planet/高考数学/2021年全国高考乙卷数学理科试题.tm

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

  <subsubsection*|一、单选题>

  <\enumerate>
    <item>设<math|2*<around*|(|z+<wide|z|\<bar\>>|)>+3*<around*|(|z-<wide|z|\<bar\>>|)>=4+6*i>，则<math|z=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|1-2*i>
      </cell>|<\cell>
        B. <math|1+2*i>
      </cell>|<\cell>
        C. <math|1+i>
      </cell>|<\cell>
        D. <math|1-i>
      </cell>>>>
    </wide-tabular>

    <item>已知集合<math|S=<around*|{|s\|s=2*n+1,n\<in\>\<b-Z\>|}>>，<math|T=<around*|{|t\|t=4*n+1,n\<in\>\<b-Z\>|}>>，则<math|S\<cap\>T=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|\<emptyset\>>
      </cell>|<\cell>
        B. <math|S>
      </cell>|<\cell>
        C. <math|T>
      </cell>|<\cell>
        D. <math|\<b-Z\>>
      </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>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>将5名北京冬奥会志愿者分配到花样滑冰、短道速滑、冰球和冰壶4个项目进行培训，每名志愿者只分配到1个项目，每个项目至少分配1名志愿者，则不同的分配方案共有<underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. 60种
      </cell>|<\cell>
        B. 120种
      </cell>|<\cell>
        C. 240种
      </cell>|<\cell>
        D. 480种
      </cell>>>>
    </wide-tabular>

    <item>把函数<math|y=f<around*|(|x|)>>图像上所有点的横坐标缩短到原来的<math|<frac|1|2>>倍，纵坐标不变，再把所得曲线向右平移<math|<frac|\<pi\>|3>>个单位长度，得到函数<math|y=sin<around*|(|x-<frac|\<pi\>|4>|)>>的图像，则<math|f<around*|(|x|)>=><underline|<space|3em>>

    <\wide-tabular>
      <tformat|<cwith|1|1|1|1|cell-valign|b>|<table|<row|<\cell>
        A. <math|sin<around*|(|<frac|x|2>-<frac|7*\<pi\>|12>|)>>
      </cell>|<\cell>
        B. <math|sin<around*|(|<frac|x|2>+<frac|\<pi\>|12>|)>>
      </cell>>|<row|<\cell>
        C. <math|sin<around*|(|2*x-<frac|7*\<pi\>|12>|)>>
      </cell>|<\cell>
        D. <math|sin<around*|(|2*x+<frac|\<pi\>|12>|)>>
      </cell>>>>
    </wide-tabular>

    <item>在区间<math|<around*|(|0,1|)>>与<math|<around*|(|1,2|)>>中各随机取1个数，则两数之和大于<math|<frac|7|4>>的概率为<underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|7|9>>
      </cell>|<\cell>
        B. <math|<frac|23|32>>
      </cell>|<\cell>
        C. <math|<frac|9|32>>
      </cell>|<\cell>
        D. <math|<frac|2|9>>
      </cell>>>>
    </wide-tabular>

    <item>魏晋时刘徽撰写的《海岛算经》是有关测量的数学著作，其中第一题是测海岛的高．如图，点E，H，G在水平线AC上，DE和FG是两个垂直于水平面且等高的测量标杆的高度，称为\P表高\Q，EG称为\P表距\Q，GC和EH都称为\P表目距\Q，GC与EH的差称为\P表目距的差\Q则海岛的高<math|AB=><underline|<space|3em>>

    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    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<frac|表高\<times\>表距|表目距的差>+表高>
      </cell>|<\cell>
        B. <math|<frac|表高\<times\>表距|表目距的差>-表高>
      </cell>>|<row|<\cell>
        C. <math|<frac|表高\<times\>表距|表目距的差>+表距>
      </cell>|<\cell>
        D. <math|<frac|表高\<times\>表距|表目距的差>-表距>
      </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>

