From 5f925cccf4b5bbda9d49ea04c090b94a83968d01 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Fri, 7 Jul 2023 14:18:21 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A5=E9=80=89=E5=BF=85=E4=B8=80?= =?UTF-8?q?=E7=A9=BA=E9=97=B4=E5=90=91=E9=87=8F=E5=8F=8A=E5=85=B6=E5=BA=94?= =?UTF-8?q?=E7=94=A8=E4=BE=8B=E9=A2=98=E4=B8=8E=E4=B9=A0=E9=A2=98=20?= =?UTF-8?q?=E5=B9=B6=E5=AF=B9=E5=BA=94=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具v2/latex编辑题目答案及提示.py | 2 +- 工具v2/关键字筛选题号.py | 6 +- 工具v2/批量收录题目.py | 2 +- 题库0.3/Problems.json | 1056 +++++++++++++++++++++++++++++ 4 files changed, 1061 insertions(+), 5 deletions(-) diff --git a/工具v2/latex编辑题目答案及提示.py b/工具v2/latex编辑题目答案及提示.py index de5068c3..04d9847b 100644 --- a/工具v2/latex编辑题目答案及提示.py +++ b/工具v2/latex编辑题目答案及提示.py @@ -1,4 +1,4 @@ -id_string = "20386" +id_string = "013298,014818,030516,030536,031162" editor = "徐慧" diff --git a/工具v2/关键字筛选题号.py b/工具v2/关键字筛选题号.py index 9246f9b3..8f5c4f65 100644 --- a/工具v2/关键字筛选题号.py +++ b/工具v2/关键字筛选题号.py @@ -1,6 +1,6 @@ keywords_dict = { "id":[""], #题号 - "content":[""], #题面内容 + "content":["两个"],"content2":["正态"], #题面内容 "objs":[""], #目标代码 "tags":[""], #标签, 如["第二单元"]等 "genre":[""], #题目类型, 填空题, 选择题, 解答题 @@ -8,7 +8,7 @@ keywords_dict = { "solution":[""], #解答与提示 "duration":[""], #解题时间(目前未设置) "usages":[""], #使用记录, 数据库中格式为 <日期>\t<届别><班别>\t正确率[\t正确率]... 例如"20230301\t2023届01班\t0.985\t0.211 - "origin":["高二下学期期末","高一下学期"], #题目来源 + "origin":[""], #题目来源 "edit":[""], #导入者及编辑者 "same":[""], #相同题目题号 "related":[""], #关联题目题号 @@ -16,7 +16,7 @@ keywords_dict = { "space":[""], #解答题下的空间(em)表示一个m的宽度 "unrelated":[""], #无关题目题号 # "content2":["双曲线"], #在字段名中加入数字表示这个字段的另一个必要条件 - "content_not":["抛"], #加_not表示不能出现该样式的词 + "content_not":["OBSOLETE"], #加_not表示不能出现该样式的词 } #同一字段名中的条件为"或"的关系, 不同字段名(可加数字表示同一字段)中的条件为"且"的关系 outputfilepath = "临时文件/题号筛选.txt" diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py index b77e2acb..ea57280c 100644 --- a/工具v2/批量收录题目.py +++ b/工具v2/批量收录题目.py @@ -1,5 +1,5 @@ #修改起始id,出处,文件名 -starting_id = 18784 #起始id设置, 来自"寻找空闲题号"功能 +starting_id = 19006 #起始id设置, 来自"寻找空闲题号"功能 raworigin = "" #题目来源的前缀(中缀在.tex文件中) filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件 editor = "王伟叶" #编辑者姓名 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 64990b87..a4daba5b 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -487809,6 +487809,1062 @@ "space": "4em", "unrelated": [] }, + "019006": { + "id": "019006", + "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$为棱$B_1C_1$上任意一点. 只考虑图上已作出线段所对应的向量, 回答以下问题:\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 ($(B_1)!0.4!(C_1)$) node [above] {$E$} coordinate (E);\n\\draw (A_1)--(B)--(E);\n\\draw [dashed] (D_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 分别写出$\\overrightarrow{AB}$的相等向量与$\\overrightarrow{A_1B}$的负向量;\\\\\n(2) 用另外两个向量的和或差表示$\\overrightarrow{BB_1}$(举两个例子);\\\\\n(3) 用三个或三个以上向量的和表示$\\overrightarrow{BE}$(举两个例子).", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019007": { + "id": "019007", + "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, 点$E$在棱$CC_1$的延长线上, 且$|C_1E|=|CC_1|$. 设$\\overrightarrow{AA_1}=\\overrightarrow {a}$, $\\overrightarrow{AB}=\\overrightarrow {b}$, $\\overrightarrow{AD}=\\overrightarrow {c}$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的线性组合表示下列向量:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{1.5}\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 ($(C)!2!