planet/高考数学/2022年全国新高考二卷数学试题.tm

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  <doc-data|<doc-title|2022年数学新高考2卷>>

  一、单选题（本大题共8小题，共40分）

  <\enumerate-numeric>
    <item>已知集合<math|A=<around|{|-1,1,2,4|}>>，<math|B=<around|{|x\|<around|\||x-1|\|>\<leq\>1|}>>，则<math|A\<cap\>B=>

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|<around|{|-1,2|}>>>|<cell|B. <math|<around|{|1,2|}>>>|<cell|C.
    <math|<around|{|1,4|}>>>|<cell|D. <math|<around|{|-1,4|}>>>>>>>

    <item><math|<around|(|2+2*i|)>*<around|(|1-2*i|)>=>

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|-2+4*i>>|<cell|B. <math|-2-4*i>>|<cell|C. <math|6+2*i>>|<cell|D.
    <math|6-2*i>>>>>>

    <item>中国的古建筑不仅是挡风遮雨的住处，更是美学和哲学的体现.
    如图是某古建筑物的剖面图，
    <math|A*A<rprime|'>>，<math|B*B<rprime|'>>，<math|C*C<rprime|'>>，<math|D*D<rprime|'>>是桁，
    <math|D*D<rsub|1>>，<math|C*C<rsub|1>>，<math|B*B<rsub|1>>，<math|A*A<rsub|1>>是脊，
    <math|O*D<rsub|1>>，<math|D*C<rsub|1>>，<math|C*B<rsub|1>>，<math|B*A<rsub|1>>是相等的步，
    相邻桁的脊步的比分别为<math|<frac|D*D<rsub|1>|O*D<rsub|1>>=0.5>，<math|<frac|C*C<rsub|1>|D*C<rsub|1>>=k<rsub|1>>，<math|<frac|B*B<rsub|1>|C*B<rsub|1>>=k<rsub|2>>，<math|<frac|A*A<rsub|1>|B*A<rsub|1>>=k<rsub|3>>，若<math|k<rsub|1>>，<math|k<rsub|2>>，<math|k<rsub|3>>是公差为<math|0.1>的等差数列，直线<math|O*A>的斜率为<math|0.725>，则<math|k<rsub|3>=>

    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    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|0.75>>|<cell|B. <math|0.8>>|<cell|C. <math|0.85>>|<cell|D.
    <math|0.9>>>>>>

    <item>已知向量<math|<wide|a|\<wide-varrightarrow\>>=<around|(|3,4|)>>，<math|<wide|b|\<wide-varrightarrow\>>=<around|(|1,0|)>>，<math|<wide|c|\<wide-varrightarrow\>>=<wide|a|\<wide-varrightarrow\>>+t<wide|b|\<wide-varrightarrow\>>>，若<math|\<less\><wide|a|\<wide-varrightarrow\>>,<wide|c|\<wide-varrightarrow\>>\<gtr\>=\<less\><wide|b|\<wide-varrightarrow\>>,<wide|c|\<wide-varrightarrow\>>\<gtr\>>，则实数<math|t=>

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|-6>>|<cell|B. <math|-5>>|<cell|C. <math|5>>|<cell|D. <math|6>>>>>>

    <item>甲乙丙丁戊<math|5>名同学站成一排参加文艺汇演，若甲不站在两端，丙和丁相邻的不同排列方式有

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|12>种>|<cell|B. <math|24>种>|<cell|C. <math|36>种>|<cell|D.
    <math|48>种>>>>>

    <item>若<math|<math-up|sin><around|(|\<alpha\>+\<beta\>|)>+<math-up|cos><around|(|\<alpha\>+\<beta\>|)>=2*<sqrt|2><math-up|cos><around|(|\<alpha\>+<frac|\<pi\>|4>|)><math-up|sin>\<beta\>>，则

