问题 解答题
一动圆和直线l:x=-
1
2
相切,并且经过点F(
1
2
,0)

(Ⅰ)求动圆的圆心θ的轨迹C的方程;
(Ⅱ)若过点P(2,0)且斜率为k的直线交曲线C于M(x1,y1),N(x2,y2)两点.
求证:OM⊥ON.
答案

( I)∵动圆和直线l:x=-

1
2
相切,并且经过点F(
1
2
,0)

∴圆心θ到F(

1
2
,0)的距离等于θ到定直线l:x=-
1
2
的距离,都等于圆的半径…(2分)

根据抛物线的定义,可得:圆心θ的轨迹C就是以F为焦点,l为准线的抛物线,…(3分)

设抛物线方程为y2=2px,其中

p
2
=
1
2
,解得p=1

∴抛物线方程是y2=2x,即为所求轨迹C的方程.…(6分)

( II)证明:设过点P(2,0)且斜率为k的直线的方程为

y=k(x-2)(k≠0)①…(7分)

代入y2=2x消去y,可得k2x2-2(k2+1)x+4k2=0.②…(8分)

由根与系数的关系,得x1x2=

4k2
k2
=4.…(9分)

结合y12=2x1y22=2x2,可得y1y2=

4x2x2
=2
x2x2
=4.…(10分)

OM
ON
=x1x2+y1y2=4-4=0,

由此可得向量

OM
ON
夹角为90°,即OM⊥ON.…(12分)

选择题
完形填空

Made in the USA: An Export Boom
In his State of the Union address two years ago, President Obama argued that in order to recover from the economic recession, one of the few things the U.S. needed to do was to export more goods around the world. That night, the president unveiled a new goal: to double U.S. exports over the next five years. It would be an increase that the president said would “support two million jobs in America.”
Most economists dismissed the promise at the time as something unrealistic, but two years later, the U.S. is on pace to meet that goal. American exports are up 34 percent since the president gave that speech, and the number continues to rise.
Competitive In A Global Market
Marlin Steel, a metal working business in Baltimore, makes parts that ship all across the world.“ We export to 36 countries,” owner Drew Greenblatt tells All Things Considered Host Guy Raz. “We're working around the clock, and we're growing.”
It's not just advanced manufacturing exports on the rise, but pork, cattle and all kinds of agricultural exports are up as well. Even American craft beer has found an export market.
Flying Dog CEO Jim Caruso says that increasingly, people all over the world are trying the beer from the Maryland-based brewery. Caruso says,“Even in those top beer-producing countries, a competitive American product is finding a market.”
Services Are Exports, Too
Another place exports are coming from is New York City—in particular, the 30th floor of a Manhattan skyscraper on 5th Avenue and 52nd Street. That's where the consulting firm Kurt Solomon lies. It doesn't actually produce a product for export; it provides management advice and strategy.
“Four out of every five Americans is now employed in the service industry,” the nation's top trade official, Ron Kirk says, “Services are a significant part of our exports, and make up about a quarter of our exported goods.” These services can include everything from legal consulting, finance, information technology and even engineering.
And There Are Other Factors
So why has there been an increase of more than 30 percent for exports in almost everything? Part of the increase, at least for the manufacturing side, is due to better technology, says Tyler Cowen, an economist. “A lot of it is being driven by smart machines,” he tells Raz,“The U.S. has high wage rates, which is a disadvantage, but if machines are doing a lot of the work, that doesn't matter.”
China factors a lot in America's export economy, too. “Wages in China have been going up as the country becomes more productive. Thus China is losing the cheap labor advantage it has held for some time.” Cowen says.
Will Jobs Grow, too?
“Companies have become more productive by dismissing workers and lowering costs.” Cowen says, “So I don't view exporting as a way of creating a very large number of jobs, but it will create more profits.”
So not every business or worker is necessarily benefiting from the export boom in the U.S., and Cowen says that could ultimately lead to a polarization(两极) of economic outcomes.
Made In the USA: An Export Boom
Outline
Details
The purpose of  increasing exports
*To help America make a 小题1:______ from the economic recession
*To help raise the nation's小题2:______ rate
The current situation
*American exports have risen 小题3:______ thirty-four percent up to now
*There has been an increase in exports in everything
*The export boom does not necessarily 小题4:_____ every business or worker
小题5:______ contributing to the export boom
小题6:______products
Even in those top beer-producing countries, people try craft beer from Flying Dog, a brewery 小题7:_____ in Maryland.
Various products
A variety of products are provided around the world,services 小题8:______ for 25%
Lower costs
*小题9:_____ take the place of labor, helping companies reduce wages
*China, who used to take小题10:______ of cheap labor, has given way to America in exports to some extent because of its increasing production