Public Assessment of the HKDSE Mathematics Examination

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1 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 Multiple-choice questions 65% 5% 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% hours The mode of public assessment of the Hong Kong Diploma of Secondary Education (HKDSE) Mathematics Exam in Module (Algebra and Calculus) is shown below. Component Weighting Duration Public Examination Conventional questions 00% hours. 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 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). IV

2 Exam Strategies A. General Strategies. Before the start of the examination Make sure that the time on your watch matches with that of the examination centre. Listen carefully to the invigilator for any errors and changes in the examination papers. Read carefully the instructions on the cover of the question-answer book or question book. Check carefully whether there are any omitted or blank pages in the examination paper according to the invigilator s instruction.. During the examination Use proper stationery. Paper : use a pen mainly, but an HB pencil for drawing. Paper : use an HB pencil. Show your work clearly and neatly. Do not get stuck on any one of the questions. Skip it and go on to another one.. After answering all the questions Do not be tempted to leave early. Check whether there are any questions that were missed. Go back to questions skipped earlier. Check whether there are any careless mistakes or not. Do not cross out anything unless you have enough time to replace it correctly. Make sure you write your candidate number on the answer book, supplementary answer sheets and multiple-choice answer sheet. B. Specific Strategies. Paper ( 4 hours) Allocate a reasonable proportion of time to each section. Sections Suggested Time Allocation Approximate Time per Question A () 40 minutes 5 minutes A () 40 minutes 5 0 minutes B 50 minutes 5 5 minutes In general, spend 5 minutes for every 4 marks. Allow 5 minutes for final checking. V

3 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 II

4 6 More about Trigonometry Trigonometry Trigonometric Ratios of Angles. cosθ a c, tanθ b a and Fig. 6. sinθ b c where c a + b. Signs of Trigonometric Functions:. cosθ x r, tanθ y x and sinθ y r where r x + y Fig. 6. Fig. 6. Special Angles θ sin θ 0 or 0-0 cos θ or 0-0 tan θ 0 or undefined 0 undefined 0 Table 6. Trigonometric Identities θ. tanθ sin. sin θ + cos θ cosθ. sin(90 - θ) cos θ 4. cos(90 - θ) sin θ 5. tan ( 90 θ) tanθ Reducing Trigonometric Ratios 80 - θ 80 + θ 60 - θ sin sin θ -sin θ -sin θ cos -cos θ -cos θ cos θ tan -tan θ tan θ -tan θ Table 6.

5 More about Trigonometry Graphs of Trigonometric Functions. (a) y sin x (b) y cos x (c) y tan x Fig. 6.4 Fig. 6.5 Trigonometric Equations. Simple Trigonometric Equations (a) a sin θ b (b) a cos θ b (c) a tan θ b. Other Trigonometric Equations Examples: (a) a tan θ sin θ (b) cos θ - sin θ 0 Fig For any real value of x, - sin x and - cos x.. The periods of sin x, cos x and tan x are 60, 60 and 80 respectively. Transformation on the Graphs of Trigonometric Functions Consider the graph of y sinx.. Translation (a) Vertical: y sinx + k (b) Horizontal: y sin(x + k ). Reflection About the x-axis: y -sin x. Reduction or Enlargement (a) Vertical: y ksinx (b) Horizontal: y sin (kx)

6 Rate and Ratio A Rate Rate is a comparison between two different kinds of quantities. For two different quantities x and y, the rate is given by x y or y x and it has a unit. For example, a worker cleans 0 cars in 5 hours. 0 cars The cleaning speed 5 hours 4 cars/hour. The cleaning time for each car 5 hours 0.5 hour/car. 0 cars B Ratio Ratio is a comparison between two or more quantities of the same kind. The ratio of a to b is a : b or a and it has no unit. The ratio for three or b more quantities such as a : b : c is called a continued ratio. For example, the weights of A, B and C are 80 kg, 50 kg and 0 kg respectively. Weight of A : Weight of B 80 : 50 8 : 5. Weight of A : Weight of B : Weight of C 80 : 50 : 0 8 : 5 :. A continued ratio cannot be expressed as a fraction. Determine whether each of the following statements is true or false.. Rate is a comparison between two different kinds of quantities.. Ratio is a comparison between two or more quantities of the same kind.. If 5m 8n, then m : n 5 : If a : b : and b : c :, then a : b : c : :. 5. The speed of a car, say 80 km/h, is an example of rate. 6. The speeds of two cars, say 80 km/h and 90 km/h, can be expressed by the ratio 8 : 9. (For answers, see the bottom of the page.) Suggested Answer (Check Your Progress ). T. T. F (m : n 8 : 5) 4. F (a : b : c : : 6) 5. T 6. T

