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江苏省苏州市2017届高三调研测试数学试题(WORD版,含答案)

江苏省苏州市2017届高三调研测试数学试题(WORD版,含答案)
江苏省苏州市2017届高三调研测试数学试题(WORD版,含答案)

苏州市2017届高三调研数学试卷

一、填空题(本大题共14小题,每小题5分,共70分)

1、已知集合{}1>=x x A ,{}

3<=x x B ,则集合=B A . 2、已知复数i

i

z 21-=

,其中i 为虚数单位,则复数z 的虚部为 . 3、在平面直角坐标系xOy 中,双曲线16

32

2=-y x 的离心率为 .

4、用分层抽样的方法从某高中校学生中抽取一个容量为45的样本,其中高一年级抽20

人,高三年级抽10人,已知该校高二年级共有学生300人,则该校学生总数为 . 5、一架飞机向目标投弹,击毁目标的概率为20.,目标未受损的概率为40.,则目标受损 但未完全击毁的概率为 .

6、阅读下面的流程图,如果输出的函数)(x f 的值在区间],[2

141内,那么输入的实数x 的 取值范围是 .

7、已知实数y x ,满足??

?

??≥+≤-≤431y x x x y ,则目标函数y x z -=2的最大值是 .

8、设n S 是等差数列{}n a 的前n 项和,若7772-==S a ,,则7a 的值为 . 9、在平面直角坐标系xOy 中,已知过点),(11M 的直线l 与圆5212

2

=-++)()(y x 相切, 且与直线01=-+y ax 垂直,则实数=a .

10、一个长方体的三条棱长分别为983,,,若在该长方体上面钻一个圆柱形的孔后其表面 积没有变化,则圆孔的半径为 . 11、已知正数y x ,满足1=+y x ,则

1

1

24++

+y x 的最小值为 . 12、若8

32π

αtan

tan =,则=-

)tan(8

π

α .

13、已知函数???>-≤-=0

50

42x e x x x f x ,,)(,若关于x 的方程05=--ax x f )(恰有三个不同的

实数解,则满足条件的所有实数a 的取值集合为 个.

14、已知C B A ,,是半径为1的圆O 上的三点,AB 为圆O 的直径,P 为圆O 内一点(含 圆周),则?+?+?的取值范围为 .

二、解答题(本大题共6小题,共90分.解答应写出必要的文字说明、证明 或演算步骤)

15、已知函数2

1

2232--=

x x x f cos sin )(. (1)求函数)(x f 的最小值,并写出取得最小值时的自变量x 的集合 (2)设ABC ?的内角C B A ,,所对的边分别为c b a ,,,且3=

c ,0=)(C f ,若

A B sin sin 2=,求b a ,的值.

16、如图,已知直四棱柱1111D C B A ABCD -的底面是菱形,F 是1BB 的中点,M 是线 段1AC 的的中点.

(1)求证:直线//MF 平面ABCD ;(2)求证:平面⊥1AFC 平面11A ACC .

17、已知椭圆)(:012222>>=+b a b

y a x C 的离心率为23

,且过点),(12-P .

(1)求椭圆C 的方程;

(2)设点Q 在椭圆C 上,且PQ 与x 轴平行,过P 点作两条直线分别交椭圆C 于),(11y x A

),(22y x B 两点,若直线PQ 平分APB ∠,求证:直线AB 的斜率是定值,并求出这个定

值.

18、某湿地公园内有一条河,现打算建一座桥(图1)将河两岸的路连接起来,剖面设计图纸(图2)如下:

其中,点E A ,为x 轴上关于原点对称的两点,曲线BCD 是桥的主体,C 为桥顶,且曲线 段BCD 在图纸上的图形对应函数的解析式为],[,2248

2

-∈+=

x x

y ,曲线段DE AB ,均 为开口向上的抛物线段,且E A ,分别为两抛物线的顶点.设计时要求:保持两曲线在各衔 接处),(D B 的切线的斜率相等.

(1)求曲线段AB 在图纸上对应函数的解析式,并写出定义域;

(2)车辆从A 经B 到C 爬坡.定义车辆上桥过程中某点P 所需要的爬坡能力为:=P M (该点P 与桥顶间的水平距离)?(设计图纸上该点P 处的切线的斜率),其中P M 的单 位:米.若该景区可提供三种类型的观光车:①游客踏乘;②蓄电池动力;③内燃机动力, 它们的爬坡能力分别为80.米,51.米,02.米,又已知图纸上一个单位长度表示实际长度 1米,试问三种类型的观光车是否都可以顺利过桥?

19、已知数列{}n a 的前n 项和为n S ,且22-=n n a S (*

∈N n ).

(1)求数列{}n a 的通项公式; (2)若数列{}n b 满足1

211212121

133221+-+--++-+=+n n n n b b b b a )( ,求数列{}n b 的 通项公式;

(3)在(2)的条件下,设n n n b c λ+=2,问是否存在实数λ,使得数列{}n c (*

∈N n )

是单调递增数列?若存在,求出λ的取值范围;若不存在,请说明你的理由.

20、已知函数x k x x f )(ln )(1--=(R ∈k ). (1)当1>x 时,求函数)(x f 的单调区间和极值;

(2)若对于任意],[2e e x ∈,都有x x f ln )(4<成立,求实数k 的取值范围; (3)若21x x ≠,且)()(21x f x f =,证明:k e x x 221<.

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