diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py index 1d8a2e31..96e6541b 100644 --- a/工具/批量收录题目.py +++ b/工具/批量收录题目.py @@ -1,8 +1,8 @@ #修改起始id,出处,文件名 -starting_id = 17423 +starting_id = 17444 raworigin = "" filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目12.tex" -editor = "20230601\t王伟叶" +editor = "20230602\t王伟叶" indexed = True IndexDescription = "试题" diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 6501011b..24839945 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,65 +1,64 @@ ans -17423 -$-\dfrac 45$ +17444 +$1$ -17424 -$(\dfrac 12,1)$ +17445 +$[2k\pi,2k\pi+\pi ]$, $k\in \mathbf{Z}$ -17425 -$160$ +17446 +$-\dfrac{1}{2}$ -17426 -$9$ +17447 +$3\pi$ -17427 -$0$ +17448 +$\dfrac{59}{4}$ -17428 -$\dfrac{\sqrt{6\pi}}{6}$ +17449 +$6$ -17429 -$5$ +17450 +$-1$ -17430 -$\dfrac{2\sqrt{3}}{3}$ +17451 +$\dfrac{\pi}{2}$ -17431 -$\dfrac{2}{3}$ +17452 +$-\mathrm{i}$ -17432 -$4\sqrt{2}-3$ +17453 +$\sqrt{2}$ -17433 -$3+\dfrac{5\sqrt{3}}{3}$ +17454 +$8$ -17434 -$(-1,0)$ +17455 +$(-2,0)\cup (0,2)$ -17435 -A +17456 +D -17436 +17457 C -17437 -B +17458 +D -17438 -A +17459 +D -17439 -(1) $\dfrac 12$; (2) $(\dfrac{\sqrt{3}}{2},1]$ +17460 +(1) 当$\alpha$为第一象限角时, $f(\alpha)+g(\alpha)=\sqrt{2(1-m^2)}$; 当$\alpha$为第一象限角时, $f(\alpha)+g(\alpha)=-\sqrt{2(1-m^2)}$; (2) $\dfrac{2}{5}$ +17461 +(1) 证明略; (2) 证明略 -17440 -(1) $4\sqrt{3}$; (2) 证明略 +17462 +(1) $\hat{y}=\hat{c}+\hat{d}\ln x$更好; (2) $\hat{y}=60\ln x+12$; (3) $30$万元 -17441 -(1) $\dfrac 57$; (2) $E[X]=1$, $D[X]=0.8$ +17463 +(1) 焦点为$(0,\dfrac{p}{2})$, 准线为$y=-\dfrac{p}{2}$; (2) $k=2$; (3) 证明略 -17442 -(1) $x^2-\dfrac{y^2}{3}=1$; (2) $1+\sqrt{2}$; (3) $(\sqrt{2},+\infty)$ - -17443 -(1) $m=9$; (2) $\dfrac{4\sqrt{3}}{9}$; (3) 不存在, 证明略 +17464 +(1) $y=f_1(x)$与$y=f_2(x)$都是$\mathbf{R}$上的``平缓函数''; (2) 证明略; (3) 证明略 \ No newline at end of file diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 20cdbabe..c6c6bf2c 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -448585,6 +448585,426 @@ "space": "4em", "unrelated": [] }, + "017444": { + "id": "017444", + "content": "已知集合$A=\\{x, x^2+1,-1\\}$中的最大元素为$2$, 则实数$x=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$1$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题1", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017445": { + "id": "017445", + "content": "函数$y=2 \\cos x$的严格减区间为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$[2k\\pi,2k\\pi+\\pi ]$, $k\\in \\mathbf{Z}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题2", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017446": { + "id": "017446", + "content": "若函数$y=f(x)$为偶函数, 且当$x<0$时, $f(x)=2^x-1$, 则$f(1)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$-\\dfrac{1}{2}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题3", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017447": { + "id": "017447", + "content": "若某圆锥高为$3$, 其侧面积与底面积之比为$2: 1$, 则该圆锥的体积为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$3\\pi$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题4", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017448": { + "id": "017448", + "content": "已知样本数据$2$、$4$、$8$、$m$的极差为$10$, 其中$m>0$, 则该组数据的方差为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\dfrac{59}{4}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题5", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017449": { + "id": "017449", + "content": "在财务审计中, 我们可以用``本$\\cdot$福特定律''来检验数据是否造假. 本$\\cdot$福特定律指出, 在一组没有人为编造的自然生成的数据 (均为正实数) 中, 首位非零的数字是$1 \\sim 9$这九个事件不是等可能的. 具体来说, 随机变量$X$是一组没有人为编造的首位非零数字, \n则$P(X=k)=\\lg \\dfrac{k+1}{k}$, $k=1,2, \\cdots, 9$. 则根据本$\\cdot$福特定律, 首位非零数字是$1$与首位非零数字是$8$的概率之比约为\\blank{50}(保留至整数).", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$6$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题6", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017450": { + "id": "017450", + "content": "若$(1-2 x)^{2023}=a_0+a_1 x+\\cdots+a_{2023} x^{2023}$, 则$\\dfrac{a_1}{2}+\\dfrac{a_2}{2^2}+\\cdots+\\dfrac{a_{2023}}{2^{2023}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$-1$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题7", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017451": { + "id": "017451", + "content": "若向量$\\overrightarrow {a}$与$\\overrightarrow {b}$不共线也不垂直, 且$\\overrightarrow {c}=\\overrightarrow {a}-(\\dfrac{\\overrightarrow {a} \\cdot \\overrightarrow {a}}{\\overrightarrow {a} \\cdot \\overrightarrow {b}}) \\overrightarrow {b}$, 则向量夹角$\\langle\\overrightarrow {a}, \\overrightarrow {c}\\rangle=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\dfrac{\\pi}{2}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题8", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017452": { + "id": "017452", + "content": "已知复数$z$在复平面内对应的点是$A$, 其共轭复数$\\overline {z}$在复平面内对应的点是$B$, $O$是坐标原点, 若$A$在第一象限, 且$OA \\perp OB$, 则$\\dfrac{z+\\overline {z}}{z-\\overline {z}}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$-\\mathrm{i}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题9", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017453": { + "id": "017453", + "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别为$F_1$、$F_2$, $\\Gamma$的渐近线与圆$x^2+y^2=a^2$在第一象限的交点为$M$, 线段$MF_2$与$\\Gamma$交于点$N$, $O$为坐标原点. 若$MF_1\\parallel ON$, 则$\\Gamma$的离心率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$\\sqrt{2}$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题10", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017454": { + "id": "017454", + "content": "若项数为$10$的数列$\\{a_n\\}$, 满足$1 \\leq|a_{i+1}-a_i| \\leq 2$($i=1,2, \\cdots, 9$), 且$a_1=a_{10} \\in[-1,0]$, 则数列$\\{a_n\\}$中最大项的最大值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$8$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题11", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017455": { + "id": "017455", + "content": "若实数$a$使得存在两两不同的实数$x$、$y$、$z$, 有$\\dfrac{x^3+a}{y+z}=\\dfrac{y^3+a}{z+x}=\\dfrac{z^3+a}{x+y}=-3$, 则实数$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "$(-2,0)\\cup (0,2)$", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题12", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017456": { + "id": "017456", + "content": "我国古代数学著作《九章算术》中有如下问题: ``今有善走男, 日增等里, 首日行走一百里, 九日共行一千二百六十里, 问日增几何?\", 该问题中, ``善走男''第$5$日所走的路程里数为\\bracket{20}.\n\\fourch{$110$}{$120$}{$130$}{$140$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "D", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题13", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017457": { + "id": "017457", + "content": "``$(\\log _a 2) x^2+(\\log _b 2) y^2=1$表示焦点在$y$轴上的椭圆''的一个充分非必要条件是\\bracket{20}.\n\\fourch{$0=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A)--(B_1)--(C);\n\\draw [dashed] (A)--(C)(B)--(D)(A)--(D_1)(A_1)--(D)(A_1)--(C)(A)--(C_1)(B_1)--(D);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$AB_1$、$AC$、$AD_1$的长度}{$AC$、$B_1D$、$A_1C$的长度}{$B_1C$、$A_1D$、$B_1D$的长度}{$AC_1$、$BD$、$CC_1$的长度}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "D", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题15", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017459": { + "id": "017459", + "content": "设关于$x$、$y$的表达式$F(x, y)=\\cos ^2 x+\\cos ^2 y-\\cos (x y)$, 当$x$、$y$取遍所有实数时, $F(x, y)$\\bracket{20}.\n\\twoch{既有最大值, 也有最小值}{有最大值, 无最小值}{无最大值, 有最小值}{既无最大值, 也无最小值}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "D", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题16", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "", + "unrelated": [] + }, + "017460": { + "id": "017460", + "content": "在平面直角坐标系$x O y$中, $A(\\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{2}}{2})$在以原点$O$为圆心半径等$1$的圆上, 将射线$OA$绕\n原点$O$逆时针方向旋转$\\alpha$后交该圆于点$B$, 设点$B$的横坐标为$f(\\alpha)$, 纵坐标$g(\\alpha)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) circle (1);\n\\draw (0,0) -- (45:1) node [above right] {$A$} coordinate (A);\n\\draw (0,0) -- (130:1) node [above left] {$B$} coordinate (B);\n\\draw (0,0) pic [draw, scale = 0.5, \"$\\alpha$\", angle eccentricity = 2] {angle = A--O--B};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果$\\sin \\alpha=m, 0=latex]\n\\draw (0,0,0) node [below] {$M$} coordinate (M);\n\\draw (2.5,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,-2) node [left] {$N$} coordinate (N);\n\\draw (M) ++ (0,1.5,0) node [left] {$A$} coordinate (A);\n\\draw (N) ++ (0,1.5,0) node [above] {$D$} coordinate (D);\n\\draw (N) ++ (1.5,0,0) node [right] {$C$} coordinate (C);\n\\draw (A)--(M)--(B)--cycle(A)--(C)--(B)(A)--(D)--(C);\n\\draw [dashed] (M)--(N)--(C)(M)--(C)(D)--(N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$AMB\\parallel$平面$DNC$;\\\\\n(2) 若$MC \\perp CB$, 求证: $BC \\perp AC$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) 证明略; (2) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题18", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017462": { + "id": "017462", + "content": "某科技公司为确定下一年度投入某种产品的研发费, 需了解年研发费$x$(单位: 万元) 对年销售量$y$(单位: 百件) 和年利润 (单位: 万元) 的影响, 现对近$6$年的年研发费$x_i$和年销售量$y_i$($i=1,2, \\cdots, 6$)数据作了初步处理, 得到下面的散点图及一些统计量的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,3.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {5/105,8/182,11/233,14/258,17/280,20/295}\n{\\filldraw ({\\i/5},{\\j/100}) circle (0.03);}; \n\\end{tikzpicture}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline $\\displaystyle\\sum_{i=1}^6(x_i-\\overline {x})^2$&$\\displaystyle\\sum_{i=1}^6(y_i-\\overline {y})^2$&$\\displaystyle\\sum_{i=1}^6(\\mu_i-\\overline {\\mu})^2$&$\\displaystyle\\sum_{i=1}^6(x_i-\\overline {x})(y_i-\\overline {y})$&$\\displaystyle\\sum_{i=1}^6(\\mu_i-\\overline {\\mu})(y_i-\\overline {y})$\\\\\n\\hline 157.5 & 16800 & 4.5 & 1254 & 270 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n表中$\\mu_i=\\ln x_i$, $\\overline {\\mu}=\\dfrac{1}{6} \\displaystyle\\sum_{i=1}^6 \\mu_i$.