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@ -303027,6 +303027,804 @@
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"remark": "",
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"space": "12ex"
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},
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"012287": {
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"id": "012287",
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"content": "已知集合$A=(-2,1]$, $B=\\mathbf{Z}$, 则$A \\cap B=$\\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": "2023届松江区一模试题1",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012288": {
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"id": "012288",
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"content": "函数$y=\\sin x \\cdot \\cos x$的最小正周期为\\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": "2023届松江区一模试题2",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012289": {
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"id": "012289",
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"content": "已知$a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, 若$a-\\mathrm{i}$与$2+b \\mathrm{i}$互为共轭复数, 则$(a+b \\mathrm{i})^2=$\\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": "2023届松江区一模试题3",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012290": {
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"id": "012290",
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"content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 若$2 S_3=3 S_2+6$, 则公差$d=$\\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": "2023届松江区一模试题4",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012291": {
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"id": "012291",
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"content": "已知函数$y=a-\\dfrac 2{2^x+1}$为奇函数, 则实数$a=$\\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": "2023届松江区一模试题5",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012292": {
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"id": "012292",
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"content": "已知圆锥的母线长为$5$, 侧面积为$20 \\pi$, 则此圆锥的体积为\\blank{50}. (结果保留$\\pi$)",
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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": "2023届松江区一模试题6",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012293": {
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"id": "012293",
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"content": "已知向量$\\overrightarrow a=(5,3)$, $\\overrightarrow b=(-1,2)$, 则$\\overrightarrow a$在$\\overrightarrow b$上的投影向量的坐标为\\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": "2023届松江区一模试题7",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012294": {
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"id": "012294",
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"content": "对任意$x \\in \\mathbf{R}$, 不等式$|x-2|+|x-3|\\geq 2 a^2+a$恒成立, 则实数$a$的取值范围为\\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": "2023届松江区一模试题8",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012295": {
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"id": "012295",
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"content": "已知集合$A=\\{x | \\dfrac 2{x-2} \\geq 1, \\ x \\in \\mathbf{R}\\}$, 设函数$y=\\log_{\\frac 12} x+a(x \\in A)$的值域为$B$, 若$B \\subseteq A$, 则实数$a$的取值范围为\\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": "2023届松江区一模试题9",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012296": {
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"id": "012296",
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"content": "已知$F_1$、$F_2$是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, 点$M$是双曲线$\\Gamma$上的任意一点(不是顶点), 过$F_1$作$\\angle F_1MF_2$的角平分线的垂线, 垂足为$N$, 线段$F_1N$的延长线交$MF_2$于点$Q$, $O$是坐标原点, 若$|ON|=\\dfrac{|F_1F_2|}6$, 则双曲线$\\Gamma$的渐近线方程为\\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": "2023届松江区一模试题10",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012297": {
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"id": "012297",
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"content": "动点$P$在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$表面上运动, 且与点$A$的距离是$\\dfrac{2 \\sqrt 3}3$, 点$P$的集合形成一条曲线, 这条曲线的长度为\\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": "2023届松江区一模试题11",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012298": {
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"id": "012298",
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"content": "已知数列$\\{a_n\\}$的各项都是正数, $a_{n+1}^2-a_{n+1}=a_n$($n \\in \\mathbf{N}^*$), 若数列$\\{a_n\\}$为严格增数列, 则首项$a_1$的取值范围是\\blank{50}; 当$a_1=\\dfrac 23$时, 记$b_n=\\dfrac{(-1)^{n-1}}{a_n-1}$, 若$k<b_1+b_2+\\cdots+b_{2022}<k+1$, 则整数$k=$\\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": "2023届松江区一模试题12",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012299": {
