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录入2025届周末卷05补充题目

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"space": "4em",
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"023166": {
"id": "023166",
"content": "已知直线 $m,n$ 与平面 $\\alpha, \\beta$, 给出下列三个命题: \\textcircled{1} 若 $m \\parallel \\alpha$, $n \\parallel \\alpha$, 则 $m \\parallel n$; \\textcircled{2}若 $m \\parallel \\alpha$, $n \\perp \\alpha$, 则 $n \\perp m$; \\textcircled{3}若 $m \\perp \\alpha$, $m \\parallel \\beta$, 则 $\\alpha \\perp \\beta$.其中真命题是\\blank{50}.",
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"genre": "填空题",
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"023167": {
"id": "023167",
"content": "二面角 $\\alpha-l-\\beta$ 大小为 $105^{\\circ}$, 若两异面直线 $a$ 和 $b$ 分别垂直于二面角的二个面, 那么这两条异面直线所成的角的大小是\\blank{50}.",
"objs": [],
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"genre": "填空题",
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"023168": {
"id": "023168",
"content": "四面体 $ABCD$ 的各棱长都相等, $M, N$ 分别为 $BC, AD$ 的中点, 则异面直线 $AM$ 与 $CN$ 所成的角的大小为\\blank{50}.",
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"023169": {
"id": "023169",
"content": "将一块斜边长为 $8 \\mathrm{cm}$ 的等腰直角三角形的三角板的斜边放在看作平面的桌面上, 并让其中一条直角边所在直线与桌面所成的角为 $30^{\\circ}$, 则该三角板的斜边上的中线与桌面所成的角的大小为\\blank{50}.",
"objs": [],
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"genre": "填空题",
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"023170": {
"id": "023170",
"content": "已知 $AB$ 和 $CD$ 都垂直于平面 $\\alpha$, $B, D$ 分别是垂足. 若 $AB=4$, $CD=8$, $BD=5$,则 $AC=$\\blank{50}.",
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"023171": {
"id": "023171",
"content": "已知平面 $\\alpha \\parallel $ 平面 $\\beta$, 点 $P \\notin \\alpha, P \\notin \\beta$, 过 $P$ 分别作两条直线 $AB, CD$, 交 $\\alpha$ 于 $A, C$两点, 交 $\\beta$ 于 $B, D$ 两点. 若 $AB=10$, $PB=6$, $AC=4$, 则 $BD=$\\blank{50}.",
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"023172": {
"id": "023172",
"content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中. 若 $AB=AD=2 \\sqrt{3}$, $CC_1=\\sqrt{2}$, 则二面角 $C_1-BD-C$ 的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{{2*sqrt(2)/sqrt(3)}}\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)--(C_1);\n\\draw [dashed] (B)--(D)--(C_1);\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
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"genre": "填空题",
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"023173": {
"id": "023173",
"content": "如图, 梯形 $ABCD$ 的底边 $AD$ 在平面 $\\alpha$ 内, 另一边 $BC$与平面 $\\alpha$ 的距离为 $5 \\mathrm{cm}$, 若 $AD: BC=7: 3$, 那么梯形对角线的交点 $O$ 到 $\\alpha$ 的距离是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3, x = {(-5:1cm)}]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (7,0,0) node [below] {$D$} coordinate (D);\n\\draw (2,5,-2) node [above] {$B$} coordinate (B);\n\\draw (5,5,-2) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw [name path = AC] (A)--(C);\n\\draw [name path = BD] (B)--(D);\n\\draw [name intersections = {of = AC and BD, by = O}];\n\\draw (O) node [right] {$O$};\n\\draw (A) ++ (-3,0,3) --++ (13,0,0) --++ (0,0,-8) coordinate (T) ++ (-13,0,0) coordinate (S) --++ (0,0,8);\n\\path [name path = ST] (S)--(T);\n\\path [name path = AB] (A)--(B);\n\\path [name path = CD] (C)--(D);\n\\draw [name intersections = {of = ST and AB, by = U}];\n\\draw [name intersections = {of = ST and CD, by = V}];\n\\draw (S)--(U)(V)--(T);\n\\draw [dashed] (U)--(V);\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"023174": {
