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收录24届长宁区一模

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工具v2/文本文件/新题收录列表.txt
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@ -232,3 +232,6 @@
20240125-153606
024239:024241,020797,024242:024250,004660
20240127-165000
024251:024271

507
题库0.3/Problems.json
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@ -129694,7 +129694,9 @@
"20220705\t王伟叶"
],
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"024259"
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@ -139052,7 +139054,9 @@
"20220709\t王伟叶"
],
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"024254"
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"space": "",
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@ -287834,7 +287838,9 @@
"20220806\t王伟叶"
],
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"related": [
"024265"
],
"remark": "",
"space": "4em",
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@ -303117,7 +303123,9 @@
"20220806\t王伟叶"
],
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"related": [
"024255"
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@ -330593,7 +330601,9 @@
"20220901\t王伟叶"
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@ -341186,7 +341196,9 @@
"20221209\t王伟叶"
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"024253"
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"remark": "",
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@ -412452,7 +412464,9 @@
"20230407\t王伟叶"
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"024263"
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@ -459761,7 +459775,9 @@
"20230503\t王伟叶"
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"related": [
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"space": "4em",
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@ -467331,7 +467347,8 @@
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@ -480264,7 +480281,8 @@
],
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@ -574760,7 +574778,8 @@
],
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@ -624532,7 +624551,9 @@
"20240105\t毛培菁"
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@ -631410,7 +631431,8 @@
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@ -639394,7 +639416,8 @@
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@ -652080,6 +652103,454 @@
"space": "4em",
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},
"024251": {
"id": "024251",
"content": "已知集合 $A=(-\\infty, 4]$, $B=\\{1,3,5,7\\}$, 则 $A \\cap B=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题1",
"edit": [
"20240127\t王伟叶"
],
"same": [],
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"030605"
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"remark": "",
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"024252": {
"id": "024252",
"content": "复数 $z$ 满足 $z=\\dfrac{1}{1-\\mathrm{i}}$ ($\\mathrm{i}$ 为虚数单位), 则 $|\\overline{z}|=$\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题2",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "",
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},
"024253": {
"id": "024253",
"content": "不等式 $\\dfrac{1}{x}>1$ 的解集为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题3",
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"20240127\t王伟叶"
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"same": [],
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"012225"
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"remark": "",
"space": "",
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"024254": {
"id": "024254",
