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HKDSE MATH CP PAPER 2 Set 23 HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION MATHEMATICS Compulsory Part S6 MOCK EXAM PAPER 2 Time allowed: 1¼ hours INSTRUCTIONS 1. Read carefully the instructions on the Answer Sheet. After the announcement of the start of the examination, you should first stick a barcode label and insert the information required in the spaces provided. No extra time will be given for sticking on the barcode label after the ‘Time is up’ announcement. 2. When told to open this book, you should check that all the questions are there. Look for the words ‘END OF PAPER’ after the last question. 3. All questions carry equal marks. 4. ANSWER ALL QUESTIONS. You are advised to use an HB pencil to mark all the answers on the Answer Sheet, so that wrong marks can be completely erased with a clean rubber. You must mark the answers clearly; otherwise you will lose marks if the answers cannot be captured. 5. You should mark only ONE answer for each question. If you mark more than one answer, you will receive NO MARKS for that question. 6. No marks will be deducted for wrong answers. Not to be taken away before the end of the examination session Set 23 PAPER 2-1 1 There are 30 questions in Section A and 15 questions in Section B. The diagrams in this paper are not necessarily drawn to scale. Choose the best answer for each question. Section A 1. 2. 25456 27345 = 3123 A. 51824 . B. 15912 . C. 151356 . D. 151824 . If 2 x − 7 = y + 6 = 2 x − 4 y + 37 , then x + y = A. B. C. D. 3. 11. 16. 20. 23. 1 = 7 A. B. C. D. 0.380 (correct to 3 decimal places). 0.3779 (correct to 4 decimal places). 0.37796 (correct to 5 significant figures). 0.377965 (correct to 6 significant figures). Set 23 PAPER 2-2 2 4. If A. B. C. D. 5. u v , then u = = 4 − 3u 4 + 3v v . 1− 6v v . 1+ 6v 2v . 2 − 3v 2v . 2 + 3v 2 2 If y − ky + 9 ( y − l ) + m , where k, l and m are constants, then m = A. 9. 2 B. C. D. 6. Which of the following must have a + b as a factor? I. a 4 − b 4 II. a 4 + b 4 2 III. a (a + b) − a + b A. B. C. D. 7. k . 4 9− k2. 9+ k2 . 9− I only II only I and III only II and III only 2 Let f ( x) = − x + bx + 11 , where b is a constant. If f (−2) = −1 , then b = A. −3. B. 4. C. 7. D. 8. Set 23 PAPER 2-3 3 8. 3 2 Let p and q be constants. If px + x − 2qx + 8 is divisible by x + 4, then 8p − q = A. B. C. D. 9. −24. −3. 3. 24. 8 x + 3 x − 11 The solution of is − 2( x − 5) 4 A. x 3. B. x −2. C. x −2. D. −2 x 3. 2 10. If 0 c 1, which of the following may represent the graph of y = (c − x) − c ? A. B. C. D. Set 23 PAPER 2-4 4 11. If A. B. C. D. 3a − 2b 2 2a − 3b = , then = a + 4b 3 4a + b 3 − . 2 2 − . 3 1 . 9 23 . 67 12. In the figure, the 1st pattern consists of 5 dots. For any positive integer n, the (n + 1)th pattern is formed by adding (3n + 4) dots to the nth pattern. Find the number of dots in the 7th pattern. A. B. C. D. 70 77 87 92 13. The cost of a shirt is $200. The shirt is sold for $24 less than its marked price. If the percentage profit is 25% before the discount, find the percentage profit after the discount. A. 10.4% B. 12% C. 13% D. 22% Set 23 PAPER 2-5 5 14. It is given that h partly varies directly as k2 and partly varies inversely as k . When k = 1, h = 41 and when k = 4, h = 5. When k = 9, h = A. −306. B. −67. 283 C. . 3 D. 552. 15. In the figure, ABD and BCD are triangles with BC = BD and AD // BC. If BAD = 84 and ABD : CBD = 2 : 1, then BCD = A. 66. B. 69. C. 74. D. 84. 