问题 填空题
已知关于x的方程x3+ax2+bx+c=0的三个实根可作为一个椭圆、一条双曲线和一条抛物线的离心率,则
b-1
a+1
的取值范围为______.
答案

依题意,关于x的方程 x3+ax2+bx+c=0有一个根是1

所以可设x3+ax2+bx+c=0=(x-1)(x2+mx+n)

根据多项式恒等的充要条件,得

m-1=a①

n-m=b②

n+c=0③

取①②两式联立得

m=a+1,n=a+b+1

构造函数 f(x)=x2+mx+n 即 f(x)=x2+(a+1)x+(a+b+1)

依题意f(x)=0的两个根x1,x2分别作为椭圆和双曲线的离心率

故 0<x1<1<x2

根据一元二次方程根的分布,可得关于实系数a,b的约束条件:

判别式=(a+1)2-4(a+b+1)=(a-1)2-4b-4>0

f(0)=a+b+1>0,f(1)=2a+b+3<0

令a为横轴,b为纵轴,建立平面直角坐标系,作出这三个不等式所对应的平面区域S,

设P(a,b)是平面区域S内的任意一点,A(-1,1),k=

b-1
a+1

则k的几何意义是直线PA的斜率.

作图,得-2<k<0

故答案为(-2,0)

阅读理解

阅读理解。

    Jack Brown, an office worker, lives in Washington. He inherited (继承) a million dollars when he was

23, but he wasn't happy at all. When his college friends were looking for their jobs, he didn't have to. Jack

decided to live a simple life like everyone else. He gave some of his money to a charity (慈善机构) to help

poor children live a better life. Today he is 36. He still wears cheap shoes and clothes and drives a small car

only, but he is very happy.

    Up to now Jack has helped some children from poor countries all over the world, by sending them each

$ 200 a mouth. The money was used for the children's study, food, medicine and clothing. Jack receives a

report each year on the children's progress (进步). They can write to each other, but usually the children

do not speak English. 

    When Jack first heard about these children, he wanted to help them. "It was nothing special," he said.

"Until I went to these countries and met the children I was helping, I didn't know anything about their life."

Once Jack went to meet a little girl in Africa. He said that the meeting was very exciting. "When I met her.

I felt very, very happy." He said. "And I saw that the money was used for a good cause. It brought me

happiness. I want to do everything I can to go on helping those children."

1. Jack has sent them each $200 a month to help some poor children ______. [ ]

A. around the world 

B. Only in England

C. Only in Africa 

D. Only in Washington

2. Jack learned about the children's progress by ______. [ ]

A. visiting them each year      

B. getting a report every year

C. staying with them        

D. talking to them in English

3. The underlined (划线的) words "for a good cause" means "______" in Chinese. [ ]

A. 施舍  

B. 怜悯   

C. 行善   

D. 赏赐

4. All the details in this passage support the main idea that ______. [ ]

A. it's the most important to help the children in Africa

B. it's necessary to write letters to poor children

C. live a simple life, and you can give others help

D. help others, and you will feel happy

单项选择题