一、填空题(本大题共12题,满分54分,第1-6题每题4分,第7-12题每题5分)
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3.
已知
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3EZ%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmath%3E)
, 则
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmi%3EZ%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%C2%AF%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%3D%3C%2Fmo%3E%3C%2Fmath%3E)
.
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5.
三角形
ABC中,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmtext%3EB%3C%2Fmtext%3E%3Cmtext%3EC%3C%2Fmtext%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E%2C%3C%2Fmn%3E%3Cmtext%3EA%3C%2Fmtext%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmtext%3E%CF%80%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmn%3E%2C%3C%2Fmn%3E%3Cmtext%3EB%3C%2Fmtext%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmtext%3E%CF%80%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
, 则
AB=
.
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7.
数列{an},an=n+c , S7<0,c的取值范围为 .
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8.
三角形三边长为5,6,7,则以边长为6的两个顶点为焦点,过另外一个顶点的双曲线的离心率为 .
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9.
已知
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E%29%3C%2Fmn%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmn%3E%2C%3C%2Fmn%3E%3Cmi%3Eg%3C%2Fmi%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E%29%3C%2Fmn%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmrow%3E%3Cmo%3E%7B%3C%2Fmo%3E%3Cmrow%3E%3Cmtable+columnalign%3D%22left%22%3E%3Cmtr+columnalign%3D%22left%22%3E%3Cmtd+columnalign%3D%22left%22%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E%29%3C%2Fmn%3E%3Cmn%3E%2C%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmtext%3E%E2%A9%BE%3C%2Fmtext%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr+columnalign%3D%22left%22%3E%3Cmtd+columnalign%3D%22left%22%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E%29%3C%2Fmn%3E%3Cmn%3E%2C%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 求
g(
x)≤2﹣
x的
x的取值范围
.
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10.
已知四棱柱
ABCD﹣
A1B1C1D1底面
ABCD为平行四边形,
AA1=3,
BD=4且
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmtext%3EA%3C%2Fmtext%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EB%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmtext%3EB%3C%2Fmtext%3E%3Cmtext%3EC%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmtext%3EA%3C%2Fmtext%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3ED%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmtext%3ED%3C%2Fmtext%3E%3Cmtext%3EC%3C%2Fmtext%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%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmath%3E)
, 求异面直线
AA1与
BD的夹角
.
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11.
正方形草地
ABCD边长1.2,
E到
AB ,
AD距离为0.2,
F到
BC ,
CD距离为0.4,有个圆形通道经过
E ,
F , 且与
AD只有一个交点,求圆形通道的周长
. (精确到0.01)
![](//tikupic.21cnjy.com/2024/06/11/9d/1f/9d1fae63986a92ddcb6dcdf5bf867294.png)
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12.
a1=2,a2=4,a3=8,a4=16,任意b1 , b2 , b3 , b4∈R,满足{ai+aj|1≤i<j≤4}={bi+bj|1≤i<j≤4},求有序数列{b1 , b2 , b3 , b4}有对.
二、选择题(本大题共4题,满分18分,第13-14题每题4分,第15-16题每题5分)
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13.
a , b , c∈R,b>c , 下列不等式恒成立的是( )
A . a+b2>a+c2
B . a2+b>a2+c
C . ab2>ac2
D . a2b>a2c
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14.
空间中有两个不同的平面α,β和两条不同的直线m , n , 则下列说法中正确的是( )
A . 若α⊥β,m⊥α,n⊥β,则m⊥n
B . 若α⊥β,m⊥α,m⊥n , 则n⊥β
C . 若α∥β,m∥α,n∥β,则m∥n
D . 若α∥β,m∥α,m∥n , 则n∥β
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15.
有四种礼盒,前三种里面分别仅装有中国结、记事本、笔袋,第四个礼盒里面三种礼品都有,现从中任选一个盒子,设事件A:所选盒中有中国结,事件B:所选盒中有记事本,事件C:所选盒中有笔袋,则( )
A . 事件A与事件B互斥
B . 事件A与事件B相互独立
C . 事件A与事件B∪C互斥
D . 事件A与事件B∩C相互独立
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16.