    <item>设<math|B>是椭圆C: <math|<frac|x<rsup|2>|a<rsup|2>>+<frac|y<rsup|2>|b<rsup|2>>=1<around*|(|a\<gtr\>b\<gtr\>0|)>>的上顶点，若<math|C>上的任意一点<math|P>都满足<math|<around*|\||PB|\|>\<leqslant\>2*b>，则<math|C>的离心率的取值范围是<underline|<space|3em>>

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<around*|[|<frac|<sqrt|2>|2>,1|)>>
      </cell>|<\cell>
        B. <math|<around*|[|<frac|1|2>,1|)>>
      </cell>|<\cell>
        C. <math|<around*|(|0,<frac|<sqrt|2>|2>|]>>
      </cell>|<\cell>
        D. <math|<around*|(|0,<frac|1|2>|]>>
      </cell>>>>
    </wide-tabular>

    <item>设<math|a=2*ln 1.01>，<math|b=ln 1.02>，<math|c=<sqrt|1.04>-1>.
    则<underline|<space|3em>>

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

  <subsubsection*|二、填空题>

  <\enumerate>
    <assign|item-nr|12><item>已知双曲线C:
    <math|<frac|x<rsup|2>|m>-y<rsup|2>=1<around*|(|m\<gtr\>0|)>>的一条渐近线为<math|<sqrt|3>*x+m*y=0>，则<math|C>的焦距为<underline|<space|3em>>.

    <item>已知向量<math|<wide|a|\<vect\>>=<around*|(|1,3|)>,<wide|b|\<vect\>>=<around*|(|3,4|)>>，若<math|<around*|(|<wide|a|\<vect\>>-\<lambda\><wide|b|\<vect\>>|)>\<bot\><wide|b|\<vect\>>>，则<math|\<lambda\>=><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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    （1）求BC；

    （2）求二面角A-PM-B的正弦值.

    <item>设<math|S<rsub|n>>为数列<math|<around*|{|a<rsub|n>|}>>的前<math|n>项和，<math|b<rsub|n>>为数列<math|<around*|{|S<rsub|n>|}>>的前<math|n>项积，已知<math|<frac|2|S<rsub|n>>+<frac|1|b<rsub|n>>=2>.

    （1）证明：数列<math|<around*|{|b<rsub|n>|}>>是等差数列；

    （2）求<math|<around*|{|a<rsub|n>|}>>的通项公式.

    <item>设函数<math|f<around*|(|x|)>=ln
    <around*|(|a-x|)>>，已知<math|x=0>是函数<math|y=x*f<around*|(|x|)>>的极值点.

    （1）求a；

    （2）设函数<math|g<around*|(|x|)>=<frac|x+f<around*|(|x|)>|x*f<around*|(|x|)>>>.
    证明：<math|g<around*|(|x|)>\<less\>1>.

    <item>已知抛物线C: <math|x<rsup|2>=2*p*y<around*|(|p\<gtr\>0|)>>的焦点为<math|F>，且<math|F>与圆<math|M>:
    <math|x<rsup|2>+<around*|(|y+4|)><rsup|2>=1>上点的距离的最小值为4.

    （1）求<math|p>；

    （2）若点<math|P>在<math|M>上，<math|PA>，<math|PB>是<math|C>的两条切线，<math|A>,<math|B>是切点，求<math|\<vartriangle\>PAB>面积的最大值.

    <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|global-title|2021年全国高考乙卷数学（理）试卷>
    <associate|page-medium|paper>
  </collection>
</initial>

<\references>
  <\collection>
    <associate|auto-1|<tuple|?|1|.Xmacs/texts/scratch/no_name_7.tm>>
    <associate|auto-2|<tuple|12|2|.Xmacs/texts/scratch/no_name_7.tm>>
    <associate|auto-3|<tuple|16|2|.Xmacs/texts/scratch/no_name_7.tm>>
  </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>