收录高三寒假作业41新题
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20240125-122103
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20240125-122103
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20240125-122347
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"space": "4em",
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"023936": {
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"id": "023936",
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"content": "若正三棱柱 $ABC-A_1B_1C_1$ 的侧棱长与底面边长相等, 则 $AB_1$ 与侧面 $ACC_1A_1$ 所成角的正弦值等于\\blank{50}.",
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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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"022721"
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"023937": {
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"id": "023937",
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"content": "如图, 正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$ , 将该正方体沿对角面 $BB_1D_1D$ 切成两块, 再将这两块拼接成一个不是正方体的四棱柱, 那么所得四棱柱的全面积为\\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 [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\\fill [pattern = north east lines] (B)--(D)--(D_1)--(B_1)--cycle;\n\\draw (B_1)--(D_1);\n\\draw [dashed] (B)--(D);\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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"023938": {
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"id": "023938",
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"content": "若某圆柱底面的半径为 $1$, 高为 $2$, 则该圆柱的表面积为\\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": "自拟题目",
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"edit": [
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"012247",
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"016770"
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"023939": {
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"id": "023939",
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"content": "已知一个正方体与一个圆柱等高, 若侧面积相等, 则这个正方体和圆柱的体积之比为\\blank{50}.",
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"genre": "填空题",
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"duration": -1,
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"origin": "自拟题目",
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"020995"
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"023940": {
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"id": "023940",
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"content": "正三棱柱 $ABC-A_1B_1C_1$ 的各棱长都 $2$, 若 $E$、$F$ 分别是 $AB$、$A_1C_1$ 的中点, 则 $EF$ 的长是\\bracket{20}.\n\\fourch{$2$}{$\\sqrt{3}$}{$\\sqrt{5}$}{$\\sqrt{7}$}",
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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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"origin": "自拟题目",
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"023941": {
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"id": "023941",
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"content": "如图, 在正方体 $ABCD-A_1B_1C_1D_1$ 中, 点 $P$ 是线段 $BC_1$ 上的一个动点, 有下列三个结论: \\textcircled{1} $A_1P \\parallel $ 平面 $ACD_1$; \\textcircled{2} $B_1D \\perp A_1P$; \\textcircled{3} 平面 $A_1PB \\perp$ 平面 $B_1CD$. 其中所有正确结论的序号是\\bracket{20}.\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 (B)--(C_1);\n\\draw ($(B)!0.6!(C_1)$) node [right] {$P$} coordinate (P);\n\\draw [dashed] (A_1)--(P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{3}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{2}}",
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"genre": "选择题",
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"duration": -1,
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"origin": "自拟题目",
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"023942": {
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"id": "023942",
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"content": "如图, $AA_1$ 是圆柱的一条母线, $AB$ 是圆柱的底面直径, $C$ 在圆柱下底面圆周上, $M$ 是线段 $A_1C$ 的中点. 已知 $AA_1=AC=4$, $BC=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (-2.5,0) node [left] {$A$} coordinate (A);\n\\draw (2.5,0) node [right] {$B$} coordinate (B);\n\\draw (A) ++ (0,4) node [left] {$A_1$} coordinate (A_1);\n\\draw (-70:2.5 and {2.5/4}) node [below] {$C$} coordinate (C);\n\\draw ($(A_1)!0.5!(C)$) node [above right] {$M$} coordinate (M);\n\\draw [dashed] (A)--(C)--(B)--cycle(A_1)--(C)(A)--(M);\n\\draw [dashed] (A) arc (180:0:2.5 and {2.5/4});\n\\draw (A)--++(0,4)(B)--++(0,4)(A_1)arc (180:-180:2.5 and {2.5/4})(A) arc (180:360:2.5 and {2.5/4});\n\\end{tikzpicture}\n\\end{center}\n(1) 求圆柱的侧面积;\\\\\n(2) 求证: $BC \\perp AM$.",
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"genre": "解答题",
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"ans": "",
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"origin": "自拟题目",
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"space": "4em",
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"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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"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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