2. Find [sin(In x)dx.(10分)

3. Find . Hint: Use the polar coordinate system. (10 分)

4. Determine the minimum area of a triangle form by the x axis, the y axis, and a line through the point (a,b). Here, a and b are positive constants. (10 分)

5. Find .

6. Find In x dx. This is the area of the region bounded by the graph of y= xInx and the x axis.(15分)

7. Find the directional derivative of the function f(x,y)=(x+y)/(x-y) at point (1,-1) in the direction v=.(10分)

8. Find the area of the region T bounded by the parabolas y=x2 and y=4-x2. (10分)

9. Find the point within a given triangle for which the sum of the squares of its perpendicular distances to the sides is least. ( 15 分)

I. Translation (20%) Please translate the following Chinese passage into English1. 隨著全球氣候變遷問題日益嚴重，歐洲多國加速推動綠能轉型。德國、法國 和丹麥等國家在風能和太陽能領域的投資顯著增加，並制定了到 2030 年大幅 降低碳排放的目標。德國計劃逐步淘汰煤炭發電，並擴建海上風電場；法國 則加強核能和可再生能源的佈局，實現能源結構多元化；丹麥致力於成為世 界上首個完全依靠綠能供電的國家。這些國家的綠能政策不僅包括技術研發 和基礎設施建設，還涉及對企業和家庭的補貼措施，鼓勵更多人使用新能源。 同時，歐盟也在推動成員國之間的能源合作，建立跨國電網，提升能源利用效率。2. 文化多樣性是指欣賞社會由許多不同群體組成，這些群體擁有不同的興趣、 技能、才能和需求。這也意味著你認可社會中的人可能有不同的宗教信仰和 政治取向。

II. Essay Writing (20%) Educators and students alike have benefited greatly from digital learning in today's rapidly changing educational landscape. Please provide 2 digital resources, such as tools and platforms, as examples to support this dynamic shift in teaching English from traditional classrooms to virtual learning environments. Please explain how the 2 resources can help learners enhance their listening, speaking, reading, or writing skills.

III. Examination Questions Design(20%) Please design GSAT mixed questions (學測混合題型) worth ten points in total based on the following passage. Point allocation must be specified, and answer keys provided.        Scientists have discovered a new planet that may be able to support human life, according to a new study.        The study, published in Monthly Notices of the Royal Astronomical Society on Thursday, named the new planet Gliese 12 b. Scientists said in the study that the planet is “the nearest, transiting temperate, Earth-sized planet found to date.”        The international team of scientists discovered the planet by using NASA’s Transiting Exoplanet Survey Satellite (TESS). Scientists say that it’s one of the few rocky planets discovered that has the potential for humans to survive on, but that the planet is 40 light years away.        Gliese 12 b is only a little smaller than Earth. It’s an “exoplanet,” which means that it’s a planet outside of our solar system. The planet orbits a small and cool red dwarf star. The planet also bears some similarities to Venus, which is often dubbed Earth’s “twin” because of the similarities they share.       A year on Gliese 12 b is just 12.8 days on Earth because the planet orbits its star so closely. The planet receives about 1.6 times more energy from its star than Earth does from the sun, scientists said.        The planet has an average estimated surface temperature only 50°F warmer 5 than Earth’s.                    “Thrillingly, this planet is the closest Earth-sized and temperature planet we know,” Thomas Wilson, one of the study’s authors, told BBC. “Planets like Gliese 12 b are very few and far between, so for us to be able to examine one this closely and learn about its atmosphere and temperature is very rare.”        In order to find out if Gliese 12 b could support human life, scientists will need to determine if it has an atmosphere similar to Earth—one that could have water on its surface, since that is essential for a planet to be habitable. But scientists said they were still unsure what kind of atmosphere it has—if any at all. Scientists said the planet was a good candidate for further atmospheric study.Source: https://reurl.cc/r9KxXk

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