Chapter 2 More about Percentages 某基金投資公司的李主任正在向陳先生推銷基金 李主任 : 我們現正代理兩種新推出的保證基金, 回報率如下 : (1) 生物科技基金 每年保證有 4% 的複式增長 ; (2) 環球債券基金 保證 3 年後有 15% 的增長 陳先生 : 我打算投資三年, 該買哪一種基金呢? 答案.. (1 + 4%) 3 1.125 < 1 + 15% 應購買 (2) 環球債券基金 1. 利息的利率一般以年利率 (interest rate per annum) 計算, 所以時期需以年為單位 2. 計算複利息 (compound interest) 時, 若利息不是每年結算一次, 則需注意在公式 A = P (1 + r%) n 中,r% 及 n 的變化 例 : 若年利率 = 6%, 年期 = 3 年, 複利息每半年結算一次, 則 r% = 6% 6 12 = 3%( 因每次只可得到全年利息的一半 ) n = 2 3 = 6( 因每年計算利息 2 次, 共有 3 年 ) 12 Progressive Mathematics 3 ( Fourth Edition )
3. 當情況複雜時, 如有兩件或以上的事件時, 我們可利用樹形圖 (tree diagram) 或列表法來協助找出所有可能結果 (possible outcome) 4. 幾何概率 (geometric probability) 是關於幾何圖形的面積或長度等量度的概率 Important Formulas: For an event E, Number of favourable outcomes 1.1 P (E) = Total number of possible outcomes 2.1 Experimental probability = Number of trials in which event E occurs Total number of trials 3.1 Geometric probability = Areas (/ length) of the region in which event E occurs Areas (/ length) of the whole figure 1. State whether each of the following statement is true (T) or false (F). (a) If a card is drawn from a pack of poker at random, then the probability of getting a red card is the same as getting a black card. (a) 1 (b) If a bag contains red balls only, then the event of getting 1 a blue ball is an impossible event. (b) 1 (c) 2 π can be the probability of an event. (c) 1 (d) The experimental probability must be different from 1 theoretical probability. (d) 1 (e) When throwing a dice twice, the total number of possible 1 outcomes is 36. (e) 1 (f) If a bag contains $5 coins only, then the event that a $2 1 coin is taken out from this bag is an impossible event. (f) Probability 61
A. Multiple-choice Question 1. The following graph represents the inequality A. x > 2. B. x 2. C. x < 2. D. x 2. 2. The following graph represents the inequality A. x > 3. B. x 3. C. x < 3. D. x 3. 3. If a > b > 0, which of the following may not be true? A. -2a < 2b B. a 3 > b 3 C. (1 - a)(1 - b) < 0 D. 1 - a < 1 - b 4. How many positive integers satisfy the inequality x 5π 2? A. 6 B. 7 C. 8 D. 9 5. Which of the following numbers does not satisfy the inequality x > 7? A. 2.65 8 B. 3 3π C. 2 9 D. 4 6. Suppose 0 < a < b < 1. Which of the following is false? A. ab > 1 B. 2a < 2b 1 C. a > 1 b D. b - a > 0 36 Progressive Mathematics 3 ( Fourth Edition )
A. Multiple-choice Question 1. How many positive integers satisfy x 2 + x + 1 < x + 2 3 4? A. 0 B. 1 C. 2 D. 3 2. a and b are solutions of the following figure, where a < b. Which of the following must be true? I. a + b > 0 II. ab < 0 III. (a + 6)(b a) > 0 A. I only B. III only C. I and II only D. I and III only 4. If ab > p, which of the following must be true? A. b < p B. a > p b C. ab + p > 0 D. ab + c > p + c, where c is any number. 5. Solve 2x - 1 3 A. x < 1 B. x > 1 C. x < 1 D. x > 1 + 1 2 > 1 (x - 1). 4 6. If a < b < 0, which of the following is true? A. ab < 0 B. b a < 0 C. a + b < 0 D. a 2 < b 2 3. If a > b and a, b are both negative, which of the following is / are true? I. 2a < 2b II. -3a < -3b III. -3a > 3b A. II only B. III only C. I and III only D. II and III only 40 Progressive Mathematics 3 ( Fourth Edition )
