# -*- coding:utf-8 -*- import random, hashlib import requests import configs import shutil from utils.diffi_label import get_item_diff from utils.img2latex import get_ocrlatex_by_url from utils.qcloud_bucket import upload_img_to_qcloud, img_inbucket_count, client, filestream_upload from concurrent.futures import ProcessPoolExecutor from concurrent.futures import ThreadPoolExecutor, ProcessPoolExecutor from utils.image_convert import svg2png from utils.label_data_Hphy import Label from bisect import bisect_left import re, time, os, json, datetime from copy import deepcopy import traceback logger = configs.myLog(__name__, log_cate="ruku_log").getlog() subject_id = { "高中数学": 3, "高中英语": 8, "高中物理": 12, "高中化学": 13, "高中生物": 14, "高中政治": 15, "高中历史": 16, "高中地理": 17, "初中数学": 41, "初中英语": 42, "初中物理": 43, "初中化学": 44, "初中生物": 45, "初中地理": 46, "初中政治": 47, "初中历史": 48, "高中语文": 9, "初中语文": 40, } class Ruku(): def __init__(self, items_list, htmlt, svg_data, wordid, callback_info, subject=""): self.items_list = items_list self.htmlt = htmlt self.svg_data = svg_data self.wordid = wordid self.callback_url = callback_info["callback_url"] self.source = callback_info["source"] self.subject = subject # items_list[0]["period"] + items_list[0]["subject"] self.callback_code = 0 self.callback_err = "" self.ltx2url = {} # latex 映射 线上可访问url def bucket_img_del(self): """ wordbin中图片上传腾讯云的原因：1.存在线上服务器的图片要定期删除，而有的题可能还没审核完；2.上传腾讯云比较便宜 删除腾讯云中的图片 根据【解析结果】文本中删除的图片信息，通过阙值判断是否从腾讯云中删除图片 也有限制：解析结果不能为空，结果中出现的图片在原试卷应该也要有！ 有风险：对于已经校对过的试卷，如果后期有人在资源库里删除了大量文本，很可能导致正在使用的图片被删除 暂时第三方上传的试卷还是先不要删除比较稳妥！或新建图片用md5命名 :return: """ items_str = str(self.items_list) raw_imgs = [] # img_source = "" http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_23.png img_source = "" items_str = re.sub(r'( 5: # 多余图片超过5张才开始删除，包含mathjax生成的图片 bucket_imgs = [i['Key'] for i in bucket_imgs] # dict:[{'Key': put_key}, {}] to_del_imgs = [bimg for bimg in bucket_imgs if bimg not in raw_imgs] if to_del_imgs: objects = { "Quiet": "true", "Object": [{'Key': item} for item in to_del_imgs] } client.delete_objects(configs.public_bucket, objects) # 批量删除 logger.info("----【paper_id:{}】删除桶数据时间：{}".format(self.wordid, time.time() - time2)) def mathjx2png(self): """ 【基于mathjax渲染输出是svg格式】 将mathjax渲染的公式转化为图片格式 mathjax渲染的svg图片提取->保存->格式转化 :return: """ file_path = configs.IMG_FOLDER + "/" + str(self.wordid) svgp_ltx = {} # svg图片本地路径 映射 latex put_key_mjmath = [] # 桶中key local_mjmath = [] # 本地图片存储位置 ltx2svgcss = {} # latex 映射 svg-css if self.callback_url and "MathJax" in str(self.svg_data["svg_html_data"]): # 再解析中存在mathjax公式渲染的标签 # 需要统计mathjax转png失败时的具体定位 # 题号所在位置 topicinfo = [[nm.end(), nm.group(1)] for nm in re.finditer('class="topic-number">(\d+)\.', str(self.svg_data["svg_html_data"]))] topicidx = [i[0] for i in topicinfo] topicno = [i[1] for i in topicinfo] time3 = time.time() all_mathjax = [[a.start(), a.group(1)] for a in re.finditer('(()*)', str(self.svg_data["svg_html_data"]))] # all_mathjax = re.findall('(()*)', # str(self.svg_data["svg_html_data"])) all_linkdata = re.findall('()', self.svg_data["svg_path"]) link_dict = {a[1]: a[0] for a in all_linkdata if a} for n, jax in enumerate(all_mathjax): svgs = re.findall("", jax[1]) latex = re.findall(''.format( shape[0] / 2, shape[1] / 2, "${}$".format(mj_ltx)) self.ltx2url[mj_ltx] = mjmath_online logger.info(json.dumps({"log_level": "info", "paper_id": self.wordid, "svg2png_time": time.time() - time4, "ltx2url": self.ltx2url}, ensure_ascii=False)) return put_key_mjmath, local_mjmath def upload_img(self): """ items_list：结构化纯文本 htmlt:ocr或word解析后的html文本 svg_data：{"svg_html_data": "", "svg_path": ""} 含svg数据的结构化整体html文本,svg中的索引数据 wordid：试卷存储id callback_url: 回调地址 入库操作包含： 1、再解析后确认入库时，将image上传,htmlt中图片； 2、学管端可以是组合（文本+图片）的解析结果。