"content": "设向量 $\\overrightarrow{a}=(1,-2)$, $\\overrightarrow{b}=(-1, m)$, 若 $\\overrightarrow{a}\\parallel \\overrightarrow{b}$, 则 $m=$\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题4",
"edit": [
"20240127\t王伟叶"
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"023028"
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"024255": {
"id": "024255",
"content": "将 $4$ 个人排成一排, 若甲和乙必须排在一起, 则共有\\blank{50}种不同排法.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题5",
"edit": [
"20240127\t王伟叶"
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"same": [],
"related": [
"010844"
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"remark": "",
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"024256": {
"id": "024256",
"content": "物体位移 $s$ 和时间 $t$ 满足函数关系 $s=100 t-5 t^2$($0<t<20$), 则当 $t=2$ 时, 物体的瞬时速度为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题6",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [
"023638",
"021372"
],
"remark": "",
"space": "",
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"024257": {
"id": "024257",
"content": "现利用随机数表法从编号为 $00,01,02, \\cdots, 18,19$ 的 20 支水笔中随机选取 6 支, 选取方法是从下列随机数表第 1 行的第 9 个数字开始由左到右依次选取两个数字, 则选出来的第 6 支水笔的编号为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cccccc}\n95226000 & 49840128 & 66175168 & 39682927 & 43772366 & 27096623 \\\\\n92580956 & 43890890 & 06482834 & 59741458 & 29778149 & 64608925\n\\end{tabular}\n\\end{center}",
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"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题7",
"edit": [
"20240127\t王伟叶"
],
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"017725"
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"remark": "",
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"024258": {
"id": "024258",
"content": "在有声世界, 声强级是表示声强度相对大小的指标. 其值 $y$ (单位: dB ) 定义为 $y=10 \\lg \\dfrac{I}{I_0}$. 其中 $I$ 为声场中某点的声强度, 其单位为 $\\text{W} / \\text{m}^2$, $I_0=10^{-12}\\text{W}/ \\text{m}^2$ 为基准值. 若 $I=10 \\text{W}/ \\text{m}^2$, 则其相应的声强级为\\blank{50}dB.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题8",
"edit": [
"20240127\t王伟叶"
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"same": [],
"related": [
"011873"
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"remark": "",
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"024259": {
"id": "024259",
"content": "若向量 $\\overrightarrow{a}=(1,0,2)$, $\\overrightarrow{b}=(0,1,-1)$, 则 $\\overrightarrow{a}$ 在 $\\overrightarrow{b}$ 方向上的投影向量为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题9",
"edit": [
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],
"same": [],
"related": [
"004208"
],
"remark": "",
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},
"024260": {
"id": "024260",
"content": "若``存在 $x>0$, 使得 $x^2+a x+1<0$''是假命题, 则实数 $a$ 的取值范围为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题10",
"edit": [
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"024261": {
"id": "024261",
"content": "若函数 $f(x)=\\sin x+a \\cos x$ 在 $(\\dfrac{2 \\pi}{3}, \\dfrac{7 \\pi}{6})$ 上是严格单调函数, 则实数 $a$ 的取值范围为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题11",
"edit": [
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],
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"remark": "",
"space": "",
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"024262": {
"id": "024262",
"content": "设 $f(x)=|\\log _2 x+a x+b|$($a>0$), 记函数 $y=f(x)$ 在区间 $[t, t+1]$($t>0$) 上的最大值为 $M_t(a, b)$, 若对任意 $b \\in \\mathbf{R}$, 都有 $M_t(a, b) \\geq a+1$, 则实数 $t$ 的最大值为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题12",
"edit": [
"20240127\t王伟叶"
],
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"remark": "",
"space": "",
"unrelated": []
},
"024263": {
"id": "024263",
