From 706a76fb58936286836b4af410df260978fdebdd Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 29 Jul 2023 10:47:17 +0800 Subject: [PATCH 1/7] =?UTF-8?q?=E6=B7=BB=E5=8A=A0mathpix=E4=BC=98=E5=8C=96?= =?UTF-8?q?=E4=B8=AD=E7=9A=84=E4=B8=A4=E4=B8=AA=E5=B8=B8=E9=94=99=E8=AF=8D?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/database_tools.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/工具v2/database_tools.py b/工具v2/database_tools.py index 1a2cda0e..02294ba6 100644 --- a/工具v2/database_tools.py +++ b/工具v2/database_tools.py @@ -976,7 +976,7 @@ def setCopy(string): # 写入剪切板内容 def RefineMathpix(raw_string): # 进一步修改mathpix得到的字符串 puctuationsfulltosemi = {"　": " ","。": ". ","．": ". ","，": ", ","：": ": ","；": "; ","（": "(","）": ")","？": "? ","“": "``","”": "''", "【": "[", "】": "]"} replacestrings = {r"\\overparen": r"\\overset\\frown", "eqslant": "eq", r"\\vec": r"\\overrightarrow ", r"\\bar": r"\\overline", r"\\lim": r"\\displaystyle\\lim", r"\\sum":r"\\displaystyle\\sum", r"\\prod":r"\\displaystyle\\prod", r"\\mid":"|", r"\^\{\\prime\}":"'",r"e\^":r"\\mathrm{e}^",r"/\s*/":r"\\parallel "} - wrongrecog = {"雉":"锥","[粗秿]圆":"椭圆","投郑":"投掷","抛郑":"抛掷","范目":"范围","揷":"插","末见":"未见","末成":"未成","针角":"钝角","幕函数":"幂函数","末知":"未知","阀值":"阈值","祖[桓晅]":"祖暅"} + wrongrecog = {"雉":"锥","[粗秿]圆":"椭圆","投郑":"投掷","抛郑":"抛掷","范目":"范围","揷":"插","末见":"未见","末成":"未成","针角":"钝角","幕函数":"幂函数","末知":"未知","阀值":"阈值","祖[桓晅]":"祖暅","图象":"图像","末使用":"未使用"} string = raw_string string = re.sub(r"\\left(?:\.?)|\\right(?:\.?)","",string) #删去括号前的\left与\right标记 for s in puctuationsfulltosemi: From 033af3766aa31ba5fd2ab7ff08a824b1f7c88a2d Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 29 Jul 2023 11:21:18 +0800 Subject: [PATCH 2/7] =?UTF-8?q?mathpix=E8=AF=86=E5=88=AB=E4=BC=98=E5=8C=96?= =?UTF-8?q?=E4=B8=AD=E6=96=B0=E5=A2=9E=E9=80=89=E6=8B=A9=E9=A2=98=E7=9A=84?= =?UTF-8?q?=E6=A8=AA=E7=BA=BF=E5=8F=98=E4=B8=BA=E6=8B=AC=E5=8F=B7=E7=9A=84?= =?UTF-8?q?=E5=8A=9F=E8=83=BD?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/database_tools.py | 1 + 1 file changed, 1 insertion(+) diff --git a/工具v2/database_tools.py b/工具v2/database_tools.py index 02294ba6..578c90b2 100644 --- a/工具v2/database_tools.py +++ b/工具v2/database_tools.py @@ -1013,6 +1013,7 @@ def RefineMathpix(raw_string): # 进一步修改mathpix得到的字符串 string = re.sub(r"(是|为|(?:=\$))\s*([,.;\n])",lambda matchobj: matchobj.group(1) + "\\blank{50}" + ("." if matchobj.group(2) == "\n" else "") + matchobj.group(2),string) #给行中的题目需要的地方加上空格 string = re.sub(r"([\u4e00-\u9fa5\$])(?:\\bracket\{20\})*[\.]*[\s]*\n\\fourch",lambda matchobj: matchobj.group(1)+"\\bracket{20}.\n\\fourch",string) #给中文或公式结尾的题目最后一行加上选择题的括号. string = re.sub(r"(%[^\n]*)\\blank\{50\}\.",lambda matchobj:matchobj.group(1),string) #注释行不加\blank{50} + string = re.sub(r"\\blank\{50\}\.\s*\n\\fourch",r"\\bracket{20}."+"\n"+r"\\fourch",string) #选择题前的空格改为括号 string = re.sub(r"[\\\\]*\n(\(\d{1,2}\))(?:(?!\n)\s)*",lambda matchobj: "\\\\\n"+matchobj.group(1)+" ",string) #新一行的小题号回车前加上换行符 string = re.sub(r"\(([^\(\)]*(?:\\in|=|\\ge|\\le|\\ne|>|<|\\parallel|\\perp)[^\(\)]*)\)\$",lambda matchobj: "$($" + matchobj.group(1) + "$)",string) #公式最后的范围陈述的括号放到公式环境外 string = re.sub(r"\$\$\(","(",string) #删去上一步造成的多余双$ From 0a9e6a0673e80017a318c89fa0eb253d1beca8db Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 30 Jul 2023 14:38:06 +0800 Subject: [PATCH 3/7] =?UTF-8?q?=E6=B7=BB=E5=8A=A019304-19790=E9=AB=98?= =?UTF-8?q?=E8=80=83=E6=95=B0=E5=AD=A6=E9=A3=8E=E6=9A=B4=E9=83=A8=E5=88=86?= =?UTF-8?q?=E9=A2=98=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 9900 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 9900 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 9829a8cb..07b3da02 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -494437,6 +494437,9906 @@ "space": "4em", "unrelated": [] }, + "019304": { + "id": "019304", + "content": "命题``存在 $x \\in \\mathbf{R}$, 使得 $x^2+2 x+2 \\leq 0$''的否定是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019305": { + "id": "019305", + "content": "设不等式 $\\dfrac{a(x-2)}{x+3}<2$ 的解集为 $A$, 且 $1 \\notin A$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019306": { + "id": "019306", + "content": "迎春杯数学竞赛后, 甲、乙、丙、丁四名同学猜测他们之中谁能获奖. 甲说:``如果我能获奖, 那么乙也能获奖.''乙说:``如果我能获奖, 那么丙也能获奖.''丙说:``如果丁没获奖, 那么我也不能获奖.''实际上, 他们之中只有一个人没有获奖, 并且甲、乙、丙说的话都是正确的. 那么没能获奖的同学是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019307": { + "id": "019307", + "content": "若关于 $x$ 的不等式 $x^2+a x-a-2>0$ 和 $2 x^2+2(2 a+1) x+4 a^2+1>0$ 的解集依次为 $A$ 和 $B$, 那么, 使得 $A=\\mathbf{R}$ 和 $B=\\mathbf{R}$ 至少有一个成立的实数 $a$\\bracket{20}.\n\\onech{可以是 $\\mathbf{R}$ 中的任何一个数}{有无穷多个, 但并不是 $\\mathbf{R}$ 中所有的实数都能满足要求}{有且仅有一个}{不存在}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019308": { + "id": "019308", + "content": "已知 $x \\in \\mathbf{R}$, 条件 $p: x^20$), 若 $p$ 是 $q$ 的充分不必要条件,则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019309": { + "id": "019309", + "content": "已知集合 $A=\\{x | x^2+p x-2 q=0\\}$, $B=\\{x | x^2+q x-4 q^2+2 p=0\\}$, 则``实数 $p= q=1$''是``$1 \\in A \\cap B$''的\\bracket{20}.\n\\twoch{充分必要条件}{充分不必要条件}{必要不充分条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019310": { + "id": "019310", + "content": "已知 $a, b, c$ 为实常数,数列 $\\{x_n\\}$ 的通项为 $x_n=a n^2+b n+c$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则``存在正整数 $k$, 使得 $x_{100+k}, x_{200+k}, x_{300+k}$ 成等差数列''的一个必要条件为\\bracket{20}.\n\\fourch{$a \\geq 0$}{$b \\leq 0$}{$c=0$}{$a-2 b+c=0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019311": { + "id": "019311", + "content": "设 $A=\\{x | x=m^2-n^2, m, n \\in \\mathbf{Z}\\}$, $B=\\{8,9,10\\}$, 则 $A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019312": { + "id": "019312", + "content": "设集合 $A=\\{1,2, m\\}$, 其中 $m$ 为实数, 令 $B=\\{a^2 | a \\in A\\}$, $C=A \\cup B$, 若 $C$ 的所有元素之和为$6$,则 $C$ 的所有元素之积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019313": { + "id": "019313", + "content": "已知集合 $A=\\{x | x=a^2-b^2, a, b \\in \\mathbf{Z}\\}$, 则下列说法不正确的是\\bracket{20}.\n\\twoch{$2023 \\in A$}{所有质数都在集合 $A$ 中}{所有奇数都在集合 $A$ 中}{所有 $4$ 的倍数都在集合 $A$ 中}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019314": { + "id": "019314", + "content": "设集合 $M=\\{(x, y) | x^2+y^2 \\geq 1\\}$, $N=\\{(x, y) | y>x+\\sqrt{2}\\}$, 求证: $N \\subset M$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019315": { + "id": "019315", + "content": "已知函数 $y=f(x)$ 的定义域为 $\\mathbf{R}$, 下列是 $f(x)$ 无最大值的一个充分条件是\\bracket{20}.\n\\twoch{$f(x)$ 是偶函数且关于点 $(1,1)$ 对称}{$f(x)$ 是偶函数且关于直线 $x=1$ 对称}{$f(x)$ 是奇函数且关于点 $(1,1)$ 对称}{$f(x)$ 是奇函数且关于直线 $x=1$ 对称}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019316": { + "id": "019316", + "content": "已知命题 $p$ : 存在实数 $a \\neq 0$, 对于任意的 $x \\in \\mathbf{R}$, 都有 $f(x+a)0$;\n命题 $q_2: f(x)$ 是 $\\mathbf{R}$ 上严格递增的函数且存在 $x_0<0$ 使得 $f(x_0)=0$,\n证明: $q_1, q_2$ 都是 $p$ 的充分条件.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019317": { + "id": "019317", + "content": "已知函数 $f(x)=x^{\\frac{1}{3}}$, $x \\in[8,64]$ 的值域为 $A$, 集合 $B=\\{x | x^2-4 x+3<0\\}$, 则 $A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019318": { + "id": "019318", + "content": "$A=\\{-1,0, a\\}$, $B=\\{x | 1<3^x<100, x \\in \\mathbf{Z}\\}$, 若 $A \\cap B \\neq \\varnothing$, 则实数 $a$ 组成的集合是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019319": { + "id": "019319", + "content": "已知全集 $U=\\{1,2,3,4,5\\}$, 集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | x=2 a, a \\in A\\}$, 则集合 $\\overline{A \\cup B}$ 中元素的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019320": { + "id": "019320", + "content": "已知互异的复数 $a, b$ 满足 $a b \\neq 0$, 集合 $\\{a, b\\}=\\{a^2, b^2\\}$, 则 $a+b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019321": { + "id": "019321", + "content": "已知 $a \\in \\mathbf{R}$, 不等式 $\\dfrac{x-3}{x+a}\\geq 1$ 的解集为 $P$, 且 $-2 \\notin P$,则 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019322": { + "id": "019322", + "content": "$A_k=\\{x | x=k t+\\dfrac{1}{k t},\\ \\dfrac{1}{k^2}\\leq t \\leq 1\\}$, 其中 $k=2,3, \\cdots, 2018$, 则所有 $A_k$ 的交集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019323": { + "id": "019323", + "content": "设 $z_1, z_2 \\in \\mathbf{C}$, 则``$z_1, z_2$ 中至少有一个数是虚数''是``$z_1-z_2$ 是虚数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019324": { + "id": "019324", + "content": "命题 $A:|x-1|<3$, 命题 $B:(x+2)(x+a)<0$. 若 $A$ 是 $B$ 的充分而不必要条件,则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-4)$}{$[4,+\\infty)$}{$(4,+\\infty)$}{$(-\\infty,-4]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019325": { + "id": "019325", + "content": "设集合 $A=\\{(x, y) | x-y \\geq 1, \\ a x+y>4,\\ x-a y \\leq 2\\}$, 则\\bracket{20}.\n\\twoch{对任意实数 $a,(2,1) \\in A$}{对任意实数 $a,(2,1) \\notin A$}{当且仅当 $a<0$ 时, $(2,1) \\notin A$}{当且仅当 $a \\leq \\dfrac{3}{2}$ 时, $(2,1) \\notin A$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019326": { + "id": "019326", + "content": "已知集合 $A=\\{x|| x-2 |0\\}$, 集合 $B=\\{x | \\dfrac{2 x-2}{x+3}<1\\}$.\\\\\n(1) 若 $a=1$, 求 $A \\cap B$;\\\\\n(2) 若 $A \\subset B$, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019327": { + "id": "019327", + "content": "对任意一个非零复数 $z$ 定义集合 $M_z=\\{\\omega | \\omega=z^{2 n-1}, n$ 是正整数$\\}$.\\\\\n(1) 设 $\\alpha$ 是方程 $x+\\dfrac{1}{x}=\\sqrt{2}$ 的一个根, 试用列举法表示集合 $M_\\alpha$. 若在 $M_\\alpha$ 中任取两个数, 求其和为零的概率 $p$;\\\\\n(2) 若复数 $\\omega \\in M_z$, 求证: $M_\\omega \\subseteq M_z$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019328": { + "id": "019328", + "content": "若对于任意的 $a \\in A$, 总有 $-a \\notin A$, 则称集合 $A$ 具有性质 $P$. 已知集合 $A=\\{a_1, a_2, a_3, \\cdots, a_k\\}$($k \\geq 2$), 其中 $a_i \\in \\mathbf{Z}$($i=1,2, \\cdots, k$), 由 $A$ 中的元素构成的集合 $T=\\{(a, b) | a \\in A,\\ b \\in A,\\ a-b \\in A\\}$, 其中 $(a, b)$ 是有序实数对, 集合 $T$ 的元素个数为 $n$.\\\\\n(1) 检验集合 $\\{0,1,2,3\\}$ 与 $\\{-1,2,3\\}$ 是否具有性质 $P$, 并对其中具有性质 $P$ 的集合写出相应的集合 $T$;\\\\\n(2) 对任何具有性质 $P$ 的集合 $A$, 证明 : $n \\leq \\dfrac{k(k-1)}{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书01集合与逻辑", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019329": { + "id": "019329", + "content": "已知 $a, b \\in \\mathbf{R}$, 且 $a b \\neq 0$, 则``$a>b$''是``$\\dfrac{1}{a}<\\dfrac{1}{b}$''成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019330": { + "id": "019330", + "content": "若 $a>b>0$, $c0$ 的解集为 $(p, q)$, 其中 $q<0$, 则关于 $x$ 的一元二次不等式 $c x^2-b x+a>0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019337": { + "id": "019337", + "content": "求关于 $x$ 的不等式 $\\dfrac{(x+2)(x-a)}{x}\\geq 0$ (其中 $a \\in \\mathbf{R}$) 的解集.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019338": { + "id": "019338", + "content": "若 $x>0$, $y>0$, 且 $\\dfrac{1}{x}+2 y=3$, 则 $\\dfrac{y}{x}$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019339": { + "id": "019339", + "content": "若存在实数 $x$, 使得不等式 $|x+3|+|x-1| \\leq a^2-3 a$ 有解, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019340": { + "id": "019340", + "content": "已知 $x>0$, $y>0$, $x+y=4$, 则使不等式 $\\dfrac{1}{x}+\\dfrac{4}{y}\\geq m$ 恒成立的实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019341": { + "id": "019341", + "content": "已知 $a>b>0$, 当代数式 $a^2+\\dfrac{16}{b(a-b)}$ 取最小值时, 点 $P(a, b)$ 的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019342": { + "id": "019342", + "content": "设 $x, y$ 是正实数, 且 $x+y=1$, 则 $\\dfrac{x^2}{x+2}+\\dfrac{y^2}{y+1}$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019343": { + "id": "019343", + "content": "在平面直角坐标系中, 定义 $P(x_1, y_1)$、$Q(x_2, y_2)$ 之间的``直角距离''为$d(P, Q)=|x_1-x_2|+|y_1-y_2|$.\\\\\n(1) 已知 $A(x, 1)$、$B(1,2)$、$C(5,2)$ 三点, 若 $d(A, B)>d(A, C)$, 求 $x$ 的取值范围;\\\\\n(2) 已知 $A(x-1, y)$、$B(0,-1)$、$C(x, y-2)$ 三点, 对任意 $x, y \\in \\mathbf{R}$, 不等式 $d(A, B) \\geq t^2-2 t-d(B, C)$ 恒成立, 求 $t$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019344": { + "id": "019344", + "content": "设关于 $x$ 的不等式 $(a x-a^2-9)(x-b) \\geq 0$ 的解集为 $A$, 其中 $a, b \\in \\mathbf{R}$.\\\\\n(1) 若 $b=6$, $A=(-\\infty,+\\infty)$, 求 $a$ 的值;\\\\\n(2) 若 $b=6$, 记 $L=d-c$ 为闭区间 $[c, d]$ 的长度. 当 $a<0$ 时, 求区间 $A$ 的长度 $L$ 的最小值;\\\\\n(3) 当 $b=2 a-8$, 且 $a<9$ 时, 求 $A$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019345": { + "id": "019345", + "content": "如图, 射线 $OA, OB$ 均为笔直的公路, 扇形 $OPQ$ 区域 (含边界) 是一种蔬菜种植园, 其中 $P, Q$ 分别在射线 $OA, OB$ 上, 扇形 $OPQ$ 的中心角 $\\angle POQ=\\dfrac{2 \\pi}{3}$, 半径 $OP=1$ 千米. 为了方便菜农经营, 打算在扇形 $OPQ$ 区域外修建一条公路 $MN$, 分别与射线 $OA, OB$ 交于 $M, N$ 两点, 并要求直线 $MN$ 与圆弧 $\\overset\\frown{PQ}$ 相切于点 $S$, 设 $\\angle POS=\\alpha$, 假设所有公路的宽度都忽略不计.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (60:1) node [right] {$Q$} coordinate (Q);\n\\draw (180:1) node [below] {$P$} coordinate (P);\n\\draw (P) arc (180:60:1);\n\\draw (115:1) node [above left] {$S$} coordinate (S);\n\\draw (S) ++ (205:{tan(65)}) node [below] {$M$} coordinate (M);\n\\draw (S) ++ (25:{tan(55)}) node [right] {$N$} coordinate (N);\n\\draw ($(O)!1.2!(M)$) node [below] {$A$} coordinate (A);\n\\draw ($(O)!1.2!(N)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(O)--(B)(O)--(S)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 试将公路 $MN$ 的长度表示为 $\\alpha$ 的函数, 并写出定义域;\\\\\n(2) 试确定 $\\alpha$ 的值, 使公路 $MN$ 长度最小, 并求出其最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019346": { + "id": "019346", + "content": "若 $x_1$、$x_2$ 是方程 $a^x=(\\dfrac{1}{a})^{-\\frac{1}{x}+1}$($a>1$) 的两个实数解, 则 $x_1+x_2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019347": { + "id": "019347", + "content": "已知不等式 $a x^2+b x+c>0$ 的解为 $-30$, $n>0$, 且 $m+n=1$, 求 $\\dfrac{1}{m}+\\dfrac{1}{n}$ 的最小值''提出各自的解题思路:\\\\\n甲: $\\dfrac{1}{m}+\\dfrac{1}{n}=\\dfrac{m+n}{m}+\\dfrac{m+n}{n}=2+\\dfrac{n}{m}+\\dfrac{m}{n}$, 可用基本不等式求解;\\\\\n乙: $\\dfrac{1}{m}+\\dfrac{1}{n}=\\dfrac{m+n}{m m}=\\dfrac{1}{m n}=\\dfrac{1}{m(1-m)}$, 可用二次函数配方法求解;\\\\\n丙: $\\dfrac{1}{m}+\\dfrac{1}{n}=(\\dfrac{1}{m}+\\dfrac{1}{n})(m+n)=2+\\dfrac{n}{m}+\\dfrac{m}{n}$, 可用基本不等式求解;\\\\\n参考上述解题思路, 可求得当 $x=$\\blank{50}时, $y=\\dfrac{a^2}{x^2}+\\dfrac{1}{100-x^2}$($00$) 有最小值.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019352": { + "id": "019352", + "content": "下列不等式一定成立的是\\bracket{20}.\n\\twoch{$\\lg (x^2+\\dfrac{1}{4})>\\lg x$($x>0$)}{$\\sin x+\\dfrac{1}{\\sin x}\\geq 2$($x \\neq k \\pi$, $k \\in \\mathbf{Z}$)}{$x^2+1 \\geq 2|x|$($x \\in \\mathbf{R}$)}{$\\dfrac{1}{x^2+1}>1$($x \\in \\mathbf{R}$)}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019353": { + "id": "019353", + "content": "若不等式 $a \\leq|x-1|-|x-3| \\leq b$ 对任意实数 $x$ 恒成立, 则 $b-a$ 的最小值为\\bracket{20}.\n\\fourch{$4$}{$2$}{$-4$}{$-2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019354": { + "id": "019354", + "content": "若不等式 $(|x-a|-b) \\sin (\\pi x+\\dfrac{\\pi}{6}) \\leq 0$ 对 $x \\in[-1,1]$ 恒成立, 则 $a+b$ 的值等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{5}{6}$}{$1$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019355": { + "id": "019355", + "content": "某热力公司每年燃料费约 $24$ 万元, 为了``环评''达标, 需要安装一块面积为 $x$ (其中 $x \\geq 0$, 单位: 平方米) 可用 15 年的太阳能板, 其工本费为 $\\dfrac{x}{2}$ (单位: 万元), 并与燃料供热互补工作, 从此, 公司每年的燃料费为 $\\dfrac{k}{20 x+100}$($x \\geq 0$, $k$ 为常数) 万元. 记 $y$ 为该公司安装太阳能板的费用与 15 年的燃料费之和.\\\\\n(1) 求 $k$ 的值, 并建立 $y$ 关于 $x$ 的函数关系式;\\\\\n(2) 求 $y$ 的最小值, 并求出此时所安装太阳能板的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019356": { + "id": "019356", + "content": "已知 $x_1$、$x_2$ 是关于 $x$ 的一元二次方程 $4 k x^2-4 k x+k+1=0$ 的两个实数根.\\\\\n(1) 是否存在实数 $k$, 使得 $(2 x_1-x_2)(x_1-2 x_2)=-\\dfrac{3}{2}$ 成立? 若存在, 求出 $k$ 的值; 若不存在, 请说明理由;\\\\\n(2) 若 $k \\in \\mathbf{Z}$, 求使 $\\dfrac{x_1}{x_2}+\\dfrac{x_2}{x_1}-2$ 的值为整数的实数 $k$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019357": { + "id": "019357", + "content": "若函数 $f(x)$ 满足: 对于任意正数 $s, t$, 都有 $f(s)>0$, $f(t)>0$, 且 $f(s)+f(t)\\dfrac{x}{2}-\\dfrac{2}{x}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书02等式与不等式", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019358": { + "id": "019358", + "content": "下列图形中, (是以 $x$ 为自变量的) 函数的图像的是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{1}};\n\\foreach \\i in {0.1,0.2,...,2.5}\n{\\filldraw (\\i,{1.5+sin((\\i-0.1)/2.4*360)*1.2}) circle (0.02);};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{2}};\n\\draw (0,0) -- (0.5,2.8) -- (1,0.4) -- (1.5,2.6) -- (2,0.8) -- (2.5,2.4) -- (2.8,1.5);\n\\draw [dashed] (2.8,1.5) -- (2.8,0);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{3}};\n\\draw (1.5,1.5) circle (1.3);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019359": { + "id": "019359", + "content": "函数 $y=\\lg (3-2^x)$ 的定义域是\\blank{50}, 值域为\\blank{50}. (用区间表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019360": { + "id": "019360", + "content": "函数 $f(x)=\\begin{cases}(\\dfrac{1}{2})^x,& x<0,\\\\\\log _{\\frac{1}{2}}x,& x \\geq 0,\\end{cases}$ 则 $f(\\dfrac{1}{4})+f(\\log _2 \\dfrac{1}{6})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019361": { + "id": "019361", + "content": "设 $D$ 是含数$1$的有限实数集, $f(x)$ 是定义在 $D$ 上的函数, 若 $f(x)$ 的图像绕原点逆时针旋转 $\\dfrac{\\pi}{6}$ 后与原图像重合, 则在以下各项中, $f(1)$ 的取值只能是\\bracket{20}.\n\\fourch{$\\sqrt{3}$}{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019362": { + "id": "019362", + "content": "设曲线 $C$ 与函数 $f(x)=\\dfrac{\\sqrt{3}}{12}x^2$($00\\end{cases}$为奇函数, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019364": { + "id": "019364", + "content": "已知 $a \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2}, 1,2,3\\}$, 若幂函数 $f(x)=x^a$ 为奇函数, 且在 $(0,+\\infty)$ 上严格减, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019365": { + "id": "019365", + "content": "已知 $y=\\begin{cases}(3-a) x-2,& x<1\\\\x^2-2 x,& x \\geq 1\\end{cases}$ 是 $\\mathbf{R}$ 上的严格增函数, 则实数 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019366": { + "id": "019366", + "content": "已知 $a, b$ 为正实数, 函数 $f(x)=a x^3+b x+2^x$ 在 $[0,1]$ 上的最大值为 4 , 则 $f(x)$ 在 $[-1,0]$ 上的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019367": { + "id": "019367", + "content": "已知 $\\omega \\in \\mathbf{R}$, 函数 $f(x)=(x-6)^2 \\sin \\omega x$, 若存在常数 $a \\in \\mathbf{R}$, 使得 $f(x+a)$ 为偶函数, 则 $\\omega$ 的值可能为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019368": { + "id": "019368", + "content": "已知函数 $f(x)$ 是定义在 $(-1,0) \\cup(0,1)$ 上的奇函数, 当 $x \\in(-1,0)$ 时, $f(x)=a x+\\dfrac{1}{x^2}$.\\\\\n(1) 求函数 $y=f(x)$ 在 $(0,1)$ 上的函数解析式;\\\\\n(2) 当 $a>-2$ 时, 判断函数 $y=f(x)$ 在 $(0,1)$ 上的单调性, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019369": { + "id": "019369", + "content": "已知函数 $f(x)=\\dfrac{2-x}{x+1}$.\\\\\n(1) 求不等式 $f(x-4)+10$ 是 $F(x_1)+F(x_2)>0$ 的充要条件.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019371": { + "id": "019371", + "content": "已知$1$是函数 $f(x)$ 的周期, 且当 $0=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [below] {$-4$}-- (0,1) node [right] {$4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [below] {$-4$}-- (-1,1) -- (0,1) node [right] {$4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [above] {$-4$}-- (0,-1) node [right] {$-4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [above] {$-4$}-- (-1,-1) -- (0,-1) node [right] {$-4$};\n\\draw (1,0) node [below] {$4$} -- (1,1) -- (0,1) node [left] {$4$};\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019379": { + "id": "019379", + "content": "下列函数 $f(x)$ 中, 满足``对任意 $x_1, x_2 \\in(0,+\\infty)$, 当 $x_1f(x_2)$''的是\\bracket{20}.\n\\fourch{$f(x)=\\dfrac{1}{x}$}{$f(x)=(x-1)^2$}{$f(x)=\\mathrm{e}^x$}{$f(x)=\\ln (x+1)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019380": { + "id": "019380", + "content": "已知定义域为 $\\mathbf{R}$ 的函数 $f(x)=\\dfrac{1-2^x}{1+2^x}$.\\\\\n(1) 试判断函数 $f(x)=\\dfrac{1-2^x}{1+2^x}$ 在 $\\mathbf{R}$ 上的单调性, 并用函数单调性的定义证明;\\\\\n(2) 若对于任意 $t \\in \\mathbf{R}$, 不等式 $f(t^2-2 t)+f(t^2-k)<0$ 恒成立, 求实数 $k$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019381": { + "id": "019381", + "content": "设 $f(x)=x^3+a x^2-2 x$($x \\in \\mathbf{R}$), 其中常数 $a \\in \\mathbf{R}$.\\\\\n(1) 判断函数 $y=f(x)$ 的奇偶性, 并说明理由;\\\\\n(2) 若不等式 $f(x)>\\dfrac{3}{2}x^3$ 在区间 $[\\dfrac{1}{2}, 1]$ 上有解, 求实数 $a$ 的取值范围;\\\\\n(3) 已知: 若对函数 $y=h(x)$ 定义域内的任意 $x$, 都有 $h(x)+h(2 m-x)=2 n$, 则函数 $y=h(x)$ 的图像有对称中心 $(m, n)$. 利用以上结论探究: 对于任意的实数 $a$, 函数 $y=f(x)$ 是否都有对称中心? 若是, 求出对称中心的坐标 (用 $a$ 表示); 若不是, 证明你的结论.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书03函数的概念和性质", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019382": { + "id": "019382", + "content": "已知函数 $f(x), g(x)$ 在数集 $D$ 上都有定义, 对于任意的 $x_1, x_2 \\in D$, 当 $x_1b$}{$a ba^2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书04导数及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019403": { + "id": "019403", + "content": "如图为函数 $y=f(x)$ 的导函数 $y=f'(x)$ 的图像, 那么函数 $y=f(x)$ 的图像可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:3] plot (\\x,{\\x*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2.4] plot (\\x,{1.1*\\x*(\\x-1)*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.4:2.2] plot (\\x,{-\\x*(\\x-1)*(\\x-2)});\n\\draw [dashed] ({(3-sqrt(3))/3},{-2/3/sqrt(3)}) -- ({(3-sqrt(3))/3},0) node [above] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{-\\x*\\x+\\x*\\x*\\x/3+1});\n\\draw [dashed] (2,{-4+8/3+1}) -- (2,0) node [above] {$2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{\\x*\\x-\\x*\\x*\\x/3-0.5});\n\\draw [dashed] (2,{4-8/3-0.5}) -- (2,0) node [below] {$2$};\n\\end{tikzpicture}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书04导数及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019404": { + "id": "019404", + "content": "某商品的成本 $C$ 和产量 $q$ 满足函数关系 $C=50000+200 q$, 该商品的销售单价 $p$ 和产量 $q$ 满足函数关系 $p=24200-\\dfrac{1}{5}q^2$. 问 : 要使利润最大, 应如何确定产量?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书04导数及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019405": { + "id": "019405", + "content": "已知函数 $f(x)=x^2 \\mathrm{e}^{-x}$.\\\\\n(1) 求 $f(x)$ 的极小值和极大值;\\\\\n(2) 当曲线 $y=f(x)$ 的切线 $l$ 的斜率为负数时, 求 $l$ 在 $x$ 轴上的截距 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书04导数及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019406": { + "id": "019406", + "content": "已知函数 $f(x)=\\dfrac{\\mathrm{e}^x}{x}-\\ln x+x-a$.\\\\\n(1) 判断函数的单调区间;\\\\\n(2) 若 $f(x) \\geq 0$, 求 $a$ 的取值范围;\\\\\n(3) 证明: 若 $f(x)$ 有两个零点 $x_1$、$x_2$, 则 $x_1 x_2<1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书04导数及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019407": { + "id": "019407", + "content": "已知欧拉公式 $\\mathrm{e}^{\\mathrm{i}x}=\\cos x+\\mathrm{i}\\sin x$ ($\\mathrm{i}$ 为虚数单位), 则 $\\mathrm{e}^{2022 \\mathrm{i}}$ 对应的点在复平面位于第\\blank{50}象限.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019408": { + "id": "019408", + "content": "设 $O$ 是复平面的原点, 满足 $|z-\\mathrm{i}|+|z-1|=\\sqrt{2}$ 的复数在复平面上所对应的点构成集合 $M$,在 $M$ 中任取不同的两点 $A$ 和 $B$, 则 $\\angle AOB$ 的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019409": { + "id": "019409", + "content": "已知 $k+2$ 个两两互不相等的复数 $z_1$、$z_2$、$\\cdots$、$z_k$、$w_1$、$w_2$, 满足 $\\overline{w_1}-\\overline{w_2}=\\dfrac{4}{w_1-w_2}$, 且 $|w_j-z_\\alpha| \\in\\{1,3\\}$ (其中 $j=1$、$2$; $\\alpha=0$、$1$、$2$、$\\cdots$、$k$), 则 $k$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019410": { + "id": "019410", + "content": "已知 $|\\overrightarrow{a}|=3$, $|\\overrightarrow{b}|=5$, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=-12$, 则向量 $\\overrightarrow{a}$ 在向量 $\\overrightarrow{b}$ 上的投影向量为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019411": { + "id": "019411", + "content": "已知 $\\lambda>0$, 平面向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足: $|\\overrightarrow{a}|=|\\overrightarrow{b}|=|\\overrightarrow{c}|=\\lambda$, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=0$, $\\overrightarrow{c}\\cdot \\overrightarrow{a}=2$, $\\overrightarrow{c}\\cdot \\overrightarrow{b}=1$, 则 $\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019412": { + "id": "019412", + "content": "如图, 在 $\\triangle ABC$ 中, $\\angle C=\\dfrac{\\pi}{2}$, $AC= BC=2$, $M$ 在 $AC$ 的中点, $P$ 在 $AB$ 上, 则 $\\overrightarrow{MP}\\cdot \\overrightarrow{CP}$ 最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$C$} coordinate (C);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (0,1) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.3!(B)$) node [above right] {$P$} coordinate (P);\n\\draw [->] (M)--(P);\n\\draw [->] (C)--(P);\n\\draw (A)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019413": { + "id": "019413", + "content": "如图所示, 正八边形 $A_1A_2A_3A_4A_5A_6A_7A_8$ 的边长为 2 , 若 $P$ 为该正八边形边上的动点, 则 $\\overrightarrow{A_1A_3}\\cdot \\overrightarrow{A_1P}$ 的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1) --++ (0:1) node [below] {$A_2$} coordinate (A_2) --++ (45:1) node [right] {$A_3$} coordinate (A_3) --++ (90:1) node [right] {$A_4$} coordinate (A_4) --++ (135:1) node [above] {$A_5$} coordinate (A_5) --++ (180:1) node [above] {$A_6$} coordinate (A_6) --++ (225:1) node [left] {$A_7$} coordinate (A_7) --++ (270:1) node [left] {$A_8$} coordinate (A_8) -- cycle;\n\\draw [->] (A_1) -- (A_3);\n\\draw [->] (A_1) -- ($(A_6)!0.5!(A_7)$) node [above left] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt{2}]$}{$[-2 \\sqrt{2}, 8+6 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 2 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 8+6 \\sqrt{2}]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019414": { + "id": "019414", + "content": "如图, 四个棱长为 1 的正方体排成一个正四棱柱, $AB$ 是一条侧棱, $P_i$($i=1,2, \\cdots, 8$) 是上底面上其余的八个点, 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AP_i}$($i=1,2, \\cdots, 8$) 的不同值的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) coordinate (B) --++ (45:{2/2}) coordinate (C)\n--++ (0,1) coordinate (C1) node [above left] {$P_8$}\n--++ (-2,0) coordinate (D1) node [above left] {$P_2$} --++ (225:{2/2}) node [above left] {$B$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) coordinate (B1) node [above left] {$P_6$} -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$P_1$} coordinate (P1);\n\\draw ($(B1)!0.5!(C1)$) node [above left] {$P_7$} coordinate (P7);\n\\draw ($(A1)!0.5!(B1)$) node [above left] {$P_3$} coordinate (P3);\n\\draw ($(P1)!0.5!(P7)$) node [above left] {$P_4$} coordinate (P4);\n\\draw ($(C1)!0.5!(D1)$) node [above left] {$P_5$} coordinate (P5);\n\\draw (P1) -- (P7) (P3) -- (P5) (P3) --++ (0,-1) coordinate (S) (P7) --++ (0,-1);\n\\draw [dashed] (P1) --++ (0,-1) --++ (2,0) (P5) --++ (0,-1) -- (S);\n\\draw [dashed] (P4) --++ (0,-1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$2$}{$4$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019415": { + "id": "019415", + "content": "在平面直角坐标系 $xOy$ 中, 已知椭圆 $C_1: \\dfrac{x^2}{36}+\\dfrac{y^2}{4}=1$ 和 $C_2: x^2+\\dfrac{y^2}{9}=1, P$ 为 $C_1$ 上的动点, $Q$ 为 $C_2$ 上的动点, $w$ 是 $\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}$ 的最大值. 记 $\\Omega=\\{(P, Q) | P$ 在 $C_1$ 上, $Q$ 在 $C_2$ 上, 且 $\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}=w\\}$, 则 $\\Omega$ 中元素个数为\\bracket{20}.\n\\fourch{$2$个}{$4$个}{$8$个}{无穷个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019416": { + "id": "019416", + "content": "已知 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 是不平行的两个向量, $k$ 是实数, 且 $\\overrightarrow{AP}=k \\overrightarrow{AB}$, 试用 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 表示 $\\overrightarrow{OP}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019417": { + "id": "019417", + "content": "已知点 $A(-2,0)$, 设 $B$、$C$ 是圆 $O: x^2+y^2=1$ 上的两个不同的动点, 且向量 $\\overrightarrow{OB}=t \\overrightarrow{OA}+(1-t) \\overrightarrow{OC}$ (其中 $t$ 为实数), 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019418": { + "id": "019418", + "content": "在 $\\triangle ABC$ 中, 点 $O$ 是 $BC$ 的中点, 过点 $O$ 的直线分别交直线 $AB, AC$ 与不同的两点 $M, N$, 若 $\\overrightarrow{AB}=m \\overrightarrow{AM}$, $\\overrightarrow{AC}=n \\overrightarrow{AN}$, $m>0$, $n>0$, 求 $\\dfrac{1}{m}+\\dfrac{4}{n}$ 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019419": { + "id": "019419", + "content": "设 $|\\overrightarrow{OA}|=1$, $|\\overrightarrow{OB}|=2$, $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=0$, $\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$, 且 $\\lambda+\\mu=1$, 则 $\\overrightarrow{OA}$ 在 $\\overrightarrow{OP}$ 上投影的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019420": { + "id": "019420", + "content": "点 $P$ 是椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$ 上的动点, 点 $Q$ 与点 $P$ 关于 $x$ 轴对称, 若有 $\\overrightarrow{F_1P}$. $\\overrightarrow{F_2P}\\leq 1$, 则向量 $\\overrightarrow{F_1P}$ 与 $\\overrightarrow{F_2Q}$ 的夹角的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019421": { + "id": "019421", + "content": "若复数 $z$ 满足 $3 z+\\overline{z}=1+\\mathrm{i}$, 其中 $\\mathrm{i}$ 为虚数单位, 则 $z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019422": { + "id": "019422", + "content": "设 $m, n$ 分别为连续两次投掷骰子得到的点数, 且向量 $\\overrightarrow{a}=(m, n)$, $\\overrightarrow{b}=(1,-1)$, 则 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为锐角的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019423": { + "id": "019423", + "content": "在直角坐标系 $xOy$ 中, 已知三点 $A(a, 1), B(2, b), C(3,4)$, 若向量 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 在向量 $\\overrightarrow{OC}$ 方向上的数量投影相同, 则 $3 a-4 b$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019424": { + "id": "019424", + "content": "若点 $P$、$Q$ 均在椭圆 $\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{a^2-1}=1$($a>1$) 上运动, $F_1$、$F_2$ 是椭圆 $\\Gamma$ 的左、右焦点, 则 $|\\overrightarrow{PF_1}+\\overrightarrow{PF_2}-2 \\overrightarrow{PQ}|$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019425": { + "id": "019425", + "content": "在复平面上, 已知直线 $l$ 上的点所对应的复数 $z$ 满足 $|z-2-\\mathrm{i}|=|z+\\mathrm{i}|$, 则直线 $l$ 的倾斜角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019426": { + "id": "019426", + "content": "已知平面上 $A_1$、$A_2$、$A_3$、$A_4$、$A_5$ 五个点, 满足 $\\overrightarrow{A_n A_{n+1}}+\\overrightarrow{A_{n+1}A_{n+2}}=0$($n=1,2,3$), $|\\overrightarrow{A_n A_{n+1}}|+ |\\overrightarrow{A_{n+1}A_{n+2}}|=n+1$($n=1,2,3$), 则 $|\\overrightarrow{A_1A_5}|$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019427": { + "id": "019427", + "content": "设 $\\overrightarrow{m}, \\overrightarrow{n}$ 为非零向量, 则``存在负数 $\\lambda$, 使得 $\\overrightarrow{m}=\\lambda \\overrightarrow{n}$''是``$\\overrightarrow{m}\\cdot \\overrightarrow{n}<0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019428": { + "id": "019428", + "content": "已知 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 均为单位向量, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=0$. 若 $|\\overrightarrow{c}-4 \\overrightarrow{a}|+|\\overrightarrow{c}-3 \\overrightarrow{b}|=5$, 则 $|\\overrightarrow{c}+\\overrightarrow{a}|$ 的取值范围是\\bracket{20}.\n\\fourch{$[3, \\sqrt{10}]$}{$[3,5]$}{$[3,4]$}{$[\\sqrt{10}, 5]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019429": { + "id": "019429", + "content": "已知 $A, B$ 为平面上的两个定点, 且 $|\\overrightarrow{AB}|=2$, 该平面上的动线段 $PQ$ 的端点 $P, Q$ 满足 $|\\overrightarrow{AP}| \\leq 5$, $\\overrightarrow{AP}\\cdot \\overrightarrow{AB}=6$, $\\overrightarrow{AQ}=-2 \\overrightarrow{AP}$, 则动线段 $PQ$ 所形成图形的面积为\\bracket{20}.\n\\fourch{$36$}{$60$}{$81$}{$108$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019430": { + "id": "019430", + "content": "已知平面向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 满足 $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 且 $\\{|\\overrightarrow{a}|,|\\overrightarrow{b}|,|\\overrightarrow{c}|\\}=\\{1,2,3\\}$, 求 $|\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}|$ 的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019431": { + "id": "019431", + "content": "已知关于 $t$ 的方程 $t^2-2 t+a=0$($a \\in \\mathbf{R}$) 有两个虚根 $t_1$、$t_2$, 且满足 $|t_1-t_2|=2 \\sqrt{3}$.\\\\\n(1) 求方程的两个根以及实数 $a$ 的值;\\\\\n(2) 若对于任意 $x \\in \\mathbf{R}$, 不等式 $\\log _a(x^2+a) \\geq-k^2+2 m k-2 k$ 对于任意的 $k \\in[2,3]$ 恒成立, 求实数 $m$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019432": { + "id": "019432", + "content": "在直角坐标平面 $xOy$ 上的一列点 $A_1(1, a_1), A_2(2, a_2), \\cdots, A_n(n, a_n), \\cdots$, 简记为 $\\{A_n\\}$. 若由 $b_n=\\overrightarrow{A_n A_{n+1}}\\cdot \\overrightarrow{j}$ 构成的数列 $\\{b_n\\}$ 满足 $b_{n+1}>b_n$, $n=1,2, \\cdots$, 其中 $\\overrightarrow{j}$ 为方向与 $y$ 轴正方向相同的单位向量, 则称 $\\{A_n\\}$ 为 $T$ 点列.\\\\\n(1) 判断 $A_1(1,1), A_2(2, \\dfrac{1}{2}), A_3(1, \\dfrac{1}{3}), \\cdots, A_n(n, \\dfrac{1}{n}), \\cdots$, 是否为 $T$ 点列, 并说明理由;\\\\\n(2) 若 $\\{A_n\\}$ 为 $T$ 点列, 则点 $A_2$ 在点 $A_1$ 的右上方. 任取其中连续三点 $A_k$、$A_{k+1}$、$A_{k+2}$. 判断 $\\triangle A_k A_{k+1}A_{k+2}$ 的形状 (锐角三角形、直角三角形、钝角三角形), 并予以证明;\\\\\n(3) 若 $\\{A_n\\}$ 为 $T$ 点列, 正整数 $1 \\leq m \\overrightarrow{A_m A_p}\\cdot \\overrightarrow{j}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书05平面向量与复数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019433": { + "id": "019433", + "content": "在单位圆中, 圆心角 $\\dfrac{2}{3}\\pi$ 所对的弧长为\\blank{50}, 含这段弧的弓形面积是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019434": { + "id": "019434", + "content": "若 $\\sin (\\pi+\\theta)=\\dfrac{1}{3}$, 且 $\\theta$ 为第三象限角, 则 $\\cos (2 \\pi-\\theta)$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019435": { + "id": "019435", + "content": "在 $\\triangle ABC$ 中, $AC=3$, $3 \\sin A=2 \\sin B$, 且 $\\cos C=\\dfrac{1}{4}$, 则 $AB=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019436": { + "id": "019436", + "content": "已知 $\\tan (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{1}{3}$, 则 $\\cos (\\dfrac{2}{3}\\pi+2 \\alpha)$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019437": { + "id": "019437", + "content": "已知锐角 $\\alpha, \\beta$, 且 $\\cos \\alpha=\\dfrac{1}{7}$, $\\sin (\\alpha+\\beta)=\\dfrac{5 \\sqrt{3}}{14}$, 则 $\\cos \\beta$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019438": { + "id": "019438", + "content": "已知 $\\theta>0$, 对任意正整数 $n$, 总存在实数 $\\varphi$, 使得 $\\cos (n \\theta+\\varphi)<\\dfrac{\\sqrt{3}}{2}$, 则 $\\theta$ 的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019439": { + "id": "019439", + "content": "如图 $AD=BC=6$, $AB=20$, $\\angle DAB= \\angle ABC=120^{\\circ}, O$ 为 $AB$ 的中点, 曲线 $CMD$ 上任意一点到点 $O$ 的距离相等, $MO \\perp AB$, 点 $P$ 是曲线 $CM$ 上的动点, 点 $Q$ 与点 $P$ 关于 $OM$ 对称.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-10,0) node [below] {$A$} coordinate (A);\n\\draw (10,0) node [below] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (B) ++ (60:6) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (120:6) node [left] {$D$} coordinate (D);\n\\draw (C) arc ({atan(3*sqrt(3)/13)}:{180-atan(3*sqrt(3)/13)}:14);\n\\draw (0,14) node [above] {$M$} coordinate (M);\n\\draw (D) -- (A) -- (B) -- (C) (O) -- (M);\n\\draw (50:14) node [above] {$P$} coordinate (P);\n\\draw (130:14) node [above] {$Q$} coordinate (Q);\n\\draw (C) -- (P) -- (M) -- (Q) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点 $P$ 与点 $C$ 重合, 求 $\\angle POB$ 的大小;\\\\\n(2) 求五边形 $DCPMQ$ 面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019440": { + "id": "019440", + "content": "一个二面角的余弦值为 $-\\dfrac{3}{5}$, 则这个二面角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019441": { + "id": "019441", + "content": "已知向量 $\\overrightarrow{m}=(\\dfrac{1}{2}, \\dfrac{1}{2}\\sin 2 x+\\dfrac{\\sqrt{3}}{2}\\cos 2 x)$, $\\overrightarrow{n}=(f(x),-1)$, 且 $\\overrightarrow{m}\\perp \\overrightarrow{n}$. 则函数 $f(x)$ 在 $x \\in[0, \\pi]$ 上的减区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019442": { + "id": "019442", + "content": "已知函数 $f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$), 若 $f(\\dfrac{\\pi}{6})=f(\\dfrac{\\pi}{3})$, 且 $f(x)$ 在区间 $(\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3})$ 内有最小值无最大值, 则实数 $\\omega=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019443": { + "id": "019443", + "content": "已知关于 $x$ 的方程 $\\sqrt{3}\\sin 2 x+\\cos 2 x=k+1$ 在区间 $[0, \\dfrac{\\pi}{2}]$ 内有相异两个实数根, 则实数 $k$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019444": { + "id": "019444", + "content": "函数 $f(x)=\\cos (\\pi x+\\varphi)$($0<\\varphi<\\dfrac{\\pi}{2}$) 的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2, samples = 100] plot (\\x,{cos(180*\\x+30)});\n\\draw (0,{sqrt(3)/2}) node [left] {$\\dfrac{\\sqrt{3}}{2}$};\n\\draw [dashed] (0,{sqrt(3)/2}) --++ ({5/3},0) -- ({5/3},0) node [below] {$x_0$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\varphi$ 及图中 $x_0$ 的值;\\\\\n(2) 设 $g(x)=f(x)+f(x+\\dfrac{1}{3})$, 求函数 $g(x)$ 在区间 $[-\\dfrac{1}{2}, \\dfrac{1}{3}]$ 上的最大值和最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019445": { + "id": "019445", + "content": "方程 $\\sin x+\\sqrt{3}\\cos x=1$ 在 $(\\pi, 2 \\pi)$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019446": { + "id": "019446", + "content": "已知函数 $f(x)=\\cos (2 x-\\dfrac{\\pi}{3})+2 \\sin (x-\\dfrac{\\pi}{4}) \\sin (x+\\dfrac{\\pi}{4})$, 函数 $f(x)$ 在区间 $[-\\dfrac{\\pi}{12}, \\dfrac{\\pi}{2}]$ 上的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019447": { + "id": "019447", + "content": "设函数 $f(x)=\\cos (\\omega x+\\dfrac{\\pi}{3})$($0<\\omega<2$), 若将 $f(x)$ 图像向左平移 $\\dfrac{4 \\pi}{5}$ 个单位后, 所得函数图像的对称轴与原函数图像的对称轴重合, 则 $\\omega=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019448": { + "id": "019448", + "content": "已知 $f(x)=A \\sin (w x+\\theta)$($w>0$), 若两个不等的实数 $x_1, x_2 \\in\\{x | f(x)=\\dfrac{A}{2}\\}$, $|x_1-x_2|_{\\min}=\\pi$, 则 $f(x)$ 的最小正周期是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019449": { + "id": "019449", + "content": "设函数 $f(x)=\\sin x-m(x \\in[0, \\dfrac{5 \\pi}{2}])$ 的零点为 $x_1, x_2, x_3$, 若 $x_1, x_2, x_3$ 成等比数列, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019450": { + "id": "019450", + "content": "已知函数 $f(x)=\\sin x+a \\cos x$, $\\dfrac{\\pi}{4}$ 是函数 $f(x)$ 的一个零点, 若 $\\alpha, \\beta \\in(0, \\dfrac{\\pi}{2})$, 且 $f(\\alpha+\\dfrac{\\pi}{4})=\\dfrac{\\sqrt{10}}{5}$, $f(\\beta+\\dfrac{3 \\pi}{4})=\\dfrac{3 \\sqrt{5}}{5}$, 则 $\\sin (\\alpha+\\beta)$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019451": { + "id": "019451", + "content": "已知 $\\triangle ABC$ 是斜三角形, 则``$A>B$''是``$|\\tan A|>|\\tan B|$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019452": { + "id": "019452", + "content": "在 $\\triangle ABC$ 中, 设三个内角 $A, B, C$ 的对边依次为 $a, b, c$. 则``$C \\in\\{\\dfrac{\\pi}{3}, \\dfrac{2 \\pi}{3}\\}$''是``$a^2+b^2= c^2+a b$''成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分条件又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019453": { + "id": "019453", + "content": "设函数 $f(x)=\\sin (x-\\dfrac{\\pi}{6})$, 若对于任意 $\\alpha \\in[-\\dfrac{5 \\pi}{6},-\\dfrac{\\pi}{2}]$, 在区间 $[0, m]$ 上总存在唯一确定的 $\\beta$, 使得 $f(\\alpha)+f(\\beta)=0$, 则 $m$ 的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{7 \\pi}{6}$}{$\\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019454": { + "id": "019454", + "content": "已知 $f(x)=\\sin w x$($w>0$).\\\\\n(1) 若 $f(x)$ 的周期是 $4 \\pi$, 求 $\\omega$, 并求此时 $f(x)=\\dfrac{1}{2}$ 的解集;\\\\\n(2) 已知 $\\omega=1$, $g(x)=f^2(x)+\\sqrt{3}f(-x) f(\\dfrac{\\pi}{2}-x)$, $x \\in[0, \\dfrac{\\pi}{4}]$, 求 $g(x)$ 的值域.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019455": { + "id": "019455", + "content": "已知 $A$、$B$、$C$ 为 $\\triangle ABC$ 的三个内角, $a$、$b$、$c$ 是其三条边, $a=2$, $\\cos C=-\\dfrac{1}{4}$.\\\\\n(1) 若 $\\sin A=2 \\sin B$, 求 $b$、$c$;\\\\\n(2) $\\cos (A-\\dfrac{\\pi}{4})=\\dfrac{4}{5}$, 求 $c$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019456": { + "id": "019456", + "content": "如图, 某海岸线可近视地看成曲线段 $A-B-C$, 其中 $AB$ 为线段, $\\overset\\frown{BC}$ 为四分之一的圆弧, $BD=39.2 \\mathrm{km}$, $\\angle BDC=22^{\\circ}$, $\\angle CBD=68^{\\circ}$, $\\angle BDA=58^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}\n\\path (0,0) coordinate (D) node [below right] {$D$};\n\\path (112:3.92) coordinate (B) node [above left] {$B$};\n\\path (0,3.63456) coordinate (C) node [above right] {$C$};\n\\path (-4.2365,0.747) coordinate (A) node [left] {$A$};\n\\draw (D) -- (C) -- (B) (A) -- (D) -- (B);\n\\draw [very thick] (C) arc (45:135:1.0383565) (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\overset\\frown{BC}$ 的长度;\\\\\n(2) 若 $AB=40 \\mathrm{km}$, 求点 $D$ 到海岸线 $A-B-C$ 的最短距离.(精确到 $0.001 \\mathrm{km}$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书06三角与三角函数", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019457": { + "id": "019457", + "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=2$ 若对任意的正整数 $p, q$, 都有 $a_{p+q}=a_p+a_q$, 则 $a_{2023}=$\\blank{50}, 若对任意的正整数 $p, q$, 都有 $a_{p+q}=a_p \\cdot a_q$, 则 $a_{2023}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019458": { + "id": "019458", + "content": "已知无穷数列 $\\{a_n\\}$ 的通项公式 $a_n=\\dfrac{9^n(n+1)}{10^n}$, 试判断此数列是否有最大项, 若有, 求出第几项最大, 若没有, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019459": { + "id": "019459", + "content": "已知 $S_n$ 为数列 $\\{a_n\\}$ 的前 $n$ 项和, 点 $(a_n, S_n)$ 在直线 $y=2 x-3 n$ 上.\\\\\n(1) 若数列 $\\{a_n+c\\}$ 成等比, 求常数的值;\\\\\n(2) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 数列 $\\{a_n\\}$ 中是否存在三项, 它们可以构成等差数列? 若存在, 请求出一组适合条件的项; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019460": { + "id": "019460", + "content": "已知数列 $\\{x_n\\}$ 满足 $x_1=\\dfrac{1}{2}$, $x_{n+1}=\\dfrac{1}{1+x_n}$, $n \\in \\mathbf{N}$, $n \\geq 1$. 猜想数列 $\\{x_{2 n}\\}$ 的单调性, 并证明你的结论.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019461": { + "id": "019461", + "content": "已知数列 $\\{a_n\\}$ 是公差不为 $0$ 的等差数列, $a_1=\\dfrac{3}{2}$, 数列 $\\{b_n\\}$ 是等比数列, 且 $b_1=a_1$, $b_2=-a_3$, $b_3=a_4$, 数列 $\\{b_n\\}$ 的前 $n$ 项和为 $S_n$, 记点 $Q_n(b_n, S_n), n \\in \\mathbf{N}$, $n \\geq 1$.\\\\\n(1) 求数列 $\\{b_n\\}$ 的通项公式;\\\\\n(2) 证明: 点 $Q_1$、$Q_2$、$Q_3$、$\\cdots$、$Q_n$、$\\cdots$ 在同一直线 $l$ 上, 并求出直线 $l$ 的方程;\\\\\n(3) 若 $A \\leq S_n-\\dfrac{1}{S_n}\\leq B$ 对 $n \\in \\mathbf{N}$, $n \\geq 1$ 恒成立, 求 $B-A$ 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019462": { + "id": "019462", + "content": "记 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和. 若 $a_1=-2$, $a_2+a_6=2$, 则 $S_{10}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019463": { + "id": "019463", + "content": "若正项等比数列 $\\{a_n\\}$ 满足: $a_3+a_5=4$, 则 $a_4$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019464": { + "id": "019464", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且满足 $S_m+S_n=S_{m+n}$, 若 $a_1=2$, 则 $a_{20}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019465": { + "id": "019465", + "content": "已知公差不为 $0$ 的等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_4, S_5, S_7 \\in\\{-10,0\\}$, 则 $S_n$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019466": { + "id": "019466", + "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_1=1$, $S_n-\\dfrac{1}{3}a_{n+1}=0$($n \\in \\mathbf{N}$, $n \\geq 1$), 则 $\\{a_n\\}$ 的通项公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019467": { + "id": "019467", + "content": "设 $\\{a_n\\}$ 是公差为 $d$ 的等差数列, $\\{b_n\\}$ 是公比为 $q$ 的等比数列. 已知数列 $\\{a_n+b_n\\}$ 的前 $n$ 项和 $S_n=n^2-n+2^n-1$($n \\in \\mathbf{N}$, $n \\geq 1$), 则 $d+q$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019468": { + "id": "019468", + "content": "已知数列满足 $a_1+2 a_2+3 a_3+\\cdots+n a_n=(2 n-1) \\cdot 3^n$, 设 $b_n=\\dfrac{4 n}{a_n}$, $S_n$ 为数列 $\\{b_n\\}$ 前 $n$ 项和, 若 $S_n<\\lambda$ ($\\lambda$ 为常数), 则 $\\lambda$ 最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019469": { + "id": "019469", + "content": "已知数列 $\\{a_n\\}$ 中, 若 $a_1=0$, $a_i=k^2$($i \\in \\mathbf{N}$, $2^k \\leq i<2^{k+1}$, $k=1,2,3, \\cdots$), 则满足 $a_i+a_{2 i}\\geq 100$ 的 $i$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019470": { + "id": "019470", + "content": "若无穷等差数列 $\\{a_n\\}$ 的首项 $a_1<0$, 公差 $d>0,\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则以下结论中一定正确的是\\bracket{20}.\n\\fourch{$S_n$ 严格递增}{$S_n$ 严格递减}{$S_n$ 有最小值}{$S_n$ 有最大值}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019471": { + "id": "019471", + "content": "已知等差数列 $\\{a_n\\}$ 单调递增且满足 $a_1+a_8=6$, 则 $a_6$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, 3)$}{$(3,6)$}{$(3,+\\infty)$}{$(6,+\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019472": { + "id": "019472", + "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则下列判断一定正确的是\\bracket{20}.\n\\twoch{若 $S_3>0$, 则 $a_{2018}>0$}{若 $S_3<0$, 则 $a_{2018}<0$}{若 $a_2>a_1$, 则 $a_{2019}>a_{2018}$}{若 $\\dfrac{1}{a_2}>\\dfrac{1}{a_1}$, 则 $a_{2019}0$;\\\\\n\\textcircled{3} 若数列 $\\{a_n\\}$ 是周期数列, 则最小正周期可能为$2$;\\\\\n\\textcircled{4} 若数列 $\\{a_n\\}$ 是常数列, 则 $d \\geq-\\dfrac{1}{4}$ 其中, 所有正确结论的个数是\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书07数列与数学归纳法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019474": { + "id": "019474", + "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=(n-k_1)(n-k_2)$, 其中 $k_1, k_2 \\in \\mathbf{Z}$.\\\\\n(1) 试写出一组 $k_1, k_2 \\in \\mathbf{Z}$ 的值, 使得数列 $\\{a_n\\}$ 中的各项均为正数;\\\\\n(2) 若 $k_1=1$、$k_2$ 为正整数, 数列 $\\{b_n\\}$ 满足 $b_n=\\dfrac{a_n}{n}$, 且对任意 $m \\in \\mathbf{N}$($m \\geq 1$, $m \\neq 3$), 均有 $b_30$) 的公共弦的长为 $2 \\sqrt{3}$, 则实数 $a$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019483": { + "id": "019483", + "content": "已知圆 $C_1: x^2+y^2-2 a x-2 y+a^2-15=0$, 圆 $C_2: x^2+y^2-4 a x-2 y+4 a^2=0 $($a>0$), 求实数 $a$ 的值或取值范围, 分别使得圆 $C_1$ 与圆 $C_2$.\\\\\n(1) 相切;\\\\\n(2) 相交;\\\\\n(3) 内含.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019484": { + "id": "019484", + "content": "已知圆 $C: x^2+(y-1)^2=5$, 直线 $l: m x-y+1-m=0$.\\\\\n(1) 求证: 对 $m \\in \\mathbf{R}$, 直线 $l$ 与圆 $C$ 总有两个不同的交点;\\\\\n(2) 设 $l$ 与圆 $C$ 交于 $A$、$B$ 两点,若 $|AB|=\\sqrt{17}$, 求 $l$ 的倾斜角;\\\\\n(3) 求直线 $l$ 中, 截圆所得的弦最长时的直线方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019485": { + "id": "019485", + "content": "已知方程 $\\dfrac{x^2}{|m|-1}+\\dfrac{y^2}{2-m}=1$ 表示焦点在 $y$ 轴上的椭圆, 则 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019486": { + "id": "019486", + "content": "已知动圆 $P$ 过点 $A(3,0)$, 且与圆 $(x+3)^2+y^2=4$ 相外切, 则动圆圆心 $P$ 的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019487": { + "id": "019487", + "content": "已知 $F$ 是抛物线 $y^2=4 x$ 的焦点, $P$ 是抛物线上的一个动点, 则 $\\triangle APF$ 周长的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019488": { + "id": "019488", + "content": "已知抛物线 $C: y^2=2 p x$($p>0$) 的焦点为 $F$, 点 $M$ 是抛物线 $C$ 上一点,圆 $M$ 与 $y$ 轴相切且被直线 $x=\\dfrac{p}{2}$ 截得的弦长为 $\\sqrt{2}p$, 若 $|MF|=\\dfrac{5}{2}$, 求抛物线 $C$ 的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019489": { + "id": "019489", + "content": "直线 $x-y-2=0$ 关于直线 $2 x-2 y+1=0$ 对称的直线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019490": { + "id": "019490", + "content": "已知过点 $(0,-2)$ 且具有斜率 $k$ 的直线 $l$ 与以点 $A(3,1)$ 和 $B(-2,5)$ 为端点的线段 $AB$ 相交, 则实数 $k$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019491": { + "id": "019491", + "content": "已知方程 $\\dfrac{x^2}{m^2+n}-\\dfrac{y^2}{3 m^2-n}=1$ 表示双曲线, 且该双曲线的焦距为 $4$, 则 $n$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019492": { + "id": "019492", + "content": "已知 $P_1$、$P_2$、$P_3$、$\\cdots$、$P_{10}$ 是抛物线 $y^2=8 x$ 上不同的点, 点 $F(2,0)$, 若 $\\overrightarrow{FP_1}+\\overrightarrow{FP_2}+\\cdots+ \\overrightarrow{FP_{10}}=\\overrightarrow{0}$, 则 $|\\overrightarrow{FP_1}|+|\\overrightarrow{FP_2}|+\\cdots+|\\overrightarrow{FP_{10}}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019493": { + "id": "019493", + "content": "椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 第一象限上一点与中心右焦点构成一个正三角形, 则此椭圆的离心率 $e=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019494": { + "id": "019494", + "content": "已知实数 $x$、$y$ 满足 $\\dfrac{x|x|}{4}+y|y|=1$, 则 $|x+2 y-4|$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019495": { + "id": "019495", + "content": "直线 $x-\\sqrt{3}y=0$ 绕原点按逆时针方向旋转 $30^{\\circ}$ 后所得的直线 $l$ 与圆 $(x-2)^2+y^2=3$ 的位置关系是\\bracket{20}.\n\\twoch{直线 $l$ 过圆心}{直线 $l$ 与圆相交, 但不过圆心}{直线 $l$ 与圆相切}{直线 $l$ 与圆无公共点}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019496": { + "id": "019496", + "content": "设抛物线 $C: x^2=8 y$ 的焦点 $F$, 准线为 $l, P(x_0, y_0)$ 为 $C$ 上一动点, $A(2,1)$, 则下列结论错误的是\\bracket{20}.\n\\onech{当 $x_0=4$ 时, $|PF|$ 的值为 $6$}{当 $x_0=2$ 时,抛物线 $C$ 在点 $P$ 处的切线方程为 $x-2 y-2=0$}{$|PA|+|PF|$ 的最小值为 $3$}{$|PA|-|PF|$ 的最大值为 $\\sqrt{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019497": { + "id": "019497", + "content": "已知椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右顶点分别为 $A_1, A_2$, 且以线段 $A_1A_2$ 为直径的圆与直线 $b x-a y+2 a b=0$ 相交,则椭圆的离心率的取值范围为\\bracket{20}.\n\\fourch{$(0, \\dfrac{\\sqrt{6}}{3})$}{$(\\dfrac{\\sqrt{6}}{3}, 1)$}{$(\\dfrac{\\sqrt{2}}{3}, 1)$}{$(0, \\dfrac{\\sqrt{2}}{3})$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019498": { + "id": "019498", + "content": "已知 $a \\in \\mathbf{R}$, 双曲线 $\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$.\\\\\n(1) 若点 $(2,1)$ 在双曲线上, 求 $\\Gamma$ 的焦点坐标;\\\\\n(2) 若 $a=1$, 直线 $y=k x+1$ 与 $\\Gamma$ 相交于 $A, B$ 两点, 且线段 $AB$ 中点的横坐标为 $1$, 求实数 $k$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019499": { + "id": "019499", + "content": "在平面直角坐标系 $xOy$ 中, 一动圆经过点 $A(\\dfrac{1}{2}, 0)$ 且与直线 $x=-\\dfrac{1}{2}$ 相切, 设该动圆圆心的轨迹为曲线 $K, P$ 是曲线 $K$ 上一点.\\\\\n(1) 求曲线 $K$ 的方程;\\\\\n(2) 过点 $A$ 且斜率为 $k$ 的直线 $l$ 与曲线 $K$ 交于 $B$、$C$ 两点, 若 $l \\parallel OP$ 且直线 $OP$ 与直线 $x=1$ 交于 $Q$ 点, 求 $\\dfrac{|AB| \\cdot|AC|}{|OP| \\cdot|OQ|}$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019500": { + "id": "019500", + "content": "设常数 $m>0$ 且 $m \\neq 1$, 椭圆 $\\Gamma: \\dfrac{x^2}{m^2}+y^2=1$, 点 $P$ 是 $\\Gamma$ 上的动点.\\\\\n(1) 若点 $P$ 的坐标为 $(2,0)$, 求 $\\Gamma$ 的焦点坐标;\\\\\n(2) 设 $m=3$, 若定点 $A$ 的坐标为 $(2,0)$, 求 $|PA|$ 的最大值与最小值;\\\\\n(3) 设 $m=\\dfrac{1}{2}$, 若 $\\Gamma$ 上的另一动点 $Q$ 满足 $OP \\perp OQ$ ($O$ 为坐标原点), 求证: $O$ 到直线 $PQ$ 的距离是定值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书08解析几何基本问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019501": { + "id": "019501", + "content": "已知两圆 $C_1:(x-2)^2+y^2=54$, $C_2:(x+2)^2+y^2=6$, 动圆 $M$ 在圆 $C_1$ 内部且和圆 $C_1$ 内切、和圆 $C_2$ 外切.\\\\\n(1) 求动圆圆心 $M$ 的轨迹 $C$ 的方程;\\\\\n(2) 过点 $A(3,0)$ 的直线与 (1) 中的曲线 $C$ 交于 $P$、$Q$ 两点, 点 $P$ 关于 $x$ 轴对称的点为 $R$, 求 $\\triangle ARQ$ 面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019502": { + "id": "019502", + "content": "某团队开发一款``猫捉老鼠''的游戏. 如图所示, $A$、$B$ 两个信号源相距 $10$ 米, $O$ 是 $AB$ 的中点, 过点 $O$ 的直线 $l$ 与直线 $AB$ 的夹角为 $45^{\\circ}$. 机器猫在直线 $l$ 上运动, 机器鼠的运动轨迹始终满足: 接收到点 $A$ 的信号比接收到点 $B$ 的信号晩 $\\dfrac{8}{v_0}$ 秒, 其中 $v_0$ (单位: 米/ 秒) 是信号传播的速度. 游戏设定: 机器鼠在距离直线 $l$ 不超过 $1.5$ 米的区域运动时, 有``被抓''的风险, 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''的风险?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019503": { + "id": "019503", + "content": "如图, 设 $F$ 是椭圆 $\\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$ 的下焦点, 直线 $y=k x-4$($k>0$) 与椭圆相交于 $A$、$B$ 两点, 与 $y$ 轴交于 $P$ 点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (O) ellipse ({sqrt(3)} and 2);\n\\draw (0,-1) node [left] {$F$} coordinate (F);\n\\draw (0,-4) node [left] {$P$} coordinate (P);\n\\draw ({3*sqrt(5)/8},{-7/4}) node [below right] {$A$} coordinate (A);\n\\draw ($(P)!2!(A)$) node [right] {$B$} coordinate (B);\n\\draw ($(P)!-0.2!(B)$) -- ($(B)!-0.2!(P)$);\n\\draw (A)--(F)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\overrightarrow{PA}=\\overrightarrow{AB}$, 求 $k$ 的值;\\\\\n(2) 求证: $\\angle AFP=\\angle BFO$;\\\\\n(3) 求 $\\triangle ABF$ 面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019504": { + "id": "019504", + "content": "已知抛物线 $\\Gamma: y^2=4 x$ 的焦点为 $F$, 若 $\\triangle ABC$ 的三个顶点都在抛物线 $\\Gamma$ 上, 且\n$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}=\\overrightarrow{0}$, 则称该三角形为``核心三角形''.\\\\\n(1) 是否存在``核心三角形'', 其中两个顶点的坐标分别为 $(0,0)$ 和 $(1,2)$? 请说明理由;\\\\\n(2) 设``核心三角形''$ABC$ 的一边 $AB$ 所在直线的斜率为 4 , 求直线 $AB$ 的方程;\\\\\n(3) 已知 $\\triangle ABC$ 是``核心三角形'', 证明: 点 $A$ 的横坐标小于 $2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019505": { + "id": "019505", + "content": "已知椭圆 $C: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$, 直线 $l$ 经过椭圆的右焦点 $F$, 交椭圆 $C$ 于 $P$、$Q$ 两点 (点 $P$ 在第二象限), 若 $Q$ 关于 $x$ 轴对称的点为 $Q'$, 且满足 $PQ \\perp FQ'$, 则直线 $l$ 的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019506": { + "id": "019506", + "content": "若曲线 $C: y^2-2 y-x+3=0$ 和直线 $l: y=k x+\\dfrac{3}{2}$ 只有一个公共点, 则实数 $k$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019507": { + "id": "019507", + "content": "已知抛物线: $y^2=2 p x$($p>0$), 若第一象限的 $A$、$B$ 两点在抛物线上, 焦点为 $F$, $|AF|= 2$, $|BF|=4$, $|AB|=3$, 则直线 $AB$ 的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019508": { + "id": "019508", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-y^2=1$($a>0$), 双曲线右支上有任意两点 $P_1(x_1, y_1), P_2(x_2, y_2)$ 满足 $x_1 x_2-y_1 y_2>0$ 恒成立, 则 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019509": { + "id": "019509", + "content": "$P$ 为双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的右支上一点, $M, N$ 分别是圆 $(x+5)^2+y^2=4$ 和 $(x-5)^2+ y^2=4$ 上的点, 则 $|PM|-|PN|$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019510": { + "id": "019510", + "content": "双曲线 $C: x^2-\\dfrac{y^2}{2}=1$, 过定点 $A(-1,0)$ 的两条垂线分别交双曲线于 $P, Q$ 两点, 直线 $PQ$ 恒过定点\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019511": { + "id": "019511", + "content": "已知直线方程 $f(x, y)=0$ 表示直线 $l$, 点 $P(x_0, y_0)$ 为定点, 若点 $P$ 不在直线 $l$ 上, 则 $f(x, y)=f(x_0, y_0)$ 一定表示\\bracket{20}.\n\\twoch{过 $P$ 而与直线 $l$ 相交的直线}{过 $P$ 而与直线 $l$ 平行的直线}{过 $P$ 而与直线 $l$ 垂直的直线}{过 $P$ 而与直线 $l$ 重合的直线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019512": { + "id": "019512", + "content": "如图, 某绿色蔬菜种植基地在 $A$ 处, 要把此处生产的蔬菜沿道路 $AA_1$ 或 $AA_2$ 运送到四边形区域 $A_1A_2A_3A_4$ 的农贸市场. 现要求在农贸市场中确定一条界线, 使位于界线一侧的点沿道路 $AA_1$ 比沿道路 $AA_2$ 运送蔬菜近, 而另一侧的点则反之, 该界线所在曲线为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [right] {$A_2$} coordinate (A_2);\n\\draw (0.5,-1.5) node [below] {$A$} coordinate (A);\n\\draw (2.5,1) node [right] {$A_3$} coordinate (A_3);\n\\draw (0.3,1.8) node [above] {$A_4$} coordinate (A_4);\n\\filldraw [pattern = north east lines] (A_1)--(A_2)--(A_3)--(A_4)--cycle;\n\\draw (A_1)--(A)--(A_2);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{直线}{椭圆}{双曲线}{抛物线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019513": { + "id": "019513", + "content": "在直角坐标平面内, 点 $A, B$ 的坐标分别为 $(-1,0),(1,0)$, 则满足 $\\tan \\angle PAB \\cdot \\tan \\angle PBA=m$ ($m$ 为非零常数) 的点 $P$ 的轨迹方程是\\bracket{20}.\n\\fourch{$x^2-\\dfrac{y^2}{m}=1$($y \\neq 0$)}{$x^2-\\dfrac{y^2}{m}=1$}{$x^2+\\dfrac{y^2}{m}=1$($y \\neq 0$)}{$x^2+\\dfrac{y^2}{m}=1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019514": { + "id": "019514", + "content": "如图, 双曲线 $\\Gamma: \\dfrac{x^2}{3}-y^2=1$ 的左、右焦点分别为 $F_1, F_2$, 过 $F_2$ 作直线 $l$ 交 $y$ 轴于点 $Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -5.5:5.5, samples = 100] plot ({sqrt(3+3*\\x*\\x)},\\x);\n\\draw [domain = -5.5:5.5, samples = 100] plot ({-sqrt(3+3*\\x*\\x)},\\x);\n\\filldraw (-2,0) circle (0.1) node [below left] {$F_1$};\n\\filldraw (2,0) circle (0.1) node [below right] {$F_2$};\n\\draw (0,-2) node [right] {$Q$};\n\\draw (-4,-6) -- (8,6);\n\\draw [dashed] (-10,{-10/sqrt(3)}) -- (10,{10/sqrt(3)});\n\\draw [dashed] (10,{-10/sqrt(3)}) -- (-10,{10/sqrt(3)});\n\\end{tikzpicture}\n\\end{center}\n(1) 当直线 $l$ 平行于 $\\Gamma$ 的一条渐近线时, 求点 $F_1$ 到直线 $l$ 的距离;\\\\\n(2) 当直线 $l$ 的斜率为 $1$ 时, 在 $\\Gamma$ 的右支上是否存在点 $P$, 满足 $\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_1Q}=0$ ? 若存在,求出 $P$ 点的坐标; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019515": { + "id": "019515", + "content": "在平面直角坐标系中, $A$、$B$ 分别为椭圆 $\\Gamma: \\dfrac{x^2}{2}+y^2=1$ 的上、下顶点, 若动直线 $l$ 过点 $P(0, b)$($b>1$), 且与椭圆 $\\Gamma$ 相交于 $C$、$D$ 两个不同点 (直线 $l$ 与 $y$ 轴不重合, 且 $C$、$D$ 两点在 $y$ 轴右侧, $C$ 在 $D$ 的上方), 直线 $AD$ 与 $BC$ 相交于点 $Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.9) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (-1,0) circle (0.03) node [above] {$F_1$};\n\\filldraw (1,0) circle (0.03) node [above] {$F_2$};\n\\draw (0,1.6) node [right] {$P(0,b)$} coordinate (P);\n\\draw (1.5,-1) coordinate (R) node [right] {$l$};\n\\draw [name path = l] ($(P)!-0.1!(R)$) -- (R);\n\\draw [name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [name intersections = {of = l and elli, by = {C,D}}];\n\\draw (C) node [above] {$C$};\n\\draw (D) node [below] {$D$};\n\\draw (0,1) node [above left] {$A$} coordinate (A);\n\\draw (0,-1) node [below left] {$B$} coordinate (B);\n\\draw [name path = BC] (B)--(C);\n\\draw [name path = AD] (A)--(D);\n\\draw [name intersections = {of = BC and AD, by = Q}];\n\\draw (Q) node [left] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 设 $\\Gamma$ 的两焦点为 $F_1$、$F_2$, 求 $\\angle F_1AF_2$ 的值;\\\\\n(2) 若 $b=3$, 且 $\\overrightarrow{PD}=\\dfrac{3}{2}\\overrightarrow{PC}$, 求点 $Q$ 的横坐标;\\\\\n(3) 是否存在这样的点 $P$, 使得点 $Q$ 的纵坐标恒为 $\\dfrac{1}{3}$ ? 若存在, 求出点 $P$ 的坐标; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019516": { + "id": "019516", + "content": "已知斜率为 $k$ 的直线 $l$ 经过抛物线 $C: y^2=4 x$ 的焦点 $F$, 且与抛物线 $C$ 交于不同的两点 $A(x_1, y_1)$、$B(x_2, y_2)$.\\\\\n(1) 若点 $A$ 和 $B$ 到抛物线准线的距离分别为 $\\dfrac{3}{2}$ 和 $3$, 求 $|AB|$;\\\\\n(2) 若 $|AF|+|AB|=2|BF|$, 求 $k$ 的值;\\\\\n(3) 点 $M(t, 0)$, $t>0$, 对任意确定的实数 $k$, 若 $\\triangle AMB$ 是以 $AB$ 为斜边的直角三角形,判断符合条件的点 $M$ 有几个, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书09直线与圆锥曲线", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019517": { + "id": "019517", + "content": "过正方体中心的平面截正方体所得的截面中, 不可能的图形是\\bracket{20}.\n\\fourch{三角形}{长方形}{对角线不相等的菱形}{六边形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019518": { + "id": "019518", + "content": "已知平面 $\\alpha$、$\\beta$、$\\gamma$ 两两垂直, 直线 $a$、$b$、$c$ 满足: $a \\subset \\alpha, b \\subset \\beta, c \\subset \\gamma$, 则直线 $a$、$b$、$c$ 不可能满足以下哪种关系\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019519": { + "id": "019519", + "content": "如果 $a$、$b$ 是异面直线, $P$ 是不在 $a$、$b$ 上的一点, 下列四个结论:\\\\\n\\textcircled{1} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都异面;\\\\\n\\textcircled{2} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都垂直;\\\\\n\\textcircled{3} 过 $P$ 一定可作平面 $\\alpha$ 与 $a$、$b$ 都平行;\\\\\n\\textcircled{4} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都平行.\\\\\n其中正确的结论有\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019520": { + "id": "019520", + "content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$、$Q$、$R$、$S$ 分别为棱 $AB$、$BC$、$BB_1$、$CD$ 的中点, 联结 $A_1S$、$B_1D$. 空间任意两点 $M$ 、 $N$, 若线段 $MN$ 上不存在点在线段 $A_1S$、$B_1D$ 上, 则称 $M$、$N$ 两点可视,则下列选项中与点 $D_1$ 可视的为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\filldraw ($(A)!0.5!(B)$) circle (0.05) node [below] {$P$};\n\\filldraw ($(C)!0.5!(B)$) circle (0.05) node [below right] {$Q$};\n\\filldraw ($(B1)!0.5!(B)$) circle (0.05) node [above right] {$R$};\n\\filldraw ($(C)!0.5!(D)$) circle (0.05) node [above] {$S$} coordinate (S);\n\\draw [dashed] (A1) -- (S) (B1) -- (D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{点 $P$}{点 $B$}{点 $R$}{点 $Q$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019521": { + "id": "019521", + "content": "已知直二面角 $\\alpha-l-\\beta$, 直线 $a \\subset$ 平面 $\\alpha$, 直线 $b \\subset$ 平面 $\\beta$, 且 $a$ 与 $l$ 不垂直, $b$ 与 $l$ 不垂直, 那么\\bracket{20}.\n\\twoch{$a$ 与 $b$ 可能垂直, 但不可能平行}{$a$ 与 $b$ 可能垂直, 也可能平行}{$a$ 与 $b$ 不可能垂直, 但可能平行}{$a$ 与 $b$ 不可能垂直, 也不可能平行}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019522": { + "id": "019522", + "content": "如图, 在棱长为 $10$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$ 为左侧面 $ADD_1A_1$ 上一点, 已知点 $P$ 到 $A_1D_1$ 的距离为 $3, P$ 到 $AA_1$ 的距离为 $2$ , 则与过点 $P$ 且与 $A_1C$ 平行的直线相交的面是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\def\\l{10}\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 [dashed] (C)--(A_1);\n\\draw [dashed] ($(A)!0.7!(A_1)$) --++ (0,0,-2) node [right] {$P$} --++ (0,3,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{面 $AA_1B_1B$}{面 $BB_1C_1C$}{面 $CC_1D_1D$}{面 $ABCD$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019523": { + "id": "019523", + "content": "如图, $ABCD$ 是矩形, $PA \\perp$ 平面 $ABCD, E$、$F$ 分别是 $AB$、$PC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,1.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(P)$) node [right] {$F$} coordinate (F);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(P)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CD \\perp PD$;\\\\\n(2) 求证: $EF\\parallel $ 平面 $PAD$;\\\\\n(3) 当平面 $PCD$ 与平面 $ABCD$ 成多大角时, 直线 $EF \\perp$ 平面 $PCD$?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019524": { + "id": "019524", + "content": "在四面体 $ABCD$ 中, 面 $ABD$ 、面 $ACD$ 是全等的直角三角形, $AD$ 是公共的斜边, 且 $AD=\\sqrt{3}$, $BD=CD=1$, 面 $ABC$ 是等边三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (D) ++ (0,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) ++ (0,{sqrt(6)/2},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AC$ 与 $BD$ 所成的角的大小;\\\\\n(2) 求二面角 $B-AD-C$ 的大小;\\\\\n(3) 过点 $A$ 作 $AO \\perp$ 平面 $BCD$, 垂足为 $O$, 求证: 四边形 $OBDC$ 是正方形; 并点 $A$ 到平面 $BCD$ 的距离;\\\\\n(4) 在直线 $AC$ 上是否存在一点 $E$, 使得 $ED$ 与面 $BCD$ 成 $30^{\\circ}$ 角? 若存在, 确定点 $E$ 的位置, 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019525": { + "id": "019525", + "content": "在正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, 底面 $ABCD$ 的边长为 $3$, $BD_1$ 与底面所成角的大小为 $\\arctan \\dfrac{2}{3}$, 则该正四棱柱的高等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{{2*sqrt(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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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)(B)--(D_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019526": { + "id": "019526", + "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为 $A_1B_1$ 的中点, $AB=BB_1=2$, $AC=2 \\sqrt{5}$, 则异面直线 $BE$ 与 $AC$ 所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{4}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 [above] {$E$} coordinate (E);\n\\draw (B)--(E);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019527": { + "id": "019527", + "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为 $A_1B_1$ 的中点, $AB=BB_1=2$, $AC=2 \\sqrt{5}$, 则二面角 $C_1-BD-C$ 的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{4}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 [above] {$E$} coordinate (E);\n\\draw (B)--(E);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019528": { + "id": "019528", + "content": "设 $m$、$n$ 是空间两条不同直线, $\\alpha$、$\\beta$ 是两个不同平面, 下面有四个命题:\\\\\n\\textcircled{1} $m \\perp \\alpha$, $n \\parallel \\beta$, $\\alpha \\parallel \\beta \\Rightarrow m \\perp n$;\\\\\n\\textcircled{2} $m \\perp n$, $\\alpha \\parallel \\beta$, $m \\perp \\alpha \\Rightarrow n \\parallel \\beta$;\\\\\n\\textcircled{3} $m \\perp n$, $\\alpha \\parallel \\beta$, $m \\parallel \\alpha \\Rightarrow n \\perp \\beta$;\\\\\n\\textcircled{4} $m \\perp \\alpha$, $m \\parallel n$, $\\alpha \\parallel \\beta \\Rightarrow n \\perp \\beta$.