(B_1)!0.5!(C_1)$) node [right] {$C$} coordinate (S);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$D$} coordinate (T);\n\\draw (B)--(S)--(T);\n\\draw [dashed] (B)--(D)--(T);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{\\sqrt{2}}{4}$}{$\\dfrac{\\sqrt{3}}{4}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017794": { + "id": "017794", + "content": "沿着正四面体$O-ABC$的三条棱$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$的方向有大小分别等于$1$, $2$和$3$的三个力$\\overrightarrow{f_1}, \\overrightarrow{f_2}, \\overrightarrow{f_3}$, 则它们的合力的大小为\\bracket{20}.\n\\fourch{$6$}{$5$}{$4$}{$3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017795": { + "id": "017795", + "content": "如图所示, 正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是正方形$ADD_1A_1$和$ABCD$的中心, $G$是$CC_1$的中点. 设$GF$、$C_1E$与$AB$所成的角分别为$\\alpha, \\beta$. 求$\\alpha+\\beta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$A$} coordinate (A);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$C$} coordinate (C);\n\\draw (B) -- (A) -- (D);\n\\draw [dashed] (B) -- (C) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (A) ++ (0,\\l,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (C) ++ (0,\\l,0) node [above left] {$C_1$} coordinate (C_1);\n\\draw (B_1) -- (A_1) -- (D_1) -- (C_1) -- cycle;\n\\draw (B) -- (B_1) (A) -- (A_1) (D) -- (D_1);\n\\draw [dashed] (C) -- (C_1);\n\\draw ($(C)!0.5!(C_1)$) node [left] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(A_1)!0.5!(D)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (G)--(F)(C_1)--(E);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017796": { + "id": "017796", + "content": "如图, 在正四棱柱$ABCD-A_1B_1C_1D_1$中, 已知$AB=1$, $BB_1=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{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,-\\m) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\m) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\n,0) node [left] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D1);\n\\draw (A) ++ (0,\\n,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);\n\\draw (B1)--(D1);\n\\draw [dashed] (A1)--(C)(B1)--(A)--(D1);\n\\end{tikzpicture}\n\\end{center} \n(1) 求异面直线$A_1C$与直线$AD_1$所成的角的大小;\\\\\n(2) 求点$C$到平面$AB_1D_1$的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017797": { + "id": "017797", + "content": "如图, 在六面体$ABCD-A_1B_1C_1D_1$中, 四边形$ABCD$是边长为$2$的正方形, 四边形$A_1B_1C_1D_1$是边长为$1$的正方形, $DD_1 \\perp$平面$A_1B_1C_1D_1$, $DD_1 \\perp$平面$ABCD$, $DD_1=2$.\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 (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (D_1) ++ (0,0,{\\l/2}) node [left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ ({\\l/2},0,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (D_1) ++ ({\\l/2},0,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C)(B)--(D);\n\\draw (A_1)--(C_1)(B_1)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $A_1C_1$与$AC$共面, $B_1D_1$与$BD$共面;\\\\\n(2) 求二面角$A-BB_1-C$的大小 (结果用反三角函数值表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017798": { + "id": "017798", + "content": "如图, 四棱锥$P-ABCD$中, $PA \\perp$底面$ABCD$. 