From c94ece235f88ce5a51a0b702c91fdfc2d5cd1066 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 14 Oct 2023 22:00:46 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B7=BB=E5=8A=A0=E9=83=A8=E5=88=862025?= =?UTF-8?q?=E5=B1=8A=E6=A0=A1=E6=9C=AC=E6=94=B9=E7=BC=96=E7=9A=84=E9=A2=98?= =?UTF-8?q?=E7=9B=AE?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 109 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 108 insertions(+), 1 deletion(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index edfd4cee..ace2edc5 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -516517,6 +516517,111 @@ "space": "4em", "unrelated": [] }, + "019870": { + "id": "019870", + "content": "$\\triangle ABC$ 所在平面 $\\alpha$ 外一点 $D$, $AB=BC=AC=BD=CD=1$, 设 $AD=x$, 当 $\\triangle BCD$ 绕 $BC$ 边旋转时, 求 $x$ 的取值范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [right] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0,0) node [left] {$A$} coordinate (A);\n\\draw (-0.5,{sqrt(3-0.25)},0) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle (B)--(D);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20231014\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019871": { + "id": "019871", + "content": "如图, 在四棱锥 $P-ABCD$ 中, 底面 $ABCD$ 是 $\\angle DAB=60$ 且边长为 $a$ 的菱形, 侧面 $PAD$ 是等边三角形, 且平面 $PAD$ 垂直于底面 $ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (1,0,{-sqrt(3)}) node [above] {$D$} coordinate (D);\n\\draw (D) ++ (2,0,0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(D)$) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle (P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $G$ 为 $AD$ 的中点, 求证: $BG \\perp$ 平面 $PAD$;\\\\\n(2) 求证: $AD \\perp PB$;\\\\\n(3) 求二面角 $A-BC-P$ 的大小.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20231014\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019872": { + "id": "019872", + "content": "已知正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$ , 则异面直线 $A_1B_1$ 与 $BC_1$ 的距离为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20231014\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "019873": { + "id": "019873", + "content": "已知正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $a$,\\\\\n(1) 若 $BD_1=2 \\sqrt{3}$, 求异面直线 $BD_1$ 与 $AC$ 之间的距离;\\\\\n(2) 若 $a=1$, 求异面直线 $A_1D$ 与 $AC$ 之间的距离.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "自拟题目", + "edit": [ + "20231014\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "019874": { + "id": "019874", + "content": "在空间四边形$ABCD$中, $AB=DC=4$, $BC=AD=3$, $AD \\perp BC$, $AB \\perp BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (5,0,0) node [right] {$C$} coordinate (C);\n\\draw ({16/5},0,{12/5}) node [below] {$B$} coordinate (B);\n\\draw ({9/5},{3*sqrt(7)/4},{27/20}) node [above] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (B) -- (D);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$BD$的长;\\\\\n(2) 求证:$BD$是$AD$、$BC$的公垂线.", + "objs": [], + "tags": [ + "第六单元" + ], + "genre": "4em", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二校本作业必修第十章-20231014修改", + "edit": [ + "20230101\t王伟叶", + "20231014\t王伟叶" + ], + "same": [], + "related": [ + "020969" + ], + "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.", @@ -542864,7 +542969,9 @@ "20230101\t王伟叶" ], "same": [], - "related": [], + "related": [ + "019874" + ], "remark": "", "space": "4em", "unrelated": []