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@ -475342,5 +475342,803 @@
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"related": [],
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"remark": "",
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"space": "12ex"
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},
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"040605": {
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"id": "040605",
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"content": "四人互相传球, 由甲开始发球, 并作为第一次传球, 经过$3$次传球后, 球仍回到甲手中, 则不同的传球方式共有\\blank{50}种.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$6$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题1",
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"edit": [
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"20230423\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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"040606": {
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"id": "040606",
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"content": "书架上某层有$8$本书, 新买$2$本插进去, 要保持原有$8$本书的顺序, 则有\\blank{50}种不同的插法. (具体数字作答)",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$90$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题2",
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"edit": [
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"20230423\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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"040607": {
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"id": "040607",
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"content": "若$(x+1)^n$的展开式中第$3$项与第$9$项的二项式系数相等, 则$n=$\\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": "$10$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题3",
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"edit": [
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"20230423\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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"040608": {
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"id": "040608",
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"content": "$7$ 个志愿者的名额分给$3$个班, 每班至少一个名额, 则有\\blank{50}种不同的分配方法. (用数字作答)",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$15$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题4",
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"edit": [
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"20230423\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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"040609": {
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"id": "040609",
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"content": "$A$、$B$、$C$、$D$、$E$五名同学站成一排合影, 若$A$不站在两端, $B$和$C$相邻, 则不同的站队方式共有\\blank{50}种. (用数字作答)",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$24$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题5",
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"edit": [
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"20230423\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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"040610": {
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"id": "040610",
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"content": "设函数$f(x)=\\dfrac{1}{3} x^2-27 \\ln x$在区间$[a, 2 a+1]$上严格减, 则实数$a$的取值范围是\\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": "$(0,\\dfrac{9\\sqrt{2}-2}4]$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题6",
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"edit": [
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"20230423\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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"040611": {
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"id": "040611",
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"content": "$6$ 位大学毕业生分配到$3$家单位, 每家单位至少录用$1$人, 则不同的分配方法共有\\blank{50}种.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$540$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题7",
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"edit": [
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"20230423\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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"040612": {
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"id": "040612",
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"content": "已知在四面体$V-ABC$中, $VA=VB=VC=2$, $AB=1$, $\\angle ACB=\\dfrac{\\pi}{6}$, 则该四面体外接球的表面积为\\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": "$\\dfrac{16\\pi}3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题8",
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"edit": [
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"20230423\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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"040613": {
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"id": "040613",
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"content": "用 1、2、3、4、5 组成没有重复数字的五位数$\\overline{a b c d e}$, 其中满足$a>b>c$, 且$c<d<e$的五位数有$n$个, 则在$1+(1+x)^1+(1+x)^2+(1+x)^3+\\cdots+(1+x)^n$的展开式中, $x^2$的系数是\\blank{50}.(用数字作答)",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "$35$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题9",
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"edit": [