\\\\\n其中真命题的编号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019529": { + "id": "019529", + "content": "已知集合 $A$、$B$、$C$, $A=\\{$ 直线 $\\}$, $B=\\{$ 平面 $\\}$, $C=A \\cup B$, 若 $a \\in A$, $b \\in B$, $c \\in C$, 下面给出四个命题:\\\\\n\\textcircled{1} $\\begin{cases}a \\perp b,\\\\c \\perp b\\end{cases}\\Rightarrow a \\parallel c$; \\textcircled{2} $\\begin{cases}a \\perp b,\\\\c \\parallel b\\end{cases}\\Rightarrow a \\perp c$; \\textcircled{3} $\\begin{cases}a \\parallel b,\\\\c \\parallel b\\end{cases}\\Rightarrow a \\parallel c$; \\textcircled{4} $\\begin{cases}a \\parallel b,\\\\c \\perp b\\end{cases}\\Rightarrow a \\perp c$.\n其中所有正确命题的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019530": { + "id": "019530", + "content": "已知正方体的棱长为 $1$, 每条棱所在的直线与平面所成的角相等, 则平面截此正方体所得截面面积的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019531": { + "id": "019531", + "content": "两条直线 $a$、$b$ 分别和异面直线 $c$、$d$ 都相交, 则直线 $a$、$b$ 的位置关系是\\bracket{20}.\n\\twoch{一定是异面直线}{一定是相交直线}{可能是平行直线}{可能是异面直线, 也可能是相交直线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019532": { + "id": "019532", + "content": "平面 $\\alpha$ 外有两条直线 $m$、$n$, 如果 $m$、$n$ 在平面 $\\alpha$ 内的射影分别是直线 $m'$、$n'$, 给出下列四个命题:\\\\\n\\textcircled{1} $m' \\perp n' \\Rightarrow m \\perp n$; \\textcircled{2} $m \\perp n \\Rightarrow m' \\perp n'$; \\textcircled{3} 若 $m'$ 和 $n'$ 相交, 则 $m$ 和 $n$ 相交或重合; \\textcircled{4} 若 $m'$ 和 $n'$ 平行, 则 $m$ 和 $n$ 平行或重合;\n其中真命题的个数是\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019533": { + "id": "019533", + "content": "如图, 已知正三棱柱 $ABC-A_1B_1C_1, AC=AA_1, E$、$F$ 分别是棱 $BC, A_1C_1$ 上的点. 记 $EF$ 与 $AA_1$ 所成的角为 $\\alpha$, $EF$ 与平面 $ABC$ 所成的角为 $\\beta$, 二面角 $F-BC-A$ 的平面角为 $\\gamma$, 则\\bracket{20}.\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 ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A_1)!0.7!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\alpha \\leq \\gamma \\leq \\beta$}{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019534": { + "id": "019534", + "content": "如图, 已知 $ABCD$ 是矩形, $SA \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} 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] {$S$} coordinate (S);\n\\filldraw ($(S)!0.3!(C)$) circle (0.03) node [left] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(S)--cycle(S)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(C)(A)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) $E$ 是 $SC$ 上一点, 求证: $BE$ 不可能垂直于平面 $SCD$;\\\\\n(2) 若 $SA=AB=2$, 求直线 $AC$ 与平面 $SCD$ 所成的角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019535": { + "id": "019535", + "content": "如图, 已知 $P$ 是平行四边形 $ABCD$ 所在平面外一点, $M$、$N$ 分别是 $AB$、$PC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,1) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(M)--(N);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$ 平面 $PAD$;\\\\\n(2) 若 $MN=BC=4$, $PA=4 \\sqrt{3}$, 求异面直线 $PA$ 与 $MN$ 所成的角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019536": { + "id": "019536", + "content": "《九章算术 -- 商功》: ``斜解立方, 得两堑堵. 斜解堑堵, 其一为阳马, 一为鳖臑. 阳马居二, 鳖臑居一, 不易之率也. 合两鳖臑三而一, 验之以基, 其形露矣.'' 刘徽注: ``此术臑者, 背节也, 或曰半阳马, 其形有似鳖肘, 故以名云. 中破阳马, 得两鳖臑, 鳖臑之起数, 数同而实据半, 故云六而一即得.'' \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--cycle(D_1)--(B)--(C_1);\n\\draw [dashed] (A)--(D)--(C)--(D_1)--(D);\n\\draw (1.5,0,0) node [below] {堑堵};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (A)--(B)--(C)--(D_1)--cycle(D_1)--(B);\n\\draw [dashed] (A)--(D)--(C)(D_1)--(D);\n\\draw (1.5,0,0) node [below] {阳马};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (B)--(C)--(C_1)--(D_1)--cycle(B)--(C_1);\n\\draw [dashed] (C)--(D_1);\n\\draw (1.5,0,0) node [below] {鳖臑};\n\\end{tikzpicture}\n\\end{center}\n如图, 在鳖臑 $ABCD$ 中, 侧棱 $AB \\perp$ 底面 $BCD$, 底面 $BCD$ 为直角三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\angle ADB=\\theta_1$, $\\angle BDC=\\theta_2$, $\\angle ADC=\\theta_3$, 求证: $\\cos \\theta_1 \\cdot \\cos \\theta_2=\\cos \\theta_3$;\\\\\n(2) 若 $AB=1$, $BC=2$, $CD=1$, 求异面直线 $AC$ 与 $BD$ 所成角的余弦值;\\\\\n(3) 若 $BD \\perp CD, AB=BD=CD=2$, 点 $P$ 在棱 $AC$ 上运动, 试求 $\\triangle PBD$ 面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书10空间直线与平面", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019537": { + "id": "019537", + "content": "下列命题中: \\textcircled{1} 底面是矩形的平行六面体是长方体; \\textcircled{2} 底面是正方形的直平行六面体是正四棱柱; \\textcircled{3} 底面正方形的直四棱柱是正方体; \\textcircled{4} 所有棱长都相等的直平行六面体是正方体. 正确的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019538": { + "id": "019538", + "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马, 设 $AA_1$ 是正六棱柱的一条侧棱, 如图, 若阳马以该正六棱柱的顶点为顶点, 以 $AA_1$ 为底面矩形的一边, 则这样的阳马的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}\n\\coordinate (A) at (0,0) node [below] {$A$};\n\\path (A) --++ (45:{sqrt(3)/2}) --++ (0.5,0) coordinate (B);\n\\path (A) --++ (45:{sqrt(3)}) coordinate (C);\n\\path (C) --++ (-1,0) coordinate (D);\n\\path (B) --++ (-2,0) coordinate (E);\n\\coordinate (F) at (-1,0);\n\\draw (F) -- (A) -- (B) -- (C);\n\\draw [dashed] (C) -- (D) -- (E) -- (F);\n\\foreach \\i in {(A),(B),(C),(F)}{\\draw \\i --++ (0,2);};\n\\foreach \\i in {(D),(E)}{\\draw [dashed] \\i --++ (0,2);};\n\\path (A) --++ (0,2) coordinate (A1) node [above] {$A_1$};\n\\path (B) --++ (0,2) coordinate (B1);\n\\path (C) --++ (0,2) coordinate (C1);\n\\path (D) --++ (0,2) coordinate (D1);\n\\path (E) --++ (0,2) coordinate (E1);\n\\path (F) --++ (0,2) coordinate (F1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- (E1) -- (F1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019539": { + "id": "019539", + "content": "如图, 已知正四棱柱 $ABCD-A_1B_1C_1D_1$, $AB=BC=2$, $AA_1=a$, 若在棱 $AA_1$ 上存在点 $M$ 使得 $MC_1 \\perp MB$, 则 $a$ 的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(A)!0.65!(A_1)$) node [left] {$M$} coordinate (M);\n\\draw (B)--(M);\n\\draw [dashed] (M)--(C_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019540": { + "id": "019540", + "content": "已知三棱柱 $ABC-A_1B_1C_1$ 的 $6$ 个顶点都在球 $O$ 的球面上, 若 $AB=3$, $AC=4, AB \\perp AC$, $AA_1=12$, 则球 $O$ 的半径为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\sqrt{17}}{2}$}{$2 \\sqrt{10}$}{$\\dfrac{13}{2}$}{$3 \\sqrt{10}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019541": { + "id": "019541", + "content": "已知一个直角三角形的两条直角边的长分别为 $1$ 和 $2$, 将这个三角形分别绕其两条直角边旋转得到两个圆锥, 则这两个圆锥的体积之比为\\bracket{20}.\n\\fourch{$1$}{$2$}{$4$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019542": { + "id": "019542", + "content": "正四棱锥 $P-ABCD$ 的底面边长为 $2 \\sqrt{3}$, 侧面积为 $8 \\sqrt{3}$, 则它的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019543": { + "id": "019543", + "content": "一个圆锥与一个球的体积相等且圆锥的底面半径是球半径的 $2$ 倍. 若圆锥的高为 $1$ ,则球的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019544": { + "id": "019544", + "content": "已知圆柱的底面半径为 $2$ , 高为 $4$, 经过圆柱两条母线的截面与圆柱的轴之间的距离为 $\\sqrt{3}$, 则该截面的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019545": { + "id": "019545", + "content": "圆锥的侧面展开图恰好是一个半圆, 则该圆锥的母线与底面所成的角的大小是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019546": { + "id": "019546", + "content": "矩形 $ABCD$ 中, $AB=4$, $BC=3$, 沿 $AC$ 将矩形 $ABCD$ 折成一个直二面角 $B-AC-D$, 则四面体 $ABCD$ 的外接球的体积为\\bracket{20}.\n\\fourch{$\\dfrac{125}{12}\\pi$}{$\\dfrac{125}{9}\\pi$}{$\\dfrac{125}{6}\\pi$}{$\\dfrac{125}{3}\\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019547": { + "id": "019547", + "content": "如图所示, 在边长为 $4$ 的正方形纸片 $ABCD$ 中, $AC$ 与 $BD$ 相交于点 $O$, 剪去 $\\triangle AOB$, 将剩余部分沿 $OC$、$OD$ 折叠,使 $OA$、 $OB$ 重合, 则以 $A(B)$、$C$、$D$、$O$ 为顶点的四面体的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A) (2,0) node [below] {$B$} coordinate (B) (2,2) node [above] {$C$} coordinate (C) (0,2) node [above] {$D$} coordinate (D);\n\\draw (1,1) node [above] {$O$} coordinate (O);\n\\fill [pattern = north east lines] (A)--(B)--(O)--cycle;\n\\draw (A) rectangle (C) (A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019548": { + "id": "019548", + "content": "课本中介绍了应用祖暅原理推导棱锥体积公式的做法. 祖暅原理也可用来求旋转体的体积. 现介绍用祖暅原理求球体体积公式的做法: 可构造一个底面半径和高都与球半径相等的圆柱, 然后在圆柱内挖去一个以圆柱下底面圆心为顶点, 圆柱上底面为底面的圆锥, 用这样一个几何体与半球应用祖暅原理, 即可求得球的体积公式. 请研究和理解球的体积公式求法的基础上, 解答以下问题: 已知椭圆的标准方程 $\\dfrac{x^2}{4}+\\dfrac{y^2}{25}=1$, 将此椭圆绕 $y$ 轴旋转一周后, 得一橄榄状的几何体, 求其体积.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) arc (180:0:2) arc (0:-180:2 and 0.5);\n\\draw [dashed] (0,0) arc (180:0:2 and 0.5) -- (0,0);\n\\fill [color = gray!30] (2,1) ellipse ({sqrt(3)} and {sqrt(3)/4});\n\\draw ({2-sqrt(3)},{1}) arc (180:360:{sqrt(3)} and {sqrt(3)/4});\n\\draw [dashed] ({2-sqrt(3)},{1}) arc (180:0:{sqrt(3)} and {sqrt(3)/4});\n\\draw [dashed] (2,0) -- (2,1) (2,0.2) node [left] {$h$};\n\\draw [dashed] (2,0) -- ({2+sqrt(3)},1) (3,0) node [below] {$R$};\n\\filldraw [even odd rule, gray!30] (7,1) ellipse (2 and 0.5) (7,1) ellipse (1 and 0.25);\n\\draw (5,0) arc (180:360:2 and 0.5) (5,2) arc (180:-180:2 and 0.5) (5,0) -- (5,2) (9,0) -- (9,2);\n\\draw [dashed] (5,0) -- (9,0) (7,0) -- (7,1) (7,0) -- (5,2) (7,0) -- (9,2) (8,0) node [below] {$R$} (7,0.4) node [left] {$h$};\n\\draw (5,1) arc (180:360:2 and 0.5);\n\\draw [dashed] (5,1) arc (180:0:2 and 0.5) (6,1) arc (180:-180:1 and 0.25);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (1 and 1.5);\n\\draw [dashed] (0,0) ellipse (0.5 and 1.5);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019549": { + "id": "019549", + "content": "正三棱锥的高为$1$, 底面边长为 $2 \\sqrt{6}$, 内有一个球与它的四个面都相切, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(6)},0,0) coordinate (A);\n\\draw ({sqrt(6)},0,0) coordinate (B);\n\\draw (0,0,{-3*sqrt(2)}) coordinate (C);\n\\draw (0,1,{-sqrt(2)}) coordinate (D);\n\\draw (D)--(A)--(B)--(C)--cycle(D)--(B);\n\\draw [dashed] (A)--(C);\n\\filldraw (0,{sqrt(6)-2},{-sqrt(2)}) coordinate (O) circle (0.03);\n\\draw [dashed] (O) circle ({sqrt(6)-2});\n\\end{tikzpicture}\n\\end{center}\n(1) 正三棱锥的表面积;\\\\\n(2) 内切球的表面积与体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019550": { + "id": "019550", + "content": "若圆柱的侧面积为 $2 \\pi$, 底面积为 $\\pi$, 则该圆柱的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019551": { + "id": "019551", + "content": "已知正四棱锥 $S-ABCD$ 中, $SA=2 \\sqrt{3}$, 它的高为 $2$, 则它的侧面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019552": { + "id": "019552", + "content": "正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长 $AB=2$, 若直线 $B_1C$ 与底面 $ABCD$ 所成的角的大小为 $\\arctan 2$, 则正四棱柱 $ABCD-A_1B_1C_1D_1$ 的侧面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019553": { + "id": "019553", + "content": "若一个圆锥的侧面展开图是面积为 $2 \\pi$ 的半圆面, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019554": { + "id": "019554", + "content": "如图, 在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$ 是 $C_1B_1$ 的中点, 若 $E$、$F$ 都是 $AB$ 上的点, 且 $|EF|=\\dfrac{a}{2}, Q$ 是 $A_1B_1$ 上的点,则四面体 $EFPQ$ 的体积是\\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 ($(B_1)!0.5!(C_1)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(A_1)!0.6!(B_1)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\filldraw ($(A)!0.2!(B)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A)!0.7!(B)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019555": { + "id": "019555", + "content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $\\sqrt{3}$, 以顶点 $A$ 为球心 $2$ 为半径的球面被正方体的表面 $ABB_1A_1$、$BCC_1B_1$ 截得的两段弧长之和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019556": { + "id": "019556", + "content": "一个球与一个正三棱柱的三个侧面和两个底面都相切,已知这个球的体积是 $\\dfrac{32}{3}\\pi$, 那么该三棱柱的体积是\\bracket{20}.\n\\fourch{$48 \\sqrt{3}$}{$16 \\sqrt{3}$}{$24 \\sqrt{3}$}{$96 \\sqrt{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019557": { + "id": "019557", + "content": "圆锥形容器的高为 $h$, 顶点向上放置时, 圆锥内水面的高为 $h_1$, 且 $h_1=\\dfrac{1}{3}h$, 若将圆锥倒置, 顶点向下放置, 水面高为 $h_2$, 则 $h_2$ 等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}h$}{$\\dfrac{19}{27}h$}{$\\dfrac{\\sqrt[3]{6}}{3}h$}{$\\dfrac{\\sqrt[3]{19}}{3}h$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019558": { + "id": "019558", + "content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$, 则下列四个命题:\\\\\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} 若点 $M$ 是平面 $A_1B_1C_1D_1$ 上到点 $D$ 和 $C_1$ 距离相等的点, 则点 $M$ 的轨迹是过点 $D_1$ 的直线.\\\\\n其中所有真命题是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B_1) -- (B) (C_1) -- (C);\n\\draw [dashed] (D_1) -- (D);\n\\draw (B)--(C_1)(A)--(C);\n\\draw [dashed] (A)--(D_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019559": { + "id": "019559", + "content": "在如图所示的组合体中, 三棱柱 $ABC-A_1B_1C_1$ 的侧面 $ABB_1A_1$ 是圆柱的轴截面, $C$ 是圆柱底面圆周上不与 $A$、$B$ 重合的一个点.\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) coordinate (O) circle (0.03);\n\\filldraw (0,2) coordinate (O_1) circle (0.03);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (1,2) node [right] {$B_1$} coordinate (B_1);\n\\draw (O) ++ (250:1 and 0.25) node [below] {$C$} coordinate (C);\n\\draw (C) ++ (0,2) node [below right] {$C_1$} coordinate (C_1);\n\\draw (A) arc (180:360:1 and 0.25) -- (B_1) (A_1)--(A) (A_1)--(B_1) (A_1)--(C_1)--(B_1) (C)--(C_1);\n\\draw (O_1) ellipse (1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25)(A)--(B)(A_1)--(C)(A_1)--(B)(A)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 圆柱的轴截面是正方形, 当点 $C$ 是弧 $AB$ 的中点时, 求异面直线 $A_1C$ 与 $AB_1$ 的所成角的大小;\\\\\n(2) $C$ 是弧 $AB$ 的中点时, 求四棱锥 $A_1-BCC_1B_1$ 与圆柱的体积比.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019560": { + "id": "019560", + "content": "已知圆锥的顶点为 $P$, 底面圆心为 $O$, 半径为 $2$.\n\\begin{center}\n\\begin{tikzpicture}\n\\node (0,0) [left] {$O$} coordinate (O);\n\\draw (-1.5,0) arc (180:360:1.5 and {1.5/3}) node [right] {$B$} coordinate (B);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and {1.5/3}) coordinate (C);\n\\draw (C) -- (0,3) node [above] {$P$} coordinate (P) -- (B); \n\\coordinate (A) at ({1.5*cos(250)},{0.5*sin(250)});\n\\draw [dashed] (A) node [below left] {$A$} -- (O) -- (B) -- cycle;\n\\coordinate (M) at ($(A)!0.5!(B)$);\n\\draw [dashed] (O) -- (P) -- (M) node [shift = {(-45:0.5)}] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 设圆锥的母线长为 $4$, 求圆锥的体积;\\\\\n(2) 设 $PO=4, OA$、$OB$ 是底面半径, 且 $\\angle AOB=90^{\\circ}, M$ 为线段 $AB$ 的中点, 求异面直线 $PM$ 与 $OB$ 所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019561": { + "id": "019561", + "content": "已知三棱锥 $P-ABC, PA \\perp$ 平面 $ABC, PA=AB=BC=2$, 直线 $PC$ 与平面 $ABC$ 所成角的大小为 $\\arctan \\dfrac{\\sqrt{2}}{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(P)--(C)--(B)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$ 平面 $PAB$;\\\\\n(2) 设 $E$ 为线段 $PC$ 的中点, 求异面直线 $AE$ 与 $BC$ 所成角的大小;\\\\\n(3) 设 $M$ 是三棱锥 $P-ABC$ 内 (包括边界) 的动点, 且满足 $|AM| \\leq \\sqrt{2}$, 求点 $M$ 所形成的几何体的全面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书11简单几何体", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019562": { + "id": "019562", + "content": "在平行六面体 $ABCD-A' B' C' D'$ 中, 点 $M$ 在对角线 $A' B$ 上, 且 $|\\overrightarrow{A' M}|=\\dfrac{1}{2}|\\overrightarrow{MB}|$, 点 $N$ 在对角线 $A' C$ 上, 且 $|\\overrightarrow{A' N}|=\\dfrac{1}{3}|\\overrightarrow{NC}|$. 记 $\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AD}=\\overrightarrow{b}$, $\\overrightarrow{AA'}=\\overrightarrow{c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{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) ++ (2.3,0,-\\m) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0.3,0,-\\m) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0.4,\\n,-0.2) node [left] {$D'$} coordinate (D');\n\\draw (A) ++ (0.4,\\n,-0.2) node [right] {$A'$} coordinate (A');\n\\draw (B) ++ (0.4,\\n,-0.2) node [above right] {$B'$} coordinate (B');\n\\draw (C) ++ (0.4,\\n,-0.2) node [above left] {$C'$} coordinate (C');\n\\draw (D') -- (A') -- (B') -- (C') -- cycle;\n\\draw (D) -- (D') (A) -- (A') (B) -- (B');\n\\draw [dashed] (C) -- (C');\n\\filldraw ($(A')!{1/3}!(B)$) node [right] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($(A')!{1/4}!(C)$) node [left] {$N$} coordinate (N) circle (0.03);\n\\draw (A')--(B);\n\\draw [dashed] (A')--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $M$、$N$、$D'$ 三点共线;\\\\\n(2) $\\overrightarrow{AP}=\\dfrac{1}{5}\\overrightarrow{a}+\\dfrac{2}{5}\\overrightarrow{b}+\\dfrac{2}{5}\\overrightarrow{c}$, 求证: 点 $P \\in$ 平面 $A' BD$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019563": { + "id": "019563", + "content": "如图, 在三棱锥 $D-ABC$ 中, $\\angle DAC=\\angle BAC= 60^{\\circ}$, $AC=1$, $AB=2$, $AD=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,{-1.5*sqrt(3)}) node [above] {$D$} coordinate (D);\n\\draw ({1/2},{sqrt(6)/3},{-sqrt(3)/6}) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)(A)--(C)(B)--(D);\n\\draw [dashed] (A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\overrightarrow{AC}\\cdot \\overrightarrow{BD}$, 并说明异面直线 $AC$ 与 $BD$ 所成的角 $\\theta$ 的大小在棱 $BD$ 长度增大时是怎样变化的;\\\\\n(2) 若 $\\overrightarrow{AC}\\cdot \\overrightarrow{BC}=0$, 判断点 $D$ 在平面 $ABC$ 上的射影是否可能在直线 $BC$ 上, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019564": { + "id": "019564", + "content": "如图, 在圆柱 $OO_1$ 中, 它的轴截面 $ABB_1A_1$ 是一个边长为 2 的正方形, 点 $C$ 为棱 $BB_1$ 的中点, 点 $C_1$ 为弧 $A_1B_1$ 的中点. 求.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) coordinate (O) circle (0.03) node [below] {$O$};\n\\filldraw (0,2) coordinate (O_1) circle (0.03) node [above] {$O_1$};\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (1,2) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$C$} coordinate (C);\n\\draw (O_1) ++ (260:1 and 0.25) node [below left] {$C_1$} coordinate (C_1);\n\\draw (A) arc (180:360:1 and 0.25) -- (B_1) (A_1)--(A) (A_1)--(B_1)(A_1)--(C_1);\n\\draw (O_1) ellipse (1 and 0.25);\n\\draw [dashed] (A)--(B)(O)--(A_1)(O)--(C_1)(O)--(C)(C_1)--(C)(A_1)--(C);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}\n(1) 异面直线 $OC$ 与 $A_1C_1$ 所成角的大小;\\\\\n(2) 直线 $CC_1$ 与圆柱 $OO_1$ 底面所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019565": { + "id": "019565", + "content": "四棱锥 $P-ABCD$ 的底面 $ABCD$ 是平行四边形, $PF \\perp$ 平面 $ABCD$, 垂足 $F$ 在 $AD$ 上, 且 $AF=\\dfrac{1}{3}FD, FB \\perp FC$, $FB=FC=2, E$ 是 $BC$ 的中点, 四面体 $P-BCF$ 的体积为 $\\dfrac{8}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [above right] {$F$} coordinate (F);\n\\draw (F) ++ (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (F) ++ ({-sqrt(2)/2},0) node [left] {$A$} coordinate (A);\n\\draw (F) ++ ({3*sqrt(2)/2},0) node [right] {$D$} coordinate (D);\n\\draw (F) ++ (0,0,{sqrt(2)}) node [below] {$E$} coordinate (E);\n\\draw (E) ++ ({-sqrt(2)},0) node [left] {$B$} coordinate (B);\n\\draw (E) ++ ({sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\draw (B)--(C)--(D)--(P)--cycle (P)--(C);\n\\draw [dashed] (B)--(A)--(D)(B)--(F)--(C)(E)--(F)--(P)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角 $B-PC-F$ 的大小;\\\\\n(2) 求点 $D$ 到平面 $PBF$ 的距离;\\\\\n(3) 设三棱锥 $F-PCD$ 的体积为 $V$, 是否存在体积为 $n V$ ($n$ 为正整数), 且各棱长均相等的直平行六面体, 使得它的所有棱长和为$24$, 若存在, 请构造出这样的一个直平行六面体; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019566": { + "id": "019566", + "content": "在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$、$F$ 分别是棱 $BC$、$CD$ 上的点, 且 $BE=CF$.\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] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.3!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.3!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (B_1)--(F)(D_1)--(E); \n\\end{tikzpicture}\n\\end{center}\n(1) 若 $BE=CF=\\dfrac{1}{2}a$, 求证: $EF\\parallel $ 平面 $AB_1D_1$;\\\\\n(2) 求证: $B_1F \\perp D_1E$;\\\\\n(3) 是否存在点 $E$、$F$, 使 $A_1C \\perp$ 面 $C_1EF$ ?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019567": { + "id": "019567", + "content": "已知向量 $\\overrightarrow{a}=(1,0,2)$, $\\overrightarrow{b}=(2,1,0)$, 则向量 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019568": { + "id": "019568", + "content": "若 $(0,-2,4)$ 和 $(-\\sqrt{15}, 1,2)$ 分别是平面 $\\alpha, \\beta$ 的一个法向量, 则 $\\alpha$ 与 $\\beta$ 构成的二面角大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019569": { + "id": "019569", + "content": "在四棱锥 $P-ABCD$ 中, 设向量 $\\overrightarrow{AB}=(4,-2,3)$, $\\overrightarrow{AD}=(-4, 1,0)$, $\\overrightarrow{AP}=(-6,2,-8)$, 则顶点 $P$ 到底面 $ABCD$ 的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019570": { + "id": "019570", + "content": "如图, 长方体 $EPFC-ADBO$ 中, $OA=AD=1$, $AE=2$, 则\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [above left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$D$} coordinate (D);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (0,0,-\\m) node [left] {$O$} coordinate (O);\n\\draw (A) -- (D) -- (B);\n\\draw [dashed] (A) -- (O) -- (B);\n\\draw (A) ++ (0,\\n,0) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (0,\\n,0) node [right] {$P$} coordinate (P);\n\\draw (B) ++ (0,\\n,0) node [above right] {$F$} coordinate (F);\n\\draw (O) ++ (0,\\n,0) node [above left] {$C$} coordinate (C);\n\\draw (E) -- (P) -- (F) -- (C) -- cycle;\n\\draw (A) -- (E) (D) -- (P) (B) -- (F);\n\\draw [dashed] (O) -- (C);\n\\draw [->] (A) -- ($(O)!1.5!(A)$) node [left] {$x$};\n\\draw [->] (B) -- ($(O)!1.5!(B)$) node [right] {$y$};\n\\draw [->] (C) -- ($(O)!1.2!(C)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}\n(1) 异面直线 $OP$ 与 $AC$ 所成的角为\\blank{50}.\\\\\n(2) 直线 $DF$ 与平面 $ODPC$ 所成的角为\\blank{50}.\\\\\n(3) 写出平面 $ABC$ 的一个法向量 $\\overrightarrow{n}=$\\blank{50}.\\\\\n(4) 二面角 $O-AB-C$ 的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019571": { + "id": "019571", + "content": "圆柱底面半径为$1$, 高为 $2$, $AB$ 为上底底面的直径, 点 $C$ 是下底底面圆弧上的一个动点, 点 $C$ 绕着下底底面旋转一周, 则 $\\triangle ABC$ 面积的范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019572": { + "id": "019572", + "content": "已知边长为 $4 \\sqrt{2}$ 的正三角形 $ABC$ 中, $E, F$ 分别为 $BC$ 和 $AC$ 的中点, $PA \\perp$ 面 $ABC$, 且 $PA=2$, 设平面 $\\alpha$ 过 $PF$ 且与 $AE$ 平行, 则 $AE$ 与平面 $\\alpha$ 间的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019573": { + "id": "019573", + "content": "正方体 $ABCD-A_1B_1C_1D_1$ 中, $E, F$ 分别是 $AA_1$ 与 $CC_1$ 的中点, 则直线 $ED$ 与 $D_1F$ 所成角的余弦值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019574": { + "id": "019574", + "content": "棱长都为 $2$ 的直平行六面体 $ABCD-A_1B_1C_1D_1$ 中, $\\angle BAD=60^{\\circ}$, 则对角线 $A_1C$ 与侧面 $DCC_1D_1$ 所成角的正弦值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{4}$}{$\\dfrac{\\sqrt{3}}{8}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019575": { + "id": "019575", + "content": "设 $A, B, C, D$ 是半径为 $1$ 的球面上的四个不同点, 且满足 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=0$, $\\overrightarrow{AC}\\cdot \\overrightarrow{AD}=0$, $\\overrightarrow{AD}\\cdot \\overrightarrow{AB}=0$, 用 $S_1, S_2, S_3$ 分别表示 $\\triangle ABC$、$\\triangle ACD$、$\\triangle ABD$ 的面积, 则 $S_1+S_2+S_3$ 的最大值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$2$}{$4$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019576": { + "id": "019576", + "content": "四棱锥 $P-ABCD$, 底面为正方形 $ABCD$, 边长为 $4$, $E$ 为 $AB$ 中点, $PE \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,0,1) node [below] {$E$} coordinate (E);\n\\draw (2,0,1) node [right] {$F$} coordinate (F);\n\\draw (E) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C)(P)--(F);\n\\draw [dashed] (P)--(E)--(F)(P)--(A)--(C)(B)--(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\triangle PAB$ 为等边三角形, 求四棱锥 $P-ABCD$ 的体积;\\\\\n(2) 若 $CD$ 的中点为 $F$, $PF$ 与平面 $ABCD$ 所成角为 $45^{\\circ}$, 求 $PD$ 与 $AC$ 所成角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019577": { + "id": "019577", + "content": "如图, 直三棱柱 $ABC-A_1B_1C_1$ 中, $\\angle BAC=90^{\\circ}$, $AA_1=AB=AC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$M$} coordinate (M);\n\\draw (C)--(B)--(B_1)--(A_1)--(C_1)--cycle(B_1)--(C_1)--(M);\n\\draw [dashed] (C_1)--(A)--(B)(C)--(A)--(M)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 设 $M$ 是棱 $BB_1$ 的中点, 求 $C$ 到平面 $MAC_1$ 的距离;\\\\\n(2) 若 $M$ 是棱 $BB_1$ 上的任意一点 (包括端点), 求二面角 $M-AC_1-A_1$ 的大小的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019578": { + "id": "019578", + "content": "在三棱锥 $P-ABC$ 中, $AB \\perp BC, AB=BC=k PA$, 点 $O, D$ 分别是 $AC, PC$ 的中点, $OP \\perp$ 底面 $ABC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [right] {$D$} coordinate (D);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(P)--(O)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OD\\parallel $ 平面 $PAB$;\\\\\n(2) 当 $k=\\dfrac{1}{2}$ 时, 求直线 $PA$ 与平面 $PBC$ 所成角的大小;\\\\\n(3) 当 $k$ 为何值时, $O$ 在平面 $PBC$ 内的射影恰好为 $\\triangle PBC$ 的重心?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书12空间向量及其应用", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019579": { + "id": "019579", + "content": "从 $5$ 个男生和 $4$ 个女生中选出 $4$ 名学生参加一次会议, 共有多少种不同的选法?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019580": { + "id": "019580", + "content": "用五种不同颜色给图中四个区域涂色, 如果每个区域涂一种颜色, 相邻区域不同色, 共有多少种不同的涂色方法?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (2,1.5);\n\\draw (0,0.75) -- (1,1.5);\n\\draw (0.3,1.25) node {$1$};\n\\draw (0.5,1.125) -- (2,1.125) (1,0) -- (1,1.125);\n\\draw (1.3,1.3125) node {$2$};\n\\draw (0.5,0.625) node {$3$} (1.5,0.625) node {$4$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019581": { + "id": "019581", + "content": "$6$ 位选手依次演讲, 其中选手甲不在第一个也不在最后一个演讲, 则不同的演讲次序共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019582": { + "id": "019582", + "content": "某食堂规定, 每份午餐可以在四种水果中任选两种, 则甲、乙两同学各自所选的两种水果相同的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019583": { + "id": "019583", + "content": "在报名的 $3$ 名男教师和 $6$ 名女教师中, 选取 $5$ 人参加义务献血, 要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}. (结果用数值表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019584": { + "id": "019584", + "content": "$6$ 个人排成一行, 其中甲、乙两人不相邻的不同排法共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019585": { + "id": "019585", + "content": "从 $3$ 名骨科、$4$ 名脑外科和 $5$ 名内科医生中选派 $5$ 人组成一个抗震救灾医疗小组, 则骨科、脑外科和内科医生都至少有 $1$ 人的选派方法种数是\\blank{50}(用数字作答)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019586": { + "id": "019586", + "content": "将序号分别为 $1,2,3,4,5$ 的 $5$ 张参观券全部分给 $4$ 人, 每人至少 $1$ 张, 如果分给同一人的 $2$ 张参观券连号, 那么不同的分法种数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019587": { + "id": "019587", + "content": "某同学有同样的画册 $2$ 本, 同样的集邮册 $3$ 本, 从中取出 $4$ 本赠送给 $4$ 位朋友每位朋友 $1$ 本, 则不同的赠送方法共有\\bracket{20}.\n\\fourch{$4$ 种}{$10$ 种}{$18$ 种}{$20$ 种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019588": { + "id": "019588", + "content": "$4$ 位同学每人从甲、乙、丙 $3$ 门课程中选修 $1$ 门, 则恰有 $2$ 人选修课程甲的不同选法共有\\bracket{20}.\n\\fourch{$12$ 种}{$24$ 种}{$30$ 种}{$36$ 种}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019589": { + "id": "019589", + "content": "求 $(1+2 x)^7$ 的展开式的第四项的系数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019590": { + "id": "019590", + "content": "设常数 $a \\in \\mathbf{R}$. 若 $(x^2+\\dfrac{a}{x})^5$ 的二项展开式中 $x^7$ 项的系数为 $-10$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019591": { + "id": "019591", + "content": "在 $(1+x+\\dfrac{1}{x^{2015}})^{10}$ 的展开式中, $x^2$ 项的系数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019592": { + "id": "019592", + "content": "已知 $(1-2 x)^7=a_0+a_1 x+a_2 x^2+\\cdots+a_7 x^7$, 则 $a_1+a_2+\\cdots+a_7= a_1+a_3+a_5+a_7= ;|a_0|+|a_1|+\\cdots+|a_7|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019593": { + "id": "019593", + "content": "$(x+\\dfrac{a}{x})(2 x-\\dfrac{1}{x})^5$ 的展开式中各项系数的和为 $2$, 则该展开式中常数项为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019594": { + "id": "019594", + "content": "若多项式 $x^2+x^{11}=a_0+a_1(x+1)+n(x+1)^2+\\cdots+a_{11}(x+1)^{11}$, 则 $n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019595": { + "id": "019595", + "content": "随机抽取 $9$ 个同学中, 至少有 $2$ 个同学在同一月出生的概率是\\blank{50}(默认每月天数相同, 结果精确到$0.001$).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019596": { + "id": "019596", + "content": "为强化安全意识, 某商场拟在未来连续 $10$ 天中随机选择 $3$ 天进行紧急疏散演练, 则选择 $3$ 天恰好为连续 $3$ 天的概率是\\blank{50}(结果用最简分数表示).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019597": { + "id": "019597", + "content": "在 $30$ 瓶饮料中, 有 $3$ 瓶已过了保质期, 从这 $30$ 瓶饮料中任取 $2$ 瓶,则至少取到 $1$ 瓶已过保质期饮料的概率为\\blank{50}. (结果用最简分数表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019598": { + "id": "019598", + "content": "有 $3$ 个兴趣小组, 甲、乙两位同学各自参加其中一个小组, 每位同学参加各个小组的可能性相同,则这两位同学参加同一个兴趣小组的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019599": { + "id": "019599", + "content": "现有 $10$ 个数, 它们能构成一个以 $1$ 为首项, $-3$ 为公比的等比数列, 若从这 $10$ 个数中随机抽取一个数, 则它小于 $8$ 的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019600": { + "id": "019600", + "content": "在二项式 $(x+1)^9$ 的展开式中任取 $2$ 项, 则取出的 $2$ 项中系数均为奇数的概率为\\blank{50}. (用分数表示结果)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019601": { + "id": "019601", + "content": "有 $8$ 本互不相同的书, 其中数学书 $3$ 本、外文书 $2$ 本、其他书 $3$ 本, 若将这些书排成一排放在书架上, 则数学书排在一起, 外文书也排在一起的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019602": { + "id": "019602", + "content": "为了检测学生的自体素质指标, 从游泳类 $1$ 项,球类 $3$ 项, 田径类 $4$ 项共 $8$ 项项目中随机抽取 $4$ 项进行检测, 则每一类都初抽到的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019603": { + "id": "019603", + "content": "电视台连续播放 $6$ 个广告, 其中含 $4$ 个不同的商业广告和 $2$ 个不同的公益广告, 要求首尾必须播放公益广告, 则共有种\\blank{50}不同的播放方式. (结果用数值表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019604": { + "id": "019604", + "content": "甲、乙两个袋中均有红、白两种颜色的小球, 这些小球除颜色外完全相同, 其中甲袋装有 $4$ 个红球、 $2$ 个白球, 乙袋装有 $1$ 个红球、$5$ 个白球. 现分别从甲、乙两袋中各随机取出一个球,则取出的两球都是红球的概率为\\blank{50}. (答案用分数表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019605": { + "id": "019605", + "content": "从 $20$ 名男同学, $10$ 名女同学中任选 $3$ 名参加体能测试, 则选到的 $3$ 名同学中既有男同学又有女同学的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019606": { + "id": "019606", + "content": "从甲、乙等 $10$ 位同学中任选 $3$ 位去参加某项活动, 则所选 $3$ 位中有甲但没有乙的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019607": { + "id": "019607", + "content": "$(1+3 x)^n$($n \\in \\mathbf{N}$, $n \\geq 6$) 的展开式中 $x^5$ 与 $x^6$ 的系数相等, 则 $n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019608": { + "id": "019608", + "content": "设二项式 $(x-\\dfrac{a}{\\sqrt{x}})^6$($a>0$) 的展开式中 $x^3$ 的系数为 $A$, 常数项为 $B$, 若 $B=4A$, 则 $a$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019609": { + "id": "019609", + "content": "如果把两条异面直线看作``一对'', 则在五棱锥的棱所在的直线中, 异面直线有\\bracket{20}.\n\\fourch{$15$ 对}{$25$ 对}{$30$ 对}{$20$ 对}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019610": { + "id": "019610", + "content": "从 $0$ 到 $9$ 这 $10$ 个数字中任取 $3$ 个数字组成一个没有重复数字的三位数, 这个数不能被 $3$ 整除的概率为\\bracket{20}.\n\\fourch{$\\dfrac{41}{60}$}{$\\dfrac{38}{54}$}{$\\dfrac{35}{54}$}{$\\dfrac{19}{54}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019611": { + "id": "019611", + "content": "在 $(\\sqrt{x}+\\dfrac{1}{\\sqrt[3]{x}})^{24}$ 的展开式中, $x$ 的幂的指数是整数的项共有\\bracket{20}.\n\\fourch{$3$ 项}{$4$ 项}{$5$ 项}{$6$ 项}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019612": { + "id": "019612", + "content": "要安排甲、乙、丙、丁、戌 $5$ 名同学参加上海世博会志愿者服务活动, 每人从事翻译、导游、礼仪、司机四项工作之一, 每项工作至少有一人参加. 甲、乙不会开车但能从事其他三项工作, 丙丁戌都能胜任四项工作, 求不同安排方案的种数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019613": { + "id": "019613", + "content": "一个口袋内装有大小相等编号为 $a_1, a_2, a_3$ 的 $3$ 个白球和 $1$ 个黑球 $b$.\\\\\n(1) 从中摸出 $2$ 个球, 求摸出 $2$ 个白球的概率;\\\\\n(2) 从中连续取两次, 每次取一球后放回, 求取出的两球恰好有 $1$ 个黑球的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019614": { + "id": "019614", + "content": "已知 $(a^2+1)^n$ 展开式中的各项系数的和等于 $(\\dfrac{16}{5}x^2+\\dfrac{1}{\\sqrt{x}})^5$ 的展开式的常数项, 而 $(a^2+1)^n$ 展开式的系数最大的项等于$54$, 求 $a$ 的值 ($a \\in \\mathbf{R}$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书13计数原理与二项式定理", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019615": { + "id": "019615", + "content": "从分别写有 $1,2,3,4,5,6$ 的 $6$ 张卡片中无放回随机抽取 $2$ 张, 则抽到的 $2$ 张卡片上的数字之积是 $4$ 的倍数的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{5}$}{$\\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019616": { + "id": "019616", + "content": "某棋手与甲、乙、丙三位棋手各比赛一盘, 各盘比赛结果相互独立.已知该棋手与甲、乙、丙比赛获胜的概率分别为 $p_1, p_2, p_3$, 且 $p_3>p_2>p_1>0$. 记该棋手连胜两盘的概率为 $p$, 则\\bracket{20}.\n\\twoch{$p$ 与该棋手和甲、乙、丙的比赛次序无关}{该棋手在第二盘与甲比赛, $p$ 最大}{该棋手在第二盘与乙比赛, $p$ 最大}{该棋手在第二盘与丙比赛, $p$ 最大}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019617": { + "id": "019617", + "content": "某社区通过公益讲座以普及社区居民的垃圾分类知识. 为了解讲座效果, 随机抽取 $10$ 位社区居民,让他们在讲座前和讲座后各回答一份垃圾分类知识问卷, 这 $10$ 位社区居民在讲座前和讲座后问卷答题的正确率如下图: 则\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.7, yscale = 0.08]\n\\draw [->] (0,55) -- (11,55);\n\\draw [->] (0,55) -- (0,56) -- (0.2,57) -- (-0.2,58) -- (0,59) -- (0,105);\n\\draw (0,55) node [below left] {$O$};\n\\foreach \\i in {1,2,...,10}\n{\\draw (\\i,55.5) -- (\\i,55) node [below] {$\\i$};};\n\\foreach \\i in {60,65,...,100}\n{\\draw [dotted] (10.5,\\i) -- (0,\\i) node [left] {$\\i\\%$};};\n\\draw (5.5,45) node {居民编号};\n\\draw (-2,80) node [rotate = 90] {正确率};\n\\filldraw (12,70) circle (0.05 and {7/16}) ++ (1,0) node {讲座后};\n\\filldraw (12,80) node {\\tiny$\\times$} node {\\tiny$+$} ++ (1,0) node {讲座前};\n\\foreach \\i/\\j/\\k in {1/65/90,2/60/85,3/70/80,4/60/90,5/65/85,6/75/85,7/90/95,8/85/100,9/80/85,10/95/100}\n{\\filldraw (\\i,\\j) node {\\tiny$\\times$} node {\\tiny$+$} (\\i,\\k) circle (0.05 and {7/16});};\n\\end{tikzpicture}\n\\end{center}\n\\onech{讲座前问卷答题的正确率的中位数小于 $70 \\%$}{讲座后问卷答题的正确率的平均数大于 $85 \\%$}{讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差}{讲座后问卷答题的正确率的极差大于讲座前正确率的极差}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019618": { + "id": "019618", + "content": "某家庭记录了未使用节水龙头 50 天的日用水量数据 (单位: $\\mathrm{m}^3$ ) 和使用了节水龙头 50 天的日用水量数据, 得到频数分布表如下:\n\\begin{center}\n未使用节水龙头 $\\mathbf{50}$ 天的日用水量频数分布表\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\\hline 日用水量 &{$[0,0.1)$}&{$[0.1,0.2)$}&{$[0.2,0.3)$}&{$[0.3,0.4)$}&{$[0.4,0.5)$}&{$[0.5,0.6)$}&{$[0.6,0.7)$}\\\\\n\\hline 频数 & 1 & 3 & 2 & 4 & 9 & 26 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n使用了节水龙头 $\\mathbf{50}$ 天的日用水量频数分布表\n\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline 日用水量 &{$[0,0.1)$}&{$[0.1,0.2)$}&{$[0.2,0.3)$}&{$[0.3,0.4)$}&{$[0.4,0.5)$}&{$[0.5,0.6)$}\\\\\n\\hline 频数 & 1 & 5 & 13 & 10 & 16 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 作出使用了节水龙头 $50$ 天的日用水量数据的频率分布直方图:\\\\\n(2) 估计该家庭使用节水龙头后, 日用水量小于 $0.35 \\mathrm{m}^3$ 的概率;\\\\\n(3) 估计该家庭使用节水龙头后, 一年能节省多少水?(一年按 $365$ 天计算, 同一组中的数据以这组数据所在区间中点的值作代表.)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019619": { + "id": "019619", + "content": "已知某单位甲、乙、丙三个部门的员工人数分别为 $24,16,16$. 现采用分层抽样的方法从中抽取 $7$ 人,进行睡眠时间的调查.\\\\\n(1) 应从甲、乙、丙三个部门的员工中分别抽取多少人?\\\\\n(2) 若抽出的 $7$ 人中有 $4$ 人睡眠不足, $3$ 人睡眠充足, 现从这 $7$ 人中随机抽取 $3$ 人做进一步的身体检查. 设 $A$ 为事件``抽取的 3 人中, 既有睡眠充足的员工, 也有睡眠不足的员工'', 求事件 $A$ 发生的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019620": { + "id": "019620", + "content": "从甲、乙等 $5$ 名同学中随机选 $3$ 名参加社区服务工作, 则甲、乙都人选的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019621": { + "id": "019621", + "content": "将 $4$ 个 $1$ 和 $2$ 个 $0$ 随机排成一行, 则 $2$ 个 $0$ 不相邻的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019622": { + "id": "019622", + "content": "有 $5$ 支彩笔 (除颜色外无差别), 颜色分别为红、黄、蓝、绿、紫. 从这 $5$ 支彩笔中任取 $2$ 支不同颜色的彩笔, 则取出的 $2$ 支彩笔中含有红色彩笔的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019623": { + "id": "019623", + "content": "从正方体的 $8$ 个顶点中任选 $4$ 个, 则这 $4$ 个点在同一个平面的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019624": { + "id": "019624", + "content": "某公司生产 $A, B, C$ 三种不同型号的轿车, 产量之比依次为 $2: 3: 4$, 为检验该公司的产品质量, 用分层抽样的方法抽取一个容量为 $n$ 的样本, 若样本中 $A$ 种型号的轿车比 $B$ 种型号的轿车少 $8$ 辆, 则 $n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019625": { + "id": "019625", + "content": "已知样本数据 $x_1, x_2, \\cdots, x_n$ ($n$ 为正整数) 的平均数与方差分别是 $a$ 和 $b$, 若 $y_i=-2 x_i+ 3$($i=1,2, \\cdots, n$), 且样本数据 $y_1, y_2, \\cdots, y_n$ 的平均数与方差分别是 $b$ 和 $a$, 则 $a-b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019626": { + "id": "019626", + "content": "分别统计了甲、乙两位同学 $16$ 周的各周课外体育运动时长(单位: $h$), 得如下茎叶图, 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccc|c|cccccccc}\n\\multicolumn{4}{c|}{甲组} & & \\multicolumn{8}{c}{乙组}\\\\ \\hline\n& & 6 & 1 & 5. \\\\ \n8 & 5 & 3 & 0 & 6. & 3 \\\\\n7 & 5 & 3 & 2 & 7. & 4 & 6 \\\\\n6 & 4 & 2 & 1 & 8. & 1 & 2 & 2 & 5 & 6 & 6 & 6 & 6\\\\\n& & 4 & 2 & 9. & 0 & 2 & 3 & 8\\\\\n& & & & 10. & 1 \n\\end{tabular}\n\\end{center}\n\\onech{甲同学周课外体育运动时长的样本中位数为 $7.4$}{乙同学周课外体育运动时长的样本平均数大于 $8$}{甲同学周课外体育运动时长大于 $8$ 的概率的估计值大于 $0.4$}{乙同学周课外体育运动时长大于 $8$ 的概率的估计值大于 $0.6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019627": { + "id": "019627", + "content": "在一组样本数据中, $1,2,3,4$ 出现的频率分别为 $p_1, p_2, p_3, p_4$, 且 $\\displaystyle\\sum_{i=1}^4 p_i=1$, 则下面四种情形中, 对应样本的标准差最大的一组是\\bracket{20}.\n\\twoch{$p_1=p_4=0.1$, $p_2=p_3=0.4$}{$p_1=p_4=0.4$, $p_2=p_3=0.1$}{$p_1=p_4=0.2$, $p_2=p_3=0.3$}{$p_1=p_4=0.3$, $p_2=p_3=0.2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019628": { + "id": "019628", + "content": "有 $6$ 个相同的球, 分别标有数字 $1,2,3,4,5,6$, 从中有放回的随机取两次, 每次取 $1$ 个球, 甲表示事件``第一次取出的球的数字是 1'', 乙表示事件``第二次取出的球的数字是 2'', 丙表示事件``两次取出的球的数字之和是 8'', 丁表示事件``两次取出的球的数字之和是 7'', 则\\bracket{20}.\n\\fourch{甲与丙相互独立}{甲与丁相互独立}{乙与丙相互独立}{丙与丁相互独立}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019629": { + "id": "019629", + "content": "某调查小组进行了一次问卷调查, 并从中随机抽取了 $12$ 份问卷, 得到测试成绩 (百分制) 的茎叶图如图.\n\\begin{center}\n\\begin{tabular}{c|cccc} \n& \\multicolumn{4}{c}{成绩} \\\\\n\\hline 5 & 2 & & & \\\\\n6 & 3 & 7 & 8 & \\\\\n7 & 2 & 6 & 6 & 6 \\\\\n8 & 2 & 8 & & \\\\\n9 & 3 & 4 & & \n\\end{tabular}\n\\end{center}\n(1) 写出该样本的第 $50$ 百分位数, 若该校共有 $3000$ 名学生, 试估计该校测试成绩在 $70$ 分以上的人数;\\\\\n(2) 从测试成绩为 $[70,90]$ 的学生中随机抽取 $2$ 人,求两位学生的测试成绩均落在 $[70,80]$ 的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019630": { + "id": "019630", + "content": "某校在一次期末数学测试中, 为统计学生的考试情况, 从学校的 $2000$ 名学生中随机抽取 $50$ 名学生的考试成绩, 被测学生成绩全部介于 $65$ 分到 $145$ 分之间 (满分 $150$ 分), 将统计结果按如下方式分成八组: 第一组 $[65,75)$, 第二组 $[75,85)$, $\\cdots$, 第八组 $[135,145]$, 如图是按上述分组方法得到的频率分布直方图的一部分.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 100]\n\\draw [->] (55,0) -- (155,0) node [below] {分数};\n\\draw [->] (55,0) -- (55,0.035) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (55,0) node [below] {$0$};\n\\foreach \\i/\\j in {65/0.004,75/0.012,85/0.016,95/0.03,105/0.02,115/0.006,125/0,135/0.004}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {75/0.012,85/0.016,95/0.03,105/0.02,115/0.006,125/0,135/0.004}\n{\\draw [dashed] (\\i,\\j) -- (55,\\j) node [left] {\\small $\\k$};};\n\\draw (145,0) node [below] {$145$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求第七组的频率, 并完成频率分布直方图;\\\\\n(2) 用样本数据估计该校的 2000 名学生这次考试成绩的平均分 (同一组中的数据用该组区间的中点值代表该组数据平均值);\\\\\n(3) 若从样本成绩属于第六组和第八组的所有学生中随机抽取 $2$ 名, 求他们的分差的绝对值小于 $10$ 分的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019631": { + "id": "019631", + "content": "甲、乙、丙三位同学进行羽毛球比赛,约定赛制如下: 累计负两场者被淘汰; 比赛前抽签决定首先比赛的两人, 另一人轮空; 每场比赛的胜者与轮空者进行下一场比赛, 负者下一场轮空, 直至有一人被淘汰; 当一人被淘汰后, 剩余的两人继续比赛, 直至其中一人被淘汰, 另一人最终获胜, 比赛结束. 经抽签, 甲、乙首先比赛, 丙轮空. 设每场比赛双方获胜的概率都为 $\\dfrac{1}{2}$.\\\\\n(1) 求甲连胜四场的概率;\\\\\n(2) 求需要进行第五场比赛的概率;\\\\\n(3) 求丙最终获胜的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书14概率与统计", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019632": { + "id": "019632", + "content": "某校从学生文艺部 $6$ 名成员 ($4$ 男 $2$ 女) 中, 挑选 $2$ 人参加学校举办的文艺汇演活动.\\\\\n(1) 求男生甲被选中的概率;\\\\\n(2) 在已知男生甲被选中的条件下,女生乙被选中的概率;\\\\\n(3) 在要求被选中的两人中必须一男一女的条件下,求女生乙被选中的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019633": { + "id": "019633", + "content": "设某工厂有两个车间生产同型号家用电器, 第一车间的合格率为 $0.85$, 第二车间的合格率为 $0.88$, 两个车间的成品都混合堆放在一个仓库, 假设第一, 二车间生产的成品比例为 $2: 3$, 今有一客户从成品仓库中随机提一台产品.\\\\\n(1) 求这台产品是合格品的概率;\\\\\n(2) 已知取到的是合格品, 求它来自第一车间生产的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019634": { + "id": "019634", + "content": "现有 $7$ 张卡片, 分别写上数字 $1,2,2,3,4,5,6$. 从这 $7$ 张卡片中随机抽取 $3$ 张, 记所抽取卡片上数字的最小值为 $X$, 则 $P(X=2)=$\\blank{50}, $E[X]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019635": { + "id": "019635", + "content": "甲、乙两名篮球运动员, 甲投篮一次命中的概率为 $\\dfrac{3}{5}$, 乙投篮一次命中的概率为 $\\dfrac{1}{2}$, 其中甲、乙两人投篮是否命中相互没有影响.\\\\\n(1) 求甲投篮三次恰好命中两次的概率;\\\\\n(2) 若甲、乙各投篮三次, 且甲、乙第一次投篮都命中, 求甲获胜 (甲投篮命中数比乙多) 的概率;\\\\\n(3) 若甲、乙各投篮两次, 设 $X$ 为甲、乙投篮命中的次数的差的绝对值, 求 $X$ 的数学期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019636": { + "id": "019636", + "content": "某学校的两个班共有 $100$ 名学生, 一次考试后数学成绩 $X$($X \\in \\mathbf{N}$) 服从正态分布 $N(100,10^2)$, 已知 $P(90 \\leq X \\leq 100)=0.4$, 估计该班学生数学成绩在 $110$ 分以上的人数为\\bracket{20}.\n\\fourch{$20$}{$10$}{$7$}{$5$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019637": { + "id": "019637", + "content": "已知 $P(B | A)=\\dfrac{1}{2}$, $P(AB)=\\dfrac{3}{10}$, 则 $P(A)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019638": { + "id": "019638", + "content": "已知随机变量 $X$ 的分布为 $\\begin{pmatrix}-1 & 0 & 1 \\\\ a & b & c\\end{pmatrix}$, 其中 $a, b, c$ 成等差数列, 则 $P(|X|=1)=$\\blank{50}, 公差 $d$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019639": { + "id": "019639", + "content": "一批产品的二等品率为 $0.02$, 从这批产品中每次随机取一件, 有放回地抽取 $100$ 次, $X$ 表示抽到的二等品件数, 则 $D[X]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019640": { + "id": "019640", + "content": "有 $N$ 件产品, 其中有 $M$ 件次品, 从中不放回地抽 $n$ 件产品, 抽到的次品数的数学期望值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019641": { + "id": "019641", + "content": "已知随机变量 $X$ 服从正态分布 $N(2, \\sigma^2)$, 且 $P(22.5)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019642": { + "id": "019642", + "content": "甲乙二人争夺一场围棋比赛的冠军, 若比赛为``三局两胜''制 (无平局), 甲在每局比赛中获胜的概率均为 $\\dfrac{2}{3}$, 且各局比赛结果相互独立, 则在甲获得冠军的条件下, 比赛进行了三局的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019643": { + "id": "019643", + "content": "设随机变量 $Y$ 满足 $Y \\sim B(4, \\dfrac{1}{2})$, 则函数 $f(x)=x^2-4 x+4Y$ 无零点的概率是\\bracket{20}.\n\\fourch{$\\dfrac{11}{16}$}{$\\dfrac{5}{16}$}{$\\dfrac{31}{32}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019644": { + "id": "019644", + "content": "设 $0=latex, xscale = 0.06, yscale = 180]\n\\draw [->] (0,0) -- (105,0) node [below] {年龄/岁};\n\\draw [->] (0,0) -- (0,0.03) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.001,10/0.002,20/0.012,30/0.017,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.001,20/0.012,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (90,0) node [below] {$90$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计该地区这种疾病患者的平均年龄 (同一组中的数据用该组区间的中点值作代表);\\\\\n(2) 估计该地区一人患这种疾病年龄在区间 $[20,70)$ 的概率;\\\\\n(3) 已知该地区这种疾病的患病率为 $0.1 \\%$, 该地区年龄位于区间 $[40,50)$ 的人口占该地区总人口的 $16 \\%$, 从该地区任选一人,若此人年龄位于区间 $[40,50)$, 求此人患该种疾病的概率. (样本数据中的患者年龄位于各区间的频率作为患者年龄位于该区间的概率, 精确到 $0.0001$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书15概率初步(续)", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019649": { + "id": "019649", + "content": "某超市统计了最近 $5$ 年的商品销售额与利润率数据, 经计算相关系数 $r= 0.862$, 则下列判断正确的是\\bracket{20}.\n\\onech{商品销售额与利润率正相关, 且具有较弱的相关关系}{商品销售额与利润率正相关, 且具有较强的相关关系}{商品销售额与利润率负相关, 且具有较弱的相关关系}{商品销售额与利润率负相关, 且具有较强的相关关系}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019650": { + "id": "019650", + "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山. 为估计一林区某种树木的总材积量, 随机选取了 $10$ 棵这种树木, 测量每棵树的根部横截面积 (单位: $\\mathrm{m}^2$ ) 和材积量 (单位: $\\mathrm{m}^3$ ), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline 样本号 $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\n\\hline 根部横截面积 $x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\n\\hline 材积量 $y_i$ & 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n并计算得 $\\displaystyle\\sum_{i=1}^{10}x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10}y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10}x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 $0.01$);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积,并得到所有这种树木的根部横截面积总和为 $186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\n附 : 相关系数 $r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})(y_i-\\overline{y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})^2 \\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})^2}}, \\sqrt{1.896}\\approx 1.377$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019651": { + "id": "019651", + "content": "变量 $x, y$ 之间的一组相关数据如表所示: 若 $x, y$ 之间的线性回归方程为 $y= \\hat{a}x+12.28$, 则 $\\hat{a}$ 的值为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\\hline$x$ & 4 & 5 & 6 & 7 \\\\\n\\hline$y$ & 8.2 & 7.8 & 6.6 & 5.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$-0.92$}{$-0.94$}{$-0.96$}{$-0.98$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019652": { + "id": "019652", + "content": "某新兴科技公司为了确定新研发的产品下一季度的营销计划, 需了解月宣传费 $x$ (单位: 万元) 对月销售量 $y$ (单位: 千件)的影响, 收集了 2020 年 3 月至 2020 年 8 月共 6 个月的月宣传费 $x$ 和月销售量 $y$ 的数据如表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline 月份 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\n\\hline 宣传费 $x$ & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 月销售量 $y$ & 0.4 & 3.5 & 5.2 & 7.0 & 8.6 & 10.7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现分别用模型 \\textcircled{1} $y=\\hat{a}x+\\hat{b}$ 和模型\\textcircled{2} $y=\\mathrm{e}^{\\hat{m}x+\\hat{n}}$ 对以上数据进行拟合, 得到回归模型, 并计算出模型的离差如表: (模型\\textcircled{1}和模型\\textcircled{2}的离差分别为 $\\hat{e}_1$ 和 $\\hat{e}_2$, 离差$=$实际值$-$预报值)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline$x$ & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline$y$ & 0.4 & 3.5 & 5.3 & 7.0 & 8.6 & 10.7 \\\\\n\\hline$\\hat{e}_1$ & -0.6 & 0.54 & 0.28 & 0.12 & -0.24 & -0.1 \\\\\n\\hline$\\hat{e}_2$ & -0.63 & 1.71 & 2.10 & 1.63 & -0.7 & -5.42 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表的离差数据, 应选择哪个模型来拟合月宣传费 $x$ 与月销售量 $y$ 的关系较为合适,简要说明理由;\\\\\n(2) 为了优化模型, 将 (1) 中选择的模型离差绝对值最大所对应的一组数据 $(x, y)$ 剔除, 根据剩余的 $5$ 组数据,求该模型的回归方程, 并预测月宣传费为 $12$ 万元时, 该公司的月销售量.\\\\\n(剔除数据前的参考数据: $\\overline{x}=7.5$, $\\overline{y}=5.9$, $\\displaystyle\\sum_{i=1}^6 x_i y_i=299.8$, $\\displaystyle\\sum_{i=1}^6 x_i^2=355$, $z=\\ln y$. $\\overline{z}\\approx-1.41$, $\\displaystyle\\sum_{i=1}^6 x_i y_i=-73.10$, $\\ln 10.7 \\approx 2.37$, $\\mathrm{e}^{4.034}\\approx 56.49$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019653": { + "id": "019653", + "content": "甲、乙两城之间的长途客车均由 $A$ 和 $B$ 两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的 500 个班次, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 准点班次数 & 末准点班次数 \\\\\n\\hline$A$ & 240 & 20 \\\\\n\\hline$B$ & 210 & 30 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表, 分别估计这两家公司甲、乙两城之间的长途客车准点的概率;\\\\\n(2) 能否有 $90 \\%$ 的把握认为甲、乙两城之间的长途客车是否准点与客车所属公司有关?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{tabular}{|c|c|c|c|}\n\\hline $P(\\chi^2 \\geq k$) & 0.100 & 0.050 & 0.010 \\\\\n\\hline $k$ & 2.706 & 3.841 & 6.635 \\\\\n\\hline\n\\end{tabular}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019654": { + "id": "019654", + "content": "某商家今年上半年各月的人均销售额 (单位: 千元) 与利润率统计表如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 月份 & 1 & 2 & 3 & 4 & 5 & 6 \\\\\n\\hline 人均销售额 & 6 & 5 & 8 & 3 & 4 & 7 \\\\\n\\hline 利润率(\\%) & 12.6 & 10.4 & 18.5 & 3.0 & 8.1 & 16.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\textcircled{1} 利润率与人均销售额成正相关关系 ; \\textcircled{2} 利润率与人均销售额成负相关关系; \\textcircled{3} 利润率与人均销售额成正比例函数关系; \\textcircled{4} 利润率与人均销售额成反比例函数关系根据表中数据, 上述说法正确的是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019655": { + "id": "019655", + "content": "已知变量 $x$ 和变量 $y$ 的 3 对随机观测数据 $(2,2),(3,-1),(5,-7)$, 则成对样本数据的样本相关系数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019656": { + "id": "019656", + "content": "已知 $x$ 与 $y$ 之间的一组数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$ & 0 & 1 & 2 & 3 \\\\\n\\hline$y$ & $m$ & 3 & 5.5 & 7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n已求得关于 $y$ 与 $x$ 的线性回归方程 $y=2.3 x+0.85$, 则 $m$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019657": { + "id": "019657", + "content": "研究变量 $x, y$ 得到一组样本数据, 进行回归分析, 有以下结论\\\\\n\\textcircled{1} 离差平方和越小的模型, 拟合的效果越好;\\\\\n\\textcircled{2} 用相关指数 $\\chi^2$ 来刻画回归效果, $\\chi^2$ 越小说明拟合效果越好;\\\\\n\\textcircled{3} 在回归直线方程 $y=-0.2 x+0.8$ 中, 当解释变量 $x$ 每增加 $1$ 个单位时, 预报变量 $y$ 平均减少 $0.2$ 个单位;\\\\\n\\textcircled{4} 若变量 $y$ 和 $x$ 之间的相关系数为 $r=-0.9462$, 则变量 $y$ 和 $x$ 之间的负相关很强.\\\\\n以上正确说法的是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019658": { + "id": "019658", + "content": "某市通过随机询问 $100$ 名性别不同的居民是否能做到``光盘''行动, 得到如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 做不到``光盘''& 能做到``光盘''\\\\\n\\hline 男 & 45 & 10 \\\\\n\\hline 女 & 30 & 15 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表, \\blank{50}(填``可以''或``不可以'')确定``该市居民能否做到``光盘''与性别有关''.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019659": { + "id": "019659", + "content": "下列关于相关系数 $r$ 的说法不正确的是\\bracket{20}.\n\\onech{相关系数 $r$ 越大两个变量间相关性越强}{相关系数 $r$ 的取值范围为 $[-1,1]$}{相关系数 $r>0$ 时两个变量正相关, $r<0$ 时两个变量负相关}{相关系数 $r=1$ 时, 样本点在同一直线上}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019660": { + "id": "019660", + "content": "为了研究某班学生的脚长 $x$ (单位: 厘米) 和身高 $y$ (单位: 厘米)的关系, 从该班随机抽取 $10$ 名学生, 根据测量数据的散点图可以看出 $y$ 与 $x$ 之间有线性相关关系, 设其回归直线方程为 $y=\\hat{a}x+\\hat{b}$. 已知 $\\displaystyle\\sum_{i=1}^{10}x_i=225$, $\\displaystyle\\sum_{i=1}^{10}y_i=1600$, $\\hat{a}=4$. 该班某学生的脚长为 $24$, 据此估计其身高为\\bracket{20}.\n\\fourch{$160$}{$163$}{$166$}{$170$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019661": { + "id": "019661", + "content": "``独立性检验''中, 在犯错误的概率不超过 $0.05$ 的前提下认为事件 $A$ 和 $B$ 有关, 则算出的数据满足\\bracket{20}.\n\\fourch{$\\chi^2<6.63$}{$\\chi^2<3.84$}{$\\chi^2>3.84$}{$\\chi^2>6.63$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019662": { + "id": "019662", + "content": "下图是某地区 2000 年至 2016 年环境基础设施投资额 $y$ (单位: 亿元) 的折线图. 为了预测该地区 2018 年的环境基础设施投资额, 建立了 $y$ 与时间变量 $t$ 的两个线性回归模型. 根据 2000 年至 2016 年的数据 (时间变量 $t$ 的值依次为 $1,2, \\cdots, 17$ ) 建立模型\\textcircled{1}: $y=-30.4+13.5 t$; 根据 2010 年至 2016 年的数据 (时间变量 $t$ 的值依次为 $1,2, \\cdots, 7$ ) 建立模型\\textcircled{2}: $y=99+17.5 t$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.8]\n\\draw [->] (0,0) -- (18,0) node [below] {年份};\n\\draw [->] (0,0) -- (0,5.5) node [left] {投资额};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {2000,2001,...,2016}\n{\\draw ({\\i-1999},0.05) --++ (0,-0.05) node [below] {$\\i$};};\n\\foreach \\i in {20,40,...,240}\n{\\draw (0.05,{\\i/50}) --++ (-0.05,0) node [left] {$\\i$};};\n\\foreach \\i/\\j in {1/11,2/19,3/25,4/35,5/37,6/42,7/42,8/47,9/53,10/56,11/122,12/129,13/148,14/171,15/184,16/209,17/220}\n{\\filldraw (\\i,{\\j/50}) circle (0.03) node [above] {$\\j$};};\n\\draw (1,0.22)--(2,0.38)--(3,0.5)--(4,0.7)--(5,0.74)--(6,0.84)--(7,0.84)--(8,0.94)--(9,1.06)--(10,1.12)--(11,2.44)--(12,2.58)--(13,2.96)--(14,3.42)--(15,3.68)--(16,4.18)--(17,4.4);\n\\end{tikzpicture}\n\\end{center}\n(1) 分别利用这两个模型, 求该地区 2018 年的环境基础设施投资额的预测值;\\\\\n(2) 你认为用哪个模型得到的预测值更可靠? 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019663": { + "id": "019663", + "content": "为保护农民种粮收益, 促进粮食生产, 确保国家粮食安全, 调动广大农民生产粮食的积极性, 国家实施了对种粮农民直接补贴的政策. 通过对 2017-2021 年的数据进行调查, 发现某地区发放粮食补贴额 $x$ (单位: 亿元) 与该地区粮食产量 $y$ (单位: 万亿吨) 之间存在着线性相关关系,统计数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 年份 & 2017 & 2018 & 2019 & 2020 & 2021 \\\\\n\\hline 补贴额 $x$ (亿元) & 9 & 10 & 12 & 11 & 8 \\\\\n\\hline 粮食产量 $y$ (万亿吨) & 25 & 26 & 31 & 37 & 21 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请根据上表所给的数据, 求出 $y$ 关于 $x$ 的线性回归直线方程 $y=\\hat{a}x+\\hat{b}$;\\\\\n(2) 通过对该地区粮食产量的分析研究, 计划 2022 年在该地区发放粮食补贴 $7$ 亿元, 请根据 (1) 中所得到的线性回归直线方程, 预测 2022 年该地区的粮食产量.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019664": { + "id": "019664", + "content": "甲、乙两台机床生产同种产品, 产品按质量分为一级品和二级品, 为了比较两台机床产品的质量, 分别用两台机床各生产了 200 件产品, 产品的质量情况统计如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 一级品 & 二级品 & 合计 \\\\\n\\hline 甲机床 & 150 & 50 & 200 \\\\\n\\hline 乙机床 & 120 & 80 & 200 \\\\\n\\hline 合计 & 270 & 130 & 400 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 甲机床、乙机床生产的产品中一级品的频率分别是多少?\\\\\n(2) 能否有 $99 \\%$ 的把握认为甲机床的产品质量与乙机床的产品质量有差异?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{tabular}{|c|c|c|c|}\n\\hline $P(\\chi^2 \\geq k$) & 0.050 & 0.010 & 0.001 \\\\\n\\hline $k$ & 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书16成对数据的统计分析", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019665": { + "id": "019665", + "content": "已知函数 $f(x)=\\begin{cases}|x|,& x \\leq m,\\\\x^2-2 m x+4 m,& x>m,\\end{cases}$其中 $m>0$. 若存在实数 $b$, 使得关于 $x$ 的方程 $f(x)=b$ 有三个不同的根, 则 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019666": { + "id": "019666", + "content": "当 $x \\in(1,2)$ 时, 不等式 $(x-1)^2<\\log _a x$ 恒成立, 则底数 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019667": { + "id": "019667", + "content": "已知 $\\overrightarrow{a}, \\overrightarrow{b}$ 是平面内两个互相垂直的单位向量, 若向量 $\\overrightarrow{c}$ 满足 $(\\overrightarrow{a}-\\overrightarrow{c}) \\cdot(\\overrightarrow{b}-\\overrightarrow{c})=0$, 则 $|\\overrightarrow{c}|$ 的最大值是\\bracket{20}.\n\\fourch{$1$}{$2$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019668": { + "id": "019668", + "content": "已知圆 $C:(x-3)^2+(y-4)^2=1$ 和两点 $A(-m, 0)$, $B(m, 0)$($m>0$). 若圆 $C$ 上存在点 $P$, 使得 $\\angle APB=90^{\\circ}$, 则 $m$ 的最大值为\\bracket{20}.\n\\fourch{$7$}{$6$}{$5$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019669": { + "id": "019669", + "content": "已知抛物线的方程为 $x^2=8 y$, 点 $F$ 是其焦点, 点 $A(-2,4)$, 在抛物线上求一点 $P$, 使 $\\triangle APF$ 的周长最小, 求此时点 $P$ 的坐标.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019670": { + "id": "019670", + "content": "已知不等式 $|a-3 x|>x-1$ 对任意 $x \\in[0,2]$ 恒成立, 则实数 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019671": { + "id": "019671", + "content": "设 $P$ 是抛物线 $y^2=4 x$ 上的一个动点, 则点 $P$ 到点 $A(-1,1)$ 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": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019672": { + "id": "019672", + "content": "已知函数 $f(x)=\\begin{cases}2^x+a,& x \\geq 0,\\\\x^2-a x,& x<0,\\end{cases}$若 $f(x)$ 的最小值是 $a$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019673": { + "id": "019673", + "content": "设 $P(x, y)$ 是曲线 $C: \\sqrt{\\dfrac{x^2}{25}}+\\sqrt{\\dfrac{y^2}{9}}=1$ 上的点, $F_1(-4,0)$, $F_2(4,0)$, 则 $|PF_1|+|PF_2|$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019674": { + "id": "019674", + "content": "若方程 $x+k=\\sqrt{1-x^2}$ 有且只有一个解,则 $k$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019675": { + "id": "019675", + "content": "从双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左焦点 $F$ 引圆 $x^2+y^2= a^2$ 的切线, 切点为 $T$, 延长 $FT$ 交双曲线右支于点 $P$, 若 $M$ 是线段 $FP$ 的中点, $O$ 为原点, 则 $|MO|-|MT|$ 的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\path [domain = -4:4, samples = 100, name path = cr, draw] plot ({sqrt(1+\\x*\\x/3)},\\x);\n\\draw [domain = -4:4, samples = 100] plot ({-sqrt(1+\\x*\\x/3)},\\x);\n\\draw (-2,0) node [below] {$F$} coordinate (F_1);\n\\draw (2,0) node [below] {$F'$} coordinate (F_2);\n\\path [name path = F1P] (F_1) --++ (30:5);\n\\path [name intersections = {of = F1P and cr, by = P}];\n\\draw (P) node [right] {$P$} -- (F_1) (P) -- (F_2);\n\\draw ($(F_1)!0.5!(P)$) node [above] {$M$} coordinate (M) -- (0,0);\n\\draw ($(F_1)!(0,0)!(P)$) node [above left] {$T$} coordinate (T) -- (0,0);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书17数形结合", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019676": { + "id": "019676", + "content": "已知方程 $2^{-x}-|\\log _2 x|=0$ 的两根分别为 $x_1, x_2$, 则\\bracket{20}.\n\\fourch{$12$}{$x_1 x_2=1$}{$00$) 及曲线 $C_2: y=\\dfrac{1}{3 x}$($x>0$), $C_1$ 上的点 $P_1$ 的横坐标为 $a_1$($0=latex, scale = 2]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1.5, samples = 100] plot (\\x,{2*\\x/(1+\\x)});\n\\draw [domain = {1/4.5}:1.5, samples = 100] plot (\\x,{1/\\x/3});\n\\draw (0.2,{1/3}) node [above left] {$P_1$} coordinate (P_1);\n\\draw (1,{1/3}) node [above right] {$Q_1$} coordinate (Q_1);\n\\draw (1,1) node [below right] {$P_2$} coordinate (P_2);\n\\draw ({1/3},1) node [above right] {$Q_2$} coordinate (Q_2);\n\\draw ({1/3},{1/2}) node [above left] {$P_3$} coordinate (P_3);\n\\draw ({2/3},{1/2}) node [above right] {$Q_3$} coordinate (Q_3);\n\\draw ({2/3},{4/5}) node [right] {$P_4$} coordinate (P_4);\n\\draw ({5/12},{4/5}) coordinate (Q_4);\n\\draw ({5/12},{10/17}) coordinate (P_5);\n\\draw ({17/30},{10/17}) coordinate (Q_5);\n\\draw (P_1)--(Q_1)--(P_2)--(Q_2)--(P_3)--(Q_3)--(P_4)--(Q_4)--(P_5)--(Q_5);\n\\draw [dashed] (P_1) --++ (0,{-1/3}) node [below] {$a_1$};\n\\draw [dashed] (Q_1) --++ (0,{-1/3}) node [below] {$a_2$};\n\\draw [dashed] (P_3) --++ (0,{-1/2}) node [below] {$a_3$};\n\\draw [dashed] (Q_3) --++ (0,{-1/2}) node [below] {$a_4$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求曲线 $C_1$ 和曲线 $C_2$ 的交点坐标;\\\\\n(2) 试求 $a_{n+1}$ 与 $a_n$ 之间的关系;\\\\\n(3) 证明: $a_{2 n-1}<\\dfrac{1}{2}|PF_2|$, 则 $\\dfrac{|PF_1|}{|PF_2|}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019685": { + "id": "019685", + "content": "在四面体 $ABCD$ 中, $AB=CD, M$、$N$ 分别是 $BC$、$AD$ 的中点, 若 $AB$ 与 $CD$ 所成的角的大小为 $60^{\\circ}$, 则 $MN$ 与 $CD$ 所成的角的大小为\\bracket{20}.\n\\fourch{$30^{\\circ}$}{$30^{\\circ}$ 或 $60^{\\circ}$}{$60^{\\circ}$}{$60^{\\circ}$ 或 $15^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019686": { + "id": "019686", + "content": "若函数 $f(x)=a \\mathrm{e}^x-x-2 a$ 有两个零点, 则实数 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, \\dfrac{1}{\\mathrm{e}})$}{$(0, \\dfrac{1}{\\mathrm{e}})$}{$(-\\infty, 0)$}{$(0,+\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019687": { + "id": "019687", + "content": "已知 $M=\\{x | x-a=0\\}$, $N=\\{x | a x-1=0\\}$, 若 $M \\cap N=N$, 则实数 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019688": { + "id": "019688", + "content": "若 $\\mathrm{C}_{10}^3=\\mathrm{C}_{10}^n$, 则正整数 $n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019689": { + "id": "019689", + "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=3^n+2$, 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": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019690": { + "id": "019690", + "content": "若关于 $x$ 的方程 $(4 x+\\dfrac{5}{x})-|5 x-\\dfrac{4}{x}|=m$ 在 $(0,+\\infty)$ 内恰有三个相异实根, 则实数 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019691": { + "id": "019691", + "content": "已知 $A(a, 0)$, 点 $P$ 为 $y^2=2 x$ 上动点, 则 $|PA|$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019692": { + "id": "019692", + "content": "已知函数 $f(x)=\\begin{cases}-\\log _2(3-x),& x<2\\\\2^{x-2}-1,& x \\geq 2\\end{cases}$, 若 $f(2-a)=1$, 则 $f(a)$ 等于\\bracket{20}.\n\\fourch{$-2$}{$-1$}{1}{2}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019693": { + "id": "019693", + "content": "已知正三棱柱的侧面展开图是边长分别为 $6$ 和 $4$ 的矩形, 则它的体积为\\bracket{20}.\n\\fourch{$\\dfrac{8 \\sqrt{3}}{3}$}{$4 \\sqrt{3}$}{$\\dfrac{2 \\sqrt{3}}{9}$}{$4 \\sqrt{3}$ 或 $\\dfrac{8 \\sqrt{3}}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019694": { + "id": "019694", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019695": { + "id": "019695", + "content": "已知 $a \\in \\mathbf{R}$, 函数 $f(x)=\\log _2(\\dfrac{1}{x}+a)$. 若关于 $x$ 的方程 $f(x)-\\log _2[(a-4) x+2 a-5]=0$ 的解集中恰好有一个元素, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019696": { + "id": "019696", + "content": "函数 $f(x)=[a x^2-(3 a+1) x+3 a+2] \\cdot \\mathrm{e}^x$ 在 $x=1$ 处取得极小值, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019697": { + "id": "019697", + "content": "若动点 $M$ 到定点 $A(0,1)$ 与定直线 $l: y=3$ 的距离之和为 $4$ .\\\\\n(1) 求点 $M$ 的轨迹方程, 并画出方程的曲线草图;\\\\\n(2) 记 (1) 得到的轨迹为曲线 $C$, 问曲线 $C$ 上关于点 $B(0, t)$($t \\in \\mathbf{R}$) 对称的不同点有几对? 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书18分类讨论", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019698": { + "id": "019698", + "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 已知 $2 \\sin C=\\tan A(1- 2 \\cos C)$, $c=2 b$, 则 $\\cos B$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019699": { + "id": "019699", + "content": "已知直线 $x=\\varphi$, $\\varphi \\in[0, \\pi]$ 是函数 $f(x)=3 \\sin x+ 4 \\cos x$ 图像的一条对称轴, 则 $\\cos \\varphi=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019700": { + "id": "019700", + "content": "在直角梯形 $ABCD$ 中, $AB=8$, $CD=4, AB \\parallel CD$, $AB \\perp AD$, $E$ 是 $BC$ 的中点, 则 $\\overrightarrow{AB}\\cdot(\\overrightarrow{AC}+\\overrightarrow{AE})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019701": { + "id": "019701", + "content": "设 $k$ 是正整数, 令 $f(x)=\\sin ^4 \\dfrac{k x}{200}+\\cos ^4 \\dfrac{k x}{200}$, 若对于任意实数 $a$ 均有 $\\{y | y= f(x), a0, b_1, b_2$, $b_3 \\in \\mathbf{R}$. 若函数 $y=f(x)$ 的图像如图所示, 则数组 $(b_1, b_2, b_3)$ 的一组值可以是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-5,1) -- (-1,1) -- (0.5,4) -- (3,-1) -- (5,-1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$(3,-1,1)$}{$(1,-2,-1)$}{$(-1,2,2)$}{$(1,-3,1)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019705": { + "id": "019705", + "content": "设 $n$ 阶方阵\n$A_n=\\begin{pmatrix}1 & 3 & 5 & \\cdots & 2 n-1 \\\\ 2 n+1 & 2 n+3 & 2 n+5 & \\cdots & 4 n-1 \\\\ 4 n+1 & 4 n+3 & 4 n+5 & \\cdots & 6 n-1 \\\\ \\cdots & \\cdots & \\cdots & \\cdots & \\cdots \\\\ 2 n(n-1)+1 & 2 n(n-1)+3 & 2 n(n-1)+5 & \\cdots & 2 n^2-1\\end{pmatrix}$, 任取 $A_n$ 中的一个元素, 记为 $x_1$; 划去 $x_1$ 所在的行和列, 将剩下的元素按原来的位置关系组成 $n-1$ 阶方阵 $A_{n-1}$, 任取 $A_{n-1}$ 中的一个元素, 记为 $x_2$; 划去 $x_2$ 所在的行和列, $\\cdots \\cdots$; 将最后剩下的一个元素记为 $x_n$. 记 $S_n=x_1+x_2+\\cdots+x_n$, 则 $S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019706": { + "id": "019706", + "content": "已知函数 $y=f(x)$, 对任意 $x \\in \\mathbf{R}$, 都有 $f(x+2) \\cdot f(x)=k$ ($k$ 为常数), 且当 $x \\in[0,2]$ 时, $f(x)=x^2+1$, 则 $f(2021)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019707": { + "id": "019707", + "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n \\cdot \\sin \\dfrac{n \\pi}{2}$, 前 $n$ 项和为 $S_n$, 求 $S_{2022}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019708": { + "id": "019708", + "content": "若数列 $\\{c_n\\}$ 满足``对任意正整数 $i, j$, $i \\neq j$, 都存在正整数 $k$, 使得 $c_k=c_i c_j$'', 则称数列 $\\{c_n\\}$ 具有``性质 $P$''. 已知数列 $\\{a_n\\}$ 为无穷数列.