From 708dbf306884a22f1d06e62701427838790ffb11 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Wed, 22 Feb 2023 19:32:56 +0800 Subject: [PATCH] 20230222 evening --- 工具/分年级专用工具/赋能卷生成.ipynb | 23 +- 工具/寻找阶段末尾空闲题号.ipynb | 8 +- 工具/文本文件/题号筛选.txt | 2 +- 工具/添加题目到数据库.ipynb | 109 +- 工具/识别题库中尚未标注的题目类型.ipynb | 154 +- 工具/题号选题pdf生成.ipynb | 11 +- 文本处理工具/剪贴板圆圈数字生成.ipynb | 2 +- 题库0.3/Problems.json | 3196 ++++++++++++++++++----- 8 files changed, 2774 insertions(+), 731 deletions(-) diff --git a/工具/分年级专用工具/赋能卷生成.ipynb b/工具/分年级专用工具/赋能卷生成.ipynb index ddca3533..33a558ed 100644 --- a/工具/分年级专用工具/赋能卷生成.ipynb +++ b/工具/分年级专用工具/赋能卷生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 7, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -18,15 +18,16 @@ "#在 模板文件 目录中保留 赋能template.tex.\n", "#在 临时文件/赋能答题纸 目录中保留output目录.\n", "\"\"\"---设置文件名---\"\"\"\n", - "filename = \"赋能17\"\n", + "filename = \"赋能18\"\n", "\n", "\"\"\"---设置题目列表---\"\"\"\n", "problems = r\"\"\"\n", - "496:498,31205,500:505\n", + "506:515\n", "\"\"\"\n", "#完成后将含有 filename 的文件移至其它目录\n", "\n", "import os,re,json,shutil\n", + "from PIL import Image\n", "\n", "\n", "def generate_number_set(string,dict):\n", @@ -63,6 +64,16 @@ "os.rename(pdffile,\"临时文件/赋能答题纸/\"+filename+\"答题纸_raw.pdf\")\n", "os.remove(\"tempanswersheet.pdf\")\n", "\n", + "#图片下方涂白\n", + "pic = Image.open(\"临时文件/赋能答题纸/output/\"+filename+\"答题纸.png\")\n", + "whiteheight = pic.size[1]-880-5\n", + "whitewidth = pic.size[0]-100\n", + "whitecover = Image.new(\"RGB\",(whitewidth,whiteheight),(255,255,255))\n", + "pic1 = pic\n", + "pic1.paste(whitecover,(2,880))\n", + "pic1.save(\"临时文件/赋能答题纸/output/\"+filename+\"答题纸.png\")\n", + "\n", + "\n", "#替换tex模板中的内容\n", "problem_list = [id for id in generate_number_set(problems.strip(),pro_dict) if id in pro_dict]\n", "\n", @@ -96,7 +107,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3.8.15 ('mathdept')", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -110,12 +121,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.15" + "version": "3.9.15" }, "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "42dd566da87765ddbe9b5c5b483063747fec4aacc5469ad554706e4b742e67b2" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/寻找阶段末尾空闲题号.ipynb b/工具/寻找阶段末尾空闲题号.ipynb index 9e254d13..6876f94f 100644 --- a/工具/寻找阶段末尾空闲题号.ipynb +++ b/工具/寻找阶段末尾空闲题号.ipynb @@ -11,8 +11,8 @@ "text": [ "首个空闲id: 14532 , 直至 020000\n", "首个空闲id: 22048 , 直至 030000\n", - "首个空闲id: 31236 , 直至 040000\n", - "首个空闲id: 40037 , 直至 999999\n" + "首个空闲id: 40037 , 直至 999999\n", + "首个空闲id: 31241 , 直至 040000\n" ] } ], @@ -46,7 +46,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -65,7 +65,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt index 12021a25..dfb30bac 100644 --- a/工具/文本文件/题号筛选.txt +++ b/工具/文本文件/题号筛选.txt @@ -1 +1 @@ -000049,000055,000062,000070,001152,001210,001287,002838,002853,002854,002863,002864,002865,002874,002875,002880,002902,002909,002913,002924,002925,002938,002954,002965,002987,003022,003601,003625,003667,003718,003730,003757,003770,003772,003801,003815,003862,003892,003907,003921,003936,003966,003980,003981,004074,004094,004116,004155,004157,004220,004284,004305,004347,004366,004368,004385,004401,004403,004422,004439,004440,004523,004525,004544,004546,004563,004631,004697,004739,004757,004760,004902,004907,005008,005016,005103,005104,005105,005193,005194,005195,005196,005197,005198,005283,005284,005285,005286,005287,005288,005296,005298,005299,005300,005301,005302,005303,005329,005330,005331,005332,005333,005362,005368,005369,005370,005371,005372,005373,005374,005375,005376,005377,005378,005379,005380,005381,005382,005383,005384,005385,005386,005424,005445,005446,005447,005448,005449,005450,005472,005473,005474,005475,005476,005477,005491,005492,005493,005494,005495,005496,005501,005502,005503,005504,005505,005506,005507,005508,005523,005524,005525,005526,005527,005528,005529,005530,005541,005542,005543,005563,005564,005565,005566,005567,005568,005569,005570,005571,005572,005573,005574,005612,005613,005614,005615,005616,005617,005618,005619,005620,005621,005653,005654,005655,005656,005657,005658,005659,005660,005687,005688,005689,005690,005691,005692,005693,005694,005695,005696,005713,005714,005718,005719,005720,005721,005722,005723,005724,005758,005759,005760,005761,005762,005763,005764,005782,005783,005784,005785,005786,007861,007892,007958,007959,007966,007984,008036,008037,008081,008082,008094,008095,008371,008372,008387,008388,008389,009476,009508,009537,010137,010149,010150,010173,010938,010977,010999,011041,011042,011043,011126,011145,011147,011148,011196,011212,011215,011233,011236,011256,011320,011404,011446,011529,011548,011594,011616,011687,011733,011911,011915,011927,011929,011947,011985,012001,012003,012022,012023,012090,012130,012132,012153,012173,012194,012215,012257,012259,012278,012300,012301,012302,012320,012344,012365,012367,012371,012374,012399,012401,012428,012430,012466,012471,012502,012543,012564,012605,012626,012628,012709,012733,012751,012783,012792,012836,012843,012849,012850,012860,012861,012863,012884,012885,012890,012895,012904,012905,012906,012907,012913,013297,013314,013342,013388,013418,013433,013447,013449,013477,013479,013508,013523,013539,013553,013554,013583,013600,013621,013768,013798,013799,013801,013829,013841,020358,020359,020365,020366,020377,020378,020379,020386,020387,020406,020407,020408,020417,020418,020419,020439,020440,020441,020442,020451,020465,020476,020477,020478,020487,020498,020499,020504,020524,020525,020526,020529,020554,020555,020561,020562,020566,020678,020679,020680,020681,021369,021388,030300,030307,030372,030379,030393,030400,030442,030667,030668,030671,030672,030673,030674,030675,030689,030690,030691,030692,030701,030722,030723,030728,030729,030755,030756 +12916:12938,12939:12961,12962:12984,12985:13007,13918:13930,13931:13943,13944:13954,13955:13960,14532:14549,14550:14569 \ No newline at end of file diff --git a/工具/添加题目到数据库.ipynb b/工具/添加题目到数据库.ipynb index 82f4c8a0..44b01408 100644 --- a/工具/添加题目到数据库.ipynb +++ b/工具/添加题目到数据库.ipynb @@ -7,10 +7,10 @@ "outputs": [], "source": [ "#修改起始id,出处,文件名\n", - "starting_id = 31236\n", - "raworigin = \"2022年北京高考\"\n", - "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\自拟题目8.tex\"\n", - "editor = \"20230220\\t王伟叶\"\n", + "starting_id = 14532\n", + "raworigin = \"2023年空中课堂高三复习课\"\n", + "filename = r\"C:\\Users\\weiye\\Documents\\wwy sync\\临时工作区\\空中课堂第五批.tex\"\n", + "editor = \"20230221\\t王伟叶\"\n", "indexed = False\n" ] }, @@ -23,7 +23,102 @@ "name": "stdout", "output_type": "stream", "text": [ - "添加题号031236, 来源: 2022年北京高考\n" + "添加题号014532, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014533, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014534, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014535, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014536, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014537, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014538, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014539, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014540, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014541, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014542, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014543, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014544, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014545, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014546, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014547, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014548, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014549, 