浙江省湖州中学2014届高三上学期期中考试数学理试题 Word版含答案
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试 卷
一、选择题：本大题共10小题，每小题5分，共50分. 1．全集{0,1,2,3}U =，{2}U C M =，则集合M =（ ）
A ．{0，1，3}
B ．{1，3}
C ．{0，3}
D ．{2}
2．若函数()f x （x R ∈）是奇函数，函数()g x （x R ∈）是偶函数，则（ ）
A ．函数()()f x g x +是奇函数
B ．函数()()f x g x ?是奇函数
C ．函数[()]f g x 是奇函数
D ． 函数[()]g f x 是奇函数 3. 下列函数中，图像的一部分如右图所示的是（ ） A ．sin()6
y x π
=+
B ．sin(2)6
y x π
=-
C ．cos(2)6y x π
=-
D ．cos(4)3
y x π
=-
4．等比数列{a n }的前n 项和为S n ，若S 2n ＝3(a 1＋a 3＋…＋a 2n －1)，a 1a 2a 3＝8，则a 10等于（ ）
A ．－1024
B ．1024
C ．－512
D ．512
5．已知函数2
()f x x bx =+的图象在点A (1，f (1))处的切线的斜率为3，数列1()f n ??
????
的前n 项
和为n S ，则2013S 的值为（ ）
A ．
2010
2011
B ．
20112012 C ．2012
2013
D ．
2013
2014
6．若实数x ，y 满足不等式组2402300x y x y x y +-≥--≥-≥??
???
， 则x ＋y 的最小值是（ ）
A ．43
B ．3
C ． 4
D ． 6
第3题图
8．命题p ：“1≠x 或3y ≠”是命题q ：“4≠+y x ”的（ ）条件
A .充分不必要
B .必要不充分
C .充要
D .既不充分也不必要
9．如图，半圆的直径6AB =，O 为圆心，C 为半圆上不同于A 、B 的任意一点，若P 为半
径OC 上的动点，则()
PA PB PC +的最小值为（ ）
A ．92
B ．9
C ．92
- D ．－9
10．若函数3
2
()f x x ax bx c =+++有两个极值点12,x x ,且11()f x x =,则关于x 的方程
23(())2()0f x af x b ++=的不同实根个数是（ ）
A ．3
B ．4
C ．5
D ．6
二、填空题：本大题共7小题，每小题4分，共28分. 11. 不等式2
(2)211x x -≤+的解集为 ____.
12. 已知数列{}n a 满足11a =，12n n n a a +=+，则10a =_________. 13．在ABC ?中，,,a b c 分别是内角,,A B C 的对边，已知1
6,4,cos 3
a c B ===，则____
b =. 14. 已知41)4
sin(=
+
π
θ， ),23(ππθ--∈，则)127cos(πθ+的值为________．
15. 若)4
sin(3)4
sin()(π
π
-
++=x x a x f 是偶函数，则=a .
16. 函数2
1()2ln 2
f x x x x a =+-+在区间(0,2)上恰有一个零点，则实数a 的取值范围是_____.
17. 已知函数2
()|21|f x x x =+-，若1a b <<-且()()f a f b =，则ab a b ++的取值范围_____.
O
P C B
A
浙江省湖州中学
2013学年第一学期高三期中考试
数 学 答 卷（理）
一、选择题：本大题共10小题，每小题5分，共50分.
二、填空题：本大题共7小题，每小题4分，共28分.
11___________________ 12_________________ 13______________________
14___________________ 15_________________ 16______________________
17___________________
三、解答题：第18、19、20题每题14分，第21、22每题15分，共72分．
18．已知 1:(),3
x
p f x -=且|()|2f a <；
:q 集合{
}
2(2)10,A x x a x x R =+++=∈，{}0B x x =>且 A B =?.
若p ∨q 为真命题，p ∧q 为假命题，求实数a 的取值范围.
