问题 解答题
已知二次函数f(x)=ax2+bx+1和函数g(x)=
bx-1
a2x+2b

(1)若f(x)为偶函数,试判断g(x)的奇偶性;
(2)若方程g(x)=x有两个不等的实根x1,x2(x1<x2),则
①试判断函数f(x)在区间(-1,1)上是否具有单调性,并说明理由;
②若方程f(x)=0的两实根为x3,x4(x3<x4),求使x3<x1<x2<x4成立的a的取值范围.
答案

(1)∵f(x)为偶函数,∴f(-x)=f(x),∴bx=0,∴b=0

g(x)=-

1
a2x
,∴函数g(x)为奇函数;(4分)

(2)①由g(x)=

bx-1
a2x+2b
=x得方程a2x2+bx+1=0(*)有不等实根

∴△=b2-4a2>0及a≠0得|

b
2a
|>1即-
b
2a
<-1或-
b
2a
>1
(7分)

又f(x)的对称轴x=-

b
2a
∉(-1,1)

故f(x)在(-1,1)上是单调函数(10分)

②x1,x2是方程(*)的根,∴a2x12+bx1+1=0

∴bx1=-a2x12-1,同理bx2=-a2x22-1

∴f(x1)=ax12+bx1+1=ax12-a2x12=(a-a2)x12

同理f(x2)=(a-a2)x22

要使x3<x1<x2<x4,只需

a>0
f(x1)<0
f(x2)<0
a>0
a-a2<0
,∴a>1

a<0
f(x1)>0
f(x2)>0
a<0
a-a2>0
,解集为φ

故a的取值范围a>1(16分)

完形填空
A lady wanted a birthday gift from her husband. For many months she had liked a beautiful   1 , and knowing her husband could   2  it, she told him that was all she wanted. On the morning of the   3 , her husband told her how   4  he was to have such a good wife, and how much he loved her. He   5  her a beautiful gift package (纸盒). She opened it and   6  a Bible.
  7 , she raised her voice and said to her husband, “With all your money, you give me a Bible?” She was so   8  that she left him.
Many years passed and the lady was very   9  in business. She owned two large companies. She   10  her husband was very old, and thought perhaps she should go to   11  him. But before she could, she   12  a letter telling her that he had died, and gave all of his possessions (财产) to her. She needed to come back and   13  things.
When she arrived at his house, sadness   14  her heart. She saw the still new
  15 , just as she had left it years before. With   16 , she opened it and began to turn the pages. A ring   17  from the Bible to the floor and a  18  could be seen. She picked it up and found it was the ring she wanted in those days. And on the card was the date of her birth, and the words “LUV U ALWAYS”.
  19  your gift is not packaged the way you want it, it’s because it is better packaged the way it is! Always appreciate little things; they usually lead you to bigger things! The best and most beautiful things in the world cannot be seen or even touched for they must be felt with the   20 .
小题1:
A.watchB.ringC.coatD.box
小题2:
A.affordB.likeC.designD.guess
小题3:
A.holidayB.performance C.birthdayD.meeting
小题4:
A.proudB.kindC.sorryD.brave
小题5:
A.lentB.askedC.gaveD.moved
小题6:
A.carriedB.foundC.missedD.held
小题7:
A.PolitelyB.LightlyC.PatientlyD.Angrily
小题8:
A.calmB.upsetC.gratefulD.careless
小题9:
A.failingB.lonelyC.successfulD.poor
小题10:
A.wonderedB.hopedC.realizedD.explained
小题11:
A.visitB.understandC.catchD.help
小题12:
A.passedB.wroteC.sentD.received
小题13:
A.take care ofB.pay forC.look forD.give away
小题14:
A.coveredB.brokeC.leftD.filled
小题15:
A.carB.BibleC.letterD.table
小题16:
A.smilesB.laughterC.tearsD.interests
小题17:
A.hid B.droppedC.disappearedD.made
小题18:
A.bookB.penC.flowerD.card
小题19:
A.If B.ButC.AlthoughD.So
小题20:
A.handB.eyeC.faceD.heart
单项选择题