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录入25届周末卷02补充题目

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@ -619863,6 +619863,326 @@
"space": "4em",
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"023126": {
"id": "023126",
"content": "判断下列命题的真假(用``T''或``F''标记):\\\\\n\\blank{20}(1) 直线 $l$ 上两点, 到平面 $\\alpha$ 的距离相等, 则 $l \\parallel \\alpha$;\\\\\n\\blank{20}(2) 直线 $l$ 与平面 $\\alpha$ 内两条直线垂直, 则 $l \\perp \\alpha$;\\\\\n\\blank{20}(3) 如果直线 $l$ 与平面 $\\alpha$ 内无数条直线垂直, 则 $l \\perp \\alpha$;\\\\\n\\blank{20}(4) 过直线上一点只有一个平面和这条直线垂直;\\\\\n\\blank{20}(5) 过直线外一点只有一个平面和这条直线垂直;\\\\\n\\blank{20}(6) 垂直于同一平面的两直线平行;\\\\\n\\blank{20}(7) 过一点只能有一条直线和一个平面垂直;\\\\\n\\blank{20}(8) 与同一平面相交成等角的直线彼此平行.",
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"023127": {
"id": "023127",
"content": "设 $a, b$ 是两条异面直线, 判断下列命题的真假(用``T''或``F''标记):\\\\\n\\blank{20}(1) 过直线 $a$ 有且只有一个平面平行于 $b$;\\\\\n\\blank{20}(2) 经过 $a$ 有且只有一个平面垂直于 $b$;\\\\\n\\blank{20}(3) 过空间任一点必可做一条直线与 $a$ 和 $b$ 都相交\\\\\n\\blank{20}(4) 有且仅有一条直线与 $a$ 和 $b$ 都垂直;\\\\\n\\blank{20}(5) 有一个平面与 $a$ 和 $b$ 都垂直.",
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"023128": {
"id": "023128",
"content": "若两条直线没有公共点, 则这两条直线的位置关系是\\blank{50}.",
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"023129": {
"id": "023129",
"content": "如图是一个正方体的平面展开图, 将它还原为正方体后:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0) rectangle (4,1);\n\\draw (1,-1) rectangle (2,2);\n\\draw (0,1) node [above left] {$C$} -- (1,0) node [below left] {$A$} -- (2,1) node [above right] {$B$} (3,1) node [above] {$M$} -- (4,0) node [below right] {$N$};\n\\draw (3,0) -- (3,1); \n\\end{tikzpicture}\n\\end{center}\n(1) 直线 $MN$ 与 $AB$ 的位置关系是\\blank{50}.\\\\\n(2) 直线 $MN$ 与 $\\mathrm{AC}$ 的位置关系是\\blank{50}.",
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"023130": {
"id": "023130",
"content": "不在平面 $\\alpha$ 上的一条直线 $l$ 与平面 $\\alpha$ 上的两条平行线垂直, 则 $l$ 与 $\\alpha$ 的位置关系为\\blank{50}.",
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"023131": {
"id": "023131",
"content": "设 $a$、$b$ 是不在平面 $M$ 上的两条直线, 且 $a \\parallel M$, 那么 $a \\parallel b$ 是 $b \\parallel M$ 的\\blank{50}条件.",
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"genre": "填空题",
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"023132": {
"id": "023132",
"content": "四面体 $PABC$ 中, 若 $PA \\perp$ 面 $BAC$, $\\angle ABC=90^{\\circ}$, 则该四面体中直角三角形个数\\blank{50}.",
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"genre": "填空题",
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"023133": {
"id": "023133",
"content": "在空间四边形 $ABCD$ 中, $AB=CD$, $AB$ 与 $CD$ 所在直线所成的角为 $\\theta$, $M$、$N$ 分别为边 $BC$、$AD$ 的中点,异面直线 $MN$、$CD$ 所成的角为 $\\alpha$. 若 $\\theta=90^{\\circ}$, 则 $\\alpha$ 为\\blank{50}; 若 $\\theta=60^{\\circ}$, 则 $\\alpha$ 为\\blank{50}.",
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"023134": {
"id": "023134",
"content": "正方体 $ABCD-A_1B_1C_1D_1$ 中, $E$ 是棱 $C_1D_1$ 的中点.\\\\\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 ($(C_1)!0.5!(D_1)$) node [above] {$E$} coordinate (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 异面直线 $AB_1$ 与 $BC_1$ 所成的角的大小为\\blank{50};\\\\\n(2) 异面直线 $CE$ 与 $AB_1$ 所成的角的大小为\\blank{50};\\\\\n(3) 异面直线 $CE$ 与 $AD_1$ 所成的角的大小为\\blank{50}.",
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"genre": "填空题",
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"023135": {
"id": "023135",
"content": "$a$、$b$、$c$ 为三条不重合的直线, $\\alpha$ 为平面, 现有下列六个命题:\\\\\n\\textcircled{1} 若$a \\parallel c$且$b \\parallel c$, 则$a \\parallel b$;\\\\\n\\textcircled{2} 若$a \\parallel \\alpha$且$b \\parallel \\alpha$, 则$a \\parallel b$;\\\\\n\\textcircled{3} 若$a \\perp b$且$c \\perp b$, 则$a \\perp c$;\\\\\n\\textcircled{4} 若$a \\parallel b$且$b \\perp \\alpha$, 则$a \\perp \\alpha$;\\\\\n\\textcircled{5} 若$a \\parallel c$且$c \\parallel \\alpha$, 则$a \\parallel \\alpha$;\\\\\n\\textcircled{6} 若$a \\perp c$且$c \\perp \\alpha$, 则$a \\parallel \\alpha$.\\\\\n其中, 正确的命题是\\blank{50}.\n(写出所有正确命题的序号)",
