问题 解答题
已知向量m=(sinA,  
1
2
)
n=(3,  sinA+
3
cosA)
共线,其中A是△ABC的内角.
(1)求角A的大小;
(2)若BC=2,求△ABC面积S的最大值,并判断S取得最大值时△ABC的形状.
答案

(1)因为

m
n
,所以sinA•(sinA+
3
cosA)-
3
2
=0

所以

1-cos2A
2
+
3
2
sin2A-
3
2
=0,

3
2
sin2A-
1
2
cos2A=1,

sin(2A-

π
6
)=1.

因为A∈(0,π),所以2A-

π
6
∈(-
π
6
,  
11π
6
).

2A-

π
6
=
π
2
A=
π
3

(2)由余弦定理,得4=b2+c2-bc.

S△ABC=

1
2
bcsinA=
3
4
bc,

而b2+c2≥2bc⇒bc+4≥2bc⇒bc≤4,(当且仅当b=c时等号成立)

所以S△ABC=

1
2
bcsinA=
3
4
bc≤
3
4
×4=
3

当△ABC的面积取最大值时,b=c.又A=

π
3

故此时△ABC为等边三角形.

完形填空

Who’s in control of your life? Who is pulling your string? For the majority of us, it’s other people—society, colleagues, friend, family or our religious community. We learned this way of operating when we were very young, of course. We were brainwashed. We discovered that feeling important and feeling accepted was a nice experience and so we learned to do everything we could to make other people like us.

So when people tell us how wonderful we are, it makes us feel good. We long for this good feeling like a drug—we are addicted to (沉溺于) it and seek it out wherever we can. Therefore, we are so eager for the approval of others that we live unhappy and limited lives, failing to do the things we really want to. Just as drug addicts and alcoholics live worsened lives to keep getting their fix (一剂毒品), we worsen our own existence to get our own constant fix of approval.

But just as with any drug, there is a price to pay. The price of the approval drug is freedom—the freedom to be ourselves. The truth is that we cannot control what other people think. People have their own agenda(心思,计谋), and they come with their own baggage and, in the end, they’re more interested in themselves than in you. Furthermore, if we try to live by the opinions of others, we will build our life on sinking sand. Everyone has a different way of thinking, and people change their opinions all the time. The person who tries to please everyone will only end up getting exhausted and probably pleasing no one in the process.

So how can we take back control? I think there’s only one way—make a conscious decision to stop caring what other people think. We should guide ourselves by means of a set of values—not values imposed(强加) from the outside by others, but innate values which come from within. If we are driven by these values and not by the changing opinions and value system of others, we will live a more authentic, effective, purposeful and happy life.

(Note: Answer the questions in NO MORE THAN TWELVE WORDS.)

小题1:What happens to people if they care too much for the approval of others?

小题2:The price of the approval of others is ______________

小题3:Why is it unreliable for us to live by the opinions of others?

小题4:What’s the main idea of this article?

单项选择题