来源: 2023年空中课堂高三复习课22\n", + "添加题号014550, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014551, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014552, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014553, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014554, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014555, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014556, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014557, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014558, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014559, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014560, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014561, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014562, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014563, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014564, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014565, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014566, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014567, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014568, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014569, 来源: 2023年空中课堂高三复习课23\n", + "添加题号014570, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014571, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014572, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014573, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014574, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014575, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014576, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014577, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014578, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014579, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014580, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014581, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014582, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014583, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014584, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014585, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014586, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014587, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014588, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014589, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014590, 来源: 2023年空中课堂高三复习课25\n", + "添加题号014591, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014592, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014593, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014594, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014595, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014596, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014597, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014598, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014599, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014600, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014601, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014602, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014603, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014604, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014605, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014606, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014607, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014608, 来源: 2023年空中课堂高三复习课26\n", + "添加题号014609, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014610, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014611, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014612, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014613, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014614, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014615, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014616, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014617, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014618, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014619, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014620, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014621, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014622, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014623, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014624, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014625, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014626, 来源: 2023年空中课堂高三复习课29\n", + "添加题号014627, 来源: 2023年空中课堂高三复习课29\n" ] } ], @@ -128,7 +223,7 @@ ], "metadata": { "kernelspec": { - "display_name": "pythontest", + "display_name": "mathdept", "language": "python", "name": "python3" }, @@ -147,7 +242,7 @@ "orig_nbformat": 4, "vscode": { "interpreter": { - "hash": "91219a98e0e9be72efb992f647fe78b593124968b75db0b865552d6787c8db93" + "hash": "ff3c292c316ba85de6f1ad75f19c731e79d694e741b6f515ec18f14996fe48dc" } } }, diff --git a/工具/识别题库中尚未标注的题目类型.ipynb b/工具/识别题库中尚未标注的题目类型.ipynb index d42470c5..f6edabb2 100644 --- a/工具/识别题库中尚未标注的题目类型.ipynb +++ b/工具/识别题库中尚未标注的题目类型.ipynb @@ -9,27 +9,139 @@ "name": "stdout", "output_type": "stream", "text": [ - "014511 填空题\n", - "014512 填空题\n", - "014513 填空题\n", - "014514 填空题\n", - "014515 填空题\n", - "014516 填空题\n", - "014517 填空题\n", - "014518 填空题\n", - "014519 填空题\n", - "014520 填空题\n", - "014521 填空题\n", - "014522 填空题\n", - "014523 选择题\n", - "014524 选择题\n", - "014525 选择题\n", - "014526 选择题\n", - "014527 解答题\n", - "014528 解答题\n", - "014529 解答题\n", - "014530 解答题\n", - "014531 解答题\n" + "014532 填空题\n", + "014533 填空题\n", + "014534 解答题\n", + "014535 填空题\n", + "014536 解答题\n", + "014537 解答题\n", + "014538 解答题\n", + "014539 填空题\n", + "014540 解答题\n", + "014541 填空题\n", + "014542 解答题\n", + "014543 填空题\n", + "014544 填空题\n", + "014545 填空题\n", + "014546 选择题\n", + "014547 填空题\n", + "014548 解答题\n", + "014549 解答题\n", + "014550 填空题\n", + "014551 填空题\n", + "014552 解答题\n", + "014553 填空题\n", + "014554 填空题\n", + "014555 解答题\n", + "014556 解答题\n", + "014557 解答题\n", + "014558 填空题\n", + "014559 填空题\n", + "014560 选择题\n", + "014561 填空题\n", + "014562 填空题\n", + "014563 填空题\n", + "014564 填空题\n", + "014565 填空题\n", + "014566 解答题\n", + "014567 解答题\n", + "014568 选择题\n", + "014569 解答题\n", + "014570 填空题\n", + "014571 填空题\n", + "014572 填空题\n", + "014573 填空题\n", + "014574 填空题\n", + "014575 填空题\n", + "014576 填空题\n", + "014577 填空题\n", + "014578 解答题\n", + "014579 填空题\n", + "014580 填空题\n", + "014581 填空题\n", + "014582 填空题\n", + "014583 填空题\n", + "014584 填空题\n", + "014585 填空题\n", + "014586 填空题\n", + "014587 填空题\n", + "014588 解答题\n", + "014589 填空题\n", + "014590 解答题\n", + "014591 填空题\n", + "014592 填空题\n", + "014593 填空题\n", + "014594 填空题\n", + "014595 解答题\n", + "014596 解答题\n", + "014597 解答题\n", + "014598 填空题\n", + "014599 填空题\n", + "014600 解答题\n", + "014601 填空题\n", + "014602 填空题\n", + "014603 填空题\n", + "014604 填空题\n", + "014605 填空题\n", + "014606 填空题\n", + "014607 填空题\n", + "014608 解答题\n", + "014609 填空题\n", + "014610 选择题\n", + "014611 填空题\n", + "014612 填空题\n", + "014613 填空题\n", + "014614 填空题\n", + "014615 填空题\n", + "014616 填空题\n", + "014617 填空题\n", + "014618 填空题\n", + "014619 填空题\n", + "014620 填空题\n", + "014621 填空题\n", + "014622 填空题\n", + "014623 填空题\n", + "014624 解答题\n", + "014625 解答题\n", + "014626 填空题\n", + "014627 解答题\n", + "031236 解答题\n", + "040001 选择题\n", + "040002 填空题\n", + "040003 选择题\n", + "040004 填空题\n", + "040005 填空题\n", + "040006 填空题\n", + "040007 填空题\n", + "040008 填空题\n", + "040009 