7. 設函數 f (x) =│log x│ ，若 a ≠b  ，且 ，求數對 (a ,b ) ?

(1)試證 f (x)  =0 在 0<x<2之間恰有一實根 t 。 

(2)三直線 x = 0， x = 2， y = 0 與曲線 y =f (x) 所圍成區域的面積為 S ，試以 t 表示。

(3)試求當 t 為多少時 S 有最小值，並求出 S 的最小值。 

9.若數列 ＜an  中每一項均為正數，設數列 之前項 n 的和為 。已知 ，試求 及 (皆以 n 的式子表示)。

10.在坐標平面上，已知 為三個非零向量，其中 ， 上的正射影長為 1 ，若， 試求的最大可能值。

II. Open-ended Question (5 points per question)1. Present 5 cloze test questions in this passage, with answers.As one of the most renowned French writers from the Romantic period, Victor Hugo’s literary impact has cemented his place in history. It took him 17 years to complete Les Misérables, a novel in which he speaks passionately about social injustice.The novel follows the lives of several characters, with its focus set on the struggles of ex-convict Jean Valjean. Arrested for stealing a loaf of bread to feed his starving family, Jean Valjean was forced to serve a harsh sentence. Although he strove to regain his faith in the world after his release, he didn’t seem to be treated fairly. He had suffered numerous ruthless insults by the time he encountered the bishop of Digne, whose generous and merciful act helped Valjean realize life was not a bed of roses and that he couldn’t turn over a new leaf until he cast out the hatred he had for the world. Yearly later, Cossette, whom he adopted and treated as though she were his real daughter, fell in love with a student revolutionary, Marius. In order not to stain the young couple with his own criminal history, Valjean chose to leave. It was not until Valjean’s final moments that Cossette and Marius learned how much he had done for them.Though criticized greatly at the time, Les Misérables has remained Hugo’s enduring work, being adapted for stage, film, as well as television and causing social change in his own time and beyond. By weaving human emotion and drama into the social and political conflict of nineteenth century France, the novel is littered with examples of conscientious actions motivated by the desire to improve the lives of others through love, forgiveness and compassion. Such is the impact of this novel that people all over the world pay tribute to this literary master.

2.Writing and Cloze Test Design:Please paraphrase the following passage into a cloze test, ensuring the length is between 200 and 300 words. Afterward, create five multiple-choice questions, each with four options, which would be appropriate for vocational high school freshmen.Minecraft YouTuber Technoblade has died at the age of 23 following a year-long battle with cancer, his family announced Friday. In a YouTube video entitled “so long nerds,” the content creator’s father shared a final message from his son with his subscribers. “Hello everyone, Technoblade here,” the message began. “If you’re watching this. I am dead.” In the video, Technoblade, who has kept his identity a secret, revealed that his real name was Alex. He also thanked fans for supporting his content, buying merchandise, and channel memberships and joked that his siblings “are going to college.”“If I had another hundred lives, I think I would choose to be Technoblade again every single time, as those were the happiest years of my life. I hope you guys enjoyed my content and that I made some of you laugh and I hope you all go on to live long, prosperous and happy lives.”In just under a decade, Technoblade amassed nearly 11 million YouTube subscribers and 3.7 million Twitter followers as he shared his passion for Minecraft, a videogame where users can build virtual worlds out of blocks and create their own storylines. He went public with his cancer diagnosis back in August 2021, telling his followers during a video of himself playing the game that he sought medical help after experiencing pain in his arm that he initially believed was repetitive stress injury from gaming.According to his father, Technoblade wrote his final message approximately eight hours before he died. Holding back tears, Technoblade’s father described his son as “the most amazing kid” anyone could ever ask for, adding: “I miss Technoblade.” The video, which has been viewed more than 19 million times, ended with a statement from Technoblade’s family, which read: “This past year had a lot of rough spots for our son as he battled stage four cancer. But he didn’t complain, and kept using his famous strategic mind to try to beat what he knew were almost impossible odds. My son’s bravery on this path was a shining lesson to all of us who were privileged to walk it with him. Thank you for sharing his journey through it all, as he did the work he loved for his beloved fans.”Social media has been flooded with tributes from fans, celebrities and fellow gamers. YouTuber Ted Nivison remembered him as: “Effortlessly funny. Endlessly talented. Gone too soon. Rest easy, Technoblade.”In a statement sent to CNN, Sarcoma Foundation of America paid tribute to Technoblade, sharing “deepest condolences with his family.” Brandi Felser, CEO of the charity, said in the statement: “Technoblade rallied his supporters to raise over $500,000 to date to support sarcoma research. This allowed SFA to provide research grants in Technoblade’s name. Through these efforts, he leaves a legacy that will impact generations.” Felser said that Technoblade will be recognized with the organization’s Courage Award, an annual honor given to individuals who inspire others “to create hope and a better life for the sarcoma community” later this year.

3.English-to-Chinese Translation:Even though renewable energy sources like wind and solar are growing rapidly around the world, demand for electricity has been rising even faster, which means countries are still burning more fossil fuels each year. Geopolitical conflicts, from the U.S.-China rivalry to war in places like Ukraine and the Gaza Strip, have made international cooperation on climate change harder. And rich countries have failed to keep their financial promises to help poor countries shift away from oil, gas and coal. The new U.N. report finds that at least 151 countries have formally pledged under the Paris climate agreement to curb their greenhouse gas emissions by 2030. If every country followed through on its stated plans, which is far from assured, then global emissions could be 3% to 11% lower at the end of the decade than they are today.

4.Chinese-to-English Translation:教育部建議每學期最高可請三天的心理健康假，且高中生無需提交任何證明即可以申請半天或全天休假，但需得到父母的許可。因為大多數高中生和職業學校學生仍然是未成年人，必需要出示父母或監護人的同意證明才能請假。該計劃是為了回應台灣年輕人對心理健康問題日益增長的擔憂。允許心理健康日有助於學生更了解自己的情緒狀態，尤其是當他們在學校精疲力盡的時候。教育部也鼓勵學生尋求專業協助，同時呼籲學校不要歧視申請心理健康假的學生，而是提供必要的協助。

5.Chinese-to-English Translation:湧入的俄羅斯旅客為土耳其的旅遊業注入強心針，遊客數增加超過 5 倍。土耳其的 18 個聯合國教科文組織世界遺產涵蓋了令人驚嘆的海濱景致、建築奇觀、古代遺址和溫泉，這裡絕對是愛冒險旅行者的理想目的地。

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