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test.py

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  1. #!/usr/bin/env/python
    2. 

  2. # -*- coding:utf-8 -*-
    3. 

  3. import datetime
    4. 

  4. import pickle
    5. 

  5. import random
    6. 

  6. # import re
    7. 

  7. import time
    8. 

  8. 

  9. import requests, os
    10. 

  10. import shutil
    11. 

  11. # html =  """
    12. 

  12. # <html><head><meta charset="utf-8" /></head><body>
    13. 

  13. # <p>四校联考2017—2018学年度第一学期期中考试</p>
    14. 

  14. # <p>高一物理试卷</p>
    15. 

  15. # <p>考试时间： 60分钟  总分： 100分   </p>
    16. 

  16. # <p>一．单项选择题（本大题共20小题，每小题3分，共60分）</p>
    17. 

  17. # <p>请阅读下列材料，回答1 – 3 小题</p>
    18. 

  18. # <p>在冬季的时候，由于路面结冰，极易造成交通事故。一辆质量为3t的汽车，以40km/h的速度沿平直公路行驶，已知橡胶轮胎与普通路面的动摩擦因数为μ<sub>1 </sub>= 0.6，与结冰地面的动摩擦因数为μ<sub>2 </sub>= 0.2（g = 10m/s<sup>2</sup>）</p>
    19. 

  19. # <p>1．汽车的重力为</p>
    20. 

  20. # <p>A．3×10<sup>2</sup>N    B．3×10<sup>3</sup>N    C．3×10<sup>4</sup>N     D．3×10<sup>5</sup>N</p>
    21. 

  21. # <p>2．在汽车正常行驶时，以汽车为参考系</p>
    22. 

  22. # <p>A．路边的树是静止的</p>
    23. 

  23. # <p>B．路边的树向后运动</p>
    24. 

  24. # <p>C．汽车里的乘客是运动的</p>
    25. 

  25. # <p>D．前方的汽车一定是运动的</p>
    26. 

  26. # <p>3．甲、乙两辆相同的汽车分别在普通路面和结冰地面上，刹车滑行做匀减速直线运动。下图中x表示位移、v表示速度，能正确描述该过程的图像是</p>
    27. 

  27. # <p><img src="files/image1.png" width="378pt" height="85.95pt" data-latex="$$" /></p>
    28. 

  28. # <p> </p>
    29. 

  29. # <p> </p>
    30. 

  30. # <p> </p>
    31. 

  31. # <p>4．在考察下列运动员的比赛成绩时，可视为质点的是 </p>
    32. 

  32. # <p>    A．马拉松             B．跳水            C．击剑            D．体操</p>
    33. 

  33. # <p>5．在运动会上，甲、乙两位同学进行百米赛跑，假如起跑加速的时间忽略，把他<img src="files/image2.png" width="1.6pt" height="1.05pt" />们的运动近似当作匀速直线运动来处理．他们同时从起跑线起跑，经过一段时间后他们的位置如图所示，分别作出在这段时间内两人运动的位移x、速度v与时间t的关系图象如下，其中能正确反映他们运动的是</p>
    34. 

  34. # <p> </p>
    35. 

  35. # <p> </p>
    36. 

  36. # <p> </p>
    37. 

  37. # <p>6．下列物理量属于标量的是</p>
    38. 

  38. # <p>A．速度      　  B．位移       C．加速度        D．质量</p>
    39. 

  39. # """
    40. 

  40. # path = r"F:\zwj\parse_2021\res_folder\13.html"
    41. 

  41. # html = open(path,"r",encoding="utf-8").read()
    42. 

  42. #
    43. 

  43. # res = requests.post(url="http://192.168.1.140:11066/word_structure", json={"sci_html_data": html})
    44. 

  44. # print(res.text)
    45. 

  45. # # 发送
    46. 

  46. # requests.get(url,params=data)
    47. 

  47. # requests.post(url,files=files)
    48. 

  48. # requests.post(url,data=data)
    49. 

  49. # 接收get
    50. 

  50. # request.args.get("user", "")  获取get请求参数
    51. 

  51. # request.values.get("key") 获取所有参数(POST\GET通用)
    52. 

  52. # 接收post
    53. 

  53. # request.form.get()   获取表单数据
    54. 

  54. # request.json.get()
    55. 

  55. # request.files.get()
    56. 

  56. # 如果请求体的数据不是表单格式的（如json格式，xml格式），可以通过request.data获取
    57. 

  57. # print(request.data)  # {"name": "zhangsan", "age": 18}
    58. 

  58. 

  59. # args是用来提取url中?后拼接的参数（查询字符串QueryString）
    60. 

  60. # city = request.args.get("city")  # 127.0.0.1:5000/index?city=shenzhen (类似字典的对象)
    61. 

  61. 

  62. # request的属性为data、form、args、cookies、headers、method、url、files
    63. 

  63. import json
    64. 

  64. 

  65. # import random
    66. 

  66. # ft_modify = str(random.random())
    67. 

  67. # print(ft_modify)
    68. 

  68. #
    69. 

  69. #
    70. 

  70. # resp = requests.post(url="http://192.168.1.145:10811/latex2img", data={"latex": "{ a > b {甲} }", "subfolder": "physical_formulas_imgs"}).text
    71. 

  71. # print(resp)
    72. 

  72. 

  73. # bs = BeautifulSoup(html, 'lxml')
    74. 

  74. # print(bs.prettify())
    75. 

  75. # tags = bs.find(name='script')
    76. 

  76. # print(bs.script.string)
    77. 

  77. 

  78. 

  79. # log_dir = r"C:\Users\Python\Desktop\test"+ r"\5fc0a554e3e93cca5b9e2193"
    80. 

  80. # if not os.path.exists(log_dir):
    81. 

  81. #     os.mkdir(log_dir)
    82. 

  82. 

  83. # aa = ".wmf"
    84. 

  84. # print(not aa.endswith(".wmf"))
    85. 

  85. # print("AKIDC9pETRbZfWBbmhoglkT4PUJGzRjmj3Ia"=="AKIDC9pETRbZfWBbmhoglkT4PUJGzRjmj3Ia")
    86. 

  86. # print("C6jlX4LKfleGdmfQvGNgj74lESRpBIEJ"=="C6jlX4LKfleGdmfQvGNgj74lESRpBIEJ")
    87. 

  87. # from utils.ruku_opera import ruku_upload_img
    88. 

  88. #
    89. 

  89. # data_dir = "F:/zwj/Text_Structure/img_folder/struct_items.pickle"
    90. 

  90. # html_dir = "F:/zwj/Text_Structure/img_folder/html_data.pickle"
    91. 

  91. # items_list = pickle.load(open(data_dir, 'rb'))
    92. 

  92. # html_data = pickle.load(open(html_dir, 'rb'))
    93. 

  93. # wordid = "5fc64a0a4994183dda7e74b9"
    94. 

  94. # res = ruku_upload_img(items_list, html_data, str(wordid))
    95. 

  95. # print(res)
    96. 

  96. 

  97. # qq=r'<img src="http://82.156.255.225:11086/ser_static\20/word/media/image1.png" width="145.5pt" height="117pt">'
    98. 

  98. # new_imgs = re.findall(r'<img src=".*?/ser_static/.*?/(image[^"]*?)"', str(qq))
    99. 

  99. # print(qq.replace("","66"))
    100. 

  100. # print(datetime.datetime.strftime(datetime.datetime.now(),'%Y%m%d%H%M%S'))
    101. 

  101. # print(time.time())
    102. 

  102. # print(random.randint(100000,999999))
    103. 

  103. # times = int(time.time() * 1000)
    104. 

  104. # randnum = range(100000,999999)
    105. 

  105. # print([str(int(time.time())) + str(i) for i in random.sample(randnum,5)])
    106. 

  106. #
    107. 

  107. # timeArray = time.localtime(int(time.time()))
    108. 

  108. # otherStyleTime=time.strftime("%Y-%m-%d %H:%M:%S", timeArray)
    109. 

  109. # print("timeArray:",timeArray)
    110. 

  110. # print("otherStyleTime:",otherStyleTime)
    111. 

  111. 

  112. 

  113. #  word解析
    114. 

  114. # html = requests.post(url="http://82.156.255.225:11086/wordbin",
    115. 

  115. #                      data={"word_id": "1111111"},
    116. 

  116. #                      files={"mydata": open(r"C:\Users\Python\Desktop\test\1\358.docx", "rb")},
    117. 

  117. #                      )
    118. 

  118. # text_list = re.findall(r"<p>([\s\S]*?)</p>", html)
    119. 

  119. # if len(text_list) - sum([1 for t in text_list if re.match(r"<img src.*?/>", t)]) <= 3:
    120. 

  120. #     pass
    121. 

  121. #     # imgs = re.findall(r'<img src="(.*?)"', html)
    122. 

  122. #     # res_dict = self.imgs2text(imgs)  # no text,all imgs
    123. 

  123. #     # return res_dict
    124. 

  124. # # text_list = [re.split(r"\n|<br />", t) for t in text_list]
    125. 

  125. # # return sum(text_list, [])
    126. 

  126. # else:
    127. 

  127. # print(html.text)
    128. 

  128. # with open(r"C:\Users\Python\Desktop\test\1\2_online.html", "w", encoding='utf-8') as file:
    129. 

  129. #     file.write(html.text)
    130. 

  130. # for i in re.finditer("f","fjhfjkyhm"):
    131. 

  131. #     print(i.start(),i.end())
    132. 

  132. # html2txt = "<p>14[答案]　A [解析]　考查运动的合成与分解知识，橡皮参与两个分运动，即水平方向的匀速运动和竖直方向的匀速运动，故橡皮运动速度的大小"
    133. 

  133. # a=re.sub(r"<p>\s*([1-9]|[1-4][0-9])\s*(【(解析?|答案?)】|(解析?|答案?)\s*[:：]|\[(答案|解析)\])", r"<p>\1､\2", html2txt)
    134. 

  134. #
    135. 

  135. # print(a)
    136. 

  136. from structure.danti_structure import single_parse
    137. 

  137. 

  138. s = '\\u2000'
    139. 

  139. s = s.encode('ascii', 'ignore').decode('ascii')
    140. 

  140. print(s)
    141. 

