planet/高考数学/2022年全国新高考一卷数学试题.tm

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  <doc-data|<doc-title|2022年数学新高考I卷>>

  <strong|一、单选题>（本大题共<with|font-series|bold|8>小题，共<with|font-series|bold|40>分）

  <\enumerate-numeric>
    <item>若集合<math|M=<around|{|x\|<sqrt|x>\<less\>4|}>>，<math|N=<around|{|x\|3*x\<geq\>1|}>>，则<math|M\<cap\>N=>(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|<around|{|x\|0\<leq\>x\<less\>2|}>>
      </cell>|<\cell>
        B. <math|<around|{|x\|<frac|1|3>\<leq\>x\<less\>2|}>>
      </cell>|<\cell>
        C. <math|<around|{|x\|3\<leq\>x\<less\>16|}>>
      </cell>|<\cell>
        D. <math|<around|{|x\|<frac|1|3>\<leq\>x\<less\>16|}>>
      </cell>>>>
    </wide-tabular>

    <item>若<math|i*<around|(|1-z|)>=1>，则<math|z+<wide|z|\<bar\>>=>(<space|2em>)

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

    <item>在<math|\<vartriangle\>A*B*C>中，点<math|D>在边<math|A*B>上，<math|B*D=2*D*A>.记<math|<wide|C*A|\<wide-varrightarrow\>>=<wide|m|\<wide-varrightarrow\>>>，<math|<wide|C*D|\<wide-varrightarrow\>>=<wide|n|\<wide-varrightarrow\>>>，则<math|<wide|C*B|\<wide-varrightarrow\>>=>(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|3*<wide|m|\<wide-varrightarrow\>>-2*<wide|n|\<wide-varrightarrow\>>>
      </cell>|<\cell>
        B. <math|-2*<wide|m|\<wide-varrightarrow\>>+3*<wide|n|\<wide-varrightarrow\>>>
      </cell>|<\cell>
        C. <math|3*<wide|m|\<wide-varrightarrow\>>+2*<wide|n|\<wide-varrightarrow\>>>
      </cell>|<\cell>
        D. <math|2*<wide|m|\<wide-varrightarrow\>>+3*<wide|n|\<wide-varrightarrow\>>>
      </cell>>>>
    </wide-tabular>

    <item>南水北调工程缓解了北方一些地区水资源短缺问题，其中一部分水蓄入某水库.已知该水库水位为海拔148.5m时，相应水面的面积为<math|140.0*km<rsup|2>>;水位为海拔<math|157.5>m时，相应水面的面积为<math|180.0*km<rsup|2>>.将该水库在这两个水位间的形状看作一个棱台，则该水库水位从海拔<math|148.5*>m上升到<math|157.5*>m时，增加的水量约为<math|<around*|(|<sqrt|7>\<approx\>2.65|)>>(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|1.0\<times\>10<rsup|9>*m<rsup|3>>
      </cell>|<\cell>
        B. <math|1.2\<times\>10<rsup|9>*m<rsup|3>>
      </cell>|<\cell>
        C. <math|1.4\<times\>10<rsup|9>*m<rsup|3>>
      </cell>|<\cell>
        D. <math|1.6\<times\>10<rsup|9>*m<rsup|3>>
      </cell>>>>
    </wide-tabular>

    <item>从<math|2>至<math|8>的<math|7>个整数中随机取<math|2>个不同的数，则这<math|2>个数互质的概率为(<space|2em>)

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

    <item>记函数<math|f<around|(|x|)>=sin
    <around|(|\<omega\>*x+<frac|\<pi\>|4>|)>+b*<around|(|\<omega\>\<gtr\>0|)>>的最小正周期为<math|T>.若<math|<frac|2*\<pi\>|3>\<less\>T\<less\>\<pi\>>，且<math|y=f<around|(|x|)>>的图像关于点<math|<around|(|<frac|3*\<pi\>|2>,2|)>>中心对称，则<math|f<around|(|<frac|\<pi\>|2>|)>=>(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|1>
      </cell>|<\cell>
        B. <math|<frac|3|2>>
      </cell>|<\cell>
        C. <math|<frac|5|2>>
      </cell>|<\cell>
        D. <math|3>
      </cell>>>>
    </wide-tabular>