填空题\n", + "040010 填空题\n", + "040011 填空题\n", + "040012 填空题\n", + "040013 填空题\n", + "040014 填空题\n", + "040015 解答题\n", + "040016 解答题\n", + "040017 解答题\n", + "040018 填空题\n", + "040019 填空题\n", + "040020 填空题\n", + "040021 填空题\n", + "040022 填空题\n", + "040023 填空题\n", + "040024 填空题\n", + "040025 填空题\n", + "040026 填空题\n", + "040027 填空题\n", + "040028 填空题\n", + "040029 填空题\n", + "040030 填空题\n", + "040031 填空题\n", + "040032 填空题\n", + "040033 填空题\n", + "040034 解答题\n", + "040035 解答题\n", + "040036 解答题\n" ] } ], diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb index 8cdfbcc1..ec70e057 100644 --- a/工具/题号选题pdf生成.ipynb +++ b/工具/题号选题pdf生成.ipynb @@ -2,16 +2,16 @@ "cells": [ { "cell_type": "code", - "execution_count": 5, + "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "开始编译教师版本pdf文件: 临时文件/高一前三次作业_教师用_20230221.tex\n", + "开始编译教师版本pdf文件: 临时文件/06_数列备选_教师用_20230221.tex\n", "0\n", - "开始编译学生版本pdf文件: 临时文件/高一前三次作业_学生用_20230221.tex\n", + "开始编译学生版本pdf文件: 临时文件/06_数列备选_学生用_20230221.tex\n", "0\n" ] } @@ -26,13 +26,14 @@ "\"\"\"---设置题目列表---\"\"\"\n", "#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n", "problems = r\"\"\"\n", - "021441,021442,021443,021444,021445,021446,021447,021448,021449,021450,021451,021452,021453,021454,021455,021469,021470,021471,021472,021473,021474,021475,021476,021477,021478,021479,021480,021481\n", + "a\n", + "\n", "\"\"\"\n", "\"\"\"---设置题目列表结束---\"\"\"\n", "\n", "\"\"\"---设置文件名---\"\"\"\n", "#目录和文件的分隔务必用/\n", - "filename = \"临时文件/高一前三次作业\"\n", + "filename = \"临时文件/06_数列备选\"\n", "\"\"\"---设置文件名结束---\"\"\"\n", "\n", "\n", diff --git a/文本处理工具/剪贴板圆圈数字生成.ipynb b/文本处理工具/剪贴板圆圈数字生成.ipynb index 082359cb..19032f37 100644 --- a/文本处理工具/剪贴板圆圈数字生成.ipynb +++ b/文本处理工具/剪贴板圆圈数字生成.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": null, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 483362ff..3064da8e 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -352688,6 +352688,1830 @@ "remark": "", "space": "12ex" }, + "014532": { + "id": "014532", + "content": "已知等差数列$\\{a_n\\}$, 前$n$项和为$S_n$, 若$a_7+a_9=18, a_4=1$, 则$a_{12}=$\\blank{50}, $S_{15}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014533": { + "id": "014533", + "content": "已知无穷等比数列$\\{a_n\\}$, 前$n$项和为$S_n$, 公比为$q$, $a_8=\\dfrac{1}{16}$, $q=\\dfrac{1}{2}$, 则$a_4=$\\blank{50}, $S_8=$\\blank{50}, $\\displaystyle \\lim _{n \\to+\\infty} S_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014534": { + "id": "014534", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=2 \\times 3^n+a$. 当常数$a=$时, 数列$\\{a_n\\}$为等比数列.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014535": { + "id": "014535", + "content": "在等差数列$\\{a_n\\}$中, 公差为常数$d$, 若$a_1<0$, $11 a_5=5 a_8$, 则该数列前$n$项和$S_n$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014536": { + "id": "014536", + "content": "已知等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$S_n=n-5 a_n-85$.\\\\\n(1) 证明: $\\{a_n-1\\}$是等比数列;\\\\\n(2) 试问: 数列$\\{S_n\\}$是否有最小项? 若有, 指出第几项最小; 若没有, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014537": { + "id": "014537", + "content": "已知数列$\\{a_n\\}$成等差数列, 其前$n$项和为$S_n$, 且$S_{10}=20$, $S_{20}=50$, 求$S_{30}$, 并在等比数列中, 写出相应的问题并完成求解.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014538": { + "id": "014538", + "content": "某地区$2020$年产生的生活垃圾为$20$万吨, 其中$6$万吨垃圾以环保方式处理, 剩余$14$万吨垃圾以填埋方式处理. 预测显示: 在以$2020$年为第一年的未来十年内, 该地区每年产生的生活垃圾量比上一年增长$5 \\%$, 同时, 通过环保方式处理的垃圾量比上一年增加$1.5$万吨, 剩余的垃圾以填埋方式处理. 根据预测, 解答下列问题:\\\\\n(1) 求$2021$年至$2023$年, 该地区三年通过填埋方式处理的垃圾共计多少万吨? (结果精确到$0.1$万吨)\\\\\n(2) 该地区在哪一年通过环保方式处理的垃圾量首次超过这一年产生的生活垃圾量的$50 \\%$?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014539": { + "id": "014539", + "content": "在数列$\\{a_n\\}$中, 若$a_1=2$, 且$a_n=a_{n-1}+\\lg \\dfrac{n}{n-1}$($n \\geq 2$), 则$a_{100}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014540": { + "id": "014540", + "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_4=10$.\\\\\n(1) 若$S_{20}=590$, 求等差数列$\\{a_n\\}$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": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014541": { + "id": "014541", + "content": "已知等差数列$\\{a_n\\}$的公差$d \\neq 0$, 其前$n$项和为$S_n$, 若$S_{10}=0$, 则$S_i$($i=1,2,3, \\cdots, 2023$)中不同的数值有\\blank{50}个.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014542": { + "id": "014542", + "content": "(1) 在等差数列$\\{a_n\\}$中, 公差为$d$, 其前$n$项和为$S_n$, 根据要求完成下列表格, (其中$m$为正整数).\n\\begin{center}\n\\begin{tabular}{|c|l|}\n\\hline 用$S_m$表示$S_{2 m}$&$S_{2 m}=2S_m+m^2 d$\\\\\n\\hline 用$S_m$表示$S_{3 m}$&$S_{3 m}=$\\\\\n\\hline 用$S_m$表示$S_{n m}$&$S_{N m}=$\\\\\n\\hline\n\\end{tabular} \n\\end{center}\n(2) 在等比数列$\\{b_n\\}$中, 公比为$q$, 类比问题(1), 请写出相应的结论.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014543": { + "id": "014543", + "content": "等差数列$\\{a_n\\}$的首项为$1$, 公差不为$0$. 若$a_2, a_3, a_6$成等比数列, 则$\\{a_n\\}$前$6$项的和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014544": { + "id": "014544", + "content": "已知数列$\\{a_n\\}$, 其前$n$项和为$S_n$, 若$S_n=1+\\dfrac{1}{4} a_n$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014545": { + "id": "014545", + "content": "若将数列$\\{2 n-1\\}$与$\\{3 n-2\\}$的公共项从小到大排列得到数列$\\{a_n\\}$, 则$\\{a_n\\}$的前$n$项和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014546": { + "id": "014546", + "content": "已知无穷等比数列$\\{a_n\\}$的公比为$q$, 前$n$项和为$S_n$, 且$\\displaystyle\\lim _{n \\to+\\infty} S_n=S$. 下列条件中, 使得$2S_n0$, $0.60$, $0.76$时$c_n$的表达式, 并利用计算器确定$n_0$的值(只需写出$n_0$的值).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014549": { + "id": "014549", + "content": "给定无穷数列$\\{a_n\\}$, 若无穷数列$\\{b_n\\}$满足: 对任意正整数$n$, 都有$|b_n-a_n| \\leq 1$, 则称$\\{b_n\\}$与$\\{a_n\\}$``接近''.\\\\\n(1) 设$\\{a_n\\}$是首项为$1$, 公比为$\\dfrac{1}{2}$的等比数列, $b_n=a_{n+1}+1$, $n \\in \\mathbf{N}$, $n \\geq 1$, 判断数列$\\{b_n\\}$是否与$\\{a_n\\}$接近, 并说明理由;\\\\\n(2) 设数列$\\{a_n\\}$的前四项为: $a_1=1, a_2=2, a_3=4, a_4=8$, $\\{b_n\\}$是一个与$\\{a_n\\}$接近的数列, 记集合$M=\\{x | x=b_i, i=1,2,3,4\\}$, 求$M$中元素的个数$m$的所有可能值;\\\\\n(3) 已知$\\{a_n\\}$是公差为$d$的等差数列, 若存在数列$\\{b_n\\}$满足: $\\{b_n\\}$与$\\{a_n\\}$接近, 且在$b_2-b_1, b_3-b_2, \\cdots, b_{201}-b_{200}$中正数的个数不小于$100$, 求$d$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课22", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014550": { + "id": "014550", + "content": "设$\\lambda \\in \\mathbf{R}$, 数列$\\{a_n\\}$的通项公式为$a_n=n^2-\\lambda n+8$, 若数列$\\{a_n\\}$为严格增数列, 则$\\lambda$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014551": { + "id": "014551", + "content": "若数列$\\{a_n\\}$的通项公式为$a_n=\\dfrac{n-3.5}{n-4.5}$, 则数列$\\{a_n\\}$的最大项是\\blank{50}; 最小项是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014552": { + "id": "014552", + "content": "若数列$\\{a_n\\}$满足$a_1=4$, $a_n=4 a_{n-1}-6$($n \\in \\mathbf{N}$, $n \\geq 2$).\\\\\n(1) 设$b_n=a_n-2$, 求证数列$\\{b_n\\}$是等比数列;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014553": { + "id": "014553", + "content": "已知数列$\\{a_n\\}$满足$a_1=2$, $a_{n+1}=\\dfrac{1+a_n}{1-a_n}$, 则该数列的前$2023$项的乘积$a_1 a_2 a_3 \\cdots a_{2023}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014554": { + "id": "014554", + "content": "若数列$\\{a_n\\}$满足$a_1=0$, $2 a_{n+1}-a_n a_{n+1}=1$, 则$a_{2023}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014555": { + "id": "014555", + "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=(n-5)(\\dfrac{1}{2})^n$, 试问: 该数列是否有最大项、最小项? 若有, 分别指出第几项最大、最小; 若没有, 试说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014556": { + "id": "014556", + "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$a_{n+1}-a_n=2(b_{n+1}-b_n)$.