planet/高考数学/2022年全国高考甲卷数学理科试题.tm

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  <doc-data|<doc-title|2022年数学全国甲卷（理科）>>

  <strong|一、单选题>（本大题共12小题，共60分）

  <\enumerate-numeric>
    <item>若<math|z=-1+<sqrt|3><math-up|i>>，则<math|<frac|z|z<wide|z|\<bar\>>-1>=>

    <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|-1+<sqrt|3><math-up|i>>>|<cell|B.
    <math|-1-<sqrt|3><math-up|i>>>|<cell|C.
    <math|-<frac|1|3>+<frac|<sqrt|3>|3><math-up|i>>>|<cell|D.
    <math|-<frac|1|3>-<frac|<sqrt|3>|3><math-up|i>>>>>>>

    <item>某社区通过公益讲座以普及社区居民的垃圾分类知识．为了解讲座效果，随机抽取<math|10>位社区居民，让他们在讲座前和讲座后各回答一份垃圾分类知识问卷，这<math|10>位社区居民在讲座前和讲座后问卷答题的正确率如下图：

    <image|<tuple|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

    则

    A. 讲座前问卷答题的正确率的中位数小于<math|70><with|font-family|rm|％>

    B. 讲座后问卷答题的正确率的平均数大于<math|85><with|font-family|rm|％>

    C. 讲座前问卷答题的正确率的标准差小于讲座后正确率的标准差

    D. 讲座后问卷答题的正确率的极差大于讲座前正确率的极差

    <item>设全集<math|U=<around|{|-2,-1,0,1,2,3|}>>，集合<math|A=<around|{|-1,2|}>,B=<around*|{|x\<mid\>x<rsup|2>-4*x+3=0|}>>，则<math|\<complement\><rsub|U>*<around|(|A\<cup\>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|{|1,3|}>>>|<cell|B. <math|<around|{|0,3|}>>>|<cell|C.
    <math|<around|{|-2,1|}>>>|<cell|D. <math|<around|{|-2,0|}>>>>>>>

    <item>如图，网格纸上绘制的是一个多面体的三视图，网格小正方形的边长为<math|1>，则该多面体的体积为

    <image|<tuple|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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|8>>|<cell|B. <math|12>>|<cell|C. <math|16>>|<cell|D. <math|20>>>>>>

    <item>函数<math|y=<around*|(|3<rsup|x>-3<rsup|-x>|)><math-up|cos>
    x>在区间<math|<around*|[|-<frac|\<pi\>|2>,<frac|\<pi\>|2>|]>>的图象大致为

    A.<image|<tuple|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
    B.<image|<tuple|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

    C. <image|<tuple|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

    <item>当<math|x=1>时，函数<math|f<around|(|x|)>=a<math-up|ln>x+<frac|b|x>>取得最大值<math|-2>，则<math|f<rprime|'><around*|(|2|)>=>

    <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|-1>>|<cell|B. <math|-<frac|1|2>>>|<cell|C.
    <math|<frac|1|2>>>|<cell|D. <math|1>>>>>>

    <item>在长方体<math|A*B*C*D-A<rsub|1>*B<rsub|1>*C<rsub|1>*D<rsub|1>>中，已知<math|B<rsub|1>*D>与平面<math|A*B*C*D>和平面<math|A*A<rsub|1>*B<rsub|1>*B>所成的角均为<math|30<rsup|\<circ\>>>，则

    A. <math|A*B=2*A*D>

    B. <math|A*B>与平面<math|A*B<rsub|1>*C<rsub|1>*D>所成的角为<math|30<rsup|\<circ\>>>

    C. <math|A*C=C*B<rsub|1>>

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

    <item>沈括的<with|font-family|rm|《>梦溪笔谈<with|font-family|rm|》>是中国古代科技史上的杰作，其中收录了计算圆弧长度的``会圆术''，如图，<math|<wide|A*B|\<invbreve\>>>是以<math|O>为圆心，<math|O*A>为半径的圆弧，<math|C>是的<math|A*B>中点，<math|D>在<math|<wide|A*B|\<invbreve\>>>上，``会圆术''给出<math|<wide|A*B|\<invbreve\>>>的弧长的近似值<math|s>的计算公式：<math|s=A*B+<frac|C*D<rsup|2>|O*A>>.当<math|O
    A=2,\<angle\>A O B=60<rsup|\<circ\>>>时，<math|s=>