四边形$ABCD$中, $AB \\perp AD$, $AB+AD=4$, $CD=\\sqrt{2}$, $\\angle CDA=45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (2,0,1) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$PAB \\perp$平面$PAD$;\\\\\n(2) 设$AB=AP$, 在线段$AD$上是否存在一个点$G$, 使得点$G$到点$P$、$B$、$C$、$D$的距离都相等? 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试16空间向量及其应用二单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017799": { + "id": "017799", + "content": "已知$\\{a_n\\}$为等差数列, 若$a_1=6$, $a_3+a_5=0$, 则数列$\\{a_n\\}$的通项公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017800": { + "id": "017800", + "content": "若等比数列$\\{a_n\\}$满足$a_2+a_4=20$, $a_3+a_5=40$, 则公比$q=$\\blank{50}; 前$n$项和$S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017801": { + "id": "017801", + "content": "数列$\\{a_n\\}$满足$a_1=3+\\dfrac{\\sqrt{6}}{2}$, $a_{n+1}=[a_n]+\\dfrac{1}{\\{a_n\\}}$, 其中$[a_n]$、$\\{a_n\\}$分别表示$a_n$的整数部分和小数部分, 则$a_{2020}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017802": { + "id": "017802", + "content": "用数学归纳法证明``$(n+1)(n+2) \\cdots (n+n-1)(n+n)=2^n \\cdot 1 \\cdot 3 \\cdot 5 \\cdot (2 n-1)$($n \\in \\mathbf{N}$, $n \\geq 1$)''时, 从``$n=k$''到``$n=k+1$'', 左边需要添加的代数式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017803": { + "id": "017803", + "content": "设$S_n$是公比为$q$的等比数列$\\{a_n\\}$的前$n$项的和, 若对任意的正整数$k$, 都有$\\displaystyle\\sum_{i=k+2}^{+\\infty} a_i=a_k$成立, 则$q=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017804": { + "id": "017804", + "content": "设等差数列$\\{a_n\\}$的各项都是正数, 其前$n$项和为$S_n$, 公差为$d$. 若数列$\\{\\sqrt{S_n}\\}$也是公差为$d$的等差数列, 则$\\{a_n\\}$的通项公式为$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017805": { + "id": "017805", + "content": "已知数列$\\{a_n\\}$满足$a_1=m$($m \\in \\mathbf{N}$, $m \\geq 1$), $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2}, & a_n \\text {为偶数}, \\\\ 3 a_n+1, & a_n \\text {为奇数}.\\end{cases}$ 若$a_6=1$, 则$m$的所有可能的取值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017806": { + "id": "017806", + "content": "已知数列$\\{a_n\\}$是以$q$为公比的等比数列. 若$b_n=-2 a_n$, 则数列$\\{b_n\\}$是\\bracket{20}.\n\\twoch{以$q$为公比的等比数列}{以$-q$为公比的等比数列}{以$2 q$为公比的等比数列}{以$-2 q$为公比的等比数列}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017807": { + "id": "017807", + "content": "设等差数列$\\{a_n\\}$的公差为$d$, $d \\neq 0$. 若$\\{a_n\\}$的前$10$项之和大于其前$21$项之和, 则\\bracket{20}.\n\\fourch{$d<0$}{$d>0$}{$a_{16}<0$}{$a_{16}>0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017808": { + "id": "017808", + "content": "已知数列$\\{a_n\\}$满足$a_n+a_{n+4}=a_{n+1}+a_{n+3}$($n \\in \\mathbf{N}$, $n \\geq 1$), 那么\\bracket{20}.\n\\fourch{$\\{a_n\\}$是等差数列}{$\\{a_{2 n-1}\\}$是等差数列}{$\\{a_{2 n}\\}$是等差数列}{$\\{a_{3 n}\\}$是等差数列}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017809": { + "id": "017809", + "content": "已知等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 公比是正数的等比数列$\\{b_n\\}$的前$n$项和为$T_n$. 