From bb4a38fdabee18505c30e0df192330e6ba61ea13 Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Mon, 8 Jan 2024 12:27:37 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A526=E5=B1=8A=E5=AF=92=E5=81=87?= =?UTF-8?q?=E4=BD=9C=E4=B8=9A9=E5=B9=B6=E5=BB=BA=E7=AB=8Brelated?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 358 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 355 insertions(+), 3 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 523288d2..c283d3a2 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -58258,7 +58258,9 @@ "20220625\t王伟叶" ], "same": [], - "related": [], + "related": [ + "023523" + ], "remark": "", "space": "", "unrelated": [] @@ -628197,6 +628199,352 @@ "space": "4em", "unrelated": [] }, + "023518": { + "id": "023518", + "content": "设实数 $a$ 满足 $\\log _2a=4$, 则 $a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023519": { + "id": "023519", + "content": "已知幂函数 $f(x)=(m-1) x^{m^2-3 m-5}$ 的图像不经过原点, 则实数 $m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [ + "031290" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "023520": { + "id": "023520", + "content": "函数 $f(x)=\\log _2(1-x^2)$ 的定义域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023521": { + "id": "023521", + "content": "若函数 $f(x)=a^x$($a>1$) 在 $[-1,2]$ 上的最大值为 $4$, 则其最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023522": { + "id": "023522", + "content": "在同一平面直角坐标系中, 函数 $y=g(x)$ 的图像与 $y=3^x$ 的图像关于直线 $y=x$ 对称, 而函数 $y=f(x)$ 的图像与 $y=g(x)$ 的图像关于 $y$ 轴对称, 若 $f(a)=-1$, 则 $a$ 的值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [ + "031708" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "023523": { + "id": "023523", + "content": "在数列 $\\{a_n\\}$ 中, 已知 $a_1=1$, $a_{n+1}=\\dfrac{n+1}{n+2}a_n$($n \\geq 1$), 则数列的通项公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [ + "001803" + ], + "remark": "", + "space": "", + "unrelated": [] + }, + "023524": { + "id": "023524", + "content": "若定义在 $\\mathbf{R}$ 上的奇函数 $f(x)$ 在 $(0,+\\infty)$ 上是严格增函数, 且 $f(-4)=0$, 则使得 $x f(x)>0$成立的 $x$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023525": { + "id": "023525", + "content": "函数 $f(x)=\\lg (2^x+2^{-x}+a-1)$ 的值域是 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023526": { + "id": "023526", + "content": "若直角坐标平面内两点 $P$、$Q$ 满足条件: \\textcircled{1} $P$、$Q$ 都在函数 $f(x)$ 的图像上; \\textcircled{2} $P$、$Q$ 关于原点对称, 则对称点 $(P, Q)$ 是函数 $f(x)$ 的一个``匹配点对''(点对 $(P, Q)$ 与 $(Q, P)$ 看作同一个``匹配点对''). 已知函数 $f(x)=\\begin{cases}2 x^2+4 x+1,& x<0,\\\\\\dfrac{2}{\\mathrm{e}^2},& x \\geq 0,\\end{cases}$ 则 $f(x)$ 的``匹配点对''有\\blank{50}个.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023527": { + "id": "023527", + "content": "函数 $y=1-\\dfrac{1}{x + 1}$ 的值域是\\bracket{20}.\n\\fourch{$(-\\infty, 1)$}{$(1,+\\infty)$}{$(-\\infty, 1) \\cup(1,+\\infty)$}{$(-\\infty, +\\infty)$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023528": { + "id": "023528", + "content": "已知函数 $f(x)=\\begin{cases}1, & x<0,\\\\0, & x=0, \\\\ -1, & x<0\\dots\\end{cases}$ 设 $F(x)=x^2 \\cdot f(x)$, 则 $F(x)$ 是\\bracket{20}.