    <item>设<math|a=0.1*e<rsup|0.1>>，<math|b=<frac|1|9>>，<math|c=-ln
    0.9>，则(<space|2em>)

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

    <item>已知正四棱锥的侧棱长为<math|l>，其各顶点都在同一个球面上，若该球的体积为<math|36*\<pi\>>，且<math|3\<leq\>l\<leq\>3*<sqrt|3>>，则该正四棱锥体积的取值范围是(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|A. <math|<around|[|18,<frac|81|4>|]>>|<\cell>
        B. <math|<around|[|<frac|27|4>,<frac|81|4>|]>>
      </cell>|<\cell>
        C. <math|<around|[|<frac|27|4>,<frac|64|3>|]>>
      </cell>|<\cell>
        D. <math|<around|[|18,27|]>>
      </cell>>>>
    </wide-tabular>
  </enumerate-numeric>

  <strong|二、多选题>（本大题共<with|font-series|bold|4>小题，共<with|font-series|bold|20>分）

  <\enumerate>
    <assign|item-nr|8><item>已知正方体<math|A*B*C*D-A<rsub|1>*B<rsub|1>*C<rsub|1>*D<rsub|1>>，则(<space|2em>)

    A. 直线<math|B*C<rsub|1>>与<math|D*A<rsub|1>>所成的角为<math|90<rsup|\<circ\>>>

    B. 直线<math|B*C<rsub|1>>与<math|C*A<rsub|1>>所成的角为<math|90<rsup|\<circ\>>>

    C. 直线<math|B*C<rsub|1>>与平面<math|B*B<rsub|1>*D<rsub|1>*D>所成的角为<math|45<rsup|\<circ\>>>

    D. 直线<math|B*C<rsub|1>>与平面<math|A*B*C*D>所成的角为<math|45<rsup|\<circ\>>>

    <item>已知函数<math|f<around|(|x|)>=x<rsup|3>-x+1>，则(<space|2em>)

    A. <math|f<around|(|x|)>>有两个极值点

    B. <math|f<around|(|x|)>>有三个零点

    C. 点<math|<around|(|0,1|)>>是曲线<math|y=f<around|(|x|)>>的对称中心

    D. 直线<math|y=2*x>是曲线<math|y=f<around|(|x|)>>的切线

    <item>已知<math|O>为坐标原点，点<math|A*<around|(|1,1|)>>在抛物线<math|C:x<rsup|2>=2p*y*<around|(|p\<gtr\>0|)>>上，过点<math|B*<around|(|0,-1|)>>的直线交<math|C>于<math|P>，<math|Q>两点，则(<space|2em>)

    A. <math|C>的准线为<math|y=-1>\ 

    B. 直线<math|A*B>与<math|C>相切

    C. <math|<around|\||O*P|\|>\<cdot\><around|\||O*Q|\|>\<gtr\><around|\||O*A|\|><rsup|2>>\ 

    D. <math|<around|\||B*P|\|>\<cdot\><around|\||B*Q|\|>\<gtr\><around|\||B*A|\|><rsup|2>>

    <item>已知函数<math|f<around|(|x|)>>及其导函数<math|f<rprime|'><around|(|x|)>>的定义域为<math|R>，记<math|g<around|(|x|)>=f<rprime|'><around|(|x|)>>.若<math|f*<around|(|<frac|3|2>-2*x|)>>，<math|g*<around|(|2+x|)>>均为偶函数，则(<space|2em>)

