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@ -360109,7 +360109,9 @@
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"id": "014511",
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"content": "函数$y=\\lg (2-x)$的定义域为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "填空题",
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"ans": "$(-\\infty,2)$",
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"solution": "",
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@ -360138,7 +360140,9 @@
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"id": "014512",
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"content": "已知集合$A=(-2,2)$, $B=(-3,-1) \\cup(1,5)$, 则$A \\cup B=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第一单元"
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],
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"genre": "填空题",
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"ans": "$(-3,5)$",
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"solution": "",
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@ -360167,7 +360171,9 @@
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"id": "014513",
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"content": "$(2 x+1)^5$的二项展开式中$x^3$项的系数是\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第八单元"
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],
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"genre": "填空题",
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"ans": "$80$",
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"solution": "",
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@ -360196,7 +360202,9 @@
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"id": "014514",
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"content": "已知向量$\\overrightarrow {a}=(-m, 1,3)$, $\\overrightarrow {b}=(2, n, 1)$, 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$m n$的值为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第六单元"
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],
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"genre": "填空题",
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"ans": "$-2$",
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"solution": "",
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@ -360225,7 +360233,9 @@
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"id": "014515",
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"content": "已知复数$z$满足$(1+\\mathrm{i}) z=4-2 \\mathrm{i}$($\\mathrm{i}$为虚数单位), 则复数$z$的模等于\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第五单元"
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],
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"genre": "填空题",
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"ans": "$\\sqrt{10}$",
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"solution": "",
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@ -360254,7 +360264,9 @@
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"id": "014516",
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"content": "某个品种的小麦麦穗长度(单位: $\\text{cm}$)的样本数据如下: $10.2$、$9.7$、$10.8$、$9.1$、$8.9$、$8.6$、$9.8$、$9.6$、$9.9$、$11.2$、$10.6$、$11.7$, 则这组数据的第$80$百分数为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第九单元"
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],
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"genre": "填空题",
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"ans": "$10.8$",
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"solution": "",
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@ -360283,7 +360295,9 @@
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"id": "014517",
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"content": "在平面直角坐标系$xOy$中, 若角$\\theta$的顶点为坐标原点, 始边与$x$轴的非负半轴重合, 终边与以点$O$为圆心的单位圆交于点$P(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 则$\\sin (2 \\theta-\\dfrac{\\pi}{2})$的值为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第三单元"
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],
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"genre": "填空题",
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"ans": "$\\dfrac{7}{25}$",
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"solution": "",
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@ -360312,7 +360326,9 @@
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"id": "014518",
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"content": "已知一个圆锥的侧面展开图是一个面积为$2 \\pi$的半圆, 则该圆锥的体积为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第六单元"
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],
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"genre": "填空题",
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"ans": "$\\dfrac{\\sqrt{3}}3\\pi$",
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"solution": "",
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@ -360341,7 +360357,9 @@
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"id": "014519",
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"content": "已知$\\triangle ABC$的三边长分别为$4$、$5$、$7$, 记$\\triangle ABC$的三个内角的正切值所组成的集合为$M$, 则集合$M$中的最大元素为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第三单元"
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],
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"genre": "填空题",
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"ans": "$\\dfrac{2\\sqrt{6}}5$",
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"solution": "",
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@ -360370,7 +360388,9 @@
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"id": "014520",
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"content": "现有$5$人参加抽奖活动, 每人依次从装有$5$张奖票(其中$3$张为中奖票)的箱子中不放回地随机抽取一张, 直到$3$张中奖票都被抽出时活动结束, 则活动恰好在第$4$人抽完后结束的概率为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第八单元"
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],
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"genre": "填空题",
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"ans": "$\\dfrac{3}{10}$",
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"solution": "",
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@ -360399,7 +360419,9 @@
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"id": "014521",