\\\\\n(1) 若 $\\{a_n\\}$ 为各项均等于 $a$($a \\neq 0$) 的常数数列, 判断数列 $\\{a_n\\}$ 是否具有``性质 $P$'', 并说明理由;\\\\\n(2) 若数列 $\\{a_n\\}$ 是等比数列, 且公比 $q=2$, $a_1=2^m$($m \\in \\mathbf{Z}$), 证明: $m \\geq-1$ 是数列 $\\{a_n\\}$ 具有``性质 $P$''的充要条件;\\\\\n(3) 是否存在公差小于零, 且具有``性质 $P$''的等差数列 $\\{a_n\\}$, 若存在试写出一个满足条件的数列 $\\{a_n\\}$ 的通项公式, 如果不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019709": { + "id": "019709", + "content": "已知 $M$、$N$ 是三棱锥 $P-ABC$ 的棱 $AB, PC$ 的中点, 记三棱锥 $P-ABC$ 的体积为 $V_1$, 三棱锥 $N-MBC$ 的体积为 $V_2$, 则 $\\dfrac{V_2}{V_1}$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019710": { + "id": "019710", + "content": "直线 $x+2 y-3=0$ 与直线 $a x+4 y+b=0$ 关于点 $A(1,0)$ 对称, 则 $b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019711": { + "id": "019711", + "content": "等差数列 $\\{a_n\\}$ 中,若 $a_4+a_6+a_8+a_{10}+a_{12}=120$, 则 $a_9-\\dfrac{1}{3}a_{11}$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019712": { + "id": "019712", + "content": "若圆 $C$ 的半径为 3 , 单位向量 $\\overrightarrow{e}$ 所在的直线与圆相切于定点 $A$, 点 $B$ 是圆上的动点, 则 $\\overrightarrow{e}\\cdot \\overrightarrow{AB}$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019713": { + "id": "019713", + "content": "已知函数 $f(x)=\\begin{cases}2-x^2,& x \\geq 0,\\\\-x,& x<0,\\end{cases}$ $x_1,x_2 \\in \\mathbf{R}$, $f(x_1)=f(x_2)=m$, 且 $x_1+x_2=0$, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019714": { + "id": "019714", + "content": "设函数 $f(x)=|x-a|-\\dfrac{2}{x}+a$, 若关于 $x$ 的方程 $f(x)=1$ 有且仅有两个不同的实数根, 则实数 $a$ 的取值构成的集合为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019715": { + "id": "019715", + "content": "设 $a, b \\in \\mathbf{R}$, 若 $a-|b|>0$, 则下列不等式中正确的是\\bracket{20}.\n\\fourch{$b-a>0$}{$a^3+b^3<0$}{$a^2-b^2<0$}{$b+a>0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019716": { + "id": "019716", + "content": "已知函数 $f(x)=\\begin{cases}x^2+4 x,& x \\geq 0,\\\\4 x-x^2,& x<0,\\end{cases}$ 若 $f(2-a^2)>f(a)$, 则实数 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-1) \\cup(2,+\\infty)$}{$(-1,2)$}{$(-2,1)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019717": { + "id": "019717", + "content": "已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 对任意 $x \\in \\mathbf{R}$, 恒有 $f(x)+f(x+2)=0$, 且当 $x \\in (0,1]$ 时 $f(x)=2^x+1$, 则 $f(0)+f(1)+f(2)+\\cdots+f(2021)=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019718": { + "id": "019718", + "content": "已知样本数据 $x_1, x_2, x_3, x_4$ 的每个数据都是自然数, 该样本的平均数为$4$, 方差为$5$, 且样本数据两两互不相同, 求样本数据中的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019719": { + "id": "019719", + "content": "设 $m \\in \\mathbf{R}$, 在平面直角坐标系中, 已知向量 $\\overrightarrow{a}=(m x, y+1)$, 向量 $\\overrightarrow{b}=(x, y-1)$, $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 动点 $M(x, y)$ 的轨迹为 $E$.\\\\\n(1) 求轨迹 $E$ 的方程,并说明该方程所表示曲线的形状;\\\\\n(2) 已知 $m=\\dfrac{1}{4}$, 证明: 存在圆心在原点的圆, 使得该圆的任意一条切线与轨迹 $E$ 恒有两个交点 $A, B$, 且 $OA \\perp OB$ ($O$ 为坐标原点), 并求出该圆的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019720": { + "id": "019720", + "content": "已知数列 $\\{a_n\\}$ 与 $\\{b_n\\}$ 满足 $a_{n+1}-a_n=\\lambda(b_{n+1}-b_n)$ ($\\lambda$ 为非零常数), $n$ 为正整数.\\\\\n(1) 若 $\\{b_n\\}$ 是等差数列, 求证: 数列 $\\{a_n\\}$ 也是等差数列;\\\\\n(2) 若 $a_1=2$, $\\lambda=3$, $b_n=\\sin \\dfrac{n \\pi}{2}$, 求数列 $\\{a_n\\}$ 的前 $2021$ 项和;\\\\\n(3) 设 $a_1=b_1=\\lambda$, $b_2=\\dfrac{\\lambda}{2}$, $b_n=\\dfrac{b_{n-1}+b_{n-2}}{2}$($n \\geq 3$, $n \\in \\mathbf{N}$), 若对 $\\{a_n\\}$ 中的任意两项 $a_i$, $a_j$, $i, j$ 为正整数且 $i \\neq j$, $|a_i-a_j|<2$ 都成立, 求实数 $\\lambda$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书19特殊与一般的思想及整体思维方法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019721": { + "id": "019721", + "content": "已知数列 $\\{a_n\\}$, 记 $A(n)=a_1+a_2+a_3+\\cdots+a_n$, $B(n)=a_2+a_3+a_4+\\cdots+ a_{n+1}$, $C(n)=a_3+a_4+a_5+\\cdots+a_{n+2}$($n=1,2,3, \\cdots$), 并且对于任意正整数 $n$, 恒有 $a_n>0$ 成立.\\\\\n(1) 若 $a_1=1$, $a_2=5$, 且对任意正整数 $n$, 三个数 $A(n), B(n), C(n)$ 组成等差数列, 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 证明: 数列 $\\{a_n\\}$ 是公比为 $q$ 的等比数列的充分必要条件是: 对任意正整数 $n$, 三个数 $A(n), B(n), C(n)$ 组成公比为 $q$ 的等比数列.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019722": { + "id": "019722", + "content": "已知函数 $f(x), g(x)$ 在数集 $D$ 上都有定义, 对于任意的 $x_1, x_2 \\in D$, 当 $x_1< x_2$ 时, $g(x_1) \\leq \\dfrac{f(x_1)-f(x_2)}{x_1-x_2}\\leq g(x_2)$ 或 $g(x_2) \\leq \\dfrac{f(x_1)-f(x_2)}{x_1-x_2}\\leq g(x_1)$ 成立, 则称 $g(x)$ 是数集 $D$ 上 $f(x)$ 的限制函数.\\\\\n(1) 求 $f(x)=-\\dfrac{1}{x}$ 在 $D=(0,+\\infty)$ 上的限制函数 $g(x)$ 的解析式;\\\\\n(2) 证明: 如果 $g(x)$ 在区间 $D_1 \\subseteq D$ 上恒为正值, 则 $f(x)$ 在 $D_1$ 上是严格增函数.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019723": { + "id": "019723", + "content": "证明下列三角恒等式.\\\\\n(1) $\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=\\dfrac{\\sin \\alpha}{1-\\cos \\alpha}$;\\\\\n(2) $\\dfrac{\\sin ^2 \\alpha-\\sin ^2 \\beta}{\\tan ^2 \\alpha-\\tan ^2 \\beta}=\\cos ^2 \\alpha \\cos ^2 \\beta$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019724": { + "id": "019724", + "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_2=e$ ($e$ 是自然对数的底数), 且 $a_{n+2}=\\sqrt{a_{n+1}\\cdot a_n}$, 令 $b_n=\\ln a_n, n$ 为正整数.\\\\\n(1) 证明: $b_{n+2}>\\sqrt{b_{n+1}b_n}$;\\\\\n(2) 证明: $\\{\\dfrac{b_{n+2}-b_{n+1}}{b_{n+1}-b_n}\\}$ 是等比数列, 且 $\\{b_n\\}$ 的通项公式是 $b_n=\\dfrac{2}{3}[1-(-\\dfrac{1}{2})^{n-1}]$;\\\\\n(3) 是否存在常数 $t$, 对任意正整数 $n$ 均有 $b_{n+1}\\geq t b_n$ 成立? 若存在, 求 $t$ 的取值范围, 否则, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019725": { + "id": "019725", + "content": "证明: $2 \\pi$ 是函数 $f(x)=\\sin x$ 的最小正周期.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019726": { + "id": "019726", + "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n+\\sqrt{3}$, 求证: 数列 $\\{a_n\\}$ 中的任意不同的三项不可能构成等比数列.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019727": { + "id": "019727", + "content": "若函数 $f(x)=\\sqrt{x^2-1}+\\sqrt{a-x^2}$ 为偶函数且非奇函数, 则实数 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019728": { + "id": "019728", + "content": "设 $\\triangle ABC$ 的内角 $A$、$B$、$C$ 所对的边分别为 $a$、$b$、$c$, 若 $b \\cos C+c \\cos B=a \\sin A$, 则 $\\triangle ABC$ 的形状为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019729": { + "id": "019729", + "content": "已知等比数列 $\\{a_n\\}$ 的各项均为正数, 且满足 $a_1 a_7=4$, 则数列 $\\{\\log _2 a_n\\}$ 的前 $n$ 项之和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019730": { + "id": "019730", + "content": "设 $\\{a_n\\}$ 是公比为 $q$($q \\neq 1$) 的等比数列, 若 $\\{a_n\\}$ 中任意两项之积仍是该数列中的项, 那么称 $\\{a_n\\}$ 是封闭数列,试写出一个满足封闭数列定义的数列的通项公式 $a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019731": { + "id": "019731", + "content": "已知偶函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 有下列四个函数: \\textcircled{1} $y=\\sin [f(x)]$; \\textcircled{2} $y=x \\cdot f(\\sin x)$; \\textcircled{3} $y= f(x) \\cdot f(\\sin x)$; \\textcircled{4} $y=[f(\\sin x)]^2$, 则其中为奇函数的个数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019732": { + "id": "019732", + "content": "已知函数 $f(x)=\\dfrac{1}{2}(x+\\dfrac{1}{x})$, $g(x)=\\dfrac{1}{2}(x-\\dfrac{1}{x})$, 若直线 $l: a x+b y+c=0$($a, b, c$ 为常数) 与 $f(x)$ 的图像交于不同的两点 $A$、$B$, 与 $g(x)$ 的图像交于不同的两点 $C$、$D$, 则 $|AC|- |BD|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019733": { + "id": "019733", + "content": "设 $x$、$y$、$z$ 是互不相等的正数, 则下列等式中不恒成立的是\\bracket{20}.\n\\twoch{$x^2+\\dfrac{1}{x^2}\\geq x+\\dfrac{1}{x}$}{$\\sqrt{x+3}-\\sqrt{x+1}\\leq \\sqrt{x+2}-\\sqrt{x}$}{$|x-y|+\\dfrac{1}{x-y}\\geq 2$}{$|x-y| \\leq|x-z|+|y-z|$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019734": { + "id": "019734", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019735": { + "id": "019735", + "content": "定义在 $\\mathbf{R}$ 上的函数 $f(x)$ 满足 $f(-x)=-f(x+4)$, 且 $f(x)$ 在 $(2,+\\infty)$ 上为严格增函数. 已知 $x_1+x_2<4$ 且 $(x_1-2)(x_2-2)<0$, 则 $f(x_1)+f(x_2)$ 的值\\bracket{20}.\n\\fourch{恒小于 0}{恒大于 0}{可能等于 0}{可正也可负}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019736": { + "id": "019736", + "content": "如果 $a$、$b$ 都是正数, 且 $a \\neq b$, 求证: $\\dfrac{a}{\\sqrt{b}}+\\dfrac{b}{\\sqrt{a}}>\\sqrt{a}+\\sqrt{b}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019737": { + "id": "019737", + "content": "给定常数 $c>0$, 定义函数 $f(x)=2|x+c+4|-|x+c|$, 数列 $a_1, a_2, a_3, \\cdots$ 满足 $a_{n+1}=f(a_n) n$ 是正整数,求证: 对任意正整数数 $n, a_{n+1}-a_n \\geq c$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019738": { + "id": "019738", + "content": "定义在 $\\mathbf{R}$ 上的函数 $f(x)$ 满足: 对于任意的 $x_1, x_2 \\in \\mathbf{R}$, 当 $x_12$, 写出函数 $f(x)$ 的单调区间, 并给出证明;\\\\\n(3) 若存在 $a \\in[-2,4]$, 使得关于 $x$ 的方程 $f(x)=t \\cdot f(a)$ 有三个不相等的实数解, 证明: $t \\in(1, \\dfrac{9}{8})$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书20代数证明问题选讲", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019740": { + "id": "019740", + "content": "已知 $\\odot O$ 与直线 $l$ 相切于 $A$ 点, 点 $P, Q$ 同时从 $A$ 点出发, $P$ 沿着直线 $l$ 向右、 $Q$ 沿着圆周按逆时针以相同的速度运动, 当 $Q$ 运动到点 $A$ 时, 点 $P$ 也停止运动, 连接 $OQ, OP$ (如图), 则阴影部分面积 $S_1, S_2$ 的大小关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (-1.5,-1) node [above] {$l$} -- (2.5,-1);\n\\filldraw (0,0) circle (0.03) node [left] {$O$} coordinate (O);\n\\draw (0,-1) node [below] {$A$} coordinate (A);\n\\draw (-30:2) node [below] {$P$} coordinate (P);\n\\draw (-30:1) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)/pi*180-90}:1) node [above right] {$Q$} coordinate (Q);\n\\fill [pattern = north east lines] (A) arc (-90:-30:1) -- (P) -- cycle;\n\\fill [pattern = north east lines] (O) -- (B) arc (-30:{sqrt(3)/pi*180-90}:1) -- cycle;\n\\draw (0,0) circle (1);\n\\draw (O)--(A);\n\\draw (P)--(O)--(Q);\n\\draw [->] (2,-1.3) --++ (0.5,0);\n\\draw [->] (-25:1.5) arc (-25:20:1.5);\n\\draw (1,-0.8) node {$S_2$} (0.7,-0.15) node {$S_1$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$S_1=S_2$}{$S_1 \\leq S_2$}{$S_1 \\geq S_2$}{先 $S_1S_2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019741": { + "id": "019741", + "content": "函数 $f(x)=2^x|\\log _{0.5}x|-1$ 的零点个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019742": { + "id": "019742", + "content": "若曲线 $|y|=2^x+1$ 与直线 $y=b$ 没有公共点, 则 $b$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019743": { + "id": "019743", + "content": "设 $a>0$, 函数 $f(x)=x+2(1-x) \\sin (a x)$, $x \\in(0,1)$, 若函数 $y=2 x-1$ 与 $y=f(x)$ 的图像有且只有两个不同的公共点, 则 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019744": { + "id": "019744", + "content": "设 $a, b \\in \\mathbf{R}$, 若函数 $f(x)=x+\\dfrac{a}{x}+b$ 在区间 $(1,2)$ 上有两个不同的零点, 则 $f(1)$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019745": { + "id": "019745", + "content": "已知 $\\cos (x-\\dfrac{\\pi}{4})=\\dfrac{\\sqrt{2}}{10}$, $x \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{4})$, 则 $\\sin (x+\\dfrac{\\pi}{12})$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019746": { + "id": "019746", + "content": "已知函数 $f(x)=x^2-4 x+a(10^{x-2}+10^{2-x})$ 有唯一零点, 则 $a$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019747": { + "id": "019747", + "content": "设 $A, B, C$ 是 $\\triangle ABC$ 的三个内角, 则 $m=\\dfrac{\\sin A}{\\sin B+\\sin C-\\sin A}+\\dfrac{\\sin B}{\\sin A+\\sin C-\\sin B}+ \\dfrac{\\sin C}{\\sin A+\\sin B-\\sin C}$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019748": { + "id": "019748", + "content": "某学校要招开学生代表大会, 规定各班每 $10$ 人推选一名代表, 当各班人数除以\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019749": { + "id": "019749", + "content": "设 $x$ 是实数, $n$ 是整数, 若 $|x-n|<\\dfrac{1}{2}$, 则称 $n$ 是数轴上与 $x$ 最接近的整数.\\\\\n(1) 设 $n$ 是正整数, $c_n$ 表示数轴上距离 $\\sqrt{n^2+n}$ 最近的正整数, 求数列 $\\{c_n\\}$ 的通项公式;\\\\\n(2) 若数列 $\\{d_n\\}$ 满足 $d_1=1$, 当 $n \\geq 2$ 时, $d_n=(1-\\dfrac{1}{2^2})(1-\\dfrac{1}{3^2}) \\cdots(1-\\dfrac{1}{n^2})$, $b_n$ 表示数轴上距离 $d_n$ 最近的正整数, 求数列 $\\{b_n\\}$ 的前 $n$ 项的和 $S_n$;\\\\\n(3) 对于 (2) 问中的 $S_n$, 令 $a_n$ 表示数轴上距离 $\\sqrt{s_n}$ 最近的正整数, 其中 $n$ 是正整数, 若 $\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_m}=2020$, 求正整数 $m$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019750": { + "id": "019750", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-y^2=1, P_1(x_1, y_1), P_2(x_2, y_2)$ 是双曲线右支上的任意两点, 且 $x_1 x_2-y_1 y_2>0$ 恒成立, 则正数 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019751": { + "id": "019751", + "content": "不等式 $\\dfrac{8}{(x+1)^3}+\\dfrac{10}{x+1}-x^3-5 x>0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019752": { + "id": "019752", + "content": "记不等式组 $\\begin{cases}x-y+1 \\geq 0,\\\\3 x-y-3 \\leq 0,\\\\x+y-1 \\geq 0\\end{cases}$所表示的平面区域为 $D$, 若对任意 $(x_0, y_0) \\in D$, 不等式 $x_0-2 y_0+c \\leq 0$ 恒成立, 则 $c$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019753": { + "id": "019753", + "content": "已知函数 $f(x)=\\begin{cases}x^2+3 x,& x \\geq 0,\\\\3 x-x^2,& x<0.\\end{cases}$ 若 $f(a^2-3)+f(2 a)>0$, 则实数 $a$ 的取值范围为 3. 已知函数 $f(x)=|\\lg x|$, 若 $01$, 证明: $\\dfrac{a x-1}{a-x}\\in(-1, 1)$ 且 $f(\\dfrac{a x-1}{a-x})-f(x)=-f(\\dfrac{1}{a})$;\\\\\n(3) 设数列 $\\{x_n\\}$ 中, $x_1 \\in(-1,1)$, $x_{n+1}=(-1)^{n+1}\\dfrac{3 x_n-1}{3-x_n}$, $n \\in \\mathbf{N}$, $n \\geq 1$, 求 $x_1$ 的取值范围,使得 $x_3 \\geq x_n$ 对任意正整数 $n$ 成立.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书21合理化归准确表达", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019763": { + "id": "019763", + "content": "在等差数列 $\\{a_n\\}$ 中, 当 $a_r=a_s(r \\neq s)$ 时, $\\{a_n\\}$ 必定是常数数列. 然而在等比数列 $\\{a_n\\}$ 中, 对某些正整数 $r$、$s(r \\neq s)$, 当 $a_r=a_s$ 时, 非常数数列 $\\{a_n\\}$ 的一个例子是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019764": { + "id": "019764", + "content": "求出一个数学问题的正确结论后, 将其作为条件之一, 提出与原来问题有关的新问题, 我们把它称为原来问题的一个``逆向''问题. 例如, 原来问题是``若正四棱锥底面边长为 $4$, 侧棱长为 $3$, 求该正四棱锥的体积''. 求出体积 $\\dfrac{16}{3}$ 后, 它的一个``逆向''问题可以是``若正四棱锥底面边长为 $4$, 体积为 $\\dfrac{16}{3}$, 求侧棱长''; 也可以是``若正四棱锥的体积为 $\\dfrac{16}{3}$, 求所有侧面面积之和的最小值''. 试给出问题``在平面直角坐标系 $xOy$ 中, 求点 $P(2,1)$ 到直线 $3 x+4 y=0$ 的距离.''的一个有意义的``逆向''问题, 并解答你所给出的``逆向''问题.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019765": { + "id": "019765", + "content": "已知抛物线 $F: x^2=4 y$.\\\\\n(1) $\\triangle ABC$ 的三个顶点在抛物线 $F$ 上, 记 $\\triangle ABC$ 的三边 $AB, BC$, $CA$ 所在直线的斜率分别为 $k_{AB}, k_{BC}, k_{CA}$, 若点 $A$ 在坐标原点, 求 $k_{AB}-k_{BC}+k_{CA}$ 的值;\\\\\n(2) 请你给出一个以 $P(2,1)$ 为顶点, 且其余各顶点均为抛物线 $F$ 上的动点的多边形, 写出多边形各边所在直线的斜率之间的关系式, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019766": { + "id": "019766", + "content": "若从点 $O$ 所作的两条不共线的射线 $OM, ON$ 上分别有点 $M_1, M_2$ 与点 $N_1$, $N_2$, 则三角形面积之比 $\\dfrac{S_{\\triangle OM_1N_1}}{S_{\\triangle OM_2N_2}}=\\dfrac{OM_1}{OM_2}\\cdot \\dfrac{ON_1}{ON_2}$, 若从点 $O$ 所作的不在同一平面内的三条射线 $OP, OQ$ 和 $OR$ 上, 分别有点 $P_1, P_2$, 点 $Q_1, Q_2$ 和点 $R_1, R_2$, 则类似的结论为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019767": { + "id": "019767", + "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_{10}=0$, 则等式 $a_1+a_2+\\cdots+a_n=a_1+a_2+\\cdots+a_{19-n}(n<19, n$ 是正整数) 成立. 类比上述性质, 相应地: 在等比数列 $\\{b_n\\}$ 中, 若 $b_9=1$, 试探究一个成立的等式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019768": { + "id": "019768", + "content": "已知数列 $a_1, a_2, \\cdots, a_{30}$, 其中 $a_1, a_2, \\cdots, a_{10}$ 是首项为 $1$, 公差为 $1$ 的等差数列; $a_{10}, a_{11}, \\cdots, a_{20}$ 是公差为 $d$ 的等差数列; $a_{20}, a_{21}, \\cdots, a_{30}$ 是公差为 $d^2$ 的等差数列 $(d \\neq 0)$.\\\\\n(1) 若 $a_{20}=40$, 求 $d$;\\\\\n(2) 试写出 $a_{30}$ 关于 $d$ 的关系式,并求 $a_{30}$ 的取值范围;\\\\\n(3) 续写已知数列, 使得 $a_{30}, a_{31}, \\cdots, a_{40}$ 是公差为 $d^3$ 的等差数列, $\\cdots \\cdots$, 依次类推, 把已知数列推广为无穷数列. 提出同 (2) 类似的问题 ((2) 应当作为特例), 并进行研究, 你能得到什么样的结论?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019769": { + "id": "019769", + "content": "函数 $y=f(x)$ 的定义域为 $D$, 若对任意 $x \\in D$, $f(|x|)=|f(x)|$, 则称 $y= f(x)$ 在 $D$ 内为对等函数.\\\\\n(1) 指出函数 $y=\\sqrt{x}$, $y=x^3$, $y=2^x$ 在其定义域内哪些为对等函数;\\\\\n(2) 试研究对数函数 $y=\\log _a x$($a>0$ 且 $a \\neq 1$) 在其定义域内是否是对等函数? 若是, 请说明理由; 若不是, 试给出其定义域的一个非空子集, 使 $y=\\log _a x$ 在所给集合内成为对等函数;\\\\\n(3) 若 $\\{0\\}\\subseteq D, y=f(x)$ 在 $D$ 内为对等函数, 试研究 $y=f(x)$($x \\in D$) 的奇偶性.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019770": { + "id": "019770", + "content": "公元前 6 世纪,古希腊的毕达哥拉斯学派通过研究正五边形和正十边形的作图, 发现了黄金分割值约为 $0.618$, 这一数值也可以表示为 $m=2 \\sin 18^{\\circ}$. 若 $m^2+n=4$, 则 $\\dfrac{1-2 \\cos ^227^{\\circ}}{m \\sqrt{n}}=$\\blank{50}. (用数字作答)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019771": { + "id": "019771", + "content": "同学们都知道, 在一次考试后, 如果按顺序去掉一些高分, 那么班级的平均分将降低; 反之, 如果按顺序去掉一些低分, 那么班级的平均分将提高. 这两个事实可以用数学语言描述为: 若有限数列 $a_1, a_2, \\cdots, a_n$ 满足 $a_1 \\leq a_2 \\leq \\cdots \\leq a_n$, 则\\blank{200}(结论用数学式子表示).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019772": { + "id": "019772", + "content": "规定记号``$\\diamond$''表示一种运算, 即 $a \\diamond b=(a^2-1)(b^2-2 b)$, $a$, $b \\in \\mathbf{R}$, 若 $k>0$, 函数 $f(x)= (k x) \\diamond x$ 的图象关于直线 $x=\\dfrac{1}{2}$ 对称, 则 $k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019773": { + "id": "019773", + "content": "如果一个数列由有限个连续的正整数组成 (数列的项数大于 $2$), 且所有项之和为 $N$, 那么称该数列为 $N$ 型标准数列, 例如, 数列 $2,3,4,5,6$ 为 $20$ 型标准数列, 则 $2668$ 型标准数列的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019774": { + "id": "019774", + "content": "已知函数 $y=f(x)$($x \\in \\mathbf{R}$), 对函数 $y=g(x)$($x \\in I$), 定义 $g(x)$ 关于 $f(x)$ 的``对称函数''为函数 $y=h(x)(x \\in I)$, $y=h(x)$ 满足: 对任意 $x \\in I$, 两个点 $(x, h(x)),(x, g(x))$ 关于点 $(x, f(x))$ 对称, 若 $h(x)$ 是 $g(x)=\\sqrt{4-x^2}$ 关于 $f(x)=3 x+b$ 的``对称函数'', 且 $h(x)>g(x)$ 恒成立, 则实数 $b$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019775": { + "id": "019775", + "content": "在平面直角坐标系中, 横、纵坐标均为整数的点叫做格点, 若函数 $y=f(x)$ 的图像恰好经过 $k$ 个格点, 则称函数 $y=f(x)$ 为 $k$ 阶格点函数, 已知函数: \\textcircled{1} $y=x^2$; \\textcircled{2} $y=2 \\sin x$; \\textcircled{3} $y=\\pi^x-1$; \\textcircled{4} $y=\\cos (x+\\dfrac{\\pi}{3})$; 其中为一阶格点函数的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019776": { + "id": "019776", + "content": "定义集合运算 $A \\otimes B=\\{z | z=x y,\\ x \\in A,\\ y \\in B\\}$. 设 $A=\\{1,2\\}$, $B=\\{0,2\\}$, 则集合 $A \\otimes B$ 的所有元素之和为\\bracket{20}.\n\\fourch{$0$}{$2$}{$3$}{$6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019777": { + "id": "019777", + "content": "对于 $\\triangle ABC$, 若存在 $\\triangle A_1B_1C_1$, 满足 $\\dfrac{\\cos A}{\\sin A_1}=\\dfrac{\\cos B}{\\sin B_1}=\\dfrac{\\cos C}{\\sin C_1}=1$, 则称 $\\triangle ABC$ 为``$V$ 类三角形''. ``$V$ 类三角形''一定满足\\bracket{20}.\n\\fourch{有一个内角为 $30^{\\circ}$}{有一个内角为 $45^{\\circ}$}{有一个内角为 $60^{\\circ}$}{有一个内角为 $75^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019778": { + "id": "019778", + "content": "定义域为 $[a, b]$ 的函数 $y=f(x)$ 图象的两个端点为 $A, B$, 向量 $\\overrightarrow{ON}=\\lambda \\overrightarrow{OA}+(1-\\lambda) \\overrightarrow{OB}$, $M(x, y)$ 是 $f(x)$ 图象上任意一点, 其中 $x=\\lambda a+(1-\\lambda) b$, $\\lambda \\in[0,1]$. 若不等式 $|MN| \\leq k$ 恒成立,则称函数 $f(x)$ 在 $[a, b]$ 上满足``$k$ 范围线性近似'', 其中最小的正实数 $k$ 称为该函数的线性近似阈值. 下列定义在 $[1,2]$ 上函数中, 线性近似阈值最小的是\\bracket{20}.\n\\fourch{$y=x^2$}{$y=\\dfrac{2}{x}$}{$y=\\sin \\dfrac{\\pi}{3}x$}{$y=x-\\dfrac{1}{x}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019779": { + "id": "019779", + "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中 (如图), $AD=AA_1=1$, $AB=2$, 点 $E$ 是棱 $AB$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D)--(E)--(C)(A)--(D_1)--(C)(D_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AD_1$ 与 $EC$ 所成角的大小;\\\\\n(2) 《九章算术》中, 将四个面都是直角三角形的四面体称为鳖臑. 试问四面体 $D_1CDE$ 是否为鳖臑? 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019780": { + "id": "019780", + "content": "如图, 在直角坐标系 $xOy$ 中, 有一组对角线长为 $a_n$ 的正方形 $A_n B_n C_n D_n$($n=1,2, \\cdots$), 其对角线 $B_n D_n$ 依次放置在 $x$ 轴上 (相邻顶点重合). 设 $\\{a_n\\}$ 是首项为 $a$, 公差为 $d$($d>0$) 的等差数列, 点 $B_1$ 的坐标为 $(d, 0)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.08]\n\\draw [->] (-4,0) -- (44,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (4,0) node [above left] {$B_1$} coordinate (B_1);\n\\draw (12,0) node [above] {$B_2$} coordinate (B_2) node [below] {$D_1$};\n\\draw (24,0) node [above] {$B_3$} coordinate (B_3) node [below] {$D_2$};\n\\draw (40,0) node [below] {$D_3$} coordinate (D_3);\n\\foreach \\i/\\j/\\k in {8/4/1,18/6/2,32/8/3}\n{\\draw (\\i,\\j) node [above] {$A_\\k$} coordinate (A_\\k) (\\i,-\\j) node [below] {$C_\\k$} coordinate (C_\\k);};\n\\draw (B_1)--(A_1)--(B_2)--(A_2)--(B_3)--(A_3)--(D_3)--(C_3)--(B_3)--(C_2)--(B_2)--(C_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当 $a=8$, $d=4$ 时, 证明: 顶点 $A_1$、$A_2$、$A_3$ 不在同一条直线上;\\\\\n(2) 在 (1) 的条件下, 证明: 所有顶点 $A_n$ 均落在抛物线 $y^2=2 x$ 上;\\\\\n(3) 为使所有顶点 $A_n$ 均落在抛物线 $y^2=2 p x$($p>0$) 上, 求$a$与$d$之间所应满足的关系式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019781": { + "id": "019781", + "content": "我们在下面的表格内填写数值: 先将第 $1$ 行的所有空格填上 $1$; 再把一个首项为 $1$, 公比为 $q$ 的数列 $\\{a_n\\}$ 依次填入第一列的空格内; 然后按照``任意一格的数是它上面一格的数与它左边一格的数之和''的规则填写其它空格.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline & 第 1 列 & 第 2 列 & 第 3 列 & $\\cdots$ & 第 $n$ 列 \\\\\n\\hline 第 1 行 & 1 & 1 & 1 & $\\cdots$ & 1 \\\\\n\\hline 第 2 行 & $q$ & & & & \\\\\n\\hline 第 3 行 & $q^2$ & & & &\\\\\n\\hline$\\cdots$ & $\\cdots$ & & & &\\\\\n\\hline 第 $n$ 行 & $q^{n-1}$ & & & &\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 设第 2 行的数依次为 $B_1, B_2, \\cdots, B_n$, 试用 $n, q$ 表示 $B_1+B_2+\\cdots+B_n$ 的值;\\\\\n(2) 设第 3 列的数依次为 $c_1, c_2, c_3, \\cdots, c_n$, 求证: 对于任意非零实数 $q, c_1+c_3>2 c_2$;\\\\\n(3) 请在以下两个问题中选择一个进行研究 (只能选择一个问题, 如果都选, 被认为选择了第一问).\\\\\n(I) 能否找到 $q$ 的值, 使得 (2) 中的数列 $c_1, c_2, c_3, \\cdots, c_n$ 的前 $m$ 项 $c_1, c_2, \\cdots, c_m $($m \\geq 3$) 成为等比数列? 若能找到, $m$ 的值有多少个? 若不能找到, 说明理由.\\\\\n(II) 能否找到 $q$ 的值,使得填完表格后,除第 1 列外, 还有不同的两列数的前三项各自依次成等比数列? 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书22数学创新问题", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019782": { + "id": "019782", + "content": "有一条长为 $120$ 米的步道 $OA$, $A$ 为垃圾投放点 $\\omega_1$, 若以 $O$ 为原点, $OA$ 为 $x$ 轴的正方向建立平面直角坐标系, 设点 $B(x, 0)$, 现要在步道之间建设另一座垃圾放点 $\\omega_2(t, 0)$, 函数 $f_t(x)$ 表示与 $B$ 点距离最近的垃圾投放点的距离.\\\\\n(1) 若 $t=60$, 求 $f_{60}(10), f_{60}(80), f_{60}(95)$ 的值, 并求出 $f_{60}(x)$ 的函数解析式子;\\\\\n(2) 若可以通过 $f_t(x)$ 的图像与坐标轴围成的面积来测算扔垃圾的便利程度, 面积越小越便利. 问 : 垃圾投放点 $\\omega_2$ 建在何处比建在 $OA$ 中点更便利?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019783": { + "id": "019783", + "content": "统计表明, 某种型号的汽车在匀速行驶中每小时的耗油量 $y$ (升)关于行驶速度 $x$ (千米/小时) 的函数解析式可以表示为: $y=\\dfrac{1}{128000}x^3-\\dfrac{3}{80}x+8$($0=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0) node [below] {$B$} coordinate (B);\n\\draw (4,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw (A) arc (-90:0:2);\n\\draw (A) ++ ({2*tan(20)},0) node [below] {$E$} coordinate (E) --++ ({2/tan(40)},2) node [above] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\angle ADE=20^{\\circ}$, 求 $EF$ 的长;\\\\\n(2) 当 $AE$ 多长时, 梯形 $EFBC$ 的面积有最大值, 最大面积为多少? (长度精确到 $0.1 \\mathrm{m}$, 面积精确到 $0.01 \\mathrm{m}^2$)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019785": { + "id": "019785", + "content": "``十四五''期间, 上海市将大力推进``五个新城''建设, 更好服务长三角一体化发展国家战略. 已知某企业 2021 年第一季度 (一年共四个季度) 的营业额为 $1.1$ 亿元, 预计以后每个季度的营业额比上个季度增加 $0.05$ 亿元, 该企业 2021 年第一季度的利润为 $0.16$ 亿元, 预计以后每个季度的利润比上个季度增长 $4 \\%$.\\\\\n(1) 求该企业自 2021 年起前 $20$ 个季度营业额的总和;\\\\\n(2) 请问该企业自 2021 年起哪个季度的利润将首次超过该季度营业额的 $18 \\%$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019786": { + "id": "019786", + "content": "2018 年世界人工智能大会已于 2018 年 9 月在上海徐汇西岸举行, 某高校的志愿者服务小组受大会展示项目的启发, 会后决定开发一款``猫捉老鼠''的游戏. 如右图: $A$、$B$ 两个信号源相距 $10$ 米, $O$ 是 $AB$ 的中点, 过 $O$ 点的直线 $l$ 与直线 $AB$ 的夹角为 $45^{\\circ}$. 机器猫在直线 $l$ 上运动, 机器鼠的运动轨迹始终满足: 接收到 $A$ 点的信号比接收到 $B$ 点的信号晩 $\\dfrac{8}{v_0}$ 秒 (注: 信号每秒传播 $v_0$ 米). 在时刻 $t_0$ 时,测得机器鼠距离 $O$ 点为 $4$ 米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}\n(1) 以 $O$ 为原点, 直线 $AB$ 为 $x$ 轴建立平面直角坐标系 (如图), 求时刻 $t_0$ 时机器鼠所在位置的坐标;\\\\\n(2) 游戏设定: 机器鼠在距离直线 $l$ 不超过 1.5 米的区域运动时, 有``被抓''的风险. 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''风险?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019787": { + "id": "019787", + "content": "甲、乙两队进行排球决赛, 现在的情形是甲队只要再赢一局就获冠军, 乙队需要再赢两局才能得冠军. 若两队在每局赢的概率都是 $0.5$, 则甲队获得冠军的概率为 (结果用数值表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019788": { + "id": "019788", + "content": "把一个大金属球表面涂漆, 共需要油漆 $2.4$ 千克. 若把这个大金属球熔化制成 $64$ 个大小都相等的小金属球, 不计损耗, 将这些小金属球表面都涂漆, 则需要油漆千克.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019789": { + "id": "019789", + "content": "一船向正北匀速行驶, 看见正西方向两座相距 $10$ 海里的灯塔恰好与该船在同一直线上, 继续航行半小时后, 看见其中一座灯塔在南偏西 $60^{\\circ}$ 方向上, 另一灯塔在南偏西 $75^{\\circ}$ 方向上,则该船的速度是海里/小时.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019790": { + "id": "019790", + "content": "如图, 某校有一块形如直角三角形 $ABC$ 的空地, 其中 $\\angle B$ 为直角, $AB$ 长 $40$ 米, $BC$ 长 $50$ 米, 现欲在此空地上建造一间健身房, 其占地形状为矩形, 且 $B$ 为矩形的一个顶点, 则该健身房的最大占地面积为\\blank{50}平方米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2.5,0) node [below] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw ($(A)!0.4!(C)$) coordinate (P) ($(B)!0.4!(C)$) coordinate (Q) ($(A)!0.4!(B)$) coordinate (R);\n\\draw (Q)--(P)--(R)(A)--(B)--(C)--cycle; \n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019791": { + "id": "019791", + "content": "某食品的保鲜时间 $y$ (单位: 小时) 与储存温度 $x$ (单位: $^{\\circ}\\mathrm{C}$ ) 满足函数关系 $y=\\mathrm{e}^{k x+b}$ ($\\mathrm{e}=2.718 \\cdots$ 为自然对数的底数, $k$、$b$ 为常数). 若该食品在 $0^{\\circ}\\mathrm{C}$ 的保鲜时间设计 $192$ 小时, 在 $22^{\\circ}\\mathrm{C}$ 的保鲜时间是 $48$ 小时,则该食品在 $33^{\\circ}\\mathrm{C}$ 的保鲜时间是\\blank{50}小时.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019792": { + "id": "019792", + "content": "古希腊数学家阿波罗尼斯在他的巨著《圆锥曲线论》中有一个著名的几何问题: 在平面上给定两点 $A(-a, 0)$, $B(a, 0)$, 动点 $P$ 满足 $\\dfrac{|PA|}{|PB|}=\\lambda$ (其中 $a$ 和 $\\lambda$ 是正常数, 且 $\\lambda \\neq 1$), 则 $P$ 的轨迹是一个圆,这个圆称之为``阿波罗尼斯圆''. 该圆的半径为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019793": { + "id": "019793", + "content": "青少年视力是社会普遍关注的问题,视力情况可借助视力表测量. 通常用五分记录法和小数记录法记录视力数据, 五分记录法的数据 $L$ 和小数记录表的数据 $V$ 的满足 $L=5+ \\lg V$. 已知某同学视力的五分记录法的数据为 $4.9$ , 则其视力的小数记录法的数据为\\bracket{20}.\n\\fourch{$1.5$}{$1.2$}{$0.8$}{$0.6$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019794": { + "id": "019794", + "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述. 两颗星的星等与亮度满足 $m_2- m_1=\\dfrac{5}{2}\\lg \\dfrac{E_1}{E_2}$, 其中星等为 $m_k$ 的星的亮度为 $E_k$($k=1,2$). 已知太阳的星等为 $-26.7$ , 天狼星的星等为 $-1.45$ , 则太阳与天狼星的亮度的比值为\\bracket{20}.\n\\fourch{$10^{10.1}$}{$10.1$}{$\\lg 10.1$}{$10^{-10.1}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019795": { + "id": "019795", + "content": "Logistic 模型是常用数学模型之一, 可应用于流行病学领城. 有学者根据公布数据建立了某地区新冠肺炎累计确诊病例数 $I(t)$ ($t$ 的单位: 天) 的 Logistic 模型: $I(t)= \\dfrac{K}{1+\\mathrm{e}^{-0.23(t-53)}}$, 其中 $K$ 为最大确诊病例数. 当 $I(t^*)=0.95K$ 时, 标志着已初步遏制疫情, 则 $(t^*-53)$ 的值约为\\bracket{20}.\n\\fourch{$10$}{$13$}{$63$}{$66$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019796": { + "id": "019796", + "content": "某群体的人均通勤时间, 是指单日内该群体中成员从居住地到工作地的平均用时, 某地在上班族 $S$ 中的成员仅以自驾或公交方式通勤, 分析显示: 当 $S$ 中 $x \\%$($0=latex]\n\\draw (-1.5,0) arc (180:360:1.5) -- (1.5,1) arc (0:180:1.5) --++ (0,-1);\n\\draw (1.5,0) arc (0:-180:1.5 and 0.5) (1.5,1) arc (0:-180:1.5 and 0.5);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and 0.5) (1.5,1) arc (0:180:1.5 and 0.5);\n\\draw [dashed] (0,0) -- (0,1) (-1.5,0) --++ (3,0) (-1.5,1) --++ (3,0);\n\\draw (1.5,0) --++ (1,0) (1.5,1) --++ (1,0);\n\\draw [<->] (2,0) -- (2,1) node [midway, fill = white] {$2\\text{cm}$};\n\\draw (-1.5,0) --++ (0,-2) (1.5,0) --++ (0,-2);\n\\draw [<->] (-1.5,-1.8) -- (1.5,-1.8) node [midway, fill = white] {$6\\text{cm}$};\n\\end{tikzpicture}\n\\end{center}\n(1) 这种``浮球''的体积是多少 (结果精确到 $0.1 \\mathrm{cm}^3$ )?\\\\\n(2) 要在这样 2500 个``浮球''表面涂一层胶质,如果每平方米需要涂胶 $100$ 克,共需胶多少?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019798": { + "id": "019798", + "content": "如图, 有一块半椭圆形钢板, 其长半轴长为 $2 r$, 短半轴长为 $r$, 计划将此钢板切割成等腰梯形的形状, 下底 $AB$ 是半椭圆的短轴, 上底 $CD$ 的端点在椭圆上, 记 $CD=2 x$, 梯形面积为 $S$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (1.5,0) arc (0:180:1.5 and 3) -- (1.5,0);\n\\draw (1.5,0) node [below right] {$B$} coordinate (B);\n\\draw (-1.5,0) node [below left] {$A$} coordinate (A);\n\\draw (50:1.5 and 3) node [above right] {$C$} coordinate (C);\n\\draw (130:1.5 and 3) node [above left] {$D$} coordinate (D);\n\\draw (B)--(C)--(D)--(A);\n\\draw (1.5,0) --++ (0.6,0) (0,3) --++ (2.1,0);\n\\draw [<->] (1.8,0) -- (1.8,3) node [midway, fill = white] {$2r$};\n\\draw (1.5,0) --++ (0,-0.6) (-1.5,0) --++ (0,-0.6);\n\\draw [<->] (-1.5,-0.3) -- (1.5,-0.3) node [midway, fill = white] {$2r$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求面积 $S$ 以 $x$ 为自变量的函数式,并写出其定义域;\\\\\n(2) 求面积 $S$ 的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书23应用题的解法", + "edit": [ + "20230730\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.", From fdef227a0d57c7e0680edb0323580d6453e5ac8a Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 30 Jul 2023 14:41:48 +0800 Subject: [PATCH 4/7] =?UTF-8?q?=E6=B7=BB=E5=8A=A031406-31975=E9=AB=98?= =?UTF-8?q?=E8=80=83=E6=95=B0=E5=AD=A6=E9=A3=8E=E6=9A=B4=E7=AC=AC=E4=BA=8C?= =?UTF-8?q?=E8=BD=AE=E5=A4=8D=E4=B9=A0=E7=94=A8=E4=B9=A6=E9=83=A8=E5=88=86?= =?UTF-8?q?=E9=A2=98=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 11400 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 11400 insertions(+) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 07b3da02..c64bd48d 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -600301,6 +600301,11406 @@ "space": "", "unrelated": [] }, + "031406": { + "id": "031406", + "content": "抛物线 $y=4 x^2$ 在点 $(1,4)$ 处的切线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031407": { + "id": "031407", + "content": "函数 $f(x)=\\dfrac{x+4}{\\ln x}$ 的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题2", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031408": { + "id": "031408", + "content": "若 $\\log _2(x+1)=3$, 则 $x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题3", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031409": { + "id": "031409", + "content": "过双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的右焦点, 且平行于渐近线的直线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题4", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031410": { + "id": "031410", + "content": "函数 $f(x)=\\log _2 \\dfrac{x}{4}$, 等比数列 $\\{a_n\\}$ 中, $a_2 \\cdot a_5 \\cdot a_8=8$, 则 $f(a_1)+f(a_2)+\\cdots+f(a_9)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题5", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031411": { + "id": "031411", + "content": "在二项式 $(x+\\dfrac{3}{x})^6$ 的展开式中,常数项的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题6", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031412": { + "id": "031412", + "content": "若复数 $z=(\\sin \\theta-\\dfrac{3}{5})+(\\cos \\theta-\\dfrac{4}{5}) \\mathrm{i}$ 是纯虚数, 则 $\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题7", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031413": { + "id": "031413", + "content": "已知向量 $\\overrightarrow{a}=(2,0)$, $\\overrightarrow{b}=(1,2)$, 则向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 方向上的投影为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题8", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031414": { + "id": "031414", + "content": "若一个圆锥的侧面展开图是面积为 $2 \\pi$ 的半圆面,则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题9", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031415": { + "id": "031415", + "content": "已知函数 $f(x)=\\mathrm{e}^{|x|}+|x|$, 若关于 $x$ 的方程 $f(x)=k$ 有两个不同的实根, 则实数 $k$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题10", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031416": { + "id": "031416", + "content": "若直线 $x-y+1=0$ 与圆 $(x-a)^2+y^2=2$ 有公共点, 则实数 $a$ 取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题11", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031417": { + "id": "031417", + "content": "设函数 $F(x)$ 和 $f(x)$ 都在区间 $D$ 上有定义,若对 $D$ 的任意子区间 $[u, v]$, 总有 $[u, v]$ 上的实数 $p$ 和 $q$, 使得不等式 $f(p) \\leq \\dfrac{F(u)-F(v)}{u-v}\\leq f(q)$ 成立, 则称 $F(x)$ 是 $f(x)$ 在区间 $D$ 上的甲函数, $f(x)$ 是 $F(x)$ 在区间 $D$ 上的乙函数. 已知 $F(x)=x^2-3 x$, $x \\in \\mathbf{R}$, 那么 $F(x)$ 的乙函数 $f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题12", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031418": { + "id": "031418", + "content": "曲线 $y=\\mathrm{e}^x$ 在点 $A(0,1)$ 处的切线斜率为\\bracket{20}.\n\\fourch{$1$}{$2$}{$e$}{$\\dfrac{1}{\\mathrm{e}}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题13", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031419": { + "id": "031419", + "content": "直线 $a, b$ 异面, $a \\parallel $ 平面 $\\alpha$, 则对于下列论断正确的是\\blank{50}.\\\\\n\\textcircled{1} 一定存在平面 $\\alpha$ 使 $b \\perp \\alpha$; \\textcircled{2} 一定存在平面 $\\alpha$ 使 $b \\parallel \\alpha$; \\textcircled{3} 一定存在平面 $\\alpha$ 使 $b \\subseteq \\alpha$; \\textcircled{4} 一定存在无数个平面 $\\alpha$ 与 $b$ 交于一定点\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{2} \\textcircled{3}}{\\textcircled{2}\\textcircled{3}\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题14", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031420": { + "id": "031420", + "content": "已知圆 $x^2+y^2=1$ 与 $x$ 轴的两个交点为 $A$、$B$, 若圆内的动点 $P$ 使 $|PA|$、$|PO|$、$|PB|$ 成等比数列, 则 $\\overrightarrow{PA}\\cdot \\overrightarrow{PB}$ 的取值范围为\\bracket{20}.\n\\fourch{$(0, \\dfrac{1}{2}]$}{$[-\\dfrac{1}{2}, 0)$}{$(-\\dfrac{1}{2}, 0)$}{$[-1,0)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题15", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031421": { + "id": "031421", + "content": "``$t \\geq 0$''是``函数 $f(x)=x^2+t x-t$ 在 ($-\\infty$, $+\\infty$) 内存在零点''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题16", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031422": { + "id": "031422", + "content": "已知函数 $f(x)=\\dfrac{b x-5}{x+a}$($x \\neq-a$, $a$、$b$ 是常数, 且 $a b \\neq-5$), 对定义域内任意 $x$($x \\neq-a$、$x \\neq-a-3$ 且 $x \\neq a+3$), 恒有 $f(3+x)+f(3-x)=4$ 成立.\\\\\n(1) 求函数 $y=f(x)$ 的解析式, 并写出函数的定义域;\\\\\n(2) 求 $x$ 的取值范围, 使得 $f(x) \\in[0,2) \\cup(2,4]$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题17", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031423": { + "id": "031423", + "content": "设 $m>3$, 对于项数为 $m$ 的有穷数列 $\\{a_n\\}$, 令 $b_k$ 为 $a_1, a_2, \\cdots, a_k$($k \\leq m$) 中的最大值, 称数列 $\\{b_n\\}$ 为 $\\{a_n\\}$ 的``创新数列''. 例如数列 $3,5,4,7$ 的创新数列为 $3,5,5,7$. 考查自然数 $1,2, \\cdots, m$($m>3$) 的所有排列, 将每种排列都视为一个有穷数列 $\\{c_n\\}$.\\\\\n(1) 若 $m=5$, 写出创新数列为 $3,5,5,5,5$ 的所有数列 $\\{c_n\\}$;\\\\\n(2) 是否存在数列 $\\{c_n\\}$ 的创新数列为等比数列? 若存在, 求出符合条件的创新数列; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练01试题18", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031424": { + "id": "031424", + "content": "已知集合 $A=\\{y | y=\\sin x,\\ x \\in \\mathbf{R}\\}$, 集合 $B=\\{y | y=\\sqrt{x},\\ x \\in \\mathbf{R}\\}$, 则 $A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031425": { + "id": "031425", + "content": "已知双曲线的两个焦点为 $F_1(-\\sqrt{5}, 0), F_2(\\sqrt{5}, 0), P$ 是此双曲线上的一点, 且 $PF_1 \\perp PF_2$, $|PF_1| \\cdot|PF_2|=2$, 则该双曲线的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031426": { + "id": "031426", + "content": "若 $a_n=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$ ($n$ 是正整数 $)$, 则 $a_{n+1}-a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031427": { + "id": "031427", + "content": "已知 $\\mathrm{C}_{10}^{2 x}-\\mathrm{C}_{10}^{x+1}=0$, 则 $x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031428": { + "id": "031428", + "content": "若圆 $x^2+y^2+m x-\\dfrac{1}{4}=0$ 与直线 $y=-1$ 相切, 且其圆心在 $y$ 轴的左侧, 则 $m$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031429": { + "id": "031429", + "content": "如图,用铁皮制作一个无盖的圆锥形容器,已知该圆锥的母线与底面所在平面的夹角为 $45^{\\circ}$, 容器的高为 $10 \\mathrm{cm}$, 制作该容器需要铁皮面积为\\blank{50}$\\mathrm{cm}^2$ (衔接部分忽略不计, 结果保留整数).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [<->] (1.2,1) -- (1.2,0) node [midway, fill = white] {$10\\text{cm}$};\n\\draw (0,0) -- (1,1) (0,0) -- (-1,1) (0,1) ellipse (1 and 0.25) (1,1) -- (0,1);\n\\draw [dashed] (0,0) -- (0,1);\n\\draw (1,1) -- (1.4,1) (0,0) -- (1.4,0);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031430": { + "id": "031430", + "content": "在一个小组中有 $8$ 名女同学和 $4$ 名男同学, 从中任意地挑选 $2$ 名同学担任交通安全宣传志愿者, 那么选到的两名都是女同学的概率是 (结果用分数表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031431": { + "id": "031431", + "content": "已知 $AC, BD$ 为圆 $O: x^2+y^2=4$ 的两条互相垂直的弦, $AC, BD$ 交于点 $M(1, \\sqrt{2})$, 则四边形 $ABCD$ 面积的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031432": { + "id": "031432", + "content": "已知 $O$ 是 $\\triangle ABC$ 的外心, $AB=2$, $AC=3$, $x+2 y=1$, 若 $\\overrightarrow{AO}=x \\cdot \\overrightarrow{AB}+y \\cdot \\overrightarrow{AC}$, ($x y \\neq 0$), 则 $\\cos \\angle BAC=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031433": { + "id": "031433", + "content": "已知函数 $f(x)=x^2-\\cos x$, 对于 $[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}]$ 上的任意 $x_1, x_2$, 有如下条件: \\textcircled{1} $x_1>x_2$; \\textcircled{2} $x_1^2>x_2^2$; \\textcircled{3} $|x_1|>x_2$. 其中能使 $f(x_1)>f(x_2)$ 恒成立的条件序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031434": { + "id": "031434", + "content": "已知 $\\{a_n\\}$ 是等比数列, $a_1=2$, $a_3=18$, $\\{b_n\\}$ 是等差数列, $b_1=2$, $b_1+b_2+b_3+b_4=a_1+ a_2+a_3>20$. 则数列 $\\{b_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031435": { + "id": "031435", + "content": "设函数 $y=f(x)$ 由方程 $x|x|+y|y|=1$ 确定, 下列结论: \\textcircled{1} $f(x)$ 是 $\\mathbf{R}$ 上的严格减函数; \\textcircled{2} 对于任意 $x \\in \\mathbf{R}$, $f(x)+x>0$ 恒成立; \\textcircled{3} 对于任意 $a \\in \\mathbf{R}$, 关于 $x$ 的方程 $f(x)=a$ 都有解. 其中正确结论的是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031436": { + "id": "031436", + "content": "用一个平面去截正方体, 所得截面不可能是\\bracket{20}.\n\\fourch{平面六边形}{菱形}{梯形}{直角三角形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031437": { + "id": "031437", + "content": "设 $a b c \\neq 0$, ``$a c>0$''是``曲线 $a x^2+b y^2=c$ 为椭圆''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031438": { + "id": "031438", + "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 且 $2 c \\cos ^2 \\dfrac{A}{2}=b+c$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{正三角形}{直角三角形}{等腰三角形}{等腰直角三角形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031439": { + "id": "031439", + "content": "设 $\\{a_n\\}$ 是各项为正数的无穷数列, $A_i$ 是边长为 $a_i, a_{i+1}$ 的矩形面积 ($i=1,2,3, \\cdots$), 则 $\\{A_n\\}$ 为等比数列的充要条件为\\bracket{20}.\n\\onech{$\\{a_n\\}$ 是等比数列}{$a_1, a_3, a_5, \\cdots a_{2 n-1}, \\cdots$ 或 $a_2, a_4, a_6, \\cdots a_{2 n}, \\cdots$ 是等比数列}{$a_1, a_3, a_5, \\cdots a_{2 n-1}, \\cdots$ 和 $a_2, a_4, a_6, \\cdots a_{2 n}, \\cdots$ 均是等比数列}{$a_1, a_3, a_5, \\cdots a_{2 n-1}, \\cdots$ 和 $a_2, a_4, a_6, \\cdots a_{2 n}, \\cdots$, 且公比相同}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031440": { + "id": "031440", + "content": "记 $\\min \\{p, q\\}=\\begin{cases}p ,& p \\leq q,\\\\q ,& p>q.\\end{cases}$ 若函数 $f(x)=\\min \\{3+\\log _{\\frac{1}{4}}x, \\log _2 x\\}$.\\\\\n(1) 用分段函数形式写出函数 $f(x)$ 的解析式;\\\\\n(2) 求 $f(x)<2$ 的解集.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031441": { + "id": "031441", + "content": "在边长为 $6 \\mathrm{cm}$ 的正方形 $ABCD$ 中, $E, F$ 分别为 $BC, CD$ 的中点, $M, N$ 分别为 $AB, CF$ 的中点, 现沿 $AE, AF, EF$ 折叠, 使 $B, C, D$ 三点重合于 $B$, 构成一个三棱锥 (如图所示).\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 (2,2) node [above] {$D$} coordinate (D);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\filldraw ($(C)!0.5!(F)$) node [right] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(A)!0.5!(B)$) node [left] {$M$} coordinate (M) circle (0.03);\n\\draw (B)--(C)--(D)--(A)--cycle(A)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,-2) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [below] {$E$} coordinate (E);\n\\draw (2,0,-1) node [right] {$F$} coordinate (F);\n\\draw ({4/3},{2/3},{-2/3}) node [above] {$B$} coordinate (B);\n\\draw (B)--(A)--(E)--(F)--cycle(B)--(E);\n\\draw [dashed] (A)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 在三棱锥上标注出 $M$、$N$ 点, 并判别 $MN$ 与平面 $AEF$ 的位置关系, 并给出证明;\\\\\n(2) $G$ 是线段 $AB$ 上一点, 且 $\\overrightarrow{AG}=\\lambda \\cdot \\overrightarrow{AB}$, 问是否存在点 $G$ 使得 $AB \\perp$ 面 $EGF$, 若存在, 求出 $\\lambda$ 的值; 若不存在, 请说明理由;\\\\\n(3) 求多面体 $E-AFMN$ 的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031442": { + "id": "031442", + "content": "函数 $y=\\sin 2 x-\\cos 2 x$ 的最小正周期是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031443": { + "id": "031443", + "content": "方程 $16^x-4^x-2=0$ 的解是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031444": { + "id": "031444", + "content": "已知 $\\alpha$ 为第二象限角, 且 $\\sin \\alpha=\\dfrac{1}{3}$, 则 $\\sin (\\dfrac{3 \\pi}{2}-\\alpha)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031445": { + "id": "031445", + "content": "$y=4 x^2$ 的焦点到准线的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031446": { + "id": "031446", + "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1$($a>0$) 的一条渐近线方程为 $x-2 y=0$, 则双曲线的离心率为 $e=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031447": { + "id": "031447", + "content": "曲线 $y=x^2-2 x+1$ 在点 $(1,0)$ 处的切线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031448": { + "id": "031448", + "content": "汽车的最佳使用年限是使年均消耗费用最低的年限 (年均消耗费用 $=$ 年均成本费 $+$ 年均维修费). 设某种汽车的购车的总费用为 $50000$ 元; 使用中每年的保险费、养路费及汽油费合计为 $6000$ 元; 前 $x$ 年的总维修费 $y$ 满足 $y=a x^2+b x$, 已知第一年的维修费为 $1000$ 元, 前二年总维修费为 $3000$ 元. 则这种汽车的最佳使用年限为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031449": { + "id": "031449", + "content": "$4$ 张卡片上分别写有数字 $1,2,3,4$, 从这 $4$ 张卡片中随机抽取 $2$ 张, 则取出的 $2$ 张卡片上的数字之和为奇数的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031450": { + "id": "031450", + "content": "已知 $(\\sqrt{x}+\\dfrac{3}{\\sqrt[3]{x}})^n$ 展开式中, 各项系数的和与各项二项式系数的和之比为 $64$, 则 $n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031451": { + "id": "031451", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2+2 n-1$, 则 $a_1+a_3+a_5+\\cdots+a_{25}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031452": { + "id": "031452", + "content": "$y=f(x)$ 是关于 $x=3$ 对称的奇函数, $f(1)=1$, $\\cos x-\\sin x=\\dfrac{3 \\sqrt{2}}{5}$,\n则 $f(\\dfrac{15 \\sin 2 x}{\\cos (x+\\dfrac{\\pi}{4})})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031453": { + "id": "031453", + "content": "对于使 $-x^2+2 x \\leq M$ 成立的所有常数 $M$ 中, 我们把 $M$ 的最小值 1 叫做 $-x^2+2 x$ 的上确界, 若 $a, b \\in$($0,+\\infty$), 且 $a+b=1$, 则 $-\\dfrac{1}{2 a}-\\dfrac{2}{b}$ 的上确界为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031454": { + "id": "031454", + "content": "设 $\\mathrm{i}$ 是虚数单位,复数 $\\dfrac{1+a \\mathrm{i}}{2-\\mathrm{i}}$ 为纯虚数,则实数 $a$ 为\\bracket{20}.\n\\fourch{$2$}{$-2$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031455": { + "id": "031455", + "content": "已知圆 $x^2+(y-1)^2=2$ 上任一点 $P(x, y)$, 其坐标均使得不等式 $x+y+m \\geq 0$ 成立, 则实数 $m$ 的取值范围是\\bracket{20}.\n\\fourch{$[1,+\\infty)$}{($-\\infty, 1]$}{$[-3,+\\infty$)}{$(-\\infty,-3]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031456": { + "id": "031456", + "content": "设 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 若 $a_1=1$, 公差 $d=2$, $S_{k+2}-S_k=24$, 则 $k$ 等于\\bracket{20}.\n\\fourch{8}{7}{6}{5}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031457": { + "id": "031457", + "content": "根据统计, 一名工人组装第 $x$ 件某产品所用的时间 (单位: 分钟) 为 $f(x)= \\begin{cases}\\dfrac{c}{\\sqrt{x}},& x0$, 那么该函数在 ($0, \\sqrt{a}]$ 上是严格减函数, 在 $[\\sqrt{a},+\\infty$) 上是严格增函数.\\\\\n(1) 如果函数 $y=x+\\dfrac{3^m}{x}$($x>0$) 的值域是 $[6,+\\infty)$, 求实数 $m$ 的值;\\\\\n(2) 若把函数 $f(x)=x^2+\\dfrac{a}{x^2}$(常数 $a>0$) 在 $[1,2]$ 上的最小值记为 $g(a)$,求 $g(a)$ 的表达式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031459": { + "id": "031459", + "content": "在平面直角坐标系中, 直线 $l: y=m x+3-4 m$, $m \\in \\mathbf{R}$ 恒过一定点, 且与以原点为圆心的圆 $C$ 恒有公共点.\\\\\n(1) 求出直线 $l$ 恒过的定点坐标;\\\\\n(2) 当圆 $C$ 的面积最小时, 求圆 $C$ 的方程;\\\\\n(3) 已知定点 $Q(-4,3)$, 直线 $l$ 与 (2) 中的圆 $C$ 交于 $M, N$ 两点, 试问 $\\overrightarrow{QM}\\cdot \\overrightarrow{QN}\\cdot \\tan \\angle MQN$ 是否存在最大值, 若存在则求出该最大值, 并求出此时直线 $l$ 的方程, 若不存在请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031460": { + "id": "031460", + "content": "若 $z_1=a+2 \\mathrm{i}$, $z_2=3-4 \\mathrm{i}$, 且 $\\dfrac{z_1}{z_2}$ 为纯虚数, 则实数 $a$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031461": { + "id": "031461", + "content": "已知 $\\tan \\dfrac{\\alpha}{2}=3$, 则 $\\cos \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031462": { + "id": "031462", + "content": "已知抛物线 $x^2=3 y$ 上两点 $A, B$ 的横坐标恰是方程 $x^2+5 x+1=0$ 的两个实根, 则直线 $AB$ 的方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031463": { + "id": "031463", + "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点,则 $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031464": { + "id": "031464", + "content": "在各项都为正数的等比数列 $\\{a_n\\}$ 中, 首项 $a_1=3$, 前三项和为 21 , 则 $a_3+a_4+a_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031465": { + "id": "031465", + "content": "一个与球心距离为 1 的平面截球所得的圆面面积为 $\\pi$, 则球的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031466": { + "id": "031466", + "content": "已知向量 $\\overrightarrow{OA}=(k, 12)$, $\\overrightarrow{OB}=(4,5)$, $\\overrightarrow{OC}=(-k, 10)$, 且 $A, B, C$ 三点共线, 则 $k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031467": { + "id": "031467", + "content": "设 $01000$ 成立的最小的 $n$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031471": { + "id": "031471", + "content": "已知平面向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足 $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $\\overrightarrow{a}^2=\\overrightarrow{a}\\cdot \\overrightarrow{b}$, $2 \\overrightarrow{c}^2=\\overrightarrow{b}\\cdot \\overrightarrow{c}$, 则 $|\\overrightarrow{c}-\\overrightarrow{a}|^2+|\\overrightarrow{c}-\\overrightarrow{b}|^2$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031472": { + "id": "031472", + "content": "``$a=-7$''是``直线 $(3+a) x+4 y=5-3 a$ 与直线 $2 x+(5+a) y=8$ 互相平行''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}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031473": { + "id": "031473", + "content": "函数 $f(x)=\\ln (x+1)-\\dfrac{2}{x}$ 的零点所在的区间是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{2}, 1)$}{$(1, \\mathrm{e}-1)$}{$(e-1,2)$}{$(2, \\mathrm{e})$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031474": { + "id": "031474", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031475": { + "id": "031475", + "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031476": { + "id": "031476", + "content": "已知 $\\alpha$ 为锐角, 且 $\\tan \\alpha=\\sqrt{2}-1$, 函数 $f(x)=2 x \\tan 2 \\alpha+\\sin (2 \\alpha+\\dfrac{\\pi}{4})$, 数列 $\\{a_n\\}$ 的首项 $a_1=1$, $a_{n+1}=f(a_n)$.\\\\\n(1) 求函数 $f(x)$ 的表达式;\\\\\n(2) 求数列 $(a_n)$ 的前 $n$ 项和 $S_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031477": { + "id": "031477", + "content": "设 $t \\neq 0$, 点 $P(t, 0)$ 是函数 $f(x)=x^3+a x$ 与 $g(x)=b x^2+c$ 的图像的一个公共点, 两函数的图像在点 $P$ 处有相同的切线.\\\\\n(1) 用 $t$ 表示 $a, b, c$;\\\\\n(2) 若函数 $y=f(x)-g(x)$ 在 $(-1,3)$ 上严格减, 求 $t$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031478": { + "id": "031478", + "content": "若 $a \\in\\{y | y=2^x,、 x<1\\}$, 则 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031479": { + "id": "031479", + "content": "若函数 $f(x)=\\log _a(x+\\sqrt{x^2+2 a^2})$ 是奇函数,则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031480": { + "id": "031480", + "content": "已知抛物线 $y^2=a x$ 的准线方程是 $x=-3$, 那么抛物线的焦点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031481": { + "id": "031481", + "content": "已知球的体积为 $36 \\pi$, 则该球的表面积等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031482": { + "id": "031482", + "content": "若直线 $m x+n y=4$ 和 $\\odot O: x^2+y^2=4$ 没有交点, 则过点 $(m, n)$ 的直线与椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 的交点个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031483": { + "id": "031483", + "content": "已知两条直线 $y=a x-2$ 和 $y=(a+2) x+1$ 互相垂直, 则 $a$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031484": { + "id": "031484", + "content": "某人射击一次击中的概率为 $0.6$, 经过 3 次射击, 此人至少有两次击中目标的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031485": { + "id": "031485", + "content": "若 $\\log _{2 a}\\dfrac{1+a^2}{1+a}<0$, 则 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031486": { + "id": "031486", + "content": "设 $\\alpha$ 为第四象限的角, 若 $\\dfrac{\\sin 3 \\alpha}{\\sin \\alpha}=\\dfrac{13}{5}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031487": { + "id": "031487", + "content": "设 $\\{a_n\\}$ 是正项数列, 其前 $n$ 项和 $S_n$ 满足 : $4S_n=(a_n-1)(a_n+3)$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031488": { + "id": "031488", + "content": "圆 $x^2+y^2=1$ 与直线 $y=k x+2$ 有两个公共点的充要条件是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031489": { + "id": "031489", + "content": "我们把形如 $y=\\dfrac{b}{|x|-a}$($a>0$, $b>0$) 的函数因其图像类似于汉字``圆''字, 故生动地称为``囧函数'', 并把其与 $y$ 轴的交点关于原点的对称点称为``囧点'', 以``囧点''为圆心凡是与``圆函数''有公共点的圆, 皆称之为``囧圆'', 则当 $a=1$, $b=1$ 时, 所有的``囧圆''中, 面积的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031490": { + "id": "031490", + "content": "现用铁丝做一个面积为 $1$ 平方米、形状为直角三角形的框架, 有下列四种长度的铁丝各一根供选择, 其中最合理 (即够用, 浪费最少) 的一根是\\bracket{20}.\n\\fourch{$4.6$米}{$4.8$米}{$5$米}{$5.2$米}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031491": { + "id": "031491", + "content": "若函数 $f(x)=\\log _a(x^3-a x)$($a>0$, $a \\neq 1$) 在区间 $(-\\dfrac{1}{2}, 0)$ 是上严格增函数, 则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{1}{4}, 1)$}{$[\\dfrac{3}{4}, 1)$}{($\\dfrac{9}{4},+\\infty$)}{$(1, \\dfrac{9}{4})$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031492": { + "id": "031492", + "content": "要得到函数 $y=\\sqrt{2}\\cos x$ 的图像, 只需将函数 $y=\\sqrt{2}\\sin (2 x+\\dfrac{\\pi}{4})$ 的图像上所有的点的\\bracket{20}.\n\\onech{横坐标缩短到原来的 $\\dfrac{1}{2}$ 倍 (纵坐标不变), 再向左平行移动 $\\dfrac{\\pi}{8}$ 个单位长度}{横坐标缩短到原来的 $\\dfrac{1}{2}$ 倍 (纵坐标不变), 再向右平行移动 $\\dfrac{\\pi}{4}$ 个单位长度}{横坐标伸长到原来的 $2$ 倍 (纵坐标不变), 再向左平行移动 $\\dfrac{\\pi}{4}$ 个单位长度}{横坐标伸长到原来的 $2$ 倍 (纵坐标不变), 再向右平行移动 $\\dfrac{\\pi}{8}$ 个单位长度}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031493": { + "id": "031493", + "content": "已知曲线 $\\Gamma: \\begin{cases}x=\\dfrac{3}{\\cos \\theta}, \\\\y = \\tan \\theta\\end{cases}$($\\theta$ 是参数), 过点$P(6,2)$作直线$l$与曲线$\\Gamma$有且仅有一个公共点, 则这样的直线 $l$ 有\\bracket{20}.\n\\fourch{$1$ 条}{$2$ 条}{$3$ 条}{$4$ 条}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031494": { + "id": "031494", + "content": "已知关于 $x$ 的不等式 $\\dfrac{(a+1) x-3}{x-1}<1$.\\\\\n(1) 当 $a=1$ 时,求该不等式的解集;\\\\\n(2) 当 $a>0$ 时,求该不等式的解集.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031495": { + "id": "031495", + "content": "已知点 $P_n(a_n, b_n)$ 满足 $a_{n+1}=a_n b_{n+1}$, $b_{n+1}=\\dfrac{b_n}{1-a_n^2}$, 且 $P_0(\\dfrac{1}{3}, \\dfrac{2}{3})$($n \\in \\mathbf{N}$).\\\\\n(1) 求点 $P_1$ 坐标, 并写出过点 $P_0, P_1$ 的直线 $l$ 的方程;\\\\\n(2) 猜测点 $P_n$($n \\geq 2$) 与直线 $l$ 的位置关系, 并加以证明;\\\\\n(3) 求数列 $\\{a_n\\}$ 与 $\\{b_n\\}$ 的通项公式 ($n$ 是正整数).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031496": { + "id": "031496", + "content": "如果双曲线 $\\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$ 上一点 $P$ 到双曲线右焦点的距离是 $2$, 那么点 $P$ 到 $y$ 轴的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031497": { + "id": "031497", + "content": "若复数 $(1+b \\mathrm{i})(2+\\mathrm{i})$ 是纯虚数 ($\\mathrm{i}$ 是虚数单位, $b$ 是实数), 则 $b=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031498": { + "id": "031498", + "content": "要得到 $y=\\cos (2 x-\\dfrac{\\pi}{4})$ 的图像, 且使平移的距离最短, 则需将 $y=\\cos 2 x$ 的图像向平移个单位即可得到.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031499": { + "id": "031499", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2-9 n$, 第 $k$ 项满足 $5=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below right] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A)-- ($(D)!1.5!(A)$) node [below left] {$x$};\n\\draw [->] (C)-- ($(D)!1.4!(C)$) node [below] {$y$};\n\\draw [->] (D1)-- ($(D)!1.5!(D1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031506": { + "id": "031506", + "content": "某商场买来一车苹果, 从中随机抽取了 $10$ 个苹果, 其重量 (单位: 克) 分别为: $150,152,153,149,148,146,151,150,152,147$, 由此估计这车苹果单个重量的期望值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031507": { + "id": "031507", + "content": "设有一组圆 $C_k:(x-k+1)^2+(y-3 k)^2=2 k^4$($k \\in \\mathbf{N}$, $k \\geq 1$). 下列四个命题:\\\\\n\\textcircled{1} 存在一条定直线与所有的圆均相切; \\textcircled{2} 存在一条定直线与所有的圆均相交; \\textcircled{3} 存在一条定直线与所有的圆均不相交; \\textcircled{4} 所有的圆均不经过原点. 其中真命题的代号是\\blank{50}. (写出所有真命题的代号)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031508": { + "id": "031508", + "content": "设 $\\alpha \\in\\{-1,1, \\dfrac{1}{2}, 3\\}$, 则使函数 $y=x^\\alpha$ 的定义域为 $\\mathbf{R}$ 且为奇函数的所有 $\\alpha$ 的值为\\bracket{20}.\n\\fourch{$1,3$}{$-1,1$}{$-1,3$}{$-1,1,3$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031509": { + "id": "031509", + "content": "设命题 $p$: $x>2$ 是 $x^2>4$ 的充要条件, 命题 $q$: 若 $\\dfrac{a}{c^2}>\\dfrac{b}{c^2}$, 则 $a>b$, 则\\bracket{20}.\n\\twoch{$p$ 与 $q$ 至少有一个为真命题}{$p$ 与 $q$ 均为真命题}{$p$ 为真命题 $q$ 为假命题}{$p, q$ 均为假命题}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031510": { + "id": "031510", + "content": "已知向量 $\\overrightarrow{a}=(2,3)$, $\\overrightarrow{b}=(-1,2)$, 若 $m \\overrightarrow{a}+n \\overrightarrow{b}$ 与 $\\overrightarrow{a}-2 \\overrightarrow{b}$ 共线, 则 $\\dfrac{m}{n}$ 等于\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{$-2$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031511": { + "id": "031511", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{2}=1$($a>\\sqrt{2}$) 的两条渐近线的夹角为 $\\dfrac{\\pi}{3}$, 则双曲线的离心率为 \\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\dfrac{2 \\sqrt{6}}{3}$}{$\\sqrt{3}$}{2}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031512": { + "id": "031512", + "content": "设函数 $f(x)=\\mathrm{e}^x-\\mathrm{e}^{-x}$.\\\\\n(1) 证明: $f(x)$ 的导数 $f'(x) \\geq 2$;\\\\\n(2) 若对所有 $x \\geq 0$ 都有 $f(x) \\geq a x$, 求 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031513": { + "id": "031513", + "content": "设常数 $t>2$, 在平面直角坐标系 $xOy$ 中, 已知点 $F(2,0)$, 直线 $l: x=t$, 曲线 $\\Gamma: y^2=8 x$($0 \\leq x \\leq t$, $y \\geq 0$), $l$ 与 $x$ 轴交于点 $A$ 、与 $\\Gamma$ 交于 $B$, $P, Q$ 分别是 $\\Gamma$ 与线段 $AB$ 上的动点.\\\\\n(1) 用 $t$ 表示点 $B$ 到点 $F$ 的距离;\\\\\n(2) 设 $t=3$, $|FQ|=2$, 线段 $OQ$ 的中点在直线 $FP$ 上, 求 $\\triangle AQP$ 的面积;\\\\\n(3) 设 $t=8$, 是否存在以 $FP, FQ$ 为邻边的矩形 $FPEQ$, 使得点 $E$ 在 $\\Gamma$ 上? 若存在, 求点 $P$ 的坐标, 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031514": { + "id": "031514", + "content": "已知集合 $M=\\{x|| x |<2\\}$, $N=\\{x | \\dfrac{x+1}{x-3}<0\\}$, 则集合 $M \\cap N$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031515": { + "id": "031515", + "content": "复数 $z=a+b \\mathrm{i}$, $a, b \\in \\mathbf{R}$, 且 $b \\neq 0$, 若 $z^2-4 b z$ 是实数, 则 $a, b$ 满足的条件是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031516": { + "id": "031516", + "content": "在 $(x^2-\\dfrac{1}{x})^8$ 的展开式中, 含 $x$ 的项的系数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031517": { + "id": "031517", + "content": "在等比数列 $\\{a_n\\}$ 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 10 项和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031518": { + "id": "031518", + "content": "若圆 $x^2+y^2-2 x-4 y=0$ 的圆心到直线 $x-y+a=0$ 的距离为 $\\dfrac{\\sqrt{2}}{2}$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031519": { + "id": "031519", + "content": "经过圆 $x^2+2 x+y^2=0$ 的圆心 $C$, 且与直线 $x+y=0$ 垂直的直线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031520": { + "id": "031520", + "content": "已知向量 $\\overrightarrow{a}=(1,2)$ 和 $\\overrightarrow{b}=(x, 1)$, 若向量 $\\overrightarrow{a}+2 \\overrightarrow{b}$ 与 $2 \\overrightarrow{a}-\\overrightarrow{b}$ 平行, 则实数 $x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031521": { + "id": "031521", + "content": "若 $\\dfrac{\\cos 2 \\alpha}{\\sin (\\alpha-\\dfrac{\\pi}{4})}=-\\dfrac{\\sqrt{2}}{2}$, 则 $\\cos \\alpha+\\sin \\alpha$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031522": { + "id": "031522", + "content": "如图, 测量河对岸的塔高 $AB$ 时, 可以选与塔底 $B$ 在同一水平面内的两个测点 $C$ 与 $D$. 测得 $\\angle BCD=15^{\\circ}$, $\\angle BDC=30^{\\circ}$, $CD=30$ 米, 并在点 $C$ 测得塔顶 $A$ 的仰角为 $60^{\\circ}$, 则 $BC=$\\blank{50}米, 塔高 $AB=$\\blank{50}米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (-2,0) node [left] {$C$} coordinate (C);\n\\draw (-0.6,-0.6) node [below] {$D$} coordinate (D);\n\\draw (0,1.5) node [above] {$A$} coordinate (A);\n\\filldraw (A) --++ (-0.1,-1.5) --++ (0.2,0) -- cycle;\n\\draw [dashed] (A)--(C)(C)--(B)(B)--(D);\n\\draw (C)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031523": { + "id": "031523", + "content": "如果函数 $f(x)=\\sin (\\pi x+\\theta)$($0<\\theta<2 \\pi$) 的最小正周期是 $T$, 且当 $x=2$ 时取得最大值,则 $T=$\\blank{50}, $\\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031524": { + "id": "031524", + "content": "已知 $a, b, c$ 都是实数, 若函数 $f(x)=\\begin{cases}x^2,& x \\leq a,\\\\\\dfrac{1}{x}+b,& a=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$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)!0.5!(C)$) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$M$} coordinate (M);\n\\draw ($(A_1)!0.9!(B_1)$) node [above] {$N$} coordinate (N);\n\\draw [dashed] (A)--(M)(N)--(O);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{平行}{相交}{异面垂直}{异面不垂直}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031528": { + "id": "031528", + "content": "设 $f(x)$ 是定义在正整数集上的函数, 且 $f(x)$ 满足:``当 $f(k) \\leq k^2$ 成立时, 总可推出 $f(k+1) \\leq(k+1)^2$ 成立''. 那么, 下列命题总成立的是\\bracket{20}.\n\\onech{若 $f(2) \\leq 4$ 成立, 则当 $k \\geq 1$ 时,均有 $f(k) \\leq k^2$ 成立}{若 $f(4) \\leq 16$ 成立, 则当 $k \\leq 4$ 时, 均有 $f(k) \\leq k^2$ 成立}{若 $f(6)>36$ 成立, 则当 $k \\geq 7$ 时, 均有 $f(k)>k^2$ 成立}{若 $f(7)=50$ 成立, 则当 $k \\leq 7$ 时,均有 $f(k)>k^2$ 成立}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031529": { + "id": "031529", + "content": "设等差数列 $\\{a_n\\}$ 的公差 $d$ 不为 $0$, $a_1=9 d$. 若 $a_k$ 是 $a_1$ 与 $a_{2 k}$ 的等比中项, 则 $k=\\bracket{20}$\\bracket{20}.\n\\fourch{$2$}{$4$}{$6$}{$8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031530": { + "id": "031530", + "content": "已知函数 $y=\\sqrt{\\dfrac{1+x}{1-x}}+\\lg (3-4 x+x^2)$ 的定义域为 $M$.\\\\\n(1) 求 $M$;\\\\\n(2) 当 $x \\in M$ 时, 求 $f(x)=a \\cdot 2^{x+2}+3 \\times 4^x$($a>-3$) 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031531": { + "id": "031531", + "content": "倾斜角为 $\\alpha$ 的直线经过抛物线 $y^2=8 x$ 的焦点 $F$, 且与抛物线交于 $A$、$B$ 两点.\\\\\n(1) 求抛物线的焦点 $F$ 的坐标及准线 $l$ 的方程;\\\\\n(2) 若 $\\alpha$ 为锐角, 作线段 $AB$ 的垂直平分线 $m$ 交 $x$ 轴于点 $P$, 证明: $|FP|-|FP| \\cos 2 \\alpha$ 为定值, 并求此定值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031532": { + "id": "031532", + "content": "已知 $\\mathrm{i}$ 是虚数单位, $z=\\dfrac{2-\\mathrm{i}}{\\mathrm{i}}$, 则 $|\\overline{z}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031533": { + "id": "031533", + "content": "已知集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | \\log _x 4=2\\}$, 则 $A \\cup B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031534": { + "id": "031534", + "content": "已知角 $\\varphi$ 的终边经过点 $P(1,-2)$, 函数 $f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$) 图像的相邻两条对称轴之间的距离等于 $\\dfrac{\\pi}{3}$, 则 $f(\\dfrac{\\pi}{12})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031535": { + "id": "031535", + "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031536": { + "id": "031536", + "content": "若函数 $y=x^3-2 m x^2+m^2 x$ 在 $x=1$ 处取得极小值, 则实数 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031537": { + "id": "031537", + "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031538": { + "id": "031538", + "content": "已知 $\\tan \\alpha=2$, 则 $\\dfrac{\\sin (\\pi+\\alpha)-\\sin (\\dfrac{\\pi}{2}+\\alpha)}{\\cos (\\dfrac{3 \\pi}{2}+\\alpha)+\\cos (\\pi-\\alpha)}$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031539": { + "id": "031539", + "content": "若直线 $(a^2+2 a) x-y+1=0$ 的倾斜角为钝角, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031540": { + "id": "031540", + "content": "在 $\\triangle ABC$ 中, 已知内角 $A=\\dfrac{\\pi}{3}$, 边 $BC=2 \\sqrt{3}$, 则 $\\triangle ABC$ 的面积 $S$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031541": { + "id": "031541", + "content": "某市高三数学抽样考试中, 对 90 分及其以上的成绩情况进行统计, 其频率分布直方图如图所示, 若 $(130,140]$ 分数段的人数为 90 人, 则 $(90,100]$ 分数段的人数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 70]\n\\draw [->] (80,0) -- (82,0) -- (83,0.0025) -- (85,-0.0025)-- (86,0) -- (150,0) node [below] {分数};\n\\draw [->] (80,0) -- (80,0.055) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (80,0) node [below left] {$O$};\n\\foreach \\i/\\j in {90/0.045,100/0.025,110/0.015,120/0.010,130/0.005}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {90/0.045,100/0.025,110/0.015,120/0.010,130/0.005}\n{\\draw [dashed] (\\i,\\j) -- (80,\\j) node [left] {$\\k$};};\n\\draw (140,0) node [below] {$140$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031542": { + "id": "031542", + "content": "已知点 $O$ 是 $\\triangle ABC$ 的外接圆圆心, 且 $AB=3$, $AC=4$, 若存在非零实数 $x, y$, 使得 $\\overrightarrow{AO}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 且 $x+y=1$, 则 $\\cos \\angle BAC=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031543": { + "id": "031543", + "content": "已知正方体 $C_1$ 的棱长为 $18 \\sqrt{2}$, 以 $C_1$ 各个面的中心为顶点的凸多面体为 $C_2$, 以 $C_2$ 各个面的中心为顶点的凸多面体为 $C_3$, 以 $C_3$ 各个面的中心为顶点的凸多面体为 $C_4$, 依此类推. 记凸多面体 $C_n$ 的棱长为 $a_n$, 则 $a_6=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031544": { + "id": "031544", + "content": "设 $a, b$ 是平面 $\\alpha$ 内两条不同的直线, $l$ 是平面 $\\alpha$ 外的一条直线,则``$l \\perp a$, $l \\perp b$''是``$l \\perp \\alpha$''的\\bracket{20}.\n\\twoch{充要条件}{充分而不必要的条件}{必要而不充分的条件}{既不充分也不必要的条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031545": { + "id": "031545", + "content": "函数 $f(x)=\\ln (x+1)-\\dfrac{2}{x}$ 的零点所在的区间是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{2}, 1)$}{$(1, \\mathrm{e}-1)$}{$(\\mathrm{e}-1,2)$}{$(2, \\mathrm{e})$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031546": { + "id": "031546", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031547": { + "id": "031547", + "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031548": { + "id": "031548", + "content": "已知 $\\alpha$ 为锐角, 且 $\\tan \\alpha=\\sqrt{2}-1$, 函数 $f(x)=2 x \\tan 2 \\alpha+\\sin (2 \\alpha+\\dfrac{\\pi}{4})$, 数列 $\\{a_n\\}$ 的首项 $a_1=1$, $a_{n+1}=f(a_n)$.\\\\\n(1) 求函数 $f(x)$ 的表达式;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031549": { + "id": "031549", + "content": "如图所示, 椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的一个焦点为 $F(1,0)$, 且过点 $(\\sqrt{2}, \\dfrac{\\sqrt{6}}{2})$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (4,-2.5) -- (4,2.5);\n\\draw (4,0) node [above right] {$N$} coordinate (N);\n\\draw (80:2 and {sqrt(3)}) node [above right] {$A$} coordinate (A);\n\\draw (-80:2 and {sqrt(3)}) node [below right] {$B$} coordinate (B);\n\\draw (A)--(N) ($(A)!-0.2!(B)$) -- ($(B)!-0.2!(A)$);\n\\draw [name path = BN] (B)--(N);\n\\draw [name intersections = {of = BN and elli, by = {T,M}}];\n\\draw (A)--(M) node [below] {$M$};\n\\filldraw (1,0) node [above right] {$F$} coordinate (F) circle (0.05);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆 $C$ 的方程;\\\\\n(2) 已知 $A, B$ 为椭圆上的点, 且直线 $AB$ 垂直于 $x$ 轴, 直线 $l$ : $x=4$ 与 $x$ 轴交于点 $N$, 直线 $AF$ 与 $BN$ 交于点 $M$.