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|2|2|1|2|cell-valign|c>|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|<math-up|tan><around|(|\<alpha\>+\<beta\>|)>=-1>>|<cell|B.
    <math|<math-up|tan><around|(|\<alpha\>+\<beta\>|)>=1>>>|<row|<cell|C.
    <math|<math-up|tan><around|(|\<alpha\>-\<beta\>|)>=-1>>|<cell|D.
    <math|<math-up|tan><around|(|\<alpha\>-\<beta\>|)>=1>>>>>>

    <item>已知正三棱台的高为<math|1>，上下底面的边长分别为<math|3*<sqrt|3>>和<math|4*<sqrt|3>>，其顶点都在同一球面上，则该球的表面积为

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|100*\<pi\>>>|<cell|B. <math|128*\<pi\>>>|<cell|C.
    <math|144*\<pi\>>>|<cell|D. <math|192*\<pi\>>>>>>>

    <item>若函数<math|f<around|(|x|)>>的定义域为<math|R>，且<math|f*<around|(|x+y|)>+f*<around|(|x-y|)>=f<around|(|x|)>*f<around|(|y|)>>，<math|f=1>，则<math|<big|int><rsup|22><rsub|k=1>f<around|(|k|)>=>

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|-3>>|<cell|B. <math|-2>>|<cell|C. <math|0>>|<cell|D. <math|1>>>>>>
  </enumerate-numeric>

  二、多选题（本大题共4小题，共20分）

  <\enumerate-numeric>
    <assign|item-nr|8><item>已知函数<math|f<around|(|x|)>=<math-up|sin><around|(|2*x+\<varphi\>|)>*<around|(|0\<less\>\<varphi\>\<less\>\<pi\>|)>>的图象关于点<math|<around|(|<frac|2*\<pi\>|3>,0|)>>对称，则

    A. <math|f<around|(|x|)>>在<math|<around|(|0,<frac|5*\<pi\>|12>|)>>单调递减

    B. <math|f<around|(|x|)>>在<math|<around|(|-<frac|\<pi\>|12>,<frac|11*\<pi\>|12>|)>>有两个极值点

    C. 直线<math|x=<frac|7*\<pi\>|6>>是曲线<math|y=f<around|(|x|)>>的一条对称轴

    D. 直线<math|y=<frac|<sqrt|3>|2>-x>是曲线<math|y=f<around|(|x|)>>的一条切线

    <item>已知<math|O>为坐标原点，过抛物线<math|C:y<rsup|2>=2*p*x*<around|(|p\<gtr\>0|)>>的焦点<math|F>的直线与<math|C>交于<math|A>，<math|B>两点，点<math|A>在第一象限，点<math|M*<around|(|p,0|)>>，若<math|<around|\||A*F|\|>=<around|\||A*M|\|>>，则

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|2|2|1|2|cell-valign|c>|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    直线<math|A*B>的斜率为<math|2*<sqrt|6>>>|<cell|B.
    <math|<around|\||O*B|\|>=<around|\||O*F|\|>>>>|<row|<cell|C.
    <math|<around|\||A*B|\|>\<gtr\>4*<around|\||O*F|\|>>>|<cell|D.
    <math|\<angle\>*O*A*M+\<angle\>*O*B*M\<less\>180*\<degree\>>>>>>>

    <item>如图，四边形<math|A*B*C*D>为正方形，<math|E*D*\<bot\>>平面<math|A*B*C*D>，<math|F*B//E*D>，
    <math|A*B=E*D=2*F*B>，记三棱锥<math|E-A*B*C>，<math|E-A*C*F>，<math|F-A*B*C>的体积分别为<math|V<rsub|1>>，<math|V<rsub|2>>，<math|V<rsub|3>>，则

    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|png>|90pt|87pt||>

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|V<rsub|3>=2*V<rsub|2>>>|<cell|B.
    <math|V<rsub|3>=2*V<rsub|1>>>|<cell|C.
    <math|V<rsub|3>=V<rsub|1>+V<rsub|2>>>|<cell|D.
    <math|2*V<rsub|3>=3*V<rsub|1>>>>>>>