    <\wide-tabular>
      <tformat|<table|<row|<\cell>
        A. <math|f<around|(|0|)>=0>
      </cell>|<\cell>
        B. <math|g*<around|(|-<frac|1|2>|)>=0>
      </cell>|C. <math|f*<around|(|-1|)>=f<around|(|4|)>>|<\cell>
        D. <math|g*<around|(|-1|)>=g<around|(|2|)>>
      </cell>>>>
    </wide-tabular>
  </enumerate>

  <strong|三、填空题>（本大题共<with|font-series|bold|4>小题，共<with|font-series|bold|20>分）

  <\enumerate>
    <assign|item-nr|12><item><math|<around|(|1-<frac|y|x>|)>*<around|(|x+y|)><rsup|8>>的展开式中<math|x<rsup|2>*y<rsup|6>>的系数为<underline|<space|2cm>>
    (用数字作答)．

    <item>写出与圆<math|x<rsup|2>+y<rsup|2>=1>和<math|<around|(|x-3|)><rsup|2>+<around|(|y-4|)><rsup|2>=16>都相切的一条直线的方程<underline|<space|2cm>>．

    <item>若曲线<math|y=<around|(|x+a|)>*e<rsup|x>>有两条过坐标原点的切线，则<math|a>的取值范围是<underline|<space|2cm>>．

    <item>已知椭圆<math|C:<frac|x<rsup|2>|a<rsup|2>>+<frac|y<rsup|2>|b<rsup|2>>=1*<around|(|a\<gtr\>b\<gtr\>0|)>>，<math|C>的上顶点为<math|A>，两个焦点为<math|F<rsub|1>>，<math|F<rsub|2>>，离心率为<math|<frac|1|2>>，过<math|F<rsub|1>>且垂直于<math|A*F<rsub|2>>的直线与<math|C>交于<math|D>，<math|E>两点，<math|<around|\||D*E|\|>=6>，则\<vartriangle\>的周长是<underline|<space|2cm>>．
  </enumerate>

  <strong|四、解答题>（本大题共<with|font-series|bold|6>小题，共<with|font-series|bold|70>分）

  <\enumerate>
    <assign|item-nr|16><item>记<math|S<rsub|n>>为数列<math|<around|{|a<rsub|n>|}>>的前<math|n>项和，已知<math|a<rsub|1>=1>，<math|<around*|{|<frac|S<rsub|n>|a<rsub|n>>|}>>是公差为<math|<frac|1|3>>的等差数列．

    <math|<around|(|1|)>>求<math|<around|{|a<rsub|n>|}>>的通项公式<math|;<around|(|2|)>>证明：<math|<frac|1|a<rsub|1>>+<frac|1|a<rsub|2>>+\<cdots\>+<frac|1|a<rsub|n>>\<less\>2>．<vspace|5cm>

    <item>记<math|\<vartriangle\>A*B*C>的内角<math|A>，<math|B>，<math|C>的对边分别为<math|a>，<math|b>，<math|c>，已知<math|<frac|cos
    A|1+sin A>=<frac|sin 2*B|1+cos 2*B>>．

    <math|<around|(|1|)>>若<math|C=<frac|2*\<pi\>|3>>，求<math|B;<around|(|2|)>>求<math|<frac|a<rsup|2>+b<rsup|2>|c<rsup|2>>>的最小值．<vspace|5cm>

    <item>如图，直三棱柱<math|A*B*C-A<rsub|1>*B<rsub|1>*C<rsub|1>>的体积为<math|4>，<math|\<vartriangle\>A<rsub|1>B
    C>的面积为<math|2<sqrt|2>>．

    <math|<around|(|1|)>>求<math|A>到平面<math|A<rsub|1>*B*C>的距离;

    <around|(|2|)>设<math|D>为<math|A<rsub|1>*C>的中点，<math|A*A<rsub|1>=A*B>，平面<math|A<rsub|1>*B*C*\<bot\>>平面<math|A*B*B<rsub|1>*A<rsub|1>>，求二面角<math|A-B*D-C>的正弦值．

