己知在锐角△ABC中，角A，B，C所对的边分别为a、b、c，向量m=（a2+b2_爱下问搜题

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问题解答题

己知在锐角△ABC中，角A，B，C所对的边分别为a、b、c，向量

m

=（a²+b²-c²，ab），

n

=（sinC，-cosC），且

m

⊥

n

．
（I）求角C的大小；
（II）当c=1时，求a²+b²的取值范围．

答案

（Ⅰ）由

m

⊥

n

得：（a²+b²-c²）sinC-ab•cosC=0，…（2分）

结合余弦定理得：sinC=

1

2

，∴C=30°（∵C是锐角）．…（5分）

（Ⅱ）由正弦定理得：

a

sinA

=

b

sinB

=

c

sinC

=

1

sin30°

=2，…（7分）

∴a=2sinA，b=2sinB=sin（150°-A）=2sin（A+30°）．

∴a²+b²=4sin²A+4 sin²（A+30°）=2（1-cos2A）+2[1-2cos（2A+60°）]=4-2cos2A-2cos60°cos2A+2sin60°sin2A

=4cos2A-cos2A+

3

sin2A=4+

3

sin2A-3cos2A=4+2

3

sin（2A-60°）．…（10分）

∵△ABC是锐角三角形，由0°＜A＜90°及 0°＜B=150°-A＜90°，得：60°＜A＜90°，120°＜2A＜180°，

从而 60°＜2A-60°＜120°，

3

2

＜sin（ 2A-60°）≤1，3＜2

3

sin（ 2A-60°）≤2

3

，故7＜4+2

3

sin（2A-60°）≤4+2

3

，

即a²+b²的取值范围是（7，4+2

3

）．…（12分）

单项选择题

GIS最基本的分析功能是()。

A.根据属性对空间要素分类

B.产生人口密度图

C.产生土地使用图

D.产生环境质量图

完形填空

The world would be a better place if we all had children’s eyes. This is not because children’s eyes are too sharp to allow any blot（污点，瑕疵）to pass without being     , but because adults have too many conflicting concerns to keep their eyes open to all wrongdoing.
Not long before, an examination, intended to     some police officers to higher positions, was held in Gansu province. Someone     with a good idea that 18 primary pupils could be invited to act as invigilators (监考人). At that age, students would do as they are told and act fearlessly.      , they caught 25 officers cheating on the spot, which has caused a stir (轰动，搅动) and      the question of whether adult invigilators would      their duties as well.
The implication of this event goes      its impact on the work style of local police officers and the examinations. Most people said what these children have done points to the sad fact of      being less trustworthy. If adults acted as invigilators, they would quite      turn a blind eye to cheating.
We Chinese have been holding the      that human nature was good at birth: as      of blot as a sheet of blank paper, which can be      with content (good or bad) in the process of growing up. Everything should be done on the basis of certain principles(原则).      , nowadays the repeated reports of cheating in examinations, selling ranks and      and embezzling (挪用) public funds paint a      picture of common social morals.    Adults have become too concerned with their own personal gains to      basic sense of principles. When no one      any principle and the only concern is personal interest, everyone will      , not only those who first break the rules.
Besides, cheats are unfair to honest examinees, and when the success of cheats      more people to cheat, the consequences will be disastrous.
In a word, we adults need to have our eyes as      as children’s on matters of principle.

B．investigate

A．Surprisingly

B．recommended

A．opportunity

C．contribute to

D．come across

A．understands