Public Assessment of the HKDSE Mathematics Examination

Public Assessment of the HKDSE Mathematics Examination. Public Assessment The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in the Compulsory Part is shown below. Public Examination Component Weighting Duration Paper Conventional questions Paper 2 Multiple-choice questions 65% 35% 2 4 hours 4 hours The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in Module (Calculus and Statistics) is shown below. Component Weighting Duration Public Examination Conventional questions 00% 2 2 hours The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in Module 2 (Algebra and Calculus) is shown below. Component Weighting Duration Public Examination Conventional questions 00% 2 2 hours 2. Standards-referenced Reporting The HKDSE makes use of standards-referenced reporting, which means candidates levels of performance will be reported with reference to a set of standards as defined by cut scores on the variable or scale for a given subject. The following diagram represents the set of standards for a given subject: Cut scores U 2 3 4 5 Variable/ scale Within the context of the HKDSE there will be five cut scores, which will be used to distinguish five levels of performance ( 5), with 5 being the highest. The Level 5 candidates with the best performance will have their results annotated with the symbols and the next top group with the symbol. A performance below the threshold cut score for Level will be labelled as Unclassified (U). III

Exam Strategies A. Introduction Paper 2 of the HKDSE Mathematics Examination consists of two sections. The duration is hour and 5 minutes. As the questions in Section B are more difficult than those in Section A, it is suggested that students reserve more time for Section B. Number of questions Suggested time allocation Section A 30 45 minutes Section B 5 30 minutes Moreover, students also have to learn some strategies for handling multiple-choice questions. B. Techniques. Direct calculation Use the given theorems, formulas and rules to calculate the answers directly. Example : (Modified from HKDSE sample paper) If the base and the height of a triangle are increased by 0% and x% respectively so that its area is increased by 2%, find the value of x. A. 2. B. 0 C. D. 32 Solution: Let b and h be the original base and height respectively. Then the new base = ( + 0%)b =.b, the new height = ( + x%)h. Original area = 2 bh New area = ( + 2. b )( x %) h 2 (. b)( + x%) h = bh( + 2%) 2. ( + x%) =. 2 + x% =. x% = 0. x% = 0% x = 0 B is the answer. IV

Comparison between HKDSE and HKCEE Syllabuses. Topics removed from and added to the syllabus Section Topics removed Topics added Number and Algebra Strand Quadratic equations in one unknown Functions and graphs More about graphs of functions Exponential and logarithmic functions More about polynomials Sum of roots and product of roots Operations of complex numbers Concepts of domains and codomains of functions Enlargement and reduction Change of base G.C.D. and L.C.M. of polynomials Operations of rational functions More about equations Using a given quadratic graph to solve another quadratic equation Arithmetic and geometric sequences and their summations Inequalities and linear programming Measures, Shape and Space Strand Locus Equations of straight lines and circles Properties of arithmetic and geometric sequences Solving quadratic inequalities in one unknown by the algebraic method Solving compound linear inequalities involving or Describing the locus of points with algebraic equations Possible intersection of two straight lines Intersection of a straight line and a circle Data Handling Strand Permutation and combination Concepts and notations of permutation and combination I

Chapter Number System and Estimation A Number System. Real number system Real numbers 6 e.g. -3, -0.24, -, 0,, 20, 0.25, - 25, 3, 0. 3, 0. 23, p, 3 Which of the following is not a rational number? A.. 43 B. 7 23 C. 2 3 D. 3 + 8 Irrational numbers e.g. p, 3 Rational numbers 6 e.g. -3, -0.24, -, 0,, 20, 0.25, - 25, 3, 0. 3, 0. 23 Ans: D Fractions Terminating decimals Recurring decimals Integers 6 e.g. - 25, 3 e.g. 0.25, -0.24 e.g. 0. 3, 0. 23 e.g. -3, -, 0,, 20 Negative integers e.g. -, -3 Zero 0 Positive integers (Natural numbers) e.g., 20 2. Complex number system The standard form of a complex number is a + bi, where () a and b are real numbers, and (2) i =. Complex numbers e.g. 2 + 3i, 5i, 8 Real numbers Imaginary numbers e.g. 8 e.g. 5i 2 Which of the following is a real number? A. (3 + 4i ) + 4i B. (3 + 4i )(3-4i ) C. i(3 + 4i ) D. 3 + 4i 3 Ans: B

Mathematics: Multiple-choice Questions Compulsory Part A Set Notation. How many elements are there in the set {3, 3, 4, 4, 4, 5}? A. 3 B. 4 C. 5 D. 6 x + 20 4. Given that S = { : x is a positive x integer}. Which of the following is not an element in S? A. 6 B. C. 5 D. 2 2. Given that P = {x : x is the month of a year}. How many elements are there in P? A. 4 B. 6 C. 9 D. 2 3. Consider the Venn Diagram of sets A and B. A B Fig. 8.4 The shaded region represents the set A. A B. B. A B. C. A \ B. D. B \ A. 5. Given that P = {apple, orange}, Q = {apple, pear, banana}, and R = {mango, orange}. Which of the following is an empty set? A. P Q B. Q R C. P \ R D. R \ Q 6. Given two sets P and Q. Which of the following may not be true? A. P Q = Q P B. P Q = Q P C. P P = P D. P \ Q = Q \ P 234