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"content": "已知四边形$ABCD$是平行四边形, 若$\\overrightarrow{AD}=2\\overrightarrow{DE}$, $\\overrightarrow{BF}\\parallel \\overrightarrow{BE}$, $\\overrightarrow{AF} \\cdot \\overrightarrow{BE}=0$, \n且$\\overrightarrow{AF} \\cdot \\overrightarrow{AC}=60$, 则$\\overrightarrow{AC}$在$\\overrightarrow{AF}$上的数量投影为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第五单元"
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],
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"genre": "填空题",
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"ans": "$10$",
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"solution": "",
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@ -360428,7 +360450,9 @@
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"id": "014522",
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"content": "已知曲线$C_1: y=\\sqrt{1-x^2}$与曲线$C_2: y=\\sqrt{2-x^2}$, 长度为$1$的线段$AB$的两端点$A$、$B$分别在曲线$C_1$、$C_2$上沿顺时针方向运动, 若点$A$从点$(-1,0)$开始运动, 点$B$到达点$(\\sqrt{2}, 0)$时停止运动, 则线段$AB$所扫过的区域的面积为\\blank{50}.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "填空题",
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"ans": "$\\dfrac{3\\pi}{8}$",
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"solution": "",
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@ -360457,7 +360481,9 @@
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"id": "014523",
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"content": "在平面直角坐标系$xOy$中, ``$m<0$''是``方程$x^2+m y^2=1$表示的曲线是双曲线''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "选择题",
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"ans": "C",
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"solution": "",
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@ -360486,7 +360512,9 @@
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"id": "014524",
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"content": "如图, 四边形$ABCD$是边长为$1$的正方形, $MD \\perp$平面$ABCD$, $NB \\perp$平面$ABCD$, 且$MD=NB=1$, 点$G$为$MC$的中点. 则下列结论中不正确的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2,z = {(240:0.5cm)}]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (0,0,-1) node [above right] {$D$} coordinate (D);\n\\draw (B)++(0,1,0) node [above right] {$N$} coordinate (N);\n\\draw (D)++(0,1,0) node [above] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(M)$) node [below left] {$G$} coordinate (G);\n\\draw (A)--(B)--(C)--(N)--(M)--cycle(A)--(N)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(M)--(C)(B)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$MC \\perp AN$}{平面$DCM\\parallel$平面$ABN$}{直线$GB$与$AM$是异面直线}{直线$GB$与平面$AMD$无公共点}",
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"objs": [],
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"tags": [],
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"tags": [
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"第六单元"
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],
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"genre": "选择题",
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"ans": "D",
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"solution": "",
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@ -360515,7 +360543,9 @@
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"id": "014525",
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"content": "已知$f(x)=\\sin (\\omega x+\\dfrac{\\pi}{6})(\\omega>0)$, 且函数$y=f(x)$恰有两个极大值点在$[0, \\dfrac{\\pi}{3}]$内, 则$\\omega$的取值范围是\\bracket{20}.\n\\fourch{$(7,13]$}{$[7,13)$}{$(7,10]$}{$[7,10)$}",
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"objs": [],
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"tags": [],
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"tags": [
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"第三单元"
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],
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"genre": "选择题",
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"ans": "B",
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"solution": "",
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@ -360544,7 +360574,9 @@
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"id": "014526",
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"content": "设$a$、$b$、$c$、$p$为实数, 若同时满足不等式$a x^2+b x+c>0$、$b x^2+c x+a>0$与$c x^2+a x+b>0$的全体实数$x$所组成的集合等于$(p,+\\infty)$. 则关于结论: \\textcircled{1} $a$、$b$、$c$至少有一个为$0$; \\textcircled{2} $p=0$. 下列判断中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都正确}{\\textcircled{1}和\\textcircled{2}都错误}{\\textcircled{1}正确, \\textcircled{2}错误}{\\textcircled{1}错误, \\textcircled{2}正确}",
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"objs": [],
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"tags": [],
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"tags": [
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"第一单元"
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],
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"genre": "选择题",
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"ans": "A",
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"solution": "",
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@ -360573,7 +360605,9 @@
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"id": "014527",
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"content": "已知$\\{a_n\\}$是等差数列, $\\{b_n\\}$是等比数列, 且$b_2=3$, $b_3=9$, $a_1=b_1$, $a_{14}=b_4$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 设$c_n=a_n+(-1)^n b_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 求数列$\\{c_n\\}$的前$2 n$项和.",
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"objs": [],
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"tags": [],
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"tags": [
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"第四单元"
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],
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"genre": "解答题",
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"ans": "(1) $2n-1$; (2) $4n^2+\\dfrac{9^n}{4}-\\dfrac 14$",
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"solution": "",
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@ -360602,7 +360636,9 @@