(C_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(E)$) coordinate (O);\n\\draw (O)--(E)--(C_1)(B_1)--(D_1);\n\\draw [dashed] (A)--(C_1)(B)--(D_1)(O)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow{AC_1}$;\\\\\n(2) $\\overrightarrow{D_1B_1}$;\\\\\n(3) $\\overrightarrow{BD_1}$;\\\\\n(4) $\\overrightarrow{AE}$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019008": { + "id": "019008", + "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 判断下列每组向量是否共面:\\\\\n(1) $\\overrightarrow{AB}$, $\\overrightarrow{D_1C_1}$;\\\\\n(2) $\\overrightarrow{AB}$, $\\overrightarrow{BC}, \\overrightarrow{A_1D_1}$;\\\\\n(3) $\\overrightarrow{AB}$, $\\overrightarrow{BC}, \\overrightarrow{DD_1}$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019009": { + "id": "019009", + "content": "如图, 在三棱锥$P-ABC$中, $M$是$BC$的中点. 用$\\overrightarrow{PA}$、$\\overrightarrow{PB}$和$\\overrightarrow{PC}$的线性组合表示$\\overrightarrow{AM}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{sqrt(8/3)},0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below right] {$M$} coordinate (M);\n\\draw (P)--(M)(P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (C)--(A)--(M);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019010": { + "id": "019010", + "content": "设$AB$是空间中任意一条线段, $O$是空间中任意一点, 求证: $M$为$AB$中点的一个充要条件是$\\overrightarrow{OM}=\\dfrac{1}{2}(\\overrightarrow{OA}+\\overrightarrow{OB})$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019011": { + "id": "019011", + "content": "如果$\\overrightarrow {a}$与$\\overrightarrow {b}$都是空间向量, 判断$||\\overrightarrow {a}|-| \\overrightarrow {b}|| \\leq|\\overrightarrow {a}+\\overrightarrow {b}| \\leq|\\overrightarrow {a}|+|\\overrightarrow {b}|$是否成立, 并说明等号何时成立.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019012": { + "id": "019012", + "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$a$, $E$是棱$CC_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [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\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{DD_1} \\cdot \\overrightarrow{DC_1}$;\\\\\n(2) 求$\\overrightarrow{AE} \\cdot \\overrightarrow{C_1A_1}$;\\\\\n(3) 求$\\overrightarrow{AE}$与$\\overrightarrow{C_1A_1}$的夹角的大小;\\\\\n(4) 判断$\\overrightarrow{AE}$与$\\overrightarrow{DB}$是否垂直.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019013": { + "id": "019013", + "content": "底面是平行四边形的棱柱称为平行六面体, 其特点是六个面都是平行四边形. 如图, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 点$M$在对角线$A_1B$上, 且$|A_1M|=\\dfrac{1}{2}|MB|$, 点$N$在对角线$A_1C$上, 且$|A_1N|=\\dfrac{1}{3}|NC|$. 求证: $M$、$N$、$D_1$三点共线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [above] {$A$} coordinate (A);\n\\draw (2,0,-2) node [above] {$B$} coordinate (B);\n\\draw (0,0,-2) node [above] {$C$} coordinate (C);\n\\foreach \\i/\\j in {A/below,B/right,C/left,D/below}\n{\\draw (\\i) ++ (-0.5,-2,0) coordinate (\\i_1) node [\\j] {$\\i_1$};};\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw (D)--(D_1)--(A_1)--(B_1)--(B)(A_1)--(A)(A_1)--(B);\n\\draw [dashed] (D_1)--(C_1)--(B_1)(C_1)--(C)(C)--(A_1);\n\\filldraw ($(A_1)!{1/3}!(B)$) circle (0.03) node [right] {$M$} coordinate (M);\n\\filldraw ($(A_1)!{1/4}!(C)$) circle (0.03) node [left] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019014": { + "id": "019014", + "content": "在正四面体$ABCD$中, 用向量的方法证明: $AB \\perp CD$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019015": { + "id": "019015", + "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$\\overrightarrow{AB} \\cdot \\overrightarrow{AP}$($i=1,2, \\cdots, 8$)的不同值的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,1,0) node [above] {$B$} coordinate (A_1);\n\\draw (B) ++ (0,1,0) node [above] {$P_6$} coordinate (B_1);\n\\draw (C) ++ (0,1,0) node [above] {$P_8$} coordinate (C_1);\n\\draw (D) ++ (0,1,0) node [above] {$P_2$} 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 ($(A)!0.5!