问题 填空题
下列命题:
①若
a
b
共线,则存在唯一的实数λ,使
b
a

②空间中,向量
a
b
c
共面,则它们所在直线也共面;
③P是△ABC所在平面外一点,O是点P在平面ABC上的射影.若PA、PB、PC两两垂直,则O是△ABC垂心.
④若A,B,C三点不共线,O是平面ABC外一点.
OM
=
1
3
OA
+
1
3
OB
+
1
3
OC
,则点M一定在平面ABC上,且在△ABC内部.
上述命题中正确的命题是______.
答案

①中的

a
0
这一条件缺少,于是①错.

对于②,因为向量可以任意平移,可知②错;

③当PA,PB,PC两两互相垂直时,则PA⊥平面PBC,则PA⊥BC,

又由PO⊥底面ABC,则PO⊥BC,进而BC⊥平面PAO,即AO⊥BC,

同理可证BO⊥AC,CO⊥AB,故O是△ABC的垂心,即③对;

④中A、B、C、M四点共面.

等式

OM
=
1
3
OA
+
1
3
OB
+
1
3
OC
两边同加
MO

1
3
MO
+
OA
)+
1
3
MO
+
OB
)+
1
3
MO
+
OC
)=
0

MA
+
MB
+
MC
=
0
MA
=-(
MB
+
MC
)则
MA
MB
MC
共面,

又M是三个有向线段的公共点,

则点M一定在平面ABC上,且在△ABC内部.

故④是真命题.

故答案为:③④

选择题
阅读理解

阅读理解。

     Is early childhood education really necessary? Early childhood education primarily focuses on learning

through playing to develop the child's physical, sensory (感觉的), communicational and social development.

Early childhood education has become a concern of the government, who pushes poor children to be formally

trained before they are old enough for the kindergarten.

     There are good reasons for the government to push early childhood education. Studies have shown that

orphan children who did not receive good care and education become developmentally delayed causing failure

in school, and even in life. Further studies show that poor children who take part in Head Start programs are

more prepared for school, less likely to end up in Special Education classes, and are less likely to receive public

help or go to jail.

     There are also negative parts of putting a child in formal education programs too early. Time Magazine online

explains that the younger the child, the less his chances of catching up with first-grade work. I have personally

witnessed many children of my generation who went to Head Start programs became frustrated and bored with

school before they finished high school. Yet parents have been sure that the earlier the child starts school, the

better off he or she is, so they push to start childhood education earlier.

     Actually while early formal education of poor children does show great gains in the early elementary years,

studies also show that this head start is really a"false start", as the gains are lost in middle and high school years.

It seems that environment is a bigger factor on life's success than early education. Head Start programs have

not achieved its original goal in closing the achievement gap in poor and middle school children. Perhaps it is

time to find other ways to close that gap.

1. The purpose of the government valuing early childhood education is _____.

A. to develop the children's communication

B. to give orphaned children good care and education

C. to get the children prepared for the kindergarten

D. to train the children formally and regularly

2. According to the studies which of the following is True?

A. Children with good early education may have a rich life.

B. The orphaned children are usually unhappy in their life.

C. Children without early education can do well at school.

D. Head Start programs are helpful to the poor children.

3. The author's attitude toward early childhood education is _____.

A. puzzling

B. objective

C. indifferent

D. approving

4. The main idea of the last passage is that _____.

A. early childhood education is very necessary

B. Head Start programs have helped the children a lot

C. environment is the most important to life's success

D. better ways should be found to help the poor children