2022年澳门四高校联合入学考试数学正卷 附加卷（pdf版 含答案）

澳門四高校聯合入學考試（語言科及數學科） Joint Admission Examination for Macao Four Higher Education Institutions (Languages and Mathematics) 2022 年試題及參考答案 2022 Examination Paper and Suggested Answer 數學附加卷 Mathematics Supplementary Paper 注意事項： 1. 考生獲發文件如下: 1.1 本考卷包括封面共 22 版 1.2 草稿紙一張 2. 請於本考卷封面填寫聯考編號、考場、樓宇、考室及座號。 3. 本考卷共有五條解答題，每題二十分，任擇三題作答。全卷滿分為六十分。 4. 必須在考卷內提供的橫間頁內作答，寫在其他地方的答案將不會獲評分。 5. 必須將解題步驟清楚寫出。只當答案和所有步驟正確而清楚地表示出來，考生方可獲 得滿分。 6. 本考卷的圖形並非按比例繪畫。 7. 考試中不可使用任何形式的計算機。 8. 請用藍色或黑色原子筆作答。 9. 考試完畢，考生須交回本考卷及草稿紙。 Instructions: 1. Each candidate is provided with the following documents: 1.1 Question paper including cover page – 22 pages 1.2 One sheet of draft paper 2. Fill in your JAE No., campus, building, room and seat no. on the front page of the examination paper. 3. There are 5 questions in this paper, each carries 20 marks. Answer any 3 questions. Full mark of this paper is 60. 4. Put your answers in the lined pages provided. Answers put elsewhere will not be marked. 5. Show all your steps in getting to the answer. Full credits will be given only if the answer and all the steps are correct and clearly shown. 6. The diagrams in this examination paper are not drawn to scale. 7. Calculators of any kind are not allowed in the examination. 8. Answer the questions with a blue or black ball pen. 9. Candidates must return the question paper and draft paper at the end of the examination. 2 任擇三題作答，每題二十分。請把答案寫在緊接每條題目之後的 3 頁橫間頁內。 Answer any 3 questions, each carries 20 marks. Write down the answers on the 3 lined pages following each question. 1. P C E B A D 如上圖所示，在三棱錐 P-ABC 中，PC 垂直 ABC，|| = 3，∠ = ， D 和 E 分別是 AB 和 BC 上的點，且 | | = | | = √2，|| = 2，|| = 1。 (a) 求 ||。[提示: 設 M 為 CE 的中點。] (6 分) (b) (i) 證明 ∠ = 。 (2 分) (ii) 求 ∠，答案以 cos 表示。 (5 分) (c) 求二面角 P-AB-C，答案以 tan 表示。 (7 分) [提示: 設 X 為 AB 上一點，使得 CX 與 AB 垂直。證明 PX 與 AB 垂直。] As shown in the above figure, P-ABC is a triangular pyramid, PC is perpendicular to ABC, || = 3, ∠ = . The points D and E are on AB and BC, respectively, and | | = | | = √2, || = 2,|| = 1. (a) Find ||. [Hint. Let M be the midpoint of CE.] (6 marks) (b) (i) Show that ∠ = . (2 marks) (ii) Find ∠ . Express your answer in terms of cos. (5 marks) (c) Find the dihedral angle P-AB-C. Express you answer in terms of tan. [Hint: Let X be the point on AB such that CX and AB are perpendicular. Show that PX and AB are perpendicular.] (7 marks) 3 2. (a) 已知函數 ()= 3 + 2，且 (1)= 0。 (i) 求方程 ()= 0 的所有解。 (3 分) (ii) 求 ′() 及 ′(′)。 (2 分) (iii) 求 f (x) 的局部極大值和局部極小值。 (3 分) (iv) 求曲線 = () 的拐點。 (2 分) (v) 運用 (i) – (iv) 的結果，繪出曲線 = ()。 (3 分) (b) 求由曲線 = ... ...