学管可以选择和修改文本，但选择的文本解析时都会替换原先的img标签内容， 最后入库时，传回学管端的都是带img标签的结果 3、传回学管端和共享题库的题目中的$$公式要转换为图片； $$左右还有公式字符串的话，在结构化过程中一起并入！！！ 4、调取查重关联功能，暂时不调自动标注； 5、传入校本题库（发送标注），将结构化后的每道题设置个与大数据资源库的关联标签，如"zyk_id":id+题号 5种图片：1>>ser_static/.*?/word/media(解析服务中最开始保存在线上服务器本地) 2>>/zyk/uploadfiles/wording/(解析服务中上传到腾讯云) 3、4>>ser_static/.*?/(new_image[^"]*?|eq_img_\d.png):批量再解析和单题再解析中， 用户在编辑页面新粘贴进来的base64图片，以及批量再解析中域公式转图片，临时存在结构化服务器本地 5、利用mathjax渲染latex的svg格式转为png，再上传到腾讯云 上传之前，将腾讯云桶里的new_image全部替换，raw_image判断下再替换 items_list:list 所有题目 :return: res_zyk = {"data":{"html": xxx, "items": xxx}, "errcode": 0, "errmsgs": ""} res_xbk = {"items": xxx, "errcode": 0, "errmsg":"ok"} """ res_xbk = {"items": [], "errcode": 0, "errmsg": "ok", "callback_type": 2, "subject_id": self.subject} res_zyk = {"errcode": 0, "errmsgs": "", "data": {"html": self.htmlt, "items": self.items_list}} # 1>>判断删除腾讯云桶内图片 if self.source not in ["teacher", "xue_guan"]: # {"xue_guan": "1", "teacher": "2", "ai": "3"} self.bucket_img_del() # --------------------------------------------------------------------- # 1.2>>将zyk/uploadfiles/wording/ 路径的缺latex的公式图片，再调mathpix接口拿到latex填充 # 2>>mathjax渲染的svg图片提取->保存->格式转化 svg_convert_res = self.mathjx2png() if type(svg_convert_res) == str: res_xbk = {"items": [], "errcode": 1, "errmsg": svg_convert_res, "callback_type": 2, "subject_id": self.subject} res_zyk = {"errcode": 1, "errmsgs": svg_convert_res, "data": {}} if self.callback_url: self.callback_user(res_xbk, self.callback_url) if self.source == "xue_guan": self.callback_php(res_xbk) # 失败时就不发送数据库端了,但学管端时发送 return res_zyk put_key_mjmath, local_mjmath = svg_convert_res # -------------------------------------------------------------------- # 3>> 统计 ocr和结构化试题 中的新图片，以防结构化入库的试题少了而导致图片上传不足，左边页面无法显示 put_key_list = [] # 桶中key localnewpic_list = [] # 本地图片存储位置 imgs_url_list = [] # 远程服务器上图片存储位置 put_key_imgsurl = [] new_imgs = re.findall(r'"\s*/>', r" data-latex=\1 />", s) return s def ltx2url_repl(one_items): """ 将每道题结构中的各字段进行匹配替换 ltx2url """ keys_items = ["stem", "key", "parse", "options", "slave"] if self.ltx2url: for k in keys_items: if k in one_items: if k == "options": one_items[k] = list(map(sub2, one_items[k])) elif k == "slave": for slave_one in one_items[k]: ltx2url_repl(slave_one) else: one_items[k] = sub2(one_items[k]) # else: # logger.info("----【word_id:{}】第{}道题{}字段有问题".format(self.wordid, one_items["topic_num"], k)) time6 = time.time() # 4>> 结构化题目中图片地址替换,需要区分下学管端还是云题库！！！！！一定会保存一份在资源库 items_res_to_zyk = self.items_list.copy() items_res_to_zyk = eval(sub1(items_res_to_zyk)) # for one_items in items_res_to_zyk: # for k in ["stem", "key", "parse", "options"]: # "analysis", # if k in one_items: # if k == "options": # one_items[k] = list(map(sub1, one_items[k])) # else: # one_items[k] = sub1(one_items[k]) # -----------难度和知识点自动标注------------------------ # diffs_xbk, items_res_to_zyk = self.get_diff(items_res_to_zyk.copy()) # 难度 for nn, one_items in enumerate(deepcopy(items_res_to_zyk)): new_one_item = {} # if "slave" not in one_items and re.findall("_{2,}", one_items["stem"]): # new_one_item["blank_num"] = len(re.findall("_{2,}", one_items["stem"])) if self.callback_url: # 查重 if self.source != "xue_guan": repeat_r, repeat_time = self.repeat_check(nn, one_items, repeat_time) if type(repeat_r) == str: res_xbk = {"items": [], "errcode": 1, "errmsg": repeat_r, "callback_type": 2, "subject_id": self.subject} res_zyk = {"errcode": 1, "errmsgs": repeat_r, "data": {}} self.callback_user(res_xbk, self.callback_url) return res_zyk new_one_item["repeat_res"] = repeat_r