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\<#5E74\>\<#5317\>\<#4EAC\>\<#5E02\>\<#9AD8\>\<#8003\>\<#6570\>\<#5B66\>\<#8BD5\>\<#9898\>>>

  <\with|font-series|bold>
    \<#4E00\>\<#3001\>\<#5355\>\<#9009\>\<#9898\>
  </with>

  1\<#FF0E\>\<#5DF2\>\<#77E5\>\<#96C6\>\<#5408\><math|A=<around*|{|x\|-1\<less\>x\<less\>1|}>>\<#FF0C\><math|B=<around*|{|x\|0\<leq\>x\<leq\>2|}>>\<#FF0C\>\<#5219\><math|A\<cup\>B=>\<#FF08\><space|1.2em>\<#FF09\>

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\><math|<around*|{|x\|-1\<less\>x\<less\>2|}>>>|<cell|B\<#FF0E\><math|<around*|{|x\|-1\<less\>x\<leq\>2|}>>>>|<row|<cell|C\<#FF0E\><math|<around*|{|x\|0\<leq\>x\<less\>1|}>>>|<cell|D\<#FF0E\><math|<around*|{|x\|0\<leq\>x\<leq\>2|}>>>>>>>

  2\<#FF0E\>\<#5728\>\<#590D\>\<#5E73\>\<#9762\>\<#5185\>\<#FF0C\>\<#590D\>\<#6570\><math|z>\<#6EE1\>\<#8DB3\><math|<around|(|1-i|)>*z=2>\<#FF0C\>\<#5219\><math|z=>\<#FF08\><space|1.2em>\<#FF09\>

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\><math|-1-i>>|<cell|B\<#FF0E\><math|-1+i>>|<cell|C\<#FF0E\><math|1-i>>|<cell|D\<#FF0E\><math|1+i>>>>>>

  3\<#FF0E\>\<#5DF2\>\<#77E5\><math|f<around|(|x|)>>\<#662F\>\<#5B9A\>\<#4E49\>\<#5728\>\<#4E0A\><math|<around|[|0,1|]>>\<#7684\>\<#51FD\>\<#6570\>\<#FF0C\>\<#90A3\>\<#4E48\>\P\<#51FD\>\<#6570\><math|f<around|(|x|)>>\<#5728\><math|<around|[|0,1|]>>\<#4E0A\>\<#5355\>\<#8C03\>\<#9012\>\<#589E\>\Q\<#662F\>\P\<#51FD\>\<#6570\><math|f<around|(|x|)>>\<#5728\><math|<around|[|0,1|]>>\<#4E0A\>\<#7684\>\<#6700\>\<#5927\>\<#503C\>\<#4E3A\><math|f<around|(|1|)>>\Q\<#7684\>\<#FF08\><space|1.2em>\<#FF09\>

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\>\<#5145\>\<#5206\>\<#800C\>\<#4E0D\>\<#5FC5\>\<#8981\>\<#6761\>\<#4EF6\>>|<cell|B\<#FF0E\>\<#5FC5\>\<#8981\>\<#800C\>\<#4E0D\>\<#5145\>\<#5206\>\<#6761\>\<#4EF6\>>>|<row|<cell|C\<#FF0E\>\<#5145\>\<#5206\>\<#5FC5\>\<#8981\>\<#6761\>\<#4EF6\>>|<cell|D\<#FF0E\>\<#65E2\>\<#4E0D\>\<#5145\>\<#5206\>\<#4E5F\>\<#4E0D\>\<#5FC5\>\<#8981\>\<#6761\>\<#4EF6\>>>>>>
  4\<#FF0E\>\<#67D0\>\<#56DB\>\<#9762\>\<#4F53\>\<#7684\>\<#4E09\>\<#89C6\>\<#56FE\>\<#5982\>\<#56FE\>\<#6240\>\<#793A\>\<#FF0C\>\<#8BE5\>\<#56DB\>\<#9762\>\<#4F53\>\<#7684\>\<#8868\>\<#9762\>\<#79EF\>\<#4E3A\>\<#FF08\><space|1.2em>\<#FF09\>

