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2017年哈三中二模--理数学

2017年哈三中二模--理数学
2017年哈三中二模--理数学

2017年哈尔滨市第三中学第二次高考模拟考试

数学试卷(理工类)

第Ⅰ卷(选择题,共60分)

一、选择题(共12小题,每小题5分,共60分,在每小题给出的四个选项中,只有一项是符合题目要求的.)

1. 已知复数2

1z i

=

-+,则 A.z 的模为2 B.z 的虚部为-1 C.z 的实部为1 D.z 的共轭复数为1+i 2. 已知集合{0,2,4,6},{n N |28}n A B ==∈<,则集合A B ?的子集个数为

A.8

B.7

C.6

D.4

3. 对于平面α和不重合的两条直线m 、n ,下列选项中正确的是

A.如果,m n αα? ,m 、n 共面,那么m n

B.如果m α?,n 与α相交,那么m 、n 是异面直线

C.如果,m n αα??,m 、n 是异面直线,那么n α

D.如果,m n m α⊥⊥,那么n α 4. 已知随机变量ξ服从正态分布()

()22,,40.84N P δξ≤=,则()0P ξ≤= A.0.16

B.0.32

C.0.68

D.0.84

5. 在区间??中随机取一个实数k ,则事件“直线y=kx 与圆()2231x y -+=相交发生的概率为 A.

12 B.14 C.16 D.1

8

6. 宋元时期数学名著《算学启蒙》中有关于“松竹并生”的问题:松长五尺,竹长两尺,松日自半,竹日自倍,松竹何日而长等。右图是源于其思想的一个程序框图,若输入的a 、b 分别为5、2,则输出的n=

A.2

B.3

C.4

D.5

7. 某几何体的三视图如图所示,则该几何体的体积为 A.10 B.20 C.40 D.60

8. 已知1sin 33πα??-= ???,则sin 26πα??

-= ???

A.7

9

-

B.

79

C.79

±

D.29

-

9. 德国著名数学家狄克雷在数学领域成就显著,以其名命名的函数()10,x f x x ?=??,为有理数

为无理数

,提前为狄克雷

函数,则关于函数()f x 有以下四个命题: ①()()1f f x =; ②函数()f x 是偶函数;

③对于任意一个非零有理数T ,()()f x T f x +=对任意x R ∈恒成立;

④存在三个点()()()()()()112233,,,,,A x f x B x f x C x f x ,使得ABC ?为等边三角形。其中真命题的个数是 A.4

B.3

C.2

D.1

10. “关于x 的方程20x mx n -+=有两个正根”是“方程221mx ny +=的曲线是椭圆”的 A.充分不必要条件

B.必要不充分条件

C.充要条件

D.既不充分也不必要条件

11. 已知双曲线()22

2210,0x y a b a b

-=>>的左、右焦点分别为12F F 、,焦距为()20c c >,抛物线22y cx =的

准线交双曲线左支于A ,B 两点,且120AOB ∠= ,其中O 为原点,则双曲线的离心率为

A.2

B.1

C.1

D.1

12. 已知函数()()2

11,,x f x kx x e g x e e ??????

=∈= ? ????????

?,若()(),f x g x 图像上分别存在点M ,N 。使得M ,N 关

于直线y=x 对称,则实数k 的取值范围为

A.1,e e ??-????

B.2,2e e ??-????

C.3,3e e ??-????

D.2,2e e ??

- ???

第Ⅱ卷(非选择题 ,共90分)

二、填空题(共4小题,每小分,共20分,将答案填在答题卡相应的位置上.)

13. 已知x ,y 满足400

x y x y x +≤??

