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江苏省丹阳市高二数学下学期期末考试文科试题


江苏省丹阳市 08-09 学年高二下学期期末测试

高二数学(文科)
一、填空题: (本题共有 14 小题,每小题 5 分,计 70 分,将答案填在答卷纸相应的空格处) 1、集合 A ? {x | x 2 ? x ? 2 ? 0, x ? Z} ,则集合 A 中所有元素之积为 ▲ ;

2、命题: “ ?x ? (0,

?
2

), sin x ? x ”的否定是





3、复数 z ?

2i ( i 为虚数单位)的实部是 1? i






4、 “ x ? 0 ”是“ x ? 0 ”的 要” 或“既不充分也不必要” )

条件; (填“充分不必要” 、 “必要不充分” 、 “充

5、用二分法求方程 x ?2x ?5 = 0 在区间(2,3)上的近似解,取区间中点 x? ? 2.5 ,那么下
3

一个有解区间为





x ?1 ? x?2 ?2e , 6、 f ( x) ? ? ,则 f ( f (2)) = 2 ? ?log3 ( x ? 1) , x ? 2





7、若 sin(? ?

?
12

)?

1 7? , 则 cos( ? ? ) 的值为 3 12





8、已知偶函数 f ( x) 在 ?1,4? 上单调递增,则 f (?? ) 或“=” )



1 f (log 2 ) ; (填“ ? ” 、 “?” 8
y 4.5 l P

9、如图,函数 y ? f ( x) 的图象在点 P 处的切线是 l , 则 f (2) ? f ?(2) = ▲ ;

O

2 4 y=f(x)

x

(第 9 题图)

10、已知 sin ? ? cos ? ?

1 ,0 ? ? ? ? ,则 tan ? =_ 5



_;

11、通过观察下述三个等式的规律,请你写出一个(包含下面三个命题)一般性的命题: ▲ ;

3 ; 2 3 2 2 2 ② sin 5? ? sin 65 ? ? sin 125 ? ? . 2 3 2 ? 2 ? 2 ? ③ sin 16 ? sin 76 ? sin 136 ? ; 2
2 2 2 ① sin 30? ? sin 90? ? sin 150 ? ?

12、定义行列式运算

a1 a3

a2 a4

= a1a4 ? a2 a3 。将函数 f ( x) ?

3 sin x 的图像向左平移 n 1 cos x
▲ ;

( n ? 0 )个单位,若所得图像对应的函数为偶函数,则 n 的最小值为 13、下列结论: ①函数 y=tan

x 在区间(- ? , ? )上是增函数; ; 2
1 2

②当 x ? (1,??) 时,函数 y ? x , y ? x 2 的图象都在直线 y ? x 的上方; ③定义在 R 上的奇函数 f ?x ? ,满足 f ?x ? 2? ? ? f ?x ? ,则 f ?6? 的值为 0; ④若函数 f ?x ? ? ? | x | ,若 f (?m2 ? 1) ? f (2) ,则实数 m ? (??,?1) ? (1,??) ; 其中所有正确结论的序号为 ▲ ;

14 、 对 于 区 间 [ a, b] 上 有 意 义 的 两 个 函 数 f ( x) 和 g ( x) , 如 果 对 任 意 x ? [a, b] , 均 有

| f ( x) ? g ( x) |? 1 ,那么我们称 f ( x) 和 g ( x) 在 [a, b] 上是接近的。若 f ( x) ? log2 (ax ? 1) 与

g ( x) ? log2 x 在闭区间 [1,2] 上是接近的,则 a 的取值范围是





二、解答题: (本大题共有 6 小题,共 90 分,解答时应写出必要的文字说明,证明过程或演 算步骤) 15、 (本题 14 分)若复数 z 同时满足 z ? z ? 2i , z ? iz ( i 为虚数单位) ,求复数 z 。

16、 (本题 14 分)已知命题 p : “函数 f ( x) ? ax2 ? 4x (a ? R) 在 (??,2] 上单调递减” ,命题

q: “ ?x ? R ,16x 2 ? 16(a ? 1) x ? 1 ? 0 ” ,若命题“ p 且 q ”为真命题,求实数 a 的取值范
围。

17、 (本题 15 分)已知 f ( x) ? ?2 sin 2 x ? 2 3 sin x cos x ? 5 (1)将函数化为 f ( x) ? A sin(?x ? ? ) ? k (其中 A ? 0, ? ? 0, 指出函数 f ( x) 的最小正周期; (2)当 x ? [

0 ? ? ? 2? )的形式,并

?
2

, ? ] 时,求 f ( x) 的范围;

18. (本题 15 分)已知函数 f ( x) ? x 2 ? 2ax ? 5 ( a ? 1 ). (1)若 f ( x) 的定义域和值域均是 ?1, ( 2 ) 若 f ( x) 在 区 间 ?? ?,

a? ,求实数 a 的值;