new_one_item["topic_type_id"] = one_items["checkType"]["id"] if "options_rank" in one_items and one_items["options_rank"]: new_one_item["options_rank"] = one_items["options_rank"] ltx2url_repl(one_items) self.comans2parse(one_items) new_one_item.update(one_items) items_res_to_xbk.append(new_one_item) # 5>> ocr-htmlt中图片地址替换成云上地址 self.htmlt = re.sub(r'(> new_image上传腾讯云 try: logger.info('----【paper_id:{}】再解析开始上传图片到cloud，并替换成线上地址----'.format(self.wordid)) stime_u = time.time() # TODO 一个进程解析，一个进程上传 executor1 = ThreadPoolExecutor(5) executor1.map(upload_img_to_qcloud, zip(put_key_list, localnewpic_list)) if imgs_url_list: executor1.map(filestream_upload, zip(put_key_imgsurl, imgs_url_list)) executor1.shutdown(wait=True) # 进程池内部的进程都执行完毕，才会关闭，然后执行后续代码 img_upload_time = time.time() - stime_u logger.info(json.dumps({"log_level": "info", "paper_id": self.wordid, "img_upload_time": img_upload_time}, ensure_ascii=False)) res_zyk = {"data": {"html": self.htmlt, "items": items_res_to_zyk}, "errcode": 0, "errmsgs": ""} res_xbk = {"items": items_res_to_xbk, "errcode": 0, "errmsg": "ok", "callback_type": 2, "subject_id": self.subject} except Exception as e: res_xbk = {"items": [], "errcode": 1, "errmsg": "公式或图片上传腾讯云失败", "callback_type": 2, "subject_id": self.subject} res_zyk = {"data":{"html": self.htmlt, "items": self.items_list}, "errcode": 1, "errmsgs": "公式或图片上传腾讯云失败"} logger.info(json.dumps({"log_level": "warn", "paper_id": self.wordid, "status": "公式或图片上传腾讯云失败", "errmsg": str(e)}, ensure_ascii=False)) else: # 本地没有新图片时 # -----------难度、知识点自动标注------------------------ # diffs_xbk, self.items_list = self.get_diff(self.items_list) # 难度 if self.callback_url: for nn, one_items in enumerate(self.items_list): new_one_item = {} # 查重 if self.source != "xue_guan": repeat_r, repeat_time = self.repeat_check(nn, one_items, repeat_time) if type(repeat_r) == str: res_xbk = {"items": [], "errcode": 1, "errmsg": repeat_r, "callback_type": 2, "subject_id": self.subject} res_zyk = {"errcode": 1, "errmsgs": repeat_r, "data": {}} self.callback_user(res_xbk, self.callback_url) return res_zyk new_one_item["repeat_res"] = repeat_r new_one_item["topic_type_id"] = one_items["checkType"]["id"] if "options_rank" in one_items and one_items["options_rank"]: new_one_item["options_rank"] = one_items["options_rank"] self.comans2parse(one_items) new_one_item.update(one_items) items_res_to_xbk.append(new_one_item) res_xbk = {"items": items_res_to_xbk, "errcode": 0, "errmsg": "ok", "callback_type": 2, "subject_id": self.subject} logger.info(json.dumps({"log_level": "info", "paper_id": self.wordid, "repeat_time": repeat_time}, ensure_ascii=False)) # 先传一份到校本题库，再返回结果 if self.callback_url: # 上传腾讯云失败，将结果保存本地一份 self.callback_err = "" self.callback_code = 0 self.save_post_file(res_xbk) have_err = self.callback_user(res_xbk, self.callback_url) if self.source == "xue_guan" or not have_err: # 失败时就不发送数据库端了,但学管端时发 self.callback_php(res_xbk) return res_zyk def comans2parse(self, one_items): """ 将小问答案没拆出来的部分放入公共解析字段中 :return: """ # 此阶段将每个题可能的报错信息去掉 if "errmsgs" in one_items and one_items["errmsgs"]: one_items["errmsgs"] = "" # 将小问答案没拆出来的部分放入公共解析字段中 if "slave" in one_items and one_items["slave"]: if one_items["key"] and all( [True if not s["key"].strip() else False for s in one_items["slave"]]): # 带小题的且小题答案都为空 if one_items["parse"].strip(): one_items["parse"] = "【答案】\n{}\n【解析】\n{}".format(one_items["key"], one_items["parse"]) else: one_items["parse"] = "【答案】\n{}".format(one_items["key"]) one_items["key"] = "" for one_slave in one_items["slave"]: if "slave" in one_slave and one_slave["slave"] and one_slave["key"] and \ all([True if not s["key"].strip() else False for s in one_slave["slave"]]): # 带小题的且小题答案都为空 if one_slave["parse"].strip(): one_slave["parse"] = "【答案】\n{}\n【解析】\n{}".format(one_slave["key"], one_slave["parse"]) else: one_slave["parse"] = "【答案】\n{}".format(one_slave["key"]) one_slave["key"] = "" def save(self): """ 入口函数 :return: """ res_xbk = {"items": [], "errcode": 1, "errmsg": "保存失败", "callback_type": 2, "subject_id": self.subject} try: res_zyk = self.upload_img() return res_zyk except Exception as e: print("------【paper_id:{}】最后的保存失败:{}-------".format(self.wordid, str(e))) print(traceback.print_exc()) logger.info(json.dumps({"log_level": "warn", "paper_id": self.wordid, "status": "入库失败", "errmsg": str(e), "task_name": "保存入库"}, ensure_ascii=False)) if self.callback_url: self.callback_code = 1 self.callback_err = "保存失败" self.callback_user(res_xbk, self.callback_url, err="1") if self.source == "xue_guan": self.callback_php(res_xbk) # 失败时就不发送数据库端了,但学管端时发 # else: # logger.info("------【paper_id:{}】保存失败-------".format(self.wordid)) # self.callback_code = 1 # self.callback_err = "保存失败" # self.callback_user(res_xbk, self.callback_url) # self.callback_php(res_xbk) return {"errcode": 1, "errmsgs": "入库失败！", "data": {}} def save_post_file(self, parse_res): """保存回调的数据解析结果""" now_time = datetime.datetime.now() time_str = datetime.datetime.strftime(now_time, '%Y_%m_%d_%H_%M_%S') # aft_modify = (str("word_name") + '__' + str(random.random())).encode("utf-8") # hh = hashlib.md5(aft_modify).hexdigest() aft_name = self.wordid + '__' + time_str + '.json' res_folder = configs.RES_FOLDER # 保存的文件夹 new_fpath = os.path.join(res_folder, aft_name) re_f = open(new_fpath, 'w', encoding='utf-8') json.dump(parse_res, re_f,ensure_ascii=False) def get_phy_kps_auto(self, one_items, kps_label_time): """ 调取物理自动标注的端口获取考点 :return: """ def phy_kps(input_data): label_auto = {} if input_data: r = requests.post(configs.kps_phy_ip, json=input_data) # json={"input_data": item} if r.status_code == 200: res = r.json()['result'] # eval(r.text) if res: label_auto.update(res) return label_auto t1 = time.time() temp_items = {"server_state": 11, "fine_state": 0, "add_parse_state": 1} one_info = {"topic_id": one_items["topic_num"], "content": one_items["stem"], "option": one_items["options"] if "options" in one_items else [], "parse": one_items["parse"], "kpl_codes": [], "ori_sences": [], "ori_kpl_codes": [], "topic_type_id": one_items["checkType"]["id"]} temp_items["info"] = one_info # 散题 # temp_items["parse"] = str(one_items["key"]) + "
" + one_items["parse"] # if "analysis" in one_items: # temp_items["parse"] += "
" + one_items["analysis"] auto_kps = phy_kps(temp_items) print("自动标注考点：", auto_kps) kps_label_time += time.time() - t1 return auto_kps, kps_label_time def img2latex(self): # 3>> 统计结构化试题data-latex为空的图片，再ocr-latex处理并替换 time5 = time.time() items_str = str(self.items_list) all_imgs_no_latex = re.findall(r'(回调结果: {} -------\n".format(self.wordid, r.status_code, r.text)) if r.status_code != 200: have_err = 1 if r.text != "解析完成！": have_err = 1 logger.info(json.dumps( {"log_level": "info", "paper_id": self.wordid, "callback_res": {"status": "回调成功" if r.status_code == 200 else "回调失败", "status_code": r.status_code, "text": r.text, "callback_url": callback_url}, "task_name": "入库回调"}, ensure_ascii=False)) except Exception as e: self.callback_err += "回调出错" self.callback_code = 1 have_err = 1 print("------【paper_id:{}】回调出错:{}-------".format(self.wordid, e)) logger.info(json.dumps( {"log_level": "warn", "paper_id": self.wordid, "callback_res": {"status": "回调失败", "callback_url": callback_url}, "errmsg": str(e), "task_name": "入库回调"}, ensure_ascii=False)) else: if r.status_code != 200: have_err = 1 self.callback_code = 1 self.callback_err = "回调出错:{}".format(r.status_code) logger.info(json.dumps( {"log_level": "warn", "paper_id": self.wordid, "callback_res": {"status": "回调失败", "status_code": r.status_code, "text": r.text, "callback_url": callback_url}, "task_name": "入库回调"}, ensure_ascii=False)) return have_err def callback_php(self, res_xbk): """ 将解析校对完的试题返回数据库端 :param res_xbk: 返回给其它端的数据 :return: """ if res_xbk["errmsg"] not in ["ok", "保存失败"]: # 公式上传腾讯云失败等 self.callback_err += res_xbk["errmsg"] if self.callback_err: self.callback_code = 1 type_check = {"errcode": self.callback_code, "errmsg": self.callback_err, "parseId": self.wordid, "callback_type": 2, "data": [k["checkType"]["id"] for k in self.items_list], } try: r2 = requests.post(configs.callback_url_taskcheck, data=json.dumps(type_check, ensure_ascii=False).encode("utf-8")) print("------【paper_id:{}】taskcheck post数据：{} 回调地址状态：{}===>回调结果: {}-------".format( self.wordid, type_check, r2.status_code, r2.text)) logger.info(json.dumps({"log_level": "info", "paper_id": self.wordid, "callback_data": type_check, "callback_res": {"status": "回调成功" if r2.status_code == 200 else "回调失败", "status_code": r2.status_code, "text": r2.text, "callback_url": configs.callback_url_taskcheck}, "task_name": "入库回调"}, ensure_ascii=False)) except Exception as e: print("------【paper_id:{}】taskcheck回调出错:{}-------".format(self.wordid, e)) logger.info(json.dumps({"log_level": "warn", "paper_id": self.wordid, "callback_data": type_check, "callback_res": {"status": "回调失败", "callback_url": configs.callback_url_taskcheck}, "errmsg": str(e), "task_name": "入库回调"}, ensure_ascii=False)) else: if r2.status_code != 200: print("------【paper_id:{}】taskcheck回调出错:{}".format(self.wordid, r2.status_code)) logger.info(json.dumps({"log_level": "warn", "paper_id": self.wordid, "callback_data": type_check, "callback_res": {"status": "回调失败", "status_code": r2.status_code, "text": r2.text, "callback_url": configs.callback_url_taskcheck}, "task_name": "入库回调"}, ensure_ascii=False)) if __name__ == '__main__': from pprint import pprint import pickle # items_list = [{'id': '60bdcd734a5335001b0a73cf', 'type': '填空题', # 'stem': '在数列\\({\\lbrace a_{n}\\rbrace }\\)中，若\\({a_{1}=1,a_{n}-a_{n-1}=n(n\\geq 2),}\\)则该数列的通项\\({a_{n}=}\\)', # 'options': [], 'key': '\\( \\frac{1}{2} \\) \n ', # 'subject': '数学', 'errmsgs': '', 'topic_num': 0, 'parse_img': '略', 'analysis': '', 'slave_img': '', # 'parse': '略 \n \n ', 'category': ['月考'], 'grade': '高一', # 'stem_img': "", 'period': '高中', # 'province': '湖北', 'susp_pic': None, 'option_str': '', 'blank_num': None, 'year': 2020, 'difficulty': '中', # 'specials': [], 'upload_time': '2021-07-16T13:51:49.561000Z', # 'key_img': "", 'options_img': [], # 'options_rank': None, 'text_status': None, 'img_status': 1, # 'source': {'type': 's', # 'related_exampaper': [{'paper_id': '5fc0d256407550d0b7d9a43c', # 'file_name': '十堰市一中 2019 级高一下4月月考 数学试题包含答案 ', # 'item_id': None}]}}] # ocr_html = r'数学
命题人:王旭辉
一、选择题(本大题共12小题，每小题5分,请将正确答案填涂在答题卡相应的位置。)
1.cos80°sin40°+sin50°cos10°的值为()
A.${\quad \frac{1}{2}}$B.${\quad \frac{\sqrt {2}}{2}}${c.\quad \frac{\sqrt {3}}{2}}${b.\quad -\frac{\sqrt {3}}{2}}$
2.已知在等比数列${\lbrace a_{n}\rbrace }$中，${a_{1}=1,\quad a_{5}=9,}${a_{3}=(\quad }$
B.±5.±3D.3
3.已知a>b>0,则下列不等式成立的是()
A.${\quad a>b>\frac{a+b}{2}>\sqrt {ab}}$B.${\quad a>\frac{a+b}{2}>\sqrt {ab}>b}$
${c.\quad a>\frac{a+b}{2}>b>\sqrt {ab}}$D.${\quad a>\sqrt {ab}>\frac{a+b}{2}>b}$
4.给出下列命题:
①棱柱的侧棱都相等,侧面都是全等的平行四边形;
②用一个平面去截棱锥，棱锥底面与截面之间的部分是棱台;
③半圆绕着它的直径所在的直线旋转一周所形成的曲面叫做球面;
④棱台的侧棱延长后交于一点，侧面是等腰梯形.