\\\\\n(I) 求证: 点 $M$ 恒在椭圆 $C$ 上;\\\\\n(II) 求 $\\triangle AMN$ 面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031550": { + "id": "031550", + "content": "已知集合 $A=\\{x|| x |<2\\}$, $B=\\{x | \\dfrac{1}{x+1}>0\\}$, 则 $A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031551": { + "id": "031551", + "content": "若函数 $f(x)=\\begin{cases}x^2+1,& x \\leq 1,\\\\\\lg x,& x>1,\\end{cases}$ 则 $f[f(10)]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031552": { + "id": "031552", + "content": "已知长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=BC=4$, $CC_1=2$, 则直线 $BC_1$ 和平面 $DBB_1D_1$ 所成角的正弦值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031553": { + "id": "031553", + "content": "若函数 $f(x)=\\sin \\dfrac{x+\\varphi}{3}$($\\varphi \\in[0,2 \\pi]$) 是偶函数, 则 $\\varphi=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031554": { + "id": "031554", + "content": "已知严格增的等差数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_3=a_2^2-4$, 则 $a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031555": { + "id": "031555", + "content": "若 $\\dfrac{\\sin \\alpha+\\cos \\alpha}{\\sin \\alpha-\\cos \\alpha}=\\dfrac{1}{2}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031556": { + "id": "031556", + "content": "不等式 $\\dfrac{1}{x-1}\\geq \\dfrac{1}{x^2-1}$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031557": { + "id": "031557", + "content": "若椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 上一点 $P$ 到焦点 $F_1$ 的距离为 6 , 则点 $P$ 到另一个焦点 $F_2$ 的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031558": { + "id": "031558", + "content": "6 个人站成前后二排, 每排三人, 其中甲不站前排, 乙不站在后排的站法种数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031559": { + "id": "031559", + "content": "已知向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 夹角为 $45^{\\circ}$, 且 $|\\overrightarrow{a}|=1$, $|2 \\overrightarrow{a}-\\overrightarrow{b}|=\\sqrt{10}$; 则 $|\\overrightarrow{b}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031560": { + "id": "031560", + "content": "若函数 $f(x)=\\begin{cases}x^2-2,& x \\leq 1,\\\\\\lg |x-m|,& x>1\\end{cases}$ 在区间 $[0,+\\infty)$ 上是严格增函数, 则实数 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031561": { + "id": "031561", + "content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$, $E, F$ 分别为线段 $AA_1, B_1C$ 上的点, 则三棱锥 $D_1-EDF$ 的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031562": { + "id": "031562", + "content": "设 $a>0$ 且 $a \\neq 1$, 则``函数 $f(x)=a^x$ 在 $\\mathbf{R}$ 上是严格减函数'', 是``函数 $g(x)=(2-a) x^3$ 在 $\\mathbf{R}$ 上是严格增函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031563": { + "id": "031563", + "content": "对于二项式 $(\\dfrac{1}{x}+x^3)^n, n$ 为正整数, 四位同学作出了四种判断: \\textcircled{1} 存在正整数 $n$, 使展开式中有常数项; \\textcircled{2} 对任意正整数 $n$, 展开式中没有常数项; \\textcircled{3} 对任意正整数 $n$, 展开式中没有 $x$ 的一次项; \\textcircled{4} 存在正整数 $n$, 展开式中有 $x$ 的一次项. 上述判断中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}与\\textcircled{3}}{\\textcircled{2}与\\textcircled{3}}{\\textcircled{2}与\\textcircled{4}}{\\textcircled{4}与\\textcircled{1}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031564": { + "id": "031564", + "content": "已知 $\\triangle ABC$ 是斜三角形, 则``$A>B$''是``$|\\tan A|>|\\tan B|$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031565": { + "id": "031565", + "content": "设 $F_1F_2$ 是椭圆 $E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点, $P$ 为直线 $x=\\dfrac{3 a}{2}$ 上一点, $\\triangle F_2PF_1$ 是底角为 $30^{\\circ}$ 的等腰三角形, 则 $E$ 的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}{$\\dfrac{3}{4}$}{$\\dfrac{4}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031566": { + "id": "031566", + "content": "如图, 在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $O$ 是 $AC, BD$ 的交点, $E, F$ 分别是 $AB$ 与 $AD$ 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 [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 ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw (A_1)--(C_1);\n\\draw [dashed] (E)--(F)(B)--(D)(A)--(C)(D_1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线 $OD_1$ 与直线 $A_1C_1$ 垂直;\\\\\n(2) 求异面直线 $EF$ 与 $A_1C_1$ 所成角的大小;\\\\\n(3) 求二面角 $B-AC-D_1$ 的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031567": { + "id": "031567", + "content": "定义向量 $\\overrightarrow{OM}=(a, b)$ 的``相伴函数''为 $f(x)=a \\sin x+b \\cos x$; 函数 $f(x)=a \\sin x+ b \\cos x$ 的``相伴向量''为 $\\overrightarrow{OM}=(a, b)$ (其中 $O$ 为坐标原点). 记平面内所有向量的``相伴函数''构成的集合为 $S$.\\\\\n(1) 设 $g(x)=3 \\sin (x+\\dfrac{\\pi}{2})+4 \\sin x$, 求证: $g(x) \\in S$;\\\\\n(2) 已知 $h(x)=\\cos (x+\\alpha)+2 \\cos x$, 且 $h(x) \\in S$, 求其``相伴向量''的模;\\\\\n(3) 已知 $M(a, b)$($b \\neq 0$) 为圆 $C:(x-2)^2+y^2=1$ 上一点, 向量 $\\overrightarrow{OM}$ 的``相伴函数''$f(x)$ 在 $x=x_0$ 处取得最大值. 当点 $M$ 在圆 $C$ 上运动时, 求 $\\tan 2 x_0$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031568": { + "id": "031568", + "content": "已知 $\\{a_n\\}$ 为等比数列, $a_4+a_7=2$, $a_5 a_6=-8$, 则 $a_1+a_{10}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031569": { + "id": "031569", + "content": "若 $\\sin (\\pi+\\theta)=\\dfrac{1}{3}$, 且 $\\theta$ 为第三象限角, 则 $\\cos (2 \\pi-\\theta)$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031570": { + "id": "031570", + "content": "已知函数 $f(x)=(a^2-1) x^2+2 x+a+1$ 是奇函数, 则实数 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031571": { + "id": "031571", + "content": "设 $a$ 是实数, $f(x)=(x^2-4)(x-a)$, 若 $f'(-1)=0$, 则函数 $y=f(x)$ 在区间 $[-2,2]$ 上的最大值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031572": { + "id": "031572", + "content": "已知向量 $\\overrightarrow{OA}=(3,-2)$, $\\overrightarrow{OB}=(-5,-1)$, 则向量 $\\dfrac{1}{2}\\overrightarrow{AB}$ 的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031573": { + "id": "031573", + "content": "若 $00\\end{cases}$ ($a$ 是常数且 $a>0$), 对于下列命题: \\textcircled{1} 函数 $f(x)$ 的最小值是 -1 ; \\textcircled{2} 函数 $f(x)$ 在 $\\mathbf{R}$ 上存在反函数; \\textcircled{3} 对任意 $x_1<0$, $x_2<0$ 且 $x_1 \\neq x_2$, 恒有 $f(\\dfrac{x_1+x_2}{2})<\\dfrac{f(x_1)+f(x_2)}{2}$. 其中正确命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031580": { + "id": "031580", + "content": "在 $\\triangle ABC$ 中, 若 $\\sin ^2A+\\sin ^2B<\\sin ^2C$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{钝角三角形}{直角三角形}{锐角三角形}{不能确定}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031581": { + "id": "031581", + "content": "$a=0$ 是直线 $a x+(2 a-1) y+1=0$ 和直线 $3 x+a y+3=0$ 垂直的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分条件}{既不充分又不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031582": { + "id": "031582", + "content": "我国古代数学名著《算法统宗》中有如下问题:``远望巍巍塔七层, 红光点点倍加增, 共灯三百八十一, 请问尖头几盏灯?''意思是 : 一座 $7$ 层塔共挂了 $381$ 盙灯, 且相邻两层中的下一层灯数是上一层灯数的 $2$ 倍, 则塔的顶层共有灯\\bracket{20}.\n\\fourch{$1$盏}{$3$盏}{$5$盏}{$9$盏}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031583": { + "id": "031583", + "content": "设函数 $f(x)=\\sin (x-\\dfrac{\\pi}{6})$, 若对于任意 $\\alpha \\in[-\\dfrac{5 \\pi}{6},-\\dfrac{\\pi}{2}]$, 在区间 $[0, m]$ 上总存在唯一确定的 $\\beta$, 使得 $f(\\alpha)+f(\\beta)=0$, 则 $m$ 的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{7 \\pi}{6}$}{$\\pi$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031584": { + "id": "031584", + "content": "某热力公司每年燃料费约 24 万元. 为了``环评''达标, 需要安装一块面积为 $x$($x \\geq 0$) (单位: 平方米) 可用 15 年的太阳能板, 其工本费为 $\\dfrac{x}{2}$ (单位: 万元), 并与燃料供热互补工作, 从此, 公司每年的燃料费为 $\\dfrac{k}{20 x+100}$ ($k$ 为常数) 万元. 记 $y$ 为该公司安装太阳能板的费用与 15 年的燃料费之和.\\\\\n(1) 求 $k$ 的值, 并建立 $y$ 关于 $x$ 的函数关系式;\\\\\n(2) 求 $y$ 的最小值, 并求出此时所安装太阳能板的面积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031585": { + "id": "031585", + "content": "设向量 $\\overrightarrow{a}=(x, 2)$, $\\overrightarrow{b}=(x+n, 2 x-1)(n$ 是正整数 $)$, 函数 $y=\\overrightarrow{a}\\cdot \\overrightarrow{b}$ 在 $x \\in[0,1]$ 上的最小值与最大值的和为 $a_n$, 又数列 $\\{b_n\\}$ 满足 $b_1=1$, $b_1+b_2+\\cdots+b_n=(\\dfrac{9}{10})^{n-1}$.\\\\\n(1) 求证: $a_n=n+1$;\\\\\n(2) 求数列 $\\{b_n\\}$ 的通项公式;\\\\\n(3) 设 $c_n=-a_n \\cdot b_n$, 试问数列 $\\{c_n\\}$ 中, 是否存在正整数 $k$, 使得对于任意的正整数 $n$, 都有 $c_n \\leq c_k$ 成立? 若存在, 求出所有满足条件的 $k$ 的值; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031586": { + "id": "031586", + "content": "函数 $y=\\cos 2 x$, $x \\in[0, \\pi]$ 的递增区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031587": { + "id": "031587", + "content": "已知函数 $f(x)=\\lg \\dfrac{1-x}{1+x}$, 若 $f(a)=b$, 则 $f(-a)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031588": { + "id": "031588", + "content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, 对角线 $AC_1$ 的长为 $l$, $\\angle DAC_1=45^{\\circ}$, $\\angle A_1AC_1=60^{\\circ}$, 则三棱锥 $C-B_1C_1D_1$ 的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031589": { + "id": "031589", + "content": "函数 $y=x^3-3 x^2$ 的严格递减区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031590": { + "id": "031590", + "content": "已知定义在 $\\mathbf{R}$ 上的奇函数, $f(x)$ 满足 $f(x+2)=-f(x)$, 则 $f(-6)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031591": { + "id": "031591", + "content": "椭圆 $\\dfrac{x^2}{4}+y^2=1$ 的两个焦点为 $F_1, F_2$, 过 $F_1$ 作垂直于 $x$ 轴的直线与椭圆相交, 一个交点为 $P$, 则 $|\\overrightarrow{PF_2}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031592": { + "id": "031592", + "content": "已知函数 $f(x)=(\\sin x-\\cos x) \\sin x$, $x \\in \\mathbf{R}$, 则 $f(x)$ 的最小正周期是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031593": { + "id": "031593", + "content": "直角坐标平面上三点 $A(1,2)$、$B(3,-2)$、$C(9,7)$, 若 $E$、$F$ 为线段 $BC$ 的三等分点, 则 $\\overrightarrow{AE}\\cdot \\overrightarrow{AF}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031594": { + "id": "031594", + "content": "方程 $|x+1|+|x-2|=3$ 的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031595": { + "id": "031595", + "content": "设直线 $l$ 过点 $(-2,0)$, 且与圆 $x^2+y^2=1$ 相切, 则 $l$ 的斜率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031596": { + "id": "031596", + "content": "在圆柱 $O_1O_2$ 内有一个球 $O$, 该球与圆柱的上、下底面及母线均相切. 记圆柱 $O_1O_2$ 的体积为 $V_1$, 球 $O$ 的体积为 $V_2$, 则 $\\dfrac{V_1}{V_2}$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031597": { + "id": "031597", + "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两个焦点为 $F_1, F_2$, 若 $P$ 为其上一点, 且 $|PF_1|= 2|PF_2|$, 则双曲线离心率的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031598": { + "id": "031598", + "content": "若 $\\lg a+\\lg b=0$ (其中 $a \\neq 1$, $b \\neq 1$), 则函数 $f(x)=a^x$ 与 $g(x)=b^x$ 的图像\\bracket{20}.\n\\fourch{关于直线 $y=x$ 对称}{关于 $x$ 轴对称}{关于 $y$ 轴对称}{关于原点对称}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031599": { + "id": "031599", + "content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, $M, N$ 分别为棱 $A_1B_1$ 和 $BB_1$ 的中点, 那么异面直线 $AM$ 和 $CN$ 所成角的余弦值是\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{10}}{2}$}{$\\dfrac{2}{5}$}{$-\\dfrac{2}{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031600": { + "id": "031600", + "content": "某一批花生种子, 如果每 $1$ 粒发芽的概率为 $\\dfrac{4}{5}$, 那么种下 $4$ 粒种子恰有 $2$ 粒发芽的概率是\\bracket{20}.\n\\fourch{$\\dfrac{16}{625}$}{$\\dfrac{96}{625}$}{$\\dfrac{192}{625}$}{$\\dfrac{256}{625}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031601": { + "id": "031601", + "content": "有一位同学写了这样一个不等式: $\\dfrac{x^2+1+c}{\\sqrt{x^2+c}}\\geq \\dfrac{1+c}{\\sqrt{c}}$($x \\in \\mathbf{R}$), 他发现, 当 $c=1,2,3$ 时, 不等式对一切实数 $x$ 都成立, 由此他作出如下猜测: \\textcircled{1} 当 $c$ 为所有正整数时, 不等式对一切实数 $x$ 都成立; \\textcircled{2} 只存在有限个正整数 $c$, 对 $x \\in \\mathbf{R}$ 不等式都成立; \\textcircled{3} 当 $c \\geq 1$ 时, 不等式对一切 $x \\in \\mathbf{R}$ 都成立; \\textcircled{4} 当 $c>0$ 时, 不等式对一切 $x \\in \\mathbf{R}$ 都成立. 则正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{3}}{\\textcircled{2}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{4}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031602": { + "id": "031602", + "content": "已知函数 $f(x)=|2 x-a|+a$.\\\\\n(1) 若不等式 $f(x)<6$ 的解集为 $(-1,3)$, 求 $a$ 的值;\\\\\n(2) 在 (1) 的条件下, 若存在 $x_0 \\in \\mathbf{R}$, 使 $f(x_0) \\leq t-f(-x_0)$, 求 $t$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031603": { + "id": "031603", + "content": "已知双曲线 $C$ 的对称轴是坐标轴, 一条渐近线的方程是 $x-\\sqrt{2}y=0$, 且该双曲线 $C$ 经过定点 $M(3,2)$.\\\\\n(1) 求双曲线 $C$ 的方程;\\\\\n(2) 直线 $l: a x-y-1=0$ 与双曲线 $C$ 交于 $P, Q$ 两点, 当实数 $a$ 为何值时, $|PQ|= 2 \\sqrt{1+a^2}$?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练11试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031604": { + "id": "031604", + "content": "已知 $\\mathrm{i}$ 是虚数单位, $\\dfrac{2-\\mathrm{i}}{z+\\mathrm{i}}=\\mathrm{i}$, 则 $|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031605": { + "id": "031605", + "content": "已知 $\\triangle ABC$ 的周长为 $4$, 且 $\\sin A+\\sin B=3 \\sin C$, 则 $AB$ 边的长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031606": { + "id": "031606", + "content": "已知等比数列 $\\{a_n\\}$ 中 $a_2=1$, 则其前 3 项的和 $S_3$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031607": { + "id": "031607", + "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031608": { + "id": "031608", + "content": "定义集合运算: $A \\ast B=\\{z | z=x y,\\ x \\in A,\\ y \\in B\\}$. 设 $A=\\{1,2\\}$, $B=\\{0,2\\}$, 则集合 $A \\ast B$ 的所有元素之和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031609": { + "id": "031609", + "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031610": { + "id": "031610", + "content": "已知函数 $f(x)=\\begin{cases}x+2,& x \\leq 0,\\\\-x+2,& x>0,\\end{cases}$ 则不等式 $f(x) \\geq x^2$ 的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031611": { + "id": "031611", + "content": "已知 $f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$), $f(\\dfrac{\\pi}{6})=f(\\dfrac{\\pi}{3})$, 且 $f(x)$ 在区间 $(\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3})$ 有最小值, 无最大值, 则 $\\omega=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031612": { + "id": "031612", + "content": "一支田径队有男女运动员 $98$ 人, 其中男运动员有 $56$ 人. 按男女比例用分层抽样的方法, 从全体运动员中抽出一个容量为 $28$ 的样本, 那么应抽取女运动员人数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031613": { + "id": "031613", + "content": "用与球心距离为 $1$ 的平面去截球,所得的截面面积为 $\\pi$, 则球的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031614": { + "id": "031614", + "content": "已知点 $O$ 是 $\\triangle ABC$ 的外接圆圆心, 且 $AB=3$, $AC=4$, 若存在非零实数 $x, y$, 使得 $\\overrightarrow{AO}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 且 $x+y=1$, 则 $\\cos \\angle BAC=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031615": { + "id": "031615", + "content": "已知无穷等比数列 $a_1, a_2, a_3, \\cdots$ 各项的和为 $\\dfrac{9}{2}$, 且 $a_2=-2$, 若 $|S_n-\\dfrac{9}{2}|<10^{-4}$, 则 $n$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031616": { + "id": "031616", + "content": "函数 $y=\\sqrt{x(x-1)}+\\sqrt{x}$ 的定义域为\\bracket{20}.\n\\fourch{$\\{x | x \\geq 0\\}$}{$\\{x | x \\geq 1\\}$}{$\\{x | x \\geq 1\\}\\cup\\{0\\}$}{$\\{x | 0 \\leq x \\leq 1\\}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031617": { + "id": "031617", + "content": "若 $f(x)=-\\dfrac{1}{2}x^2+b \\ln (x+2)$ 在 ($-1,+\\infty$) 上是减函数, 则 $b$ 的取值范围是\\bracket{20}.\n\\fourch{$[-1,+\\infty)$}{($-1,+\\infty$)}{$(-\\infty,-1]$}{($-\\infty,-1$)}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031618": { + "id": "031618", + "content": "过直线 $y=x$ 上的一点作圆 $(x-5)^2+(y-1)^2=2$ 的两条切线 $l_1, l_2$, 当直线 $l_1, l_2$ 关于 $y=x$ 对称时, 它们之间的夹角为\\bracket{20}.\n\\fourch{$30^{\\circ}$}{$45^{\\circ}$}{$60^{\\circ}$}{$90^{\\circ}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031619": { + "id": "031619", + "content": "设 $P$ 为曲线 $C: y=x^2+2 x+3$ 上的点, 且曲线 $C$ 在点 $P$ 处切线倾斜角的取值范围为 $[0, \\dfrac{\\pi}{4}]$, 则点 $P$ 横坐标的取值范围为\\bracket{20}.\n\\fourch{$[-1,-\\dfrac{1}{2}]$}{$[-1,0]$}{$[0,1]$}{$[\\dfrac{1}{2}, 1]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031620": { + "id": "031620", + "content": "已知 $\\alpha$ 为锐角, 且 $\\tan \\alpha=\\sqrt{2}-1$, 函数 $f(x)=2 x \\tan 2 \\alpha+\\sin (2 \\alpha+\\dfrac{\\pi}{4})$, 数列 $\\{a_n\\}$ 的首项 $a_1=1$, $a_{n+1}=f(a_n)$.\\\\\n(1) 求函数 $f(x)$ 的表达式;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031621": { + "id": "031621", + "content": "如图, 正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, 底面边长为 $2, BC_1$ 与底面 $ABCD$ 所成角的大小为 $\\arctan 2, M$ 是 $DD_1$ 的中点, $N$ 是 $BD$ 上的一动点, 设 $\\overrightarrow{DN}=\\lambda \\overrightarrow{DB}$($0<\\lambda<1$).\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(B)!0.6!(D)$) node [below] {$N$} coordinate (N);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$M$} coordinate (M);\n\\draw (B)--(C_1);\n\\draw [dashed] (B)--(D)(M)--(N)(A)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当 $\\lambda=\\dfrac{1}{2}$ 时,证明: $MN$ 与平面 $ABC_1D_1$ 平行;\\\\\n(2) 若点 $N$ 到平面 $BCM$ 的距离为 $d$, 试用 $\\lambda$ 表示 $d$, 并求出 $d$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练12试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031622": { + "id": "031622", + "content": "设集合 $A=\\{x |-2 \\leq x \\leq 2\\}$, $\\mathbf{Z}$ 为整数集, 则集合 $A \\cap \\mathbf{Z}$ 中含有 $2$ 的子集个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031623": { + "id": "031623", + "content": "若一个球的体积为 $4 \\sqrt{3}\\pi$, 则它的表面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031624": { + "id": "031624", + "content": "已知直线 $l$ 的参数方程为 $\\begin{cases}x=1+3 t\\\\y=2+4 t\\end{cases}$ ($t$ 为参数), 则点 ( 1,0$)$ 到直线 $l$ 的距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031625": { + "id": "031625", + "content": "若复数 $\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}}+\\dfrac{1}{2}b$($b \\in \\mathbf{R}$) 的实部的模与虚部相等, 则 $b$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031626": { + "id": "031626", + "content": "若 $(x-2)^5=a_5 x^5+a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$, 则 $a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031627": { + "id": "031627", + "content": "已知直线 $y=k x$ 与圆 $(x-5)^2+y^2=9$ 有两个公共点, 则实数 $k$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031628": { + "id": "031628", + "content": "若 $\\tan \\theta=\\dfrac{1}{3}$, 则 $\\cos 2 \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031629": { + "id": "031629", + "content": "若一个圆锥的母线与轴的夹角为 $\\arcsin \\dfrac{1}{3}$, 则该圆锥的侧面积是底面积的\\blank{50}倍.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031630": { + "id": "031630", + "content": "从集合 $\\{0,1,2,3\\}$ 的所有非空子集中, 等可能地取出一个. 则取出的非空子集中所有元素之和恰为 5 的概率 $=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031631": { + "id": "031631", + "content": "在某段时间内, 甲地不下雨的概率为 $p_1$($0=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (2,0,2) node [right] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw ($(B)!0.5!(D)$) node [below] {$N$} coordinate (N);\n\\draw (O) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw (E)--(D)(E)--(C)(E)--(O)(C)--(D)--(A)--(B)--cycle;\n\\draw (E)--(N)(M)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$BM=EN$, 且直线 $BM$、$EN$ 是相交直线}{$BM \\neq EN$, 且直线 $BM, EN$ 是相交直线}{$BM=EN$, 且直线 $BM$、$EN$ 是异面直线}{$BM \\neq EN$, 且直线 $BM, EN$ 是异面直线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031636": { + "id": "031636", + "content": "函数 $f(x)=x \\cos 2 x$ 在区间 $[0,2 \\pi]$ 上的零点个数为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031637": { + "id": "031637", + "content": "2019 年 1 月 3 日嫦娥四号探测器成功实现人类历史上首次月球背面软着陆, 是我国航天事业取得的重大成就, 实现月球背面软着陆需要解决的一个关键技术问题是地面与探测器的通讯联系. 为解决这个问题, 发射了嫦娥四号中继星``鹊桥'', 鹊桥沿着围绕地月拉格朗日 $L_2$ 点的轨道运行. $L_2$ 点是平衡点, 位于地月连线的延长线上. 设地球质量为 $M_1$, 月球质量为 $M_2$, 地月距离为 $R, L_2$ 点到月球的距离为 $r$, 根据牛顿运动定律和万有引力定律, $r$ 满足方程: $\\dfrac{M_1}{(R+r)^2}+\\dfrac{M_2}{r^2}=(R+r) \\dfrac{M_1}{R^3}$. 设 $\\alpha=\\dfrac{r}{R}$, 由于 $\\alpha$ 的值很小, 因此在近似计算中 $\\dfrac{3 \\alpha^3+3 \\alpha^4+\\alpha^5}{(1+\\alpha)^2}\\approx 3 \\alpha^3$, 则 $r$ 的近似值为\\bracket{20}.\n\\fourch{$\\sqrt{\\dfrac{M_2}{M_1}}R$}{$\\sqrt{\\dfrac{M_2}{2M_1}}R$}{$\\sqrt[3]{\\dfrac{3M_2}{M_1}}R$}{$\\sqrt[3]{\\dfrac{M_2}{3M_1}}R$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031638": { + "id": "031638", + "content": "如图, 已知点 $P$ 在圆柱 $OO_1$ 的底面圆 $O$ 上, $\\angle AOP=120^{\\circ}$, 圆 $O$ 的直径 $AB=4$, 圆柱的高 $OO_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\filldraw (0,0) node [above] {$O$} coordinate (O) circle (0.03);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\filldraw (O) ++ (0,3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O) ++ (-50:2 and 0.5) node [below] {$P$} coordinate (P);\n\\draw (A)--(A_1)--(B_1)--(B)arc (0:-180:2 and 0.5);\n\\draw (O_1) ellipse (2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw [dashed] (A)--(B)--(P)--cycle(A_1)--(P)(A_1)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的表面积和三棱锥 $A_1-APB$ 的体积;\\\\\n(2) 求点 $A$ 到平面 $A_1PO$ 的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031639": { + "id": "031639", + "content": "已知函数 $f(x)=2 x^3-a x^2+b$.\\\\\n(1) 讨论 $f(x)$ 的单调性;\\\\\n(2) 是否存在 $a, b$, 使得 $f(x)$ 在区间 $[0,1]$ 的最小值为 $-1$ 且最大值为 $1$? 若存在, 求出 $a, b$ 的所有值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练13试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031640": { + "id": "031640", + "content": "已知 $\\tan (\\alpha-\\dfrac{5 \\pi}{4})=\\dfrac{1}{5}$, 则 $\\tan \\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031641": { + "id": "031641", + "content": "曲线 $y=2 \\ln x$ 在点 $(1,0)$ 处的切线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031642": { + "id": "031642", + "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{3}=1$($a>0$) 的一条渐近线被圆 $(x-2)^2+y^2=4$ 所截得的弦长为 $2$ , 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031643": { + "id": "031643", + "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为棱 $CC_1$ 的中点, 则异面直线 $AE$ 与 $CD$ 所成角的正切值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031644": { + "id": "031644", + "content": "已知 $a, b \\in \\mathbf{R}$, 且 $a-3 b+6=0$, 则 $2^a+\\dfrac{1}{8^b}$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031645": { + "id": "031645", + "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的离心率为 $\\sqrt{3}$, 则其渐近线方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031646": { + "id": "031646", + "content": "不等式 $\\dfrac{1}{|x-1|}<\\dfrac{1}{2}$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031647": { + "id": "031647", + "content": "在直角坐标系 $xOy$ 中, 曲线 $C$ 的参数方程为 $\\begin{cases}x=2 \\cos \\theta,\\\\y=4 \\sin \\theta\\end{cases}$ ($\\theta$ 为参数), 直线 $l$ 的参数方程为 $\\begin{cases}x=1+t \\cos \\alpha,\\\\y=2+t \\sin \\alpha\\end{cases}$ ($t$ 为参数). 若曲线 $C$ 截直线 $l$ 所得线段的中点坐标为 $(1,2)$, 则直线 $l$ 的斜率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031648": { + "id": "031648", + "content": "已知圆锥的顶点为 $S$, 母线 $SA, SB$ 互相垂直, $SA$ 与圆锥底面所成角为 $30^{\\circ}$, 若 $\\triangle SAB$ 的面积为 $8$ , 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031649": { + "id": "031649", + "content": "某地每年消耗木材约 $20$ 万 $\\mathrm{m}^3$, 每 $\\mathrm{m}^3$ 价格为 $480$ 元, 为了减少木材消耗, 决定按 $t \\%$ 征收木材税, 这样每年的木材消耗量减少 $\\dfrac{5}{2}t$ 万 $\\mathrm{m}^3$, 为了既减少木材消耗又保证税金收人每年不少于 $180$ 万元, 则 $t$ 的范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031650": { + "id": "031650", + "content": "设某种动物活到 20 岁的概率为 $0.8$ , 活到 25 岁的概率为 $0.4$ . 现有一只 20 岁的该种动物, 它活到 25 岁的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031651": { + "id": "031651", + "content": "设 $a>1$, 若对于任意的 $x \\in[a, 2 a]$, 都有 $y \\in[a, a^2]$ 满足方程 $\\log _a x+\\log _a y=3$, 这时 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031652": { + "id": "031652", + "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_3+a_8+a_{13}=C$, 则其前 $n$ 项和 $S_n$ 的值等于 $5C$ 的是\\bracket{20}.\n\\fourch{$S_{15}$}{$S_{17}$}{$S_7$}{$S_8$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031653": { + "id": "031653", + "content": "设 $x \\in \\mathbf{R}$, 则``$x^3>8$''是``$|x|>2$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031654": { + "id": "031654", + "content": "将函数 $y=\\sin (2 x-\\dfrac{\\pi}{3})$ 图像上的点 $P(\\dfrac{\\pi}{4}, t)$ 向左平移 $s$($s>0$) 个单位长度得到点 $P'$. 若 $P'$ 位于函数 $y=\\sin 2 x$ 的图像上, 则\\bracket{20}.\n\\twoch{$t=\\dfrac{1}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{6}$}{$t=\\dfrac{\\sqrt{3}}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{6}$}{$t=\\dfrac{1}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{3}$}{$t=\\dfrac{\\sqrt{3}}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031655": { + "id": "031655", + "content": "计算机是将信息转换成二进制进行处理的, 所谓二进制即``逢二进一'', 如 $(1101)_2$ 表示二进制的数, 将它转换成十进制数的形式是 $1 \\times 2^3+1 \\times 2^2+0 \\times 2^1+1 \\times 2^0=13$, 那么将二进制数 $(\\underbrace{11 \\cdots 1}_{16 \\text{位}})_2$ 转换成十进制数是\\bracket{20}.\n\\fourch{$2^{17}-2$}{$2^{16}-1$}{$2^{16}-2$}{$2^{15}-1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031656": { + "id": "031656", + "content": "如图, 已知圆 $A$ 的半径是 $2$ , 圆外一定点 $N$ 与圆 $A$ 上的点的最短距离为 $6$ , 过动点 $P$ 作 $A$ 的切线 $PM$ ($M$ 为切点), 连结 $PN$ 使得 $PM: PN=\\sqrt{2}$, 试建立适当的坐标系, 求动点 $P$ 的轨迹.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale=0.3]\n\\filldraw (0,0) circle (0.05) node [below] {$A$} coordinate (A);\n\\filldraw ({(3-sqrt(354))/15},{(sqrt(6)+3*sqrt(59))/15}) circle (0.05) node [above] {$M$} coordinate (M);\n\\filldraw (6,{sqrt(24)}) node [above] {$P$} coordinate (P) circle (0.05);\n\\filldraw (8,0) node [below] {$N$} coordinate (N) circle (0.05);\n\\draw (A) circle (2) (M)--(P)--(N);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031657": { + "id": "031657", + "content": "已知二次函数 $f(x)$ 的图像经过点 $(4,3)$, 它在 $x$ 轴上截得的线段长为 $2$, 并且对 $x \\in \\mathbf{R}$ 都有 $f(2-x)=f(2+x)$.\\\\\n(1) 求 $f(x)$ 的解析式;\\\\\n(2) 若函数 $g(x)=\\dfrac{f(x)-4}{x+1}(x \\in(-4,-1) \\cup(-1,0))$, 求 $g(x)$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练14试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031658": { + "id": "031658", + "content": "集合 $A=\\{y | y=\\lg x, x>1\\}$, $B=\\{-2,-1,1,2\\}$, 则 $\\overline{A}\\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031659": { + "id": "031659", + "content": "函数 $y=\\sin ^4 x-\\sin ^2 x$ 的最小正周期是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031660": { + "id": "031660", + "content": "从甲、乙等 $5$ 名学生中随机选出 $2$ 人, 则甲被选中的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031661": { + "id": "031661", + "content": "若复数 $z$ 满足 $2 \\overline{z}-1=3+6 \\mathrm{i}$ ($\\mathrm{i}$ 是虚数单位), 则 $z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031662": { + "id": "031662", + "content": "已知等差数列 $\\{a_n\\}$ 中, $a_4+a_8=16$, $a_2=1$, 则 $a_{10}$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031663": { + "id": "031663", + "content": "若不等式 $|x|+|x-1|>m$ 的解集是 $\\mathbf{R}$, 则实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031664": { + "id": "031664", + "content": "如果一条直线与一个平面垂直, 那么, 称此直线与平面构成一个``正交线面对'', 在一个正方体中, 由两个顶点确定的直线与含有四个顶点的平面构成的``正交线面对''的个数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031665": { + "id": "031665", + "content": "已知 $\\{a_n\\}$ 是等差数列, $\\{b_n\\}$ 是等比数列, 且 $b_2=3$, $b_3=9$, $a_1=b_1$, $a_{14}=b_4$. 设 $c_n=a_n+ b_n$, 则数列 $\\{c_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031666": { + "id": "031666", + "content": "在极坐标系中, 直线 $\\rho \\cos \\theta-\\sqrt{3}\\rho \\sin \\theta-1=0$ 与圆 $\\rho=2 \\cos \\theta$ 交于 $A, B$ 两点, 则 $|AB|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031667": { + "id": "031667", + "content": "设 $A$、$B$ 是两个集合, 定义 $A-B=\\{x | x \\in A$, 且 $x \\notin B\\}$, 若 $M=\\{x|| x+1 | \\leq 2\\}$, $N= \\{x|x=| \\sin \\alpha |, \\ \\alpha \\in \\mathbf{R}\\}$, 则 $M-N=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031668": { + "id": "031668", + "content": "已知 $A$、$B$ 为 $x$ 轴上不同的两点, 点 $P$ 的横坐标为 $1$, 且 $|PA|=|PB|$, 若直线 $PA$ 的方程为 $x-y+1=0$, 则直线 $PB$ 的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031669": { + "id": "031669", + "content": "奇函数 $y=f(x)$ 在 ($-\\infty$, $+\\infty$) 上严格增, 偶函数 $y=g(x)$ 在 $[0,+\\infty)$ 上的图像与 $y= f(x)$ 的图像重合. 设 $ag(a)-g(-b)$; \\textcircled{2} $f(b)-f(-a)g(b)-g(-a)$; \\textcircled{4} $f(a)- f(-b)=latex]\n\\draw (0,0,0) node [above right] {$C$} coordinate (C);\n\\draw (-1,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\filldraw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A);\n\\draw [dashed] (P)--(C)--(D)(C)--(A)(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DC \\perp$ 平面 $PAC$;\\\\\n(2) 求证: 平面 $PAB \\perp$ 平面 $PAC$;\\\\\n(3) 设点 $E$ 为 $AB$ 的中点, 在棱 $PB$ 上是否存在点 $F$, 使得 $PA \\parallel $ 平面 $CEF$ ? 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031675": { + "id": "031675", + "content": "已知椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$ 过点 $A(2,0), B(0,1)$ 两点.\\\\\n(1) 求椭圆 $C$ 的方程及离心率;\\\\\n(2) 设 $P$ 为第三象限内一点且在椭圆 $C$ 上, 直线 $PA$ 与 $y$ 轴交于点 $M$, 直线 $PB$ 与 $x$ 轴交于点 $N$. 求证: 四边形 $ABNM$ 的面积为定值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练15试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031676": { + "id": "031676", + "content": "已知 $A(2,5), B(4,1)$. 若点 $P(x, y)$ 在线段 $AB$ 上, 则 $2 x-y$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031677": { + "id": "031677", + "content": "点 $P$ 从 $(1,0)$ 出发, 沿单位圆 $x^2+y^2=1$ 逆时针方向运动 $\\dfrac{2 \\pi}{3}$ 弧长到达 $Q$ 点, 则 $Q$ 的坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031678": { + "id": "031678", + "content": "若 $(a x-1)^5$ 的展开式中 $x^3$ 的系数是 $80$ , 则实数 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031679": { + "id": "031679", + "content": "曲线 $y^2=4 x$ 关于直线 $x=2$ 对称的曲线方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031680": { + "id": "031680", + "content": "若函数 $y=(\\dfrac{1}{2})^{|1-x|}+m$ 的图像与 $x$ 轴有公共点, 则实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031681": { + "id": "031681", + "content": "在正方体上任选 $3$ 个顶点连成三角形, 则所得的三角形是直角非等腰三角形的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031682": { + "id": "031682", + "content": "已知直线 $\\sqrt{3}x+y=0$ 和直线 $k x-y-1=0$, 若两直线的夹角为 $60^{\\circ}$, 则 $k=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031683": { + "id": "031683", + "content": "设函数 $f(x)=\\begin{cases}x^3-3 x,& x \\leq a,\\\\-2 x,& x>a,\\end{cases}$ 若 $f(x)$ 无最大值, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031684": { + "id": "031684", + "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的渐近线为正方形 $OABC$ 的边 $OA, OC$ 所在的直线, 点 $B$ 为该双曲线的焦点. 若正方形 $OABC$ 的边长为 $2$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031685": { + "id": "031685", + "content": "已知 $\\sin 2 \\alpha=-\\dfrac{1}{3}$, 则 $\\dfrac{4 \\cos ^2 \\alpha}{\\cot \\dfrac{\\alpha}{2}-\\tan \\dfrac{\\alpha}{2}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031686": { + "id": "031686", + "content": "正四棱锥底面边长为 $4$, 侧棱长为 $3$, 则其体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031687": { + "id": "031687", + "content": "关于函数 $f(x)=2 \\sin (3 x-\\dfrac{3 \\pi}{4})$, 有下列命题: \\textcircled{1} 其最小正周期为 $\\dfrac{2 \\pi}{3}$; \\textcircled{2} 其图像由 $y= 2 \\sin 3 x$ 向左平移 $\\dfrac{3}{4}$ 个单位而得到; \\textcircled{3} 在 $[\\dfrac{\\pi}{4}, \\pi]$ 上为严格增函数, 则其中真命题为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031688": { + "id": "031688", + "content": "在复平面内, 复数 $\\dfrac{1+\\mathrm{i}}{(1-\\mathrm{i})^2}$ 对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031689": { + "id": "031689", + "content": "若平面四边形 $ABCD$ 满足 $\\overrightarrow{AB}+\\overrightarrow{CD}=\\overrightarrow{0}$, $(\\overrightarrow{AB}-\\overrightarrow{AD}) \\cdot \\overrightarrow{AC}=0$, 则该四边形一定是\\bracket{20}.\n\\fourch{直角梯形}{矩形}{菱形}{正方形}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031690": { + "id": "031690", + "content": "集合 $A=\\{x | x^2-2 x-3=0\\}$, $B=\\{x | a x=1\\}$, 则``$a=-1$ 或 $\\dfrac{1}{3}$''是``$A \\cup B=A$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031691": { + "id": "031691", + "content": "已知 $f(x)$ 是定义域为 ($-\\infty$, $+\\infty$) 的奇函数, 满足 $f(1-x)=f(1+x)$. 若 $f(1)=2$, 则 $f(1)+f(2)+f(3)+\\cdots+f(100)$ 的值为\\bracket{20}.\n\\fourch{$-50$}{$0$}{$2$}{$50$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031692": { + "id": "031692", + "content": "$A, B, C$ 三个班共有 $100$ 名学生, 为调查他们的体育锻炼情况, 通过分层抽样获得了部分学生一周的锻炼时间, 数据如下表 (单位: 小时):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline$A$ 班 & 6 & 6.5 & 7 & 7.5 & 8 & & & \\\\\n\\hline$B$ 班 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \\\\\n\\hline$C$ 班 & 3 & 4.5 & 6 & 7.5 & 9 & 10.5 & 12 & 13.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 试估计 $C$ 班的学生人数;\\\\\n(2) 从 $A$ 班和 $C$ 班抽出的学生中, 各随机选取一人, $A$ 班选出的人记为甲, $C$ 班选出的人记为乙. 假设所有学生的锻炼时间相对独立, 求该周甲的锻炼时间比乙的锻炼时间长的概率;\\\\\n(3) 再从 $A, B, C$ 三个班中各随机抽取一名学生, 他们该周的锻炼时间分别是 $7,9,8.25$ (单位: 小时). 这 $3$ 个新数据与表格中的数据构成的新样本的平均数记 $\\mu_1$, 表格中数据的平均数记为 $\\mu_0$, 试判断 $\\mu_0$ 和 $\\mu_1$ 的大小. (结论不要求证明)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031693": { + "id": "031693", + "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) 若直线过点 $F_2$ 且与轨迹 $E$ 交于 $P$、$Q$ 两点, 无论直线 $l$ 绕点 $F_2$ 怎样转动, 在 $x$ 轴上总存在定点 $M(m, 0)$, 使 $MP \\perp MQ$ 恒成立, 求实数 $m$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练16试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031694": { + "id": "031694", + "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": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031695": { + "id": "031695", + "content": "若角 $\\alpha$ 的终边经过点 $P(1,-2)$, 则 $\\tan 2 \\alpha$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031696": { + "id": "031696", + "content": "设数列 $\\{a_n\\}$ 是等差数列, 且 $a_2=-6$, $a_8=6, S_n$ 是数列 $\\{a_n\\}$ 的前 $n$ 项和, 则 $S_{10}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031697": { + "id": "031697", + "content": "$(x-\\sqrt{2}y)^{10}$ 的展开式中 $x^6 y^4$ 项的系数是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031698": { + "id": "031698", + "content": "函数 $y=\\sqrt{\\log _{\\frac{1}{2}}(x^2-1)}$ 的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031699": { + "id": "031699", + "content": "有一种叫做``天天彩''的彩票, 每注售价为 $2$ 元. 设一等奖、二等奖两种奖. 一等奖中奖的概率 $0.1 \\%$,奖金为 $100$ 元;二等奖中的概率为 $10 \\%$, 奖金为 $10$ 元. 那么购买一注彩票的期望收益是\\blank{50}元.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031700": { + "id": "031700", + "content": "若过点 $A(4,0)$ 的直线 $l$ 与曲线 $(x-2)^2+y^2=1$ 有公共点, 则直线 $l$ 的斜率的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031701": { + "id": "031701", + "content": "某公司为改善职工的出行条件, 随机抽取 $50$ 名职工, 调查他们的居住地与公司的距离 $d$ (单位: 千米). 若样本数据分组为 $[0,2],(2,4],(4,6],(6,8],(8,10],(10,12]$, 由数据绘制的分布频率直方图如图所示, 则样本中职工居住地与公司的距离不超过 $4$ 千米的人数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 20]\n\\draw [->] (0,0) -- (14,0) node [below] {距离};\n\\draw [->] (0,0) -- (0,0.18) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.1,2/0.14,4/0.12,6/0.04,8/0.05,10/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (2,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.1,2/0.14,4/0.12,6/0.04,8/0.05}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (12,0) node [below] {$12$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031702": { + "id": "031702", + "content": "已知点 $A, B, C, D$ 在同一个球面上, $AB \\perp$ 平面 $BCD$, $BC \\perp CD$, 若 $AB=6$, $AC=2 \\sqrt{13}$, $AD=8$, 则 $B, C$ 两点间的球面距离是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031703": { + "id": "031703", + "content": "设函数 $y=f(x)$ 由方程 $x|x|+y|y|=1$ 确定, 下列结论正确的是\\blank{50}.\\\\\n\\textcircled{1} $f(x)$ 是 $\\mathbf{R}$ 上的严格减函数; \\textcircled{2} 对于任意 $x \\in \\mathbf{R}$, $f(x)+x>0$ 恒成立; \\textcircled{3} 对于任意 $a \\in \\mathbf{R}$, 关于 $x$ 的方程 $f(x)=a$ 都有解.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031704": { + "id": "031704", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$ 满足 $S_n=2 a_n+(-1)^n$, $n \\geq 1$. 则数列 $\\{a_n\\}$ 的前三项的和 $a_1+a_2+a_3=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031705": { + "id": "031705", + "content": "在标有 $1$、$2$、$3$、$4$、$5$ 的五张卡片中依次选取两张, 在第一张是奇数的条件下, 第二张也是奇数的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031706": { + "id": "031706", + "content": "函数 $y=\\sin (2 x+\\dfrac{\\pi}{3})$ 图像的对称轴方程可能是\\bracket{20}.\n\\fourch{$x=-\\dfrac{\\pi}{6}$}{$x=-\\dfrac{\\pi}{12}$}{$x=\\dfrac{\\pi}{6}$}{$x=\\dfrac{\\pi}{12}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031707": { + "id": "031707", + "content": "``双曲线的方程为 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$''是``双曲线的焦点坐标为 $(\\pm 5,0)$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031708": { + "id": "031708", + "content": "在同一平面直角坐标系中, 函数 $y=g(x)$ 的图像与 $y=\\mathrm{e}^x$ 的图像关于直线 $y=x$ 对称, 而函数 $y=f(x)$ 的图像与 $y=g(x)$ 的图像关于 $y$ 轴对称, 若 $f(m)=-1$, 则 $m$ 的值为\\bracket{20}.\n\\fourch{$-\\mathrm{e}$}{$-\\dfrac{1}{\\mathrm{e}}$}{$\\mathrm{e}$}{$\\dfrac{1}{\\mathrm{e}}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031709": { + "id": "031709", + "content": "若函数 $f(x)=x^3-a x$($a>0$) 的零点都在区间 $[-10,10]$ 上, 则使得方程 $f(x)=1000$ 有正整数解的实数 $a$ 的取值个数为\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031710": { + "id": "031710", + "content": "提高过江大桥的车辆通行能力可改善整个城市的交通状况. 在一般情况下, 大桥上的车流速度 $v$ (单位: 千米/小时) 是车流密度 $x$ (单位: 辆/千米) 的函数. 当桥上的车流密度达到 $200$ 辆/千米时, 造成堵塞, 此时车流速度为 $0$; 当车流密度不超过 $20$ 辆/千米时, 车流速度为 $60$ 千米/小时. 研究表明: 当 $20 \\leq x \\leq 200$ 时, 车流速度 $v$ 是车流密度 $x$ 的一次函数.\\\\\n(1) 当 $0 \\leq x \\leq 200$ 时,求函数 $v(x)$ 的表达式;\\\\\n(2) 当车流密度 $x$ 为多大时, 车流量 (单位时间内通过桥上某观测点的车辆数, 单位: 辆/小时) $f(x)=x \\cdot v(x)$ 可以达到最大, 并求出最大值. (精确到 $1$ 辆/小时)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031711": { + "id": "031711", + "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=AD=2$, $E$ 是棱 $CD$ 上的一点.\\\\\n(1) 求证: $AD_1 \\perp$ 平面 $A_1B_1D$;\\\\\n(2) 求异面直线 $B_1E, AD_1$ 所成的角;\\\\\n(3) 若 $E$ 是棱 $CD$ 的中点, 在棱 $AA_1$ 上是否存在点 $P$, 使得 $DP \\parallel $ 平面 $B_1AE$ ? 若存在, 求出线段 $AP$ 的长; 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练17试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031712": { + "id": "031712", + "content": "不等式 $(|x|+1)(2 x-1)>0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练18试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031713": { + "id": "031713", + "content": "已知圆 $x^2+y^2-6 x-7=0$ 与抛物线 $y^2=2 p x$($p>0$) 的准线相切, 则 $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练18试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031714": { + "id": "031714", + "content": "若集合 $M=\\{(x, y) | y=2^{-x}\\}$, $P=\\{y | y=\\sqrt{x-1}\\}$, 则 $M \\cap P=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练18试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031715": { + "id": "031715", + "content": "若 $\\sin 2 \\theta-1+\\mathrm{i}(\\sqrt{2}\\cos \\theta+1)$ 是纯虚数, 则 $\\theta$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练18试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031716": { + "id": "031716", + "content": "函数 $f(x)=|\\log _a x|$($00,\\end{cases}$ 则 $f[f(-3)]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031735": { + "id": "031735", + "content": "已知指数函数 $y=a^x$ (其中 $a>1$) 在闭区间 $[1,2]$ 上的最大值比最小值大 $\\dfrac{a}{3}$, 则实数 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031736": { + "id": "031736", + "content": "若点 $(\\theta, 0)$ 是函数 $f(x)=\\sin x+3 \\cos x$ 的一个对称中心, 则 $\\cos 2 \\theta+\\sin \\theta \\cos \\theta=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031737": { + "id": "031737", + "content": "一个口袋内装有大小相同的 6 个球, 其中 3 个白球, 3 个黑球, 从中一次摸出两个球, 则摸出的两个球至少一个是白球的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031738": { + "id": "031738", + "content": "已知双曲线 $M: x^2-\\dfrac{y^2}{6}=1$ 的左、右焦点分别为 $F_1$、$F_2$, 过 $F_1$ 的直线 $l$ 与双曲线 $M$ 的左、右两支分别交于点 $A$、$B$. 若 $\\triangle ABF_2$ 为等边三角形, 则 $\\triangle ABF_2$ 的边长为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031739": { + "id": "031739", + "content": "已知函数 $f(x)=x^2+2 x+3+m$, 若 $f(x) \\geq 0$ 对任意的 $x \\in[1,2]$ 恒成立, 则实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031740": { + "id": "031740", + "content": "已知菱形 $ABCD$ 的边长为 $1, \\angle DAB=\\dfrac{\\pi}{3}$, 点 $E$ 为该菱形边上任意一点, 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AE}$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031741": { + "id": "031741", + "content": "已知函数 $f(x)=\\begin{cases}x^2-x+3,& x \\leq 1,\\\\x+\\dfrac{2}{x},& x>1.\\end{cases}$ 设 $a \\in \\mathbf{R}$, 若关于 $x$ 的不等式 $f(x) \\geq|\\dfrac{x}{2}+a|$ 在 $\\mathbf{R}$ 上恒成立,则 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031742": { + "id": "031742", + "content": "已知直线 $a, b$ 分别在两个不同的平面 $\\alpha, \\beta$ 内. 则``直线 $a$ 和直线 $b$ 相交''是``平面 $\\alpha$ 和平面 $\\beta$ 相交''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031743": { + "id": "031743", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则``数列 $\\{\\dfrac{S_n}{n}\\}$ 为等差数列''\"是``数列 $\\{a_n\\}$ 为等差数列''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}{既不充分也不必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031744": { + "id": "031744", + "content": "若非零向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 的夹角为锐角 $\\theta$, 且 $\\dfrac{|\\overrightarrow{a}|}{|\\overrightarrow{b}|}=\\cos \\theta$,则称 $\\overrightarrow{a}$ 被 $\\overrightarrow{b}$``同余''. 已知 $\\overrightarrow{b}$ 被 $\\overrightarrow{a}$``同余'', 则 $\\overrightarrow{a}-\\overrightarrow{b}$ 在 $\\overrightarrow{a}$ 上的数量投影是\\bracket{20}.\n\\fourch{$\\dfrac{\\overrightarrow{a}^2-\\overrightarrow{b}^2}{|\\overrightarrow{a}|}$}{$\\dfrac{\\overrightarrow{a}^2-\\overrightarrow{b}^2}{\\overrightarrow{a}^2}$}{$\\dfrac{\\overrightarrow{b}^2-\\overrightarrow{a}^2}{|\\overrightarrow{a}|}$}{$\\dfrac{\\overrightarrow{a}^2-\\overrightarrow{b}^2}{|\\overrightarrow{b}|}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031745": { + "id": "031745", + "content": "已知实数 $x_1$、$y_1$、$x_2$、$y_2$、$x_3$、$y_3$ 满足 $x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2=2$, 则 $x_1 y_2$、$x_2 y_3$ 、 $x_3 y_1$ 三个数中,大于 1 的个数最多是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031746": { + "id": "031746", + "content": "某学校对面有一块空地要围建成一个面积为 $360 \\mathrm{m}^2$ 的矩形场地, 要求矩形场地的一面利用旧墙 (旧墙需要整修), 其它三面围墙要新建, 在旧墙对面的新墙上要留一个宽度为 $2 \\mathrm{m}$ 的进出口, 如图所示. 已知旧墙的整修费用为 45 元 $/ \\mathrm{m}$, 新建墙的造价为 180 元 $/ \\mathrm{m}$, 建 $2 \\mathrm{m}$ 宽的进出口需 2360 元的单独费用, 设利用的旧墙的长度为 $x$ (单位: $\\mathrm{m}$), 设修建此矩形场地围墙的总费用 (含建进出口的费用) 为 $y$ (单位: 元).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw [pattern = north east lines] (0,0) rectangle (5,0.2);\n\\draw (0,0) rectangle (5,0.2);\n\\draw (0.5,0) -- (0.5,-2) -- (2.2,-2) (4.5,0) -- (4.5,-2) -- (2.8,-2);\n\\draw (2.5,-1) node {$x$};\n\\draw [->] (2.2,-1) -- (0.5,-1);\n\\draw [->] (2.8,-1) -- (4.5,-1); \n\\end{tikzpicture}\n\\end{center}\n(1) 将 $y$ 表示为 $x$ 的函数;\\\\\n(2) 试确定 $x$, 使修建此矩形场地围墙的总费用 (含建进出口的费用)最少, 并求出最少总费用.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031747": { + "id": "031747", + "content": "已知直线 $l: y=-x+3$ 与椭圆 $C: m x^2+n y^2=1$($n>m>0$) 相切于点 $P(2,1)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$}; \n\\draw [name path = elli] (0,0) ellipse ({sqrt(6)} and {sqrt(3)});\n\\draw (1.1,1.9) node [above] {$l$} -- (2,1) node [above right] {$P$} coordinate (P) -- (2.9,0.1);\n\\draw [name path = l1] ({-2+1/3},2) node [above] {$l'$} -- ({2+1/3},-2);\n\\draw [name intersections = {of = l1 and elli, by = {A,B}}];\n\\draw (A) node [above] {$A$} --(P)--(B) node [below] {$B$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆 $C$ 的标准方程;\\\\\n(2) 若直线 $l': y=-x+b$ 交 $C$ 于 $A, B$ 两点, 且 $PA \\perp PB$, 求 $b$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练19试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031748": { + "id": "031748", + "content": "已知集合 $M=\\{0,1,2\\}$, $N=\\{x | x=2 a,\\ a \\in M\\}$, 则集合 $M \\cap N=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031749": { + "id": "031749", + "content": "若函数 $f(x)=a \\cdot 3^x+\\dfrac{1}{3^x}$ 为偶函数, 则实数 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031750": { + "id": "031750", + "content": "设直线 $2 x+m y=1$ 的倾斜角为 $\\alpha$, 若 $m \\in(-\\infty,-2 \\sqrt{3}) \\cup[2,+\\infty)$, 则角 $\\alpha$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031751": { + "id": "031751", + "content": "已知复数 $z_0=3+2 \\mathrm{i}$, 复数 $z$ 满足 $z \\cdot z_0=3 z+z_0$, 则复数 $z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031752": { + "id": "031752", + "content": "已知椭圆的方程为 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{3}=1$, 它的一个焦点与抛物线 $y^2=-4 x$ 的焦点重合, 则此椭圆方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031753": { + "id": "031753", + "content": "某小区共有住户 $2000$ 人, 其中老年人 $600$ 人, 中年人 $1000$ 人, 其余为青少年等人群, 为了调查该小区的新冠疫苗接种情况, 现采用分层抽样的方法从中抽取一个容量为 $400$ 的样本, 则样本中中年人的人数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031754": { + "id": "031754", + "content": "已知过点 $A(-2, m), B(m, 4)$ 的直线与直线 $2 x+y-1=0$ 垂直, 则 $m$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031755": { + "id": "031755", + "content": "已知 $f(x)=\\begin{cases}-\\cos \\pi x,& x>0,\\\\f(x+1)+1,& x \\leq 0,\\end{cases}$ 则 $f(\\dfrac{4}{3})+f(-\\dfrac{4}{3})$ 的值等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031756": { + "id": "031756", + "content": "已知抛物线 $y^2=2 p x$($p>0$) 上一点 $M(1, m)$ 到其焦点的距离为 $5$, 双曲线 $C: x^2-\\dfrac{y^2}{b^2}=1$($b>0$) 的左顶点为 $A$, 若双曲线 $C$ 的一条渐近线与直线 $AM$ 垂直, 则双曲线 $C$ 的焦距为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031757": { + "id": "031757", + "content": "已知 $\\alpha$ 为第二象限角, 且 $\\sin \\alpha=\\dfrac{\\sqrt{15}}{4}$, 则 $\\dfrac{\\sin (\\alpha+\\dfrac{\\pi}{4})}{\\sin 2 \\alpha+\\cos 2 \\alpha+1}$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031758": { + "id": "031758", + "content": "设曲线 $C$ 与函数 $f(x)=\\dfrac{\\sqrt{3}}{12}x^2$($0 \\leq x \\leq m$) 的图像关于直线 $y=\\sqrt{3}x$ 对称, 若曲线 $C$ 仍为某函数的图像, 则实数 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031759": { + "id": "031759", + "content": "在棱长为 2 的正四面体木块 $ABCD$ 的棱 $AB$ 上有一点 $P$($PA<1$), 过点 $P$ 要锯出与棱 $AB$ 垂直的四面体的截面, 当锯到某个位置因故停止, 这时量得在面 $ABD$ 上的锯痕 $PM=1$, 表面 $ABC$ 上的锯痕 $PN=\\dfrac{1}{3}$, 则锯缝 $MN=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031760": { + "id": "031760", + "content": "对于下列命题: \\textcircled{1} 若 $a>b>0$, $c>d>0$, 则 $\\dfrac{a+c}{d}>\\dfrac{b+d}{c}$; \\textcircled{2} 若 $a>b>0$, $c>d>0$, 则$a^c>b^d$, 关于上述命题描述正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}都均为真命题}{\\textcircled{1}和\\textcircled{2} 都均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031761": { + "id": "031761", + "content": "$M(x_0, y_0)$ 为圆 $x^2+y^2=a^2$($a>0$) 内异于圆心的一点, 则直线 $x_0 x+y_0 y=a^2$ 与该圆的位置关系为\\bracket{20}.\n\\fourch{相切}{相交}{相离}{相切或相交}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031762": { + "id": "031762", + "content": "从分别标有 $1,2, \\cdots, 9$ 的 $9$ 张卡片中不放回地随机抽取 $2$ 次, 每次抽取 $1$ 张. 则抽到的 $2$ 张卡片上的数奇偶性不同的概率是\\bracket{20}.\n\\fourch{$\\dfrac{5}{18}$}{$\\dfrac{4}{9}$}{$\\dfrac{5}{9}$}{$\\dfrac{7}{9}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031763": { + "id": "031763", + "content": "设函数 $f(x)=2^x-2^{-x}+\\dfrac{3}{|x|+1}$, $x \\in \\mathbf{R}$, 对于实数 $a$、$b$, 给出以下命题: 命题 $p_1$: $a+ b \\geq 0$; 命题 $p_2$: $a-b^2 \\geq 0$; 命题 $q$: $f(a)+f(b) \\geq 0$. 下列选项中正确的是\\bracket{20}.\n\\twoch{$p_1$、$p_2$ 中仅 $p_1$ 是 $q$ 的充分条件}{$p_1$、$p_2$ 中仅 $p_2$ 是 $q$ 的充分条件}{$p_1$、$p_2$ 都不是 $q$ 的充分条件}{$p_1$、$p_2$ 都是 $q$ 的充分条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031764": { + "id": "031764", + "content": "考虑到高速公路行车安全需要, 一般要求高速公路的车速 $v$ (公里/小时) 控制在 $[60,120]$ 范围内. 已知汽车以 $v$ 公里/小时的速度在高速公路上匀速行驶时, 每小时的油耗 (所需要的汽油量) 为 $\\dfrac{1}{5}(v-k+\\dfrac{4500}{v})$ 升, 其中 $k$ 为常数, 不同型号汽车 $k$ 值不同, 且满足 $60 \\leq k \\leq 120$.\\\\\n(1) 若某型号汽车以 $120$ 公里/小时的速度行驶时, 每小时的油耗为 $11.5$ 升, 欲使这种型号的汽车每小时的油耗不超过 $9$ 升, 求车速 $v$ 的取值范围;\\\\\n(2) 求不同型号汽车行驶 $100$ 千米的油耗的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031765": { + "id": "031765", + "content": "已知 $\\triangle ABC$ 的顶点 $A, B$ 在椭圆 $x^2+3 y^2=4$ 上, $C$ 在直线 $l: y=x+2$ 上, 且 $AB \\parallel l$.\\\\\n(1) 当 $AB$ 边通过坐标原点 $O$ 时,求 $AB$ 的长及 $\\triangle ABC$ 的面积;\\\\\n(2) 当 $\\angle ABC=90^{\\circ}$, 且斜边 $AC$ 的长最大时, 求 $AB$ 所在直线的方程.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书高中数学知识点再现填漏补缺小题训练20试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031766": { + "id": "031766", + "content": "已知 $z=2+\\dfrac{1}{\\mathrm{i}}$, 则 $\\overline{z}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031767": { + "id": "031767", + "content": "已知 $A=\\{-1,1, m\\}$, 集合 $B=\\{1,2\\}$, 若 $B \\subset A$, 则实数 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031768": { + "id": "031768", + "content": "已知 $\\sin \\alpha=3 \\cos \\alpha$, 则 $\\tan (\\alpha-\\dfrac{\\pi}{4})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031769": { + "id": "031769", + "content": "已知球 $O$ 的体积为 $36 \\pi$, 则该球的球面面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031770": { + "id": "031770", + "content": "若 $x_1$、$x_2$ 为方程 $a^x=(\\dfrac{1}{a})^{-\\frac{1}{x}+1}$($a>1$) 的两个实数解,则 $x_1+x_2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031771": { + "id": "031771", + "content": "已知两条直线 $l_1: a x+3 y-3=0$, $l_2: 2 x+6 y+1=0$. 若 $l_1 \\parallel l_2$, 则直线 $l_1$ 与 $l_2$ 之间的距离 $d=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031772": { + "id": "031772", + "content": "若 $S_n$ 为等比数列 $\\{a_n\\}$ 的前 $n$ 项和, $8 a_2+a_5=0$, 则 $\\dfrac{S_{10}}{S_5}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031773": { + "id": "031773", + "content": "如图, 将一根直径为 $d=54 \\mathrm{cm}$ 的圆木锯成截面为矩形的梁, 矩形的高为 $h \\mathrm{cm}$, 宽为 $b \\mathrm{cm}$, 则当 $b=$\\blank{50}$\\mathrm{cm}$ 时, 该梁的抗弯强度 $W=\\dfrac{1}{6}b h^2$ 取得最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [dashed] (0,0) circle (1);\n\\filldraw (0,0) circle (0.03) node [left] {$d$};\n\\draw (230:1) rectangle (50:1);\n\\draw [dashed] (50:1) -- (230:1);\n\\draw ({cos(50)},0) node [right] {$h$};\n\\draw (0,{-sin(50)}) node [above] {$b$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031774": { + "id": "031774", + "content": "已知 $f(x)=\\sin x \\cos x$, 关于该函数有下面四个说法:\\\\ \\textcircled{1} $f(x)$ 的最小正周期为 $2 \\pi$;\\\\\n\\textcircled{2} $f(x)$ 是区间 $[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$ 上的严格增函数;\\\\\n\\textcircled{3} 当 $x \\in[-\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3}]$ 时, $f(x)$ 的取值范围为 $[-\\dfrac{\\sqrt{3}}{4}, \\dfrac{\\sqrt{3}}{4}]$;\\\\\n\\textcircled{4} $f(x)$ 的图像可由 $g(x)=\\dfrac{1}{2}\\sin (2 x+\\dfrac{\\pi}{4})$ 的图像向右平移 $\\dfrac{\\pi}{8}$ 个单位长度得到.\\\\\n以上四种说法中, 正确结论的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031775": { + "id": "031775", + "content": "设函数 $f(x)=\\begin{cases}-a x+1,& x=latex, z = {(235:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$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$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C_1);\n\\draw (A_1)--(B) (D_1)--(B_1)--(C);\n\\draw [dashed] (A_1)--(D)--(B)(D_1)--(C);\n\\def\\lambda{0.4}\n\\draw ($(A_1)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(C)$);\n\\draw [dashed] ($(B)!\\lambda!(C)$) -- ($(D)!\\lambda!(C)$) -- ($(D)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(D_1)$);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$S$ 是定值, $l$ 不是定值}{$S$ 不是定值, $l$ 是定值}{$S$ 和 $l$ 都是定值}{$S$ 和 $l$ 都不是定值}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031781": { + "id": "031781", + "content": "设函数 $f(x)=3 \\sin x+2 \\cos x+1$, 若实数 $a, b, c$ 使得 $a f(x)+b f(x-c)=1$ 对任意实数 $x$ 恒成立, 则 $\\dfrac{b \\cos c}{a}$ 的值等于\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}{1}{-1}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031782": { + "id": "031782", + "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=BC=BB_1=4, M, N$ 分别为 $A_1B_1, AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (1.9,0,{sqrt(0.39)}) node [right] {$A$} coordinate (A);\n\\draw ({-sqrt(0.39)},0,1.9) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw (C)--(A)--(A_1)--(C_1)--cycle(A_1)--(B_1)--(C_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (B)--(N)--(M)--cycle(C)--(B)--(A)(B)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 平面 $BCC_1B_1$;\\\\\n(2) 若 $AB \\perp MN$, 求异面直线 $MN$ 与 $A_1C_1$ 所成的角的余弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031783": { + "id": "031783", + "content": "如图, 某市拟在长为 $10 \\mathrm{km}$ 的道路 $OP$ 的一侧修建一条运动赛道, 赛道的前一部分为曲线段 $OSM$, 该曲线段为函数 $y=4 \\sin \\omega x(\\omega>0)$, $x \\in[0,4]$ 图像, 且图像的最高点为 $S(3,4)$, 赛道的后一部分为折线段 $MNP, \\angle MNP=120^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5,samples=200,>=latex]\n\\draw [->] (-1,0) -- (11,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain=0:4] plot (\\x,{4*sin(\\x*30)});\n\\draw (3,4) node [above]{$S$};\n\\draw (4,{2*sqrt(3)}) node [above] {$M$} coordinate (M);\n\\draw (10,0) node [below] {$P$} coordinate (P);\n\\coordinate (D) at ($(4,3)!{2/sqrt(3)*sin(40)}!(8,0)$);\n\\path (P) arc (0:20:4) coordinate (N);\n\\draw (M) --(N) node [above right] {$N$}--(10,0);\n\\draw [dashed] (3,0) node [below] {$3$} --++ (0,4) --++ (-3,0) node [left] {$4$};\n\\draw [dashed] (M) -- (4,0) node [below] {$4$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\omega$ 的值和 $M, P$ 两点间的距离;\\\\\n(2) 求折线段赛道 $MNP$ 长度的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031784": { + "id": "031784", + "content": "已知椭圆 $E: \\dfrac{x^2}{8}+\\dfrac{y^2}{4}=1$ 的左焦点为 $F$, 下顶点为 $A$, 斜率为 $k$ 的直线 $l$ 经过点 $P(0,-3)$.\\\\\n(1) 若 $l$ 与直线 $AF$ 垂直, 求 $l$ 的方程;\\\\\n(2) 若直线 $l$ 与椭圆 $E$ 相交于不同的两点 $B, C$, 直线 $AB, AC$ 分别与直线 $y=-3$ 交于 $M, N$, 且 $|PM|+|PN| \\leq 16$, 求 $k$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031785": { + "id": "031785", + "content": "已知函数 $f(x)=2^x$.\\\\\n(1) 求函数 $y=3 f(x)-f(-x)-2$ 的零点;\\\\\n(2) 证明: 当 $a \\leq 16$ 时, 函数 $F(x)=f(2 x)+a \\cdot f(-x)$ 是 ($1,+\\infty$) 上的严格递增函数;\\\\\n(3) 设 $g(x)=\\dfrac{1}{1+a \\cdot f(x)}-\\dfrac{1}{1+a \\cdot f(x-1)}$, 若对任意 $x \\in(-\\infty, 0]$, $g(x) \\geq g(0)$ 恒成立, 求正实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031786": { + "id": "031786", + "content": "已知无穷数列 $\\{a_n\\}$ 的每一项均为正整数, 且 $a_{n+1}=\\begin{cases}\\dfrac{a_n}{2},& a_n \\text{是偶数},\\\\a_n+3,& a_n \\text{是奇数},\\end{cases}$ 记 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$.\\\\\n(1) 若 $a_1=20$, 求 $S_{10}$ 的值;\\\\\n(2) 若 $S_3=23$, 求 $a_1$ 的值;\\\\\n(3) 证明: 数列 $\\{a_n\\}$ 中存在某一项 $a_i$ ($i$ 为正整数), 满足 $a_i \\leq 6$, 并由此验证 $1$ 或 $3$ 是数列 $\\{a_n\\}$ 中的项.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷01试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031787": { + "id": "031787", + "content": "已知 $z=-1+2 \\mathrm{i}$, 则 $|\\overline{z}|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031788": { + "id": "031788", + "content": "双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031789": { + "id": "031789", + "content": "函数 $f(x)=\\cos ^2 x$ 的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031790": { + "id": "031790", + "content": "已知 $\\alpha \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2}, 1,2\\}$, 若幂函数 $f(x)=x^\\alpha$ 为偶函数, 且在 ($0,+\\infty$) 上严格递减, 则 $\\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031791": { + "id": "031791", + "content": "若实数 $a, b$ 满足 $a+b=2$, 则 $3^a+3^b$ 的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031792": { + "id": "031792", + "content": "已知二项式 $(x-\\dfrac{a}{x})^5$ 的展开式中 $x^3$ 的系数为 10 , 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031793": { + "id": "031793", + "content": "已知圆锥的底面半径为 $6$ ,侧面积为 $60 \\pi$, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031794": { + "id": "031794", + "content": "设 $\\{a_n\\}$ 是公比为 $q$ 的等比数列, $S_n$ 是它的前 $n$ 项和, 若 $\\{S_n\\}$ 是等差数列, 则 $q=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031795": { + "id": "031795", + "content": "设 $\\triangle ABC$ 的内角 $A, B, C$ 的对边分别为 $a, b, c$, 且满足 $a \\cos B-b \\cos A=\\dfrac{3}{5}c$, 则 $\\dfrac{\\tan A}{\\tan B}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031796": { + "id": "031796", + "content": "在平面直角坐标系 $xOy$ 中, 点 $A$、$B$ 在抛物线 $y^2=4 x$ 上, 满足 $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=-4, F$ 是抛物线的焦点,则 $S_{\\triangle OFA}\\cdot S_{\\triangle OFB}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031797": { + "id": "031797", + "content": "若实数 $a, b$ 满足 $a-4 \\sqrt{b}=2 \\sqrt{a-b}$, 且 $a, b$ 不同时为零, 则 $a$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031798": { + "id": "031798", + "content": "定义: $\\displaystyle\\prod_{i=1}^n x_i=x_1 \\cdot x_2 \\cdot \\cdots \\cdot x_n$, 其中 $n$ 为正整数. 已知 $a_1=\\dfrac{\\pi}{6}$, $a_n \\in(0, \\dfrac{\\pi}{2})$ 且 $\\tan ^2 a_{n+1}=\\tan ^2 a_n+1$, $n$ 为正整数,若 $\\displaystyle\\prod_{i=1}^m \\sin \\alpha_k=\\dfrac{1}{10}$, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031799": { + "id": "031799", + "content": "函数 $f(x)=a^x$($a>0$ 且 $a \\neq 1$) 对于任意的实数 $x, y$ 都有\\bracket{20}.\n\\fourch{$f(x y)=f(x) f(y)$}{$f(x y)=f(x)+f(y)$}{$f(x+y)=f(x) f(y)$}{$f(x+y)=f(x)+f(y)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031800": { + "id": "031800", + "content": "若 $A$、$B$ 是 $x$ 轴上的两点, 点 $P$ 的横坐标为 $2$ 且 $|PA|=|PB|$. 若直线 $PA$ 的方程为 $x-y+1=0$, 则直线 $PB$ 的方程是\\bracket{20}.\n\\fourch{$x+y-5=0$}{$2 x-y-1=0$}{$2 y-x-4=0$}{$2 x+y-7=0$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031801": { + "id": "031801", + "content": "无穷等比数列 $\\{a_n\\}$ 的公比为 $q$, 前 $n$ 项和为 $S_n$, 且 $\\displaystyle\\sum_{n=1}^{+\\infty}a_n=S$, 下列条件中, 使得 $2S_n0$, $0.60$, $0.7x$ 恒成立, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031804": { + "id": "031804", + "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, 点 $D, E$ 分别为 $AC$ 和 $B_1C_1$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B_1)!0.5!(C_1)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B)--(E);\n\\draw [dashed] (A)--(C)(A)--(E)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\parallel $ 平面 $ABB_1A_1$;\\\\\n(2) 若 $AB \\perp BC$, $AB=BC=AA_1=2$, 求点 $D$ 到平面 $ABE$ 的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031805": { + "id": "031805", + "content": "某农场有一块农田, 如图所示, 它的边界由圆 $O$ 的一段圆弧 $MPN$ ($P$ 为此圆弧的中点) 和线段 $MN$ 构成, 已知圆 $O$ 的半径为 40 米, 点 $P$ 到 $MN$ 的距离为 50 米, 先规划在此农田上修建两个温室大棚, 大棚 I 内的地块形状为矩形 $ABCD$, 大棚 II 内的地块形状为 $\\triangle CDP$, 要求 $A, B$ 均在线段 $MN$ 上, $C, D$ 均在圆弧上. 设 $OC$ 与 $MN$ 所成的角为 $\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2, 0) coordinate (R) arc (0: 180: 2) coordinate (L);\n\\draw (R) arc (0: {-asin(0.25)}: 2) node [right] {$N$} coordinate (N);\n\\draw (L) arc (180: {180+asin(0.25)}: 2) node [left] {$M$} coordinate (M);\n\\draw (0, 0) node [below left] {$O$} coordinate (O);\n\\draw [dashed] (-2, 0) -- (2, 0) (0, 2) node [above] {$P$} coordinate (P) -- ($(M)!(P)!(N)$);\n\\draw (40: 2) node [above right] {$C$} coordinate (C);\n\\draw (140: 2) node [above left] {$D$} coordinate (D);\n\\draw ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (M) -- (N);\n\\draw (A) -- (D) -- (C) -- (B);\n\\draw (C) -- (P) -- (D) (O) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用 $\\theta$ 分别表示矩形 $ABCD$ 和 $\\triangle CDP$ 的面积, 并确定 $\\sin \\theta$ 的取值范围;\\\\\n(2) 若大棚 I 内种植甲种蔬菜, 大棚 II 内种植乙种蔬菜, 且甲、乙两种蔬菜的单位面积年产值之比为 $4: 3$. 求当 $\\theta$ 为何值时, 能使甲、乙两种蔬菜的年总产值最大.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031806": { + "id": "031806", + "content": "甲、乙两名篮球运动员, 甲投篮一次命中的概率为 $\\dfrac{3}{5}$, 乙投篮一次命中的概率为 $\\dfrac{1}{2}$, 其中甲、乙两人投篮是否命中相互没有影响.\\\\\n(1) 求甲投篮三次恰好命中两次的概率;\\\\\n(2) 若甲、乙各投篮三次, 且甲、乙第一次投篮都命中, 求甲获胜 (甲投篮命中数比乙多) 的概率;\\\\\n(3) 若甲、乙各投篮两次, 设 $X$ 为甲、乙投篮命中的次数的差的绝对值, 求 $X$ 的数学期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031807": { + "id": "031807", + "content": "设数列 $A$: $a_1, a_2, \\cdots, a_N$($N \\geq 2$), 如果对小于 $n$($2 \\leq n \\leq N$) 的每个正整数 $k$ 都有 $a_k< a_n$, 则称 $n$ 是数列 $A$ 的一个``$G$ 点''. 记 $G(A)$ 是数列 $A$ 的所有``$G$ 点''组成的集合.\\\\\n(1) 对数列 $A$: $-3,-1,3,2,6,0$, 写出 $G(A)$ 的所有元素;\\\\\n(2) 若数列 $A$: $a_1, a_2, \\cdots, a_{100}$, 满足 $a_n=\\begin{cases}3n-1,& n \\text{为奇数},\\\\2^n,& n \\text{为偶数},\\end{cases}$ 求 $G(A)$ 的所有元素之和;\\\\\n(3) 证明: 若数列 $A$ 满足 $a_n-a_{n-1}\\leq 1$($n=2,3, \\cdots, N$), 则 $G(A)$ 的元素个数不小于 $a_N-a_1$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷02试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031808": { + "id": "031808", + "content": "已知集合 $A=\\{x | x^2-x<0\\}$, $B=(0, a)$($a>0$), 若 $A \\subseteq B$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031809": { + "id": "031809", + "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 若当 $x \\geq 0$ 时, $f(x)=\\log _3(3+x)$, 则 $f(-6)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031810": { + "id": "031810", + "content": "已知双曲线 $y^2-\\dfrac{x^2}{m^2}=1$($m>0$) 的一条渐近线方程为 $x+\\sqrt{3}y=0$, 则 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031811": { + "id": "031811", + "content": "在大小相同的 6 个球中, 2 个是红球, 4 个是白球, 若从中任意选取 3 个, 则所选的 3 个球中至少有一个红球的概率是 (结果用分数表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031812": { + "id": "031812", + "content": "已知等比数列 $\\{a_n\\}$ ($n$ 是正整数) 满足 $a_2 a_6=4(a_4-1)$, 则 $a_4=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031813": { + "id": "031813", + "content": "若 $x_1$、$x_2$ 为方程 $2^x=(\\dfrac{1}{2})^{-\\frac{1}{x}+1}$ 的两个实数解, 则 $x_1+x_2=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031814": { + "id": "031814", + "content": "已知 $\\alpha, \\beta$ 为锐角, 且 $\\cos (\\alpha+\\beta)=\\dfrac{\\sin \\alpha}{\\sin \\beta}$, 则 $\\tan \\alpha$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031815": { + "id": "031815", + "content": "已知关于 $x$ 的一元二次不等式 $a x^2+2 x+b>0$ 的解集为 $\\{x | x \\neq c\\}$, 则 $\\dfrac{a^2+b^2+7}{a+c}$(其中 $a+c \\neq 0$) 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031816": { + "id": "031816", + "content": "已知 $a-a^{-1}=1$, 则 $a^{12}+a^{-12}$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031817": { + "id": "031817", + "content": "关于函数 $f(x)=\\lg \\dfrac{x}{x^2+1}$, 有下列结论: \\textcircled{1} 函数 $f(x)$ 的定义域是 ($0,+\\infty$); \\textcircled{2} 函数 $f(x)$ 是奇函数; \\textcircled{3} 函数 $f(x)$ 的最小值为 $-\\lg 2$; \\textcircled{4} 当 $01$ 时,函数 $f(x)$ 是减函数. 其中正确结论的序号是\\blank{50}. (写出所有你认为正确的结论的序号)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031818": { + "id": "031818", + "content": "方程 $|a x-1|=x$ 的解集为 $A$, 若 $A \\subset[0,2]$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031819": { + "id": "031819", + "content": "已知椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的左右顶点分别为 $A, B$, 过点 $C(0,1)$ 斜率为 $k$($k>1$) 的直线 $l$ 与椭圆交于 $M, N$, 记直线 $AM, BN$ 的斜率为 $k_1, k_2$, 且 $k_1=2 k_2$, 则 $k=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-2,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\filldraw (-1,0) node [below] {$F_1$} coordinate (F_1) circle (0.03);\n\\filldraw (1,0) node [below] {$F_2$} coordinate (F_2) circle (0.03);\n\\filldraw (0,1) node [right] {$C$} coordinate (C) circle (0.03);\n\\draw [name path = MN] (C) ++ (0.8,1.2) --++ (-2.6,-3.9);\n\\draw [name intersections = {of = MN and elli, by = {M,N}}];\n\\draw (M)--(A)(N)--(B);\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031820": { + "id": "031820", + "content": "下列函数中, 既不是奇函数, 也不是偶函数的是\\bracket{20}.