·································································································································· ··································································································································
班级 学号______ 姓名 试
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19．已知函数2()sin
2cos 24
x x f x =+. （1）写出如何由函数sin y x =的图像变换得到()f x 的图像；
（2）在ABC ?中，角A B C 、、所对的边分别是a b c 、、，若C b B c a cos cos )2(=-，求
)(A f 的取值范围.
20．已知函数2
()(x
x
f x a x a =+∈R ，1)a >， （1）求函数f (x )的值域；
（2）记函数()(),[2,)g x f x x =-∈-+∞，若()g x 的最小值与a 无关，求a 的取值范围；
（3）若m >，直接写出（不需给出演算步骤）关于x 的方程()f x m =的解集．
21.已知数列{}n a 的前n 项和11()22
n n n S a -=--+（n 为正整数）．
（1）令2n n n b a =，求证数列{}n b 是等差数列，并求数列{}n a 的通项公式；
（2）令1n n n c a n +=，12 n n T c c c =+++，试比较n T 与
521
n
n +的大小，并予以证明．
·································································································································· ··································································································································
班级 学号______ 姓 试
·································································································································· ·································································································································· ----------------------------------------装-----------------------------------------订----------------------------------线-------------------
22．已知实数a 满足02a <≤，1a ≠，设函数3211()32
a f x x x ax +=-+． （1）当2a =时，求()f x 的极小值；
（2）若函数32
()(24)ln g x x bx b x x =+-++（b R ∈）的极小值点与()f x 的极小值点相同．求证：()g x 的极大值小于等于
54
．
浙江省湖州中学
2013学年第一学期高三期中考试
数 学 答 卷（理）
一、选择题：本大题共10小题，每小题5分，共50分.
二、填空题：本大题共7小题，每小题4分，共28分.
11___[1,7]-_______ 12____1023______ 13___________6_____
14_______ 15____3-_____ 16____2ln 24a ≤-或32a =-__
17_______(1,1)-____
三、解答题：第18、19、20题每题14分，第21、22每题15分，共72分．
18．已知 1:(),3
x
p f x -=且|()|2f a <；
:q 集合{
}
2(2)10,A x x a x x R =+++=∈，{}0B x x =>且 A B =?.
若p ∨q 为真命题，p ∧q 为假命题，求实数a 的取值范围.
解：对p ：所以1|()| |
|23
a
f a -=<．若命题p 为真，则有75<<-a ；...........2分
对
q ：∵}0x |x {B >=且 ?=?B A
∴若命题q 为真，则方程01x )2a (x )x (g 2
=+++=无解或只有非正根．
∴04)2a (2
<-+=?或0(0)0202
g a ?
??≥?≥??+?-...........................5分
∵p, q 中有且只有一个为真命题
∴ (1) p 真，q 假：则有4a 54a 7
a 5-≤<-?
?
?-≤<<-，即有；......................8分 ·································································································································· ··································································································································
班 学号______ 姓名 试
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(2) p 假，q 真：则有7a 4a 5
a 7a ≥?
?
?->-≤≥，即有或；
∴4a 5-≤<-或7a ≥． ........................14分
19．已知函数2()sin
2cos 24
x x f x =+. （1）写出如何由函数sin y x =的图像变换得到()f x 的图像；
（2）在ABC ?中，角A B C 、、所对的边分别是a b c 、、，若C b B c a cos cos )2(=-，求
)(A f 的取值范围.
解：()142sin 212cos 2sin
+??
?
??+=++=πx x x x f ……………………3分 （Ⅰ） 24sin sin()sin()424
x y x y x y π
π
π=????→=+?????→=+左移
横坐标伸长为
原来的倍个单位
1sin()sin()12424
x x y y ππ→=+???→=++上移
个单位 ……7分 （Ⅱ）由()C b B c a cos cos 2=-，利用三角形中的正弦定理知：1cos 2=B ∵π<
π
=
B ……………………10分
()142sin 2+??
?
??+=πA A f ，
∵320π<