    <image|<tuple|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

    <tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A.
    <math|<frac|11-3*<sqrt|3>|2>>>|<cell|B.
    <math|<frac|11-4*<sqrt|3>|2>>>|<cell|C.
    <math|<frac|9-3*<sqrt|3>|2>>>|<cell|D. <math|<frac|9-4*<sqrt|3>|2>>>>>>>

    <item>甲、乙两个圆锥的母线长相等，侧面展开图的圆心角之和为<math|2*\<pi\>>，侧面积分别为<math|S<rsub|<math-up|甲>>>和<math|S<rsub|<math-up|乙>>>，体积分别为<math|V<rsub|<math-up|甲>>>和<math|V<rsub|<math-up|乙>>>.若<math|<frac|S<rsub|<math-up|甲>>|S<rsub|<math-up|乙>>><with|math-font-family|rm|=>2>，则<math|<frac|V<rsub|<math-up|甲>>|V<rsub|<math-up|乙>>><with|math-font-family|rm|=>>

    <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|<sqrt|5>>>|<cell|B. <math|2*<sqrt|2>>>|<cell|C.
    <math|<sqrt|10>>>|<cell|D. <math|<frac|5*<sqrt|10>|4>>>>>>>

    <item>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|)>>的左顶点为<math|A>，点<math|P>，<math|Q>均在<math|C>上，且关于<math|y>轴对称．若直线<math|A*P,A*Q>的斜率之积为<math|<frac|1|4>>，则<math|C>的离心率为

    <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|<frac|<sqrt|3>|2>>>|<cell|B. <math|<frac|<sqrt|2>|2>>>|<cell|C.
    <math|<frac|1|2>>>|<cell|D. <math|<frac|1|3>>>>>>>

    <item>设函数<math|f<around|(|x|)>=<math-up|sin><around*|(|\<omega\>*x+<frac|\<pi\>|3>|)>>在区间<math|<around|(|0,\<pi\>|)>>恰有三个极值点、两个零点，则<math|\<omega\>>的取值范围是

    <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*|[|<frac|5|3>,<frac|13|6>|)>>>|<cell|B.
    <math|<around*|[|<frac|5|3>,<frac|19|6>|)>>>|<cell|C.
    <math|<around*|(|<frac|13|6>,<frac|8|3>|]>>>|<cell|D.
    <math|<around*|(|<frac|13|6>,<frac|19|6>|]>>>>>>>

    <item>已知<math|a=<frac|31|32>,b=<math-up|cos><frac|1|4>,c=4<math-up|sin><frac|1|4>>，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|c\<gtr\>b\<gtr\>a>>|<cell|B. <math|b\<gtr\>a\<gtr\>c>>|<cell|C.
    <math|a\<gtr\>b\<gtr\>c>>|<cell|D. <math|a\<gtr\>c\<gtr\>b>>>>>>
  </enumerate-numeric>

  二、填空题（本大题共4小题，共20分）

  <\enumerate-numeric>
    <assign|item-nr|12><item>设向量<math|<wide|a|\<wide-varrightarrow\>>>，<math|<wide|b|\<wide-varrightarrow\>>>的夹角的余弦值为<math|<frac|1|3>>，且<math|<around*|\||<wide|a|\<wide-varrightarrow\>>|\|>=1>，<math|<around*|\||<wide|a|\<wide-varrightarrow\>>|\|>=3>，则<math|<around*|(|2*<wide|a|\<wide-varrightarrow\>>+<wide|b|\<wide-varrightarrow\>>|)>\<cdot\><wide|b|\<wide-varrightarrow\>>=><underline|<space|2cm>>．

    <item>若双曲线<math|y<rsup|2>-<frac|x<rsup|2>|m<rsup|2>>=1*<around|(|m\<gtr\>0|)>>的渐近线与圆<math|x<rsup|2>+y<rsup|2>-4*y+3=0>相切，则<math|m=><underline|<space|2cm>>．

    <item>从正方体的<math|8>个顶点中任选<math|4>个，则这<math|4>个点在同一个平面的概率为
    <underline|<space|2cm>>．

    <item>已知<math|\<vartriangle\>*A*B*C>中，点<math|D>在边<math|B*C>上，<math|\<angle\>*A*D*B=120<rsup|\<circ\>>,A*D=2,C*D=2*B*D>.当<math|<frac|A*C|A*B>>取得最小值时，<math|B*D=><underline|<space|2cm>>．
  </enumerate-numeric>

  三、解答题（本大题共7小题，共80分）

  \ (一) \ 必考题：共 60 分.