\n\\fourch{$y=x+\\mathrm{e}^x$}{$y=x+\\dfrac{1}{x}$}{$y=2^x+\\dfrac{1}{2^x}$}{$y=\\sqrt{1+x^2}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031821": { + "id": "031821", + "content": "已知 $A$ 为 $\\triangle ABC$ 的一个内角, 且 $\\sin A+\\cos A=\\dfrac{\\sqrt{2}}{3}$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{锐角三角形}{钝角三角形}{直角三角形}{不确定}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031822": { + "id": "031822", + "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 是棱 $CC_1$ 的中点, $F$ 是侧面 $B_1BCC_1$ 上的动点, 并且 $A_1F \\parallel $ 平面 $AED_1$, 则动点 $F$ 的轨迹是\\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 [above] {$B_1$} 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 ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D_1)--cycle;\n\\filldraw (2,1.3,-0.3) node [right] {$F$} coordinate (F) circle (0.03);\n\\draw [dashed] (A_1)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{圆}{椭圆}{抛物线}{线段}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031823": { + "id": "031823", + "content": "设函数 $f(x)=a_1 \\cdot \\sin (x+\\alpha_1)+a_2 \\cdot \\sin (x+\\alpha_2)+\\cdots+a_n \\cdot \\sin (x+\\alpha_n)$, 其中 $a_i, \\alpha_i (i=1,2, \\cdots, n \\in \\mathbf{N}, n \\geq 2$) 为已知实常数, $x \\in \\mathbf{R}$. 下列关于函数 $f(x)$ 的性质判断正确的个数是\\bracket{20}.\\\\\n\\textcircled{1} 若 $f(0)=f(\\dfrac{\\pi}{2})=0$, 则 $f(x)=0$ 对任意实数 $x$ 恒成立;\\\\\n\\textcircled{2} 若 $f(0)=0$, 则函数 $f(x)$ 为奇函数;\\\\\n\\textcircled{3} 若 $f(\\dfrac{\\pi}{2})=0$, 则函数 $f(x)$ 为偶函数;\\\\\n\\textcircled{4} 当 $f^2(0)+f^2(\\dfrac{\\pi}{2}) \\neq 0$ 时, 若 $f(x_1)=f(x_2)=0$, 则 $x_1-x_2=k \\pi$($k \\in \\mathbf{Z}$)\n\\fourch{$4$}{$3$}{$2$}{$1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031824": { + "id": "031824", + "content": "已知数列 $\\{a_n\\}$ 是等差数列, 且 $a_1=2$, $a_1+a_2+a_3=12$.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 令 $b_n=2^{a_n}+9$, 数列 $\\{b_n\\}$ 前 $n$ 项和为 $S_n$, 若 $S_n \\geq 2022$, 求正整数 $n$ 的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031825": { + "id": "031825", + "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E, P$ 分别是 $BC, A_1D_1$ 的中点, $M, N$ 分别是 $AE, CD_1$ 的中点, $AD=AA_1=a$, $AB=2 a$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{4}\n\\def\\m{2}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(D_1)$) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(E)$) node [above left] {$M$} coordinate (M);\n\\draw [dashed] (A)--(E)--(P)--cycle(M)--(N)(C)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 面 $ADD_1A_1$;\\\\\n(2) 求三棱锥 $P-DEN$ 的体积.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031826": { + "id": "031826", + "content": "某项选拔共有三轮考核, 每轮设有一个问题, 能正确回答问题者进人下一轮考试, 否则即被淘汰, 已知某选手能正确回答第一、二、三轮的问题的概率分别为 $\\dfrac{4}{5}$、$\\dfrac{3}{5}$、$\\dfrac{2}{5}$, 且各轮问题能否正确回答互不影响.\\\\\n(1) 求该选手被淘汰的概率;\\\\\n(2) 该选手在选拔中回答问题的个数记为 $\\xi$, 求随机变量 $\\xi$ 的分布列与期望. (注: 本小题结果可用分数表示)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031827": { + "id": "031827", + "content": "已知函数 $f(x)=x^4+a x^3+2 x^2+b$($x \\in \\mathbf{R}$), 其中 $a, b \\in \\mathbf{R}$.\\\\\n(1) 当 $a=-\\dfrac{10}{3}$ 时, 讨论函数 $f(x)$ 的单调性;\\\\\n(2) 若函数 $f(x)$ 仅在 $x=0$ 处有极值, 求 $a$ 的取值范围;\\\\\n(3) 若对于任意的 $a \\in[-2,2]$, 不等式 $f(x) \\leq 1$ 在 $[-1,1]$ 上恒成立,求 $b$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031828": { + "id": "031828", + "content": "在平面直角坐标系 $xOy$ 中, 曲线 $C$ 上的点 $S(x, y)$ 到点 $M(\\sqrt{3}, 0)$ 的距离与它到直线 $x= \\dfrac{4}{\\sqrt{3}}$ 的距离之比为 $\\dfrac{\\sqrt{3}}{2}$, 圆 $O$ 的方程为 $x^2+y^2=4$, 曲线 $C$ 与 $x$ 轴的正半轴的交点为 $A$, 过原点 $O$ 且异于坐标轴的直线与曲线 $C$ 交于 $B, C$ 两点, 直线 $AB$ 与圆 $O$ 的另一交点为 $P$, 直线 $PD$ 与圆 $O$ 的另一交点为 $Q$, 其中 $D(-\\dfrac{6}{5}, 0)$, 设直线 $AB, AC$ 的斜率分别为 $k_1, k_2$.\\\\\n(1) 求曲线 $C$ 的方程, 并证明 $S(x, y)$ 到点 $M$ 的距离 $d \\in[2-\\sqrt{3}, 2+\\sqrt{3}]$;\\\\\n(2) 求 $k_1 k_2$ 的值;\\\\\n(3) 记直线 $PQ, BC$ 的斜率分别为 $k_{PQ}, k_{BC}$, 是否存在常数 $\\lambda$, 使得 $k_{PQ}=\\lambda k_{BC}$ ? 若存在, 求 $\\lambda$ 的值;若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷03试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031829": { + "id": "031829", + "content": "已知复数 $z=3-4 \\mathrm{i}$, 则 $|z|=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031830": { + "id": "031830", + "content": "不等式 $|x-2|<3$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031831": { + "id": "031831", + "content": "已知圆的方程为 $x^2+y^2+2 x-4 y=0$, 则该圆的圆心坐标为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031832": { + "id": "031832", + "content": "设函数 $f(x)=\\begin{cases}2-x^2,& x \\leq 2,\\\\\\dfrac{1}{2}f(x-2),& x>2,\\end{cases}$ 则 $f(6)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031833": { + "id": "031833", + "content": "已知向量 $\\overrightarrow{a}=(2,-3)$, $\\overrightarrow{b}=(3, \\lambda)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $\\lambda$ 等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031834": { + "id": "031834", + "content": "函数 $f(x)=2^x+\\dfrac{9}{2^x+1}$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031835": { + "id": "031835", + "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_{2022}=a_{20}+a_{22}=2$, 则 $a_1$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031836": { + "id": "031836", + "content": "将 6 名冬奥会志愿者分配到花样滑冰、冰球和冰壶等 5 个不同项目进行培训, 每名志愿者只分配到 1 个项目, 每个项目至少分配 1 名志愿者, 则不同的分配方案共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031837": { + "id": "031837", + "content": "设集合 $A=\\{1,2, m\\}$, 其中 $m$ 为实数, 令 $B=\\{t | t=a^2, a \\in A\\}$, $C=A \\cup B$, 若 $C$ 的所有元素之和为 6 ,则 $C$ 的所有元素之积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031838": { + "id": "031838", + "content": "已知某校的午餐由``主食''与``配菜''两部分组成, 主食和配菜均有若干种不同的选择. 某个学期的统计结果显示, 学生购买主食 $A$ 的概率为 $25 \\%$, 而在购买主食 $A$ 的学生中, 又有 $70 \\%$ 的学生会购买配菜 $B$ 作为搭配,则学生同时购买主食 $A$ 与配菜 $B$ 的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031839": { + "id": "031839", + "content": "设函数 $f(x)=\\cos x+\\log _2 x$($x>0$), 若正实数 $a$ 满足 $f(a)=f(2 a)$, 则 $f(2 a)-f(4 a)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031840": { + "id": "031840", + "content": "设正整数 $n=a_0 \\cdot 2^0+a_1 \\cdot 2+\\cdots+a_{k-1}\\cdot 2^{k-1}+a_k \\cdot 2^k$, 其中 $a_i \\in\\{0,1\\}$, $i=0,1,2, \\cdots, k$, 记 $\\omega(n)=a_0+a_1+\\cdots a_k$. 给出下列结论: \\textcircled{1} $\\omega(2 n)=\\omega(n)$; \\textcircled{2} $\\omega(2 n+3)=\\omega(n)+1$; \\textcircled{3} $\\omega(8 n+5)=\\omega(4 n+3)$; \\textcircled{4} $\\omega(2^n-1)=n$. 其中正确结论的序号为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031841": { + "id": "031841", + "content": "已知 $a\\dfrac{1}{b}$}{$a^2>b^2$}{$2-a>2-b$}{$2^a>2^b$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031842": { + "id": "031842", + "content": "已知曲线 $\\Gamma$ 的普通方程为: $x^2+y^2=4$($x \\leq 0$, $y \\geq 0$), 则曲线 $\\Gamma$ 的参数方程为\\bracket{20}.\n\\twoch{$\\begin{cases}x=2 \\cos \\theta,\\\\y=2 \\sin \\theta,\\end{cases} \\theta \\in[0, \\pi]$}{$\\begin{cases}x=2 \\sin \\theta,\\\\y=2 \\cos \\theta,\\end{cases}\\theta \\in[0, \\pi]$}{$\\begin{cases}x=2 \\sin \\theta,\\\\y=2 \\cos \\theta,\\end{cases}\\theta \\in[\\dfrac{\\pi}{2}, \\pi]$}{$\\begin{cases}x=2 \\cos \\theta,\\\\y=2 \\sin \\theta,\\end{cases}\\theta \\in[\\dfrac{\\pi}{2}, \\pi]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031843": { + "id": "031843", + "content": "如图所示某加油站地下圆柱体储油罐示意图, 已知储油罐长度为 $d$, 圆柱底面半径为 $r$ ($d, r$ 为常量), 油面高度为 $h$, 油面宽度为 $w$, 储油量为 $v(h, w, v$ 为变量), 则下列说法: \\textcircled{1} $w$ 是 $v$ 的函数; \\textcircled{2} $v$ 是 $w$ 的函数; \\textcircled{3} $h$ 是 $w$ 的函数; \\textcircled{4} $w$ 是 $h$ 的函数. 其中正确说法的序号为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(0.91)},-0.3) coordinate (A);\n\\draw ({sqrt(0.91)},-0.3) coordinate (B);\n\\draw (B)++(0,0,-5) coordinate (C);\n\\draw (A)++(0,0,-5) coordinate (D);\n\\draw (A)--(B)--(C);\n\\draw [dashed] (C)--(D)--(A);\n\\draw (0,-0.3) -- (0,-1) node [midway, right] {$h$};\n\\fill [gray!30] (A)--(B)--(C)--(D)--cycle;\n\\draw [domain = 0:360] plot ({cos(\\x)},{sin(\\x)},0);\n\\draw [domain = -45:135] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw [domain = 135:315, dashed] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw (-45:1) --++ (0,0,-5) (135:1) --++ (0,0,-5);\n\\draw (-45:1) ++ (0,-0.1) --++ (0,-0.8) (-45:1) ++ (0,0,-5) ++ (0,-0.1) --++ (0,-0.8);\n\\draw [<->] (-45:1) ++ (0,-0.5) --++ (0,0,-5) node [midway, fill = white] {$d$};\n\\draw ($(C)!0.5!(D)$) node [above] {$w$};\n\\draw (0,0) --++ (135:1) node [midway, below left] {$r$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031844": { + "id": "031844", + "content": "等差数列 $a_1, a_2, \\cdots, a_n$($n \\in \\mathbf{N}$, $n \\geq 3$) 满足 $|a_1|+|a_2|+\\cdots+|a_n|=|a_1+1|+|a_2+1| +\\cdots+|a_n+1|=|a_1-2|+|a_2-2|+\\cdots+|a_n-2|=2019$, 则\\bracket{20}.\n\\fourch{$n$ 的最大值为 $50$}{$n$ 的最小值为 $50$}{$n$ 的最大值为 $51$}{$n$ 的最小值为 $51$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031845": { + "id": "031845", + "content": "如图, 正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长为 $1$, 高为 $2$, $M$ 为线段 $AB$ 的中点. 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (C_1)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥 $A_1-MBC_1$ 的体积;\\\\\n(2) 异面直线 $AC$ 与 $MC_1$ 所成角的大小 (结果用反三角函数值表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031846": { + "id": "031846", + "content": "已知函数 $f(x)=a x+\\dfrac{1}{x-1}$, $a \\in \\mathbf{R}$.\\\\\n(1) 当 $a=2$ 时,求不等式 $f(x+2) \\leq f(x)+4$ 的解集;\\\\\n(2) 若函数 $f(x)$ 在区间 $[2,5]$ 上严格减, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031847": { + "id": "031847", + "content": "如图, 有一块边长为 $3 \\mathrm{m}$ 的正方形铁皮 $ABCD$, 其中阴影部分 $ATN$ 是一个半径为 $2 \\mathrm{m}$ 的扇形, 设这个扇形部分已经腐蚀不能使用, 但其余部分均完好. 工人师傅想在末腐蚀的部分截下一块其边落在 $BC$ 与 $CD$ 上的矩形铁皮 $PQCR$, 其中点 $P$ 在圆弧 $NT$ 上. 设 $\\angle TAP=\\theta$, 矩形 $PQCR$ 的面积为 $S \\mathrm{m}^2$.\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{scope}[even odd rule]\n\\clip (0,0) rectangle (3,3) (0.55,0.15) circle (0.15);\n\\fill [gray!30] (0,0) -- (2,0) arc (0:90:2) -- cycle;\n\\end{scope}\n\\draw (0,0) node [below left] {$A$} -- (3,0) node [below right] {$B$} -- (3,3) node [above right] {$C$} -- (0,3) node [above left] {$D$} -- cycle;\n\\draw (0,0) -- (40:2) node [above right] {$P$} -- (3,{2*sin(40)}) node [right] {$Q$} (40:2) -- ({2*cos(40)},3) node [above] {$R$};\n\\draw (0,0) -- (2,0) node [below] {$T$} arc (0:90:2) node [left] {$N$} -- cycle;\n\\draw (0.4,0) arc (0:40:0.4) (0.55,0.15) node {$\\theta$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $S$ 关于 $\\theta$ 的函数表达式;\\\\\n(2) 求 $S$ 的最大值及取最大值时 $\\theta$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031848": { + "id": "031848", + "content": "在平面直角坐标系 $xOy$ 中, 已知椭圆 $E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{5}=1$($a>\\sqrt{5}$) 的左. 右顶点为 $A$, $B$, $G$ 为 $E$ 的上项点, $\\overrightarrow{AG}\\cdot \\overrightarrow{GB}=4$. 设过点 $T(t, m)$ 的直线 $TA, TB$ 与椭圆分别交于点 $M(x_1, y_1)$, $N(x_2, y_2)$, 其中 $y_1>0$, $y_2<0$, $m>0$.\\\\\n(1) 求椭圆 $E$ 的右焦点 $F$ 的坐标;\\\\\n(2) 设 $x_1=2$, $x_2=\\dfrac{1}{3}$, 求点 $T$ 的坐标;\\\\\n(3) 若点 $T$ 为直线 $x=9$ 上的动点,求证: 直线 $MN$ 必过 $x$ 轴上的定点 $D(1,0)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷04试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031849": { + "id": "031849", + "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$. 若对于任意的正整数 $n$, 都有 $\\dfrac{a_{n+1}}{a_n}\\in[\\dfrac{1}{2}, 2]$, 则称数列 $\\{a_n\\}$ 是``$M$ 型数列''.\\\\\n(1) 若 $\\{a_n\\}$ 是``$M$ 型数列'', 且 $a_1=1$, $a_2=2$, $a_3=m$, $a_4=5$, 求 $m$ 的取值范围;\\\\\n(2) 若数列 $\\{a_n\\}$ 为等差数列, 公差为 $d$, 且 $00$) 的准线相切, 则 $p=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031855": { + "id": "031855", + "content": "若关于 $x$ 的方程 $2 \\sqrt{3}\\cos ^2 x-\\sin 2 x=\\sqrt{3}-m$ 在区间 $[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{6}]$ 上有且只有一个解, 则 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031856": { + "id": "031856", + "content": "在正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=3$, 直线 $AC_1$ 与平面 $BCC_1B_1$ 所成角大小为 $30^{\\circ}$, 则该正四棱柱的外接球表面积大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031857": { + "id": "031857", + "content": "等比数列 $\\{a_n\\}$ 的各项均为正数, 且 $a_{10}a_{11}+a_9 a_{12}=2 \\mathrm{e}^5$, 则 $\\ln a_1+\\ln a_2+\\cdots+\\ln a_{20}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031858": { + "id": "031858", + "content": "如果用反证法证明:``数列 $\\{a_n\\}$ 的各项均小于 2'', 有下列四种不同的假设:\\\\\n\\textcircled{1} 数列 $\\{a_n\\}$ 的各项均大于 $2$;\\\\\n\\textcircled{2} 数列 $\\{a_n\\}$ 的各项均大于或等于 $2$;\\\\\n\\textcircled{3} 数列 $\\{a_n\\}$ 中存在一项 $a_k$, $a_k \\geq 2$;\\\\\n\\textcircled{4} 数列 $\\{a_n\\}$ 中存在一项 $a_k$, $a_k>2$.\\\\\n其中正确的序号为\\blank{50}.(填写出所有假设正确的序号)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031859": { + "id": "031859", + "content": "对于函数 $f(x)$, 若在定义域 $D$ 上存在实数 $x$, 满足 $f(-x)=-f(x)$, 则称 $f(x)$ 为 $D$ 上``局部奇函数''. 已知 $f(x)=2^x+m$ 是定义在 $[-1,1]$ 上的``局部奇函数'', 则实数 $m$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031860": { + "id": "031860", + "content": "已知数列 $\\{a_n\\}$ 满足: 当 $n \\geq 3$ 时, $a_n=2 a_{n-1}$ 或 $a_n=a_{n-1}+a_{n-2}$, 若 $a_1=1$, $a_2=2$, 则此数列前 2015 项中, 奇数项最多有\\blank{50}项.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031861": { + "id": "031861", + "content": "设 $f(x)=\\begin{cases}2^x,& x \\leq 0,\\\\\\log _2 x,& x>0,\\end{cases}$ 若对任意 $y \\in$($2,+\\infty$), 都存在唯一的实数 $x$, 满足 $f(f(x))=2 a^2 y^2+a y$, 则正数 $a$ 的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031862": { + "id": "031862", + "content": "直线 $l_1: x+a y+2=0$ 和直线 $l_2:(a-2) x+3 y+6 a=0$, 则``$a=3$''是``$l_1 \\parallel l_2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031863": { + "id": "031863", + "content": "$y=x^2$ 与 $y=\\ln (x+a)$ 有一条斜率为 $2$ 的公切线, 则 $a=$\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2}\\ln 2$}{$\\dfrac{1}{2}\\ln 2$}{$-\\ln 2$}{$\\ln 2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031864": { + "id": "031864", + "content": "下列命题正确的是\\bracket{20}.\n\\twoch{若 $a b \\neq 0$, 则 $\\dfrac{b}{a}+\\dfrac{a}{b}\\geq 2$}{若 $a<0$, 则 $a+\\dfrac{4}{a}\\geq-4$}{若 $a>0$, $b>0$, 则 $\\lg a+\\lg b \\geq 2 \\sqrt{\\lg a \\cdot \\lg b}$}{若 $x \\neq k \\pi$, $k \\in \\mathbf{Z}$, 则 $\\sin ^2 x+\\dfrac{4}{\\sin ^2 x}\\geq 5$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031865": { + "id": "031865", + "content": "$P$ 为双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 左支上一点, $F_1, F_2$ 为其左右焦点, 若 $\\dfrac{|PF_2|^2}{|PF_1|}$ 的最小值为 $10 a$, 则双曲线的离心率为\\bracket{20}.\n\\fourch{$4+\\sqrt{5}$}{$4-\\sqrt{5}$}{$4 \\pm \\sqrt{5}$}{4}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031866": { + "id": "031866", + "content": "如图, $A, B$ 是单位圆 $O$ 上的动点, $C$ 是圆与 $x$ 轴正半轴的交点, 设 $\\angle COA=\\alpha$.\n\\begin{center}\n\\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 (0,0) circle (1);\n\\draw (1,0) node [below right] {$C$} coordinate (C);\n\\draw (0.6,0.8) node [above right] {$A$} coordinate (A);\n\\draw ({60+atan(4/3)}:1) node [above left] {$B$} coordinate (B);\n\\draw (0,0) -- (A)(0,0) --(B) --(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 当点 $A$ 的坐标为 $(\\dfrac{3}{5}, \\dfrac{4}{5})$ 时, 求 $\\dfrac{\\cos 2 \\alpha}{1+\\sin 2 \\alpha}$ 的值;\\\\\n(2) 若 $0 \\leq \\alpha \\leq \\dfrac{\\pi}{3}$, 且当点 $A, B$ 在圆上沿逆时针方向移动时, 总有 $\\angle AOB=\\dfrac{\\pi}{3}$, 试求 $BC$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031867": { + "id": "031867", + "content": "在直角 $\\triangle ABC$ 中, $\\angle ABC=90^{\\circ}$, $AC=2 \\sqrt{3}$, $AB=\\sqrt{3}, D, E$ 分别为 $AC, BD$ 的中点, 连结 $AE$ 并延长交 $BC$ 于点 $F$, 将 $\\triangle ABD$ 沿 $BD$ 折起, 使平面 $ABD \\perp$ 平面 $BCD$, 如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw ({2*sqrt(3)},0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [above right] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [above] {$E$} coordinate (E);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--cycle(B)--(D)(A)--(F);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw ({sqrt(3)},0,{-sqrt(3)/2}) node [above right] {$D$} coordinate (D);\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,0) node [below] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [above left] {$E$} coordinate (E);\n\\draw (E) ++ (0,{3/2},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A)--(F);\n\\draw [dashed] (B)--(D)--(C)(D)--(A)(A)--(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AE \\perp CD$;\\\\\n(2) 求平面 $AEF$ 与平面 $ADC$ 所成二面角的正弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031868": { + "id": "031868", + "content": "研究表明, 过量的排放会导致全球气候变暖等环境问题, 减少碳排放具有深远的意义. 中国明确提出节能减排的目标与各项措施, 在公路交通运输领域, 新能源汽车逐步取代燃油车是措施之一, 中国某地区从 2015 年至 2021 年每年汽车总销量如图, 每年新能源汽车销量占比如表. (注: 汽车总销量指新能源汽车销量与非新能源汽车销量之和)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 年份 & 2015 & 2016 & 2017 & 2018 & 2019 & 2020 & 2021 \\\\\n\\hline 新能源汽车销量占比 & $1.5 \\%$ & $2 \\%$ & $3 \\%$ & $5 \\%$ & $8 \\%$ & $9 \\%$ & $20 \\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[>=latex, yscale = 0.5]\n\\draw [->] (0,0) -- (8,0) node [below] {年份};\n\\draw [->] (0,0) -- (0,8) node [right] {汽车总销量(万辆)};\n\\foreach \\i/\\j/\\k in {2015/1/4.1,2016/2/5.6,2017/3/5.6,2018/4/5.5,2019/5/5.7,2020/6/6,2021/7/5.8}\n{\\draw (\\j,\\k) node [above] {$\\k$};\n\\draw (\\j,0) ++ (-0.3,0) --++ (0,\\k) --++ (0.6,0) --++ (0,-\\k);\n\\draw (\\j,0) node [below] {$\\i$};};\n\\end{tikzpicture}\n\\end{center}\n(1) 从 2015 年至 2021 年中随机选取一年, 求这一年该地区汽车总销量不小于 $5.5$ 万辆的概率;\\\\\n(2) 从 2015 年至 2021 年随机选取两年, 设 $X$ 表示新能源汽车销量超过 $0.5$ 万辆的年份的个数,求 $X$ 的分布列和数学期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031869": { + "id": "031869", + "content": "已知关于 $x$ 的函数 $y=f(x)$, $y=g(x)$ 与 $h(x)=k x+b$($k, b \\in \\mathbf{R}$) 在区间 $D$ 上恒有 $f(x) \\geq h(x) \\geq g(x)$.\\\\\n(1) 若 $f(x)=x^2+2 x$, $g(x)=-x^2+2 x$, $D=$($-\\infty$, $+\\infty$), 求 $h(x)$ 的表达式;\\\\\n(2) 若 $f(x)=x^2-x+1$, $g(x)=k \\ln x$, $h(x)=k x-k$, $D \\in$($0,+\\infty$), 求 $k$ 的取值范围;\\\\\n(3) 若 $f(x)=x^4-2 x^2$, $g(x)=4 x^2-8$, $h(x)=4(t^3-t) x-3 t^4+2 t^2$($0<|t| \\leq \\sqrt{2}$), $D=[m, n] \\subset[-\\sqrt{2}, \\sqrt{2}]$, 求证: $|m-n| \\leq \\sqrt{7}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031870": { + "id": "031870", + "content": "数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若对任意的正整数 $n$, 总存在正整数 $m$, 使得 $S_n=a_m$, 则称数列 $\\{a_n\\}$ 是``$E$ 数列''.\\\\\n(1) 数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=3^n, n$ 为正整数, 判断数列 $\\{a_n\\}$ 是否为``$E$ 数列'', 并说明理由;\\\\\n(2) 数列 $\\{b_n\\}$ 是等差数列, 其首项 $b_1=1$, 公差 $d<0$, 数列 $\\{b_n\\}$ 是``$E$ 数列'', 求 $d$ 的值;\\\\\n(3) 证明: 对任意的等差数列 $\\{a_n\\}$, 总存在两个``$E$ 数列''$\\{b_n\\}$ 和 $\\{c_n\\}$, 使得 $a_n=b_n+c_n$, 对一切 $n$ 为正整数成立.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷05试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031871": { + "id": "031871", + "content": "已知复数 $z=-1+\\mathrm{i}$($\\mathrm{i}$ 为虚数单位), 计算: $\\dfrac{z \\cdot \\overline{z}}{z-\\overline{z}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031872": { + "id": "031872", + "content": "已知直线 $l$ 的方程为 $2 x-y-3=0$, 则直线 $l$ 的倾斜角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031873": { + "id": "031873", + "content": "在 $(x-\\dfrac{1}{2 x})^{10}$ 的二项展开式中, $x^4$ 的系数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031874": { + "id": "031874", + "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_2 a_8=2 a_3 a_6$, $S_5=-62$, 则 $a_1$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031875": { + "id": "031875", + "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ 的一条渐近线与圆 $x^2+y^2-6 x+4 y=0$ 相切, 则该双曲线的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031876": { + "id": "031876", + "content": "不等式``$|x-m|<1$''是不等式``$\\log _2 x>1$''成立的充分不必要条件, 则 $m$ 的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031877": { + "id": "031877", + "content": "已知某拍卖行组织拍卖的 $10$ 幅名画中, 有 $2$ 幅是赝品. 某人在这次拍卖中随机买入了两幅画, 则此人买人的这两幅画中恰有一幅是赝品的事件的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031878": { + "id": "031878", + "content": "已知角 $\\varphi$ 的终边经过点 $P(1,-1)$, 点 $A(x_1, y_1), B(x_2, y_2)$ 是函数 $f(x)=\\sin (\\omega x+\\varphi) $($\\omega>0$) 图像上的任意两点, 若 $|f(x_1)-f(x_2)|=2$ 时, $|x_1-x_2|$ 的最小值为 $\\dfrac{\\pi}{3}$, 则 $f(\\dfrac{\\pi}{2})$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031879": { + "id": "031879", + "content": "已知圆锥底面半径与球的半径都是 $1 \\mathrm{cm}$, 如果圆锥的体积恰好也与球的体积相等, 那么这个圆锥的母线长为\\blank{50}$\\mathrm{cm}$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031880": { + "id": "031880", + "content": "函数 $f(x)=\\dfrac{1}{x-1}-2 \\sin \\pi x$ 在区间 $[-2,4]$ 上的所有零点之和等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031881": { + "id": "031881", + "content": "某单位 $6$ 个员工借助互联网开展工作, 每个员工上网的概率都是 $0.5$ (相互独立), 则至少有 $3$ 人同时上网的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031882": { + "id": "031882", + "content": "设 $M$ 是 $\\triangle ABC$ 内一点, $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=2 \\sqrt{3}$, $\\angle BAC=30^{\\circ}$, 定义 $f(M)=(m, n, p)$, 其中 $m, n, p$ 分别是 $\\triangle MBC, \\triangle MAC, \\triangle MAB$ 的面积, 若 $f(M)=(\\dfrac{1}{2}, x, y)$, $\\dfrac{1}{x}+\\dfrac{4}{y}=a$ 则 $\\dfrac{a^2+2}{a}$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031883": { + "id": "031883", + "content": "已知 $l, m$ 是两条不同的直线, $\\alpha$ 是一个平面, 有下列四个命题中真命题的是\\bracket{20}.\n\\onech{若 $l \\subset \\beta$, 且 $m \\perp l$, 则 $m \\perp \\alpha$}{若 $l \\perp \\alpha$, 且 $m \\parallel \\alpha$, 则 $l \\perp m$}{若 $l \\parallel m$,, 且 $m \\subset \\alpha$, 则 $l \\parallel \\alpha$}{若 $l, m$ 与平面 $\\alpha$ 所成的角相等, 则 $l \\parallel m$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031884": { + "id": "031884", + "content": "已知 $02$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031885": { + "id": "031885", + "content": "若圆柱被一平面所截, 其截面椭圆的离心率为 $\\dfrac{2 \\sqrt{2}}{3}$, 则此截面与圆柱底面所成的锐二面角的大小是\\bracket{20}.\n\\fourch{$\\arcsin \\dfrac{1}{3}$}{$\\arccos \\dfrac{1}{3}$}{$\\arcsin \\dfrac{2}{3}$}{$\\arccos \\dfrac{2}{3}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031886": { + "id": "031886", + "content": "如图所示, $f_i(x)$($i=1,2,3,4$) 是定义在 $[0,1]$ 上的四个函数, 其中满足性质:``对 $[0,1]$ 中任意的 $x_1$ 和 $x_2$, 任意 $\\lambda \\in[0,1]$, $f[\\lambda x_1+(1-\\lambda) x_2] \\leq \\lambda f(x_1)+(1-\\lambda) f(x_2)$ 恒成立''的只有\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$};\n\\draw (0.65,1.3) node {$f_1(x)$};\n\\draw [domain = 0:1, samples = 100] plot (\\x,{2*(\\x-0.7)*(\\x-0.7)-0.18});\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$};\n\\draw (0.65,1.3) node {$f_2(x)$};\n\\draw [domain = 0:1, samples = 100] plot (\\x,{-2*(\\x-0.7)*(\\x-0.7)+0.9});\n\\draw [dashed] (1,0.72) -- (1,0);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$};\n\\draw (0.65,1.3) node {$f_3(x)$};\n\\draw (0,0) -- (1,1);\n\\draw [dashed] (1,1) -- (1,0);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.2,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) node [below] {$1$};\n\\draw (0.65,1.3) node {$f_4(x)$};\n\\draw [domain = 0:1, samples = 100] plot (\\x,{0.5+0.5*sin(180*\\x-90)});\n\\draw [dashed] (1,1) -- (1,0);\n\\end{tikzpicture}\n\\end{center} \n\\fourch{$f_1(x), f_3(x)$}{$f_2(x)$}{$f_2(x), f_3(x)$}{$f_4(x)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031887": { + "id": "031887", + "content": "如图, 四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面 $ABCD$ 是平行四边形, 且 $AB=1$, $BC=2$, $\\angle ABC=60^{\\circ}$, $E$ 为 $BC$ 的中点, $AA_1 \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{{sqrt(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 (C) ++ ({1/2},0,{-sqrt(3)/2}) node [right] {$D$} coordinate (D);\n\\draw (B) ++ ({1/2},0,{-sqrt(3)/2}) 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 (B_1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D)--(A_1)(A_1)--(C)(A_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\perp$ 平面 $A_1AE$;\\\\\n(2) 若 $DE=A_1E$, 试求异面直线 $AE$ 与 $A_1D$ 所成角的余弦值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031888": { + "id": "031888", + "content": "已知 $a>0$, 函数 $f(x)=\\dfrac{1-a x}{x}$, $x \\in (0,+\\infty)$. 设 $0b>0$) 的焦距与长轴长的比为 $\\dfrac{\\sqrt{2}}{2}$, $x$ 轴被曲线 $C_2: y= x^2-b$ 截得的线段长等于 $C_1$ 的短轴长. $C_2$ 与 $y$ 轴的交点为 $M$, 过坐标原点 $O$ 的直线 $l$ 与 $C_2$ 相交于点 $A$、$B$, 直线 $MA, MB$ 分别与 $C_1$ 相交于点 $D$、$E$.\\\\\n(1) 求 $C_1, C_2$ 的方程;\\\\\n(2) 求证: $MA \\perp MB$;\\\\\n(3) 记 $\\triangle MAB, \\triangle MDE$ 的面积分别为 $S_1$、$S_2$, 若 $\\dfrac{S_1}{S_2}=\\lambda$, 求 $\\lambda$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031891": { + "id": "031891", + "content": "已知 $\\{a_n\\}$ 是首项为 $2$, 公比为 $\\dfrac{1}{2}$ 的等比数列, $S_n$ 为它的前 $n$ 项和.\\\\\n(1) 用 $S_n$ 表示 $S_{n+1}$;\\\\\n(2) 比较 $\\dfrac{4}{a_n}$ 与 $n^2$ 的大小;\\\\\n(3) 是否存在自然数 $c$ 和 $k$, 使得 $\\dfrac{S_{k+1}-c}{S_k-c}>2$ 成立.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷06试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031892": { + "id": "031892", + "content": "已知全集 $U=\\mathbf{R}$, 集合 $A=\\{x | y=\\sqrt{1-\\dfrac{1}{x}}\\}$, 则 $\\overline{A}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031893": { + "id": "031893", + "content": "函数 $y=2 \\cos ^2(\\pi x)-1$ 的最小正周期为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031894": { + "id": "031894", + "content": "在等差数列 $\\{a_n\\}$ 中,若 $a_1+a_7+a_8+a_{12}=12$, 则此数列的前 $13$ 项之和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031895": { + "id": "031895", + "content": "不等式 $\\dfrac{x+2}{x+1}>1$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031896": { + "id": "031896", + "content": "若椭圆 $\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$ 的离心率为 $\\dfrac{\\sqrt{10}}{5}$, 则 $m$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031897": { + "id": "031897", + "content": "半径为 $4$ 的圆内接三角形 $ABC$ 的面积是 $\\dfrac{1}{16}$, 角 $A$、$B$、$C$ 所对应的边依次为 $a$、$b$、$c$, 则 $a b c$ 的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031898": { + "id": "031898", + "content": "将一个正方形绕着它的一边所在的直线旋转一周, 所得几何体的体积为 $27 \\pi \\mathrm{cm}^3$, 则该几何体的侧面积为\\blank{50}$\\mathrm{cm}^2$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031899": { + "id": "031899", + "content": "已知函数 $y=f(x)$ 是奇函数, 当 $x<0$ 时, $f(x)=2^x-a x$, 且 $f(2)=2$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031900": { + "id": "031900", + "content": "若函数 $y=f(x-2)$ 的图像与函数 $y=\\log _3 \\sqrt{x}+2$ 的图像关于直线 $y=x$ 对称, 则 $f(x)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031901": { + "id": "031901", + "content": "已知 $a$、$b$ 为不垂直的异面直线, $\\alpha$ 是一个平面,则 $a$、$b$ 在 $\\alpha$ 上的射影有可能是: \\textcircled{1} 两条平行直线 ; \\textcircled{2} 两条互相垂直的直线; \\textcircled{3} 同一条直线; \\textcircled{4}一条直线及其外一点. 在上面结论中, 正确结论的编号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031902": { + "id": "031902", + "content": "甲、乙两人玩猜子游戏, 每次甲出 1 子, 2 子或 3 子, 由乙猜. 若乙猜中, 则甲所出之子归乙, 若乙末猜中, 则乙付给甲 1 子. 已知甲出 1 子、 2 子或 3 子的概率分别为 $\\dfrac{6}{13}, \\dfrac{4}{13}, \\dfrac{3}{13}$, 乙每次猜 1 子, 2 子, 3 子的概率均为 $\\dfrac{1}{3}$, 则乙每次赢得子数的期望为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031903": { + "id": "031903", + "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且 $f(2)=0$, 当 $x>0$ 时, $\\dfrac{f(x)}{x}$ 的导数小于零恒成立, 则不等式 $x^2 f(x)>0$ 的解集是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031904": { + "id": "031904", + "content": "下列函数中, 同时具有性质: \\textcircled{1} 图像过点 $(0,1)$, \\textcircled{2} 在区间 ($0,+\\infty$) 上是严格减函数, \\textcircled{3} 是偶函数, 这样的函数是\\bracket{20}.\n\\fourch{$f(x)=x^2$}{$f(x)=\\log _2(|x|+2)$}{$f(x)=(\\dfrac{1}{2})$}{$f(x)=2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031905": { + "id": "031905", + "content": "若 $S_n=1-2+3-4+\\cdots+(-1)^{n-1}$, 则 $S_{17}+S_{33}+S_{50}$ 等于\\bracket{20}.\n\\fourch{$1$}{$-1$}{$0$}{$2$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031906": { + "id": "031906", + "content": "某商场买来一车苹果, 从中随机抽取了 $10$ 个苹果, 其重量 (单位: 克) 分别为: $150,152,153,149,148,146,151,150,152,147$, 由此估计这车苹果单个重量的期望值为\\bracket{20}.\n\\fourch{$150.2$ 克}{$149.8$ 克}{$149.4$ 克}{$147.8$ 克}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031907": { + "id": "031907", + "content": "给出定义: 若 $m-\\dfrac{1}{2}=latex, scale = 0.6]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw (0,4,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(S)--cycle(S)--(B);\n\\draw [dashed] (A)--(F)--(E)(A)--(D)--(C)(D)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF \\parallel $ 平面 $SAD$;\\\\\n(2) 设 $SD=2CD$, 求二面角 $A-EF-D$ 的大小.\n第 18 题图\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031910": { + "id": "031910", + "content": "某商场经销某商品, 顾客可采用一次性付款或分期付款购买. 根据以往资料统计, 顾客采用一次性付款的概率是 $0.6$ , 经销一件该商品, 若顾客采用一次性付款, 商场获得利润 $200$ 元; 若顾客采用分期付款, 商场获得利润 $250$ 元.\\\\\n(1) 求 $3$ 位购买该商品的顾客中至少有 $1$ 位采用一次性付款的概率;\\\\\n(2) 求 $3$ 位顾客每人购买 $1$ 件该商品,商场获得利润不超过 $650$ 元的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031911": { + "id": "031911", + "content": "如图, 椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), $A_1$、$A_2$ 为椭圆 $C$ 的左、右顶点. 设 $F_1$ 为椭圆 $C$ 的左焦点, 则当且仅当椭圆 $C$ 上的点 $P$ 在椭圆的左、右顶点时 $|PF_1|$ 取得最小值与最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-1,0) node [below] {$F_1$};\n\\draw (-1,0) -- (70:2 and {sqrt(3)});\n\\end{tikzpicture}\n\\end{center}\n(1) 若椭圆 $C$ 上的点到焦点距离的最大值为 $3$, 最小值为 $1$. 求椭圆 $C$ 的标准方程;\\\\\n(2) 若直线 $l: y=k x+2$ 与 (1) 中所述椭圆 $C$ 相交于 $A, B$ 两点 ($A, B$ 不是左右顶点), 且满足 $AA_2 \\perp BA_2$, 求 $k$ 的值.\\\\\n(3) 请对问题 (2) 作进一步的研究, 提出一个你认为有价值的问题, 并作出解答.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031912": { + "id": "031912", + "content": "在数列 $\\{a_n\\}$ 中, $a_1=1$, $a_2=2$, 且 $a_{n+1}=(1+q) a_n-q a_{n-1}$($n \\geq 2$, $q \\neq 0$).\\\\\n(1) 设 $b_n=a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n \\geq 1$), 证明 $\\{b_n\\}$ 是等比数列;\\\\\n(2) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 若 $a_3$ 是 $a_6$ 与 $a_9$ 的等差中项, 求 $q$ 的值, 并证明: 对任意的 $n \\in \\mathbf{N}$, $n \\geq 1, a_n$ 是 $a_{n+3}$ 与 $a_{n+6}$ 的等差中项.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷07试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031913": { + "id": "031913", + "content": "已知集合 $M=\\{3,2^a\\}$, $N=\\{a, b\\}$, 若 $M \\cap N=\\{1\\}$, 则 $M \\cup N=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031914": { + "id": "031914", + "content": "不等式 $\\dfrac{x(x-1)}{|x|}\\leq 0$ 的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031915": { + "id": "031915", + "content": "$(x^3-\\dfrac{1}{x^2})^5$ 的二项展开式中的常数项为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031916": { + "id": "031916", + "content": "已知 $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=\\sqrt{2}$, 且 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为 $45^{\\circ}$, 要使 $\\lambda \\overrightarrow{b}-\\overrightarrow{a}$ 与 $\\overrightarrow{a}$ 垂直, 则 $\\lambda=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031917": { + "id": "031917", + "content": "已知 $\\alpha \\in(0,2 \\pi)$, 若复数 $z=\\sin \\alpha \\cos \\alpha-(1-\\cos 2 \\alpha) \\mathrm{i}$ 是纯虚数, 则 $\\alpha=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031918": { + "id": "031918", + "content": "方程 $\\log _2(x+14)+\\log _2(x+2)=3+\\log _2(x+6)$ 的解是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031919": { + "id": "031919", + "content": "已知数列 $\\{a_n\\}$ 的首项 $a_1=2$, 其前 $n$ 项和为 $S_n$. 若 $S_{n+1}=2S_n+1$, 则 $a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031920": { + "id": "031920", + "content": "在 $1,2,3,4$ 四个数中随机地抽取一个数记为 $a$, 再在剩余的三个数中随机地抽取一个数记为 $b$, 则``$\\dfrac{a}{b}$ 是整数''的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031921": { + "id": "031921", + "content": "已知直线 $a, b$ 及平面 $\\alpha$, 下列命题中: \\textcircled{1} $\\begin{cases}a\\perp b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \\textcircled{2} $\\begin{cases}a\\perp b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$; \\textcircled{3} $\\begin{cases}a\\parallel b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \n\\textcircled{4} $\\begin{cases}a\\parallel b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$. 正确命题的序号为\\blank{50}(把你认为正确的序号都填上).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031922": { + "id": "031922", + "content": "已知向量 $\\overrightarrow{m}=(1, \\cos \\omega x)$, $\\overrightarrow{n}=(\\sin \\omega x, \\sqrt{3})$($\\omega>0$), 函数 $f(x)=\\overrightarrow{m}\\cdot \\overrightarrow{n}$, 且 $f(x)$ 图像上一个最高点的坐标为 $(\\dfrac{\\pi}{12}, 2)$, 与之相邻的一个最低点的坐标为 $(\\dfrac{7 \\pi}{12},-2)$, 则 $f(x)$ 的解析式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031923": { + "id": "031923", + "content": "已知直线 $l: y=\\sqrt{3}(x-1)$ 交 $x$ 轴于 $A$ 点, 过原点 $O$ 作 $l$ 的垂线, 垂足为 $C$, 现将线段 $CA$ 绕原点 $O$ 旋转 $\\dfrac{\\pi}{2}$, 则在旋转过程中线段 $CA$ 所扫过的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031924": { + "id": "031924", + "content": "已知 $a_1, a_2, \\cdots, a_n ; b_1, b_2, \\cdots, b_n$ ($n$ 是正整数), 令 $L_1=b_1+b_2+\\cdots+b_n$, $L_2=b_2+b_3+\\cdots+b_n, \\cdots$, $L_n=b_n$. 某人用下图分析得到恒等式: $a_1 b_1+a_2 b_2+ \\cdots+a_n b_n=a_1L_1+c_2L_2+c_3L_3+\\cdots+c_k L_k+\\cdots+ c_n L_n$, 则 $c_k=$\\blank{50}($2 \\leq k \\leq n$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1.2, yscale = 1.5]\n\\draw [->] (-0.5,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {0/0.5/0.6,0.6/0.8/0.7,1.3/1.2/0.6,3.2/2.5/0.6,3.8/2.9/0.7}\n{\\draw (\\i,0) --++ (0,\\j) --++ (\\k,0);};\n\\draw (1.9,1.2) -- (1.9,0) (4.5,2.9) -- (4.5,0);\n\\draw (2.55,1) node {$\\cdots$};\n\\foreach \\i/\\j/\\k in {0.3/0.5/1,0.95/0.8/2,1.6/1.2/3,3.5/2.5/n-1,4.15/2.9/n}\n{\\draw [<->] (\\i,0) -- (\\i,\\j) node [midway, fill = white] {$a_{\\k}$};};\n\\foreach \\i/\\j/\\k in {0/0.6/1,0.6/1.3/2,1.3/1.9/3,3.2/3.8/n-1,3.8/4.5/n}\n{\\draw ({(\\i+\\j)/2},-0.2) node {$b_{\\k}$};};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031925": { + "id": "031925", + "content": "集合 $A=\\{x | \\dfrac{x-2}{x+1}<0\\}$, $B=\\{x |(x-a)(x-b)<0\\}$, 若``$a=-2$''是``$A \\cap B \\neq \\varnothing$''的充分条件, 则 $b$ 的取值范围是\\bracket{20}.\n\\fourch{$b<-1$}{$b>-1$}{$b \\geq-1$}{$-11$) 和双曲线 $\\dfrac{x^2}{n}-y^2=1$($n>0$), 点 $P$ 是它们的一个交点, 则 $\\triangle PF_1F_2$ 的形状是\\bracket{20}.\n\\twoch{直角三角形}{锐角三角形}{钝角三角形}{随 $m$、$n$ 的变化而变化}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031928": { + "id": "031928", + "content": "已知函数 $f(x)=x^3-x+1$, 则\\bracket{20}.\n\\twoch{$f(x)$ 有三个极值点}{$f(x)$ 有三个零点}{点 $(0,1)$ 是曲线 $y=f(x)$ 的对称中心}{直线 $y=2 x$ 是曲线 $y=f(x)$ 的切线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031929": { + "id": "031929", + "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB \\perp AC$, $AB=AC=AA_1=2, E$ 是 $BC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B_1)--(C);\n\\draw [dashed] (A)--(C)(A)--(E)(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥 $C-A_1B_1BA$ 的体积;\\\\\n(2) 求异面直线 $AE$ 与 $A_1C$ 所成的角.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031930": { + "id": "031930", + "content": "已知 $a \\in \\mathbf{R}$, 函数 $f(x)=x|x-a|$.\\\\\n(1) 当 $a=2$ 时,求使 $f(x) \\geq x$ 成立的 $x$ 的集合;\\\\\n(2) 求函数 $y=f(x)$ 在区间 $[1,2]$ 上的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031931": { + "id": "031931", + "content": "在某地区进行某种疾病调查, 随机调查了 $100$ 位这种疾病患者的年龄, 得到如下样本数据频率分布直方图.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.06, yscale = 200]\n\\draw [->] (0,0) -- (105,0) node [below] {年龄(岁)};\n\\draw [->] (0,0) -- (0,0.026) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.001,10/0.002,20/0.012,30/0.017,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.001,20/0.012,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (90,0) node [below] {$90$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计该地区这种疾病患者的平均年龄; (同一组数据用该区间的中点值作代表)\\\\\n(2) 估计该地区以为这种疾病患者年龄位于区间 $[20,70)$ 的概率;\\\\\n(3) 已知该地区这种疾病患者的患病率为 $0.1 \\%$, 该地区年龄位于区间 $[40,50)$ 的人口数占该地区总人口数的 $16 \\%$, 从该地区选出 $1$ 人, 若此人的年龄位于区间 $[40,50)$, 求此人患这种疾病的概率 (精确到 $0.0001$).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷08试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031932": { + "id": "031932", + "content": "已知函数 $f(x)=x^2+3 x$, 数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且对一切正整数 $n$, 点 $P_n(n, S_n)$ 都在函数 $f(x)$ 的图像上.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 设 $A=\\{x | x=a_n, n \\in \\mathbf{N}, n \\geq 1\\}$, $B=\\{x | x=2(a_n-1),\\ n \\in \\mathbf{N},\\ n \\geq 1\\}$, 等差数列 $\\{b_n\\}$ 的任一项 $b_n \\in A \\cap B$, 其中 $b_1$ 是 $A \\cap B$ 中最小的数, 且 $88=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,{sqrt(3)}) node [above] {$P$} coordinate (P);\n\\draw (1,0) node [below right] {$A$} coordinate (A);\n\\draw (190:1) node [left] {$C$} coordinate (C);\n\\draw (A)--(P)--(C);\n\\draw (0,0) circle (1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031942": { + "id": "031942", + "content": "已知 $\\theta \\in(\\pi, 2 \\pi)$, 则复数 $1+\\cos \\theta+\\mathrm{i}\\sin \\theta$ 的模为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031943": { + "id": "031943", + "content": "已知函数 $f(x)=\\begin{cases}x^{2023},& x \\leq a,\\\\x^{2022},& x>a,\\end{cases}$ 若存在实数 $b$, 使得函数 $g(x)= f(x)-b$ 有两个零点, 则 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031944": { + "id": "031944", + "content": "关于 $x$ 的方程 $2 x|x|-a|x|=1$ 有三个不同的实数解, 则实数 $a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031945": { + "id": "031945", + "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_2=2$, 则 $\\dfrac{1}{a_1^2+1}+\\dfrac{1}{a_3^2+1}$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031946": { + "id": "031946", + "content": "已知 $a \\in \\mathbf{R}$, 则``$a>1$''是``$\\dfrac{1}{a}<1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031947": { + "id": "031947", + "content": "已知平面 $\\alpha$、$\\beta$、$\\gamma$ 两两垂直, 直线 $a$、$b$、$c$ 满足: $a \\subset \\alpha$, $b \\subset \\beta$, $c \\subset \\gamma$, 则直线 $a$、$b$、$c$ 不可能满足以下哪种关系\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031948": { + "id": "031948", + "content": "函数 $f(x)=\\sin (x-\\dfrac{\\pi}{3})-\\dfrac{1}{2}$, $x \\in[t, t+40]$ 零点的个数不可能是\\bracket{20}.\n\\fourch{$11$ 个}{$12$ 个}{$13$ 个}{$14$ 个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031949": { + "id": "031949", + "content": "已知定义在 $[0,10)$ 的函数 $f(x)$ 满足 $f(x+2)=f(x)+a, f(x)$ 在 $[0,2]$ 上的解析式为 $f(x)=\\begin{cases}\\dfrac{a}{x+2}+1,& 0 \\leq x \\leq 1,\\\\\\dfrac{a}{3}x+1,& 1=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,1) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(M)--(N);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel$ 平面 $PAD$;\\\\\n(2) 若 $MN=BC=4$, $PA=4 \\sqrt{3}$, 求异面直线 $PA$ 与 $MN$ 所成的角的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031953": { + "id": "031953", + "content": "已知函数 $f(x)$, 如果存在给定的有序实数对 $(m, n)$, 使得 $f(m+x) \\cdot f(m-x)=n$ 恒成立, 则称 $f(x)$ 为``$S$ - 函数''.\\\\\n(1) 判断函数 $f_1(x)=x$ 是否是``$S$ - 函数'';\\\\\n(2) 若 $f_2(x)=\\tan x$ 是一个``$S$ 一函数'', 求出所有满足条件的有序实数对 $(m, n)$;\\\\\n(3) 若存在有序实数对 $(1, \\mathrm{e}^2)$, 使得指数函数 $f_3(x)=a^x$ 是``$S$ - 函数'', 且对任意的 $x>0$ 都有 $f_3(x) \\geq \\dfrac{1}{2}x^3-t x^2+x+1$ 恒成立, 求 $t$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031954": { + "id": "031954", + "content": "已知数列 $\\{a_n\\}$, $a_1=4$, $a_2=p$, 且 $a_n+a_{n+1}+a_{n+2}=7 \\times(\\dfrac{1}{2})^{n-1}$.\\\\\n(1) 求 $a_4$;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$;\\\\\n(3) 是否存在实数 $p$, 使得 $\\{a_n\\}$ 为严格减数列?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷09试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031955": { + "id": "031955", + "content": "已知 $A=\\{x | \\sqrt{x}<4\\}$, $B=\\{x | 3 x \\geq 1\\}$, 则 $A \\cap B=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031956": { + "id": "031956", + "content": "已知复数 $z$ 满足 $\\dfrac{2}{z-5}=1+\\mathrm{i}$, 则 $z=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031957": { + "id": "031957", + "content": "已知向量 $\\overrightarrow{a}=(1,0,3)$, $\\overrightarrow{b}=(3,1,0)$, 则向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 的夹角为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031958": { + "id": "031958", + "content": "在二项式 $(2 x-\\dfrac{1}{x})^5$ 的展开式中,含有 $x^3$ 项的系数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031959": { + "id": "031959", + "content": "顺次联结圆 $x^2+y^2=9$ 与双曲线 $x y=3$ 的交点, 得到一个凸四边形, 则该四边形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031960": { + "id": "031960", + "content": "已知圆柱的高为 $4$, 底面积为 $9 \\pi$, 则圆柱的侧面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031961": { + "id": "031961", + "content": "若 $x, y \\in$($0,+\\infty$), 且 $3 x-2 x y=1$, 则 $\\dfrac{y}{x}$ 的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031962": { + "id": "031962", + "content": "在等比数列 $\\{a_n\\}$ ($n$ 为正整数) 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 $10$ 项和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031963": { + "id": "031963", + "content": "在三角形 $\\triangle ABC$ 中, $D$ 是 $BC$ 的中点, $E$ 在边 $AB$ 上, $BE=2EA$, $AD$ 与 $CE$ 交于点 $O$, 若 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=6 \\overrightarrow{AO}\\cdot EC$, 则 $\\dfrac{AB}{AC}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031964": { + "id": "031964", + "content": "设非零常数 $d$ 是等差数列 $x_1, x_2, x_3, \\cdots, x_{19}$ 的公差, 随机变量 $\\xi$ 等可能地取值 $x_1, x_2$, $x_3, \\cdots, x_{19}$, 则方差 $D [\\xi]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031965": { + "id": "031965", + "content": "椭圆 $\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点分别为 $F_1, F_2$, 焦距为 $2 c$, 若直线 $y=\\sqrt{3}(x+c)$ 与椭圆 $\\Gamma$ 的一个交点 $M$ 满足 $\\angle MF_1F_2=2 \\angle MF_2F_1$, 则该椭圆的离心率等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031966": { + "id": "031966", + "content": "对于三次函数 $f(x)=a x^3+b x^2+c x+d$($a \\neq 0$), 定义: 设 $f''(x)$ 是函数 $y=f(x)$ 的导数 $y=f'(x)$ 的导数, 若方程 $f''(x)=0$ 有实数解 $x_0$, 则称点 $(x_0, f(x_0))$ 为函数 $y=f(x)$ 的``拐点''. 有同学发现``任何一个三次函数都有拐点, 任何一个三次函数都有对称中心; 且拐点就是对称中心.''请你将这一发现为条件, 解答问题: 若已知函数 $f(x)=x^3-\\dfrac{3}{2}x^2+ 3 x-\\dfrac{1}{4}$, 则 $f(\\dfrac{1}{2023})+f(\\dfrac{2}{2023})+f(\\dfrac{3}{2023})+\\cdots+f(\\dfrac{2022}{2023})=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031967": { + "id": "031967", + "content": "已知直线的方程为 $4 x+2 y+c=0$, 则该直线的一个法向量为\\bracket{20}.\n\\fourch{$(2,-1)$}{$(2,1)$}{$(-1,2)$}{$(1,2)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031968": { + "id": "031968", + "content": "关于函数 $f(x)=\\sin |x|+|\\sin x|$ 有下述四个结论 : \\textcircled{1} $f(x)$ 是偶函数; \\textcircled{2} $f(x)$ 在区间 $(\\dfrac{\\pi}{2}, \\pi)$ 单调递增; \\textcircled{3} $f(x)$ 在 $[-\\pi, \\pi]$ 有 $4$ 个零点; \\textcircled{4} $f(x)$ 的最大值为 $2$ . 其中所有正确结论的编号是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031969": { + "id": "031969", + "content": "如图, 上海海关大楼的上面可以看作一个正四棱柱, 四个侧面有四个时钟, 请问相邻的两个面时钟的时针相互垂直的次数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}[x = {(-10:0.9)}]\n\\draw (0,0) -- (-2,0) -- (-2,2) -- (0,2) -- cycle;\n\\draw (-1,1) circle (0.8);\n\\draw [->] (-1,1) --++ (-45:0.5);\n\\draw [->] (-1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (-1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\begin{scope}[x = {(40:0.7)}]\n\\draw (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;\n\\draw (1,1) circle (0.8);\n\\draw [->] (1,1) --++ (-45:0.5);\n\\draw [->] (1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0$}{$2$}{$4$}{$12$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031970": { + "id": "031970", + "content": "设函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 满足 $f(x+1)=2 f(x)$, 且当 $x \\in(0,1]$ 时, $f(x)=x(x-1)$, 若对任意 $x \\in(-\\infty, m]$, 都有 $f(x) \\geq-\\dfrac{8}{9}$, 则第 15 题图 $m$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, \\dfrac{9}{4}]$}{$(-\\infty, \\dfrac{7}{3}]$}{$(-\\infty, \\dfrac{5}{2}]$}{$(-\\infty, \\dfrac{8}{3}]$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031971": { + "id": "031971", + "content": "在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=AC=1$, $\\angle BAC=90^{\\circ}$, 且异面直线 $A_1B$ 与 $B_1C_1$ 所成的角等于 $60^{\\circ}$, 设 $AA_1=a$.\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 (0,0,2) node [left] {$B$} coordinate (B);\n\\foreach \\i in {A,B,C}\n{\\draw (\\i) ++ (0,2,0) coordinate (\\i_1);};\n\\draw (A_1) node [above] {$A_1$} (B_1) node [left] {$B_1$} (C_1) node [right] {$C_1$};\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (B)--(A_1)(B)--(A)--(C)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $a$ 的值;\\\\\n(2) 求三棱锥 $B_1-A_1BC$ 的体积.\n第 17 题图\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "031972": { + "id": "031972", + "content": "某省食品药品监管局对 $16$ 个大学食堂的``进货渠道合格性''和``食品安全''进行量化评估, 满分为 $10$ 分, 大部分大学食堂的评分在 $7 \\sim 10$ 分之间, 以下表格记录了它们的评分情况:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 分数段 &{$[0,7)$}&{$[7,8)$}&{$[8,9)$}&{$[9,10]$}\\\\\n\\hline 食堂个数 & 1 & 3 & 8 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 现从 $16$ 个大学食堂中随机抽取 $3$ 个, 求至多有 $1$ 个大学食堂的评分不低于 $9$ 分的概率;\\\\\n(2) 以这 $16$ 个大学食堂的评分数据评估全国的大学食堂的评分情况,若从全国的大学食堂中任选 $3$ 个, 记 $X$ 表示抽到评分不低于 $9$ 分的食堂个数, 求 $X$ 的分布列及数学期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031973": { + "id": "031973", + "content": "以坐标原点 $O$ 为圆心的单位圆与 $x$ 轴正半轴交于点 $A$, 点 $B, P$ 在单位圆上, 且 $B(-\\dfrac{\\sqrt{5}}{5}, \\dfrac{2 \\sqrt{5}}{5}), \\angle AOB=\\alpha$.\\\\\n(1) 求 $\\sin (2 \\alpha+\\dfrac{\\pi}{4})$ 的值;\\\\\n(2) 若四边形 $OAQP$ 是平行四边形. 设 $\\angle POA=\\theta$($0 \\leq \\theta \\leq 2 \\pi$), 点 $Q(m, n)$, 且 $f(\\theta)= m+\\sqrt{3}n$, 求关于 $\\theta$ 的函数 $f(\\theta)$ 的解析式, 并求其单调增区间.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031974": { + "id": "031974", + "content": "如图, 在直角坐标系 $xOy$ 中, 有一组对角线长为 $a_n$ 的正方形 $A_n B_n C_n D_n$($n=1,2, \\cdots$), 其对角线 $B_n D_n$ 依次放置在 $x$ 轴上 (相邻顶点重合). 设 $\\{a_n\\}$ 是首项为 $a$, 公差为 $d$($d>0$) 的等差数列, 点 $B_1$ 的坐标为 $(d, 0)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.08]\n\\draw [->] (-4,0) -- (44,0) node [below] {$x$};\n\\draw [->] (0,-15) -- (0,15) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (4,0) node [above left] {$B_1$} coordinate (B_1);\n\\draw (12,0) node [above] {$B_2$} coordinate (B_2) node [below] {$D_1$};\n\\draw (24,0) node [above] {$B_3$} coordinate (B_3) node [below] {$D_2$};\n\\draw (40,0) node [below] {$D_3$} coordinate (D_3);\n\\foreach \\i/\\j/\\k in {8/4/1,18/6/2,32/8/3}\n{\\draw (\\i,\\j) node [above] {$A_\\k$} coordinate (A_\\k) (\\i,-\\j) node [below] {$C_\\k$} coordinate (C_\\k);};\n\\draw (B_1)--(A_1)--(B_2)--(A_2)--(B_3)--(A_3)--(D_3)--(C_3)--(B_3)--(C_2)--(B_2)--(C_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当 $a=8$, $d=4$ 时,证明: 顶点 $A_1$、$A_2$、$A_3$ 不在同一条直线上;\\\\\n(2) 在 (1) 的条件下, 证明: 所有顶点 $A_n$ 均落在抛物线 $y^2=2 x$ 上;\\\\\n(3) 为使所有顶点 $A_n$ 均落在抛物线 $y^2=2 p x$($p>0$) 上, 求 $a$ 与 $d$ 之间所应满足的关系式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "高考数学风暴第二轮复习用书上海高考数学模拟试卷10试题1", + "edit": [ + "20230730\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "031975": { + "id": "031975", + "content": "已知函数 $f(x)=\\log _a \\dfrac{1-x}{1+x}$($0 Date: Sun, 30 Jul 2023 16:08:37 +0800 Subject: [PATCH 5/7] =?UTF-8?q?=E8=B5=8B=E4=BA=88=E9=83=A8=E5=88=86?= =?UTF-8?q?=E9=A2=98=E7=9B=AE=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 1376 ++++++++++++++++++++++++++++++----------- 1 file changed, 1032 insertions(+), 344 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index c64bd48d..f64a63f9 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -494441,7 +494441,9 @@ "id": "019304", "content": "命题``存在 $x \\in \\mathbf{R}$, 使得 $x^2+2 x+2 \\leq 0$''的否定是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494461,7 +494463,9 @@ "id": "019305", "content": "设不等式 $\\dfrac{a(x-2)}{x+3}<2$ 的解集为 $A$, 且 $1 \\notin A$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494481,7 +494485,9 @@ "id": "019306", "content": "迎春杯数学竞赛后, 甲、乙、丙、丁四名同学猜测他们之中谁能获奖. 甲说:``如果我能获奖, 那么乙也能获奖.''乙说:``如果我能获奖, 那么丙也能获奖.''丙说:``如果丁没获奖, 那么我也不能获奖.''实际上, 他们之中只有一个人没有获奖, 并且甲、乙、丙说的话都是正确的. 那么没能获奖的同学是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494501,7 +494507,9 @@ "id": "019307", "content": "若关于 $x$ 的不等式 $x^2+a x-a-2>0$ 和 $2 x^2+2(2 a+1) x+4 a^2+1>0$ 的解集依次为 $A$ 和 $B$, 那么, 使得 $A=\\mathbf{R}$ 和 $B=\\mathbf{R}$ 至少有一个成立的实数 $a$\\bracket{20}.\n\\onech{可以是 $\\mathbf{R}$ 中的任何一个数}{有无穷多个, 但并不是 $\\mathbf{R}$ 中所有的实数都能满足要求}{有且仅有一个}{不存在}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494521,7 +494529,9 @@ "id": "019308", "content": "已知 $x \\in \\mathbf{R}$, 条件 $p: x^20$), 若 $p$ 是 $q$ 的充分不必要条件,则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494541,7 +494551,9 @@ "id": "019309", "content": "已知集合 $A=\\{x | x^2+p x-2 q=0\\}$, $B=\\{x | x^2+q x-4 q^2+2 p=0\\}$, 则``实数 $p= q=1$''是``$1 \\in A \\cap B$''的\\bracket{20}.\n\\twoch{充分必要条件}{充分不必要条件}{必要不充分条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494561,7 +494573,9 @@ "id": "019310", "content": "已知 $a, b, c$ 为实常数,数列 $\\{x_n\\}$ 的通项为 $x_n=a n^2+b n+c$, $n \\in \\mathbf{N}$, $n\\ge 1$, 则``存在正整数 $k$, 使得 $x_{100+k}, x_{200+k}, x_{300+k}$ 成等差数列''的一个必要条件为\\bracket{20}.\n\\fourch{$a \\geq 0$}{$b \\leq 0$}{$c=0$}{$a-2 b+c=0$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494581,7 +494595,9 @@ "id": "019311", "content": "设 $A=\\{x | x=m^2-n^2, m, n \\in \\mathbf{Z}\\}$, $B=\\{8,9,10\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494601,7 +494617,9 @@ "id": "019312", "content": "设集合 $A=\\{1,2, m\\}$, 其中 $m$ 为实数, 令 $B=\\{a^2 | a \\in A\\}$, $C=A \\cup B$, 若 $C$ 的所有元素之和为$6$,则 $C$ 的所有元素之积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494621,7 +494639,9 @@ "id": "019313", "content": "已知集合 $A=\\{x | x=a^2-b^2, a, b \\in \\mathbf{Z}\\}$, 则下列说法不正确的是\\bracket{20}.\n\\twoch{$2023 \\in A$}{所有质数都在集合 $A$ 中}{所有奇数都在集合 $A$ 中}{所有 $4$ 的倍数都在集合 $A$ 中}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494641,7 +494661,9 @@ "id": "019314", "content": "设集合 $M=\\{(x, y) | x^2+y^2 \\geq 1\\}$, $N=\\{(x, y) | y>x+\\sqrt{2}\\}$, 求证: $N \\subset M$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -494661,7 +494683,9 @@ "id": "019315", "content": "已知函数 $y=f(x)$ 的定义域为 $\\mathbf{R}$, 下列是 $f(x)$ 无最大值的一个充分条件是\\bracket{20}.\n\\twoch{$f(x)$ 是偶函数且关于点 $(1,1)$ 对称}{$f(x)$ 是偶函数且关于直线 $x=1$ 对称}{$f(x)$ 是奇函数且关于点 $(1,1)$ 对称}{$f(x)$ 是奇函数且关于直线 $x=1$ 对称}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494681,7 +494705,9 @@ "id": "019316", "content": "已知命题 $p$ : 存在实数 $a \\neq 0$, 对于任意的 $x \\in \\mathbf{R}$, 都有 $f(x+a)0$;\n命题 $q_2: f(x)$ 是 $\\mathbf{R}$ 上严格递增的函数且存在 $x_0<0$ 使得 $f(x_0)=0$,\n证明: $q_1, q_2$ 都是 $p$ 的充分条件.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -494701,7 +494727,9 @@ "id": "019317", "content": "已知函数 $f(x)=x^{\\frac{1}{3}}$, $x \\in[8,64]$ 的值域为 $A$, 集合 $B=\\{x | x^2-4 x+3<0\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494721,7 +494749,9 @@ "id": "019318", "content": "$A=\\{-1,0, a\\}$, $B=\\{x | 1<3^x<100, x \\in \\mathbf{Z}\\}$, 若 $A \\cap B \\neq \\varnothing$, 则实数 $a$ 组成的集合是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494741,7 +494771,9 @@ "id": "019319", "content": "已知全集 $U=\\{1,2,3,4,5\\}$, 集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | x=2 a, a \\in A\\}$, 则集合 $\\overline{A \\cup B}$ 中元素的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494761,7 +494793,9 @@ "id": "019320", "content": "已知互异的复数 $a, b$ 满足 $a b \\neq 0$, 集合 $\\{a, b\\}=\\{a^2, b^2\\}$, 则 $a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494781,7 +494815,9 @@ "id": "019321", "content": "已知 $a \\in \\mathbf{R}$, 不等式 $\\dfrac{x-3}{x+a}\\geq 1$ 的解集为 $P$, 且 $-2 \\notin P$,则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494801,7 +494837,9 @@ "id": "019322", "content": "$A_k=\\{x | x=k t+\\dfrac{1}{k t},\\ \\dfrac{1}{k^2}\\leq t \\leq 1\\}$, 其中 $k=2,3, \\cdots, 2018$, 则所有 $A_k$ 的交集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -494821,7 +494859,9 @@ "id": "019323", "content": "设 $z_1, z_2 \\in \\mathbf{C}$, 则``$z_1, z_2$ 中至少有一个数是虚数''是``$z_1-z_2$ 是虚数''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494841,7 +494881,9 @@ "id": "019324", "content": "命题 $A:|x-1|<3$, 命题 $B:(x+2)(x+a)<0$. 若 $A$ 是 $B$ 的充分而不必要条件,则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-4)$}{$[4,+\\infty)$}{$(4,+\\infty)$}{$(-\\infty,-4]$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494861,7 +494903,9 @@ "id": "019325", "content": "设集合 $A=\\{(x, y) | x-y \\geq 1, \\ a x+y>4,\\ x-a y \\leq 2\\}$, 则\\bracket{20}.\n\\twoch{对任意实数 $a,(2,1) \\in A$}{对任意实数 $a,(2,1) \\notin A$}{当且仅当 $a<0$ 时, $(2,1) \\notin A$}{当且仅当 $a \\leq \\dfrac{3}{2}$ 时, $(2,1) \\notin A$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494881,7 +494925,9 @@ "id": "019326", "content": "已知集合 $A=\\{x|| x-2 |0\\}$, 集合 $B=\\{x | \\dfrac{2 x-2}{x+3}<1\\}$.\\\\\n(1) 若 $a=1$, 求 $A \\cap B$;\\\\\n(2) 若 $A \\subset B$, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -494901,7 +494947,9 @@ "id": "019327", "content": "对任意一个非零复数 $z$ 定义集合 $M_z=\\{\\omega | \\omega=z^{2 n-1}, n$ 是正整数$\\}$.\\\\\n(1) 设 $\\alpha$ 是方程 $x+\\dfrac{1}{x}=\\sqrt{2}$ 的一个根, 试用列举法表示集合 $M_\\alpha$. 若在 $M_\\alpha$ 中任取两个数, 求其和为零的概率 $p$;\\\\\n(2) 若复数 $\\omega \\in M_z$, 求证: $M_\\omega \\subseteq M_z$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -494921,7 +494969,9 @@ "id": "019328", "content": "若对于任意的 $a \\in A$, 总有 $-a \\notin A$, 则称集合 $A$ 具有性质 $P$. 已知集合 $A=\\{a_1, a_2, a_3, \\cdots, a_k\\}$($k \\geq 2$), 其中 $a_i \\in \\mathbf{Z}$($i=1,2, \\cdots, k$), 由 $A$ 中的元素构成的集合 $T=\\{(a, b) | a \\in A,\\ b \\in A,\\ a-b \\in A\\}$, 其中 $(a, b)$ 是有序实数对, 集合 $T$ 的元素个数为 $n$.\\\\\n(1) 检验集合 $\\{0,1,2,3\\}$ 与 $\\{-1,2,3\\}$ 是否具有性质 $P$, 并对其中具有性质 $P$ 的集合写出相应的集合 $T$;\\\\\n(2) 对任何具有性质 $P$ 的集合 $A$, 证明 : $n \\leq \\dfrac{k(k-1)}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -494941,7 +494991,9 @@ "id": "019329", "content": "已知 $a, b \\in \\mathbf{R}$, 且 $a b \\neq 0$, 则``$a>b$''是``$\\dfrac{1}{a}<\\dfrac{1}{b}$''成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -494961,7 +495013,9 @@ "id": "019330", "content": "若 $a>b>0$, $c0$ 的解集为 $(p, q)$, 其中 $q<0$, 则关于 $x$ 的一元二次不等式 $c x^2-b x+a>0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495101,7 +495167,9 @@ "id": "019337", "content": "求关于 $x$ 的不等式 $\\dfrac{(x+2)(x-a)}{x}\\geq 0$ (其中 $a \\in \\mathbf{R}$) 的解集.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495121,7 +495189,9 @@ "id": "019338", "content": "若 $x>0$, $y>0$, 且 $\\dfrac{1}{x}+2 y=3$, 则 $\\dfrac{y}{x}$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495141,7 +495211,9 @@ "id": "019339", "content": "若存在实数 $x$, 使得不等式 $|x+3|+|x-1| \\leq a^2-3 a$ 有解, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495161,7 +495233,9 @@ "id": "019340", "content": "已知 $x>0$, $y>0$, $x+y=4$, 则使不等式 $\\dfrac{1}{x}+\\dfrac{4}{y}\\geq m$ 恒成立的实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495181,7 +495255,9 @@ "id": "019341", "content": "已知 $a>b>0$, 当代数式 $a^2+\\dfrac{16}{b(a-b)}$ 取最小值时, 点 $P(a, b)$ 的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495201,7 +495277,9 @@ "id": "019342", "content": "设 $x, y$ 是正实数, 且 $x+y=1$, 则 $\\dfrac{x^2}{x+2}+\\dfrac{y^2}{y+1}$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495221,7 +495299,9 @@ "id": "019343", "content": "在平面直角坐标系中, 定义 $P(x_1, y_1)$、$Q(x_2, y_2)$ 之间的``直角距离''为$d(P, Q)=|x_1-x_2|+|y_1-y_2|$.\\\\\n(1) 已知 $A(x, 1)$、$B(1,2)$、$C(5,2)$ 三点, 若 $d(A, B)>d(A, C)$, 求 $x$ 的取值范围;\\\\\n(2) 已知 $A(x-1, y)$、$B(0,-1)$、$C(x, y-2)$ 三点, 对任意 $x, y \\in \\mathbf{R}$, 不等式 $d(A, B) \\geq t^2-2 t-d(B, C)$ 恒成立, 求 $t$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495241,7 +495321,9 @@ "id": "019344", "content": "设关于 $x$ 的不等式 $(a x-a^2-9)(x-b) \\geq 0$ 的解集为 $A$, 其中 $a, b \\in \\mathbf{R}$.\\\\\n(1) 若 $b=6$, $A=(-\\infty,+\\infty)$, 求 $a$ 的值;\\\\\n(2) 若 $b=6$, 记 $L=d-c$ 为闭区间 $[c, d]$ 的长度. 当 $a<0$ 时, 求区间 $A$ 的长度 $L$ 的最小值;\\\\\n(3) 当 $b=2 a-8$, 且 $a<9$ 时, 求 $A$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495261,7 +495343,9 @@ "id": "019345", "content": "如图, 射线 $OA, OB$ 均为笔直的公路, 扇形 $OPQ$ 区域 (含边界) 是一种蔬菜种植园, 其中 $P, Q$ 分别在射线 $OA, OB$ 上, 扇形 $OPQ$ 的中心角 $\\angle POQ=\\dfrac{2 \\pi}{3}$, 半径 $OP=1$ 千米. 为了方便菜农经营, 打算在扇形 $OPQ$ 区域外修建一条公路 $MN$, 分别与射线 $OA, OB$ 交于 $M, N$ 两点, 并要求直线 $MN$ 与圆弧 $\\overset\\frown{PQ}$ 相切于点 $S$, 设 $\\angle POS=\\alpha$, 假设所有公路的宽度都忽略不计.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (60:1) node [right] {$Q$} coordinate (Q);\n\\draw (180:1) node [below] {$P$} coordinate (P);\n\\draw (P) arc (180:60:1);\n\\draw (115:1) node [above left] {$S$} coordinate (S);\n\\draw (S) ++ (205:{tan(65)}) node [below] {$M$} coordinate (M);\n\\draw (S) ++ (25:{tan(55)}) node [right] {$N$} coordinate (N);\n\\draw ($(O)!1.2!(M)$) node [below] {$A$} coordinate (A);\n\\draw ($(O)!1.2!(N)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(O)--(B)(O)--(S)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 试将公路 $MN$ 的长度表示为 $\\alpha$ 的函数, 并写出定义域;\\\\\n(2) 试确定 $\\alpha$ 的值, 使公路 $MN$ 长度最小, 并求出其最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495281,7 +495365,9 @@ "id": "019346", "content": "若 $x_1$、$x_2$ 是方程 $a^x=(\\dfrac{1}{a})^{-\\frac{1}{x}+1}$($a>1$) 的两个实数解, 则 $x_1+x_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495301,7 +495387,9 @@ "id": "019347", "content": "已知不等式 $a x^2+b x+c>0$ 的解为 $-30$, $n>0$, 且 $m+n=1$, 求 $\\dfrac{1}{m}+\\dfrac{1}{n}$ 的最小值''提出各自的解题思路:\\\\\n甲: $\\dfrac{1}{m}+\\dfrac{1}{n}=\\dfrac{m+n}{m}+\\dfrac{m+n}{n}=2+\\dfrac{n}{m}+\\dfrac{m}{n}$, 可用基本不等式求解;\\\\\n乙: $\\dfrac{1}{m}+\\dfrac{1}{n}=\\dfrac{m+n}{m m}=\\dfrac{1}{m n}=\\dfrac{1}{m(1-m)}$, 可用二次函数配方法求解;\\\\\n丙: $\\dfrac{1}{m}+\\dfrac{1}{n}=(\\dfrac{1}{m}+\\dfrac{1}{n})(m+n)=2+\\dfrac{n}{m}+\\dfrac{m}{n}$, 可用基本不等式求解;\\\\\n参考上述解题思路, 可求得当 $x=$\\blank{50}时, $y=\\dfrac{a^2}{x^2}+\\dfrac{1}{100-x^2}$($00$) 有最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495401,7 +495497,9 @@ "id": "019352", "content": "下列不等式一定成立的是\\bracket{20}.\n\\twoch{$\\lg (x^2+\\dfrac{1}{4})>\\lg x$($x>0$)}{$\\sin x+\\dfrac{1}{\\sin x}\\geq 2$($x \\neq k \\pi$, $k \\in \\mathbf{Z}$)}{$x^2+1 \\geq 2|x|$($x \\in \\mathbf{R}$)}{$\\dfrac{1}{x^2+1}>1$($x \\in \\mathbf{R}$)}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495421,7 +495519,9 @@ "id": "019353", "content": "若不等式 $a \\leq|x-1|-|x-3| \\leq b$ 对任意实数 $x$ 恒成立, 则 $b-a$ 的最小值为\\bracket{20}.\n\\fourch{$4$}{$2$}{$-4$}{$-2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495441,7 +495541,9 @@ "id": "019354", "content": "若不等式 $(|x-a|-b) \\sin (\\pi x+\\dfrac{\\pi}{6}) \\leq 0$ 对 $x \\in[-1,1]$ 恒成立, 则 $a+b$ 的值等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{5}{6}$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495461,7 +495563,9 @@ "id": "019355", "content": "某热力公司每年燃料费约 $24$ 万元, 为了``环评''达标, 需要安装一块面积为 $x$ (其中 $x \\geq 0$, 单位: 平方米) 可用 15 年的太阳能板, 其工本费为 $\\dfrac{x}{2}$ (单位: 万元), 并与燃料供热互补工作, 从此, 公司每年的燃料费为 $\\dfrac{k}{20 x+100}$($x \\geq 0$, $k$ 为常数) 万元. 记 $y$ 为该公司安装太阳能板的费用与 15 年的燃料费之和.\\\\\n(1) 求 $k$ 的值, 并建立 $y$ 关于 $x$ 的函数关系式;\\\\\n(2) 求 $y$ 的最小值, 并求出此时所安装太阳能板的面积.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495481,7 +495585,9 @@ "id": "019356", "content": "已知 $x_1$、$x_2$ 是关于 $x$ 的一元二次方程 $4 k x^2-4 k x+k+1=0$ 的两个实数根.\\\\\n(1) 是否存在实数 $k$, 使得 $(2 x_1-x_2)(x_1-2 x_2)=-\\dfrac{3}{2}$ 成立? 若存在, 求出 $k$ 的值; 若不存在, 请说明理由;\\\\\n(2) 若 $k \\in \\mathbf{Z}$, 求使 $\\dfrac{x_1}{x_2}+\\dfrac{x_2}{x_1}-2$ 的值为整数的实数 $k$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495501,7 +495607,9 @@ "id": "019357", "content": "若函数 $f(x)$ 满足: 对于任意正数 $s, t$, 都有 $f(s)>0$, $f(t)>0$, 且 $f(s)+f(t)\\dfrac{x}{2}-\\dfrac{2}{x}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495521,7 +495629,9 @@ "id": "019358", "content": "下列图形中, (是以 $x$ 为自变量的) 函数的图像的是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{1}};\n\\foreach \\i in {0.1,0.2,...,2.5}\n{\\filldraw (\\i,{1.5+sin((\\i-0.1)/2.4*360)*1.2}) circle (0.02);};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{2}};\n\\draw (0,0) -- (0.5,2.8) -- (1,0.4) -- (1.5,2.6) -- (2,0.8) -- (2.5,2.4) -- (2.8,1.5);\n\\draw [dashed] (2.8,1.5) -- (2.8,0);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1.5,0) node [below] {\\textcircled{3}};\n\\draw (1.5,1.5) circle (1.3);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495541,7 +495651,9 @@ "id": "019359", "content": "函数 $y=\\lg (3-2^x)$ 的定义域是\\blank{50}, 值域为\\blank{50}. (用区间表示)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495561,7 +495673,9 @@ "id": "019360", "content": "函数 $f(x)=\\begin{cases}(\\dfrac{1}{2})^x,& x<0,\\\\\\log _{\\frac{1}{2}}x,& x \\geq 0,\\end{cases}$ 则 $f(\\dfrac{1}{4})+f(\\log _2 \\dfrac{1}{6})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495581,7 +495695,9 @@ "id": "019361", "content": "设 $D$ 是含数$1$的有限实数集, $f(x)$ 是定义在 $D$ 上的函数, 若 $f(x)$ 的图像绕原点逆时针旋转 $\\dfrac{\\pi}{6}$ 后与原图像重合, 则在以下各项中, $f(1)$ 的取值只能是\\bracket{20}.\n\\fourch{$\\sqrt{3}$}{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495601,7 +495717,9 @@ "id": "019362", "content": "设曲线 $C$ 与函数 $f(x)=\\dfrac{\\sqrt{3}}{12}x^2$($00\\end{cases}$为奇函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495641,7 +495761,9 @@ "id": "019364", "content": "已知 $a \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2}, 1,2,3\\}$, 若幂函数 $f(x)=x^a$ 为奇函数, 且在 $(0,+\\infty)$ 上严格减, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495661,7 +495783,9 @@ "id": "019365", "content": "已知 $y=\\begin{cases}(3-a) x-2,& x<1\\\\x^2-2 x,& x \\geq 1\\end{cases}$ 是 $\\mathbf{R}$ 上的严格增函数, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495681,7 +495805,9 @@ "id": "019366", "content": "已知 $a, b$ 为正实数, 函数 $f(x)=a x^3+b x+2^x$ 在 $[0,1]$ 上的最大值为 4 , 则 $f(x)$ 在 $[-1,0]$ 上的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -495701,7 +495827,9 @@ "id": "019367", "content": "已知 $\\omega \\in \\mathbf{R}$, 函数 $f(x)=(x-6)^2 \\sin \\omega x$, 若存在常数 $a \\in \\mathbf{R}$, 使得 $f(x+a)$ 为偶函数, 则 $\\omega$ 的值可能为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{2}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495721,7 +495849,9 @@ "id": "019368", "content": "已知函数 $f(x)$ 是定义在 $(-1,0) \\cup(0,1)$ 上的奇函数, 当 $x \\in(-1,0)$ 时, $f(x)=a x+\\dfrac{1}{x^2}$.\\\\\n(1) 求函数 $y=f(x)$ 在 $(0,1)$ 上的函数解析式;\\\\\n(2) 当 $a>-2$ 时, 判断函数 $y=f(x)$ 在 $(0,1)$ 上的单调性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495741,7 +495871,9 @@ "id": "019369", "content": "已知函数 $f(x)=\\dfrac{2-x}{x+1}$.\\\\\n(1) 求不等式 $f(x-4)+10$ 是 $F(x_1)+F(x_2)>0$ 的充要条件.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495781,7 +495915,9 @@ "id": "019371", "content": "已知$1$是函数 $f(x)$ 的周期, 且当 $0=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [below] {$-4$}-- (0,1) node [right] {$4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [below] {$-4$}-- (-1,1) -- (0,1) node [right] {$4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [above] {$-4$}-- (0,-1) node [right] {$-4$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$a$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$b$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [above] {$-4$}-- (-1,-1) -- (0,-1) node [right] {$-4$};\n\\draw (1,0) node [below] {$4$} -- (1,1) -- (0,1) node [left] {$4$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495941,7 +496091,9 @@ "id": "019379", "content": "下列函数 $f(x)$ 中, 满足``对任意 $x_1, x_2 \\in(0,+\\infty)$, 当 $x_1f(x_2)$''的是\\bracket{20}.\n\\fourch{$f(x)=\\dfrac{1}{x}$}{$f(x)=(x-1)^2$}{$f(x)=\\mathrm{e}^x$}{$f(x)=\\ln (x+1)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -495961,7 +496113,9 @@ "id": "019380", "content": "已知定义域为 $\\mathbf{R}$ 的函数 $f(x)=\\dfrac{1-2^x}{1+2^x}$.