"content": "下列函数中既是奇函数又是增函数的是\\bracket{20}.\n\\fourch{$f(x)=2 x$}{$f(x)=x^2$}{$f(x)=\\ln x$}{$f(x)=\\mathrm{e}^x$}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题13",
"edit": [
"20240127\t王伟叶"
],
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"related": [
"014817"
],
"remark": "",
"space": "",
"unrelated": []
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"024264": {
"id": "024264",
"content": "``$P(A \\cap B)=P(A) P(B)$''是``事件 $A$ 与事件 $\\overline{B}$ 互相独立''\\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}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题14",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "",
"unrelated": []
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"024265": {
"id": "024265",
"content": "设点 $P$ 是以原点为圆心的单位圆上的动点, 它从初始位置 $P_0(1,0)$ 出发, 沿单位圆按逆时针方向转动角 $\\alpha$($0<\\alpha<\\dfrac{\\pi}{2}$) 后到达点 $P_1$, 然后继续沿单位圆按逆时针方向转动角 $\\dfrac{\\pi}{4}$ 到达 $P_2$. 若点 $P_2$ 的横坐标为 $-\\dfrac{3}{5}$, 则点 $P_1$ 的纵坐标 \\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{2}}{10}$}{$\\dfrac{\\sqrt{2}}{5}$}{$\\dfrac{3 \\sqrt{2}}{5}$}{$\\dfrac{7 \\sqrt{2}}{10}$}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题15",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [
"010252"
],
"remark": "",
"space": "",
"unrelated": []
},
"024266": {
"id": "024266",
"content": "豆腐发酵后衣面长出一层白线线的长毛就成了毛豆腐. 将三角形豆腐 $ABC$ 悬空挂在发酵空间内, 记发酵后毛豆腐所构成的儿何体为 $T$. 若忽略三角形豆腐 $ABC$ 的原度, 设 $AB=3$, $BC=4$, $AC=5$, 点 $P$ 在 $\\triangle ABC$ 内部. 假设对于任意点 $P$, 满足 $PQ \\leq 1$ 的点 $Q$ 都在 $T$ 内,且对于 $T$ 内任意一点 $Q$, 都存在点 $P$, 满足 $PQ \\leq 1$, 则 $T$ 的体积为\\bracket{20}.\n\\fourch{$12+7 \\pi$}{$12+\\dfrac{22 \\pi}{3}$}{$14+7 \\pi$}{$14+\\dfrac{22 \\pi}{3}$}",
"objs": [],
"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题16",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "",
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"024267": {
"id": "024267",
"content": "已知等差数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 公差 $d=2$.\\\\\n(1) 若 $S_{10}=100$, 求 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 从集合 $\\{a_1, a_2, a_3, a_4, a_5, a_6\\}$ 中任取 3 个元素, 记这 3 个元索能成等差数列为事件 $A$,求事件 $A$ 发生的概率 $P(A)$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题17",
"edit": [
"20240127\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"024268": {
"id": "024268",
"content": "如图, 在三棱锥 $A-BCD$ 中, 平面 $ABD \\perp$ 平面 $BCD$, $AB=AD$, $O$为 $BD$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw (1,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$A$} coordinate (A);\n\\draw (1,0,2) node [below] {$C$} coordinate (C);\n\\draw (B)--(C)--(D)--(A)--cycle(A)--(C);\n\\draw ($(B)!0.5!(D)$) node [above left] {$O$} coordinate (O);\n\\draw [dashed] (B)--(D)(A)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AO \\perp CD$;\\\\\n(2) 若 $BD \\perp DC$, $BD=DC$, $AO=BO$, 求异面直线 $BC$ 与 $AD$ 所成的角的大小.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "2024届长宁区一模试题18",
"edit": [
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"related": [
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"remark": "",
"space": "4em",
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"024269": {
"id": "024269",