(B)$) --++ (0,1,0) node [above] {$P_3$} --++ (0,0,-2) node [above] {$P_5$} ($(B)!0.5!(C)$) --++ (0,1,0) node [above]{$P_7$}--++ (-2,0,0) node [above] {$P_1$} ++ (1,0,0) node [above] {$P_4$};\n\\draw [dashed] (A) ++ (0,0,-1) --++ (2,0,0) (A) ++ (0,0,-1) --++ (0,1,0) (A) ++ (1,0,0) --++ (0,0,-2) --++ (0,1,0) ++ (0,0,1) --++ (0,-1,0);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{1 }{2}{4 }{8 }", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019016": { + "id": "019016", + "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $E$是棱$AA_1$的中点, $O$是面对角线$BC_1$与$B_1C$的交点. 试判断向量$\\overrightarrow{EO}$与$\\overrightarrow{AB}$、\n$\\overrightarrow{AD}$是否共面.\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 [below] {$D$} 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 ($(A)!0.5!(A_1)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C_1)$) node [right] {$O$} coordinate (O);\n\\draw [dashed] (E)--(O);\n\\draw (B)--(C_1)(C)--(B_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019017": { + "id": "019017", + "content": "利用向量证明: 如果一条直线垂直于一个平面上的两条相交直线, 那么这条直线垂直于这个平面.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019018": { + "id": "019018", + "content": "如图, 在正四面体$ABCD$中, $N$是面$ABC$的中心.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{sqrt(8/3)},0) node [above] {$D$} coordinate (D);\n\\filldraw ($1/3*(A)+1/3*(B)+1/3*(C)$) circle (0.03) node [right] {$N$} coordinate (N);\n\\draw (D)--(A)--(B)--(C)--cycle(D)--(B);\n\\draw [dashed] (C)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 在此四面体的棱所对应的向量中找出两组各三个不共面的向量, 并把其他棱对应的向量分别表示成这两组向量的线性组合(互为负向量的不必另行表示), 要求第一组三个向量所在直线相交于一点, 第二组三个向量所在的直线不相交于一点(答案不唯一);\\\\\n(2) 在(1)的条件下, 把$\\overrightarrow{DN}$也分别表示为这两组向量的线性组合.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019019": { + "id": "019019", + "content": "如图, 在底面为平行四边形的四棱柱$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别在棱$BB_1$和$DD_1$上, 且$|BE|=\\dfrac{1}{3}|BB_1|$, $|DF|=\\dfrac{2}{3}|DD_1|$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\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) ++ (0.3,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0.3,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0.3,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0.3,\\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 ($(B)!{1/3}!(B_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(D)!{2/3}!(D_1)$) node [left] {$F$} coordinate (F);\n\\draw (A)--(E)--(C_1);\n\\draw [dashed] (A)--(F)--(C_1)(A)--(C_1)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $A$、$E$、$C_1$、$F$四点共面;\\\\\n(2) 若$\\overrightarrow{EF}=x \\overrightarrow{AB}+y \\overrightarrow{AD}+z \\overrightarrow{AA_1}$($x$、$y$、$z \\in \\mathbf{R}$), 求$x+y+z$的值.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019020": { + "id": "019020", + "content": "在空间直角坐标系$O-x y z$中作出点$P(7,6,4)$, 并求该点关于坐标平面$x O y$的对称点$P'$的坐标.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019021": { + "id": "019021", + "content": "如图, 给定正方体$ABCD-A_1B_1C_1D_1$.