7 Statistics Unless otherwise specified, numerical answers should be either exact or correct to significant figures. The diagrams are not necessarily drawn to scale. Section A(). (a) Find the range and the inter-quartile range of,, 5, 5, 9, 0. (b) Find the standard deviation of, 4, 7, 8, 0. (Working steps are required.) (6 marks) Suggested Solution (a) Range 0-8 A Q and Q 9 M Inter-quartile range 9-6 A (b) Mean 5 Standard deviation 6. 4 M ( 6. 4) + ( ) + ( ) + ( ) + ( ) 5 M.58 (cor. to sig. fig.) A Divide the data into two parts:,,5,5,9, 0 Q Q 9 We should find the mean first, then find the standard deviation.. Determine the sampling method used in each of the following cases. (a) There are 550 computers produced in a factory. Each computer is assigned a distinct number from 00 to 550. Ten distinct numbers are generated randomly so as to select 0 computers to form a sample. First determine whether it is probability sampling or nonprobability sampling. (b) A group of children are classified into 4 age groups: A, B, C and D. Different numbers of children are selected from each group. 75

8 More about Probability 64 6 (b) P(pass) 5 5 P(6 marks pass) P(6 marks pass) P(pass) (c) P(all answers are incorrect) 64 5 M For (b), the candidate misses the case of getting 0 mark and therefore got an incorrect P(pass). For (c), the candidate does not recognize that there are only 4 choices left and hence the situation is different from that of (a)(i). Candidates should recognize all of the possible outcomes when calculating probabilities. (b) P(pass) P(6 marks) + P( marks) P(6 marks pass) P(6 marks pass) P(pass) 5 5 (c) P(all answers are incorrect) M M A M A For (b), the candidate can make use of the result in (a) correctly to find P(pass). For (c), the candidate recognizes that there are only 4 choices in the quiz on the second attempt. 5

9 Mathematics: Conventional Questions Compulsory Part Book 0. The true bearing of ship B from island A is 0. The distance between A and B is 0 km at noon. If the ship sails due north with a speed of 0 km/h, find (a) the shortest distance between the ship and the island; (b) the time at which the distance is the shortest. Hint 5. A geographer stands at Z and finds that the true bearing of X from Z is 5. If he walks 80 m west to Y, he finds that the compass bearing of X from Y is N45 E. (a) What kind of triangle is XYZ? Explain your answer. (b) Find the width of the river. Hint 6 Fig Fig. 7.5 Section B. In the figure, X 0, Z 68 and XY 4 cm. Find the length of YZ. Fig In the figure, BC 7, AC and A 7. Find B. Fig In the figure, AB.5 cm, AC 6. cm and C 5. Find q. Fig Find D if AB cm and BD cm. 4 Fig. 7.55

10 Mathematics: Conventional Questions Compulsory Part Book Section A () Hint Arrange the data in ascending order first. Hint Find the total number of members before finding Q and Q. Hint Try to substitute a value for y and find the standard deviation of the new set of data. Hint 4 Sam should choose three digits each time. Section A () Hint 5 Use the method of completing the square. Section B Hint 6 Consider the change of Σ(x - x) after 5 compact discs are added. Hint 7 Substitute x. Hint 8 Compare the standard scores in Paper A and Paper B. Hint 9 Number Total number percentage Hint 0 First find out the percentages below and above the mean separately. Then sum up the results. 9

11 Rate and Ratio Chapter Rate and Ratio Section A() 40 worksheets. (a) The marking rate 0.5 hour 80 worksheets / hour (b) The marking rate 40 worksheets 0 minutes. worksheets / hour (cor. to sig. fig.) 70 0 packs. (a) The carriage rate vans 700 packs / van 70 0 eggs (b) The carriage rate vans 8400 eggs / van 0 km. (a) (i) The speed 4 h 80 km / h m (ii) The speed s. m / s (cor. to sig. fig.) (b) The distance 80 km / h.5 h 0 km 4. (a) The distance 900 m 600 m 0 0 (b) The time taken 0 minutes minutes 0 pages 5. (a) The writing speed (550 60) hours.09 page / hour (cor. to sig. fig.) 4 6. The time needed 0 minutes minutes The time saved (0-4) minutes 6 minutes 7. The time 5 seconds 8. The flow rate 56 seconds 8 5 ( 5 00) m 8 s 50 m /s In 8 seconds, the volume of water passing through is ( 5 00) m. 9. The total distance he travels km / h 60 h + 70 km / h 8 60 h 9 km The total time he spends hour hour His average speed 9 km hour 77.4 km / h (cor. to sig. fig.) (b) The writing time for each page 550 minutes 0 pages 55 minutes / page Average Speed Total Distance Total Time 97