    <item>若实数<math|x>，<math|y>满足<math|x<rsup|2>+y<rsup|2>-x*y=1>，则

    <tabular*|<tformat|<cwith|1|-1|1|-1|cell-valign|c>|<twith|table-width|1par>|<twith|table-hmode|exact>|<cwith|1|1|1|-1|cell-halign|l>|<table|<row|<cell|A.
    <math|x+y\<leq\>1>>|<cell|B. <math|x+y\<geq\>-2>>|<cell|C.
    <math|x<rsup|2>+y<rsup|2>\<geq\>1>>|<cell|D.
    <math|x<rsup|2>+y<rsup|2>\<leq\>2>>>>>>
  </enumerate-numeric>

  三、填空题（本大题共4小题，共20分）

  <\enumerate-numeric>
    <assign|item-nr|12><item>随机变量<math|X>服从正态分布<math|N*<around|(|2,\<sigma\><rsup|2>|)>>，若<math|P*<around|(|2\<less\>x\<leq\>2.5|)>=0.36>，则<math|P*<around|(|X\<gtr\>2.5|)>=><underline|<space|2cm>>．

    <item>曲线<math|y=<math-up|ln><around|\||x|\|>>经过坐标原点的两条切线方程分别为<underline|<space|1cm>>，<underline|<space|1cm>>．

    <item>设点<math|A*<around|(|-2,3|)>>，<math|B*<around|(|0,a|)>>，直线<math|A*B>关于直线<math|y=a>的对称直线为<math|l>，已知<math|l>与圆<math|C:<around|(|x+3|)><rsup|2>+<around|(|y+2|)><rsup|2>=1>有公共点，则<math|a>的取值范围为<underline|<space|2cm>>．

    <item>已知直线<math|l>与椭圆<math|<frac|x<rsup|2>|6>+<frac|y<rsup|2>|3>=1>在第一象限交于<math|A>，<math|B>两点，<math|l>与<math|x>轴<math|y>轴分别相交于<math|M>，<math|N>两点，且<math|<around|\||M*A|\|>=<around|\||N*B|\|>>，<math|<around|\||M*N|\|>=2*<sqrt|3>>，则直线<math|l>的方程为<underline|<space|2cm>>．
  </enumerate-numeric>

  四、解答题（本大题共6小题，共70分）

  <\enumerate-numeric>
    <assign|item-nr|16><item>已知<math|<around|{|a<rsub|n>|}>>为等差数列，<math|<around|{|b<rsub|n>|}>>为公比为<math|2>的等比数列，且<math|a<rsub|2>-b<rsub|2>=a<rsub|3>-b<rsub|3>=b<rsub|4>-a<rsub|4>>．

    (1)证明：<math|a<rsub|1>=b<rsub|1>>;

    (2)求集合<math|<around|{|k\|b<rsub|k>=a<rsub|m>+a<rsub|1>,1\<leq\>m\<leq\>500|}>>中元素个数．<vspace|5cm>

    <item>记<math|\<bigtriangleup\>*A*B*C>的三个内角分别为<math|A>，<math|B>，<math|C>，其对边分别为<math|a>，<math|b>，<math|c>，分别以<math|a>，<math|b>，<math|c>为边长的三个正三角形的面积依次为<math|S<rsub|1>>，<math|S<rsub|2>>，<math|S<rsub|3>>，且<math|S<rsub|1>-S<rsub|2>+S<rsub|3>=<frac|<sqrt|3>|2>>，<math|<math-up|sin>B=<frac|1|3>>．

    (1)求<math|\<bigtriangleup\>*A*B*C>的面积;

    (2)若<math|<math-up|sin>A<math-up|sin>C=<frac|<sqrt|2>|3>>，求<math|b>．<vspace|5cm>

    <item>在某地区进行某种疾病调查，随机调查了<math|100>位这种疾病患者的年龄，得到如下样本数据频率分布直方图．

    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3440281442269381C3E1B4E9965D6FB37DFB76066BFBF6EDDAB1A3A7A1A1A1A1A1D130FF0F2F56B875C17D01940000000049454E44AE426082>|png>|.5par|||>

