Merge commit '96b0'
This commit is contained in:
commit
7b15ea0d6b
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@ -693598,14 +693598,14 @@
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},
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"032052": {
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"id": "032052",
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"content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$平面$ABCD$, 且四边形$ABCD$为直角梯形, $\\angle ABC=\\angle BAD=\\dfrac{\\pi}{2}$, $PA=AD=2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D) (Q)--(C);\n\\draw [dashed] (B)--(A)--(D) (A)--(P); \n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$P-ABCD$的表面积;\\\\\n(2) 若$P, A, C, D$四点在同一球面上, 求该球的体积.",
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"content": "如图, 在四棱锥$P-ABCD$中, 已知$PA \\perp$平面$ABCD$, 且四边形$ABCD$为直角梯形, $\\angle ABC=\\angle BAD=\\dfrac{\\pi}{2}$, $PA=AD=2$, $AB=BC=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1,0,0) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(B) (P)--(C) (P)--(D) (B)--(C)--(D) (Q)--(C);\n\\draw [dashed] (B)--(A)--(D) (A)--(P); \n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$PC$与$AB$所成角的大小; \\\\\n(2) 求四棱锥$P-ABCD$的表面积;\\\\\n(3) 若$P, A, C, D$四点在同一球面上, 求该球的体积.",
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"objs": [],
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"tags": [
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"第六单元",
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"2023届高三-第二轮复习讲义-09-立体几何综合"
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],
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"genre": "4em",
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"ans": "(1) $\\dfrac 92+\\dfrac{\\sqrt{5}}2+\\sqrt{3}$; (2) $\\dfrac{8\\sqrt{2}}3\\pi$",
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"ans": "(1);(2) $\\dfrac 92+\\dfrac{\\sqrt{5}}2+\\sqrt{3}$; (3) $\\dfrac{8\\sqrt{2}}3\\pi$",
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"solution": "",
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"duration": -1,
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"usages": [],
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@ -694324,6 +694324,206 @@
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"space": "4em",
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"unrelated": []
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},
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"032079": {
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"id": "032079",
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"content": "设平面$\\alpha$与平面$\\beta$相交于直线$l$, 直线$a \\subset \\alpha$, 直线$b \\subset \\beta$, $a \\cap b=M$, 则$M$\\blank{50}$l$(用符号表示).",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032080": {
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"id": "032080",
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"content": "下列四个条件中, 能确定一个平面的是\\blank{50}(填写编号).\\\\\n \\textcircled{1}空间任意三点; \\textcircled{2}空间两条平行直线; \\textcircled{3}一条直线和一个点; \\textcircled{4}两两相交且不共点的三条直线.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032081": {
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"id": "032081",
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"content": "在长方体$ABCD-A_1B_1C_1D_1$中, 若$AB=3, BC=4, AA_1=2$, 则异面直线$B_1B$与$DC$之间的距离为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032082": {
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"id": "032082",
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"content": "如图, 圆锥的底面半径$OA=2$, 高$PO=6$, 点$C$是底面直径$AB$所对弧的中点, 点$D$是母线$PA$的中点, 则异面直线$CD$与$AB$所成角的大小是\\blank{50}.\n \\begin{center}\n \\begin{tikzpicture}[scale = 0.8]\n \\draw (-2,0) node [left] {$A$} coordinate (A);\n \\draw (2,0) node [right] {$B$} coordinate (B);\n \\draw (0,0) node [below right] {$O$} coordinate (O);\n \\draw (0,6)node [above] {$P$} coordinate (P);\n \\draw ({2*cos(255)},{0.5*sin(255)}) node [below] {$C$} coordinate (C);\n \\draw ($(P)!0.5!(A)$) node [left] {$D$} coordinate (D);\n \\draw (A) arc (180:360:2 and 0.5) -- (P) -- (A) (P) -- (C);\n \\draw [dashed] (A) arc (180:0:2 and 0.5) -- (A) (P)--(O)--(C) (D) -- (C);\n \\end{tikzpicture}\n \\end{center}",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032083": {
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"id": "032083",
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"content": "已知$AB\\cap \\alpha =B$, $l \\subset \\alpha$, $B\\notin l$, 则$AB$与$l$的位置关系是\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032084": {
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"id": "032084",
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"content": "在正方体$ABCD-A_1B_1C_1D_1$的所有棱所在的直线中, 与直线$AB$垂直且异面的直线共有\\blank{50}条.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032085": {
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"id": "032085",
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"content": "如果圆锥的底面积为$\\pi$, 母线长为$2$, 那么该圆锥的高为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032086": {
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"id": "032086",
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"content": "若平面$\\alpha$截半径为$2$的球$O$所得的截面圆的面积为$\\pi$, 则球心$O$到平面$\\alpha$的距离为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032087": {
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"id": "032087",
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"content": "若体积为$8$的正方体的各个顶点均在一球面上, 则该球的体积为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"032088": {
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"id": "032088",
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"content": "如图, 在正四棱锥$P-ABCD$中, $PA=AB=2\\sqrt{2}$, 点$E,F$分别为线段$PB,PD$的中点, 平面$AEF$与棱$PC$的交点为$G$.\\\\\n (1)若$M,N,Q$分别为线段$AD,AB,BE$的中点, 求证: 平面$MNQ \\mathop{//}$平面$AEGF$.\\\\\n (2)求异面直线$AE$与$PF$所成角的大小;\\\\\n (3)求平面$AEGF$与平面$ABCD$所成二面角的大小;\\\\\n \\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 1.5]\n \\draw (-1.414,0,1.414) node [left] {$A$} coordinate (A);\n \\draw (1.414,0,1.414) node [right] {$B$} coordinate (B);\n \\draw (1.414,0,-1.414) node [right] {$C$} coordinate (C);\n \\draw (-1.414,0,-1.414) node [below] {$D$} coordinate (D);\n \\draw (0,2,0) node [above] {$P$} coordinate (P);\n \\draw (P) -- (A) (P) -- (B) (P) -- (C) (A) -- (B) -- (C);\n \\draw [dashed] (P) -- (D) (A) -- (D) -- (C);\n \\draw ($(P)!0.5!(D)$) node [below] {$F$} coordinate (F);\n \\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n \\draw ($(P)!0.25!(C)$) node [right] {$G$} coordinate (G);\n \\draw ($(A)!0.5!(D)$) node [below] {$M$} coordinate (M);\n \\draw ($(A)!0.5!(B)$) node [below] {$N$} coordinate (N);\n \\draw ($(B)!0.5!(E)$) node [right] {$Q$} coordinate (Q);\n \\draw [dashed] (G) -- (F) -- (A) (M) -- (N);\n \\draw (G) -- (E) -- (A) (N) -- (Q);\n \\end{tikzpicture} \n \\end{center}",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "2024届高三二轮讲义新题补充",
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"edit": [
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"20240122\t毛培菁"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"040001": {
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"id": "040001",
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"content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}",
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