"content": "汽车转弯时遵循阿克曼转向几何原理, 即转向时所有车轮中垂线交于一点, 该点称为转向中心: 如图 1, 某汽车四轮中心分别为 $A$、$B$ 、 $C$、$D$, 向左转向, 左前轮转向角为 $\\alpha$, 右前轮转向角为 $\\beta$, 转向中心为 $O$. 设该汽车左右轮距 $AB$ 为 $w$ 米, 前后轴距 $AD$ 为 $l$ 米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{1.57}\n\\def\\w{2.68}\n\\draw (0,0) node [below left] {$D$} coordinate (D);\n\\draw (\\l,0) node [below right] {$C$} coordinate (C);\n\\draw (C)++(0,\\w) node [above right] {$B$} coordinate (B);\n\\draw (D)++(0,\\w) node [above left] {$A$} coordinate (A);\n\\draw (A)--(B)(C)--(D);\n\\draw ($(C)!0.5!(D)$) coordinate (MB) ($(A)!0.5!(B)$) coordinate (MU);\n\\draw (MB)--(MU);\n\\draw (D)++ ({-sqrt(3)*\\w},0) node [below] {$O$} coordinate (O);\n\\draw [dashed] (O)--(D)(O)--(A)(O)--(B)(A)--(D)(B)--(C);\n\\def\\t{atan(\\w/(sqrt(3)*\\w+\\l))}\n\\draw [ultra thick] (D) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (C) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (A) ++ (120:0.4) --++ (-60:0.8);\n\\draw [ultra thick] (B) ++ ({90+\\t}:0.4) --++ ({\\t-90}:0.8);\n\\draw ($(O)!0.5!(C)$) ++ (0,-1) node [below] {图 1};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{1.57}\n\\def\\w{2.68}\n\\draw (0,0) node [below left] {$D$} coordinate (D);\n\\draw (\\l,0) node [below] {$C$} coordinate (C);\n\\draw (C)++(0,\\w) node [above] {$B$} coordinate (B);\n\\draw (D)++(0,\\w) node [above left] {$A$} coordinate (A);\n\\draw (A)--(B)(C)--(D);\n\\draw ($(C)!0.5!(D)$) coordinate (MB) ($(A)!0.5!(B)$) coordinate (MU);\n\\draw (MB)--(MU);\n\\def\\t{atan(\\w/(sqrt(3)*\\w+\\l))}\n\\draw [ultra thick] (D) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (C) ++ (0,0.4) --++ (0,-0.8);\n\\draw [ultra thick] (A) ++ (120:0.4) --++ (-60:0.8);\n\\draw [ultra thick] (B) ++ ({90+\\t}:0.4) --++ ({\\t-90}:0.8);\n\\draw (-1.5,-1) node [left] {$T$} coordinate (T);\n\\draw (-1.5,1.5) node [below left] {$M$} coordinate (M);\n\\draw (-3.5,1.5) node [below] {$N$} coordinate (N);\n\\draw (N) ++ (0,3.5) node [above] {$E$} coordinate (E);\n\\draw (M) ++ (3.5,3.5) node [above right] {$F$} coordinate (F);\n\\draw (T) ++ (3.5,0) node [right] {$S$} coordinate (S);\n\\draw (T)--(M)--(N)(S)--(F)--(E);\n\\draw ($(O)!0.5!(C)$) ++ (1,-1.5) node [below] {图 2};\n\\end{tikzpicture}\n\\end{center}\n(1) 试用 $w$、$l$ 和 $\\alpha$ 表示 $\\tan \\beta$;\\\\\n(2) 如图 2, 有一直角弯道, $M$ 为内直角顶点, $EF$ 为上路边, 路宽均为 $3.5$ 米, 汽车行驶其中, 左轮 $A$、$D$ 与路边 $FS$ 相距 2 米. 试依据如下假设, 对问题*做出判断, 并说明理由.\n假设: \\textcircled{1} 转向过程中, 左前轮转向角 $\\alpha$ 的值始终为 $30^{\\circ}$; \\textcircled{2}设转向中心 $O$ 到路边 $EF$ 的距离为 $d$, 若 $OB<d$ 且 $OM<OD$, 则汽车可以通过, 否则不能通过; \\textcircled{3} $w=1.570$, $l=2.680$. 问题*: 可否选择恰当转向位置, 使得汽车通过这一弯道?",
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"usages": [],
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"024270": {
"id": "024270",
"content": "已知椭圆 $\\Gamma: \\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$, $F_1$、$F_2$ 为 $\\Gamma$ 的左、右焦点, 点 $A$ 在 $\\Gamma$ 上, 直线 $l$ 与圆 $C: x^2+y^2=2$ 相切.\\\\\n(1) 求 $\\triangle \\mathrm{AF}_1F_2$ 的周长;\\\\\n(2) 若直线 $l$ 经过 $\\Gamma$ 的右顶点, 求直线 $l$ 的方程:\\\\\n(3) 设点 $D$ 在直线 $y=2$上, $O$ 为原点, 若 $OA \\perp OD$, 求证: 直线 $AD$ 与圆 $C$ 相切.",
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"duration": -1,
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"origin": "2024届长宁区一模试题20",
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"024271": {
"id": "024271",
"content": "若函数 $y=f(x)$ 与 $y=g(x)$ 满足: 对任意 $x_1, x_2 \\in \\mathbf{R}$, 都有\n$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$, 则称函数 $y=f(x)$ 是函数 $y=g(x)$ 的``约束函数''. 已知函数 $y=f(x)$ 是函数 $y=g(x)$ 的``约束函数''.\\\\\n(1) 若 $f(x)=x^2$, 判断函数 $y=g(x)$ 的奇偶性, 并说明理由:\\\\\n(2) 若 $f(x)=a x+x^3$($a>0$), $g(x)=\\sin x$, 求实数 $a$ 的取值范围;\\\\\n(3) 若 $y=g(x)$ 为严格减函数, $f(0)<f(1)$, 且函数 $y=f(x)$ 的图像是连续曲线, 求证: $y=f(x)$ 是 $(0,1)$ 上的严格增函数.",
"objs": [],
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"genre": "解答题",
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"origin": "2024届长宁区一模试题21",
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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.",
@ -671757,7 +672228,9 @@
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@ -690550,7 +691023,9 @@
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