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"20230423\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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"040614": {
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"id": "040614",
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"content": "已知函数$f(x)$的导函数$f'(x)$的图像如图所示, 给出以下结论:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3.2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [above right] {$O$};\n\\foreach \\i in {-2,-1,1,2,3}\n{\\draw (\\i,0) -- (\\i,0.1) node [above] {$\\i$};};\n\\draw (-3,0.5) -- (-1,-0.5) -- (1,0) -- (3.2,-0.8);\n\\draw [dashed] (-1,0) -- (-1,-0.5) (3,0) -- (3,-0.75);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)$在区间$(-1,1)$上严格增;\\\\\n\\textcircled{2} $f(x)$的图像在$x=-2$处的切线斜率等于$0$;\\\\\n\\textcircled{3} $f(x)$在$x=1$处取得极大值;\\\\\n\\textcircled{4} $f'(x)$在$x=-1$处取得极小值.\\\\\n正确的序号是\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "\\textcircled{2}\\textcircled{4}",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题10",
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"edit": [
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"20230423\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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"040615": {
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"id": "040615",
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"content": "平面直角坐标系$xOy$中, 已知点$M(2,-1)$, 若直线$l: 3 x-4 y+5=0$上总存在$P$、$Q$两点, 使得$\\angle PMQ \\geq \\dfrac{\\pi}{2}$恒成立, 则线段$PQ$长度的取值范围是\\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": "$[6,+\\infty)$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题11",
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"edit": [
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"20230423\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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"040616": {
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"id": "040616",
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"content": "设$x_1$、$x_2$是函数$f(x)=a x^2-\\mathrm{e}^x$ ($a \\in \\mathbf{R}$)的两个极值点, 若$\\dfrac{x_2}{x_1} \\geq 2$, 则$a$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": "$\\log_2 \\mathrm{e}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题12",
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"edit": [
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"20230423\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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"040617": {
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"id": "040617",
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"content": "下列求导运算正确的是\\bracket{20}.\n\\twoch{$(\\ln x+\\dfrac{3}{x})'=\\dfrac{1}{x}+\\dfrac{3}{x^2}$}{$(x^2 \\mathrm{e}^x)'=2 x \\mathrm{e}^x$}{$(3^x \\cos 2 x)'=3^x(\\ln 3 \\cdot \\cos 2 x-2 \\sin 2 x)$}{$(\\ln \\dfrac{1}{2}+\\log _2 x)'=2+\\dfrac{1}{x \\ln 2}$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "C",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题13",
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"edit": [
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"20230423\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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"040618": {
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"id": "040618",
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"content": "函数$f(x)=\\dfrac{x-\\sin x}{\\mathrm{e}^x+\\mathrm{e}^{-x}}$在$[-\\pi, \\pi]$上的图像大致为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{(\\x-sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{(\\x-1.5*sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0,0.1) -- (0.3,0.1) node [right] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{-(\\x-sin(\\x/pi*180))/(exp(\\x)+exp(-\\x))});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, xscale = 0.4, yscale = 5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-0.2) -- (0,0.2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain =-pi:pi, samples = 100] plot (\\x,{sqrt(abs(\\x))/10-0.1});\n\\draw (0.3,0.1) -- (0,0.1) node [left] {$0.1$};\n\\draw (pi,0.02) -- (pi,0) node [below] {$\\pi$};\n\\draw (-pi,0.02) -- (-pi,0) node [below] {$-\\pi$};\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": "A",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题14",
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"edit": [
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"20230423\t王伟叶"
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"same": [],
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"remark": "",
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"space": ""
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},
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"040619": {
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"id": "040619",
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"content": "设$(2 x-1)^5=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5$, 则$|a_1|+2|a_2|+3|a_3|+4|a_4|+5|a_5|=$\\bracket{20}.\n\\fourch{$80$}{$242$}{$405$}{$810$}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "D",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题15",
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"edit": [
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"20230423\t王伟叶"
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],
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"same": [],
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"remark": "",
