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>>

    <image|<tuple|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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）．

    <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）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>