摘要:17.已知抛物线的顶点在原点.焦点F在x轴的正半轴上.且过点P(2.2).过F的直线交抛物线于A(x1.y1).B(x2.y2)两点.(1)求抛物线的方程,(2)设直线l是抛物线的准线.求证:以AB为直径的圆与直线l相切.

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1.C   2.A   3.B   4.D   5.C   6.B   7.D   8.C   9.B  10.A

  11.120°   12.3x+y-1=0   13.   14.10    15.100    16.(1),(4)

17.解:(1)设抛物线,将(2,2)代入,得p=1. …………4分

∴y2=2x为所求的抛物线的方程.………………………………………………………5分

(2)联立 消去y,得到. ………………………………7分

设AB的中点为,则

∴ 点到准线l的距离.…………………………………9分

,…………………………11分

,故以AB为直径的圆与准线l相切.…………………… 12分

(注:本题第(2)也可用抛物线的定义法证明)

18.解:(1)在△ACF中,,即.………………………………5分

.又,∴.…………………… 7分

(2)

. ……………………………14分

(注:用坐标法证明,同样给分)

19.

解法一:(1)连OM,作OH⊥SM于H.

∵SM为斜高,∴M为BC的中点,∴BC⊥OM.

∵BC⊥SM,∴BC⊥平面SMO.

又OH⊥SM,∴OH⊥平面SBC.……… 2分

由题意,得

设SM=x,

,解之,即.………………… 5分

(2)设面EBC∩SD=F,取AD中点N,连SN,设SN∩EF=Q.

∵AD∥BC,∴AD∥面BEFC.而面SAD∩面BEFC=EF,∴AD∥EF.

又AD⊥SN,AD⊥NM,AD⊥面SMN.

从而EF⊥面SMN,∴EF⊥QS,且EF⊥QM.

∴∠SQM为所求二面角的平面角,记为α.……… 7分

由平几知识,得

,∴

,即所求二面角为. ……………… 10分

(3)存在一点P,使得OP⊥平面EBC.取SD的中点F,连FC,可得梯形EFCB,

取AD的中点G,连SG,GM,得等腰三角形SGM,O为GM的中点,

设SG∩EF=H,则H是EF的中点.

连HM,则HM为平面EFCB与平面SGM的交线.

又∵BC⊥SO,BC⊥GM,∴平面EFCB⊥平面SGM. …………… 12分

在平面SGM中,过O作OQ⊥HM,由两平面垂直的性质,可知OQ⊥平面EFCB.

而OQ平面SOM,在平面SOM中,延长OQ必与SM相交于一点,

故存在一点P,使得OP⊥平面EBC. ……………………… 14分

 

∵底面边长为1,∴

.    ……………… 1分

平面SBC的一个法向量

∴y=2h,n=(0,2h,1).… 3分

=(0,1,0),由题意,得.解得

∴斜高. …………………………………………………… 5分

(2)n=(0,2h,1)=

由对称性,面SAD的一个法向量为n1. ………………………………6分

设平面EBC的一个法向量n2=(x,y,1),由

,得

 解得.………………… 8分

设所求的锐二面角为α,则

,∴.…………… 10分

(3)存在满足题意的点.证明如下:

. ………………………… 11分

,令与n2共线,则. ……………… 13分

.故存在P∈SM,使OP⊥面EBC.……………………… 14分

20. 解:(1)当n为奇数时,an≥a,于是,. ………………3分

         当n为偶数时,a-1≥1,且an≥a2,于是

=. …………6分

(2)∵,∴公比.……9分

. …………………………………………10分

(注:如用求和公式,漏掉q=1的讨论,扣1分)

 . ……………12分

.……15分21.解:(1)∵,∴,∴. 1分

,即,∴. …3分

①当,即时,上式不成立.………………………………………………4分

②当,即时,.由条件,得到

,解得. ……………………………………………5分

,解得.…………………………………………6分

 m的取值范围是. ………………………………………7分

(2)有一个实根.………………………………………………………………………………9分

,即

,则

. ………………………10分

 △>0,故有相异两实根

,∴ 显然

,∴,∴. …………12分

于是

                    

为三次函数的极小值点,故与x轴只有一个交点.

∴  方程只有一个实根.…………………………15分

第四节:书面表达(满分12分)

       某期中学生报的“热点话题”专栏讨论的主题是“上大学是高中生的唯一选择吗?”你班同学就此话题进行了热烈讨论。请根据以下内容用英语给该专栏的编辑写一封信,介绍讨论结果,并发表你的个人意见。

观点

80%的学生认为应该上大学

20%的学生认为不必上大学

原因

能够增长知识

提高自身的品质

利于择业…

大学的学费昂贵

毕业后就业困难…

注意: 1. 信的开头已给出,但不计入总词数;

                  2. 词数:100左右。

Dear Editor,

       The students in our class had a discussion about whether the only way for senior high students to succeed is to go to college.

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

                                                                              Yours sincerely,

                                                                     Li Hua

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完形填空 I :阅读下面短文,掌握其大意,然后从各题所给的四个选项中,选出最佳选项。(共12个小题;每小题1分,满分12分)
A businessman was on a business trip to Tokyo. One day he went to    29   some gifts for his employees (雇员) back in his country. He    30   a supermarket. A lady smiled and   welcomed him. He was very touched, and felt good .Because he was not able to    31    the warmth of her smile, he was watching her as he was    32  . She was giving the same    33    to all the people who walked into the supermarket.
The businessman started thinking if she hated doing the same thing. So he asked, “My dear lady, are you    34    of doing this job?” The lady smiled and said, “No, sir, I have    35   here for the last 10 years and I love my job.” The businessman was surprised and asked, “Why have you stayed here for ten years, and why do you like your job   36    much?” The lady said, “Because I am doing something good for my    37   .” The businessman found this interesting. He asked, “Why?” The lady said, “As most of our customers are    38   , they spend foreign currency here. So our country has a lot of money and becomes richer. People who are happy with our service will 39  _____ more often and spend more money in our country.”
Surprised by her    40   , the businessman thanked her. After returning to his own country he worked hard to introduce the same attitude (态度) to his workers, and today his company is one of the best companies in the world.
29. A. buy                         B. order         C. make                    D. pay
30. A. opened                    B. cleaned       C. entered                  D. checked 
31. A. require                  B. enjoy               C. hide                         D. forget  
32. A. traveling                   B. working             C. singing                            D. shopping
33. A. feeling                  B. smile         C. warmth                    D. watching 
34. A. fond                         B. tired          C. proud                             D. afraid
35. A. stood                         B. learned       C. worked                    D. lived 
36. A. so                         B. very          C. even                        D. enough
37. A. home                        B. country        C. supermarket           D. boss 
38. A. millionaires            B. managers      C. strangers              D. foreigners 
39. A. praise                        B. visit          C. talk                         D. save 
40. A. action                        B. service        C. attitude                    D. help 

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