录入2025届周末卷04补充题目
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"023153": {
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"id": "023153",
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"content": "棱长为 $1$ 的正四面体 $ABCD$ 中, $E, F , G$ 分别是棱 $AB, AC, AD$ 的中点.\\\\\n(1) 点 $A$ 和平面 $BCD$ 的距离为\\blank{50};\\\\\n(2) 直线 $EF$ 和平面 $BCD$ 的距离为\\blank{50};\\\\\n(3) 平面 $EFG$ 与平面 $BCD$ 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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"023154": {
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"id": "023154",
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"content": "如图, 平行四边形的一个顶点 $A$ 在平面 $\\alpha$ 内, 其余顶点在 $\\alpha$ 的同侧, 已知其中有两个顶点到 $\\alpha$ 的距离分别为 $1$ 和 $2$, 那么剩下的一个顶点到平面 $\\alpha$ 的距离可能是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (1,1,-1) node [right] {$B$} coordinate (B);\n\\draw (-2,2,0) node [left] {$D$} coordinate (D);\n\\draw (-1,3,-1) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw (A) ++ (-3,0,2) --++ (5,0,0) --++ (0,0,-4) coordinate (V) ++ (-5,0,0) coordinate (U) (A)++(-3,0,2) --++ (0,0,-4);\n\\path [name path = AD] (A)--(D);\n\\path [name path = AB] (A)--(B);\n\\path [name path = UV] (U)--(V);\n\\draw [name intersections = {of = UV and AD, by = S}];\n\\draw [name intersections = {of = UV and AB, by = T}];\n\\draw (U)--(S)(T)--(V);\n\\draw [dashed] (S)--(T);\n\\draw (A) ++ (-3,0,2) node [above right = 0 and 0.2] {$\\alpha$};\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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"023155": {
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"id": "023155",
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"content": "将等腰直角 $\\triangle ABC$ 绕斜边 $AB$ 翻折到平面 $ABC'$, 已知二面角 $C-AB-C'$ 的大小为 $60^{\\circ}$, 则 $CC'$ 与平面 $ABC$ 所成的角大小为\\blank{50}.",
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"genre": "填空题",
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"ans": "",
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"023156": {
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"id": "023156",
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"content": "如图所示, 正方体 $D-A_1B_1C_1D_1$ 的棱长为 $1$, $P$ 为 $BC$ 的中点, $Q$ 为线段 $CC_1$ 上的动点, 过点 $A, P, Q$ 的平面截该正方体所得的截面记为 $S$, 则下列命题正确的是\\blank{50}.(写出所有正确命题的编号)\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 ($(B)!0.5!(C)$) circle (0.03) node [below right] {$P$} coordinate (P);\n\\filldraw ($(C)!0.7!(C_1)$) circle (0.03) node [right] {$Q$} coordinate (Q);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 当 $0<CQ<\\dfrac{1}{2}$ 时, $S$ 为四边形;\\\\\n\\textcircled{2} 当 $CQ=\\dfrac{1}{2}$ 时, $S$ 为等腰梯形;\\\\\n\\textcircled{3} 当 $CQ=\\dfrac{3}{4}$ 时, $S$ 与 $C_1D_1$ 的交点 $R$ 满足 $C_1R=\\dfrac{1}{3}$;\\\\\n\\textcircled{4} 当 $\\dfrac{3}{4}<CQ<1$ 时, $S$ 为六边形;\\\\\n\\textcircled{5} 当 $CQ=1$ 时, $S$ 的面积为 $\\dfrac{\\sqrt{6}}{2}$.",
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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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"duration": -1,
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"023157": {
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"id": "023157",
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"content": "若 $l, m$ 是两条不同的直线, $m$ 垂直于平面 $\\alpha$, 则``$l \\perp m$''是``$l \\parallel \\alpha$''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}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"remark": "",
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"space": "",
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"023158": {
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"id": "023158",
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"content": "线段 $OA, OB, OC$ 不共面, $\\angle AOB=\\angle AOC=\\angle COA=60^{\\circ}$, $OA=1$, $OB=2$, $OC=3$, 则 $\\triangle ABC$ 是\\bracket{20}.\n\\fourch{等边三角形}{非等边的等腰三角形}{锐角三角形}{钝角三角形}",
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"objs": [],