2? 上 是 减 函 数 , 且 对 任 意 的 x1 , x2 ? ?1, a ? 1? , 总 有

f ( x1 ) ? f ( x 2 ) ? 4 ,求实数 a 的取值范围.
19、 (本题 16 分)如图,宽为 a 的走廊与另一宽为 b 的走廊垂直相连,设细杆 AC 的长为 l ,

?ACD ? ? (1)试用 a, b, ? 表示 l ; (2)当 b ? a 时,求当细杆 AC 能水平通过拐角时 l 的最大值; (3)当 l =8a 时,问细杆 AC 能水平通过拐角,则另一走廊宽 b 至少是多少?
D

?
B

C

b

A

a

20( 、本题 16 分) 已知函数 f ( x) ? ax ? 3, g ( x) ? bx (1)试求 b, c 所满足的关系式;

?1

1 ? cx ? 2 (a, b ? R)且g (? ) ? g (1) ? f (0). 2

(2)若 b ? 1 , F ( x) ? f ( x) ? g ( x) 在 x ? [ ,?? ) 为增函数,求 a 的取值范围。 (3)若 b ? 0 ,方程 f ( x) ? g ( x) 在 x ? (0,??) 有唯一解,求 a 的取值范围;

1 2

高二数学(文科)参考答案
一、填空题: (本题共有 14 小题,每小题 5 分,计 70 分) 1、0; 要; 5、 (2, 2.5) (注: [2, 2.5) 、 (2, 2.5] 、 [ 2, 2.5] 也对) 8、? 12、 6、2 7、 ? 2、 ?x ? (0,

?
2

) ,使 sin x ? x ;

3、 ? 1 ;

4、充分不必

1 ; 3

9、

5? ; 6

9 ; 8

10、 ?

4 3 2 2 ? 2 ? ; 11、sin ? ? sin (? ? 60 ) ? sin (? ? 120 ) ? ; 2 3

13、①③④;

14、 [0,1] ;

二、解答题: (本大题共有 6 小题,共 90 分) 15 、 ( 本 题 14 分 ) 解 : 设

z ? a ? bi ( 其 中 a, b ? R ) , 则

z ? a ? bi ----------------------------2 分

由 题 意 得 :

?a ? bi ? (a ? bi) ? 2i ? ?a ? bi ? i(a ? bi)

即 :

?2bi ? 2i ? ?a ? bi ? ai ? b

------------------------------10 分

∴?

?2b ? 2 ?a ? ?b

∴?

?a ? ?1 --------------------------------------------12 分 ?b ? 1

∴ 分

z ? ?1 ? i --------------------------------------------------------------14

16 、( 本 题

14

分 ) 解 : P

为 真 : ①

当 a?0

不 符 合 题 意 ;

-------------------------------------2 分 ② 当

a?0

时 ,

f ( x) ? ?4 x 在 (??,2] 上 单 调 递 减 , 故

a ? 0 ------------------------------4 分
③ 当 a ? 0 时,只需对称轴 x ? ? ∴

?4 2 2 ? 在区间 (??,2] 的右侧,即 ?2 2a a a

0 ? a ?1

------------------------------------------------------7 分 -----------------------------------------------------8 分

综合①②③: a ? [0,1]

q 为 真 : 命 题 等 价 于 : 方 程 16x 2 ? 16(a ? 1) x ? 1 ? 0 无 实 根 。
---------------------10 分

? ? [16(a ? 1)]2 ? 4 ?16 ? 0
-----------------------------------12 分



1 3 ?a? 2 2



命题“ p 且 q ”为真命题



?0 ? a ? 1 ? ?1 3 ?a? ? 2 ?2



1 ? a ?1。 2

---------------------------------------------14 分

17、 (本题 15 分) 解: (1)f ( x) ? ?2 ? 分

1 ? cos 2 x ? 3 sin 2 x ? 5 ------------------------2 2

? cos2x ? 3 sin 2x ? 4
---------------4 分 =

? 2(

3 1 sin 2 x ? cos2 x) ? 4 2 2

2 sin( 2 x ?

?
6

)?4

--------------------------------------------------------6 分 函 数

f ( x)













T?

2? ?? 2

----------------------------------------------------7 分 (2) ∵

?
2

? x??



7? ? 13? ? 2x ? ? 6 6 6

---------------------9 分



sin( 2 x ?

?

1 ) ? [?1, ] 6 2

-------------------------------------------13 分



f ( x)









[2,5]



-------------------------------------------------------------------------15 分
2 2 18、 (本题 15 分)解:(1)∵ f ( x) ? ( x ? a) ? 5 ? a ( a ? 1 ),



f ( x)



?1,

a?













----------------------------------------------------------------------- 2 分 又 定 义 域 和 值 域 均 为

?1,

a?