\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 [right] {$C_1$} 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\\end{tikzpicture}\n\\end{center}\n(1) 求对角线$CA_1$与$CA$所成角的余弦值;\\\\\n(2) 求证: $CA_1 \\perp BD$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019022": { + "id": "019022", + "content": "正四棱柱$ABCD-A_1B_1C_1D_1$中, 底面$ABCD$是边长为$4$的正方形, $A_1C_1$与$B_1D_1$交于点$N$, $BC_1$与$B_1C$交于点$M$, 且$AM \\perp BN$.\\\\\n(1) 用向量方法求线段$AA_1$的长;\\\\\n(2) 对于$n$个向量$\\overrightarrow {a_1}$、$\\overrightarrow {a_2}$、$\\cdots$、$\\overrightarrow {a_n}$, 若存在不全为零的$n$个实数$\\lambda_1$、$\\lambda_2$、$\\cdots$、$\\lambda_n$, 使得$\\lambda_1 \\overrightarrow {a_1}+\\lambda_2 \\overrightarrow {a_2}+\\cdots+\\lambda_n \\overrightarrow {a_n}=\\overrightarrow{0}$, 则称这$n$个向量$\\overrightarrow {a_1}$、$\\overrightarrow {a_2}$、$\\cdots$、$\\overrightarrow {a_n}$线性相关, 否则称其线性无关. 试判断三个向量$\\overrightarrow{AM}$、$\\overrightarrow{BN}$、$\\overrightarrow{CD}$是否线性相关, 并说明理由.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019023": { + "id": "019023", + "content": "证明: 平面上的一条直线和这个平面的一条斜线垂直的一个充要条件是它和这条斜线在平面上的投影垂直.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019024": { + "id": "019024", + "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 求证:\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\\end{tikzpicture}\n\\end{center}\n(1) $AC_1 \\perp$平面$A_1BD$, $AC_1 \\perp$平面$CD_1B_1$;\\\\\n(2) 平面$A_1BD\\parallel$平面$CD_1B_1$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019025": { + "id": "019025", + "content": "若空间向量$\\overrightarrow {a}=(1,-2,1)$, $\\overrightarrow {b}=(1,0,2)$, 则下列向量中可作为向量$\\overrightarrow {a}$、$\\overrightarrow {b}$所在平面的一个法向量的是\\bracket{20}.\n\\fourch{$(4,-1,2)$}{$(-4,-1,2)$}{$(-4,1,2)$}{$(4,-1,-2)$}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019026": { + "id": "019026", + "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过点$D$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$(2,3,2)$, 则平面$ABC_1D_1$的一个法向量的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{3}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [above 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 [->] (C) -- ($(D)!1.2!(C)$) node [below] {$y$};\n\\draw [->] (A) -- ($(D)!1.5!(A)$) node [right] {$x$};\n\\draw [->] (D_1) -- ($(D)!1.3!(D_1)$) node [right] {$z$};\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019027": { + "id": "019027", + "content": "如图所示, 正方体$ABCD-A_1B_1C_1D_1$的棱长为$2, E$、$F$分别是$BB_1$、$DD_1$的中点. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(D_1)$) node [left] {$F$} coordinate (F);\n\\draw (A)--(E);\n\\draw [dashed] (E)--(D)(B_1)--(F)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) $FC_1\\parallel$平面$ADE$;\\\\\n(2) 平面$ADE\\parallel$平面$B_1C_1F$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019028": { + "id": "019028", + "content": "若平面$\\alpha$的一个法向量为$(\\sqrt{2}, \\sin \\theta, \\cos \\theta)$, 平面$\\beta$的一个法向量为$(\\dfrac{\\sqrt{2}}{2}, \\cos \\theta, \\sin \\theta)$, 且$\\alpha \\perp \\beta$, 则$\\theta=$\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019029": { + "id": "019029", + "content": "如图, 在四棱柱$ABCD-A_1B_1C_1D_1$中, $A_1D \\perp$底面$ABCD$, 底面$ABCD$是边长为$1$的正方形, 侧棱长$|AA_1|=2$. 