  <\enumerate-numeric>
    <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|<frac|2*S<rsub|n>|n>+n=2*a<rsub|n>+1>．

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

    (2)若<math|a<rsub|4>,a<rsub|7>,a<rsub|9>>成等比数列，求<math|S<rsub|n>>的最小值．<vspace|5cm>

    <item>在四棱锥<math|P-A*B*C*D>中，<math|P*D*\<bot\>>底面<math|A*B*C*D,C*D//A*B,A*D=D*C=C*B=1,A*B=2,D*P=<sqrt|3>>．

    (1)证明：<math|B*D*\<bot\>*P*A>；

    (2)求<math|P*D>与平面<math|P*A*B>所成的角的正弦值．

    <image|<tuple|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

    <item>甲、乙两个学校进行体育比赛，比赛共设三个项目，每个项目胜方得<math|10>分，负方得<math|0>分，没有平局．三个项目比赛结束后，总得分高的学校获得冠军．已知甲学校在三个项目中获胜的概率分别为<math|5>，<math|0.4>，<math|0.8>，各项目的比赛结果相互独立．

    (1)求甲学校获得冠军的概率；

    (2)用<math|X>表示乙学校的总得分，求<math|X>的分布列与期望．<vspace|5cm>

    <item>设抛物线<math|C:y<rsup|2>=2*p*x*<around|(|p\<gtr\>0|)>>的焦点为<math|F>，点<math|D*<around|(|p,0|)>>，过<math|F>的直线交<math|C>于<math|M>，<math|N>两点．当直线<math|M*D>垂直于<math|x>轴时，<math|<around|\||M*F|\|>=3>．

    (1)求<math|C>的方程；

    (2)设直线<math|M*D,N*D>与<math|C>的另一个交点分别为<math|A>，<math|B>，记直线<math|M*N,A*B>的倾斜角分别为<math|\<alpha\>,\<beta\>>.当<math|\<alpha\>-\<beta\>>取得最大值时，求直线<math|A*B>的方程．<vspace|5cm>

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

    (1)若<math|f<around*|(|x|)>\<ge\>0>，求<math|a>的取值范围；

    (2)证明：若<math|f<around*|(|x|)>>有两个零点<math|x<rsub|1>,x<rsub|2>>，则<math|x<rsub|1>*x<rsub|2>\<less\>1>．<vspace|5cm>

    \;
  </enumerate-numeric>

  (二) 选考题：共 10 分

  <\enumerate>
    <assign|item-nr|21><item>在直角坐标系<math|x*O*y>中，曲线<math|C<rsub|1>>的参数方程为<math|<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|1|1|cell-rborder|0ln>|<table|<row|<cell|x=<frac|2+t|6>>>|<row|<cell|y=<sqrt|t>>>>>>|\<nobracket\>>(t>为参数<math|>)，曲线<math|C<rsub|2>>的参数方程为<math|<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|1|1|cell-rborder|0ln>|<table|<row|<cell|x=-<frac|2+s|6>>>|<row|<cell|y=-<sqrt|s>>>>>>|\<nobracket\>>(s>为参数<math|>)．

    (1)写出<math|C<rsub|1>>的普通方程；

    (2)以坐标原点为极点，<math|x>轴正半轴为极轴建立极坐标系，曲线<math|C<rsub|3>>的极坐标方程为<math|2<math-up|cos>\<theta\>-<math-up|sin>\<theta\>=0>，求<math|C<rsub|3>>与<math|C<rsub|1>>交点的直角坐标，及<math|C<rsub|3>>与<math|C<rsub|2>>交点的直角坐标．<vspace|5cm>

    <item>已知<math|a>，<math|b>，<math|c>均为正数，且<math|a<rsup|2>+b<rsup|2>+4*c<rsup|2>=3>，证明：

    (1)<math|a+b+2*c\<le\>3>；

    (2)若<math|b=2*c>，则<math|<frac|1|a>+<frac|1|c>\<ge\>3>．<vspace|5cm>
  </enumerate>
</body>