\n\\onech{奇函数, 在 $(-\\infty, +\\infty)$ 上为严格减函数}{奇函数, 在 $(-\\infty, +\\infty)$ 上为严格增函数}{偶函数, 在 $(-\\infty, 0)$ 上严格减, 在$(0, +\\infty)$ 上严格增}{偶函数, 在 $(-\\infty, 0)$ 上严格增, 在 $(0,+\\infty)$ 上严格减}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023529": { + "id": "023529", + "content": "设 $a>b>c>0$, 则 $2 a^2+\\dfrac{1}{a b}+\\dfrac{1}{a(a-b)}-10 a c+25 c^2$ 取得最小值时, $a$ 的值为\\bracket{20}.\n\\fourch{$\\sqrt{2}$}{$2$}{$4$}{$2 \\sqrt{5}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "023530": { + "id": "023530", + "content": "已知函数 $f(x)=a x^2+2 a x+1$.\\\\\n(1) 若实数 $a=1$ , 请写出函数 $y=3^{f(x)}$ 的单调区间 (不需要过程); \\\\\n(2) 已知函数 $y=f(x)$ 在区间 $[-3,2]$ 上的最大值为 $2$, 求实数 $a$ 的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023531": { + "id": "023531", + "content": "设函数 $f(x)=|2 x-a|$, $g(x)=x+2$.\\\\\n(1) 当 $a=1$ 时, 求不等式 $f(x) \\leq g(x)$ 的解集;\\\\\n(2) 求证: $f(\\dfrac{b}{2}), f(-\\dfrac{b}{2}), f(\\dfrac{1}{2})$ 中至少有一个不小于 $\\dfrac{1}{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023532": { + "id": "023532", + "content": "研究表明: 在一节 40 分钟的网课中, 学生的注意力指数 $y$ 与听课时间 $x$ (单位: 分钟)之间的变化曲线如图所示.\n当 $x \\in[0,16]$ 时, 曲线是二次函数图像的一部分 (顶点坐标为 $(12,84)$ ) 当 $x \\in[16,40]$时, 曲线是函数 $y=\\log _{0.8}(x+a)+80$ 图像的一部分. 当学生的注意力指数不高于 68 时,称学生处于``久佳听课状态''.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 0.06]\n\\draw [->] (0,0) -- (50,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,96) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (12,0) node [below] {$12$} -- (12,84) -- (0,84) node [left] {$84$};\n\\draw [dashed] (16,0) node [below] {$16$} -- (16,80) -- (0,80) node [left] {$80$};\n\\draw [dashed] (40,0) node [below] {$40$} -- (40,{ln(25)/ln(0.8)+80});\n\\draw [domain = 16:40, samples = 100] plot (\\x,{ln(\\x-15)/ln(0.8)+80});\n\\draw [domain = 0:16, samples = 100] plot (\\x,{-0.25*(\\x-12)*(\\x-12)+84});\n\\end{tikzpicture}\n\\end{center}\n(1) 求函数 $y=f(x)$ 的解析式;\\\\\n(2) 在一节 40 分钟的网课中, 学生处于``欠佳听课状态''的时间有多长? (精确到1分钟)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023533": { + "id": "023533", + "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 满足 $a_1=\\dfrac{1}{2}$, $S_{n+1}=S_n+\\dfrac{2 a_n}{2 a_n+1}$.\\\\\n(1) 证明数列 $\\{\\dfrac{1}{a_n}\\}$ 是等差数列, 并求出数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 若数列 $\\{b_n\\}$ 满足 $b_n=(2 n + 1)^2a_na_{n+1}$, 求数列 $\\{b_n\\}$ 的前 $n$ 项和 $T_n$;\\\\\n(3) 若数列 $\\{C_n\\}$ 满足 $C_n=(2^n-8) a_n$, 则数列 $C_n$ 最大项和最小项是否存在? 若存在请指出, 若不存在请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "023534": { + "id": "023534", + "content": "若函数 $f(x)$ 的定义域为 $D$, 集合 $M \\subseteq D$, 若存在非零实数 $t$ 使得任意 $x \\in M$ 都有 $x+t \\in D$, 且 $f(x+t)>f(x)$, 则称 $f(x)$ 为 $M$ 上的 $t-$增长函数.\\\\\n(1) 已知函数 $g(x)=x$ , 判断 $g(x)$ 是否为区间 $[-1,0]$ 上的 $\\dfrac{3}{2}-$ 增长函数, 并说明理由;\\\\\n(2) 已知函数 $f(x)=|x|$, 且 $f(x)$ 是区间 $[-4,-2]$ 上的 $n-$ 增长函数, 求正整数 $n$ 的最小值;\\\\\n(3) 如果 $f(x)$ 是定义域为 $\\mathbf{R}$ 的奇函数, 当 $x \\geq 0$ 时, $f(x)=|x-a^2|-a^2$, 且 $f(x)$ 为 $\\mathbf{R}$ 上的 $4-$增长函数, 求实数 $a$ 的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "26届寒假作业补充题目", + "edit": [ + "20240108\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "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