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  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\><math|<frac|3|2>+<frac|<sqrt|3>|2>>>|<cell|B\<#FF0E\><math|3+<sqrt|3>>>|<cell|C\<#FF0E\><math|<frac|3|2>+<sqrt|3>>>|<cell|D\<#FF0E\><math|3+<frac|<sqrt|3>|2>>>>>>>
  5\<#FF0E\>\<#82E5\>\<#53CC\>\<#66F2\>\<#7EBF\><math|<frac|x<rsup|2>|a<rsup|2>>-<frac|y<rsup|2>|b<rsup|2>>=1>\<#79BB\>\<#5FC3\>\<#7387\>\<#4E3A\><math|2>\<#FF0C\>\<#8FC7\>\<#70B9\><math|<around*|(|<sqrt|2>,<sqrt|3>|)>>\<#FF0C\>\<#5219\>\<#8BE5\>\<#53CC\>\<#66F2\>\<#7EBF\>\<#7684\>\<#65B9\>\<#7A0B\>\<#4E3A\>\<#FF08\><space|1.2em>\<#FF09\>

  A\<#FF0E\><math|2*x<rsup|2>-y<rsup|2>=1>
  B\<#FF0E\><math|x<rsup|2>-<frac|y<rsup|2>|3>=1>
  C\<#FF0E\><math|5*x<rsup|2>-3*y<rsup|2>=1>
  D\<#FF0E\><math|<frac|x<rsup|2>|2>-<frac|y<rsup|2>|6>=1>

  6\<#FF0E\>\<#300A\>\<#4E2D\>\<#56FD\>\<#5171\>\<#4EA7\>\<#515A\>\<#515A\>\<#65D7\>\<#515A\>\<#5FBD\>\<#5236\>\<#4F5C\>\<#548C\>\<#4F7F\>\<#7528\>\<#7684\>\<#82E5\>\<#5E72\>\<#89C4\>\<#5B9A\>\<#300B\>\<#6307\>\<#51FA\>\<#FF0C\>\<#4E2D\>\<#56FD\>\<#5171\>\<#4EA7\>\<#515A\>\<#515A\>\<#65D7\>\<#4E3A\>\<#65D7\>\<#9762\>\<#7F00\>\<#6709\>\<#91D1\>\<#9EC4\>\<#8272\>\<#515A\>\<#5FBD\>\<#56FE\>\<#6848\>\<#7684\>\<#7EA2\>\<#65D7\>\<#FF0C\>\<#901A\>\<#7528\>\<#89C4\>\<#683C\>\<#6709\>\<#4E94\>\<#79CD\>.\<#8FD9\>\<#4E94\>\<#79CD\>\<#89C4\>\<#683C\>\<#515A\>\<#65D7\>\<#7684\>\<#957F\><math|a<rsub|1>,a<rsub|2>,a<rsub|3>,a<rsub|4>,a<rsub|5>>
  (\<#5355\>\<#4F4D\>:cm)\<#6210\>\<#7B49\>\<#5DEE\>\<#6570\>\<#5217\>\<#FF0C\>\<#5BF9\>\<#5E94\>\<#7684\>\<#5BBD\>\<#4E3A\><math|b<rsub|1>,b<rsub|2>,b<rsub|3>,b<rsub|4>,b<rsub|5>>
  (\<#5355\>\<#4F4D\>: cm),\<#4E14\>\<#957F\>\<#4E0E\>\<#5BBD\>\<#4E4B\>\<#6BD4\>\<#90FD\>\<#76F8\>\<#7B49\>\<#FF0C\>\<#5DF2\>\<#77E5\><math|a<rsub|1>=288>\<#FF0C\><math|a<rsub|5>=96>\<#FF0C\><math|b<rsub|1>=192>\<#FF0C\>\<#5219\><math|b<rsub|3>=>\<#FF08\><space|1.2em>\<#FF09\>

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  7\<#FF0E\>\<#51FD\>\<#6570\><math|f<around|(|x|)>=cos x-cos
  2*x>\<#662F\>\<#FF08\><space|1.2em>\<#FF09\>

