mirror of https://gitee.com/XmacsLabs/planet.git
138 lines
983 KiB
Tcl
138 lines
983 KiB
Tcl
<TeXmacs|2.1.3>
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<style|<tuple|generic|std-latex|chinese>>
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<\body>
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<doc-data|<doc-title|2021年全国新高考II卷数学试题>>
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<\with|font-series|bold>
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一、单选题
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</with>
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1．复数<math|<frac|2-i|1-3*i>>在复平面内对应的点所在的象限为（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．第一象限>|<cell|B．第二象限>|<cell|C．第三象限>|<cell|D．第四象限>>>>>\
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2．设集合<math|U=<around|{|1,2,3,4,5,6|}>,A=<around|{|1,3,6|}>,B=<around|{|2,3,4|}>>，则<math|A\<cap\><around*|(|\<complement\><rsub|U>*B|)>=>（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|<around|{|3|}>>>|<cell|B．<math|<around|{|1,6|}>>>|<cell|C．<math|<around|{|5,6|}>>>|<cell|D．<math|<around|{|1,3|}>>>>>>>\
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3．抛物线<math|y<rsup|2>=2*p*x*<around|(|p\<gtr\>0|)>>的焦点到直线<math|y=x+1>的距离为<math|<sqrt|2>>，则<math|p=>（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．1>|<cell|B．2>|<cell|C．<math|2*<sqrt|2>>>|<cell|D．4>>>>>\
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4．北斗三号全球卫星导航系统是我国航天事业的重要成果．在卫星导航系统中，地球静止同步卫星的轨道位于地球赤道所在平面，轨道高度为<math|36000>km（轨道高度是指卫星到地球表面的距离）．将地球看作是一个球心为<em|O>，半径<em|r>为<math|6400>km的球，其上点<em|A>的纬度是指<math|O*A>与赤道平面所成角的度数．地球表面上能直接观测到一颗地球静止同步轨道卫星点的纬度最大值为<math|\<alpha\>>，记卫星信号覆盖地球表面的表面积为<math|S=2*\<pi\>*r<rsup|2>*<around|(|1-cos
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\<alpha\>|)>>（单位：km<rsup|<math|2>>），则<em|S>占地球表面积的百分比约为（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．26%>|<cell|B．34%>|<cell|C．42%>|<cell|D．50%>>>>>\
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5．正四棱台的上､下底面的边长分别为2，4，侧棱长为2，则其体积为（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|20+12*<sqrt|3>>>|<cell|B．<math|28*<sqrt|2>>>|<cell|C．<math|<frac|56|3>>>|<cell|D．<math|<frac|28*<sqrt|2>|3>>>>>>>\
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6．某物理量的测量结果服从正态分布<math|N<around*|(|10,\<sigma\><rsup|2>|)>>，下列结论中不正确的是（<space|1.2em>）
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A．<math|\<sigma\>>越小，该物理量在一次测量中在<math|<around|(|9.9,10.1|)>>的概率越大
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B．<math|\<sigma\>>越小，该物理量在一次测量中大于10的概率为0.5
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C．<math|\<sigma\>>越小，该物理量在一次测量中小于9.99与大于10.01的概率相等
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D．<math|\<sigma\>>越小，该物理量在一次测量中落在<math|<around|(|9.9,10.2|)>>与落在<math|<around|(|10,10.3|)>>的概率相等
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7．已知<math|a=log<rsub|5>2>，<math|b=log<rsub|8>3>，<math|c=<frac|1|2>>，则下列判断正确的是（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|c\<less\>b\<less\>a>>|<cell|B．<math|b\<less\>a\<less\>c>>|<cell|C．<math|a\<less\>c\<less\>b>>|<cell|D．<math|a\<less\>b\<less\>c>>>>>>\
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8．已知函数<math|f<around|(|x|)>>的定义域为<math|<math-bf|R>>，<math|f*<around|(|x+2|)>>为偶函数，<math|f*<around|(|2*x+1|)>>为奇函数，则（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|f*<around*|(|-<frac|1|2>|)>=0>>|<cell|B．<math|f*<around|(|-1|)>=0>>|<cell|C．<math|f<around|(|2|)>=0>>|<cell|D．<math|f<around|(|4|)>=0>>>>>>\
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<with|font-series|bold|二、多选题>
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9．下列统计量中，能度量样本<math|x<rsub|1>,x<rsub|2>,\<cdots\>,x<rsub|n>>的离散程度的是（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．样本<math|x<rsub|1>,x<rsub|2>,\<cdots\>,x<rsub|n>>的标准差>|<cell|B．样本<math|x<rsub|1>,x<rsub|2>,\<cdots\>,x<rsub|n>>的中位数>>|<row|<cell|C．样本<math|x<rsub|1>,x<rsub|2>,\<cdots\>,x<rsub|n>>的极差>|<cell|D．样本<math|x<rsub|1>,x<rsub|2>,\<cdots\>,x<rsub|n>>的平均数>>>>>\