    <image|<tuple|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

    <item>一支医疗团队研究某地的一种地方性疾病与当地居民的卫生习惯<math|(>卫生习惯分为良好和不够良好两类<math|)>的关系，在已患该疾病的病例中随机调查了<math|100>例<math|(>称为病例组<math|)>，同时在未患该疾病的人群中随机调查了<math|100>人<math|(>称为对照组<math|)>，得到如下数据：

    <center|<center|><block*|<tformat|<table|<row|<cell|>|<cell|不够良好>|<cell|良好>>|<row|<cell|病例组>|<cell|40>|<cell|60>>|<row|<cell|对照组>|<cell|10>|<cell|90>>>>>>

    <math|<around|(|1|)>>能否有<math|99%>的把握认为患该疾病群体与未患该疾病群体的卫生习惯有差异\<cdot\>

    <around|(|2|)>从该地的人群中任选一人，<math|A>表示事件``选到的人卫生习惯不够良好''，<math|B>表示事件``选到的人患有该疾病''，<math|<frac|P<around|(|B\|A|)>|P<around|(|<wide|B|\<bar\>>\|A|)>>>与<math|<frac|P<around|(|B\|<wide|A|\<bar\>>|)>|P<around|(|<wide|B|\<bar\>>\|<wide|A|\<bar\>>|)>>>的比值是卫生习惯不够良好对患该疾病风险程度的一项度量指标，记该指标为<math|R>．

    <math|<around|(|<math|i>|)>>证明：<math|R=<frac|P<around|(|A\|B|)>|P<around|(|<wide|A|\<bar\>>\|B|)>>>．<frac|P<around|(|<wide|A|\<bar\>>\|<wide|B|\<bar\>>|)>|P<around|(|A\|<wide|B|\<bar\>>|)>>;

    <around|(|ii|)>利用该调查数据，给出<math|P<around|(|A\|B|)>>，<math|P<around|(|A\|<wide|B|\<bar\>>|)>>的估计值，并利用<math|<around|(|i|)>>的结果给出<math|R>的估计值．

    附：<math|K<rsup|2>=<frac|n*<around|(|a*d-b*c|)><rsup|2>|<around|(|a+b|)>*<around|(|c+d|)>*<around|(|a+c|)>*<around|(|b+d|)>>>，<block|<tformat|<cwith|2|2|1|1|cell-halign|c>|<table|<row|<cell|<math|P<around*|(|K<rsup|2>\<geqslant\>k|)>>>|<cell|0.050>|<cell|0.010>|<cell|0.001>>|<row|<cell|<math|k>>|<cell|3.841>|<cell|6.635>|<cell|10.828>>>>><vspace|5cm>

    <item>已知点<math|A*<around|(|2,1|)>>在双曲线<math|C:<frac|x<rsup|2>|a<rsup|2>>-<frac|y<rsup|2>|a<rsup|2>-1>=1*<around|(|a\<gtr\>1|)>>上，直线<math|l>交<math|C>于<math|P>，<math|Q>两点，直线<math|A*P>，<math|A*Q>的斜率之和为<math|0>．

    <math|<around|(|1|)>>求<math|l>的斜率<math|;<around|(|2|)>>若<math|tan
    \<angle\>*P*A*Q=2<sqrt|2>>，求<math|\<vartriangle\>P*A*Q>的面积．<vspace|5cm>

    <item>已知函数<math|f<around|(|x|)>=e<rsup|x>-a*x>和<math|g<around|(|x|)>=a*x-ln
    x>有相同的最小值．

    <math|<around|(|1|)>>求<math|a>;

    <around|(|2|)>证明：存在<math|y=b>直线，其与两条曲线<math|y=f<around|(|x|)>>和<math|y=g<around|(|x|)>>共有三个不同的交点，并且从左到右的三个交点的横坐标成等差数列．<vspace|5cm>
  </enumerate>

  \;
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