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"ans": "",
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"duration": -1,
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},
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"023159": {
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"id": "023159",
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"content": "已知正方体的棱长为 $1$, 每条棱所在直线与平面 $\\alpha$ 所成的角都相等, 则平面 $\\alpha$ 截此正方体所得截面面积的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{3 \\sqrt{3}}{4}$}{$\\dfrac{2 \\sqrt{3}}{3}$}{$\\dfrac{3 \\sqrt{2}}{4}$}{$\\dfrac{\\sqrt{3}}{2}$}",
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"objs": [],
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"ans": "",
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"duration": -1,
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"023160": {
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"id": "023160",
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"content": "如图, $SA \\perp$ 底面 $ABC$, $AB \\perp BC$, $DE$ 垂直平分 $SC$ 且分别交 $AC$、$SC$ 于 $D$、$E$, 又 $SA=AB$, $SB=BC$, 求以 $BD$ 为棱, 以 $BDE$、$BDC$ 为面的二面角的大小. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,1,0) node [above] {$S$} coordinate (S);\n\\draw ({sqrt(3)/3},0,{sqrt(6)/3}) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,0) node [right] {$C$} coordinate (C);\n\\draw ($(S)!0.5!(C)$) node [above] {$E$} coordinate (E);\n\\draw ($(A)!{1/3}!(C)$) node [above right] {$D$} coordinate (D);\n\\draw (S)--(B)--(C)--cycle(E)--(B)--(A)--(S);\n\\draw [dashed] (A)--(C)(E)--(D)--(B);\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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"space": "4em",
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"023161": {
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"id": "023161",
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"content": "河堤斜面与水平面所成角为 $60^{\\circ}$, 堤面上有一条直道 $CD$, 它与堤脚的水平线 $AB$ 的夹角为 $30^{\\circ}$, 沿着这条直道从堤脚上行走到 $10$ 米时, 人升高了多少(精确到 0.1 米)? \n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(30:1cm)}, y={(150:1cm)}, z = {(90:1cm)}, scale = 1.5]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,2,0) node [left] {$A$} coordinate (A);\n\\draw (B) ++ (1,0,0) coordinate (S);\n\\draw (A) ++ (1,0,0) coordinate (T);\n\\draw (B) ++ ({1/2},0,{sqrt(3)/2}) coordinate (U);\n\\draw (A) ++ ({1/2},0,{sqrt(3)/2}) coordinate (V);\n\\path [name path = BU] (B)--(U);\n\\path [name path = ST] (S)--(T);\n\\draw [name intersections = {of = BU and ST, by = P}];\n\\draw (A)--(V)--(U)--(B)--cycle;\n\\draw (B)--(S)--(P);\n\\draw [dashed] (A)--(T)--(P);\n\\draw ($(A)!0.3!(B)$) node [below left] {$C$} coordinate (C);\n\\draw (C) --++ ({1/4},{-sqrt(3)/2},{sqrt(3)/4}) node [right] {$D$} coordinate (D);\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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"space": "4em",
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"023162": {
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"id": "023162",
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"content": "如图, 已知二面角 $\\alpha-PQ-\\beta$ 为 $60^{\\circ}$, 点 $A$ 和点 $B$ 分别在平面 $\\alpha$ 和平面 $\\beta$ 内, 点 $C$ 在棱 $PQ$ 上, $\\angle ACP=\\angle BCP=30^{\\circ}$, $CA=CB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-3,0,0) node [left] {$P$} coordinate (P);\n\\draw (1,0,0) node [right] {$Q$} coordinate (Q);\n\\draw (P)++(0,0,2) coordinate (P_1);\n\\draw (Q)++(0,0,2) coordinate (Q_1);\n\\draw (P)++(0,{sqrt(3)},1) coordinate (P_2);\n\\draw (Q)++(0,{sqrt(3)},1) coordinate (Q_2);\n\\path [name path = PQ] (P)--(Q);\n\\draw (P)--(P_1)--(Q_1)--(Q)--(Q_2)--(P_2)--(P);\n\\draw (0,0,0) node [above] {$C$} coordinate (C);\n\\draw (C) --++ ({-sqrt(3)},0,1) node [left] {$A$} coordinate (A);\n\\draw (C) --++ ({-sqrt(3)},{sqrt(3)/2},{1/2}) node [left] {$B$} coordinate (B);\n\\draw [name path = AB] (A)--(B);\n\\draw [name intersections = {of = AB and PQ, by = S}];\n\\draw (P)--(S)(C)--(Q);\n\\draw [dashed] (S)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AB \\perp PQ$;\\\\\n(2) 求点 $B$ 到平面 $\\alpha$ 的距离;\\\\\n(3) 求二面角 $B-AC-P$ 的大小.",