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"space": ""
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},
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"040620": {
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"id": "040620",
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"content": "点$P$为抛物线$C: y^2=4 x$准线上的点, 若存在过$P$的直线交抛物线$C$于$A$、$B$两点, 且$|PA|=|AB|$, 则称点$P$为``$\\Omega$点'', 那么下列结论中正确的是\\bracket{20}.\n\\onech{准线上的所有点都不是``$\\Omega$点''}{准线上的所有点都是``$\\Omega$点''}{准线上仅有有限个点是``$\\Omega$点''}{准线上有无穷多个点(不是所有的点)是``$\\Omega$点''}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "B",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题16",
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"edit": [
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"20230423\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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"040621": {
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"id": "040621",
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"content": "如图, 已知四棱锥$P-ABCD$的底面是菱形, 对角线$AC$、$BD$交于点$O$, $OA=3$, $OB=4$, $OP=3$, $OP \\perp$底面$ABCD$, 设点$M$满足$\\overrightarrow{PM}=\\dfrac{1}{2} \\overrightarrow{MC}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [below] {$O$} coordinate (O);\n\\draw ({-3/sqrt(2)},0,{3/sqrt(2)}) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,{2*sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ($(A)!2!(O)$) node [right] {$C$} coordinate (C);\n\\draw ($(B)!2!(O)$) node [left] {$D$} coordinate (D);\n\\draw (O) ++ (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!{1/3}!(C)$) node [right] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(B)(M)--(B)(P)--(D)--(A);\n\\draw [dashed] (A)--(C)(B)--(D)(P)--(O)(D)--(M)(D)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$PA$与平面$BDM$所成角的正弦值;\\\\\n(2) 求点$P$到平面$BDM$的距离.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $\\dfrac{\\sqrt{10}}{10}$; (2) $\\dfrac{3\\sqrt{5}}{5}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题17",
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"edit": [
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"20230423\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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"040622": {
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"id": "040622",
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"content": "对于代数式$(2 x-\\dfrac{1}{\\sqrt{x}})^5$,\\\\\n(1) 求其展开式中含$x^2$的项的系数;\\\\\n(2) 设该代数式的展开式中前三项的二项式系数的和为$M$, $(1+a x)^4$的展开式中各项系数的和为$N$, 若$M=N$, 求实数$a$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": "(1) $80$; (2) $1$或$-3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题18",
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"edit": [
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"20230423\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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"040623": {
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"id": "040623",
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"content": "已知直线$l: y=k x$($k \\neq 0$)与圆$C: x^2+y^2-2 x-3=0$相交于$A$、$B$两点.\\\\\n(1) 若$|AB|=\\sqrt{13}$, 求$k$;\\\\\n(2) 在$x$轴上是否存在点$M$, 使得当$k$变化时, 总有直线$MA$、$MB$的斜率之和为 $0$ , 若存在, 求出点$M$的坐标; 若不存在, 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": "(1) $\\pm \\sqrt{3}$; (2) 存在, 点$M$的坐标为$(-3,0)$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题19",
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"edit": [
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"20230423\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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"040624": {
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"id": "040624",
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"content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的离心率为$\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 若直线$l: y=k x-2$与椭圆$C$交于两个不同点$D$、$E$, 以线段$DE$为直径的圆经过原点, 求实数$k$的值;\\\\\n(3) 设$A$、$B$为椭圆$C$的左、右顶点, $H$为椭圆$C$上除$A$、$B$外任意一点, 线段$BH$的垂直平分线分别交直线$BH$和直线$AH$于点$P$和点$Q$, 分别过点$P$和$Q$作$x$轴的垂线, 垂足分别为$M$和$N$, 求证: 线段$MN$的长为定值.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $\\dfrac{x^2}4+y^2=1$; (2) $k=\\pm 2$; (3) 定值为$\\dfrac 23$, 证明略",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题20",
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"edit": [
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"20230423\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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"040625": {
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"id": "040625",
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"content": "已知函数$f(x)=x-\\ln x-3$.\\\\\n(1) 求曲线$y=f(x)$在$x=1$处的切线方程;\\\\\n(2) 函数$f(x)$在区间$(k, k+1)$ ($k \\in \\mathbf{N}$)上有零点, 求$k$的值;\\\\\n(3) 记函数$g(x)=x^2-b x-3-f(x)$, 设$x_1$、$x_2$($x_1<x_2$)是函数$g(x)$的两个极值点, 若$b \\geq 2$, 且$g(x_1)-g(x_2) \\geq m$恒成立, 求实数$m$的最大值.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $y=-2$; (2) $k=0$或$4$; (3) $\\dfrac 34-\\ln 2$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304华二高二期中考试试题21",
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"edit": [
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"20230423\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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"040626": {
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"id": "040626",