已知$a_1=1$, $b_1=3$, $a_3+b_3=17$, $T_3-S_3=12$, 求$\\{a_n\\}$, $\\{b_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017810": { + "id": "017810", + "content": "我们把所有真约数 (除其本身之外的正约数) 的和等于它本身的正整数叫做``完全数''. 如$6$、$28$、$496$都是``完全数'': $6=1+2+3$; $28=1+2+4+7+14$; $496=1+2+4+8+16+31+62+124+248$. 试判断命题``若$2^n-1$是质数, 则$2^{n-1}(2^n-1)$($n \\in \\mathbf{N}$, $n \\geq 2)$是``完全数'''的真假, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017811": { + "id": "017811", + "content": "已知数列$\\{b_n\\}$是公差不为$0$的等差数列, $b_1=\\dfrac{3}{2}$, 数列$\\{a_n\\}$是等比数列, 且$a_1=b_1$, $a_2=-b_3$, $a_3=b_4$, 数列$\\{a_n\\}$的前$n$项和为$S_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若$\\lambda \\leq S_n-\\dfrac{1}{S_n} \\leq \\mu$对任意$n \\in \\mathbf{N}$($n \\geq 1$)都成立, 求$\\mu-\\lambda$的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017812": { + "id": "017812", + "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$与函数$f(x)$, $\\{a_n\\}$是首项$a_1=15$, 公差$d \\neq 0$的等差数列, $\\{b_n\\}$满足: $b_n=f(a_n)$.\\\\\n(1) 若$a_4, a_7, a_8$成等比数列, 求$d$的值;\\\\\n(2) 若$d=2$, $f(x)=|x-21|$, 求$\\{b_n\\}$的前$n$项和$S_n$;\\\\\n(3) 若$d=-1$, $f(x)=\\mathrm{e}^x$, $T_n=b_1 \\cdot b_2 \\cdot b_3 \\cdots b_n$, 问$n$为何值时, $T_n$的值最大?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试17数列和数学归纳法单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017813": { + "id": "017813", + "content": "已知函数$f(x)=\\mathrm{e}^x$, 若$f'(x_0)=1$, 则$x_0=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017814": { + "id": "017814", + "content": "已知函数$f(x)=x+\\tan x$, 则$f'(\\dfrac{\\pi}{3})$的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017815": { + "id": "017815", + "content": "已知定义在$\\mathbf{R}$上的函数$f(x)$, 其导函数为$f'(x)$, 满足$f'(x)>2$, 且$f(2)=1$, 则不等式$f(x)>2 x-3$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017816": { + "id": "017816", + "content": "已知函数$f(x)=(x+1) \\ln x$, 则曲线$y=f(x)$在$(1, f(1))$处的切线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017817": { + "id": "017817", + "content": "已知曲线$f(x)=x \\mathrm{e}^x-\\dfrac{1}{\\mathrm{e}} \\ln x+1$在点$(x_0, f(x_0))$处的切线的斜率为$\\dfrac{1}{\\mathrm{e}}$, $x_0+\\ln x_0=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017818": { + "id": "017818", + "content": "设曲线$y=\\dfrac{1}{2} x^2$在点$A(1, \\dfrac{1}{2})$处的切线与曲线$y=x \\ln x$在点$P$处的切线互相平行, 则点$P$的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017819": { + "id": "017819", + "content": "如图为函数$f(x)$的导函数的图像, 则下列判断正确的是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-4,0) -- (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\\foreach \\i in {-3,-2,-1,1,2,3,4}\n{\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};};\n\\draw [domain = -3:1] plot (\\x,{sin(45*\\x+45)*2});\n\\draw [domain = 1:5] plot (\\x,{sin(90*\\x)*2});\n\\draw [dashed] (1,0) -- (1,2) (3,0) -- (3,-2);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在$(-3,1)$上单调增;\n\\textcircled{2} $x=-1$是$f(x)$的极小值点;\n\\textcircled{3} $f(x)$在$(2,4)$上单调减, 在$(-1,2)$上单调增;\n\\textcircled{4} $x=2$是$f(x)$的极小值点.