添加2026届高三上学期测验卷03部分试题
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@ -519973,6 +519973,286 @@
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"space": "4em",
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"unrelated": []
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},
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"019916": {
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"id": "019916",
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"content": "已知集合 $A=\\{x, x^2\\}$($x \\in \\mathbf{R}$), 若 $1 \\in A$, 则 $x=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019917": {
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"id": "019917",
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"content": "若复数 $z=\\dfrac{2+\\mathrm{i}}{1-2 \\mathrm{i}}$ ($\\mathrm{i}$ 为虚数单位), 则 $z$ 的模 $|z|=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019918": {
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"id": "019918",
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"content": "已知函数 $f(x)=\\lg \\dfrac{1-x}{1+x}$, 则该函数的定义域是\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019919": {
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"id": "019919",
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"content": "已知 $\\sin (\\pi-\\theta)=-\\dfrac{1}{3}$, 则 $\\cos (\\dfrac{\\pi}{2}-\\theta)=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019920": {
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"id": "019920",
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"content": "在 $\\triangle ABC$ 中, $\\angle A=90^{\\circ}$, $AB=3$, $AC=4$, 将 $\\triangle ABC$ 绕边 $AC$ 所在直线旋转一周得到儿何体 $\\Gamma$, 则 $\\Gamma$ 的侧面积为\\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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019921": {
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"id": "019921",
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"content": "某歌手电视大奖赛中, 七位评委对某选手打出如下分数: $7.9,8.1,8.4,8.5,8.5,8.7,9.9$, 则其第 50 百分位数为\\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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019922": {
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"id": "019922",
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"content": "已知 $\\triangle ABC$ 的外接圆圆心为 $O$, $\\angle A=120^{\\circ}$, 若 $\\overrightarrow{AO}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$($x, y \\in \\mathbf{R}$), 则 $x+y$ 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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019923": {
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"id": "019923",
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"content": "关于 $x$ 的方程 $x^2+a x+b-3=0$($a, b \\in \\mathbf{R}$) 在 $[1,2]$ 上有实根, 则 $a^2+(b-4)^2$ 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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019924": {
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"id": "019924",
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"content": "为了得到函数 $y=\\sin x-\\sqrt{3}\\cos x$($x \\in \\mathbf{R}$) 的图像, 可以将函数 $y=2 \\sin x$($x \\in \\mathbf{R}$) 的图像\\bracket{20}.\n\\fourch{向右平移 $\\dfrac{\\pi}{6}$ 个单位}{向左平移 $\\dfrac{\\pi}{3}$ 个单位}{向右平移 $\\dfrac{\\pi}{3}$ 个单位}{向左平移 $\\dfrac{\\pi}{6}$ 个单位}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019925": {
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"id": "019925",