16. In the figure, ABC is an isosceles triangle with AC = BC. D and E are points lying on AB and BC respectively. AE and CD intersect at F. If AF = 64 cm, FE = 36 cm and CFE = 2ABC, then AC = A. B. C. D. 75 cm. 80 cm. 81 cm. 82 cm. Set 23 PAPER 2-6 6 17. OAB is a sector of circle with centre O. OAB is folded such that AO and BO join together to form a right circular cone. If AO = BO = 20 cm and the angle of the sector OAB is 216, then the volume of the circular cone is A. 240 cm3. B. 768 cm3. C. 960 cm3. D. 1024 cm3. 18. In the figure, ABCD is a parallelogram. B and D are points on AS and PC respectively such that PD : DC = 2 : 3 and AB : BS = 4 : 1. If PS intersects AD and BC at Q and R respectively, then the area of DQP : the area of BRS = A. 2 : 1. B. 4 : 1. C. 8 : 3. D. 64 : 9. 19. In the figure, AC = A. 2x . B. 2x. C. 2 3 x . D. 2 x 2 + 18 . Set 23 PAPER 2-7 7 20. In the figure, ABCD is a parallelogram. O is the centre of the circle ABD and ODC is a straight line. AD and OB intersect at E. If BCD = 38, then BED = A. B. C. D. 104. 109. 114. 142. O A D E B C 21. In the figure, AB produced and DC produced meet at E. It is given that AB = 3 , CD = 4 and AD = 6 . If ADC = 60, then BEC = A. B. C. D. 30. 40. 48. 66. 22. In the figure, AB and CD are two vertical poles standing on the horizontal ground. The angle of elevation of C from A is 50 and the angle of depression of D from A is 35. If AB = 2 m, then AC = 2 tan 35 C A. m. sin 50 2 tan 35 B. m. cos 50 2 C. m. sin 50 tan 35 A 2 D. m. cos 50 tan 35 B Set 23 PAPER 2-8 8 D 23. The rectangular coordinates of points P and Q are (2k, −k + 2) and (−2, 8) respectively, where k is a constant. If the length of the line segment joining P and Q is 2 5 , then the polar coordinates of P are A. (4, 0) . B. (4 2 , 135) . C. (4 2 , 225) . D. (4 2 , 315) . 24. The straight line L and the straight line 2 x + 7 y − 28 = 0 do not intersect each other. If the x-intercept of L is 5, then the equation of L is A. 2 x + 7 y − 10 = 0 . B. 2 x + 7 y − 35 = 0 . C. 7 x − 2 y + 10 = 0 . D. 7 x − 2 y − 35 = 0 . 25. In the figure, the straight line L1: y = ax + b and the straight line L2: y = px + q intersect at a point on the negative x-axis. Which of the following must be true? I. ab 0 II. pq 0 III. aq = bp A. I only B. I and II only C. II and III only D. I, II and III Set 23 PAPER 2-9 9 26. S is a line segment of length 5 cm. Let P be a moving point such that the distance from P to S is always 2 cm. Find the area of the region bounded by the locus of P correct to the nearest 0.1 cm2. A. 32.6 cm2 B. 36.0 cm2 C. 45.1 cm2 D. 63.6 cm2 27. The bar chart below shows the numbers of ice-creams of three flavours sold at an ice-cream shop on a certain day. Number of ice-creams sold 80 70 60 50 40 30 Vanilla Chocolate Strawberry One of the staff makes the following claims: I. On that day, the total number of vanilla ice-creams and strawberry ice-creams sold is less than the number of chocolate ice-creams sold. II. On that day, the number of chocolate ice-creams sold is two times the number of strawberry ice-creams sold. III. On that day, the number of vanilla ice-creams sold is 50% more than the number of strawberry ice-creams sold. Which of the above claim(s) is/are true? A. I only B. II only C. I and III only D. II and III only Set 23 PAPER 2-10 10 28. There are 12 black balls and k white balls in a bag. Peter repeats drawing a ball at a time randomly from the bag with replacement for 90 times. If he draws a white ball for 60 times, which of the following is the best estimate of the value of k? A. 8 B. 18 C. 24 D. 60 29. The frequency distribution table below shows the distribution of the heights of 25 ten-year-old boys. Height (cm) Frequency 126 − 130 2 131 − 135 3 136 − 140 8 141 − 145 7 146 − 150 5 Find an estimate of the standard deviation of the heights of the 25 boys correct to the nearest 0.1 cm. A. 5.8 cm B. 6.2 cm C. 34.0 cm D. 38.0 cm 30. Consider the following positive integers: 3 6 6 7 13 13 17 18 20 x y z If the mean of x and y is 4, and the mean of all the above data is 10, then the median of the above data is A. 8. B. 9. C. 11. D. 15. Set 23 PAPER 2-11 11 Section B 31. x 2 A. B. C. D. 1 1 − 2 = + 2 x − 3 3x − 2 x − 1 2 . ( x + 3)(3 x + 1) 2 . ( x − 3)(3 x − 1) 2x − 4 . ( x + 1)( x − 3)(3 x − 1) 2x + 4 . ( x − 1)( x + 3)(3 x + 1) 32. 