山东省武城县第二中学2014-2015学年高一上学期期末复习数学试题(八)
高一数学期末复习测试（八）
一、选择题 1．集合2{|
0},{|20}1
x
A x
B x x x x =>-≤-，则A B = （ ） A ．R
B ．?
C ．(,0)(1,2)-∞
D ．(1,2]
2．将函数sin(2)6
y x π
=-
的图象向左平移
4
π
个单位，所得图象的一条对称轴方程为（ ） A ．6
x π
=
B ．3
x π
=
C ．12x π=
D ．12
x π
=-
3．已知等差数列{}n a 满足24354,10a a a a +=+=，则它的前10项和10S =（ ）
A ．23
B ．95
C ．135
D ．138
4．已知(3,2),(,1)a b x y =-=- ，且//a b ，若,x y R +∈，则32
x y
+的最小值为（ ）
A ．8
B ．
83
C ．
53
D ．24
5．ABC ?中，D 在AC
上，,5,sin AB BD BC BD ABC ⊥==∠=，则CD 的长为（ ）
A
B
．
C ．5
D ．4
6．已知,,a b c R ∈，那么下列命题正确的是（ ）
A ．若a b <，则2
2
ac bc <
B ．若0a b >>，0c <，则c c a b
< C ．若a b >，则22
()()a c b c +>+
D ．若0ab >，则
2b a
a b
+≥ 7．如右图，ABC ?中，4AB BC ==，30ABC ∠=?，AD
为
BC 边上的高，则AD AC ?
＝（ ）
A ．0
B ．8
C ．4
D ．－4
8
．已知函数()cos (0)f x wx wx w +>的图象与直线2y =-的两个相邻公共点之间的距离等于π，则()f x 的单调减区间是（ ）
A ．2
[,]63
k k π
πππ+
+ k z ∈
B ．[,]36
k k π
π
ππ-
+ k z ∈
A
C
D
C ．4
[2,2]33
k k π
πππ+
+ k z ∈ D ．5
[2,2]12
12
k k π
πππ-
+
k z ∈ 9．ABC ?中，1AB BC ==，60B ∠=?，则2AB BC + 与AB
的夹角大小是（ ）
A ．
23
π B ．
2
π C ．
3
π D ．
6
π 10．函数()sin()(0,0,||)2
f x A x A π
ω?ω?=+>><
的部分图象如图所示，若
12(,)63
x x ππ
∈-
，且12()()f x f x =，则12()f x x +＝（ ） A ．1
B ．
12
C
．
2
D
．
2
二、填空题
11．若平面向量,a b 满足||1a b +=
，a b + 平行于x 轴，且
(2,1)b =-
，则a =
.
12．将函数()2sin()(0)3
f x wx w π
=->的图象向右平移
3w
π
个单位，得到函数()y g x =的图象，若()y g x =在[0,
]4
π
上为增函数，则w 的最大值为
.
13．函数213
1
()log 1x x x f x x x ?-+≤?=?>??对任意的x R ∈，2
3()4f x m m ≤-恒成立，则实数m 的取
值范围是
.
14．已知||1OA = ，||2OB = ，2
3AOB π∠=，1124
OC OA OB =+ ，则OA 与OC 的夹角
为
.
15．已知数列{}n a 满足11a =，*log (1),(2,)n n a n n n N =+≥∈，定义：使乘积123k a a a a ?????为正整 数的*
()k k N ∈叫“易整数”，则在[1,2015]内所有“易整数”的和为 .
三、解答题
16．已知,,a b c 为ABC ?
的三个内角的对边，且sin cos a b A c A ++.
（I ）求角c ；
（II ）如图，设D 为BC 的中点，且2AD =，求ABC ?面积的最大值.
A
B
C
17．设{}n a 为等差数列，{}n b 为各项都是正整数的等比数列，且111a b ==，13250a b =，
82345a b a a +=++ *n N ∈.
（I ）求{},{}n n a b 的通项公式； （II ）若数列{}n d 满足21
log 811()2
n b n n d d ++-+=（*
n N ∈）且116d =，试求{}n d 的通项公
式及其前2n 项和2n S .
18．已知()f x a b =? ，其中(2cos ,2),(cos ,1)a x x b x ==
x R ∈.
（I ）求函数()y f x =的单调减区间；
（II ）在ABC ?中，角,,A B C 所对边分别为,,a b c ，()1,f A a =-，且向量
(3,sin )m B = 与(2,sin )n C =
共线，求边,b c 的值.
19．在等差数列{}n a 中，已知公差为2，2a 是1a 与4a 的等比中项.
（I ）求数列{}n a 的通项公式；
（II ）设(1)2
n n n b a +=，记1234(1)n
n n T b b b b b =-+-+-???+-，求n T .
20．为了保护环境，发展低碳经济，某单位在国家科研部门的支持下，进行技术攻关，新上了把二氧化碳处理转化为一种可利用的化工产品的项目，经测算，该项目月处理成本y （元）与月处理量
x （吨）之间的函数关系可近似地表示为
3
221805040,[120,144)3120080000,[144,500]2
x x x x y x x x ?-+∈??=??-+∈??，且每处理一吨二氧化碳可得到可利用的化工产品
价值为200元，若该项目不获利，国家将给予补偿.
（I ）当[2
00,300]x ∈时，判断该项目能否获利？如果获利，求出最大利润；如果不获利，
则国家每月至少需要补贴多少元才能使该项目不亏损？
（II ）该项目每月处理量为多少吨时，才能使每吨的平均处理成本最低？
21．单调递增数列{}n a 的前行项和为n S ，且满足244n n S a n =+.
（I ）求数列{}n a 的通项公式； （II ）数列{}n b 满足：1221
log log 2
n n n a b a ++=，求数列{}n b 的前n 项和n T .
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