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"content": "已知 $k \\in \\mathbf{R}$, 函数 $f(x)=|x^2-4|+x^2+k x$ 的定义域为 $\\mathbf{R}$, 若函数 $f(x)$ 在区间 $(0,4)$ 上有两个不同的零点, 则 $k$ 的取值范围是\\bracket{20} .\n\\fourch{$-7<k<-2$}{$k<-7$ 或 $k>-2$}{$-7<k<0$}{$-2<k<0$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"019926": {
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"id": "019926",
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"content": "已知 $a$、$b$、$c$ 分别是 $\\triangle ABC$ 中角 $A$、$B$、$C$ 的对边, $a=4 \\sqrt{3}$, $b=6$, $\\cos A=-\\dfrac{1}{3}$.\\\\\n(1) 求 $c$;\\\\\n(2) 求 $\\cos 2B$ 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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019927": {
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"id": "019927",
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"content": "为了调查甲、乙两个网站受欢迎的程度, 随机选取了 14 天, 统计上午 $8: 00-10: 00$ 间各自的点击量, 得如下数据:\\\\\n甲: $73,24,58,72,64,38,66,70,20,41,55,67,8,25$;\\\\\n乙: $12,37,21,5,54,42,61,45,19,6,19,36,42,14$;\\\\\n(1) 用茎叶图表示上面的数据;\\\\\n(2) 甲网站点击量在 $[10,40]$ 间的频率是多少?\\\\\n(3) 从统计的角度考虑, 你认为哪个网站更受欢迎? 请说明理由.",
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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": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019928": {
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"id": "019928",
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"content": "如图, 在四面体 $ABCD$ 中, $\\triangle ABC$ 是边长为 $2$ 的等边三角形, $\\triangle DBC$ 为直角三角形, 其中 $D$ 为直角顶点, $\\angle DCB=60^{\\circ}$. $E$、$F$、$G$、$H$ 分别是线段 $AB$、$AC$、$CD$、$DB$ 上的动点, 且四边形 $EFGH$ 为平行四边形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$C$} coordinate (C);\n\\draw ({-sqrt(3)},0,0) node [left] {$B$} coordinate (B);\n\\draw ({(sqrt(2)-sqrt(3))/2},1,{1/2-sqrt(3/2)}) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.4!(D)$) node [above left] {$H$} coordinate (H);\n\\draw ($(C)!0.4!(D)$) node [below right] {$G$} coordinate (G);\n\\draw ($(C)!0.4!(A)$) node [above left] {$F$} coordinate (F);\n\\draw ($(B)!0.4!(A)$) node [above left] {$E$} coordinate (E);\n\\draw (E)--(F)--(G);\n\\draw [dashed] (E)--(H)--(G);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\parallel $ 平面 $EFGH$;\\\\\n(2) 设二面角 $A-BC-D$ 的平面角为 $\\theta$, 求 $\\theta$ 在区间 $[0, \\dfrac{\\pi}{2}]$ 变化的过程中, 线段 $DA$ 在平面 $BCD$ 上的投影所扫过的平面区域的面积;\\\\\n(3) 设 $\\lambda=\\dfrac{AE}{AB}$($\\lambda \\in(0,1)$), 且平面 $ABC \\perp$ 平面 $BCD$, 则当 $\\lambda$ 为何值时, 多面体 $ADEFGH$ 的体积恰好为 $\\dfrac{1}{4}$ ?",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"019929": {
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"id": "019929",
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"content": "若定义域为 $\\mathbf{R}$ 的函数 $y=h(x)$ 满足: 对于任意 $x \\in \\mathbf{R}$, 都有 $h(x+2 \\pi)=h(x)+h(2 \\pi)$, 则称函数 $y=h(x)$ 具有性质 $P$.\\\\\n(1) 设函数 $y=f(x)$, $y=g(x)$ 的表达式分别为 $f(x)=\\sin x+x$, $g(x)=\\cos x$, 判断函数 $y=f(x)$与 $y=g(x)$ 是否具有性质 $P$, 说明理由;\\\\\n(2) 设函数 $y=f(x)$ 的表达式为 $f(x)=\\sin (\\omega x+\\varphi)$, 是否存在 $0<\\omega<1$ 以及 $-\\pi<\\varphi<\\pi$, 使得函数 $y=\\sin (\\omega x+\\varphi)$ 具有性质 $P$ ? 若存在, 求出 $\\omega$、$\\varphi$ 的值; 若不存在, 说明理由;\\\\\n(3) 设函数 $y=f(x)$ 具有性质 $P$, 且在 $[0,2 \\pi]$ 上的值域恰为 $[f(0), f(2 \\pi)]$; 以 $2 \\pi$ 为周期的函数 $y=g(x)$ 的表达式为 $g(x)=\\sin (f(x))$, 且在开区间 $(0,2 \\pi)$ 上有且仅有一个零点, 求证: $f(2 \\pi)=2 \\pi$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "自拟题目",
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"edit": [
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"20231102\t余利成"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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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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