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录入三道自拟题目

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wangweiye7840 2024-01-22 11:30:21 +08:00
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题库0.3/Problems.json
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"space": "",
"unrelated": []
},
"023632": {
"id": "023632",
"content": "证明: 设 $\\overrightarrow{a}$ 与 $\\overrightarrow{b}$、$\\overrightarrow{c}$ 是向量, $\\lambda$ 是实数, 则\\\\\n(1) 向量数量积的交换律: $\\overrightarrow{a}\\cdot \\overrightarrow{b}=\\overrightarrow{b}\\cdot \\overrightarrow{a}$;\\\\\n(2) 向量数量积对数乘的结合律: $(\\lambda \\overrightarrow{a}) \\cdot \\overrightarrow{b}=\\overrightarrow{a}\\cdot(\\lambda \\overrightarrow{b})$;\\\\\n(3) 向量数量积对加法的分配律: $\\overrightarrow{a}\\cdot(\\overrightarrow{b}+\\overrightarrow{c})=\\overrightarrow{a}\\cdot \\overrightarrow{b}+\\overrightarrow{a}\\cdot \\overrightarrow{c}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20240122\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023633": {
"id": "023633",
"content": "如图, 给定边长为 $6$ 的正三角形 $ABC$. 求 $\\overrightarrow{AB}\\cdot \\overrightarrow{AC}$ 和 $\\overrightarrow{AB}\\cdot \\overrightarrow{BC}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$C$} coordinate (C);\n\\draw (60:2) node [above] {$A$} coordinate (A);\n\\draw [->] (A)--(B);\n\\draw [->] (A)--(C);\n\\draw [->] (B)--(C);\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20240122\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"023634": {
"id": "023634",
"content": "已知向量 $\\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ 满足 $\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, $(\\overrightarrow{a}-\\overrightarrow{b}) \\cdot(\\overrightarrow{a}-\\overrightarrow{c})=0$, $|\\overrightarrow{b}-\\overrightarrow{c}|=9$, 求 $|\\overrightarrow{a}|$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "自拟题目",
"edit": [
"20240122\t赵琍琍"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"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.",