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"objs": [],
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"genre": "解答题",
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"ans": "",
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"space": "4em",
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},
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"023163": {
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"id": "023163",
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"content": "如图, 在几何体 $P-ABCD$ 中, 底面 $ABCD$ 是正方形, $PA \\perp$ 底面 $ABCD$, $PA=AB$, $Q$ 是 $PC$ 中点, $AC, BD$ 交于 $O$ 点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-125:0.5cm)}]\n\\draw (0,0,0) node [left] {$A$} 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 [left] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw ($(C)!0.5!(P)$) node [above] {$Q$} coordinate (Q);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C)(B)--(Q)--(D);\n\\draw [dashed] (P)--(A)--(C)(B)--(D)(B)--(A)--(D)(Q)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角 $Q-BD-C$ 的大小;\\\\\n(2) 求二面角 $B-QD-C$ 的大小.",
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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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"023164": {
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"id": "023164",
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"content": "如图的几何图形 $P-ABCD$ 的底面为直角梯形, $AB \\parallel DC$, $\\angle DAB=90^{\\circ}$, $PA \\perp$ 底面 $ABCD$, 且 $PA=AD=DC=\\dfrac{1}{2}AB=1$, $M$ 是 $PB$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,1) node [below] {$D$} coordinate (D);\n\\draw (1,0,1) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [above] {$M$} coordinate (M);\n\\draw (P)--(D)--(C)--(B)--cycle(M)--(C)(P)--(C);\n\\draw [dashed] (D)--(A)--(B)(P)--(A)--(C)(A)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 若过三点 $M, D, C$ 的平面交 $PA$ 于点 $N$, 判断四边形 $MNDC$ 的形状, 并说明理由;\\\\\n(2) 求二面角 $M-AC-B$ 的大小;\\\\\n(3) 求点 $M$ 到平面 $PAC$ 的距离.",
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"objs": [],
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"genre": "解答题",
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"ans": "",
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"023165": {
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"id": "023165",
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"content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AD=AA_1=1$, $AB=2$, 点 $E$ 在棱 $AB$ 上移动.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.6]\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 ($(B)!{sqrt(3)/2}!(A)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A_1)--(D)(E)--(C)--(D_1)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, scale = 1.6]\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 ($(B)!{1/2}!(A)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A)--(D_1)(E)--(D)--(B_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 判断异面直线 $D_1E$ 与 $A_1D$ 所成角的大小是否随点 $E$ 的移动而改变? 并证明你的结论;\\\\\n(2) 求点 $E$ 移动到何位置时, 二面角 $D_1-EC-D$ 的大小为 $\\dfrac{\\pi}{4}$;\\\\\n(3) 若点 $E$ 是棱 $AB$ 的中点, 判断直线 $AD_1$ 与平面 $B_1DE$ 的位置关系, 并证明你的结论.",
|
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"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "25届周末卷补充题目",
|
||||
"edit": [
|
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"20240106\t杨懿荔"
|
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],
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||||
"same": [],
|
||||
"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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"030001": {
|
||||
"id": "030001",
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"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.",
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Reference in New Issue