添加2024届高三上学期测验5部分题目并建立关联
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"022858"
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"022858",
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"022918"
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"20231124\t毛培菁"
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"related": [
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"022915"
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"20231124\t毛培菁"
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],
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"same": [],
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"related": [],
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"related": [
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"022917"
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],
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"remark": "",
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"space": "4em",
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"space": "4em",
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"unrelated": []
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},
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"022914": {
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"id": "022914",
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"content": "设函数 $f(x)=A \\sin (\\omega x-\\dfrac{\\pi}{6})(\\omega>0, A>0)$, $x \\in[0,2 \\pi]$, 若 $f(x)$ 恰有 4 个零点,则下述结论中:\\\\\n\\textcircled{1} 若 $f(x_0) \\geq f(x)$ 恒成立, 则 $x_0$ 的值有且仅有 2 个;\\\\\n\\textcircled{2} $f(x)$ 在 $[0, \\dfrac{8 \\pi}{19}]$ 上单调递增;\\\\\n\\textcircled{3} 存在 $\\omega$ 和 $x_1$, 使得 $f(x_1) \\leq f(x) \\leq f(x_1+\\dfrac{\\pi}{2})$ 对任意 $x \\in[0,2 \\pi]$ 恒成立;\\\\\n\\textcircled{4} ``$A \\geq 1$''是``方程 $f(x)=-\\dfrac{1}{2}$ 在 $[0,2 \\pi]$ 内恰有五个解''的必要条件.\\\\\n所有正确结论的编号是\\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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"20231207\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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"022915": {
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"id": "022915",
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"content": "设 $a, b, c \\in \\mathbf{R}$, 则``$a, b, c$ 构成等比数列''是``$b^2=a c$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}",
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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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"20231207\t毛培菁"
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],
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"same": [],
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"related": [
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"022851"
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],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"022916": {
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"id": "022916",
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"content": "某人驾驶一艘小游艇位于湖面 $A$ 处, 测得岸边一座电视塔的塔底在北偏东 $21^{\\circ}$ 方向, 且塔顶的仰角为 $18^{\\circ}$, 此人驾驶游艇向正东方向行驶 1000 米后到达 $B$ 处, 此时测得塔底位于北偏西 $39^{\\circ}$ 方向, 则该塔的高度约为\\bracket{20}.\n\\fourch{265 米}{279 米}{292 米}{306 米}",
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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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"20231207\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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"022917": {
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"id": "022917",
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"content": "已知抛物线 $\\Gamma: y^2=8 x$ 和圆 $\\Omega: x^2+y^2-4 x=0$, 抛物线 $\\Gamma$ 的焦点为 $F$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -5:5] plot ({\\x*\\x/8},\\x);\n\\draw (2,0) circle (2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求 $\\Omega$ 的圆心到 $\\Gamma$ 的准线的距离;\\\\\n(2) 若点 $T(x, y)$ 在抛物线 $\\Gamma$ 上, 且满足 $x \\in[1,4]$, 过点 $T$ 作圆 $\\Omega$ 的两条切线, 记切点为 $A$、$B$, 求四边形 $TAFB$ 的面积的取值范围;\\\\\n(3) 如图, 若直线 $l$ 与抛物线 $\\Gamma$ 和圆 $\\Omega$ 依次交于 $M$、$P$ 、 $Q$、$N$ 四点, 证明:``$|MP|=|QN|=\\dfrac{1}{2}|PQ|$''的充要条件是``直线 $l$ 的方程为 $x=2$''.",
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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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"20231207\t毛培菁"
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],
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"same": [],
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"related": [
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"022887"
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],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"022918": {
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"id": "022918",
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"content": "现定义: 设 $a$ 是非零实常数, 若对于任意的 $x \\in D$, 都有 $f(a-x)=f(a+x)$, 则称函数 $y=f(x)$ 为``关于 $a$ 的偶型函数''.\\\\\n(1) 请以三角函数为例, 写出一个``关于 2 的偶型函数''的解析式 (无需证明);\\\\\n(2) 设定义域为 $\\mathbf{R}$ 的``关于 $a$ 的偶型函数''$y=f(x)$ 在区间 ($-\\infty, a$) 上单调递增, 求证: $y=f(x)$ 在区间 ($a,+\\infty$) 上单调递减;\\\\\n(3) 设定义域为 $\\mathbf{R}$ 的``关于 $\\dfrac{1}{2}$ 的偶型函数''$y=f(x)$ 是奇函数. 若 $n \\in \\mathrm{N}$, $n \\geq 1$, 请猜测 $f(n)$ 的值, 并用数学归纳法证明你的结论.",
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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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"20231207\t毛培菁"
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],
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"same": [],
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"related": [
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"011006"
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],
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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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