其中正确命题的序号是()
A.①②④B.①②③C.②③).③
5.已知向量a${i=(1,2),\vec {b}=(2,-2),\quad \vec {c}=(\lambda ,-1)}$，若${\vec {c}//(2\vec {a}+\vec {b})}$则λ=()
A.-2B.-1${c.\quad -\frac{1}{2}}$D.${\quad \frac{1}{2}}$
6.已知△ABC中,a=1,${b=\sqrt {3},}$A=30°,则B等于()
A.30°B.30°或150°C.60°D.60°或120°
7.在△ABC中,a、b、c分别为角A、B、C的对边,若b=2,c=1,C=30°,则a=()()
A.${\quad \sqrt {3}}$B.3C.${\quad \sqrt {5}}$D.1
8.若(a+b+c)(b+c-a)=3bc,且sinA=2sinBcosC,那么△ABC是()
A.直角三角形B.等边三角形
C.等腰三角形D.等腰直角三角形
9.在等差数列${\lbrace a_{n}\rbrace }$中，${S_{15}>0,\quad S_{16}<0,}$则使${a_{n}>0}$成立的n的最大值为()
A.6B.7C.8D.S
10.已知等比数列${\lbrace a_{n}\rbrace }$的前n项和为${S_{n},}${S_{5}=2,\quad S_{10}=6,}${a_{16}+a_{17}+a_{18}+a_{19}+a_{20}=}$
()
A.54B.48C.32D.16
11.设点D为△ABC中BC边上的中点，O为AD边上靠近点A的三等分点,则()
A.${\quad \overrightarrow {BO}=-\frac{1}{6}\overrightarrow {AB}+\frac{1}{2}\overrightarrow {AC}}$B.${\quad \overrightarrow {BO}=\frac{1}{6}\overrightarrow {AB}-\frac{1}{2}\overrightarrow {AC}}$
${\therefore \overrightarrow {BO}=\frac{5}{6}\overrightarrow {AB}-\frac{1}{6}\overrightarrow {AC}}$D.${\quad \overrightarrow {BO}=-\frac{5}{6}\overrightarrow {AB}+\frac{1}{6}\overrightarrow {AC}}$
12.△ABC中,角A、B、C的对边分别为a、b、c,且2a+b=2ccosB,若△ABC的面
积为${S=\sqrt {3}c,}$则ab的最小值为()
A.12B.24C.28D.48
二、填空题(本大题共4小题，每小题5分.请将正确答案填写在答题卡相应位置。)
13.在数列${\lbrace a_{n}\rbrace }$中，若${a_{1}=1,a_{n}-a_{n-1}=n(n\geq 2),}$则该数列的通项${a_{n}=}$
14.已知圆锥的侧面展开图是一个半径为6cm,圆心角为${\frac{2\pi}{3}}$的扇形,则此圆锥的体积为
15.已知平面向量${\vec {a}}${\vec {b}}$的夹角为${\frac{\pi}{3},\quad \vec {a}=(\sqrt {3},-1),\quad |\vec {b}|=1,}$则|${2\vec {a}-\vec {b}|=}$
16.已知α、β为锐角,sinα=2,tan(β-α)=-,则tanβ=______
三.解答题(本大题共6小题，共70分.请将正确答案写在答题卡相应位置。解答应写出
文字说明、证明过程或演算步骤.)
17.(10分)(1)解不等式(x-1)(x-a)≥0
(2)已知${f(x)=\frac{x^{2}+6x+9}{x+1},}$其中x>-1,求f(x)的最小值.
18.(12分)已知函数j${{(x)=2\sin x\cos (x+\frac{\pi}{3})+\frac{\sqrt {3}}{2}}}$
(1)求函数f(x)的最小正周期;
(2)若f(x)+m≤0对${x\∈\lbrack 0,\frac{\pi}{2}\rbrack }$亘成立,求实数m的取值范围.
19.(12分)已知等比数列${\lbrace a_{n}\rbrace }$的前n项和为${S_{n},}$且满足${S_{3}=7,\quad S_{6}=63,}$
(1)求数列${\lbrace a_{n}\rbrace }$的通项公式;
(2)若${b_{n}=a_{n}+\log _{2}a_{n},}$求数列${\lbrace b_{n}\rbrace }$的前n项和${T_{n}}$
0.(12分)已知数列${\lbrace a_{n}\rbrace }$满足${a_{1}=\frac{3}{2},}${l_{n}=\frac{a_{n-1}}{2}+\frac{1}{2^{n-1}}(n\geq 2,n\∈N^{*})}$
(1)求证:数列${\lbrace 2^{n}a_{n}\rbrace }$是等差数列,并求出数列${\lbrace a_{n}\rbrace }$的通项公式;
(2)求数列${\lbrace a_{n}\rbrace }$的前n项和${S_{n}.}$
21.(12分)如图,在△ABC中，${C=\frac{\pi}{4},\overrightarrow {CA}\cdot \overrightarrow {CB}=48,}$点D在BC边上，且
${AD=5\sqrt {2},\cos \angle ADB=\frac{3}{5}}$
(I)求AC,CD的长;(II)求cos∠BAD的值.