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"id": "012299",
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"content": "下面四个条件中, 使$a>b$成立的充要条件为\\bracket{20}.\n\\fourch{$a^2>b^2$}{$a^3>b^3$}{$a>b-1$}{$a>b+1$}",
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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": "2023届松江区一模试题13",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012300": {
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"id": "012300",
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"content": "函数$y=(x^2-1) \\mathrm{e}^x$的图像可能为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{5*(-1-\\x)*exp(-\\x*\\x-1)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{0.4*(\\x+0.5)*(\\x+2.5)*(\\x-1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{(\\x*\\x-1)*exp(\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{0.4*(\\x+1)*(\\x+3)*(\\x-1)});\n\\end{tikzpicture}}",
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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": "2023届松江区一模试题14",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012301": {
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"id": "012301",
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"content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述, 两颗星的星等与亮度满足$m_2-m_1=\\dfrac 52 \\lg \\dfrac{E_1}{E_2}$, 其中星等为$m_k$的星的亮度为$E_k$($k=1$、$2$), 已知太阳的星等是$-26.7$, 天狼星的星等是$-1.45$, 则太阳与天狼星的亮度的比值为\\bracket{20}.\n\\fourch{$10^{10.1}$}{$10.1$}{$\\lg 10.1$}{$10^{-10.1}$}",
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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": "2023届松江区一模试题15",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012302": {
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"id": "012302",
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"content": "已知函数$f(x)=\\begin{cases}|x+2|, & x<0, \\\\x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$, 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围为\\bracket{20}.\n\\fourch{$(-2, \\dfrac 12)$}{$(-6, \\dfrac 12)$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac 12,+\\infty)$}",
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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": "2023届松江区一模试题16",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012303": {
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"id": "012303",
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"content": "如图, 已知$AB \\perp$平面$BCD$, $BC \\perp CD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$D$} coordinate (D);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) (A) -- (D) (A) -- (C);\n\\draw [dashed] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$ACD \\perp$平面$ABC$;\\\\\n(2) 若$AB=1$, $CD=BC=2$, 求直线$AD$与平面$ABC$所成角的大小.",
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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": "2023届松江区一模试题17",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012304": {
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"id": "012304",
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"content": "在$\\triangle ABC$中, 内角$A$、$B$、$C$所对边分别为$a$、$b$、$c$, 已知$b \\sin A=a \\cos (B-\\dfrac{\\pi}6)$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$c=2a$, $\\triangle ABC$的面积为$\\dfrac{2 \\sqrt 3}3$, 求$\\triangle ABC$的周长.",
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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": "2023届松江区一模试题18",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012305": {
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"id": "012305",
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"content": "某地准备在山谷中建一座桥梁, 桥址位置的坚直截面图如图所示, 谷底$O$在水平线$MN$上、桥$AB$与$MN$平行, $OO'$为铅垂线($O'$在$AB$上). 经测量, 山谷左侧的轮廓曲线$AO$上任一点$D$到$MN$的距离$h_1$(米)与$D$到$OO'$的距离$a$(米) 之间满足关系式$h_1=\\dfrac 1{40} a^2$, 山谷右侧的轮廓曲线$BO$上任一点$F$到$MN$的距离$h_2$(米)与$F$到$OO'$的距离$b$(米)之间满足关系式$h_2=-\\dfrac 1{800} b^3+6 b$. 已知点$B$到$OO'$的距离为$40$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-8,16) node [left] {$A$} coordinate (A);\n\\draw (4,16) node [right] {$B$} coordinate (B);\n\\draw (-10,0) node [below] {$M$} coordinate (M);\n\\draw (6,0) node [below] {$N$} coordinate (N);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,16) node [above] {$O'$} coordinate (O');\n\\draw (-6,16) node [above] {$C$} coordinate (C);\n\\draw (-6,9) node [left] {$D$} coordinate (D);\n\\draw (2,16) node [above] {$E$} coordinate (E);\n\\draw (2,12) node [right] {$F$} coordinate (F);\n\\draw [ultra thick] (A) -- (B) (C) -- (D) (E) -- (F);\n\\draw (M) -- (N);\n\\draw [dashed] (O) -- (O');\n\\draw [domain = -8:0] plot (\\x,{0.25*pow(\\x,2)});\n\\draw [domain = 0:4.2] plot (\\x,{16-pow(\\x-4,2)});\n\\draw [dashed] (D) --++ (0,-9) node [midway,left] {$h_1$} coordinate (h_1) (D) --++ (6,0) node [midway,above] {$a$} coordinate (a);\n\\draw [dashed] (F) --+ (0,-12) node [midway,right] {$h_2$} coordinate (h_2) (F) --++ (-2,0) node [midway,above] {$b$} coordinate (b);\n\\end{tikzpicture}\n\\end{center}\n(1) 求谷底$O$到桥面$AB$的距离和桥$AB$的长度;\\\\\n(2) 计划在谷底两侧建造平行于$OO'$的桥墩$CD$和$EF$, 且$CE$为$80$米, 其中$C$、$E$在$AB$上(不包括端点), 桥墩$EF$每米造价为$k$(万元)、桥墩$CD$每米造价为$\\dfrac 32 k$(万元)($k>0$). 问$O'E$为多少米时, 桥墩$CD$与$EF$的总造价最低?",