\\\\\n(1) 试判断函数 $f(x)=\\dfrac{1-2^x}{1+2^x}$ 在 $\\mathbf{R}$ 上的单调性, 并用函数单调性的定义证明;\\\\\n(2) 若对于任意 $t \\in \\mathbf{R}$, 不等式 $f(t^2-2 t)+f(t^2-k)<0$ 恒成立, 求实数 $k$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -495981,7 +496135,9 @@ "id": "019381", "content": "设 $f(x)=x^3+a x^2-2 x$($x \\in \\mathbf{R}$), 其中常数 $a \\in \\mathbf{R}$.\\\\\n(1) 判断函数 $y=f(x)$ 的奇偶性, 并说明理由;\\\\\n(2) 若不等式 $f(x)>\\dfrac{3}{2}x^3$ 在区间 $[\\dfrac{1}{2}, 1]$ 上有解, 求实数 $a$ 的取值范围;\\\\\n(3) 已知: 若对函数 $y=h(x)$ 定义域内的任意 $x$, 都有 $h(x)+h(2 m-x)=2 n$, 则函数 $y=h(x)$ 的图像有对称中心 $(m, n)$. 利用以上结论探究: 对于任意的实数 $a$, 函数 $y=f(x)$ 是否都有对称中心? 若是, 求出对称中心的坐标 (用 $a$ 表示); 若不是, 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496001,7 +496157,9 @@ "id": "019382", "content": "已知函数 $f(x), g(x)$ 在数集 $D$ 上都有定义, 对于任意的 $x_1, x_2 \\in D$, 当 $x_1b$}{$a ba^2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496421,7 +496619,9 @@ "id": "019403", "content": "如图为函数 $y=f(x)$ 的导函数 $y=f'(x)$ 的图像, 那么函数 $y=f(x)$ 的图像可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:3] plot (\\x,{\\x*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2.4] plot (\\x,{1.1*\\x*(\\x-1)*(\\x-2)});\n\\draw (2,0.2) -- (2,0) node [below] {$2$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.4:2.2] plot (\\x,{-\\x*(\\x-1)*(\\x-2)});\n\\draw [dashed] ({(3-sqrt(3))/3},{-2/3/sqrt(3)}) -- ({(3-sqrt(3))/3},0) node [above] {$1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{-\\x*\\x+\\x*\\x*\\x/3+1});\n\\draw [dashed] (2,{-4+8/3+1}) -- (2,0) node [above] {$2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [above left] {$O$};\n\\draw [domain = -1:2.85] plot (\\x,{\\x*\\x-\\x*\\x*\\x/3-0.5});\n\\draw [dashed] (2,{4-8/3-0.5}) -- (2,0) node [below] {$2$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496441,7 +496641,9 @@ "id": "019404", "content": "某商品的成本 $C$ 和产量 $q$ 满足函数关系 $C=50000+200 q$, 该商品的销售单价 $p$ 和产量 $q$ 满足函数关系 $p=24200-\\dfrac{1}{5}q^2$. 问 : 要使利润最大, 应如何确定产量?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496461,7 +496663,9 @@ "id": "019405", "content": "已知函数 $f(x)=x^2 \\mathrm{e}^{-x}$.\\\\\n(1) 求 $f(x)$ 的极小值和极大值;\\\\\n(2) 当曲线 $y=f(x)$ 的切线 $l$ 的斜率为负数时, 求 $l$ 在 $x$ 轴上的截距 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496481,7 +496685,9 @@ "id": "019406", "content": "已知函数 $f(x)=\\dfrac{\\mathrm{e}^x}{x}-\\ln x+x-a$.\\\\\n(1) 判断函数的单调区间;\\\\\n(2) 若 $f(x) \\geq 0$, 求 $a$ 的取值范围;\\\\\n(3) 证明: 若 $f(x)$ 有两个零点 $x_1$、$x_2$, 则 $x_1 x_2<1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496501,7 +496707,9 @@ "id": "019407", "content": "已知欧拉公式 $\\mathrm{e}^{\\mathrm{i}x}=\\cos x+\\mathrm{i}\\sin x$ ($\\mathrm{i}$ 为虚数单位), 则 $\\mathrm{e}^{2022 \\mathrm{i}}$ 对应的点在复平面位于第\\blank{50}象限.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496521,7 +496729,9 @@ "id": "019408", "content": "设 $O$ 是复平面的原点, 满足 $|z-\\mathrm{i}|+|z-1|=\\sqrt{2}$ 的复数在复平面上所对应的点构成集合 $M$,在 $M$ 中任取不同的两点 $A$ 和 $B$, 则 $\\angle AOB$ 的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496541,7 +496751,9 @@ "id": "019409", "content": "已知 $k+2$ 个两两互不相等的复数 $z_1$、$z_2$、$\\cdots$、$z_k$、$w_1$、$w_2$, 满足 $\\overline{w_1}-\\overline{w_2}=\\dfrac{4}{w_1-w_2}$, 且 $|w_j-z_\\alpha| \\in\\{1,3\\}$ (其中 $j=1$、$2$; $\\alpha=0$、$1$、$2$、$\\cdots$、$k$), 则 $k$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496561,7 +496773,9 @@ "id": "019410", "content": "已知 $|\\overrightarrow{a}|=3$, $|\\overrightarrow{b}|=5$, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=-12$, 则向量 $\\overrightarrow{a}$ 在向量 $\\overrightarrow{b}$ 上的投影向量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496581,7 +496795,9 @@ "id": "019411", "content": "已知 $\\lambda>0$, 平面向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足: $|\\overrightarrow{a}|=|\\overrightarrow{b}|=|\\overrightarrow{c}|=\\lambda$, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=0$, $\\overrightarrow{c}\\cdot \\overrightarrow{a}=2$, $\\overrightarrow{c}\\cdot \\overrightarrow{b}=1$, 则 $\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496601,7 +496817,9 @@ "id": "019412", "content": "如图, 在 $\\triangle ABC$ 中, $\\angle C=\\dfrac{\\pi}{2}$, $AC= BC=2$, $M$ 在 $AC$ 的中点, $P$ 在 $AB$ 上, 则 $\\overrightarrow{MP}\\cdot \\overrightarrow{CP}$ 最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$C$} coordinate (C);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (0,1) node [left] {$M$} coordinate (M);\n\\draw ($(A)!0.3!(B)$) node [above right] {$P$} coordinate (P);\n\\draw [->] (M)--(P);\n\\draw [->] (C)--(P);\n\\draw (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496621,7 +496839,9 @@ "id": "019413", "content": "如图所示, 正八边形 $A_1A_2A_3A_4A_5A_6A_7A_8$ 的边长为 2 , 若 $P$ 为该正八边形边上的动点, 则 $\\overrightarrow{A_1A_3}\\cdot \\overrightarrow{A_1P}$ 的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1) --++ (0:1) node [below] {$A_2$} coordinate (A_2) --++ (45:1) node [right] {$A_3$} coordinate (A_3) --++ (90:1) node [right] {$A_4$} coordinate (A_4) --++ (135:1) node [above] {$A_5$} coordinate (A_5) --++ (180:1) node [above] {$A_6$} coordinate (A_6) --++ (225:1) node [left] {$A_7$} coordinate (A_7) --++ (270:1) node [left] {$A_8$} coordinate (A_8) -- cycle;\n\\draw [->] (A_1) -- (A_3);\n\\draw [->] (A_1) -- ($(A_6)!0.5!(A_7)$) node [above left] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt{2}]$}{$[-2 \\sqrt{2}, 8+6 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 2 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 8+6 \\sqrt{2}]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496641,7 +496861,9 @@ "id": "019414", "content": "如图, 四个棱长为 1 的正方体排成一个正四棱柱, $AB$ 是一条侧棱, $P_i$($i=1,2, \\cdots, 8$) 是上底面上其余的八个点, 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AP_i}$($i=1,2, \\cdots, 8$) 的不同值的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) coordinate (B) --++ (45:{2/2}) coordinate (C)\n--++ (0,1) coordinate (C1) node [above left] {$P_8$}\n--++ (-2,0) coordinate (D1) node [above left] {$P_2$} --++ (225:{2/2}) node [above left] {$B$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,1) coordinate (B1) node [above left] {$P_6$} -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) coordinate (D) --++ (2,0) (D) --++ (0,1);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$P_1$} coordinate (P1);\n\\draw ($(B1)!0.5!(C1)$) node [above left] {$P_7$} coordinate (P7);\n\\draw ($(A1)!0.5!(B1)$) node [above left] {$P_3$} coordinate (P3);\n\\draw ($(P1)!0.5!(P7)$) node [above left] {$P_4$} coordinate (P4);\n\\draw ($(C1)!0.5!(D1)$) node [above left] {$P_5$} coordinate (P5);\n\\draw (P1) -- (P7) (P3) -- (P5) (P3) --++ (0,-1) coordinate (S) (P7) --++ (0,-1);\n\\draw [dashed] (P1) --++ (0,-1) --++ (2,0) (P5) --++ (0,-1) -- (S);\n\\draw [dashed] (P4) --++ (0,-1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$2$}{$4$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496661,7 +496883,9 @@ "id": "019415", "content": "在平面直角坐标系 $xOy$ 中, 已知椭圆 $C_1: \\dfrac{x^2}{36}+\\dfrac{y^2}{4}=1$ 和 $C_2: x^2+\\dfrac{y^2}{9}=1, P$ 为 $C_1$ 上的动点, $Q$ 为 $C_2$ 上的动点, $w$ 是 $\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}$ 的最大值. 记 $\\Omega=\\{(P, Q) | P$ 在 $C_1$ 上, $Q$ 在 $C_2$ 上, 且 $\\overrightarrow{OP}\\cdot \\overrightarrow{OQ}=w\\}$, 则 $\\Omega$ 中元素个数为\\bracket{20}.\n\\fourch{$2$个}{$4$个}{$8$个}{无穷个}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496681,7 +496905,9 @@ "id": "019416", "content": "已知 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 是不平行的两个向量, $k$ 是实数, 且 $\\overrightarrow{AP}=k \\overrightarrow{AB}$, 试用 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 表示 $\\overrightarrow{OP}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496701,7 +496927,9 @@ "id": "019417", "content": "已知点 $A(-2,0)$, 设 $B$、$C$ 是圆 $O: x^2+y^2=1$ 上的两个不同的动点, 且向量 $\\overrightarrow{OB}=t \\overrightarrow{OA}+(1-t) \\overrightarrow{OC}$ (其中 $t$ 为实数), 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496721,7 +496949,9 @@ "id": "019418", "content": "在 $\\triangle ABC$ 中, 点 $O$ 是 $BC$ 的中点, 过点 $O$ 的直线分别交直线 $AB, AC$ 与不同的两点 $M, N$, 若 $\\overrightarrow{AB}=m \\overrightarrow{AM}$, $\\overrightarrow{AC}=n \\overrightarrow{AN}$, $m>0$, $n>0$, 求 $\\dfrac{1}{m}+\\dfrac{4}{n}$ 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496741,7 +496971,9 @@ "id": "019419", "content": "设 $|\\overrightarrow{OA}|=1$, $|\\overrightarrow{OB}|=2$, $\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=0$, $\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$, 且 $\\lambda+\\mu=1$, 则 $\\overrightarrow{OA}$ 在 $\\overrightarrow{OP}$ 上投影的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496761,7 +496993,9 @@ "id": "019420", "content": "点 $P$ 是椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$ 上的动点, 点 $Q$ 与点 $P$ 关于 $x$ 轴对称, 若有 $\\overrightarrow{F_1P}$. $\\overrightarrow{F_2P}\\leq 1$, 则向量 $\\overrightarrow{F_1P}$ 与 $\\overrightarrow{F_2Q}$ 的夹角的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496781,7 +497015,9 @@ "id": "019421", "content": "若复数 $z$ 满足 $3 z+\\overline{z}=1+\\mathrm{i}$, 其中 $\\mathrm{i}$ 为虚数单位, 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496801,7 +497037,9 @@ "id": "019422", "content": "设 $m, n$ 分别为连续两次投掷骰子得到的点数, 且向量 $\\overrightarrow{a}=(m, n)$, $\\overrightarrow{b}=(1,-1)$, 则 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为锐角的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496821,7 +497059,9 @@ "id": "019423", "content": "在直角坐标系 $xOy$ 中, 已知三点 $A(a, 1), B(2, b), C(3,4)$, 若向量 $\\overrightarrow{OA}, \\overrightarrow{OB}$ 在向量 $\\overrightarrow{OC}$ 方向上的数量投影相同, 则 $3 a-4 b$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496841,7 +497081,9 @@ "id": "019424", "content": "若点 $P$、$Q$ 均在椭圆 $\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{a^2-1}=1$($a>1$) 上运动, $F_1$、$F_2$ 是椭圆 $\\Gamma$ 的左、右焦点, 则 $|\\overrightarrow{PF_1}+\\overrightarrow{PF_2}-2 \\overrightarrow{PQ}|$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496861,7 +497103,9 @@ "id": "019425", "content": "在复平面上, 已知直线 $l$ 上的点所对应的复数 $z$ 满足 $|z-2-\\mathrm{i}|=|z+\\mathrm{i}|$, 则直线 $l$ 的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496881,7 +497125,9 @@ "id": "019426", "content": "已知平面上 $A_1$、$A_2$、$A_3$、$A_4$、$A_5$ 五个点, 满足 $\\overrightarrow{A_n A_{n+1}}+\\overrightarrow{A_{n+1}A_{n+2}}=0$($n=1,2,3$), $|\\overrightarrow{A_n A_{n+1}}|+ |\\overrightarrow{A_{n+1}A_{n+2}}|=n+1$($n=1,2,3$), 则 $|\\overrightarrow{A_1A_5}|$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -496901,7 +497147,9 @@ "id": "019427", "content": "设 $\\overrightarrow{m}, \\overrightarrow{n}$ 为非零向量, 则``存在负数 $\\lambda$, 使得 $\\overrightarrow{m}=\\lambda \\overrightarrow{n}$''是``$\\overrightarrow{m}\\cdot \\overrightarrow{n}<0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496921,7 +497169,9 @@ "id": "019428", "content": "已知 $\\overrightarrow{a}$、$\\overrightarrow{b}$ 均为单位向量, 且 $\\overrightarrow{a}\\cdot \\overrightarrow{b}=0$. 若 $|\\overrightarrow{c}-4 \\overrightarrow{a}|+|\\overrightarrow{c}-3 \\overrightarrow{b}|=5$, 则 $|\\overrightarrow{c}+\\overrightarrow{a}|$ 的取值范围是\\bracket{20}.\n\\fourch{$[3, \\sqrt{10}]$}{$[3,5]$}{$[3,4]$}{$[\\sqrt{10}, 5]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496941,7 +497191,9 @@ "id": "019429", "content": "已知 $A, B$ 为平面上的两个定点, 且 $|\\overrightarrow{AB}|=2$, 该平面上的动线段 $PQ$ 的端点 $P, Q$ 满足 $|\\overrightarrow{AP}| \\leq 5$, $\\overrightarrow{AP}\\cdot \\overrightarrow{AB}=6$, $\\overrightarrow{AQ}=-2 \\overrightarrow{AP}$, 则动线段 $PQ$ 所形成图形的面积为\\bracket{20}.\n\\fourch{$36$}{$60$}{$81$}{$108$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -496961,7 +497213,9 @@ "id": "019430", "content": "已知平面向量 $\\overrightarrow{a}$、$\\overrightarrow{b}$、$\\overrightarrow{c}$ 满足 $\\overrightarrow{a}\\perp \\overrightarrow{b}$, 且 $\\{|\\overrightarrow{a}|,|\\overrightarrow{b}|,|\\overrightarrow{c}|\\}=\\{1,2,3\\}$, 求 $|\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}|$ 的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -496981,7 +497235,9 @@ "id": "019431", "content": "已知关于 $t$ 的方程 $t^2-2 t+a=0$($a \\in \\mathbf{R}$) 有两个虚根 $t_1$、$t_2$, 且满足 $|t_1-t_2|=2 \\sqrt{3}$.\\\\\n(1) 求方程的两个根以及实数 $a$ 的值;\\\\\n(2) 若对于任意 $x \\in \\mathbf{R}$, 不等式 $\\log _a(x^2+a) \\geq-k^2+2 m k-2 k$ 对于任意的 $k \\in[2,3]$ 恒成立, 求实数 $m$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497001,7 +497257,9 @@ "id": "019432", "content": "在直角坐标平面 $xOy$ 上的一列点 $A_1(1, a_1), A_2(2, a_2), \\cdots, A_n(n, a_n), \\cdots$, 简记为 $\\{A_n\\}$. 若由 $b_n=\\overrightarrow{A_n A_{n+1}}\\cdot \\overrightarrow{j}$ 构成的数列 $\\{b_n\\}$ 满足 $b_{n+1}>b_n$, $n=1,2, \\cdots$, 其中 $\\overrightarrow{j}$ 为方向与 $y$ 轴正方向相同的单位向量, 则称 $\\{A_n\\}$ 为 $T$ 点列.\\\\\n(1) 判断 $A_1(1,1), A_2(2, \\dfrac{1}{2}), A_3(1, \\dfrac{1}{3}), \\cdots, A_n(n, \\dfrac{1}{n}), \\cdots$, 是否为 $T$ 点列, 并说明理由;\\\\\n(2) 若 $\\{A_n\\}$ 为 $T$ 点列, 则点 $A_2$ 在点 $A_1$ 的右上方. 任取其中连续三点 $A_k$、$A_{k+1}$、$A_{k+2}$. 判断 $\\triangle A_k A_{k+1}A_{k+2}$ 的形状 (锐角三角形、直角三角形、钝角三角形), 并予以证明;\\\\\n(3) 若 $\\{A_n\\}$ 为 $T$ 点列, 正整数 $1 \\leq m \\overrightarrow{A_m A_p}\\cdot \\overrightarrow{j}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497021,7 +497279,9 @@ "id": "019433", "content": "在单位圆中, 圆心角 $\\dfrac{2}{3}\\pi$ 所对的弧长为\\blank{50}, 含这段弧的弓形面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497041,7 +497301,9 @@ "id": "019434", "content": "若 $\\sin (\\pi+\\theta)=\\dfrac{1}{3}$, 且 $\\theta$ 为第三象限角, 则 $\\cos (2 \\pi-\\theta)$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497061,7 +497323,9 @@ "id": "019435", "content": "在 $\\triangle ABC$ 中, $AC=3$, $3 \\sin A=2 \\sin B$, 且 $\\cos C=\\dfrac{1}{4}$, 则 $AB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497081,7 +497345,9 @@ "id": "019436", "content": "已知 $\\tan (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{1}{3}$, 则 $\\cos (\\dfrac{2}{3}\\pi+2 \\alpha)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497101,7 +497367,9 @@ "id": "019437", "content": "已知锐角 $\\alpha, \\beta$, 且 $\\cos \\alpha=\\dfrac{1}{7}$, $\\sin (\\alpha+\\beta)=\\dfrac{5 \\sqrt{3}}{14}$, 则 $\\cos \\beta$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497121,7 +497389,9 @@ "id": "019438", "content": "已知 $\\theta>0$, 对任意正整数 $n$, 总存在实数 $\\varphi$, 使得 $\\cos (n \\theta+\\varphi)<\\dfrac{\\sqrt{3}}{2}$, 则 $\\theta$ 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497141,7 +497411,9 @@ "id": "019439", "content": "如图 $AD=BC=6$, $AB=20$, $\\angle DAB= \\angle ABC=120^{\\circ}, O$ 为 $AB$ 的中点, 曲线 $CMD$ 上任意一点到点 $O$ 的距离相等, $MO \\perp AB$, 点 $P$ 是曲线 $CM$ 上的动点, 点 $Q$ 与点 $P$ 关于 $OM$ 对称.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-10,0) node [below] {$A$} coordinate (A);\n\\draw (10,0) node [below] {$B$} coordinate (B);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (B) ++ (60:6) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (120:6) node [left] {$D$} coordinate (D);\n\\draw (C) arc ({atan(3*sqrt(3)/13)}:{180-atan(3*sqrt(3)/13)}:14);\n\\draw (0,14) node [above] {$M$} coordinate (M);\n\\draw (D) -- (A) -- (B) -- (C) (O) -- (M);\n\\draw (50:14) node [above] {$P$} coordinate (P);\n\\draw (130:14) node [above] {$Q$} coordinate (Q);\n\\draw (C) -- (P) -- (M) -- (Q) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点 $P$ 与点 $C$ 重合, 求 $\\angle POB$ 的大小;\\\\\n(2) 求五边形 $DCPMQ$ 面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497161,7 +497433,9 @@ "id": "019440", "content": "一个二面角的余弦值为 $-\\dfrac{3}{5}$, 则这个二面角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497181,7 +497455,9 @@ "id": "019441", "content": "已知向量 $\\overrightarrow{m}=(\\dfrac{1}{2}, \\dfrac{1}{2}\\sin 2 x+\\dfrac{\\sqrt{3}}{2}\\cos 2 x)$, $\\overrightarrow{n}=(f(x),-1)$, 且 $\\overrightarrow{m}\\perp \\overrightarrow{n}$. 则函数 $f(x)$ 在 $x \\in[0, \\pi]$ 上的减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497201,7 +497477,9 @@ "id": "019442", "content": "已知函数 $f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$), 若 $f(\\dfrac{\\pi}{6})=f(\\dfrac{\\pi}{3})$, 且 $f(x)$ 在区间 $(\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3})$ 内有最小值无最大值, 则实数 $\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497221,7 +497499,9 @@ "id": "019443", "content": "已知关于 $x$ 的方程 $\\sqrt{3}\\sin 2 x+\\cos 2 x=k+1$ 在区间 $[0, \\dfrac{\\pi}{2}]$ 内有相异两个实数根, 则实数 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497241,7 +497521,9 @@ "id": "019444", "content": "函数 $f(x)=\\cos (\\pi x+\\varphi)$($0<\\varphi<\\dfrac{\\pi}{2}$) 的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.2:2, samples = 100] plot (\\x,{cos(180*\\x+30)});\n\\draw (0,{sqrt(3)/2}) node [left] {$\\dfrac{\\sqrt{3}}{2}$};\n\\draw [dashed] (0,{sqrt(3)/2}) --++ ({5/3},0) -- ({5/3},0) node [below] {$x_0$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\varphi$ 及图中 $x_0$ 的值;\\\\\n(2) 设 $g(x)=f(x)+f(x+\\dfrac{1}{3})$, 求函数 $g(x)$ 在区间 $[-\\dfrac{1}{2}, \\dfrac{1}{3}]$ 上的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497261,7 +497543,9 @@ "id": "019445", "content": "方程 $\\sin x+\\sqrt{3}\\cos x=1$ 在 $(\\pi, 2 \\pi)$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497281,7 +497565,9 @@ "id": "019446", "content": "已知函数 $f(x)=\\cos (2 x-\\dfrac{\\pi}{3})+2 \\sin (x-\\dfrac{\\pi}{4}) \\sin (x+\\dfrac{\\pi}{4})$, 函数 $f(x)$ 在区间 $[-\\dfrac{\\pi}{12}, \\dfrac{\\pi}{2}]$ 上的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497301,7 +497587,9 @@ "id": "019447", "content": "设函数 $f(x)=\\cos (\\omega x+\\dfrac{\\pi}{3})$($0<\\omega<2$), 若将 $f(x)$ 图像向左平移 $\\dfrac{4 \\pi}{5}$ 个单位后, 所得函数图像的对称轴与原函数图像的对称轴重合, 则 $\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497321,7 +497609,9 @@ "id": "019448", "content": "已知 $f(x)=A \\sin (w x+\\theta)$($w>0$), 若两个不等的实数 $x_1, x_2 \\in\\{x | f(x)=\\dfrac{A}{2}\\}$, $|x_1-x_2|_{\\min}=\\pi$, 则 $f(x)$ 的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497341,7 +497631,9 @@ "id": "019449", "content": "设函数 $f(x)=\\sin x-m(x \\in[0, \\dfrac{5 \\pi}{2}])$ 的零点为 $x_1, x_2, x_3$, 若 $x_1, x_2, x_3$ 成等比数列, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497361,7 +497653,9 @@ "id": "019450", "content": "已知函数 $f(x)=\\sin x+a \\cos x$, $\\dfrac{\\pi}{4}$ 是函数 $f(x)$ 的一个零点, 若 $\\alpha, \\beta \\in(0, \\dfrac{\\pi}{2})$, 且 $f(\\alpha+\\dfrac{\\pi}{4})=\\dfrac{\\sqrt{10}}{5}$, $f(\\beta+\\dfrac{3 \\pi}{4})=\\dfrac{3 \\sqrt{5}}{5}$, 则 $\\sin (\\alpha+\\beta)$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497381,7 +497675,9 @@ "id": "019451", "content": "已知 $\\triangle ABC$ 是斜三角形, 则``$A>B$''是``$|\\tan A|>|\\tan B|$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497401,7 +497697,9 @@ "id": "019452", "content": "在 $\\triangle ABC$ 中, 设三个内角 $A, B, C$ 的对边依次为 $a, b, c$. 则``$C \\in\\{\\dfrac{\\pi}{3}, \\dfrac{2 \\pi}{3}\\}$''是``$a^2+b^2= c^2+a b$''成立的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分条件又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497421,7 +497719,9 @@ "id": "019453", "content": "设函数 $f(x)=\\sin (x-\\dfrac{\\pi}{6})$, 若对于任意 $\\alpha \\in[-\\dfrac{5 \\pi}{6},-\\dfrac{\\pi}{2}]$, 在区间 $[0, m]$ 上总存在唯一确定的 $\\beta$, 使得 $f(\\alpha)+f(\\beta)=0$, 则 $m$ 的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{7 \\pi}{6}$}{$\\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497441,7 +497741,9 @@ "id": "019454", "content": "已知 $f(x)=\\sin w x$($w>0$).\\\\\n(1) 若 $f(x)$ 的周期是 $4 \\pi$, 求 $\\omega$, 并求此时 $f(x)=\\dfrac{1}{2}$ 的解集;\\\\\n(2) 已知 $\\omega=1$, $g(x)=f^2(x)+\\sqrt{3}f(-x) f(\\dfrac{\\pi}{2}-x)$, $x \\in[0, \\dfrac{\\pi}{4}]$, 求 $g(x)$ 的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497461,7 +497763,9 @@ "id": "019455", "content": "已知 $A$、$B$、$C$ 为 $\\triangle ABC$ 的三个内角, $a$、$b$、$c$ 是其三条边, $a=2$, $\\cos C=-\\dfrac{1}{4}$.\\\\\n(1) 若 $\\sin A=2 \\sin B$, 求 $b$、$c$;\\\\\n(2) $\\cos (A-\\dfrac{\\pi}{4})=\\dfrac{4}{5}$, 求 $c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497481,7 +497785,9 @@ "id": "019456", "content": "如图, 某海岸线可近视地看成曲线段 $A-B-C$, 其中 $AB$ 为线段, $\\overset\\frown{BC}$ 为四分之一的圆弧, $BD=39.2 \\mathrm{km}$, $\\angle BDC=22^{\\circ}$, $\\angle CBD=68^{\\circ}$, $\\angle BDA=58^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}\n\\path (0,0) coordinate (D) node [below right] {$D$};\n\\path (112:3.92) coordinate (B) node [above left] {$B$};\n\\path (0,3.63456) coordinate (C) node [above right] {$C$};\n\\path (-4.2365,0.747) coordinate (A) node [left] {$A$};\n\\draw (D) -- (C) -- (B) (A) -- (D) -- (B);\n\\draw [very thick] (C) arc (45:135:1.0383565) (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\overset\\frown{BC}$ 的长度;\\\\\n(2) 若 $AB=40 \\mathrm{km}$, 求点 $D$ 到海岸线 $A-B-C$ 的最短距离.(精确到 $0.001 \\mathrm{km}$)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497501,7 +497807,9 @@ "id": "019457", "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=2$ 若对任意的正整数 $p, q$, 都有 $a_{p+q}=a_p+a_q$, 则 $a_{2023}=$\\blank{50}, 若对任意的正整数 $p, q$, 都有 $a_{p+q}=a_p \\cdot a_q$, 则 $a_{2023}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497521,7 +497829,9 @@ "id": "019458", "content": "已知无穷数列 $\\{a_n\\}$ 的通项公式 $a_n=\\dfrac{9^n(n+1)}{10^n}$, 试判断此数列是否有最大项, 若有, 求出第几项最大, 若没有, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497541,7 +497851,9 @@ "id": "019459", "content": "已知 $S_n$ 为数列 $\\{a_n\\}$ 的前 $n$ 项和, 点 $(a_n, S_n)$ 在直线 $y=2 x-3 n$ 上.\\\\\n(1) 若数列 $\\{a_n+c\\}$ 成等比, 求常数的值;\\\\\n(2) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 数列 $\\{a_n\\}$ 中是否存在三项, 它们可以构成等差数列? 若存在, 请求出一组适合条件的项; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497561,7 +497873,9 @@ "id": "019460", "content": "已知数列 $\\{x_n\\}$ 满足 $x_1=\\dfrac{1}{2}$, $x_{n+1}=\\dfrac{1}{1+x_n}$, $n \\in \\mathbf{N}$, $n \\geq 1$. 猜想数列 $\\{x_{2 n}\\}$ 的单调性, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497581,7 +497895,9 @@ "id": "019461", "content": "已知数列 $\\{a_n\\}$ 是公差不为 $0$ 的等差数列, $a_1=\\dfrac{3}{2}$, 数列 $\\{b_n\\}$ 是等比数列, 且 $b_1=a_1$, $b_2=-a_3$, $b_3=a_4$, 数列 $\\{b_n\\}$ 的前 $n$ 项和为 $S_n$, 记点 $Q_n(b_n, S_n), n \\in \\mathbf{N}$, $n \\geq 1$.\\\\\n(1) 求数列 $\\{b_n\\}$ 的通项公式;\\\\\n(2) 证明: 点 $Q_1$、$Q_2$、$Q_3$、$\\cdots$、$Q_n$、$\\cdots$ 在同一直线 $l$ 上, 并求出直线 $l$ 的方程;\\\\\n(3) 若 $A \\leq S_n-\\dfrac{1}{S_n}\\leq B$ 对 $n \\in \\mathbf{N}$, $n \\geq 1$ 恒成立, 求 $B-A$ 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -497601,7 +497917,9 @@ "id": "019462", "content": "记 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和. 若 $a_1=-2$, $a_2+a_6=2$, 则 $S_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497621,7 +497939,9 @@ "id": "019463", "content": "若正项等比数列 $\\{a_n\\}$ 满足: $a_3+a_5=4$, 则 $a_4$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497641,7 +497961,9 @@ "id": "019464", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且满足 $S_m+S_n=S_{m+n}$, 若 $a_1=2$, 则 $a_{20}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497661,7 +497983,9 @@ "id": "019465", "content": "已知公差不为 $0$ 的等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_4, S_5, S_7 \\in\\{-10,0\\}$, 则 $S_n$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497681,7 +498005,9 @@ "id": "019466", "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_1=1$, $S_n-\\dfrac{1}{3}a_{n+1}=0$($n \\in \\mathbf{N}$, $n \\geq 1$), 则 $\\{a_n\\}$ 的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497701,7 +498027,9 @@ "id": "019467", "content": "设 $\\{a_n\\}$ 是公差为 $d$ 的等差数列, $\\{b_n\\}$ 是公比为 $q$ 的等比数列. 已知数列 $\\{a_n+b_n\\}$ 的前 $n$ 项和 $S_n=n^2-n+2^n-1$($n \\in \\mathbf{N}$, $n \\geq 1$), 则 $d+q$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497721,7 +498049,9 @@ "id": "019468", "content": "已知数列满足 $a_1+2 a_2+3 a_3+\\cdots+n a_n=(2 n-1) \\cdot 3^n$, 设 $b_n=\\dfrac{4 n}{a_n}$, $S_n$ 为数列 $\\{b_n\\}$ 前 $n$ 项和, 若 $S_n<\\lambda$ ($\\lambda$ 为常数), 则 $\\lambda$ 最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497741,7 +498071,9 @@ "id": "019469", "content": "已知数列 $\\{a_n\\}$ 中, 若 $a_1=0$, $a_i=k^2$($i \\in \\mathbf{N}$, $2^k \\leq i<2^{k+1}$, $k=1,2,3, \\cdots$), 则满足 $a_i+a_{2 i}\\geq 100$ 的 $i$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -497761,7 +498093,9 @@ "id": "019470", "content": "若无穷等差数列 $\\{a_n\\}$ 的首项 $a_1<0$, 公差 $d>0,\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则以下结论中一定正确的是\\bracket{20}.\n\\fourch{$S_n$ 严格递增}{$S_n$ 严格递减}{$S_n$ 有最小值}{$S_n$ 有最大值}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497781,7 +498115,9 @@ "id": "019471", "content": "已知等差数列 $\\{a_n\\}$ 单调递增且满足 $a_1+a_8=6$, 则 $a_6$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, 3)$}{$(3,6)$}{$(3,+\\infty)$}{$(6,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497801,7 +498137,9 @@ "id": "019472", "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则下列判断一定正确的是\\bracket{20}.\n\\twoch{若 $S_3>0$, 则 $a_{2018}>0$}{若 $S_3<0$, 则 $a_{2018}<0$}{若 $a_2>a_1$, 则 $a_{2019}>a_{2018}$}{若 $\\dfrac{1}{a_2}>\\dfrac{1}{a_1}$, 则 $a_{2019}0$;\\\\\n\\textcircled{3} 若数列 $\\{a_n\\}$ 是周期数列, 则最小正周期可能为$2$;\\\\\n\\textcircled{4} 若数列 $\\{a_n\\}$ 是常数列, 则 $d \\geq-\\dfrac{1}{4}$ 其中, 所有正确结论的个数是\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -497841,7 +498181,9 @@ "id": "019474", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=(n-k_1)(n-k_2)$, 其中 $k_1, k_2 \\in \\mathbf{Z}$.\\\\\n(1) 试写出一组 $k_1, k_2 \\in \\mathbf{Z}$ 的值, 使得数列 $\\{a_n\\}$ 中的各项均为正数;\\\\\n(2) 若 $k_1=1$、$k_2$ 为正整数, 数列 $\\{b_n\\}$ 满足 $b_n=\\dfrac{a_n}{n}$, 且对任意 $m \\in \\mathbf{N}$($m \\geq 1$, $m \\neq 3$), 均有 $b_30$) 的公共弦的长为 $2 \\sqrt{3}$, 则实数 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498021,7 +498379,9 @@ "id": "019483", "content": "已知圆 $C_1: x^2+y^2-2 a x-2 y+a^2-15=0$, 圆 $C_2: x^2+y^2-4 a x-2 y+4 a^2=0 $($a>0$), 求实数 $a$ 的值或取值范围, 分别使得圆 $C_1$ 与圆 $C_2$.\\\\\n(1) 相切;\\\\\n(2) 相交;\\\\\n(3) 内含.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498041,7 +498401,9 @@ "id": "019484", "content": "已知圆 $C: x^2+(y-1)^2=5$, 直线 $l: m x-y+1-m=0$.\\\\\n(1) 求证: 对 $m \\in \\mathbf{R}$, 直线 $l$ 与圆 $C$ 总有两个不同的交点;\\\\\n(2) 设 $l$ 与圆 $C$ 交于 $A$、$B$ 两点,若 $|AB|=\\sqrt{17}$, 求 $l$ 的倾斜角;\\\\\n(3) 求直线 $l$ 中, 截圆所得的弦最长时的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498061,7 +498423,9 @@ "id": "019485", "content": "已知方程 $\\dfrac{x^2}{|m|-1}+\\dfrac{y^2}{2-m}=1$ 表示焦点在 $y$ 轴上的椭圆, 则 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498081,7 +498445,9 @@ "id": "019486", "content": "已知动圆 $P$ 过点 $A(3,0)$, 且与圆 $(x+3)^2+y^2=4$ 相外切, 则动圆圆心 $P$ 的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498101,7 +498467,9 @@ "id": "019487", "content": "已知 $F$ 是抛物线 $y^2=4 x$ 的焦点, $P$ 是抛物线上的一个动点, 则 $\\triangle APF$ 周长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498121,7 +498489,9 @@ "id": "019488", "content": "已知抛物线 $C: y^2=2 p x$($p>0$) 的焦点为 $F$, 点 $M$ 是抛物线 $C$ 上一点,圆 $M$ 与 $y$ 轴相切且被直线 $x=\\dfrac{p}{2}$ 截得的弦长为 $\\sqrt{2}p$, 若 $|MF|=\\dfrac{5}{2}$, 求抛物线 $C$ 的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498141,7 +498511,9 @@ "id": "019489", "content": "直线 $x-y-2=0$ 关于直线 $2 x-2 y+1=0$ 对称的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498161,7 +498533,9 @@ "id": "019490", "content": "已知过点 $(0,-2)$ 且具有斜率 $k$ 的直线 $l$ 与以点 $A(3,1)$ 和 $B(-2,5)$ 为端点的线段 $AB$ 相交, 则实数 $k$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498181,7 +498555,9 @@ "id": "019491", "content": "已知方程 $\\dfrac{x^2}{m^2+n}-\\dfrac{y^2}{3 m^2-n}=1$ 表示双曲线, 且该双曲线的焦距为 $4$, 则 $n$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498201,7 +498577,9 @@ "id": "019492", "content": "已知 $P_1$、$P_2$、$P_3$、$\\cdots$、$P_{10}$ 是抛物线 $y^2=8 x$ 上不同的点, 点 $F(2,0)$, 若 $\\overrightarrow{FP_1}+\\overrightarrow{FP_2}+\\cdots+ \\overrightarrow{FP_{10}}=\\overrightarrow{0}$, 则 $|\\overrightarrow{FP_1}|+|\\overrightarrow{FP_2}|+\\cdots+|\\overrightarrow{FP_{10}}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498221,7 +498599,9 @@ "id": "019493", "content": "椭圆 $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 第一象限上一点与中心右焦点构成一个正三角形, 则此椭圆的离心率 $e=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498241,7 +498621,9 @@ "id": "019494", "content": "已知实数 $x$、$y$ 满足 $\\dfrac{x|x|}{4}+y|y|=1$, 则 $|x+2 y-4|$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498261,7 +498643,9 @@ "id": "019495", "content": "直线 $x-\\sqrt{3}y=0$ 绕原点按逆时针方向旋转 $30^{\\circ}$ 后所得的直线 $l$ 与圆 $(x-2)^2+y^2=3$ 的位置关系是\\bracket{20}.\n\\twoch{直线 $l$ 过圆心}{直线 $l$ 与圆相交, 但不过圆心}{直线 $l$ 与圆相切}{直线 $l$ 与圆无公共点}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498281,7 +498665,9 @@ "id": "019496", "content": "设抛物线 $C: x^2=8 y$ 的焦点 $F$, 准线为 $l, P(x_0, y_0)$ 为 $C$ 上一动点, $A(2,1)$, 则下列结论错误的是\\bracket{20}.\n\\onech{当 $x_0=4$ 时, $|PF|$ 的值为 $6$}{当 $x_0=2$ 时,抛物线 $C$ 在点 $P$ 处的切线方程为 $x-2 y-2=0$}{$|PA|+|PF|$ 的最小值为 $3$}{$|PA|-|PF|$ 的最大值为 $\\sqrt{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498301,7 +498687,9 @@ "id": "019497", "content": "已知椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右顶点分别为 $A_1, A_2$, 且以线段 $A_1A_2$ 为直径的圆与直线 $b x-a y+2 a b=0$ 相交,则椭圆的离心率的取值范围为\\bracket{20}.\n\\fourch{$(0, \\dfrac{\\sqrt{6}}{3})$}{$(\\dfrac{\\sqrt{6}}{3}, 1)$}{$(\\dfrac{\\sqrt{2}}{3}, 1)$}{$(0, \\dfrac{\\sqrt{2}}{3})$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498321,7 +498709,9 @@ "id": "019498", "content": "已知 $a \\in \\mathbf{R}$, 双曲线 $\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$.\\\\\n(1) 若点 $(2,1)$ 在双曲线上, 求 $\\Gamma$ 的焦点坐标;\\\\\n(2) 若 $a=1$, 直线 $y=k x+1$ 与 $\\Gamma$ 相交于 $A, B$ 两点, 且线段 $AB$ 中点的横坐标为 $1$, 求实数 $k$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498341,7 +498731,9 @@ "id": "019499", "content": "在平面直角坐标系 $xOy$ 中, 一动圆经过点 $A(\\dfrac{1}{2}, 0)$ 且与直线 $x=-\\dfrac{1}{2}$ 相切, 设该动圆圆心的轨迹为曲线 $K, P$ 是曲线 $K$ 上一点.\\\\\n(1) 求曲线 $K$ 的方程;\\\\\n(2) 过点 $A$ 且斜率为 $k$ 的直线 $l$ 与曲线 $K$ 交于 $B$、$C$ 两点, 若 $l \\parallel OP$ 且直线 $OP$ 与直线 $x=1$ 交于 $Q$ 点, 求 $\\dfrac{|AB| \\cdot|AC|}{|OP| \\cdot|OQ|}$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498361,7 +498753,9 @@ "id": "019500", "content": "设常数 $m>0$ 且 $m \\neq 1$, 椭圆 $\\Gamma: \\dfrac{x^2}{m^2}+y^2=1$, 点 $P$ 是 $\\Gamma$ 上的动点.\\\\\n(1) 若点 $P$ 的坐标为 $(2,0)$, 求 $\\Gamma$ 的焦点坐标;\\\\\n(2) 设 $m=3$, 若定点 $A$ 的坐标为 $(2,0)$, 求 $|PA|$ 的最大值与最小值;\\\\\n(3) 设 $m=\\dfrac{1}{2}$, 若 $\\Gamma$ 上的另一动点 $Q$ 满足 $OP \\perp OQ$ ($O$ 为坐标原点), 求证: $O$ 到直线 $PQ$ 的距离是定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498381,7 +498775,9 @@ "id": "019501", "content": "已知两圆 $C_1:(x-2)^2+y^2=54$, $C_2:(x+2)^2+y^2=6$, 动圆 $M$ 在圆 $C_1$ 内部且和圆 $C_1$ 内切、和圆 $C_2$ 外切.\\\\\n(1) 求动圆圆心 $M$ 的轨迹 $C$ 的方程;\\\\\n(2) 过点 $A(3,0)$ 的直线与 (1) 中的曲线 $C$ 交于 $P$、$Q$ 两点, 点 $P$ 关于 $x$ 轴对称的点为 $R$, 求 $\\triangle ARQ$ 面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498401,7 +498797,9 @@ "id": "019502", "content": "某团队开发一款``猫捉老鼠''的游戏. 如图所示, $A$、$B$ 两个信号源相距 $10$ 米, $O$ 是 $AB$ 的中点, 过点 $O$ 的直线 $l$ 与直线 $AB$ 的夹角为 $45^{\\circ}$. 机器猫在直线 $l$ 上运动, 机器鼠的运动轨迹始终满足: 接收到点 $A$ 的信号比接收到点 $B$ 的信号晩 $\\dfrac{8}{v_0}$ 秒, 其中 $v_0$ (单位: 米/ 秒) 是信号传播的速度. 游戏设定: 机器鼠在距离直线 $l$ 不超过 $1.5$ 米的区域运动时, 有``被抓''的风险, 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''的风险?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498421,7 +498819,9 @@ "id": "019503", "content": "如图, 设 $F$ 是椭圆 $\\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$ 的下焦点, 直线 $y=k x-4$($k>0$) 与椭圆相交于 $A$、$B$ 两点, 与 $y$ 轴交于 $P$ 点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (O) ellipse ({sqrt(3)} and 2);\n\\draw (0,-1) node [left] {$F$} coordinate (F);\n\\draw (0,-4) node [left] {$P$} coordinate (P);\n\\draw ({3*sqrt(5)/8},{-7/4}) node [below right] {$A$} coordinate (A);\n\\draw ($(P)!2!(A)$) node [right] {$B$} coordinate (B);\n\\draw ($(P)!-0.2!(B)$) -- ($(B)!-0.2!(P)$);\n\\draw (A)--(F)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\overrightarrow{PA}=\\overrightarrow{AB}$, 求 $k$ 的值;\\\\\n(2) 求证: $\\angle AFP=\\angle BFO$;\\\\\n(3) 求 $\\triangle ABF$ 面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498441,7 +498841,9 @@ "id": "019504", "content": "已知抛物线 $\\Gamma: y^2=4 x$ 的焦点为 $F$, 若 $\\triangle ABC$ 的三个顶点都在抛物线 $\\Gamma$ 上, 且\n$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}=\\overrightarrow{0}$, 则称该三角形为``核心三角形''.\\\\\n(1) 是否存在``核心三角形'', 其中两个顶点的坐标分别为 $(0,0)$ 和 $(1,2)$? 请说明理由;\\\\\n(2) 设``核心三角形''$ABC$ 的一边 $AB$ 所在直线的斜率为 4 , 求直线 $AB$ 的方程;\\\\\n(3) 已知 $\\triangle ABC$ 是``核心三角形'', 证明: 点 $A$ 的横坐标小于 $2$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498461,7 +498863,9 @@ "id": "019505", "content": "已知椭圆 $C: \\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$, 直线 $l$ 经过椭圆的右焦点 $F$, 交椭圆 $C$ 于 $P$、$Q$ 两点 (点 $P$ 在第二象限), 若 $Q$ 关于 $x$ 轴对称的点为 $Q'$, 且满足 $PQ \\perp FQ'$, 则直线 $l$ 的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498481,7 +498885,9 @@ "id": "019506", "content": "若曲线 $C: y^2-2 y-x+3=0$ 和直线 $l: y=k x+\\dfrac{3}{2}$ 只有一个公共点, 则实数 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498501,7 +498907,9 @@ "id": "019507", "content": "已知抛物线: $y^2=2 p x$($p>0$), 若第一象限的 $A$、$B$ 两点在抛物线上, 焦点为 $F$, $|AF|= 2$, $|BF|=4$, $|AB|=3$, 则直线 $AB$ 的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498521,7 +498929,9 @@ "id": "019508", "content": "已知双曲线 $\\dfrac{x^2}{a^2}-y^2=1$($a>0$), 双曲线右支上有任意两点 $P_1(x_1, y_1), P_2(x_2, y_2)$ 满足 $x_1 x_2-y_1 y_2>0$ 恒成立, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498541,7 +498951,9 @@ "id": "019509", "content": "$P$ 为双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的右支上一点, $M, N$ 分别是圆 $(x+5)^2+y^2=4$ 和 $(x-5)^2+ y^2=4$ 上的点, 则 $|PM|-|PN|$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498561,7 +498973,9 @@ "id": "019510", "content": "双曲线 $C: x^2-\\dfrac{y^2}{2}=1$, 过定点 $A(-1,0)$ 的两条垂线分别交双曲线于 $P, Q$ 两点, 直线 $PQ$ 恒过定点\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498581,7 +498995,9 @@ "id": "019511", "content": "已知直线方程 $f(x, y)=0$ 表示直线 $l$, 点 $P(x_0, y_0)$ 为定点, 若点 $P$ 不在直线 $l$ 上, 则 $f(x, y)=f(x_0, y_0)$ 一定表示\\bracket{20}.\n\\twoch{过 $P$ 而与直线 $l$ 相交的直线}{过 $P$ 而与直线 $l$ 平行的直线}{过 $P$ 而与直线 $l$ 垂直的直线}{过 $P$ 而与直线 $l$ 重合的直线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498601,7 +499017,9 @@ "id": "019512", "content": "如图, 某绿色蔬菜种植基地在 $A$ 处, 要把此处生产的蔬菜沿道路 $AA_1$ 或 $AA_2$ 运送到四边形区域 $A_1A_2A_3A_4$ 的农贸市场. 现要求在农贸市场中确定一条界线, 使位于界线一侧的点沿道路 $AA_1$ 比沿道路 $AA_2$ 运送蔬菜近, 而另一侧的点则反之, 该界线所在曲线为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [right] {$A_2$} coordinate (A_2);\n\\draw (0.5,-1.5) node [below] {$A$} coordinate (A);\n\\draw (2.5,1) node [right] {$A_3$} coordinate (A_3);\n\\draw (0.3,1.8) node [above] {$A_4$} coordinate (A_4);\n\\filldraw [pattern = north east lines] (A_1)--(A_2)--(A_3)--(A_4)--cycle;\n\\draw (A_1)--(A)--(A_2);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{直线}{椭圆}{双曲线}{抛物线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498621,7 +499039,9 @@ "id": "019513", "content": "在直角坐标平面内, 点 $A, B$ 的坐标分别为 $(-1,0),(1,0)$, 则满足 $\\tan \\angle PAB \\cdot \\tan \\angle PBA=m$ ($m$ 为非零常数) 的点 $P$ 的轨迹方程是\\bracket{20}.\n\\fourch{$x^2-\\dfrac{y^2}{m}=1$($y \\neq 0$)}{$x^2-\\dfrac{y^2}{m}=1$}{$x^2+\\dfrac{y^2}{m}=1$($y \\neq 0$)}{$x^2+\\dfrac{y^2}{m}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498641,7 +499061,9 @@ "id": "019514", "content": "如图, 双曲线 $\\Gamma: \\dfrac{x^2}{3}-y^2=1$ 的左、右焦点分别为 $F_1, F_2$, 过 $F_2$ 作直线 $l$ 交 $y$ 轴于点 $Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-6) -- (0,6) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -5.5:5.5, samples = 100] plot ({sqrt(3+3*\\x*\\x)},\\x);\n\\draw [domain = -5.5:5.5, samples = 100] plot ({-sqrt(3+3*\\x*\\x)},\\x);\n\\filldraw (-2,0) circle (0.1) node [below left] {$F_1$};\n\\filldraw (2,0) circle (0.1) node [below right] {$F_2$};\n\\draw (0,-2) node [right] {$Q$};\n\\draw (-4,-6) -- (8,6);\n\\draw [dashed] (-10,{-10/sqrt(3)}) -- (10,{10/sqrt(3)});\n\\draw [dashed] (10,{-10/sqrt(3)}) -- (-10,{10/sqrt(3)});\n\\end{tikzpicture}\n\\end{center}\n(1) 当直线 $l$ 平行于 $\\Gamma$ 的一条渐近线时, 求点 $F_1$ 到直线 $l$ 的距离;\\\\\n(2) 当直线 $l$ 的斜率为 $1$ 时, 在 $\\Gamma$ 的右支上是否存在点 $P$, 满足 $\\overrightarrow{F_1P}\\cdot \\overrightarrow{F_1Q}=0$ ? 若存在,求出 $P$ 点的坐标; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498661,7 +499083,9 @@ "id": "019515", "content": "在平面直角坐标系中, $A$、$B$ 分别为椭圆 $\\Gamma: \\dfrac{x^2}{2}+y^2=1$ 的上、下顶点, 若动直线 $l$ 过点 $P(0, b)$($b>1$), 且与椭圆 $\\Gamma$ 相交于 $C$、$D$ 两个不同点 (直线 $l$ 与 $y$ 轴不重合, 且 $C$、$D$ 两点在 $y$ 轴右侧, $C$ 在 $D$ 的上方), 直线 $AD$ 与 $BC$ 相交于点 $Q$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.9) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\filldraw (-1,0) circle (0.03) node [above] {$F_1$};\n\\filldraw (1,0) circle (0.03) node [above] {$F_2$};\n\\draw (0,1.6) node [right] {$P(0,b)$} coordinate (P);\n\\draw (1.5,-1) coordinate (R) node [right] {$l$};\n\\draw [name path = l] ($(P)!-0.1!(R)$) -- (R);\n\\draw [name path = elli] (0,0) ellipse ({sqrt(2)} and 1);\n\\draw [name intersections = {of = l and elli, by = {C,D}}];\n\\draw (C) node [above] {$C$};\n\\draw (D) node [below] {$D$};\n\\draw (0,1) node [above left] {$A$} coordinate (A);\n\\draw (0,-1) node [below left] {$B$} coordinate (B);\n\\draw [name path = BC] (B)--(C);\n\\draw [name path = AD] (A)--(D);\n\\draw [name intersections = {of = BC and AD, by = Q}];\n\\draw (Q) node [left] {$Q$};\n\\end{tikzpicture}\n\\end{center}\n(1) 设 $\\Gamma$ 的两焦点为 $F_1$、$F_2$, 求 $\\angle F_1AF_2$ 的值;\\\\\n(2) 若 $b=3$, 且 $\\overrightarrow{PD}=\\dfrac{3}{2}\\overrightarrow{PC}$, 求点 $Q$ 的横坐标;\\\\\n(3) 是否存在这样的点 $P$, 使得点 $Q$ 的纵坐标恒为 $\\dfrac{1}{3}$ ? 若存在, 求出点 $P$ 的坐标; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498681,7 +499105,9 @@ "id": "019516", "content": "已知斜率为 $k$ 的直线 $l$ 经过抛物线 $C: y^2=4 x$ 的焦点 $F$, 且与抛物线 $C$ 交于不同的两点 $A(x_1, y_1)$、$B(x_2, y_2)$.\\\\\n(1) 若点 $A$ 和 $B$ 到抛物线准线的距离分别为 $\\dfrac{3}{2}$ 和 $3$, 求 $|AB|$;\\\\\n(2) 若 $|AF|+|AB|=2|BF|$, 求 $k$ 的值;\\\\\n(3) 点 $M(t, 0)$, $t>0$, 对任意确定的实数 $k$, 若 $\\triangle AMB$ 是以 $AB$ 为斜边的直角三角形,判断符合条件的点 $M$ 有几个, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498701,7 +499127,9 @@ "id": "019517", "content": "过正方体中心的平面截正方体所得的截面中, 不可能的图形是\\bracket{20}.\n\\fourch{三角形}{长方形}{对角线不相等的菱形}{六边形}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498721,7 +499149,9 @@ "id": "019518", "content": "已知平面 $\\alpha$、$\\beta$、$\\gamma$ 两两垂直, 直线 $a$、$b$、$c$ 满足: $a \\subset \\alpha, b \\subset \\beta, c \\subset \\gamma$, 则直线 $a$、$b$、$c$ 不可能满足以下哪种关系\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498741,7 +499171,9 @@ "id": "019519", "content": "如果 $a$、$b$ 是异面直线, $P$ 是不在 $a$、$b$ 上的一点, 下列四个结论:\\\\\n\\textcircled{1} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都异面;\\\\\n\\textcircled{2} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都垂直;\\\\\n\\textcircled{3} 过 $P$ 一定可作平面 $\\alpha$ 与 $a$、$b$ 都平行;\\\\\n\\textcircled{4} 过 $P$ 一定可作直线 $l$ 与 $a$、$b$ 都平行.\\\\\n其中正确的结论有\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498761,7 +499193,9 @@ "id": "019520", "content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$、$Q$、$R$、$S$ 分别为棱 $AB$、$BC$、$BB_1$、$CD$ 的中点, 联结 $A_1S$、$B_1D$. 空间任意两点 $M$ 、 $N$, 若线段 $MN$ 上不存在点在线段 $A_1S$、$B_1D$ 上, 则称 $M$、$N$ 两点可视,则下列选项中与点 $D_1$ 可视的为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (45:{2/2}) node [right] {$C$} coordinate (C)\n--++ (0,2) node [above right] {$C_1$} coordinate (C1)\n--++ (-2,0) node [above left] {$D_1$} coordinate (D1) --++ (225:{2/2}) node [left] {$A_1$} coordinate (A1) -- cycle;\n\\draw (A) ++ (2,2) node [right] {$B_1$} coordinate (B1) -- (B) (B1) --++ (45:{2/2}) (B1) --++ (-2,0);\n\\draw [dashed] (A) --++ (45:{2/2}) node [left] {$D$} coordinate (D) --++ (2,0) (D) --++ (0,2);\n\\filldraw ($(A)!0.5!(B)$) circle (0.05) node [below] {$P$};\n\\filldraw ($(C)!0.5!(B)$) circle (0.05) node [below right] {$Q$};\n\\filldraw ($(B1)!0.5!(B)$) circle (0.05) node [above right] {$R$};\n\\filldraw ($(C)!0.5!(D)$) circle (0.05) node [above] {$S$} coordinate (S);\n\\draw [dashed] (A1) -- (S) (B1) -- (D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{点 $P$}{点 $B$}{点 $R$}{点 $Q$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498781,7 +499215,9 @@ "id": "019521", "content": "已知直二面角 $\\alpha-l-\\beta$, 直线 $a \\subset$ 平面 $\\alpha$, 直线 $b \\subset$ 平面 $\\beta$, 且 $a$ 与 $l$ 不垂直, $b$ 与 $l$ 不垂直, 那么\\bracket{20}.\n\\twoch{$a$ 与 $b$ 可能垂直, 但不可能平行}{$a$ 与 $b$ 可能垂直, 也可能平行}{$a$ 与 $b$ 不可能垂直, 但可能平行}{$a$ 与 $b$ 不可能垂直, 也不可能平行}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498801,7 +499237,9 @@ "id": "019522", "content": "如图, 在棱长为 $10$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$ 为左侧面 $ADD_1A_1$ 上一点, 已知点 $P$ 到 $A_1D_1$ 的距离为 $3, P$ 到 $AA_1$ 的距离为 $2$ , 则与过点 $P$ 且与 $A_1C$ 平行的直线相交的面是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\def\\l{10}\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 [dashed] (C)--(A_1);\n\\draw [dashed] ($(A)!0.7!(A_1)$) --++ (0,0,-2) node [right] {$P$} --++ (0,3,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{面 $AA_1B_1B$}{面 $BB_1C_1C$}{面 $CC_1D_1D$}{面 $ABCD$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -498821,7 +499259,9 @@ "id": "019523", "content": "如图, $ABCD$ 是矩形, $PA \\perp$ 平面 $ABCD, E$、$F$ 分别是 $AB$、$PC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,1.5,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(P)$) node [right] {$F$} coordinate (F);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(P)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $CD \\perp PD$;\\\\\n(2) 求证: $EF\\parallel $ 平面 $PAD$;\\\\\n(3) 当平面 $PCD$ 与平面 $ABCD$ 成多大角时, 直线 $EF \\perp$ 平面 $PCD$?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498841,7 +499281,9 @@ "id": "019524", "content": "在四面体 $ABCD$ 中, 面 $ABD$ 、面 $ACD$ 是全等的直角三角形, $AD$ 是公共的斜边, 且 $AD=\\sqrt{3}$, $BD=CD=1$, 面 $ABC$ 是等边三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (D) ++ (0,0,1) node [below] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) ++ (0,{sqrt(6)/2},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AC$ 与 $BD$ 所成的角的大小;\\\\\n(2) 求二面角 $B-AD-C$ 的大小;\\\\\n(3) 过点 $A$ 作 $AO \\perp$ 平面 $BCD$, 垂足为 $O$, 求证: 四边形 $OBDC$ 是正方形; 并点 $A$ 到平面 $BCD$ 的距离;\\\\\n(4) 在直线 $AC$ 上是否存在一点 $E$, 使得 $ED$ 与面 $BCD$ 成 $30^{\\circ}$ 角? 若存在, 确定点 $E$ 的位置, 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -498861,7 +499303,9 @@ "id": "019525", "content": "在正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, 底面 $ABCD$ 的边长为 $3$, $BD_1$ 与底面所成角的大小为 $\\arctan \\dfrac{2}{3}$, 则该正四棱柱的高等于\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{3}\n\\def\\m{3}\n\\def\\n{{2*sqrt(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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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)(B)--(D_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498881,7 +499325,9 @@ "id": "019526", "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为 $A_1B_1$ 的中点, $AB=BB_1=2$, $AC=2 \\sqrt{5}$, 则异面直线 $BE$ 与 $AC$ 所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{4}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 [above] {$E$} coordinate (E);\n\\draw (B)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498901,7 +499347,9 @@ "id": "019527", "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为 $A_1B_1$ 的中点, $AB=BB_1=2$, $AC=2 \\sqrt{5}$, 则二面角 $C_1-BD-C$ 的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{4}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 [above] {$E$} coordinate (E);\n\\draw (B)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498921,7 +499369,9 @@ "id": "019528", "content": "设 $m$、$n$ 是空间两条不同直线, $\\alpha$、$\\beta$ 是两个不同平面, 下面有四个命题:\\\\\n\\textcircled{1} $m \\perp \\alpha$, $n \\parallel \\beta$, $\\alpha \\parallel \\beta \\Rightarrow m \\perp n$;\\\\\n\\textcircled{2} $m \\perp n$, $\\alpha \\parallel \\beta$, $m \\perp \\alpha \\Rightarrow n \\parallel \\beta$;\\\\\n\\textcircled{3} $m \\perp n$, $\\alpha \\parallel \\beta$, $m \\parallel \\alpha \\Rightarrow n \\perp \\beta$;\\\\\n\\textcircled{4} $m \\perp \\alpha$, $m \\parallel n$, $\\alpha \\parallel \\beta \\Rightarrow n \\perp \\beta$.\\\\\n其中真命题的编号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498941,7 +499391,9 @@ "id": "019529", "content": "已知集合 $A$、$B$、$C$, $A=\\{$ 直线 $\\}$, $B=\\{$ 平面 $\\}$, $C=A \\cup B$, 若 $a \\in A$, $b \\in B$, $c \\in C$, 下面给出四个命题:\\\\\n\\textcircled{1} $\\begin{cases}a \\perp b,\\\\c \\perp b\\end{cases}\\Rightarrow a \\parallel c$; \\textcircled{2} $\\begin{cases}a \\perp b,\\\\c \\parallel b\\end{cases}\\Rightarrow a \\perp c$; \\textcircled{3} $\\begin{cases}a \\parallel b,\\\\c \\parallel b\\end{cases}\\Rightarrow a \\parallel c$; \\textcircled{4} $\\begin{cases}a \\parallel b,\\\\c \\perp b\\end{cases}\\Rightarrow a \\perp c$.\n其中所有正确命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498961,7 +499413,9 @@ "id": "019530", "content": "已知正方体的棱长为 $1$, 每条棱所在的直线与平面所成的角相等, 则平面截此正方体所得截面面积的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -498981,7 +499435,9 @@ "id": "019531", "content": "两条直线 $a$、$b$ 分别和异面直线 $c$、$d$ 都相交, 则直线 $a$、$b$ 的位置关系是\\bracket{20}.\n\\twoch{一定是异面直线}{一定是相交直线}{可能是平行直线}{可能是异面直线, 也可能是相交直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499001,7 +499457,9 @@ "id": "019532", "content": "平面 $\\alpha$ 外有两条直线 $m$、$n$, 如果 $m$、$n$ 在平面 $\\alpha$ 内的射影分别是直线 $m'$、$n'$, 给出下列四个命题:\\\\\n\\textcircled{1} $m' \\perp n' \\Rightarrow m \\perp n$; \\textcircled{2} $m \\perp n \\Rightarrow m' \\perp n'$; \\textcircled{3} 若 $m'$ 和 $n'$ 相交, 则 $m$ 和 $n$ 相交或重合; \\textcircled{4} 若 $m'$ 和 $n'$ 平行, 则 $m$ 和 $n$ 平行或重合;\n其中真命题的个数是\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{3 个}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499021,7 +499479,9 @@ "id": "019533", "content": "如图, 已知正三棱柱 $ABC-A_1B_1C_1, AC=AA_1, E$、$F$ 分别是棱 $BC, A_1C_1$ 上的点. 记 $EF$ 与 $AA_1$ 所成的角为 $\\alpha$, $EF$ 与平面 $ABC$ 所成的角为 $\\beta$, 二面角 $F-BC-A$ 的平面角为 $\\gamma$, 则\\bracket{20}.\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 ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A_1)!0.7!(C_1)$) node [above] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\alpha \\leq \\gamma \\leq \\beta$}{$\\alpha \\leq \\beta \\leq \\gamma$}{$\\beta \\leq \\alpha \\leq \\gamma$}{$\\beta \\leq \\gamma \\leq \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499041,7 +499501,9 @@ "id": "019534", "content": "如图, 已知 $ABCD$ 是矩形, $SA \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} 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] {$S$} coordinate (S);\n\\filldraw ($(S)!0.3!(C)$) circle (0.03) node [left] {$E$} coordinate (E);\n\\draw (B)--(C)--(D)--(S)--cycle(S)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(C)(A)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) $E$ 是 $SC$ 上一点, 求证: $BE$ 不可能垂直于平面 $SCD$;\\\\\n(2) 若 $SA=AB=2$, 求直线 $AC$ 与平面 $SCD$ 所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499061,7 +499523,9 @@ "id": "019535", "content": "如图, 已知 $P$ 是平行四边形 $ABCD$ 所在平面外一点, $M$、$N$ 分别是 $AB$、$PC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,1) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(M)--(N);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$ 平面 $PAD$;\\\\\n(2) 若 $MN=BC=4$, $PA=4 \\sqrt{3}$, 求异面直线 $PA$ 与 $MN$ 所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499081,7 +499545,9 @@ "id": "019536", "content": "《九章算术 -- 商功》: ``斜解立方, 得两堑堵. 斜解堑堵, 其一为阳马, 一为鳖臑. 阳马居二, 鳖臑居一, 不易之率也. 合两鳖臑三而一, 验之以基, 其形露矣.'' 刘徽注: ``此术臑者, 背节也, 或曰半阳马, 其形有似鳖肘, 故以名云. 中破阳马, 得两鳖臑, 鳖臑之起数, 数同而实据半, 故云六而一即得.'' \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--cycle(D_1)--(B)--(C_1);\n\\draw [dashed] (A)--(D)--(C)--(D_1)--(D);\n\\draw (1.5,0,0) node [below] {堑堵};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (A)--(B)--(C)--(D_1)--cycle(D_1)--(B);\n\\draw [dashed] (A)--(D)--(C)(D_1)--(D);\n\\draw (1.5,0,0) node [below] {阳马};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) coordinate (A);\n\\draw (3,0,0) coordinate (B);\n\\draw (3,0,-2) coordinate (C);\n\\draw (0,0,-2) coordinate (D);\n\\draw (C) ++ (0,2,0) coordinate (C_1);\n\\draw (D) ++ (0,2,0) coordinate (D_1);\n\\fill [gray!30] (B)--(C)--(D_1)--cycle;\n\\draw (B)--(C)--(C_1)--(D_1)--cycle(B)--(C_1);\n\\draw [dashed] (C)--(D_1);\n\\draw (1.5,0,0) node [below] {鳖臑};\n\\end{tikzpicture}\n\\end{center}\n如图, 在鳖臑 $ABCD$ 中, 侧棱 $AB \\perp$ 底面 $BCD$, 底面 $BCD$ 为直角三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\angle ADB=\\theta_1$, $\\angle BDC=\\theta_2$, $\\angle ADC=\\theta_3$, 求证: $\\cos \\theta_1 \\cdot \\cos \\theta_2=\\cos \\theta_3$;\\\\\n(2) 若 $AB=1$, $BC=2$, $CD=1$, 求异面直线 $AC$ 与 $BD$ 所成角的余弦值;\\\\\n(3) 若 $BD \\perp CD, AB=BD=CD=2$, 点 $P$ 在棱 $AC$ 上运动, 试求 $\\triangle PBD$ 面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499101,7 +499567,9 @@ "id": "019537", "content": "下列命题中: \\textcircled{1} 底面是矩形的平行六面体是长方体; \\textcircled{2} 底面是正方形的直平行六面体是正四棱柱; \\textcircled{3} 底面正方形的直四棱柱是正方体; \\textcircled{4} 所有棱长都相等的直平行六面体是正方体. 正确的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499121,7 +499589,9 @@ "id": "019538", "content": "《九章算术》中, 称底面为矩形而有一侧棱垂直于底面的四棱锥为阳马, 设 $AA_1$ 是正六棱柱的一条侧棱, 如图, 若阳马以该正六棱柱的顶点为顶点, 以 $AA_1$ 为底面矩形的一边, 则这样的阳马的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}\n\\coordinate (A) at (0,0) node [below] {$A$};\n\\path (A) --++ (45:{sqrt(3)/2}) --++ (0.5,0) coordinate (B);\n\\path (A) --++ (45:{sqrt(3)}) coordinate (C);\n\\path (C) --++ (-1,0) coordinate (D);\n\\path (B) --++ (-2,0) coordinate (E);\n\\coordinate (F) at (-1,0);\n\\draw (F) -- (A) -- (B) -- (C);\n\\draw [dashed] (C) -- (D) -- (E) -- (F);\n\\foreach \\i in {(A),(B),(C),(F)}{\\draw \\i --++ (0,2);};\n\\foreach \\i in {(D),(E)}{\\draw [dashed] \\i --++ (0,2);};\n\\path (A) --++ (0,2) coordinate (A1) node [above] {$A_1$};\n\\path (B) --++ (0,2) coordinate (B1);\n\\path (C) --++ (0,2) coordinate (C1);\n\\path (D) --++ (0,2) coordinate (D1);\n\\path (E) --++ (0,2) coordinate (E1);\n\\path (F) --++ (0,2) coordinate (F1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- (E1) -- (F1) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$8$}{$12$}{$16$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499141,7 +499611,9 @@ "id": "019539", "content": "如图, 已知正四棱柱 $ABCD-A_1B_1C_1D_1$, $AB=BC=2$, $AA_1=a$, 若在棱 $AA_1$ 上存在点 $M$ 使得 $MC_1 \\perp MB$, 则 $a$ 的取值范围是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(A)!0.65!(A_1)$) node [left] {$M$} coordinate (M);\n\\draw (B)--(M);\n\\draw [dashed] (M)--(C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499161,7 +499633,9 @@ "id": "019540", "content": "已知三棱柱 $ABC-A_1B_1C_1$ 的 $6$ 个顶点都在球 $O$ 的球面上, 若 $AB=3$, $AC=4, AB \\perp AC$, $AA_1=12$, 则球 $O$ 的半径为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\sqrt{17}}{2}$}{$2 \\sqrt{10}$}{$\\dfrac{13}{2}$}{$3 \\sqrt{10}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499181,7 +499655,9 @@ "id": "019541", "content": "已知一个直角三角形的两条直角边的长分别为 $1$ 和 $2$, 将这个三角形分别绕其两条直角边旋转得到两个圆锥, 则这两个圆锥的体积之比为\\bracket{20}.\n\\fourch{$1$}{$2$}{$4$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499201,7 +499677,9 @@ "id": "019542", "content": "正四棱锥 $P-ABCD$ 的底面边长为 $2 \\sqrt{3}$, 侧面积为 $8 \\sqrt{3}$, 则它的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499221,7 +499699,9 @@ "id": "019543", "content": "一个圆锥与一个球的体积相等且圆锥的底面半径是球半径的 $2$ 倍. 若圆锥的高为 $1$ ,则球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499241,7 +499721,9 @@ "id": "019544", "content": "已知圆柱的底面半径为 $2$ , 高为 $4$, 经过圆柱两条母线的截面与圆柱的轴之间的距离为 $\\sqrt{3}$, 则该截面的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499261,7 +499743,9 @@ "id": "019545", "content": "圆锥的侧面展开图恰好是一个半圆, 则该圆锥的母线与底面所成的角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499281,7 +499765,9 @@ "id": "019546", "content": "矩形 $ABCD$ 中, $AB=4$, $BC=3$, 沿 $AC$ 将矩形 $ABCD$ 折成一个直二面角 $B-AC-D$, 则四面体 $ABCD$ 的外接球的体积为\\bracket{20}.\n\\fourch{$\\dfrac{125}{12}\\pi$}{$\\dfrac{125}{9}\\pi$}{$\\dfrac{125}{6}\\pi$}{$\\dfrac{125}{3}\\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499301,7 +499787,9 @@ "id": "019547", "content": "如图所示, 在边长为 $4$ 的正方形纸片 $ABCD$ 中, $AC$ 与 $BD$ 相交于点 $O$, 剪去 $\\triangle AOB$, 将剩余部分沿 $OC$、$OD$ 折叠,使 $OA$、 $OB$ 重合, 则以 $A(B)$、$C$、$D$、$O$ 为顶点的四面体的体积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A) (2,0) node [below] {$B$} coordinate (B) (2,2) node [above] {$C$} coordinate (C) (0,2) node [above] {$D$} coordinate (D);\n\\draw (1,1) node [above] {$O$} coordinate (O);\n\\fill [pattern = north east lines] (A)--(B)--(O)--cycle;\n\\draw (A) rectangle (C) (A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499321,7 +499809,9 @@ "id": "019548", "content": "课本中介绍了应用祖暅原理推导棱锥体积公式的做法. 祖暅原理也可用来求旋转体的体积. 现介绍用祖暅原理求球体体积公式的做法: 可构造一个底面半径和高都与球半径相等的圆柱, 然后在圆柱内挖去一个以圆柱下底面圆心为顶点, 圆柱上底面为底面的圆锥, 用这样一个几何体与半球应用祖暅原理, 即可求得球的体积公式. 请研究和理解球的体积公式求法的基础上, 解答以下问题: 已知椭圆的标准方程 $\\dfrac{x^2}{4}+\\dfrac{y^2}{25}=1$, 将此椭圆绕 $y$ 轴旋转一周后, 得一橄榄状的几何体, 求其体积.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) arc (180:0:2) arc (0:-180:2 and 0.5);\n\\draw [dashed] (0,0) arc (180:0:2 and 0.5) -- (0,0);\n\\fill [color = gray!30] (2,1) ellipse ({sqrt(3)} and {sqrt(3)/4});\n\\draw ({2-sqrt(3)},{1}) arc (180:360:{sqrt(3)} and {sqrt(3)/4});\n\\draw [dashed] ({2-sqrt(3)},{1}) arc (180:0:{sqrt(3)} and {sqrt(3)/4});\n\\draw [dashed] (2,0) -- (2,1) (2,0.2) node [left] {$h$};\n\\draw [dashed] (2,0) -- ({2+sqrt(3)},1) (3,0) node [below] {$R$};\n\\filldraw [even odd rule, gray!30] (7,1) ellipse (2 and 0.5) (7,1) ellipse (1 and 0.25);\n\\draw (5,0) arc (180:360:2 and 0.5) (5,2) arc (180:-180:2 and 0.5) (5,0) -- (5,2) (9,0) -- (9,2);\n\\draw [dashed] (5,0) -- (9,0) (7,0) -- (7,1) (7,0) -- (5,2) (7,0) -- (9,2) (8,0) node [below] {$R$} (7,0.4) node [left] {$h$};\n\\draw (5,1) arc (180:360:2 and 0.5);\n\\draw [dashed] (5,1) arc (180:0:2 and 0.5) (6,1) arc (180:-180:1 and 0.25);\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (1 and 1.5);\n\\draw [dashed] (0,0) ellipse (0.5 and 1.5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499341,7 +499831,9 @@ "id": "019549", "content": "正三棱锥的高为$1$, 底面边长为 $2 \\sqrt{6}$, 内有一个球与它的四个面都相切, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(6)},0,0) coordinate (A);\n\\draw ({sqrt(6)},0,0) coordinate (B);\n\\draw (0,0,{-3*sqrt(2)}) coordinate (C);\n\\draw (0,1,{-sqrt(2)}) coordinate (D);\n\\draw (D)--(A)--(B)--(C)--cycle(D)--(B);\n\\draw [dashed] (A)--(C);\n\\filldraw (0,{sqrt(6)-2},{-sqrt(2)}) coordinate (O) circle (0.03);\n\\draw [dashed] (O) circle ({sqrt(6)-2});\n\\end{tikzpicture}\n\\end{center}\n(1) 正三棱锥的表面积;\\\\\n(2) 内切球的表面积与体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499361,7 +499853,9 @@ "id": "019550", "content": "若圆柱的侧面积为 $2 \\pi$, 底面积为 $\\pi$, 则该圆柱的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499381,7 +499875,9 @@ "id": "019551", "content": "已知正四棱锥 $S-ABCD$ 中, $SA=2 \\sqrt{3}$, 它的高为 $2$, 则它的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499401,7 +499897,9 @@ "id": "019552", "content": "正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长 $AB=2$, 若直线 $B_1C$ 与底面 $ABCD$ 所成的角的大小为 $\\arctan 2$, 则正四棱柱 $ABCD-A_1B_1C_1D_1$ 的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499421,7 +499919,9 @@ "id": "019553", "content": "若一个圆锥的侧面展开图是面积为 $2 \\pi$ 的半圆面, 则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499441,7 +499941,9 @@ "id": "019554", "content": "如图, 在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $P$ 是 $C_1B_1$ 的中点, 若 $E$、$F$ 都是 $AB$ 上的点, 且 $|EF|=\\dfrac{a}{2}, Q$ 是 $A_1B_1$ 上的点,则四面体 $EFPQ$ 的体积是\\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 ($(B_1)!0.5!(C_1)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(A_1)!0.6!(B_1)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\filldraw ($(A)!0.2!(B)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(A)!0.7!(B)$) node [below] {$F$} coordinate (F) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499461,7 +499963,9 @@ "id": "019555", "content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $\\sqrt{3}$, 以顶点 $A$ 为球心 $2$ 为半径的球面被正方体的表面 $ABB_1A_1$、$BCC_1B_1$ 截得的两段弧长之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499481,7 +499985,9 @@ "id": "019556", "content": "一个球与一个正三棱柱的三个侧面和两个底面都相切,已知这个球的体积是 $\\dfrac{32}{3}\\pi$, 那么该三棱柱的体积是\\bracket{20}.\n\\fourch{$48 \\sqrt{3}$}{$16 \\sqrt{3}$}{$24 \\sqrt{3}$}{$96 \\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499501,7 +500007,9 @@ "id": "019557", "content": "圆锥形容器的高为 $h$, 顶点向上放置时, 圆锥内水面的高为 $h_1$, 且 $h_1=\\dfrac{1}{3}h$, 若将圆锥倒置, 顶点向下放置, 水面高为 $h_2$, 则 $h_2$ 等于\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}h$}{$\\dfrac{19}{27}h$}{$\\dfrac{\\sqrt[3]{6}}{3}h$}{$\\dfrac{\\sqrt[3]{19}}{3}h$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499521,7 +500029,9 @@ "id": "019558", "content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$, 则下列四个命题:\\\\\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} 若点 $M$ 是平面 $A_1B_1C_1D_1$ 上到点 $D$ 和 $C_1$ 距离相等的点, 则点 $M$ 的轨迹是过点 $D_1$ 的直线.\\\\\n其中所有真命题是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (A_1) ++ (\\l,0,0) node [below right] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (\\l,0,-\\l) node [right] {$C_1$} coordinate (C_1);\n\\draw (A_1) ++ (0,0,-\\l) node [left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw [dashed] (A_1) -- (D_1) -- (C_1);\n\\draw (A_1) ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D_1) ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A_1) -- (A) (B_1) -- (B) (C_1) -- (C);\n\\draw [dashed] (D_1) -- (D);\n\\draw (B)--(C_1)(A)--(C);\n\\draw [dashed] (A)--(D_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{3}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499541,7 +500051,9 @@ "id": "019559", "content": "在如图所示的组合体中, 三棱柱 $ABC-A_1B_1C_1$ 的侧面 $ABB_1A_1$ 是圆柱的轴截面, $C$ 是圆柱底面圆周上不与 $A$、$B$ 重合的一个点.\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) coordinate (O) circle (0.03);\n\\filldraw (0,2) coordinate (O_1) circle (0.03);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (1,2) node [right] {$B_1$} coordinate (B_1);\n\\draw (O) ++ (250:1 and 0.25) node [below] {$C$} coordinate (C);\n\\draw (C) ++ (0,2) node [below right] {$C_1$} coordinate (C_1);\n\\draw (A) arc (180:360:1 and 0.25) -- (B_1) (A_1)--(A) (A_1)--(B_1) (A_1)--(C_1)--(B_1) (C)--(C_1);\n\\draw (O_1) ellipse (1 and 0.25);\n\\draw [dashed] (A) arc (180:0:1 and 0.25)(A)--(B)(A_1)--(C)(A_1)--(B)(A)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 圆柱的轴截面是正方形, 当点 $C$ 是弧 $AB$ 的中点时, 求异面直线 $A_1C$ 与 $AB_1$ 的所成角的大小;\\\\\n(2) $C$ 是弧 $AB$ 的中点时, 求四棱锥 $A_1-BCC_1B_1$ 与圆柱的体积比.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499561,7 +500073,9 @@ "id": "019560", "content": "已知圆锥的顶点为 $P$, 底面圆心为 $O$, 半径为 $2$.\n\\begin{center}\n\\begin{tikzpicture}\n\\node (0,0) [left] {$O$} coordinate (O);\n\\draw (-1.5,0) arc (180:360:1.5 and {1.5/3}) node [right] {$B$} coordinate (B);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and {1.5/3}) coordinate (C);\n\\draw (C) -- (0,3) node [above] {$P$} coordinate (P) -- (B); \n\\coordinate (A) at ({1.5*cos(250)},{0.5*sin(250)});\n\\draw [dashed] (A) node [below left] {$A$} -- (O) -- (B) -- cycle;\n\\coordinate (M) at ($(A)!0.5!(B)$);\n\\draw [dashed] (O) -- (P) -- (M) node [shift = {(-45:0.5)}] {$M$};\n\\end{tikzpicture}\n\\end{center}\n(1) 设圆锥的母线长为 $4$, 求圆锥的体积;\\\\\n(2) 设 $PO=4, OA$、$OB$ 是底面半径, 且 $\\angle AOB=90^{\\circ}, M$ 为线段 $AB$ 的中点, 求异面直线 $PM$ 与 $OB$ 所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499581,7 +500095,9 @@ "id": "019561", "content": "已知三棱锥 $P-ABC, PA \\perp$ 平面 $ABC, PA=AB=BC=2$, 直线 $PC$ 与平面 $ABC$ 所成角的大小为 $\\arctan \\dfrac{\\sqrt{2}}{2}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(P)--(C)--(B)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(A)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$ 平面 $PAB$;\\\\\n(2) 设 $E$ 为线段 $PC$ 的中点, 求异面直线 $AE$ 与 $BC$ 所成角的大小;\\\\\n(3) 设 $M$ 是三棱锥 $P-ABC$ 内 (包括边界) 的动点, 且满足 $|AM| \\leq \\sqrt{2}$, 求点 $M$ 所形成的几何体的全面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499601,7 +500117,9 @@ "id": "019562", "content": "在平行六面体 $ABCD-A' B' C' D'$ 中, 点 $M$ 在对角线 $A' B$ 上, 且 $|\\overrightarrow{A' M}|=\\dfrac{1}{2}|\\overrightarrow{MB}|$, 点 $N$ 在对角线 $A' C$ 上, 且 $|\\overrightarrow{A' N}|=\\dfrac{1}{3}|\\overrightarrow{NC}|$. 记 $\\overrightarrow{AB}=\\overrightarrow{a}$, $\\overrightarrow{AD}=\\overrightarrow{b}$, $\\overrightarrow{AA'}=\\overrightarrow{c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{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) ++ (2.3,0,-\\m) node [right] {$B$} coordinate (B);\n\\draw (D) ++ (0.3,0,-\\m) node [left] {$C$} coordinate (C);\n\\draw (D) -- (A) -- (B);\n\\draw [dashed] (D) -- (C) -- (B);\n\\draw (D) ++ (0.4,\\n,-0.2) node [left] {$D'$} coordinate (D');\n\\draw (A) ++ (0.4,\\n,-0.2) node [right] {$A'$} coordinate (A');\n\\draw (B) ++ (0.4,\\n,-0.2) node [above right] {$B'$} coordinate (B');\n\\draw (C) ++ (0.4,\\n,-0.2) node [above left] {$C'$} coordinate (C');\n\\draw (D') -- (A') -- (B') -- (C') -- cycle;\n\\draw (D) -- (D') (A) -- (A') (B) -- (B');\n\\draw [dashed] (C) -- (C');\n\\filldraw ($(A')!{1/3}!(B)$) node [right] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($(A')!{1/4}!(C)$) node [left] {$N$} coordinate (N) circle (0.03);\n\\draw (A')--(B);\n\\draw [dashed] (A')--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $M$、$N$、$D'$ 三点共线;\\\\\n(2) $\\overrightarrow{AP}=\\dfrac{1}{5}\\overrightarrow{a}+\\dfrac{2}{5}\\overrightarrow{b}+\\dfrac{2}{5}\\overrightarrow{c}$, 求证: 点 $P \\in$ 平面 $A' BD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499621,7 +500139,9 @@ "id": "019563", "content": "如图, 在三棱锥 $D-ABC$ 中, $\\angle DAC=\\angle BAC= 60^{\\circ}$, $AC=1$, $AB=2$, $AD=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1.5,0,{-1.5*sqrt(3)}) node [above] {$D$} coordinate (D);\n\\draw ({1/2},{sqrt(6)/3},{-sqrt(3)/6}) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)(A)--(C)(B)--(D);\n\\draw [dashed] (A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\overrightarrow{AC}\\cdot \\overrightarrow{BD}$, 并说明异面直线 $AC$ 与 $BD$ 所成的角 $\\theta$ 的大小在棱 $BD$ 长度增大时是怎样变化的;\\\\\n(2) 若 $\\overrightarrow{AC}\\cdot \\overrightarrow{BC}=0$, 判断点 $D$ 在平面 $ABC$ 上的射影是否可能在直线 $BC$ 上, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499641,7 +500161,9 @@ "id": "019564", "content": "如图, 在圆柱 $OO_1$ 中, 它的轴截面 $ABB_1A_1$ 是一个边长为 2 的正方形, 点 $C$ 为棱 $BB_1$ 的中点, 点 $C_1$ 为弧 $A_1B_1$ 的中点. 求.