求证:\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 [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\m) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\m) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (1,{sqrt(3)},0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (1,{sqrt(3)},0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (1,{sqrt(3)},0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (1,{sqrt(3)},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\\end{tikzpicture}\n\\end{center}\n(1) $AC \\perp$平面$A_1BD$;\\\\\n(2) 乎面$A_1C_1D \\perp$平面$A_1BD$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019030": { + "id": "019030", + "content": "如图, 在长方体$ABCD-A' B' C' D'$中, $|AB|=2$, $|AD|=|AA'|=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\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] {$A$} coordinate (A);\n\\draw (B') ++ (0,\\n,0) node [right] {$B$} coordinate (B);\n\\draw (C') ++ (0,\\n,0) node [above right] {$C$} coordinate (C);\n\\draw (D') ++ (0,\\n,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A') -- (A) (B') -- (B) (C') -- (C);\n\\draw [dashed] (D') -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求顶点$B'$到平面$D' AC$的距离;\\\\\n(2) 求直线$BC'$到平面$D' AC$的距离.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019031": { + "id": "019031", + "content": "设正方体$ABCD-A_1B_1C_1D_1$的棱长为$a$, 求平行平面$A_1BD$与$CD_1B_1$之间的距离.\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 (D_1)--(B_1)--(C)(A_1)--(B);\n\\draw [dashed] (B)--(D)--(A_1)(C)--(D_1);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019032": { + "id": "019032", + "content": "正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, 若点$O$是底面$A_1B_1C_1D_1$的中心, 则点$O$到平面$ABC_1D_1$的距离为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019033": { + "id": "019033", + "content": "若正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, 则直线$A_1B_1$到平面$ABC_1D_1$的距离为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019034": { + "id": "019034", + "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 若$|AB|=|BC|=1$, $|AA_1|=2$, 则平行平面$AB_1D_1$与平面$C_1DB$之间的距离为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019035": { + "id": "019035", + "content": "已知正方形$ABCD$的边长$4$, $E$、$F$分别是边$AB$、$AD$的中点, $CG \\perp$平面$ABCD$, 若$|CG|=2$, 则点$B$到平面$EFG$的距离为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019036": { + "id": "019036", + "content": "若正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, 求点$C$到平面$BDC_1$的距离. (用两种方法求解)", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019037": { + "id": "019037", + "content": "在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别为棱$AA_1$、$BB_1$的中点, $M$为棱$A_1B_1$上的一点, 点$N$为$ME$的中点. 设常数$\\lambda$满足$0<\\lambda<2$, 若$|A_1M|=\\lambda$, 则点$N$到平面$D_1EF$的距离为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019038": { + "id": "019038", + "content": "如图, 在四面体$ABCD$中, $E$是棱$BC$的中点, $|CA|=|CB|=|CD|=|BD|=2$, $|AB|=|AD|=\\sqrt{2}$, 求点$E$到平面$ACD$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,-1) node [above] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\filldraw (E) circle (0.03);\n\\draw (A)--(B)--(C)--cycle;\n\\draw [dashed] (A)--(D)--(C)(D)--(B);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019039": { + "id": "019039", + "content": "如图, 在正方体$ABCD-A' B' C' D'$中, $E$、$F$分别是$AD$、$AB$的中点. 求直线$B' E$与$C' F$所成角的大小.\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] {$A$} coordinate (A);\n\\draw (B') ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C') ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D') ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A') -- (A) (B') -- (B) (C') -- (C);\n\\draw [dashed] (D') -- (D);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [above] {$F$} coordinate (F);\n\\draw [dashed] (B')--(E)(C')--(F);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019040": { + "id": "019040", + "content": "如图, 在四棱锥$P-ABCD$中, $PA \\perp$平面$ABCD$, $AB \\perp AD$, $BC\\parallel AD$, $|PA|=|AB|=|BC|=1$, $|CD|=\\sqrt{2}$, $\\angle CDA=45^{\\circ}$. 