\\\\ \n(1) 设$\\{a_n\\}$的第$n_0$项是数列$\\{a_n\\}$的最大项, 即$a_{n_0} \\geq a_n$对一切正整数$n$恒成立, 求证: $\\{b_n\\}$的第$n_0$项是数列$\\{b_n\\}$的最大项;\\\\\n(2) 设$\\lambda \\in \\mathbf{R}$, $a_1=3 \\lambda<0$, $b_n=\\lambda^n$, 求$\\lambda$的取值范围, 使得对任意正整数$m$、$n$, $a_n \\neq 0$, $\\dfrac{a_m}{a_n} \\in(\\dfrac{1}{6}, 6)$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014557": { + "id": "014557", + "content": "对于无穷数列$\\{a_n\\}$与$\\{b_n\\}$, 记$A=\\{x | x=a_n\\}, B=\\{x | x=b_n\\}$, 若同时满足条件: \\textcircled{1} $\\{a_n\\}$, $\\{b_n\\}$均为严格增数列; \\textcircled{2} $A \\cap B=\\varnothing$且$A \\cup B=\\{x | x \\in \\mathbf{N}, x \\geq 1\\}$, 则称$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列.\\\\\n(1) 若$a_n=2 n-1$, $b_n=4 n-2$, 判断$\\{a_n\\}$与$\\{b_n\\}$是否为无穷互补数列, 并说明理由;\\\\\n(2) 若$a_n=2^n$且$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, 求数列$\\{b_n\\}$的前$2023$项的和;\\\\\n(3) 若$\\{a_n\\}$与$\\{b_n\\}$是无穷互补数列, $\\{a_n\\}$为等差数列且$a_{10}=23$, 求$\\{a_n\\}$与$\\{b_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014558": { + "id": "014558", + "content": "定义``等和数列'': 在一个数列中, 如果每一项与它的后一项的和都为同一个常数, 那么这个数列叫做等和数列, 这个常数叫做该数列的公和. 已知数列$\\{a_n\\}$是等和数列, 且$a_1=2$, 公和为$5$, 那么$a_{28}=$\\blank{50}, 这个数列的前$n$项和$S_n$的计算公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014559": { + "id": "014559", + "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, 若$a_1=1$, $\\{\\dfrac{S_n}{a_n}\\}$是公差为$\\dfrac{1}{3}$的等差数列, 则数列$\\{a_n\\}$的通项公式为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014560": { + "id": "014560", + "content": "已知数列$\\{a_n\\}$的前$n$项的和$S_n$满足$S_{n+1}+S_n=n$. 对于以下两个命题: \\textcircled{1} 若$a_1=-1$, 则$S_{203}=1010$; \\textcircled{2} 数列$\\{a_{n+1}+a_n\\}$是常数列, 下列说法正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}都正确}{\\textcircled{1}正确, \\textcircled{2}不正确}{\\textcircled{1}\\textcircled{2}都不正确}{\\textcircled{1}不正确, \\textcircled{2}正确}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014561": { + "id": "014561", + "content": "已知数列$\\{a_n\\}$, $\\{b_n\\}$满足$a_1=b_1=1$, $a_{n+1}=a_n+b_n+\\sqrt{a_n^2+b_n^2}$, $b_{n+1}=a_n+b_n-\\sqrt{a_n^2+b_n^2}$, 设$c_n=3^n(\\dfrac{1}{a_n}+\\dfrac{1}{b_n})$, 则数列$\\{c_n\\}$的前$2023$项之和为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014562": { + "id": "014562", + "content": "若$a_n=2 n-7$, 下列说法中, 所有正确说法的序号是\\blank{50}.\\\\\n\\textcircled{1} 数列$\\{a_n\\}$是严格增数列;\\\\\n\\textcircled{2} 数列$\\{n a_n\\}$是严格增数列;\\\\\n\\textcircled{3} 数列$\\{\\dfrac{a_n}{n}\\}$是严格增数列;\\\\\n\\textcircled{4} 数列$\\{\\dfrac{1}{a_n}\\}$是严格减数列.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014563": { + "id": "014563", + "content": "已知数列$\\{a_n\\}$中$a_1=1$, $a_2=2$, $a_3=4$, 满足$a_{n+1}=-a_{n-2}$($n \\in \\mathbf{N}$, $n \\geq 3$), 则$a_{12}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014564": { + "id": "014564", + "content": "已知数列$\\{a_n\\}$的通项公式$a_n=\\dfrac{1}{(n+1)^2}$, 记$f(n)=(1-a_1)(1-a_2) \\cdots(1-a_n)$, 则$f(n)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014565": { + "id": "014565", + "content": "已知数列$\\{a_n\\}$满足$a_1=1, n a_{n+1}=(n+1) a_n+1$, 设$t \\in \\mathbf{R}$, 若对于任意的$a \\in[-2,2]$, 不等式$\\dfrac{a_{n+1}}{n+1}<3-a \\cdot 2^t$恒成立, 则$t$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014566": { + "id": "014566", + "content": "下图是某神奇``黄金分割草''的生长图, 第$1$阶段生长为坚直向上长为$1$米的枝干, 第$2$阶段在枝头生长出两根新的枝干, 新枝干的长度是原来的$\\dfrac{\\sqrt{5}-1}{2}$, 且与旧枝成$120^\\circ$角, 第$3$阶段又在每个枝头各长出两根新的枝干, 新枝干的长度是原来的$\\dfrac{\\sqrt{5}-1}{2}$, 且与旧枝成$120^\\circ$角, $\\cdots$, 依次生长, 直到永远.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{0.618}\n\\draw (0,0) -- (0,1);\n\\draw (3,0) -- (3,1) --++ (30:\\l) (3,1)--++(150:\\l);\n\\draw (6,0) -- (6,1) --++ (30:\\l) coordinate (A1) (6,1)--++(150:\\l) coordinate (A2) (A2)--++ (210:{\\l*\\l}) (A2) --++ (90:{\\l*\\l}) (A1) --++ (-30:{\\l*\\l}) (A1) --++ (90:{\\l*\\l});\n\\draw (9,0) -- (9,1) --++ (30:\\l) coordinate (A1) (9,1)--++(150:\\l) coordinate (A2) (A2)--++ (210:{\\l*\\l}) coordinate (B1) (A2) --++ (90:{\\l*\\l}) coordinate (B2) (A1) --++ (-30:{\\l*\\l}) coordinate (B4) (A1) --++ (90:{\\l*\\l}) coordinate (B3) (B1) --++ (-90:{\\l*\\l*\\l}) (B1) --++ (150:{\\l*\\l*\\l}) (B2) --++ (150:{\\l*\\l*\\l}) (B2) --++ (30:{\\l*\\l*\\l}) (B3) --++ (150:{\\l*\\l*\\l}) (B3) --++ (30:{\\l*\\l*\\l}) (B4) --++ (30:{\\l*\\l*\\l}) (B4) --++ (-90:{\\l*\\l*\\l});\n\\draw (12,1) node {$\\cdots$};\n\\draw (0,0) node [below] {第$1$阶段};\n\\draw (3,0) node [below] {第$2$阶段};\n\\draw (6,0) node [below] {第$3$阶段};\n\\draw (9,0) node [below] {第$4$阶段};\n\\draw (12,0) node [below] {$\\cdots\\cdots$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求第$3$阶段``黄金分割草''的高度;\\\\\n(2) 求第$13$阶段``黄金分割草''的所有枝干的长度之和;(结果精确到$0.01$米)", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014567": { + "id": "014567", + "content": "已知数列$\\{a_n\\}$中, $a_1=a$($a \\in \\mathbf{R}$, $a \\neq-\\dfrac{1}{2}$), $a_n=2 a_{n-1}+\\dfrac{\\mathbf{1}}{n}+\\dfrac{\\mathbf{1}}{n(n+1)}$($n \\geq 2$, $n \\in \\mathbf{N}$), 又数列$\\{b_n\\}$满足: $b_n=a_n+\\dfrac{1}{n+1}$.\\\\\n(1) 求证: 数列$\\{b_n\\}$是等比数列;\\\\\n(2) 若数列$\\{a_n\\}$是严格增数列, 求实数$a$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014568": { + "id": "014568", + "content": "若数列$\\{a_n\\}$满足$a_1>0$, $a_{n+1} a_n-a_n^2=1$, 且$\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\cdots+\\dfrac{1}{a_{2023}}=2023$, 则\\bracket{20}.\n\\twoch{$20230$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n \\geq m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$, 求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_n\\}$的前$2m$项和;\\\\\n(3) 是否存在$p$、$q$, 使得$b_m=3 m+2$? 如果存在, 求出所有满足条件的$p$、$q$的值; 如果不存在, 请说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课23", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014570": { + "id": "014570", + "content": "书架上层放有$6$本不同的数学书, 下层放有$5$本不同的语文书.\\\\\n(1) 若从中任取一本书, 则共有\\blank{50}种不同的取法;\\\\\n(2) 若从中任取数学书与语文书各一本, 则共有\\blank{50}种不同的取法.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014571": { + "id": "014571", + "content": "将$5$名北京冬奥会志愿者分配到花样滑冰、短道速滑、冰球和冰壶$4$个项目进行培训, 若每名志愿者只分配到$1$个项目, 且每个项目至少分配$1$名志愿者, 则不同的分配方案共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014572": { + "id": "014572", + "content": "已知$n$是大于等于$3$的正整数, 且$\\mathrm{C}_n^2+\\mathrm{C}_n^3=\\mathrm{P}_n^2$, 则$n$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014573": { + "id": "014573", + "content": "设$a \\in \\mathbf{R}$, 在$(2 x+\\dfrac{a}{x})^7$的二项展开式中, 若$x^{-3}$项的系数是$84$, 则$a$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014574": { + "id": "014574", + "content": "用$0,1,2,3,4$这$5$个数字, 组成四位数.\\\\\n(1) 共可以组成\\blank{50}个没有重复数字的四位数;\\\\\n(2) 共可以组成个\\blank{50}没有重复数字的四位偶数.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014575": { + "id": "014575", + "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人, 并且男、女生中各有$1$人是队长.\\\\\n(1) 这$10$名同学站成一排拍照, 若女生都不相邻, 则共有\\blank{50}种不同的排法;\\\\\n(2) 现从这$10$名同学中选出$4$名同学安排到$4$个小区参加志愿者活动, 每个小区一人, 若选出的同学中男生女生都要有, 则共有\\blank{50}种不同的安排方法.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014576": { + "id": "014576", + "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人, 并且男、女生中各有$1$人是队长. 这$10$名同学站成一排拍照, 若男生队长不站排头, 女生队长不站排尾, 则共有\\blank{50}种不同的排法.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014577": { + "id": "014577", + "content": "某社团共有$10$人, 其中男生$6$人, 女生$4$人, 并且男、女生中各有$1$人是队长. 现从这$10$名同学中选出$4$名同学去参加志愿者活动, 既要有队长, 又要有女生的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014578": { + "id": "014578", + "content": "设$(2 x+1)^n=a_0+a_1 x+a_2 x^2+\\cdots+a_n x^n$.\\\\\n(1) 若$a_0+a_1+a_2+\\cdots+a_n=6561$, 求$a_3$的值;\\\\\n(2) 若$n=8$, 求$(a_0+a_2+\\cdots+a_8)^2-(a_1+a_3+\\cdots+a_7)^2$的值;\\\\\n(3) 若$n=15$, 求$a_0, a_1, \\cdots, a_n$中的最大项.