3 47 − 815 + 816 = A. D000000B00016. B. E000000C00016. C. D0000000B00016. D. E0000000C00016. 33. The figure shows the graph of y = log a x and the graph of y = log b x on the same rectangular coordinate system, where a and b are positive constants. If a vertical line L cuts the x-axis, the graph of y = log a x and the graph of y = log b x at P, Q and R respectively, which of the following is/are true? a−b0 a+b2 QR = log b a − 1 III PQ y I II A. B. C. D. L R y = logb x y = loga x Q II only I and II only I and III only II and III only O P x 2 34. Let f ( x) = ax − 4 x + a , where a is a non-zero constant. If the maximum value of f (x) is 3, then a = A. 4. B. 1. C. −1. D. −4. Set 23 PAPER 2-12 12 40i is a purely imaginary number, where k is a real number, then k = 3+i −5. −4. 4. 5. 35. If k + A. B. C. D. 36. The sum of all the positive terms in the arithmetic sequence 999, 991, 983, is A. 62 372. B. 62 375. C. 62 874. D. 62 875. 2 x + 2 y 8 37. Which of the following shaded regions represents the solution of x 0 ? 0 y 3 A. B. C. D. Set 23 PAPER 2-13 13 38. Which of the following figures show the graph of y = 1− cos x ? A. B. C. D. 39. In the figure, AO is a vertical pole standing on the horizontal ground OBC. The bearing of C from B is A. B. C. D. A N30W. N60W. S30E. S60E. 60 West O C 45 B South Set 23 PAPER 2-14 14 40. In the figure, EB is the tangent to the circle ABCD at B and DAE is a straight line. If AEB = 76 and ACB = 21, then ACD = A. B. C. D. D 55. 62. 76. 83. C 21 A E 76 B 41. The coordinates of the points P, Q and R are (3, 2), (15, 2) and (3, 7) respectively. If C is the inscribed circle of PQR, then the equation of C is 2 2 A. x + y − 5 x − 4 y + 37 = 0 . 9 B. x 2 + y 2 − 9 x − y + 59 = 0 . 2 2 2 C. x + y − 10 x − 8 y + 37 = 0 . D. x 2 + y 2 − 18x − 9 y + 59 = 0 . 42. A queue is formed by 5 boys and 5 girls. If boys and girls stand alternately in the queue, how many different queues can be formed? A. 14 400 B. 28 800 C. 30 240 D. 86 400 Set 23 PAPER 2-15 15 43. There are two cities, X and Y. In a morning, the probabilities of raining in X and Y are 2 and 5 1 respectively. Given that at most one of the two cities is raining in the morning, find the 4 probability that it is raining in X in the morning. 3 A. 10 1 B. 3 2 C. 5 4 D. 9 44. A computer salesman designs a questionnare to collect the opinions about a top-selling computer from its users. The salesman has 10 friends who are users of the computer. He randomly selects three of his friends and only these three friends are selected as a sample to fill in the questionaire. Which of the following is/are disadvantage(s) of this sampling method? I. The sample size is too small. II. He selects three of his friends by himself. III. Not all the users of the computer have an equal chance of being selected. A. I only B. I and III only C. II and III only D. I, II and III 45. Let a1, b1 and c1 be the mean, the variance and the inter-quartile range of a group of numbers {x1, x2, x3, …, x40} respectively while a2, b2 and c2 be the mean, the variance and the inter-quartile range of the group of numbers {x1, x2, x3, …, x40, x41, x42, x43, x44} respectively. If x41 = x42 = x43 = x44 = a2, which of the following must be true? I. a1 = a2 II. b1 b2 III. c1 c2 A. I and II only B. I and III only C. II and III only D. I, II and III END OF PAPER Set 23 PAPER 2-16 16