  141. LATEX_SYMS = ['\\leftrightharpoondowndown', '\\barleftarrowrightarrowba', '\\shortrightarrowleftarrow', '\\leftarrowshortrightarrow', '\\rightarrowshortleftarrow', '\\concavediamondtickright', '\\smallblacktriangleright', '\\updownharpoonrightright', '\\downupharpoonsleftright', '\\updownharpoonsleftright', '\\invwhitelowerhalfcircle', '\\nvtwoheadrightarrowtail', '\\invwhiteupperhalfcircle', '\\smallblacktriangleleft', '\\updownharpoonleftright', '\\downtrianglerightblack', '\\leftrightarrowtriangle', '\\blackcircleulquadwhite', '\\nvtwoheadleftarrowtail', '\\leftrightharpoondownup', '\\leftrightharpoonupdown', '\\concavediamondtickleft', '\\partialmeetcontraction', '\\updownharpoonrightleft', '\\downharpoonsleftright', '\\leftrightharpoonsdown', '\\circlebottomhalfblack', '\\closedvarcupsmashprod', '\\underrightharpoondown', '\\barovernorthwestarrow', '\\twoheadrightarrowtail', '\\blackdiamonddownarrow', '\\rightleftharpoonsdown', '\\updownharpoonleftleft', '\\downtriangleleftblack', '\\rangledownzigzagarrow', '\\whiteinwhitetriangle', '\\leftrightarrowcircle', '\\CapitalDifferentialD', '\\curvearrowrightminus', '\\bigblacktriangledown', '\\dashrightharpoondown', '\\barrightarrowdiamond', '\\leftrightharpoondown', '\\underleftharpoondown', '\\rightarrowbackapprox', '\\twoheaduparrowcircle', '\\whitesquaretickright', '\\blackrighthalfcircle', '\\twoheadleftarrowtail', '\\blackcircledownarrow', '\\blackcircledrightdot', '\\leftrightharpoonupup', '\\NestedGreaterGreater', '\\downrightcurvedarrow', '\\circlerighthalfblack', '\\rightdowncurvedarrow', '\\nvtwoheadrightarrow', '\\leftrightharpoonsup', '\\downharpoonrightbar', '\\leftrightsquigarrow', '\\twoheadleftdbkarrow', '\\blockthreeqtrshaded', '\\mbfitsansvarepsilon', '\\dashleftharpoondown', '\\upharpoonsleftright', '\\longrightsquigarrow', '\\rightleftharpoonsup', '\\whitearrowupfrombar', '\\whitesquaretickleft', '\\diamondleftarrowbar', '\\blackcircledtwodots', '\\bardownharpoonright', '\\leftdowncurvedarrow', '\\circlelefthalfblack', '\\blackinwhitediamond', '\\rightharpoonsupdown', '\\leftarrowbackapprox', '\\rightharpoondownbar', '\\blacklefthalfcircle', '\\barrightharpoondown', '\\rightarrowbsimilar', '\\longleftrightarrow', '\\blackinwhitesquare', '\\circledwhitebullet', '\\smalltriangleright', '\\nvtwoheadleftarrow', '\\barleftharpoondown', '\\rightarrowtriangle', '\\leftrightharpoonup', '\\PrecedesSlantEqual', '\\longleftsquigarrow', '\\overleftrightarrow', '\\upharpoonrightdown', '\\curvearrowleftplus', '\\acwopencirclearrow', '\\DownRightTeeVector', '\\rightharpoonaccent', '\\measuredrightangle', '\\leftharpoondownbar', '\\leftharpoonsupdown', '\\RightDownVectorBar', '\\blacktriangleright', '\\circleonrightarrow', '\\parallelogramblack', '\\trianglerightblack', '\\RightDownTeeVector', '\\rightrightharpoons', '\\bsimilarrightarrow', '\\DownRightVectorBar', '\\rightpentagonblack', '\\rightupdownharpoon', '\\circletophalfblack', '\\ccwundercurvearrow', '\\SucceedsSlantEqual', '\\downharpoonleftbar', '\\bigblacktriangleup', '\\Longleftrightarrow', '\\inversewhitecircle', '\\errbarblackdiamond', '\\rightharpoonupdash', '\\bardownharpoonleft', '\\smalltriangledown', '\\varhexagonlrbonds', '\\circleonleftarrow', '\\leftharpoonaccent', '\\mbfitsansvartheta', '\\barupharpoonright', '\\measuredangleleft', '\\varcarriagereturn', '\\mbfitsansvarsigma', '\\rightarrowonoplus', '\\DownLeftTeeVector', '\\cwopencirclearrow', '\\mbfsansvarepsilon', '\\twoheadrightarrow', '\\rightleftharpoons', '\\bsimilarleftarrow', '\\triangleleftblack', '\\upharpoonrightbar', '\\LeftDownTeeVector', '\\vardoublebarwedge', '\\circleurquadblack', '\\blacktriangleleft', '\\kernelcontraction', '\\LeftDownVectorBar', '\\rightarrowsimilar', '\\blackpointerright', '\\rightarrowdiamond', '\\uprightcurvearrow', '\\leftarrowtriangle', '\\leftupdownharpoon', '\\cwundercurvearrow', '\\diamondrightblack', '\\mbfitsansvarkappa', '\\smalltriangleleft', '\\errbarblackcircle', '\\rightharpoonupbar', '\\barrightharpoonup', '\\errbarblacksquare', '\\blacktriangledown', '\\leftharpoonupdash', '\\DownLeftVectorBar', '\\whitepointerright', '\\upharpoonleftdown', '\\acwgapcirclearrow', '\\leftarrowbsimilar', '\\similarrightarrow', '\\leftrightharpoons', '\\mbfitsansomicron', '\\mbfitsansepsilon', '\\rightrightarrows', '\\nvrightarrowtail', '\\rightarrowsupset', '\\twoheadleftarrow', '\\leftarrowsimilar', '\\barleftharpoonup', '\\diamondleftarrow', '\\rightarrowapprox', '\\whitepointerleft', '\\mbfitsanspartial', '\\RightUpTeeVector', '\\downdownharpoons', '\\ointctrclockwise', '\\acwunderarcarrow', '\\RightTriangleBar', '\\blackpointerleft', '\\RightUpVectorBar', '\\leftrightharpoon', '\\squarerightblack', '\\upharpoonleftbar', '\\NotRightTriangle', '\\downarrowuparrow', '\\vartriangleright', '\\sphericalangleup', '\\leftharpoonupbar', '\\diamondleftblack', '\\leftleftharpoons', '\\rightharpoondown', '\\APLrightarrowbox', '\\rightcurvedarrow', '\\similarleftarrow', '\\varointclockwise', '\\upharpoonrightup', '\\twoheaddownarrow', '\\uparrowdownarrow', '\\ntrianglerighteq', '\\mbfitsansupsilon', '\\nvleftrightarrow', '\\leftarrowonoplus', '\\barupharpoonleft', '\\uprevequilibrium', '\\rightleftharpoon', '\\downharpoonright', '\\rightthreearrows', '\\circlearrowright', '\\cwgapcirclearrow', '\\rightsquigarrow', '\\mbfitsansvarphi', '\\ntrianglelefteq', '\\underrightarrow', '\\unicodeellipsis', '\\leftharpoondown', '\\ulblacktriangle', '\\diamondbotblack', '\\urblacktriangle', '\\leftthreearrows', '\\downzigzagarrow', '\\vrectangleblack', '\\LeftUpTeeVector', '\\lrblacktriangle', '\\APLleftarrowbox', '\\rightbarharpoon', '\\squarecrossfill', '\\blockhalfshaded', '\\vartriangleleft', '\\acwleftarcarrow', '\\mbfsansvartheta', '\\diamondtopblack', '\\measanglerdtose', '\\LeftUpVectorBar', '\\rightwhitearrow', '\\measangleultonw', '\\upharpoonleftup', '\\measangledltosw', '\\multimapdotboth', '\\blacktriangleup', '\\nvleftarrowtail', '\\varhexagonblack', '\\nleftrightarrow', '\\circlearrowleft', '\\rightthreetimes', '\\barrightharpoon', '\\LeftTriangleBar', '\\leftcurvedarrow', '\\measangleldtosw', '\\enleadertwodots', '\\cwrightarcarrow', '\\mbfitvarepsilon', '\\measanglerutone', '\\leftarrowapprox', '\\mbfitsanslambda', '\\mbfsansvarsigma', '\\measangleurtone', '\\twoheadmapsfrom', '\\squareleftblack', '\\APLdownarrowbox', '\\smalltriangleup', '\\hrectangleblack', '\\downharpoonleft', '\\measangledrtose', '\\APLnotbackslash', '\\enclosetriangle', '\\downarrowbarred', '\\circledownarrow', '\\llblacktriangle', '\\mbfitsansvarrho', '\\trianglerighteq', '\\mbfsansvarkappa', '\\rightleftarrows', '\\circledparallel', '\\bigtriangleleft', '\\leftrightarrows', '\\measanglelutonw', '\\leftarrowsubset', '\\equalrightarrow', '\\NotGreaterTilde', '\\bigtriangledown', '\\widebridgeabove', '\\circledrightdot', '\\acwoverarcarrow', '\\curvearrowright', '\\uparrowoncircle', '\\NotLeftTriangle', '\\leftwhitearrow', '\\mbfitsansdelta', '\\leftbarharpoon', '\\mbfitsanssigma', '\\trianglelefteq', '\\revequilibrium', '\\blockrighthalf', '\\squarebotblack', '\\lparenextender', '\\rbrackextender', '\\underleftarrow', '\\ocommatopright', '\\vardiamondsuit', '\\bigslopedwedge', '\\rightleftarrow', '\\mbfitsansnabla', '\\mdlgblkdiamond', '\\leftthreetimes', '\\rightdasharrow', '\\downupharpoons', '\\upharpoonright', '\\rightharpoonup', '\\rightwavearrow', '\\wedgedoublebar', '\\mbfsanspartial', '\\lbrackextender', '\\rightanglemdot', '\\rparenextender', '\\looparrowright', '\\hookrightarrow', '\\leftrightarrow', '\\RightVectorBar', '\\equalleftarrow', '\\downwhitearrow', '\\mbfsansomicron', '\\circlevertfill', '\\RightTeeVector', '\\acwcirclearrow', '\\mdlgblklozenge', '\\downdownarrows', '\\Longrightarrow', '\\longmappedfrom', '\\mdlgwhtdiamond', '\\Longmappedfrom', '\\blackhourglass', '\\blockqtrshaded', '\\triangleserifs', '\\rightarrowplus', '\\mbfitsansvarpi', '\\longrightarrow', '\\enclosediamond', '\\mbfitsansomega', '\\sphericalangle', '\\leftsquigarrow', '\\updownharpoons', '\\mbfsansepsilon', '\\fdiagovnearrow', '\\overrightarrow', '\\NestedLessLess', '\\concavediamond', '\\mbfitsanskappa', '\\squareneswfill', '\\updownarrowbar', '\\rightarrowtail', '\\leftleftarrows', '\\mbfitsanstheta', '\\doublebarwedge', '\\carriagereturn', '\\NotGreaterLess', '\\mbfsansupsilon', '\\ntriangleright', '\\mbfitsansgamma', '\\squaretopblack', '\\rightouterjoin', '\\circledtwodots', '\\vbraceextender', '\\APLboxquestion', '\\curvearrowleft', '\\Leftrightarrow', '\\twoheaduparrow', '\\barleftharpoon', '\\rdiagovsearrow', '\\squarenwsefill', '\\harrowextender', '\\dashrightarrow', '\\mdlgwhtlozenge', '\\mbfitsansalpha', '\\mdlgblkcircle', '\\blocklefthalf', '\\bigtalloblong', '\\rightfishtail', '\\ntriangleleft', '\\bdtriplevdash', '\\mbfsansvarrho', '\\triangleminus', '\\mbfsansvarphi', '\\longleftarrow', '\\biginterleave', '\\parallelogram', '\\mdsmblkcircle', '\\mbfitvarsigma', '\\rightpentagon', '\\leftarrowplus', '\\SucceedsTilde', '\\mbfitsanszeta', '\\botsemicircle', '\\vysmblkcircle', '\\equalparallel', '\\mdlgwhtcircle', '\\PrecedesTilde', '\\Hermaphrodite', '\\circledbslash', '\\squarelrblack', '\\dashleftarrow', '\\mdsmblksquare', '\\rightarrowgtr', '\\triangletimes', '\\LeftTeeVector', '\\rightarrowbar', '\\llparenthesis', '\\mbfvarepsilon', '\\shortlefttack', '\\emptysetoarrl', '\\varlrtriangle', '\\smallsetminus', '\\vysmblksquare', '\\threedotcolon', '\\wasytherefore', '\\blacktriangle', '\\vysmwhtcircle', '\\threeunderdot', '\\leftwavearrow', '\\mdlgblksquare', '\\divideontimes', '\\leftdasharrow', '\\mbfitsansiota', '\\vysmwhtsquare', '\\varsubsetneqq', '\\mbfitvartheta', '\\leftouterjoin', '\\fallingdotseq', '\\enclosesquare', '\\dotsminusdots', '\\draftingarrow', '\\shortdowntack', '\\Longleftarrow', '\\APLboxupcaret', '\\overleftarrow', '\\LeftVectorBar', '\\twoheadmapsto', '\\downdasharrow', '\\circledbullet', '\\upequilibrium', '\\emptysetocirc', '\\leftharpoonup', '\\mbfitsansbeta', '\\RightArrowBar', '\\upharpoonleft', '\\uparrowbarred', '\\mdlgwhtsquare', '\\upbackepsilon', '\\squareulblack', '\\DifferentialD', '\\errbardiamond', '\\enclosecircle', '\\wideangledown', '\\APLuparrowbox', '\\mbfsanslambda', '\\leftarrowtail', '\\mitvarepsilon', '\\inversebullet', '\\squarellblack', '\\mdsmwhtsquare', '\\rrparenthesis', '\\fullouterjoin', '\\triangleright', '\\mbfitvarkappa', '\\cwcirclearrow', '\\measuredangle', '\\mdsmwhtcircle', '\\hookleftarrow', '\\rightanglesqr', '\\pentagonblack', '\\looparrowleft', '\\bigtriangleup', '\\rightdotarrow', '\\leftarrowless', '\\squareurblack', '\\sixteenthnote', '\\topsemicircle', '\\RightTriangle', '\\mbfsansthree', '\\mbfitsansphi', '\\hexagonblack', '\\doublebarvee', '\\LeftTriangle', '\\squareulquad', '\\varsubsetneq', '\\triangleplus', '\\unicodecdots', '\\Longmapsfrom', '\\upvarepsilon', '\\ogreaterthan', '\\squareurquad', '\\leftdbkarrow', '\\mdblkdiamond', '\\squarehvfill', '\\NotLessTilde', '\\downarrowbar', '\\closedvarcap', '\\DownArrowBar', '\\barleftarrow', '\\longmapsfrom', '\\mbfitsanschi', '\\mbfsansgamma', '\\upwhitearrow', '\\varepsilonup', '\\mdwhtlozenge', '\\smwhtdiamond', '\\downfishtail', '\\dottedcircle', '\\wedgemidvert', '\\lozengeminus', '\\mbfsansdelta', '\\bigwhitestar', '\\mathsterling', '\\veedoublebar', '\\mbfitpartial', '\\mdblklozenge', '\\varspadesuit', '\\odotslashdot', '\\mbfsansalpha', '\\circledequal', '\\mbfitsanstau', '\\medblackstar', '\\cupleftarrow', '\\circlellquad', '\\rbracklrtick', '\\dottedsquare', '\\npreccurlyeq', '\\mbfitepsilon', '\\mbfsanskappa', '\\circleulquad', '\\lrtriangleeq', '\\squarellquad', '\\circlelrquad', '\\blacklozenge', '\\eqqslantless', '\\nsucccurlyeq', '\\mbfitupsilon', '\\notbackslash', '\\mbfsansseven', '\\blocklowhalf', '\\errbarcircle', '\\mbfitsansvar', '\\mbfitsanspsi', '\\smwhtlozenge', '\\rdiagovfdiag', '\\smblklozenge', '\\mdwhtdiamond', '\\lbracklltick', '\\PropertyLine', '\\medwhitestar', '\\revangleubar', '\\intclockwise', '\\risingdotseq', '\\longdivision', '\\LeftArrowBar', '\\varheartsuit', '\\triangleubar', '\\rbrackurtick', '\\leftfishtail', '\\multimapboth', '\\mbfsanstheta', '\\upupharpoons', '\\trianglecdot', '\\mbfitomicron', '\\fdiagovrdiag', '\\smblkdiamond', '\\lbrackultick', '\\invsmileface', '\\subsetapprox', '\\nvrightarrow', '\\triangledown', '\\mbfsanssigma', '\\hyphenbullet', '\\rightdbltail', '\\closedvarcup', '\\mbfsansvarpi', '\\errbarsquare', '\\Diamondblack', '\\leftdotarrow', '\\updownarrows', '\\wedgeonwedge', '\\squarelrquad', '\\bigslopedvee', '\\triangleleft', '\\triangleodot', '\\emptysetobar', '\\downuparrows', '\\rightbkarrow', '\\cntclockoint', '\\oturnedcomma', '\\circleurquad', '\\mbfitsansrho', '\\dingasterisk', '\\mbfsanseight', '\\hermitmatrix', '\\GreaterTilde', '\\supsetapprox', '\\mbfitsanseta', '\\mbfsansomega', '\\emptysetoarr', '\\underbracket', '\\mbfsansnabla', '\\sqsubsetneq', '\\circleddash', '\\mbfsansiota', '\\squarehfill', '\\corresponds', '\\approxident', '\\midbarwedge', '\\leftdbltail', '\\APLnotslash', '\\lgblksquare', '\\vertoverlay', '\\quarternote', '\\diamondcdot', '\\mbfsansfive', '\\mitvarsigma', '\\smwhtcircle', '\\circledless', '\\nvleftarrow', '\\mitvarkappa', '\\blacksquare', '\\blkvertoval', '\\ruledelayed', '\\circledvert', '\\succnapprox', '\\nwovnearrow', '\\otimesrhrim', '\\Rrightarrow', '\\mbfitsanspi', '\\RIGHTCIRCLE', '\\mbfvarsigma', '\\mbfitvarphi', '\\mdwhtsquare', '\\blacksmiley', '\\mbfsanszero', '\\curlyeqsucc', '\\mbfitsansnu', '\\neovsearrow', '\\smwhitestar', '\\whthorzoval', '\\mbfsansbeta', '\\wasylozenge', '\\mdwhtcircle', '\\mbfitsansxi', '\\wideangleup', '\\diamondsuit', '\\preccurlyeq', '\\mbfitlambda', '\\varclubsuit', '\\updownarrow', '\\lcurvyangle', '\\caretinsert', '\\smblksquare', '\\eqslantless', '\\varbarwedge', '\\capricornus', '\\smwhtsquare', '\\squarevfill', '\\mbfvartheta', '\\rightarrowx', '\\leftbkarrow', '\\vartriangle', '\\shortuptack', '\\backtrprime', '\\overbracket', '\\whtvertoval', '\\mbfitsansmu', '\\mitvartheta', '\\curlyeqprec', '\\smblkcircle', '\\restriction', '\\ballotcheck', '\\mbfsansfour', '\\seovnearrow', '\\nrightarrow', '\\asteraccent', '\\mbfvarkappa', '\\Planckconst', '\\succcurlyeq', '\\blkhorzoval', '\\RIGHTcircle', '\\mbfsansnine', '\\threedangle', '\\circledstar', '\\neovnwarrow', '\\blockuphalf', '\\multimapinv', '\\mdblksquare', '\\ExponetialE', '\\precnapprox', '\\sagittarius', '\\rcurvyangle', '\\nsqsubseteq', '\\lgwhtcircle', '\\lgblkcircle', '\\sqsupsetneq', '\\revemptyset', '\\equilibrium', '\\mbfitvarrho', '\\updasharrow', '\\circledcirc', '\\simminussim', '\\GreaterLess', '\\intextender', '\\mdblkcircle', '\\eqqslantgtr', '\\lgwhtsquare', '\\gleichstark', '\\backepsilon', '\\otimeslhrim', '\\mbfsanszeta', '\\Updownarrow', '\\RRightarrow', '\\nsqsupseteq', '\\varisinobar', '\\arrowbullet', '\\msansseven', '\\mbfitsansx', '\\eighthnote', '\\mbfitsansf', '\\mbfsanseta', '\\rparenlend', '\\minusrdots', '\\APLcomment', '\\mbfitsanss', '\\mbfdigamma', '\\gtreqqless', '\\plussubtwo', '\\lessapprox', '\\upoldkoppa', '\\lltriangle', '\\mitupsilon', '\\mbfittheta', '\\mbfitsansa', '\\baruparrow', '\\vrectangle', '\\varnothing', '\\mbfsanspsi', '\\mbfitsansl', '\\Diamonddot', '\\rparenuend', '\\mbfitsansq', '\\lblkbrbrak', '\\subsetcirc', '\\mbfsanstau', '\\supsetplus', '\\LEFTCIRCLE', '\\curlywedge', '\\rbracklend', '\\sqrtbottom', '\\mbfitsansw', '\\hrectangle', '\\backdprime', '\\mbfitsansp', '\\sqsupseteq', '\\mitpartial', '\\mbfsanstwo', '\\rbraceuend', '\\rblkbrbrak', '\\mbfitsanst', '\\postalmark', '\\oplusrhrim', '\\eqslantgtr', '\\mbfitsanso', '\\mbfitsansy', '\\commaminus', '\\upfishtail', '\\interleave', '\\Rparenless', '\\rightangle', '\\tripleplus', '\\upvarkappa', '\\DDownarrow', '\\mathdollar', '\\msanseight', '\\pointright', '\\nleftarrow', '\\mbfitsansn', '\\varthetaup', '\\LEFTcircle', '\\varhexagon', '\\conictaper', '\\circledgtr', '\\upvartheta', '\\turnediota', '\\underparen', '\\mbfsansvar', '\\mbfitsansi', '\\smashtimes', '\\rightslice', '\\doubleplus', '\\Rightarrow', '\\mathsfbfit', '\\mbfitsansb', '\\circlehbar', '\\varsigmaup', '\\lparenless', '\\precapprox', '\\mbfitsansk', '\\mbfsanssix', '\\mbfitgamma', '\\mbfitvarpi', '\\subsetplus', '\\mbfitomega', '\\mbfitsansv', '\\mbfpartial', '\\varepsilon', '\\underbrace', '\\ultriangle', '\\msansthree', '\\supsetcirc', '\\vardiamond', '\\Lleftarrow', '\\mbfepsilon', '\\mbfitkappa', '\\talloblong', '\\circledast', '\\twolowline', '\\octothorpe', '\\Proportion', '\\cupovercap', '\\urtriangle', '\\rightarrow', '\\mbfitsansz', '\\lbracklend', '\\succapprox', '\\rightimply', '\\mbfitdelta', '\\mbfomicron', '\\lparenlend', '\\veeonwedge', '\\subsetneqq', '\\rbrackuend', '\\lbrackubar', '\\lparenuend', '\\longmapsto', '\\wideutilde', '\\fourthroot', '\\opluslhrim', '\\mbfitnabla', '\\rbrackubar', '\\mappedfrom', '\\mitomicron', '\\mitepsilon', '\\mbfupsilon', '\\lmoustache', '\\upuparrows', '\\lbraceuend', '\\mbfsansrho', '\\supsetneqq', '\\mbfitsansh', '\\rbracelend', '\\SetDelayed', '\\lbracelend', '\\mbfitsansj', '\\LLeftarrow', '\\lrtriangle', '\\Longmapsto', '\\mbfsansphi', '\\mbfitsansm', '\\mbfsansone', '\\sqsubseteq', '\\mbfitalpha', '\\mbfitsansc', '\\minusfdots', '\\leftarrowx', '\\upvarsigma', '\\mbfitsigma', '\\mbfitsansd', '\\mbfitsansr', '\\mbfitsansg', '\\veemidvert', '\\mbfitsanse', '\\mbfsanschi', '\\UpArrowBar', '\\complement', '\\capovercup', '\\rmoustache', '\\Ddownarrow', '\\mbfitsansu', '\\Mappedfrom', '\\lesseqqgtr', '\\Eulerconst', '\\lbrackuend', '\\CheckedBox', '\\mbfitsans', '\\msansnine', '\\Lparengtr', '\\dasharrow', '\\backslash', '\\mathcolon', '\\trapezium', '\\turnednot', '\\modtwosum', '\\bigcupdot', '\\capbarcup', '\\LessTilde', '\\varveebar', '\\ngeqslant', '\\llbracket', '\\supsetneq', '\\underline', '\\rvboxline', '\\gtrapprox', '\\righttail', '\\nsubseteq', '\\lvboxline', '\\downarrow', '\\Downarrow', '\\hknwarrow', '\\medbullet', '\\mathslash', '\\supseteqq', '\\tieconcat', '\\therefore', '\\mbfsansxi', '\\rangledot', '\\fracslash', '\\typecolon', '\\hatapprox', '\\Leftarrow', '\\triangles', '\\subseteqq', '\\sumbottom', '\\boxcircle', '\\leftarrow', '\\neswarrow', '\\msansfour', '\\upupsilon', '\\angleubar', '\\overbrace', '\\subsetneq', '\\mbfsansmu', '\\mbfsansnu', '\\intbottom', '\\biohazard', '\\epsilonup', '\\lbracemid', '\\turnangle', '\\approxeqq', '\\nleqslant', '\\mbfitiota', '\\leftslice', '\\mitlambda', '\\bagmember', '\\spadesuit', '\\conjquant', '\\semicolon', '\\wedgeodot', '\\rparengtr', '\\checkmark', '\\accurrent', '\\lesseqgtr', '\\triangleq', '\\mbflambda', '\\nparallel', '\\overparen', '\\mbfitzeta', '\\rightmoon', '\\midbarvee', '\\upsilonup', '\\wideparen', '\\msansfive', '\\radiation', '\\supsetdot', '\\geqqslant', '\\upepsilon', '\\rightturn', '\\backsimeq', '\\mitvarrho', '\\boxbslash', '\\vlongdash', '\\hknearrow', '\\dashcolon', '\\updigamma', '\\ampersand', '\\nsupseteq', '\\bigotimes', '\\varniobar', '\\pitchfork', '\\DashVDash', '\\increment', '\\nwsearrow', '\\mathratio', '\\subsetdot', '\\cupbarcap', '\\Angstroem', '\\impliedby', '\\leqqslant', '\\olessthan', '\\upvarbeta', '\\blockfull', '\\heartsuit', '\\upomicron', '\\hourglass', '\\mbfsanspi', '\\rrbracket', '\\partialup', '\\clockoint', '\\rbracemid', '\\mbfvarrho', '\\backprime', '\\mbfvarphi', '\\fourvdots', '\\mitvarphi', '\\gtreqless', '\\langledot', '\\otimeshat', '\\smileface', '\\longdashv', '\\disjquant', '\\mbfitbeta', '\\lightning', '\\msanszero', '\\sqlozenge', '\\mbffrakw', '\\mathbfit', '\\mitsansf', '\\mbffrakr', '\\clubsuit', '\\emptyset', '\\mbfsansg', '\\forksnot', '\\vartheta', '\\BbbGamma', '\\mitsansr', '\\msansone', '\\upvarrho', '\\mitvarpi', '\\Bbbthree', '\\mbffraku', '\\mitalpha', '\\mbfitrho', '\\mitsanse', '\\mbfsansd', '\\twonotes', '\\Bbbeight', '\\mbfsigma', '\\leftturn', '\\hksearow', '\\Rvzigzag', '\\mbffrakm', '\\ComplexJ', '\\smeparsl', '\\mbffrakq', '\\mbfalpha', '\\mitsansz', '\\mitsansk', '\\upvarphi', '\\mbffrakk', '\\mitsansj', '\\mitsansp', '\\approxeq', '\\subseteq', '\\sterling', '\\biguplus', '\\mapsdown', '\\astrosun', '\\hkswarow', '\\eqvparsl', '\\gesdotol', '\\mbfitvar', '\\lefttail', '\\lnapprox', '\\mitsanss', '\\varphiup', '\\profsurf', '\\mitsansh', '\\llcorner', '\\mathsfit', '\\sqsubset', '\\msanssix', '\\mbfdelta', '\\bigsqcup', '\\mitsansa', '\\mitsanst', '\\succnsim', '\\mbfsansp', '\\UUparrow', '\\gnapprox', '\\pullback', '\\mbfsanse', '\\underbar', '\\ComplexI', '\\bigwedge', '\\mittheta', '\\bbrktbrk', '\\mitsansd', '\\linefeed', '\\curlyvee', '\\notasymp', '\\supseteq', '\\mbfitchi', '\\mbfnabla', '\\backcong', '\\mitdelta', '\\mathbold', '\\invlazys', '\\triangle', '\\ulcorner', '\\mathfrak', '\\diameter', '\\lesdotor', '\\mbffrako', '\\Bbbseven', '\\mitsansn', '\\Mapsfrom', '\\mitsansu', '\\mbfsansh', '\\mathring', '\\zproject', '\\divslash', '\\varkappa', '\\mitgamma', '\\mbffrakh', '\\MapsDown', '\\cuberoot', '\\barwedge', '\\mbffrakl', '\\mbffrakf', '\\mbfsansj', '\\varrhoup', '\\mbfsansf', '\\precnsim', '\\mbffrakv', '\\mbfsansv', '\\varheart', '\\APLinput', '\\geqslant', '\\profline', '\\mbfsansl', '\\mbfvarpi', '\\coloneqq', '\\strictfi', '\\Bbbgamma', '\\mitnabla', '\\Question', '\\notowner', '\\bullseye', '\\mbfsansc', '\\varspade', '\\mbfgamma', '\\wedgebar', '\\Coloneqq', '\\precneqq', '\\mbffrakj', '\\mtteight', '\\medspace', '\\mbffraks', '\\mbffraki', '\\mbffraka', '\\doteqdot', '\\mbffrake', '\\strictif', '\\boxtimes', '\\mitsansl', '\\upstigma', '\\ringplus', '\\ngtrless', '\\lfbowtie', '\\horizbar', '\\mbfsansx', '\\mbfsansy', '\\mitkappa', '\\Uuparrow', '\\lambdaup', '\\mathsfbf', '\\veeonvee', '\\drbkarow', '\\mbffrakx', '\\plustrif', '\\dotequiv', '\\scpolint', '\\mbfsansk', '\\rtriltri', '\\mitsansv', '\\mitsansb', '\\rppolint', '\\mitsansx', '\\mitsanso', '\\simrdots', '\\dottimes', '\\smallint', '\\mbfsanss', '\\parallel', '\\capwedge', '\\mbfsanst', '\\mitsansi', '\\boxonbox', '\\mitsansq', '\\rfbowtie', '\\lrcorner', '\\mbffraky', '\\mbffrakc', '\\mitsansc', '\\boxminus', '\\mbfsansb', '\\elinters', '\\mbfsansi', '\\mbftheta', '\\mbfsansu', '\\viewdata', '\\bsolhsub', '\\timesbar', '\\msanstwo', '\\nlessgtr', '\\mbfitphi', '\\mbffrakn', '\\isinobar', '\\mathcent', '\\intlarhk', '\\mttseven', '\\mbfitpsi', '\\mitsansg', '\\mttthree', '\\mbfsansr', '\\pointint', '\\overline', '\\dotminus', '\\mbffrakb', '\\varsigma', '\\mbfkappa', '\\mbfiteta', '\\mbffrakd', '\\bigtimes', '\\mbfsanso', '\\acidfree', '\\mitsansy', '\\mbfittau', '\\mbfomega', '\\mbffrakz', '\\mbffrakp', '\\wedgedot', '\\mbfsansz', '\\intprodr', '\\mbfsansa', '\\uplambda', '\\mbfsansq', '\\mbffrakt', '\\sinewave', '\\nlesssim', '\\rvzigzag', '\\pentagon', '\\minusdot', '\\mbfsansm', '\\bigsqcap', '\\mapsfrom', '\\multimap', '\\lvzigzag', '\\intercal', '\\leqslant', '\\sqsupset', '\\vectimes', '\\boxslash', '\\mitsansw', '\\question', '\\notslash', '\\mbffrakg', '\\bigoplus', '\\mitomega', '\\cirfnint', '\\succneqq', '\\tieinfty', '\\steaming', '\\mbfsansn', '\\setminus', '\\Angstrom', '\\eqqcolon', '\\varisins', '\\revangle', '\\mitsigma', '\\Lvzigzag', '\\aquarius', '\\leftmoon', '\\dotequal', '\\mbfsansw', '\\mitsansm', '\\urcorner', '\\mttfour', '\\plushat', '\\epsilon', '\\trslash', '\\varbeta', '\\lesdoto', '\\simneqq', '\\gesdoto', '\\preceqq', '\\Swarrow', '\\mbfitxi', '\\boxdiag', '\\upkoppa', '\\varprod', '\\isindot', '\\mitiota', '\\npolint', '\\upsilon', '\\mitbeta', '\\supmult', '\\bigstar', '\\ballotx', '\\dualmap', '\\suphsub', '\\Digamma', '\\mitsans', '\\ggcurly', '\\circled', '\\uparrow', '\\matheth', '\\rbrbrak', '\\imageof', '\\plusdot', '\\bigodot', '\\congdot', '\\diamond', '\\fisheye', '\\Bbbnine', '\\eqcolon', '\\mbfitnu', '\\dbloint', '\\intprod', '\\upsigma', '\\npreceq', '\\mbfscrd', '\\Rbrbrak', '\\percent', '\\mbfscrs', '\\mbfscrn', '\\Scorpio', '\\mbfscrx', '\\mttnine', '\\uptheta', '\\searrow', '\\smallin', '\\upgamma', '\\lesssim', '\\rsqhook', '\\mbfscrl', '\\rsolbar', '\\hexagon', '\\medcirc', '\\obrbrak', '\\kappaup', '\\succsim', '\\lbrbrak', '\\topfork', '\\sadface', '\\gammaup', '\\omegaup', '\\lrtimes', '\\mbfscry', '\\digamma', '\\recycle', '\\ubrbrak', '\\nexists', '\\upvarpi', '\\ddotseq', '\\mbfzeta', '\\ltquest', '\\mbfscrb', '\\pushout', '\\supdsub', '\\nsubset', '\\neptune', '\\upomega', '\\mbfsans', '\\upsampi', '\\vzigzag', '\\Lbrbrak', '\\mbfscre', '\\ngtrsim', '\\alphaup', '\\nearrow', '\\succneq', '\\shuffle', '\\nwarrow', '\\simplus', '\\mitzeta', '\\nsucceq', '\\hzigzag', '\\mbffrak', '\\Coloneq', '\\thetaup', '\\dotplus', '\\project', '\\Upsilon', '\\Nwarrow', '\\rftimes', '\\eqqplus', '\\simless', '\\nvinfty', '\\Bbbfive', '\\upkappa', '\\frownie', '\\questeq', '\\Nearrow', '\\because', '\\precsim', '\\implies', '\\Bbbzero', '\\lsqhook', '\\partial', '\\plussim', '\\precneq', '\\mbfscrp', '\\Diamond', '\\subrarr', '\\twocups', '\\mbfbeta', '\\lessdot', '\\rrangle', '\\fbowtie', '\\cirscir', '\\mbfiota', '\\upalpha', '\\mbfscrc', '\\eqqless', '\\thermod', '\\succeqq', '\\mbfscrw', '\\jupiter', '\\Neptune', '\\Jupiter', '\\deltaup', '\\mbfscrv', '\\lftimes', '\\gtrless', '\\olcross', '\\mbfscrj', '\\lllnest', '\\coloneq', '\\veeodot', '\\leadsto', '\\mbfscrh', '\\between', '\\ddagger', '\\gtquest', '\\varclub', '\\mbfscrr', '\\scorpio', '\\suphsol', '\\squoval', '\\sigmaup', '\\obslash', '\\mbfscrg', '\\boxplus', '\\smallni', '\\annuity', '\\overbar', '\\napprox', '\\notowns', '\\mbfitmu', '\\varpiup', '\\mbfscri', '\\Bbbfour', '\\mbfitpi', '\\warning', '\\mbfscro', '\\sptilde', '\\greater', '\\maltese', '\\Searrow', '\\mbfscrf', '\\mbfscrk', '\\llcurly', '\\mathcal', '\\nsupset', '\\swarrow', '\\natural', '\\varsdef', '\\closure', '\\bumpeqq', '\\Uparrow', '\\mbfscrt', '\\submult', '\\gggnest', '\\mbfscra', '\\twocaps', '\\dbkarow', '\\updelta', '\\lessgtr', '\\benzenr', '\\mbfscrm', '\\backsim', '\\mbfscru', '\\llangle', '\\suplarr', '\\supedot', '\\mbfscrq', '\\mbfscrz', '\\varstar', '\\lozenge', '\\Mercury', '\\fatsemi', '\\revnmid', '\\yinyang', '\\mttfive', '\\pluseqq', '\\mttzero', '\\diceiii', '\\mercury', '\\trprime', '\\subedot', '\\mapsup', '\\fourth', '\\mbfitm', '\\ltrivb', '\\origof', '\\inttop', '\\dotsim', '\\mbfith', '\\mfrakg', '\\ltimes', '\\iotaup', '\\apprge', '\\msanst', '\\msansh', '\\subset', '\\msansn', '\\mfrakc', '\\taurus', '\\mbfitz', '\\mfrakt', '\\sumint', '\\mapsto', '\\preceq', '\\mbfitd', '\\supsim', '\\sqrint', '\\msanse', '\\eqless', '\\gtrdot', '\\isinvb', '\\second', '\\APLlog', '\\stareq', '\\Lambda', '\\succeq', '\\subsup', '\\veedot', '\\Square', '\\stigma', '\\mbfitq', '\\mbfitx', '\\supset', '\\Rbrack', '\\elsdot', '\\varphi', '\\mfrakp', '\\APLinv', '\\bigvee', '\\anchor', '\\mitvar', '\\wedgeq', '\\dprime', '\\mfrakm', '\\mbfitr', '\\barcap', '\\mitpsi', '\\capdot', '\\mttone', '\\lfloor', '\\bigbot', '\\mbfito', '\\msansk', '\\cupdot', '\\mbfite', '\\rtimes', '\\ltlarr', '\\msansq', '\\mitrho', '\\btimes', '\\Uranus', '\\msansf', '\\gemini', '\\mfrake', '\\msansx', '\\msanso', '\\mbfrho', '\\lparen', '\\varrho', '\\models', '\\mbfitk', '\\oslash', '\\boxast', '\\bigcup', '\\mfrakj', '\\msansu', '\\sslash', '\\lambda', '\\mathrm', '\\Supset', '\\msansd', '\\hslash', '\\gesdot', '\\gtrarr', '\\mfraka', '\\Bbbsix', '\\mbfitc', '\\cancer', '\\mbfitn', '\\barcup', '\\mbfitb', '\\mfrako', '\\msansj', '\\msansa', '\\mathbb', '\\nvdash', '\\simgtr', '\\mfraks', '\\intcup', '\\laplac', '\\eqqsim', '\\iiiint', '\\msansp', '\\mfrakv', '\\egsdot', '\\supsub', '\\cirbot', '\\bigcap', '\\mbfitw', '\\danger', '\\oiiint', '\\tminus', '\\mfrakd', '\\iinfin', '\\pisces', '\\mbfchi', '\\mbfits', '\\subsub', '\\candra', '\\eqcirc', '\\eparsl', '\\mfrakz', '\\mathtt', '\\qprime', '\\msansm', '\\msanss', '\\angdnr', '\\mfrakf', '\\eqeqeq', '\\barvee', '\\CIRCLE', '\\iddots', '\\invnot', '\\diceiv', '\\Bbbsum', '\\intbar', '\\vbrtri', '\\mbfpsi', '\\betaup', '\\dagger', '\\mbfitp', '\\bigtop', '\\pounds', '\\upzeta', '\\invneg', '\\mfrakw', '\\zetaup', '\\intcap', '\\Bbbone', '\\boxbar', '\\spddot', '\\forall', '\\bowtie', '\\mfrakx', '\\mbfity', '\\MapsUp', '\\rbrace', '\\rfloor', '\\upbeta', '\\exclam', '\\lgroup', '\\rbrack', '\\veebar', '\\nsimeq', '\\mfraky', '\\mathit', '\\msansz', '\\saturn', '\\midcir', '\\mtttwo', '\\mfrakb', '\\circeq', '\\bullet', '\\coprod', '\\mbfitg', '\\assert', '\\mbfita', '\\apprle', '\\Bbbtwo', '\\Mapsto', '\\nunlhd', '\\smiley', '\\mfrakn', '\\angles', '\\mbfphi', '\\nasymp', '\\Circle', '\\square', '\\mbfitu', '\\msansl', '\\niobar', '\\pencil', '\\varnis', '\\mbfiti', '\\Stigma', '\\Taurus', '\\mfrakh', '\\lowint', '\\bumpeq', '\\mbfvar', '\\uminus', '\\lbrace', '\\scurel', '\\msansv', '\\rgroup', '\\ominus', '\\parsim', '\\prurel', '\\mfrakq', '\\msansg', '\\lbrack', '\\daleth', '\\supsup', '\\mbfitl', '\\mfrakl', '\\mitphi', '\\neuter', '\\uranus', '\\Gemini', '\\mbfitj', '\\gtlpar', '\\exists', '\\msansr', '\\topcir', '\\period', '\\mbfitf', '\\rparen', '\\Rparen', '\\msansc', '\\mitchi', '\\Subset', '\\mfrakr', '\\Otimes', '\\utilde', '\\propto', '\\otimes', '\\ovhook', '\\invamp', '\\mfraku', '\\approx', '\\nequiv', '\\mbfscr', '\\diceii', '\\mbfitv', '\\gesles', '\\mathbf', '\\subsim', '\\boxbox', '\\Lbrack', '\\mfraki', '\\upiota', '\\lesdot', '\\ddddot', '\\Vvdash', '\\nunrhd', '\\mbftau', '\\mathsf', '\\gtrsim', '\\swords', '\\lesges', '\\mbfeta', '\\atsign', '\\rangle', '\\dicevi', '\\msansb', '\\miteta', '\\sumtop', '\\topbot', '\\measeq', '\\Exclam', '\\mbfitt', '\\boxdot', '\\mittau', '\\msansy', '\\Lparen', '\\Bumpeq', '\\female', '\\cupvee', '\\eqqgtr', '\\droang', '\\mathds', '\\mfrakk', '\\msansi', '\\msansw', '\\Saturn', '\\mttsix', '\\cirmid', '\\langle', '\\mitnu', '\\cdotp', '\\Vdash', '\\oplus', '\\perps', '\\mbfpi', '\\hfill', '\\mbfxi', '\\vdots', '\\uplus', '\\uptau', '\\tplus', '\\Colon', '\\Sampi', '\\chiup', '\\skull', '\\Sqcap', '\\imath', '\\lsime', '\\breve', '\\operp', '\\upvar', '\\smile', '\\csupe', '\\sqint', '\\ularc', '\\virgo', '\\Venus', '\\upphi', '\\times', '\\mscrt', '\\nrres', '\\isins', '\\mscrx', '\\delta', '\\Omega', '\\adots', '\\Qoppa', '\\eqdot', '\\slash', '\\ndres', '\\Micro', '\\psurj', '\\sqcup', '\\mbfnu', '\\BbbPi', '\\libra', '\\lrarc', '\\check', '\\zpipe', '\\unrhd', '\\mscrr', '\\Equiv', '\\lceil', '\\mscrg', '\\house', '\\omega', '\\rhoup', '\\upint', '\\mscru', '\\upold', '\\mscrm', '\\zhide', '\\mscrj', '\\Libra', '\\gtcir', '\\tauup', '\\fltns', '\\cdots', '\\uprho', '\\alpha', '\\wedge', '\\mscrw', '\\Bbbpi', '\\qoppa', '\\mscrf', '\\jmath', '\\mbfit', '\\lnsim', '\\upeta', '\\pluto', '\\dddot', '\\kappa', '\\upchi', '\\varpi', '\\nsucc', '\\iiint', '\\forkv', '\\tcohm', '\\mscrs', '\\Theta', '\\mscrn', '\\arceq', '\\phiup', '\\mscri', '\\sqcap', '\\mscra', '\\eqdef', '\\Vvert', '\\sigma', '\\sampi', '\\mfrak', '\\mscrk', '\\psiup', '\\earth', '\\csube', '\\Equal', '\\theta', '\\oiint', '\\Sqcup', '\\awint', '\\Dashv', '\\unlhd', '\\sphat', '\\llarc', '\\lblot', '\\asymp', '\\nless', '\\doteq', '\\gsiml', '\\mscrv', '\\minus', '\\amalg', '\\Wedge', '\\Doteq', '\\DashV', '\\mscrb', '\\dicei', '\\grave', '\\asteq', '\\simeq', '\\mscre', '\\prime', '\\Gamma', '\\ldots', '\\mscrl', '\\equiv', '\\ncong', '\\VDash', '\\dashv', '\\Euler', '\\mscrp', '\\Koppa', '\\strns', '\\nabla', '\\mbfmu', '\\gimel', '\\sharp', '\\infty', '\\mitpi', '\\comma', '\\mscry', '\\ltcir', '\\tilde', '\\mitmu', '\\dicev', '\\xbsol', '\\uppsi', '\\Delta', '\\vdash', '\\gescc', '\\colon', '\\barin', '\\nhpar', '\\mscro', '\\lescc', '\\third', '\\veeeq', '\\upand', '\\DDDot', '\\mscrd', '\\notni', '\\mscrh', '\\angle', '\\gamma', '\\msans', '\\nprec', '\\Pluto', '\\mscrz', '\\nsime', '\\gnsim', '\\eqsim', '\\mscrq', '\\ocirc', '\\frown', '\\gneqq', '\\eqgtr', '\\forks', '\\lsimg', '\\Sigma', '\\notin', '\\lneqq', '\\equal', '\\Aries', '\\aleph', '\\etaup', '\\ddots', '\\rblot', '\\acute', '\\mitxi', '\\koppa', '\\rceil', '\\gsime', '\\aries', '\\urarc', '\\disin', '\\mscrc', '\\mttm', '\\fint', '\\dlsh', '\\eqeq', '\\mbfb', '\\lang', '\\leqq', '\\upxi', '\\Bbbl', '\\mbfn', '\\mitq', '\\mitr', '\\iint', '\\mbfu', '\\csup', '\\mitj', '\\mbfl', '\\geqq', '\\mttn', '\\Bbbw', '\\Bbbp', '\\piup', '\\mbfh', '\\Bbbc', '\\siml', '\\Bbbu', '\\mtty', '\\XBox', '\\sdef', '\\Bbbt', '\\BbbS', '\\tcmu', '\\Bbbk', '\\BbbO', '\\mitx', '\\Bbbz', '\\csub', '\\mbfq', '\\Game', '\\dres', '\\tosa', '\\circ', '\\Mars', '\\psur', '\\BbbU', '\\mitk', '\\xiup', '\\hash', '\\upmu', '\\rsub', '\\mttg', '\\mttx', '\\hide', '\\mitu', '\\semi', '\\intx', '\\rbag', '\\BbbG', '\\nisd', '\\nmid', '\\isin', '\\mbfi', '\\Bbbs', '\\mbfk', '\\mbfc', '\\plus', '\\mitm', '\\lnot', '\\flat', '\\tinj', '\\beta', '\\ring', '\\mbfv', '\\Rdsh', '\\gets', '\\Bbbh', '\\mtta', '\\BbbJ', '\\gtcc', '\\nuup', '\\BbbM', '\\mbfd', '\\mitn', '\\tsur', '\\BbbA', '\\prod', '\\Bbbv', '\\mbfj', '\\mits', '\\BbbT', '\\pinj', '\\mttj', '\\Ldsh', '\\Bbbn', '\\mite', '\\beth', '\\DDot', '\\obot', '\\ddot', '\\mita', '\\buni', '\\ffun', '\\mitp', '\\male', '\\zcmp', '\\mitg', '\\BbbW', '\\mitd', '\\Bbbd', '\\simg', '\\BbbI', '\\cong', '\\BbbP', '\\lneq', '\\lbag', '\\mbfx', '\\mttb', '\\Bbbj', '\\girl', '\\mbfs', '\\land', '\\Bbbb', '\\star', '\\mbff', '\\miti', '\\dint', '\\Bbby', '\\rimg', '\\quad', '\\BbbB', '\\BbbY', '\\mitl', '\\mito', '\\mitw', '\\limg', '\\mttp', '\\BbbQ', '\\sans', '\\Prec', '\\spot', '\\mttf', '\\BbbN', '\\rang', '\\BbbK', '\\mbfy', '\\upin', '\\pfun', '\\vert', '\\Bbbx', '\\Vbar', '\\mbfm', '\\BbbH', '\\mttz', '\\xsol', '\\mbfe', '\\mbfa', '\\mbfg', '\\VERT', '\\mbfz', '\\uppi', '\\BbbE', '\\mitf', '\\odiv', '\\lvec', '\\odot', '\\mity', '\\smte', '\\ltcc', '\\nleq', '\\Rbag', '\\mbfp', '\\zeta', '\\Bbbe', '\\prec', '\\mttk', '\\mttu', '\\sqrt', '\\tona', '\\BbbX', '\\duni', '\\dash', '\\owns', '\\Lbag', '\\Bbbo', '\\less', '\\nexi', '\\mitc', '\\fcmp', '\\cent', '\\Finv', '\\succ', '\\mttd', '\\iota', '\\dsub', '\\Bbbg', '\\surj', '\\Vert', '\\Bbbf', '\\BbbF', '\\mbfo', '\\Bbbm', '\\mttt', '\\Bbbq', '\\mitv', '\\Same', '\\Bbbr', '\\finj', '\\BbbR', '\\mbfr', '\\mitz', '\\ngeq', '\\mttl', '\\mtti', '\\cdot', '\\mlcp', '\\Perp', '\\mttw', '\\euro', '\\BbbV', '\\ngtr', '\\BbbC', '\\upnu', '\\Barv', '\\BbbZ', '\\mbfw', '\\mttr', '\\comp', '\\mtts', '\\Zbar', '\\Bbbi', '\\muup', '\\Join', '\\mttq', '\\LVec', '\\gneq', '\\rres', '\\mtte', '\\mtth', '\\obar', '\\dsol', '\\BbbD', '\\perp', '\\towa', '\\late', '\\tfun', '\\Bbba', '\\mttv', '\\drsh', '\\mtto', '\\mitb', '\\oint', '\\mbft', '\\mitt', '\\mscr', '\\mttc', '\\BbbL', '\\nsim', '\\Succ', '\\toea', '\\Bot', '\\mbf', '\\cup', '\\Lsh', '\\cos', '\\Mho', '\\Phi', '\\fun', '\\Top', '\\vee', '\\lll', '\\yen', '\\nis', '\\leo', '\\dot', '\\geq', '\\boy', '\\max', '\\Vec', '\\log', '\\int', '\\rel', '\\div', '\\lor', '\\Rsh', '\\ggg', '\\Not', '\\cat', '\\rhd', '\\QED', '\\bar', '\\glj', '\\RHD', '\\eth', '\\mho', '\\psi', '\\rho', '\\mit', '\\Leo', '\\top', '\\lhd', '\\var', '\\tau', '\\leq', '\\mtt', '\\LHD', '\\not', '\\sin', '\\min', '\\mid', '\\neq', '\\neg', '\\sun', '\\sim', '\\Psi', '\\cap', '\\Yup', '\\gla', '\\phi', '\\iff', '\\exi', '\\sum', '\\tan', '\\cir', '\\chi', '\\inj', '\\nni', '\\ell', '\\ast', '\\bij', '\\Sun', '\\vec', '\\smt', '\\Vee', '\\bot', '\\Cup', '\\Cap', '\\Dot', '\\eta', '\\lat', '\\nin', '\\hat']
    142. 