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014579": { + "id": "014579", + "content": "在一次运动会上有四项比赛的冠军分别在甲、乙、丙三人中产生, 那么不同的夺冠情况共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014580": { + "id": "014580", + "content": "已知有$4$名男生, $6$名女生, 若从这$10$人中任选$3$人, 则恰有$1$名男生和$2$名女生的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014581": { + "id": "014581", + "content": "若在$(x+\\dfrac{1}{x})^{10}$的二项展开式中第$k$项的系数最大, 则$(2 x+\\dfrac{1}{\\sqrt{x}})^k$的二项展开式中, 常数项是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014582": { + "id": "014582", + "content": "甲乙丙丁戊$5$名同学站成一排参加文艺汇演, 甲不站在两端, 丙和丁相邻的不同排列方式共有\\blank{50}种.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014583": { + "id": "014583", + "content": "已知$x$是小于等于$7$的正整数, 若$\\mathrm{C}_{13}^{2 x-1}=\\mathrm{C}_{13}^{x+2}$, 则$x$的值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014584": { + "id": "014584", + "content": "从$7$个人中选$4$人负责元旦三天假期的值班工作, 若第一天安排$2$人, 第二天和第三天均安排$1$人, 且人员不重复, 则共有\\blank{50}种不同的安排方式. (结果用数值表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014585": { + "id": "014585", + "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": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014586": { + "id": "014586", + "content": "设$(x-1)(x+1)^5=a_0+a_1 x+a_2 x^2+a_3 x^3+\\cdots+a_6 x^6$, 则$a_3=$\\blank{50}.(结果用数值表示)", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014587": { + "id": "014587", + "content": "若$O$是正六边形$A_1A_2A_3A_4A_5A_6$的中心, $B=\\{\\overrightarrow{OA_i} | i=1,2,3,4,5,6\\}$, $\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c} \\in B$, 且$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$互不相等, 要使得$(\\overrightarrow {a}+\\overrightarrow {b}) \\cdot \\overrightarrow {c}=0$, 则有序向量组$(\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c})$的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014588": { + "id": "014588", + "content": "在高三一班元旦晩会上, 有$6$个演唱节目, $4$个舞蹈节目.\\\\\n(1) 若$4$个舞蹈节目排在一起, 则不同的节目安排顺序共有多少种?\\\\\n(2) 若要求每$2$个舞蹈节目之间至少安排$1$个演唱节目, 则不同的节目安排顺序共有多少种?\\\\\n(3) 若已定好节目单, 后来情况有变, 需加上诗歌朗诵和快板$2$个节目, 但不能改变原来节目的相对顺序, 则共有多少种不同的节目演出顺序?", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014589": { + "id": "014589", + "content": "已知数列$\\{a_n\\}$共有$11$项, $a_1=0$, $a_{11}=4$, 且$|a_{k+1}-a_k|=1$($k=1,2,3, \\cdots, 10$), 满足这样条件的不同数列的个数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014590": { + "id": "014590", + "content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n, m$是正整数, 且$m \\leq n$)的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的两个性质: \\textcircled{1} $\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; \\textcircled{2} $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能推广, 则写出推广的形式并给出证明; 若不能, 则说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课25", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014591": { + "id": "014591", + "content": "从甲、乙等$5$名同学中随机选$3$名参加社区服务工作, 则甲、乙两人中至少有一人入选的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014592": { + "id": "014592", + "content": "$100$件产品中有$5$件次品, 不放回地抽取$2$次, 每次抽出$1$件, 已知第一次抽出的是次品, 则第二次抽出正品的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014593": { + "id": "014593", + "content": "袋中有大小与质地均相同的$12$个小球, 分别为红球、黑球、黄球、绿球, 从中任取$1$个球, 若得到红球的概率是$\\dfrac{1}{4}$, 得到黑球或黄球的概率$\\dfrac{5}{12}$, 得到黄球或绿球的概率是$\\dfrac{1}{2}$, 则得到绿球的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014594": { + "id": "014594", + "content": "已知随机变量$X$服从正态分布$N(2, \\sigma^2)$, 若$P(22.5)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014595": { + "id": "014595", + "content": "实力相当的甲、乙两人参加乒乓球比赛, 规定$5$局$3$胜制(即$5$局内谁先赢$3$局就算胜出并停止比赛).\\\\\n(1) 试求甲打完$5$局才获胜的概率;\\\\\n(2) 按比赛规则甲获胜的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014596": { + "id": "014596", + "content": "某工厂有三个车间生产同一产品, 第一车间的次品率为$0.05$, 第一车间的次品率为$0.03$, 第一车间的次品率为$0.01$, 各车间的产品数量分别为$1500$件、$2000$件、$1500$件, 出厂时, 三个车间的产品完全混合, 现从中任取$1$件产品, 求该产品是次品的概率.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014597": { + "id": "014597", + "content": "甲、乙两个学校进行体育比赛, 比赛共设三个项目, 每个项目胜方得$10$分, 负方得$0$分, 没有平局. 三个项目比赛结束后, 总得分高的学校获得冠军. 已知甲学校在三个项目中获胜的概率分别为$0.5$、$0.4$、$0.8$, 各项目的比赛结果相互独立.\n(1) 求甲学校获得冠军的概率;\\\\\n(2) 用$X$表示乙学校的总得分, 求$X$的分布与期望.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014598": { + "id": "014598", + "content": "已知随机变量$X$服从二项分布$B(12,0.25)$, 且$E[a X-3]=3$($a \\in \\mathbf{R}$), 则$D[a X-3]=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014599": { + "id": "014599", + "content": "有若干个大小与质地均相同的红球和白球. 已知甲袋中有$6$个红球, $4$个白球, 乙袋中有$8$个红球, $6$个白球, 若随机取一个袋子, 再从该袋中随机取一个球, 则该球是红球的概率是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014600": { + "id": "014600", + "content": "袋中装有大小与质地均相同的$2$个白球和$3$个黑球.\\\\\n(1) 从中有放回地摸两次, 每次摸$1$个球, 求两球颜色不同的概率;\\\\\n(2) 从中不放回地摸两次, 每次摸$1$个球, 记$X$为摸出两球中白球的个数, 求$X$的期望和方差.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014601": { + "id": "014601", + "content": "哥德巴赫猜想是指``每个大于$2$的偶数都可以表示为两个素数的和'', 例如$10=7+3$, $16=13+3$. 在不超过$32$的所有素数中, 随机选取两个不同的数, 其和等于$32$的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014602": { + "id": "014602", + "content": "袋中有大小与质地均相同的$5$个红球, $4$个白球, 现随机地从中取出一个球, 记录颜色后, 将其放回袋中, 并随之放入$2$个与之颜色相同的球, 再从袋中第二次取出一球, 则第二次取出的是白球的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014603": { + "id": "014603", + "content": "某实验测试的规则如下: 每位学生最多可做$3$次实验, 一旦实验成功, 则停止实验, 否则做完$3$次为止. 设某学生每次实验成功的概率为$p$($01.39$, 则$p$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014604": { + "id": "014604", + "content": "已知两个随机变量$X, Y$, 其中$X \\sim B(4, \\dfrac{1}{4})$, $Y \\sim N(\\mu, \\sigma^2)$($\\sigma>0$), 若$E[X]=E[Y]$, 且$P(|Y|<1)=0.4$, 则$P(Y>3)=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014605": { + "id": "014605", + "content": "某射击小组共有$25$名射手, 其中一级射手$5$人, 二级射手$10$人, 三级射手$10$人, 若一、二、三级射手能通过选拔进入比赛的概率分别是$0.9$, $0.8$, $0.4$, 则从中任选一名射手能通过选拔进入比赛的概率为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014606": { + "id": "014606", + "content": "在四次独立重复试验中, 事件$A$在每次试验中发生的概率相同, 若事件$A$至少发生一次的概率为$\\dfrac{65}{81}$, 则事件$A$发生次数$X$的期望是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014607": { + "id": "014607", + "content": "某批零件的尺寸$X$服从正态分布$N(10, \\sigma^2)$, 且满足$p(X<9)=\\dfrac{1}{6}$, 零件的尺寸与$10$的误差不超过$1$即合格, 从这批产品中随机抽取$n$件, 若要保证抽取的合格零件不少于$2$件的概率不低于$0.9$, 则$n$的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014608": { + "id": "014608", + "content": "甲、乙两人各拿两颗骰子做抛掷游戏, 规则如下: 若掷出的点数之和为$3$的倍数, 原掷骰子的人再继续扶; 若掷出的点数之和不是$3$的倍数, 就由对方接着掷. 第一次由甲开始掷, 求第$n$次由甲掷的概率$P_n$(用含$n$的式子表示).", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课26", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014609": { + "id": "014609", + "content": "如图, $\\triangle ABC$中, $AD \\perp BC$于$D$, $\\angle BAC=45^{\\circ}$, $BD=2$, $CD=1$, 则$\\triangle ABC$的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (-2,0) node [left] {$B$} coordinate (B);\n\\draw (1,0) node [right] {$C$} coordinate (C);\n\\draw (0,0) node [below] {$D$} coordinate (D);\n\\draw (0,{(sqrt(17)+3)/2}) node [above] {$A$} coordinate (A);\n\\draw (B)--(A)--(C)--cycle(A)--(D);\n\\end{tikzpicture}\n\\end{center}", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014610": { + "id": "014610", + "content": "已知实数$a>1$, 则方程$a^x+x-2=0$的根$x_0$满足\\bracket{20}.