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"023136": {
"id": "023136",
"content": "设 $A$ 是不在边长为 $a$ 的正三角形 $BCD$ 所在平面上的一点, $M, N$ 分别是 $\\triangle ABC$ 与 $\\triangle ACD$ 的重心, 则线段 $MN$ 的长是\\blank{50}.",
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"023137": {
"id": "023137",
"content": "若直线 $a$ 不平行于平面 $\\alpha$, 且 $a$ 不在平面 $\\alpha$ 上, 则下列结论成立的是\\bracket{20}.\n\\twoch{$\\alpha$ 内的所有直线与 $a$ 异面}{$\\alpha$ 内不存在与 $a$ 平行的直线}{$\\alpha$ 内存在唯一的直线与 $a$ 平行}{$\\alpha$ 内的直线与 $a$ 都相交}",
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"genre": "选择题",
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"023138": {
"id": "023138",
"content": "如图, $\\alpha \\cap \\beta=l$, $C \\in \\alpha$, $E \\in \\beta$, $D \\in \\beta$, 过 $C, D, E$ 三点作平面 $CDE$, 画出该平面分别与平面 $\\alpha$、平面 $\\beta$ 的交线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (O) -- (2,0,0) node [below left = 0 and 0.4] {$\\beta$} --++(0,0,2) --++ (-2,0,0) coordinate (P) -- cycle;\n\\draw (0,0,0) --++ (-1,2,0) node [below = 0.3] {$\\alpha$} coordinate (Q) --++ (0,0,2) --++ (1,-2,0);\n\\filldraw (0.5,0,1.5) circle (0.03) node [above] {$E$} coordinate (E);\n\\filldraw (1.5,0,1.5) circle (0.03) node [above] {$D$} coordinate (D);\n\\filldraw ($(O)!0.7!($(P)!0.7!(Q)$)$) circle (0.03) node [left] {$C$} coordinate (C);\n\\end{tikzpicture}\n\\end{center}",
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"023139": {
"id": "023139",
"content": "如图, 从一点 $A$ 向相交平面 $\\alpha$ 与 $\\beta$ 分别作垂线 $AB, AC$ 于 $B, C$, 又从 $C$ 作 $CD \\perp \\alpha$ 于 $D, \\alpha \\cap \\beta=l$, 求证: $l \\perp BD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,1) coordinate (O);\n\\draw (-1,0,1) coordinate (S);\n\\draw (3,0,1) coordinate (T);\n\\draw (-0.3,-0.9,1) coordinate (Q);\n\\draw ($(Q)!3!(O)$) coordinate (P);\n\\foreach \\i in {O,S,T,Q,P}\n{\\draw (\\i) --++ (0,0,-2) coordinate (\\i_1);};\n\\draw (S)--(T)(Q)--(P)(O_1)--(P_1)(O_1)--(T_1);\n\\path [name path = S1O1] (S_1)--(O_1);\n\\path [name path = PQ] (P)--(Q);\n\\draw [name intersections = {of = S1O1 and PQ, by = U}];\n\\draw (S_1)--(U);\n\\path [name path = Q1O1] (Q_1)--(O_1);\n\\path [name path = ST] (S)--(T);\n\\draw [name intersections = {of = Q1O1 and ST, by = V}];\n\\draw (Q_1)--(V);\n\\draw ($(O)!0.5!(P_1)$) ++ (0.1,0.3,0) node [below left] {$C$} coordinate (C) --++ (1.2,-0.4,0) node [right] {$A$} coordinate (A);\n\\draw (C) ++ (0,-1.2,0) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,-0.8,0) node [right] {$B$} coordinate (B);\n\\draw (C)--(D)--(B)--(A);\n\\draw (T_1) node [below left = 0 and 0.2] {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}",
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"023140": {
"id": "023140",
"content": "空间三条线段 $AC=AB=BD=a$, $AC \\perp AB$, $AB \\perp BD$, 且 $AC$ 与 $BD$ 所成角为 $60^{\\circ}$, 求 $AB$ 与 $CD$ 所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [above left] {$B$} coordinate (B);\n\\draw (2,0,-2) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,{sqrt(3)},-1) node [left] {$C$} coordinate (C);\n\\draw (C)--(A)--(B)--(D);\n\\draw (A) ++ (-0.5,0,0.5) --++ (3,0,0) --++ (0,0,-3) --++ (-3,0,0) -- cycle;\n\\end{tikzpicture}\n\\end{center}",
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"023141": {
"id": "023141",
"content": "如图, 长方体 $ABCD-A_1B_1C_1D_1$ 中, $AB=BC=4$, $AA_1=5$, $P$ 是棱 $CC_1$ 上一点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\def\\l{4}\n\\def\\m{4}\n\\def\\n{5}\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 (A_1)--(B);\n\\draw [dashed] (A_1)--(D)--(B);\n\\draw ($(C)!{16/25}!(C_1)$) node [right] {$P$} coordinate (P);\n\\draw [dashed] (A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 当 $AP \\parallel $平面 $A_1B_1C_1D_1$ 时, 求点 $P$ 在哪个位置;\\\\ (2) 当 $AP \\perp$ 平面 $A_1BD$ 时, 求点 $P$ 在哪个位置.",
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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.",