  142. import re
    143. 

  143. def latex_Standardize(ltx):
    144. 

  144.     """
    145. 

  145.     latex格式清洗规范化
    146. 

  146.     :return:
    147. 

  147.     """
    148. 

  148.     ltx = ltx.replace("：", ":").replace("，", ",").replace("。", ".").replace("；", ";").replace("？", "?") \
    149. 

  149.         .replace("！", "!").replace("．", ".").replace("､", "、").replace("∶", ":").replace("…", "...") \
    150. 

  150.         .replace("）", ")").replace("（", "(").replace("·", " \cdot ").replace("′", "'").replace("α", "\\alpha")
    151. 

  151.     LATEX_SYMS.sort(key=lambda i: len(i), reverse=True)
    152. 

  152. 

  153.     sub2ltx = {}
    154. 

  154.     num = 1
    155. 

  155.     for syb in LATEX_SYMS:
    156. 

  156.         if syb in ltx:
    157. 

  157.             ltx = ltx.replace(syb, "【{}】".format(num))
    158. 

  158.             sub2ltx["【{}】".format(num)] = syb + " "  # 加空格
    159. 

  159.             num += 1
    160. 

  160.         if "\\" in ltx:
    161. 

  161.             continue
    162. 

  162.         else:
    163. 

  163.             break
    164. 

  164.     for k1, v1 in sub2ltx.items():
    165. 

  165.         ltx = ltx.replace(k1, v1).replace("  ", " ")
    166. 

  166. 

  167.     return ltx
    168. 

  168. # aa=r"{\sqrt{5gR-\dfrac{5EqR}{m}}\leqv0<\sqrt{6gR-\frac{5EqR}{m}}}"
    169. 

  169. # tt=latex_Standardize(aa)
    170. 

  170. # print(int(80/1.27+1))
    171. 

  171. 

  172. ltx2url = {'1': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300192641.png" width="7.0" height="12.5" data-latex="$1$" />', '{A\\cdot (-\\infty ,\\frac{1}{2}\\rbrack }': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300336259.png" width="73.5" height="21.5" data-latex="${A\\cdot (-\\infty ,\\frac{1}{2}\\rbrack }$" />', '{(-\\infty ,\\frac{1}{2})}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300700852.png" width="54.0" height="21.5" data-latex="${(-\\infty ,\\frac{1}{2})}$" />', '{y=bx+a}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300642867.png" width="66.5" height="15.5" data-latex="${y=bx+a}$" />', '{X\\sim B(20,\\frac{1}{2}),}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300184091.png" width="89.5" height="21.5" data-latex="${X\\sim B(20,\\frac{1}{2}),}$" />', '{R^{2}}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300615124.png" width="17.5" height="15.5" data-latex="${R^{2}}$" />', '{AC_{1}\\perp }': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300906466.png" width="42.5" height="15.5" data-latex="${AC_{1}\\perp }$" />', '{B_{1}CD_{1}}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300589222.png" width="46.5" height="14.5" data-latex="${B_{1}CD_{1}}$" />', '{2\\sqrt {2}}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300377309.png" width="27.0" height="17.5" data-latex="${2\\sqrt {2}}$" />', '{B_{1}D_{1}}': '<img src="http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/zyk/uploadfiles/wording/612f60307ddb8b2765e50512/MJMATH-1630572300826869.png" width="36.0" height="14.5" data-latex="${B_{1}D_{1}}$" />'}
    173. 

  173. def sub2(s):
    174. 

  174.     if s:
    175. 

  175.         all_ltx = re.findall(r'\$.*?\$', s)  # 查找试题结构中的公式
    176. 

  176.         all_ltx.extend(re.findall('\\\\\(.*?\\\\\)', s))
    177. 

  177.         all_ltx = list(set(all_ltx))
    178. 

  178.         for ltx in all_ltx:
    179. 

  179.             new_ltx = ltx.replace("$", "").replace("\\(", "").replace("\\)", "")
    180. 

  180.             if new_ltx in ltx2url:  # 将latex换为其渲染图片的线上可访问地址
    181. 

  181.                 s = s.replace(ltx, ltx2url[new_ltx])
    182. 

  182.     return s
    183. 

  183. one_items = {"options_img": [
    184. 

  184.         "<img src=\"http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_0.png\" width=\"22\" height=\"14\">$1$",
    185. 

  185.         "<img src=\"http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_1.png\" width=\"26\" height=\"14\">",
    186. 

  186.         "<img src=\"http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_2.png\" width=\"25\" height=\"14\">",
    187. 

  187.         "<img src=\"http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_3.png\" width=\"10\" height=\"14\">"
    188. 

  188.     ],}
    189. 

  189. # k = "options_img"
    190. 

  190. # if k == "options_img":
    191. 

  191. #     one_items[k] = list(map(sub2, one_items[k]))
    192. 

  192. # print(one_items)
    193. 

  193. # items_str = "<img src='http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/612f60307ddb8b2765e50512/img_45.png'"
    194. 

  194. # rr=re.search(r'<img src=(["\']).*?/(imgpaper/lqy_upload/[^"\']*?)(\1)', items_str)
    195. 

  195. # print(rr.group(3))
    196. 

  196. 

  197. 

  198. def sub2(ss):
    199. 

  199. 

  200.     if re.search(r'^(img|/?h[123456]|/?su[bp]>|t0b>|br\s*/?>'
    201. 

  201.                  r'|/?(p|span|font|article|ul|ol|div|table|t?body|html|head|t[drh])(\s*|\s+style=.*?")>'
    202. 

  202.                  r'|/?[a-z]+ style=.*?">)', ss.group(1)) is None:
    203. 

  203.         print(ss.group(1))
    204. 

  204.         return "&lt;{}".format(ss.group(1))
    205. 

  205.     else:
    206. 

  206.         return "<{}".format(ss.group(1))
    207. 

  207. aabb = """
    208. 

  208. 

  209. <p>2020-2021年福州高级中学高二上期末考试试卷</p>\n<p> </p>\n<p>物理试卷答案</p>\n<p>一､选择题</p>
    210. 

  210. <t0b></p>\n<p>二､实验题</p>\n<p>21.【答案】0.772 ×1 12 <imgsrc50 data-latex="$$"/> 偏小</p>
    211. 

  211. <p>22.【答案】如图 V<sub>1</sub> R<sub>1 </sub>0.11</p>
    212. 

  212. """
    213. 

  213. 

  214. # html2txt = re.sub("<([^<]{1,30})", sub2, aabb)
    215. 

  215. # print(html2txt)
    216. 

  216. # import hashlib
    217. 

  217. # name_list = random.sample(range(100000, 999999), 1)
    218. 

  218. # word_id_temp = str(int(time.time())) + str(name_list[0])
    219. 

  219. # print(word_id_temp)
    220. 

  220. # md = hashlib.md5()
    221. 

  221. # md.update(word_id_temp.encode("utf-8"))
    222. 

  222. # word_id = str(md.hexdigest())
    223. 

  223. # print(word_id)
    224. 

  224. # html2txt = """
    225. 

  225. # 6.超声波测速是一种常用的测速手段。如图所示,安装有超声波发射和接收装置的测速仪B
    226. 

  226. # 固定在道路某处,A为测速仪B正前方的一辆小汽车,两者相距为338m。某时刻B发出超
    227. 

  227. # 声波,同时A由静止开始做匀加速直线运动。当B接收到反射回来的超声波信号时,A、B相
    228. 

  228. # 距为346m,已知声速为340m/s,则下列说法正确的是
    229. 

  229. # A.A车加速度的大小为${2m//s^{2}}$
    230. 

  230. # <img src='http://zxhx-1302712961.cos.ap-shanghai.myqcloud.com/imgpaper/lqy_upload/6195e6d034b3123bf9d1fbf7/img_31.png' width='174' height='64' />
    231. 

  231. # B.超声波追上A车时,A车前进了4m
    232. 

  232. # C.经过2s,测速仪B接收到返回的超声波
    233. 

  233. # D.测速仪B接收到返回的超声波时,A车的速度为16m/s
    234. 

  234. # """
    235. 

  235. # html2txt = re.sub('w\s*w\s*w\..*?(\.\s*c\s*o\s*m|\.cn)+|(?<!["“=\'])http:.*?\.(com|cn|org)', "",
    236. 

  236. #                           html2txt)
    237. 

  237. # a = {"aa": 1}
    238. 

  238. # b = {"bb":a}
    239. 

  239. # def nn(ss):
    240. 

  240. #     ss["aa"]=2
    241. 

  241. #     return ss
    242. 

  242. #
    243. 

  243. # a = nn(a)
    244. 

  244. # print(b)
    245. 

  245. # from bisect import bisect_left
    246. 

  246. # f1 = open(r"F:\zwj\Text_Structure\new_tiku_structure_v3_sci\svg_data.pickle", 'rb')
    247. 

  247. # svg_html_data = pickle.load(f1)
    248. 

  248. # topicinfo = [[nm.end(),nm.group(1)] for nm in re.finditer('class="topic-number">(\d+)\.', str(svg_html_data))]
    249. 

  249. # topicidx = [i[0] for i in topicinfo]
    250. 

  250. # topicno = [i[1] for i in topicinfo]
    251. 

  251. # print(topicidx)
    252. 

  252. # time3 = time.time()
    253. 

  253. # all_mathjax1 = [[a.start(), a.group(1)] for a in
    254. 

  254. #                 re.finditer('(<span class="MathJax_Preview".*?</script>(</span>)*)', str(svg_html_data))]
    255. 

  255. # print("all_mathjax1:", len(all_mathjax1), all_mathjax1)
    256. 

  256. # for n, jax in enumerate(all_mathjax1):
    257. 

  257. #     svgs = re.findall("<svg .*?</svg>", jax[1])
    258. 

  258. #     latex = re.findall('<script .+?">(((?!(</)).)*?)</script>(</span>)*', jax[1])
    259. 

  259. #     if latex and svgs:
    260. 

  260. #         print('+++++++++++++++++++++++:',latex[0][0])
    261. 

  261. #     else:
    262. 

  262. #         id_idx = bisect_left(topicidx, jax[0])
    263. 

  263.         # logger.info("----【word_id:{}】存在mathjax公式信息有问题:latex:{}\nsvg:{}".format(self.wordid, latex, svgs))
    264. 

  264.         # return "第{}题的latex：{}不规范".format(topicno[id_idx-1],latex[0][0]))
    265. 

  265. 