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10．如图，在正方体中，<em|O>为底面的中心，<em|P>为所在棱的中点，<em|M>，<em|N>为正方体的顶点．则满足<math|M*N*\<bot\>*O*P>的是（<space|1.2em>）
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11．已知直线<math|l:a*x+b*y-r<rsup|2>=0>与圆<math|C:x<rsup|2>+y<rsup|2>=r<rsup|2>>，点<math|A*<around|(|a,b|)>>，则下列说法正确的是（<space|1.2em>）
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A．若点<em|A>在圆<em|C>上，则直线<em|l>与圆<em|C>相切\
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B．若点<em|A>在圆<em|C>内，则直线<em|l>与圆<em|C>相离
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C．若点<em|A>在圆<em|C>外，则直线<em|l>与圆<em|C>相离\
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D．若点<em|A>在直线<em|l>上，则直线<em|l>与圆<em|C>相切
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12．设正整数<math|n=a<rsub|0>\<cdot\>2<rsup|0>+a<rsub|1>\<cdot\>2+\<cdots\>+a<rsub|k-1>\<cdot\>2<rsup|k-1>+a<rsub|k>\<cdot\>2<rsup|k>>，其中<math|a<rsub|i>\<in\><around*|{|0,1|}>>，记<math|\<omega\><around|(|n|)>=a<rsub|0>+a<rsub|1>+\<cdots\>+a<rsub|k>>．则（<space|1.2em>）
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<tabular|<tformat|<twith|table-width|1par>|<twith|table-hmode|exact>|<table|<row|<cell|A．<math|\<omega\>*<around|(|2*n|)>=\<omega\><around|(|n|)>>>|<cell|B．<math|\<omega\>*<around|(|2*n+3|)>=\<omega\><around|(|n|)>+1>>>|<row|<cell|C．<math|\<omega\>*<around|(|8*n+5|)>=\<omega\>*<around|(|4*n+3|)>>>|<cell|D．<math|\<omega\>*<around*|(|2<rsup|n>-1|)>=n>>>>>>\
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<with|font-series|bold|三、填空题>
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13．若双曲线<math|<frac|x<rsup|2>|a<rsup|2>>-<frac|y<rsup|2>|b<rsup|2>>=1>的离心率为2，则此双曲线的渐近线方程<underline|<space|2cm>>．
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14．写出一个同时具有下列性质①②③的函数<math|f<around|(|x|)>>:
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<underline|<space|2cm>>．
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①<math|f*<around*|(|x<rsub|1>*x<rsub|2>|)>=f<around*|(|x<rsub|1>|)>*f<around*|(|x<rsub|2>|)>>；②当<math|x\<in\>
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<around*|(|0,+\<infty\>|)>>时，<math|f<rprime|'><around|(|x|)>\<gtr\>0>；③<math|f<rprime|'><around|(|x|)>>是奇函数．
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15．已知向量<math|<wide|a|\<wide-varrightarrow\>>+<wide|b|\<wide-varrightarrow\>>+<wide|c|\<wide-varrightarrow\>>=<wide|0|\<wide-varrightarrow\>>>，<math|<around*|\||<wide|a|\<wide-varrightarrow\>>|\|>=1>，<math|<around*|\||<wide|b|\<wide-varrightarrow\>>|\|>=<around*|\||<wide|c|\<wide-varrightarrow\>>|\|>=2>，<math|<wide|a|\<wide-varrightarrow\>>\<cdot\><wide|b|\<wide-varrightarrow\>>+<wide|b|\<wide-varrightarrow\>>\<cdot\><wide|c|\<wide-varrightarrow\>>+<wide|c|\<wide-varrightarrow\>>\<cdot\><wide|a|\<wide-varrightarrow\>>=><underline|<space|2cm>>．
|
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16．已知函数<math|f<around|(|x|)>=<around*|\||e<rsup|x>-1|\|>,x<rsub|1>\<less\>0,x<rsub|2>\<gtr\>0>，函数<math|f<around|(|x|)>>的图象在点<math|A*<around*|(|x<rsub|1>,f<around*|(|x<rsub|1>|)>|)>>和点<math|B*<around*|(|x<rsub|2>,f<around*|(|x<rsub|2>|)>|)>>的两条切线互相垂直，且分别交<em|y>轴于<em|M>，<em|N>两点，则<math|<frac|<around|\||A*M|\|>|<around|\||B*N|\|>>>取值范围是<underline|<space|2cm>>．
|
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|
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<with|font-series|bold|四、解答题>
|
||
|
||
17．记<math|S<rsub|n>>是公差不为0的等差数列<math|<around*|{|a<rsub|n>|}>>的前<em|n>项和，若<math|a<rsub|3>=S<rsub|5>,a<rsub|2>*a<rsub|4>=S<rsub|4>>．
|
||
|
||
（1）求数列<math|<around*|{|a<rsub|n>|}>>的通项公式<math|a<rsub|n>>；
|
||
|
||
（2）求使<math|S<rsub|n>\<gtr\>a<rsub|n>>成立的<em|n>的最小值．<vspace|3cm>
|
||
|
||
18．在<math|\<triangle\>A*B*C>中，角<math|A>、<math|B>、<math|C>所对的边长分别为<math|a>、<math|b>、<math|c>，<math|b=a+1>，<math|c=a+2>.