现定义如下:当
x∈(
n ,
n+1)时(
n∈N),若
f(
x+1)=
f'(
x),则称
f(
x)为延展函数.现有,当
x∈(0,1)时,
g(
x)=
ex与
h(
x)=
x10均为延展函数,则以下结论( )
①存在y=kx+b(k , b∈R;k , b≠0)与y=g(x)有无穷个交点
②存在y=kx+b(k , b∈R;k , b≠0)与y=h(x)有无穷个交点
A . ①②都成立
B . ①②都不成立
C . ①成立②不成立
D . ①不成立②成立
三、解答题(本大题共5题,共14+14+14+18+18=78分)
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17.
已知
f(
x)=sin(ω
x+
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmtext%3E%CF%80%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
),ω>0.
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(1)
设ω=1,求解:y=f(x),x∈[0,π]的值域;
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(2)
a>π(a∈R),f(x)的最小正周期为π,若在x∈[π,a]上恰有3个零点,求a的取值范围.
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18.
如图,
PA、
PB、
PC为圆锥三条母线,
AB=
AC .
![](//tikupic.21cnjy.com/2024/06/17/41/64/41644b35eda1d453f8f0283466a65156_m_101x121.png)
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(2)
若圆锥侧面积为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsqrt%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsqrt%3E%3Cmtext%3E%CF%80%3C%2Fmtext%3E%3C%2Fmath%3E)
, BC为底面直径,
BC=2,求二面角
B﹣
PA﹣
C的大小.
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19.
水果分为一级果和二级果,共136箱,其中一级果102箱,二级果34箱.
-
(1)
随机挑选两箱水果,求恰好一级果和二级果各一箱的概率;
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(2)
进行分层抽样,共抽8箱水果,求一级果和二级果各几箱;
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(3)
抽取若干箱水果,其中一级果共120个,单果质量平均数为303.45克,方差为603.46;二级果48个,单果质量平均数为240.41克,方差为648.21;求168个水果的方差和平均数,并预估果园中单果的质量.
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20.
在平面直角坐标系
xOy中,已知点
A为椭圆
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmtext%3E%CE%93%3C%2Fmtext%3E%3Cmn%3E%3A%3C%2Fmn%3E%3Cmfrac%3E%3Cmrow%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmath%3E)
上一点,
F1、
F2分别为椭圆的左、右焦点.
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-
(2)
设Γ的上、下顶点分别为M1、M2 , 记△AF1F2的面积为S1 , △AM1M2的面积为S2 , 若S1≥S2 , 求|OA|的取值范围.
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(3)
若点
A在
x轴上方,设直线
AF2与Γ交于点
B , 与
y轴交于点
K ,
KF1延长线与Γ交于点
C , 是否存在
x轴上方的点
C , 使得
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EA%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EB%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EC%3C%2Fmtext%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%3Cmtext%3E%CE%BB%3C%2Fmtext%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EA%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EB%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmover+accent%3D%22true%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmrow%3E%3Cmtext%3EF%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmtext%3EC%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmo+stretchy%3D%22true%22%3E%E2%86%92%3C%2Fmo%3E%3C%2Fmover%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmtext%3E%CE%BB%3C%2Fmtext%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E%29%3C%2Fmn%3E%3C%2Fmath%3E)
成立?若存在,请求出点
C的坐标;若不存在,请说明理由.
-
21.
记M(a)={t|t=f(x)﹣f(a),x≥a},L(a)={t|t=f(x)﹣f(a),x≤a}.
-
(1)
若f(x)=x2+1,求M(1)和L(1);
-
(2)
若f(x)=x3﹣3x2 , 求证:对于任意a∈R,都有M(a)⊆[﹣4,+∞),且存在a , 使得﹣4∈M(a).
-
(3)
已知定义在R上f(x)有最小值,求证“f(x)是偶函数“的充要条件是“对于任意正实数c , 均有M(﹣c)=L(c)”.