  266. items = [{'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63f5af8b6d986d416f3add4a', 'stem': '已知集合<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692911639.png" width="105" height="21" data-type="math" data-latex="$A=\\{4, x, 2 y\\}$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692465453.png" width="151" height="23" data-type="math" data-latex="$B=\\left\\{-2, x^{2}, 1-y\\right\\}$" />，若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692467324.png" width="49" height="16" data-type="math" data-latex="$ A=B $" />，则实数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692309038.png" width="9" height="11" data-type="math" data-latex="$x$" />的取值集合为', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692541601.png" width="72" height="21" data-type="math" data-latex="$\\{ - 1 , 0 , 2 \\}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692680081.png" width="56" height="21" data-type="math" data-latex="$\\{ - 2 , 2 \\}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692243272.png" width="72" height="21" data-type="math" data-latex="$ \\{-1,0,2\\} $" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692820820.png" width="72" height="21" data-type="math" data-latex="$ \\{-2,1,2\\} $" />'], 'key': 'B', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692467324.png" width="49" height="16" data-type="math" data-latex="$ A=B $" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692400052.png" width="56" height="17" data-type="math" data-latex="$-2 \\in A$" />.当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692196715.png" width="56" height="16" data-type="math" data-latex="$x=一2$" />时，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692470678.png" width="78" height="18" data-type="math" data-latex="$2y=1-y$" />，得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692487205.png" width="43" height="26" data-type="math" data-latex="$y=\\frac{1}{3}$" />;若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692418278.png" width="62" height="18" data-type="math" data-latex="$2y=-2$" />，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692660904.png" width="42" height="15" data-type="math" data-latex="$x=2$" />.故实数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692309038.png" width="9" height="11" data-type="math" data-latex="$x$" />的取值集合为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692680081.png" width="56" height="21" data-type="math" data-latex="$\\{ - 2 , 2 \\}$" />.', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 1, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0d4', 'stem': '已知<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692893411.png" width="81" height="18" data-type="math" data-latex="${z=1-2i,}$" />且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692356290.png" width="28" height="26" data-type="math" data-latex="${\\frac{a+z}{a\\cdot \\bar {z}}}$" />为实数,则实数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692507544.png" width="27" height="11" data-type="math" data-latex="$a=$" />', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692541947.png" width="22" height="16" data-type="math" data-latex="$-2$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692879950.png" width="22" height="16" data-type="math" data-latex="$-1$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692337190.png" width="7" height="12" data-type="math" data-latex="$1$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692233154.png" width="7" height="12" data-type="math" data-latex="$2$" />'], 'key': 'A', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692524584.png" width="118" height="31" data-type="math" data-latex="${\\frac{a+z}{a\\cdot z}=\\frac{a+1-2i}{a(1+2i)}=}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692481516.png" width="79" height="30" data-type="math" data-latex="$ {\\frac{a-3-2(a+2)i}{5a}} $" />为实数,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241721.png" width="54" height="16" data-type="math" data-latex="$a=-2$" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 2, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0d5', 'stem': '在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692687336.png" width="29" height="12" data-type="math" data-latex="$2022$" />年某地销售的汽车中随机选取<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692784076.png" width="29" height="12" data-type="math" data-latex="$1000$" />台，对销售价格与销售数量进行统计,这<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692784076.png" width="29" height="12" data-type="math" data-latex="$1000$" />台车辆的销售价格都不小于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861368.png" width="7" height="12" data-type="math" data-latex="$5$" />万元，小于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692831368.png" width="14" height="12" data-type="math" data-latex="$30$" />万元，将销售价格分为五组:<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692753161.png" width="52" height="21" data-type="math" data-latex="$[5，10)$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692509133.png" width="53" height="21" data-type="math" data-latex="$[10,15)$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692394906.png" width="53" height="21" data-type="math" data-latex="$[15,20)$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692486839.png" width="53" height="21" data-type="math" data-latex="$[20,25)$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692347325.png" width="53" height="21" data-type="math" data-latex="$[25,30)$" />（单位：万元).统计后制成的频率分布直方图如图所示.在选取的<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692784076.png" width="29" height="12" data-type="math" data-latex="$1000$" />台汽车中，销售价格在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692524736.png" width="61" height="21" data-type="math" data-latex="$[10，20)$" />内的车辆台数为<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/imgpaper/lqy_upload/93e2033ba27ece61fe9b752a099e2ff6.png" width="281" height="226">', 'options': ['800', '600', '700', '750'], 'key': 'C', 'parse': '由频率分布直方图知，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692523325.png" width="417" height="16" data-type="math" data-latex="$ 0.015 \\times 5+0.02 \\times 5+0.025 \\times 5+a \\times 5+0.08 \\times 5=1 $" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692201458.png" width="63" height="15" data-type="math" data-latex="$a=0.06$" />，所以销售价格在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692524736.png" width="61" height="21" data-type="math" data-latex="$[10，20)$" />内的频率为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692144577.png" width="172" height="21" data-type="math" data-latex="$ (0.06+0.08) \\times 5=0.7 $" />，故销售价格在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692226584.png" width="68" height="21" data-type="math" data-latex="$[10，20）$" />内的车辆台数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692265585.png" width="127" height="15" data-type="math" data-latex="$0.7 \\times 1000=700$" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 3, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0d6', 'stem': '已知直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />交抛物线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692553624.png" width="94" height="21" data-type="math" data-latex="$C: y^{2}=18 x$" />于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692954342.png" width="48" height="16" data-type="math" data-latex="$M，N$" />两点，且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692720907.png" width="28" height="12" data-type="math" data-latex="$MN$" />的中点为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692319827.png" width="45" height="21" data-type="math" data-latex="$ (5，3) $" />，则直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />的斜率为', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692530956.png" width="12" height="26" data-type="math" data-latex="$\\frac{9}{5}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692199317.png" width="12" height="26" data-type="math" data-latex="$\\frac{3}{2}$" />', '3', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692100504.png" width="18" height="26" data-type="math" data-latex="$\\frac{18}{5}$" />'], 'key': 'C', 'parse': '易知直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />的斜率存 在，设直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />的斜率为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692983976.png" width="8" height="15" data-type="math" data-latex="$k$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692653640.png" width="77" height="21" data-type="math" data-latex="$M\\left(x_{1}, y_{1}\\right)$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692112590.png" width="73" height="21" data-type="math" data-latex="$ N\\left(x_{2}, y_{2})\\right. $" />,<br/>则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692348658.png" width="93" height="49" data-type="math" data-latex="$ \\left\\{\\begin{array}{l}y_{1}^{2}=18 x_{1} \\\\ y_{2}^{2}=18 x_{2}\\end{array}\\right. $" />网式相减得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692593928.png" width="168" height="22" data-type="math" data-latex="$y_{1}^{2}-y_{2}^{2}=18\\left(x_{1}-x_{2}\\right)$" />整理得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692855362.png" width="103" height="28" data-type="math" data-latex="$\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\\frac{18}{y_{1}+y_{2}}$" />因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692720907.png" width="28" height="12" data-type="math" data-latex="$MN$" />的中点为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692985121.png" width="38" height="21" data-type="math" data-latex="$(5,3)$" />.所以<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692507625.png" width="147" height="27" data-type="math" data-latex="$ k=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\\frac{18}{6}=3 $" />.即直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />的斜率 为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363502.png" width="7" height="12" data-type="math" data-latex="$3$" />.', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 4, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63f5b14e6d986d416f3add4b', 'stem': '已知体积为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363502.png" width="7" height="12" data-type="math" data-latex="$3$" />的正三棱锥<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692695748.png" width="73" height="17" data-type="math" data-latex="$P-ABC$" />，底面边长为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692721174.png" width="32" height="21" data-type="math" data-latex="$2 \\sqrt{3}$" />，其内切球为球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692998371.png" width="11" height="12" data-type="math" data-latex="$O$" />，若在此三棱锥中再放人球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692356884.png" width="18" height="16" data-type="math" data-latex="$O^{\\prime}$" />，使其与三个侧面及内切球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692998371.png" width="11" height="12" data-type="math" data-latex="$O$" />均相切，则球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692356884.png" width="18" height="16" data-type="math" data-latex="$O^{\\prime}$" />的 半径为', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692279599.png" width="22" height="30" data-type="math" data-latex="${\\frac{\\sqrt {3}}{3}}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692442815.png" width="12" height="26" data-type="math" data-latex="${\\frac{1}{9}}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692465802.png" width="22" height="30" data-type="math" data-latex="${\\frac{\\sqrt {2}}{3}}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494931.png" width="22" height="30" data-type="math" data-latex="${\\frac{\\sqrt {3}}{9}}$" />'], 'key': 'D', 'parse': ' 设内切球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692998371.png" width="11" height="12" data-type="math" data-latex="$O$" />的半径为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692344756.png" width="7" height="11" data-type="math" data-latex="$r$" />，球<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692356884.png" width="18" height="16" data-type="math" data-latex="$O^{\\prime}$" />的半径为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692393910.png" width="11" height="12" data-type="math" data-latex="$R$" />.因为正三棱锥<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692695748.png" width="73" height="17" data-type="math" data-latex="$P-ABC$" />的体积为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363502.png" width="7" height="12" data-type="math" data-latex="$3$" />，底面边长为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692721174.png" width="32" height="21" data-type="math" data-latex="$2 \\sqrt{3}$" />,所以可求得侧棱长为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861396.png" width="23" height="21" data-type="math" data-latex="$\\sqrt{7}$" />.进而可得三棱锥<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692695748.png" width="73" height="17" data-type="math" data-latex="$P-ABC$" />的表面积为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692463087.png" width="32" height="21" data-type="math" data-latex="$9 \\sqrt{3}$" />由<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692475645.png" width="131" height="26" data-type="math" data-latex="$V=\\frac{1}{3} \\times 9 \\sqrt{3} \\times r$" />.得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692287866.png" width="53" height="30" data-type="math" data-latex="$r=\\frac{\\sqrt{3}}{3}$" />用一平行于底面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692201717.png" width="33" height="13" data-type="math" data-latex="$A B C$" />的平面去截此三棱锥，得到一个高为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692853995.png" width="28" height="30" data-type="math" data-latex="$\\frac{2 \\sqrt{3}}{3}$" />的棱台，那么截下去的棱锥的高是原棱锥的<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692347699.png" width="12" height="26" data-type="math" data-latex="$\\frac{1}{3}$" />，根据相似关系，截下去的棱锥的体积为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692576356.png" width="12" height="26" data-type="math" data-latex="$\\frac{1}{9}$" />,根据等体积法，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692926024.png" width="127" height="26" data-type="math" data-latex="$\\frac{1}{3} \\times \\sqrt{3} \\times R=\\frac{1}{9}$" />，解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692868002.png" width="58" height="30" data-type="math" data-latex="$R=\\frac{\\sqrt{3}}{9}$" />.', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 5, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023 金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0d8', 'stem': '古希腊毕达 哥拉斯学派的数学家研究过各种多边形数。如三角形数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692292263.png" width="66" height="18" data-type="math" data-latex="$1,3,6,10$" />，第<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692564339.png" width="9" height="11" data-type="math" data-latex="$n$" />个三角形数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692389315.png" width="144" height="30" data-type="math" data-latex="$\\frac{n(n+1)}{2}=\\frac{1}{2} n^{2}+\\frac{1}{2} n$" />记第<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692564339.png" width="9" height="11" data-type="math" data-latex="$n$" />个边形数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692667551.png" width="111" height="21" data-type="math" data-latex="$N(n, k)(k \\geqslant 3)$" />，以下列出了部分<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692983976.png" width="8" height="15" data-type="math" data-latex="$k$" />边形数中第<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692564339.png" width="9" height="11" data-type="math" data-latex="$n$" />个数的表达式：<br/>三角形数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692704580.png" width="153" height="26" data-type="math" data-latex="$N(n, 3)=\\frac{1}{2} n^{2}+\\frac{1}{2} n$" /><br/>正方形数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692804317.png" width="97" height="22" data-type="math" data-latex="$N(n, 4)=n^{2}$" /><br/>五边形数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692308599.png" width="153" height="26" data-type="math" data-latex="$N(n, 5)=\\frac{3}{2} n^{2}-\\frac{1}{2} n$" /><br/>六边形数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692846201.png" width="137" height="22" data-type="math" data-latex="$N(n, 6)=2 n^{2}-n$" /><br/>可以推测<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692146927.png" width="67" height="21" data-type="math" data-latex="$N(n，R)$" />的表达式，由此计算<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692203527.png" width="90" height="21" data-type="math" data-latex="$N(20,23)=$" />', 'options': ['4020', '4010', '4210', '4120'], 'key': 'B', 'parse': '由题可归纳<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692871473.png" width="186" height="27" data-type="math" data-latex="$N(n, k)=\\frac{k-2}{2} n^{2}+\\frac{4-k}{2} n$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692596066.png" width="328" height="27" data-type="math" data-latex="$N(20,23)=\\frac{23-2}{2} \\times 20^{2}+\\frac{4-23}{2} \\times 20=4010$" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 6, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡 市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0d9', 'stem': '如图，程序框图的算法 思路源于欧几里得在公元前<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692924061.png" width="21" height="12" data-type="math" data-latex="$300$" />年左右提 出的“转相除法”执行该程序框图，若输人<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692504579.png" width="156" height="16" data-type="math" data-latex="$m=2022，n=1314$" />，则输出<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692272682.png" width="15" height="11" data-type="math" data-latex="$m$" />的值<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/imgpaper/lqy_upload/5a609d78a01b64ca2342d97750830e04.png">', 'options': ['6', '12', '18', '24'], 'key': 'A', 'parse': '题中程序框图为用辗转相除法求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692687336.png" width="29" height="12" data-type="math" data-latex="$2022$" />和<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692821966.png" width="29" height="12" data-type="math" data-latex="$1314$" />的最大公约数.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692838520.png" width="798" height="18" data-type="math" data-latex="$2022=1314+708,1314=708+606 708=606+102,606=102\\times5+96，102=96+6，96=6\\times16+0$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692687336.png" width="29" height="12" data-type="math" data-latex="$2022$" />和<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692821966.png" width="29" height="12" data-type="math" data-latex="$1314$" />的最大公约数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692978083.png" width="7" height="12" data-type="math" data-latex="$6$" />.', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 7, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [[201511102671126, 0.93]], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0da', 'stem': '若二项式<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692428299.png" width="165" height="36" data-type="math" data-latex="$\\left(2 x-\\frac{1}{\\sqrt{x}}\\right)^{n}\\left(n \\in \\mathbf{N}^{*}\\right)$" />的展开式中只有第<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861368.png" width="7" height="12" data-type="math" data-latex="$5$" />项的二项式系数最大则展开式中<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692820023.png" width="17" height="18" data-type="math" data-latex="$x^{2}$" />项的系数为', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692399702.png" width="48" height="16" data-type="math" data-latex="$-1120$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692937761.png" width="48" height="16" data-type="math" data-latex="$-1792$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692244446.png" width="29" height="12" data-type="math" data-latex="$1792$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692146538.png" width="29" height="12" data-type="math" data-latex="$1120$" />'], 'key': 'D', 'parse': '因为展开式中只有第<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861368.png" width="7" height="12" data-type="math" data-latex="$5$" />项的二项式系数最大,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692932765.png" width="42" height="15" data-type="math" data-latex="$n=8$" />.通项为,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692470859.png" width="266" height="31" data-type="math" data-latex="$ {T_{r+1}=C_{8}^{y}(2x)^{8-r}(-\\frac{1}{\\sqrt {x}})^{r}=C_{8}^{r}2^{-r}.} $" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692935432.png" width="91" height="27" data-type="math" data-latex="${(-1)^{r}x^{8-\\frac{3}{2}r},}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692420441.png" width="87" height="27" data-type="math" data-latex="${8-\\frac{3}{7}r=2,}$" />得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692695772.png" width="39" height="15" data-type="math" data-latex="$r=4$" />,所 以展开式中<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692372008.png" width="17" height="18" data-type="math" data-latex="${x^{2}}$" />项的系数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692357939.png" width="144" height="23" data-type="math" data-latex="${C_{8}^{1}2^{4}(-1)^{4}=1120.}$" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 8, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63f5b14e6d986d416f3add4c', 'stem': '已知函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692768099.png" width="341" height="24" data-type="math" data-latex="$ f(x)=A \\sin (\\omega x+\\varphi)\\left(A>0, \\omega>0,|\\varphi|<\\frac{\\pi}{2}\\right) $" />函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />的图象,若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />图象相邻对称轴间<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692667794.png" width="57" height="22" data-type="math" data-latex="${|\\varphi |<\\frac{\\pi}{2}}$" />的图象向右平移<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692421621.png" width="13" height="22" data-type="math" data-latex="${\\frac{\\pi}{6}}$" />个单位长度后得到的距离为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692251125.png" width="18" height="22" data-type="math" data-latex="${\\frac{\\pi}{2},}$" />对任意<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692309038.png" width="9" height="11" data-type="math" data-latex="$x$" />,都有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692944679.png" width="117" height="21" data-type="math" data-latex="${g(-x)+g(x)=}$" />', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />的最大值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692986009.png" width="23" height="21" data-type="math" data-latex="$\\sqrt{3}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692400724.png" width="33" height="21" data-type="math" data-latex="$ f(x) $" />的图象关于点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692141256.png" width="52" height="26" data-type="math" data-latex="$\\left(\\frac{2 \\pi}{3}, 0\\right)$" />中心对称', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692400724.png" width="33" height="21" data-type="math" data-latex="$ f(x) $" />的图象关于直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692321851.png" width="46" height="22" data-type="math" data-latex="$x=\\frac{\\pi}{6}$" />对称', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692400724.png" width="33" height="21" data-type="math" data-latex="$ f(x) $" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692664458.png" width="68" height="26" data-type="math" data-latex="$ \\left.[-\\frac{5 \\pi}{12}, \\frac{\\pi}{12}]\\right. $" />上单调递址'], 'key': 'D', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692120904.png" width="131" height="21" data-type="math" data-latex="$g(-x)+g(x)=0$" />.所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />为奇函数.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />图象相邻对称轴间的距离为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692742948.png" width="13" height="22" data-type="math" data-latex="$\\frac{\\pi}{2}$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />的周<br/>期为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692796648.png" width="9" height="11" data-type="math" data-latex="$ \\pi $" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692819293.png" width="203" height="24" data-type="math" data-latex="$g(x)=A \\sin \\left(\\omega x-\\frac{\\omega \\pi}{6}+\\varphi\\right)$" />,所 以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692960988.png" width="42" height="15" data-type="math" data-latex="$\\omega=2$" />.<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692661896.png" width="166" height="22" data-type="math" data-latex="$-\\frac{\\omega \\pi}{6}+\\varphi=k \\pi(k \\in \\mathbf{Z})$" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692476778.png" width="57" height="22" data-type="math" data-latex="$|\\varphi|<\\frac{\\pi}{2}$" />,所 以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692665110.png" width="47" height="22" data-type="math" data-latex="$\\varphi=\\frac{\\pi}{3}$" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692899786.png" width="78" height="23" data-type="math" data-latex="$f(0)=\\sqrt{3}$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692673223.png" width="45" height="16" data-type="math" data-latex="$A=2$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692503506.png" width="158" height="24" data-type="math" data-latex="$f(x)=2 \\sin \\left(2 x+\\frac{\\pi}{3}\\right)$" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692631702.png" width="90" height="21" data-type="math" data-latex="$f(x)_{\\max }=2$" />，所以A错误.<br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692616192.png" width="151" height="22" data-type="math" data-latex="$2 x+\\frac{\\pi}{3}=k \\pi(k \\in \\mathbf{Z})$" />，得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692383996.png" width="157" height="26" data-type="math" data-latex="$x=-\\frac{\\pi}{6}+\\frac{k \\pi}{2}(k \\in \\mathbf{Z})$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />图象的对称中心为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692293347.png" width="97" height="26" data-type="math" data-latex="$ (-\\frac{\\pi}{6}+\\frac{k \\pi}{2},0) $" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692641206.png" width="56" height="21" data-type="math" data-latex="$(k \\in \\mathbf{Z})$" />，故B错误.<br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692794098.png" width="103" height="22" data-type="math" data-latex="$2 x+\\frac{\\pi}{3}=\\frac{\\pi}{2}+$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692496881.png" width="75" height="21" data-type="math" data-latex="$k \\pi(k \\in \\mathbf{Z})$" />.得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692656582.png" width="148" height="26" data-type="math" data-latex="$x=\\frac{\\pi}{12}+\\frac{k \\pi}{2}(k \\in \\mathbf{Z})$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />图象的对称轴 为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692656582.png" width="148" height="26" data-type="math" data-latex="$x=\\frac{\\pi}{12}+\\frac{k \\pi}{2}(k \\in \\mathbf{Z})$" />故C错误.<br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692669120.png" width="94" height="22" data-type="math" data-latex="$-\\frac{\\pi}{2}+2 k \\pi \\leqslant$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692751692.png" width="195" height="22" data-type="math" data-latex="$2 x+\\frac{\\pi}{3} \\leqslant \\frac{\\pi}{2}+2 k \\pi(k \\in \\mathbf{Z})$" />,得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692483253.png" width="244" height="26" data-type="math" data-latex="$-\\frac{5 \\pi}{12}+k \\pi \\leqslant x \\leqslant \\frac{\\pi}{12}+k \\pi(k \\in \\mathbf{Z})$" />.所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />的单调递增区间为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692379912.png" width="81" height="26" data-type="math" data-latex="$\\left[-\\frac{5 \\pi}{12}+k \\pi\\right.$" />+<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692228037.png" width="79" height="21" data-type="math" data-latex="$k \\pi](k \\in \\mathbf{Z})$" />，故D正确', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 9, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0dc', 'stem': '己知函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692700420.png" width="222" height="56" data-type="math" data-latex="$ f(x)=\\left\\{\\begin{array}{l}x^{2}-2 a x+9, x \\leqslant 1 \\\\ 2 x+\\frac{2}{x-1}, x>1\\end{array}\\right. $" />，若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />的最小值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692978083.png" width="7" height="12" data-type="math" data-latex="$6$" />,则实数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692619448.png" width="8" height="11" data-type="math" data-latex="$a$" />的取值范围是', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692922024.png" width="34" height="21" data-type="math" data-latex="$[1,2]$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692713852.png" width="62" height="23" data-type="math" data-latex="$[-\\sqrt{3}, 3]$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692871988.png" width="62" height="23" data-type="math" data-latex="$[-\\sqrt{3}, 2]$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692344232.png" width="48" height="21" data-type="math" data-latex="$[一2,2]$" />'], 'key': 'C', 'parse': ' 因为当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692505403.png" width="42" height="15" data-type="math" data-latex="$x>1$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692198077.png" width="458" height="36" data-type="math" data-latex="$ {2x+\\frac{2}{x{-1}}=2(x-1)+\\frac{2}{x-1}+2\\geq 2\\sqrt {2(x-1)\\times \\frac{2}{x-1}}+2=6} $" />,当且仅当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692660904.png" width="42" height="15" data-type="math" data-latex="$x=2$" />时,等号成立,所以当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692505403.png" width="42" height="15" data-type="math" data-latex="$x>1$" />时 ，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692713223.png" width="92" height="21" data-type="math" data-latex="${f(x)_{\\min }=6,}$" />,当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692863265.png" width="42" height="17" data-type="math" data-latex="$x≤1$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />的最小值大于或等于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692978083.png" width="7" height="12" data-type="math" data-latex="$6$" />.当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692740729.png" width="41" height="17" data-type="math" data-latex="$a≥1$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692583189.png" width="58" height="21" data-type="math" data-latex="$(-∞,1]$" />上单调递减,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692961410.png" width="189" height="21" data-type="math" data-latex="$ f(x)_{\\min}=f(1)=10-2a $" />.由<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692200429.png" width="112" height="45" data-type="math" data-latex="${\\left\\lbrace \\begin{array}{l}{10-2a\\geq 6},\\\\{1\\leq a},\\end{array}\\right.}$" />得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494789.png" width="73" height="17" data-type="math" data-latex="$1≤a≤2$" />.当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692841221.png" width="41" height="15" data-type="math" data-latex="$a<1$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692554638.png" width="199" height="22" data-type="math" data-latex="${f(x)_{\\min }=f(a)=-a^{2}+9.}$" />由<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692646817.png" width="111" height="45" data-type="math" data-latex="${\\left\\lbrace \\begin{array}{l}{-a^{2}+9\\geq 6}\\\\{a<1},\\end{array}\\right.}$" />得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692282293.png" width="106" height="21" data-type="math" data-latex="${-\\sqrt {3}\\leq a<1.}$" />综合可得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692743535.png" width="98" height="23" data-type="math" data-latex="${a∈\\lbrack -\\sqrt {3},2\\rbrack .}$" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 10, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0dd', 'stem': '设函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692393910.png" width="11" height="12" data-type="math" data-latex="$R$" />上的导函数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692319341.png" width="38" height="21" data-type="math" data-latex="$f^{\\prime}(x)$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692438298.png" width="132" height="21" data-type="math" data-latex="$f(x)+f(-x)=0$" />,对任意<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692969401.png" width="92" height="21" data-type="math" data-latex="$x \\in(0,+\\infty)$" />，都有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692663243.png" width="105" height="21" data-type="math" data-latex="$f(x) f^{\\prime}(x)>x$" />,且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692705000.png" width="63" height="21" data-type="math" data-latex="$f(1)=2$" />，则不等式<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692318931.png" width="192" height="22" data-type="math" data-latex="$ [f(x-1)]^{2}<x^2-2x+4 $" />的解集为', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692780157.png" width="141" height="21" data-type="math" data-latex="$(-\\infty, 0) \\cup(2,+\\infty)$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692543653.png" width="38" height="21" data-type="math" data-latex="$(0,2)$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692403350.png" width="38" height="21" data-type="math" data-latex="$(1,3)$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692666793.png" width="141" height="21" data-type="math" data-latex="$(-\\infty, 1) \\cup(3,+\\infty)$" />'], 'key': 'B', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692438298.png" width="132" height="21" data-type="math" data-latex="$f(x)+f(-x)=0$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />为奇函数.