22.(12分)数列{a,}中,a,=1,当n≥2时,其前n项和S,满足S,2=a,(S,--)${(u_{n})^{\prime }//t_{1},\quad u_{1}-}${4b_{n}//A_{n}-u_{n}\cdot (\beta _{n}-\bar {2})}$
(1)求${S_{n}}$的表达式;
(2)求数列${\lbrace a_{n}\rbrace }$的通项公式;
(3)设${b_{n}=\frac{S_{n}}{2n+1}.}$求数列${\lbrace b_{n}\rbrace }$的前n项和${T_{n}.}$
数学答案
一、选择题
二、填空题
${13.\frac{n^{2}+n}{2}\quad 14.\frac{16\sqrt {2}}{3}\pi\quad 15.\sqrt {13}\quad 16.\frac{73}{9}}$
三、解答题
17.(1)当a>1时,原不等式解集是{x|x≥a,或x≤1};
当a=1时,原不等式解集是R:
当a<1时,原不等式解集是(x|x|1或X]a}_5分
(2)∵x>-1,则x+1>0,
由基本不等式得${f(x)=\frac{x^{2}+6x+9}{x+1}=}$+1x+1${(x+1)+\frac{4}{x+i}+4}$
${=2\sqrt {(x+1)\cdot \frac{4}{x+1}}+4=8}$(当且仅当${x+1=\frac{4}{x+1}}$时,即当x=1时取得等号)
因此，函数${f(x)=\frac{x^{2}+6x+9}{x+1}(x>-1}$)的最小值为810分
18.解:(1)因为${f(x)=2\sin x\cos (x+}$
${=2\sin x(\cos x\cos \frac{\pi}{3}-\sin x\sin x\sin }$
${=2\sin x(\frac{1}{2}\cos x-\frac{\sqrt {3}}{2}\sin x}$
${\div \sin x\cos x-\sqrt {3}\sin ^{2}x+\frac{\sqrt {3}}{2}}$
${=\frac{1}{2}\sin 2x+\frac{\sqrt {3}}{2}\cos 2x}$
${=\sin (2x+\frac{\pi}{3})}$
所以f(x)的最小正周期为${f=\frac{2\pi}{2}=\pi}$
(2)"f(x)+m≤0对${x\∈\lbrack 0,\frac{\pi}{2}\rbrack }$恒成立"等价于"${f(x)_{\max }+m\leq 0^{n}}$
因为${x\∈\lbrack 0,\frac{\pi}{2}\rbrack }$
所以${2x+\frac{\pi}{3}\∈\lbrack \frac{\pi}{3},\frac{4\pi}{3}\rbrack }$
${2x+\frac{\pi}{3}=\frac{\pi}{2},}${x=\frac{\pi}{12}}$
f(x)的最大值为${f(\frac{\pi}{12})=1.}$
所以实数m的取值范围为${(-9O,-1\rbrack }$12分
19.(1)由题意知S${\because _{6}\neq 2s_{3},q\neq 1\cdots }$
${\therefore S_{3}=\frac{a_{1}(7-q^{3})}{1-q}=7}$
........................3分
${s_{6}=\frac{a_{1}(1-q^{6})}{1-q}}$
解得${\left\lbrace \begin{array}{l}{a_{1}=7}\\{q=2}\end{array}\right.\cdots \cdots }$5分${\therefore a_{n}=2^{n-7}\cdots \cdots }$.6分
2)由(1)知${b_{n}=2^{n-7}+n-1\cdots \cdots \cdots }$..7分
∴T。=(1+2+-..+${y^{-x-3}_{n}}$+[1+2+--+(n-1)].................9分
${=2^{n}+\frac{n^{2}-n}{2}-1\cdots }$.........12分
20.(1)因为${a_{n}=\frac{a_{n-1}}{2}+\frac{1}{2^{n-1}}(n\geq 2,n\∈N^{*})}$，所以${2^{n}a_{n}=2^{n-1}a_{a}+2}$，即
${2^{n}a_{n}-2^{n-1}a_{n-1}=2,}$
所以数列${\lbrace 2^{n}a_{n}\rbrace }$是等差数列,且公差d=2,其首项${2a_{1}=3}$
所以${2^{7}a_{n}=3+(n-1)\times 2=2n+7}$，解得${a_{n}=\frac{2n+i}{2^{n}}}$
${2)\quad S_{n}=\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+\cdots +\frac{2n-1}{2^{p-1}}+\frac{2n+1}{z^{\theta }},}$
${\frac{S_{n}}{2}=\frac{3}{2^{2}}+\frac{5}{2^{3}}+\frac{7}{2^{4}}+\cdots +\frac{2n-1}{2^{n}}+\frac{2n+1}{2^{n}+1},}$
①-②,得${\frac{S_{e}}{z}=\frac{3}{z}+}${+\frac{7}{2^{3}}+\cdots +\frac{1}{2^{s}}}${=\frac{3}{2}+}${z^{e}+x^{2}}$ ${1-\frac{7}{2}}$
${=\frac{5}{x}-\frac{2n+5}{a+1},}$
所以${S_{n}=5-\frac{2n+5}{2^{n}}}$12分
21.(I)在△ABD中,∵${\cos \angle ADB=\frac{3}{5},\therefore \sin \angle ADB=\frac{4}{5}.}$
sin∠CAD=sin(∠ADB-${\angle AcD,=\sin \angle ADB\cos \frac{\pi}{4}-\cos \angle ADB\sin \frac{\pi}{4}}$