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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": "2023届松江区一模试题19",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012306": {
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"id": "012306",
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"content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的长轴长为$2 \\sqrt 3$, 离心率为$\\dfrac{\\sqrt 6}3$, 斜率为$k$的直线$l$与椭圆$\\Gamma$有两个不同的交点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 若直线$l$的方程为$y=x+t$, 椭圆上点$M(-\\dfrac 32, \\dfrac 12)$关于直线$l$的对称点$N$(与$M$不重合)在椭圆$\\Gamma$上, 求$t$的值;\\\\\n(3) 设$P(-2,0)$, 直线$PA$与椭圆$\\Gamma$的另一个交点为$C$, 直线$PB$与椭圆$\\Gamma$的另一个交点为$D$、若点$C$、$D$和点$Q(-\\dfrac 74, \\dfrac 12)$三点共线, 求$k$的值.",
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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": "2023届松江区一模试题20",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012307": {
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"id": "012307",
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"content": "己知定义在$\\mathbf{R}$上的函数$f(x)=\\mathrm{e}^{k x+b}$($\\mathrm{e}$是自然对数的底数) 满足$f(x)=f'(x)$且$f(-1)=1$, 删除无穷数列$f(1)$、$f(2)$、$f(3)$、$\\cdots$、$f(n)$、$\\cdots$中的第$3$项、第$6$项、$\\cdots$、第$3n$项, $\\cdots$, ($n \\in \\mathbf{N}$, $n\\ge 1$), 余下的项按原来顺序组成一个新数列$\\{t_n\\}$, 记数列$\\{t_n\\}$前$n$项和为$T_n$.\\\\\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 已知数列$\\{t_n\\}$的通项公式是$t_n=f(g(n))$, $n \\in \\mathbf{N}$, $n\\ge 1$, 求函数$g(n)$的解析式;\n(3) 设集合$X$是实数集$\\mathbf{R}$的非空子集, 如果正实数$a$满足: 对任意$x_1$、$x_2 \\in X$, 都有$|x_1-x_2|\\leq a$, 则称$a$为集合$X$的一个``阈度'', 记集合$H=\\{w | w=\\dfrac{T_n}{f(\\dfrac{3 n}2-\\dfrac{1+3(-1)^n}4)}, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 试问集合$H$存在``阈度''吗? 若存在, 求出集合$H$``阈度''的取值范围, 若不存在, 试说明理由.",
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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": "2023届松江区一模试题21",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012308": {
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"id": "012308",
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"content": "已知集合$A=\\{x | 0<x \\leq 4\\}$, $B=\\{-1,2,3,4,5\\}$, 则$A \\cap B=$\\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": "2023届崇明区一模试题1",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012309": {
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"id": "012309",
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"content": "不等式$\\dfrac{2 x+1}{x-2}<0$的解集为\\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": "2023届崇明区一模试题2",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012310": {
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"id": "012310",
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"content": "已知复数$z_1=2+a \\mathrm{i}$, $z_2=3+\\mathrm{i}$, 若$z_1,z_2$是纯虚数, 则实数$a=$\\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": "2023届崇明区一模试题3",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012311": {
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"id": "012311",
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"content": "已知对数函数$y=\\log_a x$($a>0$, $a \\neq 1$)的图像经过点$(4,2)$, 则实数$a=$\\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": "2023届崇明区一模试题4",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012312": {
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"id": "012312",
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"content": "设等比数列$\\{a_n\\}$满足$a_1+a_2=-1$, $a_1-a_3=-3$, 则$a_4=$\\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": "2023届崇明区一模试题5",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012313": {
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"id": "012313",
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"content": "已知方程组$\\begin{cases}x+m y=2, \\\\ m x+16 y=8\\end{cases}$无解, 则实数$m=$\\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": "2023届崇明区一模试题6",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012314": {
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"id": "012314",
