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录入26届高一寒假作业并建立related

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wangweiye7840 2024-01-08 10:51:46 +08:00
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题库0.3/Problems.json
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@ -26113,7 +26113,9 @@
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@ -626177,6 +626201,426 @@
"space": "4em",
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"023424": {
"id": "023424",
"content": "函数 $y=x^{-2}$ 在区间 $[\\dfrac{1}{2}, 2]$ 上的最大值是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
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"023425": {
"id": "023425",
"content": "若幂函数 $y=f(x)$ 的图像过点 $(3, \\sqrt[4]{27})$, 则 $y=f^{-1}(x)$ 的解析式是\\blank{50}.",
"objs": [],
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"genre": "填空题",
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"023426": {
"id": "023426",
"content": "若函数 $y=f(x)$ 的图像与函数 $g(x)=(\\dfrac{1}{2})^x$ 的图像关于直线 $y=x$ 对称, 则 $f(2 x-x^2)$ 的严格减区间为\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"023427": {
"id": "023427",
"content": "若 $y=x^{a^2-4 a-9}$ 是偶函数, 且在 ($0,+\\infty$) 是减函数, 则整数 $a$ 的值是\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
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"023428": {
"id": "023428",
"content": "若幂函数 $y=x^{(-1)^k\\dfrac{n}{m}}$($m, n, k \\in \\mathbf{N}$, $m$、$n$、$k>0$, $m, n$ 互质)的图像在一、二象限, 不过原点, 则 $k, m, n$ 奇偶性为\\blank{50}.",
"objs": [],
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"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
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"remark": "",
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},
"023429": {
"id": "023429",
"content": "已知函数 $y=f(x)$ 的定义域是 $[1,2)$, 则函数 $f(2^x)$ 的定义域是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "26届寒假作业补充题目",
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"remark": "",
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},
"023430": {
"id": "023430",
"content": "当 $a>0$ 且 $a \\neq 1$ 时, 函数 $f(x)=a^{x-2}-3$ 必过定点\\blank{50}.",
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"023431": {
"id": "023431",
"content": "函数 $y=(\\dfrac{1}{2})^{\\sqrt{-x^2+x+2}}$ 的值域是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
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"023432": {
"id": "023432",
"content": "函数 $y=\\dfrac{1-3 x}{2+x}$ 的图像的对称中心是\\blank{50}.",
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"023433": {
"id": "023433",
"content": "设函数 $f(x)=a+\\dfrac{1}{3^x-1}$ 为奇函数, 则实数 $a$ 的值是\\blank{50}.",
"objs": [],
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"023434": {
"id": "023434",
"content": "已知 $f(x)=\\dfrac{\\mathrm{e}^x-\\mathrm{e}^{-x}}{2}$, 则下列正确的是\\bracket{20}.\n\\twoch{奇函数, 在 $\\mathbf{R}$ 上为增函数}{偶函数, 在 $\\mathbf{R}$ 上为增函数}{奇函数, 在 $\\mathbf{R}$ 上为减函数}{偶函数, 在 $\\mathbf{R}$ 上为减函数}",
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"tags": [],
"genre": "选择题",
"ans": "",
"solution": "",
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"023435": {
"id": "023435",
"content": "如图所示, 幂函数 $y=x^{\\alpha}$ 在第一象限的图像, 比较 $0, \\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, 1$ 的大小关系为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:{sqrt(3)}, samples = 100] plot (\\x,{\\x*\\x}) node [above] {$y=x^{\\alpha_1}$};\n\\draw [domain = {1/3}:3, samples = 100] plot (\\x,{1/\\x}) node [above right] {$y=x^{\\alpha_2}$};\n\\draw [domain = {1/sqrt(3)}:3, samples = 100] plot (\\x,{1/\\x/\\x}) node [right] {$y=x^{\\alpha_3}$};\n\\draw [domain = 0:3, samples = 100] plot (\\x,{sqrt(\\x)}) node [above] {$y=x^{\\alpha_4}$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$\\alpha_1<\\alpha_3<0<\\alpha_4<\\alpha_2<1$}{$0<\\alpha_1<\\alpha_2<\\alpha_3<\\alpha_4<1$}{$\\alpha_2<\\alpha_4<0<\\alpha_3<1<\\alpha_1$}{$\\alpha_3<\\alpha_2<0<\\alpha_4<1<\\alpha_1$}",
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"genre": "选择题",
"ans": "",
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"023436": {
"id": "023436",
"content": "函数 $f(x)=\\begin{cases}2^{-x}-1,& x \\leq 0,\\\\x^{\\frac{1}{2}},& x>0,\\end{cases}$ 求满足 $f(x)>1$ 的 $x$ 的取值范围.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"023437": {
"id": "023437",
"content": "若函数 $y=\\log _2(k x^2+4 k x+3)$ 的定义域为 $\\mathbf{R}$, 求实数 $k$ 的取值范围.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"edit": [
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"023438": {
"id": "023438",
"content": "画出下列函数的图像, 并指出该函数图像可经由哪个幂函数的图像经过一系列的图像变换得到(写清楚图像变换的过程). \\\\\n(1) $y=(x-1)^{-\\frac{7}{5}}-2$;\\\\\n(2) $y=\\dfrac{1}{|x|-1}$.",
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"023439": {
"id": "023439",
"content": "对于幂函数 $f(x)=x^{\\frac{4}{5}}$, 若 $0<x_1<x_2$, 则 $f(\\dfrac{x_1+x_2}{2}), \\dfrac{f(x_1)+f(x_2)}{2}$ 的大小关系是\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"023440": {
"id": "023440",
"content": "已知定义域在 $\\mathbf{R}$ 上的奇函数 $f(x)$ 在区间 $(-\\infty, 0]$ 上是严格减函数, 则满足 $f(4^x-4)+f(2^{x+1}-4^x) \\geq 0$ 的 $x$ 的集合是\\blank{50}.",
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"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
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"023441": {
"id": "023441",
"content": "设 $a>0$, $a \\neq 1$, 则函数 $f(x)=a^x+x^2-2 x-2 a+1$ 的零点个数为\\blank{50}.",
"objs": [],
"tags": [],
"genre": "填空题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
"related": [
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"remark": "",
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"023442": {
"id": "023442",
"content": "已知函数 $f(x)=\\log _2\\dfrac{x+1}{x-1}+\\log _2(x-1)+\\log _2(p-x), p$ 为常数, 且 $p \\in \\mathbf{R}$.\\\\\n(1) 求函数 $f(x)$ 的定义域; \\\\\n(2) 求函数 $f(x)$ 的值域.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"related": [
"002839"
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"remark": "",
"space": "4em",
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"023443": {
"id": "023443",
"content": "设函数 $f(x)=\\lg (x+\\sqrt{x^2+1})$.\\\\\n(1) 确定函数 $f(x)$ 的定义域;\\\\\n(2) 判断函数 $f(x)$ 的奇偶性;\\\\\n(3) 证明函数 $f(x)$ 在其定义域上是严格增函数;\\\\\n(4) 求函数 $f(x)$ 的反函数.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"same": [],
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"remark": "",
"space": "4em",
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"030001": {
"id": "030001",
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",