令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692470019.png" width="144" height="22" data-type="math" data-latex="$ {g(x)=[f(x)]^{2}-x^{2}} $" />，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692531443.png" width="217" height="21" data-type="math" data-latex="${g^{\\prime }(x)=2f(x)f^{\\prime }(x)-2x>0,}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692133260.png" width="32" height="21" data-type="math" data-latex="${g(x)}$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692718649.png" width="61" height="21" data-type="math" data-latex="$(0,+∞)$" />上单调递增.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692602133.png" width="32" height="21" data-type="math" data-latex="$ {g(x)} $" />为偶 函数,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692133260.png" width="32" height="21" data-type="math" data-latex="${g(x)}$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692302778.png" width="61" height="21" data-type="math" data-latex="$(-∞,0)$" />上单调递减.不等式<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692318931.png" width="192" height="22" data-type="math" data-latex="$ [f(x-1)]^{2}<x^2-2x+4 $" />等价于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692768580.png" width="199" height="22" data-type="math" data-latex="$ {[f(x-1)]^{2}-(x-1)^{2}\\leq 3,} $" />因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692705000.png" width="63" height="21" data-type="math" data-latex="$f(1)=2$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692287043.png" width="165" height="22" data-type="math" data-latex="${g(1)=\\lbrack f(1)\\rbrack ^{2}-1=3}$" />,所以不等式[<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692731741.png" width="195" height="22" data-type="math" data-latex="${f(x-1)^{-1}-(x-1)^{2}\\leq 3}$" />等价于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692512759.png" width="121" height="21" data-type="math" data-latex="$ {g(x-1)<g(1),} $" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692360176.png" width="117" height="16" data-type="math" data-latex="$-1<x-1<1$" />,即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692676568.png" width="73" height="15" data-type="math" data-latex="$0<x<2$" />.', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 11, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 29, 'options_rank': 2, 'id': '63ef42086d986d416f3ad0de', 'stem': '已知<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692668680.png" width="19" height="17" data-type="math" data-latex="$S_{n}$" />是 数列<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692622830.png" width="36" height="21" data-type="math" data-latex="$\\left\\{a_{n}\\right\\}$" />的前<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692564339.png" width="9" height="11" data-type="math" data-latex="$n$" />项和，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692527227.png" width="48" height="17" data-type="math" data-latex="$a_{1}=1$" />.<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692487841.png" width="165" height="21" data-type="math" data-latex="$S_{n}+S_{n+1}=n(n+2)$" />，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692622464.png" width="51" height="31" data-type="math" data-latex="$\\frac { S _ { 2024 } } { S_ { 2023 } } =$" />', 'options': ['<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692181840.png" width="31" height="26" data-type="math" data-latex="$\\frac{1013}{1012}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692113066.png" width="31" height="27" data-type="math" data-latex="$\\frac{2023}{2024}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692689214.png" width="31" height="26" data-type="math" data-latex="$\\frac{2025}{2023}$" />', '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692865940.png" width="31" height="26" data-type="math" data-latex="$\\frac{1013}{1011}$" />'], 'key': 'A', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692256368.png" width="183" height="21" data-type="math" data-latex="${S_{n}+S_{n}+1=n(n+2),}$" />所以当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692857106.png" width="42" height="15" data-type="math" data-latex="$n=1$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692924616.png" width="182" height="17" data-type="math" data-latex="${S_{1}+S_{2}=2a_{1}+a_{2}=3.}$" />因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692709976.png" width="53" height="18" data-type="math" data-latex="${a_{1}=1,}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692269163.png" width="53" height="17" data-type="math" data-latex="${a_{2}=1.}$" />当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692607148.png" width="42" height="17" data-type="math" data-latex="$n≥2$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692780124.png" width="212" height="21" data-type="math" data-latex="${S_{n-1}+S_{n}=(n-1)(n+1),}$" />两式相减得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692965312.png" width="151" height="18" data-type="math" data-latex="${a_{n}+a_{n+1}=2n+1.}$" />因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692777705.png" width="202" height="21" data-type="math" data-latex="${a_{n}+a_{n+1}=2n+1(n\\geq 2)}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692653352.png" width="266" height="21" data-type="math" data-latex="$ {a_{n+1}-(n+1)=-(a_{n}-n)(n\\geq 2)} $" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692811699.png" width="97" height="18" data-type="math" data-latex="${a_{2}-2=-1,}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692724479.png" width="67" height="21" data-type="math" data-latex="${\\lbrace a_{n}-n\\rbrace }$" />从第二项起是公比为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692879950.png" width="22" height="16" data-type="math" data-latex="$-1$" />的等比数列,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692428645.png" width="66" height="15" data-type="math" data-latex="${a_{n}=n+}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692264606.png" width="121" height="22" data-type="math" data-latex="${(-1)^{n-1}(n\\geq 2),}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692336182.png" width="207" height="46" data-type="math" data-latex="${a_{n}=\\left\\lbrace \\begin{array}{l}{1,n=1},\\\\{n+(-1)^{n-1},n\\geq 2},\\end{array}\\right.}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692691831.png" width="341" height="17" data-type="math" data-latex="$ {S_{2023}=1+2+3+\\cdots +2023=2023\\times1012 } $" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692867935.png" width="351" height="17" data-type="math" data-latex="$S_{2024}=1+2+3+..+2024-1=2023×1013$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241530.png" width="91" height="31" data-type="math" data-latex="$ {\\frac{S_{2024}}{S_{2023}}=\\frac{1013}{1012}.} $" />', 'type': '选择题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 12, 'blank_num': None, 'slave': [], 'answer_type': '1', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市 理三入学试题', 'item_id': None}]}, 'checkType': {'id': 29, 'btt_id': 1, 'name': '单选题'}}, {'difficulty': '中', 'repeat_res': [[201511101050509, 0.97], [201511100901123, 0.95], [201511101342101, 0.95], [201511101394601, 0.95], [201511102662188, 0.95], [201511100941726, 0.94], [201511101050511, 0.94], [201511101050514, 0.94], [201511101223854, 0.94], [201511101368370, 0.94]], 'topic_type_id': 31, 'id': '63ef4bcd6d986d416f3ad11a', 'stem': '已知向量<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692780017.png" width="186" height="31" data-type="math" data-latex="$ \\overrightarrow a=(4,-2), \\overrightarrow b=(2, \\lambda) , $" />若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692223918.png" width="109" height="31" data-type="math" data-latex="$ \\overrightarrow a \\perp(\\overrightarrow a-\\overrightarrow b)$" /> , 则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692928676.png" width="28" height="15" data-type="math" data-latex="$ \\lambda= $" />', 'options': [], 'key': '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692583897.png" width="22" height="16" data-type="math" data-latex="$-6$" />', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692145046.png" width="109" height="31" data-type="math" data-latex="$\\overrightarrow a \\perp(\\overrightarrow a-\\overrightarrow b)$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692929350.png" width="383" height="31" data-type="math" data-latex="$ \\overrightarrow {a} \\cdot(\\overrightarrow {a}-\\overrightarrow {b})=|\\overrightarrow {a}|^{2}-\\overrightarrow a \\cdot \\overrightarrow b=20-(8-2 \\lambda)=0 $" />,得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692133452.png" width="60" height="16" data-type="math" data-latex="${\\lambda =-6.}$" />', 'type': '填空题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 13, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': '3', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 31, 'btt_id': 5, 'name': '单空题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 31, 'id': '63ef4bcd6d986d416f3ad11b', 'stem': '方程<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692311300.png" width="136" height="27" data-type="math" data-latex="$\\frac{50}{10^{x}+1}-5=10^{x+1}$" />的实数解为<span style="color:RGB(50%,40%,30%); blank space">____</span>', 'options': [], 'key': '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692625466.png" width="28" height="26" data-type="math" data-latex="${\\lg\\frac{3}{2}}$" />', 'parse': '令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692616905.png" width="56" height="15" data-type="math" data-latex="$ 10^x=t $" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692362255.png" width="108" height="27" data-type="math" data-latex="${\\frac{50}{t+1}-5=10t,}$" />整理得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692926054.png" width="252" height="22" data-type="math" data-latex="${2t^{2}+3t-9=(t+3)(2t-3)=0}$" />,解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692212546.png" width="51" height="16" data-type="math" data-latex="$t=-3$" />(舍去)或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692941077.png" width="46" height="26" data-type="math" data-latex="${t=\\frac{3}{2}.}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692316931.png" width="62" height="26" data-type="math" data-latex="${10^{x}=\\frac{3}{2}}$" />，故<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692266930.png" width="67" height="26" data-type="math" data-latex="${x=\\lg\\frac{3}{2}.}$" />', 'type': '填空题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 14, 'blank_num': 1, 'options_rank': None, 'slave': [], 'answer_type': '2', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 31, 'btt_id': 5, 'name': '单空题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 31, 'id': '63ef4bcd6d986d416f3ad11c', 'stem': '在长方体<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692420065.png" width="159" height="18" data-type="math" data-latex="$A B C D-A_{1} B_{1} C_{1} D_{1}$" />中，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692414072.png" width="60" height="16" data-type="math" data-latex="$AD=1$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692347318.png" width="66" height="18" data-type="math" data-latex="$A A_{1}=2$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692137984.png" width="58" height="16" data-type="math" data-latex="$AB=3$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241190.png" width="15" height="12" data-type="math" data-latex="$M$" />是棱<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692956854.png" width="35" height="14" data-type="math" data-latex="$C_{1} D_{1}$" />上一点，且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692245458.png" width="72" height="17" data-type="math" data-latex="$ D_{1} M=1 $" />，则异面直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692432775.png" width="22" height="12" data-type="math" data-latex="$CD$" />与<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692359843.png" width="26" height="12" data-type="math" data-latex="$BM$" />所成角的余弦值为<span style="color:RGB(50%,40%,30%); blank space">___</span>', 'options': [], 'key': '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692704947.png" width="12" height="26" data-type="math" data-latex="${\\frac{2}{3}}$" />', 'parse': '因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692753795.png" width="71" height="21" data-type="math" data-latex="$AB//CD$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692651416.png" width="57" height="16" data-type="math" data-latex="$∠ABM$" />即异面直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692432775.png" width="22" height="12" data-type="math" data-latex="$CD$" />与<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692359843.png" width="26" height="12" data-type="math" data-latex="$BM$" />所成角.连接<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692130701.png" width="26" height="13" data-type="math" data-latex="$AM$" />(图略),因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692174150.png" width="137" height="19" data-type="math" data-latex="$ {AD=1,AA_{1}=2,} $" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692137984.png" width="58" height="16" data-type="math" data-latex="$AB=3$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692119624.png" width="153" height="22" data-type="math" data-latex="${AM=\\sqrt {6},BM=3.}$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692915127.png" width="64" height="16" data-type="math" data-latex="$△ABM$" />中<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692448705.png" width="90" height="22" data-type="math" data-latex="${,AM=\\sqrt {6},}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692710952.png" width="112" height="16" data-type="math" data-latex="$AB=BM=3$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692708742.png" width="206" height="32" data-type="math" data-latex="${\\cos \\angle ABM=\\frac{3^{2}+3^{2}-(\\sqrt {6})^{2}}{2\\times 3\\times 3}=}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692384540.png" width="17" height="26" data-type="math" data-latex="${\\frac{2}{3}.}$" />故异面直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692432775.png" width="22" height="12" data-type="math" data-latex="$CD$" />与<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692359843.png" width="26" height="12" data-type="math" data-latex="$BM$" />所成角的余弦值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692384540.png" width="17" height="26" data-type="math" data-latex="${\\frac{2}{3}.}$" />', 'type': '填空题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 15, 'blank_num': 1, 'options_rank': None, 'slave': [], 'answer_type': '2', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试 题', 'item_id': None}]}, 'checkType': {'id': 31, 'btt_id': 5, 'name': '单空题'}}, {'difficulty': '中', 'repeat_res': [[201511101303546, 0.94]], 'topic_type_id': 31, 'id': '63f5b14e6d986d416f3add4d', 'stem': '已知双曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692245464.png" width="88" height="30" data-type="math" data-latex="$x^{2}-\\frac{y^{2}}{3}=1$" />的左、右焦点分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692791535.png" width="15" height="14" data-type="math" data-latex="$F_{1}$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692301458.png" width="15" height="14" data-type="math" data-latex="$F_{2}$" />,过右焦点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692301458.png" width="15" height="14" data-type="math" data-latex="$F_{2}$" />的直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />与双曲线的右支交于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692866613.png" width="33" height="19" data-type="math" data-latex="$A,B$" />两点，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692415678.png" width="66" height="18" data-type="math" data-latex="$\\triangle A F_{1} F_{2}$" />和<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692813261.png" width="67" height="18" data-type="math" data-latex="$\\triangle B F_{1} F_{2}$" />的内切圆面积之和的取值范围是<span style="color:RGB(50%,40%,30%); blank space">____</span>', 'options': [], 'key': '<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692981017.png" width="63" height="26" data-type="math" data-latex="${\\lbrack 2\\pi,\\frac{10\\pi}{3})}$" />', 'parse': '设直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692494804.png" width="4" height="15" data-type="math" data-latex="$l$" />的倾斜角为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692538260.png" width="8" height="15" data-type="math" data-latex="$θ$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692180090.png" width="88" height="26" data-type="math" data-latex="${\\theta ∈(\\frac{\\pi}{3},\\frac{2\\pi}{3}).}$" />设<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692982620.png" width="117" height="19" data-type="math" data-latex="${AF_{1},AF_{2},F_{1}F_{2}}$" />与圆的切点分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692121522.png" width="62" height="18" data-type="math" data-latex="$M,N,E$" />,因为<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692471927.png" width="282" height="21" data-type="math" data-latex="$ {|F_{1}E|-|F_{2}E|=2,|F_{1}E|+|F_{2}E|=4} $" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692337943.png" width="161" height="21" data-type="math" data-latex="${|F_{1}E|=3,|F_{2}E|=1.}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692812656.png" width="66" height="18" data-type="math" data-latex="${\\triangle AF_{1}F_{2}}$" /> 和<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692997794.png" width="67" height="18" data-type="math" data-latex="${\\triangle BF_{1}F_{2}}$" />的内切圆圆心分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692113847.png" width="54" height="18" data-type="math" data-latex="$ {O_{1},O_{2},} $" />半径分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692736946.png" width="43" height="15" data-type="math" data-latex="${r_{1},r_{2},}$" />因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692374618.png" width="161" height="18" data-type="math" data-latex="$ \\triangle O_{1} E F_{2} \\backsim \\triangle F_{2} E O_{2} $" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692808702.png" width="103" height="33" data-type="math" data-latex="${\\frac{|O_{1}E|}{|F_{2}E|}=\\frac{|EF_{2}|}{|EO_{2}|},}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692571566.png" width="57" height="27" data-type="math" data-latex="$ {\\frac{r_{1}}{1}=\\frac{1}{r_{2}}} $" />，即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692559289.png" width="63" height="17" data-type="math" data-latex="$ r_1r_2 =1 $" />.因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692229618.png" width="133" height="26" data-type="math" data-latex="${\\angle AF_{2}E∈(\\frac{\\pi}{3},\\frac{2\\pi}{3})}$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692575234.png" width="135" height="22" data-type="math" data-latex="${\\angle O_{1}F_{2}E∈(\\frac{\\pi}{6},\\frac{\\pi}{3})}$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692426078.png" width="267" height="30" data-type="math" data-latex="$ {\\tan \\angle O_{1}F_{2}E=\\frac{r_{1}}{1}=r_{1}∈(\\frac{\\sqrt {3}}{3},\\sqrt {3}),} $" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692931438.png" width="127" height="27" data-type="math" data-latex="$ {r_1r_2=1,r_{2}=\\frac{1}{r_1}} $" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692495352.png" width="198" height="32" data-type="math" data-latex="$ {r_1^{2}+r_2^{2}=r_1^{2}+\\frac{1}{r_1^2}\\in[2,\\frac{10}{3})} $" />,所以所求面积之和的取值范围是<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692586796.png" width="68" height="26" data-type="math" data-latex="${\\lbrack 2\\pi,\\frac{10\\pi}{3}).}$" /><br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/new_image1677034069689453.png">', 'type': '填空题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 16, 'blank_num': 1, 'options_rank': None, 'slave': [], 'answer_type': '2', 'slave_no': None, 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 31, 'btt_id': 5, 'name': '单空题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f58beb6d986d416f3adcd1', 'stem': '在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692864558.png" width="54" height="16" data-type="math" data-latex="$\\triangle A B C$" />中，设角<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692544872.png" width="54" height="19" data-type="math" data-latex="$A.B,C$" />所对的边分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692804547.png" width="39" height="18" data-type="math" data-latex="$a,b,c$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692782752.png" width="53" height="21" data-type="math" data-latex="$b=\\sqrt{3}$" />,且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692628909.png" width="136" height="24" data-type="math" data-latex="$b=2 a \\sin \\left(C+\\frac{\\pi}{6}\\right)$" />.<br/>(1)求角<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692979367.png" width="10" height="13" data-type="math" data-latex="$A$" />;<br/>(2)若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692834019.png" width="12" height="12" data-type="math" data-latex="$D$" />为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692289829.png" width="21" height="13" data-type="math" data-latex="$AB$" />的中点，且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692243733.png" width="60" height="15" data-type="math" data-latex="$CD=1$" />，求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692864558.png" width="54" height="16" data-type="math" data-latex="$\\triangle A B C$" />的面积', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '（1）因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692628909.png" width="136" height="24" data-type="math" data-latex="$b=2 a \\sin \\left(C+\\frac{\\pi}{6}\\right)$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692276882.png" width="197" height="24" data-type="math" data-latex="$\\sin B=2 \\sin A \\sin \\left(C+\\frac{\\pi}{6}\\right)$" />，<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692557833.png" width="312" height="33" data-type="math" data-latex="$\\sin (A+C)=2 \\sin A\\left(\\frac{\\sqrt{3}}{2} \\sin C+\\frac{1}{2} \\cos C\\right)$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692627325.png" width="388" height="21" data-type="math" data-latex="$\\sin A\\cos C\\cos A\\sin C=\\sqrt 3sin A\\sin C+\\sin A\\cos C$" />,.................................3分<br/>即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692296926.png" width="191" height="21" data-type="math" data-latex="$\\cos A \\sin C=\\sqrt{3} \\sin A \\sin$" />C.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692209644.png" width="69" height="19" data-type="math" data-latex="$\\sin C \\neq 0$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692284053.png" width="126" height="21" data-type="math" data-latex="$\\cos A=\\sqrt{3} \\sin A$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692607239.png" width="87" height="30" data-type="math" data-latex="$\\tan A=\\frac{\\sqrt{3}}{3}$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692541556.png" width="75" height="21" data-type="math" data-latex="$A \\in(0, \\pi)$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692540609.png" width="49" height="22" data-type="math" data-latex="$A=\\frac{\\pi}{6}$" />.<br/>（2在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692243563.png" width="56" height="16" data-type="math" data-latex="$\\triangle A C D$" />中，由余弦定理得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692919653.png" width="120" height="27" data-type="math" data-latex="$ 1=3+\\frac{c^{2}}{4}-\\frac{3}{2} c $" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692401147.png" width="114" height="19" data-type="math" data-latex="$c^{2}-6 c+8=0$" />,解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692466952.png" width="39" height="15" data-type="math" data-latex="$c=2$" />或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692174699.png" width="39" height="15" data-type="math" data-latex="$c=4$" />.<br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692466952.png" width="39" height="15" data-type="math" data-latex="$c=2$" />时，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692372537.png" width="188" height="30" data-type="math" data-latex="$S_{\\triangle A B C}=\\frac{1}{2} b c \\sin A=\\frac{\\sqrt{3}}{2}$" />;当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692174699.png" width="39" height="15" data-type="math" data-latex="$c=4$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692998843.png" width="97" height="22" data-type="math" data-latex="$S_{\\triangle A B C}=\\sqrt{3}$" />', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 17, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f58beb6d986d416f3adcd2', 'stem': '在数字化时代，电子书阅读给人们的阅读方式、认知模式与思维习惯带来了改变，电子书阅读的快速增长也再次引发人们对相关问题的思考.某地对本地群众（中老年人与年轻人）的年龄与阅读习惯（经常电子阅读与经常纸质阅读<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692353370.png" width="6" height="21" data-type="math" data-latex="$)$" />进行了调查统计，得到如下列联表:<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/imgpaper/lqy_upload/fc44b2003e799f0107ed5a78aba1e0c9.png" width="636" height="120"><br/>设从经常电子阅读的人中任取<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692337190.png" width="7" height="12" data-type="math" data-latex="$1$" />人，记抽取到的中老年人数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692643012.png" width="12" height="18" data-type="math" data-latex="$\\xi ;$" />从经常纸质阅读的人中任取<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692337190.png" width="7" height="12" data-type="math" data-latex="$1$" />人，记抽取到的中老年人数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692967183.png" width="8" height="15" data-type="math" data-latex="$\\eta$" />巳知<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692372361.png" width="174" height="27" data-type="math" data-latex="$P(\\xi=0)=\\frac{23}{17} P(\\eta=0)$" />.<br/>(1)求列联表中<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692659523.png" width="76" height="18" data-type="math" data-latex="$x,y,M,N$" />的值，并判断是否有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692243725.png" width="32" height="16" data-type="math" data-latex="$95\\%$" />的把握认为阅读习惯与年龄有关.<br/>(2)从年轻人中按阅读习惯用分层抽样的方法抽出<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692161348.png" width="14" height="12" data-type="math" data-latex="$10$" />人，再从抽出的<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692161348.png" width="14" height="12" data-type="math" data-latex="$10$" />人中用简单随机抽样的方法抽取<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363502.png" width="7" height="12" data-type="math" data-latex="$3$" />人，若其中经常电子阅读的人数为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692531209.png" width="12" height="12" data-type="math" data-latex="$X$" />，求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692531209.png" width="12" height="12" data-type="math" data-latex="$X$" />的分布列及数学期望.<br/>参考公式及数据:<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692287797.png" width="177" height="36" data-type="math" data-latex="$K^{2}=\\frac{n(a d-b c)^{2}}{(a+b)(c+d)(a+c)(b+d)}$" />其中<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692817513.png" width="131" height="16" data-type="math" data-latex="$ n=a+b+c+d $" />.<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/new_image1677036249595799.png" width="657" height="76">', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1)因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692372361.png" width="174" height="27" data-type="math" data-latex="$P(\\xi=0)=\\frac{23}{17} P(\\eta=0)$" />.所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692784572.png" width="117" height="38" data-type="math" data-latex="$\\frac{\\mathrm{C}_{50}^{1}}{\\mathrm{C}_{85}^{1}}=\\frac{23}{17} \\times \\frac{\\mathrm{C}_{x}^{1}}{\\mathrm{C}_{115}^{1}}$" /><br/>解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692823103.png" width="256" height="18" data-type="math" data-latex="$x=50,y=65,M=100,N=100$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692202433.png" width="354" height="32" data-type="math" data-latex="$ K^{2}=\\frac{200 \\times(50 \\times 65-35 \\times 50)^{2}}{100 \\times 100 \\times 115 \\times 85}=\\frac{1800}{391} \\approx 4.604>3.841 $" />，<br/>所以有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692243725.png" width="32" height="16" data-type="math" data-latex="$95\\%$" />的把握认为阅读习惯与年龄有关.<br/>(2)由题意可知，抽出的<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692161348.png" width="14" height="12" data-type="math" data-latex="$10$" />人中，经常电子阅读的有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861368.png" width="7" height="12" data-type="math" data-latex="$5$" />人，经常纸质阅读的有<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692861368.png" width="7" height="12" data-type="math" data-latex="$5$" />人，从中取<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363502.png" width="7" height="12" data-type="math" data-latex="$3$" />人，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692531209.png" width="12" height="12" data-type="math" data-latex="$X$" />的可能<br/>取值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692908732.png" width="58" height="18" data-type="math" data-latex="$0,1,2,3$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692461775.png" width="173" height="38" data-type="math" data-latex="$P(X=0)=\\frac{\\mathrm{C}_{5}^{0} \\mathrm{C}_{5}^{3}}{\\mathrm{C}_{10}^{3}}=\\frac{1}{12}$" />;<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692337776.png" width="173" height="38" data-type="math" data-latex="$P(X=1)=\\frac{\\mathrm{C}_{5}^{1} \\mathrm{C}_{5}^{2}}{\\mathrm{C}_{10}^{3}}=\\frac{5}{12}$" /><br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692680365.png" width="173" height="38" data-type="math" data-latex="$P(X=2)=\\frac{\\mathrm{C}_{5}^{2} \\mathrm{C}_{5}^{1}}{\\mathrm{C}_{10}^{3}}=\\frac{5}{12}$" />;<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692812480.png" width="173" height="38" data-type="math" data-latex="$P(X=3)=\\frac{\\mathrm{C}_{5}^{3} \\mathrm{C}_{5}^{0}}{\\mathrm{C}_{10}^{3}}=\\frac{1}{12}$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692531209.png" width="12" height="12" data-type="math" data-latex="$X$" />的分布列为<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/new_image1677036249120720.png" width="662" height="85"><br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692802766.png" width="358" height="26" data-type="math" data-latex="$E(X)=0 \\times \\frac{1}{12}+1 \\times \\frac{5}{12}+2 \\times \\frac{5}{12}+3 \\times \\frac{1}{12}=\\frac{3}{2}$" />.', 'type': '解答题', 'subject': ' 数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 18, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答 题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f591e26d986d416f3add0d', 'stem': '在四棱锥<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692738346.png" width="88" height="17" data-type="math" data-latex="$P-ABCD$" />中，底面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692926093.png" width="44" height="13" data-type="math" data-latex="$ ABCD $" />是矩形,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692123625.png" width="45" height="16" data-type="math" data-latex="$ P A \\perp $" />平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692456972.png" width="44" height="13" data-type="math" data-latex="$ABCD$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692131920.png" width="108" height="16" data-type="math" data-latex="$PA=AD=8$" />.以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692920548.png" width="21" height="13" data-type="math" data-latex="$AC$" />的中点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692998371.png" width="11" height="12" data-type="math" data-latex="$O$" />为球心，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692920548.png" width="21" height="13" data-type="math" data-latex="$AC$" />为直径的球面交<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692215217.png" width="22" height="12" data-type="math" data-latex="$PD$" />于点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241190.png" width="15" height="12" data-type="math" data-latex="$M$" />.<br/>(1)证明:<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241190.png" width="15" height="12" data-type="math" data-latex="$M$" />为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692215217.png" width="22" height="12" data-type="math" data-latex="$PD$" />的中点.<br/>(2)若二面角<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692359883.png" width="99" height="17" data-type="math" data-latex="$B-AM-C$" />的余弦值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692285703.png" width="22" height="30" data-type="math" data-latex="$\\frac{\\sqrt{3}}{3}$" />,求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692289829.png" width="21" height="13" data-type="math" data-latex="$AB$" />.<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/imgpaper/lqy_upload/61271304cc28dab8c31e2282e8febef6.png" width="167" height="162">', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1)因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692920548.png" width="21" height="13" data-type="math" data-latex="$AC$" />是所作球面的直径,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692495012.png" width="86" height="16" data-type="math" data-latex="$A M \\perp M C$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692905084.png" width="21" height="13" data-type="math" data-latex="$P A$" />平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692456972.png" width="44" height="13" data-type="math" data-latex="$ABCD$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692181090.png" width="77" height="16" data-type="math" data-latex="$P A \\perp C D$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692631903.png" width="78" height="16" data-type="math" data-latex="$C D \\perp A D$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692389095.png" width="46" height="15" data-type="math" data-latex="$ C D\\perp $" />平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692110135.png" width="33" height="13" data-type="math" data-latex="$PAD$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692980038.png" width="87" height="19" data-type="math" data-latex="$C D \\perp A M,$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692432811.png" width="49" height="16" data-type="math" data-latex="$A M \\perp$" />平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692296044.png" width="34" height="12" data-type="math" data-latex="$PCD$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692643914.png" width="82" height="16" data-type="math" data-latex="$A M \\perp P D$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692821443.png" width="77" height="16" data-type="math" data-latex="$PA=AD$" />，所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692241190.png" width="15" height="12" data-type="math" data-latex="$M$" />为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692215217.png" width="22" height="12" data-type="math" data-latex="$PD$" />的中点.