\\\\\n(1) 根据散点图判断$\\hat{y}=\\hat{a}+\\hat{b} x$与$\\hat{y}=\\hat{c}+\\hat{d} \\ln x$哪一个更适宜作为年研发费$x$的回归方程类型; (给出判断即可, 不必说明理由)\\\\\n(2) 根据 (1) 的判断结果及表中数据, 建立$y$关于$x$的回归方程;\\\\\n(3) 已知这种产品的年利润$z=0.5 y-x$, 根据 (2) 的结果, 当年研发费为多少时, 年利润$z$的预报值最大?\\\\\n附: 对于一组数据$(w_1, v_1),(w_2, v_2), \\cdots,(w_n, v_n)$, 其回归直线$\\hat{v}=\\hat{\\alpha}+\\widehat{\\beta} w$的斜率和截距的最小二乘估计分别为$\\widehat{\\beta}=\\dfrac{\\displaystyle\\sum_{i=1}^n(w_i-\\overline {w})(v_i-\\overline {v})}{\\displaystyle\\sum_{i=1}^n(w_i-\\overline {w})^2}$, $\\widehat{\\alpha}=\\overline {v}-\\widehat{\\beta} \\overline {w}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $\\hat{y}=\\hat{c}+\\hat{d}\\ln x$更好; (2) $\\hat{y}=60\\ln x+12$; (3) $30$万元", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题19", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017463": { + "id": "017463", + "content": "贝塞尔曲线是计算机图形学和相关领域中重要的参数曲线. 法国数学象卡斯特利奥对贝塞尔曲线进行了图形化应用的测试, 提出了 De Casteljau 算法: 已知三个定点, 根据对应的比例, 使用递推画法, 可以画出抛物线. 反之, 已知抛物线上三点的切线, 也有相应成比例的结论. 如图所示, 抛物线$\\Gamma: x^2=2 p y$, 其中$p>0$为一给定的实数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.5:1.5, samples = 100] plot (\\x,{\\x*\\x});\n\\foreach \\i/\\j/\\k in {-1.2/above/A,-0.3/above/B,0.7/right/C}\n{\\filldraw (\\i,{\\i*\\i}) node [\\j] {$\\k$} coordinate (\\k) circle (0.03);};\n\\draw (A) ++ (1,-2.4) coordinate (A_1);\n\\draw (B) ++ (1,-0.6) coordinate (B_1);\n\\draw (C) ++ (1,1.4) coordinate (C_1);\n\\path [name path = la, draw] ($(A)!-0.2!(A_1)$) -- ($(A)!1.3!(A_1)$);\n\\path [name path = lb, draw] ($(B)!-0.7!(B_1)$) -- ($(B)!1.3!(B_1)$);\n\\path [name path = lc, draw] ($(C)!-1.5!(C_1)$) -- ($(C)!0.5!(C_1)$);\n\\path [name intersections = {of = la and lb, by = D}];\n\\path [name intersections = {of = la and lc, by = E}];\n\\path [name intersections = {of = lb and lc, by = F}];\n\\foreach \\i/\\j in {D/below left,E/left,F/below}\n{\\filldraw (\\i) node [\\j] {$\\i$} circle (0.03);};\n\\end{tikzpicture}\n\\end{center}\n(1) 写出抛物线$\\Gamma$的焦点坐标及准线方程;\\\\\n(2) 若直线$l: y=k x-2 p k+2 p$与抛物线只有一个公共点, 求实数$k$的值;\\\\\n(3) 如图, $A, B, C$是$H$上不同的三点, 过三点的三条切线分别两两交于点$D, E, F$, 证明: $\\dfrac{|AD|}{|DE|}=\\dfrac{|EF|}{|FC|}=\\dfrac{|DB|}{|BF|}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) 焦点为$(0,\\dfrac{p}{2})$, 准线为$y=-\\dfrac{p}{2}$; (2) $k=2$; (3) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题20", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, + "017464": { + "id": "017464", + "content": "设$y=f(x)$是定义域为$\\mathbf{R}$的函数, 如果对任意的$x_1$、$x_2 \\in \\mathbf{R}$($x_1 \\neq x_2$), $|f(x_1)-f(x_2)|<|x_1-x_2|$均成立, 则称$y=f(x)$是``平缓函数''.\\\\\n(1) 若$f_1(x)=\\dfrac{1}{x^2+1}$, $f_2(x)=\\sin x$, 试判断$y=f_1(x)$和$y=f_2(x)$是否为``平缓函数''? 并说明理由; (参考公式: $x>0$时, $\\sin x1$使得函数$y=A \\cdot g(x)$为``平缓函数\". 现定义数列$\\{x_n\\}$满足: $x_1=0$, $x_n=g(x_{n-1})$($n=2,3,4, \\cdots$), 试证明: 对任意的正整数$n$, $g(x_n) \\leq \\dfrac{A|g(0)|}{A-1}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "(1) $y=f_1(x)$与$y=f_2(x)$都是$\\mathbf{R}$上的``平缓函数''; (2) 证明略; (3) 证明略", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023届复旦附中三模试题21", + "edit": [ + "20230602\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "4em", + "unrelated": [] + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",