? f (1) ? a ? ? f (a ) ? 1

-------------------------------------------------------4 分 即

? 1 ? 2a ? 5 ? a ? 2 2 ?a ? 2a ? 5 ? 1







a ? 2 .---------------------------------------------------------6 分
(2) ∵

f ( x)

在 区 间

?? ?,

2? 上 是 减 函 数 , ∴

a?2



----------------------------------------- 8 分 又 x ? a ? ?1, ∴

a ? 1?,且, (a ? 1) ? a ? a ? 1

f ( x) max ? f (1) ? 6 ? 2a



f ( x) min ? f (a) ? 5 ? a 2 .--------------------------------------------11 分
∵对任意的 x1 , x2 ? ?1, ∴

a ? 1? ,总有 f ( x1 ) ? f ( x 2 ) ? 4 ,

f ( x) max ? f ( x) min ? 4



--------------------------------------------------------------------------------13 分 即

(6 ? 2a) ? (5 ? a 2 ) ? 4







?1 ? a ? 3



----------------------------------------------------14 分 又

a?2



∴ 2 ? a ? 3 .------------------------------------------------------------------------------------15 分 19、 (本题 16 分)解: (1) AB ?

a , cos ?

BC ?
∴l ?

b sin ?

----------------------2 分

∴ 分

l ? AB ? BC ?

a b ? cos? sin?

a b ? ? , ? ? (0, ) -----------3 cos ? sin ? 2

(2) l ? a(

1 1 sin ? ? cos ? ? )?a cos ? sin ? sin ? cos ?

令 t ? sin ? ? cos ? ?

2 sin(? ?

?
4

)

∵0 ? ? ?

?
2



t ? (1, 2 ]

sin ? cos? ?

t 2 ?1 2



l?a

t 2at ? t 2 ?1 t 2 ?1 2

t ? (1, 2 ]

-------------------------------------5



而l/ ?

2at ? 2a(1 ? t 2 ) 在 t ? (1, 2 ] 是减函数,且当 t 大于 1 且无限趋 ? 0 ,∴ l ? 2 2 2 t ?1 (t ? 1)
1 时 ,





l ? ??





l ?[4 2,??)

-----------------------------------------7 分 ∴ 细 杆 AC 能 水 平 通 过 拐 角 时 l 的 最 大 值 为 4 2 。

----------------------------------------------8 分 (3)由(1)可得: b ? 8a sin ? ? 分

a sin ? , cos ?

? ? ? (0, )
2

------------------9

b / ? 8a cos? ?
----11 分

a a(8 cos3 ? ? 1) a(2 cos? ? 1)(4 cos2 ? ? 2 cos? ? 1) ? ? cos2 ? cos2 ? cos2 ?

/ o s ?? 令b ? 0, 则c

1 ? ,? ? 2 3

-------------------------------------------13

分 当0 ? ? ?

?
3

时, b ? 0 ;

/



?
3

?? ?

?
2 3

时, b ? 0

/





??

?





(b) m ? 3 3 i a

n

---------------------------------------------15 分 ∴ 另 一 走 廊 的 宽 至 少 为

3 3a



------------------------------------------------16 分

20、 (本题 16 分)解: (1)由 g (? ) ? g (1) ? f (0) ,得 (?2b ? 4c) ? (b ? c) ? ?3 ∴b、c 所满足的关系式为 b ? c ? 1 ? 0 . ------------------------------------------2 分 (2)由 b ? 1 , b ? c ? 1 ? 0 ,可得 c ? 0 , 分 ∵ ∴

1 2

g ( x) ?

1 x

F ( x ) ? ax ?

1 ? 3 -----3 x

1 F ( x) 在 x ? [ ,?? ) 上递增 2

∴ 分 ∴ 分

F / ( x) ? a ?

1 1 1 1 ? 0 在 x ? [ ,?? ) 上恒成立 即: a ? 2 在 x ? [ ,?? ) 上恒成立—5 2 2 2 x x

a?(

1 ) max x2

----------------------------------------------------------6

而y? 分 ∴

1 1 在 x ? [ ,?? ) 上单调递减 2 2 x



(

1 ) max ? 4 x2

-----------------------7

y

a?4

------------------------------------------------------------------8 分 (3)由 b ? 0 , b ? c ? 1 ? 0 ,可得 c ? ?1 .-------------------------9 分 方程 f ( x) ? g ( x) ,即 ax ? 3 ? ? x ?2 ,可化为 a ? 3x ?1 ? x ?3 ,----------10 分 令 x ?1 ? t ,则由题意可得, a ? 3t ? t 3 在 t ? (0,?? ) 上有唯一解,---------11 分 令 h(t ) ? 3t ? t 3 (t ? 0) ,由 h?(t ) ? 3 ? 3t 2 ? 0 ,可得 t ? 1 ,-------------12 分 当 0 ? t ? 1 时,由 h?(t ) ? 0 ,可知 h(t ) 是增函数; 当 t ? 1 时,由 h?(t ) ? 0 ,可知 h(t ) 是减函数.故当 t ? 1 时, h(t ) 取极大值 2 .-----14 分 O x

由函数 h(t ) 的图象可知,当 a ? 2 或 a ? 0 时,方程 f ( x) ? g ( x) 有且仅有一个正实数解. 故 所 求

a













{a | a ? 2



a ? 0} . -----------------------------------------16 分


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