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"content": "已知$\\mathrm{C}_9^x=\\mathrm{C}_9^{2 x}$, 则正整数$x=$\\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": "$3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题1",
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"edit": [
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"20230423\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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"040627": {
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"id": "040627",
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"content": "$\\mathrm{C}_4^4+\\mathrm{C}_5^4+\\mathrm{C}_6^4+\\mathrm{C}_7^4+\\mathrm{C}_8^4+\\mathrm{C}_9^4+\\mathrm{C}_{10}^4=$\\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": "$462$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题2",
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"edit": [
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"20230423\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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"040628": {
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"id": "040628",
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"content": "函数$f(x)=\\dfrac{1}{2} x^2-2 x+\\ln x$的驻点为\\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": "$1$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题3",
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"edit": [
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"20230423\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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"040629": {
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"id": "040629",
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"content": "$(x+\\dfrac{2}{\\sqrt{x}})^6$的二项展开式中常数项是\\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": "$240$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题4",
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"edit": [
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"20230423\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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"040630": {
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"id": "040630",
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"content": "函数$f(x)=\\dfrac{1}{3} x^3+3 x^2+5 x+2$的极大值为\\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": "$\\dfrac{31}3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题5",
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"edit": [
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"20230423\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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"040631": {
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"id": "040631",
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"content": "已知男、女学生共有$8$人, 若从男生中任选$2$人, 从女生中任选$1$人, 共有$30$种不同的选法, 则女生的总人数为\\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": "$2$或$3$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题6",
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"edit": [
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"20230423\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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"040632": {
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"id": "040632",
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"content": "已知函数$f(x)=(x-1) \\cdot \\mathrm{e}^x$, 则$\\displaystyle\\lim _{x \\to 1} \\dfrac{f(x)-f(1)}{x-1}=$\\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": "$\\mathrm{e}$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题7",
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"edit": [
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"20230423\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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"040633": {
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"id": "040633",
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"content": "$(2+3 x)^{10}$的二项展开式中系数最大的项为\\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": "$2449440x^6$",
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"solution": "",
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|
"duration": -1,
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|
|
"usages": [],
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|
|
"origin": "202304建平高二期中考试试题8",
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|
"edit": [
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"20230423\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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"040634": {
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"id": "040634",
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|
"content": "一场晩会共有$5$个唱歌节目和$3$个舞蹈节目, 随机排序形成一个节目单, 则节目单中前$3$个节目有$2$个舞蹈节目的概率为\\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": "$\\dfrac{15}{56}$",
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|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题9",
|
|
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|
|
"edit": [
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|
"20230423\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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|
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|
|
"space": ""
|
|
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|
|
},
|
|
|
|
|
"040635": {
|
|
|
|
|
"id": "040635",
|
|
|
|
|
