planet/高考数学/2021年全国高考乙卷数学文科试题.tm

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  <doc-data|<doc-title|2021年全国高考乙卷数学（文）试卷>>

  <subsubsection*|一、单选题>

  <\enumerate>
    <item>已知全集<math|U=<around*|{|1,2,3,4,5|}>>，集合<math|M=<around*|{|1,2|}>,N=<around*|{|3,4|}>>，则<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>设<math|i*z=4+3*i>，则<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>已知命题<math|p>：<math|\<exists\>x\<in\>\<bbb-R\>,sin
    x\<less\>1>；命题<math|q>：<math|\<forall\>x\<in\>\<bbb-R\>,e<rsup|<around*|\||x|\|>>\<geqslant\>1>，则下列命题中为真命题的是<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>函数<math|f<around*|(|x|)>=sin
    <frac|x|3>+cos<frac|x|3>>的最小正周期和最大值分别是<underline|<space|3em>>

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

    <item>若<math|x>，<math|y>满足约束条件<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>,>>>>>>则<math|z=3*x+y>的最小值为<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>在区间0, <math|<frac|1|2>>随机取1个数，则取到的数小于<math|<frac|1|3>>的概率为<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>下列函数中最小值为4的是<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>设函数<math|f<around*|(|x|)>=<frac|1-x|1+x>>，则下列函数中为奇函数的是<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>在正方体<math|ABCD-A<rsub|1>B<rsub|1>C<rsub|1>D<rsub|1>>中，<math|P>为<math|B<rsub|1>D<rsub|1>>的中点，则直线<math|PB>与<math|AD<rsub|1>>所成的角为<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>设<math|B>是椭圆<math|C:<frac|x<rsup|2>|5>+y=1>的上顶点，点<math|P>在<math|C>上，则<math|<around*|\||PB|\|>>的最大值为<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>设<math|a\<neq\>0>，若<math|x=a>为函数<math|f<around*|(|x|)>=a*<around*|(|x-a|)><rsup|2><around*|(|x-b|)>>的极大值点，则<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*|二、填空题>

  <\enumerate>
    <assign|item-nr|12><item>已知向量<math|<wide|a|\<vect\>>=<around*|(|2,5|)>,<wide|b|\<vect\>>=<around*|(|\<lambda\>,4|)>>，若<math|<wide|a|\<vect\>>//<wide|b|\<vect\>>>，则<math|\<lambda\>=><underline|<space|3em>>.

    <item>双曲线<math|<frac|x<rsup|2>|4>-<frac|y<rsup|2>|5>=1>的右焦点到直线<math|x+2*y-8=0>的距离为<underline|<space|3em>>.

    <item>记<math|\<triangle\>ABC>的内角<math|A>，<math|B>，<math|C>的对边分别为a，b，c，面积为<math|<sqrt|3>>，<math|B=60<rsup|\<circ\>>>，<math|a<rsup|2>+c<rsup|2>=3*a*c>，则<math|b=><underline|<space|3em>>.

    <item>以图①为正视图，在图②③④⑤中选两个分别作为侧视图和俯视图，组成某个三棱锥的三视图，则所选侧视图和俯视图的编号依次为<underline|<space|3em>>（写出符合要求的一组答案即可）．

    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|png>|0.618par|||>
  </enumerate>

  <subsubsection*|三、解答题>

  <\enumerate>
    <assign|item-nr|16><item>某厂研制了一种生产高精产品的设备，为检验新设备生产产品的某项指标有无提高，用一台旧设备和一台新设备各生产了10件产品，得到各件产品该项指标数据如下：

    <block*|<tformat|<cwith|1|1|1|1|cell-valign|b>|<cwith|2|2|6|6|cell-halign|r>|<table|<row|<cell|旧设备>|<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|新设备>|<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>>>>>

    旧设备和新设备生产产品的该项指标的样本平均数分别记为和，样本方差分别记为和．

    （1）求<math|<wide|x|\<bar\>>,<wide|y|\<bar\>>,s<rsup|2><rsub|1>,s<rsup|2><rsub|2>>；

    （2）判断新设备生产产品的该项指标的均值较旧设备是否有显著提高（如果<math|<wide|y|\<bar\>>-<wide|x|\<bar\>>\<geqslant\>2<sqrt|<frac|s<rsup|2><rsub|1>+s<rsup|2><rsub|2>|10>>>，则认为新设备生产产品的该项指标的均值较旧设备有显著提高，否则不认为有显著提高）．

    <item>如图，四棱锥<math|P-ABCD>的底面是矩形，<math|PD\<perp\>底面ABCD>，<math|M>为<math|BC>的中点，且<math|PB\<perp\>AM>.

    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|png>|0.3par|||>

    （1）证明：<math|平面PAM\<perp\>平面PBD>；

    （2）若<math|PD=DC=1>，求四棱锥<math|P-ABCD>的体积.

    <item>设<math|<around*|{|a<rsub|n>|}>>是首项为1的等比数列，数列<math|<around*|{|b<rsub|n>|}>>满足<math|b<rsub|n>=<frac|n*a<rsub|n>|3>>.
    已知<math|a<rsub|1>>，<math|3*a<rsub|2>>，<math|9*a<rsub|3>>成等差数列.

    （1）求<math|<around*|{|a<rsub|n>|}>>和<math|<around*|{|b<rsub|n>|}>>的通项公式；

    （2）记<math|S<rsub|n>>和<math|T<rsub|n>>分别为<math|<around*|{|a<rsub|n>|}>>和<math|<around*|{|b<rsub|n>|}>>的前<math|n>项和.
    证明：<math|T<rsub|n>\<less\><frac|S<rsub|n>|2>>.

    <item>已知抛物线<math|C>：<math|y<rsup|2>=2*p*x<around*|(|p\<gtr\>0|)>>的焦点<math|F>到准线的距离为2.

    （1）求<math|C>的方程；

    （2）已知<math|O>为坐标原点，点<math|P>在<math|C>上，点<math|Q>满足<math|<wide|PQ|\<vect\>>=9<wide|QF|\<vect\>>>，求直线<math|OQ>斜率的最大值.

    <item>已知函数<math|f<around*|(|x|)>=x<rsup|3>-x<rsup|2>+a*x+1>.

    （1）讨论<math|f<around*|(|x|)>>的单调性；

    （2）求曲线<math|y=f<around*|(|x|)>>过坐标原点的切线与曲线<math|y=f<around*|(|x|)>>的公共点的坐标.

    <item>在直角坐标系<math|x O y>中，<math|\<odot\>C>的圆心为<math|C<around*|(|2,1|)>>，半径为1.

    （1）写出<math|\<odot\>C>的一个参数方程；

    （2）过点<math|F<around*|(|4,1|)>>作<math|\<odot\>C>的两条切线，以坐标原点为极点，<math|x>轴正半轴为极轴建立极坐标系，求这两条切线的极坐标方程.

    <item>已知函数<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|]>>>.

    （1）当<math|a=1>时，求不等式<math|f<around*|(|x|)>\<geqslant\>6>的解集；

    （2）若<math|f<around*|(|x|)>\<gtr\>-\<alpha\>>，求<math|a>的取值范围.
  </enumerate>
</body>

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

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

<\auxiliary>
  <\collection>
    <\associate|toc>
      <with|par-left|<quote|2tab>|一、单选题
      <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>|二、填空题
      <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>|三、解答题
      <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>