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  8\<#FF0E\>\<#67D0\>\<#4E00\>\<#65F6\>\<#95F4\>\<#6BB5\>\<#5185\>\<#FF0C\>\<#4ECE\>\<#5929\>\<#7A7A\>\<#964D\>\<#843D\>\<#5230\>\<#5730\>\<#9762\>\<#4E0A\>\<#7684\>\<#96E8\>\<#6C34\>\<#FF0C\>\<#672A\>\<#7ECF\>\<#84B8\>\<#53D1\>\<#3001\>\<#6E17\>\<#6F0F\>\<#3001\>\<#6D41\>\<#5931\>\<#800C\>\<#5728\>\<#6C34\>\<#5E73\>\<#9762\>\<#4E0A\>\<#79EF\>\<#805A\>\<#7684\>\<#6DF1\>\<#5EA6\>\<#FF0C\>\<#79F0\>\<#4E3A\>\<#8FD9\>\<#4E2A\>\<#65F6\>\<#6BB5\>\<#7684\>\<#964D\>\<#96E8\>\<#91CF\>\<#FF08\>\<#5355\>\<#4F4D\>\<#FF1A\>mm\<#FF09\>\<#FF0E\>24h\<#964D\>\<#96E8\>\<#91CF\>\<#7684\>\<#7B49\>\<#7EA7\>\<#5212\>\<#5206\>\<#5982\>\<#4E0B\>\<#FF1A\>

  <block*|<tformat|<table|<row|<cell|\<#7B49\>\<#7EA7\>>|<cell|24h\<#964D\>\<#96E8\>\<#91CF\>\<#FF08\>\<#7CBE\>\<#786E\>\<#5230\>0.1\<#FF09\>>>|<row|<cell|......>|<cell|......>>|<row|<cell|\<#5C0F\>\<#96E8\>>|<cell|<math|0.1\<sim\>9.9>>>|<row|<cell|\<#4E2D\>\<#96E8\>>|<cell|<math|10.0\<sim\>24.9>>>|<row|<cell|\<#5927\>\<#96E8\>>|<cell|<math|25.0\<sim\>49.9>>>|<row|<cell|\<#66B4\>\<#96E8\>>|<cell|<math|50.0\<sim\>99.9>>>|<row|<cell|.....>|<cell|......>>>>>

  <image|<tuple|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

  \<#5728\>\<#7EFC\>\<#5408\>\<#5B9E\>\<#8DF5\>\<#6D3B\>\<#52A8\>\<#4E2D\>\<#FF0C\>\<#67D0\>\<#5C0F\>\<#7EC4\>\<#81EA\>\<#5236\>\<#4E86\>\<#4E00\>\<#4E2A\>\<#5E95\>\<#9762\>\<#76F4\>\<#5F84\>\<#4E3A\>200
  mm\<#FF0C\>\<#9AD8\>\<#4E3A\>300 mm\<#7684\>\<#5706\>\<#9525\>\<#5F62\>\<#96E8\>\<#91CF\>\<#5668\>.\<#82E5\>\<#4E00\>\<#6B21\>\<#964D\>\<#96E8\>\<#8FC7\>\<#7A0B\>\<#4E2D\>\<#FF0C\>\<#8BE5\>\<#96E8\>\<#91CF\>\<#5668\>\<#6536\>\<#96C6\>\<#7684\>24h\<#7684\>\<#96E8\>\<#6C34\>\<#9AD8\>\<#5EA6\>\<#662F\>150
  mm\<#FF08\>\<#5982\>\<#56FE\>\<#6240\>\<#793A\>)\<#FF0C\>\<#5219\>\<#8FD9\>24h\<#964D\>\<#96E8\>\<#91CF\>\<#7684\>\<#7B49\>\<#7EA7\>\<#662F\>\<#FF08\><space|1.2em>\<#FF09\>

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  9\<#FF0E\>\<#5DF2\>\<#77E5\>\<#76F4\>\<#7EBF\><math|y=k*x+m>\<#FF08\><math|m>\<#4E3A\>\<#5E38\>\<#6570\>\<#FF09\>\<#4E0E\>\<#5706\><math|x<rsup|2>+y<rsup|2>=4>\<#4EA4\>\<#4E8E\>\<#70B9\><math|M*\<#FF0C\>*N>\<#FF0C\>\<#5F53\><math|k>\<#53D8\>\<#5316\>\<#65F6\>\<#FF0C\>\<#82E5\><math|<around|\||M*N|\|>>\<#7684\>\<#6700\>\<#5C0F\>\<#503C\>\<#4E3A\>2\<#FF0C\>\<#5219\><math|m=>\<#FF08\><space|1.2em>\<#FF09\>

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\><math|\<pm\>1>>|<cell|B\<#FF0E\><math|\<pm\><sqrt|2>>>|<cell|C\<#FF0E\><math|\<pm\><sqrt|3>>>|<cell|D\<#FF0E\><math|\<pm\>2>>>>>>\ 

  10\<#FF0E\>\<#5DF2\>\<#77E5\><math|<around*|{|a<rsub|n>|}>>\<#662F\>\<#5404\>\<#9879\>\<#5747\>\<#4E3A\>\<#6574\>\<#6570\>\<#7684\>\<#9012\>\<#589E\>\<#6570\>\<#5217\>\<#FF0C\>\<#4E14\><math|a<rsub|1>\<geq\>3>\<#FF0C\>\<#82E5\><math|a<rsub|1>+a<rsub|2>+\<cdot\>\<cdot\>\<cdot\>+a<rsub|n>=100>\<#FF0C\>\<#5219\><math|n>\<#7684\>\<#6700\>\<#5927\>\<#503C\>\<#4E3A\>\<#FF08\><space|1.2em>\<#FF09\>

  <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A\<#FF0E\>9>|<cell|B\<#FF0E\>10>|<cell|C\<#FF0E\>11>|<cell|D\<#FF0E\>12>>>>>\ 

  <with|font-series|bold|\<#4E8C\>\<#3001\>\<#586B\>\<#7A7A\>\<#9898\>>

  11\<#FF0E\>\<#5728\><math|<around|(|x<rsup|3>-<frac|1|x>|)><rsup|4>>\<#7684\>\<#5C55\>\<#5F00\>\<#5F0F\>\<#4E2D\>\<#FF0C\>\<#5E38\>\<#6570\>\<#9879\>\<#4E3A\><underline|<space|2cm>>\<#FF0E\>

  12\<#FF0E\>\<#82E5\>\<#70B9\><math|A*<around|(|cos \<theta\>,sin
  \<theta\>|)>>\<#5173\>\<#4E8E\><math|y>\<#8F74\>\<#5BF9\>\<#79F0\>\<#70B9\>\<#4E3A\><math|B*<around|(|cos
  <around|(|\<theta\>+<frac|\<pi\>|6>|)>,sin
  <around|(|\<theta\>+<frac|\<pi\>|6>|)>|)>>\<#FF0C\>\<#5199\>\<#51FA\><math|\<theta\>>\<#7684\>\<#4E00\>\<#4E2A\>\<#53D6\>\<#503C\>\<#4E3A\><underline|<space|2cm>>\<#FF0E\>

  13\<#FF0E\>\<#5DF2\>\<#77E5\>\<#51FD\>\<#6570\><math|f<around|(|x|)>=<around*|\||lg
  x|\|>-k*x-2>\<#FF0C\>\<#7ED9\>\<#51FA\>\<#4E0B\>\<#5217\>\<#56DB\>\<#4E2A\>\<#7ED3\>\<#8BBA\>\<#FF1A\>

  \<#2460\>\<#82E5\><math|k=0>\<#FF0C\><math|f<around|(|x|)>>\<#6070\>
  \<#6709\>2\<#4E2A\>\<#96F6\>\<#70B9\>\<#FF1B\>

  \<#2461\>\<#5B58\>\<#5728\>\<#8D1F\>\<#6570\><math|k>\<#FF0C\>\<#4F7F\>\<#5F97\><math|f<around|(|x|)>>\<#6070\>\<#6709\>\<#4E2A\>1\<#96F6\>\<#70B9\>\<#FF1B\>

  \<#2462\>\<#5B58\>\<#5728\>\<#8D1F\>\<#6570\><math|k>\<#FF0C\>\<#4F7F\>\<#5F97\><math|f<around|(|x|)>>\<#6070\>\<#6709\>\<#4E2A\>3\<#96F6\>\<#70B9\>\<#FF1B\>

  \<#2463\>\<#5B58\>\<#5728\>\<#6B63\>\<#6570\><math|k>\<#FF0C\>\<#4F7F\>\<#5F97\><math|f<around|(|x|)>>\<#6070\>\<#6709\>\<#4E2A\>3\<#96F6\>\<#70B9\>\<#FF0E\>

  \<#5176\>\<#4E2D\>\<#6240\>\<#6709\>\<#6B63\>\<#786E\>\<#7ED3\>\<#8BBA\>\<#7684\>\<#5E8F\>\<#53F7\>\<#662F\><underline|<space|2cm>>\<#FF0E\>

  <with|font-series|bold|\<#4E09\>\<#3001\>\<#53CC\>\<#7A7A\>\<#9898\>>

  14\<#FF0E\>\<#5DF2\>\<#77E5\>\<#629B\>\<#7269\>\<#7EBF\><math|y<rsup|2>=4*x>\<#7684\>\<#7126\>\<#70B9\>\<#4E3A\><math|F>\<#FF0C\>\<#70B9\><math|M>\<#5728\>\<#629B\>\<#7269\>\<#7EBF\>\<#4E0A\>\<#FF0C\><math|M*N>\<#5782\>\<#76F4\><math|x>\<#8F74\>\<#4E0E\>\<#4E8E\>\<#70B9\><math|N>.\<#82E5\><math|<around|\||M*F|\|>=6>\<#FF0C\>\<#5219\>\<#70B9\><math|M>\<#7684\>\<#6A2A\>\<#5750\>\<#6807\>\<#4E3A\><underline|<space|2cm>>\<#FF1B\>
  <math|\<triangle\>M*N*F>\<#7684\>\<#9762\>\<#79EF\>\<#4E3A\><underline|<space|2cm>>\<#FF0E\>

  15\<#FF0E\>\<#5DF2\>\<#77E5\>\<#5411\>\<#91CF\><math|<wide|a|\<wide-varrightarrow\>>,<wide|b|\<wide-varrightarrow\>>,<wide|c|\<wide-varrightarrow\>>>\<#5728\>\<#6B63\>\<#65B9\>\<#5F62\>\<#7F51\>\<#683C\>\<#4E2D\>\<#7684\>\<#4F4D\>\<#7F6E\>\<#5982\>\<#56FE\>\<#6240\>\<#793A\>\<#FF0E\>\<#82E5\>\<#7F51\>\<#683C\>\<#7EB8\>\<#4E0A\>\<#5C0F\>\<#6B63\>\<#65B9\>\<#5F62\>\<#7684\>\<#8FB9\>\<#957F\>\<#4E3A\>1\<#FF0C\>\<#5219\><math|<around|(|<wide|a|\<wide-varrightarrow\>>+<wide|b|\<wide-varrightarrow\>>|)>\<cdot\><wide|c|\<wide-varrightarrow\>>=><underline|<space|2cm>>\<#FF1B\><math|<wide|a|\<wide-varrightarrow\>>\<cdot\><wide|b|\<wide-varrightarrow\>>=><underline|<space|2cm>>\<#FF0E\>

  <center|<image|<tuple|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  16\<#FF0E\>\<#5728\><math|\<triangle\>A*B*C>\<#4E2D\>\<#FF0C\><math|c=2b*cos
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  17\<#FF0E\>\<#5982\>\<#56FE\>\<#FF1A\>\<#5728\>\<#6B63\>\<#65B9\>\<#4F53\><math|A*B*C*D-A<rsub|1>*B<rsub|1>*C<rsub|1>*D<rsub|1>>\<#4E2D\>\<#FF0C\><math|E>\<#4E3A\><math|A<rsub|1>*D<rsub|1>>\<#4E2D\>\<#70B9\>\<#FF0C\><math|B<rsub|1>*C<rsub|1>>\<#4E0E\>\<#5E73\>\<#9762\><math|C*D*E>\<#4EA4\>\<#4E8E\>\<#70B9\><math|F>\<#FF0E\>

  <center|<image|<tuple|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</body>

<initial|<\collection>
</collection>>