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"content": "已知角$\\alpha$的终边与单位圆$x^2+y^2=1$交于点$P(\\dfrac 12, y)$, 则$\\sin (\\dfrac{\\pi}2+\\alpha)=$\\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": "2023届崇明区一模试题7",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012315": {
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"id": "012315",
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"content": "将半径为$2$的半圆形纸片卷成一个无盖的圆锥筒, 则该圆锥筒的高为\\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": "2023届崇明区一模试题8",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012316": {
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"id": "012316",
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"content": "已知函数$f(x)=x^2$, 则曲线$y=f(x)$在点$P(1,1)$处的切线方程是\\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": "2023届崇明区一模试题9",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012317": {
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"id": "012317",
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"content": "设函数$f(x)=\\sin (\\omega x-\\dfrac{\\pi}6)+k$($\\omega>0$), 若$f(x) \\leq f(\\dfrac{\\pi}3)$对任意的实数$x$都成立, 则$\\omega$的最小值为\\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": "2023届崇明区一模试题10",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012318": {
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"id": "012318",
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"content": "在边长为$2$的正六边形$ABCDEF$中, 点$P$为其内部或边界上一点, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{BP}$的取值范围为\\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": "2023届崇明区一模试题11",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012319": {
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"id": "012319",
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"content": "已知椭圆$\\Gamma_1$与双曲线$\\Gamma_2$的离心率互为倒数, 且它们有共同的焦点$F_1$、$F_2$, $P$是$\\Gamma_1$与$\\Gamma_2$在第一象限的交点, 当$\\angle F_1PF_2=\\dfrac{\\pi}6$时, 双曲线$\\Gamma_2$的离心率为\\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": "2023届崇明区一模试题12",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012320": {
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"id": "012320",
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"content": "下列函数中, 既是奇函数又在区间$(0,1)$上是严格增函数的是\\bracket{20}.\n\\fourch{$y=\\sqrt x$}{$y=-x^3$}{$y=\\lg x$}{$y=\\sin x$}",
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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": "2023届崇明区一模试题13",
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|
"edit": [
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"20221210\t王伟叶"
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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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},
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"012321": {
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"id": "012321",
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|
"content": "设$x \\in \\mathbf{R}$, 则``$x+\\dfrac 1x>2$''是``$x \\neq 1$''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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|
"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": "2023届崇明区一模试题14",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012322": {
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"id": "012322",
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"content": "设函数$f(x)=\\sin (x-\\dfrac{\\pi}6)$, 若对于任意$\\alpha \\in[-\\dfrac{5 \\pi}6,-\\dfrac{\\pi}2]$, 在区间$[0, m]$上总存在唯一确定的$\\beta$, 使得$f(\\alpha)+f(\\beta)=0$, 则$m$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}6$}{$\\dfrac{\\pi}2$}{$\\dfrac{7 \\pi}6$}{$\\pi$}",
|
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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": "2023届崇明区一模试题15",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012323": {
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"id": "012323",
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"content": "已知曲线$C:(x^2+y^2)^3=16 x^2 y^2$, 命题$p$: 曲线$C$仅过一个横坐标与纵坐标都是整数的点; 命题$q$: 曲线$C$上的点到原点的最大距离是$2$, 则下列说法正确的是\\bracket{20}.\n\\twoch{$p$、$q$都是真命题}{$p$是真命题, $q$是假命题}{$p$是假命题, $q$是真命题}{$p$、$q$都是假命题}",
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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": "2023届崇明区一模试题16",
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"edit": [
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"20221210\t王伟叶"
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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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},
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"012324": {
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"id": "012324",
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|