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017820": { + "id": "017820", + "content": "已知函数$f(x)=\\dfrac{1+\\ln x}{x}$在区间$(a, a+\\dfrac{2}{3})$($a>0$)上存在极值, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017821": { + "id": "017821", + "content": "如图, 已知圆柱和半径为$\\sqrt{3}$的半球$O$, 圆柱的下底面在半球$O$底面所在平面上, 圆柱的上底面内接于球$O$, 则该圆柱体积的最大值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw [dashed] (O) ellipse (1.6 and 0.4);\n\\draw (2,0) arc (0:180:2);\n\\draw (2,0) arc (360:180:2 and 0.5);\n\\draw [dashed] (2,0) arc (0:180:2 and 0.5);\n\\draw [dashed] (1.6,1.2) arc (0:180:1.6 and 0.4);\n\\draw (1.6,1.2) arc (360:180:1.6 and 0.4);\n\\draw [dashed] (1.6,0) -- (1.6,1.2) (-1.6,0) -- (-1.6,1.2);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017822": { + "id": "017822", + "content": "已知函数$f(x)$图像的对称轴为$x=1$, 当$x>1$时, $f(x)=\\dfrac{x}{\\ln x}$, 若$f^2(x)-2 m f(x)+4 m=0$有$8$个不同的实数解, 则实数$m$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017823": { + "id": "017823", + "content": "设函数$f(x)=\\ln x+1$, 则$\\displaystyle\\lim _{\\Delta x \\to 0} \\dfrac{f(1+5 \\Delta x)-f(1)}{\\Delta x}=$\\bracket{20}.\n\\fourch{$1$}{$5$}{$\\dfrac{1}{5}$}{$0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017824": { + "id": "017824", + "content": "有一机器人的运动方程为$s(t)=t^2+3 t$, ($t$是时间, $s$是位移), 则该机器人在时刻$t=2$时的瞬时速度为\\bracket{20}.\n\\fourch{$5$}{$7$}{$10$}{$13$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017825": { + "id": "017825", + "content": "函数$y=f(x)$的图像如图所示, $f'(x)$是$f(x)$的导函数, 则下列数值排序正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (0,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 1.2:3.5] plot (\\x,{\\x*\\x*\\x/12+0.2});\n\\draw [dashed] (2,0) node [below] {$2$} --++ (0,{8/12+0.2}) --++ (-2,0);\n\\draw [dashed] (3,0) node [below] {$3$} --++ (0,{27/12+0.2}) --++ (-3,0);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$f'(2)=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw [dashed] (A) rectangle (C);\n\\draw (A) --++ (2,0.5) node [right] {$E$} coordinate (E);\n\\draw (A) --++ (0.5,2) node [above] {$F$};\n\\draw (E) arc ({atan(1/4)}:{atan(4)}:{sqrt(4+0.25)});\n\\draw (1,1) node {I};\n\\draw (2,0) node [above left] {II};\n\\draw (0,2) node [below right] {II};\n\\draw (2,2) node [below left] {III};\n\\end{tikzpicture}\n\\end{center}\n(1) 要使观赏区的年收人不低于$5$万元, 求$\\theta$的最大值;\\\\\n(2) 试问: 当$\\theta$为多少时, 年总收人最大?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017829": { + "id": "017829", + "content": "已知函数$f(x)=\\ln x$.\\\\\n(1) 当$x \\in[1,+\\infty)$时, 证明: 函数$f(x)$的图像恒在函数$g(x)=\\dfrac{2}{3} x^3-\\dfrac{1}{2} x^2$的图像的下方;\\\\\n(2) 讨论方程$f(x)+k x=0$的根的个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试18导数及其应用单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017830": { + "id": "017830", + "content": "个位数是$0$, 且各位上的数字互不相同的三位数共有\\blank{50}个.