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) coordinate (O) circle (0.03) node [below] {$O$};\n\\filldraw (0,2) coordinate (O_1) circle (0.03) node [above] {$O_1$};\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,2) node [left] {$A_1$} coordinate (A_1);\n\\draw (1,2) node [right] {$B_1$} coordinate (B_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$C$} coordinate (C);\n\\draw (O_1) ++ (260:1 and 0.25) node [below left] {$C_1$} coordinate (C_1);\n\\draw (A) arc (180:360:1 and 0.25) -- (B_1) (A_1)--(A) (A_1)--(B_1)(A_1)--(C_1);\n\\draw (O_1) ellipse (1 and 0.25);\n\\draw [dashed] (A)--(B)(O)--(A_1)(O)--(C_1)(O)--(C)(C_1)--(C)(A_1)--(C);\n\\draw [dashed] (A) arc (180:0:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}\n(1) 异面直线 $OC$ 与 $A_1C_1$ 所成角的大小;\\\\\n(2) 直线 $CC_1$ 与圆柱 $OO_1$ 底面所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499661,7 +500183,9 @@ "id": "019565", "content": "四棱锥 $P-ABCD$ 的底面 $ABCD$ 是平行四边形, $PF \\perp$ 平面 $ABCD$, 垂足 $F$ 在 $AD$ 上, 且 $AF=\\dfrac{1}{3}FD, FB \\perp FC$, $FB=FC=2, E$ 是 $BC$ 的中点, 四面体 $P-BCF$ 的体积为 $\\dfrac{8}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [above right] {$F$} coordinate (F);\n\\draw (F) ++ (0,4,0) node [above] {$P$} coordinate (P);\n\\draw (F) ++ ({-sqrt(2)/2},0) node [left] {$A$} coordinate (A);\n\\draw (F) ++ ({3*sqrt(2)/2},0) node [right] {$D$} coordinate (D);\n\\draw (F) ++ (0,0,{sqrt(2)}) node [below] {$E$} coordinate (E);\n\\draw (E) ++ ({-sqrt(2)},0) node [left] {$B$} coordinate (B);\n\\draw (E) ++ ({sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\draw (B)--(C)--(D)--(P)--cycle (P)--(C);\n\\draw [dashed] (B)--(A)--(D)(B)--(F)--(C)(E)--(F)--(P)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角 $B-PC-F$ 的大小;\\\\\n(2) 求点 $D$ 到平面 $PBF$ 的距离;\\\\\n(3) 设三棱锥 $F-PCD$ 的体积为 $V$, 是否存在体积为 $n V$ ($n$ 为正整数), 且各棱长均相等的直平行六面体, 使得它的所有棱长和为$24$, 若存在, 请构造出这样的一个直平行六面体; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499681,7 +500205,9 @@ "id": "019566", "content": "在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$、$F$ 分别是棱 $BC$、$CD$ 上的点, 且 $BE=CF$.\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] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.3!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.3!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (B_1)--(F)(D_1)--(E); \n\\end{tikzpicture}\n\\end{center}\n(1) 若 $BE=CF=\\dfrac{1}{2}a$, 求证: $EF\\parallel $ 平面 $AB_1D_1$;\\\\\n(2) 求证: $B_1F \\perp D_1E$;\\\\\n(3) 是否存在点 $E$、$F$, 使 $A_1C \\perp$ 面 $C_1EF$ ?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499701,7 +500227,9 @@ "id": "019567", "content": "已知向量 $\\overrightarrow{a}=(1,0,2)$, $\\overrightarrow{b}=(2,1,0)$, 则向量 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499721,7 +500249,9 @@ "id": "019568", "content": "若 $(0,-2,4)$ 和 $(-\\sqrt{15}, 1,2)$ 分别是平面 $\\alpha, \\beta$ 的一个法向量, 则 $\\alpha$ 与 $\\beta$ 构成的二面角大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499741,7 +500271,9 @@ "id": "019569", "content": "在四棱锥 $P-ABCD$ 中, 设向量 $\\overrightarrow{AB}=(4,-2,3)$, $\\overrightarrow{AD}=(-4, 1,0)$, $\\overrightarrow{AP}=(-6,2,-8)$, 则顶点 $P$ 到底面 $ABCD$ 的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499761,7 +500293,9 @@ "id": "019570", "content": "如图, 长方体 $EPFC-ADBO$ 中, $OA=AD=1$, $AE=2$, 则\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [above left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$D$} coordinate (D);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (0,0,-\\m) node [left] {$O$} coordinate (O);\n\\draw (A) -- (D) -- (B);\n\\draw [dashed] (A) -- (O) -- (B);\n\\draw (A) ++ (0,\\n,0) node [left] {$E$} coordinate (E);\n\\draw (D) ++ (0,\\n,0) node [right] {$P$} coordinate (P);\n\\draw (B) ++ (0,\\n,0) node [above right] {$F$} coordinate (F);\n\\draw (O) ++ (0,\\n,0) node [above left] {$C$} coordinate (C);\n\\draw (E) -- (P) -- (F) -- (C) -- cycle;\n\\draw (A) -- (E) (D) -- (P) (B) -- (F);\n\\draw [dashed] (O) -- (C);\n\\draw [->] (A) -- ($(O)!1.5!(A)$) node [left] {$x$};\n\\draw [->] (B) -- ($(O)!1.5!(B)$) node [right] {$y$};\n\\draw [->] (C) -- ($(O)!1.2!(C)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}\n(1) 异面直线 $OP$ 与 $AC$ 所成的角为\\blank{50}.\\\\\n(2) 直线 $DF$ 与平面 $ODPC$ 所成的角为\\blank{50}.\\\\\n(3) 写出平面 $ABC$ 的一个法向量 $\\overrightarrow{n}=$\\blank{50}.\\\\\n(4) 二面角 $O-AB-C$ 的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499781,7 +500315,9 @@ "id": "019571", "content": "圆柱底面半径为$1$, 高为 $2$, $AB$ 为上底底面的直径, 点 $C$ 是下底底面圆弧上的一个动点, 点 $C$ 绕着下底底面旋转一周, 则 $\\triangle ABC$ 面积的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499801,7 +500337,9 @@ "id": "019572", "content": "已知边长为 $4 \\sqrt{2}$ 的正三角形 $ABC$ 中, $E, F$ 分别为 $BC$ 和 $AC$ 的中点, $PA \\perp$ 面 $ABC$, 且 $PA=2$, 设平面 $\\alpha$ 过 $PF$ 且与 $AE$ 平行, 则 $AE$ 与平面 $\\alpha$ 间的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -499821,7 +500359,9 @@ "id": "019573", "content": "正方体 $ABCD-A_1B_1C_1D_1$ 中, $E, F$ 分别是 $AA_1$ 与 $CC_1$ 的中点, 则直线 $ED$ 与 $D_1F$ 所成角的余弦值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499841,7 +500381,9 @@ "id": "019574", "content": "棱长都为 $2$ 的直平行六面体 $ABCD-A_1B_1C_1D_1$ 中, $\\angle BAD=60^{\\circ}$, 则对角线 $A_1C$ 与侧面 $DCC_1D_1$ 所成角的正弦值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{2}}{2}$}{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{3}}{4}$}{$\\dfrac{\\sqrt{3}}{8}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499861,7 +500403,9 @@ "id": "019575", "content": "设 $A, B, C, D$ 是半径为 $1$ 的球面上的四个不同点, 且满足 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}=0$, $\\overrightarrow{AC}\\cdot \\overrightarrow{AD}=0$, $\\overrightarrow{AD}\\cdot \\overrightarrow{AB}=0$, 用 $S_1, S_2, S_3$ 分别表示 $\\triangle ABC$、$\\triangle ACD$、$\\triangle ABD$ 的面积, 则 $S_1+S_2+S_3$ 的最大值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$2$}{$4$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -499881,7 +500425,9 @@ "id": "019576", "content": "四棱锥 $P-ABCD$, 底面为正方形 $ABCD$, 边长为 $4$, $E$ 为 $AB$ 中点, $PE \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,0,2) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (0,0,1) node [below] {$E$} coordinate (E);\n\\draw (2,0,1) node [right] {$F$} coordinate (F);\n\\draw (E) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C)(P)--(F);\n\\draw [dashed] (P)--(E)--(F)(P)--(A)--(C)(B)--(A)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\triangle PAB$ 为等边三角形, 求四棱锥 $P-ABCD$ 的体积;\\\\\n(2) 若 $CD$ 的中点为 $F$, $PF$ 与平面 $ABCD$ 所成角为 $45^{\\circ}$, 求 $PD$ 与 $AC$ 所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499901,7 +500447,9 @@ "id": "019577", "content": "如图, 直三棱柱 $ABC-A_1B_1C_1$ 中, $\\angle BAC=90^{\\circ}$, $AA_1=AB=AC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$M$} coordinate (M);\n\\draw (C)--(B)--(B_1)--(A_1)--(C_1)--cycle(B_1)--(C_1)--(M);\n\\draw [dashed] (C_1)--(A)--(B)(C)--(A)--(M)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 设 $M$ 是棱 $BB_1$ 的中点, 求 $C$ 到平面 $MAC_1$ 的距离;\\\\\n(2) 若 $M$ 是棱 $BB_1$ 上的任意一点 (包括端点), 求二面角 $M-AC_1-A_1$ 的大小的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499921,7 +500469,9 @@ "id": "019578", "content": "在三棱锥 $P-ABC$ 中, $AB \\perp BC, AB=BC=k PA$, 点 $O, D$ 分别是 $AC, PC$ 的中点, $OP \\perp$ 底面 $ABC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [right] {$D$} coordinate (D);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(C)(P)--(O)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $OD\\parallel $ 平面 $PAB$;\\\\\n(2) 当 $k=\\dfrac{1}{2}$ 时, 求直线 $PA$ 与平面 $PBC$ 所成角的大小;\\\\\n(3) 当 $k$ 为何值时, $O$ 在平面 $PBC$ 内的射影恰好为 $\\triangle PBC$ 的重心?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499941,7 +500491,9 @@ "id": "019579", "content": "从 $5$ 个男生和 $4$ 个女生中选出 $4$ 名学生参加一次会议, 共有多少种不同的选法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499961,7 +500513,9 @@ "id": "019580", "content": "用五种不同颜色给图中四个区域涂色, 如果每个区域涂一种颜色, 相邻区域不同色, 共有多少种不同的涂色方法?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (2,1.5);\n\\draw (0,0.75) -- (1,1.5);\n\\draw (0.3,1.25) node {$1$};\n\\draw (0.5,1.125) -- (2,1.125) (1,0) -- (1,1.125);\n\\draw (1.3,1.3125) node {$2$};\n\\draw (0.5,0.625) node {$3$} (1.5,0.625) node {$4$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -499981,7 +500535,9 @@ "id": "019581", "content": "$6$ 位选手依次演讲, 其中选手甲不在第一个也不在最后一个演讲, 则不同的演讲次序共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500001,7 +500557,9 @@ "id": "019582", "content": "某食堂规定, 每份午餐可以在四种水果中任选两种, 则甲、乙两同学各自所选的两种水果相同的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500021,7 +500579,9 @@ "id": "019583", "content": "在报名的 $3$ 名男教师和 $6$ 名女教师中, 选取 $5$ 人参加义务献血, 要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500041,7 +500601,9 @@ "id": "019584", "content": "$6$ 个人排成一行, 其中甲、乙两人不相邻的不同排法共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500061,7 +500623,9 @@ "id": "019585", "content": "从 $3$ 名骨科、$4$ 名脑外科和 $5$ 名内科医生中选派 $5$ 人组成一个抗震救灾医疗小组, 则骨科、脑外科和内科医生都至少有 $1$ 人的选派方法种数是\\blank{50}(用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500081,7 +500645,9 @@ "id": "019586", "content": "将序号分别为 $1,2,3,4,5$ 的 $5$ 张参观券全部分给 $4$ 人, 每人至少 $1$ 张, 如果分给同一人的 $2$ 张参观券连号, 那么不同的分法种数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500101,7 +500667,9 @@ "id": "019587", "content": "某同学有同样的画册 $2$ 本, 同样的集邮册 $3$ 本, 从中取出 $4$ 本赠送给 $4$ 位朋友每位朋友 $1$ 本, 则不同的赠送方法共有\\bracket{20}.\n\\fourch{$4$ 种}{$10$ 种}{$18$ 种}{$20$ 种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -500121,7 +500689,9 @@ "id": "019588", "content": "$4$ 位同学每人从甲、乙、丙 $3$ 门课程中选修 $1$ 门, 则恰有 $2$ 人选修课程甲的不同选法共有\\bracket{20}.\n\\fourch{$12$ 种}{$24$ 种}{$30$ 种}{$36$ 种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -500141,7 +500711,9 @@ "id": "019589", "content": "求 $(1+2 x)^7$ 的展开式的第四项的系数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -500161,7 +500733,9 @@ "id": "019590", "content": "设常数 $a \\in \\mathbf{R}$. 若 $(x^2+\\dfrac{a}{x})^5$ 的二项展开式中 $x^7$ 项的系数为 $-10$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500181,7 +500755,9 @@ "id": "019591", "content": "在 $(1+x+\\dfrac{1}{x^{2015}})^{10}$ 的展开式中, $x^2$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500201,7 +500777,9 @@ "id": "019592", "content": "已知 $(1-2 x)^7=a_0+a_1 x+a_2 x^2+\\cdots+a_7 x^7$, 则 $a_1+a_2+\\cdots+a_7= a_1+a_3+a_5+a_7= ;|a_0|+|a_1|+\\cdots+|a_7|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500221,7 +500799,9 @@ "id": "019593", "content": "$(x+\\dfrac{a}{x})(2 x-\\dfrac{1}{x})^5$ 的展开式中各项系数的和为 $2$, 则该展开式中常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500241,7 +500821,9 @@ "id": "019594", "content": "若多项式 $x^2+x^{11}=a_0+a_1(x+1)+n(x+1)^2+\\cdots+a_{11}(x+1)^{11}$, 则 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500261,7 +500843,9 @@ "id": "019595", "content": "随机抽取 $9$ 个同学中, 至少有 $2$ 个同学在同一月出生的概率是\\blank{50}(默认每月天数相同, 结果精确到$0.001$).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500281,7 +500865,9 @@ "id": "019596", "content": "为强化安全意识, 某商场拟在未来连续 $10$ 天中随机选择 $3$ 天进行紧急疏散演练, 则选择 $3$ 天恰好为连续 $3$ 天的概率是\\blank{50}(结果用最简分数表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500301,7 +500887,9 @@ "id": "019597", "content": "在 $30$ 瓶饮料中, 有 $3$ 瓶已过了保质期, 从这 $30$ 瓶饮料中任取 $2$ 瓶,则至少取到 $1$ 瓶已过保质期饮料的概率为\\blank{50}. (结果用最简分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500321,7 +500909,9 @@ "id": "019598", "content": "有 $3$ 个兴趣小组, 甲、乙两位同学各自参加其中一个小组, 每位同学参加各个小组的可能性相同,则这两位同学参加同一个兴趣小组的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500341,7 +500931,9 @@ "id": "019599", "content": "现有 $10$ 个数, 它们能构成一个以 $1$ 为首项, $-3$ 为公比的等比数列, 若从这 $10$ 个数中随机抽取一个数, 则它小于 $8$ 的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500361,7 +500953,9 @@ "id": "019600", "content": "在二项式 $(x+1)^9$ 的展开式中任取 $2$ 项, 则取出的 $2$ 项中系数均为奇数的概率为\\blank{50}. (用分数表示结果)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500381,7 +500975,9 @@ "id": "019601", "content": "有 $8$ 本互不相同的书, 其中数学书 $3$ 本、外文书 $2$ 本、其他书 $3$ 本, 若将这些书排成一排放在书架上, 则数学书排在一起, 外文书也排在一起的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500401,7 +500997,9 @@ "id": "019602", "content": "为了检测学生的自体素质指标, 从游泳类 $1$ 项,球类 $3$ 项, 田径类 $4$ 项共 $8$ 项项目中随机抽取 $4$ 项进行检测, 则每一类都初抽到的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500421,7 +501019,9 @@ "id": "019603", "content": "电视台连续播放 $6$ 个广告, 其中含 $4$ 个不同的商业广告和 $2$ 个不同的公益广告, 要求首尾必须播放公益广告, 则共有种\\blank{50}不同的播放方式. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500441,7 +501041,9 @@ "id": "019604", "content": "甲、乙两个袋中均有红、白两种颜色的小球, 这些小球除颜色外完全相同, 其中甲袋装有 $4$ 个红球、 $2$ 个白球, 乙袋装有 $1$ 个红球、$5$ 个白球. 现分别从甲、乙两袋中各随机取出一个球,则取出的两球都是红球的概率为\\blank{50}. (答案用分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500461,7 +501063,9 @@ "id": "019605", "content": "从 $20$ 名男同学, $10$ 名女同学中任选 $3$ 名参加体能测试, 则选到的 $3$ 名同学中既有男同学又有女同学的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500481,7 +501085,9 @@ "id": "019606", "content": "从甲、乙等 $10$ 位同学中任选 $3$ 位去参加某项活动, 则所选 $3$ 位中有甲但没有乙的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500501,7 +501107,9 @@ "id": "019607", "content": "$(1+3 x)^n$($n \\in \\mathbf{N}$, $n \\geq 6$) 的展开式中 $x^5$ 与 $x^6$ 的系数相等, 则 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500521,7 +501129,9 @@ "id": "019608", "content": "设二项式 $(x-\\dfrac{a}{\\sqrt{x}})^6$($a>0$) 的展开式中 $x^3$ 的系数为 $A$, 常数项为 $B$, 若 $B=4A$, 则 $a$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -500541,7 +501151,9 @@ "id": "019609", "content": "如果把两条异面直线看作``一对'', 则在五棱锥的棱所在的直线中, 异面直线有\\bracket{20}.\n\\fourch{$15$ 对}{$25$ 对}{$30$ 对}{$20$ 对}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -500561,7 +501173,9 @@ "id": "019610", "content": "从 $0$ 到 $9$ 这 $10$ 个数字中任取 $3$ 个数字组成一个没有重复数字的三位数, 这个数不能被 $3$ 整除的概率为\\bracket{20}.\n\\fourch{$\\dfrac{41}{60}$}{$\\dfrac{38}{54}$}{$\\dfrac{35}{54}$}{$\\dfrac{19}{54}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -500581,7 +501195,9 @@ "id": "019611", "content": "在 $(\\sqrt{x}+\\dfrac{1}{\\sqrt[3]{x}})^{24}$ 的展开式中, $x$ 的幂的指数是整数的项共有\\bracket{20}.\n\\fourch{$3$ 项}{$4$ 项}{$5$ 项}{$6$ 项}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -500601,7 +501217,9 @@ "id": "019612", "content": "要安排甲、乙、丙、丁、戌 $5$ 名同学参加上海世博会志愿者服务活动, 每人从事翻译、导游、礼仪、司机四项工作之一, 每项工作至少有一人参加. 甲、乙不会开车但能从事其他三项工作, 丙丁戌都能胜任四项工作, 求不同安排方案的种数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -500621,7 +501239,9 @@ "id": "019613", "content": "一个口袋内装有大小相等编号为 $a_1, a_2, a_3$ 的 $3$ 个白球和 $1$ 个黑球 $b$.\\\\\n(1) 从中摸出 $2$ 个球, 求摸出 $2$ 个白球的概率;\\\\\n(2) 从中连续取两次, 每次取一球后放回, 求取出的两球恰好有 $1$ 个黑球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -500641,7 +501261,9 @@ "id": "019614", "content": "已知 $(a^2+1)^n$ 展开式中的各项系数的和等于 $(\\dfrac{16}{5}x^2+\\dfrac{1}{\\sqrt{x}})^5$ 的展开式的常数项, 而 $(a^2+1)^n$ 展开式的系数最大的项等于$54$, 求 $a$ 的值 ($a \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501001,7 +501623,9 @@ "id": "019632", "content": "某校从学生文艺部 $6$ 名成员 ($4$ 男 $2$ 女) 中, 挑选 $2$ 人参加学校举办的文艺汇演活动.\\\\\n(1) 求男生甲被选中的概率;\\\\\n(2) 在已知男生甲被选中的条件下,女生乙被选中的概率;\\\\\n(3) 在要求被选中的两人中必须一男一女的条件下,求女生乙被选中的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501021,7 +501645,9 @@ "id": "019633", "content": "设某工厂有两个车间生产同型号家用电器, 第一车间的合格率为 $0.85$, 第二车间的合格率为 $0.88$, 两个车间的成品都混合堆放在一个仓库, 假设第一, 二车间生产的成品比例为 $2: 3$, 今有一客户从成品仓库中随机提一台产品.\\\\\n(1) 求这台产品是合格品的概率;\\\\\n(2) 已知取到的是合格品, 求它来自第一车间生产的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501041,7 +501667,9 @@ "id": "019634", "content": "现有 $7$ 张卡片, 分别写上数字 $1,2,2,3,4,5,6$. 从这 $7$ 张卡片中随机抽取 $3$ 张, 记所抽取卡片上数字的最小值为 $X$, 则 $P(X=2)=$\\blank{50}, $E[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501061,7 +501689,9 @@ "id": "019635", "content": "甲、乙两名篮球运动员, 甲投篮一次命中的概率为 $\\dfrac{3}{5}$, 乙投篮一次命中的概率为 $\\dfrac{1}{2}$, 其中甲、乙两人投篮是否命中相互没有影响.\\\\\n(1) 求甲投篮三次恰好命中两次的概率;\\\\\n(2) 若甲、乙各投篮三次, 且甲、乙第一次投篮都命中, 求甲获胜 (甲投篮命中数比乙多) 的概率;\\\\\n(3) 若甲、乙各投篮两次, 设 $X$ 为甲、乙投篮命中的次数的差的绝对值, 求 $X$ 的数学期望.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501081,7 +501711,9 @@ "id": "019636", "content": "某学校的两个班共有 $100$ 名学生, 一次考试后数学成绩 $X$($X \\in \\mathbf{N}$) 服从正态分布 $N(100,10^2)$, 已知 $P(90 \\leq X \\leq 100)=0.4$, 估计该班学生数学成绩在 $110$ 分以上的人数为\\bracket{20}.\n\\fourch{$20$}{$10$}{$7$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501101,7 +501733,9 @@ "id": "019637", "content": "已知 $P(B | A)=\\dfrac{1}{2}$, $P(AB)=\\dfrac{3}{10}$, 则 $P(A)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501121,7 +501755,9 @@ "id": "019638", "content": "已知随机变量 $X$ 的分布为 $\\begin{pmatrix}-1 & 0 & 1 \\\\ a & b & c\\end{pmatrix}$, 其中 $a, b, c$ 成等差数列, 则 $P(|X|=1)=$\\blank{50}, 公差 $d$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501141,7 +501777,9 @@ "id": "019639", "content": "一批产品的二等品率为 $0.02$, 从这批产品中每次随机取一件, 有放回地抽取 $100$ 次, $X$ 表示抽到的二等品件数, 则 $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501161,7 +501799,9 @@ "id": "019640", "content": "有 $N$ 件产品, 其中有 $M$ 件次品, 从中不放回地抽 $n$ 件产品, 抽到的次品数的数学期望值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501181,7 +501821,9 @@ "id": "019641", "content": "已知随机变量 $X$ 服从正态分布 $N(2, \\sigma^2)$, 且 $P(22.5)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501201,7 +501843,9 @@ "id": "019642", "content": "甲乙二人争夺一场围棋比赛的冠军, 若比赛为``三局两胜''制 (无平局), 甲在每局比赛中获胜的概率均为 $\\dfrac{2}{3}$, 且各局比赛结果相互独立, 则在甲获得冠军的条件下, 比赛进行了三局的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501221,7 +501865,9 @@ "id": "019643", "content": "设随机变量 $Y$ 满足 $Y \\sim B(4, \\dfrac{1}{2})$, 则函数 $f(x)=x^2-4 x+4Y$ 无零点的概率是\\bracket{20}.\n\\fourch{$\\dfrac{11}{16}$}{$\\dfrac{5}{16}$}{$\\dfrac{31}{32}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501241,7 +501887,9 @@ "id": "019644", "content": "设 $0=latex, xscale = 0.06, yscale = 180]\n\\draw [->] (0,0) -- (105,0) node [below] {年龄/岁};\n\\draw [->] (0,0) -- (0,0.03) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {0/0.001,10/0.002,20/0.012,30/0.017,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {0/0.001,20/0.012,40/0.023,50/0.020,60/0.017,70/0.006,80/0.002}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (90,0) node [below] {$90$};\n\\end{tikzpicture}\n\\end{center}\n(1) 估计该地区这种疾病患者的平均年龄 (同一组中的数据用该组区间的中点值作代表);\\\\\n(2) 估计该地区一人患这种疾病年龄在区间 $[20,70)$ 的概率;\\\\\n(3) 已知该地区这种疾病的患病率为 $0.1 \\%$, 该地区年龄位于区间 $[40,50)$ 的人口占该地区总人口的 $16 \\%$, 从该地区任选一人,若此人年龄位于区间 $[40,50)$, 求此人患该种疾病的概率. (样本数据中的患者年龄位于各区间的频率作为患者年龄位于该区间的概率, 精确到 $0.0001$)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501341,7 +501997,9 @@ "id": "019649", "content": "某超市统计了最近 $5$ 年的商品销售额与利润率数据, 经计算相关系数 $r= 0.862$, 则下列判断正确的是\\bracket{20}.\n\\onech{商品销售额与利润率正相关, 且具有较弱的相关关系}{商品销售额与利润率正相关, 且具有较强的相关关系}{商品销售额与利润率负相关, 且具有较弱的相关关系}{商品销售额与利润率负相关, 且具有较强的相关关系}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501361,7 +502019,9 @@ "id": "019650", "content": "某地经过多年的环境治理, 已将荒山改造成了绿水青山. 为估计一林区某种树木的总材积量, 随机选取了 $10$ 棵这种树木, 测量每棵树的根部横截面积 (单位: $\\mathrm{m}^2$ ) 和材积量 (单位: $\\mathrm{m}^3$ ), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline 样本号 $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 总和 \\\\\n\\hline 根部横截面积 $x_i$ & 0.04 & 0.06 & 0.04 & 0.08 & 0.08 & 0.05 & 0.05 & 0.07 & 0.07 & 0.06 & 0.6 \\\\\n\\hline 材积量 $y_i$ & 0.25 & 0.40 & 0.22 & 0.54 & 0.51 & 0.34 & 0.36 & 0.46 & 0.42 & 0.40 & 3.9 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n并计算得 $\\displaystyle\\sum_{i=1}^{10}x_i^2=0.038$, $\\displaystyle\\sum_{i=1}^{10}y_i^2=1.6158$, $\\displaystyle\\sum_{i=1}^{10}x_i y_i=0.2474$.\\\\\n(1) 估计该林区这种树木平均一棵的根部横截面积与平均一棵的材积量;\\\\\n(2) 求该林区这种树木的根部横截面积与材积量的样本相关系数 (精确到 $0.01$);\\\\\n(3) 现测量了该林区所有这种树木的根部横截面积,并得到所有这种树木的根部横截面积总和为 $186 \\mathrm{m}^2$. 已知树木的材积量与其根部横截面积近似成正比. 利用以上数据给出该林区这种树木的总材积量的估计值.\n附 : 相关系数 $r=\\dfrac{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})(y_i-\\overline{y})}{\\sqrt{\\displaystyle\\sum_{i=1}^n(x_i-\\overline{x})^2 \\displaystyle\\sum_{i=1}^n(y_i-\\overline{y})^2}}, \\sqrt{1.896}\\approx 1.377$.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501381,7 +502041,9 @@ "id": "019651", "content": "变量 $x, y$ 之间的一组相关数据如表所示: 若 $x, y$ 之间的线性回归方程为 $y= \\hat{a}x+12.28$, 则 $\\hat{a}$ 的值为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\\hline$x$ & 4 & 5 & 6 & 7 \\\\\n\\hline$y$ & 8.2 & 7.8 & 6.6 & 5.4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$-0.92$}{$-0.94$}{$-0.96$}{$-0.98$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501401,7 +502063,9 @@ "id": "019652", "content": "某新兴科技公司为了确定新研发的产品下一季度的营销计划, 需了解月宣传费 $x$ (单位: 万元) 对月销售量 $y$ (单位: 千件)的影响, 收集了 2020 年 3 月至 2020 年 8 月共 6 个月的月宣传费 $x$ 和月销售量 $y$ 的数据如表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline 月份 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\n\\hline 宣传费 $x$ & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 月销售量 $y$ & 0.4 & 3.5 & 5.2 & 7.0 & 8.6 & 10.7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现分别用模型 \\textcircled{1} $y=\\hat{a}x+\\hat{b}$ 和模型\\textcircled{2} $y=\\mathrm{e}^{\\hat{m}x+\\hat{n}}$ 对以上数据进行拟合, 得到回归模型, 并计算出模型的离差如表: (模型\\textcircled{1}和模型\\textcircled{2}的离差分别为 $\\hat{e}_1$ 和 $\\hat{e}_2$, 离差$=$实际值$-$预报值)\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline$x$ & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline$y$ & 0.4 & 3.5 & 5.3 & 7.0 & 8.6 & 10.7 \\\\\n\\hline$\\hat{e}_1$ & -0.6 & 0.54 & 0.28 & 0.12 & -0.24 & -0.1 \\\\\n\\hline$\\hat{e}_2$ & -0.63 & 1.71 & 2.10 & 1.63 & -0.7 & -5.42 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表的离差数据, 应选择哪个模型来拟合月宣传费 $x$ 与月销售量 $y$ 的关系较为合适,简要说明理由;\\\\\n(2) 为了优化模型, 将 (1) 中选择的模型离差绝对值最大所对应的一组数据 $(x, y)$ 剔除, 根据剩余的 $5$ 组数据,求该模型的回归方程, 并预测月宣传费为 $12$ 万元时, 该公司的月销售量.\\\\\n(剔除数据前的参考数据: $\\overline{x}=7.5$, $\\overline{y}=5.9$, $\\displaystyle\\sum_{i=1}^6 x_i y_i=299.8$, $\\displaystyle\\sum_{i=1}^6 x_i^2=355$, $z=\\ln y$. $\\overline{z}\\approx-1.41$, $\\displaystyle\\sum_{i=1}^6 x_i y_i=-73.10$, $\\ln 10.7 \\approx 2.37$, $\\mathrm{e}^{4.034}\\approx 56.49$)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501421,7 +502085,9 @@ "id": "019653", "content": "甲、乙两城之间的长途客车均由 $A$ 和 $B$ 两家公司运营, 为了解这两家公司长途客车的运行情况, 随机调查了甲、乙两城之间的 500 个班次, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 准点班次数 & 末准点班次数 \\\\\n\\hline$A$ & 240 & 20 \\\\\n\\hline$B$ & 210 & 30 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据上表, 分别估计这两家公司甲、乙两城之间的长途客车准点的概率;\\\\\n(2) 能否有 $90 \\%$ 的把握认为甲、乙两城之间的长途客车是否准点与客车所属公司有关?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$,\n\\begin{tabular}{|c|c|c|c|}\n\\hline $P(\\chi^2 \\geq k$) & 0.100 & 0.050 & 0.010 \\\\\n\\hline $k$ & 2.706 & 3.841 & 6.635 \\\\\n\\hline\n\\end{tabular}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501441,7 +502107,9 @@ "id": "019654", "content": "某商家今年上半年各月的人均销售额 (单位: 千元) 与利润率统计表如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 月份 & 1 & 2 & 3 & 4 & 5 & 6 \\\\\n\\hline 人均销售额 & 6 & 5 & 8 & 3 & 4 & 7 \\\\\n\\hline 利润率(\\%) & 12.6 & 10.4 & 18.5 & 3.0 & 8.1 & 16.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\textcircled{1} 利润率与人均销售额成正相关关系 ; \\textcircled{2} 利润率与人均销售额成负相关关系; \\textcircled{3} 利润率与人均销售额成正比例函数关系; \\textcircled{4} 利润率与人均销售额成反比例函数关系根据表中数据, 上述说法正确的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501461,7 +502129,9 @@ "id": "019655", "content": "已知变量 $x$ 和变量 $y$ 的 3 对随机观测数据 $(2,2),(3,-1),(5,-7)$, 则成对样本数据的样本相关系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501481,7 +502151,9 @@ "id": "019656", "content": "已知 $x$ 与 $y$ 之间的一组数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$x$ & 0 & 1 & 2 & 3 \\\\\n\\hline$y$ & $m$ & 3 & 5.5 & 7 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n已求得关于 $y$ 与 $x$ 的线性回归方程 $y=2.3 x+0.85$, 则 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501501,7 +502173,9 @@ "id": "019657", "content": "研究变量 $x, y$ 得到一组样本数据, 进行回归分析, 有以下结论\\\\\n\\textcircled{1} 离差平方和越小的模型, 拟合的效果越好;\\\\\n\\textcircled{2} 用相关指数 $\\chi^2$ 来刻画回归效果, $\\chi^2$ 越小说明拟合效果越好;\\\\\n\\textcircled{3} 在回归直线方程 $y=-0.2 x+0.8$ 中, 当解释变量 $x$ 每增加 $1$ 个单位时, 预报变量 $y$ 平均减少 $0.2$ 个单位;\\\\\n\\textcircled{4} 若变量 $y$ 和 $x$ 之间的相关系数为 $r=-0.9462$, 则变量 $y$ 和 $x$ 之间的负相关很强.\\\\\n以上正确说法的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501521,7 +502195,9 @@ "id": "019658", "content": "某市通过随机询问 $100$ 名性别不同的居民是否能做到``光盘''行动, 得到如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 做不到``光盘''& 能做到``光盘''\\\\\n\\hline 男 & 45 & 10 \\\\\n\\hline 女 & 30 & 15 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n根据上表, \\blank{50}(填``可以''或``不可以'')确定``该市居民能否做到``光盘''与性别有关''.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501541,7 +502217,9 @@ "id": "019659", "content": "下列关于相关系数 $r$ 的说法不正确的是\\bracket{20}.\n\\onech{相关系数 $r$ 越大两个变量间相关性越强}{相关系数 $r$ 的取值范围为 $[-1,1]$}{相关系数 $r>0$ 时两个变量正相关, $r<0$ 时两个变量负相关}{相关系数 $r=1$ 时, 样本点在同一直线上}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501561,7 +502239,9 @@ "id": "019660", "content": "为了研究某班学生的脚长 $x$ (单位: 厘米) 和身高 $y$ (单位: 厘米)的关系, 从该班随机抽取 $10$ 名学生, 根据测量数据的散点图可以看出 $y$ 与 $x$ 之间有线性相关关系, 设其回归直线方程为 $y=\\hat{a}x+\\hat{b}$. 已知 $\\displaystyle\\sum_{i=1}^{10}x_i=225$, $\\displaystyle\\sum_{i=1}^{10}y_i=1600$, $\\hat{a}=4$. 该班某学生的脚长为 $24$, 据此估计其身高为\\bracket{20}.\n\\fourch{$160$}{$163$}{$166$}{$170$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501581,7 +502261,9 @@ "id": "019661", "content": "``独立性检验''中, 在犯错误的概率不超过 $0.05$ 的前提下认为事件 $A$ 和 $B$ 有关, 则算出的数据满足\\bracket{20}.\n\\fourch{$\\chi^2<6.63$}{$\\chi^2<3.84$}{$\\chi^2>3.84$}{$\\chi^2>6.63$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501601,7 +502283,9 @@ "id": "019662", "content": "下图是某地区 2000 年至 2016 年环境基础设施投资额 $y$ (单位: 亿元) 的折线图. 为了预测该地区 2018 年的环境基础设施投资额, 建立了 $y$ 与时间变量 $t$ 的两个线性回归模型. 根据 2000 年至 2016 年的数据 (时间变量 $t$ 的值依次为 $1,2, \\cdots, 17$ ) 建立模型\\textcircled{1}: $y=-30.4+13.5 t$; 根据 2010 年至 2016 年的数据 (时间变量 $t$ 的值依次为 $1,2, \\cdots, 7$ ) 建立模型\\textcircled{2}: $y=99+17.5 t$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.8]\n\\draw [->] (0,0) -- (18,0) node [below] {年份};\n\\draw [->] (0,0) -- (0,5.5) node [left] {投资额};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {2000,2001,...,2016}\n{\\draw ({\\i-1999},0.05) --++ (0,-0.05) node [below] {$\\i$};};\n\\foreach \\i in {20,40,...,240}\n{\\draw (0.05,{\\i/50}) --++ (-0.05,0) node [left] {$\\i$};};\n\\foreach \\i/\\j in {1/11,2/19,3/25,4/35,5/37,6/42,7/42,8/47,9/53,10/56,11/122,12/129,13/148,14/171,15/184,16/209,17/220}\n{\\filldraw (\\i,{\\j/50}) circle (0.03) node [above] {$\\j$};};\n\\draw (1,0.22)--(2,0.38)--(3,0.5)--(4,0.7)--(5,0.74)--(6,0.84)--(7,0.84)--(8,0.94)--(9,1.06)--(10,1.12)--(11,2.44)--(12,2.58)--(13,2.96)--(14,3.42)--(15,3.68)--(16,4.18)--(17,4.4);\n\\end{tikzpicture}\n\\end{center}\n(1) 分别利用这两个模型, 求该地区 2018 年的环境基础设施投资额的预测值;\\\\\n(2) 你认为用哪个模型得到的预测值更可靠? 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501621,7 +502305,9 @@ "id": "019663", "content": "为保护农民种粮收益, 促进粮食生产, 确保国家粮食安全, 调动广大农民生产粮食的积极性, 国家实施了对种粮农民直接补贴的政策. 通过对 2017-2021 年的数据进行调查, 发现某地区发放粮食补贴额 $x$ (单位: 亿元) 与该地区粮食产量 $y$ (单位: 万亿吨) 之间存在着线性相关关系,统计数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 年份 & 2017 & 2018 & 2019 & 2020 & 2021 \\\\\n\\hline 补贴额 $x$ (亿元) & 9 & 10 & 12 & 11 & 8 \\\\\n\\hline 粮食产量 $y$ (万亿吨) & 25 & 26 & 31 & 37 & 21 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 请根据上表所给的数据, 求出 $y$ 关于 $x$ 的线性回归直线方程 $y=\\hat{a}x+\\hat{b}$;\\\\\n(2) 通过对该地区粮食产量的分析研究, 计划 2022 年在该地区发放粮食补贴 $7$ 亿元, 请根据 (1) 中所得到的线性回归直线方程, 预测 2022 年该地区的粮食产量.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -501641,7 +502327,9 @@ "id": "019664", "content": "甲、乙两台机床生产同种产品, 产品按质量分为一级品和二级品, 为了比较两台机床产品的质量, 分别用两台机床各生产了 200 件产品, 产品的质量情况统计如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 一级品 & 二级品 & 合计 \\\\\n\\hline 甲机床 & 150 & 50 & 200 \\\\\n\\hline 乙机床 & 120 & 80 & 200 \\\\\n\\hline 合计 & 270 & 130 & 400 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 甲机床、乙机床生产的产品中一级品的频率分别是多少?\\\\\n(2) 能否有 $99 \\%$ 的把握认为甲机床的产品质量与乙机床的产品质量有差异?\\\\\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{tabular}{|c|c|c|c|}\n\\hline $P(\\chi^2 \\geq k$) & 0.050 & 0.010 & 0.001 \\\\\n\\hline $k$ & 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", From 570e4edd00e0176225c7be012e5fb2407994d674 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 30 Jul 2023 16:15:08 +0800 Subject: [PATCH 6/7] =?UTF-8?q?=E8=B5=8B=E4=BA=88=E9=83=A8=E5=88=86?= =?UTF-8?q?=E9=A2=98=E7=9B=AE=E4=BB=A5=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 673 +++++++++++++++++++++++++++++++----------- 1 file changed, 508 insertions(+), 165 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index f64a63f9..1587efff 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -501283,7 +501283,9 @@ "id": "019615", "content": "从分别写有 $1,2,3,4,5,6$ 的 $6$ 张卡片中无放回随机抽取 $2$ 张, 则抽到的 $2$ 张卡片上的数字之积是 $4$ 的倍数的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{1}{3}$}{$\\dfrac{2}{5}$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501303,7 +501305,9 @@ "id": "019616", "content": "某棋手与甲、乙、丙三位棋手各比赛一盘, 各盘比赛结果相互独立.已知该棋手与甲、乙、丙比赛获胜的概率分别为 $p_1, p_2, p_3$, 且 $p_3>p_2>p_1>0$. 记该棋手连胜两盘的概率为 $p$, 则\\bracket{20}.\n\\twoch{$p$ 与该棋手和甲、乙、丙的比赛次序无关}{该棋手在第二盘与甲比赛, $p$ 最大}{该棋手在第二盘与乙比赛, $p$ 最大}{该棋手在第二盘与丙比赛, $p$ 最大}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501383,7 +501387,9 @@ "id": "019620", "content": "从甲、乙等 $5$ 名同学中随机选 $3$ 名参加社区服务工作, 则甲、乙都人选的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501443,7 +501449,9 @@ "id": "019623", "content": "从正方体的 $8$ 个顶点中任选 $4$ 个, 则这 $4$ 个点在同一个平面的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -501503,7 +501511,9 @@ "id": "019626", "content": "分别统计了甲、乙两位同学 $16$ 周的各周课外体育运动时长(单位: $h$), 得如下茎叶图, 则下列结论中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{cccc|c|cccccccc}\n\\multicolumn{4}{c|}{甲组} & & \\multicolumn{8}{c}{乙组}\\\\ \\hline\n& & 6 & 1 & 5. \\\\ \n8 & 5 & 3 & 0 & 6. & 3 \\\\\n7 & 5 & 3 & 2 & 7. & 4 & 6 \\\\\n6 & 4 & 2 & 1 & 8. & 1 & 2 & 2 & 5 & 6 & 6 & 6 & 6\\\\\n& & 4 & 2 & 9. & 0 & 2 & 3 & 8\\\\\n& & & & 10. & 1 \n\\end{tabular}\n\\end{center}\n\\onech{甲同学周课外体育运动时长的样本中位数为 $7.4$}{乙同学周课外体育运动时长的样本平均数大于 $8$}{甲同学周课外体育运动时长大于 $8$ 的概率的估计值大于 $0.4$}{乙同学周课外体育运动时长大于 $8$ 的概率的估计值大于 $0.6$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501543,7 +501553,10 @@ "id": "019628", "content": "有 $6$ 个相同的球, 分别标有数字 $1,2,3,4,5,6$, 从中有放回的随机取两次, 每次取 $1$ 个球, 甲表示事件``第一次取出的球的数字是 1'', 乙表示事件``第二次取出的球的数字是 2'', 丙表示事件``两次取出的球的数字之和是 8'', 丁表示事件``两次取出的球的数字之和是 7'', 则\\bracket{20}.\n\\fourch{甲与丙相互独立}{甲与丁相互独立}{乙与丙相互独立}{丙与丁相互独立}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "概率" + ], "genre": "选择题", "ans": "", "solution": "", @@ -501603,7 +501616,9 @@ "id": "019631", "content": "甲、乙、丙三位同学进行羽毛球比赛,约定赛制如下: 累计负两场者被淘汰; 比赛前抽签决定首先比赛的两人, 另一人轮空; 每场比赛的胜者与轮空者进行下一场比赛, 负者下一场轮空, 直至有一人被淘汰; 当一人被淘汰后, 剩余的两人继续比赛, 直至其中一人被淘汰, 另一人最终获胜, 比赛结束. 经抽签, 甲、乙首先比赛, 丙轮空. 设每场比赛双方获胜的概率都为 $\\dfrac{1}{2}$.\\\\\n(1) 求甲连胜四场的概率;\\\\\n(2) 求需要进行第五场比赛的概率;\\\\\n(3) 求丙最终获胜的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -502389,7 +502404,9 @@ "id": "019667", "content": "已知 $\\overrightarrow{a}, \\overrightarrow{b}$ 是平面内两个互相垂直的单位向量, 若向量 $\\overrightarrow{c}$ 满足 $(\\overrightarrow{a}-\\overrightarrow{c}) \\cdot(\\overrightarrow{b}-\\overrightarrow{c})=0$, 则 $|\\overrightarrow{c}|$ 的最大值是\\bracket{20}.\n\\fourch{$1$}{$2$}{$\\sqrt{2}$}{$\\dfrac{\\sqrt{2}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502409,7 +502426,10 @@ "id": "019668", "content": "已知圆 $C:(x-3)^2+(y-4)^2=1$ 和两点 $A(-m, 0)$, $B(m, 0)$($m>0$). 若圆 $C$ 上存在点 $P$, 使得 $\\angle APB=90^{\\circ}$, 则 $m$ 的最大值为\\bracket{20}.\n\\fourch{$7$}{$6$}{$5$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "圆" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502469,7 +502489,9 @@ "id": "019671", "content": "设 $P$ 是抛物线 $y^2=4 x$ 上的一个动点, 则点 $P$ 到点 $A(-1,1)$ 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": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502489,7 +502511,9 @@ "id": "019672", "content": "已知函数 $f(x)=\\begin{cases}2^x+a,& x \\geq 0,\\\\x^2-a x,& x<0,\\end{cases}$若 $f(x)$ 的最小值是 $a$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502509,7 +502533,9 @@ "id": "019673", "content": "设 $P(x, y)$ 是曲线 $C: \\sqrt{\\dfrac{x^2}{25}}+\\sqrt{\\dfrac{y^2}{9}}=1$ 上的点, $F_1(-4,0)$, $F_2(4,0)$, 则 $|PF_1|+|PF_2|$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502549,7 +502575,9 @@ "id": "019675", "content": "从双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的左焦点 $F$ 引圆 $x^2+y^2= a^2$ 的切线, 切点为 $T$, 延长 $FT$ 交双曲线右支于点 $P$, 若 $M$ 是线段 $FP$ 的中点, $O$ 为原点, 则 $|MO|-|MT|$ 的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\path [domain = -4:4, samples = 100, name path = cr, draw] plot ({sqrt(1+\\x*\\x/3)},\\x);\n\\draw [domain = -4:4, samples = 100] plot ({-sqrt(1+\\x*\\x/3)},\\x);\n\\draw (-2,0) node [below] {$F$} coordinate (F_1);\n\\draw (2,0) node [below] {$F'$} coordinate (F_2);\n\\path [name path = F1P] (F_1) --++ (30:5);\n\\path [name intersections = {of = F1P and cr, by = P}];\n\\draw (P) node [right] {$P$} -- (F_1) (P) -- (F_2);\n\\draw ($(F_1)!0.5!(P)$) node [above] {$M$} coordinate (M) -- (0,0);\n\\draw ($(F_1)!(0,0)!(P)$) node [above left] {$T$} coordinate (T) -- (0,0);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502629,7 +502657,9 @@ "id": "019679", "content": "已知实数 $x, y$ 满足 $\\dfrac{x|x|}{4}-y|y|=1$, 求 $|x-2 y+\\sqrt{5}|$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -502709,7 +502739,9 @@ "id": "019683", "content": "设常数 $a \\geq 0$, 函数 $f(x)=\\dfrac{2^x+a}{2^x-a}$, 讨论函数 $f(x)$ 的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -502729,7 +502761,10 @@ "id": "019684", "content": "设 $F_1, F_2$ 为椭圆 $\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$ 的两个焦点, 点 $P$ 为椭圆上一点, 已知点 $P, F_1,F_2$ 是一个直角三角形的三个顶点, 且 $|PF_1|>|PF_2|$, 则 $\\dfrac{|PF_1|}{|PF_2|}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "椭圆" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503169,7 +503204,9 @@ "id": "019706", "content": "已知函数 $y=f(x)$, 对任意 $x \\in \\mathbf{R}$, 都有 $f(x+2) \\cdot f(x)=k$ ($k$ 为常数), 且当 $x \\in[0,2]$ 时, $f(x)=x^2+1$, 则 $f(2021)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503229,7 +503266,9 @@ "id": "019709", "content": "已知 $M$、$N$ 是三棱锥 $P-ABC$ 的棱 $AB, PC$ 的中点, 记三棱锥 $P-ABC$ 的体积为 $V_1$, 三棱锥 $N-MBC$ 的体积为 $V_2$, 则 $\\dfrac{V_2}{V_1}$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503329,7 +503368,9 @@ "id": "019714", "content": "设函数 $f(x)=|x-a|-\\dfrac{2}{x}+a$, 若关于 $x$ 的方程 $f(x)=1$ 有且仅有两个不同的实数根, 则实数 $a$ 的取值构成的集合为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503349,7 +503390,9 @@ "id": "019715", "content": "设 $a, b \\in \\mathbf{R}$, 若 $a-|b|>0$, 则下列不等式中正确的是\\bracket{20}.\n\\fourch{$b-a>0$}{$a^3+b^3<0$}{$a^2-b^2<0$}{$b+a>0$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503489,7 +503532,9 @@ "id": "019722", "content": "已知函数 $f(x), g(x)$ 在数集 $D$ 上都有定义, 对于任意的 $x_1, x_2 \\in D$, 当 $x_1< x_2$ 时, $g(x_1) \\leq \\dfrac{f(x_1)-f(x_2)}{x_1-x_2}\\leq g(x_2)$ 或 $g(x_2) \\leq \\dfrac{f(x_1)-f(x_2)}{x_1-x_2}\\leq g(x_1)$ 成立, 则称 $g(x)$ 是数集 $D$ 上 $f(x)$ 的限制函数.\\\\\n(1) 求 $f(x)=-\\dfrac{1}{x}$ 在 $D=(0,+\\infty)$ 上的限制函数 $g(x)$ 的解析式;\\\\\n(2) 证明: 如果 $g(x)$ 在区间 $D_1 \\subseteq D$ 上恒为正值, 则 $f(x)$ 在 $D_1$ 上是严格增函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503629,7 +503674,9 @@ "id": "019729", "content": "已知等比数列 $\\{a_n\\}$ 的各项均为正数, 且满足 $a_1 a_7=4$, 则数列 $\\{\\log _2 a_n\\}$ 的前 $n$ 项之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503869,7 +503916,9 @@ "id": "019741", "content": "函数 $f(x)=2^x|\\log _{0.5}x|-1$ 的零点个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503909,7 +503958,10 @@ "id": "019743", "content": "设 $a>0$, 函数 $f(x)=x+2(1-x) \\sin (a x)$, $x \\in(0,1)$, 若函数 $y=2 x-1$ 与 $y=f(x)$ 的图像有且只有两个不同的公共点, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503929,7 +503981,9 @@ "id": "019744", "content": "设 $a, b \\in \\mathbf{R}$, 若函数 $f(x)=x+\\dfrac{a}{x}+b$ 在区间 $(1,2)$ 上有两个不同的零点, 则 $f(1)$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504129,7 +504183,9 @@ "id": "019754", "content": "在 $\\triangle ABC$ 中, $\\angle C=\\dfrac{\\pi}{2}$, $AC=BC=2$, 则 $M$ 为 $AC$ 中点, $P$ 在 $AB$ 上, 则 $\\overrightarrow{MP}\\cdot \\overrightarrow{CP}$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504169,7 +504225,9 @@ "id": "019756", "content": "已知 $x_1, x_2, y_1, y_2$ 满足 $x_1^2+y_1^2=1$, $x_2^2+y_2^2=1$, $x_1 x_2+y_1 y_2=\\dfrac{1}{2}$, 则 $\\dfrac{|x_1+y_1-1|}{\\sqrt{2}}+ \\dfrac{|x_2+y_2-1|}{\\sqrt{2}}$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504209,7 +504267,9 @@ "id": "019758", "content": "已知函数 $y=f(x)$ 的定义域为 $\\mathbf{R}$, 下列是 $f(x)$ 无最大值的充分条件是\\bracket{20}.\n\\twoch{$f(x)$ 是偶函数且关于点 $(1,1)$ 对称}{$f(x)$ 是偶函数且关于直线 $x=1$ 对称}{$f(x)$ 是奇函数且关于点 $(1,1)$ 对称}{$f(x)$ 是奇函数且关于直线 $x=1$ 对称}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -504269,7 +504329,9 @@ "id": "019761", "content": "已知椭圆 $x^2+2 y^2=1$, 过原点的两条直线 $l_1$ 和 $l_2$ 分别与椭圆交于 $A, B$ 和 $C, D$, 记得到的平行四边形 $ACBD$ 的面积为 $S$.\\\\\n(1) 设 $A(x_1, y_1)$, $C(x_2, y_2)$, 用 $A, C$ 的坐标表示点 $C$ 到直线 $l_1$ 的距离, 并证明 $S=2|x_1 y_2-x_2 y_1|$;\\\\\n(2) 设 $l_1, l_2$ 的斜率之积为 $-\\dfrac{1}{2}$, 求面积 $S$ 的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -504349,7 +504411,9 @@ "id": "019765", "content": "已知抛物线 $F: x^2=4 y$.\\\\\n(1) $\\triangle ABC$ 的三个顶点在抛物线 $F$ 上, 记 $\\triangle ABC$ 的三边 $AB, BC$, $CA$ 所在直线的斜率分别为 $k_{AB}, k_{BC}, k_{CA}$, 若点 $A$ 在坐标原点, 求 $k_{AB}-k_{BC}+k_{CA}$ 的值;\\\\\n(2) 请你给出一个以 $P(2,1)$ 为顶点, 且其余各顶点均为抛物线 $F$ 上的动点的多边形, 写出多边形各边所在直线的斜率之间的关系式, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -504389,7 +504453,9 @@ "id": "019767", "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_{10}=0$, 则等式 $a_1+a_2+\\cdots+a_n=a_1+a_2+\\cdots+a_{19-n}(n<19, n$ 是正整数) 成立. 类比上述性质, 相应地: 在等比数列 $\\{b_n\\}$ 中, 若 $b_9=1$, 试探究一个成立的等式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -504449,7 +504515,9 @@ "id": "019770", "content": "公元前 6 世纪,古希腊的毕达哥拉斯学派通过研究正五边形和正十边形的作图, 发现了黄金分割值约为 $0.618$, 这一数值也可以表示为 $m=2 \\sin 18^{\\circ}$. 若 $m^2+n=4$, 则 $\\dfrac{1-2 \\cos ^227^{\\circ}}{m \\sqrt{n}}=$\\blank{50}. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504509,7 +504577,9 @@ "id": "019773", "content": "如果一个数列由有限个连续的正整数组成 (数列的项数大于 $2$), 且所有项之和为 $N$, 那么称该数列为 $N$ 型标准数列, 例如, 数列 $2,3,4,5,6$ 为 $20$ 型标准数列, 则 $2668$ 型标准数列的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504769,7 +504839,9 @@ "id": "019786", "content": "2018 年世界人工智能大会已于 2018 年 9 月在上海徐汇西岸举行, 某高校的志愿者服务小组受大会展示项目的启发, 会后决定开发一款``猫捉老鼠''的游戏. 如右图: $A$、$B$ 两个信号源相距 $10$ 米, $O$ 是 $AB$ 的中点, 过 $O$ 点的直线 $l$ 与直线 $AB$ 的夹角为 $45^{\\circ}$. 机器猫在直线 $l$ 上运动, 机器鼠的运动轨迹始终满足: 接收到 $A$ 点的信号比接收到 $B$ 点的信号晩 $\\dfrac{8}{v_0}$ 秒 (注: 信号每秒传播 $v_0$ 米). 在时刻 $t_0$ 时,测得机器鼠距离 $O$ 点为 $4$ 米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\filldraw (-1,0) circle (0.03) node [below] {$A$} coordinate (A);\n\\filldraw (1,0) circle (0.03) node [below] {$B$} coordinate (B);\n\\draw (-1.6,-1.6) -- (1.6,1.6) node [right] {$l$};\n\\draw (0.8,0.8) node [fill = white] {\\rotatebox{45}{猫}};\n\\draw ({4/3},0.8) node {鼠};\n\\end{tikzpicture}\n\\end{center}\n(1) 以 $O$ 为原点, 直线 $AB$ 为 $x$ 轴建立平面直角坐标系 (如图), 求时刻 $t_0$ 时机器鼠所在位置的坐标;\\\\\n(2) 游戏设定: 机器鼠在距离直线 $l$ 不超过 1.5 米的区域运动时, 有``被抓''的风险. 如果机器鼠保持目前的运动轨迹不变, 是否有``被抓''风险?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -504789,7 +504861,10 @@ "id": "019787", "content": "甲、乙两队进行排球决赛, 现在的情形是甲队只要再赢一局就获冠军, 乙队需要再赢两局才能得冠军. 若两队在每局赢的概率都是 $0.5$, 则甲队获得冠军的概率为 (结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "概率" + ], "genre": "解答题", "ans": "", "solution": "", @@ -504849,7 +504924,9 @@ "id": "019790", "content": "如图, 某校有一块形如直角三角形 $ABC$ 的空地, 其中 $\\angle B$ 为直角, $AB$ 长 $40$ 米, $BC$ 长 $50$ 米, 现欲在此空地上建造一间健身房, 其占地形状为矩形, 且 $B$ 为矩形的一个顶点, 则该健身房的最大占地面积为\\blank{50}平方米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2.5,0) node [below] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw ($(A)!0.4!(C)$) coordinate (P) ($(B)!0.4!(C)$) coordinate (Q) ($(A)!0.4!(B)$) coordinate (R);\n\\draw (Q)--(P)--(R)(A)--(B)--(C)--cycle; \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504909,7 +504986,9 @@ "id": "019793", "content": "青少年视力是社会普遍关注的问题,视力情况可借助视力表测量. 通常用五分记录法和小数记录法记录视力数据, 五分记录法的数据 $L$ 和小数记录表的数据 $V$ 的满足 $L=5+ \\lg V$. 已知某同学视力的五分记录法的数据为 $4.9$ , 则其视力的小数记录法的数据为\\bracket{20}.\n\\fourch{$1.5$}{$1.2$}{$0.8$}{$0.6$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -504929,7 +505008,9 @@ "id": "019794", "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述. 两颗星的星等与亮度满足 $m_2- m_1=\\dfrac{5}{2}\\lg \\dfrac{E_1}{E_2}$, 其中星等为 $m_k$ 的星的亮度为 $E_k$($k=1,2$). 已知太阳的星等为 $-26.7$ , 天狼星的星等为 $-1.45$ , 则太阳与天狼星的亮度的比值为\\bracket{20}.\n\\fourch{$10^{10.1}$}{$10.1$}{$\\lg 10.1$}{$10^{-10.1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -504969,7 +505050,9 @@ "id": "019796", "content": "某群体的人均通勤时间, 是指单日内该群体中成员从居住地到工作地的平均用时, 某地在上班族 $S$ 中的成员仅以自驾或公交方式通勤, 分析显示: 当 $S$ 中 $x \\%$($0=latex]\n\\draw (1.5,0) arc (0:180:1.5 and 3) -- (1.5,0);\n\\draw (1.5,0) node [below right] {$B$} coordinate (B);\n\\draw (-1.5,0) node [below left] {$A$} coordinate (A);\n\\draw (50:1.5 and 3) node [above right] {$C$} coordinate (C);\n\\draw (130:1.5 and 3) node [above left] {$D$} coordinate (D);\n\\draw (B)--(C)--(D)--(A);\n\\draw (1.5,0) --++ (0.6,0) (0,3) --++ (2.1,0);\n\\draw [<->] (1.8,0) -- (1.8,3) node [midway, fill = white] {$2r$};\n\\draw (1.5,0) --++ (0,-0.6) (-1.5,0) --++ (0,-0.6);\n\\draw [<->] (-1.5,-0.3) -- (1.5,-0.3) node [midway, fill = white] {$2r$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求面积 $S$ 以 $x$ 为自变量的函数式,并写出其定义域;\\\\\n(2) 求面积 $S$ 的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -601033,7 +601119,9 @@ "id": "031408", "content": "若 $\\log _2(x+1)=3$, 则 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601093,7 +601181,10 @@ "id": "031411", "content": "在二项式 $(x+\\dfrac{3}{x})^6$ 的展开式中,常数项的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "二项式定理" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601133,7 +601224,9 @@ "id": "031413", "content": "已知向量 $\\overrightarrow{a}=(2,0)$, $\\overrightarrow{b}=(1,2)$, 则向量 $\\overrightarrow{b}$ 在向量 $\\overrightarrow{a}$ 方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601153,7 +601246,9 @@ "id": "031414", "content": "若一个圆锥的侧面展开图是面积为 $2 \\pi$ 的半圆面,则该圆锥的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601353,7 +601448,9 @@ "id": "031424", "content": "已知集合 $A=\\{y | y=\\sin x,\\ x \\in \\mathbf{R}\\}$, 集合 $B=\\{y | y=\\sqrt{x},\\ x \\in \\mathbf{R}\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601673,7 +601770,9 @@ "id": "031440", "content": "记 $\\min \\{p, q\\}=\\begin{cases}p ,& p \\leq q,\\\\q ,& p>q.\\end{cases}$ 若函数 $f(x)=\\min \\{3+\\log _{\\frac{1}{4}}x, \\log _2 x\\}$.\\\\\n(1) 用分段函数形式写出函数 $f(x)$ 的解析式;\\\\\n(2) 求 $f(x)<2$ 的解集.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -601713,7 +601812,9 @@ "id": "031442", "content": "函数 $y=\\sin 2 x-\\cos 2 x$ 的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601753,7 +601854,9 @@ "id": "031444", "content": "已知 $\\alpha$ 为第二象限角, 且 $\\sin \\alpha=\\dfrac{1}{3}$, 则 $\\sin (\\dfrac{3 \\pi}{2}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601853,7 +601956,9 @@ "id": "031449", "content": "$4$ 张卡片上分别写有数字 $1,2,3,4$, 从这 $4$ 张卡片中随机抽取 $2$ 张, 则取出的 $2$ 张卡片上的数字之和为奇数的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601873,7 +601978,9 @@ "id": "031450", "content": "已知 $(\\sqrt{x}+\\dfrac{3}{\\sqrt[3]{x}})^n$ 展开式中, 各项系数的和与各项二项式系数的和之比为 $64$, 则 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602093,7 +602200,9 @@ "id": "031461", "content": "已知 $\\tan \\dfrac{\\alpha}{2}=3$, 则 $\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602173,7 +602282,9 @@ "id": "031465", "content": "一个与球心距离为 1 的平面截球所得的圆面面积为 $\\pi$, 则球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602193,7 +602304,9 @@ "id": "031466", "content": "已知向量 $\\overrightarrow{OA}=(k, 12)$, $\\overrightarrow{OB}=(4,5)$, $\\overrightarrow{OC}=(-k, 10)$, 且 $A, B, C$ 三点共线, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602233,7 +602346,9 @@ "id": "031468", "content": "一支田径队有男女运动员 $98$ 人, 其中男运动员有 $56$ 人. 按男女比例用分层抽样的方法, 从全体运动员中抽出一个容量为 $28$ 的样本, 那么应抽取女运动员人数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602273,7 +602388,9 @@ "id": "031470", "content": "已知集合 $A=\\{x | x=2 n-1,\\ n \\in \\mathbf{N}, \\ n \\geq 1\\}$, $B=\\{x | x=2^n, n \\in \\mathbf{N},\\ n \\geq 1\\}$, 将 $A \\cup B$ 中的所有元素按从小到大的顺序排列构成一个数列 $\\{a_n\\}$, 设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 则使得 $S_n>1000$ 成立的最小的 $n$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602293,7 +602410,9 @@ "id": "031471", "content": "已知平面向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足 $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $\\overrightarrow{a}^2=\\overrightarrow{a}\\cdot \\overrightarrow{b}$, $2 \\overrightarrow{c}^2=\\overrightarrow{b}\\cdot \\overrightarrow{c}$, 则 $|\\overrightarrow{c}-\\overrightarrow{a}|^2+|\\overrightarrow{c}-\\overrightarrow{b}|^2$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602493,7 +602612,9 @@ "id": "031481", "content": "已知球的体积为 $36 \\pi$, 则该球的表面积等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602973,7 +603094,9 @@ "id": "031505", "content": "(如图) 以长方体 $ABCD-A_1B_1C_1D_1$ 的顶点 $D$ 为坐标原点, 过 $D$ 的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若 $\\overrightarrow{DB_1}$ 的坐标为 $(4,3,2)$, 则 $\\overrightarrow{AC_1}$ 的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below right] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [below right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (A)-- ($(D)!1.5!(A)$) node [below left] {$x$};\n\\draw [->] (C)-- ($(D)!1.4!(C)$) node [below] {$y$};\n\\draw [->] (D1)-- ($(D)!1.5!(D1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603373,7 +603496,9 @@ "id": "031525", "content": "设椭圆 $\\dfrac{x^2}{2}+y^2=1$ 的左右焦点为 $F_1, F_2$, 点 $P$ 在该椭圆上, 则使得 $\\triangle PF_1F_2$ 为等腰三角形的点 $P$ 的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603453,7 +603578,9 @@ "id": "031529", "content": "设等差数列 $\\{a_n\\}$ 的公差 $d$ 不为 $0$, $a_1=9 d$. 若 $a_k$ 是 $a_1$ 与 $a_{2 k}$ 的等比中项, 则 $k=\\bracket{20}$\\bracket{20}.\n\\fourch{$2$}{$4$}{$6$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603513,7 +603640,9 @@ "id": "031532", "content": "已知 $\\mathrm{i}$ 是虚数单位, $z=\\dfrac{2-\\mathrm{i}}{\\mathrm{i}}$, 则 $|\\overline{z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603913,7 +604042,9 @@ "id": "031552", "content": "已知长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=BC=4$, $CC_1=2$, 则直线 $BC_1$ 和平面 $DBB_1D_1$ 所成角的正弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603953,7 +604084,9 @@ "id": "031554", "content": "已知严格增的等差数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_3=a_2^2-4$, 则 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604053,7 +604186,9 @@ "id": "031559", "content": "已知向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 夹角为 $45^{\\circ}$, 且 $|\\overrightarrow{a}|=1$, $|2 \\overrightarrow{a}-\\overrightarrow{b}|=\\sqrt{10}$; 则 $|\\overrightarrow{b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604073,7 +604208,9 @@ "id": "031560", "content": "若函数 $f(x)=\\begin{cases}x^2-2,& x \\leq 1,\\\\\\lg |x-m|,& x>1\\end{cases}$ 在区间 $[0,+\\infty)$ 上是严格增函数, 则实数 $m$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604153,7 +604290,9 @@ "id": "031564", "content": "已知 $\\triangle ABC$ 是斜三角形, 则``$A>B$''是``$|\\tan A|>|\\tan B|$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604213,7 +604352,9 @@ "id": "031567", "content": "定义向量 $\\overrightarrow{OM}=(a, b)$ 的``相伴函数''为 $f(x)=a \\sin x+b \\cos x$; 函数 $f(x)=a \\sin x+ b \\cos x$ 的``相伴向量''为 $\\overrightarrow{OM}=(a, b)$ (其中 $O$ 为坐标原点). 记平面内所有向量的``相伴函数''构成的集合为 $S$.\\\\\n(1) 设 $g(x)=3 \\sin (x+\\dfrac{\\pi}{2})+4 \\sin x$, 求证: $g(x) \\in S$;\\\\\n(2) 已知 $h(x)=\\cos (x+\\alpha)+2 \\cos x$, 且 $h(x) \\in S$, 求其``相伴向量''的模;\\\\\n(3) 已知 $M(a, b)$($b \\neq 0$) 为圆 $C:(x-2)^2+y^2=1$ 上一点, 向量 $\\overrightarrow{OM}$ 的``相伴函数''$f(x)$ 在 $x=x_0$ 处取得最大值. 当点 $M$ 在圆 $C$ 上运动时, 求 $\\tan 2 x_0$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604253,7 +604394,9 @@ "id": "031569", "content": "若 $\\sin (\\pi+\\theta)=\\dfrac{1}{3}$, 且 $\\theta$ 为第三象限角, 则 $\\cos (2 \\pi-\\theta)$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604293,7 +604436,9 @@ "id": "031571", "content": "设 $a$ 是实数, $f(x)=(x^2-4)(x-a)$, 若 $f'(-1)=0$, 则函数 $y=f(x)$ 在区间 $[-2,2]$ 上的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604433,7 +604578,9 @@ "id": "031578", "content": "已知点 $A(-2,0)$、$B(3,0)$, 动点 $P(x, y)$ 满足 $\\overrightarrow{PA}\\cdot \\overrightarrow{PB}=x^2$, 则点 $P$ 的轨迹是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604473,7 +604620,9 @@ "id": "031580", "content": "在 $\\triangle ABC$ 中, 若 $\\sin ^2A+\\sin ^2B<\\sin ^2C$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{钝角三角形}{直角三角形}{锐角三角形}{不能确定}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604513,7 +604662,9 @@ "id": "031582", "content": "我国古代数学名著《算法统宗》中有如下问题:``远望巍巍塔七层, 红光点点倍加增, 共灯三百八十一, 请问尖头几盏灯?''意思是 : 一座 $7$ 层塔共挂了 $381$ 盙灯, 且相邻两层中的下一层灯数是上一层灯数的 $2$ 倍, 则塔的顶层共有灯\\bracket{20}.\n\\fourch{$1$盏}{$3$盏}{$5$盏}{$9$盏}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604533,7 +604684,9 @@ "id": "031583", "content": "设函数 $f(x)=\\sin (x-\\dfrac{\\pi}{6})$, 若对于任意 $\\alpha \\in[-\\dfrac{5 \\pi}{6},-\\dfrac{\\pi}{2}]$, 在区间 $[0, m]$ 上总存在唯一确定的 $\\beta$, 使得 $f(\\alpha)+f(\\beta)=0$, 则 $m$ 的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{2}$}{$\\dfrac{7 \\pi}{6}$}{$\\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604553,7 +604706,9 @@ "id": "031584", "content": "某热力公司每年燃料费约 24 万元. 为了``环评''达标, 需要安装一块面积为 $x$($x \\geq 0$) (单位: 平方米) 可用 15 年的太阳能板, 其工本费为 $\\dfrac{x}{2}$ (单位: 万元), 并与燃料供热互补工作, 从此, 公司每年的燃料费为 $\\dfrac{k}{20 x+100}$ ($k$ 为常数) 万元. 记 $y$ 为该公司安装太阳能板的费用与 15 年的燃料费之和.\\\\\n(1) 求 $k$ 的值, 并建立 $y$ 关于 $x$ 的函数关系式;\\\\\n(2) 求 $y$ 的最小值, 并求出此时所安装太阳能板的面积.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604593,7 +604748,9 @@ "id": "031586", "content": "函数 $y=\\cos 2 x$, $x \\in[0, \\pi]$ 的递增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604733,7 +604890,9 @@ "id": "031593", "content": "直角坐标平面上三点 $A(1,2)$、$B(3,-2)$、$C(9,7)$, 若 $E$、$F$ 为线段 $BC$ 的三等分点, 则 $\\overrightarrow{AE}\\cdot \\overrightarrow{AF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604753,7 +604912,9 @@ "id": "031594", "content": "方程 $|x+1|+|x-2|=3$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604973,7 +605134,9 @@ "id": "031605", "content": "已知 $\\triangle ABC$ 的周长为 $4$, 且 $\\sin A+\\sin B=3 \\sin C$, 则 $AB$ 边的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604993,7 +605156,9 @@ "id": "031606", "content": "已知等比数列 $\\{a_n\\}$ 中 $a_2=1$, 则其前 3 项的和 $S_3$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605093,7 +605258,9 @@ "id": "031611", "content": "已知 $f(x)=\\sin (\\omega x+\\dfrac{\\pi}{3})$($\\omega>0$), $f(\\dfrac{\\pi}{6})=f(\\dfrac{\\pi}{3})$, 且 $f(x)$ 在区间 $(\\dfrac{\\pi}{6}, \\dfrac{\\pi}{3})$ 有最小值, 无最大值, 则 $\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605113,7 +605280,9 @@ "id": "031612", "content": "一支田径队有男女运动员 $98$ 人, 其中男运动员有 $56$ 人. 按男女比例用分层抽样的方法, 从全体运动员中抽出一个容量为 $28$ 的样本, 那么应抽取女运动员人数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605133,7 +605302,9 @@ "id": "031613", "content": "用与球心距离为 $1$ 的平面去截球,所得的截面面积为 $\\pi$, 则球的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605173,7 +605344,9 @@ "id": "031615", "content": "已知无穷等比数列 $a_1, a_2, a_3, \\cdots$ 各项的和为 $\\dfrac{9}{2}$, 且 $a_2=-2$, 若 $|S_n-\\dfrac{9}{2}|<10^{-4}$, 则 $n$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605293,7 +605466,9 @@ "id": "031621", "content": "如图, 正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, 底面边长为 $2, BC_1$ 与底面 $ABCD$ 所成角的大小为 $\\arctan 2, M$ 是 $DD_1$ 的中点, $N$ 是 $BD$ 上的一动点, 设 $\\overrightarrow{DN}=\\lambda \\overrightarrow{DB}$($0<\\lambda<1$).\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.75]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(B)!0.6!(D)$) node [below] {$N$} coordinate (N);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$M$} coordinate (M);\n\\draw (B)--(C_1);\n\\draw [dashed] (B)--(D)(M)--(N)(A)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 当 $\\lambda=\\dfrac{1}{2}$ 时,证明: $MN$ 与平面 $ABC_1D_1$ 平行;\\\\\n(2) 若点 $N$ 到平面 $BCM$ 的距离为 $d$, 试用 $\\lambda$ 表示 $d$, 并求出 $d$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -605333,7 +605508,9 @@ "id": "031623", "content": "若一个球的体积为 $4 \\sqrt{3}\\pi$, 则它的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605353,7 +605530,9 @@ "id": "031624", "content": "已知直线 $l$ 的参数方程为 $\\begin{cases}x=1+3 t\\\\y=2+4 t\\end{cases}$ ($t$ 为参数), 则点 ( 1,0$)$ 到直线 $l$ 的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605373,7 +605552,9 @@ "id": "031625", "content": "若复数 $\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}}+\\dfrac{1}{2}b$($b \\in \\mathbf{R}$) 的实部的模与虚部相等, 则 $b$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605393,7 +605574,10 @@ "id": "031626", "content": "若 $(x-2)^5=a_5 x^5+a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$, 则 $a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "二项式定理" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605453,7 +605637,9 @@ "id": "031629", "content": "若一个圆锥的母线与轴的夹角为 $\\arcsin \\dfrac{1}{3}$, 则该圆锥的侧面积是底面积的\\blank{50}倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605473,7 +605659,10 @@ "id": "031630", "content": "从集合 $\\{0,1,2,3\\}$ 的所有非空子集中, 等可能地取出一个. 则取出的非空子集中所有元素之和恰为 5 的概率 $=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "概率" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605553,7 +605742,9 @@ "id": "031634", "content": "``$(2 x+1) x=0''$ 是``$x=0''$ 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}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605573,7 +605764,9 @@ "id": "031635", "content": "如图, 点 $N$ 为正方形 $ABCD$ 的中心, $\\triangle ECD$ 为正三角形, 平面 $ECD \\perp$ 平面 $ABCD, M$ 是线段 $ED$ 的中点,则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (2,0,2) node [right] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$D$} coordinate (D);\n\\draw ($(C)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw ($(B)!0.5!(D)$) node [below] {$N$} coordinate (N);\n\\draw (O) ++ (0,{sqrt(3)},0) node [above] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(E)$) node [left] {$M$} coordinate (M);\n\\draw (E)--(D)(E)--(C)(E)--(O)(C)--(D)--(A)--(B)--cycle;\n\\draw (E)--(N)(M)--(B);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$BM=EN$, 且直线 $BM$、$EN$ 是相交直线}{$BM \\neq EN$, 且直线 $BM, EN$ 是相交直线}{$BM=EN$, 且直线 $BM$、$EN$ 是异面直线}{$BM \\neq EN$, 且直线 $BM, EN$ 是异面直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605593,7 +605786,9 @@ "id": "031636", "content": "函数 $f(x)=x \\cos 2 x$ 在区间 $[0,2 \\pi]$ 上的零点个数为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605633,7 +605828,9 @@ "id": "031638", "content": "如图, 已知点 $P$ 在圆柱 $OO_1$ 的底面圆 $O$ 上, $\\angle AOP=120^{\\circ}$, 圆 $O$ 的直径 $AB=4$, 圆柱的高 $OO_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\filldraw (0,0) node [above] {$O$} coordinate (O) circle (0.03);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,3) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,3) node [right] {$B_1$} coordinate (B_1);\n\\filldraw (O) ++ (0,3) node [above] {$O_1$} coordinate (O_1) circle (0.03);\n\\draw (O) ++ (-50:2 and 0.5) node [below] {$P$} coordinate (P);\n\\draw (A)--(A_1)--(B_1)--(B)arc (0:-180:2 and 0.5);\n\\draw (O_1) ellipse (2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw [dashed] (A)--(B)--(P)--cycle(A_1)--(P)(A_1)--(B)(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的表面积和三棱锥 $A_1-APB$ 的体积;\\\\\n(2) 求点 $A$ 到平面 $A_1PO$ 的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -605693,7 +605890,10 @@ "id": "031641", "content": "曲线 $y=2 \\ln x$ 在点 $(1,0)$ 处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605753,7 +605953,9 @@ "id": "031644", "content": "已知 $a, b \\in \\mathbf{R}$, 且 $a-3 b+6=0$, 则 $2^a+\\dfrac{1}{8^b}$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605873,7 +606075,9 @@ "id": "031650", "content": "设某种动物活到 20 岁的概率为 $0.8$ , 活到 25 岁的概率为 $0.4$ . 现有一只 20 岁的该种动物, 它活到 25 岁的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605933,7 +606137,9 @@ "id": "031653", "content": "设 $x \\in \\mathbf{R}$, 则``$x^3>8$''是``$|x|>2$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605953,7 +606159,9 @@ "id": "031654", "content": "将函数 $y=\\sin (2 x-\\dfrac{\\pi}{3})$ 图像上的点 $P(\\dfrac{\\pi}{4}, t)$ 向左平移 $s$($s>0$) 个单位长度得到点 $P'$. 若 $P'$ 位于函数 $y=\\sin 2 x$ 的图像上, 则\\bracket{20}.\n\\twoch{$t=\\dfrac{1}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{6}$}{$t=\\dfrac{\\sqrt{3}}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{6}$}{$t=\\dfrac{1}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{3}$}{$t=\\dfrac{\\sqrt{3}}{2}$, $s$ 的最小值为 $\\dfrac{\\pi}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -606033,7 +606241,9 @@ "id": "031658", "content": "集合 $A=\\{y | y=\\lg x, x>1\\}$, $B=\\{-2,-1,1,2\\}$, 则 $\\overline{A}\\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606093,7 +606303,9 @@ "id": "031661", "content": "若复数 $z$ 满足 $2 \\overline{z}-1=3+6 \\mathrm{i}$ ($\\mathrm{i}$ 是虚数单位), 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606153,7 +606365,9 @@ "id": "031664", "content": "如果一条直线与一个平面垂直, 那么, 称此直线与平面构成一个``正交线面对'', 在一个正方体中, 由两个顶点确定的直线与含有四个顶点的平面构成的``正交线面对''的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606553,7 +606767,9 @@ "id": "031684", "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的渐近线为正方形 $OABC$ 的边 $OA, OC$ 所在的直线, 点 $B$ 为该双曲线的焦点. 若正方形 $OABC$ 的边长为 $2$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606593,7 +606809,9 @@ "id": "031686", "content": "正四棱锥底面边长为 $4$, 侧棱长为 $3$, 则其体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606693,7 +606911,9 @@ "id": "031691", "content": "已知 $f(x)$ 是定义域为 ($-\\infty$, $+\\infty$) 的奇函数, 满足 $f(1-x)=f(1+x)$. 若 $f(1)=2$, 则 $f(1)+f(2)+f(3)+\\cdots+f(100)$ 的值为\\bracket{20}.\n\\fourch{$-50$}{$0$}{$2$}{$50$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -606753,7 +606973,9 @@ "id": "031694", "content": "不等式 $\\dfrac{x-1}{x+2}>1$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606773,7 +606995,9 @@ "id": "031695", "content": "若角 $\\alpha$ 的终边经过点 $P(1,-2)$, 则 $\\tan 2 \\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606833,7 +607057,9 @@ "id": "031698", "content": "函数 $y=\\sqrt{\\log _{\\frac{1}{2}}(x^2-1)}$ 的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606973,7 +607199,9 @@ "id": "031705", "content": "在标有 $1$、$2$、$3$、$4$、$5$ 的五张卡片中依次选取两张, 在第一张是奇数的条件下, 第二张也是奇数的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607573,7 +607801,9 @@ "id": "031735", "content": "已知指数函数 $y=a^x$ (其中 $a>1$) 在闭区间 $[1,2]$ 上的最大值比最小值大 $\\dfrac{a}{3}$, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607633,7 +607863,9 @@ "id": "031738", "content": "已知双曲线 $M: x^2-\\dfrac{y^2}{6}=1$ 的左、右焦点分别为 $F_1$、$F_2$, 过 $F_1$ 的直线 $l$ 与双曲线 $M$ 的左、右两支分别交于点 $A$、$B$. 若 $\\triangle ABF_2$ 为等边三角形, 则 $\\triangle ABF_2$ 的边长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607653,7 +607885,9 @@ "id": "031739", "content": "已知函数 $f(x)=x^2+2 x+3+m$, 若 $f(x) \\geq 0$ 对任意的 $x \\in[1,2]$ 恒成立, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607673,7 +607907,9 @@ "id": "031740", "content": "已知菱形 $ABCD$ 的边长为 $1, \\angle DAB=\\dfrac{\\pi}{3}$, 点 $E$ 为该菱形边上任意一点, 则 $\\overrightarrow{AB}\\cdot \\overrightarrow{AE}$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607693,7 +607929,9 @@ "id": "031741", "content": "已知函数 $f(x)=\\begin{cases}x^2-x+3,& x \\leq 1,\\\\x+\\dfrac{2}{x},& x>1.\\end{cases}$ 设 $a \\in \\mathbf{R}$, 若关于 $x$ 的不等式 $f(x) \\geq|\\dfrac{x}{2}+a|$ 在 $\\mathbf{R}$ 上恒成立,则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607773,7 +608011,9 @@ "id": "031745", "content": "已知实数 $x_1$、$y_1$、$x_2$、$y_2$、$x_3$、$y_3$ 满足 $x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2=2$, 则 $x_1 y_2$、$x_2 y_3$ 、 $x_3 y_1$ 三个数中,大于 1 的个数最多是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -607853,7 +608093,9 @@ "id": "031749", "content": "若函数 $f(x)=a \\cdot 3^x+\\dfrac{1}{3^x}$ 为偶函数, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607993,7 +608235,9 @@ "id": "031756", "content": "已知抛物线 $y^2=2 p x$($p>0$) 上一点 $M(1, m)$ 到其焦点的距离为 $5$, 双曲线 $C: x^2-\\dfrac{y^2}{b^2}=1$($b>0$) 的左顶点为 $A$, 若双曲线 $C$ 的一条渐近线与直线 $AM$ 垂直, 则双曲线 $C$ 的焦距为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608013,7 +608257,9 @@ "id": "031757", "content": "已知 $\\alpha$ 为第二象限角, 且 $\\sin \\alpha=\\dfrac{\\sqrt{15}}{4}$, 则 $\\dfrac{\\sin (\\alpha+\\dfrac{\\pi}{4})}{\\sin 2 \\alpha+\\cos 2 \\alpha+1}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608033,7 +608279,9 @@ "id": "031758", "content": "设曲线 $C$ 与函数 $f(x)=\\dfrac{\\sqrt{3}}{12}x^2$($0 \\leq x \\leq m$) 的图像关于直线 $y=\\sqrt{3}x$ 对称, 若曲线 $C$ 仍为某函数的图像, 则实数 $m$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608073,7 +608321,9 @@ "id": "031760", "content": "对于下列命题: \\textcircled{1} 若 $a>b>0$, $c>d>0$, 则 $\\dfrac{a+c}{d}>\\dfrac{b+d}{c}$; \\textcircled{2} 若 $a>b>0$, $c>d>0$, 则$a^c>b^d$, 关于上述命题描述正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}都均为真命题}{\\textcircled{1}和\\textcircled{2} 都均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608133,7 +608383,9 @@ "id": "031763", "content": "设函数 $f(x)=2^x-2^{-x}+\\dfrac{3}{|x|+1}$, $x \\in \\mathbf{R}$, 对于实数 $a$、$b$, 给出以下命题: 命题 $p_1$: $a+ b \\geq 0$; 命题 $p_2$: $a-b^2 \\geq 0$; 命题 $q$: $f(a)+f(b) \\geq 0$. 下列选项中正确的是\\bracket{20}.\n\\twoch{$p_1$、$p_2$ 中仅 $p_1$ 是 $q$ 的充分条件}{$p_1$、$p_2$ 中仅 $p_2$ 是 $q$ 的充分条件}{$p_1$、$p_2$ 都不是 $q$ 的充分条件}{$p_1$、$p_2$ 都是 $q$ 的充分条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608153,7 +608405,9 @@ "id": "031764", "content": "考虑到高速公路行车安全需要, 一般要求高速公路的车速 $v$ (公里/小时) 控制在 $[60,120]$ 范围内. 已知汽车以 $v$ 公里/小时的速度在高速公路上匀速行驶时, 每小时的油耗 (所需要的汽油量) 为 $\\dfrac{1}{5}(v-k+\\dfrac{4500}{v})$ 升, 其中 $k$ 为常数, 不同型号汽车 $k$ 值不同, 且满足 $60 \\leq k \\leq 120$.\\\\\n(1) 若某型号汽车以 $120$ 公里/小时的速度行驶时, 每小时的油耗为 $11.5$ 升, 欲使这种型号的汽车每小时的油耗不超过 $9$ 升, 求车速 $v$ 的取值范围;\\\\\n(2) 求不同型号汽车行驶 $100$ 千米的油耗的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -608233,7 +608487,9 @@ "id": "031768", "content": "已知 $\\sin \\alpha=3 \\cos \\alpha$, 则 $\\tan (\\alpha-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608273,7 +608529,9 @@ "id": "031770", "content": "若 $x_1$、$x_2$ 为方程 $a^x=(\\dfrac{1}{a})^{-\\frac{1}{x}+1}$($a>1$) 的两个实数解,则 $x_1+x_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608313,7 +608571,9 @@ "id": "031772", "content": "若 $S_n$ 为等比数列 $\\{a_n\\}$ 的前 $n$ 项和, $8 a_2+a_5=0$, 则 $\\dfrac{S_{10}}{S_5}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608373,7 +608633,9 @@ "id": "031775", "content": "设函数 $f(x)=\\begin{cases}-a x+1,& x2,\\end{cases}$ 则 $f(6)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609553,7 +609836,9 @@ "id": "031834", "content": "函数 $f(x)=2^x+\\dfrac{9}{2^x+1}$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609613,7 +609898,9 @@ "id": "031837", "content": "设集合 $A=\\{1,2, m\\}$, 其中 $m$ 为实数, 令 $B=\\{t | t=a^2, a \\in A\\}$, $C=A \\cup B$, 若 $C$ 的所有元素之和为 6 ,则 $C$ 的所有元素之积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609633,7 +609920,9 @@ "id": "031838", "content": "已知某校的午餐由``主食''与``配菜''两部分组成, 主食和配菜均有若干种不同的选择. 