"content": "已知关于$x$的不等式$x-\\ln x-a>0$对任意$x \\in(0,+\\infty)$恒成立, 则实数$a$的取值范围是\\blank{50}.",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "填空题",
|
|
|
|
|
"ans": "$(-\\infty,1)$",
|
|
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|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题10",
|
|
|
|
|
"edit": [
|
|
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|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
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|
|
"same": [],
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|
|
"related": [],
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|
|
"remark": "",
|
|
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|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040636": {
|
|
|
|
|
"id": "040636",
|
|
|
|
|
"content": "若$(1+x)^8+(2+x)^8=a_0+a_1(1-x)^1+a_2(1-x)^2+\\cdots+a_8(1-x)^8$对任意$x \\in \\mathbf{R}$恒成立, 则$a_4=$\\blank{50}.",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "填空题",
|
|
|
|
|
"ans": "$6790$",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题11",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
|
|
|
"same": [],
|
|
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|
|
"related": [],
|
|
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|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040637": {
|
|
|
|
|
"id": "040637",
|
|
|
|
|
"content": "已知$A(a, 1-a^2)$, $B(b, 1-b^2)$, 其中$a b<0$, 过$A$、$B$分别作二次函数$y=1-x^2$的切线, 则两条切线与$x$轴围成的三角形面积的最小值为\\blank{50}.",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "填空题",
|
|
|
|
|
"ans": "$\\dfrac{8\\sqrt{3}}9$",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题12",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
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|
|
"same": [],
|
|
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|
|
"related": [],
|
|
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|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
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|
|
},
|
|
|
|
|
"040638": {
|
|
|
|
|
"id": "040638",
|
|
|
|
|
"content": "在古典概率模型中, $\\Omega$是样本空间, $x$是样本点, $A$是随机事件, 则下列表述正确的\\bracket{20}.\n\\fourch{$x \\in \\Omega$}{$x \\subseteq \\Omega$}{$A \\in \\Omega$}{$\\Omega \\subseteq A$}",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "选择题",
|
|
|
|
|
"ans": "A",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题13",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
|
|
|
"same": [],
|
|
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|
|
"related": [],
|
|
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|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040639": {
|
|
|
|
|
"id": "040639",
|
|
|
|
|
"content": "已知$A$、$B$为两个随机事件, 则``$A$、$B$为互斥事件''是``$A$、$B$为对立事件''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{非充分非必要条件}",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "选择题",
|
|
|
|
|
"ans": "B",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题14",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
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|
|
"same": [],
|
|
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|
|
"related": [],
|
|
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|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040640": {
|
|
|
|
|
"id": "040640",
|
|
|
|
|
"content": "下列关于排列数$\\mathrm{P}_n^{m-1}$和组合数$\\mathrm{C}_n^{m-1}$的计算中正确的是\\bracket{20}.\n\\twoch{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(m-1) !}$}{$\\mathrm{P}_n^{m-1}=\\dfrac{n !}{(n-m-1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m+1) !}$}{$\\mathrm{C}_n^{m-1}=\\dfrac{n !}{(m-1) !(n-m-1) !}$}",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "选择题",
|
|
|
|
|
"ans": "C",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题15",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
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|
|
"same": [],
|
|
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|
|
"related": [],
|
|
|
|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040641": {
|
|
|
|
|
"id": "040641",
|
|
|
|
|
"content": "已知$x \\in \\mathbf{N}$, $y \\in \\mathbf{N}$, $x<y$, 则方程$x^y=y^x$的解的组数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{无穷多个}",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "选择题",
|
|
|
|
|
"ans": "B",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题16",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
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|
|
"same": [],
|
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|
|
"related": [],
|
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|
|
"remark": "",
|
|
|
|
|
"space": ""
|
|
|
|
|
},
|
|
|
|
|
"040642": {
|
|
|
|
|
"id": "040642",
|
|
|
|
|
"content": "已知函数$f(x)=-x^3+a x^2+(1-2 a) x+a$.\\\\\n(1) 求函数$f(x)$在$x=1$处的切线方程;\\\\\n(2) 若函数$f(x)$在$\\mathbf{R}$上严格减, 求实数$a$的取值范围.",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "解答题",
|
|
|
|
|
"ans": "(1) $y=-2x+2$; (2) $[3-\\sqrt{6},3+\\sqrt{6}]$",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题17",
|
|
|
|
|
"edit": [
|
|
|
|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
|
|
|
"same": [],
|
|
|
|
|
"related": [],
|
|
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|
|
"remark": "",
|
|
|
|
|
"space": "12ex"
|
|
|
|
|
},
|
|
|
|
|
"040643": {
|
|
|
|
|
"id": "040643",
|
|
|
|
|
"content": "甲、乙两人进行乒乓球决赛, 采用五局三胜制, 对于每局比赛, 甲获胜的概率为$\\dfrac{2}{3}$, 乙获胜的概率为$\\dfrac{1}{3}$, 且每局比赛的结果互相独立.\\\\\n(1) 在乒乓球比赛中, 如果一方全胜最终获得比赛的胜利, 那么将其形象地称之为``剃光头''. 求甲、乙的这场乒兵球决赛``剃光头''的概率;\\\\\n(2) 在乒乓球比赛中, 如果实力较弱的一方最终获得比赛的胜利, 那么将其称之为``爆冷门'', 求甲、乙的这场乒乓球决赛``爆冷门''的概率.",
