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(理数)09届江门市普通高中高三调研测试

江门市2009届普通高中高三调研测试

数学（理科）

本卷共21题，满分150分，测试用时120分钟．

参考公式：如果事件A 、B 互斥，那么P （A+B ）=P （A)+P （B ）．

+=+n

a b a )(+-b a

C n 1

11--+n n n ab

C n

b +，其中=k

n C !

)

1()1(k k n n n +-- .

一、选择题：本大题共8小题，每小题5分，满分40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．已知全集U=R ，集合}1|{≤=y y A ,}0)2.)(1(|{>-+=x x x B ，则A B C U

)(= A ．（-∞，-1] B ．（2，+∞） C ．[-1，1] D ．φ

2．如图1，点A(2，0)、B(0，2)，若复数z=a+bi （a 、b ∈R ， i 为虚数单位）对应复平面内的点Z 是线段AB 的中点，则a+b=

A ．2 B.1 C ．0 D ．-1 3．3

)(x

k x -

（k 是常数）的展开式中，常数项的数值是3，则k 的值是

A ．1±

B ．-1

C ．3±

D ．-3

4．已知E 是平行四边形ABCD 的边AD 的中点，若a BA =,b BC =，则CE = A ．a b 2

B. a b 2

C. b a 2

D.b a 21-

5．一个直三棱柱，底面是等腰直角三角形，三视图如图2，则这个直三棱柱的体积是 A ．

2 B ．2 C ．

1 D ．1

6．圆(x-1)2

+(y+1)2

=4上的点到直线3x+4y=4的最大距离与最小距离的差是 A ．1 B ．2 C ．3 D ．4

7．已知直线x+y=5和直线x+2y=8相交于点P ，在不等式组???

?≤+≤+≥≥8

2500y x y x y x 确定的平面区域内，

目标函数z=x+ay 只在点P 取得最大值，则常数a 的值可能是 A ．2

3 C ．2 D ．1

8．已知函数m

x x x x f ++-=33)(2

3，对任意x ∈[-1，2]，a x f ≤|)(|恒成立，则a 的取值范围是 A ．2

9≥

a B ．a ≥5 C ．a ≥7 D ．a ≥9

二、填空题：本大题共7小题，你只需要作答6小题，每小题5分，满分30分．

（一）必做题（9～12题）

9．已知直线l 经过点P(-1，2)，且与抛物线y=x 2

的准线平行，则直线l 的方程是________．

10．阅读右边程序框图，若输入m=2008，则输出S=____． 11．已知一组数据：2、3、a 、b 、11、12（其中a 、b 是常数），它们的平均数是6。若要使这组数据的标准差最小，则常数a=____．

12．已知真命题：矩形ABCD 的对角线AC 与边AB 、AD 的

夹角分别为α、β，则1

cos cos 2

2=+βα.将这个真命题推广，可以得到一个关于空间长方体的真命题是__________ ___________________________________________.

（二）选做题（13～15题，你只需要从中选做两题，如果三题全做，则只计算前两题的分数）

13．（坐标系与参数方程选做题）经过极点的圆ρ=4cos θ与极轴的另一个交点为P ，经过点P 的圆的切线的极坐标方程是_____. 14．（不等式选讲选做题）已知a 是常数，x=1是不等式++||a x 10||>-a x 的解，则

a 的取值范围是______． 15．（几何证明选讲选做题）如图3，⊙O 内的两条弦AB 、 CD 相交于P ，已知PA=PB=2PC=4，∠APC=600

，则⊙O 的半径r=_____．

三、解答题：本大题共6小题，满分80分．解答须写出文字说明、证明过

程和演算步骤．

16．（本小题满分13分）已知定义在区间[0，+∞）上的函数==)(x f y )3

sin(φπ

+x A （其

中A>0，2

||π

φ<

)，对0>?

x ，f(x)

(2)函数y=f(x)-2的零点从小到大依次为a 1、a 2、a 3……，求a n ．

17．（本小题满分13分）一位学生每天骑车上学要经过A 、B 、C 、D 四个交通岗，每个交通岗连续循环亮绿灯、黄灯和红灯。A 、B 、D 岗一次亮绿灯、黄灯和红灯的时间都是20秒、5秒和35秒，C 岗一次亮绿灯、黄灯和红灯的时间是25秒、5秒和20秒，该学生到达每个交通岗遇到信号灯的种类和时机是随机的、相互独立。假设该学生每次上学遇到绿灯的次数为δ．

(1)求δ的分布列；

(2)求该学生上学路上遇到绿灯的平均次数．

18．（本小题满分14分）如图4，四棱锥P-ABCD 中，底面ABCD 是正方形，侧棱PD

垂直底面ABCD ，E 为PB 上一点，已知PD=AD=CD=1，PB PE 3

1=.