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"id": "014528",
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"content": "如图所示, 四棱锥$P-ABCD$中, 底面$ABCD$为菱形, 且$PA \\perp$平面$ABCD$, 又棱$PA=AB=2, E$为棱$CD$的中点, $\\angle ABC=60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (C)++(-2,0,0) node [below] {$B$} coordinate (B);\n\\draw ($(C)!0.5!(D)$) node [below right] {$E$} coordinate (E);\n\\draw (A)++(0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(E)(A)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 直线$AE \\perp$平面$PAB$;\\\\\n(2) 求直线$AE$与平面$PCD$所成角的正切值.",
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"objs": [],
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"tags": [],
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"tags": [
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"第六单元"
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],
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"genre": "解答题",
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|
"ans": "(1) 证明略; (2) $\\dfrac{2\\sqrt{3}}3$",
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"solution": "",
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@ -360631,7 +360667,10 @@
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"id": "014529",
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"content": "某展览会有四个展馆, 分别位于矩形$ABCD$的四个顶点$A$、$B$、$C$、$D$处, 现要修建如图中实线所示的步道(宽度忽略不计, 长度可变)把这四个展馆连在一起, 其中$AB=8$百米, $AD=6$百米, 且$AE=DE=BF=CF$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0) node [below] {$B$} coordinate (B);\n\\draw (4,3) node [above] {$C$} coordinate (C);\n\\draw (0,3) node [above] {$D$} coordinate (D);\n\\draw [dashed] (A)--(B) node [midway,below] {$8$} -- (C)--(D)--cycle node [midway, left] {$6$};\n\\draw (0.8,1.5) node [below right] {$E$} coordinate (E);\n\\draw (3.2,1.5) node [below left] {$F$} coordinate (F);\n\\draw (A)--(E)--(D)(B)--(F)--(C)(E)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 试从各段步道的长度与图中各角的弧度数中选择某一变量作为自变量$x$, 并求出步道的总长$y$(单位: 百米)关于$x$的函数关系式;\\\\\n(2) 求步道的最短总长度(精确到$0.01$百米).",
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"objs": [],
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"tags": [],
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"tags": [
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|
"第三单元",
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"第二单元"
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|
],
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"genre": "解答题",
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|
"ans": "(1) 如设$AE=x$百米, 则$y=4x+8-2\\sqrt{x^2-9}$($3<x<5$); 如设$\\angle MAE=x$, 则$y=\\dfrac{12}{\\cos x}+8-6\\tan x$($0<x<\\arctan\\dfrac 43$)等; (2) 约$18.39$百米",
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"solution": "",
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@ -360660,7 +360699,9 @@
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"id": "014530",
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"content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的离心率为$\\dfrac{\\sqrt{2}}{2}$, 以其四个顶点为顶点的四边形的面积等于$8 \\sqrt{2}$. 动直线$l_1$、$l_2$都过点$M(0, m)$($0<m<1$), 斜率分别为$k$、$-3k$, $l_1$与椭圆$C$交于点$A$、$P$, $l_2$与椭圆$C$交于点$B$、$Q$, 点$P$、$Q$分别在第一、四象限且$PQ \\perp x$轴.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-3.5,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\path [draw, name path = elli] (0,0) ellipse ({2*sqrt(2)} and 2);\n\\path [name path = PQ] (1.5,-2) -- (1.5,2);\n\\path [name intersections = {of = elli and PQ, by = {P,Q}}];\n\\draw (P) node [above] {$P$} --(Q) node [below] {$Q$};\n\\draw ($(P)!0.25!(Q)$) ++ (-1.5,0) node [left] {$M$} coordinate (M);\n\\path [name path = AP] (M)--($(P)!3!(M)$);\n\\path [name path = BQ] (M)--($(Q)!1.5!(M)$);\n\\path [name intersections = {of = AP and elli, by = A}];\n\\path [name intersections = {of = BQ and elli, by = B}];\n\\draw (A) node [left] {$A$} --(B) node [above] {$B$}(A)--(P)(B)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$C$的标准方程;\\\\\n(2) 若直线$l_1$与$x$轴交于点$N$, 求证: $|NP|=2|NM|$;\\\\\n(3) 求直线$AB$的斜率的最小值, 并求直线$AB$的斜率取最小值时的直线$l_1$的方程.",
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"objs": [],
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"tags": [],
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"tags": [
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"第七单元"
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],
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"genre": "解答题",
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"ans": "(1) $\\dfrac{x^2}8+\\dfrac{y^2}4=1$; (2) 证明略; (3) 斜率的最小值为$\\dfrac{\\sqrt{6}}2$, 此时直线$l_1$的方程为$y=\\dfrac{\\sqrt{6}}6x+\\dfrac{2\\sqrt{7}}7$",
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"solution": "",
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@ -360689,7 +360730,9 @@
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"id": "014531",
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"content": "已知集合$A$和定义域为$\\mathbf{R}$的函数$y=f(x)$, 若对任意$t \\in A, x \\in \\mathbf{R}$, 都有$f(x+t)-f(x) \\in A$, 则称$f(x)$是关于$A$的同变函数.\\\\\n(1) 当$A=(0,+\\infty)$与$(0,1)$时, 分别判断$f(x)=2^x$是否为关于$A$的同变函数, 并说明理由;\\\\\n(2) 若$f(x)$是关于$\\{2\\}$的同变函数, 且当$x \\in[0,2)$时, $f(x)=\\sqrt{2 x}$, 试求$f(x)$在$[2 k, 2 k+2)$($k \\in \\mathbf{Z}$)上的表达式, 并比较$f(x)$与$x+\\dfrac{1}{2}$的大小;\\\\\n(3) 若$n$为正整数, 且$f(x)$是关于$[2^{-n}, 2^{1-n}]$的同变函数, 求证: $f(x)$既是关于$\\{m \\cdot 2^{-n}\\}$($m \\in \\mathbf{Z}$)的同变函数, 也是关于$[0,+\\infty)$的同变函数.",
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"objs": [],
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"tags": [],
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"tags": [
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"第二单元"
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],
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"genre": "解答题",
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"ans": "(1) $f(x)=2^x$是关于$(0,+\\infty)$的同变函数, 不是关于$(0,1)$的同变函数; (2) $f(x)=\\sqrt{2(x-2k)}+2k$, 当$x=2k+\\dfrac 12$($k\\in\\mathbf{Z}$)时, $f(x)=x+\\dfrac 12$, 当$x\\ne 2k+\\dfrac 12$($k\\in\\mathbf{Z}$)时, $f(x)<x+\\dfrac 12$; (3) 证明略",
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"solution": "",
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