|
|
|
|
|
"objs": [],
|
|
|
|
|
"tags": [],
|
|
|
|
|
"genre": "解答题",
|
|
|
|
|
"ans": "(1) $\\dfrac 13$; (2) $\\dfrac{17}{81}$",
|
|
|
|
|
"solution": "",
|
|
|
|
|
"duration": -1,
|
|
|
|
|
"usages": [],
|
|
|
|
|
"origin": "202304建平高二期中考试试题18",
|
|
|
|
|
"edit": [
|
|
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|
|
"20230423\t王伟叶"
|
|
|
|
|
],
|
|
|
|
|
"same": [],
|
|
|
|
|
"related": [],
|
|
|
|
|
"remark": "",
|
|
|
|
|
"space": "12ex"
|
|
|
|
|
},
|
|
|
|
|
"040644": {
|
|
|
|
|
"id": "040644",
|
|
|
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"content": "``得地率''是指可供人活动的区域的占地面积与总占地面积之比. ``得地率''越高, 也就意味着人们可活动的区域更大, 因此在设计活动场地时, 通常会将``得地率''作为一个重要的指标进行考虑. 上海某大型购物商场欲将地下一层的一块半圆形空地改建为亲子乐园, 建造一个供亲子游玩的海洋球池和两个供人们休息和娱乐, 且大小完全相同的休息区. 除海洋球池和休息区外的剩余空地全部用气垫筑起``高墙'', 以保护亲子乐园中的人们. 如图所示, 设半圆形空地的圆心为$O$, 半径为$R$, $MN$为直径, 矩形海洋球池$ABCD$的顶点$A$、$B$在$MN$上, 顶点$C$、$D$在半圆的圆周上, 矩形休息区$BEFG$和$AHIJ$的顶点$E$、$H$在$MN$上, 顶点$F$、$I$在半圆的圆周上, 顶点$G$、$J$分别在线段$BC$、$AD$上. 已知$\\angle EOF=\\dfrac{\\pi}{6}$, 设$\\angle BOC=\\theta$, 其中$\\theta \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}]$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\t{50}\n\\draw (3,0) node [below] {$N$} coordinate (N);\n\\draw (-3,0) node [below] {$M$} coordinate (M);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw (N) arc (0:180:3) -- cycle;\n\\draw (30:3) node [above right] {$F$} coordinate (F);\n\\draw (150:3) node [above left] {$I$} coordinate (I);\n\\draw (F) -- ($(M)!(F)!(N)$) node [below] {$E$} coordinate (E);\n\\draw (I) -- ($(M)!(I)!(N)$) node [below] {$H$} coordinate (H);\n\\draw (\\t:3) node [above right] {$C$} coordinate (C);\n\\draw ({180-\\t}:3) node [above left] {$D$} coordinate (D);\n\\draw (C) -- ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw (D) -- ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (C)--(D);\n\\draw (I) -- ($(A)!(I)!(D)$) node [right] {$J$} coordinate (J);\n\\draw (F) -- ($(B)!(F)!(C)$) node [left] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(C)$) node {海洋球池};\n\\draw ($(A)!0.5!(I)$) node {息};\n\\draw ($(A)!0.5!(I)$) ++ (0,0.4) node {休};\n\\draw ($(A)!0.5!(I)$) ++ (0,-0.4) node {区};\n\\draw ($(B)!0.5!(F)$) node {息};\n\\draw ($(B)!0.5!(F)$) ++ (0,0.4) node {休};\n\\draw ($(B)!0.5!(F)$) ++ (0,-0.4) node {区};\n\\end{tikzpicture}\n\\end{center}\n(1) 求当$\\theta=\\dfrac{\\pi}{4}$时该亲子乐园可供人活动的区域面积$S$, 并求出此时的``得地率''(结果精确到$1 \\%)$;\\\\\n(2) 求当$\\theta$为多大时, 该亲子乐园的``得地率''最大?",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $\\theta=\\dfrac\\pi 4$时, $S=\\dfrac{2-\\sqrt{2}+\\sqrt{3}}2R^2$, ``得地率''约为$74\\%$; (2) $\\theta = \\arcsin\\dfrac{1+\\sqrt{33}}8$时, ``得地率''最大",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题19",
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"edit": [
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"20230423\t王伟叶"
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"remark": "",
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"space": "12ex"
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},
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"040645": {
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"id": "040645",
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"content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点分别为$F_1$、$F_2$. 椭圆$\\Gamma$上有互异的且不在$x$轴上的三点$A$、$B$、$C$满足直线$AC$经过$F_1$, 直线$BC$经过$F_2$.\\\\\n(1) 若椭圆$\\Gamma$的长轴长为 $4$ , 离心率为$\\dfrac{1}{2}$, 求$b$的值;\\\\\n(2) 若点$C$的坐标为$(0,1)$, $\\triangle ABC$的面积$S=\\dfrac{64}{49} \\sqrt{3}$, 求$a$的值;\\\\\n(3) 若$a=\\sqrt{2}$, $b=1$, 直线$AB$经过点$(\\dfrac{3}{2}, 0)$, 求$C$的坐标.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $b=\\sqrt{3}$; (2) $a=2$; (3) $C$的坐标为$(-\\dfrac 43,-\\dfrac 13)$或$(-\\dfrac 43,\\dfrac 13)$",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题20",
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"edit": [
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"20230423\t王伟叶"
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"space": "12ex"
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},
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"040646": {
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"id": "040646",
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"content": "已知定义在$\\mathbf{R}$上的函数$f(x)$的导函数为$f'(x)$, 若$|f'(x)| \\leq 1$对任意$x \\in \\mathbf{R}$恒成立, 则称函数$f(x)$为``线性控制函数''.\\\\\n(1) 判断函数$f(x)=\\sin x$和$g(x)=\\mathrm{e}^x$是否为``线性控制函数'', 并说明理由;\\\\\n(2) 若函数$f(x)$为``线性控制函数'', 且$f(x)$在$\\mathbf{R}$上严格增, 设$A$、$B$为函数$f(x)$图像上互异的两点, 设直线$AB$的斜率为$k$, 判断命题``$0<k \\leq 1$''的真假, 并说明理由;\\\\\n(3) 若函数$f(x)$为``线性控制函数'', 且$f(x)$是以$T$($T>0$)为周期的周期函数, 证明: 对任意$x_1$、$x_2$都有$|f(x_1)-f(x_2)| \\leq T$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "(1) $f(x)$是``线性控制函数'', $g(x)$不是``线性控制函数'', 理由略; (2) 是真命题, 理由略; (3) 证明略",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "202304建平高二期中考试试题21",
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"edit": [
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"20230423\t王伟叶"
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"same": [],
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"remark": "",
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"space": "12ex"
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}
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}
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