求直线$PB$与平面$PCD$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(A);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019041": { + "id": "019041", + "content": "确定下列直线与平面的位置关系, 如果相交, 求它们所成角的大小:\\\\\n(1) 直线的一个方向向量为$(1,-2,9)$, 平面的一个法向量为$(3,-4,7)$;\\\\\n(2) 直线的一个方向向量为$(-1,1,2)$, 平面的一个法向量为$(2,1,-1)$;\\\\\n(3) 直线的一个方向向量为$(-2,-7,3)$, 平面的一个法向量为$(4,-2,-2)$;\\\\\n(4) 直线的一个方向向量为$(\\dfrac{\\sqrt{2}}{2},-1, \\sqrt{2})$, 平面的一个法向量为$(-1, \\sqrt{2},-2)$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019042": { + "id": "019042", + "content": "在正方体$ABCD-A' B' C' D'$中, $M$、$N$分别是$AD$、$AB$的中点, $E$、$F$分别是$BC$、$CD$的中点. 求:\\\\\n(1) 异面直线$B' M$与$C' N$所成的角的大小;\\\\\n(2) 直线$A' D$与平面$EFD' B'$所成角的大小.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019043": { + "id": "019043", + "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 直线$l$与$AB$、$AD$、$AA_1$所成角都相等, 求$l$与$AB$所成角的大小.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019044": { + "id": "019044", + "content": "已知正四面体$ABCD$, 点$E$在棱$AC$上, 且$|AE|=3|EC|$, 则直线$AB$与平面$DBE$所成角的大小为\\blank{50}.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019045": { + "id": "019045", + "content": "如图, 在正四棱锥$P-ABCD$中, $|PA|=|AB|=2 \\sqrt{2}$, $E$、$F$分别为$PB$、$PD$的中点, 平面$AEF$与棱$PC$交于点$G$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$A$} coordinate (A);\n\\draw (-2,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,-2) node [below] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (D)--(A)--(B)--(P)--cycle;\n\\draw (P)--(A);\n\\draw [dashed] (P)--(C)(B)--(C)--(D);\n\\draw ($(P)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!{1/3}!(C)$) node [below] {$G$} coordinate (G);\n\\draw (F)--(A)--(E);\n\\draw [dashed] (F)--(G)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求平面$AEGF$与平面$ABCD$所成二面角的大小;\\\\\n(2) 若点$M$是棱$AB$上的动点, 确定点$M$的位置, 使得平面$PCM$与平面$DCM$所成的锐二面角是$60^{\\circ}$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019046": { + "id": "019046", + "content": "已知正三棱锥$A-BCD$的棱长都为$a$, 求二面角$A-BC-D$的大小.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019047": { + "id": "019047", + "content": "已知正方体$ABCD-A' B' C' D'$, 求二面角$B-AC-D'$的大小.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019048": { + "id": "019048", + "content": "如图, 圆锥的轴截面为等腰直角三角形$SAB$, $Q$为底面圆周上一点. 若二面角$A-SB-Q$大小为$\\arctan \\dfrac{\\sqrt{6}}{3}$, 则$\\angle AOQ$的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale=1.5]\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (0,1) node [above] {$S$} coordinate (S);\n\\draw (-120:1 and 0.25) node [below] {$Q$} coordinate (Q);\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw (A)--(S)--(B)arc (360:180:1 and 0.25);\n\\draw (S)--(Q);\n\\draw [dashed] (S)--(O)(A)--(B)(O)--(Q)--(B)(A) arc (180:0:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019049": { + "id": "019049", + "content": "如图, 一个工作台的下半部分是个正四棱柱$ABCD-A_1B_1C_1D_1$, 其底面边长为$4$, 高为$1$, 工作台的上半部分是一个底面半径$D_1E$为$\\sqrt{2}$, 母线$EE_1$长为$3$的圆柱体的四分之一. 若点$P$是圆弧$E_1F_1$上的动点, 当平面$PA_1C_1$与平面$ABC$所成的钝二面角最大时, 求点$P$在圆弧上的位置.