问题 解答题

已知抛物线y=ax2-1上存在关于直线x+y=0成轴对称的两点,试求实数a的取值范围.

答案

设抛物线上关于直线l对称的两相异点为P(x1,y1)、Q(x2,y2),线段PQ的中点为M(x0,y0),设

直线PQ的方程为y=x+b,由于P、Q两点存在,

所以方程组

y=x+b
y=ax2-1
有两组不同的实数解,即得方程ax2-x-(1+b)=0.①

判别式△=1+4a(1+b)>0.②

由①得x0=

x1+x2
2
=
1
2a
,y0=x0+b=
1
2a
+b.

∵M∈l,∴0=x0+y0=

1
2a
+
1
2a
+b,

即b=-

1
a
,代入②解得a>
3
4

故实数a的取值范围(

3
4
,+∞)

阅读理解

To most of us, school means classes, teachers, schedules, grades and tests. But for the children at Sudbury Valley School in Massachusetts, school is very different.

Firstly, there are no lessons. All the children, aged between 4 and 19, do whatever they want. There are no teachers, only “staff members”. The idea behind this is that you do not need to make children learn, because children want to learn anyway. “You do not need to say to a three-year-old, ‘Go explore your environment.’ You can’t stop them!”says Daniel Greenberg, a founder of the school. “But if you make children do what you want all day… they will lose all taste of learning.” At Sudbury Valley School, you will find children talking, reading, painting, cooking, working on computers, studying French, playing the piano, climbing trees, or just running around. Two boys spent three years just fishing!

The other way that Sudbury Valley School is different is that the children can decide the rules. Every week, there is a school meeting where both children and staff have one vote (投票)each—even the four-year-olds. They decide the school rules, how to spend the school money, and even which staff they want and do not want anymore.

When the school first opened in 1968, people said it would never work. But today, the school has 200 students, and 80% of its students go on to college. Even the two boys who went fishing all the time have successful careers today. One of them is a musician and the other is a computer scientist.

小题1:What's special about the school?

A.There is nobody but students there.

B.All the students are of the same age.

C.There are no classes for the students.

D.There are no rules in the school.小题2:What do the students do in the school?

A.They learn everything there from the staff members.

B.They grow up there without learning anything.

C.They spend all the time with their parents.

D.They do whatever they are interested in.小题3:The school seems to believe that children________.

A.learn best when they are doing what they want

B.are too young to learn anything by themselves

C.can look after themselves without any help

D.can't go to college without any hard work小题4:The last paragraph wants to tell us that_         .

A.the writer doesn't like the idea of the school

B.the school is quite successful in its work

C.most of the parents don't believe in the school

D.younger students learn better there than older ones小题5:The underlined word career in the last paragraph most probably means_.

A. family   B. money    C. lifestyle   d. job

判断题