收录高三寒假作业38新题
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@ -112,3 +112,6 @@
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20240125-120258
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023911:023914,019177,023915,020862,023916
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20240125-120516
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023917:023924
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@ -644961,6 +644961,166 @@
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"space": "4em",
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"unrelated": []
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},
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"023917": {
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"id": "023917",
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"content": "如图所示, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, 若 $M$、$N$ 分别为棱 $C_1D_1$、$C_1C$ 的中点, 则以下所有正确结论的序号是\\blank{50}.\\\\\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 [below] {$D$} 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 ($(C)!0.5!(C_1)$) node [right] {$N$} coordinate (N);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$M$} coordinate (M);\n\\draw (B_1)--(M)(B)--(N);\n\\draw [dashed] (A)--(M);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 直线 $AM$ 与 $CC_1$ 是相交直线;\\\\\n\\textcircled{2} 直线 $AM$ 与 $BN$ 是平行直线;\\\\\n\\textcircled{3} 直线 $BN$ 与 $MB_1$ 是异面直线;\\\\\n\\textcircled{4} 直线 $AM$ 与 $DD_1$ 是异面直线.",
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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": "自拟题目",
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"edit": [
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"20240125\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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"023918": {
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"id": "023918",
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"content": "如图是表示一个正方体表面的一种平面展开图, 图中的四条线段 $AB$、$CD$、$EF$ 和 $GH$ 在原正方体中相互异面的有\\blank{50}对.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1]\n\\draw (0,0) -- (3,0) (0,1) -- (4,1) (1,-1) -- (2,-1) (2,2) node [above left] {$C$} -- (4,2);\n\\draw (0,0) -- (0,1) (1,-1) -- (1,1) (2,-1) -- (2,2) (3,0) -- (3,2) (4,1) -- (4,2);\n\\draw (2,-1) node [below right] {$F$} -- (1,0) node [below left] {$E$} coordinate (E) (2,2) -- (3,1) node [below right] {$D$};\n\\draw (0,0) node [below left] {$H$} -- (1,1) node [above] {$G$};\n\\draw (3,2) node [above] {$A$} -- (4,1) node [below right] {$B$};\n\\draw [dashed] (1,0) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n\\draw [dashed] (1,0) ++ (0,1) --++ (45:1/2) --++ (1,0) --++ (225:1/2);\n\\draw [dashed] (1,0) ++ (45:1/2) --++ (0,1) ++ (1,0) --++ (0,-1);\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": "自拟题目",
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"edit": [
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"20240125\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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"023919": {
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"id": "023919",
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"content": "如图, 棱长为 $1$ 的正方体 $ABCD-A_1B_1C_1D_1$, 若 $E$ 为棱 $BC$ 的中点, $F$ 为棱 $DD_1$ 的中点. 则异面直线 $EF$ 与 $BD_1$ 所成角的余弦值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-120:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$B$} coordinate (B);\n\\draw (B) ++ (\\l,0,0) node [below right] {$C$} coordinate (C);\n\\draw (B) ++ (\\l,0,-\\l) node [right] {$D$} coordinate (D);\n\\draw (B) ++ (0,0,-\\l) node [left] {$A$} coordinate (A);\n\\draw (B) -- (C) -- (D);\n\\draw [dashed] (B) -- (A) -- (D);\n\\draw (B) ++ (0,\\l,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above right] {$D_1$} coordinate (D_1);\n\\draw (A) ++ (0,\\l,0) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B_1) -- (C_1) -- (D_1) -- (A_1) -- cycle;\n\\draw (B) -- (B_1) (C) -- (C_1) (D) -- (D_1);\n\\draw [dashed] (A) -- (A_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F)(B)--(D_1);\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": "自拟题目",
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"edit": [
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"20240125\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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"023920": {
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"id": "023920",
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"content": "如图, 直线 $a$、$b$ 相交于点 $O$ 且 $a$、$b$ 成 $60^{\\circ}$ 角, 过点 $O$ 与 $a$、$b$ 都成 $60^{\\circ}$ 的直线有\\blank{50}条.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (30:2) node [right] {$a$} coordinate (a);\n\\draw (-30:2) node [right] {$b$} coordinate (b);\n\\draw (b) -- ($(b)!2!(O)$) (a) -- ($(a)!2!(O)$);\n\\draw pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = b--O--a};\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": "自拟题目",
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"edit": [
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"20240125\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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"023921": {
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"id": "023921",
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"content": "在正方体 $ABCD-A_1B_1C_1D_1$ 中, 若 $E$、$F$ 分别是 $AA_1$ 与 $CC_1$ 的中点, 则直线 $ED$ 与 $D_1F$ 所成角的大小是\\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": "自拟题目",
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"edit": [
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"20240125\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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"023922": {
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"id": "023922",
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"content": "若空间中有两条直线, 则``这两条直线为异面直线''是``这两条直线没有公共点''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}",
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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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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20240125\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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"023923": {
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"id": "023923",
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"content": "过正方体 $ABCD-A_1B_1C_1D_1$ 的顶点 $A$ 作直线 $l$, 使 $l$ 与棱 $AB$、$AD$、$AA_1$ 所成的角都相等, 这样的直线 $l$ 可以作 \\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}",
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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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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20240125\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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"023924": {
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"id": "023924",
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"content": "如图, 若 $P$ 是 $\\triangle ABC$ 所在平面外一点, $PA \\neq PB$, $PN \\perp AB, N$ 为垂足. $M$ 为 $AB$ 的中点, 求证: $PN$ 与 $MC$ 为异面直线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (1.4,-0.8) node [below] {$B$} coordinate (B);\n\\draw (1.5,1.3) node [above] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw ($(A)!0.7!(M)$) node [below left] {$N$} coordinate (N);\n\\draw (A)--(B)--(C)--(P)--cycle(B)--(P)(P)--(N);\n\\draw [dashed] (C)--(M)(A)--(C);\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": "自拟题目",
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"edit": [
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"20240125\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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"030001": {
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"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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