In the figure, DPQR is an isosceles triangle with PQ = PR. S is a point on QR such that the orthocentre of DPQR lies on PS. (a) Show that DPQS DPRS. (b) Are the orthocentre, circumcentre and incentre of DPQR collinear? Explain your answer. 考題趨勢 DSE 卷二常見 三角形的中心 題型 ( 常涉及其他課題的知識 ) DSE 沒有 三角形不等式 題型 Solution: (6 marks) (a) PS is an altitude. 垂心為頂垂線的交點 PS QR PSQ = PSR = 90 (2 M) PQ = PR (given) PS = PS (common side) DPQS DPRS (RHS) (1 M) (b) QS = RS (corr. sides, Ds) (1 M) Since PS is the perpendicular bisector of QR, the circumcentre lies on PS. QPS = RPS (corr. s, Ds) (1 M) Since PS is the angle bisector of QPR, the incentre lies on PS. The orthocentre, circumcentre and incentre of DPQR are collinear. (1 M) 參考 CE 2011 Paper 1 Q16 1. In the figure, AB = AC. P is a point on BC such that the incentre of DABC lies on AP. (a) Show that DABP DACP. (b) Are the orthocentre, circumcentre and incentre of DABC collinear? Explain your answer. (6 marks) 2. In the figure, CD is a median of DABC and AD = CD. (a) Find p + q. (b) Is DABC a right-angled triangle? Explain your answer. (6 marks) Open-ended Question 休憩室 Deductive Geometry (II) 107
Score sheet (Assessment 3) Target I: To understand algebraic expression (Chapter 1, 3) Target II: To apply percentages including growth and depreciation (Chapters 2) Target III: To explore and study 3-D figures and their measurements (Chapters 4, 9) Target IV: To study the probability and central tendency (Chapter 5, 6) Target V: To recognize the deductive approach in handling geometric proofs (Chapters 7, 8) Target VI: To apply trigonometric ratios, bearing and gradient (Chapters 10) Target VII: To study the coordinate geometry of straight lines (Chapter 11) Question Target Section A 1 (30%) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Section B 16 (37%) 17 18 19 20 21 22 Section C 23 (33%) 24 25 Scores I II III IV V VI VII Total 0 5 0 4 0 7 0 6 0 9 0 6 0 9 0 46 6 9 5 7 8 12 7 11 10 16 7 10 10 16 47 81 10 11 8 9 13 15 12 14 17 19 11 13 17 19 82 100 Key: Not Yet able to Beginning to develop the ability to Generally able to 190 Progressive Mathematics 3 ( Fourth Edition )
Assessment 3 (Revision for Chapters 1 11) Time allowed: 1 hour and 30 minutes Full marks: 100 Answer ALL questions Section A: Multiple-choice Question (30 marks) Each question carries 2 marks. 1. Which of the following is false? A. a 2-25a = a(a - 5) B. a 2-49 = (a + 7)(a - 7) C. a 2-2a - 35 = (a + 5)(a - 7) D. a 3 + 10a 2 + 25a = a(a + 5) 2 2. Which of the following has/have x - 3 as a factor? I. x 3-27 II. x 2 + 4x - 21 III. x 4-81 A. I and II only B. I and III only C. II and III only D. I, II and III 5. How many positive integers satisfy x < 6 6 8? A. 3 B. 4 C. 5 D. 6 6. In the figure, a solid is formed by cutting out a cylinder from a cube with side length a. The base radius and the height of the cylinder are r and a respectively. Which of the following can be expressed by 6a 2 + 2pr(a - r)? 3. Suppose S = vt. If v is increased by 10% and t is decreased by 8%, find the percentage change in S. A. -2.8% B. +1.2% C. +2% D. +2.5% 4. Tom is 10% heavier than Peter and Jack is 8% lighter than Peter. By how much percent is Tom heavier than Jack? A. 16.4% (cor. to 1 d. p.) B. 17.2% (cor. to 1 d. p.) C. 18% D. 19.6% (cor. to 1 d. p.) A. Sum of the lengths of all edges of the solid B. Total surface area of the solid C. Base perimeter of the solid D. Volume of the solid Assessment 3 191