(1)求证：DE ⊥PB;

(2)求DE 与侧面PCD 所成角的余弦值；

(3)若过DE 且与PB 垂直的平面与BC （或其延长线）相交于F ，求PF 的长．

19．（本小题满分14分）椭圆:

C 12

b y

a x

)0(>>b a 的离心率2

e ，以椭圆的四个顶点为顶点的四边形的周长为54. (1)求椭圆的方程；

(2)若直线2:+=kx y l 与曲线C 恒有两个不同的交点A 和B ，求三角形OAB （其中O 为坐标原点）面积的最大值．

20．（本小题满分14分）已知函数c

bx ax x x f +++=2

ln 2)(，x>0，a 、b 、c 是常数，曲线y=f(x ）在点(1，f(1))处的切线方程为y=3x+1.

(1)若x=2是函数f(x)的一个极值点，求f(x)的单调递减区间； (2)若b>1，试证明对任意x 1、x 2>0，)2(

21x x f +)]()([2

21x f x f +≥恒成立． 21．（本小题满分12分）数列}{n a 满足a 1=k （k 是常数），*N n ∈?，k n a a n n -+=+1. (1)若k=1，求数列}{n a 的通项：

(2)若a 2008是数列}{n a 的最小项，求k 的取值范围；

(3)若数列}{n b 对任意，n ∈N*都有++212b b +322b

2+-=+n n n a b ，求数列}{n b 的前n 项和S n .

参考答案

说明：以下答案仅供参考，学生合理有效的解答均应相应给分。一、选择题 CABD DCBA 二、填空题 9．y=2 10．

2009 11.4 12．参考答案：长方体ABCD-A'B'C'D'

的对角线AC'与棱AB 、AD 、AA'（或“面ABCD 、ABB'A'、ADD'A'，或“面上对角线AC 、AB'、

AD'”）的夹角分别为α、β、γ，则βα22cos cos +1cos 2=+γ（或2

cos cos cos 2

22=++γβα) 13. ρcos θ=4 14.(-∞, -5)∪(5, +∞) 15.72 三、解答题

16．(1)由对0>?x ，f(x)≤f(1)=2可知:f （x)max =2 ...2分，所以A=2....3分 f(1)=2，即2)3

sin(2=+φπ

.......4分

所以1)3

sin(=+φπ,=+φπ

2ππ+k ，k ∈Z ........5分

所以6

2π

πφ+

=k ，k ∈Z ，因为2||π

φ<

，所以6

φ=

? ......6分

(2)令f(x ）-2=0……7分，得：1)6

sin(=+π

πx ....8分

x 2

2π

π+

k ，k ∈Z...9分，所以x=6k+1，k ∈Z.........10分

所以a 1=1，61=-+n n a a ，所以56-=n a n ，n ∈N* ........13分 17．(1)该学生每次到达A 、B 、D 岗遇到绿灯的概率35

520201++=p 3

...1分

每次到达C 岗遇到绿灯的概率20

525252++=

p 2

1= ....2分

依题意δ=0、1、2、3、4，==)0(ξP ?-31)1(p 27

4)1(2=

-p

==)1(ξP +?-231)1(p p 2

113)1(p p C -??27

10)1(2=-?p

==)2(ξP ??11

3p C 2

322

1)1(C p p +?-)1(12

1p p -??31)1(2=-?p ?==2

3)3(C P ξ?-?)1(12

1p p ??+3

32p C p ?-0

1)1(

p 54

7)1(2=-p

==)4(ξP 54

123

?p p .......7分（每正确列式并计算1个给1分）

所以，δ的分布列为

....9分

(2）)0(0=?=ξξP E )1(1=?+ξP )2(2=?+ξP )3(3=?+ξP )4(4=?+ξP ...11分

3 =，即该学生上学路上遇到绿灯的平均次数为

3 .......13分

18．(1)以D 为原点，DA 、DC 、DP 为单位正交基底建立空间直角坐标系Oxyz....1分则D(0, 0, 0), A(1, 0, 0), B(1, 1, 0), C(0, 1, 0), P(0, 0, 1) ...... 2分

)1,1,1(-=PB ，DP DE =DP PE =+=+

PB 31

,31,31( .....4分 =

?PB DE 13

113

1?+

?0)2(3

2=-?+

.....5分，所以DE ⊥PB .....6分

(2))0,0,1(=DA 是侧面PCD 的一个法向量 ...7分设DE 与侧面PCD 所成的角为θ，则DA <，>=

DE θπ

-2

DA =DE 6

1= ....9分，所以6

5cos =θ ...10分

(3)依题意，设F(a ，1，0)，则PB ⊥EF..........11分

DF EF =)0,1,(a DE

=-)32,31,31(-)3

,32,31(--=a ...........12分解=?EF PB )3

1(1-?a )1(3

21-+?

+0)3

2(=-

?，得a=-1..............13分

所以=PF 3)10()01()0(2

22=-+-+-a ..........14分

19．(1)由题意：=+224b a 5542

2=+?b a ……2分

c e =

-a

b a

a ?2

b 2= ...........4分解得a=2，b=1......分，椭圆的方程为

=+y

……6分

（2）将2.kx y =代入14

=+y x 得22).41(x k +0428=++

kx .....7分由直线l 与椭圆交于不同的两点得：

-=?2

)28(k 0)41(162

>+k ，即4

①.........8分

设A(x 1，y 1)、B(x 2，y 2)，则=+21x x 2

4128k

、2

21414k

x x +=

…9分

设直线l 与y 轴交于点D ，则：2||=OD ...10分

=-=???OAD OBD OAB S S S ?||21OD 2

212

2121)

14(14224)(22||+-?=-+?=-k k x x x x x x …11分

令t=4k 2+1，t>2，则4k 2=t-1

181

)411(22222222

≥+--?=-?=?t t

t S OAB

…12分当且仅当t=4是取“=”，此时4

3.2=

k ,满足①.........13分

故三角形OAB 面积的最大值为1 ……14分

20．(1)x x f 2

)(/

ax ++2 .....1分，由已知得??