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017831": { + "id": "017831", + "content": "$20$部5G手机中, 有$15$部为一等品, $5$部为二等品, 现从中取$4$部手机, 求取到的手机中至多有$1$部为二等品的概率\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017832": { + "id": "017832", + "content": "观察下列各式: $\\mathrm{C}_1^0=4^0$; $\\mathrm{C}_3^0+\\mathrm{C}_3^1=4^1$; $\\mathrm{C}_5^0+\\mathrm{C}_5^1+\\mathrm{C}_5^2=4^2$; $\\mathrm{C}_7^0+\\mathrm{C}_7^1+\\mathrm{C}_7^2+\\mathrm{C}_7^3=4^3$; $\\cdots$; 照此规律, 当$n \\in \\mathbf{N}$($n \\geq 1$)时, $C_{2 n-1}^0+C_{2 n-1}^2+\\cdots+C_{2 n-1}^{n-1}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017833": { + "id": "017833", + "content": "满足$a, b \\in\\{-1,0,1,2\\}$, 且关于$x$的方程$a x^2+2 x+b=0$有实数解, 则有序数对$(a, b)$的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017834": { + "id": "017834", + "content": "设虚数$a=2+\\mathrm{i}$, 化简$1-\\mathrm{C}_{12}^1 a+\\mathrm{C}_{12}^2 a^2-\\mathrm{C}_{12}^3 a^3+\\cdots+\\mathrm{C}_{12}^{12} a^{12}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017835": { + "id": "017835", + "content": "从单词``equation''中选取$5$个不同的字母排成一排, 含有``qu''(其中``qu''相连且顺序不变) 的不同排列共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017836": { + "id": "017836", + "content": "如图, 在由二项式系数所构成的杨辉三角形中, 第\\blank{50}行中从左至右第$14$与第$15$个数的比为$2: 3$.\n\\begin{center}\n\\begin{tabular}{cc}\n第0行 & 1 \\\\\n第1行 & 1 \\quad 1 \\\\\n第2行 & 1 \\quad 2 \\quad 1 \\\\\n第3行 & 1 \\quad 3 \\quad 3 \\quad 1 \\\\\n第4行 & 1 \\quad 4 \\quad 6 \\quad 4 \\quad 1 \\\\\n第5行 & 1 \\quad 5 \\quad 10 \\quad 10 \\quad 5 \\quad 1 \\\\\n$\\cdots$ & $\\cdots$ \\quad $\\cdots$ \\quad $\\cdots$ \\quad $\\cdots$\n\\end{tabular}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017837": { + "id": "017837", + "content": "设常数$a \\in \\mathbf{R}$, 若$(x^2+\\dfrac{a}{x})^5$的二项展开式中含$x^7$项的系数为$-10$, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017838": { + "id": "017838", + "content": "一次国际会议, 从某大学外语系选出$11$名翻译, 其中$5$人只会英语, $4$人只会日语, 两人既会英语, 也会日语, 现从这$11$人中选出$4$名当英语翻译, $4$名当日语翻译, 不同的选法有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017839": { + "id": "017839", + "content": "设集合$A=\\{(x_1, x_2, x_3, x_4, x_5) | x_i \\in\\{-1,0,1\\},\\ i=1,2,3,4,5\\}$, 那么集合$A$中满足条件``$1 \\leq|x_1|+|x_2|+|x_3|+|x_4|+|x_5| \\leq 3$''的元素个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017840": { + "id": "017840", + "content": "某班级要从$4$名男生、 $2$名女生中选派$4$人参加某项社区服务, 如果要求至少有$1$名女生, 那么不同的选派方案种数为\\bracket{20}.\n\\fourch{$14$}{$24$}{$28$}{$48$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017841": { + "id": "017841", + "content": "$(1+\\dfrac{1}{x^2})(1+x)^6$的展开式中含$x^2$的项的系数为\\bracket{20}.\n\\fourch{$15$}{$20$}{$30$}{$35$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017842": { + "id": "017842", + "content": "使$(3 x+\\dfrac{1}{x \\sqrt{x}})^n$($n \\in \\mathbf{N}$, $n \\geq 1$)的展开式中含有常数项的最小的$n$为\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$7$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017843": { + "id": "017843", + "content": "设$(\\dfrac{\\sqrt{2}}{2}+x)^{2 n}=a_0+a_1 x+a_2 x^2+\\cdots+a_{2 n-1} x^{2 n-1}+a_{2 n} x^{2 n}$, 则$\\displaystyle\\lim_{n\\to\\infty}[(a_0+a_2+a_4+\\cdots+a_{2 n})^2-(a_1+a_3+a_5+\\cdots+a_{2 n-1})^2]=$\\bracket{20}.