\n\\fourch{$x_0<-1$}{$-11$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014611": { + "id": "014611", + "content": "已知不等式$2 x-1>m(x^2-1)$对满足$|m| \\leq 2$的一切实数$m$恒成立, 则$x$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014612": { + "id": "014612", + "content": "已知点$A$、$B$、$C$是圆$O$上不同的三点, 线段$OC$与线段$AB$交于点$D$(圆心$O$与点$D$不重合), 若$\\overrightarrow{OC}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$($\\lambda, \\mu \\in \\mathbf{R}$), 则$\\lambda+\\mu$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014613": { + "id": "014613", + "content": "已知$\\{a_n\\}$是首项为$a$, 公差为$1$的等差数列, $b_n=\\dfrac{1+a_n}{a_n}$, 若对于任意正整数$n$, 都有$b_n \\geq b_8$成立, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014614": { + "id": "014614", + "content": "已知$a \\in \\mathbf{R}$, 若关于$x$的方程$\\lg a x=2 \\lg (x-1)$有解, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014615": { + "id": "014615", + "content": "已知$a \\neq 0$, $a \\in \\mathbf{R}$, 若关于$x$的不等式$(a x-1)(x^2-a x-1) \\geq 0$对于任意$x>0$都成立, 则$a=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014616": { + "id": "014616", + "content": "已知$m \\in \\mathbf{R}$, $f(x)=\\begin{cases}2 x^2-x,& x \\leq 0, \\\\ -x^2+x,& x>0,\\end{cases}$ $g(x)=f(x)-m$, 若函数$y=g(x)$恰有三个零点$x_1, x_2, x_3$, 则$x_1 x_2 x_3$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014617": { + "id": "014617", + "content": "已知抛物线$C: y^2=x$的焦点为$F$, $A(x_0, y_0)$是抛物线$C$上一点, 若$|AF|=|\\dfrac{5}{4} x_0|$, 则$x_0=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014618": { + "id": "014618", + "content": "已知$k \\in \\mathbf{R}$, 若不等式$x^2-k x+k-1>0$对任意$x \\in(1,2)$恒成立, 则$k$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014619": { + "id": "014619", + "content": "在矩形$ABCD$中, 边$AB, AD$的长分别为$2,1$. 若点$M, N$分别是边$BC, CD$上的点, 且满足$\\dfrac{|\\overrightarrow{BM}|}{|\\overrightarrow{BC}|}=\\dfrac{|\\overrightarrow{CN}|}{|\\overrightarrow{CD}|}$, 则$\\overrightarrow{AM} \\cdot \\overrightarrow{AN}$的最小值是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014620": { + "id": "014620", + "content": "已知$a \\in \\mathbf{R}$, $A=\\{x | x^2-4 x+3<0\\}$, $B=\\{x | x^2-6 x+8<0\\}$, $C=\\{x | 2 x^2-9 x+a<0\\}$, 若$A \\cap B \\subseteq C$, 则$a$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014621": { + "id": "014621", + "content": "关于$x$的方程$(x^2-1)^2-|x^2-1|+k=0$, 给出下列四个命题:\\\\\n\\textcircled{1} 存在实数$k$, 使得方程恰有$2$个不同的实根;\\\\\n\\textcircled{2} 存在实数$k$, 使得方程恰有$4$个不同的实根;\\\\\n\\textcircled{3} 存在实数$k$, 使得方程恰有$5$个不同的实根;\\\\\n\\textcircled{4} 存在实数$k$, 使得方程恰有$8$个不同的实根, 其中所有真命题的序号是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014622": { + "id": "014622", + "content": "方程$x^2+2 x-1=0$的解可视为函数$y=x+2$的图像与函数$y=\\dfrac{1}{x}$的图像交点的横坐标. 若方程$x^4+a x-4=0$的各个实根$x_1, x_2, \\cdots, x_k$($k \\leq 4$)所对应的点$(x_i, \\dfrac{4}{x_i})$($i=1,2,3, \\cdots, k$)均在直线$y=x$的同侧, 则$a$取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014623": { + "id": "014623", + "content": "已知$a \\in \\mathbf{R}$, $f(x)=\\dfrac{1}{3} x^3-\\dfrac{1}{2} a x^2+x$, 若函数$y=f(x)$在区间$(\\dfrac{1}{2}, 3)$内既有极大值点又有极小值点, 则$a$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014624": { + "id": "014624", + "content": "设数列$\\{a_n\\}$的首项$a_1$为常数, 且$a_1 \\neq \\dfrac{3}{5}$, 又$a_{n+1}=3^n-2 a_n$. 若$\\{a_n\\}$是严格递增数列, 求$a_1$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014625": { + "id": "014625", + "content": "设$y=f(x)$是定义在$\\mathbf{R}$上的偶函数, 其图像关于直线$x=1$成轴对称, 对任意的$x_1, x_2 \\in[0, \\dfrac{1}{2}]$, 都有$f(x_1+x_2)=f(x_1) \\cdot f(x_2)$且$f(1)=a>0$. 设数列$\\{a_n\\}$满足$a_n=f(2 n+\\dfrac{1}{2 n})$($n \\geq 1$, $n \\in \\mathbf{N}$), 求$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "014626": { + "id": "014626", + "content": "设偶函数$y=f(x)$是定义在$\\mathbf{R}$上的可导函数, 且$f(1)=0$. 当$x<0$时, 有$x f'(x)-f(x)>0$恒成立, 则不等式$f(x)>0$的解集为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "014627": { + "id": "014627", + "content": "设$f(x)=\\dfrac{2}{x}+a \\ln x-2$($a>0$), 设函数$y=f(x)$.\\\\\n(1) 若对于任意$x \\in(0,+\\infty)$都有$f(x)>2(a-1)$成立, 求实数$a$的取值范围;\\\\\n(2) 设$b \\in \\mathbf{R}$, 记$g(x)=f(x)+x-b$. 当$a=1$时, 函数$y=g(x)$在区间$[\\mathrm{e}^{-1}, \\mathrm{e}]$上有两个零点, 求$b$的取值范围.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2023年空中课堂高三复习课29", + "edit": [ + "20230221\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, "020001": { "id": "020001", "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.", @@ -429262,7 +431086,7 @@ "content": "已知函数 $f(x)=\\mathrm{e}^x \\ln (1+x)$.\\\\\n(1) 求曲线$y=f(x)$在点$(0, f(0))$处的切线方程;\\\\\n(2) 设$g(x)=f'(x)$, 讨论函数$g(x)$在$[0,+\\infty)$上的单调性;\\\\\n(3) 证明: 对任意$s,t\\in (0,+\\infty)$, 均成立$f(s+t)>f(s)+f(t)$.", "objs": [], "tags": [], - "genre": "", + "genre": "解答题", "ans": "", "solution": "", "duration": -1, @@ -429274,691 +431098,7 @@ "same": [], "related": [], "remark": "", - "space": "" - }, - "040001": { - "id": "040001", - "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040002": { - "id": "040002", - "content": "将参数方程$\\begin{cases}x=1+2 \\cos \\theta, \\\\ y=2 \\sin \\theta\\end{cases}$($\\theta$为参数)化为普通方程, 所得方程是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040003": { - "id": "040003", - "content": "下列参数($t$为参数)方程中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt{|t|}, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sin t, \\\\ y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040004": { - "id": "040004", - "content": "参数方程$\\begin{cases}x=t+\\dfrac{1}{t}, \\\\ y=t-\\dfrac{1}{t}\\end{cases}$表示的曲线是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040005": { - "id": "040005", - "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040006": { - "id": "040006", - "content": "将参数方程$\\begin{cases}x=\\sin \\theta+\\cos \\theta, \\\\ y=\\sin \\theta-\\cos \\theta,\\end{cases}$ $\\theta \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$($\\theta$为参数)化为普通方程是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040007": { - "id": "040007", - "content": "经过点$P(2,1)$, 且倾斜角为$\\dfrac{2 \\pi}{3}$的直线$l$的参数方程是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040008": { - "id": "040008", - "content": "已知直线$l$的参数方程为: $\\begin{cases}x=1+\\dfrac{1}{2} t, \\\\ y=2-\\dfrac{\\sqrt{3}}{2} t\\end{cases}$($t$为参数), 则直线$l$的倾斜角的大小为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040009": { - "id": "040009", - "content": "已知$A(3,1), F$是抛物线$y^2=4 x$的焦点, $P$是抛物线上的一个动点, 则$\\triangle APF$周长的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040010": { - "id": "040010", - "content": "已知长度为$7$的线段$AB$的两个端点在抛物线$x^2=4 y$上运动, 则线段$AB$的中点$G$到$x$轴的距离的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040011": { - "id": "040011", - "content": "过抛物线$C: y^2=4 x$的焦点$F$的直线交$C$于$A$、$B$两点, 过$A$、$B$两点分别作$C$的准线的垂线, 垂足为$A_1$、$B_1$, 以线段$A_1B_1$为直径的圆$E$过点$M(-2,3)$, 则圆$E$的方程为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040012": { - "id": "040012", - "content": "在平面直角坐标系$x O y$中, $O$为坐标原点, 定点$A(-2,3)$, 动点$B$在曲线$x^2+4 y^2=4$上运动, 以$OA$、$OB$为两边作平行四边形$OACB$, 则动点$C$的轨迹方程为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040013": { - "id": "040013", - "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左、右焦点分别是$F_1$、$F_2$, 点$P$是椭圆$C$上的一点且在第一象限, $\\triangle PF_1F_2$的周长为$4+2 \\sqrt{3}$. 过点$P$作椭圆$C$的切线$l$, 分别与$x$轴和$y$轴交于$A$、$B$两点, $O$为坐标原点. 当点$P$在椭圆$C$上移动时, $\\triangle AOB$面积的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040014": { - "id": "040014", - "content": "已知椭圆$C: \\dfrac{x^2}{2}+y^2=1$, 过点$A(0,2)$的直线$l$交椭圆$C$于不同的两点$P$、$Q$. 若$\\overrightarrow{AQ}=\\lambda \\overrightarrow{AP}$, 则实数$\\lambda$的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040015": { - "id": "040015", - "content": "在平面直角坐标系$x O y$中, 若直线$y=k x+1$与抛物线$x^2=2 y$相交于$A$、$B$两点.\\\\\n(1) 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值;\\\\\n(2) 若$\\triangle AOB$的面积为$2$, 求实数$k$的值.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040016": { - "id": "040016", - "content": "已知两圆$C_1: (x-2)^2+y^2=54$, $C_2: (x+2)^2+y^2=6$, 动圆$M$在圆$C_1$内部且和圆$C_1$内切、和圆$C_2$外切.\\\\\n(1) 求动圆圆心$M$的轨迹$C$的方程;\\\\\n(2) 过点$A(3,0)$的直线与(1)中的曲线$C$交于$P$、$Q$两点, 点$P$关于$x$轴对称的点为$R$, 求$\\triangle ARQ$面积的最大值.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040017": { - "id": "040017", - "content": "已知斜率为$k$的直线$l$经过抛物线$C: y^2=4 x$的焦点$F$, 且与抛物线$C$交于不同的两点$A(x_1, y_1)$、$B(x_2, y_2)$.