一、单项选择题:本题共8道小题,每小题5分,共40分.在每小题给出的四个选项中,只有一项是符合题目要求的.
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1.
已知A= {3,4,5,6},B={2≤x<6},则A∩B=( )
A . {2,3,4}
B . {3,4,5}
C . {2,3,4,5}
D . {3,4,5,6}
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2.
复数z的共轭复数是1+
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
i(其中i为虚数单位),则z的虛部是( )
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3.
已知向量
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E)
=(-3,1),
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmo%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E)
=(1,-2),则向量
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E)
与
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmo%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmath%3E)
夹角的大小为( )
A . 30°
B . 45°
C . 60°
D . 135°
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4.
已知一组数据如下:1,2,5,6,11,则该组数据的方差为( )
A . 12.4
B . 12.3
C . 12.2
D . 12.1
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5.
已知sin2α=cos(
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmi%3E%CF%80%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
+α),α∈(
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmi%3E%CF%80%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,π),则tanα的值为( )
A . -
B . -1
C .
D . -2
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6.
在某个时期,某湖泊中的蓝藻每天以6.25%的增长率呈指数增长,已知经过30天以后,该湖泊的蓝藻数大约为原来的6倍,那么经过60天后该湖泊的蓝藻数大约为原来的( )
A . 18倍
B . 24倍
C . 36倍
D . 48倍
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7.
已知函数f(x)=sinx+
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
cosx,则“x
0=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmi%3E%CF%80%3C%2Fmi%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
"是“f(x)在x=x
0处取得最大值”的( )
A . 充分不必要条件
B . 必要不充分条件
C . 充要条件
D . 既不充分也不必要条件
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8.
已知实数a,b,c满足a>b>0>c,则下列不等式中成立的是( )
二、多选题:本题共4小题,每小题5分,共20分。在每小题给出的选项中,有多项符合题目要求。全部选对的得5分,部分选对的得2分,有选错的得0分。
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9.
人口普查是世界各国广泛采用的一种搜集人口资料的方法,根据人口普查可以科学地研究制定社会、经济、科教等各项发展政策.下图是我国七次人口普查的全国人口及年均增长率情况,则下列说法正确的是( )
A . 年均增长率逐次减小
B . 第二次至第七次普查的人口年均增长率的极差是1.56%
C . 这七次普查的人口数逐次增加,且第四次增幅最小
D . 第七次普查的人口数最多,且第三次增幅最大
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11.
已知实数x,y,z满足2
x=log
2y=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmi%3Ez%3C%2Fmi%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,则下列关系式中可能成立的是( )
A . y=z>x
B . z=x>y
C . y>z>x
D . z>y>x
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12.
如图,在四面体ABCD中,AB=CD=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,AC=AD=BC=BD=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,若用一个与AB,CD都平行的平面α截该四面体,下列说法中正确的是( )
A . 异面直线AB与CD所成的角为90°
B . 平面α截四面体ABCD所得截面周长不变
C . 平面α截四面体ABCD所得截面不可能为正方形
D . 该四面体的外接球表面积为6π
三、填空题:本题共4小题,每小题5分,共20分.
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13.
求值:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsup%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3C%2Fmath%3E)
+log
48+e
ln2=
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14.
甲、乙、丙三名射击运动员中靶概率分别为0.8、0.9、0.7,每人各射击一次,三人中靶与否互不影响,则三人中至少有一人中靶的概率为
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15.
如图,在边长为2的正方体ABCD-A
1B
1C
1D
1中,点E,F分别为AD,AB的中点,则直线EF与平面BCD
1所成角的大小为
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16.
已知函数f(x)=ln(x
2+1),g(x)=4(m-1)sin(2x+
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmi%3E%CF%80%3C%2Fmi%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
)-m
2 , 若x
1 ![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%80%3C%2Fmo%3E%3C%2Fmath%3E)
∈[-1,3],
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%80%3C%2Fmo%3E%3C%2Fmath%3E)
x
2∈[0,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmi%3E%CF%80%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
],f(x
1)≥g(x
2),则m的取值范围是
四、解答题:本题共6小题,共70分.解答应写出文字说明、证明过程或演算步骤。
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17.
已知z,z1 , z2均为复数,在复平面内,z1对应的点的坐标为(3,4),z2对应的向量坐标为(0,1),且zz1=-1+7i(其中i为虚数单位)。
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-
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18.
在△ABC中,角A,B,C的对边分别为a,b,c,(sinA-sinC)2=sin2B-sinAsinC。
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(2)
若b=1,△ABC的面积为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,求△ABC的周长。
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19.
某校高一年级学生打算利用周六休息时间做义工,为了了解高一年级学生做义工时长的情况,随机抽取了高一年级100名学生进行调查,将收集到的做义工时间(单位:小时)数据分成6组:[0,1),[,2), [2,3),[3,4),[4,5),[5,6],(时间均在[0,6]内),已知上述时间数据的第70百分位数为3.5
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(1)
求m,n的值,并估计这100位学生做义工时间的平均值(同一组中的数据用该组区间的中点值作代表);
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(2)
现从第二组,第四组中,采用按比例分层抽样的方法抽取6人,再从6人中随机抽取2人,求两个人来自于不同组的概率。
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20.
如图,在△ABC中,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3ED%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3EB%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,点E为AC中点,点F为BC的三等分点,且靠近点C,设
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmi%3EB%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmstyle+mathvariant%3D%22bold%22+mathsize%3D%22normal%22%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmstyle%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmi%3EA%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
=
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmstyle+mathvariant%3D%22bold%22+mathsize%3D%22normal%22%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmstyle%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
。
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-
(2)
如果∠ACB=60°,AC=2,且CD⊥EF,求|
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmi%3ED%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
|。
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21.
如图1所示,在矩形ABCD中,AB=4,BC=6
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsqrt%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,点E为线段AB上一点,AE=1,现将△BCE沿CE折起,将点B折到点B'位置,使得点B'在平面AECD上的射影在线段AD上,得到如图2所示的四棱锥B'-AECD
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(1)
在图2中,线段B'C上是否存在点F,使得EF∥平面B'AD?若存在,求
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%27%3C%2Fmo%3E%3Cmi%3EF%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%27%3C%2Fmo%3E%3Cmi%3EC%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的值,若不存在,请说明理由;
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22.
已知函数f(x)=|ex-1|
-
(1)
试判断函数f(x)的单调性,并画出函数f(x)图象的草图;
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(2)
若关于x的方程2f2(x)-4mf(x)+5m-2=0有两个不相等的实数根,求m的取值范围。