${=\frac{4}{5}\times \frac{\sqrt {2}}{2}-\frac{3}{5}\times \frac{\sqrt {2}}{2}=\frac{\sqrt {2}}{10}}$
在△ADC中,由正弦定理得${\frac{Ac}{\sin \angle ADc}=\frac{cD}{\sin \angle CAD}=\frac{AD}{\sin \angle ACD}}${\frac{AC}{\frac{4}{5}}=\frac{cD}{\sqrt {2}}=\frac{5\sqrt {2}}{\sqrt {2}}}$
解得${AC=8,CD=\sqrt {2},}$
${\Pi)\because \overrightarrow {CA}\cdot \overrightarrow {CB}=48,}${8\cdot cB\cdot \frac{\sqrt {2}}{2}=48.}$解得${c_{B}=6\sqrt {2},}$∴BD=CB-CD=5√2
在△ABC中,A${B=\sqrt {8^{2}+(6\sqrt {2})-2\times 8\times 6\sqrt {2}}=2\sqrt {10}}$，在△ABD中
${\infty \angle BAD=\frac{(2\sqrt {10})^{2}+(5\sqrt {2})}{2\times 2\sqrt {10}\times 5\sqrt {2}}=\frac{\sqrt {5}}{5}}$
22.(1)由${\sin ^{2}=5}=S_{n}-5_{5}}&{T_{6}^{z}=-7}}${s_{w}^{2}=(S_{w}-S_{6-1})(s_{w}-\frac{1}{2})=s_{e}^{\vec {x}}-\frac{1}{2}S_{\varphi }-S_{6}=+\frac{7}{2}s}$
${\therefore \frac{1}{s_{n}}-\frac{1}{S_{n-1}}=2(n\supseteq 2)}$
${\therefore \left\lbrace \begin{array}{l}{\frac{1}{s}\rbrace }$是以${\frac{1}{s_{7}}}$为首项,以2为公差的等差数列，
${\therefore \frac{1}{s_{n}}=2n-1,}$
${s_{n}=\frac{1}{2n-1}(n\∈N^{*}.}$
${a_{n}=\left\lbrace \begin{array}{l}{7,n=7}\\{\frac{1}{2p-1}-\frac{7}{2n-3},n\geq }\end{array}\right.}$
(3)${b_{n}=\frac{7}{(2n-7)(2n+i)}=}$
${\therefore 7_{8}=\frac{1}{2}(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{2\pi-1}-\frac{1}{2\pi+1})=\frac{1}{2}(-\frac{1}{2n+1})=\frac{\pi}{2\pi}}$12/' # svg_data = {'svg_html_data': ['
1.在数列{an}中，若a1=1,an−an−1=n(n≥2),则该数列的通项an=
'], 'svg_path': ''} # # wordid = "5fc0d256407550d0b7d9a43c" # # res=Ruku(items_list, ocr_html, svg_data, wordid).upload_img() # pprint(res) one_item = {'blank_num': None, 'category': ['月考'], 'checkType': {'id': 2, 'name': '多选题'}, 'difficulty': '中', 'errmsgs': '', 'grade': '高二', 'id': '61a86291144d756b26cdd313', 'img_status': None, 'key': 'B、D', 'key_img': None, 'options': ['车厢与斜面间的动摩擦因数为
', '车厢与斜面间的动摩擦因数为Y', '轻绳OA的拉力大小为2mg', '轻绳OA拉力大小为mg'], 'options_img': [], 'options_rank': 1, 'parse': '', 'parse_img': None, 'period': '高中', 'province': '山东', 'source': {'related_exampaper': [{'file_name': '山东选调物理', 'item_id': None, 'paper_id': '6195e6d034b3123bf9d1fbf7'}], 'type': 's'}, 'specials': [], 'stem': '如图所示,一车厢沿固定足够长斜面向上运动,斜面倾角θ=30°,车厢内有一质量为m的小
球通过轻绳系于车厢顶部,稳定时,轻绳OA与垂直斜面方向的夹角也为θ,重力加速度为g,
下列说法正确的是
", 'stem_img': None, 'subject': '物理', 'topic_num': 9, 'type': '多选题', 'upload_time': '2021-12-03T14:41:31.570000', 'year': 2021} f1 = open(r"G:\zwj\WL\Text_Structure\new_tiku_structure_v3_art\struct_items.pickle", 'rb').read() items_list = pickle.loads(f1) # pprint(items_list) f2 = open(r"G:\zwj\WL\Text_Structure\new_tiku_structure_v3_art\svg_data.pickle", 'rb').read() svg_data = pickle.loads(f2) res = Ruku(items_list, "", svg_data, "0213441313131", {"callback_url": "1223", "source": ""}, "高中政治").upload_img() # a,t = Ruku("","","","").get_phy_kps_auto(one_item, 0) print(res)