|
||
|
||
（1）若<math|2*sin C=3*sin A>，求<math|\<triangle\>A*B*C>的面积；
|
||
|
||
（2）是否存在正整数<math|a>，使得<math|\<triangle\>A*B*C>为钝角三角形?若存在，求出<math|a>的值；若不存在，说明理由．<vspace|3cm>
|
||
|
||
19．在四棱锥<math|Q-A*B*C*D>中，底面<math|A*B*C*D>是正方形，若<math|A*D=2,Q*D=Q*A=<sqrt|5>,Q*C=3>．
|
||
|
||
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|png>|.2par|||>>
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（1）证明：平面<math|Q*A*D*\<bot\>>平面<math|A*B*C*D>；
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（2）求二面角<math|B-Q*D-A>的平面角的余弦值．<vspace|3cm>
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20．已知椭圆<em|C>的方程为<math|<frac|x<rsup|2>|a<rsup|2>>+<frac|y<rsup|2>|b<rsup|2>>=1*<around|(|a\<gtr\>b\<gtr\>0|)>>，右焦点为<math|F<around|(|<sqrt|2>,0|)>>，且离心率为<math|<frac|<sqrt|6>|3>>．
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（1）求椭圆<em|C>的方程；
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（2）设<em|M>，<em|N>是椭圆<em|C>上的两点，直线<math|M*N>与曲线<math|x<rsup|2>+y<rsup|2>=b<rsup|2>*<around|(|x\<gtr\>0|)>>相切．证明：<em|M>，<em|N>，<em|F>三点共线的充要条件是<math|<around|\||M*N|\|>=<sqrt|3>>．<vspace|3cm>
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21．一种微生物群体可以经过自身繁殖不断生存下来，设一个这种微生物为第0代，经过一次繁殖后为第1代，再经过一次繁殖后为第2代......，该微生物每代繁殖的个数是相互独立的且有相同的分布列，设<em|X>表示1个微生物个体繁殖下一代的个数，<math|P*<around|(|X=i|)>=p<rsub|i>*<around|(|i=0,1,2,3|)>>．
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（1）已知<math|p<rsub|0>=0.4,p<rsub|1>=0.3,p<rsub|2>=0.2,p<rsub|3>=0.1>，求<math|E<around|(|X|)>>；
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（2）设<em|p>表示该种微生物经过多代繁殖后临近灭绝的概率，<em|p>是关于<em|x>的方程：<math|p<rsub|0>+p<rsub|1>*x+p<rsub|2>*x<rsup|2>+p<rsub|3>*x<rsup|3>=x>的一个最小正实根，求证：当<math|E<around|(|X|)>\<leq\>1>时，<math|p=1>，当<math|E<around|(|X|)>\<gtr\>1>时，<math|p\<less\>1>；
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（3）根据你的理解说明（2）问结论的实际含义．<vspace|3cm>
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22．已知函数<math|f<around|(|x|)>=<around|(|x-1|)>*e<rsup|x>-a*x<rsup|2>+b>．
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（1）讨论<math|f<around|(|x|)>>的单调性；
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（2）从下面两个条件中选一个，证明：<math|f<around|(|x|)>>只有一个零点.
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①<math|<frac|1|2>\<less\>a\<leq\><frac|e<rsup|2>|2>,b\<gtr\>2*a>；
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②<math|0\<less\>a\<less\><frac|1|2>,b\<leq\>2*a>．<vspace|3cm>
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</body> |