From 926b3a5e7601f5be20a323d4afd4e576ffafde05 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Tue, 11 Apr 2023 22:21:48 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B7=BB=E5=8A=A022=E5=B1=8A=E7=AC=AC=E4=B8=89?= =?UTF-8?q?=E8=BD=AE=E8=AE=B2=E4=B9=89=E9=A2=98=E7=9B=AE73=E9=81=93?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/修改题目数据库.py | 2 +- 工具/批量收录题目.py | 8 +- 题库0.3/Problems.json | 1393 +++++++++++++++++++++++++++++++++++++++- 3 files changed, 1395 insertions(+), 8 deletions(-) diff --git a/工具/修改题目数据库.py b/工具/修改题目数据库.py index 029e5c31..0d0b7678 100644 --- a/工具/修改题目数据库.py +++ b/工具/修改题目数据库.py @@ -1,6 +1,6 @@ import os,re,json """这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭, 随后运行第二个代码块""" -problems = "40466,40473,40482" +problems = "40477" def generate_number_set(string,dict): string = re.sub(r"[\n\s]","",string) diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index c7fd3626..f17c5ecf 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,9 +1,9 @@ #修改起始id,出处,文件名 -starting_id = 14826 +starting_id = 14847 raworigin = "" -filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目9.tex" -editor = "202304010\t王伟叶" -indexed = True +filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目6.tex" +editor = "202304011\t王伟叶" +indexed = False import os,re,json diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index cd5093ce..d946c23c 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -365974,6 +365974,1393 @@ "remark": "", "space": "12ex" }, + "014847": { + "id": "014847", + "content": "已知$n$是正整数, 设抛物线: $y=n(n+1) x^2-(2 n+1) x+1$的图像在$x$轴上截得的线段的长度为$a_n$, 求$\\displaystyle\\lim_{n\\to\\infty}(a_1+a_2+\\cdots+a_n)$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014848": { + "id": "014848", + "content": "设函数$f(x)=x(\\dfrac{1}{2})^x+\\dfrac{1}{x+1}$, $O$为坐标原点, $A_n$为函数$y=f(x)$图象上横坐标$n$($n \\in \\mathbf{N}$, $n\\ge 1$)的点, 向量$\\overrightarrow{OA_n}$与向量$\\overrightarrow {i}=(1,0)$的夹角为$\\theta_n$, 求满足$\\tan \\theta_1+\\tan \\theta_2+\\cdots+\\tan \\theta_n<\\dfrac{5}{3}$的最大整数$n$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014849": { + "id": "014849", + "content": "设$g(k)$是关于$x$不等式$\\log _2 x+\\log _2(3 \\sqrt{2^{2 k+2}}-x) \\geq 2 k+3$($k \\in \\mathbf{N}$, $k\\ge 1$)的整数解的个数, 求$g(k)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014850": { + "id": "014850", + "content": "设二次函数$f(x)=(k-4) x^2+k x$, 且对任意实数$x$, 都有$f(x) \\leq 6 x+2$恒成立, 数列$\\{a_n\\}$满足$a_{n+1}=f(a_n)$.