\n\\fourch{$-1$}{$0$}{$1$}{$\\dfrac{\\sqrt{2}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017844": { + "id": "017844", + "content": "从$5$名男生、 $3$名女生中选$5$名担任$5$门不同学科的课代表, 求符合下列条件的不同选取方法数.\\\\\n(1) $5$门课代表中必须有女生;\\\\\n(2) 英语课代表由女生担任.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017845": { + "id": "017845", + "content": "若$(x+2)^n=x^n+\\cdots+a x^3+b x^2+c x+d$($n \\in \\mathbf{N}$, $n \\geq 3$), 且$c: d=4: 1$, 求$a: b$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017846": { + "id": "017846", + "content": "已知函数$f(x)=\\sqrt{2} \\sin \\dfrac{\\pi}{3} x$, $A=\\{1,2,3,4,5, \\cdots, 10\\}$, 现从$A$中随机取两个不同数$a$、$b$, 求满足$f(a) \\cdot f(b)=0$的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017847": { + "id": "017847", + "content": "若$m, n \\in\\{x | x=a_2 \\times 10^2+a_1 \\times 10+a_0, \\ a_i \\in\\{1,2,3,4,5,6,7\\}(i=0,1,2)\\}$, 且$m+n=636$, 求实数对$(m, n)$表示平面上不同点的个数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试19计数原理单元测试试题18", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017848": { + "id": "017848", + "content": "将一枚质地均匀的硬币抛掷$2$次, 设事件$A$为``第一次出现正面'', 事件$B$为``第二次出现正面''. 求$P(A | B)$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题1", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017849": { + "id": "017849", + "content": "一个袋子里装有大小与质地相同的$8$个红球、$6$个白球, 甲、乙两人依次随机不放回地摸出$1$个球, 在甲摸到红球的条件下, 乙摸到白球的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题2", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017850": { + "id": "017850", + "content": "郑一颗骰子并观察出现的点数. 已知出现的点数不超过$3$, 求出现的点数是偶数的概率\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题3", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017851": { + "id": "017851", + "content": "某超市仓库中的家用空调的$60 \\%$来自 A 公司, $40 \\%$来自 B 公司, A、B 两个公司的一等品率分别是$80 \\%$和$90 \\%$. 现从仓库中任提取一台空调, 则它是一等品的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题4", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017852": { + "id": "017852", + "content": "一个不透明的袋中装有分别写上数字$1$、$2$、$3$、$4$、$5$、$6$的$6$张卡片, 现随机取出$2$张卡片, 用$X$表示卡片上数字较大的数, 则$X$的分布为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题5", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017853": { + "id": "017853", + "content": "某手机芯片生产厂, 第一车间的次品率为$0.15$, 第二车间的次品率为$0.12$, 两个车间生产的芯片都混合放在一个仓库里, 假设第一、二车间生产的芯片数量比例为$2: 3$, 今有一客户从手机芯片仓库中随机买一个芯片, 则该芯片是合格品的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题6", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017854": { + "id": "017854", + "content": "已知随机变量$X \\sim B(n, p)$, $E[X]=120$, $D[X]=48$, 则$n=$\\blank{50}, $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题7", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017855": { + "id": "017855", + "content": "将大小质地完全相同的球分别装人三个盒子, 每盒$10$个. 其中, 第一个盒子中有$7$个球标有字母$A$, $3$个球标有字母$B$; 第二个盒子中有红球和白球各$5$个; 第三个盒子中有红球$8$个, 白球$2$个. 试验按如下规则进行: 先在第一个盒子中任取一个球, 若取得标有字母$A$的球, 则在第二个盒子中任取一个球; 若在第一个盒子中取得标有字母$B$的球, 则在第三个盒子中任取一个球. 