录入25届几何复习卷（2张）补充题并建立related
This commit is contained in:
parent
9085821cec
commit
f3d8a72bf2
|
|
@ -107529,7 +107529,9 @@
|
|||
"20220701\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"related": [
|
||||
"023365"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
|
|
@ -397126,7 +397128,9 @@
|
|||
"20230202\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"related": [
|
||||
"023370"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
|
|
@ -455327,7 +455331,9 @@
|
|||
"20230503\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"related": [
|
||||
"023364"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
|
|
@ -457443,7 +457449,9 @@
|
|||
"20230503\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"related": [
|
||||
"023366"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
|
|
@ -458631,7 +458639,9 @@
|
|||
"20230503\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"related": [
|
||||
"023366"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
|
|
@ -499686,7 +499696,8 @@
|
|||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"023190"
|
||||
"023190",
|
||||
"023369"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
|
|
@ -624812,6 +624823,218 @@
|
|||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023361": {
|
||||
"id": "023361",
|
||||
"content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, 设 $A_1C$ 与平面 $ABC_1D_1$ 交于 $Q$. 求证: $B$、$Q$、$D_1$ 三点共线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (B)--(C_1);\n\\draw [dashed] (A)--(D_1)(C)--(A_1);\n\\filldraw ($(A)!0.5!(C_1)$) node [below] {$Q$} coordinate (Q) circle (0.03);\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023362": {
|
||||
"id": "023362",
|
||||
"content": "长方体 $ABCD-A_1B_1C_1D_1$ 中, 已知 $AB=a$, $BC=b$, $AA_1=c$, 且 $a<b$. 求异面直线 $D_1B$与 $AC$ 所成角的余弦值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2.5}\n\\def\\m{3}\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 [dashed] (D_1)--(B)(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": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023363": {
|
||||
"id": "023363",
|
||||
"content": "在四面体 $ABCD$ 中, $AB=AC=AD=BC=1$, $CD=\\sqrt{2}$, 且 $\\angle BCD=90^{\\circ}$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,0) node [right] {$D$} coordinate (D);\n\\draw ($(B)!{1/3}!(D)$) ++ (0,0,{sqrt(2/3)}) node [below] {$C$} coordinate (C);\n\\draw ({sqrt(3)/2},{1/2},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) $A$ 到平面 $BCD$ 的距离;\\\\\n(2) $AC$ 与平面 $BCD$ 所成的角.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023364": {
|
||||
"id": "023364",
|
||||
"content": "已知边长为 $6$ 的正方形 $ABCD$ 所在平面外一点 $P$, $PD \\perp$ 平面 $ABCD$, $PD=8$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,6) node [below] {$A$} coordinate (A);\n\\draw (6,0,6) node [below] {$B$} coordinate (B);\n\\draw (6,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,8,0) node [left] {$P$} coordinate (P);\n\\draw (A)--(B)--(C);\n\\draw (P)--(B)(P)--(A)(P)--(C);\n\\draw [dashed] (B)--(D)(A)--(C)(A)--(D)--(C)(P)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 连接 $PB, AC$, 证明: $PB \\perp AC$;\\\\\n(2) 连接 $PA$, 求 $PA$ 与平面 $PBD$ 所成的角的大小;\\\\\n(3) 求点 $D$ 到平面 $PAC$ 的距离.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"016734"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023365": {
|
||||
"id": "023365",
|
||||
"content": "如图, $P$ 为 $\\triangle ABC$ 所在平面外一点, $PA \\perp$ 平面 $ABC, \\angle ABC=90^{\\circ}$, $AE \\perp PB$ 于 $E$, $AF \\perp PC$ 于 $F$. 求证: $PC \\perp$ 平面 $AEF$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round]\n\\draw (0,0) coordinate (B) (3,0) coordinate (C) (1,1) coordinate (A) node [below] {$A$} (1,4) coordinate (P);\n\\draw (B) node [below left] {$B$} -- (C) node [below right] {$C$} -- (P) node [above] {$P$} -- cycle;\n\\draw ($(B)!0.6!0:(P)$) coordinate (E) node [left] {$E$} ($(C)!0.65!0:(P)$) coordinate (F) node [right] {$F$};\n\\draw (E) -- (F);\n\\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (C) (A) -- (E) (A) -- (F); \n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"003462"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023366": {
