planet/高考数学/2021年北京市高考数学试题.tm

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  <doc-data|<doc-title|2021年北京市高考数学试题>>

  <\with|font-series|bold>
    一、单选题
  </with>

  1．已知集合<math|A=<around*|{|x\|-1\<less\>x\<less\>1|}>>，<math|B=<around*|{|x\|0\<leq\>x\<leq\>2|}>>，则<math|A\<cup\>B=>（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|<around*|{|x\|-1\<less\>x\<less\>2|}>>>|<cell|B．<math|<around*|{|x\|-1\<less\>x\<leq\>2|}>>>>|<row|<cell|C．<math|<around*|{|x\|0\<leq\>x\<less\>1|}>>>|<cell|D．<math|<around*|{|x\|0\<leq\>x\<leq\>2|}>>>>>>>

  2．在复平面内，复数<math|z>满足<math|<around|(|1-i|)>*z=2>，则<math|z=>（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|-1-i>>|<cell|B．<math|-1+i>>|<cell|C．<math|1-i>>|<cell|D．<math|1+i>>>>>>

  3．已知<math|f<around|(|x|)>>是定义在上<math|<around|[|0,1|]>>的函数，那么\P函数<math|f<around|(|x|)>>在<math|<around|[|0,1|]>>上单调递增\Q是\P函数<math|f<around|(|x|)>>在<math|<around|[|0,1|]>>上的最大值为<math|f<around|(|1|)>>\Q的（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．充分而不必要条件>|<cell|B．必要而不充分条件>>|<row|<cell|C．充分必要条件>|<cell|D．既不充分也不必要条件>>>>>
  4．某四面体的三视图如图所示，该四面体的表面积为（<space|1.2em>）