Statistics 39. The following histogram shows the heights of plants in a greenhouse. 30 Heights of plants in a greenhouse In the following questions, unless otherwise specified, all the answers should be exact or correct to 3 significant figures. Frequency 20 0 0 20 40 60 80 Height (cm) 00 Fig. 9.24 Find the standard deviation of the heights. (Give the answer correct to 3 significant figures.) A. 24.0 cm B. 26.2 cm C. 32.5 cm D. 46.4 cm 40. The following table shows the time spent on watching television programmes by a group of students last Sunday. Time (in hours) 0-2 3-5 6-8 9 - Number of students 20 5 2 Table 9.4 Find the variance. (Give the answer correct to 3 significant figures.) A. 2.38 B. 3.84 C. 4.2 D. 5.66 Level. Which of the following is a/are discrete datum/data? I. Price of a book II. Number of books in a locker III. Weight of a school bag A. II only B. I and II only C. I and III only D. I, II and III 2. During a body check, the following data of a group of clients are obtained. Which of them is a qualitative datum? A. Blood pressure B. Temperature C. Blood type D. Pulse 265

Chapter 20 Inequalities and Linear Programming A Linear Inequalities in One Unknown. Basic properties of inequalities: (a) If a > b and b > c, then a > c. (b) If a > b, then a + c > b + c. (c) (i) If a > b and c > 0, then ac > bc. (ii) If a > b and c < 0, then ac < bc. (d) (i) If a > b > 0, then <. a b (ii) If a < b < 0, then >. a b (e) If a 0, then a 2 > 0. Note: Properties (a) (d) are also applicable to inequalities with the sign or. 2. The solution of a linear inequality can be represented on a number line. (a) x > 4 (b) x 6 x x Fig. 20. Fig. 20.2 Note: In the above figures, the symbol represents the number which is not included in the solution; the symbol represents the number which is a part of the solution. x 9 Solve the inequality < 3. 2 A. x < -3 B. x > -3 C. x < 3 D. x > 3 2 Ans: D If a < 0 < b, which of the following may not be true? A. < a b B. a 2 < b 2 C. a + < b + D. 9a < 2b Ans: B 270

Trigonometry () 7. cos 2 2 + cos 2 4 +... + cos 2 88 + cos 2 90 = A. 0 B. 22 C. 23 D. 45 8. sin 4 θ - cos 4 θ + = Level 2. Find the maximum value of 3 - cos2x. A. 2 B. 3 C. 4 D. 5 A. sin 2 θ B. sin 4 θ C. 2sin 2 θ D. 2cos 2 θ 9. Solve cos 2 θ - 2sin θ + 2 = 0 for 0 θ 90. A. θ = 0 B. θ = 35 C. θ = 60 D. θ = 90 2. What is the period of the function y = sin(2x + 0 )? A. 70 B. 80 C. 350 D. 360 3. In the figure, ABCD is a cyclic quadrilateral. 0. If rcos θ - 6 = 0 and rsin θ - 8 = 0, where 0 θ 90, then r = A. 0. B. 2. C. 48. D. 00. Fig. 5.22 Which of the following must be true? I. cos A = cos C II. sin B = sin D III. cos B + cos D = 0 A. I and II only B. I and III only C. II and III only D. I, II and III 205

Mathematics: Multiple-choice Questions Compulsory Part Chapter Number System and Estimation Check Point. D 3 + 8 = 3 + 2 2 Since 2 2 is not a rational number, 3 + 2 2 is not a rational number. Alternative Solution For A,. 43 is a recurring decimal, which is a rational number. For B, 7 is a fraction, which is a rational number. 23 For C, 2 3 = 36 = 6, which is a rational number. A, B and C are not the answer. \ D is the answer. \ 2. B (3 + 4i )(3-4i ) = 3 2 - (4i) 2 3. B 4. D = 9-6i 2 = 9 + 6 = 25, which is a real number. ( 2 3 2 )( 3 + 2 ) = 2 3( 3) + 2 3( 2 ) 2 ( 3) 2 ( 2 ) = 6 + 2 6 6 2 = 4 + 6 5. C Step : Calculate the sum of the units digits. + 3 + + 0 = 5 Step 2: Estimate the sum of the remaining digits. 0.57 + 0.68 + 0.34 + 0.45 = (0.57 + 0.45) + (0.68 + 0.34) + = 2 Step 3: Combine the results above. Estimated value = 5 + 2 = 7 6. A Percentage error Maximum absolute error = Measured value = 0.05 3.4 00% =.47% Fundamental Exercise A. Number System. D Let x = - 2 and y = - 3. xy = 2 3 00% =, which is not a natural number. 6 2. D Since M is an odd integer, 3M must be an odd integer. The sum of two odd integers must be an even integer. \ 3M + 7 must be an even integer. \ Alternative Solution For A, if M =, M 2 = 5.5, which is not an integer. For B, if M = 3, M 2 = 9, which is not even. 2