问题 解答题

已知集合A={x|x2+a≤(a+1)x,a∈R}.

(Ⅰ)求A;

(Ⅱ)若a>0,以a为首项,a为公比的等比数列前n项和记为Sn,对于任意的n∈N+,均有Sn∈A,求a的取值范围.

答案

(Ⅰ)A={x|x2+a≤(a+1)x,a∈R}={x|(x-1)(x-a)≤0,a∈R}.

(1)a≥1时,A={x|1≤x≤a};

(2)a<1时,A={x|a≤x≤1}

(Ⅱ)(i)当a≥1时,A={x|1≤x≤a}.

而S2=a+a2>a,S2∉A,故a≥1时,不存在满足条件的a;

(ii)当0<a<1时,A={a≤x≤1},Sn=

a(1-an)
1-a
Sn-a=
a(1-an)
1-a
-a=
a2-an+1
1-a
≥0
,∴Sn≥a,

又an>0,∴Sn

a
1-a

对任意的n∈N+,Sn∈A,只须a满足

0<a<1
a
1-a
≤1
,解得0<a≤
1
2

综上所述,a的取值范围是0<a≤

1
2

阅读理解

读下面的一篇短文,然后根据短文内容,将问题的最佳答案填写在题目前的括号里。

    One day we invited some friends to dinner. When it was about six o'clock, my wife found that we had

little bread.

    So she asked our five-year-old daughter, Kathy, to buy some.

    "Here's a dollar," my wife said," get two loaves (条) of bread, if they have it. If they don't have it, get

anything. But hurry!" Kathy hurried off, and we waited and waited. By a quarter past six, all the friends had

arrived. But Kathy had not returned. Finally we saw her dancing around the corner with a bright new hula

hoop (呼啦圈) whirling (旋转) round her middle.

    "Kathy!" Her mother cried. "Where did you get that hula hoop, and where is the bread?"

    "Well, when I got to the shop, they had sold all their bread," answered Kathy. "And you said 'If they don't

have that, get anything!'"

1. One day Kathy's parents invited some friends to ______. [ ]

A. has breakfast

B. have lunch

C. has supper

D. have a party

2. How much did Kathy's mother give her? [ ]

A. a dollar

B. two dollars

C. five dollars

D. six dollars

3. Kathy ______ in a hurry. [ ]

A. played games

B. went to the shop

C. went to dance

D. came back home

4. What did Kathy buy at last? [ ]

A. other food

B. nothing

C. hula hoop

D. two loaves of bread

5. Why did she do this? [ ]

A. because the shop had sold all their bread.

B. because her mother asked her to get anything if they hadn't had bread.

C. because she loved playing with hula hoop.

D. all above.

单项选择题 B1型题