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）．

    <image|<tuple|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
  </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>.

    <image|<tuple|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

    （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>