-≥??≥?,若目标函数z=x+2y 的最大值为n

,则n

x ? ?展开式的常数项为______。

14. 在ABC ?中,a ,b ,c 分别是角A ,B ,C 的对边,已知c=2,若222

sin sin sin sin sin A B A

B C +-=,则

a+b 的取值范围是______。

15. 已知(

)[](]21,11,1,2x f x x x ∈-=-∈??,则()21f x dx =-? 。

16. 已知函数()f x 的定义域为R ,若存在常数k ,使得()2017k

f x x ≤对所有实数x 均成立,

则称函数()f x 为“期望函数”,给出下列函数: ①()2f x x = ②()x f x xe = ③()2

1

x

f x x x =

-+

④()1

x

x

f x e =

+

三、解答题(本大题共6小题,共70分,解答应写出文字说明,证明过程或演算步骤.) 17.(本小题满分12分)设n S 是数列{}n a 的前n 项和,已知*113,23()n n a a S n N +==+∈. (Ⅰ)求数列{}n a 的通项公式;

(Ⅱ)令()21n n b n a =-,求数列{b }n 的前n 项和n T .

18. (本小题满分12分)近年来,空气质量成为人们越来越关注的话题。空气质量指数(Air Quality Index ,简称AQI )是定量描述空气质量状况的指数,空气质量按照AQI 大小分为六级,0~50为优;51~100为良;101~150为轻度污染;151~200为中度污染;201~300为重度污染;大于300为严重污染。环保部门记录了2017年某月哈尔滨市10天的AQI 的茎叶图如下.

(Ⅰ)利用该样本估计该地本月空气质量优良()100AQI ≤的天数;(按这个月总共30天计算)

(Ⅱ)现工作人员从这10天中空气质量为优良的日子里随机抽取2天进行某项研究,求抽取的2天中至少有一天空气质量是优的概率;

(Ⅲ)将频率现为概率,从本月中随机抽取3天,记空气质量优良的天数为ξ,求ξ的概率分布列和数学期望.

19. (本小题满分12分)如图,四棱锥P-ABCD 底面为正方形,已知PD ⊥平面ABCD ,PD=AD ,点M 为线段PA 上任意一点(不含端点),点N 在线段BD 上,且PM=DN. (Ⅰ)求证:直线MN 平面PCD ;

(Ⅱ)若M 为线段PA 中点,求直线PB 与平面AMN 所成的角的余弦值.

20. (本小题满分12分)已知圆O :224x y +=与x 轴交于A 、B 两点,点M 为圆O 上异于A 、B 的任意一点,圆O 在点M 处的切线与圆O 在点A 、B 处的切线分别交于点C 、D ,直线AD 和BC 交鱼点P ,设P 点的轨迹为曲线E.

(Ⅰ)求曲线E 的方程;

(Ⅱ)由线E 与y 轴正半轴交点为H ,则曲线E 是否存在直角顶点为H 的内接等腰直角三角形Rt GHK ?,若存在,求出所有满足条件的Rt GHK ?的两条直角边所在直线的方程,若不存在,请说明理由.

21. (本小题满分12分)定义:设()f x 为(a ,b )上的可导函数,若()'f x 为增函数,则称()f x 为(a ,b )上的凸函数.

(Ⅰ)判断函数3y x =与1

lg y x

=是否为凸函数;

(Ⅱ)设()f x 为(a ,b )上的凸函数,求证:若()1211,01,2,n i n λλλλ+++=>=………,,则()(),1,2

,,i x a b i n ?∈=…,恒有()()()()11221122n n n n f x f x f x f x x x λλλλλλ++≥+++…+?成立; (Ⅲ)设*,,0,,a b c n N n b >∈≥,求证:532532532n n n n n n a b c a b c b c a c a b ---++≥++.

22. (本小题满分12分)

圆锥曲线C 的极坐标方程为:()

221sin 2ρθ+=.

(Ⅰ)以极点为原点,极轴为x 轴非负半轴建立平面直角坐标系,求曲线C 的直角坐标方程,并求曲线C 在直角坐标系下的焦点坐标及在极坐标下的焦点坐标; (Ⅱ)直线l 的极坐标方程为()3

R π

θρ=∈,若曲线C 上的点M 到直线l 的距离最大,求点M 的坐标(直

角坐标和极坐标均可). 23. (本小题满分12分)

(Ⅰ)已知对于任意非零实数a 和b ,不等式()311a b a b a x x ++-≥-++恒成立,试求实数x 的取值范围;

(Ⅱ)已知不等式211x -<的解集为M ,若,a b M ∈,试比较11ab +与11

a b

+的大小.(并说明理由).