"id": "023174",
"content": "在长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=2$, $BB_1=BC=1, E$ 为线段 $D_1C_1$ 内一点, 直线 $DC$ 到平面 $ABE$ 的距离是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
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"solution": "",
"duration": -1,
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"023175": {
"id": "023175",
"content": "已知 $l \\parallel \\alpha$, $l \\parallel \\beta$, $\\alpha \\bigcap \\beta=a$, 那么 $l$ 与 $a$ 的位置关系是\\bracket{20}.\n\\fourch{异面}{平行}{垂直}{相交但不垂直}",
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"tags": [],
"genre": "选择题",
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"023176": {
"id": "023176",
"content": "已知平面 $\\alpha \\parallel $ 平面 $\\beta$, 对于命题:\\\\\n\\textcircled{1} 对于 $\\beta$, 存在 $\\alpha$ 上有无数条直线 $\\parallel \\beta$.\\\\\n\\textcircled{2} 对于 $\\alpha$ 上一条给定的直线 $m$, 存在 $\\beta$ 上无数条直线与 $m$ 异面.\n下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}",
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"solution": "",
"duration": -1,
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"023177": {
"id": "023177",
"content": "已知三个不同的平面两两之间均有公共点. 对于命题:\\\\\n\\textcircled{1} 若存在某一个点同时在这三个平面上, 则必存在另一个点也同时在这三个平面上;\\\\\n\\textcircled{2} 若不存在任意点同时在这三个平面上, 则这三个平面两两之间的公共点的集合为三条互相平行的直线.\n下列判断正确的是\\bracket{20}.\n\\twoch{\\textcircled{1}和\\textcircled{2}均为真命题}{\\textcircled{1}和\\textcircled{2}均为假命题}{\\textcircled{1}为真命题, \\textcircled{2}为假命题}{\\textcircled{1}为假命题, \\textcircled{2}为真命题}",
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"023178": {
"id": "023178",
"content": "如图, 在长方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $E$ 是棱 $DD_1$ 的中点, $BD_1$ 与底面 $ABCD$ 所成的角为 $60^{\\circ}$, $AB=AD=1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{{2*sqrt(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 [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 [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 ($(D)!0.5!(D_1)$) node [left] {$E$} coordinate (E);\n\\draw (A)--(B_1);\n\\draw [dashed] (A)--(C)--(E)--cycle(B)--(D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD_1 \\parallel $ 平面 $EAC$;\\\\\n(2) 求 $AB_1$ 与平面 $BB_1D_1D$ 所成的角的正弦值.",
"objs": [],
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"genre": "解答题",
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"023179": {
"id": "023179",
"content": "正方体 $ABCD-A_1B_1C_1D_1$ 中, 求:\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 (A_1)--(B)(D_1)--(B_1)(B)--(C_1);\n\\draw [dashed] (A)--(C)--(D_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) $BA_1$ 与平面 $ABC_1D_1$ 所成角大小;\\\\\n(2) $B_1D_1$ 与平面 $ACD_1$ 所成角大小.",
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"023180": {
"id": "023180",
"content": "如图, 四边形 $ABCD$ 是直角梯形, $\\angle DAB=90^{\\circ}$, $PC \\perp$ 平面 $ABCD$, $BD \\perp AC$, $AB=2 \\sqrt{3}$, $AD=\\sqrt{6}$, $PC=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0,0) node [above right] {$D$} coordinate (D);\n\\draw (D) ++ (0,0,{sqrt(6)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ ({2*sqrt(3)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(P)--cycle;\n\\draw [dashed] (A)--(D)--(C)--(B)(P)--(D)(P)--(C)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 直线 $BD \\perp PA$;\\\\\n(2) 求二面角 $P-AB-D$ 的大小;\\\\\n(3) 证明:直线 $PA$、$CB$ 是异面直线;\\\\\n(4) 求异面直线 $PA$、$CB$ 所成的角的余弦值.",
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"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.",