\\\\\n(1) 若点$A$和$B$到抛物线准线的距离分别为$\\dfrac{3}{2}$和$3$, 求$|AB|$;\\\\\n(2) 若$|AF|+|AB|=2|BF|$, 求$k$的值;\\\\\n(3) 点$M(t, 0), t>0$, 对任意确定的实数$k$, 若$\\triangle AMB$是以$AB$为斜边的直角三角形, 判断符合条件的点$M$有几个, 并说明理由.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2024届高二下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040018": { - "id": "040018", - "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) $45^{\\circ}=$\\blank{50};\n(2) $30^{\\circ}=$\\blank{50};\n(3) $18^{\\circ}=$\\blank{50};\n(4) $60^{\\circ}=$\\blank{50};\n(5) $75^{\\circ}=$\\blank{50};\n(6) $12^{\\circ}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040019": { - "id": "040019", - "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) $\\dfrac{\\pi}{3}=$\\blank{50};\n(2) $\\dfrac{\\pi}{5}=$\\blank{50};\n(3) $\\dfrac{\\pi}{4}=$\\blank{50};\n(4) $\\dfrac{5 \\pi}{12}=$\\blank{50};\n(5) $\\dfrac{2 \\pi}{9}=$\\blank{50};\n(6) $\\dfrac{3 \\pi}{10}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040020": { - "id": "040020", - "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+90^{\\circ}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+270^{\\circ}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+210^{\\circ}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\times 180^{\\circ}+45^{\\circ}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\times 90^{\\circ}+30^{\\circ}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040021": { - "id": "040021", - "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{\\pi}{3}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{11 \\pi}{6}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi-\\dfrac{7 \\pi}{6}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\pi-\\dfrac{\\pi}{4}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\cdot \\dfrac{\\pi}{2}+\\dfrac{5 \\pi}{18}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040022": { - "id": "040022", - "content": "下面的各个角$\\beta$与角$\\alpha(0^{\\circ} \\leq \\alpha<360^{\\circ})$的终边重合, 请你写出相应的角$\\alpha$.\\\\\n(1) 设$\\beta=1410^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-120^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=2010^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n. (4) 设$\\beta=-420^{\\circ}$, 则角$\\alpha=$\\blank{100}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040023": { - "id": "040023", - "content": "下面的各个角与角$\\alpha(\\alpha \\in[0,2 \\pi))$的终边重合, 请你写出相应的角$\\alpha$.\\\\0\n(1) 设$\\beta=\\dfrac{22}{3} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-\\dfrac{13}{6} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=10$, 则角$\\alpha=$\\blank{100};\\\\\n(4) 设$\\beta=-10$, 则角$\\alpha=$\\blank{100}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040024": { - "id": "040024", - "content": "在等差数列$\\{a_n\\}$中, $a_5=6, a_{10}=12$, 则$a_{15}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040025": { - "id": "040025", - "content": "若数列$\\{a_n\\}$为等差数列, $a_5=9, a_{11}=-3$, 则$a_8=$\\blank{50}, 公差$d=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040026": { - "id": "040026", - "content": "等差数列$\\{a_n\\}$中, $a_1=51, a_2=49$.\\\\\n(1) 设$-2021$是数列$\\{a_n\\}$的的第$m$项, 则$m=$\\blank{50};\\\\\n(2) 数列$\\{a_n\\}$中的偶数项依次构成数列$\\{b_n\\}$, 则$\\{b_n\\}$的第$k$项$b_k=$\\blank{50};\\\\\n(3) 设数列$\\{a_n\\}$在区间$[-999,0]$内共有$t$项, 则$t=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040027": { - "id": "040027", - "content": "等差数列$\\{a_n\\}$的公差小于 0 , 且有$a_2 \\cdot a_4=12, a_2+a_4=8$, 则通项$a_n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040028": { - "id": "040028", - "content": "等差数列$\\{a_n\\}$中, $a_3+a_4+a_{10}+a_{11}=20$, 则$a_5+a_7+a_9=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040029": { - "id": "040029", - "content": "在首项为 40 , 公差为$-7$的等差数列$\\{a_n\\}$中, 绝对值最小的项的序数为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040030": { - "id": "040030", - "content": "设常数$d \\in \\mathbf{R}$. 已知等差数列$\\{a_n\\}$的公差是$d$, 首项$a_1=1$. 若$a_8$是第一个比$29$大的项, 则$d$的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040031": { - "id": "040031", - "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 根据$S_n$, 求$\\{a_n\\}$的通项公式.\n(1) 若$S_n=n^2$, 则$a_n=$\\blank{50};\\\\\n(2) 若$S_n=n^2+1$, 则$a_n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040032": { - "id": "040032", - "content": "设常数$m, n \\in \\mathbf{R}$. 已知关于$x$的方程$(x^2-4 x+m)(x^2-4 x+n)=0$的四个根组成一个首项为$1$的等差数列, 则数对$(m, n)$为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040033": { - "id": "040033", - "content": "数列$\\{a_n\\}$对于任意正整数$p, q$, 恒有$a_p+a_q=a_{p+q}$, 若$a_1=2$, 则$a_{100}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040034": { - "id": "040034", - "content": "已知数列$\\{a_n\\}$中, $a_n=3^n-n$, 求证: 数列$\\{a_n\\}$是严格增数列.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040035": { - "id": "040035", - "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n, S_n=\\begin{cases}n^2,& n=2 k-1, \\\\ n^2+1,& n=2 k,\\end{cases}$ ($k \\in \\mathbf{N}$, $k\\ge 1$), 求$\\{a_n\\}$的通项公式.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" - }, - "040036": { - "id": "040036", - "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别是$a_n=2 n+1, b_n=3 n$, $n \\in \\mathbf{N}$, $n\\ge 1$. 将集合$\\{x | x=a_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\} \\cap \\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$中的元素从小到大依次排列, 构成数列$c_1, c_2, \\cdots, c_n, \\cdots$, 求数列$\\{c_n\\}$的通项公式.", - "objs": [], - "tags": [], - "genre": "", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一下学期周末卷01", - "edit": [ - "20230218\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "" + "space": "12ex" }, "031237": { "id": "031237", @@ -430063,5 +431203,689 @@ ], "remark": "", "space": "" + }, + "040001": { + "id": "040001", + "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\ y=t^2-2\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{一支双曲线}{线段}{圆弧}{射线}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040002": { + "id": "040002", + "content": "将参数方程$\\begin{cases}x=1+2 \\cos \\theta, \\\\ y=2 \\sin \\theta\\end{cases}$($\\theta$为参数)化为普通方程, 所得方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040003": { + "id": "040003", + "content": "下列参数($t$为参数)方程中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt{|t|}, \\\\ y=t\\end{cases}$}{$\\begin{cases}x=\\sin t, \\\\ y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\ y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", + "objs": [], + "tags": [], + "genre": "选择题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040004": { + "id": "040004", + "content": "参数方程$\\begin{cases}x=t+\\dfrac{1}{t}, \\\\ y=t-\\dfrac{1}{t}\\end{cases}$表示的曲线是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040005": { + "id": "040005", + "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\ y=\\sqrt{2} \\sin \\theta\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040006": { + "id": "040006", + "content": "将参数方程$\\begin{cases}x=\\sin \\theta+\\cos \\theta, \\\\ y=\\sin \\theta-\\cos \\theta,\\end{cases}$ $\\theta \\in[\\dfrac{3 \\pi}{4}, \\dfrac{5 \\pi}{4}]$($\\theta$为参数)化为普通方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040007": { + "id": "040007", + "content": "经过点$P(2,1)$, 且倾斜角为$\\dfrac{2 \\pi}{3}$的直线$l$的参数方程是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040008": { + "id": "040008", + "content": "已知直线$l$的参数方程为: $\\begin{cases}x=1+\\dfrac{1}{2} t, \\\\ y=2-\\dfrac{\\sqrt{3}}{2} t\\end{cases}$($t$为参数), 则直线$l$的倾斜角的大小为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040009": { + "id": "040009", + "content": "已知$A(3,1), F$是抛物线$y^2=4 x$的焦点, $P$是抛物线上的一个动点, 则$\\triangle APF$周长的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040010": { + "id": "040010", + "content": "已知长度为$7$的线段$AB$的两个端点在抛物线$x^2=4 y$上运动, 则线段$AB$的中点$G$到$x$轴的距离的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040011": { + "id": "040011", + "content": "过抛物线$C: y^2=4 x$的焦点$F$的直线交$C$于$A$、$B$两点, 过$A$、$B$两点分别作$C$的准线的垂线, 垂足为$A_1$、$B_1$, 以线段$A_1B_1$为直径的圆$E$过点$M(-2,3)$, 则圆$E$的方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040012": { + "id": "040012", + "content": "在平面直角坐标系$x O y$中, $O$为坐标原点, 定点$A(-2,3)$, 动点$B$在曲线$x^2+4 y^2=4$上运动, 以$OA$、$OB$为两边作平行四边形$OACB$, 则动点$C$的轨迹方程为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040013": { + "id": "040013", + "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左、右焦点分别是$F_1$、$F_2$, 点$P$是椭圆$C$上的一点且在第一象限, $\\triangle PF_1F_2$的周长为$4+2 \\sqrt{3}$. 