如果第二次取出的是红球, 那么试验成功, 则试验成功的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题8", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017856": { + "id": "017856", + "content": "已知随机事件$A$、$B$满足$P(B | A)=\\dfrac{1}{2}$, $P(A \\cap B)=\\dfrac{3}{8}$, 则$P(A)=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{16}$}{$\\dfrac{13}{16}$}{$\\dfrac{3}{4}$}{$\\dfrac{1}{4}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题9", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017857": { + "id": "017857", + "content": "小东步行上学途中要经过两个交通路口, 第一个路口遇见红灯的概率为$0.5$, 两个路口连续遇到红灯的概率为$0.35$, 则甲在第一个路口遇到红灯的条件下, 第二个路口遇到红灯的概率为\\bracket{20}.\n\\fourch{$0.6$}{$0.7$}{$0.8$}{$0.9$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题10", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017858": { + "id": "017858", + "content": "一个不透明的袋中装有除颜色外完全相同的$5$个红球和$2$个白球, 若采用不放回地依次取$2$个小球, 则在第$1$次取到红球的条件下, 第$2$次取到红球的概率是\\bracket{20}.\n\\fourch{$\\dfrac{3}{5}$}{$\\dfrac{3}{10}$}{$\\dfrac{2}{3}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题11", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017859": { + "id": "017859", + "content": "某射击选手每次射击击中目标的概率是$0.8$, 这名选手在$10$次独立的射击中, 恰有$8$次击中目标的概率为\\bracket{20}.\n\\fourch{$\\mathrm{C}_{10}^8 \\times 0.8^8 \\times 0.2^2$}{$0.8^8 \\times 0.2^2$}{$\\mathrm{C}_{10}^2 \\times 0.2^8 \\times 0.8^2$}{$0.2^8 \\times 0.8^2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题12", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017860": { + "id": "017860", + "content": "已知随机变量$X \\sim B(6, \\dfrac{1}{3})$, 则$P(X=2)=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{16}$}{$\\dfrac{4}{243}$}{$\\dfrac{13}{243}$}{$\\dfrac{80}{243}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题13", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017861": { + "id": "017861", + "content": "将编号为$1$、$2$、$3$、$4$的球随机放入编号为$1$、$2$、$3$、$4$的四个盒子里, 当球的编号与盒子的编号相同时称为一个配对. 用$X$表示配对数, 求$E[X]$、$D[X]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题14", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017862": { + "id": "017862", + "content": "已知甲袋子中装有$7$张分别写上数字$1$、$2$、$3$、$4$、$5$、$6$、$7$的卡片, 乙袋子中装有$6$张分别写上数字$8$、$9$、$10$、$11$、$12$、$14$的卡片. 现从甲袋中任取$2$张卡片放人乙袋中, 再从乙袋中任取$2$张卡片, 求从乙袋中取出的$2$张卡片上的数字都是奇数的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题15", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017863": { + "id": "017863", + "content": "一个不透明的袋中装有$6$个黄色、 $4$个白色的乒乓球(只有颜色不同, 大小质地完全相同), 不放回抽取, 每次任取一球, 取两次, 求:\\\\\n(1) 第二次才取到黄球的概率;\\\\\n(2) 取出的两个球的其中之一是黄球时, 另一个也是黄球的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题16", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017864": { + "id": "017864", + "content": "某蜂蜜生产公司的瓶装蜂蜜标识质量是$500 \\text{g}$, 已知每瓶蜂蜜的实际质量服从$\\mu=500$、$\\sigma^2=1.5^2$的正态分布. 该公司董事长承诺: 有$99 \\%$的把握保证顾客随意买一瓶蜂蜜其质量误差不超过$4.5 \\text{g}$, 试问董事长的话是否有道理?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高中数学质量测试与监控单元知识测试20概率初步续单元测试试题17", + "edit": [ + "20230606\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",