某个学期的统计结果显示, 学生购买主食 $A$ 的概率为 $25 \\%$, 而在购买主食 $A$ 的学生中, 又有 $70 \\%$ 的学生会购买配菜 $B$ 作为搭配,则学生同时购买主食 $A$ 与配菜 $B$ 的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609733,7 +610022,9 @@ "id": "031843", "content": "如图所示某加油站地下圆柱体储油罐示意图, 已知储油罐长度为 $d$, 圆柱底面半径为 $r$ ($d, r$ 为常量), 油面高度为 $h$, 油面宽度为 $w$, 储油量为 $v(h, w, v$ 为变量), 则下列说法: \\textcircled{1} $w$ 是 $v$ 的函数; \\textcircled{2} $v$ 是 $w$ 的函数; \\textcircled{3} $h$ 是 $w$ 的函数; \\textcircled{4} $w$ 是 $h$ 的函数. 其中正确说法的序号为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(0.91)},-0.3) coordinate (A);\n\\draw ({sqrt(0.91)},-0.3) coordinate (B);\n\\draw (B)++(0,0,-5) coordinate (C);\n\\draw (A)++(0,0,-5) coordinate (D);\n\\draw (A)--(B)--(C);\n\\draw [dashed] (C)--(D)--(A);\n\\draw (0,-0.3) -- (0,-1) node [midway, right] {$h$};\n\\fill [gray!30] (A)--(B)--(C)--(D)--cycle;\n\\draw [domain = 0:360] plot ({cos(\\x)},{sin(\\x)},0);\n\\draw [domain = -45:135] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw [domain = 135:315, dashed] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw (-45:1) --++ (0,0,-5) (135:1) --++ (0,0,-5);\n\\draw (-45:1) ++ (0,-0.1) --++ (0,-0.8) (-45:1) ++ (0,0,-5) ++ (0,-0.1) --++ (0,-0.8);\n\\draw [<->] (-45:1) ++ (0,-0.5) --++ (0,0,-5) node [midway, fill = white] {$d$};\n\\draw ($(C)!0.5!(D)$) node [above] {$w$};\n\\draw (0,0) --++ (135:1) node [midway, below left] {$r$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609773,7 +610064,9 @@ "id": "031845", "content": "如图, 正四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面边长为 $1$, 高为 $2$, $M$ 为线段 $AB$ 的中点. 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (C_1)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥 $A_1-MBC_1$ 的体积;\\\\\n(2) 异面直线 $AC$ 与 $MC_1$ 所成角的大小 (结果用反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609873,7 +610166,9 @@ "id": "031850", "content": "已知 $a, b \\in \\mathbf{R}, \\mathrm{i}$ 是虚数单位, 若 $a+\\mathrm{i}=2-b \\mathrm{i}$, 则 $(a+b \\mathrm{i})^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609893,7 +610188,9 @@ "id": "031851", "content": "函数 $y=\\sqrt{1-\\lg x}$ 的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609933,7 +610230,9 @@ "id": "031853", "content": "已知函数 $y=f(x)+x$ 是偶函数, 且 $f(2)=1$, 则 $f(-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610253,7 +610552,9 @@ "id": "031869", "content": "已知关于 $x$ 的函数 $y=f(x)$, $y=g(x)$ 与 $h(x)=k x+b$($k, b \\in \\mathbf{R}$) 在区间 $D$ 上恒有 $f(x) \\geq h(x) \\geq g(x)$.\\\\\n(1) 若 $f(x)=x^2+2 x$, $g(x)=-x^2+2 x$, $D=$($-\\infty$, $+\\infty$), 求 $h(x)$ 的表达式;\\\\\n(2) 若 $f(x)=x^2-x+1$, $g(x)=k \\ln x$, $h(x)=k x-k$, $D \\in$($0,+\\infty$), 求 $k$ 的取值范围;\\\\\n(3) 若 $f(x)=x^4-2 x^2$, $g(x)=4 x^2-8$, $h(x)=4(t^3-t) x-3 t^4+2 t^2$($0<|t| \\leq \\sqrt{2}$), $D=[m, n] \\subset[-\\sqrt{2}, \\sqrt{2}]$, 求证: $|m-n| \\leq \\sqrt{7}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -610333,7 +610634,10 @@ "id": "031873", "content": "在 $(x-\\dfrac{1}{2 x})^{10}$ 的二项展开式中, $x^4$ 的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "二项式定理" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610553,7 +610857,9 @@ "id": "031884", "content": "已知 $02$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610733,7 +611039,9 @@ "id": "031893", "content": "函数 $y=2 \\cos ^2(\\pi x)-1$ 的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610773,7 +611081,9 @@ "id": "031895", "content": "不等式 $\\dfrac{x+2}{x+1}>1$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610813,7 +611123,9 @@ "id": "031897", "content": "半径为 $4$ 的圆内接三角形 $ABC$ 的面积是 $\\dfrac{1}{16}$, 角 $A$、$B$、$C$ 所对应的边依次为 $a$、$b$、$c$, 则 $a b c$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610853,7 +611165,9 @@ "id": "031899", "content": "已知函数 $y=f(x)$ 是奇函数, 当 $x<0$ 时, $f(x)=2^x-a x$, 且 $f(2)=2$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611153,7 +611467,9 @@ "id": "031914", "content": "不等式 $\\dfrac{x(x-1)}{|x|}\\leq 0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611393,7 +611709,9 @@ "id": "031926", "content": "为评估一种农作物的种植效果, 选了 $n$ 块地作试验田. 这 $n$ 块地的亩产量 (单位: $\\mathrm{kg}$) 分别为 $x_1, x_2, \\cdots, x_n$,下面给出的指标中可以用来评估这种农作物亩产量稳定程度的是\\bracket{20}.\n\\twoch{$x_1, x_2, \\cdots, x_n$ 的平均数}{$x_1, x_2, \\cdots, x_n$ 的标准差}{$x_1, x_2, \\cdots, x_n$ 的最大值}{$x_1, x_2, \\cdots, x_n$ 的中位数}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -611553,7 +611871,9 @@ "id": "031934", "content": "已知复数 $z$ 满足 $\\dfrac{1+3 \\mathrm{i}}{z}=2 \\mathrm{i}$ (其中 $\\mathrm{i}$ 为虚数单位), 则 $|\\overline{z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611573,7 +611893,9 @@ "id": "031935", "content": "不等式 $\\dfrac{2 x+5}{x-2}<1$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611613,7 +611935,9 @@ "id": "031937", "content": "已知 $a \\in\\{-2,-1,-\\dfrac{1}{2}, \\dfrac{1}{2}, 1,2,3\\}$, 若幂函数 $f(x)=x^a$ 为奇函数, 且在 ($0,+\\infty$) 上严格减, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611633,7 +611957,9 @@ "id": "031938", "content": "方程 $(\\log _3 x)^2-\\log _33 x=1$ 的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611793,7 +612119,9 @@ "id": "031946", "content": "已知 $a \\in \\mathbf{R}$, 则``$a>1$''是``$\\dfrac{1}{a}<1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -611813,7 +612141,9 @@ "id": "031947", "content": "已知平面 $\\alpha$、$\\beta$、$\\gamma$ 两两垂直, 直线 $a$、$b$、$c$ 满足: $a \\subset \\alpha$, $b \\subset \\beta$, $c \\subset \\gamma$, 则直线 $a$、$b$、$c$ 不可能满足以下哪种关系\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -611873,7 +612203,9 @@ "id": "031950", "content": "已知 $A$、$B$、$C$ 为 $\\triangle ABC$ 的三个内角, $a$、$b$、$c$ 是其三条边, $a=2$, $\\cos C=-\\dfrac{1}{4}$.\\\\\n(1) 若 $\\sin A=2 \\sin B$, 求 $b$、$c$;\\\\\n(2) $\\cos (A-\\dfrac{\\pi}{4})=\\dfrac{4}{5}$, 求 $c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611913,7 +612245,9 @@ "id": "031952", "content": "如图, 已知 $P$ 是平行四边形 $ABCD$ 所在平面外一点, $M$、$N$ 分别是 $AB$、$PC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,1) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(C)$) node [above] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(M)--(N);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel$ 平面 $PAD$;\\\\\n(2) 若 $MN=BC=4$, $PA=4 \\sqrt{3}$, 求异面直线 $PA$ 与 $MN$ 所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -612013,7 +612347,9 @@ "id": "031957", "content": "已知向量 $\\overrightarrow{a}=(1,0,3)$, $\\overrightarrow{b}=(3,1,0)$, 则向量 $\\overrightarrow{a}, \\overrightarrow{b}$ 的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612073,7 +612409,9 @@ "id": "031960", "content": "已知圆柱的高为 $4$, 底面积为 $9 \\pi$, 则圆柱的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612153,7 +612491,10 @@ "id": "031964", "content": "设非零常数 $d$ 是等差数列 $x_1, x_2, x_3, \\cdots, x_{19}$ 的公差, 随机变量 $\\xi$ 等可能地取值 $x_1, x_2$, $x_3, \\cdots, x_{19}$, 则方差 $D [\\xi]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612273,7 +612614,9 @@ "id": "031970", "content": "设函数 $f(x)$ 的定义域为 $\\mathbf{R}$, 满足 $f(x+1)=2 f(x)$, 且当 $x \\in(0,1]$ 时, $f(x)=x(x-1)$, 若对任意 $x \\in(-\\infty, m]$, 都有 $f(x) \\geq-\\dfrac{8}{9}$, 则第 15 题图 $m$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty, \\dfrac{9}{4}]$}{$(-\\infty, \\dfrac{7}{3}]$}{$(-\\infty, \\dfrac{5}{2}]$}{$(-\\infty, \\dfrac{8}{3}]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", From a76c028fa67b694d8110befb4a5e035d91652ed8 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sun, 30 Jul 2023 21:50:40 +0800 Subject: [PATCH 7/7] =?UTF-8?q?=E9=80=9A=E8=BF=87=E5=AD=97=E7=AC=A6?= =?UTF-8?q?=E4=B8=B2=E6=AF=94=E5=AF=B9=E8=87=AA=E5=8A=A8=E8=B5=8B=E4=BA=88?= =?UTF-8?q?=E4=BA=86=E4=B8=80=E4=BA=9B=E9=A2=98=E7=9B=AE=E4=BB=A5=E5=8D=95?= =?UTF-8?q?=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 960 +++++++++++++++++++++++++++++++----------- 1 file changed, 720 insertions(+), 240 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 1587efff..265b39e4 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -502364,7 +502364,9 @@ "id": "019665", "content": "已知函数 $f(x)=\\begin{cases}|x|,& x \\leq m,\\\\x^2-2 m x+4 m,& x>m,\\end{cases}$其中 $m>0$. 若存在实数 $b$, 使得关于 $x$ 的方程 $f(x)=b$ 有三个不同的根, 则 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502449,7 +502451,9 @@ "id": "019669", "content": "已知抛物线的方程为 $x^2=8 y$, 点 $F$ 是其焦点, 点 $A(-2,4)$, 在抛物线上求一点 $P$, 使 $\\triangle APF$ 的周长最小, 求此时点 $P$ 的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -502617,7 +502621,9 @@ "id": "019677", "content": "设函数 $f(x)=\\begin{cases}\\log _2(x+1),& x \\geq 0,\\\\\\sqrt{-x},& x<0,\\end{cases}$则满足 $f(x+1)<2$ 的 $x$ 的取值范围为\\bracket{20}.\n\\fourch{$(-4,3)$}{$(-5,2)$}{$(-3,4)$}{$(-\\infty,-3) \\cup(4,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502719,7 +502725,9 @@ "id": "019682", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_3+S_6=2S_9$, 则数列的公比 $q$ 是\\bracket{20}.\n\\fourch{$-\\dfrac{\\sqrt[3]{3}}{2}$}{$\\dfrac{\\sqrt[3]{3}}{2}$}{$-\\dfrac{\\sqrt[3]{4}}{2}$}{$\\dfrac{\\sqrt[3]{4}}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502824,7 +502832,9 @@ "id": "019687", "content": "已知 $M=\\{x | x-a=0\\}$, $N=\\{x | a x-1=0\\}$, 若 $M \\cap N=N$, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502844,7 +502854,9 @@ "id": "019688", "content": "若 $\\mathrm{C}_{10}^3=\\mathrm{C}_{10}^n$, 则正整数 $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502864,7 +502876,9 @@ "id": "019689", "content": "设数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n=3^n+2$, 则数列 $\\{a_n\\}$ 的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -502924,7 +502938,9 @@ "id": "019692", "content": "已知函数 $f(x)=\\begin{cases}-\\log _2(3-x),& x<2\\\\2^{x-2}-1,& x \\geq 2\\end{cases}$, 若 $f(2-a)=1$, 则 $f(a)$ 等于\\bracket{20}.\n\\fourch{$-2$}{$-1$}{1}{2}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -502964,7 +502980,9 @@ "id": "019694", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503004,7 +503022,9 @@ "id": "019696", "content": "函数 $f(x)=[a x^2-(3 a+1) x+3 a+2] \\cdot \\mathrm{e}^x$ 在 $x=1$ 处取得极小值, 求 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503044,7 +503064,9 @@ "id": "019698", "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 已知 $2 \\sin C=\\tan A(1- 2 \\cos C)$, $c=2 b$, 则 $\\cos B$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503084,7 +503106,9 @@ "id": "019700", "content": "在直角梯形 $ABCD$ 中, $AB=8$, $CD=4, AB \\parallel CD$, $AB \\perp AD$, $E$ 是 $BC$ 的中点, 则 $\\overrightarrow{AB}\\cdot(\\overrightarrow{AC}+\\overrightarrow{AE})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503226,7 +503250,9 @@ "id": "019707", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n \\cdot \\sin \\dfrac{n \\pi}{2}$, 前 $n$ 项和为 $S_n$, 求 $S_{2022}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503288,7 +503314,9 @@ "id": "019710", "content": "直线 $x+2 y-3=0$ 与直线 $a x+4 y+b=0$ 关于点 $A(1,0)$ 对称, 则 $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503308,7 +503336,9 @@ "id": "019711", "content": "等差数列 $\\{a_n\\}$ 中,若 $a_4+a_6+a_8+a_{10}+a_{12}=120$, 则 $a_9-\\dfrac{1}{3}a_{11}$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503348,7 +503378,9 @@ "id": "019713", "content": "已知函数 $f(x)=\\begin{cases}2-x^2,& x \\geq 0,\\\\-x,& x<0,\\end{cases}$ $x_1,x_2 \\in \\mathbf{R}$, $f(x_1)=f(x_2)=m$, 且 $x_1+x_2=0$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503412,7 +503444,9 @@ "id": "019716", "content": "已知函数 $f(x)=\\begin{cases}x^2+4 x,& x \\geq 0,\\\\4 x-x^2,& x<0,\\end{cases}$ 若 $f(2-a^2)>f(a)$, 则实数 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(-\\infty,-1) \\cup(2,+\\infty)$}{$(-1,2)$}{$(-2,1)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503432,7 +503466,9 @@ "id": "019717", "content": "已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 对任意 $x \\in \\mathbf{R}$, 恒有 $f(x)+f(x+2)=0$, 且当 $x \\in (0,1]$ 时 $f(x)=2^x+1$, 则 $f(0)+f(1)+f(2)+\\cdots+f(2021)=$\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503492,7 +503528,9 @@ "id": "019720", "content": "已知数列 $\\{a_n\\}$ 与 $\\{b_n\\}$ 满足 $a_{n+1}-a_n=\\lambda(b_{n+1}-b_n)$ ($\\lambda$ 为非零常数), $n$ 为正整数.\\\\\n(1) 若 $\\{b_n\\}$ 是等差数列, 求证: 数列 $\\{a_n\\}$ 也是等差数列;\\\\\n(2) 若 $a_1=2$, $\\lambda=3$, $b_n=\\sin \\dfrac{n \\pi}{2}$, 求数列 $\\{a_n\\}$ 的前 $2021$ 项和;\\\\\n(3) 设 $a_1=b_1=\\lambda$, $b_2=\\dfrac{\\lambda}{2}$, $b_n=\\dfrac{b_{n-1}+b_{n-2}}{2}$($n \\geq 3$, $n \\in \\mathbf{N}$), 若对 $\\{a_n\\}$ 中的任意两项 $a_i$, $a_j$, $i, j$ 为正整数且 $i \\neq j$, $|a_i-a_j|<2$ 都成立, 求实数 $\\lambda$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503554,7 +503592,9 @@ "id": "019723", "content": "证明下列三角恒等式.\\\\\n(1) $\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=\\dfrac{\\sin \\alpha}{1-\\cos \\alpha}$;\\\\\n(2) $\\dfrac{\\sin ^2 \\alpha-\\sin ^2 \\beta}{\\tan ^2 \\alpha-\\tan ^2 \\beta}=\\cos ^2 \\alpha \\cos ^2 \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503574,7 +503614,9 @@ "id": "019724", "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=1$, $a_2=e$ ($e$ 是自然对数的底数), 且 $a_{n+2}=\\sqrt{a_{n+1}\\cdot a_n}$, 令 $b_n=\\ln a_n, n$ 为正整数.\\\\\n(1) 证明: $b_{n+2}>\\sqrt{b_{n+1}b_n}$;\\\\\n(2) 证明: $\\{\\dfrac{b_{n+2}-b_{n+1}}{b_{n+1}-b_n}\\}$ 是等比数列, 且 $\\{b_n\\}$ 的通项公式是 $b_n=\\dfrac{2}{3}[1-(-\\dfrac{1}{2})^{n-1}]$;\\\\\n(3) 是否存在常数 $t$, 对任意正整数 $n$ 均有 $b_{n+1}\\geq t b_n$ 成立? 若存在, 求 $t$ 的取值范围, 否则, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503594,7 +503636,9 @@ "id": "019725", "content": "证明: $2 \\pi$ 是函数 $f(x)=\\sin x$ 的最小正周期.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503614,7 +503658,9 @@ "id": "019726", "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n+\\sqrt{3}$, 求证: 数列 $\\{a_n\\}$ 中的任意不同的三项不可能构成等比数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503634,7 +503680,9 @@ "id": "019727", "content": "若函数 $f(x)=\\sqrt{x^2-1}+\\sqrt{a-x^2}$ 为偶函数且非奇函数, 则实数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503654,7 +503702,9 @@ "id": "019728", "content": "设 $\\triangle ABC$ 的内角 $A$、$B$、$C$ 所对的边分别为 $a$、$b$、$c$, 若 $b \\cos C+c \\cos B=a \\sin A$, 则 $\\triangle ABC$ 的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -503776,7 +503826,9 @@ "id": "019734", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 对任意正整数 $n, a_{n+1}=3S_n$, 则下列关于 $\\{a_n\\}$ 的论断中正确的是\\bracket{20}.\n\\twoch{一定是等差数列}{一定是等比数列}{可能是等差数列, 但不会是等比数列}{可能是等比数列, 但不会是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -503816,7 +503868,9 @@ "id": "019736", "content": "如果 $a$、$b$ 都是正数, 且 $a \\neq b$, 求证: $\\dfrac{a}{\\sqrt{b}}+\\dfrac{b}{\\sqrt{a}}>\\sqrt{a}+\\sqrt{b}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -503856,7 +503910,9 @@ "id": "019738", "content": "定义在 $\\mathbf{R}$ 上的函数 $f(x)$ 满足: 对于任意的 $x_1, x_2 \\in \\mathbf{R}$, 当 $x_10$ 恒成立, 则正数 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -504163,7 +504225,9 @@ "id": "019753", "content": "已知函数 $f(x)=\\begin{cases}x^2+3 x,& x \\geq 0,\\\\3 x-x^2,& x<0.\\end{cases}$ 若 $f(a^2-3)+f(2 a)>0$, 则实数 $a$ 的取值范围为 3. 已知函数 $f(x)=|\\lg x|$, 若 $0=latex,scale = 1.5]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$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)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (D)--(E)--(C)(A)--(D_1)--(C)(D_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线 $AD_1$ 与 $EC$ 所成角的大小;\\\\\n(2) 《九章算术》中, 将四个面都是直角三角形的四面体称为鳖臑. 试问四面体 $D_1CDE$ 是否为鳖臑? 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -601141,7 +601209,9 @@ "id": "031409", "content": "过双曲线 $\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$ 的右焦点, 且平行于渐近线的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601204,7 +601274,9 @@ "id": "031412", "content": "若复数 $z=(\\sin \\theta-\\dfrac{3}{5})+(\\cos \\theta-\\dfrac{4}{5}) \\mathrm{i}$ 是纯虚数, 则 $\\tan (\\theta-\\dfrac{\\pi}{4})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601470,7 +601542,9 @@ "id": "031425", "content": "已知双曲线的两个焦点为 $F_1(-\\sqrt{5}, 0), F_2(\\sqrt{5}, 0), P$ 是此双曲线上的一点, 且 $PF_1 \\perp PF_2$, $|PF_1| \\cdot|PF_2|=2$, 则该双曲线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601490,7 +601564,9 @@ "id": "031426", "content": "若 $a_n=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$ ($n$ 是正整数 $)$, 则 $a_{n+1}-a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601510,7 +601586,9 @@ "id": "031427", "content": "已知 $\\mathrm{C}_{10}^{2 x}-\\mathrm{C}_{10}^{x+1}=0$, 则 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601650,7 +601728,9 @@ "id": "031434", "content": "已知 $\\{a_n\\}$ 是等比数列, $a_1=2$, $a_3=18$, $\\{b_n\\}$ 是等差数列, $b_1=2$, $b_1+b_2+b_3+b_4=a_1+ a_2+a_3>20$. 则数列 $\\{b_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601730,7 +601810,9 @@ "id": "031438", "content": "在 $\\triangle ABC$ 中, 角 $A, B, C$ 的对边分别为 $a, b, c$, 且 $2 c \\cos ^2 \\dfrac{A}{2}=b+c$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{正三角形}{直角三角形}{等腰三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -601876,7 +601958,9 @@ "id": "031445", "content": "$y=4 x^2$ 的焦点到准线的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -601896,7 +601980,9 @@ "id": "031446", "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1$($a>0$) 的一条渐近线方程为 $x-2 y=0$, 则双曲线的离心率为 $e=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602000,7 +602086,9 @@ "id": "031451", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2+2 n-1$, 则 $a_1+a_3+a_5+\\cdots+a_{25}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602020,7 +602108,9 @@ "id": "031452", "content": "$y=f(x)$ 是关于 $x=3$ 对称的奇函数, $f(1)=1$, $\\cos x-\\sin x=\\dfrac{3 \\sqrt{2}}{5}$,\n则 $f(\\dfrac{15 \\sin 2 x}{\\cos (x+\\dfrac{\\pi}{4})})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602060,7 +602150,9 @@ "id": "031454", "content": "设 $\\mathrm{i}$ 是虚数单位,复数 $\\dfrac{1+a \\mathrm{i}}{2-\\mathrm{i}}$ 为纯虚数,则实数 $a$ 为\\bracket{20}.\n\\fourch{$2$}{$-2$}{$-\\dfrac{1}{2}$}{$\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602100,7 +602192,9 @@ "id": "031456", "content": "设 $S_n$ 为等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 若 $a_1=1$, 公差 $d=2$, $S_{k+2}-S_k=24$, 则 $k$ 等于\\bracket{20}.\n\\fourch{8}{7}{6}{5}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602180,7 +602274,9 @@ "id": "031460", "content": "若 $z_1=a+2 \\mathrm{i}$, $z_2=3-4 \\mathrm{i}$, 且 $\\dfrac{z_1}{z_2}$ 为纯虚数, 则实数 $a$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602222,7 +602318,9 @@ "id": "031462", "content": "已知抛物线 $x^2=3 y$ 上两点 $A, B$ 的横坐标恰是方程 $x^2+5 x+1=0$ 的两个实根, 则直线 $AB$ 的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602242,7 +602340,9 @@ "id": "031463", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点,则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602262,7 +602362,9 @@ "id": "031464", "content": "在各项都为正数的等比数列 $\\{a_n\\}$ 中, 首项 $a_1=3$, 前三项和为 21 , 则 $a_3+a_4+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602326,7 +602428,9 @@ "id": "031467", "content": "设 $0b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602492,7 +602600,9 @@ "id": "031475", "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602572,7 +602682,9 @@ "id": "031479", "content": "若函数 $f(x)=\\log _a(x+\\sqrt{x^2+2 a^2})$ 是奇函数,则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602592,7 +602704,9 @@ "id": "031480", "content": "已知抛物线 $y^2=a x$ 的准线方程是 $x=-3$, 那么抛物线的焦点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602714,7 +602828,9 @@ "id": "031486", "content": "设 $\\alpha$ 为第四象限的角, 若 $\\dfrac{\\sin 3 \\alpha}{\\sin \\alpha}=\\dfrac{13}{5}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602734,7 +602850,9 @@ "id": "031487", "content": "设 $\\{a_n\\}$ 是正项数列, 其前 $n$ 项和 $S_n$ 满足 : $4S_n=(a_n-1)(a_n+3)$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602754,7 +602872,9 @@ "id": "031488", "content": "圆 $x^2+y^2=1$ 与直线 $y=k x+2$ 有两个公共点的充要条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602814,7 +602934,9 @@ "id": "031491", "content": "若函数 $f(x)=\\log _a(x^3-a x)$($a>0$, $a \\neq 1$) 在区间 $(-\\dfrac{1}{2}, 0)$ 是上严格增函数, 则 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$[\\dfrac{1}{4}, 1)$}{$[\\dfrac{3}{4}, 1)$}{($\\dfrac{9}{4},+\\infty$)}{$(1, \\dfrac{9}{4})$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -602874,7 +602996,9 @@ "id": "031494", "content": "已知关于 $x$ 的不等式 $\\dfrac{(a+1) x-3}{x-1}<1$.\\\\\n(1) 当 $a=1$ 时,求该不等式的解集;\\\\\n(2) 当 $a>0$ 时,求该不等式的解集.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -602914,7 +603038,9 @@ "id": "031496", "content": "如果双曲线 $\\dfrac{x^2}{4}-\\dfrac{y^2}{2}=1$ 上一点 $P$ 到双曲线右焦点的距离是 $2$, 那么点 $P$ 到 $y$ 轴的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602934,7 +603060,9 @@ "id": "031497", "content": "若复数 $(1+b \\mathrm{i})(2+\\mathrm{i})$ 是纯虚数 ($\\mathrm{i}$ 是虚数单位, $b$ 是实数), 则 $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -602974,7 +603102,9 @@ "id": "031499", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=n^2-9 n$, 第 $k$ 项满足 $5\\sqrt{2}$) 的两条渐近线的夹角为 $\\dfrac{\\pi}{3}$, 则双曲线的离心率为 \\bracket{20}.\n\\fourch{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\dfrac{2 \\sqrt{6}}{3}$}{$\\sqrt{3}$}{2}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603276,7 +603416,9 @@ "id": "031514", "content": "已知集合 $M=\\{x|| x |<2\\}$, $N=\\{x | \\dfrac{x+1}{x-3}<0\\}$, 则集合 $M \\cap N$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603296,7 +603438,9 @@ "id": "031515", "content": "复数 $z=a+b \\mathrm{i}$, $a, b \\in \\mathbf{R}$, 且 $b \\neq 0$, 若 $z^2-4 b z$ 是实数, 则 $a, b$ 满足的条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603316,7 +603460,9 @@ "id": "031516", "content": "在 $(x^2-\\dfrac{1}{x})^8$ 的展开式中, 含 $x$ 的项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603336,7 +603482,9 @@ "id": "031517", "content": "在等比数列 $\\{a_n\\}$ 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 10 项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603356,7 +603504,9 @@ "id": "031518", "content": "若圆 $x^2+y^2-2 x-4 y=0$ 的圆心到直线 $x-y+a=0$ 的距离为 $\\dfrac{\\sqrt{2}}{2}$, 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603376,7 +603526,9 @@ "id": "031519", "content": "经过圆 $x^2+2 x+y^2=0$ 的圆心 $C$, 且与直线 $x+y=0$ 垂直的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603396,7 +603548,9 @@ "id": "031520", "content": "已知向量 $\\overrightarrow{a}=(1,2)$ 和 $\\overrightarrow{b}=(x, 1)$, 若向量 $\\overrightarrow{a}+2 \\overrightarrow{b}$ 与 $2 \\overrightarrow{a}-\\overrightarrow{b}$ 平行, 则实数 $x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603416,7 +603570,9 @@ "id": "031521", "content": "若 $\\dfrac{\\cos 2 \\alpha}{\\sin (\\alpha-\\dfrac{\\pi}{4})}=-\\dfrac{\\sqrt{2}}{2}$, 则 $\\cos \\alpha+\\sin \\alpha$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603456,7 +603612,9 @@ "id": "031523", "content": "如果函数 $f(x)=\\sin (\\pi x+\\theta)$($0<\\theta<2 \\pi$) 的最小正周期是 $T$, 且当 $x=2$ 时取得最大值,则 $T=$\\blank{50}, $\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603518,7 +603676,9 @@ "id": "031526", "content": "平面 $\\alpha \\parallel $ 平面 $\\beta$ 的一个充分条件是\\bracket{20}.\n\\onech{存在一条直线 $a$, $a \\parallel \\alpha$, $a \\parallel \\beta$}{存在一条直线 $a$, $a \\subset \\alpha$, $a \\parallel \\beta$}{存在两条平行直线 $a$, $b$, $a \\subset \\alpha$, $b \\subset \\beta$, $a \\parallel \\beta$, $b \\parallel \\alpha$}{存在两条异面直线 $a$, $b$, $a \\subset \\alpha$, $b \\subset \\beta$, $a \\parallel \\beta$, $b \\parallel \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603538,7 +603698,9 @@ "id": "031527", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $O$ 是底面正方形 $ABCD$ 的中心, $M$ 是 $DD_1$ 的中点, $N$ 是 $A_1B_1$ 上的动点, 则直线 $NO$、$AM$ 的位置关系是\\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 ($(A)!0.5!(C)$) node [left] {$O$} coordinate (O) circle (0.03);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$M$} coordinate (M);\n\\draw ($(A_1)!0.9!(B_1)$) node [above] {$N$} coordinate (N);\n\\draw [dashed] (A)--(M)(N)--(O);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{平行}{相交}{异面垂直}{异面不垂直}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603600,7 +603762,9 @@ "id": "031530", "content": "已知函数 $y=\\sqrt{\\dfrac{1+x}{1-x}}+\\lg (3-4 x+x^2)$ 的定义域为 $M$.\\\\\n(1) 求 $M$;\\\\\n(2) 当 $x \\in M$ 时, 求 $f(x)=a \\cdot 2^{x+2}+3 \\times 4^x$($a>-3$) 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -603662,7 +603826,9 @@ "id": "031533", "content": "已知集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | \\log _x 4=2\\}$, 则 $A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603702,7 +603868,9 @@ "id": "031535", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603722,7 +603890,9 @@ "id": "031536", "content": "若函数 $y=x^3-2 m x^2+m^2 x$ 在 $x=1$ 处取得极小值, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603742,7 +603912,9 @@ "id": "031537", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603762,7 +603934,9 @@ "id": "031538", "content": "已知 $\\tan \\alpha=2$, 则 $\\dfrac{\\sin (\\pi+\\alpha)-\\sin (\\dfrac{\\pi}{2}+\\alpha)}{\\cos (\\dfrac{3 \\pi}{2}+\\alpha)+\\cos (\\pi-\\alpha)}$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603802,7 +603976,9 @@ "id": "031540", "content": "在 $\\triangle ABC$ 中, 已知内角 $A=\\dfrac{\\pi}{3}$, 边 $BC=2 \\sqrt{3}$, 则 $\\triangle ABC$ 的面积 $S$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -603882,7 +604058,9 @@ "id": "031544", "content": "设 $a, b$ 是平面 $\\alpha$ 内两条不同的直线, $l$ 是平面 $\\alpha$ 外的一条直线,则``$l \\perp a$, $l \\perp b$''是``$l \\perp \\alpha$''的\\bracket{20}.\n\\twoch{充要条件}{充分而不必要的条件}{必要而不充分的条件}{既不充分也不必要的条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603902,7 +604080,9 @@ "id": "031545", "content": "函数 $f(x)=\\ln (x+1)-\\dfrac{2}{x}$ 的零点所在的区间是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{2}, 1)$}{$(1, \\mathrm{e}-1)$}{$(\\mathrm{e}-1,2)$}{$(2, \\mathrm{e})$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603922,7 +604102,9 @@ "id": "031546", "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>b>0$) 的右焦点为 $F(c, 0)$, 直线 $y=k(x-c)$ 与双曲线的右支有两个交点, 则\\bracket{20}.\n\\fourch{$|k|>\\dfrac{b}{a}$}{$|k|<\\dfrac{b}{a}$}{$|k|>\\dfrac{c}{a}$}{$|k|<\\dfrac{c}{a}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603942,7 +604124,9 @@ "id": "031547", "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的严格增函数, 函数 $y=f(x-1)$ 的图像关于点 $(1,0)$ 对称. 若对任意的 $x, y \\in \\mathbf{R}$, $f(x^2-6 x+21)+f(y^2-8 y)<0$ 恒成立, 则当 $x>3$ 时, $x^2+y^2$ 的取值范围是\\bracket{20}.\n\\fourch{$(3,7)$}{$(9,25)$}{$(13,49)$}{$(9,49)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -603982,7 +604166,9 @@ "id": "031549", "content": "如图所示, 椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的一个焦点为 $F(1,0)$, 且过点 $(\\sqrt{2}, \\dfrac{\\sqrt{6}}{2})$.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (4,-2.5) -- (4,2.5);\n\\draw (4,0) node [above right] {$N$} coordinate (N);\n\\draw (80:2 and {sqrt(3)}) node [above right] {$A$} coordinate (A);\n\\draw (-80:2 and {sqrt(3)}) node [below right] {$B$} coordinate (B);\n\\draw (A)--(N) ($(A)!-0.2!(B)$) -- ($(B)!-0.2!(A)$);\n\\draw [name path = BN] (B)--(N);\n\\draw [name intersections = {of = BN and elli, by = {T,M}}];\n\\draw (A)--(M) node [below] {$M$};\n\\filldraw (1,0) node [above right] {$F$} coordinate (F) circle (0.05);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆 $C$ 的方程;\\\\\n(2) 已知 $A, B$ 为椭圆上的点, 且直线 $AB$ 垂直于 $x$ 轴, 直线 $l$ : $x=4$ 与 $x$ 轴交于点 $N$, 直线 $AF$ 与 $BN$ 交于点 $M$.\\\\\n(I) 求证: 点 $M$ 恒在椭圆 $C$ 上;\\\\\n(II) 求 $\\triangle AMN$ 面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604002,7 +604188,9 @@ "id": "031550", "content": "已知集合 $A=\\{x|| x |<2\\}$, $B=\\{x | \\dfrac{1}{x+1}>0\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604106,7 +604294,9 @@ "id": "031555", "content": "若 $\\dfrac{\\sin \\alpha+\\cos \\alpha}{\\sin \\alpha-\\cos \\alpha}=\\dfrac{1}{2}$, 则 $\\tan 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604146,7 +604336,9 @@ "id": "031557", "content": "若椭圆 $\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$ 上一点 $P$ 到焦点 $F_1$ 的距离为 6 , 则点 $P$ 到另一个焦点 $F_2$ 的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604230,7 +604422,9 @@ "id": "031561", "content": "正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$, $E, F$ 分别为线段 $AA_1, B_1C$ 上的点, 则三棱锥 $D_1-EDF$ 的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604312,7 +604506,9 @@ "id": "031565", "content": "设 $F_1F_2$ 是椭圆 $E: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点, $P$ 为直线 $x=\\dfrac{3 a}{2}$ 上一点, $\\triangle F_2PF_1$ 是底角为 $30^{\\circ}$ 的等腰三角形, 则 $E$ 的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}{$\\dfrac{3}{4}$}{$\\dfrac{4}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -604332,7 +604528,9 @@ "id": "031566", "content": "如图, 在棱长为 $a$ 的正方体 $ABCD-A_1B_1C_1D_1$ 中, $O$ 是 $AC, BD$ 的交点, $E, F$ 分别是 $AB$ 与 $AD$ 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 [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 ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw (A_1)--(C_1);\n\\draw [dashed] (E)--(F)(B)--(D)(A)--(C)(D_1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线 $OD_1$ 与直线 $A_1C_1$ 垂直;\\\\\n(2) 求异面直线 $EF$ 与 $A_1C_1$ 所成角的大小;\\\\\n(3) 求二面角 $B-AC-D_1$ 的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -604374,7 +604572,9 @@ "id": "031568", "content": "已知 $\\{a_n\\}$ 为等比数列, $a_4+a_7=2$, $a_5 a_6=-8$, 则 $a_1+a_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604416,7 +604616,9 @@ "id": "031570", "content": "已知函数 $f(x)=(a^2-1) x^2+2 x+a+1$ 是奇函数, 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604518,7 +604720,9 @@ "id": "031575", "content": "若正四棱锥的底面对角线的长为 $2 \\sqrt{6}$, 体积为 $4 \\sqrt{3}$, 则侧面与底面所成的二面角等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604770,7 +604974,9 @@ "id": "031587", "content": "已知函数 $f(x)=\\lg \\dfrac{1-x}{1+x}$, 若 $f(a)=b$, 则 $f(-a)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604790,7 +604996,9 @@ "id": "031588", "content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, 对角线 $AC_1$ 的长为 $l$, $\\angle DAC_1=45^{\\circ}$, $\\angle A_1AC_1=60^{\\circ}$, 则三棱锥 $C-B_1C_1D_1$ 的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604830,7 +605038,9 @@ "id": "031590", "content": "已知定义在 $\\mathbf{R}$ 上的奇函数, $f(x)$ 满足 $f(x+2)=-f(x)$, 则 $f(-6)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604850,7 +605060,9 @@ "id": "031591", "content": "椭圆 $\\dfrac{x^2}{4}+y^2=1$ 的两个焦点为 $F_1, F_2$, 过 $F_1$ 作垂直于 $x$ 轴的直线与椭圆相交, 一个交点为 $P$, 则 $|\\overrightarrow{PF_2}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604934,7 +605146,9 @@ "id": "031595", "content": "设直线 $l$ 过点 $(-2,0)$, 且与圆 $x^2+y^2=1$ 相切, 则 $l$ 的斜率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -604974,7 +605188,9 @@ "id": "031597", "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的两个焦点为 $F_1, F_2$, 若 $P$ 为其上一点, 且 $|PF_1|= 2|PF_2|$, 则双曲线离心率的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605014,7 +605230,9 @@ "id": "031599", "content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, $M, N$ 分别为棱 $A_1B_1$ 和 $BB_1$ 的中点, 那么异面直线 $AM$ 和 $CN$ 所成角的余弦值是\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{10}}{2}$}{$\\dfrac{2}{5}$}{$-\\dfrac{2}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605114,7 +605332,9 @@ "id": "031604", "content": "已知 $\\mathrm{i}$ 是虚数单位, $\\dfrac{2-\\mathrm{i}}{z+\\mathrm{i}}=\\mathrm{i}$, 则 $|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605178,7 +605398,9 @@ "id": "031607", "content": "抛物线 $y^2=2 p x$($p>0$) 的准线经过双曲线 $x^2-y^2=1$ 的左焦点, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605218,7 +605440,9 @@ "id": "031609", "content": "设等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $S_{m-1}=5$, $S_m=-11$, $S_{m+1}=21$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605386,7 +605610,9 @@ "id": "031617", "content": "若 $f(x)=-\\dfrac{1}{2}x^2+b \\ln (x+2)$ 在 ($-1,+\\infty$) 上是减函数, 则 $b$ 的取值范围是\\bracket{20}.\n\\fourch{$[-1,+\\infty)$}{($-1,+\\infty$)}{$(-\\infty,-1]$}{($-\\infty,-1$)}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -605488,7 +605714,9 @@ "id": "031622", "content": "设集合 $A=\\{x |-2 \\leq x \\leq 2\\}$, $\\mathbf{Z}$ 为整数集, 则集合 $A \\cap \\mathbf{Z}$ 中含有 $2$ 的子集个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605597,7 +605825,9 @@ "id": "031627", "content": "已知直线 $y=k x$ 与圆 $(x-5)^2+y^2=9$ 有两个公共点, 则实数 $k$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605617,7 +605847,9 @@ "id": "031628", "content": "若 $\\tan \\theta=\\dfrac{1}{3}$, 则 $\\cos 2 \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605702,7 +605934,9 @@ "id": "031632", "content": "在封闭的直三棱柱 $ABC-A_1B_1C_1$ 内有一个体积为 $V$ 的球. 若 $AB \\perp BC$, $AB=6$, $BC= 8$, $AA_1=3$, 则 $V$ 的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605870,7 +606104,9 @@ "id": "031640", "content": "已知 $\\tan (\\alpha-\\dfrac{5 \\pi}{4})=\\dfrac{1}{5}$, 则 $\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605913,7 +606149,9 @@ "id": "031642", "content": "若双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{3}=1$($a>0$) 的一条渐近线被圆 $(x-2)^2+y^2=4$ 所截得的弦长为 $2$ , 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605933,7 +606171,9 @@ "id": "031643", "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 为棱 $CC_1$ 的中点, 则异面直线 $AE$ 与 $CD$ 所成角的正切值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -605975,7 +606215,9 @@ "id": "031645", "content": "双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 的离心率为 $\\sqrt{3}$, 则其渐近线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606015,7 +606257,9 @@ "id": "031647", "content": "在直角坐标系 $xOy$ 中, 曲线 $C$ 的参数方程为 $\\begin{cases}x=2 \\cos \\theta,\\\\y=4 \\sin \\theta\\end{cases}$ ($\\theta$ 为参数), 直线 $l$ 的参数方程为 $\\begin{cases}x=1+t \\cos \\alpha,\\\\y=2+t \\sin \\alpha\\end{cases}$ ($t$ 为参数). 若曲线 $C$ 截直线 $l$ 所得线段的中点坐标为 $(1,2)$, 则直线 $l$ 的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606117,7 +606361,9 @@ "id": "031652", "content": "在等差数列 $\\{a_n\\}$ 中, 若 $a_3+a_8+a_{13}=C$, 则其前 $n$ 项和 $S_n$ 的值等于 $5C$ 的是\\bracket{20}.\n\\fourch{$S_{15}$}{$S_{17}$}{$S_7$}{$S_8$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -606263,7 +606509,9 @@ "id": "031659", "content": "函数 $y=\\sin ^4 x-\\sin ^2 x$ 的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606283,7 +606531,9 @@ "id": "031660", "content": "从甲、乙等 $5$ 名学生中随机选出 $2$ 人, 则甲被选中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606325,7 +606575,9 @@ "id": "031662", "content": "已知等差数列 $\\{a_n\\}$ 中, $a_4+a_8=16$, $a_2=1$, 则 $a_{10}$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606345,7 +606597,9 @@ "id": "031663", "content": "若不等式 $|x|+|x-1|>m$ 的解集是 $\\mathbf{R}$, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606387,7 +606641,9 @@ "id": "031665", "content": "已知 $\\{a_n\\}$ 是等差数列, $\\{b_n\\}$ 是等比数列, 且 $b_2=3$, $b_3=9$, $a_1=b_1$, $a_{14}=b_4$. 设 $c_n=a_n+ b_n$, 则数列 $\\{c_n\\}$ 的前 $n$ 项和 $S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606507,7 +606763,9 @@ "id": "031671", "content": "函数 $f(x)=\\begin{cases}\\sin (\\pi x^2),&-1=latex]\n\\draw (0,0,0) node [above right] {$C$} coordinate (C);\n\\draw (-1,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\filldraw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\draw (P)--(D)--(A)--(B)--cycle(P)--(A);\n\\draw [dashed] (P)--(C)--(D)(C)--(A)(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $DC \\perp$ 平面 $PAC$;\\\\\n(2) 求证: 平面 $PAB \\perp$ 平面 $PAC$;\\\\\n(3) 设点 $E$ 为 $AB$ 的中点, 在棱 $PB$ 上是否存在点 $F$, 使得 $PA \\parallel $ 平面 $CEF$ ? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -606587,7 +606849,9 @@ "id": "031675", "content": "已知椭圆 $C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$ 过点 $A(2,0), B(0,1)$ 两点.\\\\\n(1) 求椭圆 $C$ 的方程及离心率;\\\\\n(2) 设 $P$ 为第三象限内一点且在椭圆 $C$ 上, 直线 $PA$ 与 $y$ 轴交于点 $M$, 直线 $PB$ 与 $x$ 轴交于点 $N$. 求证: 四边形 $ABNM$ 的面积为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -606647,7 +606911,9 @@ "id": "031678", "content": "若 $(a x-1)^5$ 的展开式中 $x^3$ 的系数是 $80$ , 则实数 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606667,7 +606933,9 @@ "id": "031679", "content": "曲线 $y^2=4 x$ 关于直线 $x=2$ 对称的曲线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606687,7 +606955,9 @@ "id": "031680", "content": "若函数 $y=(\\dfrac{1}{2})^{|1-x|}+m$ 的图像与 $x$ 轴有公共点, 则实数 $m$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606727,7 +606997,9 @@ "id": "031682", "content": "已知直线 $\\sqrt{3}x+y=0$ 和直线 $k x-y-1=0$, 若两直线的夹角为 $60^{\\circ}$, 则 $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606747,7 +607019,9 @@ "id": "031683", "content": "设函数 $f(x)=\\begin{cases}x^3-3 x,& x \\leq a,\\\\-2 x,& x>a,\\end{cases}$ 若 $f(x)$ 无最大值, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606789,7 +607063,9 @@ "id": "031685", "content": "已知 $\\sin 2 \\alpha=-\\dfrac{1}{3}$, 则 $\\dfrac{4 \\cos ^2 \\alpha}{\\cot \\dfrac{\\alpha}{2}-\\tan \\dfrac{\\alpha}{2}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -606851,7 +607127,9 @@ "id": "031688", "content": "在复平面内, 复数 $\\dfrac{1+\\mathrm{i}}{(1-\\mathrm{i})^2}$ 对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -607017,7 +607295,9 @@ "id": "031696", "content": "设数列 $\\{a_n\\}$ 是等差数列, 且 $a_2=-6$, $a_8=6, S_n$ 是数列 $\\{a_n\\}$ 的前 $n$ 项和, 则 $S_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607037,7 +607317,9 @@ "id": "031697", "content": "$(x-\\sqrt{2}y)^{10}$ 的展开式中 $x^6 y^4$ 项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607099,7 +607381,9 @@ "id": "031700", "content": "若过点 $A(4,0)$ 的直线 $l$ 与曲线 $(x-2)^2+y^2=1$ 有公共点, 则直线 $l$ 的斜率的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607179,7 +607463,9 @@ "id": "031704", "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$ 满足 $S_n=2 a_n+(-1)^n$, $n \\geq 1$. 则数列 $\\{a_n\\}$ 的前三项的和 $a_1+a_2+a_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607221,7 +607507,9 @@ "id": "031706", "content": "函数 $y=\\sin (2 x+\\dfrac{\\pi}{3})$ 图像的对称轴方程可能是\\bracket{20}.\n\\fourch{$x=-\\dfrac{\\pi}{6}$}{$x=-\\dfrac{\\pi}{12}$}{$x=\\dfrac{\\pi}{6}$}{$x=\\dfrac{\\pi}{12}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -607321,7 +607609,9 @@ "id": "031711", "content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=AD=2$, $E$ 是棱 $CD$ 上的一点.\\\\\n(1) 求证: $AD_1 \\perp$ 平面 $A_1B_1D$;\\\\\n(2) 求异面直线 $B_1E, AD_1$ 所成的角;\\\\\n(3) 若 $E$ 是棱 $CD$ 的中点, 在棱 $AA_1$ 上是否存在点 $P$, 使得 $DP \\parallel $ 平面 $B_1AE$ ? 若存在, 求出线段 $AP$ 的长; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -607341,7 +607631,9 @@ "id": "031712", "content": "不等式 $(|x|+1)(2 x-1)>0$ 的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607361,7 +607653,9 @@ "id": "031713", "content": "已知圆 $x^2+y^2-6 x-7=0$ 与抛物线 $y^2=2 p x$($p>0$) 的准线相切, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607381,7 +607675,9 @@ "id": "031714", "content": "若集合 $M=\\{(x, y) | y=2^{-x}\\}$, $P=\\{y | y=\\sqrt{x-1}\\}$, 则 $M \\cap P=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607441,7 +607737,9 @@ "id": "031717", "content": "在 $\\triangle ABC$ 中, $\\angle A=60^{\\circ}$, $b=1$, 这个三角形的面积为 $\\sqrt{3}$, 则 $\\triangle ABC$ 外接圆的直径是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607461,7 +607759,9 @@ "id": "031718", "content": "$(x^2-\\dfrac{1}{2 x})^9$ 展开式中 $x^9$ 的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607501,7 +607801,9 @@ "id": "031720", "content": "抛物线 $y^2=2 p x$ 与直线 $a x+y-4=0$ 的一个交点是 $(1,2)$, 则抛物线的焦点到直线的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -607561,7 +607863,9 @@ "id": "031723", "content": "已知 $f(x+1)=-f(x)$, 且 $f(x)=\\begin{cases}1,& -10,\\\\f(x+1)+1,& x \\leq 0,\\end{cases}$ 则 $f(\\dfrac{4}{3})+f(-\\dfrac{4}{3})$ 的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608447,7 +608777,9 @@ "id": "031766", "content": "已知 $z=2+\\dfrac{1}{\\mathrm{i}}$, 则 $\\overline{z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608467,7 +608799,9 @@ "id": "031767", "content": "已知 $A=\\{-1,1, m\\}$, 集合 $B=\\{1,2\\}$, 若 $B \\subset A$, 则实数 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608509,7 +608843,9 @@ "id": "031769", "content": "已知球 $O$ 的体积为 $36 \\pi$, 则该球的球面面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608551,7 +608887,9 @@ "id": "031771", "content": "已知两条直线 $l_1: a x+3 y-3=0$, $l_2: 2 x+6 y+1=0$. 若 $l_1 \\parallel l_2$, 则直线 $l_1$ 与 $l_2$ 之间的距离 $d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608715,7 +609053,9 @@ "id": "031779", "content": "已知双曲线 $x^2-\\dfrac{y^2}{m}=1$ 的渐近线方程为 $y= \\pm \\sqrt{3}x$, 则该双曲线的离心率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{2}{$\\dfrac{\\sqrt{10}}{3}$}{$\\dfrac{3 \\sqrt{10}}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608735,7 +609075,9 @@ "id": "031780", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=a$, 任作平面 $\\alpha$ 与对角线 $AC_1$ 垂直, 使得 $\\alpha$ 与正方体的每个面都有公共点, 记这样得到的截面多边形的面积为 $S$, 周长为 $l$, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(235:0.5cm)}]\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 [dashed] (A)--(C_1);\n\\draw (A_1)--(B) (D_1)--(B_1)--(C);\n\\draw [dashed] (A_1)--(D)--(B)(D_1)--(C);\n\\def\\lambda{0.4}\n\\draw ($(A_1)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(B_1)$) -- ($(B)!\\lambda!(C)$);\n\\draw [dashed] ($(B)!\\lambda!(C)$) -- ($(D)!\\lambda!(C)$) -- ($(D)!\\lambda!(D_1)$) -- ($(A_1)!\\lambda!(D_1)$);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$S$ 是定值, $l$ 不是定值}{$S$ 不是定值, $l$ 是定值}{$S$ 和 $l$ 都是定值}{$S$ 和 $l$ 都不是定值}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -608777,7 +609119,9 @@ "id": "031782", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=BC=BB_1=4, M, N$ 分别为 $A_1B_1, AC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (1.9,0,{sqrt(0.39)}) node [right] {$A$} coordinate (A);\n\\draw ({-sqrt(0.39)},0,1.9) node [left] {$C$} coordinate (C);\n\\draw (A) ++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw (C)--(A)--(A_1)--(C_1)--cycle(A_1)--(B_1)--(C_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M);\n\\draw [dashed] (B)--(N)--(M)--cycle(C)--(B)--(A)(B)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 平面 $BCC_1B_1$;\\\\\n(2) 若 $AB \\perp MN$, 求异面直线 $MN$ 与 $A_1C_1$ 所成的角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -608837,7 +609181,9 @@ "id": "031785", "content": "已知函数 $f(x)=2^x$.\\\\\n(1) 求函数 $y=3 f(x)-f(-x)-2$ 的零点;\\\\\n(2) 证明: 当 $a \\leq 16$ 时, 函数 $F(x)=f(2 x)+a \\cdot f(-x)$ 是 ($1,+\\infty$) 上的严格递增函数;\\\\\n(3) 设 $g(x)=\\dfrac{1}{1+a \\cdot f(x)}-\\dfrac{1}{1+a \\cdot f(x-1)}$, 若对任意 $x \\in(-\\infty, 0]$, $g(x) \\geq g(0)$ 恒成立, 求正实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -608877,7 +609223,9 @@ "id": "031787", "content": "已知 $z=-1+2 \\mathrm{i}$, 则 $|\\overline{z}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -608981,7 +609329,9 @@ "id": "031792", "content": "已知二项式 $(x-\\dfrac{a}{x})^5$ 的展开式中 $x^3$ 的系数为 10 , 则 $a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609023,7 +609373,9 @@ "id": "031794", "content": "设 $\\{a_n\\}$ 是公比为 $q$ 的等比数列, $S_n$ 是它的前 $n$ 项和, 若 $\\{S_n\\}$ 是等差数列, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609043,7 +609395,9 @@ "id": "031795", "content": "设 $\\triangle ABC$ 的内角 $A, B, C$ 的对边分别为 $a, b, c$, 且满足 $a \\cos B-b \\cos A=\\dfrac{3}{5}c$, 则 $\\dfrac{\\tan A}{\\tan B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609083,7 +609437,9 @@ "id": "031797", "content": "若实数 $a, b$ 满足 $a-4 \\sqrt{b}=2 \\sqrt{a-b}$, 且 $a, b$ 不同时为零, 则 $a$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609223,7 +609579,9 @@ "id": "031804", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, 点 $D, E$ 分别为 $AC$ 和 $B_1C_1$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$D$} coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B_1)!0.5!(C_1)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B)--(E);\n\\draw [dashed] (A)--(C)(A)--(E)--(D)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\parallel $ 平面 $ABB_1A_1$;\\\\\n(2) 若 $AB \\perp BC$, $AB=BC=AA_1=2$, 求点 $D$ 到平面 $ABE$ 的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609305,7 +609663,9 @@ "id": "031808", "content": "已知集合 $A=\\{x | x^2-x<0\\}$, $B=(0, a)$($a>0$), 若 $A \\subseteq B$, 则实数 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609325,7 +609685,9 @@ "id": "031809", "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 若当 $x \\geq 0$ 时, $f(x)=\\log _3(3+x)$, 则 $f(-6)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609345,7 +609707,9 @@ "id": "031810", "content": "已知双曲线 $y^2-\\dfrac{x^2}{m^2}=1$($m>0$) 的一条渐近线方程为 $x+\\sqrt{3}y=0$, 则 $m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609385,7 +609749,9 @@ "id": "031812", "content": "已知等比数列 $\\{a_n\\}$ ($n$ 是正整数) 满足 $a_2 a_6=4(a_4-1)$, 则 $a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609425,7 +609791,9 @@ "id": "031814", "content": "已知 $\\alpha, \\beta$ 为锐角, 且 $\\cos (\\alpha+\\beta)=\\dfrac{\\sin \\alpha}{\\sin \\beta}$, 则 $\\tan \\alpha$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609445,7 +609813,9 @@ "id": "031815", "content": "已知关于 $x$ 的一元二次不等式 $a x^2+2 x+b>0$ 的解集为 $\\{x | x \\neq c\\}$, 则 $\\dfrac{a^2+b^2+7}{a+c}$(其中 $a+c \\neq 0$) 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609527,7 +609897,9 @@ "id": "031819", "content": "已知椭圆 $\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$ 的左右顶点分别为 $A, B$, 过点 $C(0,1)$ 斜率为 $k$($k>1$) 的直线 $l$ 与椭圆交于 $M, N$, 记直线 $AM, BN$ 的斜率为 $k_1, k_2$, 且 $k_1=2 k_2$, 则 $k=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [name path = elli] (0,0) ellipse (2 and {sqrt(3)});\n\\draw (-2,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\filldraw (-1,0) node [below] {$F_1$} coordinate (F_1) circle (0.03);\n\\filldraw (1,0) node [below] {$F_2$} coordinate (F_2) circle (0.03);\n\\filldraw (0,1) node [right] {$C$} coordinate (C) circle (0.03);\n\\draw [name path = MN] (C) ++ (0.8,1.2) --++ (-2.6,-3.9);\n\\draw [name intersections = {of = MN and elli, by = {M,N}}];\n\\draw (M)--(A)(N)--(B);\n\\draw (M) node [above] {$M$} (N) node [below] {$N$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609547,7 +609919,9 @@ "id": "031820", "content": "下列函数中, 既不是奇函数, 也不是偶函数的是\\bracket{20}.\n\\fourch{$y=x+\\mathrm{e}^x$}{$y=x+\\dfrac{1}{x}$}{$y=2^x+\\dfrac{1}{2^x}$}{$y=\\sqrt{1+x^2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609567,7 +609941,9 @@ "id": "031821", "content": "已知 $A$ 为 $\\triangle ABC$ 的一个内角, 且 $\\sin A+\\cos A=\\dfrac{\\sqrt{2}}{3}$, 则 $\\triangle ABC$ 的形状是\\bracket{20}.\n\\fourch{锐角三角形}{钝角三角形}{直角三角形}{不确定}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609587,7 +609963,9 @@ "id": "031822", "content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 是棱 $CC_1$ 的中点, $F$ 是侧面 $B_1BCC_1$ 上的动点, 并且 $A_1F \\parallel $ 平面 $AED_1$, 则动点 $F$ 的轨迹是\\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 [above] {$B_1$} 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 ($(C)!0.5!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D_1)--cycle;\n\\filldraw (2,1.3,-0.3) node [right] {$F$} coordinate (F) circle (0.03);\n\\draw [dashed] (A_1)--(F);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{圆}{椭圆}{抛物线}{线段}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -609627,7 +610005,9 @@ "id": "031824", "content": "已知数列 $\\{a_n\\}$ 是等差数列, 且 $a_1=2$, $a_1+a_2+a_3=12$.\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 令 $b_n=2^{a_n}+9$, 数列 $\\{b_n\\}$ 前 $n$ 项和为 $S_n$, 若 $S_n \\geq 2022$, 求正整数 $n$ 的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609647,7 +610027,9 @@ "id": "031825", "content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $E, P$ 分别是 $BC, A_1D_1$ 的中点, $M, N$ 分别是 $AE, CD_1$ 的中点, $AD=AA_1=a$, $AB=2 a$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{4}\n\\def\\m{2}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,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 ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(D_1)$) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(E)$) node [above left] {$M$} coordinate (M);\n\\draw [dashed] (A)--(E)--(P)--cycle(M)--(N)(C)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN \\parallel $ 面 $ADD_1A_1$;\\\\\n(2) 求三棱锥 $P-DEN$ 的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609687,7 +610069,9 @@ "id": "031827", "content": "已知函数 $f(x)=x^4+a x^3+2 x^2+b$($x \\in \\mathbf{R}$), 其中 $a, b \\in \\mathbf{R}$.\\\\\n(1) 当 $a=-\\dfrac{10}{3}$ 时, 讨论函数 $f(x)$ 的单调性;\\\\\n(2) 若函数 $f(x)$ 仅在 $x=0$ 处有极值, 求 $a$ 的取值范围;\\\\\n(3) 若对于任意的 $a \\in[-2,2]$, 不等式 $f(x) \\leq 1$ 在 $[-1,1]$ 上恒成立,求 $b$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -609816,7 +610200,9 @@ "id": "031833", "content": "已知向量 $\\overrightarrow{a}=(2,-3)$, $\\overrightarrow{b}=(3, \\lambda)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $\\lambda$ 等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609858,7 +610244,9 @@ "id": "031835", "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_{2022}=a_{20}+a_{22}=2$, 则 $a_1$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -609982,7 +610370,9 @@ "id": "031841", "content": "已知 $a\\dfrac{1}{b}$}{$a^2>b^2$}{$2-a>2-b$}{$2^a>2^b$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610086,7 +610476,9 @@ "id": "031846", "content": "已知函数 $f(x)=a x+\\dfrac{1}{x-1}$, $a \\in \\mathbf{R}$.\\\\\n(1) 当 $a=2$ 时,求不等式 $f(x+2) \\leq f(x)+4$ 的解集;\\\\\n(2) 若函数 $f(x)$ 在区间 $[2,5]$ 上严格减, 求实数 $a$ 的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -610210,7 +610602,9 @@ "id": "031852", "content": "集合 $A=\\{x|| x-2 | \\leq 3, x \\in \\mathbf{R}\\}$, $B=\\{y | y=-x^2,-1 \\leq x \\leq 2\\}$, 则 $\\overline{A \\cap B}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610252,7 +610646,9 @@ "id": "031854", "content": "已知圆 $x^2+y^2-6 x-7=0$ 与抛物线 $y^2=2 p x$($p>0$) 的准线相切, 则 $p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610292,7 +610688,9 @@ "id": "031856", "content": "在正四棱柱 $ABCD-A_1B_1C_1D_1$ 中, $AA_1=3$, 直线 $AC_1$ 与平面 $BCC_1B_1$ 所成角大小为 $30^{\\circ}$, 则该正四棱柱的外接球表面积大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610312,7 +610710,9 @@ "id": "031857", "content": "等比数列 $\\{a_n\\}$ 的各项均为正数, 且 $a_{10}a_{11}+a_9 a_{12}=2 \\mathrm{e}^5$, 则 $\\ln a_1+\\ln a_2+\\cdots+\\ln a_{20}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610372,7 +610772,9 @@ "id": "031860", "content": "已知数列 $\\{a_n\\}$ 满足: 当 $n \\geq 3$ 时, $a_n=2 a_{n-1}$ 或 $a_n=a_{n-1}+a_{n-2}$, 若 $a_1=1$, $a_2=2$, 则此数列前 2015 项中, 奇数项最多有\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610392,7 +610794,9 @@ "id": "031861", "content": "设 $f(x)=\\begin{cases}2^x,& x \\leq 0,\\\\\\log _2 x,& x>0,\\end{cases}$ 若对任意 $y \\in$($2,+\\infty$), 都存在唯一的实数 $x$, 满足 $f(f(x))=2 a^2 y^2+a y$, 则正数 $a$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610472,7 +610876,9 @@ "id": "031865", "content": "$P$ 为双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$) 左支上一点, $F_1, F_2$ 为其左右焦点, 若 $\\dfrac{|PF_2|^2}{|PF_1|}$ 的最小值为 $10 a$, 则双曲线的离心率为\\bracket{20}.\n\\fourch{$4+\\sqrt{5}$}{$4-\\sqrt{5}$}{$4 \\pm \\sqrt{5}$}{4}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610512,7 +610918,9 @@ "id": "031867", "content": "在直角 $\\triangle ABC$ 中, $\\angle ABC=90^{\\circ}$, $AC=2 \\sqrt{3}$, $AB=\\sqrt{3}, D, E$ 分别为 $AC, BD$ 的中点, 连结 $AE$ 并延长交 $BC$ 于点 $F$, 将 $\\triangle ABD$ 沿 $BD$ 折起, 使平面 $ABD \\perp$ 平面 $BCD$, 如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (0,{sqrt(3)}) node [above] {$A$} coordinate (A);\n\\draw ({2*sqrt(3)},0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [above right] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(D)$) node [above] {$E$} coordinate (E);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--cycle(B)--(D)(A)--(F);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw ({sqrt(3)},0,{-sqrt(3)/2}) node [above right] {$D$} coordinate (D);\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,0) node [below] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [above left] {$E$} coordinate (E);\n\\draw (E) ++ (0,{3/2},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--cycle;\n\\draw (A)--(F);\n\\draw [dashed] (B)--(D)--(C)(D)--(A)(A)--(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AE \\perp CD$;\\\\\n(2) 求平面 $AEF$ 与平面 $ADC$ 所成二面角的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -610594,7 +611002,9 @@ "id": "031871", "content": "已知复数 $z=-1+\\mathrm{i}$($\\mathrm{i}$ 为虚数单位), 计算: $\\dfrac{z \\cdot \\overline{z}}{z-\\overline{z}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610614,7 +611024,9 @@ "id": "031872", "content": "已知直线 $l$ 的方程为 $2 x-y-3=0$, 则直线 $l$ 的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610657,7 +611069,9 @@ "id": "031874", "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 若 $a_2 a_8=2 a_3 a_6$, $S_5=-62$, 则 $a_1$ 的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610677,7 +611091,9 @@ "id": "031875", "content": "已知双曲线 $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ 的一条渐近线与圆 $x^2+y^2-6 x+4 y=0$ 相切, 则该双曲线的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610697,7 +611113,9 @@ "id": "031876", "content": "不等式``$|x-m|<1$''是不等式``$\\log _2 x>1$''成立的充分不必要条件, 则 $m$ 的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -610837,7 +611255,9 @@ "id": "031883", "content": "已知 $l, m$ 是两条不同的直线, $\\alpha$ 是一个平面, 有下列四个命题中真命题的是\\bracket{20}.\n\\onech{若 $l \\subset \\beta$, 且 $m \\perp l$, 则 $m \\perp \\alpha$}{若 $l \\perp \\alpha$, 且 $m \\parallel \\alpha$, 则 $l \\perp m$}{若 $l \\parallel m$,, 且 $m \\subset \\alpha$, 则 $l \\parallel \\alpha$}{若 $l, m$ 与平面 $\\alpha$ 所成的角相等, 则 $l \\parallel m$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -610919,7 +611339,9 @@ "id": "031887", "content": "如图, 四棱柱 $ABCD-A_1B_1C_1D_1$ 的底面 $ABCD$ 是平行四边形, 且 $AB=1$, $BC=2$, $\\angle ABC=60^{\\circ}$, $E$ 为 $BC$ 的中点, $AA_1 \\perp$ 平面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{{sqrt(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 (C) ++ ({1/2},0,{-sqrt(3)/2}) node [right] {$D$} coordinate (D);\n\\draw (B) ++ ({1/2},0,{-sqrt(3)/2}) 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 (B_1);\n\\draw (C) ++ (0,\\n,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\n,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A)--(E)--(D)--(A_1)(A_1)--(C)(A_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $DE \\perp$ 平面 $A_1AE$;\\\\\n(2) 若 $DE=A_1E$, 试求异面直线 $AE$ 与 $A_1D$ 所成角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611019,7 +611441,9 @@ "id": "031892", "content": "已知全集 $U=\\mathbf{R}$, 集合 $A=\\{x | y=\\sqrt{1-\\dfrac{1}{x}}\\}$, 则 $\\overline{A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611061,7 +611485,9 @@ "id": "031894", "content": "在等差数列 $\\{a_n\\}$ 中,若 $a_1+a_7+a_8+a_{12}=12$, 则此数列的前 $13$ 项之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611103,7 +611529,9 @@ "id": "031896", "content": "若椭圆 $\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$ 的离心率为 $\\dfrac{\\sqrt{10}}{5}$, 则 $m$ 的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611187,7 +611615,9 @@ "id": "031900", "content": "若函数 $y=f(x-2)$ 的图像与函数 $y=\\log _3 \\sqrt{x}+2$ 的图像关于直线 $y=x$ 对称, 则 $f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611247,7 +611677,9 @@ "id": "031903", "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且 $f(2)=0$, 当 $x>0$ 时, $\\dfrac{f(x)}{x}$ 的导数小于零恒成立, 则不等式 $x^2 f(x)>0$ 的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611367,7 +611799,9 @@ "id": "031909", "content": "如图, 在四棱锥 $S-ABCD$ 中, 底面 $ABCD$ 为正方形, 侧棱 $SD \\perp$ 底面 $ABCD, E, F$ 分别是 $AB, SC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (2,0,2) node [below] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw (0,4,0) node [above] {$S$} coordinate (S);\n\\draw ($(S)!0.5!(C)$) node [above right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(S)--cycle(S)--(B);\n\\draw [dashed] (A)--(F)--(E)(A)--(D)--(C)(D)--(S);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF \\parallel $ 平面 $SAD$;\\\\\n(2) 设 $SD=2CD$, 求二面角 $A-EF-D$ 的大小.\n第 18 题图\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611427,7 +611861,9 @@ "id": "031912", "content": "在数列 $\\{a_n\\}$ 中, $a_1=1$, $a_2=2$, 且 $a_{n+1}=(1+q) a_n-q a_{n-1}$($n \\geq 2$, $q \\neq 0$).\\\\\n(1) 设 $b_n=a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n \\geq 1$), 证明 $\\{b_n\\}$ 是等比数列;\\\\\n(2) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(3) 若 $a_3$ 是 $a_6$ 与 $a_9$ 的等差中项, 求 $q$ 的值, 并证明: 对任意的 $n \\in \\mathbf{N}$, $n \\geq 1, a_n$ 是 $a_{n+3}$ 与 $a_{n+6}$ 的等差中项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611447,7 +611883,9 @@ "id": "031913", "content": "已知集合 $M=\\{3,2^a\\}$, $N=\\{a, b\\}$, 若 $M \\cap N=\\{1\\}$, 则 $M \\cup N=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611489,7 +611927,9 @@ "id": "031915", "content": "$(x^3-\\dfrac{1}{x^2})^5$ 的二项展开式中的常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611509,7 +611949,9 @@ "id": "031916", "content": "已知 $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=\\sqrt{2}$, 且 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$ 的夹角为 $45^{\\circ}$, 要使 $\\lambda \\overrightarrow{b}-\\overrightarrow{a}$ 与 $\\overrightarrow{a}$ 垂直, 则 $\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611529,7 +611971,9 @@ "id": "031917", "content": "已知 $\\alpha \\in(0,2 \\pi)$, 若复数 $z=\\sin \\alpha \\cos \\alpha-(1-\\cos 2 \\alpha) \\mathrm{i}$ 是纯虚数, 则 $\\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611549,7 +611993,9 @@ "id": "031918", "content": "方程 $\\log _2(x+14)+\\log _2(x+2)=3+\\log _2(x+6)$ 的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611569,7 +612015,9 @@ "id": "031919", "content": "已知数列 $\\{a_n\\}$ 的首项 $a_1=2$, 其前 $n$ 项和为 $S_n$. 若 $S_{n+1}=2S_n+1$, 则 $a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611609,7 +612057,9 @@ "id": "031921", "content": "已知直线 $a, b$ 及平面 $\\alpha$, 下列命题中: \\textcircled{1} $\\begin{cases}a\\perp b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \\textcircled{2} $\\begin{cases}a\\perp b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$; \\textcircled{3} $\\begin{cases}a\\parallel b,\\\\b \\parallel \\alpha\\end{cases}\\Rightarrow a \\parallel \\alpha$; \n\\textcircled{4} $\\begin{cases}a\\parallel b,\\\\b \\perp \\alpha\\end{cases}\\Rightarrow a \\perp \\alpha$. 正确命题的序号为\\blank{50}(把你认为正确的序号都填上).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611751,7 +612201,9 @@ "id": "031928", "content": "已知函数 $f(x)=x^3-x+1$, 则\\bracket{20}.\n\\twoch{$f(x)$ 有三个极值点}{$f(x)$ 有三个零点}{点 $(0,1)$ 是曲线 $y=f(x)$ 的对称中心}{直线 $y=2 x$ 是曲线 $y=f(x)$ 的切线}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -611771,7 +612223,9 @@ "id": "031929", "content": "如图, 在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB \\perp AC$, $AB=AC=AA_1=2, E$ 是 $BC$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (D);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,{sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [below right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(C_1)--(A_1)--cycle(A_1)--(B_1)--(B)(B_1)--(C_1)(B_1)--(C);\n\\draw [dashed] (A)--(C)(A)--(E)(A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥 $C-A_1B_1BA$ 的体积;\\\\\n(2) 求异面直线 $AE$ 与 $A_1C$ 所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611791,7 +612245,9 @@ "id": "031930", "content": "已知 $a \\in \\mathbf{R}$, 函数 $f(x)=x|x-a|$.\\\\\n(1) 当 $a=2$ 时,求使 $f(x) \\geq x$ 成立的 $x$ 的集合;\\\\\n(2) 求函数 $y=f(x)$ 在区间 $[1,2]$ 上的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -611915,7 +612371,9 @@ "id": "031936", "content": "若 $\\log _a 2 b=-1$, 则 $a+b$ 的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -611999,7 +612457,9 @@ "id": "031940", "content": "设无穷等比数列 $\\{a_n\\}$ 的公比是 $q$, 前 $n$ 项和为 $S_n$, 若 $3 a_1-\\displaystyle\\sum_{i=1}^{+\\infty}a_i=0$, 则 $q=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612059,7 +612519,9 @@ "id": "031943", "content": "已知函数 $f(x)=\\begin{cases}x^{2023},& x \\leq a,\\\\x^{2022},& x>a,\\end{cases}$ 若存在实数 $b$, 使得函数 $g(x)= f(x)-b$ 有两个零点, 则 $a$ 的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612099,7 +612561,9 @@ "id": "031945", "content": "已知等差数列 $\\{a_n\\}$ 满足 $a_2=2$, 则 $\\dfrac{1}{a_1^2+1}+\\dfrac{1}{a_3^2+1}$ 的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612287,7 +612751,9 @@ "id": "031954", "content": "已知数列 $\\{a_n\\}$, $a_1=4$, $a_2=p$, 且 $a_n+a_{n+1}+a_{n+2}=7 \\times(\\dfrac{1}{2})^{n-1}$.\\\\\n(1) 求 $a_4$;\\\\\n(2) 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$;\\\\\n(3) 是否存在实数 $p$, 使得 $\\{a_n\\}$ 为严格减数列?", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -612307,7 +612773,9 @@ "id": "031955", "content": "已知 $A=\\{x | \\sqrt{x}<4\\}$, $B=\\{x | 3 x \\geq 1\\}$, 则 $A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612327,7 +612795,9 @@ "id": "031956", "content": "已知复数 $z$ 满足 $\\dfrac{2}{z-5}=1+\\mathrm{i}$, 则 $z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612369,7 +612839,9 @@ "id": "031958", "content": "在二项式 $(2 x-\\dfrac{1}{x})^5$ 的展开式中,含有 $x^3$ 项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612451,7 +612923,9 @@ "id": "031962", "content": "在等比数列 $\\{a_n\\}$ ($n$ 为正整数) 中, 若 $a_1=1$, $a_4=\\dfrac{1}{8}$, 则该数列的前 $10$ 项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612514,7 +612988,9 @@ "id": "031965", "content": "椭圆 $\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$) 的左、右焦点分别为 $F_1, F_2$, 焦距为 $2 c$, 若直线 $y=\\sqrt{3}(x+c)$ 与椭圆 $\\Gamma$ 的一个交点 $M$ 满足 $\\angle MF_1F_2=2 \\angle MF_2F_1$, 则该椭圆的离心率等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -612554,7 +613030,9 @@ "id": "031967", "content": "已知直线的方程为 $4 x+2 y+c=0$, 则该直线的一个法向量为\\bracket{20}.\n\\fourch{$(2,-1)$}{$(2,1)$}{$(-1,2)$}{$(1,2)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -612636,7 +613114,9 @@ "id": "031971", "content": "在直三棱柱 $ABC-A_1B_1C_1$ 中, $AB=AC=1$, $\\angle BAC=90^{\\circ}$, 且异面直线 $A_1B$ 与 $B_1C_1$ 所成的角等于 $60^{\\circ}$, 设 $AA_1=a$.\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 (0,0,2) node [left] {$B$} coordinate (B);\n\\foreach \\i in {A,B,C}\n{\\draw (\\i) ++ (0,2,0) coordinate (\\i_1);};\n\\draw (A_1) node [above] {$A_1$} (B_1) node [left] {$B_1$} (C_1) node [right] {$C_1$};\n\\draw (B)--(C)--(C_1)--(A_1)--(B_1)--cycle(B_1)--(C_1);\n\\draw [dashed] (B)--(A_1)(B)--(A)--(C)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $a$ 的值;\\\\\n(2) 求三棱锥 $B_1-A_1BC$ 的体积.\n第 17 题图\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "",