From 89163531ae733092b1fa3f99316c3965f57a1cde Mon Sep 17 00:00:00 2001 From: WangWeiye Date: Tue, 13 Dec 2022 12:27:03 +0800 Subject: [PATCH] 20221213 noon --- 工具/修改题目数据库.ipynb | 6 +- 工具/关键字筛选题号.ipynb | 12 +- 工具/批量添加题库字段数据.ipynb | 494 ++++++++- 工具/文本文件/metadata.txt | 1759 +++++++++++++++++++++++++++++-- 工具/文本文件/题号筛选.txt | 2 +- 工具/讲义生成.ipynb | 12 +- 题库0.3/Problems.json | 1741 ++++++++++++++++++++++-------- 7 files changed, 3445 insertions(+), 581 deletions(-) diff --git a/工具/修改题目数据库.ipynb b/工具/修改题目数据库.ipynb index 7850cff9..a4b8495e 100644 --- a/工具/修改题目数据库.ipynb +++ b/工具/修改题目数据库.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 6, + "execution_count": 5, "metadata": {}, "output_type": "execute_result" } @@ -19,7 +19,7 @@ "source": [ "import os,re,json\n", "\"\"\"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\"\"\"\n", - "problems = \"30498:30499\"\n", + "problems = \"12193,12197,12198,12200,12202\"\n", "\n", "def generate_number_set(string,dict):\n", " string = re.sub(r\"[\\n\\s]\",\"\",string)\n", diff --git a/工具/关键字筛选题号.ipynb b/工具/关键字筛选题号.ipynb index 1d40b272..a1589bd7 100644 --- a/工具/关键字筛选题号.ipynb +++ b/工具/关键字筛选题号.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -11,7 +11,7 @@ "0" ] }, - "execution_count": 1, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -21,7 +21,7 @@ "\n", "\"\"\"---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---\"\"\"\n", "keywords_dict_table = [\n", - " {\"_nottags\":[\"单元\",\"暂\"]}\n", + " {\"origin\":[\"2022\"],\"origin2\":[\"春季\"]}\n", "]\n", "\"\"\"---关键字设置完毕---\"\"\"\n", "# 示例： keywords_dict_table = [\n", @@ -89,7 +89,7 @@ ], "metadata": { "kernelspec": { - "display_name": "mathdept", + "display_name": "Python 3.8.15 ('mathdept')", "language": "python", "name": "python3" }, @@ -103,12 +103,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.15" + "version": "3.8.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" + "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" } } }, diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb index 7c2c8e4d..eb05dd0c 100644 --- a/工具/批量添加题库字段数据.ipynb +++ b/工具/批量添加题库字段数据.ipynb @@ -9,48 +9,452 @@ "name": "stdout", "output_type": "stream", "text": [ - "题号: 012287 , 字段: ans 中已修改数据: $\\{-1,0,1\\}$\n", - "题号: 012288 , 字段: ans 中已修改数据: $\\pi$\n", - "题号: 012289 , 字段: ans 中已修改数据: $3+4\\mathrm{i}$\n", - "题号: 012290 , 字段: ans 中已修改数据: $2$\n", - "题号: 012291 , 字段: ans 中已修改数据: $1$\n", - "题号: 012292 , 字段: ans 中已修改数据: $16\\pi$\n", - "题号: 012293 , 字段: ans 中已修改数据: $(-\\dfrac 15,\\dfrac 25)$\n", - "题号: 012294 , 字段: ans 中已修改数据: $[-1,\\dfrac 12]$\n", - "题号: 012295 , 字段: ans 中已修改数据: $(4,5]$\n", - "题号: 012296 , 字段: ans 中已修改数据: $y=\\pm 2\\sqrt{2} x$\n", - "题号: 012297 , 字段: ans 中已修改数据: $\\dfrac{5\\sqrt{3}}6\\pi$\n", - "题号: 012298 , 字段: ans 中已修改数据: $(0,2)$, $-6$\n", - "题号: 012299 , 字段: ans 中已修改数据: B\n", - "题号: 012300 , 字段: ans 中已修改数据: C\n", - "题号: 012301 , 字段: ans 中已修改数据: A\n", - "题号: 012302 , 字段: ans 中已修改数据: A\n", - "题号: 012303 , 字段: ans 中已修改数据: (1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{5}}3$\n", - "题号: 012304 , 字段: ans 中已修改数据: (1) $\\dfrac\\pi 3$; (2) $2\\sqrt{3}+2$\n", - "题号: 012305 , 字段: ans 中已修改数据: (1) $120$米; (2) $20$米时, 总造价最低\n", - "题号: 012306 , 字段: ans 中已修改数据: (1) $\\dfrac{x^2}3+y^2=1$; (2) $\\dfrac 12$; (3) $2$\n", - "题号: 012307 , 字段: ans 中已修改数据: (1) $f(x)=\\mathrm{e}^{x+1}$; (2) $g(n)=\\begin{cases}\\dfrac{3n-1}{2}, & n\\text{为奇数},\\\\ \\dfrac{3n-2}{2}, & n\\text{为偶数};\\end{cases}$ (3) 存在``阈度'', 取值范围为$[\\dfrac{\\mathrm{e}^4+1}{\\mathrm{e}^4-\\mathrm{e}},+\\infty)$\n", - "题号: 012308 , 字段: ans 中已修改数据: $\\{2,3,4\\}$\n", - "题号: 012309 , 字段: ans 中已修改数据: $(-\\dfrac 12,2)$\n", - "题号: 012310 , 字段: ans 中已修改数据: $6$\n", - "题号: 012311 , 字段: ans 中已修改数据: $2$\n", - "题号: 012312 , 字段: ans 中已修改数据: $-8$\n", - "题号: 012313 , 字段: ans 中已修改数据: $-4$\n", - "题号: 012314 , 字段: ans 中已修改数据: $\\dfrac 12$\n", - "题号: 012315 , 字段: ans 中已修改数据: $\\sqrt{3}$\n", - "题号: 012316 , 字段: ans 中已修改数据: $y=2x-1$\n", - "题号: 012317 , 字段: ans 中已修改数据: $2$\n", - "题号: 012318 , 字段: ans 中已修改数据: $[-4,12]$\n", - "题号: 012319 , 字段: ans 中已修改数据: $2+\\sqrt{3}$\n", - "题号: 012320 , 字段: ans 中已修改数据: D\n", - "题号: 012321 , 字段: ans 中已修改数据: A\n", - "题号: 012322 , 字段: ans 中已修改数据: B\n", - "题号: 012323 , 字段: ans 中已修改数据: A\n", - "题号: 012324 , 字段: ans 中已修改数据: (1) $\\dfrac 43$; (2) $\\arccos \\dfrac{\\sqrt{6}}6$\n", - "题号: 012325 , 字段: ans 中已修改数据: (1) $[k\\pi-\\dfrac{3\\pi}8,k\\pi+\\dfrac\\pi 8]$($k\\in \\mathbf{Z}$); (2) $[-\\dfrac{\\sqrt{2}}2,\\dfrac{\\sqrt{2}}2]$\n", - "题号: 012326 , 字段: ans 中已修改数据: (1) $y=-0.02x^2+100$($0\\le x\\le 50$); (2) $f(x)=\\begin{cases}x(-0.02x^2+100), & 30\\le x\\le 50, \\\\ x(-x+100), & 50b$''是``$a^3+1>b^3+1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -298881,7 +298908,9 @@ "id": "012088", "content": "演讲比赛共有$9$位评委分别给出某选手的原始评分, 评定该选手的成绩时, 从$9$个原始评分中去掉$1$个最高分、$1$个最低分, 得到$7$个有效评分. $7$个有效评分与$9$个原始评分相比, 不变的数字特征是\\bracket{20}.\n\\fourch{中位数}{平均数}{方差}{极差}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -298913,7 +298942,9 @@ "id": "012089", "content": "已知$\\omega$是常数, 若函数$y=|\\sin (\\omega x+\\dfrac \\pi 3)|$图像的一条对称轴是直线$x=\\dfrac\\pi 6$. 则$\\omega$的值不可能在区间\\bracket{20}中.\n\\fourch{$(0,2]$}{$(2,4]$}{$(4,6]$}{$(6,8]$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -298945,7 +298976,9 @@ "id": "012090", "content": "对于两个定义在$\\mathbf{R}$上的函数$y=f(x)$与$y=g(x)$, 构造新的函数$y=h(x)$如下: 对任意$x_0\\in \\mathbf{R}$, $h(x_0)=f(x_0)+g(x_0)$. 现已知$y=h(x)$是严格增函数, 对于以下两个命题: \n\\textcircled{1} $y=f(x)$与$y=g(x)$中至少有一个是严格增函数;\n\\textcircled{2} $y=f(x)$与$y=g(x)$中至少有一个无最大值.\n其中\\bracket{20}.\n\\fourch{\\textcircled{1}和\\textcircled{2}都是真命题}{只有\\textcircled{1}是真命题}{只有\\textcircled{2}是真命题}{没有真命题}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -298977,7 +299010,9 @@ "id": "012091", "content": "如图, 设$P-ABCD$是底面为矩形的四棱锥, $PA\\perp$平面$ABCD$. $PA=AB=2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,2) node [left] {$B$} coordinate (B);\n\\draw (3,0,2) node [right] {$C$} coordinate (C);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [left] {$P$} coordinate (P);\n\\draw (P) -- (B) -- (C) -- (D) -- (P) (P) -- (C);\n\\draw [dashed] (A) -- (P) (A) -- (B) (A) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PC\\perp BD$, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若直线$PD$与平面$PAB$所成的角的大小为$\\arctan 2$, 求直线$PC$与平面$ABCD$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 83$; (2) $\\arctan \\dfrac{\\sqrt{5}}5$", "solution": "", @@ -299009,7 +299044,9 @@ "id": "012092", "content": "设$a$是实常数, 并记$f(x)=x^3+ax^2+2x$.\\\\\n(1) 当$a=-\\dfrac{5}{2}$时, 求函数$y=f(x)$的单调减区间;\\\\ \n(2) 是否存在$a$, 使得函数$y=f(x)$在实数范围内有且仅有三个零点, 且三个零点可按某种顺序排列后成等差数列? 若存在, 求所有满足条件的$a$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $[\\dfrac 23,1]$; (2) 存在, $a=-3$或$3$", "solution": "", @@ -299041,7 +299078,9 @@ "id": "012093", "content": "如图, 某市郊外景区内一条笔直的公路$a$经过三个景点$A$、$B$、$C$. 景区管委会又开发了风景优美的景点$D$.经测量景点$D$位于景点$A$的北偏东$30^\\circ$方向$8$千米处, 且位于景点$B$的正北方向, 还位于景点$C$的北偏西$75^\\circ$方向上.已知$AB=5$千米.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.25]\n \\draw [->] (6,8) -- (10,8) node [right] {东};\n \\draw [->] (6,8) -- (6,12) node [above] {北};\n \\draw [->] (0,0) node [below] {$A$} coordinate (A) -- (0,8) node [left] {$N$} coordinate (N);\n \\draw (A) --++ (60:8) node [above] {$D$} coordinate (D);\n \\draw (4,3) node [below] {$B$} coordinate (B) -- (D);\n \\draw [name path = linea] (A) -- ($(A)!2.2!(B)$) node [right] {$a$} coordinate (a);\n \\path [name path = DC] (D) --++ (-15:4);\n \\path [name intersections = {of = linea and DC, by = C}];\n \\draw (D) -- (C) node [below] {$C$};\n \\draw (60:2) arc (60:90:2);\n \\draw (75:4) node {$30^\\circ$};\n \\end{tikzpicture}\n\\end{center}\n(1) 景区管委会准备由景点$D$向景点$B$修一条笔直的公路, 不考虑其他因素, 求出这条公路的长(结果精确到$0.1$千米);\\\\\n(2) 求景点$C$与景点$D$之间的距离(结果精确到$0.1$千米).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) 约为$3.9\\text{km}$; (2) 约为$4.0\\text{km}$", "solution": "", @@ -299073,7 +299112,9 @@ "id": "012094", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2^n+\\lambda n$, 其中常数$\\lambda\\in \\mathbf{R}$.\\\\\n(1) 若$a_3=4a_2$, 求$\\lambda$的值;\\\\\n(2) 若$\\{a_n\\}$前$10$项的和为$1551$, 试分析$\\{a_n\\}$的单调性;\\\\\n(3) 对于常数$t$, 记集合$C_t=\\{n|a_n=t\\}$, 试求当$\\lambda$与$t$变化时, 集合$C_t$中元素个数的最大值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $-\\dfrac 85$; (2) 前四项严格递减, 从第四项起严格递增; (3) 元素个数的最大值为$2$", "solution": "", @@ -299105,7 +299146,9 @@ "id": "012095", "content": "已知椭圆$E$的方程为$\\dfrac{x^2}{12}+\\dfrac{y^2}{4}=1$, $F_1(-2\\sqrt{2},0)$与$F_2(2\\sqrt{2},0)$是$E$的两个焦点, $A(0,-2)$是$E$的下顶点.\\\\\n(1) 设斜率为$1$的直线$l$过点$F_1$, 且与$E$交于$M,N$两点, 求弦$MN$的长;\\\\\n(2) 若$E$上一点$P$满足$|F_1P|=3|F_2P|$, 求$\\triangle F_1F_2P$的面积;\\\\\n(3) 是否存在椭圆$E$上, 且位于第一象限的点$Q$, 使得射线$QA$平分$\\angle F_1QF_2$? 若存在, 请写出一个满足条件的点$Q$的坐标并加以验证; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $2\\sqrt{3}$; (2) $2\\sqrt{5}$; (3) $Q$的坐标为$(3,1)$, 验证过程略", "solution": "", @@ -299137,7 +299180,9 @@ "id": "012096", "content": "已知集合$A=\\{x|\\dfrac{2x}{x-1}\\le 1\\}$, $B=\\{-1,0,1,2\\}$, 则$A\\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{-1,0\\}$", "solution": "", @@ -299169,7 +299214,9 @@ "id": "012097", "content": "设$a\\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位. 若$(a-\\mathrm{i})(1-2\\mathrm{i})$为纯虚数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -299201,7 +299248,9 @@ "id": "012098", "content": "在空间直角坐标系中, 点$A(1,2,-3)$关于$xOz$平面对称的点的坐标是$\\blank{50}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$(1,-2,-3)$", "solution": "", @@ -299233,7 +299282,9 @@ "id": "012099", "content": "已知$m\\in \\mathbf{R}$, 直线$l_1:\\sqrt{3}x-y+7=0$, $l_2:mx+y-1=0$. 若$l_1\\parallel l_2$, 则$l_1$与$l_2$之间的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$3$", "solution": "", @@ -299265,7 +299316,9 @@ "id": "012100", "content": "为了解某校高三年级男生的体重, 从该校高三年级男生中抽取$17$名, 测得他们的体重数据如下(按从小到大的顺序排列, 单位: kg): $56\\ 56 \\ 57\\ 58\\ 59\\ 59\\ 61\\ 63\\ 64\\ 65\\ 66\\ 68\\ 69\\ 70\\ 73\\ 74\\ 83$\\\\\n据此估计该校高三年级男生体重的第$75$百分位数为\\blank{50}kg.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "$69$", "solution": "", @@ -299297,7 +299350,9 @@ "id": "012101", "content": "设$a,b$为实数. 若关于$x$的方程$x^2+abx+b=0$的解集为$\\{1,3\\}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$-\\dfrac 43$", "solution": "", @@ -299329,7 +299384,9 @@ "id": "012102", "content": "已知常数$m>0$. 在$(x+\\dfrac mx)^6$的二项展开式中, $x^2$项的系数是$\\dfrac{1}{x^2}$项的系数的$4$倍, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 12$", "solution": "", @@ -299361,7 +299418,9 @@ "id": "012103", "content": "在平面上, 已知$\\overrightarrow{a}$, $\\overrightarrow{b}$为两个不平行的单位向量, $O$为定点, 集合$\\Omega = \\{P|\\overrightarrow{OP}=\\lambda \\overrightarrow{a}+\\mu \\overrightarrow{b}, \\ 0\\le \\lambda \\le 1, \\ 0\\le \\mu \\le 2\\}$. 若$\\Omega$中所有点构成图形的面积为$1$, 则$\\overrightarrow{a}$与$\\overrightarrow{b}$夹角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$\\dfrac\\pi 6$或$\\dfrac{5\\pi}6$", "solution": "", @@ -299393,7 +299452,9 @@ "id": "012104", "content": "已知定义在$(-3,3)$上的奇函数$y=f(x)$的导函数是$f'(x)$, 当$x\\ge 0$时, $y=f(x)$的图像如图所示, 则关于$x$的不等式$\\dfrac{f'(x)}{x}>0$的解集为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth, line cap = round, line join = round,scale = 0.6]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3] plot (\\x,{1-pow(\\x-1,2)});\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) (3,-3) -- (3,0) node [below right] {$3$};\n\\draw (2,0) node [below left] {$2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$(-3,-1)\\cup (0,1)$", "solution": "", @@ -299425,7 +299486,9 @@ "id": "012105", "content": "第$14$届国际数学教育大会(ICME-14)于$2021$年$7$月$12$日至$18$日在上海举办, 已知张老师和李老师都在$7$天中随机选择了连续的$3$天参会, 则两位老师所选的日期恰好都不相同的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "$\\dfrac 6{25}$", "solution": "", @@ -299457,7 +299520,9 @@ "id": "012106", "content": "已知$A$、$B$、$C$是半径为$1$的球面上的三点, 若$AB=AC=1$, 则$BC$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -299489,7 +299554,9 @@ "id": "012107", "content": "设$a\\in \\mathbf{R}$, $m\\in \\mathbf{Z}$. 若存在唯一的$m$使得关于$x$的不等式组$\\dfrac 12 x^2-\\dfrac 12=stealth, line cap = round, line join = round]\n\\draw (0,0) node {$1$};\n\\draw (0,-0.5) node {$2$};\n\\draw (0,-1) node {$3$};\n\\draw (0.3,0) node {$6$};\n\\draw (0.6,0) node {$8$};\n\\draw (0.3,-0.5) node {$1$} (0.6,-0.5) node {$2$} (0.9,-0.5) node {$2$};\n\\draw (0.3,-1) node {$1$};\n\\draw (0.15,0.25) -- (0.15,-1.25);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$21$}{$21.5$}{$22$}{$22.5$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -299553,7 +299622,9 @@ "id": "012109", "content": "已知双曲线$\\Gamma_1:\\dfrac{x^2}{a_1^2}-\\dfrac{y^2}{b_1^2}=1$($a_1>0$, $b_1>0$)与$\\Gamma_2:\\dfrac{x^2}{a_2^2}-\\dfrac{y^2}{b_2^2}=1$($a_2>0$, $b_2>0$)有共同的渐近线, 则它们一定有相等的\\bracket{20}.\n\\fourch{实轴长}{虚轴长}{焦距}{离心率}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -299585,7 +299656,9 @@ "id": "012110", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$S_{2023}=a_{2023}$, 则$\\{a_n\\}$不可能是\\bracket{20}. \n\\twoch{公差大于$0$的等差数列}{公差小于$0$的等差数列}{公比大于$0$的等比数列}{公比小于$0$的等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -299617,7 +299690,9 @@ "id": "012111", "content": "已知$\\omega \\in \\mathbf{R}$, $\\varphi \\in [0, 2\\pi)$. 若对任意实数$x$均有$\\sin x\\ge \\cos (\\omega x+\\varphi)$, 则满足条件的有序实数对$(\\omega , \\varphi)$的个数为\\bracket{20}. \n\\fourch{$1$个}{$2$个}{$3$个}{无数个}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -299649,7 +299724,9 @@ "id": "012112", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_4=10$.\\\\\n(1) 若$S_{20}=590$, 求$\\{a_n\\}$的公差;\\\\\n(2) 若$a_1\\in \\mathbf{Z}$, 且$S_7$是数列$\\{S_n\\}$中最大的项, 求$a_1$所有可能的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) 公差为$3$; (2) $a_1$的所有可能的值为$18,19,20$", "solution": "", @@ -299681,7 +299758,9 @@ "id": "012113", "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, $AC$、$BD$相交于点$O$.\n\\begin{center}\n\\begin{tikzpicture}[>=stealth, line cap = round, line join = round, scale = 1.7]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {\\footnotesize $A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {\\footnotesize $B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {\\footnotesize $C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {\\footnotesize $D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {\\footnotesize $A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [above] {\\footnotesize $B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {\\footnotesize $C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {\\footnotesize $D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1)-- (C1);\n\\draw [dashed] (A1) -- (D) (C1) -- (D) (A) -- (C) (B) -- (D);\n\\draw ($(A)!0.5!(C)$) node [below] {\\footnotesize $O$} coordinate (O);\n\\draw [dashed] (O) -- (B1);\n\\end{tikzpicture}\n\\end{center}\n(1)证明: 直线$B_1O$与平面$A_1C_1D$平行;\\\\\n(2)求三棱锥$O-A_1C_1D$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\dfrac 13$", "solution": "", @@ -299713,7 +299792,9 @@ "id": "012114", "content": "闲置房出租是增加社会住房供给量, 满足人们居住需求的重要途径. 王先生有一套住房以每月$7000$元的价格出租, 但合同租期本月到期. 房客直接向王先生提出希望从下月起续租三年, 并愿意每月支付$8000$元的租金. 王先生通过中介公司了解到: 该房屋所在小区的类似住宅, 目前的租金为每月$8000$-$9000$元, 在委托中介公司后, 一般$2$-$4$周左右可以\n找到承租人, 同时每次租赁交易成功后, 中介公司向出租方和承租方各收取一个月租金的$50\\%$作为中介费. 对于是否同意房客续租, 王先生需要作出决策.\\\\\n(1) 除了上述了解到的情况, 还有哪些因素王先生可能需要考虑? 写出这些因素(不超过$5$个);\\\\ \n(2) 为了简化问题, 请对相关因素作出合情假设, 由此帮助王先生作出决策, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) (示例)王先生可能需要考虑的因素有:\\\\\n\\textcircled{1} 未来月租金的变化;\\\\\n\\textcircled{2} 找到承租人的时长的变化;\\\\\n\\textcircled{3} 未来租客的租期长短;\\\\\n\\textcircled{4} 房屋是否未来三年内可以用于出租;\\\\\n\\textcircled{5} 换租客的过程中是否需要重新装修;\n\\textcircled{6} 寻租过程中的时间、精力成本等.\\\\\n(2) 言之有理即可", "solution": "", @@ -299745,7 +299826,9 @@ "id": "012115", "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$F_1,F_2$, 直线$l$的斜率为$k$, 在$y$轴上的截距为$m$.\\\\\n(1) 设$k=1$, 若$\\Gamma$的焦距为$2$, $l$过点$F_1$, 求$l$的方程;\\\\\n(2) 设$m=0$. 若$P(\\sqrt{3},\\dfrac 12)$是$\\Gamma$上的一点, 且$|\\overrightarrow{PF_1}|+|\\overrightarrow{PF_2}|=4$, $l$与$\\Gamma$交于不同的两点$A,B$, $Q$为$\\Gamma$的上顶点, 求$\\triangle ABQ$面积的最大值;\\\\\n(3) 设$\\overrightarrow{n}$是$l$的一个法向量, $M$是$l$上一点, 对于坐标平面内的点$N$, 定义$\\delta_N=\\dfrac{\\overrightarrow{n}\\cdot \\overrightarrow{MN}}{|\\overrightarrow{n}|}$. 用$a,b,k,m$表示$\\delta_{F_1}\\cdot \\delta_{F_2}$, 并利用$\\delta_{F_1}\\cdot \\delta_{F_2}$与$b^2$的大小关系, 提出一个关于$l$与$\\Gamma$位置关系的真命题, 给出该命题的证明.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $y=x+1$; (2) 面积的最大值为$2$, 此时$l:y=0$; (3) 当$\\delta_{F_1}\\cdot \\delta_{F_2}>b^2$($=b^2$, $1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -299962,7 +300065,9 @@ "id": "012125", "content": "方程$1+\\log_2x=\\log_2(x^2-3)$的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -299981,7 +300086,9 @@ "id": "012126", "content": "平面直角坐标系中, 满足到$F_1(-1, 0)$的距离比到$F_2(1, 0)$的距离大$1$的点的轨迹为曲线$T$, 点$P_n(n, y_n)$(其中$y_n>0$, $n\\in \\mathbf{N}$, $n\\ge 1$)是曲线$T$上的点, 原点$O$到直线$P_nF_2$的距离为$d_n$, 则$\\displaystyle\\lim_{n\\to \\infty} d_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300000,7 +300107,9 @@ "id": "012127", "content": "如图所示矩形$ABCD$中, $AB=2$, $AD=1$, 分别将边$BC$与$DC$等分成$8$份, 并将等分点自下而上依次记作$E_1,E_2,\\cdots,E_7$, 自左到右依次记作$F_1,F_2,\\cdots,F_7$, 满足$\\overrightarrow{AE_i}\\cdot \\overrightarrow{AF_j}\\le 2$, (其中$i,j\\in \\mathbf{N}$, $1\\le i, j\\le 7$)的有序数对$(i,j)$共有\\blank{50}对.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [below left] {$A$} coordinate (A) rectangle (2,1) node [above right] {$C$} coordinate (C);\n\\draw (2,0) node [below right] {$B$} coordinate (B) (0,1) node [above left] {$D$} coordinate (D);\n\\foreach \\i in {1,2,...,7} {\\draw [->] (A) -- ($(B)!{(\\i)/8}!(C)$) node [right] {\\small $E_{\\i}$};};\n\\foreach \\i in {1,2,...,7} {\\draw [->] (A) -- ($(D)!{(\\i)/8}!(C)$) node [above] {\\small $F_{\\i}$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300019,7 +300128,9 @@ "id": "012128", "content": "已知函数$y=f(x)$在定义域$\\mathbf{R}$上是单调函数, 值域为$(-\\infty ,\\ 0)$, 满足$f(-1)=-\\dfrac 13$, 且对于任意$x,\\ y\\in \\mathbf{R}$, 都有$f(x+y)=-f(x)f(y)$. $y=f(x)$的反函数为$y=f^{-1}(x)$, 若将$y=kf(x)$(其中常数$k>0$)的反函数的图像向上平移1个单位, 将得到函数$y=f^{-1}(x)$的图像, 则实数$k$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300039,7 +300150,9 @@ "id": "012129", "content": "设$a>b>0$, $c\\ne 0$, 则下列不等式中, 恒成立的是\\bracket{20}.\n\\fourch{$\\dfrac 1a>\\dfrac 1b$}{$ac^2>bc^2$}{$ac>bc$}{$\\dfrac ca<\\dfrac cb$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300058,7 +300171,9 @@ "id": "012130", "content": "下列函数中, 值域为$(0\\text,+\\infty)$的是\\bracket{20}.\n\\fourch{$y=x^2$}{$y=\\dfrac 2x$}{$y=2^x$}{$y=|\\log_2x|$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300077,7 +300192,9 @@ "id": "012131", "content": "从正方体的$8$个顶点中选取$4$个作为顶点, 可得到四面体的个数为\\bracket{20}.\n\\fourch{$\\mathrm{C}_8^4-12$}{$\\mathrm{C}_8^4-8$}{$\\mathrm{C}_8^4-6$}{$\\mathrm{C}_8^4-4$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300096,7 +300213,9 @@ "id": "012132", "content": "设集合$A=\\{y|y=a^x, \\ x>0\\}$(其中常数$a>0$, $a\\ne 1$), $B=\\{y|y=x^k,\\ x\\in A\\}$(其中常数$k\\in \\mathbf{Q}$), 则``$k<0$''是``$A\\cap B=\\varnothing$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300115,7 +300234,9 @@ "id": "012133", "content": "如图所示, 在直三棱柱$ABC-A_1B_1C_1$中, 底面是等腰直角三角形, $\\angle ACB=90^{^\\circ}$, $CA=CB=CC_1=2$. 点$D,D_1$分别是棱$AC,A_1C_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(C)$) node [above left] {$D$} coordinate (D);\n\\draw ($(A_1)!0.5!(C_1)$) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A) -- (B) -- (B_1) -- (C_1) -- (A_1) -- (A);\n\\draw (D_1) -- (B_1) (A_1) -- (B_1);\n\\draw [dashed] (D) -- (B) (A) -- (C) -- (B) (D) -- (D_1) (C) -- (C_1) (C_1) -- (B);\n\\end{tikzpicture}\n\\end{center}\n(1)\t求证: $D,B,B_1,D_1$四点共面;\\\\\n(2)\t求直线$BC_1$与平面$DBB_1D_1$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300134,7 +300255,9 @@ "id": "012134", "content": "设常数$k\\in \\mathbf{R}$, $f(x)=k\\cos ^2x+\\sqrt 3\\sin x\\cos x$, $x\\in \\mathbf{R}$.\\\\\n(1) 若$f(x)$是奇函数, 求实数$k$的值;\\\\\n(2) 设$k=1$, $\\triangle ABC$中, 内角$A,B,C$的对边分别为$a,b,c$. 若$f(A)=1$, $a=\\sqrt 7$, $b=3$, 求$\\triangle ABC$的面积$S$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300153,7 +300276,10 @@ "id": "012135", "content": "某校运会上无人机飞行表演, 在水平距离$x\\in [10,24]$(单位: 米)内的飞行轨迹如图所示, $y$表示飞行高度(单位: 米).其中当$x\\in [10,20]$时, 轨迹为开口向上的抛物线的一段(端点为$MQ$), 当$x\\in [20,24]$时, 轨迹为线段$QN$, 经测量, 起点$M(10,24)$, 终点$N(24,24)$, 最低点$P(14,8)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.1]\n\\draw [->] (-5,0) -- (30,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,50) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 10:20] plot (\\x,{pow(\\x-14,2)+8});\n\\draw (20,44) node [above] {$Q$} coordinate (O) -- (24,24) node [right] {$N$} (10,24) node [above] {$M$} (14,8) node [below] {$P$} (0,24) node [left] {$A$} coordinate (A);\n\\draw [dashed] (0,24) --++ ({atan(2*sqrt(180)-28)}:25) coordinate (T);\n\\draw [dashed] (0,24) --++ (24,24);\n\\draw (A) pic [\"$\\theta$\", draw, angle eccentricity = 1.5, angle radius = 10] {angle = T--A--O}; \n\\end{tikzpicture}\n\\end{center}\n(1) 求$y$关于$x$的函数解析式;\\\\\n(2) 在$A(0,24)$处有摄像机跟踪拍摄, 为确保始终拍到无人机, 求拍摄视角$\\theta$的最小值.(精确到$0.1^\\circ$)", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300172,7 +300298,9 @@ "id": "012136", "content": "设$A_1,A_2$分别是椭圆$\\Gamma :\\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左、右顶点, 点$B$为椭圆的上顶点.\\\\\n(1) 若$\\overrightarrow{A_1B}\\cdot \\overrightarrow{A_2B}=-4$, 求椭圆$\\Gamma$的方程;\\\\\n(2) 设$a=\\sqrt 2$, $F_2$是椭圆的右焦点, 点$Q$是椭圆第二象限部分上一点, 若线段$F_2Q$的中点$M$在$y$轴上, 求$\\triangle F_2BQ$的面积;\\\\ \n(3) 设$a=3$, 点$P$是直线$x=6$上的动点, 点$C$和$D$是椭圆上异于左右顶点的两点, 且$C$, $D$分别在直线$PA_1$和$PA_2$上, 求证: 直线$CD$恒过一定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300191,7 +300319,9 @@ "id": "012137", "content": "设数列$\\{a_n\\}$与$\\{b_n\\}$满足: $\\{a_n\\}$的各项均为正数, $b_n=\\cos a_n$, $n\\in \\mathbf{N}$, $n\\ge 1$.\\\\\n(1) 设$a_2=\\dfrac{3 \\pi}4$, $a_3=\\dfrac{\\pi}3$, 若$\\{b_n\\}$是无穷等比数列, 求数列$\\{b_n\\}$的通项公式;\\\\ \n(2) 设$0=latex, z = {(215:0.5)}]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) ++ (0,\\l,0) coordinate (A1);\n\\draw (B) ++ (0,\\l,0) coordinate (B1);\n\\draw (C) ++ (0,\\l,0) coordinate (C1);\n\\draw (D) ++ (0,\\l,0) coordinate (D1);\n\\draw ($(A)!0.5!(B)$) coordinate (M);\n\\draw ($(B)!0.5!(C)$) coordinate (N) node [right] {$D$};\n\\draw ($(C)!0.5!(D)$) coordinate (P);\n\\draw ($(D)!0.5!(A)$) coordinate (Q);\n\\draw (M) ++ (0,\\l) coordinate (M1);\n\\draw (N) ++ (0,\\l) coordinate (N1);\n\\draw (P) ++ (0,\\l) coordinate (P1) node [above] {$B$};\n\\draw (Q) ++ (0,\\l) coordinate (Q1) node [above] {$A$};\n\\draw ($(A)!0.5!(A1)$) coordinate (A2);\n\\draw ($(B)!0.5!(B1)$) coordinate (B2);\n\\draw ($(C)!0.5!(C1)$) coordinate (C2) node [right] {$C$};\n\\draw ($(D)!0.5!(D1)$) coordinate (D2);\n\\draw (M) -- (A2) -- (Q) -- cycle (M) -- (B2) -- (N) -- cycle;\n\\draw (N) -- (C2) -- (N1) -- (B2);\n\\draw (N1) -- (M1) -- (Q1) -- (P1) -- cycle;\n\\draw (B2) -- (M1) -- (A2);\n\\draw (A2) -- (Q1);\n\\draw [dashed] (C2) -- (P1) -- (D2) -- (P) -- cycle;\n\\draw [dashed] (N) -- (P) -- (Q);\n\\draw [dashed] (Q) -- (D2) -- (Q1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300457,7 +300613,9 @@ "id": "012151", "content": "为求解方程$x^5-1=0$的虚根, 可以把原方程变形为 $(x-1)(x^4+x^3+x^2+x+1)=0$, 再变形为$(x-1)(x^2+a x+1)(x^2+b x+1)=0$, 由此可得原方程的一个虚根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300476,7 +300634,9 @@ "id": "012152", "content": "若向量$\\overrightarrow a=(2,0)$, $\\overrightarrow b=(1,1)$ , 则下列结论正确的是\\bracket{20}.\n\\fourch{$\\overrightarrow a \\cdot \\overrightarrow b=1$}{$|\\overrightarrow a|=|\\overrightarrow b|$}{$(\\overrightarrow a-\\overrightarrow b) \\perp \\overrightarrow b$}{$\\overrightarrow a \\parallel \\overrightarrow b$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300495,7 +300655,9 @@ "id": "012153", "content": "函数 $f(x)=\\dfrac{4^x-1}{2^x}$ 的图像关于\\bracket{20}.\n\\fourch{原点对称}{直线$y=x$对称}{直线$y=-x$对称}{$y$轴对称}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300514,7 +300676,9 @@ "id": "012154", "content": "直线$l: y=k(x+\\dfrac 12)$与圆$C: x^2+y^2=1$的位置关系为\\bracket{20}.\n\\fourch{相交或相切}{相交或相离}{相切}{相交}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300533,7 +300697,9 @@ "id": "012155", "content": "若 $\\overrightarrow{a_1}$、$\\overrightarrow{a_2}$、$\\overrightarrow{a_3}$均为单位向量, 则$\\overrightarrow{a_1}=(\\dfrac{\\sqrt 3}3, \\dfrac{\\sqrt 6}3)$是$\\overrightarrow{a_1}+\\overrightarrow{a_2}+\\overrightarrow{a_3}=(\\sqrt 3, \\sqrt 6)$的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300552,7 +300718,9 @@ "id": "012156", "content": "向量$\\overrightarrow a=(\\sin 2 x-1, \\cos x), \\overrightarrow b=(1,2 \\cos x)$, 设函数$f(x)=\\overrightarrow a \\cdot \\overrightarrow b$, 求函数$f(x)$的最小正周期及$x \\in[0, \\dfrac{\\pi}2]$时的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300571,7 +300739,9 @@ "id": "012157", "content": "某甜品店制作一种蛋筒冰激凌, 上部分是半球形, 下半部分呈圆锥形(如图), 现把半径为$10\\text{cm}$ 的圆形蛋皮等分成$5$个扇形, 用一个蛋皮围成圆锥的侧面 (蛋皮厚度忽略不计), 求该蛋筒冰激凌的表面积和体积. (精确到$0.01$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\fill [gray!20] (0,{-2*sqrt(6)}) -- (1,0) arc (360:180:1 and 0.3) -- cycle;\n\\draw (0,{-2*sqrt(6)}) -- (1,0) (0,{-2*sqrt(6)}) -- (-1,0);\n\\draw (1,0) arc (360:180:1 and 0.3);\n\\draw [dashed] (1,0) arc (0:180:1 and 0.3);\n\\draw (1,0) arc (0:180:1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300590,7 +300760,9 @@ "id": "012158", "content": "已知抛物线 $F:x^2=4y$.\\\\\n(1) $\\triangle ABC$ 的三个顶点在抛物线 $F$ 上, 记$\\triangle ABC$的三边 $AB$、$BC$、$CA$所在直线的斜率分别为$k_{AB}$、$k_{BC}$、$k_{CA}$, 若点 $A$在坐标原点, 求 $k_{AB}-k_{BC}+k_{CA}$的值;\\\\\n(2) 请你给出一个以$P(2,1)$为顶点, 且其余各顶点均为抛物线$F$上的动点的多边形, 写出多边形各边所在直线的斜率之间的关系式, 并说明理由. 说明: 第(2)题将根据结论的一般性程度给与不同的评分.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300609,7 +300781,9 @@ "id": "012159", "content": "定义域为 $\\mathbf{R}$, 且对任意实数 $x_1$、$x_2$都满足不等式$f(\\dfrac{x_1+x_2}2) \\le \\dfrac{f(x_1)+f(x_2)}2$的所有函数$f(x)$组成的集合记为$M$, 例如$f(x)=k x+b \\in M$.\\\\\n(1) 已知函数 $f(x)=\\begin{cases}x, & x \\ge 0,\\\\ \\dfrac 12 x & x<0,\\end{cases}$ 证明: $f(x) \\in M$;\\\\\n(2) 写出一个函数$f(x)$, 使得$f(x) \\not\\in M$, 并说明理由;\\\\\n(3) 写出一个函数$f(x) \\in M$, 使得数列极限$\\displaystyle\\lim_{n \\to \\infty} \\dfrac{f(n)}{n^2}=1$, $\\displaystyle\\lim_{n \\to \\infty} \\dfrac{f(-n)}{-n}=1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300628,7 +300802,9 @@ "id": "012160", "content": "对于给定首项$x_0>\\sqrt[3]a(a>0)$, 由递推式$x_{n+1}=\\dfrac 12(x_n+\\sqrt {\\dfrac a{x_n}})$($n \\in \\mathbf{N}$, $n\\ge 1$)得到数列$\\{x_n\\}$, 且对于任意的$n \\in \\mathbf{N}$, $n\\ge 1$, 都有 $x_n>\\sqrt[3]a$, 用数列$\\{x_n\\}$可以计算$\\sqrt[3]a$的近似值.\\\\\n(1) 取$x_0=5$, $a=100$, 计算 $x_1$、$x_2$、$x_3$的值(精确到$0.01$), 并且归纳出$x_n$、$x_{n+1}$的大小关系;\\\\\n(2) 当$n \\ge 1$时, 证明: $x_n-x_{n+1}<\\dfrac 12(x_{n-1}-x_n)$;\\\\\n(3) 当$x_0 \\in [5,10]$时, 用数列$\\{x_n\\}$计算$\\sqrt [3]{100}$的近似值, 要求满足$|x_n-x_{n+1}|<10^{-4}$, 请你估计$n$, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300647,7 +300823,9 @@ "id": "012161", "content": "设集合$A=\\{1,2,3\\}$, 集合$B=\\{3,4\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300666,7 +300844,9 @@ "id": "012162", "content": "不等式$|x-1|<3$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300685,7 +300865,9 @@ "id": "012163", "content": "若复数$z$满足$2\\overline{z}-1=3+6\\mathrm{i}$($\\mathrm{i}$是虚数单位), 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300704,7 +300886,9 @@ "id": "012164", "content": "若$\\cos \\alpha=\\dfrac 13$, 则$\\sin (\\alpha-\\dfrac{\\pi}2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300723,7 +300907,9 @@ "id": "012165", "content": "若关于$x$、$y$的方程组$\\begin{cases}x+2 y=4, \\\\3 x+a y=6\\end{cases}$无解, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300742,7 +300928,9 @@ "id": "012166", "content": "若等差数列$\\{a_n\\}$的前$5$项的和为$25$, 则$a_1+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300761,7 +300949,9 @@ "id": "012167", "content": "若$P$、$Q$是圆$x^2+y^2-2 x+4 y+4=0$上的动点, 则$|PQ|$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300780,7 +300970,9 @@ "id": "012168", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=3^n$, 则$\\displaystyle\\lim_{n \\to \\infty} \\dfrac{a_1+a_2+a_3+\\cdots+a_n}{a_n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300799,7 +300991,9 @@ "id": "012169", "content": "若$(x+\\dfrac 2x)^n$的二项展开式的各项系数之和为$729$, 则该展开式中常数项的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300816,9 +301010,11 @@ }, "012170": { "id": "012170", - "content": "设粗圆$\\dfrac{x^2}2+y^2=1$的左、右焦点分别为$F_1$、$F_2$, 点$P$在该椭圆上, 则使得$\\triangle F_1F_2P$是等腰三角形的点$P$的个数是\\blank{50}.", + "content": "设椭圆$\\dfrac{x^2}2+y^2=1$的左、右焦点分别为$F_1$、$F_2$, 点$P$在该椭圆上, 则使得$\\triangle F_1F_2P$是等腰三角形的点$P$的个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300837,7 +301033,9 @@ "id": "012171", "content": "设$a_1$、$a_2$、$\\cdots$、$a_6$为$1$、$2$、$3$、$4$、$5$、$6$的一个排列, 则满足$|a_1-a_2|+|a_3-a_4|+|a_5-a_6|=3$的不同排列的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300856,7 +301054,9 @@ "id": "012172", "content": "设$a$、$b\\in \\mathbf{R}$, 若函数$f(x)=x+\\dfrac ax+b$在区间$(1,2)$上有两个不同的零点, 则$f(1)$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -300875,7 +301075,9 @@ "id": "012173", "content": "函数$f(x)=(x-1)^2$的单调递增区间是\\bracket{20}.\n\\fourch{$[0,+\\infty)$}{$[1,+\\infty)$}{$(-\\infty, 0]$}{$(-\\infty, 1]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300894,7 +301096,9 @@ "id": "012174", "content": "设$a \\in \\mathbf{R}$, ``$a>0$''是``$\\dfrac 1a>0$''的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既非充分也非必要}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300913,7 +301117,9 @@ "id": "012175", "content": "过正方体中心的平面截正方体所得的截面中, 不可能的图形是\\bracket{20}\n\\fourch{三角形}{长方形}{对角线不相等的菱形}{六边形}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300932,7 +301138,9 @@ "id": "012176", "content": "如图所示, 正八边形$A_1A_2A_3A_4A_5A_6A_7A_8$的边长为$2$, 若$P$为该正八边形边上的动点, 则$\\overrightarrow{A_1A_3} \\cdot \\overrightarrow{A_1P}$的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1) --++ (0:1) node [below] {$A_2$} coordinate (A_2) --++ (45:1) node [right] {$A_3$} coordinate (A_3) --++ (90:1) node [right] {$A_4$} coordinate (A_4) --++ (135:1) node [above] {$A_5$} coordinate (A_5) --++ (180:1) node [above] {$A_6$} coordinate (A_6) --++ (225:1) node [left] {$A_7$} coordinate (A_7) --++ (270:1) node [left] {$A_8$} coordinate (A_8) -- cycle;\n\\draw [->] (A_1) -- (A_3);\n\\draw [->] (A_1) -- ($(A_6)!0.5!(A_7)$) node [above left] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt 2]$}{$[-2 \\sqrt 2, 8+6 \\sqrt 2]$}{$[-8-6 \\sqrt 2, 2 \\sqrt 2]$}{$[-8-6 \\sqrt 2, 8+6 \\sqrt 2]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -300951,7 +301159,9 @@ "id": "012177", "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中,$AB=BC=2$, $AA_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (B);\n\\draw [dashed] (A1) -- (D) (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求异面直线$A_1C$与$DD_1$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300970,7 +301180,9 @@ "id": "012178", "content": "设$a \\in \\mathbf{R}$, 函数$f(x)=\\dfrac{2^x+a}{2^x+1}$.\\\\\n(1) 求$a$的值, 使得$f(x)$为奇函数;\\\\\n(2) 若$f(x)<\\dfrac{a+2}2$对任意$x \\in \\mathbf{R}$成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -300989,7 +301201,10 @@ "id": "012179", "content": "某景区欲建造两条圆形观景步道$M_1$、$M_2$(宽度忽略不计), 如图所示, 已知$AB \\perp AC$, $AB=AC=AD=60$(单位: 米), 要求圆$M_1$与$AB$、$AD$分别相切于点$B$、$D$, 圆$M_2$与$AC$、$AD$分别相切于点$C$、$D$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$C$} coordinate (C);\n\\draw (60:2) node [below right] {$D$} coordinate (D);\n\\draw (C) -- (A) -- (B) (A) -- (D);\n\\filldraw (2,{2/sqrt(3)}) circle (0.02) coordinate (O1);\n\\draw (O1) circle ({2/sqrt(3)});\n\\draw (O1) ++ (30:{2/sqrt(3)}) node [above right] {$M_1$};\n\\filldraw (C) ++ ({2*tan(15)},0) circle (0.02) coordinate (O2);\n\\draw (O2) circle ({2*tan(15)});\n\\draw (O2) ++ (75:{2*tan(15)}) node [above right] {$M_2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle BAD=60^{\\circ}$, 求圆$M_1$、$M_2$的半径; (结果精确到$0.1$米)\\\\\n(2) 若观景步道$M_1$与$M_2$的造价分别为每米$0.8$千元与每米$0.9$千元, 如何设计圆$M_1$、$M_2$的大小, 使总造价最低? 最低总造价是多少? (结果精确到$0.1$千元)", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301008,7 +301223,9 @@ "id": "012180", "content": "已知双曲线$\\Gamma: x^2-\\dfrac{y^2}{b^2}=1$($b>0$), 直线$l: y=k x+m$($km \\neq 0$), $l$与$\\Gamma$交于$P$、$Q$两点,$P'$为$P$关于$y$轴的对称点, 直线$P'Q$与$y$轴交于点$N(0,n)$.\\\\\n(1) 若点$(2,0)$是$\\Gamma$的一个焦点, 求$\\Gamma$的渐近线方程;\\\\\n(2) 若$b=1$, 点$P$的坐标为$(-1,0)$, 且$\\overrightarrow{NP'}=\\dfrac 32 \\overrightarrow{P'Q}$, 求$k$的值;\\\\\n(3) 若$m=2$, 求$n$关于$b$的表达式.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301027,7 +301244,11 @@ "id": "012181", "content": "已知函数$f(x)=\\log_2 \\dfrac{1+x}{1-x}$.\\\\\n(1) 解方程$f(x)=1$;\\\\\n(2) 设$x \\in(-1,1)$, $a \\in(1,+\\infty)$, 证明:$\\dfrac{a x-1}{a-x} \\in(-1,1)$, 且$f(\\dfrac{a x-1}{a-x})-f(x)=-f(\\dfrac 1a)$;\\\\\n(3) 设数列$\\{x_n\\}$中, $x_1 \\in(-1,1)$, $x_{n+1}=(-1)^{n+1} \\dfrac{3 x_n-1}{3-x_n}$, $n \\in \\mathbf{N}$, $n\\ge 1$. 求$x_1$的取值范围, 使得$x_3 \\ge x_n$对任意$n \\in \\mathbf{N}$, $n\\ge 1$成立.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301046,7 +301267,9 @@ "id": "012182", "content": "不等式$|x|>1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301065,7 +301288,9 @@ "id": "012183", "content": "计算:$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{3 n-1}{n+2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301084,7 +301309,9 @@ "id": "012184", "content": "设集合$A=\\{x|0=latex, scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [above right] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(C1)$) node [above left] {$O$} coordinate (O);\n\\draw (O) -- (B1) (A) -- (B1) (A1) -- (C1);\n\\draw [dashed] (O) -- (A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301179,7 +301414,9 @@ "id": "012189", "content": "某校组队参加辩论赛, 从$6$名学生中选出$4$人分别担任一、二、三、四辩, 若其中学生甲必须参赛且不担任四辩, 则不同的安排方法种数为\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301198,7 +301435,9 @@ "id": "012190", "content": "设$a \\in \\mathbf{R}$, 若$(x^2+\\dfrac 2x)^9$与$(x+\\dfrac a{x^2})^9$的二项展开式中的常数项相等, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301217,7 +301456,9 @@ "id": "012191", "content": "设$m \\in \\mathbf{R}$, 若$z$是关于$x$的方程$x^2+m x+m^2-1=0$的一个虚根, 则$|\\overline{z}|$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301236,7 +301477,10 @@ "id": "012192", "content": "设$a>0$, 函数$f(x)=x+2(1-x) \\sin (ax)$, $x \\in(0,1)$, 若函数$y=2 x-1$与$y=f(x)$的图像有且仅有两个不同的公共点, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301253,9 +301497,11 @@ }, "012193": { "id": "012193", - "content": "如图, 正方形$ABCD$的边长为$20$米, 圆$O$的半径为$1$米, 圆心是正方形的中心, 点$P$、$Q$分别在线段$AD$、$CB$上, 若线段$PQ$与圆$O$有公共点, 则称点$Q$在点$P$的 ``盲区'' 中, 已知点$P$以$1.5$米/秒的速度从$A$出发向$D$移动, 同时, 点$Q$以$1$米/秒的速度从$C$出发向$B$移动, 则在点$P$从$A$移动到$D$的过程中, 点$Q$在点$P$的盲区中的时长约为\\blank{50}秒.(精确到$0.1$秒)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (0,2) node [above right] {$C$} coordinate (C) --++ (-2,0) node [above left] {$D$} coordinate (D) -- cycle;\n\\filldraw (1,1) circle (0.01);\n\\draw (1,1) circle (0.1);\n\\draw (1.1,0.9) node [below right] {$O$};\n\\draw (0,0.45) -- (2,1.7);\n\\end{tikzpicture}\n\\end{center}", + "content": "如图, 正方形$ABCD$的边长为$20$米, 圆$O$的半径为$1$米, 圆心是正方形的中心, 点$P$、$Q$分别在线段$AD$、$CB$上, 若线段$PQ$与圆$O$有公共点, 则称点$Q$在点$P$的 ``盲区'' 中, 已知点$P$以$1.5$米/秒的速度从$A$出发向$D$移动, 同时, 点$Q$以$1$米/秒的速度从$C$出发向$B$移动, 则在点$P$从$A$移动到$D$的过程中, 点$Q$在点$P$的盲区中的时长约为\\blank{50}秒.(精确到$0.1$秒)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A) --++ (2,0) node [below right] {$B$} coordinate (B) --++ (0,2) node [above right] {$C$} coordinate (C) --++ (-2,0) node [above left] {$D$} coordinate (D) -- cycle;\n\\filldraw (1,1) circle (0.01);\n\\draw (1,1) circle (0.1);\n\\draw (1.1,0.9) node [below right] {$O$};\n\\draw (0,0.45) node [left] {$P$} -- (2,1.7) node [right] {$Q$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301263,7 +301509,8 @@ "usages": [], "origin": "2018届春季高考试题12", "edit": [ - "20221209\t王伟叶" + "20221209\t王伟叶", + "20221213\t余利成" ], "same": [], "related": [], @@ -301274,7 +301521,9 @@ "id": "012194", "content": "下列函数中, 为偶函数的是\\bracket{20}.\n\\fourch{$y=x^{-2}$}{$y=x^{\\frac 13}$}{$y=x^{-\\frac 12}$}{$y=x^3$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301293,7 +301542,9 @@ "id": "012195", "content": "如图, 在直三棱柱$ABC-A_1B_1C_1$的棱所在的直线中, 与直线$BC_1$异面的直线的条数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (2,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [below left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) -- (C_1);\n\\draw (A) -- (B) -- (C) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{1}{2}{3}{4}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301312,7 +301563,9 @@ "id": "012196", "content": "设$S_n$为数列$\\{a_n\\}$的前$n$项和, ``$\\{a_n\\}$是递增数列''是``$\\{S_n\\}$是递增数列''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301329,9 +301582,12 @@ }, "012197": { "id": "012197", - "content": "已知$A$、$B$为平面上的两个定点, 且$|\\overrightarrow{AB}|=2$, 该平面上的动线段$PQ$的端点$P$、$Q$, 满足$|\\overrightarrow{AP}|\\leq 5$, $\\overrightarrow{AP} \\cdot \\overrightarrow{AB}=6$, $\\overrightarrow{AQ}=-2 \\overrightarrow{AP}$, 则动线段$PQ$所形成图形的面积为\\bracket{20}.\n\\fourch{$36$}{$60$}{$72$}{$108$}", + "content": "已知$A$、$B$为平面上的两个定点, 且$|\\overrightarrow{AB}|=2$, 该平面上的动线段$PQ$的端点$P$、$Q$满足$|\\overrightarrow{AP}|\\leq 5$, $\\overrightarrow{AP} \\cdot \\overrightarrow{AB}=6$, $\\overrightarrow{AQ}=-2 \\overrightarrow{AP}$, 则动线段$PQ$所形成图形的面积为\\bracket{20}.\n\\fourch{$36$}{$60$}{$72$}{$108$}", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301339,7 +301595,8 @@ "usages": [], "origin": "2018届春季高考试题16", "edit": [ - "20221209\t王伟叶" + "20221209\t王伟叶", + "20221213\t余利成" ], "same": [], "related": [], @@ -301348,9 +301605,11 @@ }, "012198": { "id": "012198", - "content": "已知$y=\\cos x$.\\\\\n(1) 若$f(\\alpha)=\\dfrac 13$, 且$\\alpha \\in[0, \\pi]$, 求$f(\\alpha-\\dfrac{\\pi}3)$的值;\\\\\n(2) 求函数$y=f(2x)-2 f(x)$的最小值.", + "content": "已知$f(x)=\\cos x$.\\\\\n(1) 若$f(\\alpha)=\\dfrac 13$, 且$\\alpha \\in[0, \\pi]$, 求$f(\\alpha-\\dfrac{\\pi}3)$的值;\\\\\n(2) 求函数$y=f(2x)-2 f(x)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301358,7 +301617,8 @@ "usages": [], "origin": "2018届春季高考试题17", "edit": [ - "20221209\t王伟叶" + "20221209\t王伟叶", + "20221213\t余利成" ], "same": [], "related": [], @@ -301369,7 +301629,9 @@ "id": "012199", "content": "已知$a \\in \\mathbf{R}$, 双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$.\\\\\n(1) 若点$(2,1)$在$\\Gamma$上, 求$\\Gamma$的焦点坐标;\\\\\n(2) 若$a=1$, 直线$y=k x+1$与$\\Gamma$相交于$A$、$B$两点, 且线段$AB$中点的横坐标为$1$, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301386,9 +301648,12 @@ }, "012200": { "id": "012200", - "content": "利用``平行于圆锥母线的平面截圆锥面, 所得截线是抛物线''的几何原理, 某快餐店用两个射灯(射出的光锥为圆锥)广告牌上投影出其标识, 如图1所示, 图2是投影射出的抛物线的平面图, 图3是一个射灯投影的直观图, 在图2与图3中, 点$O$、$A$、$B$在抛物线上, $OC$是抛物线的对称轴, $OC \\perp AB$于$C$, $AB=3$米, $OC=4.5$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\clip (-2,0.5) rectangle (1.2,-2);\n\\fill [orange, opacity = 0.5, x = {(10:1)}, domain = -2:0.08] plot (\\x,{-2.5*pow(\\x+1,2)});\n\\fill [orange, opacity = 0.5, x = {(10:1)}, domain = -1.06:1.1] plot (\\x,{-2.5*pow(\\x,2)});\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (3,0) node [right] {$B$} coordinate (B) (1.5,0) node [below] {$C$} coordinate (C) -- (1.5,4.5) node [above] {$O$} coordinate (O);\n\\draw [samples = 200,thick,domain = 0:3.2] plot (\\x,{4.5-2* pow(\\x-1.5,2)}); \n\\draw [samples = 200,dashed,domain = 0:3.2] plot (-\\x,{4.5-2* pow(\\x-1.5,2)}); \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$C$} coordinate (C);\n\\draw (1.5,0) node [right] {$E$} coordinate (E) arc (360:180:1.5 and 0.5) node [left] {$D$} coordinate (D);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and 0.5) --++ (3,0);\n\\draw ({1.5*cos(100)},{0.5*sin(100)}) node [above left] {$A$} coordinate (A);\n\\draw ({1.5*cos(280)},{0.5*sin(280)}) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A) -- (B);\n\\draw (D) -- (0,4) node [above] {$S$} coordinate (S) -- (E);\n\\draw [dashed] (S) -- (C) (C) -- ($(S)!0.5!(E)$) node [above right] {$O$} coordinate (O);\n\\draw (O) .. controls ({0.75+0.3*cos(-80)},{2+0.3*sin(-80)}).. (B);\n\\draw [dashed] (O) .. controls ({0.75-0.1*cos(-80)},{2-0.1*sin(-80)}).. (A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求抛物线的焦点到准线的距离;\\\\\n(2) 在图3中, 已知$OC$平行于圆锥的母线$SD$, $AB$、$DE$是圆锥底面的直径, 求圆锥的母线与轴的夹角的大小(精确到$0.01^{\\circ}$).", + "content": "利用``平行于圆锥母线的平面截圆锥面, 所得截线是抛物线''的几何原理, 某快餐店用两个射灯(射出的光锥为圆锥)在广告牌上投影出其标识, 如图1所示, 图2是投影射出的抛物线的平面图, 图3是一个射灯投影的直观图, 在图2与图3中, 点$O$、$A$、$B$在抛物线上, $OC$是抛物线的对称轴, $OC \\perp AB$于$C$, $AB=3$米, $OC=4.5$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}\n\\clip (-2,0.5) rectangle (1.2,-2);\n\\fill [orange, opacity = 0.5, x = {(10:1)}, domain = -2:0.08] plot (\\x,{-2.5*pow(\\x+1,2)});\n\\fill [orange, opacity = 0.5, x = {(10:1)}, domain = -1.06:1.1] plot (\\x,{-2.5*pow(\\x,2)});\n\\end{scope}\n\\draw (-0.4,-2) node [below] {图1};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (3,0) node [right] {$B$} coordinate (B) (1.5,0) node [below] {$C$} coordinate (C) -- (1.5,4.5) node [above] {$O$} coordinate (O);\n\\draw [samples = 200,thick,domain = 0:3.2] plot (\\x,{4.5-2* pow(\\x-1.5,2)}); \n\\draw [samples = 200,dashed,domain = 0:3.2] plot (-\\x,{4.5-2* pow(\\x-1.5,2)}); \n\\draw (0,-1) node [below] {图2};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$C$} coordinate (C);\n\\draw (1.5,0) node [right] {$E$} coordinate (E) arc (360:180:1.5 and 0.5) node [left] {$D$} coordinate (D);\n\\draw [dashed] (1.5,0) arc (0:180:1.5 and 0.5) --++ (3,0);\n\\draw ({1.5*cos(100)},{0.5*sin(100)}) node [above left] {$A$} coordinate (A);\n\\draw ({1.5*cos(280)},{0.5*sin(280)}) node [below] {$B$} coordinate (B);\n\\draw [dashed] (A) -- (B);\n\\draw (D) -- (0,4) node [above] {$S$} coordinate (S) -- (E);\n\\draw [dashed] (S) -- (C) (C) -- ($(S)!0.5!(E)$) node [above right] {$O$} coordinate (O);\n\\draw (O) .. controls ({0.75+0.3*cos(-80)},{2+0.3*sin(-80)}).. (B);\n\\draw [dashed] (O) .. controls ({0.75-0.1*cos(-80)},{2-0.1*sin(-80)}).. (A);\n\\draw (0,-1) node [below] {图3};\n\\end{tikzpicture}\n\\end{center}\n(1) 求抛物线的焦点到准线的距离;\\\\\n(2) 在图3中, 已知$OC$平行于圆锥的母线$SD$, $AB$、$DE$是圆锥底面的直径, 求圆锥的母线与轴的夹角的大小(精确到$0.01^{\\circ}$).", "objs": [], - "tags": [], + "tags": [ + "第六单元", + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301407,7 +301672,9 @@ "id": "012201", "content": "设$a>0$, 函数$f(x)=\\dfrac 1{1+a \\cdot 2^x}$.\\\\\n(1) 若$a=1$, 求$f(x)$的反函数$f^{-1}(x)$;\\\\\n(2) 求函数$y=f(x) \\cdot f(-x)$的最大值(用$a$表示);\\\\\n(3) 设$g(x)=f(x)-f(x-1)$, 若对任意$x \\in(-\\infty, 0]$, $g(x) \\ge g(0)$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301424,9 +301691,11 @@ }, "012202": { "id": "012202", - "content": "若$\\{c_n\\}$是递增数列, 数列$\\{a_n\\}$满足: 对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 存在$m \\in \\mathbf{N}$, $m\\ge 1$, 使得$\\dfrac{a_m-c_n}{a_m-c_{n+1}} \\leq 0$, 则称$\\{a_n\\}$是$\\{c_n\\}$的``分隔数列''.\\\\\n(1) 设$c_n=2 n$, $a_n=n+1$, 证明: 数列$\\{a_n\\}$是$\\{c_n\\}$的分隔数列;\\\\\n(2) 设$c_n=n-4$, $S_n$是$\\{c_n\\}$的前$n$项和, $d_n=c_{3 n-2}$, 判断数列$\\{S_n\\}$是否是数列$\\{d_n\\}$的分隔数列, 并说明理由;\\\\\n(3) 设$c_n=a q^{n-1}$, $T_n$是$\\{c_n\\}$的前$n$项和, 若数列$\\{T_n\\}$是$\\{c_n\\}$的分隔数列, 求实数$a$、$q$的取值范围.", + "content": "若$\\{c_n\\}$是严格递增数列, 数列$\\{a_n\\}$满足: 对任意$n \\in \\mathbf{N}$, $n\\ge 1$, 存在$m \\in \\mathbf{N}$, $m\\ge 1$, 使得$\\dfrac{a_m-c_n}{a_m-c_{n+1}} \\leq 0$, 则称$\\{a_n\\}$是$\\{c_n\\}$的``分隔数列''.\\\\\n(1) 设$c_n=2 n$, $a_n=n+1$, 证明: 数列$\\{a_n\\}$是$\\{c_n\\}$的分隔数列;\\\\\n(2) 设$c_n=n-4$, $S_n$是$\\{c_n\\}$的前$n$项和, $d_n=c_{3 n-2}$, 判断数列$\\{S_n\\}$是否是数列$\\{d_n\\}$的分隔数列, 并说明理由;\\\\\n(3) 设$c_n=a q^{n-1}$, $T_n$是$\\{c_n\\}$的前$n$项和, 若数列$\\{T_n\\}$是$\\{c_n\\}$的分隔数列, 求实数$a$、$q$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301445,7 +301714,9 @@ "id": "012203", "content": "已知集合$A=\\{1,2,3,4,5\\}$, $B=\\{3,5,6\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301464,7 +301735,9 @@ "id": "012204", "content": "计算:$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{2 n^2-3 n+1}{n^2-4 n+1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301483,7 +301756,9 @@ "id": "012205", "content": "不等式$|x+1|<5$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301502,7 +301777,9 @@ "id": "012206", "content": "函数$f(x)=x^2$($x>0$)的反函数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301521,7 +301798,9 @@ "id": "012207", "content": "设$\\mathrm{i}$为虚数单位, $3 \\overline{z}-\\mathrm{i}=6+5 \\mathrm{i}$, 则$|z|$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301540,7 +301819,9 @@ "id": "012208", "content": "已知二元线性方程组$\\begin{cases}2 x+2 y=-1, \\\\4 x+a^2 y=a\\end{cases}$有无穷多解, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301559,7 +301840,9 @@ "id": "012209", "content": "在$(x+\\dfrac 1{\\sqrt x})^6$的二项展开式中, 常数项的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301578,7 +301861,9 @@ "id": "012210", "content": "在$\\triangle ABC$中, $AC=3$, $3 \\sin A=2 \\sin B$, 且$\\cos C=\\dfrac 14$, 则$AB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301597,7 +301882,9 @@ "id": "012211", "content": "首届中国国际进口博览会在上海举行, 某高校拟派$4$人参与连续$5$天的志愿者活动, 其中甲连续参加$2$天, 其余每人各参加$1$天, 则所有不同的安排种数为\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301616,7 +301903,9 @@ "id": "012212", "content": "如图, 正方形$OABC$的边长为$a$($a>1$), 函数$y=3 x^2$的图像交$AB$于点$Q$, 函数$y=x^{-\\frac 12}$与$BC$交于点$P$, 则当$|AQ|+|CP|$最小时, $a$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) node [below] {$C$} coordinate (C) -- (2,2) node [above right] {$B$} coordinate (B) -- (0,2) node [left] {$A$} coordinate (A);\n\\draw [samples = 100, domain = 0.2:2.2] plot (\\x,{pow(\\x,-0.5)});\n\\draw [samples = 100, domain = 0:0.9] plot (\\x,{3*pow(\\x,2)});\n\\draw (2,{sqrt(2)/2}) node [below left] {$P$} coordinate (P);\n\\draw ({sqrt(2/3)},2) node [above right] {$Q$} coordinate (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301635,7 +301924,9 @@ "id": "012213", "content": "已知$P$为椭圆$\\dfrac{x^2}4+\\dfrac{y^2}2=1$上的任意一点, $Q$与$P$关于$x$轴对称, $F_1$、$F_2$为椭圆的左、右焦点, 若有$\\overrightarrow{F_1 P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则向量$\\overrightarrow{F_1 P}$与$\\overrightarrow{F_2 Q}$的夹角范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301654,7 +301945,9 @@ "id": "012214", "content": "已知$t \\in \\mathbf{R}$, 集合$A=[t, t+1] \\cup[t+4, t+9]$, $0 \\not\\in A$, 若存在正数$\\lambda$, 对任意$a \\in A$, 都有$\\dfrac{\\lambda}a \\in A$, 则$t$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301673,7 +301966,10 @@ "id": "012215", "content": "下列函数中, 值域为$[0,+\\infty)$的是\\bracket{20}.\n\\fourch{$y=2^x$}{$y=x^{\\frac 12}$}{$y=\\tan x$}{$y=\\cos x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301692,7 +301988,9 @@ "id": "012216", "content": "已知$a$、$b \\in \\mathbf{R}$, 则``$a^2>b^2$''是``$|a|>|b|$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301711,7 +302009,9 @@ "id": "012217", "content": "已知平面$\\alpha$、$\\beta$、$\\gamma$两两垂直, 直线$a$、$b$、$c$满足: $a \\subseteq \\alpha$, $b \\subseteq \\beta$, $c \\subseteq \\gamma$, 则直线$a$、$b$、$c$不可能是\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301730,7 +302030,9 @@ "id": "012218", "content": "平面直角坐标系中, 两动圆$O_1$、$O_2$的圆心分别为$(a_1, 0)$、$(a_2, 0)$, 且两圆均过定点$(1,0)$, 两圆与$y$轴正半轴分别交于点$(0, y_1)$、$(0, y_2)$, 若$\\ln y_1+\\ln y_2=0$, 点$(\\dfrac 1{a_1}, \\dfrac 1{a_2})$的轨迹为$\\Gamma$, 则$\\Gamma$所在的曲线可能是\\bracket{20}.\n\\fourch{直线}{圆}{椭圆}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -301749,7 +302051,9 @@ "id": "012219", "content": "如图, 正三棱锥$P-ABC$中, 侧棱长为$2$, 底面边长为$\\sqrt 3$, $M$、$N$分别是$PB$和$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw (1,{sqrt(3)},{sqrt(3)/3}) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [right] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw (A) -- (B) -- (C) (A) -- (P) -- (C) (M) -- (N) (P) -- (B);\n\\draw [dashed] (A) -- (C); \n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$MN$与$AC$所成角的大小;\\\\\n(2) 求三棱锥$P-ABC$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301768,7 +302072,9 @@ "id": "012220", "content": "已知数列$\\{a_n\\}$中, $a_1=3$, 前$n$项和为$S_n$.\\\\\n(1) 若$\\{a_n\\}$为等差数列, 且$a_4=15$, 求$S_n$;\\\\\n(2) 若$\\{a_n\\}$为等比数列, 且$\\displaystyle\\lim_{n\\to\\infty} S_n<12$, 求公比$q$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301787,7 +302093,9 @@ "id": "012221", "content": "改革开放$40$年, 我国卫生事业取得巨大成就, 卫生总费用增长了数十倍. 卫生总费用包括个人现在支出、社会支出、政府支出, 下表为$2012$年至$2015$年我国卫生费用中个人现金支出、社会支出和政府支出的费用(单位：亿元)和在卫生总费用中的占比. \n\\begin{center}\n\\begin{tabular}{|p{.05\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|p{.1\\textwidth}<\\centering|}\n\\hline\n& & \\multicolumn{2}{c|}{个人现金卫生支出} & \\multicolumn{2}{c|}{社会卫生支出} & \\multicolumn{2}{c|}{政府卫生支出} \\\\ \\hline\n年份& 卫生总费用(亿元)& 绝对数(亿元) & 占卫生总费用比重($\\%$) & 绝对数(亿元) & 占卫生总费用比重($\\%$)& 绝对数(亿元) & 占卫生总费用比重($\\%$)\\\\ \\hline\n$2012$ & $28119.00$ & $9656.32$ & $34.34$ & $10030.70$ & $35.67$ & $8431.98$ & $29.99$ \\\\ \\hline\n$2013$ & $31668.95$ & $10729.34$ & $33.88$ & $11393.79$ & $35.98$ & $9545.81$ & $30.14$ \\\\ \\hline\n$2014$ & $35312.40$ & $11295.41$ & $31.99$ & $13437.75$ & $38.05$ & $10579.23$ & $29.96$ \\\\ \\hline\n$2015$ & $40974.64$ & $11992.65$ & $29.27$ & $16506.71$ & $40.29$ & $12475.28$ & $30.45$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(数据来源于国家统计年鉴)\\\\\n(1) 指出$2012$年到$2015$年之间我国卫生总费用中个人现金支出占比和社会支出占比的变化趋势;\\\\\n(2) 设$t=1$表示$1978$年, 第$t$年卫生总费用与年份$t$之间拟合函数$f(t)=\\dfrac{357876.6053}{1+\\mathrm{e}^{6.4420-0.1136t}}$, 研究函数$f(t)$的单调性, 并预测我国卫生总费用首次超过$12$万亿的年份.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301807,7 +302115,9 @@ "id": "012222", "content": "已知抛物线$y^2=4 x$, $F$为焦点,$P$为抛物线准线$l$上一动点, 线段$PF$与抛物线交于点$Q$, 定义$d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$P$坐标为$(-1,-\\dfrac 83)$, 求$d(P)$;\\\\\n(2) 求证: 存在常数$a$, 使得$2 d(P)=|FP|+a$关于点$P$恒成立;\\\\\n(3) 设$P_1$、$P_2$、$P_3$为抛物线准线$l$上的三点, 且$|P_1P_2|=|P_2P_3|$, 试比较$d(P_1)+d(P_3)$与$2d(P_2)$的大小.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301829,7 +302139,10 @@ "id": "012223", "content": "若$\\{a_n\\}$是等差数列, 公差$d \\in(0, \\pi]$, 数列$\\{b_n\\}$满足: $b_n=\\sin (a_n)$, $n \\in \\mathbf{N}$, $n\\ge 1$, 记$S=\\{x|x=b_n,\\ n \\in \\mathbf{N},\\ n\\ge 1\\}$.\\\\\n(1) 设$a_1=0$, $d=\\dfrac 23 \\pi$, 求集合$S$;\\\\\n(2) 设$a_1=\\dfrac{\\pi}2$, 试求$d$的值, 使得集合$S$恰有两个元素;\\\\\n(3) 若集合$S$恰有三个元素, 且$b_{n+T}=b_n$, 其中$T$为不超过$7$的正整数, 求$T$所有可能值.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -301848,7 +302161,9 @@ "id": "012224", "content": "集合$A=\\{1,3\\}, B=\\{1,2, a\\}$, 若$A \\subseteq B$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301867,7 +302182,9 @@ "id": "012225", "content": "不等式$\\dfrac 1x>3$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301886,7 +302203,9 @@ "id": "012226", "content": "函数$y=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301905,7 +302224,9 @@ "id": "012227", "content": "已知复数$z$满足$z+2 \\overline{z}=6+\\mathrm{i}$, 则$z$的实部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301924,7 +302245,9 @@ "id": "012228", "content": "已知$3 \\sin 2 x=2 \\sin x$, $x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301943,7 +302266,9 @@ "id": "012229", "content": "若函数$y=a \\cdot 3^x+\\dfrac 1{3^x}$为偶函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301962,7 +302287,9 @@ "id": "012230", "content": "已知直线$l_1: x+a y=1$, $l_2: a x+y=1$, 若$l_1\\parallel l_2$, 则$l_1$与$l_2$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -301981,7 +302308,9 @@ "id": "012231", "content": "已知二项式$(2 x+\\sqrt x)^5$, 则展开式中$x^3$的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302000,7 +302329,9 @@ "id": "012232", "content": "三角形$ABC$中,$D$是$BC$中点,$AB=2, AC=3, BC=4$, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{AB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302019,7 +302350,9 @@ "id": "012233", "content": "已知$A=\\{-3,-2,-1,0,1,2,3\\}$, $a$、$b \\in A$, 则$|a|<|b|$的情况有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302038,7 +302371,9 @@ "id": "012234", "content": "已知平面上有$A_1$、$A_2$、$A_3$、$A_4$、$A_5$五个点, 满足$\\overrightarrow{A_n A_{n+1}} \\cdot \\overrightarrow{A_{n+1} A_{n+2}}=0$($n=1,2,3$), $|\\overrightarrow{A_n A_{n+1}}|\\cdot|\\overrightarrow{A_{n+1} A_{n+2}}|=n+1$($n=1,2,3$), 则$|\\overrightarrow{A_1 A_5}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302057,7 +302392,9 @@ "id": "012235", "content": "已知$f(x)=\\sqrt {x-1}$, 其反函数为$f^{-1}(x)$, 若$f^{-1}(x)-a=f(x+a)$有实数根, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302076,7 +302413,9 @@ "id": "012236", "content": "计算:$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{3^n+5^n}{3^{n-1}+5^{n-1}}=$\\bracket{20}.\n\\fourch{$3$}{$\\dfrac 53$}{$\\dfrac 35$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302095,7 +302434,9 @@ "id": "012237", "content": "``$\\alpha=\\beta$''是``$\\sin ^2 \\alpha+\\cos ^2 \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302114,7 +302455,9 @@ "id": "012238", "content": "已知椭圆$\\dfrac{x^2}2+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆的一部分}{双曲线的一部分}{圆的一部分}{抛物线的一部分}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302134,7 +302477,9 @@ "id": "012239", "content": "数列$\\{a_n\\}$各项均为实数, 对任意$n \\in \\mathbf{N}$, $n\\ge 1$满足$a_{n+3}=a_n$, 且行列式$\\begin{vmatrix}a_n & a_{n+1} \\\\ a_{n+2} & a_{n+3}\\end{vmatrix}=c$为定值, 则下列选项中不可能的是\\bracket{20}.\n\\fourch{$a_1=1$, $c=1$}{$a_1=2$, $c=2$}{$a_1=-1$, $c=4$}{$a_1=2$, $c=0$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302153,7 +302498,9 @@ "id": "012240", "content": "已知四棱锥$P-ABCD$, 底面$ABCD$为正方形, 边长为$3$, $PD \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (3,0,3) node [right] {$B$} coordinate (B);\n\\draw (0,0,3) node [left] {$A$} coordinate (A);\n\\draw (D) ++ (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$PC=5$, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若直线$AD$与$BP$的夹角为$60^{\\circ}$, 求$PD$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302172,7 +302519,9 @@ "id": "012241", "content": "已知各项均为正数的数列$\\{a_n\\}$, 其前$n$项和为$S_n$, $a_1=1$.\\\\\n(1) 若数列$\\{a_n\\}$为等差数列, $S_{10}=70$, 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$\\{a_n\\}$为等比数列, $a_4=\\dfrac 18$, 求满足$S_n>100 a_n$时$n$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302191,7 +302540,9 @@ "id": "012242", "content": "有一条长为$120$米的步行道$OA$, $A$是垃圾投放点$\\omega_1$, 若以$O$为原点, $OA$为$x$轴正半轴建立直角坐标系, 设点$B(x, 0)$, 现要建设另一座垃圾投放点$\\omega_2(t, 0)$, 函数$f_t(x)$表示与$B$点距离最近的垃圾投放点的距离.\\\\\n(1) 若$t=60$, 求$f_{60}(10)$、$f_{60}(80)$、$f_{60}(95)$的值, 并写出$f_{60}(x)$的函数解析式;\\\\\n(2) 若可以通过$f_t(x)$与坐标轴围成的面积来测算扔垃圾的便利程度, 面积越小越便利. 问: 垃圾投放点$\\omega_2$建在何处才能比建在中点时更加便利?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302210,7 +302561,9 @@ "id": "012243", "content": "已知抛物线$y^2=x$上的动点$M(x_0, y_0)$, 过$M$分别作两条直线交抛物线于$P$、$Q$两点, 交直线$x=t$于$A$、$B$两点.\\\\\n(1) 若点$M$纵坐标为$\\sqrt 2$, 求$M$与焦点的距离;\\\\\n(2) 若$t=-1$, $P(1,1)$, $Q(1,-1)$, 求证:$y_A \\cdot y_B$为常数;\\\\\n(3) 是否存在$t$, 使得$y_A \\cdot y_B=1$且$y_P \\cdot y_Q$为常数? 若存在, 求出$t$的所有可能值, 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302229,7 +302582,9 @@ "id": "012244", "content": "已知非空集合$A \\subseteq \\mathbf{R}$, 函数$y=f(x)$的定义域为$D$, 若对任意$t \\in A$且$x \\in D$, 不等式$f(x) \\leq f(x+t)$恒成立, 则称函数$f(x)$具有$A$性质.\\\\\n(1) 当$A=\\{-1\\}$, 判断$f(x)=-x$、$g(x)=2 x$是否具有$A$性质;\\\\\n(2) 当$A=(0,1)$, $f(x)=x+\\dfrac 1x$, $x \\in[a,+\\infty)$, $a>0$, 若$f(x)$具有$A$性质, 求$a$的取值范围;\\\\\n(3) 当$A=\\{-2, m\\}$, $m \\in \\mathbf{Z}$, 若$D=\\mathbf{Z}$且具有$A$性质的函数均为常值函数, 求所有符合条件的$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302249,7 +302604,9 @@ "id": "012245", "content": "已知等差数列$\\{a_n\\}$的首项为$3$, 公差为$2$, 则$a_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302268,7 +302625,9 @@ "id": "012246", "content": "已知$z=1-3 \\mathrm{i}$, 则$|\\overline{z}-\\mathrm{i}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302287,7 +302646,9 @@ "id": "012247", "content": "已知圆柱的底面半径为$1$, 高为$2$, 则圆柱的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302306,7 +302667,9 @@ "id": "012248", "content": "不等式$\\dfrac{2 x+5}{x-2}<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302325,7 +302688,9 @@ "id": "012249", "content": "直线$x=-2$与直线$\\sqrt 3 x-y+1=0$的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302344,7 +302709,9 @@ "id": "012250", "content": "若方程组$\\begin{cases}a_1 x+b_1 y=c_1, \\\\a_2 x+b_2 y=c_2\\end{cases}$无解, 则$\\begin{vmatrix}a_1 & b_1 \\\\ a_2 & b_2\\end{vmatrix}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302363,7 +302730,9 @@ "id": "012251", "content": "已知$(1+x)^n$的展开式中, 唯有$x^3$的系数最大, 则$x^3$的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302382,7 +302751,9 @@ "id": "012252", "content": "已知函数$f(x)=3^x+\\dfrac a{3^x+1}$($a>0$)的最小值为$5$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302401,7 +302772,9 @@ "id": "012253", "content": "在无穷等比数列$\\{a_n\\}$中, $\\displaystyle\\lim_{n\\to\\infty}(a_1-a_n)=4$, 则$a_2$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302420,7 +302793,9 @@ "id": "012254", "content": "某人某天需要运动总时长大于等于$60$分钟, 现有五项运动可以选择, 如下表所示, 共有\\blank{50}种运动方式组合.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline $A$运动 & $B$运动 & $C$运动 & $D$运动 & $E$运动 \\\\\n\\hline $7$点--$8$点 & $8$点--$9$点 & $9$点--$10$点 & $10$点--$11$点 & $11$点--$12$点 \\\\\n\\hline $30$分钟 & $20$分钟 & $40$分钟 & $30$分钟 & $30$分钟 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302439,7 +302814,9 @@ "id": "012255", "content": "已知椭圆$x^2+\\dfrac{y^2}{b^2}=1$($00$, 存在实数$\\varphi$, 使得对任意$n \\in \\mathbf{N}$, $n\\ge 1$, $\\cos (n \\theta+\\varphi)<\\dfrac{\\sqrt 3}2$, 则$\\theta$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302477,7 +302856,10 @@ "id": "012257", "content": "下列函数中, 在定义域内存在反函数的是\\bracket{20}.\n\\fourch{$f(x)=x^2$}{$f(x)=\\sin x$}{$f(x)=2^x$}{$f(x)=1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302496,7 +302878,9 @@ "id": "012258", "content": "已知集合$A=\\{x|x>-1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x|x^2-x-2 \\ge 0,\\ x \\in \\mathbf{R}\\}$, 则下列关系中, 正确的是\\bracket{20}\n\\fourch{$A \\subseteq B$}{$\\overline{A} \\subseteq \\overline{B}$}{$A \\cap B=\\varnothing$}{$A \\cup B=\\mathbf{R}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302515,7 +302899,9 @@ "id": "012259", "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 下列是$f(x)$无最大值的充分条件的是\\bracket{20}.\n\\twoch{$f(x)$为偶函数且关于点$(1,1)$对称}{$f(x)$为偶函数且关于直线$x=1$对称}{$f(x)$为奇函数且关于点$(1,1)$对称}{$f(x)$为奇函数且关于直线$x=1$对称}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302534,7 +302920,9 @@ "id": "012260", "content": "在$\\triangle ABC$中, $D$为$BC$中点, $E$为$AD$中点, 则以下结论: \\textcircled{1} 存在$\\triangle ABC$, 使得$\\overrightarrow{AB} \\cdot \\overrightarrow{CE}=0$; \\textcircled{2} 存在三角形$\\triangle ABC$, 使得$\\overrightarrow{CE} \\parallel (\\overrightarrow{CB}+\\overrightarrow{CA})$; 它们的成立情况是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302553,7 +302941,9 @@ "id": "012261", "content": "四棱锥$P-ABCD$, 底面为正方形$ABCD$, 边长为$4$, $E$为$AB$中点, $PE \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (4,0,4) node [right] {$C$} coordinate (C);\n\\draw (0,0,4) node [left] {$B$} coordinate (B);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E) ++ (0,{2*sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (E) ++ (4,0,0) node [right] {$F$} coordinate (F) -- (P);\n\\draw (P) -- (B) (P) -- (C) (P) -- (D) (B) -- (C) -- (D);\n\\draw [dashed] (E) -- (F) (P) -- (A) (A) -- (B) (A) -- (D) (A) -- (C) (P) -- (E);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\triangle PAB$为等边三角形, 求四棱锥$P-ABCD$的体积;\\\\ \n(2) 若$CD$的中点为$F$, $PF$与平面$ABCD$所成角为$45^{\\circ}$, 求$PC$与$AD$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302572,7 +302962,9 @@ "id": "012262", "content": "已知$A$、$B$、$C$为$\\triangle ABC$的三个内角, $a$、$b$、$c$是其三条边, $a=2$, $\\cos C=-\\dfrac 14$.\\\\\n(1) 若$\\sin A=2 \\sin B$, 求$b$、$c$;\\\\\n(2) 若$\\cos (A-\\dfrac{\\pi}4)=\\dfrac 45$, 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302591,7 +302983,9 @@ "id": "012263", "content": "设$O$是地面上的一点, 团队在$O$点西侧、东侧$20$千米处设有$A$、$B$两站点.\\\\\n(1) 测量距离发现一点$P$满足$|PA|-|PB|=20$千米, 可知$P$在以$A$、$B$为焦点的双曲线上, 以$O$点为原点, 东侧为$x$轴正半轴, 北侧为$y$轴正半轴, 建立平面直角坐标系, 若$P$在$O$点北偏东$60^{\\circ}$处, 求双曲线标准方程和$P$点坐标;\\\\\n(2) 团队又在$O$点南侧、北侧$15$千米处设有$C$、$D$两站点, 测量距离发现一点$Q$满足$|QA|-|QB|=30$千米, $|QC|-|QD|=10$千米, 求$|OQ|$和$Q$点方位. (精确到$1$米, $1^{\\circ}$)", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302610,7 +303004,9 @@ "id": "012264", "content": "已知函数$f(x)=\\sqrt {|x+a|-a}-x$.\\\\\n(1) 若$a=1$, 求函数的定义域;\\\\\n(2) 若$a \\neq 0$, 且$f(a x)=a$有$2$个不同实数根, 求$a$的取值范围;\\\\\n(3) 是否存在实数$a$, 使得函数$f(x)$在定义域内具有单调性? 若存在, 求出$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302629,7 +303025,9 @@ "id": "012265", "content": "已知数列$\\{a_n\\}$满足$a_n \\ge 0$, 对任意$n \\ge 2$, $a_n$和$a_{n+1}$中存在一项使其为另一项与$a_{n-1}$的等差中项.\\\\\n(1) 已知$a_1=5$, $a_2=3$, $a_4=2$, 求$a_3$的所有可能取值;\\\\\n(2) 已知$a_1=a_4=a_7=0$, $a_2$、$a_5$、$a_8$为正数, 求证:$a_2$、$a_5$、$a_8$成等比数列, 并求出公比$q$;\\\\\n(3) 已知数列中恰有$3$项为$0$, 即$a_r=a_s=a_t=0$, $20)$的右支上, 若$x_1 x_2>y_1 y_2$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302857,7 +303277,9 @@ "id": "012277", "content": "已知函数$y=f(x)$为定义域为$\\mathbf{R}$的奇函数, 其图像关于$x=1$对称, 且当$x \\in (0, 1]$时, $f(x)=\\ln x$, 若将方程$f(x)=x+1$的正实数根从小到大依次记为$x_1,x_2,x_3,\\cdots,x_n$, 则$\\displaystyle\\lim_{n\\to\\infty}(x_{n+1}-x_n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -302876,7 +303298,9 @@ "id": "012278", "content": "下列函数定义域为$\\mathbf{R}$的是\\bracket{20}.\n\\fourch{$y=x^{-\\frac 12}$}{$y=x^{-1}$}{$y=x^{\\dfrac 13}$}{$y=x^{\\dfrac 12}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302895,7 +303319,9 @@ "id": "012279", "content": "若$a>b>c>d$, 则下列不等式恒成立的是\\bracket{20}.\n\\fourch{$a+d>b+c$}{$a+c>b+d$}{$a c>b d$}{$a d>b c$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302914,7 +303340,9 @@ "id": "012280", "content": "上海海关大楼的顶部为逐级收拢的四面钟楼, 如图, 四个大钟分布在正四棱柱的四个侧面, 则每天$0$点至$12$点(包含$0$点, 不含$12$点)相邻两钟面上的时针相互垂直的次数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}[x = {(-10:0.9)}]\n\\draw (0,0) -- (-2,0) -- (-2,2) -- (0,2) -- cycle;\n\\draw (-1,1) circle (0.8);\n\\draw [->] (-1,1) --++ (-45:0.5);\n\\draw [->] (-1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (-1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\begin{scope}[x = {(40:0.7)}]\n\\draw (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;\n\\draw (1,1) circle (0.8);\n\\draw [->] (1,1) --++ (-45:0.5);\n\\draw [->] (1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$0$}{$2$}{$4$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302933,7 +303361,9 @@ "id": "012281", "content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 前$n$项积为$T_n$, 则下列选项判断正确的是\\bracket{20}.\n\\twoch{若$S_{2022}>S_{2021}$, 则数列$\\{a_n\\}$是递增数列}{若$T_{2022}>T_{2021}$, 则数列$\\{a_n\\}$是递增数列}{若数列$\\{S_n\\}$是递增数列, 则$a_{2022} \\ge a_{2021}$}{若数列$\\{T_n\\}$是递增数列, 则$a_{2022} \\ge a_{2021}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -302952,7 +303382,9 @@ "id": "012282", "content": "如图, 圆柱下底面与上底面的圆心分别为$O$、$O_1$, $AA_1$为圆柱的母线, 底面半径为$1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) arc (180:360:1 and 0.3) node [right] {$A$} coordinate (A);\n\\draw (-1,3) arc (180:360:1 and 0.3) node [right] {$A_1$} coordinate (A_1);\n\\draw (-1,3) arc (180:0:1 and 0.3);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.3);\n\\draw (-1,0) -- (-1,3) (1,0) -- (1,3) -- (0,3) node [left] {$O_1$} coordinate (O_1);\n\\draw [dashed] (O_1) --++ (0,-3) node [left] {$O$} coordinate (O)--++ (1,0);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AA_1=4$, $M$为$AA_1$的中点, 求直线$MO_1$与上底面所成角的大小;\\\\\n(2) 若圆柱过$OO_1$的截面为正方形, 求圆柱的体积与侧面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302971,7 +303403,9 @@ "id": "012283", "content": "已知在数列$\\{a_n\\}$中, $a_2=1$, 其前$n$项和为$S_n$.\\\\\n(1) 若$\\{a_n\\}$是等比数列, $S_2=3$, 求$\\displaystyle\\lim_{n\\to\\infty} S_n$;\\\\\n(2) 若$\\{a_n\\}$是等差数列, $S_{2 n} \\geq n$, 求其公差$d$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -302988,9 +303422,11 @@ }, "012284": { "id": "012284", - "content": "为有效塑造城市景观、提升城市环境品质, 上海市正在努力推进新一轮架空线入地工程的建设. 如图是一处要架空线入地的矩形地块$ABCD$, $AB=30\\text{m}$, $AD=15\\text{m}$. 为保护$D$处的一棵古树, 有关部门划定了以$D$为圆心、$DA$为半径的四分之一圆的地块为历史古迹封闭区. 若空线入线口为$AB$边上的点$E$, 出线口为$CD$边上的点$F$, 施工要求$EF$与封闭区边界相切, $EF$右侧的四边形地块$BCFE$将作为绿地保护生态区.(计算长度精确到$0.1\\text{m}$, 计算面积精确到$0.01 \\text{m}^2$)\\\\\n(1) 若$\\angle ADE=20^{\\circ}$, 求$EF$的长;\\\\\n(2) 当入线口$E$在$AB$上的什么位置时, 生态区的面积最大? 最大面积是多少?", + "content": "为有效塑造城市景观、提升城市环境品质, 上海市正在努力推进新一轮架空线入地工程的建设. 如图是一处要架空线入地的矩形地块$ABCD$, $AB=30\\text{m}$, $AD=15\\text{m}$. 为保护$D$处的一棵古树, 有关部门划定了以$D$为圆心、$DA$为半径的四分之一圆的地块为历史古迹封闭区. 若空线入线口为$AB$边上的点$E$, 出线口为$CD$边上的点$F$, 施工要求$EF$与封闭区边界相切, $EF$右侧的四边形地块$BCFE$将作为绿地保护生态区.(计算长度精确到$0.1\\text{m}$, 计算面积精确到$0.01 \\text{m}^2$)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0) node [below] {$A$} coordinate (A) -- (2,0) node [below] {$B$} coordinate (B) -- (2,1) node [above] {$C$} coordinate (C) -- (0,1) node [above] {$D$} coordinate (D) -- cycle;\n\\draw (0,0) arc (-90:0:1);\n\\draw ({tan(20)},0) node [below] {$E$} coordinate (E) -- ({1/cos(50)},1) node [above] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle ADE=20^{\\circ}$, 求$EF$的长;\\\\\n(2) 当入线口$E$在$AB$上的什么位置时, 生态区的面积最大? 最大面积是多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303009,7 +303445,9 @@ "id": "012285", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>1$), $A$、$B$两点分别为$\\Gamma$的左顶点、下顶点, $C$、$D$两点均在直线$l: x=a$上, 且$C$在第一象限.\\\\\n(1) 设$F$是椭圆$\\Gamma$的右焦点, 且$\\angle AFB=\\dfrac{\\pi}6$, 求$\\Gamma$的标准方程;\\\\\n(2) 若$C$、$D$两点纵坐标分别为$2$、$1$, 请判断直线$AD$与直线$BC$的交点是否在椭圆$\\Gamma$上, 并说明理由;\\\\ \n(3) 设直线$AD$、$BC$分别交椭圆$\\Gamma$于点$P$、点$Q$, 若$P$、$Q$关于原点对称, 求$|CD|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303028,7 +303466,9 @@ "id": "012286", "content": "已知函数$f(x)$的定义域为$\\mathbf{R}$, 现有两种对$f(x)$变换的操作:\n$\\varphi$变换: 得到$f(x)-f(x-t)$; $\\omega$变换: 得到$|f(x+t)-f(x)|$, 其中$t$为大于$0$的常数.\\\\\n(1) 设$f(x)=2^x$, $t=1$, $g(x)$为$f(x)$作$\\varphi$变换后的结果, 解方程: $g(x)=2$;\\\\\n(2) 设$f(x)=x^2$, $h(x)$为$f(x)$作$\\omega$变换后的结果, 解不等式: $f(x) \\ge h(x)$;\\\\\n(3) 设$f(x)$在$(-\\infty, 0)$上是严格增函数, $f(x)$先作$\\varphi$变换后得到$u(x)$, $u(x)$再作$\\omega$变换后得到$h_1(x)$; $f(x)$先作$\\omega$变换后得到$v(x)$, $v(x)$再作$\\varphi$变换后得到$h_2(x)$. 若$h_1(x)=h_2(x)$恒成立, 证明: 函数$f(x)$在$\\mathbf{R}$上是严格增函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -303047,7 +303487,9 @@ "id": "012287", "content": "已知集合$A=(-2,1]$, $B=\\mathbf{Z}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$\\{-1,0,1\\}$", "solution": "", @@ -303066,7 +303508,9 @@ "id": "012288", "content": "函数$y=\\sin x \\cdot \\cos x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\pi$", "solution": "", @@ -303085,7 +303529,9 @@ "id": "012289", "content": "已知$a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, 若$a-\\mathrm{i}$与$2+b \\mathrm{i}$互为共轭复数, 则$(a+b \\mathrm{i})^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$3+4\\mathrm{i}$", "solution": "", @@ -303104,7 +303550,9 @@ "id": "012290", "content": "记$S_n$为等差数列$\\{a_n\\}$的前$n$项和, 若$2 S_3=3 S_2+6$, 则公差$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -303123,7 +303571,9 @@ "id": "012291", "content": "已知函数$y=a-\\dfrac 2{2^x+1}$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$1$", "solution": "", @@ -303142,7 +303592,9 @@ "id": "012292", "content": "已知圆锥的母线长为$5$, 侧面积为$20 \\pi$, 则此圆锥的体积为\\blank{50}. (结果保留$\\pi$)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$16\\pi$", "solution": "", @@ -303161,7 +303613,9 @@ "id": "012293", "content": "已知向量$\\overrightarrow a=(5,3)$, $\\overrightarrow b=(-1,2)$, 则$\\overrightarrow a$在$\\overrightarrow b$上的投影向量的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$(-\\dfrac 15,\\dfrac 25)$", "solution": "", @@ -303180,7 +303634,9 @@ "id": "012294", "content": "对任意$x \\in \\mathbf{R}$, 不等式$|x-2|+|x-3|\\geq 2 a^2+a$恒成立, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$[-1,\\dfrac 12]$", "solution": "", @@ -303199,7 +303655,10 @@ "id": "012295", "content": "已知集合$A=\\{x | \\dfrac 2{x-2} \\geq 1, \\ x \\in \\mathbf{R}\\}$, 设函数$y=\\log_{\\frac 12} x+a(x \\in A)$的值域为$B$, 若$B \\subseteq A$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第一单元" + ], "genre": "填空题", "ans": "$(4,5]$", "solution": "", @@ -303218,7 +303677,9 @@ "id": "012296", "content": "已知$F_1$、$F_2$是双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点, 点$M$是双曲线$\\Gamma$上的任意一点(不是顶点), 过$F_1$作$\\angle F_1MF_2$的角平分线的垂线, 垂足为$N$, 线段$F_1N$的延长线交$MF_2$于点$Q$, $O$是坐标原点, 若$|ON|=\\dfrac{|F_1F_2|}6$, 则双曲线$\\Gamma$的渐近线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$y=\\pm 2\\sqrt{2} x$", "solution": "", @@ -303237,7 +303698,9 @@ "id": "012297", "content": "动点$P$在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$表面上运动, 且与点$A$的距离是$\\dfrac{2 \\sqrt 3}3$, 点$P$的集合形成一条曲线, 这条曲线的长度为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\dfrac{5\\sqrt{3}}6\\pi$", "solution": "", @@ -303256,7 +303719,9 @@ "id": "012298", "content": "已知数列$\\{a_n\\}$的各项都是正数, $a_{n+1}^2-a_{n+1}=a_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 若数列$\\{a_n\\}$为严格增数列, 则首项$a_1$的取值范围是\\blank{50}; 当$a_1=\\dfrac 23$时, 记$b_n=\\dfrac{(-1)^{n-1}}{a_n-1}$, 若$kb$成立的充要条件为\\bracket{20}.\n\\fourch{$a^2>b^2$}{$a^3>b^3$}{$a>b-1$}{$a>b+1$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -303295,7 +303762,9 @@ "id": "012300", "content": "函数$y=(x^2-1) \\mathrm{e}^x$的图像可能为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{5*(-1-\\x)*exp(-\\x*\\x-1)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{0.4*(\\x+0.5)*(\\x+2.5)*(\\x-1.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{(\\x*\\x-1)*exp(\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\clip (-2.25,-2.25) rectangle (2.25,2.25);\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (-1,0.2) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.2) -- (1,0) node [below] {$1$};\n\\draw (0.2,1) -- (0,1) node [left] {$1$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2.5:2.5,ultra thick,samples = 100] plot (\\x,{0.4*(\\x+1)*(\\x+3)*(\\x-1)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "C", "solution": "", @@ -303314,7 +303783,9 @@ "id": "012301", "content": "在天文学中, 天体的明暗程度可以用星等或亮度来描述, 两颗星的星等与亮度满足$m_2-m_1=\\dfrac 52 \\lg \\dfrac{E_1}{E_2}$, 其中星等为$m_k$的星的亮度为$E_k$($k=1$、$2$), 已知太阳的星等是$-26.7$, 天狼星的星等是$-1.45$, 则太阳与天狼星的亮度的比值为\\bracket{20}.\n\\fourch{$10^{10.1}$}{$10.1$}{$\\lg 10.1$}{$10^{-10.1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -303333,7 +303804,9 @@ "id": "012302", "content": "已知函数$f(x)=\\begin{cases}|x+2|, & x<0, \\\\x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$, 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围为\\bracket{20}.\n\\fourch{$(-2, \\dfrac 12)$}{$(-6, \\dfrac 12)$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac 12,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -303352,7 +303825,9 @@ "id": "012303", "content": "如图, 已知$AB \\perp$平面$BCD$, $BC \\perp CD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$D$} coordinate (D);\n\\draw ({sqrt(2)},0,{sqrt(2)}) node [below] {$C$} coordinate (C);\n\\draw (0,1,0) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) (A) -- (D) (A) -- (C);\n\\draw [dashed] (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$ACD \\perp$平面$ABC$;\\\\\n(2) 若$AB=1$, $CD=BC=2$, 求直线$AD$与平面$ABC$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) 证明略; (2) $\\arccos \\dfrac{\\sqrt{5}}3$", "solution": "", @@ -303371,7 +303846,9 @@ "id": "012304", "content": "在$\\triangle ABC$中, 内角$A$、$B$、$C$所对边分别为$a$、$b$、$c$, 已知$b \\sin A=a \\cos (B-\\dfrac{\\pi}6)$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$c=2a$, $\\triangle ABC$的面积为$\\dfrac{2 \\sqrt 3}3$, 求$\\triangle ABC$的周长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac\\pi 3$; (2) $2\\sqrt{3}+2$", "solution": "", @@ -303390,7 +303867,9 @@ "id": "012305", "content": "某地准备在山谷中建一座桥梁, 桥址位置的竖直截面图如图所示, 谷底$O$在水平线$MN$上、桥$AB$与$MN$平行, $OO'$为铅垂线($O'$在$AB$上). 经测量, 山谷左侧的轮廓曲线$AO$上任一点$D$到$MN$的距离$h_1$(米)与$D$到$OO'$的距离$a$(米) 之间满足关系式$h_1=\\dfrac 1{40} a^2$, 山谷右侧的轮廓曲线$BO$上任一点$F$到$MN$的距离$h_2$(米)与$F$到$OO'$的距离$b$(米)之间满足关系式$h_2=-\\dfrac 1{800} b^3+6 b$. 已知点$B$到$OO'$的距离为$40$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (-8,16) node [left] {$A$} coordinate (A);\n\\draw (4,16) node [right] {$B$} coordinate (B);\n\\draw (-10,0) node [below] {$M$} coordinate (M);\n\\draw (6,0) node [below] {$N$} coordinate (N);\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (0,16) node [above] {$O'$} coordinate (O');\n\\draw (-6,16) node [above] {$C$} coordinate (C);\n\\draw (-6,9) node [left] {$D$} coordinate (D);\n\\draw (2,16) node [above] {$E$} coordinate (E);\n\\draw (2,12) node [right] {$F$} coordinate (F);\n\\draw [ultra thick] (A) -- (B) (C) -- (D) (E) -- (F);\n\\draw (M) -- (N);\n\\draw [dashed] (O) -- (O');\n\\draw [domain = -8:0] plot (\\x,{0.25*pow(\\x,2)});\n\\draw [domain = 0:4.2] plot (\\x,{16-pow(\\x-4,2)});\n\\draw [dashed] (D) --++ (0,-9) node [midway,left] {$h_1$} coordinate (h_1) (D) --++ (6,0) node [midway,above] {$a$} coordinate (a);\n\\draw [dashed] (F) --+ (0,-12) node [midway,right] {$h_2$} coordinate (h_2) (F) --++ (-2,0) node [midway,above] {$b$} coordinate (b);\n\\end{tikzpicture}\n\\end{center}\n(1) 求谷底$O$到桥面$AB$的距离和桥$AB$的长度;\\\\\n(2) 计划在谷底两侧建造平行于$OO'$的桥墩$CD$和$EF$, 且$CE$为$80$米, 其中$C$、$E$在$AB$上(不包括端点), 桥墩$EF$每米造价为$k$(万元)、桥墩$CD$每米造价为$\\dfrac 32 k$(万元)($k>0$). 问$O'E$为多少米时, 桥墩$CD$与$EF$的总造价最低?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $120$米; (2) $20$米时, 总造价最低", "solution": "", @@ -303410,7 +303889,9 @@ "id": "012306", "content": "已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的长轴长为$2 \\sqrt 3$, 离心率为$\\dfrac{\\sqrt 6}3$, 斜率为$k$的直线$l$与椭圆$\\Gamma$有两个不同的交点$A$、$B$.\\\\\n(1) 求椭圆$\\Gamma$的方程;\\\\\n(2) 若直线$l$的方程为$y=x+t$, 椭圆上点$M(-\\dfrac 32, \\dfrac 12)$关于直线$l$的对称点$N$(与$M$不重合)在椭圆$\\Gamma$上, 求$t$的值;\\\\\n(3) 设$P(-2,0)$, 直线$PA$与椭圆$\\Gamma$的另一个交点为$C$, 直线$PB$与椭圆$\\Gamma$的另一个交点为$D$、若点$C$、$D$和点$Q(-\\dfrac 74, \\dfrac 12)$三点共线, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}3+y^2=1$; (2) $\\dfrac 12$; (3) $2$", "solution": "", @@ -303429,7 +303910,10 @@ "id": "012307", "content": "己知定义在$\\mathbf{R}$上的函数$f(x)=\\mathrm{e}^{k x+b}$($\\mathrm{e}$是自然对数的底数) 满足$f(x)=f'(x)$且$f(-1)=1$, 删除无穷数列$f(1)$、$f(2)$、$f(3)$、$\\cdots$、$f(n)$、$\\cdots$中的第$3$项、第$6$项、$\\cdots$、第$3n$项, $\\cdots$, ($n \\in \\mathbf{N}$, $n\\ge 1$), 余下的项按原来顺序组成一个新数列$\\{t_n\\}$, 记数列$\\{t_n\\}$前$n$项和为$T_n$.\\\\\n(1) 求函数$f(x)$的解析式;\\\\\n(2) 已知数列$\\{t_n\\}$的通项公式是$t_n=f(g(n))$, $n \\in \\mathbf{N}$, $n\\ge 1$, 求函数$g(n)$的解析式;\\\\\n(3) 设集合$X$是实数集$\\mathbf{R}$的非空子集, 如果正实数$a$满足: 对任意$x_1$、$x_2 \\in X$, 都有$|x_1-x_2|\\leq a$, 则称$a$为集合$X$的一个``阈度'', 记集合$H=\\{w | w=\\dfrac{T_n}{f(\\dfrac{3 n}2-\\dfrac{1+3(-1)^n}4)}, \\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$, 试问集合$H$存在``阈度''吗? 若存在, 求出集合$H$``阈度''的取值范围, 若不存在, 试说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "(1) $f(x)=\\mathrm{e}^{x+1}$; (2) $g(n)=\\begin{cases}\\dfrac{3n-1}{2}, & n\\text{为奇数},\\\\ \\dfrac{3n-2}{2}, & n\\text{为偶数};\\end{cases}$ (3) 存在``阈度'', 取值范围为$[\\dfrac{\\mathrm{e}^4+1}{\\mathrm{e}^4-\\mathrm{e}},+\\infty)$", "solution": "", @@ -303449,7 +303933,9 @@ "id": "012308", "content": "已知集合$A=\\{x | 00$, $a \\neq 1$)的图像经过点$(4,2)$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -303525,7 +304017,9 @@ "id": "012312", "content": "设等比数列$\\{a_n\\}$满足$a_1+a_2=-1$, $a_1-a_3=-3$, 则$a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "$-8$", "solution": "", @@ -303544,7 +304038,9 @@ "id": "012313", "content": "已知方程组$\\begin{cases}x+m y=2, \\\\ m x+16 y=8\\end{cases}$无解, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "$-4$", "solution": "", @@ -303563,7 +304059,9 @@ "id": "012314", "content": "已知角$\\alpha$的终边与单位圆$x^2+y^2=1$交于点$P(\\dfrac 12, y)$, 则$\\sin (\\dfrac{\\pi}2+\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$\\dfrac 12$", "solution": "", @@ -303582,7 +304080,9 @@ "id": "012315", "content": "将半径为$2$的半圆形纸片卷成一个无盖的圆锥筒, 则该圆锥筒的高为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "$\\sqrt{3}$", "solution": "", @@ -303601,7 +304101,9 @@ "id": "012316", "content": "已知函数$f(x)=x^2$, 则曲线$y=f(x)$在点$P(1,1)$处的切线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "$y=2x-1$", "solution": "", @@ -303620,7 +304122,9 @@ "id": "012317", "content": "设函数$f(x)=\\sin (\\omega x-\\dfrac{\\pi}6)+k$($\\omega>0$), 若$f(x) \\leq f(\\dfrac{\\pi}3)$对任意的实数$x$都成立, 则$\\omega$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "$2$", "solution": "", @@ -303639,7 +304143,9 @@ "id": "012318", "content": "在边长为$2$的正六边形$ABCDEF$中, 点$P$为其内部或边界上一点, 则$\\overrightarrow{AD} \\cdot \\overrightarrow{BP}$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "$[-4,12]$", "solution": "", @@ -303658,7 +304164,9 @@ "id": "012319", "content": "已知椭圆$\\Gamma_1$与双曲线$\\Gamma_2$的离心率互为倒数, 且它们有共同的焦点$F_1$、$F_2$, $P$是$\\Gamma_1$与$\\Gamma_2$在第一象限的交点, 当$\\angle F_1PF_2=\\dfrac{\\pi}6$时, 双曲线$\\Gamma_2$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "$2+\\sqrt{3}$", "solution": "", @@ -303677,7 +304185,9 @@ "id": "012320", "content": "下列函数中, 既是奇函数又在区间$(0,1)$上是严格增函数的是\\bracket{20}.\n\\fourch{$y=\\sqrt x$}{$y=-x^3$}{$y=\\lg x$}{$y=\\sin x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "D", "solution": "", @@ -303696,7 +304206,9 @@ "id": "012321", "content": "设$x \\in \\mathbf{R}$, 则``$x+\\dfrac 1x>2$''是``$x \\neq 1$''的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -303715,7 +304227,9 @@ "id": "012322", "content": "设函数$f(x)=\\sin (x-\\dfrac{\\pi}6)$, 若对于任意$\\alpha \\in[-\\dfrac{5 \\pi}6,-\\dfrac{\\pi}2]$, 在区间$[0, m]$上总存在唯一确定的$\\beta$, 使得$f(\\alpha)+f(\\beta)=0$, 则$m$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}6$}{$\\dfrac{\\pi}2$}{$\\dfrac{7 \\pi}6$}{$\\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "B", "solution": "", @@ -303734,7 +304248,9 @@ "id": "012323", "content": "已知曲线$C:(x^2+y^2)^3=16 x^2 y^2$, 命题$p$: 曲线$C$仅过一个横坐标与纵坐标都是整数的点; 命题$q$: 曲线$C$上的点到原点的最大距离是$2$, 则下列说法正确的是\\bracket{20}.\n\\twoch{$p$、$q$都是真命题}{$p$是真命题, $q$是假命题}{$p$是假命题, $q$是真命题}{$p$、$q$都是假命题}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "A", "solution": "", @@ -303753,7 +304269,9 @@ "id": "012324", "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=\\sqrt 2$, $A_1C$与底面$ABCD$所成角为$45^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{{sqrt(2)}}\n\\def\\m{{sqrt(2)}}\n\\def\\n{2}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (D1) -- (B1) (A1) -- (B);\n\\draw [dashed] (A) -- (D1) (A1) -- (D) (A1) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求四棱锥$A_1-ABCD$的体积;\\\\\n(2) 求异面直线$A_1B$与$B_1D_1$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac 43$; (2) $\\arccos \\dfrac{\\sqrt{6}}6$", "solution": "", @@ -303772,7 +304290,9 @@ "id": "012325", "content": "已知函数$f(x)=\\sin x \\cos x-\\sin ^2 x+\\dfrac 12$.\\\\\n(1) 求$f(x)$的单调递增区间;\\\\\n(2) 在$\\triangle ABC$中, $a$、$b$、$c$为角$A$、$B$、$C$的对边, 且满足$b \\cos 2 A=b \\cos A-a \\sin B$, 且$0=latex,scale = 0.03]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (100,0) node [below right] {$A$} coordinate (A);\n\\draw (0,100) node [above left] {$B$} coordinate (B);\n\\draw (50,50) node [above right] {$C$} coordinate (C);\n\\draw (C) -- (A) (A) -- (O) -- (B);\n\\draw [domain = 0:50, samples = 100] plot (\\x,{100-\\x*\\x/50});\n\\draw (40,68) node [above right] {$D$} coordinate (D);\n\\draw (D) -- ($(O)!(D)!(A)$) node [below] {$E$} coordinate (E);\n\\draw (D) -- ($(O)!(D)!(B)$) node [left] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 试建立平面直角坐标系, 求曲线段$BC$的方程;\\\\\n(2) 求面积$S$关于$x$的函数解析式$S=f(x)$;\\\\\n(3) 试确定点$D$的位置, 使得游乐场的面积$S$最大.(结果精确到$0.1$米)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "(1) $y=-0.02x^2+100$($0\\le x\\le 50$); (2) $f(x)=\\begin{cases}x(-0.02x^2+100), & 30\\le x\\le 50, \\\\ x(-x+100), & 501$)的右焦点为$F$, 左右顶点分别为$A$、$B$, 直线$l$过点$B$且与$x$轴垂直, 点$P$是椭圆上异于$A$、$B$的点, 直线$AP$交直线$l$于点$D$.\\\\\n(1) 若$E$是椭圆的上顶点, 且$\\triangle AEF$是直角三角形, 求椭圆的标准方程;\\\\\n(2) 若$a=2$, $\\angle PAB=45^{\\circ}$, 求$\\triangle PAF$的面积;\\\\\n(3) 判断以$BD$为直径的圆与直线$PF$的位置关系, 并加以证明.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "(1) $\\dfrac{x^2}{\\frac{1+\\sqrt{5}}2}+y^2=1$; (2) $\\dfrac{4+2\\sqrt{3}}5$; (3) 相切, 证明略", "solution": "", @@ -303829,7 +304353,9 @@ "id": "012328", "content": "己知数列$\\{a_n\\}$满足$|a_i-a_{i+1}|\\leq|a_{i+1}-a_{i+2}|$($i=1,2, \\cdots, n-2$).\\\\\n(1) 若数列$\\{a_n\\}$的前$4$项分别为$4$、$2$、$a_3$、$1$, 求$a_3$的取值范围;\\\\\n(2) 已知数列$\\{a_n\\}$中各项互不相同, 令$b_m=|a_m-a_{m+1}|$($m=1,2, \\cdots, n-1$), 求证: 数列$\\{a_n\\}$是等差数列的充要条件是数列$\\{b_m\\}$是常数列;\\\\\n(3) 已知数列$\\{a_n\\}$是$m$($m \\in \\mathbf{N}$且$m \\geq 3$)个连续正整数$1,2, \\cdots, m$的一个排列, 若$\\displaystyle\\sum_{k=1}^{m-1}|a_k-a_{k+1}|=m+2$, 求$m$的所有取值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "(1) $[4,+\\infty)$; (2) 证明略; (3) $4$或$5$", "solution": "", @@ -303848,7 +304374,9 @@ "id": "012329", "content": "已知集合$A=\\{1,2,k\\}, B=\\{2,5\\}$, 若$A \\cup B=\\{1,2,3,5\\}$, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303867,7 +304395,9 @@ "id": "012330", "content": "函数$y=\\sqrt {x+1}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303886,7 +304416,9 @@ "id": "012331", "content": "抛物线$y^2=8 x$的焦点坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303905,7 +304437,9 @@ "id": "012332", "content": "若复数$z$满足$\\mathrm{i} z=1+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303924,7 +304458,9 @@ "id": "012333", "content": "函数$f(x)=\\sin (2 x+\\dfrac{\\pi}4)$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303943,7 +304479,9 @@ "id": "012334", "content": "方程$4^x-2^{x+1}=0$的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303962,7 +304500,9 @@ "id": "012335", "content": "若$(2 x-1)^5=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+a_5 x^5$, 则$a_0+a_1+a_2+a_3+a_4+a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -303981,7 +304521,9 @@ "id": "012336", "content": "若$f(x)=\\dfrac{(x+2)(x+m)}x$为奇函数, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304000,7 +304542,9 @@ "id": "012337", "content": "函数$y=\\log_2 x+\\dfrac 4{\\log_2 x} (x \\in[2,4])$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304019,7 +304563,9 @@ "id": "012338", "content": "若复数$z$满足$|z-\\mathrm{i}|\\leq \\sqrt 2$($\\mathrm{i}$为虚数单位), 则$z$在复平面内所对应的图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304038,7 +304584,9 @@ "id": "012339", "content": "某校要从$2$名男生和$4$名女生中选出$4$人担任某游泳赛事的志愿者工作, 则在选出的志愿者中, 男、女都有的概率为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304057,7 +304605,9 @@ "id": "012340", "content": "若不等式$x^2-k x+k-1>0$对$x \\in(1,2)$恒成立, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304076,7 +304626,10 @@ "id": "012341", "content": "已知等差数列$\\{a_n\\}$的首项及公差均为正数, 令$b_n=\\sqrt {a_n}+\\sqrt {a_{2012-n}}$($n \\in \\mathbf{N}$, $1\\le n<2012$), 当$b_k$是数列$\\{b_n\\}$的最大项时, $k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元", + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304095,7 +304648,9 @@ "id": "012342", "content": "若矩阵$\\begin{pmatrix}a_{11} & a_{12} \\\\a_{21} & a_{22}\\end{pmatrix}$满足: $a_{11}$、$a_{12}$、$a_{21}$、$a_{22} \\in\\{-1,1\\}$, 且$\\begin{vmatrix}a_{11} & a_{12} \\\\a_{21} & a_{22}\\end{vmatrix}=0$, 则这样的互不相等的矩阵共有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304114,7 +304669,9 @@ "id": "012343", "content": "已知粗圆$C_1: \\dfrac{x^2}{12}+\\dfrac{y^2}4=1$, $C_2: \\dfrac{x^2}{16}+\\dfrac{y^2}8=1$, 则\\bracket{20}.\n\\fourch{$C_1$与$C_2$顶点相同}{$C_1$与$C_2$长轴长相同}{$C_1$与$C_2$短轴长相同}{$C_1$与$C_2$焦距相等}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304133,7 +304690,9 @@ "id": "012344", "content": "记函数$y=f(x)$的反函数为$y=f^{-1}(x)$, 如果函数$y=f(x)$的图像过点$(1,0)$, 那么函数$y=f^{-1}(x)+1$的图像过点\\bracket{20}.\n\\fourch{$(0,0)$}{$(0,2)$}{$(1,1)$}{$(2,0)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304152,7 +304711,9 @@ "id": "012345", "content": "已知空间三条直线$l$、$m$、$n$, 若$l$与$m$异面, 且$l$与$n$异面, 则\\bracket{20}.\n\\twoch{$m$与$n$异面}{$m$与$n$相交}{$m$与$n$平行}{$m$与$n$异面、相交、平行均有可能}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304171,7 +304732,9 @@ "id": "012346", "content": "设$O$为$\\triangle ABC$所在平面上一点, 若实数$x$、$y$、$z$满足$x \\overrightarrow{OA}+y \\overrightarrow{OB}+z \\overrightarrow{OC}=\\overrightarrow 0$($x^2+y^2+z^2 \\neq 0$), 则``$xyz=0$''是``点$O$在$\\triangle ABC$的边所在直线上''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304190,7 +304753,9 @@ "id": "012347", "content": "如图, 正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, $M$为线段$AB$的中点, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.3]\n\\def\\l{1}\n\\def\\m{1}\n\\def\\n{2}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (M) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥$C_1-MBC$的体积;\\\\\n(2) 异面直线$CD$与$MC_1$所成角的大小. (结果用反三角函数值表示)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304209,7 +304774,9 @@ "id": "012348", "content": "某环线地铁按内、外环线同时运行, 内、外环线的长均为$30$千米. (忽略内、外环线长度差异)\\\\\n(1) 当$9$列列车同时在内环线上运行时, 要使内环线乘客最长候车时间为$10$分钟, 求内环线列车的最小平均速度;\\\\\n(2) 新调整的方案要求内环线列车平均速度为$25$千米/小时, 外环线列车平均速度为$30$千米/小时, 现内、外环线共有$18$列列车全部投入运行, 要使内、外环线乘客的最长候车时间之差不超过$1$分钟, 问: 内、外环线应名投入几列列车运行?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304228,7 +304795,9 @@ "id": "012349", "content": "已知双曲线$C_1: x^2-\\dfrac{y^2}4=1$.\\\\\n(1) 求与双曲线$C_1$有相同的焦点, 且过点$P(4, \\sqrt 3)$的双曲线$C_2$的标准方程;\\\\\n(2) 直线$l: y=x+m$分别交双曲线$C_1$的两条渐近线于$A$、$B$两点, 当$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$时, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304247,7 +304816,9 @@ "id": "012350", "content": "已知数列$\\{a_n\\}$、$\\{b_n\\}$、$\\{c_n\\}$满足$(a_{n+1}-a_n)(b_{n+1}-b_n)=c_n$($n \\in \\mathbf{N}$, $n\\ge 1$).\\\\\n(1) 设$c_n=3 n+6$, $\\{a_n\\}$是公差为$3$的等差数列, 当$b_1=1$时, 求$b_2$、$b_3$的值;\\\\\n(2) 设$c_n=n^3$, $a_n=n^2-8 n$, 求正整数$k$, 使得一切$n \\in \\mathbf{N}$, $n\\ge 1$, 均有$b_n \\geq b_k$;\\\\\n(3) 设$c_n=2^n+n$, $a_n=\\dfrac{1+(-1)^n}2$, 当$b_1=1$时, 求数列$\\{b_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304266,7 +304837,9 @@ "id": "012351", "content": "定义向量$\\overrightarrow{OM}=(a, b)$的``相伴函数''为$f(x)=a \\sin x+b \\cos x$; 函数$f(x)=a \\sin x+b \\cos x$的``相伴向量''为$\\overrightarrow{OM}=(a, b)$(其中$O$为坐标原点), 记平面内所有向量的``相伴函数''构成的集合为$S$.\\\\\n(1) 设$g(x)=3 \\sin (x+\\dfrac{\\pi}2)+4 \\sin x$, 求证: $g(x) \\in S$;\\\\\n(2) 已知$h(x)=\\cos (x+\\alpha)+2 \\cos x$, 且$h(x) \\in S$, 求其``相伴向量''的模;\\\\\n(3) 已知$M(a, b)$($b \\neq 0$)为圆$C:(x-2)^2+y^2=1$上一点, 向量$\\overrightarrow{OM}$的``相伴函数''$f(x)$在$x=x_0$处取得最大值, 当点$M$在圆$C$上运动时, 求$\\tan 2 x_0$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304285,7 +304858,9 @@ "id": "012352", "content": "函数$y=\\log_2(x+2)$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304304,7 +304879,9 @@ "id": "012353", "content": "方程$2^x=8$的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304323,7 +304900,9 @@ "id": "012354", "content": "抛物线$y^2=8 x$的准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304342,7 +304921,9 @@ "id": "012355", "content": "函数$y=2 \\sin x$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304361,7 +304942,9 @@ "id": "012356", "content": "已知向量$\\overrightarrow a=(1, k)$, $\\overrightarrow b=(9, k-6)$, 若$\\overrightarrow a \\parallel \\overrightarrow b$, 则实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304380,7 +304963,9 @@ "id": "012357", "content": "函数$y=4 \\sin x+3 \\cos x$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304399,7 +304984,9 @@ "id": "012358", "content": "复数$2+3 \\mathrm{i}$($\\mathrm{i}$是虚数单位)的模是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304418,7 +305005,9 @@ "id": "012359", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对边长分别为$a$、$b$、$c$, 若$a=5$, $c=8$, $B=60^{\\circ}$, 则$b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304437,7 +305026,9 @@ "id": "012360", "content": "在如图所示的正方体$ABCD-A_1B_1C_1D_1$中, 异面直线$A_1B$与$B_1C$所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{1.5}\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,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304456,7 +305047,9 @@ "id": "012361", "content": "从$4$名男同学和$6$名女同学中随机选取$3$人参加某社团活动, 选出的$3$人中男女同学都有的概率为\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304475,7 +305068,9 @@ "id": "012362", "content": "若等差数列的前$6$项和为$23$, 前$9$项和为$57$, 则数列的前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304494,7 +305089,9 @@ "id": "012363", "content": "$36$的所有正约数之和可按如下方法得到: 因为$36=2^2 \\times 3^2$, 所以$36$的所有正约数之和为$(1+3+3^2)+(2+2 \\times 3+2 \\times 3^2)+(2^2+2^2 \\times 3+2^2 \\times 3^2)=(1+2+2^2)(1+3+3^2)=91$, 参照上述方法, 可求得 $2000$的所有正约数之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304513,7 +305110,9 @@ "id": "012364", "content": "展开式为$a d-b c$的行列式是\\bracket{20}.\n\\fourch{$\\begin{vmatrix}a & b \\\\d & c\\end{vmatrix}$}{$\\begin{vmatrix}a & c \\\\b & d\\end{vmatrix}$}{$\\begin{vmatrix}a & d \\\\b & c\\end{vmatrix}$}{$\\begin{vmatrix}b & a \\\\d & c\\end{vmatrix}$}", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304532,7 +305131,9 @@ "id": "012365", "content": "设$f^{-1}(x)$为函数$f(x)=\\sqrt x$的反函数, 下列结论正确的是\\bracket{20}.\n\\fourch{$f^{-1}(2)=2$}{$f^{-1}(2)=4$}{$f^{-1}(4)=2$}{$f^{-1}(4)=4$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304551,7 +305152,9 @@ "id": "012366", "content": "直线$2 x-3 y+1=0$的一个方向向量是\\bracket{20}.\n\\fourch{$(2,-3)$}{$(2,3)$}{$(-3,2)$}{$(3,2)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304570,7 +305173,9 @@ "id": "012367", "content": "函数$f(x)=x^{-\\frac 12}$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.12:3] plot (\\x,{pow(\\x,-0.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3] plot (\\x,{pow(\\x,0.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-1.75,0) -- (1.75,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:1.75] plot (\\x,{pow(\\x,0.5)});\n\\draw [domain = 0:1.75] plot (-\\x,{pow(\\x,0.5)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-1.75,0) -- (1.75,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.12:1.75] plot (\\x,{pow(\\x,-0.5)});\n\\draw [domain = 0.12:1.75] plot (-\\x,{pow(\\x,-0.5)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -304589,7 +305194,9 @@ "id": "012368", "content": "如果$a=latex,scale = 0.4]\n\\def\\l{2*sqrt(3)}\n\\def\\h{6}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw (B) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304760,7 +305383,9 @@ "id": "012377", "content": "如图, 某校有一块形如直角三角形$ABC$的空地, 其中$\\angle B$为直角, $A B$长$40$米, $BC$长$50$米, 现欲在此空地上建造一间健身房, 其占地形状为矩形, 且$B$为矩形的一个顶点, 求该健身房的最大占地面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2.5,0) node [right] {$C$} coordinate (C);\n\\draw (0,2) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304779,7 +305404,9 @@ "id": "012378", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=-n^2+n$, 数列$\\{b_n\\}$满足$b_n=2^{a_n}$, 求$\\displaystyle \\lim_{n \\to \\infty}(b_1+b_2+\\cdots+b_n)$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304798,7 +305425,9 @@ "id": "012379", "content": "已知椭圆$C$的两个焦点分别为$F_1(-1,0)$、$F_2(1,0)$, 短轴的两个端点分别为$B_1$、$B_2$.\\\\\n(1) 若$\\triangle F_1B_1B_2$为等边三角形, 求椭圆$C$的方程;\\\\\n(2) 若椭圆$C$的短轴长为$2$, 过点$F_2$的直线$l$与椭圆$C$相交于$P$、$Q$两点, 且$\\overrightarrow{F_1P} \\perp \\overrightarrow{F_1Q}$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304817,7 +305446,9 @@ "id": "012380", "content": "已知抛物线$C: y^2=4 x$的焦点为$F$.\\\\\n(1) 点$A$、$P$满足$\\overrightarrow{AP}=-2 \\overrightarrow{FA}$, 当点$A$在抛物线$C$上运动时, 求动点$P$的轨迹方程;\\\\\n(2) 在$x$轴上是否存在点$Q$, 使得点$Q$关于直线$y=2 x$的对称点在抛物线$C$上? 如果存在, 求所有满足条件的点$Q$的坐标; 如果不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304836,7 +305467,9 @@ "id": "012381", "content": "在平面直角坐标系$xOy$中, 点$A$在$y$轴正半轴上, 点$P_n$在$x$轴上, 其横坐标为$x_n$, 且$\\{x_n\\}$是首项为$1$、公比为$2$的等比数列, 记$\\angle P_nAP_{n+1}=\\theta_n$, $n \\in \\mathbf{N}$, $n\\ge 1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.35]\n\\draw [->] (-1,0) -- (11,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,3,4} {\\draw ({pow(2,\\i-1)},0) -- (0,4); \\draw ({pow(2,\\i-1)},0) node [below] {$P_\\i$};};\n\\draw (10,0) node [below] {$\\cdots$};\n\\draw (0,4) node [left] {$A$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\theta_3=\\arctan \\dfrac 13$, 求点$A$的坐标;\\\\\n(2) 若点$A$的坐标为$(0,8 \\sqrt 2)$, 求$\\theta_n$的最大值及相应$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304855,7 +305488,9 @@ "id": "012382", "content": "已知真命题: ``函数$y=f(x)$的图像关于点$P(a, b)$成中心对称图形''的充要条件为``函数$y=f(x+a)-b$是奇函数''.\n(1) 将函数$g(x)=x^3-3 x^2$的图像向左平移$1$个单位, 再向上平移$2$个单位, 求此时图像对应的函数解析式, 并利用题设中的真命题求函数$g(x)$图像对称中心的坐标;\\\\\n(2) 求函数$h(x)=\\log_2 \\dfrac{2 x}{4-x}$图像对称中心的坐标;\\\\\n(3) 已知命题: ``函数$y=f(x)$的图像关于某直线成轴对称图形''的充要条件为``存在实数$a$和$b$, 使得函数$y=f(x+a)-b$是偶函数'' , 判断该命题的真假, 如果是真命题, 请给予证明; 如果是假命题, 请说明理由, 并类比题设的真命题对它进行修改, 使之成为真命题(不必证明).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -304874,7 +305509,9 @@ "id": "012383", "content": "若$4^x=16$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304893,7 +305530,9 @@ "id": "012384", "content": "计算:$\\mathrm{i}(1+\\mathrm{i})=$\\blank{50}.($\\mathrm{i}$为虚数单位)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304912,7 +305551,9 @@ "id": "012385", "content": "$1$、$1$、$2$、$2$、$5$这五个数的中位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304931,7 +305572,9 @@ "id": "012386", "content": "若函数$f(x)=x^3+a$为奇函数, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304950,7 +305593,9 @@ "id": "012387", "content": "点$O(0,0)$到直线$x+y-4=0$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304969,7 +305614,9 @@ "id": "012388", "content": "函数$y=\\dfrac 1{x+1}$的反函数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -304988,7 +305635,9 @@ "id": "012389", "content": "已知等差数列$\\{a_n\\}$的首项为$1$, 公差为$2$, 则该数列的前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305007,7 +305656,9 @@ "id": "012390", "content": "已知$\\cos \\alpha=\\dfrac 13$, 则$\\cos 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305026,7 +305677,9 @@ "id": "012391", "content": "已知$a$、$b \\in (0,+\\infty)$, 若$a+b=1$, 则$ab$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305045,7 +305698,9 @@ "id": "012392", "content": "在$10$件产品中, 有$3$件次品, 从中随机取出$5$件, 则恰含$1$件次品的概率是\\blank{50}.(结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305064,7 +305719,9 @@ "id": "012393", "content": "某货船在$O$处看灯塔$M$在北偏东$30^{\\circ}$方向, 它以每小时$18$海里的速度向正北方向航行, 经过$40$分钟到达$B$处, 看到灯塔$M$在北偏东$75^{\\circ}$方向, 此时货船到灯塔$M$的距离为\\blank{50}海里.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305083,7 +305740,9 @@ "id": "012394", "content": "已知函数$f(x)=\\dfrac{x-2}{x-1}$与$g(x)=m x+1-m$的图像相交于$A$、$B$两点, 若动点$P$满足$|\\overrightarrow{PA}+\\overrightarrow{PB}|=2$, 则$P$的轨迹方程为\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305102,7 +305761,9 @@ "id": "012395", "content": "两条异面直线所成的角的范围是\\bracket{20}.\n\\fourch{$(0, \\dfrac{\\pi}2)$}{$(0, \\dfrac{\\pi}2]$}{$[0, \\dfrac{\\pi}2)$}{$[0, \\dfrac{\\pi}2]$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305121,7 +305782,9 @@ "id": "012396", "content": "复数$2+\\mathrm{i}$($\\mathrm{i}$为虚数单位) 的共轭复数为 \\bracket{20}.\n\\fourch{$2-\\mathrm{i}$}{$-2+i$}{$-2-\\mathrm{i}$}{$1+2 \\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305140,7 +305803,9 @@ "id": "012397", "content": "如图是下列函数中某个函数的部分图像, 则该函数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = -1:4,samples = 100] plot (\\x,{sin(\\x/pi*360)});\n\\draw [dashed] ({pi/4},1) -- (0,1) node [left] {$1$};\n\\draw [dashed] ({3*pi/4},-1) -- (0,-1) node [left] {$-1$};\n\\draw ({pi/2},0) node [below left] {$\\frac\\pi 2$};\n\\draw (pi,0) node [below right] {$\\pi$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$y=\\sin x$}{$y=\\sin 2 x$}{$y=\\cos x$}{$y=\\cos 2 x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305159,7 +305824,9 @@ "id": "012398", "content": "在$(x+1)^4$的二项展开式中,$x^2$项的系数为\\bracket{20}.\n\\fourch{$6$}{$4$}{$2$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305178,7 +305845,10 @@ "id": "012399", "content": "下列函数中, 在$\\mathbf{R}$上为增函数的是\\bracket{20}.\n\\fourch{$y=x^2$}{$y=|x|$}{$y=\\sin x$}{$y=x^3$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305197,7 +305867,9 @@ "id": "012400", "content": "$\\begin{vmatrix}\\cos \\theta & -\\sin \\theta \\\\\\sin \\theta & \\cos \\theta\\end{vmatrix}=$\\bracket{20}.\n\\fourch{$\\cos 2 \\theta$}{$\\sin 2 \\theta$}{$1$}{$-1$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305216,7 +305888,9 @@ "id": "012401", "content": "设$x_0$为函数$f(x)=2^x+x-2$的零点, 则$x_0 \\in$\\bracket{20}.\n\\fourch{$(-2,-1)$}{$(-1,0)$}{$(0,1)$}{$(1,2)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305235,7 +305909,9 @@ "id": "012402", "content": "若$a>b$, $c \\in \\mathbf{R}$, 则下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$\\dfrac 1a<\\dfrac 1b$}{$a^2>b^2$}{$a|c|>b|c|$}{$\\dfrac a{c^2+1}>\\dfrac b{c^2+1}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305254,7 +305930,9 @@ "id": "012403", "content": "若两个球的体积之比为$8: 27$, 则它们的表面积之比为\\bracket{20}.\n\\fourch{$2: 3$}{$4: 9$}{$8: 27$}{$2 \\sqrt 2: 3 \\sqrt 3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305273,7 +305951,9 @@ "id": "012404", "content": "已知数列$\\{a_n\\}$是以$q$为公比的等比数列, 若$b_n=-2 a_n$, 则数列$\\{b_n\\}$是\\bracket{20}.\n\\twoch{以$q$为公比的等比数列}{以$-q$为公比的等比数列}{以$2 q$为公比的等比数列}{以$-2 q$为公比的等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305292,7 +305972,9 @@ "id": "012405", "content": "若点$P$的坐标为$(a, b)$, 曲线$C$的方程为$F(x, y)=0$, 则``$F(a, b)=0$''是``点$P$在曲线$C$上''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305311,7 +305993,10 @@ "id": "012406", "content": "如图, 在底面半径和高均为$1$的圆锥中, $AB$、$CD$是底面圆$O$的两条互相垂直的直径, $E$是母线$PB$的中点, 已知过$CD$与$E$的平面与圆锥侧面的交线是以$E$为顶点的抛物线的一部分, 则该抛物线的焦点到圆锥顶点$P$的距离为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\def\\r{1}\n\\def\\h{1}\n\\draw ({-\\r},0,0) node [left] {$A$} coordinate (A) -- (0,\\h,0) node [above] {$P$} coordinate (P) -- (\\r,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,0) node [below left] {$O$} coordinate (O);\n\\draw (A) arc (180:360:{\\r} and {\\r/4});\n\\draw [dashed] (A) arc (180:0:{\\r} and {\\r/4});\n\\draw [dashed] (A) -- (B) (O) -- (P);\n\\draw ($(P)!0.5!(B)$) node [above right] {$E$} coordinate (E);\n\\draw [dashed] (O) -- (E);\n\\draw ({\\r*cos(-70)},{\\r/4*sin(-70)}) node [below] {$C$} coordinate (C);\n\\draw ({\\r*cos(110)},{\\r/4*sin(110)}) node [below] {$D$} coordinate (D);\n\\draw (C) .. controls +({\\r/10},{\\r/10}) and +({\\r*cos(-70)/3},{\\r/4*sin(-70)/3}) .. (E);\n\\draw [dashed] (D) .. controls +({\\r/10},{\\r/10}) and +({-\\r*cos(-70)/3},{-\\r/4*sin(-70)/3}) .. (E) (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$\\dfrac{\\sqrt 3}2$}{$\\dfrac{\\sqrt 6}2$}{$\\dfrac{\\sqrt{10}}4$}", "objs": [], - "tags": [], + "tags": [ + "第六单元", + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305330,7 +306015,10 @@ "id": "012407", "content": "已知不等式$\\dfrac{x-2}{x+1}<0$的解集为$A$, 函数$y=\\lg (x-1)$的定义域为集合$B$, 求$A \\cap B$.", "objs": [], - "tags": [], + "tags": [ + "第一单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305349,7 +306037,9 @@ "id": "012408", "content": "已知函数$f(x)=x^2-4 x+a, x \\in[-3,3]$, 若$f(1)=2$, 求$y=f(x)$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305368,7 +306058,9 @@ "id": "012409", "content": "如图, 在体积为$\\dfrac 13$的三棱锥$P-ABC$中, $PA$与平面$ABC$垂直, $AP=AB=1$, $\\angle BAC=\\dfrac{\\pi}2$, $E$、$F$分别是$PB$、$AB$的中点, 求异面直线$EF$与$PC$所成的角的大小. (结果用反三角函数值表示)\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (B) -- (P) -- (C);\n\\draw (B) -- (C);\n\\draw [dashed] (B) -- (A) -- (C) (A) -- (P);\n\\draw ($(A)!0.5!(B)$) node [right] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(B)$) node [above left] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305387,7 +306079,9 @@ "id": "012410", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1(a>1)$的左焦点为$F$, 上顶点为$B$.\\\\\n(1) 若直线$FB$的一个方向向量为$(1, \\dfrac{\\sqrt 3}3)$, 求实数$a$的值;\\\\\n(2) 若$a=\\sqrt 2$, 直线$l: y=k x-2$与椭圆$C$相交于$M$、$N$两点, 且$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=3$, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305406,7 +306100,10 @@ "id": "012411", "content": "已知数列$\\{a_n\\}$满足$a_n>0$, 双曲线$C_n: \\dfrac{x^2}{a_n}-\\dfrac{y^2}{a_{n+1}}=1$($n \\in \\mathbf{N}$, $n\\ge 1$).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {sqrt(2)}:4,samples = 100] plot (\\x,{sqrt(\\x*\\x-2)});\n\\draw [domain = {sqrt(2)}:4,samples = 100] plot (-\\x,{sqrt(\\x*\\x-2)});\n\\draw [domain = {sqrt(2)}:4,samples = 100] plot (\\x,{-sqrt(\\x*\\x-2)});\n\\draw [domain = {sqrt(2)}:4,samples = 100] plot (-\\x,{-sqrt(\\x*\\x-2)});\n\\draw (-4,-4) -- (4,4) (-4,4) -- (4,-4);\n\\draw (2,2) node [left] {$Q_n$} coordinate (Q_n);\n\\draw ({sqrt(6)},2) node [right] {$P_n$} coordinate (P_n);\n\\filldraw [pattern = north west lines] (O) -- (P_n) -- (Q_n);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$a_1=1$, $a_2=2$, 双曲线$C_n$的焦距为$2 c_n$, $c_n=\\sqrt {4 n-1}$, 求$\\{a_n\\}$的通项公式;\\\\\n(2) 如图, 在双曲线$C_n$的右支上取点$P_n(x_{P_n}, n)$, 过$P_n$作$y$轴的垂线, 在第一象限内交$C_n$的渐近线于点$Q_n$, 联结$O P_n$, 记$\\triangle OP_nQ_n$的面积为$S_n$, 若$\\displaystyle\\lim_{n \\to \\infty} a_n=2$, 求$\\displaystyle \\lim_{n \\to \\infty} S_n$.(关于数列极限的运算, 还可参考如下性质: 若$\\displaystyle \\lim_{n \\to \\infty} u_n=A(u_n \\geq 0)$, 则$\\displaystyle \\lim_{n \\to \\infty} \\sqrt {u_n}=\\sqrt A$)", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305425,7 +306122,9 @@ "id": "012412", "content": "已知直角三角形$A B C$的两直角边$A C$、$B C$的边长分别为$b$、$a$, 如图, 过$A C$边的$n$等分点$A_i$作$A C$边的垂线$d_i$, 过$B C$边的$n$等分点$B_i$和顶点$A$作直线$l_i$, 记$d_i$与$l_i$的交点为$P_i$($i=1,2, \\cdots, n-1$), 是否存在一条圆锥曲线, 对任意的正整数$n \\geq 2$, 点$P_i$($i=1,2, \\cdots, n-1$)都在这条曲线上? 说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$C$} coordinate (C);\n\\draw (4,3) node [right] {$B$} coordinate (B);\n\\draw (A) -- (C) -- (B) (A) -- (B);\n\\foreach \\i in {1,2,3,4,5} {\\draw ($(A)!{\\i/6}!(C)$) --++ (0,{0.5*\\i});};\n\\foreach \\i in {1,2,3,4,5} {\\draw ($(B)!{\\i/6}!(C)$) -- (A);};\n\\foreach \\i/\\j in {1/1,2/2,4/i} {\\draw ($(A)!{\\i/6}!(C)$) node [below] {$A_\\j$}; \\draw ($(C)!{\\i/6}!(B)$) node [right] {$B_\\j$};};\n\\foreach \\i/\\j in {3,5} {\\draw ($(A)!{\\i/6}!(C)$) node [below] {$\\cdots$}; \\draw ($(C)!{\\i/6}!(B)$) node [right] {$\\cdots$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305444,7 +306143,9 @@ "id": "012413", "content": "某人造卫星在地球赤道平面绕地球飞行, 甲、乙两个监测点分别位于赤道上东经$131^{\\circ}$和$147^{\\circ}$, 在某时刻测得甲监测点到卫星的距离为$1537.45$千米, 乙监测点到卫星的距离为$887.64$千米, 假设地球赤道是一个半径为$6378$千米的圆, 求此时卫星所在位置的高度(结果精确到$0.01$千米)和经度(结果精确到$0.01^{\\circ}$).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305463,7 +306164,9 @@ "id": "012414", "content": "如果存在非零常数$c$, 对于函数$y=f(x)$定义域$\\mathbf{R}$上的任意$x$, 都有$f(x+c)>f(x)$成立, 那么称函数为``$Z$函数''.\\\\\n(1) 求证: 若$y=f(x)$($x \\in \\mathbf{R}$)是单调函数, 则它是``$Z$函数'';\\\\\n(2) 若函数$g(x)=a x^3+b x^2$是``$Z$函数'', 求实数$a$、$b$满足的条件.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305482,7 +306185,9 @@ "id": "012415", "content": "设全集为$U=\\{1,2,3\\}$, 若集合$A=\\{1,2\\}$, 则$\\complement_UA=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305501,7 +306206,9 @@ "id": "012416", "content": "计算:$\\dfrac{1+\\mathrm{i}}{\\mathrm{i}}=$\\blank{50}.(其中$\\mathrm{i}$为虚数单位)", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305520,7 +306227,9 @@ "id": "012417", "content": "函数$y=\\sin (2 x+\\dfrac{\\pi}4)$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305539,7 +306248,9 @@ "id": "012418", "content": "计算:$\\displaystyle\\lim_{n \\to \\infty} \\dfrac{n^2-3}{2 n^2+n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305558,7 +306269,9 @@ "id": "012419", "content": "以$(2,6)$为圆心, $1$为半径的圆的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305577,7 +306290,9 @@ "id": "012420", "content": "已知向量$\\overrightarrow a=(1,3)$, $\\overrightarrow b=(m,-1)$, 若$\\overrightarrow a \\perp \\overrightarrow b$, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305596,7 +306311,9 @@ "id": "012421", "content": "函数$y=x^2-2 x+4, x \\in[0,2]$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305615,7 +306332,9 @@ "id": "012422", "content": "若线性方程组的增广矩阵为$\\begin{pmatrix}a & 0 & 2 \\\\0 & 1 & b\\end{pmatrix}$, 解为$\\begin{cases}x=2, \\\\y=1,\\end{cases}$ 则$a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305634,7 +306353,9 @@ "id": "012423", "content": "方程$\\lg (2 x+1)+\\lg x=1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305653,7 +306374,9 @@ "id": "012424", "content": "在$(x+\\dfrac 1{x^2})^9$的二项展开式中, 常数项的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305672,7 +306395,9 @@ "id": "012425", "content": "用数字组成无重复数字的三位数, 其中奇数的个数为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305691,7 +306416,9 @@ "id": "012426", "content": "已知点$A(1,0)$, 直线$l: x=-1$, 两个动圆均过点$A$且与$l$相切, 其圆心分别为$C_1$、$C_2$, 若动点$M$满足$2 \\overrightarrow{C_2M}=\\overrightarrow{C_2C_1}+\\overrightarrow{C_2A}$, 则$M$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -305710,7 +306437,9 @@ "id": "012427", "content": "若$a<0\\dfrac 1b$}{$-a>b$}{$a^2>b^2$}{$a^30$的解集为\\bracket{20}\n\\fourch{$(-\\infty, \\dfrac 34)$}{$(-\\infty, \\dfrac 23)$}{$(-\\infty, \\dfrac 23) \\cup(1,+\\infty)$}{$(\\dfrac 23, 1)$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305767,7 +306500,9 @@ "id": "012430", "content": "下列函数中, 是奇函数且在$(0,+\\infty)$上单调递增的为\\bracket{20}.\n\\fourch{$y=x^2$}{$y=x^{\\frac 13}$}{$y=x^{-1}$}{$y=x^{-\\frac 12}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305786,7 +306521,9 @@ "id": "012431", "content": "直线$3 x-4 y-5=0$的倾斜角为\\bracket{20}.\n\\fourch{$\\arctan \\dfrac 34$}{$\\pi-\\arctan \\dfrac 34$}{$\\arctan \\dfrac 43$}{$\\pi-\\arctan \\dfrac 43$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305805,7 +306542,9 @@ "id": "012432", "content": "底面半径为 1 , 母线长为 2 的圆锥的体积为\\bracket{20}.\n\\fourch{$2 \\pi$}{$\\sqrt 3 \\pi$}{$\\dfrac{2 \\pi}3$}{$\\dfrac{\\sqrt 3 \\pi}3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305824,7 +306563,9 @@ "id": "012433", "content": "以$(-3,0)$和$(3,0)$为焦点, 长轴长为$8$的椭圆方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}+\\dfrac{y^2}{25}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}7=1$}{$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}7+\\dfrac{y^2}{16}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305843,7 +306584,9 @@ "id": "012434", "content": "在复平面上, 满足$|z-1|=|z+\\mathrm{i}|$($\\mathrm{i}$为虚数单位) 的复数$z$对应的点的轨迹为\\bracket{20}.\n\\fourch{椭圆}{圆}{线段}{直线}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305862,7 +306605,9 @@ "id": "012435", "content": "若无穷等差数列$\\{a_n\\}$的首项$a_1>0$, 公差$d<0$, $\\{a_n\\}$的前$n$项和为$S_n$, 则\\bracket{20}.\n\\fourch{$S_n$单调递减}{$S_n$单调递增}{$S_n$有最大值}{$S_n$有最小值}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305879,9 +306624,11 @@ }, "012436": { "id": "012436", - "content": "已知$a>0$, $b>0$, 若$a+b=4$, 则\\bracket{20}。\n\\fourch{$a^2+b^2$有最小值}{$\\sqrt {a b}$有最小值}{$\\dfrac 1a+\\dfrac 1b$有最大值}{$\\dfrac 1{\\sqrt a+\\sqrt b}$有最大值}", + "content": "已知$a>0$, $b>0$, 若$a+b=4$, 则\\bracket{20}.\n\\fourch{$a^2+b^2$有最小值}{$\\sqrt {a b}$有最小值}{$\\dfrac 1a+\\dfrac 1b$有最大值}{$\\dfrac 1{\\sqrt a+\\sqrt b}$有最大值}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305900,7 +306647,9 @@ "id": "012437", "content": "组合数$\\mathrm{C}_n^m+2\\mathrm{C}_n^{m-1}+\\mathrm{C}_n^{m-2}$($n \\geq m \\geq 2$, $m, n \\in \\mathbf{N}$)恒等于\\bracket{20}.\n\\fourch{$\\mathrm{C}_{n+2}^m$}{$\\mathrm{C}_{n+2}^{m+1}$}{$\\mathrm{C}_{n+1}^m$}{$\\mathrm{C}_{n+1}^{m+1}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305919,7 +306668,9 @@ "id": "012438", "content": "设集合$P_1=\\{x | x^2+a x+1>0\\}, P_2=\\{x| x^2+a x+2>0\\}, Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a, b \\in \\mathbf{R}$, 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意的$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意的$b$, $Q_1$不是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -305938,7 +306689,9 @@ "id": "012439", "content": "如图, 在正四棱柱$ABCD-A_1B_1C_1D_1$中, $AB=1$, $D_1B$和平面$ABCD$所成的角的大小为$\\arctan \\dfrac{3 \\sqrt 2}4$, 求该四棱柱的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305957,7 +306710,9 @@ "id": "012440", "content": "已知$a$为实数, 函数$f(x)=\\dfrac{x^2+a x+4}x$是奇函数, 求$f(x)$在$(0,+\\infty)$上的最小值及取到最小值时所对应的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305976,7 +306731,9 @@ "id": "012441", "content": "某船在海平面$A$处测得灯塔$B$在北偏东$30^{\\circ}$方向, 与$A$相距$6.0$海里, 船由$A$向正北方向航行$8.1$海里到达$C$处, 这时灯塔$B$与船相距多少海里(精确到$0.1$海里)?$B$ 在船的什么方向(精确到$1^\\circ$)?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -305995,7 +306752,9 @@ "id": "012442", "content": "已知点$F_1$、$F_2$依次为双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a, b>0$)的左右焦点, $|F_1F_2|=6$, $B_1(0,-b)$, $B_2(0, b)$.\\\\\n(1) 若$a=\\sqrt 5$, 以$\\overrightarrow d=(3,-4)$为方向向量的直线$l$经过$B_1$, 求$F_2$到$l$的距离;\\\\\n(2) 若双曲线$C$上存在点$P$, 使得$\\overrightarrow{PB_1} \\cdot \\overrightarrow{PB_2}=-2$, 求实数$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306012,9 +306771,12 @@ }, "012443": { "id": "012443", - "content": "已知函数$f(x)=|2^{x-2}-2|$($x \\in \\mathbf{R}$).\n(1) 解不等式$f(x)<2$;\\\\\n(2) 数列$\\{a_n\\}$满足$a_n=f(n)$($n \\in \\mathbf{N}$, $n\\ge 1$), $S_n$为$\\{a_n\\}$的前$n$项和, 若对任意的$n \\geq 4$, 不等式$S_n+\\dfrac 12 \\geq k a_n$恒成立, 求实数$k$的取值范围.", + "content": "已知函数$f(x)=|2^{x-2}-2|$($x \\in \\mathbf{R}$).\\\\\n(1) 解不等式$f(x)<2$;\\\\\n(2) 数列$\\{a_n\\}$满足$a_n=f(n)$($n \\in \\mathbf{N}$, $n\\ge 1$), $S_n$为$\\{a_n\\}$的前$n$项和, 若对任意的$n \\geq 4$, 不等式$S_n+\\dfrac 12 \\geq k a_n$恒成立, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306033,7 +306795,9 @@ "id": "012444", "content": "对于集合$A$、$B, `` A \\neq B$''是``$A \\cap B \\subset A \\cup B$''的\\bracket{20}.\n\\fourch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306052,7 +306816,9 @@ "id": "012445", "content": "对于任意实数$a$、$b$, $(a-b)^2 \\geq k a b$均成立, 则实数$k$的取值范围是\\bracket{20}.\n\\twoch{$\\{-4,0\\}$}{$[-4,0]$}{$(-\\infty, 0]$}{$(-\\infty,-4] \\cup[0,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306071,7 +306837,9 @@ "id": "012446", "content": "已知数列$\\{a_n\\}$满足$a_n+a_{n+4}=a_{n+1}+a_{n+3}$($n \\in \\mathbf{N}$, $n\\ge 1$), 那么\\bracket{20}.\n\\fourch{$\\{a_n\\}$是等差数列}{$\\{a_{2 n-1}\\}$是等差数列}{$\\{a_{2 n}\\}$是等差数列}{$\\{a_{3 n}\\}$是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306090,7 +306858,10 @@ "id": "012447", "content": "关于$x$的实系数一元二次方程$x^2+p x+2=0$的两个虚数根为$z_1$、$z_2$, 若$z_1$、$z_2$在复平面上对应的点是经过原点的椭圆的两个焦点, 则该椭圆的长轴长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元", + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306109,7 +306880,9 @@ "id": "012448", "content": "已知圆心为$O$, 半径为$1$的圆上有三点$A$、$B$、$C$, 若$7 \\overrightarrow{OA}+5 \\overrightarrow{OB}+8 \\overrightarrow{OC}=\\overrightarrow 0$, 则$|\\overrightarrow{BC}|=$\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306128,7 +306901,9 @@ "id": "012449", "content": "函数$f(x)$与$g(x)$的图像拼成如图所示``$Z$''字形折线段$ABOCD$, 不含$A(0,1)$, $B(1,1)$, $O(0,0)$, $C(-1,-1)$, $D(0,-1)$五个点, 若$f(x)$的图像关于原点对称的图形即为$g(x)$的图像, 则其中一个函数的解析式可以为\\blank{50}.\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 right] {$O$} coordinate (O);\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (0,-1) node [below left] {$-1$} (0,1) node [below right] {$1$};\n\\draw (0,1) node [left] {$A$} coordinate (A) -- (1,1) node [right] {$B$} coordinate (B) -- (-1,-1) node [left] {$C$} coordinate (C) -- (0,-1) node [right] {$D$} coordinate (D);\n\\foreach \\i in {A,B,C,D,O} {\\filldraw [white] (\\i) circle (0.03); \\draw (\\i) circle (0.03);};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306147,7 +306922,10 @@ "id": "012450", "content": "对于函数$f(x)$、$g(x)$, 若存在函数$h(x)$, 使得$f(x)=g(x) \\cdot h(x)$, 则称$f(x)$是$g(x)$的``$h(x)$关联函数''.\\\\\n(1) 已知$f(x)=\\sin x$, $g(x)=\\cos x$, 是否存在定义域为$\\mathbf{R}$的函数$h(x)$, 使得$f(x)$是$g(x)$的``$h(x)$关联函数''? 