2019年澳门四高校联合入学考试数学附加卷

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) 2019年試題及參考答案 2019 Examination Paper and Suggested Answer 數學附加卷 Mathematics Supplementary Paper 2 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. 3 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. 如上圖所示，PAB 是邊長為 1 的等邊三角形，ABCD 是一長方形， | | 2BC  ，平面 PAB 與平面 ABC 垂直，M 和 N 分別為 AD 和 AB 的中點。 (a) (i) 證明 PN 垂直平面 ABC，從而求三棱錐 P-BCM 的體積。 (5 分) (ii) 求三角形 PBM 的面積，從而求點 C 至平面 PBM 的距離。 (6 分) (b) (i) 證明 NMC 是一直角。[提示: 考慮三角形 NMC 的邊長。] (4 分) (ii) 求二面角 N MC P  。[提示: 證明 PM MC 。] (5 分) In the above figure, PAB is an equilateral triangle with edge length 1, ABCD is a rectangl