  <center|<image|<tuple|<#89504E470D0A1A0A0000000D49484452000002D7000002D008060000005B8B5CE5000000097048597300000B1300000B1301009A9C180000200049444154789CECDDE9731BD79536F0A7577463DF0102202951D4E64DB6A766525353F9927F7CBEBCC9388E97C896B59992B860DF77A0D1DBFBC1EA1BCA5162CAA20488787E552CBB2252B9A4A18B47B7EF3947F27DDF077C5C8C74EEDF2FFA35BFF5FBBCCEEFF9BD7FEBF7DC04FFEEFBFAAD9FEDEBBEBFDFFA39BDE9D7BC8B9FE145BF97B7FD9AB7F1A63F93B779EDBF0B97F5E7F2A2FF3FBFE775F77BBCE9F7F2BE7E0E17F536AF9D75FD0CB7C145F7DA37F9D9FD9EAF79DD7ADEC6FBD8C72EBAD6B7FDBDDFE6CFFEDB78177F5E2EFBBDEB2AFC99DEF4BDF14D5E8BAFFFB3AFBED97FA8CBFA8F7A155E1C97F51FE6A29FB7093FB375FEF77FDF7FF03E249BF0DA202222220090D7BD0022222222A2AB82E19A88888888E89248BEEFFDEA99F2DBDC1D7BEDFFC5B97FBF4A77A92FFB5AC86559D7DDE477E1F7FC1C37F5F572DEFBB8174D444444EBC0936B22222222A24B22FDD22D848888888888DE164FAE89888888882E09C33511111111D12561B82622222222BA240CD744444444449784E19A88888888E892305C13111111115D12866B22222222A24BC2704D44444444744918AE89888888882E09C33511111111D12561B82622222222BA240CD744444444449784E19A88888888E892305C13111111115D12866B22222222A24BA2AE7B014444F42ADFF76159162CCB82E338F07DFF8DBEDEF33CF8BE0FDFF7214912645986244917FA5A4992A0AAEA2B1F8AA2FC9E6FE39D715D178EE3C0755DB14659E65911116D06866B22A20DE3FB3EE6F33946A311168B8508CB1715844FCFF34438BE68F85455158661C0344DF1B189E17AB95CC2B22C98A609599619AE896863BCF3707DFE0DC1711C71DAF0A65FFFEB371649922E7C120340BCB904FFDCB48DD8F77DB8AE0BCFF32049D21BBD1912D1D5109C36BBAE8BC9648266B389C96402D7757F73DF0CBE16006CDB866DDB705D17BAAE23140A5D783FD1751DF1781CF1781C894402AAAA42D7F5B7FEDE2E936559180E87188FC7884422884422300C03866120140AAD7B7944B4E5DECBC9B5EFFBF03C0FA3D1089D4E07BD5EEF953782DFFADAE02338BD09C2F145DF2C244942229140229140341A85699A0887C36FFB6D5D2AD775319D4E319BCDA0691AA2D1E8C6AD9188DE2DDFF7E1380E56AB159ACD261E3E7C884EA773E1701D7CDE72B9C462B180E3388846A38846A350D58B6DF786612093C9209D4E636F6F0FA150089148E432BEBD4B331A8DF0E4C9131C1D1D41D334689A8642A1809B376FE2F0F070DDCB23A22DF75EC2B5E779701C07C3E1102F5EBCC08B172FC429ED6F056CCFF35EF9705D178AA2887B761721CB322A950ACAE532F2F93C6459DEB8E0EABA2E66B3193A9D0EC2E130344DDBB83512D1BB15EC958BC502F57A1D0F1F3EC4D9D9D985C2B5E779B06D1B8EE3603A9D623299C0B66DA45229A4D3E90B9F3E472211140A05140A05E8BA8E7C3E7F19DFDAA51A0E8778F2E409FEF297BF88C397C3C343844221866B225ABBF7722D243889592C16188FC7180C06222CFF96D56A05CBB2C43F2DCB42381C462C16BB70F8946519B1580CB1584C3C42DC34B66DA3D168E0F1E3C7D0340DD96C567C6432191886B1EE2512D13B16141FAAAA8A542A854AA5024DD32EB45F06A17A369BC1711CF47A3D4CA753944A25ECEEEE22168B5D680D8661209D4E239D4E23914840D3B4CBF8D62E55703A3F9D4EC5CFA6DBEDE2F8F8183FFDF41392C92452A9140F2888682DDE5BB85E2E97582E97E21EE045599685F1788CE9742A3EB2D9EC1B9FECAE562BCC6633CCE773D8B6FD7BBE9577CAB22C9C9C9CE0EBAFBF866DDB482693C8E7F3F8FCF3CF71EFDE3D866BA22D1074EA300C03D7AF5F472412C16C36BBD0153ACBB2D0ED76D1ED76F1E8D12358960500B87DFB36FEF4A73F2197CB5D680D8AA2C0344D188681643279E150FE3E298A825028F4CA7B80E779383E3E866559B87BF72EEEDEBDCB704D446BF15E4FAE1DC70180373A09095A528D46230C06030C0603689A867C3E7FE1C79C41F1A3E338B06DFB4227E6EFDB6AB542A3D1C00F3FFC80C96482582C8642A180482482FDFD7D442211711D8688AEA6A096445555ECEFEF637F7FFFC25F3B9FCF717676866AB58AF97C8E9393134C26135CBB760DFFFDDFFF8DDDDDDD77B8F2F74B9665E8BA8E70382C7E669224A1D168A05EAF43D775140A055190A928CA1B15C01311BD8D779ED4645916D5DBC562118AA2BCD11DBEE3E363844221789E274EB1D3E9346EDDBA853B77EE5CE8F7902409E17018E17018F1781CD168F4777D2FEFD2F9132BDFF745F165AFD7C3B7DF7E8B6EB78B4AA58252A9B4EEA512D106725D17F3F91CFD7E1FC3E110B3D9EC77F7C9FE50047BFBAF0B36178B057EFAE927F4FB7D542A15542A958D6B27484457D73B0FD7922441D775288A024DD3904C26C509F645844221AC562B4C26138C4623288A82743A8D9B376FE2CB2FBFBCF0EF13B4B60B2ACB378D2449D0344DF4940D1EC7F6FB7D7CF7DD77180E87D0759DE19A885ECBF33CCCE7730C06030C87434CA7532C97CB2B1FAE23910832990C4CD304F08F1EE10F1E3C40BD5E87244928954A0CD744F4DEBC9770AD280A144581AEEB6F5C4C188BC5A0AAAAE81462DB365455453C1EBFF01DC20FC1F942265555914EA791CBE5309FCFC56954B55A45341A15FD67D9CF958802E74FAE83E133B66D63B95C62369B61B15840D3B42B77B54C555598A6894422217A5DD76A354C26130C8743D4EB751C1D1D21954A211

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|<frac|3|2>+<frac|<sqrt|3>|2>>>|<cell|B．<math|3+<sqrt|3>>>|<cell|C．<math|<frac|3|2>+<sqrt|3>>>|<cell|D．<math|3+<frac|<sqrt|3>|2>>>>>>>
  5．若双曲线<math|<frac|x<rsup|2>|a<rsup|2>>-<frac|y<rsup|2>|b<rsup|2>>=1>离心率为<math|2>，过点<math|<around*|(|<sqrt|2>,<sqrt|3>|)>>，则该双曲线的方程为（<space|1.2em>）

  A．<math|2*x<rsup|2>-y<rsup|2>=1> B．<math|x<rsup|2>-<frac|y<rsup|2>|3>=1>
  C．<math|5*x<rsup|2>-3*y<rsup|2>=1> D．<math|<frac|x<rsup|2>|2>-<frac|y<rsup|2>|6>=1>

  6．《中国共产党党旗党徽制作和使用的若干规定》指出，中国共产党党旗为旗面缀有金黄色党徽图案的红旗，通用规格有五种.这五种规格党旗的长<math|a<rsub|1>,a<rsub|2>,a<rsub|3>,a<rsub|4>,a<rsub|5>>
  (单位:cm)成等差数列，对应的宽为<math|b<rsub|1>,b<rsub|2>,b<rsub|3>,b<rsub|4>,b<rsub|5>>
  (单位: cm),且长与宽之比都相等，已知<math|a<rsub|1>=288>，<math|a<rsub|5>=96>，<math|b<rsub|1>=192>，则<math|b<rsub|3>=>（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．64>|<cell|B．96>|<cell|C．128>|<cell|D．160>>>>>
  7．函数<math|f<around|(|x|)>=cos x-cos 2*x>是（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．奇函数，且最大值为2>|<cell|B．偶函数，且最大值为2>>|<row|<cell|C．奇函数，且最大值为<math|<frac|9|8>>>|<cell|D．偶函数，且最大值为<math|<frac|9|8>>>>>>>\ 

  8．某一时间段内，从天空降落到地面上的雨水，未经蒸发、渗漏、流失而在水平面上积聚的深度，称为这个时段的降雨量（单位：mm）．24h降雨量的等级划分如下：

  <block*|<tformat|<table|<row|<cell|等级>|<cell|24h降雨量（精确到0.1）>>|<row|<cell|......>|<cell|......>>|<row|<cell|小雨>|<cell|<math|0.1\<sim\>9.9>>>|<row|<cell|中雨>|<cell|<math|10.0\<sim\>24.9>>>|<row|<cell|大雨>|<cell|<math|25.0\<sim\>49.9>>>|<row|<cell|暴雨>|<cell|<math|50.0\<sim\>99.9>>>|<row|<cell|.....>|<cell|......>>>>>

  <image|<tuple|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

  在综合实践活动中，某小组自制了一个底面直径为200
  mm，高为300 mm的圆锥形雨量器.若一次降雨过程中，该雨量器收集的24h的雨水高度是150
  mm（如图所示)，则这24h降雨量的等级是（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．小雨>|<cell|B．中雨>|<cell|C．大雨>|<cell|D．暴雨>>>>>\ 

  9．已知直线<math|y=k*x+m>（<math|m>为常数）与圆<math|x<rsup|2>+y<rsup|2>=4>交于点<math|M*，*N>，当<math|k>变化时，若<math|<around|\||M*N|\|>>的最小值为2，则<math|m=>（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|\<pm\>1>>|<cell|B．<math|\<pm\><sqrt|2>>>|<cell|C．<math|\<pm\><sqrt|3>>>|<cell|D．<math|\<pm\>2>>>>>>\ 

  10．已知<math|<around*|{|a<rsub|n>|}>>是各项均为整数的递增数列，且<math|a<rsub|1>\<geq\>3>，若<math|a<rsub|1>+a<rsub|2>+\<cdot\>\<cdot\>\<cdot\>+a<rsub|n>=100>，则<math|n>的最大值为（<space|1.2em>）

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．9>|<cell|B．10>|<cell|C．11>|<cell|D．12>>>>>\ 

  <with|font-series|bold|二、填空题>

  11．在<math|<around|(|x<rsup|3>-<frac|1|x>|)><rsup|4>>的展开式中，常数项为<underline|<space|2cm>>．

  12．若点<math|A*<around|(|cos \<theta\>,sin
  \<theta\>|)>>关于<math|y>轴对称点为<math|B*<around|(|cos
  <around|(|\<theta\>+<frac|\<pi\>|6>|)>,sin
  <around|(|\<theta\>+<frac|\<pi\>|6>|)>|)>>，写出<math|\<theta\>>的一个取值为<underline|<space|2cm>>．

  13．已知函数<math|f<around|(|x|)>=<around*|\||lg
  x|\|>-k*x-2>，给出下列四个结论：

  ①若<math|k=0>，<math|f<around|(|x|)>>恰 有2个零点；

  ②存在负数<math|k>，使得<math|f<around|(|x|)>>恰有个1零点；

  ③存在负数<math|k>，使得<math|f<around|(|x|)>>恰有个3零点；

  ④存在正数<math|k>，使得<math|f<around|(|x|)>>恰有个3零点．

  其中所有正确结论的序号是<underline|<space|2cm>>．

  <with|font-series|bold|三、双空题>

  14．已知抛物线<math|y<rsup|2>=4*x>的焦点为<math|F>，点<math|M>在抛物线上，<math|M*N>垂直<math|x>轴与于点<math|N>.若<math|<around|\||M*F|\|>=6>，则点<math|M>的横坐标为<underline|<space|2cm>>；
  <math|\<triangle\>M*N*F>的面积为<underline|<space|2cm>>．

  15．已知向量<math|<wide|a|\<wide-varrightarrow\>>,<wide|b|\<wide-varrightarrow\>>,<wide|c|\<wide-varrightarrow\>>>在正方形网格中的位置如图所示．若网格纸上小正方形的边长为1，则<math|<around|(|<wide|a|\<wide-varrightarrow\>>+<wide|b|\<wide-varrightarrow\>>|)>\<cdot\><wide|c|\<wide-varrightarrow\>>=><underline|<space|2cm>>；<math|<wide|a|\<wide-varrightarrow\>>\<cdot\><wide|b|\<wide-varrightarrow\>>=><underline|<space|2cm>>．

  <center|<image|<tuple|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

  <with|font-series|bold|四、解答题>

  16．在<math|\<triangle\>A*B*C>中，<math|c=2b*cos
  B>，<math|C=<frac|2*\<pi\>|3>>．

  （1）求<math|\<angle\>*B>；

  （2）再从条件①、条件②、条件③这三个条件中选择一个作为已知，使<math|\<triangle\>A*B*C>存在且唯一确定，求<math|B*C>边上中线的长．

  条件①：<math|c=<sqrt|2>*b>；

  条件②：<math|\<triangle\>A*B*C>的周长为<math|4+2*<sqrt|3>>；

  条件③：<math|\<triangle\>A*B*C>的面积为<math|<frac|3*<sqrt|3>|4>>；<vspace|3cm>

  17．如图：在正方体<math|A*B*C*D-A<rsub|1>*B<rsub|1>*C<rsub|1>*D<rsub|1>>中，<math|E>为<math|A<rsub|1>*D<rsub|1>>中点，<math|B<rsub|1>*C<rsub|1>>与平面<math|C*D*E>交于点<math|F>．

  <center|<image|<tuple|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

  （1）求证：<math|F>为<math|B<rsub|1>*C<rsub|1>>的中点；

  （2）点<math|M>是棱<math|A<rsub|1>*B<rsub|1>>上一点，且二面角<math|M-F*C-E>的余弦值为<math|<frac|<sqrt|5>|3>>，求<math|<frac|A<rsub|1>*M|A<rsub|1>*B<rsub|1>>>的值．<vspace|3cm>

  18．在核酸检测中, \P<em|k>合1\Q
  混采核酸检测是指：先将<em|k>个人的样本混合在一起进行1次检测,如果这<em|k>个人都没有感染新冠病毒，则检测结果为阴性，得到每人的检测结果都为阴性，检测结束:如果这<em|k>个人中有人感染新冠病毒，则检测结果为阳性，此时需对每人再进行1次检测,得到每人的检测结果，检测结束.

  现对100人进行核酸检测，假设其中只有2人感染新冠病毒，并假设每次检测结果准确.

  (I)将这100人随机分成10组，每组10人，且对每组都采用\P10合1\Q混采核酸检测.

  (i)如果感染新冠病毒的2人在同一组，求检测的总次数;

  (ii)已知感染新冠病毒的2人分在同一组的概率为<math|<frac|1|11>>.设<em|X>是检测的总次数，求<em|X>的

  分布列与数学期望<math|E(X)>.

  (II)将这100人随机分成20组，每组5人，且对每组都采用\P5合1\Q混采核酸检测.设<math|Y>是检测的总次数，试判断数学期望<math|E(<em|Y>)>与(I)中<math|E(X)>的大小.(结论不要求证明)<vspace|3cm>

  19．已知函数<math|f<around|(|x|)>=<frac|3-2*x|x<rsup|2>+a>>．

  （1）若<math|a=0>，求曲线<math|y=f<around|(|x|)>>在点<math|<around*|(|1,f<around|(|1|)>|)>>处的切线方程；

  （2）若<math|f<around|(|x|)>>在<math|x=-1>处取得极值，求<math|f<around|(|x|)>>的单调区间，以及其最大值与最小值．<vspace|3cm>

  20．已知椭圆<math|<frac|x<rsup|2>|a<rsup|2>>+<frac|y<rsup|2>|b<rsup|2>>=1*<around|(|a\<gtr\>b\<gtr\>0|)>>一个顶
  点<math|A<around*|(|0,-2|)>>，以椭圆<math|E>的四个顶点为顶点的四边形面积为<math|4*<sqrt|5>>．

  （1）求椭圆<math|<em|E>>的方程；

  （2）过点<math|<em|P>(0,-3)>的直线<math|<em|l>>斜率为<math|k>的直线与椭圆<math|E>交于不同的两点<em|B>，<em|C>，直线<em|AB>，<math|A
  C>分别与直线交<math|y=-3>交于点<em|M，N>，当<math|<around*|\||P
  M|\|>+<around*|\||P N|\|>\<leq\>15>时，求<math|k>的取值范围．<vspace|3cm>

  21．设<em|p>为实数.若无穷数列<math|<around*|{|a<rsub|n>|}>>满足如下三个性质，则称<math|<around*|{|a<rsub|n>|}>>为<math|R<rsub|p>>数列：

  ①<math|a<rsub|1>+p\<geq\>0>，且<math|a<rsub|2>+p=0>；

  ②<math|a<rsub|4*n-1>\<less\>a<rsub|4*n>,<around|(|n=1,2,\<cdot\>\<cdot\>\<cdot\>|)>>；

  ③<math|a<rsub|m+n>\<in\><around*|{|a<rsub|m>+a<rsub|n>+p,a<rsub|m>+a<rsub|n>+p+1|}>>，<math|<around|(|m,n=1,2,\<cdot\>\<cdot\>\<cdot\>|)>>．

  （1）如果数列<math|<around*|{|a<rsub|n>|}>>的前4项为<math|2,-2,-2,-1>，那么<math|<around*|{|a<rsub|n>|}>>是否可能为<math|R<rsub|2>>数列？说明理由；

  （2）若数列<math|<around*|{|a<rsub|n>|}>>是<math|R<rsub|0>>数列，求<math|a<rsub|5>>；

  （3）设数列<math|<around*|{|a<rsub|n>|}>>的前<math|n>项和为<math|S<rsub|n>>.是否存在<math|R<rsub|p>>数列<math|<around*|{|a<rsub|n>|}>>，使得<math|S<rsub|n>\<geq\>S<rsub|10>>恒成立？如果存在，求出所有的<em|p>；如果不存在，说明理由．<vspace|3cm>
</body>

<\references>
  <\collection>
    <associate|auto-1|<tuple|1|?>>
  </collection>
</references>