<br/>(2)以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692979367.png" width="10" height="13" data-type="math" data-latex="$A$" />为坐标原点，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692558272.png" width="21" height="13" data-type="math" data-latex="$AB $" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692177889.png" width="22" height="13" data-type="math" data-latex="$AD$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692951115.png" width="21" height="13" data-type="math" data-latex="$AP$" />所在直线分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692309038.png" width="9" height="11" data-type="math" data-latex="$x$" />轴，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692517878.png" width="8" height="15" data-type="math" data-latex="$y$" />轴，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692923326.png" width="8" height="11" data-type="math" data-latex="$z$" />轴，建立如图所示的空间直角坐标系,<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/new_image1677037456281060.png" /><br/>设<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692893644.png" width="56" height="16" data-type="math" data-latex="$AB=t$" />，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692313394.png" width="221" height="21" data-type="math" data-latex="$M(0,4,4),C(t,8,0),B(t,0,0)$" />.<br/>设平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692434306.png" width="37" height="13" data-type="math" data-latex="$ABM$" />的法向量为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692228386.png" width="120" height="26" data-type="math" data-latex="$ \\overrightarrow{n}=\\left(x_{1}, y_{1}, z_{1}\\right) $" />,因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692568683.png" width="103" height="32" data-type="math" data-latex="$ \\overrightarrow{A B}=(t, 0,0) $" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692208042.png" width="111" height="32" data-type="math" data-latex="$ \\overrightarrow{A M}=(0,4,4) $" />.<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692174852.png" width="197" height="64" data-type="math" data-latex="$ \\{ \\begin{array}  { l  }  { m \\cdot \\overrightarrow{A B} = t x _ { 1 } = 0 } \\\\ { n \\cdot \\overrightarrow{A M}  = 4 y _ { 1 } + 4 z _ { 1 } = 0 } \\end{array} $" />，令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692917617.png" width="48" height="18" data-type="math" data-latex="$y_1=1$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692576263.png" width="113" height="26" data-type="math" data-latex="$ {\\overrightarrow n=(0,1,-1).} $" /><br/>设平面<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692723216.png" width="37" height="13" data-type="math" data-latex="$ACM$" />的法向量为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692498355.png" width="120" height="26" data-type="math" data-latex="$ \\overrightarrow{m}=\\left(x_{2}, y_{2}, z_{2}\\right) $" />,因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692565622.png" width="103" height="32" data-type="math" data-latex="$\\overrightarrow{A C}=(t, 8,0)$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692127522.png" width="111" height="32" data-type="math" data-latex="$\\overrightarrow{A M}=(0,4,4)$" />,<br/>所 以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692683751.png" width="216" height="64" data-type="math" data-latex="$ \\left\\{\\begin{array}{l}\\boldsymbol{m} \\cdot \\overrightarrow{A C}=t x_{2}+8 y_{2}=0, \\\\ \\boldsymbol{m} \\cdot \\overrightarrow{A M}=4 y_{2}+4 z_{2}=0,\\end{array}\\right. $" /> 令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692703177.png" width="48" height="18" data-type="math" data-latex="$y_{2}=1$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692778316.png" width="128" height="27" data-type="math" data-latex="$ \\overrightarrow{m}=\\left(-\\frac{8}{t}, 1,-1\\right) $" />.<br/>设二面角<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692359883.png" width="99" height="17" data-type="math" data-latex="$B-AM-C$" />为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692890547.png" width="11" height="11" data-type="math" data-latex="$\\alpha$" />,<br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692189447.png" width="364" height="56" data-type="math" data-latex="$ \\cos \\alpha=|\\cos \\langle \\overrightarrow{{n}},  \\overrightarrow {m}\\rangle|=\\frac{|\\overrightarrow{{n}} \\cdot  \\overrightarrow{{m}}|}{| \\overrightarrow{n}|| \\overrightarrow{m}|}=\\frac{2}{\\sqrt{2} \\times \\sqrt{\\frac{64}{t^{2}}+2}}=\\frac{\\sqrt{3}}{3} $" /><br/>解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692405406.png" width="38" height="15" data-type="math" data-latex="$t=4$" />，即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692363704.png" width="58" height="16" data-type="math" data-latex="$AB=4$" />.', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 19, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新 乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f5a69f6d986d416f3add2b', 'stem': '已知椭圆<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692963021.png" width="116" height="27" data-type="math" data-latex="${C:\\frac{x^{2}}{4}+y^{2}=1}$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692403240.png" width="60" height="16" data-type="math" data-latex="$△PAB$" />的三个顶点都在椭圆<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692883308.png" width="11" height="12" data-type="math" data-latex="$C$" />上,且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692452016.png" width="10" height="12" data-type="math" data-latex="$P$" />为椭圆<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692883308.png" width="11" height="12" data-type="math" data-latex="$C$" />的左顶点,直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692289829.png" width="21" height="13" data-type="math" data-latex="$AB$" />经过点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692979609.png" width="52" height="21" data-type="math" data-latex="$(-1,0)$" />.<br/>(1)求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692403240.png" width="60" height="16" data-type="math" data-latex="$△PAB$" />面积的最大值.<br/>(2)若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692403240.png" width="60" height="16" data-type="math" data-latex="$△PAB$" />三边所在的直线斜率都存在,且分别记为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692986113.png" width="103" height="18" data-type="math" data-latex="${k_{PA},k_{PB},k_{AB}}$" />试判断<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692534534.png" width="122" height="21" data-type="math" data-latex="$ {k_{AB}(k_{PA}+k_{PB})} $" />是否为定值.若是,求出该定值;若不是,请说明理由.', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1)因为椭圆C的方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692155025.png" width="88" height="27" data-type="math" data-latex="$\\frac{x^{2}}{4}+y^{2}=1$" />，所以椭圆C的左顶点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692573978.png" width="64" height="21" data-type="math" data-latex="$P(-2,0)$" />.<br/>设直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692289829.png" width="21" height="13" data-type="math" data-latex="$AB$" />的方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692819613.png" width="87" height="18" data-type="math" data-latex="$x=my-1$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692792916.png" width="71" height="21" data-type="math" data-latex="$A\\left(x_{1}, y_{1}\\right)$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692325239.png" width="71" height="21" data-type="math" data-latex="$B\\left(x_{2}, y_{2}\\right)$" />,<br/>联立方程组<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692201326.png" width="108" height="56" data-type="math" data-latex="$\\left\\{\\begin{array}{l} x=m y-1 \\\\ \\frac{x^{2}}{4}+y^{2}=1 \\end{array}\\right.$" />，得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692310636.png" width="204" height="23" data-type="math" data-latex="$\\left(m^{2}+4\\right) y^{2}-2 m y-3=0$" /><br/>则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692280415.png" width="116" height="27" data-type="math" data-latex="$y_{1}+y_{2}=\\frac{2 m}{m^{2}+4}$" />.<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692125714.png" width="94" height="28" data-type="math" data-latex="$y_{1} y_{2}=\\frac{-3}{m^{2}+4}$" />.<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692230490.png" width="431" height="45" data-type="math" data-latex="$| A B | = \\sqrt { ( 1 + m ^ { 2 } ) [ ( y _ { 1 } + y _ { 2 } ) ^ { 2 } - 4 y _ { 1 } y _ { 2 } ] } = 4 \\sqrt { \\frac { ( 1 + m ^ { 2 } ) ( m ^ { 2 } + 3 ) } { ( m ^ { 2 } + 4 ) ^ { 2 } } }$" /><br/>因为点<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692452016.png" width="10" height="12" data-type="math" data-latex="$P$" />到直线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692289829.png" width="21" height="13" data-type="math" data-latex="$AB$" />的距离<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692638743.png" width="82" height="32" data-type="math" data-latex="$d=\\frac{1}{\\sqrt{1+m^{2}}}$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692180031.png" width="559" height="49" data-type="math" data-latex="$ S_{\\triangle P A B}=\\frac{1}{2}|A B| d=2 \\sqrt{\\frac{m^{2}+3}{\\left(m^{2}+4\\right)^{2}}}=2 \\sqrt{\\frac{m^{2}+3}{\\left(m^{2}+3\\right)^{2}+2\\left(m^{2}+3\\right)+1}}=2 \\sqrt{\\frac{1}{m^{2}+3+\\frac{1}{m^{2}+3}+2}} .$" /><br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692940498.png" width="84" height="21" data-type="math" data-latex="$m^{2}+3 \\geqslant 3$" />.所以 当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692889125.png" width="84" height="19" data-type="math" data-latex="$m^{2}+3=3$" />.即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692361716.png" width="47" height="15" data-type="math" data-latex="$m=0$" />时，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692140775.png" width="54" height="16" data-type="math" data-latex="$\\triangle P A B$" />面积取得最大值，最大值为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692607776.png" width="22" height="30" data-type="math" data-latex="$\\frac{\\sqrt{3}}{2}$" /><br/>(2)由（1）知<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692892362.png" width="154" height="28" data-type="math" data-latex="$k_{P A}=\\frac{y_{1}}{x_{1}+2}=\\frac{y_{1}}{m y_{1}+1}$" />.<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692954097.png" width="154" height="28" data-type="math" data-latex="$k_{P B}=\\frac{y_{2}}{x_{2}+2}=\\frac{y_{2}}{m y_{2}+1}$" />.<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692283462.png" width="69" height="24" data-type="math" data-latex="$k_{A B}=\\frac{1}{m}$" />.<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692897079.png" width="483" height="33" data-type="math" data-latex="$ k_{A B}\\left(k_{P A}+k_{P B}\\right)=\\frac{1}{m}\\left(\\frac{y_{1}}{m y_{1}+1}+\\frac{y_{2}}{m y_{2}+1}\\right)=\\frac{1}{m} \\times \\frac{2 m y_{1} y_{2}+y_{1}+y_{2}}{m^{2} y_{1} y_{2}+m\\left(y_{1}+y_{2}\\right)+1} $" /><br/><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692890196.png" width="288" height="53" data-type="math" data-latex="$=\\frac{1}{m} \\times \\frac{\\frac{-6 m}{m^{2}+4}+\\frac{2 m}{m^{2}+4}}{\\frac{-3 m^{2}}{m^{2}+4}+\\frac{2 m^{2}}{m^{2}+4}+1}=\\frac{1}{m} \\times \\frac{-4 m}{4}=-1$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692667863.png" width="126" height="21" data-type="math" data-latex="$k_{A B}\\left(k_{P A}+k_{P B}\\right)$" />为定值，且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692331810.png" width="171" height="21" data-type="math" data-latex="$k_{A B}\\left(k_{P A}+k_{P B}\\right)=-1$" />.', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 20, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f5ae7b6d986d416f3add46', 'stem': '已知函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692272561.png" width="189" height="27" data-type="math" data-latex="$ {f(x)=\\frac{e^{x}}{x^{2}}-ax+2a\\ln x.} $" /><br/>(1)判断<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />极值点的个数;<br/>(2)当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692665570.png" width="39" height="16" data-type="math" data-latex="$a≤e$" />时,证明:<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692887761.png" width="64" height="21" data-type="math" data-latex="$f(x)≥0$" />.', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1)因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692272561.png" width="189" height="27" data-type="math" data-latex="$ {f(x)=\\frac{e^{x}}{x^{2}}-ax+2a\\ln x.} $" />c,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692833351.png" width="158" height="27" data-type="math" data-latex="${f^{\\prime }(x)=\\frac{x-2}{x}(\\frac{e^{x}}{x^{2}}-a).}$" /><br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692918878.png" width="135" height="27" data-type="math" data-latex="${g(x)=\\frac{e^{x}}{x^{2}}(x>0),}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692270528.png" width="116" height="31" data-type="math" data-latex="${g^{\\prime }(x)=\\frac{(x-2)e^{x}}{x^{3}},}$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692377580.png" width="32" height="21" data-type="math" data-latex="$g(x)$" />在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692543653.png" width="38" height="21" data-type="math" data-latex="$(0,2)$" />上单调递减,在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692493765.png" width="61" height="21" data-type="math" data-latex="$(2,+∞)$" />上单调递增,所以,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692286994.png" width="153" height="27" data-type="math" data-latex="${g(x)_{\\min }=g(2)=\\frac{e^{2}}{4}.}$" /><br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692561833.png" width="49" height="27" data-type="math" data-latex="${a\\leq \\frac{e^{2}}{4}}$" />时，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692987710.png" width="81" height="27" data-type="math" data-latex="${\\frac{e^{x}}{x^{2}}-a\\geq 0}$" />，若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692975565.png" width="69" height="21" data-type="math" data-latex="$x∈(0,2)$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692332158.png" width="69" height="21" data-type="math" data-latex="${f^{\\prime }(x)<0}$" />,若<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692692193.png" width="92" height="21" data-type="math" data-latex="$x∈(2,+∞)$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692245583.png" width="75" height="21" data-type="math" data-latex="${f^{\\prime }(x)>0,}$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692733977.png" width="33" height="21" data-type="math" data-latex="$f(x)$" />只有一个极值点.<br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692477298.png" width="49" height="27" data-type="math" data-latex="${a>\\frac{e^{2}}{4}}$" />时,存在<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692447388.png" width="186" height="21" data-type="math" data-latex="$ {x_{1}∈(0,2),x_{2}∈(2,+\\infty )} $" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692466904.png" width="86" height="27" data-type="math" data-latex="${\\frac{e^{x}}{x^{2}}-a=0.}$" /><br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692185635.png" width="168" height="21" data-type="math" data-latex="${x∈(0,x_{1})\\cup (x_{2},+\\infty )}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692385014.png" width="88" height="27" data-type="math" data-latex="${,\\frac{e^{x}}{x^{2}}-a>0}$" /> ；当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692272484.png" width="88" height="21" data-type="math" data-latex="${x∈(x_{1},x_{2})}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692838653.png" width="93" height="27" data-type="math" data-latex="${,\\frac{e^{x}}{x^{2}}-a<0.}$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692478937.png" width="327" height="21" data-type="math" data-latex="${x∈(0,x_{1}),f^{\\prime }(x)<0,x∈(x_{1},2),f^{\\prime }(x)>0;}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692807710.png" width="78" height="21" data-type="math" data-latex="$x∈(2,x_2)$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692933042.png" width="69" height="21" data-type="math" data-latex="$f\'(x)<0$" />;<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692657217.png" width="106" height="21" data-type="math" data-latex="$x∈(x_2,+∞),$" /> <img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692566182.png" width="69" height="21" data-type="math" data-latex="$ f\'(x)>0 $" />.<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692339374.png" width="33" height="21" data-type="math" data-latex="$ {f(x)} $" />有三个极值点.<br/>综上,当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692561833.png" width="49" height="27" data-type="math" data-latex="${a\\leq \\frac{e^{2}}{4}}$" /><img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692810813.png" width="41" height="21" data-type="math" data-latex="${,f(x)}$" />只有一个极值点;当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692477298.png" width="49" height="27" data-type="math" data-latex="${a>\\frac{e^{2}}{4}}$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692339374.png" width="33" height="21" data-type="math" data-latex="$ {f(x)} $" />三个极值点<br/>(2)证明:<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693784608.png" width="383" height="27" data-type="math" data-latex="${f(x)=\\frac{e^{x}}{x^{2}}-ax+2a\\ln x=e^{x-2\\ln x}-a(x-2\\ln x).}$" /><br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693363447.png" width="127" height="21" data-type="math" data-latex="$h(x)=x-2\\ln x$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693571196.png" width="91" height="26" data-type="math" data-latex="${h^{\\prime }(x)=\\frac{x-2}{x}}$" />,所以当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692975565.png" width="69" height="21" data-type="math" data-latex="$x∈(0,2)$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693183030.png" width="112" height="21" data-type="math" data-latex="${h^{\\prime }(x)<0,h(x)}$" />单调递减,<br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692692193.png" width="92" height="21" data-type="math" data-latex="$x∈(2,+∞)$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693474794.png" width="120" height="21" data-type="math" data-latex="${,h^{\\prime }(x)>0,h(x)}$" />单调递增,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693657481.png" width="180" height="21" data-type="math" data-latex="$h(x)≥h(2)=2-2\\ln 2$" />,<br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693582979.png" width="99" height="16" data-type="math" data-latex="$t=x-2\\ln x$" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692887761.png" width="64" height="21" data-type="math" data-latex="$f(x)≥0$" />等价 于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693113056.png" width="87" height="19" data-type="math" data-latex="$ {e^{t}-at\\geq 0.} $" /><br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693857805.png" width="94" height="21" data-type="math" data-latex="$t=h(x)>0$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693251008.png" width="82" height="19" data-type="math" data-latex="$ {e^{t}-at\\ge 0} $" />等价于<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693882918.png" width="53" height="27" data-type="math" data-latex="${a\\leq \\frac{e^{t}}{t}.}$" /><br/>令<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693810512.png" width="176" height="27" data-type="math" data-latex="$ {\\varphi (t)=\\frac{e^{t}}{t},t\\geq 2-2\\ln2} $" />,则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693182216.png" width="111" height="32" data-type="math" data-latex="${\\varphi ^{\\prime }(t)=\\frac{(t-1)e^{t}}{t^{2}},}$" /><br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693131408.png" width="123" height="21" data-type="math" data-latex="$t∈[2-2\\ln 2,1)$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693537182.png" width="114" height="21" data-type="math" data-latex="$ {,\\varphi ^{\\prime }(t)<0,\\varphi (t)} $" />单调递减<br/>当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693744276.png" width="88" height="21" data-type="math" data-latex="$t∈(1,+∞)$" />时<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693422103.png" width="114" height="21" data-type="math" data-latex="$ {,\\varphi\' (t)>0,\\varphi (t)} $" />单调递增,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693777610.png" width="146" height="21" data-type="math" data-latex="${\\varphi (t)_{\\min }=\\varphi (1)=e.}$" /><br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692665570.png" width="39" height="16" data-type="math" data-latex="$a≤e$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693207707.png" width="63" height="21" data-type="math" data-latex="$a≤φ(t)$" />,故<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692887761.png" width="64" height="21" data-type="math" data-latex="$f(x)≥0$" />.', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 21, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}, {'difficulty': '中', 'repeat_res': [], 'topic_type_id': 33, 'id': '63f5ac366d986d416f3add3c', 'stem': '在直角坐标系<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693704088.png" width="32" height="18" data-type="math" data-latex="$xOy$" />中,以坐标原点为极点,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692309038.png" width="9" height="11" data-type="math" data-latex="$x$" />轴的正半轴为极轴建立极坐标系,曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693604369.png" width="16" height="14" data-type="math" data-latex="${C_{1}}$" />及坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693699030.png" width="137" height="26" data-type="math" data-latex="$ {\\rho\\cos (\\theta +\\frac{\\pi}{4})=\\sqrt {2}} $" />,曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693374201.png" width="16" height="14" data-type="math" data-latex="${C_{2}}$" />的极坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693980891.png" width="159" height="21" data-type="math" data-latex="${\\rho ^{2}-2o\\cos \\theta -3=0.}$" /><br/>(1)求曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693604369.png" width="16" height="14" data-type="math" data-latex="${C_{1}}$" />和曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693374201.png" width="16" height="14" data-type="math" data-latex="${C_{2}}$" />的直角坐标方程;<br/>(2)设<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693417203.png" width="51" height="21" data-type="math" data-latex="$P(3,1)$" />,曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693604369.png" width="16" height="14" data-type="math" data-latex="${C_{1}}$" />与曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693374201.png" width="16" height="14" data-type="math" data-latex="${C_{2}}$" />的交点为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692866613.png" width="33" height="19" data-type="math" data-latex="$A,B$" />,求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693519860.png" width="66" height="33" data-type="math" data-latex="${\\frac{|PA|+|PB|}{|AB|}}$" />的值.', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1) 曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693909625.png" width="16" height="14" data-type="math" data-latex="$C_{1}$" />的极坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693691780.png" width="321" height="30" data-type="math" data-latex="$\\rho \\cos \\left(\\theta+\\frac{\\pi}{4}\\right)=\\frac{\\sqrt{2}}{2} \\rho \\cos \\theta-\\frac{\\sqrt{2}}{2} \\rho \\sin \\theta=\\sqrt{2}$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693110164.png" width="79" height="18" data-type="math" data-latex="$x=\\rho \\cos \\theta$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693257697.png" width="76" height="18" data-type="math" data-latex="$y=\\rho \\sin \\theta$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693221464.png" width="102" height="18" data-type="math" data-latex="$x-y-2=0$" />.<br/>即曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693909625.png" width="16" height="14" data-type="math" data-latex="$C_{1}$" />的直角坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693221464.png" width="102" height="18" data-type="math" data-latex="$x-y-2=0$" />.<br/>因为曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693453497.png" width="16" height="14" data-type="math" data-latex="$C_{2}$" />的极坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693170552.png" width="154" height="21" data-type="math" data-latex="$\\rho^{2}-2 \\rho \\cos \\theta-3=0$" />.<br/>且<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693110164.png" width="79" height="18" data-type="math" data-latex="$x=\\rho \\cos \\theta$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693113472.png" width="96" height="21" data-type="math" data-latex="$x^{2}+y^{2}=\\rho^{2}$" />，<br/>所以曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693453497.png" width="16" height="14" data-type="math" data-latex="$C_{2}$" />的直角坐标方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693569168.png" width="157" height="21" data-type="math" data-latex="$x^{2}+y^{2}-2 x-3=0$" />,<br/>即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693289962.png" width="131" height="22" data-type="math" data-latex="$(x-1)^{2}+y^{2}=4$" />.<br/>(2)因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693417203.png" width="51" height="21" data-type="math" data-latex="$P(3,1)$" />在曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693142369.png" width="16" height="14" data-type="math" data-latex="$C_1$" />上,所以曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693604369.png" width="16" height="14" data-type="math" data-latex="${C_{1}}$" />的参数方程为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693549307.png" width="106" height="64" data-type="math" data-latex="$\\{ \\begin{array}  { l  }  { x = 3 + \\frac { \\sqrt { 2 } } { 2 } t } \\\\ { y = 1 + \\frac { \\sqrt { 2 } } { 2 } t } \\end{array}$" />,<br/>将曲线<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693909625.png" width="16" height="14" data-type="math" data-latex="$C_{1}$" />的参数方程代入<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693289962.png" width="131" height="22" data-type="math" data-latex="$(x-1)^{2}+y^{2}=4$" />中，得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693198640.png" width="136" height="21" data-type="math" data-latex="$t^{2}+3 \\sqrt{2} t+1=0$" /><br/>设<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692866613.png" width="33" height="19" data-type="math" data-latex="$A,B$" />对应的参数分别为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693491003.png" width="13" height="16" data-type="math" data-latex="$t_{1}$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693982453.png" width="13" height="16" data-type="math" data-latex="$t_{2}$" />，则<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693270490.png" width="118" height="21" data-type="math" data-latex="$t_{1}+t_{2}=-3 \\sqrt{2}$" />，<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693734945.png" width="60" height="17" data-type="math" data-latex="$t_{1} t_{2}=1$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693986743.png" width="231" height="23" data-type="math" data-latex="$|P A|+|P B|=\\left|t_{1}+t_{2}\\right|=3 \\sqrt{2}$" />.<br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693437723.png" width="472" height="36" data-type="math" data-latex="$ |A B|=\\left|t_{1}-t_{2}\\right|=\\sqrt{\\left(t_{1}+t_{2}\\right)^{2}-4 t_{1} t_{2}}=\\sqrt{(3 \\sqrt{2})^{2}-4}=\\sqrt{14} ,$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693490856.png" width="171" height="34" data-type="math" data-latex="$\\frac{|P A|+|P B|}{|A B|}=\\frac{3 \\sqrt{2}}{\\sqrt{14}}=\\frac{3 \\sqrt{7}}{7}$" />.', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 22, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}, {'difficulty': '中', 'repeat_res': [[201511101239911, 0.97], [201511101391798, 0.96], [201511102710736, 0.96], [201511100563796, 0.95], [201511100856582, 0.95], [201511100863225, 0.95], [201511101355340, 0.95], [201511102654919, 0.95], [201511102718933, 0.95], [201511100930077, 0.94]], 'topic_type_id': 33, 'id': '63f5ac366d986d416f3add3d', 'stem': '已知函数<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693118570.png" width="186" height="21" data-type="math" data-latex="$f(x)=|x+a|+|x-2a|$" />.<br/>(1)当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693804799.png" width="41" height="15" data-type="math" data-latex="$a=1$" />时,求不等式<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693711743.png" width="64" height="21" data-type="math" data-latex="$f(x)≤5$" />的解集;<br/>(2)若对任意<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693831211.png" width="45" height="15" data-type="math" data-latex="$x\\in R$" />,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693811207.png" width="112" height="22" data-type="math" data-latex="$ {f(x)\\geq 2a^{2}-2} $" />恒成立,求<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692619448.png" width="8" height="11" data-type="math" data-latex="$a$" />的取值范围.', 'options': [], 'key': '(1)见解析<br/>(2)见解析', 'parse': '(1)由题知,当<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693804799.png" width="41" height="15" data-type="math" data-latex="$a=1$" />时,<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693238884.png" width="177" height="21" data-type="math" data-latex="$f(x)=|x+1|+|x-2|$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693808032.png" width="202" height="68" data-type="math" data-latex="${f(x)=\\left\\lbrace \\begin{array}{l}{-2x+1,x<-1}\\\\{3,-1\\leq x<2},\\\\{2x-1,x\\geq 2},\\end{array}\\right.}$" /><br/>因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693711743.png" width="64" height="21" data-type="math" data-latex="$f(x)≤5$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693753771.png" width="112" height="45" data-type="math" data-latex="${\\left\\lbrace \\begin{array}{l}{x<-1},\\\\{-2x+1\\leq 5}\\end{array}\\right.}$" />或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693876806.png" width="106" height="45" data-type="math" data-latex="${\\left\\lbrace \\begin{array}{l}{-1\\leq x<2}\\\\{3\\leq 5}\\end{array}\\right.}$" /> 或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693977469.png" width="103" height="45" data-type="math" data-latex="${\\left\\lbrace \\begin{array}{l}{x\\geq 2},\\\\{2x-1\\leq 5},\\end{array}\\right.}$" /><br/>解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693496244.png" width="101" height="17" data-type="math" data-latex="$-2≤x<-1$" />或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693212199.png" width="87" height="17" data-type="math" data-latex="$-1≤x<2$" />或<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693550214.png" width="73" height="17" data-type="math" data-latex="$2≤x≤3$" />,<br/>所以不等式<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693711743.png" width="64" height="21" data-type="math" data-latex="$f(x)≤5$" />的解集为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693801265.png" width="48" height="21" data-type="math" data-latex="$[-2,3]$" />.<br/>(2)因为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693966085.png" width="381" height="21" data-type="math" data-latex="$f(x)=|x+a|+|x-2a|≥|x+a-x+2a|=|3a|$" />,所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693334291.png" width="111" height="21" data-type="math" data-latex="${f(x)_{\\min }=|3a|,}$" /><br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693863619.png" width="111" height="22" data-type="math" data-latex="${|3a|\\geq 2a^{2}-2,}$" />所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693400316.png" width="151" height="24" data-type="math" data-latex="${2|a|^{2}-3|a|-2\\leq 0,}$" />即<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693785314.png" width="166" height="21" data-type="math" data-latex="$(2|a|+1)(|a|-2)≤0$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693960586.png" width="51" height="21" data-type="math" data-latex="$|a|≤2$" />,解得<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693153685.png" width="86" height="17" data-type="math" data-latex="$-2≤a≤2$" />,<br/>所以<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047692619448.png" width="8" height="11" data-type="math" data-latex="$a$" />的取值范围为<img src="http://zxhx-pro-1302712961.cos.ap-beijing.myqcloud.com/zyk/uploadfiles/wording/63ef1ecb11a1cdad550f42e7/MJMATH-1677047693648969.png" width="48" height="21" data-type="math" data-latex="$[-2,2]$" />.', 'type': '解答题', 'subject': '数学', 'grade': '', 'category': [], 'year': 2023, 'period': '高中', 'province': '', 'specials': [], 'upload_time': '2023-02-22T14:09:59.218000', 'errmsgs': '', 'topic_num': 23, 'blank_num': None, 'options_rank': None, 'slave': [], 'answer_type': None, 'slave_no': '1-2', 'source': {'type': 's', 'related_exampaper': [{'paper_id': '63ef1ecb11a1cdad550f42e7', 'file_name': '2023金太阳新乡市理三入学试题', 'item_id': None}]}, 'checkType': {'id': 33, 'btt_id': 8, 'name': '解答题'}}]
    267. 

  267. 

  268. 

  269. def Hmath_kps(input_data):
    270. 

  270.     label_auto = []
    271. 

  271.     try:
    272. 

  272.         r = requests.post("http://172.16.2.5:13356/auto_labels", json=input_data)  # json={"input_data": item}
    273. 

  273.         if r.status_code == 200:
    274. 

  274.             res = r.json()['data']  # eval(r.text)
    275. 

  275.             print("--------------------", res)
    276. 

  276.             if res:
    277. 

  277.                 label_auto = res
    278. 

  278.     except Exception as e:
    279. 

  279.         return "保存入库高中数学标注服务异常"
    280. 

  280.     return label_auto
    281. 

  281. 

  282. 

  283. t1 = time.time()
    284. 

  284. subject_id = {
    285. 

  285.     "高中数学": 3,
    286. 

  286.     "高中英语": 8,
    287. 

  287.     "高中物理": 12,
    288. 

  288.     "高中化学": 13,
    289. 

  289.     "高中生物": 14,
    290. 

  290.     "高中政治": 15,
    291. 

  291.     "高中历史": 16,
    292. 

  292.     "高中地理": 17,
    293. 

  293.     "初中数学": 41,
    294. 

  294.     "初中英语": 42,
    295. 

  295.     "初中物理": 43,
    296. 

  296.     "初中化学": 44,
    297. 

  297.     "初中生物": 45,
    298. 

  298.     "初中地理": 46,
    299. 

  299.     "初中政治": 47,
    300. 

  300.     "初中历史": 48,
    301. 

  301.     "高中语文": 9,
    302. 

  302.     "初中语文": 40,
    303. 

  303. }
    304. 

  304. items_info = {"subject_id": subject_id["高中数学"],
    305. 

  305.               "topics": [{"topic_id": one_items["id"],
    306. 

  306.                           "topic_type_id": one_items["checkType"]["id"],
    307. 

  307.                           "content": one_items["stem"],
    308. 

  308.                           "parse": one_items["parse"],
    309. 

  309.                           "option": one_items["options"],
    310. 

  310.                           "slave": one_items["slave"]} for one_items in items]
    311. 

  311.               }
    312. 

  312. 

  313. auto_kps = Hmath_kps(items_info)
    314. 

  314. print(auto_kps)
    315. 

  315. if type(auto_kps) == str:
    316. 

  316.     pass
    317. 

  317. else:
    318. 

  318.     for nn, one_items in enumerate(items):
    319. 

  319.         one_items.update({"auto_mark_result": auto_kps[nn]})
    320.