\\\\\n(1) 求函数$f(x)$的解析式和值域;\\\\\n(2) 试写出一个区间$(a, b)$, 使得当$a_1 \\in(a, b)$时, 数列$\\{a_n\\}$是递增数列, 并说明理由;\\\\\n(3) 已知$a_1=\\dfrac{1}{3}$, 是否存在非零整数$\\lambda$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 都有$\\log _3(\\dfrac{1}{\\frac{1}{2}-a_1})+\\log _3(\\dfrac{1}{\\frac{1}{2}-a_2})+\\cdots+\\log _3(\\dfrac{1}{\\frac{1}{2}-a_n})>-1+(-1)^{n-1}\\cdot 2 \\lambda+n \\log _32$恒成立, 若存在, 求之; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014851": { + "id": "014851", + "content": "已知函数$f(x)=\\dfrac{4 x-2}{x+1}$($x \\neq-1$, $x \\in \\mathbf{R}$), 数列$\\{a_n\\}$满足$a_1=a$($a \\neq-1$, $a \\in \\mathbf{R}$), $a_{n+1}=f(a_n)$($n\\in \\mathbf{N}$, $n\\ge 1$). 若数列$\\{a_n\\}$是常数列, 求$a$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014852": { + "id": "014852", + "content": "函数$f(x)$是定义在$[0,1]$上的增函数, 满足$f(x)=2 f(\\dfrac{x}{2})$且$f(1)=1$, 在每个区间$(\\dfrac{1}{2^i}, \\dfrac{1}{2^{i-1}}]$($i=1,2, \\cdots$)上, $y=f(x)$的图像都是斜率为同一常数$k$的直线的一部分. 求$f(0)$及$f(\\dfrac{1}{2})$, $f(\\dfrac{1}{4})$的值, 并归纳出$f(\\dfrac{1}{2^i})$($i=1,2, \\cdots$)的表达式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014853": { + "id": "014853", + "content": "数列$\\{a_n\\}$的通项公式为$a_n=n+\\dfrac{c}{n}$(其中$c$为实常数), 若数列$\\{a_n\\}$是递增数列, 求$c$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014854": { + "id": "014854", + "content": "函数$f(x)=x^2+m$, 其中$m$为实常数, 定义数列$\\{a_n\\}$如下: $a_1=0$, $a_{n+1}=f(a_n)$, $n \\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 当$m=1$时, 求$a_2, a_3, a_4$的值;\\\\\n(2) 是否存在实数$m$, 使$a_2, a_3, a_4$成等比数列? 若存在, 请求出实数$m$的值, 并求出等比数列的公比; 若不存在, 请说明理由;\\\\\n(3) 设$m=-1$, $f^{-1}(x)$为$f(x)$在$x \\in[0,+\\infty)$的反函数, 数列$\\{b_n\\}$满足: $b_1=1$, $b_{n+1}=f^{-1}(b_n^2)$($n \\in \\mathbf{N}$, $n\\ge 1$), 记$S_n=b_1^2+b_2^2+\\cdots+b_n^2$, 求使$S_n>2020$成立的最小正整数$n$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014855": { + "id": "014855", + "content": "设数列$a_n=-n^2+10 n+11$($n \\in \\mathbf{N}$, $n\\ge 1$)的前$n$项和为$S_n$, 则当$S_n$取得最大值时, $n$的值为\\bracket{20}.\n\\fourch{$10$}{$11$}{$10$或$11$}{$12$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014856": { + "id": "014856", + "content": "根据市场调查结果, 预测某种家用商品从年初开始的$n$个月内累积的需求量$S_n$(万件) 近似地满足关系式$S_n=\\dfrac{n}{90}(21 n-n^2-5)$($n=1,2, \\cdots, 12$), 按此预测, 在本年度内, 需求量超过$1.5$万件的月份是\\bracket{20}.\n\\fourch{$5$、$6$月}{$6$、$7$月}{$7$、$8$月}{$8$、$9$月}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014857": { + "id": "014857", + "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_4 \\geq 10$, $S_5 \\leq 15$, 则$a_4$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014858": { + "id": "014858", + "content": "设$\\{a_n\\}$是公比为$q$的等比数列, 且满足条件$a_1>1$, $a_{2006} a_{2007}-1>0$, $\\dfrac{a_{2006}-1}{a_{2007}-1}<0$. 设$T_n=a_1 a_2 a_3 \\cdots a_n$, 则使$T_n<1$成立的最小正整数$n$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014859": { + "id": "014859", + "content": "数列$\\{a_n\\}$中, $a_1=8$, $a_4=2$, 且满足$a_{n+2}=2 a_{n+1}-a_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=\\dfrac{1}{n(12-a_n)}$($n \\in \\mathbf{N}$, $n\\ge 1$), 其前$n$项和为$S_n$, 问是否存在最大的整数$m$, 使得对任意正整数$n$, 恒有$S_n>\\dfrac{m}{32}$成立? 