???=++==++=041)2(322)1(

b a f b a f ....3分解得??

?=-=3

1b a …4分，解0322

)('=+-=x x

x f )0(>x 得x=2 .......5分

当00；当x>2，f'(x ）<0 ..........6分

所以f(x)的单调递减区间是[2，+∞）............7分

(2)由已知得)2(21x x f ++-)([2

1x f =)](2x f 212

214)(ln

x x x x +2

2.1)(4

x x a --

......9分

f'(1)=2+2a+b=3，2

1b a -=

，当b>1时a<0.....11分

由基本不等式2

1x x 221x x +≤得2

214)(x x x x ≥+，04)(ln 2

21≥+x

x x x ....13分所以)2(21x x f ++-)([2111x f =)](2x f -+212

4)(ln x x x x 0)(4

21≥-x x a 恒成立..........14分 21.(1)n ≥2时，a n -a n-1=n-（k+1）

所以)(1--=n n n a a a )(2.1---+n n a a 1

12)(a a a +-++ ............1分 )1[()]1([-++-=n k n )]1(+-k )]1(2[+-++k =+k k n k n 22

222

++- ...2分

当n=1时，2121==k a ?+?+-k k 21212，所以对任意n ∈N *

,-=22

n a n k n k 2212++，当k=1时，对任意n ∈N *

，22

+-=n n a n ..........4分

(2)2

a n =

n k )212(

+-21

2=+k 2)212(?+-k n -+k 22

)

12(+k ，是?+=212k n 为对称轴

的二次函数 .....5分，依题意2

|2008212|

≤-+k ....6分，解得2007≤k ≤2008...7分 (3)n=1时，b 1=a 2=1；n ≥2时，11

2+-=n n n a b k n a n -=

-，1

.--=n n k n b .....8分

所以?

??--=1

21n k n b n

2=≥n n ……………9分，n=1时，S 1=b 1=1；n ≥2时， 1

--=

n n k n S +--+

-2

1n k n 122+-+

k ……10分，n n k n S 22-=+--+-121n k n 21

22+-+k 两式相减计算得k S n -=41

2--+-

n k

n ....11分，又因为--==

k S 4110

21k

-+，

所以，对任意n ∈N *，k S n -=4?-+--1

2n k n ……12分

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八年级下册期末测试题（100分）第I卷（共55分） I. 根据音标和句意选择正确的单词 (5分) （）1. Unluckily, a /∫?p/ sank in the Yangtze River on June 1st. A. ship B. sheep C. pop D. drop （）2. /'wee?/ you like Indian food or Western food, you’ll find it all in Singapore. A. Weather B. Wheel C. Whether D. Waste （）3. We should never give up trying to /?'?i:v/ our dreams. A. ancient B. against C. anyway D. achieve （）4. I’ve recently been to a very /?n'ju:?u?l/ museum in India. A. unusual B. underwear C. unfair D. unlucky （）5. A few weeks ago, I found the /mɑ:rks/of another man’s feet on the sand. A. make B. marks C. marry D. matter II．单项选择（20分）（）6. She ______to an English program while her parents ______TV.（兰州） A. was listening; is watching B. listened; were watching C. was listening; watched D. was listening; were watching （）7. —The Great Wall is famous all over the world. How long is it? —__________. A. It’s 8,859 kilometers long B. It’s about 200 meters deep C. It’s 6 meters wide D. It’s 8 meters high （）8. I play basketball every day because I want to play as as Yao Ming. A. well B. better C. best D. good （）9. Could you please the piano so loudly? Your brother is having a rest now. A. not to play B. not play C. don’t play D. to not play （）10. I think it would start a habit if you copy my homework. A. good B. bad C. well D. wonderful （）11. —How long may I your dictionary? —For one week. But you mustn’t to others. A. keep; lend B. borrow; lend C. lend; be borrowed D. have; borrow （）12. —you the e-mail yet? —No, I haven’t. But I will do it later. A. Have; written B. Did; write C. Will, write D. Do; write （）13. To my , the square was so clean after the big meeting. A. surprise B surprised C. surprising D. surprises （）14. —I have too much housework to do at home! —ask your husband for help? A. You could B. Why don’t C. Why not D. How about （）15. We shouldn’t our hopes. Everything will be better. A. put up B. cheer up C. fix up D. give up （）16. Nancy was angry and she sat there without saying a word. A. all the time B. in silence C. in danger D. over and over （）17. The driver saw an old man on the side of the road . A lie B. lay C. lying D. to lie （）18. You may fail in the exam you study hard. A. Until B. if C. because D. unless