若存在, 写出$h(x)$的解析式; 若不存在, 说明理由;\\\\\n(2) 已知函数$f(x)$、$g(x)$的定义域为$[1,+\\infty)$, 当$x \\in[n, n+1)$($n \\in \\mathbf{N}$, $n\\ge 1$)时, $f(x)=2^{n-1} \\sin \\dfrac xn-1$, 若存在函数$h_1(x)$及$h_2(x)$, 使得$f(x)$是$g(x)$的``$h_1(x)$关联函数'', 且$g(x)$是$f(x)$的``$h_2(x)$关联函数'', 求方程$g(x)=0$的解.", "objs": [], - "tags": [], + "tags": [ + "第三单元", + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306166,7 +306944,9 @@ "id": "012451", "content": "复数$3+4 \\mathrm{i}$($\\mathrm{i}$为虚数单位)的实部是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306185,7 +306965,9 @@ "id": "012452", "content": "若$\\log_2(x+1)=3$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306204,7 +306986,9 @@ "id": "012453", "content": "直线$y=x-1$与直线$y=2$的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306223,7 +307007,9 @@ "id": "012454", "content": "函数$f(x)=\\sqrt {x-2}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306242,7 +307028,9 @@ "id": "012455", "content": "三阶行列式$\\begin{vmatrix}1 & -3 & 5 \\\\4 & 0 & 0 \\\\-1 & 2 & 1\\end{vmatrix}$中, 元素$5$的代数余子式的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "暂无对应" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306261,7 +307049,9 @@ "id": "012456", "content": "函数$f(x)=\\dfrac 1x+a$的反函数的图像经过点$(2,1)$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306280,7 +307070,9 @@ "id": "012457", "content": "在$\\triangle ABC$中, 若$A=30^{\\circ}$, $B=45^{\\circ}$, $BC=\\sqrt 6$, 则$AC=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306299,7 +307091,9 @@ "id": "012458", "content": "$4$个人排成一排照相, 不同排列方式的种数为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306318,7 +307112,9 @@ "id": "012459", "content": "无穷等比数列$\\{a_n\\}$的首项为$2$, 公比为$\\dfrac 13$, 则$\\{a_n\\}$的各项和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306337,7 +307133,9 @@ "id": "012460", "content": "若$2+\\mathrm{i}$($\\mathrm{i}$为虚数单位)是关于$x$的实系数一元二次方程$x^2+a x+5=0$的一个虚根, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306356,7 +307154,9 @@ "id": "012461", "content": "函数$y=x^2-2 x+1$在区间$[0, m]$上的最小值为$0$, 最大值为$1$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306375,7 +307175,10 @@ "id": "012462", "content": "在平面直角坐标系$xOy$中, 点$A$、$B$是圆$x^2+y^2-6 x+5=0$上的两个动点, 且满足$|AB|=2 \\sqrt 3$, 则$|\\overrightarrow{OA}+\\overrightarrow{OB}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元", + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306394,7 +307197,9 @@ "id": "012463", "content": "满足$\\sin \\alpha>0$且$\\tan \\alpha<0$的角$\\alpha$属于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限;}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306413,7 +307218,9 @@ "id": "012464", "content": "半径为 1 的球的表面积为\\bracket{20}.\n\\fourch{$\\pi$}{$\\dfrac 43 \\pi$}{$2 \\pi$}{$4 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306432,7 +307239,9 @@ "id": "012465", "content": "在$(1+x)^6$的二项展开式中,$x^2$项的系数为\\bracket{20}.\n\\fourch{$2$}{$6$}{$15$}{$20$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306451,7 +307260,9 @@ "id": "012466", "content": "幂函数$y=x^{-2}$的大致图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {sqrt(2)/2}:2] plot (\\x,{pow(\\x,-2)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2] plot (\\x,{pow(\\x,{1/2})});\n\\draw [domain = 0:2] plot (-\\x,{pow(\\x,{1/2})});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {sqrt(2)/2}:2] plot (\\x,{pow(\\x,-2)});\n\\draw [domain = {sqrt(2)/2}:2] plot (-\\x,{pow(\\x,-2)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {sqrt(2)/2}:2] plot (\\x,{pow(\\x,-2)});\n\\draw [domain = {sqrt(2)/2}:2] plot (-\\x,{-pow(\\x,-2)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306470,7 +307281,9 @@ "id": "012467", "content": "已知向量$\\overrightarrow a=(1,0), \\overrightarrow b=(1,2)$, 则向量$\\overrightarrow b$在向量$\\overrightarrow a$方向上的数量投影为\\bracket{20}.\n\\fourch{$1$}{$2$}{$(1,0)$}{$(0,2)$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306489,7 +307302,9 @@ "id": "012468", "content": "设直线$l$与平面$\\alpha$平行, 直线$m$在平面$\\alpha$上, 那么\\bracket{20}.\n\\twoch{直线$l$平行于直线$m$}{直线$l$与直线$m$异面}{直线$l$与直线$m$没有公共点}{直线$l$与直线$m$不垂直}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306508,7 +307323,9 @@ "id": "012469", "content": "用数学归纳法证明等式$1+2+3+\\ldots+2 n=2 n^2+n$($n \\in \\mathbf{N}$, $n\\ge 1$)的第(ii)步中, 假设$n=k$时原等式成立, 那么在$n=k+1$时, 需要证明的等式为\\bracket{20}.\n\\onech{$1+2+3+\\ldots+2 k+2(k+1)=2 k^2+k+2(k+1)^2+(k+1)$}{$1+2+3+\\ldots+2 k+2(k+1)=2(k+1)^2+(k+1)$}{$1+2+3+\\ldots+2 k+(2 k+1)+2(k+1)=2 k^2+k+2(k+1)^2+(k+1)$}{$1+2+3+\\ldots+2 k+(2 k+1)+2(k+1)=2(k+1)^2+(k+1)$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306527,7 +307344,9 @@ "id": "012470", "content": "关于双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$与$\\dfrac{y^2}{16}-\\dfrac{x^2}4=1$的焦距和渐近线, 下列说法正确的是\\bracket{20}.\n\\twoch{焦距相等, 渐近线相同}{焦距相等, 渐近线不相同}{焦距不相等, 渐近线相同}{焦距不相等, 渐近线不相同}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306546,7 +307365,9 @@ "id": "012471", "content": "设函数$y=f(x)$的定义域为$\\mathbf{R}$, 则``$f(0)=0$''是``$y=f(x)$为奇函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306565,7 +307386,9 @@ "id": "012472", "content": "下列关于实数$a$、$b$的不等式中, 不恒成立的是\\bracket{20}.\n\\fourch{$a^2+b^2 \\geq 2 a b$}{$a^2+b^2 \\geq-2 a b$}{$(\\dfrac{a+b}2)^2 \\geq a b$}{$(\\dfrac{a+b}2)^2 \\geq-a b$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306584,7 +307407,9 @@ "id": "012473", "content": "设单位向量$\\overrightarrow{e_1}$与$\\overrightarrow{e_2}$既不平行也不垂直, 对非零向量$\\overrightarrow a=x_1 \\overrightarrow{e_1}+y_1 \\overrightarrow{e_2}, \\overrightarrow b=x_2 \\overrightarrow{e_1}+y_2 \\overrightarrow{e_2}$, 有结论: \\textcircled{1} 若$x_1 y_2-x_2 y_1=0$, 则$\\overrightarrow a / / \\overrightarrow b$; \\textcircled{2} 若$x_1 x_2+y_1 y_2=0$, 则$\\overrightarrow a \\perp \\overrightarrow b$; 关于以上两个结论, 正确的判断是\\bracket{20}.\n\\fourch{\\textcircled{1}成立, \\textcircled{2}不成立}{\\textcircled{1}不成立, \\textcircled{2}成立}{\\textcircled{1}成立, \\textcircled{2}成立}{\\textcircled{1}不成立, \\textcircled{2}不成立}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306603,7 +307428,9 @@ "id": "012474", "content": "对于椭圆$C_{(a, b)}: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a, b>0$, $a \\neq b$), 若点$(x_0, y_0)$满足$\\dfrac{x_0^2}{a^2}+\\dfrac{y_0^2}{b^2}<1$, 则称该点在椭圆$C_{(a, b)}$内, 在平面直角坐标系中, 若点$A$在过点$(2,1)$的任意椭圆$C_{(a, b)}$内或椭圆$C_{(a, b)}$上, 则满足条件的点$A$构成的图形为\\bracket{20}.\n\\fourch{三角形及其内部}{矩形及其内部}{圆及其内部}{椭圆及其内部}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306622,7 +307449,9 @@ "id": "012475", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$的体积为$9 \\sqrt 3$, 底面边长为$3$, 求异面直线$BC_1$与$AC$所成的角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{3}\n\\def\\h{4}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,\\h) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\draw (B) -- (C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306641,7 +307470,9 @@ "id": "012476", "content": "已知函数$f(x)=\\sin x+\\sqrt 3 \\cos x$, 求$f(x)$的最小正周期及最大值, 并指出$f(x)$取得最大值时$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306660,7 +307491,9 @@ "id": "012477", "content": "如图, 汽车前灯反射镜与轴截面的交线是抛物线的一部分, 灯口所在的圆面与反射镜的轴垂直, 灯泡位于抛物线的焦点$F$处, 已知灯口直径是$24 \\text{cm}$, 灯深$10 \\text{cm}$, 求灯泡与反射镜的顶点$O$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.15]\n\\filldraw (3.6,0) circle (0.1) node [right] {$F$} coordinate (F);\n\\filldraw (10,0) circle (0.1);\n\\draw [domain = -12:12] plot ({pow(\\x,2)/14.4},\\x);\n\\draw (10,0) ellipse (3 and 12);\n\\draw (0,0) node [left] {$O$} coordinate (O) --++ (0,-14);\n\\draw (10,-12) --++ (0,-2);\n\\draw [<->] (0,-13) -- (10,-13) node [midway, below] {$10\\text{cm}$};\n\\draw (10,-12) --++ (5,0) (10,12) --++ (5,0);\n\\draw [<->] (14,-12) -- (14,12) node [midway, right] {$24\\text{cm}$};\n\\draw [dashed] (10,0) --++ (0,-12);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306679,7 +307512,9 @@ "id": "012478", "content": "已知数列$\\{a_n\\}$是公差为$2$的等差数列.\\\\\n(1) 若$a_1$、$a_3$、$a_4$成等比数列, 求$a_1$的值;\\\\\n(2) 设$a_1=-19$, 数列$\\{a_n\\}$的前$n$项和为$S_n$, 数列$\\{b_n\\}$满足$b_1=1$, $b_{n+1}-b_n=(\\dfrac 12)^n$, 记$c_n=S_n+2^{n-1} \\cdot b_n$($n \\in \\mathbf{N}$, $n\\ge 1$), 求数列$\\{c_n\\}$的最小值$c_{n_0}$. (即$c_{n_0} \\leq c_n$对任意$n \\in \\mathbf{N}$, $n\\ge 1$成立)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306698,7 +307533,9 @@ "id": "012479", "content": "对于函数$f(x)$与$g(x)$, 记集合$D_{f>g}=\\{x \\mid f(x)>g(x)\\}$.\\\\\n(1) 设$f(x)=2|x|$, $g(x)=x+3$, 求$D_{f>g}$;\\\\\n(2) 设$f_1(x)=x-1$, $f_2(x)=(\\dfrac 13)^x+a \\cdot 3^x+1$, $h(x)=0$, 如果$D_{f_1>h} \\cup D_{f_2>h}=\\mathbf{R}$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -306717,7 +307554,9 @@ "id": "012480", "content": "若函数$f(x)=\\sin (x+\\varphi)$是偶函数, 则$\\varphi$的一个值是\\bracket{20}.\n\\fourch{$0$}{$\\dfrac{\\pi}2$}{$\\pi$}{$2 \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306736,7 +307575,9 @@ "id": "012481", "content": "在复平面上, 满足$|z-1|=4$的复数$z$所对应的点的轨迹是\\bracket{20}.\n\\fourch{两个点}{一条线段}{两条直线}{一个圆}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306755,7 +307596,9 @@ "id": "012482", "content": "已知函数$f(x)$的图像是折线段$ABCDE$, 如图, 其中$A(1,2)$、$B(2,1)$、$C(3,2)$、$D(4,1)$、$E(5,2)$, 若直线$y=k x+b$($k, b \\in \\mathbf{R}$)与$f(x)$的图像恰有$4$个不同的公共点, 则$k$的取值范围是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-0.5,0) -- (5.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,2) node [above] {$A$} coordinate (A) -- (2,1) node [below] {$B$} coordinate (B) -- (3,2) node [above] {$C$} coordinate (C) -- (4,1) node [below] {$D$} coordinate (D) -- (5,2) node [above] {$E$} coordinate (E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$(-1,0) \\cup(0,1)$}{$(-\\dfrac 13, \\dfrac 13)$}{$(0,1]$}{$[0, \\dfrac 13]$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -306774,7 +307617,9 @@ "id": "012483", "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的长半轴的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306793,7 +307638,9 @@ "id": "012484", "content": "已知圆锥的母线长为$10$, 母线与轴的夹角为$30^{\\circ}$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306812,7 +307659,9 @@ "id": "012485", "content": "小明用数列$\\{a_n\\}$记录某地区$2015$年$12$月份$31$天中每天是否下过雨, 方法为: 当第$k$天下过雨时, 记$a_k=1$, 当第$k$天没下过雨时, 记$a_k=-1$($1 \\leq k \\leq 31$); 他用数列$\\{b_n\\}$记录该地区该月每天气象台预报是否有雨, 方法为: 当预报第$k$天有雨时, 记$b_k=1$, 当预报第$k$天没有雨时, 记$b_k=-1$($1 \\leq k \\leq 31$); 记录完毕后, 小明计算出$a_1 b_1+a_2 b_2+a_3 b_3+\\ldots+a_{31} b_{31}=25$, 那么该月气象台预报准确的总天数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -306831,7 +307680,9 @@ "id": "012486", "content": "对于数列$\\{a_n\\}$与$\\{b_n\\}$, 若对数列$\\{c_n\\}$的每一项$c_k$, 均有$c_k=a_k$或$c_k=b_k$, 则称数列$\\{c_n\\}$是$\\{a_n\\}$与$\\{b_n\\}$的一个``并数列''.\\\\\n(1) 设数列$\\{a_n\\}$与$\\{b_n\\}$的前三项分别为$a_1=1$, $a_2=3$, $a_3=5$, $b_1=1$, $b_2=2$, $b_3=3$, 若数列$\\{c_n\\}$是$\\{a_n\\}$与$\\{b_n\\}$的一个``并数列'', 求所有可能的有序数组$(c_1, c_2, c_3)$;\\\\\n(2) 已知数列$\\{a_n\\}$、$\\{c_n\\}$均为等差数列, $\\{a_n\\}$的公差为$1$, 首项为正整数$t$, $\\{c_n\\}$的前$10$项和为$-30$, 前$20$项和为$-260$, 若存在唯一的数列$\\{b_n\\}$, 使得$\\{c_n\\}$是$\\{a_n\\}$与$\\{b_n\\}$的一个``并数列'', 求$t$的值所构成的集合.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326076,7 +326927,9 @@ "id": "030485", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, 哪些棱所在的直线与直线$BD_1$是异面直线?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\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,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326095,7 +326948,9 @@ "id": "030486", "content": "如图, 点$P$是矩形$ABCD$所在平面外的一点, $M$、$N$分别是$AB$、$PC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\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,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (D) ++ (0.2,1.5,0) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (P) -- (D);\n\\draw ($(P)!0.5!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (M) -- (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $MN\\parallel$平面$PAD$;\\\\ \n(2) 若$\\triangle PAD$是等边三角形, 求异面直线$MN$与$BC$所成的角的大小;\\\\ \n(3) 设$Q$是线段$DC$上的一点, 若平面$PAD\\parallel$平面$MNQ$, 求点$Q$的位置.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326114,7 +326969,9 @@ "id": "030487", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2$, $E$为$DD_1$上一点. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.8]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(D)!0.5!(D1)$) node [left] {$E$} coordinate (E);\n\\draw [dashed] (A) -- (C) (B) -- (D);\n\\draw [dashed] (A) -- (E) -- (C);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$D_1DBB_1\\perp$平面$EAC$;\\\\\n(2) 若$DE=2$, 求$AE$与平面$D_1DBB_1$所成的角的大小;\\\\\n(3) 若$E$为$DD_1$的中点, 且$B_1D\\perp$平面$EAC$, 求$DD_1$的长度.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326133,7 +326990,9 @@ "id": "030488", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=3$, $BC=4$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{4}\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 (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求点$A_1$到平面$B_1BDD_1$的距离;\\\\\n(2) 若$A_1B$和平面$B_1BDD_1$所成的角的大小为$\\arcsin \\dfrac{12}{25}$, 求$AA_1$的长度.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326152,7 +327011,9 @@ "id": "030489", "content": "如图, 在斜三棱柱$ABC-A_1B_1C_1$中, $\\angle A_1AC=\\angle ACB=\\dfrac{\\pi }2$, $\\angle AA_1C=\\dfrac{\\pi }6$, 侧棱$BB_1$与底面所成的角为$\\dfrac{\\pi }3$, $AA_1=4\\sqrt 3$, $BC=4$. 求斜三棱柱$ABC-A_1B_1C_1$的体积$V$. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw ({-2*sqrt(2)},0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-2*sqrt(2)}) node [above right] {$C$} coordinate (C);\n\\draw (A) -- (B);\n\\draw [dashed] (A) -- (C) -- (B);\n\\draw (A) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed] (C) -- (C1) (A1) -- (C);\n\\draw (A) -- (A1) (B) -- (B1) (A1) -- (B1) (A1) -- (C1) -- (B1); \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326171,7 +327032,9 @@ "id": "030490", "content": "如图, 圆锥$P-O$的底面直径和高均是$a$, 过$PO$的中点$O'$作平行于底面的截面, 以该截面为底面挖去一个圆柱, 求剩下几何体的体积和表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (0,1) node [left] {$O'$} coordinate (O1);\n\\draw (P) -- (-1,0) (P) -- (1,0);\n\\draw [dashed] (P) --++ (0,-2) --++ (1,0);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25) (-0.5,1) arc (180:0:0.5 and 0.125);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-0.5,0) --++ (0,1) (0.5,0) --++ (0,1);\n\\draw (-0.5,1) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (O) ellipse (0.5 and 0.125);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326190,7 +327053,9 @@ "id": "030491", "content": "如图, 三棱锥$P-MNQ$中, $PM\\perp NQ$, $PM\\perp MN$, $NQ\\perp MN$. 若$MN=NQ=1$, 二面角$P-NQ-M$的大小为$\\dfrac{\\pi }4$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$M$} coordinate (M);\n\\draw ({sqrt(2)},0,0) node [right] {$Q$} coordinate (Q);\n\\draw ({sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$N$} coordinate (N);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (M) -- (N) -- (Q) (P) -- (M) (P) -- (N) (P) -- (Q);\n\\draw [dashed] (M) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥$P-MNQ$的体积;\\\\\n(2) 点$M$到平面$PNQ$的距离. \n(3) 若点$E$为棱$PN$的中点, 点$F$为棱$PQ$的中点, 那么三棱锥$M-EFN$的体积是多少?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326209,7 +327074,9 @@ "id": "030492", "content": "已知四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为2, 且$\\angle ADC=\\dfrac{2\\pi }3$. 设$\\overrightarrow{DA}=\\overrightarrow{a}$, $\\overrightarrow{DC}=\\overrightarrow{b}$, $\\overrightarrow{DD_1}=\\overrightarrow{c}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0.5,{sqrt(14)/2},-0.5) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0.5,{sqrt(14)/2},-0.5) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0.5,{sqrt(14)/2},-0.5) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0.5,{sqrt(14)/2},-0.5) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1)设$E$是棱$A_1D_1$的中点.\\\\\n\\textcircled{1} 试用$\\overrightarrow{a},\\overrightarrow{b},\\overrightarrow{c}$的线性组合表示$\\overrightarrow{EB}$;\\\\\n\\textcircled{2} 若$\\angle ADD_1=\\angle CDD_1=\\alpha$, $\\alpha \\in (\\dfrac{\\pi }3,\\dfrac{\\pi }2]$, 求$|\\overrightarrow{EB}|$的取值范围;\\\\\n(2)求证: 当且仅当$\\angle ADD_1=\\angle CDD_1$时, $AC\\perp$平面$DBB_1D_1$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326228,7 +327095,9 @@ "id": "030493", "content": "已知直四棱柱$ABCD-A_1B_1C_1D_1$, 各棱长均为$2$, 且$\\angle ADC=\\dfrac{2\\pi }3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (3,0,{-sqrt(3)}) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (1,0,{-sqrt(3)}) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,2,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,2,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,2,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,2,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$E$} coordinate (E);\n\\draw ($(A)!{2/3}!(B)$) node [below] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F);\n\\end{tikzpicture}\n\\end{center}\n(1)设$E$是$A_1D_1$中点, 点$F$满足$\\overrightarrow{AF}=2\\overrightarrow{FB}$.\\\\\n\\textcircled{1} 求异面直线$EF$与$DD_1$所成角的大小;\\\\\n\\textcircled{2} 求直线$EF$与平面$DBB_1D_1$所成角的大小;\\\\ \n(2)求平面$DBB_1D_1$与平面$BDC_1$所成锐二面角的大小;\\\\\n(3)求四面体$A_1C_1BD$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -326447,7 +327316,9 @@ "id": "030501", "content": "已知抛物线$y^2=4 x$, $F$为焦点,$P$为抛物线准线$l$上一动点, 线段$PF$与抛物线交于点$Q$, 定义$d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$P$坐标为$(-1,-\\dfrac 83)$, 求$d(P)$;\\\\\n(2) 求证: 存在常数$a$, 使得$2 d(P)=|FP|+a$关于点$P$恒成立.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "",