planet/高考数学/2021年全国高考乙卷数学文科试题.tm

<TeXmacs|2.1.2>

<style|<tuple|generic|chinese>>

<\body>
  <doc-data|<doc-title|2021\<#5E74\>\<#5168\>\<#56FD\>\<#9AD8\>\<#8003\>\<#4E59\>\<#5377\>\<#6570\>\<#5B66\>\<#FF08\>\<#6587\>\<#FF09\>\<#8BD5\>\<#5377\>>>

  <subsubsection*|\<#4E00\>\<#3001\>\<#5355\>\<#9009\>\<#9898\>>

  <\enumerate>
    <item>\<#5DF2\>\<#77E5\>\<#5168\>\<#96C6\><math|U=<around*|{|1,2,3,4,5|}>>\<#FF0C\>\<#96C6\>\<#5408\><math|M=<around*|{|1,2|}>,N=<around*|{|3,4|}>>\<#FF0C\>\<#5219\><math|C<rsub|u><around*|(|M\<cup\>N|)>=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<around*|{|5|}>>
      </cell>|<\cell>
        B. <math|<around*|{|1,2|}>>
      </cell>|<\cell>
        C. <math|<around*|{|3,4|}>>
      </cell>|<\cell>
        D. <math|<around*|{|1,2,3,4|}>>
      </cell>>>>
    </wide-tabular>

    <item>\<#8BBE\><math|i*z=4+3*i>\<#FF0C\>\<#5219\><math|z=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|-3-4*i>
      </cell>|<\cell>
        B. <math|-3+4*i>
      </cell>|<\cell>
        C. <math|3-4*i>
      </cell>|<\cell>
        D. <math|3+4*i>
      </cell>>>>
    </wide-tabular>

    <item>\<#5DF2\>\<#77E5\>\<#547D\>\<#9898\><math|p>\<#FF1A\><math|\<exists\>x\<in\>\<bbb-R\>,sin
    x\<less\>1>\<#FF1B\>\<#547D\>\<#9898\><math|q>\<#FF1A\><math|\<forall\>x\<in\>\<bbb-R\>,e<rsup|<around*|\||x|\|>>\<geqslant\>1>\<#FF0C\>\<#5219\>\<#4E0B\>\<#5217\>\<#547D\>\<#9898\>\<#4E2D\>\<#4E3A\>\<#771F\>\<#547D\>\<#9898\>\<#7684\>\<#662F\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|p\<wedge\>q>
      </cell>|<\cell>
        B. <math|\<neg\>p\<wedge\>q>
      </cell>|<\cell>
        C. <math|p\<wedge\>\<neg\>q>
      </cell>|<\cell>
        D. <math|\<neg\><around*|(|p\<vee\>q|)>>
      </cell>>>>
    </wide-tabular>

    <item>\<#51FD\>\<#6570\><math|f<around*|(|x|)>=sin
    <frac|x|3>+cos<frac|x|3>>\<#7684\>\<#6700\>\<#5C0F\>\<#6B63\>\<#5468\>\<#671F\>\<#548C\>\<#6700\>\<#5927\>\<#503C\>\<#5206\>\<#522B\>\<#662F\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|3\<pi\>>\<#548C\><math|<sqrt|2>>
      </cell>|<\cell>
        B. <math|3*\<pi\>>\<#548C\>2
      </cell>|<\cell>
        C. <math|6*\<pi\>>\<#548C\><math|<sqrt|2>>
      </cell>|<\cell>
        D. <math|6*\<pi\>>\<#548C\>2
      </cell>>>>
    </wide-tabular>

    <item>\<#82E5\><math|x>\<#FF0C\><math|y>\<#6EE1\>\<#8DB3\>\<#7EA6\>\<#675F\>\<#6761\>\<#4EF6\><with|math-display|true|<choice|<tformat|<table|<row|<cell|<math|x+y\<geqslant\>4>,>>|<row|<cell|<math|x-y\<leqslant\>2>,>>|<row|<cell|<math|y\<leqslant\>3>,>>>>>>\<#5219\><math|z=3*x+y>\<#7684\>\<#6700\>\<#5C0F\>\<#503C\>\<#4E3A\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. 18
      </cell>|<\cell>
        B. 10
      </cell>|<\cell>
        C. 6
      </cell>|<\cell>
        D. 4
      </cell>>>>
    </wide-tabular>

    <item><math|cos<rsup|2><frac|\<pi\>|12>-cos<rsup|2><frac|5*\<pi\>|12>=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|1|2>>
      </cell>|<\cell>
        B. <math|<frac|<sqrt|3>|3>>
      </cell>|<\cell>
        C. <math|<frac|<sqrt|2>|2>>
      </cell>|<\cell>
        D. <math|<frac|<sqrt|3>|2>>
      </cell>>>>
    </wide-tabular>

    <item>\<#5728\>\<#533A\>\<#95F4\>0, <math|<frac|1|2>>\<#968F\>\<#673A\>\<#53D6\>1\<#4E2A\>\<#6570\>\<#FF0C\>\<#5219\>\<#53D6\>\<#5230\>\<#7684\>\<#6570\>\<#5C0F\>\<#4E8E\><math|<frac|1|3>>\<#7684\>\<#6982\>\<#7387\>\<#4E3A\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|3|4>>
      </cell>|<\cell>
        B. <math|<frac|2|3>>
      </cell>|<\cell>
        C. <math|<frac|1|3>>
      </cell>|<\cell>
        D. <math|<frac|1|6>>
      </cell>>>>
    </wide-tabular>

    <item>\<#4E0B\>\<#5217\>\<#51FD\>\<#6570\>\<#4E2D\>\<#6700\>\<#5C0F\>\<#503C\>\<#4E3A\>4\<#7684\>\<#662F\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|y=x<rsup|2>+2*x+4>
      </cell>|<\cell>
        B. <math|y=<around*|\||sin x|\|>+<frac|4|<around*|\||sin x|\|>>>
      </cell>>|<row|<\cell>
        C. <math|y=2<rsup|x>+2<rsup|2-x>>
      </cell>|<\cell>
        D. <math|y=ln x+<frac|4|ln x>>
      </cell>>>>
    </wide-tabular>

    <item>\<#8BBE\>\<#51FD\>\<#6570\><math|f<around*|(|x|)>=<frac|1-x|1+x>>\<#FF0C\>\<#5219\>\<#4E0B\>\<#5217\>\<#51FD\>\<#6570\>\<#4E2D\>\<#4E3A\>\<#5947\>\<#51FD\>\<#6570\>\<#7684\>\<#662F\><underline|<space|3em>>.

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|f<around*|(|x-1|)>-1>
      </cell>|<\cell>
        B. <math|f<around*|(|x-1|)>+1>
      </cell>|<\cell>
        C. <math|f<around*|(|x+1|)>-1>
      </cell>|<\cell>
        D. <math|f<around*|(|x+1|)>+1>
      </cell>>>>
    </wide-tabular>

    <item>\<#5728\>\<#6B63\>\<#65B9\>\<#4F53\><math|ABCD-A<rsub|1>B<rsub|1>C<rsub|1>D<rsub|1>>\<#4E2D\>\<#FF0C\><math|P>\<#4E3A\><math|B<rsub|1>D<rsub|1>>\<#7684\>\<#4E2D\>\<#70B9\>\<#FF0C\>\<#5219\>\<#76F4\>\<#7EBF\><math|PB>\<#4E0E\><math|AD<rsub|1>>\<#6240\>\<#6210\>\<#7684\>\<#89D2\>\<#4E3A\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|\<pi\>|2>>
      </cell>|<\cell>
        B. <math|<frac|\<pi\>|3>>
      </cell>|<\cell>
        C. <math|<frac|\<pi\>|4>>
      </cell>|<\cell>
        D. <math|<frac|\<pi\>|6>>
      </cell>>>>
    </wide-tabular>

    <item>\<#8BBE\><math|B>\<#662F\>\<#692D\>\<#5706\><math|C:<frac|x<rsup|2>|5>+y=1>\<#7684\>\<#4E0A\>\<#9876\>\<#70B9\>\<#FF0C\>\<#70B9\><math|P>\<#5728\><math|C>\<#4E0A\>\<#FF0C\>\<#5219\><math|<around*|\||PB|\|>>\<#7684\>\<#6700\>\<#5927\>\<#503C\>\<#4E3A\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|5|2>>
      </cell>|<\cell>
        B. <math|<sqrt|6>>
      </cell>|<\cell>
        C. <math|<sqrt|5>>
      </cell>|<\cell>
        D. 2
      </cell>>>>
    </wide-tabular>

    <item>\<#8BBE\><math|a\<neq\>0>\<#FF0C\>\<#82E5\><math|x=a>\<#4E3A\>\<#51FD\>\<#6570\><math|f<around*|(|x|)>=a*<around*|(|x-a|)><rsup|2><around*|(|x-b|)>>\<#7684\>\<#6781\>\<#5927\>\<#503C\>\<#70B9\>\<#FF0C\>\<#5219\><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|a\<less\>b>
      </cell>|<\cell>
        B. <math|a\<gtr\>b>
      </cell>|<\cell>
        C. <math|a*b\<less\>a<rsup|2>>
      </cell>|<\cell>
        D. <math|a*b\<gtr\>a<rsup|2>>
      </cell>>>>
    </wide-tabular>
  </enumerate>

  <subsubsection*|\<#4E8C\>\<#3001\>\<#586B\>\<#7A7A\>\<#9898\>>

  <\enumerate>
    <assign|item-nr|12><item>\<#5DF2\>\<#77E5\>\<#5411\>\<#91CF\><math|<wide|a|\<vect\>>=<around*|(|2,5|)>,<wide|b|\<vect\>>=<around*|(|\<lambda\>,4|)>>\<#FF0C\>\<#82E5\><math|<wide|a|\<vect\>>//<wide|b|\<vect\>>>\<#FF0C\>\<#5219\><math|\<lambda\>=><underline|<space|3em>>.

    <item>\<#53CC\>\<#66F2\>\<#7EBF\><math|<frac|x<rsup|2>|4>-<frac|y<rsup|2>|5>=1>\<#7684\>\<#53F3\>\<#7126\>\<#70B9\>\<#5230\>\<#76F4\>\<#7EBF\><math|x+2*y-8=0>\<#7684\>\<#8DDD\>\<#79BB\>\<#4E3A\><underline|<space|3em>>.

    <item>\<#8BB0\><math|\<triangle\>ABC>\<#7684\>\<#5185\>\<#89D2\><math|A>\<#FF0C\><math|B>\<#FF0C\><math|C>\<#7684\>\<#5BF9\>\<#8FB9\>\<#5206\>\<#522B\>\<#4E3A\>a\<#FF0C\>b\<#FF0C\>c\<#FF0C\>\<#9762\>\<#79EF\>\<#4E3A\><math|<sqrt|3>>\<#FF0C\><math|B=60<rsup|\<circ\>>>\<#FF0C\><math|a<rsup|2>+c<rsup|2>=3*a*c>\<#FF0C\>\<#5219\><math|b=><underline|<space|3em>>.

    <item>\<#4EE5\>\<#56FE\>\<#2460\>\<#4E3A\>\<#6B63\>\<#89C6\>\<#56FE\>\<#FF0C\>\<#5728\>\<#56FE\>\<#2461\>\<#2462\>\<#2463\>\<#2464\>\<#4E2D\>\<#9009\>\<#4E24\>\<#4E2A\>\<#5206\>\<#522B\>\<#4F5C\>\<#4E3A\>\<#4FA7\>\<#89C6\>\<#56FE\>\<#548C\>\<#4FEF\>\<#89C6\>\<#56FE\>\<#FF0C\>\<#7EC4\>\<#6210\>\<#67D0\>\<#4E2A\>\<#4E09\>\<#68F1\>\<#9525\>\<#7684\>\<#4E09\>\<#89C6\>\<#56FE\>\<#FF0C\>\<#5219\>\<#6240\>\<#9009\>\<#4FA7\>\<#89C6\>\<#56FE\>\<#548C\>\<#4FEF\>\<#89C6\>\<#56FE\>\<#7684\>\<#7F16\>\<#53F7\>\<#4F9D\>\<#6B21\>\<#4E3A\><underline|<space|3em>>\<#FF08\>\<#5199\>\<#51FA\>\<#7B26\>\<#5408\>\<#8981\>\<#6C42\>\<#7684\>\<#4E00\>\<#7EC4\>\<#7B54\>\<#6848\>\<#5373\>\<#53EF\>\<#FF09\>\<#FF0E\>

    <image|<tuple|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
  </enumerate>

  <subsubsection*|\<#4E09\>\<#3001\>\<#89E3\>\<#7B54\>\<#9898\>>

  <\enumerate>
    <assign|item-nr|16><item>\<#67D0\>\<#5382\>\<#7814\>\<#5236\>\<#4E86\>\<#4E00\>\<#79CD\>\<#751F\>\<#4EA7\>\<#9AD8\>\<#7CBE\>\<#4EA7\>\<#54C1\>\<#7684\>\<#8BBE\>\<#5907\>\<#FF0C\>\<#4E3A\>\<#68C0\>\<#9A8C\>\<#65B0\>\<#8BBE\>\<#5907\>\<#751F\>\<#4EA7\>\<#4EA7\>\<#54C1\>\<#7684\>\<#67D0\>\<#9879\>\<#6307\>\<#6807\>\<#6709\>\<#65E0\>\<#63D0\>\<#9AD8\>\<#FF0C\>\<#7528\>\<#4E00\>\<#53F0\>\<#65E7\>\<#8BBE\>\<#5907\>\<#548C\>\<#4E00\>\<#53F0\>\<#65B0\>\<#8BBE\>\<#5907\>\<#5404\>\<#751F\>\<#4EA7\>\<#4E86\>10\<#4EF6\>\<#4EA7\>\<#54C1\>\<#FF0C\>\<#5F97\>\<#5230\>\<#5404\>\<#4EF6\>\<#4EA7\>\<#54C1\>\<#8BE5\>\<#9879\>\<#6307\>\<#6807\>\<#6570\>\<#636E\>\<#5982\>\<#4E0B\>\<#FF1A\>

    <block*|<tformat|<cwith|1|1|1|1|cell-valign|b>|<cwith|2|2|6|6|cell-halign|r>|<table|<row|<cell|\<#65E7\>\<#8BBE\>\<#5907\>>|<cell|9.8>|<cell|10.3>|<cell|10.0>|<cell|10.2>|<cell|9.9>|<cell|9.8>|<cell|10.0>|<cell|10.1>|<cell|10.2>|<cell|9.7>>|<row|<cell|\<#65B0\>\<#8BBE\>\<#5907\>>|<cell|10.1>|<cell|10.4>|<cell|10.1>|<cell|10.0>|<cell|10.1>|<cell|10.3>|<cell|10.6>|<cell|10.5>|<cell|10.4>|<cell|10.5>>>>>

    \<#65E7\>\<#8BBE\>\<#5907\>\<#548C\>\<#65B0\>\<#8BBE\>\<#5907\>\<#751F\>\<#4EA7\>\<#4EA7\>\<#54C1\>\<#7684\>\<#8BE5\>\<#9879\>\<#6307\>\<#6807\>\<#7684\>\<#6837\>\<#672C\>\<#5E73\>\<#5747\>\<#6570\>\<#5206\>\<#522B\>\<#8BB0\>\<#4E3A\>\<#548C\>\<#FF0C\>\<#6837\>\<#672C\>\<#65B9\>\<#5DEE\>\<#5206\>\<#522B\>\<#8BB0\>\<#4E3A\>\<#548C\>\<#FF0E\>

    \<#FF08\>1\<#FF09\>\<#6C42\><math|<wide|x|\<bar\>>,<wide|y|\<bar\>>,s<rsup|2><rsub|1>,s<rsup|2><rsub|2>>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#5224\>\<#65AD\>\<#65B0\>\<#8BBE\>\<#5907\>\<#751F\>\<#4EA7\>\<#4EA7\>\<#54C1\>\<#7684\>\<#8BE5\>\<#9879\>\<#6307\>\<#6807\>\<#7684\>\<#5747\>\<#503C\>\<#8F83\>\<#65E7\>\<#8BBE\>\<#5907\>\<#662F\>\<#5426\>\<#6709\>\<#663E\>\<#8457\>\<#63D0\>\<#9AD8\>\<#FF08\>\<#5982\>\<#679C\><math|<wide|y|\<bar\>>-<wide|x|\<bar\>>\<geqslant\>2<sqrt|<frac|s<rsup|2><rsub|1>+s<rsup|2><rsub|2>|10>>>\<#FF0C\>\<#5219\>\<#8BA4\>\<#4E3A\>\<#65B0\>\<#8BBE\>\<#5907\>\<#751F\>\<#4EA7\>\<#4EA7\>\<#54C1\>\<#7684\>\<#8BE5\>\<#9879\>\<#6307\>\<#6807\>\<#7684\>\<#5747\>\<#503C\>\<#8F83\>\<#65E7\>\<#8BBE\>\<#5907\>\<#6709\>\<#663E\>\<#8457\>\<#63D0\>\<#9AD8\>\<#FF0C\>\<#5426\>\<#5219\>\<#4E0D\>\<#8BA4\>\<#4E3A\>\<#6709\>\<#663E\>\<#8457\>\<#63D0\>\<#9AD8\>\<#FF09\>\<#FF0E\>

    <item>\<#5982\>\<#56FE\>\<#FF0C\>\<#56DB\>\<#68F1\>\<#9525\><math|P-ABCD>\<#7684\>\<#5E95\>\<#9762\>\<#662F\>\<#77E9\>\<#5F62\>\<#FF0C\><math|PD\<perp\>\<#5E95\>\<#9762\>ABCD>\<#FF0C\><math|M>\<#4E3A\><math|BC>\<#7684\>\<#4E2D\>\<#70B9\>\<#FF0C\>\<#4E14\><math|PB\<perp\>AM>.

    <image|<tuple|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

    \<#FF08\>1\<#FF09\>\<#8BC1\>\<#660E\>\<#FF1A\><math|\<#5E73\>\<#9762\>PAM\<perp\>\<#5E73\>\<#9762\>PBD>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#82E5\><math|PD=DC=1>\<#FF0C\>\<#6C42\>\<#56DB\>\<#68F1\>\<#9525\><math|P-ABCD>\<#7684\>\<#4F53\>\<#79EF\>.

    <item>\<#8BBE\><math|<around*|{|a<rsub|n>|}>>\<#662F\>\<#9996\>\<#9879\>\<#4E3A\>1\<#7684\>\<#7B49\>\<#6BD4\>\<#6570\>\<#5217\>\<#FF0C\>\<#6570\>\<#5217\><math|<around*|{|b<rsub|n>|}>>\<#6EE1\>\<#8DB3\><math|b<rsub|n>=<frac|n*a<rsub|n>|3>>.
    \<#5DF2\>\<#77E5\><math|a<rsub|1>>\<#FF0C\><math|3*a<rsub|2>>\<#FF0C\><math|9*a<rsub|3>>\<#6210\>\<#7B49\>\<#5DEE\>\<#6570\>\<#5217\>.

    \<#FF08\>1\<#FF09\>\<#6C42\><math|<around*|{|a<rsub|n>|}>>\<#548C\><math|<around*|{|b<rsub|n>|}>>\<#7684\>\<#901A\>\<#9879\>\<#516C\>\<#5F0F\>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#8BB0\><math|S<rsub|n>>\<#548C\><math|T<rsub|n>>\<#5206\>\<#522B\>\<#4E3A\><math|<around*|{|a<rsub|n>|}>>\<#548C\><math|<around*|{|b<rsub|n>|}>>\<#7684\>\<#524D\><math|n>\<#9879\>\<#548C\>.
    \<#8BC1\>\<#660E\>\<#FF1A\><math|T<rsub|n>\<less\><frac|S<rsub|n>|2>>.

    <item>\<#5DF2\>\<#77E5\>\<#629B\>\<#7269\>\<#7EBF\><math|C>\<#FF1A\><math|y<rsup|2>=2*p*x<around*|(|p\<gtr\>0|)>>\<#7684\>\<#7126\>\<#70B9\><math|F>\<#5230\>\<#51C6\>\<#7EBF\>\<#7684\>\<#8DDD\>\<#79BB\>\<#4E3A\>2.

    \<#FF08\>1\<#FF09\>\<#6C42\><math|C>\<#7684\>\<#65B9\>\<#7A0B\>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#5DF2\>\<#77E5\><math|O>\<#4E3A\>\<#5750\>\<#6807\>\<#539F\>\<#70B9\>\<#FF0C\>\<#70B9\><math|P>\<#5728\><math|C>\<#4E0A\>\<#FF0C\>\<#70B9\><math|Q>\<#6EE1\>\<#8DB3\><math|<wide|PQ|\<vect\>>=9<wide|QF|\<vect\>>>\<#FF0C\>\<#6C42\>\<#76F4\>\<#7EBF\><math|OQ>\<#659C\>\<#7387\>\<#7684\>\<#6700\>\<#5927\>\<#503C\>.

    <item>\<#5DF2\>\<#77E5\>\<#51FD\>\<#6570\><math|f<around*|(|x|)>=x<rsup|3>-x<rsup|2>+a*x+1>.

    \<#FF08\>1\<#FF09\>\<#8BA8\>\<#8BBA\><math|f<around*|(|x|)>>\<#7684\>\<#5355\>\<#8C03\>\<#6027\>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#6C42\>\<#66F2\>\<#7EBF\><math|y=f<around*|(|x|)>>\<#8FC7\>\<#5750\>\<#6807\>\<#539F\>\<#70B9\>\<#7684\>\<#5207\>\<#7EBF\>\<#4E0E\>\<#66F2\>\<#7EBF\><math|y=f<around*|(|x|)>>\<#7684\>\<#516C\>\<#5171\>\<#70B9\>\<#7684\>\<#5750\>\<#6807\>.

    <item>\<#5728\>\<#76F4\>\<#89D2\>\<#5750\>\<#6807\>\<#7CFB\><math|x O
    y>\<#4E2D\>\<#FF0C\><math|\<odot\>C>\<#7684\>\<#5706\>\<#5FC3\>\<#4E3A\><math|C<around*|(|2,1|)>>\<#FF0C\>\<#534A\>\<#5F84\>\<#4E3A\>1.

    \<#FF08\>1\<#FF09\>\<#5199\>\<#51FA\><math|\<odot\>C>\<#7684\>\<#4E00\>\<#4E2A\>\<#53C2\>\<#6570\>\<#65B9\>\<#7A0B\>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#8FC7\>\<#70B9\><math|F<around*|(|4,1|)>>\<#4F5C\><math|\<odot\>C>\<#7684\>\<#4E24\>\<#6761\>\<#5207\>\<#7EBF\>\<#FF0C\>\<#4EE5\>\<#5750\>\<#6807\>\<#539F\>\<#70B9\>\<#4E3A\>\<#6781\>\<#70B9\>\<#FF0C\><math|x>\<#8F74\>\<#6B63\>\<#534A\>\<#8F74\>\<#4E3A\>\<#6781\>\<#8F74\>\<#5EFA\>\<#7ACB\>\<#6781\>\<#5750\>\<#6807\>\<#7CFB\>\<#FF0C\>\<#6C42\>\<#8FD9\>\<#4E24\>\<#6761\>\<#5207\>\<#7EBF\>\<#7684\>\<#6781\>\<#5750\>\<#6807\>\<#65B9\>\<#7A0B\>.

    <item>\<#5DF2\>\<#77E5\>\<#51FD\>\<#6570\><math|<with|math-display|true|<frac|<around*|\||OB|\|>|<around*|\||OA|\|>>=<frac|\<rho\><rsub|1>|\<rho\><rsub|2>>=<frac|1|4>\<times\>2*sin
    \<alpha\><around*|(|<sqrt|3>*cos \<alpha\>+sin
    \<alpha\>|)>=<frac|1|4><around*|[|2*sin<around*|(|2*\<alpha\>-<frac|\<pi\>|6>|)>+1|]>>>.

    \<#FF08\>1\<#FF09\>\<#5F53\><math|a=1>\<#65F6\>\<#FF0C\>\<#6C42\>\<#4E0D\>\<#7B49\>\<#5F0F\><math|f<around*|(|x|)>\<geqslant\>6>\<#7684\>\<#89E3\>\<#96C6\>\<#FF1B\>

    \<#FF08\>2\<#FF09\>\<#82E5\><math|f<around*|(|x|)>\<gtr\>-\<alpha\>>\<#FF0C\>\<#6C42\><math|a>\<#7684\>\<#53D6\>\<#503C\>\<#8303\>\<#56F4\>.
  </enumerate>
</body>

<\initial>
  <\collection>
    <associate|page-medium|paper>
  </collection>
</initial>

<\references>
  <\collection>
    <associate|auto-1|<tuple|?|1>>
    <associate|auto-2|<tuple|12|2>>
    <associate|auto-3|<tuple|16|2>>
  </collection>
</references>

<\auxiliary>
  <\collection>
    <\associate|toc>
      <with|par-left|<quote|2tab>|\<#4E00\>\<#3001\>\<#5355\>\<#9009\>\<#9898\>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-1>>

      <with|par-left|<quote|2tab>|\<#4E8C\>\<#3001\>\<#586B\>\<#7A7A\>\<#9898\>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-2>>

      <with|par-left|<quote|2tab>|\<#4E09\>\<#3001\>\<#89E3\>\<#7B54\>\<#9898\>
      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
      <no-break><pageref|auto-3>>
    </associate>
  </collection>
</auxiliary>