数学试卷(理工类)

1——6:BDAABC

7——12:BAADCB 13.240

14(]2,4

15.

4/32

π

+ 16.③④

17.解:(Ⅰ)当2n ≥时,由123n n a S +=+,得123n n a s -=+,(1分) 两式相减,得1

111222,3,3n n n n n n n n m

a a a s s a a a a ++-+-=-=∴=∴=(3分) 当n=1时,12113,23239a a s a ==+=+=,则

2

1

3a a =. ∴数列{}n a 是以3为首项,3为公比的等比数列(5分)

1333n n n a -∴=?=(6分)

(Ⅱ)由(1)得()()2123n n n b n a n a =-=-?

()()232

3

4

1

1333532133133353213

n n n n T n T n +∴=?+?+?++-?=?+?+?++-?……

错位相减得()()23112132323232136223n n n n T n n ++∴-=?+?+?++?--?=---?…

()113 3 (12)n n T n +=-?+分

18. 解:(Ⅰ)从茎叶图中可发现该样本中空气质量优的天数为2,空气质量的天数为4,故该样本中空气质量优良的频率为63105=,从而估计该月空气质量优良的天数为3

30185

?= (2)3/5

(3)由(1)估计某天空气质量优良的概率为3

,5ξ的所有可能取值为0,1,2,3.

()()()3

2

2

123328323632540,1,251255512555125F P C P C ξξξ??????========= ? ? ?

?????? ()332735125P ξ??

=== ???

显然333,,3 1.855B E ξξ??

-=?= ???

19.延长AN ,交CD 于点G ,由相似知

A

N B N A M N G

N D M P

==

,,MN PCD PG PCD MN PCD ?? 平面平面,则直线平面;

(Ⅱ)由于DA DC DP ⊥⊥,以DA ,DC ,DP 为x ,y ,z 轴建立空间直角坐标系,设A (1,0,0),则B

(1,1,0)C (0,1,0),P (0,0,1),1111,0,

,,,02222M N ???? ? ????

?

则()1,1,1PB =-

,平面AMN 的法向量为()1,1,1m =

.

则向量PB 与m 的夹角为θ,则1

cos 3

θ=,则PB 与平面AMN

.

20. (Ⅰ)设 ()00,M x y ,则M 处的切线为004x y y y +=,则000042422,,2,x x C D y y ????

+-- ? ?????,

(6分)则()()

00

004224:4224x y x y P x y x y -?

=+???+?=-?-?

,则()22:104x E y y +=≠.

(Ⅱ)由于直线GH 不与坐标轴平行或垂直,可设:1GH l y kx =+,则224401

:11KH

x y l y x k y kx ?+-==-+?

=+?

,得()221480k x kx ++=,则于0?>恒成立,设两个根为12,x x ,

则GH =,同理

HK =由GH=HK 知:()

22441k k k +=+,得:

(1)k>0时,得()()

21310k k k --+=得:k=1

或k =

(2)k<0时,得()()

21310k k k +++=得:k=-1

或k =综上,共分三种情况

(1)两条直角边所在直线方程为:1y x =±+; (2

)两条直角边所在直线方程为:1y =+ (3

)两条直角边所在直线方程为:1y =

+(12分) 21. (Ⅰ)(1

)10,2m ?<< ??

单调递增,??

单调递减,?+∞????单调递增;

(2

)0,m ?≤ ??

单调递减,?+∞????

单调递增; (3)()1

,0,2m ≥+∞单调递增(6分)

(Ⅱ)14

e

--(12分)

22.(Ⅰ)曲线C 直角坐标方程:2

212

x y +=,焦点直角坐标:()()121,0,1,0F F -

焦点极坐标:()()121,,1,0F F π

(Ⅱ)M M ? ????

或 23. (Ⅰ)334a b a b a b a b a ++-≥++-=,当且仅当()()30a b a b +-≥时取等号. 只需:()411a a x x ≥++-,由于0a ≠,只需114x x ++-≤, 故x 的取值范围为:[]2,2-

(Ⅱ)解得:()0,1,,M a M b M =∈∈知:()()11111110a b ab a b ab a b ab ab --+--+--==>,即1111a b a b

+>+.

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