"content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=\\sqrt 2$, $A_1C$与底面$ABCD$所成角为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{{sqrt(2)}}\n\\def\\m{{sqrt(2)}}\n\\def\\n{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,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (D1) -- (B1) (A1) -- (B);\n\\draw [dashed] (A) -- (D1) (A1) -- (D) (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求异面直线$A_1B$与$B_1D_1$所成角的大小.",
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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": "2023届崇明区一模试题17",
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"edit": [
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|
"20221210\t王伟叶"
|
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],
|
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012325": {
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"id": "012325",
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|
"content": "已知函数$f(x)=\\sin x \\cos x-\\sin ^2 x+\\dfrac 12$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\\n(2) 在$\\triangle ABC$中, $a$、$b$、$c$为角$A$、$B$、$C$的对边, 且满足$b \\cos 2 A=b \\cos A-a \\sin B$, 且$0<A<\\dfrac{\\pi}2$, 求$f(B)$的取值范围.",
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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,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "2023届崇明区一模试题18",
|
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|
"edit": [
|
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"20221210\t王伟叶"
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|
],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
|
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},
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"012326": {
|
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"id": "012326",
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|
"content": "某公园有一块如图所示的区域$OACB$, 该场地由线段$OA$、$OB$、$AC$及曲线段$BC$围成. 经测量, $\\angle AOB=90^{\\circ}$, $OA=OB=100$米, 曲线$BC$是以$OB$为对称轴的抛物线的一部分, 点$C$到$OA$、$OB$的距离都是$50$米. 现拟在该区域建设一个矩形游乐场$OEDF$, 其中点$D$在线段$AC$或曲线段$BC$上, 点$E$、$F$分别在线段$OA$、$OB$上, 且该游乐场最短边长不低于$30$米, 设$DF=x$米, 游乐场的面积为$S$平方米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.03]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (100,0) node [below right] {$A$} coordinate (A);\n\\draw (0,100) node [above left] {$B$} coordinate (B);\n\\draw (50,50) node [above right] {$C$} coordinate (C);\n\\draw (C) -- (A) (A) -- (O) -- (B);\n\\draw [domain = 0:50, samples = 100] plot (\\x,{100-\\x*\\x/50});\n\\draw (40,68) node [above right] {$D$} coordinate (D);\n\\draw (D) -- ($(O)!(D)!(A)$) node [below] {$E$} coordinate (E);\n\\draw (D) -- ($(O)!(D)!(B)$) node [left] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 试建立平面直角坐标系, 求曲线段$BC$的方程;\\\\\n(2) 求面积$S$关于$x$的函数解析式$S=f(x)$;\\\\\n(3) 试确定点$D$的位置, 使得游乐场的面积$S$最大.(结果精确到$0.1$米)",
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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": "2023届崇明区一模试题19",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012327": {
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"id": "012327",
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"content": "已知椭圆$\\dfrac{x^2}{a^2}+y^2=1$($a>1$)的右焦点为$F$, 左右顶点分别为$A$、$B$, 直线$l$过点$B$且与$x$轴垂直, 点$P$是椭圆上异于$A$、$B$的点, 直线$AP$交直线$l$于点$D$.\\\\\n(1) 若$E$是椭圆的上顶点, 且$\\triangle AEF$是直角三角形, 求椭圆的标准方程;\\\\\n(2) 若$a=2$, $\\angle PAB=45^{\\circ}$, 求$\\triangle PAF$的面积;\\\\\n(3) 判断以$BD$为直径的圆与直线$PF$的位置关系, 并加以证明.",
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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": "2023届崇明区一模试题20",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"012328": {
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"id": "012328",
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"content": "己知数列$\\{a_n\\}$满足$|a_i-a_{i+1}|\\leq|a_{i+1}-a_{i+2}|$($i=1,2, \\cdots, n-2$).\\\\\n(1) 若数列$\\{a_n\\}$的前$4$项分别为$4$、$2$、$a_3$、$1$, 求$a_3$的取值范围;\\\\\n(2) 已知数列$\\{a_n\\}$中各项互不相同, 令$b_m=|a_m-a_{m+1}|$($m=1,2, \\cdots, n-1$), 求证: 数列$\\{a_n\\}$是等差数列的充要条件是数列$\\{b_m\\}$是常数列;\\\\\n(3) 已知数列$\\{a_n\\}$是$m$($m \\in \\mathbf{N}$且$m \\geq 3$)个连续正整数$1,2, \\cdots, m$的一个排列, 若$\\displaystyle\\sum_{k=1}^{m-1}|a_k-a_{k+1}|=m+2$, 求$m$的所有取值.",
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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": "2023届崇明区一模试题21",
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"edit": [
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"20221210\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"020001": {
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"id": "020001",
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"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",
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