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale=0.5]\n\\def\\l{4}\n\\def\\m{4}\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 right] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1);\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A_1)--(C_1);\n\\draw (D_1) ++ (0,0,{sqrt(2)}) node [left] {$E$} coordinate (E);\n\\draw (D_1) ++ ({sqrt(2)},0,0) node [above right] {$F$} coordinate (F);\n\\draw [domain = 0:90, samples = 100] plot ({sqrt(2)*cos(\\x)},1,{sqrt(2)*sin(\\x)-4}) plot ({sqrt(2)*cos(\\x)},4,{sqrt(2)*sin(\\x)-4});\n\\draw (E) ++ (0,3,0) node [left] {$E_1$} coordinate (E_1);\n\\draw (F) ++ (0,3,0) node [right] {$F_1$} coordinate (F_1);\n\\draw (0,4,-4) node [above] {$D_2$} coordinate (D_2);\n\\draw (A_1)--(E)--(E_1)--(D_2)--(F_1)--(F)--(C_1);\n\\draw [dashed] (E)--(D_1)--(F)(D_1)--(D_2);\n\\filldraw ({sqrt(2)*cos(40)},4,{sqrt(2)*sin(40)-4}) circle (0.03) node [below] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019050": { + "id": "019050", + "content": "已知四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为$2$, 且$\\angle ADC=\\dfrac{2 \\pi}{3}$. 设$\\overrightarrow{DA}=\\overrightarrow {a}$, $\\overrightarrow{DC}=\\overrightarrow {b}$, $\\overrightarrow{DD_1}=\\overrightarrow {c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\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) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ ({sqrt(2)/2},{sqrt(3)},{-sqrt(2)/2}) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ ({sqrt(2)/2},{sqrt(3)},{-sqrt(2)/2}) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ ({sqrt(2)/2},{sqrt(3)},{-sqrt(2)/2}) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ ({sqrt(2)/2},{sqrt(3)},{-sqrt(2)/2}) 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\\end{tikzpicture}\n\\end{center}\n(1) 设$E$是棱$A_1D_1$的中点.\\\\\n(I) 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$的线性组合表示$\\overrightarrow{EB}$;\\\\\n(II) 若$\\angle ADD_1=\\angle CDD_1=\\alpha$, $\\alpha \\in(\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2}]$, 求$|\\overrightarrow{EB}|$的取值范围;\\\\\n(2) 求证: 当且仅当$\\angle ADD_1=\\angle CDD_1$时, $AC \\perp$平面$DBB_1D_1$.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019051": { + "id": "019051", + "content": "已知直四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为$2$, 且$\\angle ADC=\\dfrac{2 \\pi}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\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) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\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 ($(A_1)!0.5!(D_1)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!{2/3}!(B)$) node [below] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 设$E$是$A_1D_1$中点, 点$F$满足$\\overrightarrow{AF}=2 \\overrightarrow{FB}$.\\\\\n(I) 求异面直线$EF$与$DD_1$所成角的大小;\\\\\n(II) 求直线$EF$与平面$DBB_1D_1$所成角的大小;\\\\\n(2) 求平面$DBB_1D_1$与平面$BDC_1$所成锐二面角的大小;\\\\\n(3) 求四面体$A_1C_1BD$的体积.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019052": { + "id": "019052", + "content": "试利用空间向量知识证明以下命题:\n设实数$a_i$、$b_i$, 其中$i \\in\\{1,2,3\\}$, 则不等式$(\\displaystyle\\sum_{i=1}^3 a_i b_i)^2 \\leq \\displaystyle \\sum_{i=1}^3 a_i^2 \\cdot \\displaystyle\\sum_{i=1}^3 b_i^2$恒成立.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019053": { + "id": "019053", + "content": "如图, 在正四面体$ABCD$中, 顶点$A$在底面$BCD$上的投影为点$H$, 点$M$为线段$AH$的中点. 求证: $BM$、$CM$、$DM$两两垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(D)+1/3*(B)+1/3*(C)$) ++ (0,{sqrt(8/3)},0) node [above] {$A$} coordinate (A);\n\\filldraw ($1/3*(D)+1/3*(B)+1/3*(C)$) circle (0.03) node [right] {$H$} coordinate (H);\n\\draw (D)--(A)--(B)--(C)--cycle(A)--(C);\n\\draw [dashed] (D)--(B);\n\\draw ($(A)!0.5!(H)$) node [right] {$M$} coordinate (M);\n\\draw [dashed] (A)--(H) (C)--(M)(B)--(M)(D)--(M);\n\\filldraw (M) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "空中课堂选择性必修第一册空间向量及其应用例题与习题", + "edit": [ + "20230707\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.",