过点$P$作椭圆$C$的切线$l$, 分别与$x$轴和$y$轴交于$A$、$B$两点, $O$为坐标原点. 当点$P$在椭圆$C$上移动时, $\\triangle AOB$面积的最小值为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040014": { + "id": "040014", + "content": "已知椭圆$C: \\dfrac{x^2}{2}+y^2=1$, 过点$A(0,2)$的直线$l$交椭圆$C$于不同的两点$P$、$Q$. 若$\\overrightarrow{AQ}=\\lambda \\overrightarrow{AP}$, 则实数$\\lambda$的取值范围为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040015": { + "id": "040015", + "content": "在平面直角坐标系$x O y$中, 若直线$y=k x+1$与抛物线$x^2=2 y$相交于$A$、$B$两点.\\\\\n(1) 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值;\\\\\n(2) 若$\\triangle AOB$的面积为$2$, 求实数$k$的值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "040016": { + "id": "040016", + "content": "已知两圆$C_1: (x-2)^2+y^2=54$, $C_2: (x+2)^2+y^2=6$, 动圆$M$在圆$C_1$内部且和圆$C_1$内切、和圆$C_2$外切.\\\\\n(1) 求动圆圆心$M$的轨迹$C$的方程;\\\\\n(2) 过点$A(3,0)$的直线与(1)中的曲线$C$交于$P$、$Q$两点, 点$P$关于$x$轴对称的点为$R$, 求$\\triangle ARQ$面积的最大值.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "040017": { + "id": "040017", + "content": "已知斜率为$k$的直线$l$经过抛物线$C: y^2=4 x$的焦点$F$, 且与抛物线$C$交于不同的两点$A(x_1, y_1)$、$B(x_2, y_2)$.\\\\\n(1) 若点$A$和$B$到抛物线准线的距离分别为$\\dfrac{3}{2}$和$3$, 求$|AB|$;\\\\\n(2) 若$|AF|+|AB|=2|BF|$, 求$k$的值;\\\\\n(3) 点$M(t, 0), t>0$, 对任意确定的实数$k$, 若$\\triangle AMB$是以$AB$为斜边的直角三角形, 判断符合条件的点$M$有几个, 并说明理由.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2024届高二下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "040018": { + "id": "040018", + "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) $45^{\\circ}=$\\blank{50};\n(2) $30^{\\circ}=$\\blank{50};\n(3) $18^{\\circ}=$\\blank{50};\n(4) $60^{\\circ}=$\\blank{50};\n(5) $75^{\\circ}=$\\blank{50};\n(6) $12^{\\circ}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040019": { + "id": "040019", + "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) $\\dfrac{\\pi}{3}=$\\blank{50};\n(2) $\\dfrac{\\pi}{5}=$\\blank{50};\n(3) $\\dfrac{\\pi}{4}=$\\blank{50};\n(4) $\\dfrac{5 \\pi}{12}=$\\blank{50};\n(5) $\\dfrac{2 \\pi}{9}=$\\blank{50};\n(6) $\\dfrac{3 \\pi}{10}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040020": { + "id": "040020", + "content": "请将下列的角的单位从角度制化为弧度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+90^{\\circ}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+270^{\\circ}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$k \\times 360^{\\circ}+210^{\\circ}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\times 180^{\\circ}+45^{\\circ}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\times 90^{\\circ}+30^{\\circ}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040021": { + "id": "040021", + "content": "请将下列的角的单位从弧度制化为角度制:\\\\\n(1) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{\\pi}{3}=$\\blank{50};\\\\\n(2) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi+\\dfrac{11 \\pi}{6}=$\\blank{50};\\\\\n(3) 设$k \\in \\mathbf{Z}$, 则角$2 k \\pi-\\dfrac{7 \\pi}{6}=$\\blank{50};\\\\\n(4) 设$k \\in \\mathbf{Z}$, 则角$k \\pi-\\dfrac{\\pi}{4}=$\\blank{50};\\\\\n(5) 设$k \\in \\mathbf{Z}$, 则角$k \\cdot \\dfrac{\\pi}{2}+\\dfrac{5 \\pi}{18}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040022": { + "id": "040022", + "content": "下面的各个角$\\beta$与角$\\alpha(0^{\\circ} \\leq \\alpha<360^{\\circ})$的终边重合, 请你写出相应的角$\\alpha$.\\\\\n(1) 设$\\beta=1410^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-120^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=2010^{\\circ}$, 则角$\\alpha=$\\blank{100};\\\\\n. (4) 设$\\beta=-420^{\\circ}$, 则角$\\alpha=$\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040023": { + "id": "040023", + "content": "下面的各个角与角$\\alpha(\\alpha \\in[0,2 \\pi))$的终边重合, 请你写出相应的角$\\alpha$.\\\\0\n(1) 设$\\beta=\\dfrac{22}{3} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(2) 设$\\beta=-\\dfrac{13}{6} \\pi$, 则角$\\alpha=$\\blank{100};\\\\\n(3) 设$\\beta=10$, 则角$\\alpha=$\\blank{100};\\\\\n(4) 设$\\beta=-10$, 则角$\\alpha=$\\blank{100}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040024": { + "id": "040024", + "content": "在等差数列$\\{a_n\\}$中, $a_5=6, a_{10}=12$, 则$a_{15}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040025": { + "id": "040025", + "content": "若数列$\\{a_n\\}$为等差数列, $a_5=9, a_{11}=-3$, 则$a_8=$\\blank{50}, 公差$d=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040026": { + "id": "040026", + "content": "等差数列$\\{a_n\\}$中, $a_1=51, a_2=49$.\\\\\n(1) 设$-2021$是数列$\\{a_n\\}$的的第$m$项, 则$m=$\\blank{50};\\\\\n(2) 数列$\\{a_n\\}$中的偶数项依次构成数列$\\{b_n\\}$, 则$\\{b_n\\}$的第$k$项$b_k=$\\blank{50};\\\\\n(3) 设数列$\\{a_n\\}$在区间$[-999,0]$内共有$t$项, 则$t=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040027": { + "id": "040027", + "content": "等差数列$\\{a_n\\}$的公差小于 0 , 且有$a_2 \\cdot a_4=12, a_2+a_4=8$, 则通项$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040028": { + "id": "040028", + "content": "等差数列$\\{a_n\\}$中, $a_3+a_4+a_{10}+a_{11}=20$, 则$a_5+a_7+a_9=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040029": { + "id": "040029", + "content": "在首项为 40 , 公差为$-7$的等差数列$\\{a_n\\}$中, 绝对值最小的项的序数为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040030": { + "id": "040030", + "content": "设常数$d \\in \\mathbf{R}$. 已知等差数列$\\{a_n\\}$的公差是$d$, 首项$a_1=1$. 若$a_8$是第一个比$29$大的项, 则$d$的取值范围是\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040031": { + "id": "040031", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 根据$S_n$, 求$\\{a_n\\}$的通项公式.\n(1) 若$S_n=n^2$, 则$a_n=$\\blank{50};\\\\\n(2) 若$S_n=n^2+1$, 则$a_n=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040032": { + "id": "040032", + "content": "设常数$m, n \\in \\mathbf{R}$. 已知关于$x$的方程$(x^2-4 x+m)(x^2-4 x+n)=0$的四个根组成一个首项为$1$的等差数列, 则数对$(m, n)$为\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040033": { + "id": "040033", + "content": "数列$\\{a_n\\}$对于任意正整数$p, q$, 恒有$a_p+a_q=a_{p+q}$, 若$a_1=2$, 则$a_{100}=$\\blank{50}.", + "objs": [], + "tags": [], + "genre": "填空题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "" + }, + "040034": { + "id": "040034", + "content": "已知数列$\\{a_n\\}$中, $a_n=3^n-n$, 求证: 数列$\\{a_n\\}$是严格增数列.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "040035": { + "id": "040035", + "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n, S_n=\\begin{cases}n^2,& n=2 k-1, \\\\ n^2+1,& n=2 k,\\end{cases}$ ($k \\in \\mathbf{N}$, $k\\ge 1$), 求$\\{a_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" + }, + "040036": { + "id": "040036", + "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$的通项公式分别是$a_n=2 n+1, b_n=3 n$, $n \\in \\mathbf{N}$, $n\\ge 1$. 将集合$\\{x | x=a_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\} \\cap \\{x | x=b_n,\\ n \\in \\mathbf{N}, \\ n\\ge 1\\}$中的元素从小到大依次排列, 构成数列$c_1, c_2, \\cdots, c_n, \\cdots$, 求数列$\\{c_n\\}$的通项公式.", + "objs": [], + "tags": [], + "genre": "解答题", + "ans": "", + "solution": "", + "duration": -1, + "usages": [], + "origin": "2025届高一下学期周末卷01", + "edit": [ + "20230218\t王伟叶" + ], + "same": [], + "related": [], + "remark": "", + "space": "12ex" } } \ No newline at end of file