若存在, 求$m$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014860": { + "id": "014860", + "content": "对于任意$n \\in \\mathbf{N}$且$n>1$, 求证: $(1+\\dfrac{1}{3})(1+\\dfrac{1}{5}) \\cdots(1+\\dfrac{1}{2 n-1})>\\dfrac{\\sqrt{2 n+1}}{2}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014861": { + "id": "014861", + "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$S_5S_8$. 则下列结论中错误的是\\bracket{20}.\n\\twoch{$d<0$}{$a_7=0$}{$S_9>S_5$}{$S_6$, $S_7$均为$S_n$的最大值}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014862": { + "id": "014862", + "content": "某纯净水制造厂在净化水过程中, 每增加一次过滤可减少水中杂质$20 \\%$, 要使水中杂质减少到原来的$5 \\%$以下, 则至少需过滤的次数为\\bracket{20}.\n\\fourch{$11$}{$12$}{$13$}{$14$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014863": { + "id": "014863", + "content": "已知等比数列$\\{a_n\\}$中$a_2=1$, 则其前$3$项的和$S_3$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014864": { + "id": "014864", + "content": "已知$a_n=9 n-8$($n \\in \\mathbf{N}$, $n\\ge 1$), 且对任意$m \\in \\mathbf{N}$, $m\\ge 1$, 数列$\\{a_n\\}$中落入区间$(9^m, 9^{2 m})$内的项的个数为$b_m$, 则数列$\\{b_m\\}$的前$m$项和$S_m=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014865": { + "id": "014865", + "content": "已知等比数列$\\{a_n\\}$满足$a_n>a_{n+1}$, 且$a_3+a_6=18$, $a_4 \\cdot a_5=32$. 求数列$\\{a_n\\}$中所有小于$1$的项的各项和$S$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014866": { + "id": "014866", + "content": "设数列$\\{a_n\\}$的各项都是正数, 其前$n$项和为$S_n$, 且$a_n^2=2S_n-a_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$b_n=3^n+(-1)^{n-1} \\lambda \\cdot 2^{a_n}$($\\lambda$为非零整数, $n$为正整数), 试确定$\\lambda$的值, 使得对任意正整数$n$, 恒有$b_{n+1}>b_n$成立.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数列与不等式", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014867": { + "id": "014867", + "content": "已知$F_1, F_2$分别是椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{12}=1$的左、右焦点, 点$P$是椭圆上的任意一点, 则$\\dfrac{|PF_1|-|PF_2|}{|PF_1|}$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014868": { + "id": "014868", + "content": "若当$P(m, n)$为圆$x^2+(y-1)^2=1$上任意一点时, 不等式$m+n+c \\geq 0$恒成立, 则$c$的取值范围是\\bracket{20}.\n\\fourch{$-1-\\sqrt{2} \\leq c \\leq \\sqrt{2}-1$}{$\\sqrt{2}-1 \\leq c \\leq \\sqrt{2}+1$}{$c \\leq-\\sqrt{2}-1$}{$c \\geq \\sqrt{2}-1$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014869": { + "id": "014869", + "content": "在平面直角坐标系$x O y$中, $A(-1,0)$, $B(1,0)$, $C(0,1)$, 经过原点的直线$l$将$\\triangle ABC$分成左、右两部分, 记左、右两部分的面积分别为$S_1$、$S_2$, 求$\\dfrac{(1+S_1)^2}{1-S_2^2}$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014870": { + "id": "014870", + "content": "已知曲线$C: \\dfrac{x^2}{3}+\\dfrac{y^2}{4}=1$. 设曲线$C$与$y$轴交于$D, E$两点, 点$Q(0, m)$在线段$DE$上, 点$P$在曲线$C$上运动. 若当点$P$的坐标为$(0,2)$时, $|\\overrightarrow{QP}|$取得最小值, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014871": { + "id": "014871", + "content": "已知抛物线$y^2=4 x$的焦点为$F$, 过$F$作互相垂直的两条直线$l_1, l_2$, $l_1$与抛物线交于$A$、$B$两点, $l_2$与抛物线交于$C$、$D$两点, $M$、$N$分别是线段$AB$、$CD$的中点, 求$\\triangle FMN$面积的最小值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014872": { + "id": "014872", + "content": "已知平面上的曲线$C$及点$P$, 在$C$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到曲线$C$的距离, 记作$d(P, C)$. 则点$P(0,3)$到曲线$C: x^2-y^2=1$的距离$d(P, C)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014873": { + "id": "014873", + "content": "设$F_1$是椭圆$\\dfrac{x^2}{4}+y^2=1$的左焦点, $O$为坐标原点, 点$P$在椭圆上, 则$\\overrightarrow{PF_1} \\cdot \\overrightarrow{PO}$的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014874": { + "id": "014874", + "content": "平面直角坐标系$xOy$中, 过椭圆$M: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点$F$作直线$x+y-\\sqrt{3}=0$交$M$于$A, B$两点, $P$为$AB$的中点, 且$OP$的斜率为$\\dfrac{1}{2}$.\\\\\n(1) 求$M$的方程;\\\\\n(2)$C, D$为$M$上的两点, 若四边形$ABCD$的对角线$CD \\perp AB$, 求四边形$ABCD$面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014875": { + "id": "014875", + "content": "已知椭圆$C: \\dfrac{x^2}{3}+y^2=1$. $O$为原点, 直线$l: y=k x+t$($k \\neq 0$)与椭圆$C$交于$A, B$两点, 若存在点$P(0,-\\dfrac{1}{2})$, 使得$\\overrightarrow{BA} \\cdot \\overrightarrow{PA}=\\overrightarrow{AB} \\cdot \\overrightarrow{PB}$.\\\\\n(1) 求证: $3 k^2+1=4 t$;\\\\\n(2) 求$\\triangle AOB$面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-解析几何与函数", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014876": { + "id": "014876", + "content": "已知函数$f(x)=a+(1-a)(\\cos x+\\sin x)$, 当$x \\in[0, \\dfrac{\\pi}{2}]$时, $-2 \\leq f(x) \\leq 2$恒成立, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014877": { + "id": "014877", + "content": "给定常数$c>0$, 定义函数$f(x)=2|x+c+4|-|x+c|$, 数列$a_1, a_2, a_3, \\cdots$满足$a_{n+1}=f(a_n)$, $n$是正整数.\\\\\n(1) 若$a_1=-c-2$, 求$a_2$及$a_3$;\\\\\n(2) 求证: 对任意正整数$n$, $a_{n+1}-a_n \\geq c$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014878": { + "id": "014878", + "content": "给定常数$a$, 已知函数$f(x)=\\sqrt{x}(x-a)$的单调性如下:\n当$a \\leq 0$时, 函数$f(x)$在区间$[0,+\\infty)$上单调递增; 当$a>0$时, 函数$f(x)$在区间$[0, \\dfrac{a}{3}]$上单调递减, 在区间$[\\dfrac{a}{3},+\\infty)$上单调递增. 设$g(a)$为$f(x)$在区间$[0,2]$上的最小值.\\\\\n(1) 写出$g(a)$的表达式, 并写出$g(a)$的单调区间(不要求证明);\\\\\n(2) 求实数$a$的取值范围, 使得$-6 \\sqrt{3} \\leq g(a) \\leq-2$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014879": { + "id": "014879", + "content": "有两个相同的直三棱柱, 高为$\\dfrac{1}{a}$, 底面三角形的三边长分别为$3 a, 4 a, 5 a$($a>0$). 用它们拼成一个三棱柱或四棱柱, 在所有可能的情形中, 全面积最小的是一个四棱柱, 则实数$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014880": { + "id": "014880", + "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{4}=1$上的动点$P(x, y)$与定点$M(m, 0)$($00$), 数列$\\{a_n a_{n+1}\\}$是公比为$q$($q>0$)的等比数列, $b_n=a_{2 n-1}+a_{2 n}$, 记$S_n=b_1+b_2+\\cdots+b_n$. 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{1}{S_n}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014882": { + "id": "014882", + "content": "正数列$\\{a_n\\}$的前$n$项和$S_n$满足: $r S_n=a_n a_{n+1}-1$, $a_1=a>0$, 常数$r \\in \\mathbf{N}$.\\\\\n(1) 求证: $a_{n+2}-a_n$是一个定值;\\\\\n(2) 若数列$\\{a_n\\}$是一个周期数列, 求该数列的周期;\\\\\n(3) 若数列$\\{a_n\\}$是一个有理数等差数列, 求$S_n$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-分类讨论", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014883": { + "id": "014883", + "content": "已知点$A(7,4)$, $B(-8,2)$, 在$x$轴上求点$C$, 使经过点$C$, 且以$A, B$为焦点的椭圆的长轴长最短, 则$C$点的坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014884": { + "id": "014884", + "content": "函数$y=\\cos x+2|\\cos x|$($x \\in[0,2 \\pi]$)的图像与直线$y=k$有且仅有两个不同的公共点, 则实数$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014885": { + "id": "014885", + "content": "已知集合$A=\\{x |(x-5)(x+1)<0\\}$, 集合$B=\\{x | x^2-p x-10<0\\}$, 且$A \\subseteq B$, 则实数$p$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014886": { + "id": "014886", + "content": "函数$y=\\dfrac{1}{1-x}$的图像与函数$y=2 \\sin \\pi x$($-2 \\leq x \\leq 4$)的图像所有公共点的横坐标之和等于\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014887": { + "id": "014887", + "content": "设有函数$f(x)=a+\\sqrt{-x^2-4 x}$和$g(x)=\\dfrac{4}{3} x+1$, 已知$x \\in[-4,0]$时恒有$f(x) \\leq g(x)$成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014888": { + "id": "014888", + "content": "若关于$x$的方程$m=x-x^2$在区间$[-2,2]$上只有一个实数根, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014889": { + "id": "014889", + "content": "若关于$x$的方程$2 m=\\sqrt{x}-x$有两个不同的实数根, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014890": { + "id": "014890", + "content": "若关于$x$的方程$(2 m-1) x=x^2+1$在区间$[-2,1]$上有两个不同的实数根, 求实数$m$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014891": { + "id": "014891", + "content": "已知函数$f(x)$的定义域是$\\mathbf{R}$, 满足对任意$x\\in \\mathbf{R}$, 都成立$f(x+1)=\\dfrac{1-f(x)}{1+f(x)}$.\n(1) 证明: $2$是函数$f(x)$的一个周期;\\\\\n(2) 当$x \\in[0,1)$时, $f(x)=x$, 求$f(x)$在$[-1,0)$上的解析式;\\\\\n(3) 设$a>0$, 对于 (2) 中的函数$f(x)$, 关于$x$的方程$f(x)=a x$有$100$个根, 求正实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014892": { + "id": "014892", + "content": "已知点$A(1,1)$, 点$F$是抛物线$y=\\dfrac{1}{9} x^2$的焦点, 点$P$是抛物线上的一个动点, 当$|PA|+|PF|$最小时, $P$点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014893": { + "id": "014893", + "content": "不等式$3 x^2<\\log _a x$在区间$(0, \\dfrac{1}{3})$上恒成立, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014894": { + "id": "014894", + "content": "由方程$x|x|+y|y|=1$确定的函数$y=f(x)$在$(-\\infty,+\\infty)$上是\\bracket{20}.\n\\fourch{增函数}{减函数}{先增后减}{先减后增}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014895": { + "id": "014895", + "content": "关于$x$的方程$(x^2-1)^2-|x^2-1|+k=0$, 给出下列四个命题, 其中是假命题的是\\blank{50}.\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$3$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$5$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程恰有$8$个不同的实根.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014896": { + "id": "014896", + "content": "设平面点集$A=\\{(x, y) |(y-x)(y-\\dfrac{1}{x}) \\geq 0\\}$, $B=\\{(x, y) |(x-1)^2+(y-1)^2 \\leq 1\\}$, 则$A \\cap B$所表示的平面图形的面积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014897": { + "id": "014897", + "content": "若关于$x$的方程$\\dfrac{|x|}{x-3}=k x^2$有四个不同的实数根, 求实数$k$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014898": { + "id": "014898", + "content": "若关于$x$的方程$a=x^2-3|x|+2$有四个实数根, 求实数$a$的取值范围;", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014899": { + "id": "014899", + "content": "若关于$x$的方程$x^2+a x+2=0$在$(0,2)$内只有一个实数根, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-数形结合", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014900": { + "id": "014900", + "content": "对于定义域为$\\mathbf{R}$的函数$f(x)$, 若存在非零实数$x_0$, 使函数$f(x)$在$(-\\infty, x_0)$和$(x_0,+\\infty)$上均有零点, 则称$x_0$为函数$f(x)$的一个``界点''. 下列四个函数中, 不存在``界点''的是\\bracket{20}.\n\\fourch{$f(x)=x^2+\\pi x$}{$f(x)=2^x-x^2$}{$f(x)=2-|x-1|$}{$f(x)=x-\\sin x$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014901": { + "id": "014901", + "content": "关于$x$的方程$x^2-1-|x^2-1|+k=0$, 给出下列四个命题, 其中假命题的序号是\\blank{50}.\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$1$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$2$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程有无数个实根.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014902": { + "id": "014902", + "content": "已知数列$a_n=n+1$, $b_n=\\begin{cases}1,& n=1, \\\\ (-\\dfrac{1}{10})(\\dfrac{9}{10})^{n-2}, & n \\geq 2,\\end{cases}$ $c_n=-a_n \\cdot b_n$($n \\in \\mathbf{N}$, $n\\ge 1$). 在数列$\\{c_n\\}$中, 是否存在正整数$k$, 使得对于任意的正整数$n$, 都有$c_n \\leq c_k$成立? 若存在, 写出所有满足条件的正整数$k$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014903": { + "id": "014903", + "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$), 设$P$是双曲线$C$上任意一点, $O$为坐标原点, 设$F$为双曲线右焦点过右焦点$F$的动直线$l$交双曲线于$A$、$B$两点, 是否存在这样的$a, b$的值, 使得$\\triangle OAB$为等边三角形? 若存在, 求出所有满足条件的$a, b$的值; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014904": { + "id": "014904", + "content": "已知函数$f(x)=2 x+1$, $x \\in \\mathbf{N}$, $x\\ge 1$. 若存在正整数$x_0$, $n$, 使$f(x_0)+f(x_0+1)+\\cdots+f(x_0+n)=63$成立, 则称$(x_0, n)$为函数$f(x)$的一个``生成点''. 函数$f(x)$的``生成点''共有\\bracket{20}.\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014905": { + "id": "014905", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 点$(n, \\dfrac{S_n}{n})$在直线$y=\\dfrac{1}{2} x+\\dfrac{11}{2}$上. 数列$\\{b_n\\}$满足$b_{n+2}-2 b_{n+1}+b_n=0$($n \\in \\mathbf{N}$, $n\\ge 1$)且$b_3=11$, 前$9$项和为$153$.\\\\\n(1) 求数列$\\{a_n\\}$、$\\{b_n\\}$的通项公式;\\\\\n(2) 设$f(n)=\\begin{cases}a_n,& n=2 l-1, \\\\ b_n, &n=2 l\\end{cases}$($l \\in \\mathbf{N}$, $l\\ge 1$), 问是否存在正整数$m$, 使得$f(m+15)=5 f(m)$成立? 若不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014906": { + "id": "014906", + "content": "在平面直角坐标系$xOy$中, 原点为$O$, 抛物线$C$的方程为$x^2=4 y$, 线段$AB$是抛物线$C$的一条动弦. 当$|AB|=8$时, 设圆$D: x^2+(y-1)^2=r^2$($r>0$), 若存在且仅存在两条动弦$AB$, 满足直线$AB$与圆$D$相切, 求半径$r$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014907": { + "id": "014907", + "content": "定义函数$y=f(x)$, $x \\in D$($D$为定义域)图像上的点到坐标原点的距离为函数的$y=f(x)$, $x \\in D$的模. 若模存在最大值, 则称之为函数$y=f(x)$, $x \\in D$的长距; 若模存在最小值, 则称之为函数$y=f(x)$, $x \\in D$的短距.\\\\\n(1) 分别判断函数$f_1(x)=\\dfrac{1}{x}$与$f_2(x)=\\sqrt{-x^2-4 x+5}$是否存在长距与短距;\\\\\n(2) 对于任意$x \\in[1,2]$, 是否存在实数$a$, 使得函数$f(x)=\\sqrt{2 x|x-a|}$的短距不小于$2$且长距不大于$4$? 若存在, 求出$a$的取值范围; 若不存在, 说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-存在性问题中的构造和证明", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014908": { + "id": "014908", + "content": "若干个能唯一确定一个数列的量称为该数列的``基本量''. 设$\\{a_n\\}$是公比为$q$的无穷\n等比数列, 下列$\\{a_n\\}$的四组量中, 一定能成为该数列``基本量''的是第\\blank{50}组.(其中$n$为大于$1$的整数, $S_n$为$\\{a_n\\}$的前$n$项和)\n\\textcircled{1} $S_1$与$S_2$; \\textcircled{2} $a_2$与$S_3$; \\textcircled{3} $a_1$与$a_n$; \\textcircled{4} $q$与$a_n$.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014909": { + "id": "014909", + "content": "设$g(x)$是定义在$\\mathbf{R}$上, 以$1$为周期的函数. 若函数$f(x)=x+g(x)$在区间$[3,4]$上的值域为$[-2,5]$. 则\\\\\n(1) $f(x)$在区间$[4,5]$上的值域为\\blank{50};\\\\\n(2) $f(x)$在区间$[-8,8]$上的值域为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014910": { + "id": "014910", + "content": "在平面直角坐标系$xOy$中, 对于直线$l: a x+b y+c=0$和点$P_1(x_1, y_1)$, $P_2(x_2, y_2)$, 记$\\eta=(a x_1+b y_1+c)(a x_2+b y_2+c)$. 若$\\eta<0$, 则称点$P_1, P_2$被直线$l$分割. 若曲线$C$与直线$l$没有公共点, 且曲线$C$上存在点$P_1, P_2$被直线$l$分割, 则称直线$l$为曲线$C$的一条分割线.\\\\\n(1) 设点$A$的坐标为$(2,2)$, 直线$l: x+y-1=0$.\\\\\n(i) 求证: 点$A$、$B(-1,0)$被直线$l$分割;\\\\\n(ii) 求证: 存在一点$C$, 点$C$、$A$不被直线$l$分割;\\\\\n(2) 设曲线$C_1: x y=1$是否存在分割线? 若存在, 写出一条分割线, 并证明其为曲线$C_1$的分割线; 若不存在, 说明理由;\\\\\n(3) 求证: 曲线$C_2: |x y|=1$恰存在两条分割线.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014911": { + "id": "014911", + "content": "若实数$x, y, m$满足$\\lg (x-m)>\\lg (y-m)$, 则称$x$比$y$``更真''于$m$. 若$4 x-x^2-1$比$x-1$``更真''于$1$, 则$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014912": { + "id": "014912", + "content": "方程$x^2+\\sqrt{2} x-1=0$的解可视为函数$y=x+\\sqrt{2}$的图像与函数$y=\\dfrac{1}{x}$的图像交点的横坐标. 