|
||||
"id": "023366",
|
||||
"content": "已知球的两个平行截面的面积分别是 $5 \\pi$、$16 \\pi$, 且这两个平行平面间的距离为 $1$, 那么该球的半径是多少?",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"016884",
|
||||
"016830"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023367": {
|
||||
"id": "023367",
|
||||
"content": "如图, 已知点 $P$ 在圆柱 $OO_1$ 的底面圆 $O$ 上, $AB$ 为圆 $O$ 的直径.\n\\begin{center}\n\\begin{tikzpicture}[thick]\n\\draw (0,0) node [left] {$A$} -- (0,3) node [left] {$A_1$} -- (4,3) node [right] {$B_1$} -- (4,0) node [right] {$B$};\n\\draw (0,0) arc (180:360:2 and 0.5) (0,3) arc (180:360:2 and 0.5) (0,3) arc (180:0:2 and 0.5);\n\\draw [dashed] (0,0) arc (180:0:2 and 0.5);\n\\filldraw (2,0) circle (0.05) node [below left] {$O$} coordinate (O) (2,3) circle (0.05) node [above] {$O_1$} coordinate (O1);\n\\draw [dashed] (0,0) -- (4,0) -- (0,3) -- ({2+2*cos(-75)},{0.5*sin(-75)}) node [below] {$P$} coordinate (P) (O) -- (P) -- (0,3) (4,0) -- (P) -- (0,0);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BP \\perp A_1P$;\\\\\n(2) 若圆柱 $OO_1$ 的体积为 $12 \\pi$, $OA=2$, $A_1P=3 \\sqrt{2}$, 求二面角 $A_1-BP-A$ 的大小.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023368": {
|
||||
"id": "023368",
|
||||
"content": "已知平行六面体 $ABCD-A_1B_1C_1D_1$ 中, $A_1A \\perp$ 平面 $ABCD, AB=4$, $AD=2$. 若 $B_1D \\perp BC$, 直线 $B_1D$ 与平面 $ABCD$ 所成的角等于 $30^{\\circ}$, 求平行六面体 $ABCD-A_1B_1C_1D_1$ 的体积.\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+\\m/2},0,{-\\m*sqrt(3)/2}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({\\m/2},0,{-\\m*sqrt(3)/2}) 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 [dashed] (D)--(B_1);\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023369": {
|
||||
"id": "023369",
|
||||
"content": "如图, 在斜三棱柱 $ABC-A_1B_1C_1$ 中, $\\angle A_1AC=\\angle ACB=\\dfrac{\\pi}{2}$, $\\angle AA_1C=\\dfrac{\\pi}{6}$, 侧棱 $BB_1$ 与底面所成的角为 $\\dfrac{\\pi}{3}$, $AA_1=4 \\sqrt{3}$, $BC=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw ({-2*sqrt(2)},0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-2*sqrt(2)}) node [above right] {$C$} coordinate (C);\n\\draw (A) -- (B);\n\\draw [dashed] (A) -- (C) -- (B);\n\\draw (A) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed] (C) -- (C1) (A1) -- (C);\n\\draw (A) -- (A1) (B) -- (B1) (A1) -- (B1) (A1) -- (C1) -- (B1); \n\\end{tikzpicture}\n\\end{center}\n(1) 求斜三棱柱 $ABC-A_1B_1C_1$ 的体积 $V$;\\\\\n(2) 求斜三棱柱 $ABC-A_1B_1C_1$ 的表面积 $S$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"018719",
|
||||
"030489"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"023370": {
|
||||
"id": "023370",
|
||||
"content": "如图, $AB$ 是圆 $O$ 的直径, 点 $C$ 是圆 $O$ 上异于 $A, B$ 的点, $PO$ 垂直于圆 $O$ 所在的平面, 且 $PO=OB=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (-50:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(P)--(B)(P)--(C);\n\\draw [dashed] (A)--(B)(P)--(O)(A)--(C)--(B)(P)--(D)--(O);\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above right] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (-100:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw (A)--(P)--(B);\n\\draw ($(P)!0.6!(B)$) node [above right] {$E$} coordinate (E);\n\\draw [dashed] (O)--(E)--(C)(P)--(O)(B)--(C)(A)--(B)(O)--(C);\n\\draw [dashed] ;\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $D$ 为线段 $AC$ 的中点, 求证: $AC \\perp$ 平面 $PDO$;\\\\\n(2) 求三棱锥 $P-ABC$ 体积的最大值;\\\\\n(3) 若 $BC=\\sqrt{2}$, 点 $E$ 在线段 $PB$ 上, 求 $CE+OE$ 的最小值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "自拟题目",
|
||||
"edit": [
|
||||
"20240107\t杨懿荔"
|
||||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"014357"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"030001": {
|
||||
"id": "030001",
|
||||
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
|
||||
|
|
@ -641244,7 +641467,8 @@
|
|||
],
|
||||
"same": [],
|
||||
"related": [
|
||||
"023190"
|
||||
"023190",
|
||||
"023369"
|
||||
],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
|
|
|
|||
Reference in New Issue