    (1)估计该地区这种疾病患者的平均年龄;(同一组数据用该区间的中点值作代表<math|)>;

    (2)估计该地区以为这种疾病患者年龄位于区间<math|<around|[|20,70|)>>的概率;

    (3)已知该地区这种疾病患者的患病率为<math|0.1%>，该地区年龄位于区间<math|<around|[|40,50|)>>的人口数占该地区总人口数的<math|16%>，从该地区选出<math|1>人，若此人的年龄位于区间<math|<around|[|40,50|)>>，求此人患这种疾病的概率<math|(>精确到<math|0.0001)>．<vspace|5cm>

    <item>如图，<math|P*O>是三棱锥<math|P-A*B*C>的高，<math|P*A=P*B>，<math|A*B*\<bot\>*A*C>，<math|E>是<math|P*B>的中点．

    (1)证明：<math|O*E//>平面<math|P*A*C>;

    (2)若<math|\<angle\>*A*B*O=\<angle\>*C*B*O=30*\<degree\>>，<math|P*O=3>，<math|P*A=5>，求二面角<math|C-A*E-B>正弦值．

    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E88F49E8FF8F6A813C7C701BFB0E450200AE5D4AC51D79853ADD13682384D7E113E8FE7C00FEBBCC0E3AAC022086484382DEE3FF8D9831FF06357520D1103A5D1D0141244FC0C2C21C1616E635BF6BE919C3C3A7B71A3D12500742774B40A011049108083482201201814610442220D0089A959595EAF64140A05DA3999DFBA7BA7D101068D768EED81E214CF60B083C81FF074C0224CFBD4507310000000049454E44AE426082>|png>|101pt|60pt||>

    <vspace|5cm>

    <item>设双曲线<math|C:<frac|x<rsup|2>|a<rsup|2>>-<frac|y<rsup|2>|b<rsup|2>>=1*<around|(|a\<gtr\>0,b\<gtr\>0|)>>的右焦点为<math|F*<around|(|2,0|)>>，渐近线方程为<math|y=<sqrt|3>*x>.

    (1)求<math|C>的方程;

    (2)经过<math|F>的直线与<math|C>的渐近线分别交于<math|A>，<math|B>两点，点<math|P*<around|(|x<rsub|1>,y<rsub|1>|)>>，<math|Q<around|(|x<rsub|2>,y<rsub|2>|)>>在<math|C>上，且<math|x<rsub|1>\<gtr\>x<rsub|2>\<gtr\>0>，<math|y<rsub|1>\<gtr\>0>.过<math|P>且斜率为<math|-<sqrt|3>>的直线与过<math|Q>且斜率为<math|<sqrt|3>>的直线交于点<math|M>，从下面三个条件①②③中选择两个条件，证明另一个条件成立：<math-up|①><math|M>在<math|A*B>上<math|;<math-up|②>P*Q//A*B;<math-up|③><around|\||A*M|\|>=<around|\||B*M|\|>>．<vspace|5cm>

    <item>已知函数<math|f<around|(|x|)>=x*e<rsup|a*x>-e<rsup|x>>．

    (1)当<math|a=1>时，讨论<math|f<around|(|x|)>>的单调性;

    (2)当<math|x\<gtr\>0>时，<math|f<around|(|x|)>\<less\>-1>，求实数<math|a>的取值范围;

    (3)设<math|n\<in\>N<rsup|\<ast\>>>，证明：<math|<frac|1|<sqrt|1<rsup|2>+1>>+<frac|1|<sqrt|2<rsup|2>+2>>+\<cdots\>+<frac|1|<sqrt|n<rsup|2>+n>>\<gtr\><math-up|ln><around|(|n+1|)>>．<vspace|5cm>
  </enumerate-numeric>
</body>