若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2, \\cdots, k$)均在直线$y=x$的同侧, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014913": { + "id": "014913", + "content": "已知平面上的线段$l$及点$P$. 任取$l$上一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$. 设线段$l_1: y=2$($-2 \\leq x \\leq 2)$.\\\\\n(1) 分别求点$P_1(0,3)$、$P_2(4,3)$到线段$l_1$的距离$d(P_1, l_1)$、$d(P_2, l_1)$;\\\\\n(2) 求点的集合$D=\\{P | d(P, l)=1\\}$;\\\\\n(3) 设$A(0,2)$, 写出$\\Omega=\\{P|d(P, l_1)=| PA |\\}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014914": { + "id": "014914", + "content": "在直角坐标平面$xOy$上的一列点$A_1(1, a_1)$, $A_2(2, a_2)$, $\\cdots$, $A_n(n, a_n)$, $\\cdots$, 简记为$\\{A_n\\}$. 若由$b_n=\\overline{A_n A_{n+1}} \\cdot \\overrightarrow {j}$构成的数列$\\{b_n\\}$满足$b_{n+1}>b_n$, $n=1,2, \\cdots$, 其中$\\overrightarrow {j}$为方向与$y$轴正方向相同的单位向量, 则称$\\{A_n\\}$为$T$点列.\\\\\n(1) 判断$A_1(1,1)$, $A_2(2, \\dfrac{1}{2})$, $A_3(3, \\dfrac{1}{3})$, $\\cdots$, $A_n(n, \\dfrac{1}{n})$, $\\cdots$是否为$T$点列, 并说明理由;\\\\\n(2) 若$\\{A_n\\}$为$T$点列, 且点$A_2$在点$A_1$的右上方. 任取其中连续三点$A_k, A_{k+1}, A_{k+2}$, 判断$\\triangle A_k A_{k+1} A_{k+2}$的形状(锐角三角形、直角三角形、钝角三角形), 并予以证明;\\\\\n(3) 若$\\{A_n\\}$为$T$点列, 从小到大排列的四个不同的正整数$m,n,p,q$满足$m+q=n+p$, 求证: $\\overrightarrow{A_n A_q} \\cdot \\overrightarrow {j}>\\overrightarrow{A_m A_p} \\cdot \\overrightarrow {j}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-阅读新情境并作转化", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014915": { + "id": "014915", + "content": "已知$a, b, c$均为正实数, 求证: $\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c} \\geq \\dfrac{1}{\\sqrt{a b}}+\\dfrac{1}{\\sqrt{b c}}+\\dfrac{1}{\\sqrt{c a}}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-论证的训练", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014916": { + "id": "014916", + "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是奇函数且不等式$|f(x_1)+f(x_2)| \\geq|g(x_1)+g(x_2)|$恒成立, 问$g(x)$是否也为奇函数? 证明你的结论.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-论证的训练", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014917": { + "id": "014917", + "content": "已知函数$f(x)$、$g(x)$的定义域均为$\\mathbf{R}$, $x_1$、$x_2$是在$\\mathbf{R}$上任意选取的两个实数. 若$f(x)$是周期函数且不等式$|f(x_1)-f(x_2)| \\geq|g(x_1)-g(x_2)|$恒成立, 问$g(x)$是否为周期函数? 证明你的结论.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-论证的训练", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014918": { + "id": "014918", + "content": "若$a, b, c \\in \\mathbf{R}$, 满足$2 a+b+2 \\leq 0$.\\\\\n(1) 求证: 关于$t$的方程$t^2+a t+b-2=0$有实数解;\\\\\n(2) 求证: 关于$x$的方程$x^2+\\dfrac{1}{x^2}+a(x+\\dfrac{1}{x})+b=0$有正实数解.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2022届第三轮复习讲义-论证的训练", + "edit": [ + "202304011\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014919": { + "id": "014919", + "content": "对于无穷数列$\\{a_n\\}$, 若存在正常数$M$, 使得对任意正整数$n$, 总成立$|a_n|